distributed coordination theoryfor ground and aerial robot

Distributed Coordination Theory for Ground and Aerial

Robot Teams

by

Ashton Roza

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

c© Copyright 2019 by Ashton Roza

Abstract

Distributed Coordination Theory for Ground and Aerial Robot Teams

Ashton Roza

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2019

This thesis investigates distributed coordination problems for two important classes

of robots. One class corresponds to ground-based mobile robots, each modelled as a

kinematic unicycle. The second corresponds to flying robots, each propelled by a thrust

vector and endowed with an actuation mechanism producing torques about three or-

thogonal body axes. The following coordination problems are studied in this thesis:

rendezvous, formation control, linear and circular formation flocking and formation path

following.

For rendezvous of kinematic unicycles, a smooth, time-independent control law is

presented that drives the unicycles to a common position from arbitrary initial conditions,

under the assumption that the sensing digraph is time-invariant and contains a globally

reachable node. The proposed feedback is very simple and is local and distributed. For

rendezvous of flying robots, a control strategy is presented that makes the centres of

mass of the vehicles converge to an arbitrarily small neighborhood of one another. The

convergence is global, and each vehicle can compute its own control input using local and

distributed feedback.

For formation control, the objective is to make an ensemble of kinematic unicycles

achieve pre-defined inter-agent spacings with parallel heading angles. We consider scenar-

ios where the formation either stops or moves with a final collective motion. In the latter

case, problems of linear and circular formation flocking and formation path following are

ii

studied. A control law is presented in each case that solves the problem for almost all

initial conditions. For stopping and flocking formations, the proposed control laws are

local and distributed while for formation path following, the control laws additionally

require each agent to measure its displacement from the path. The idea used to solve

the formation control problems is to rigidly attach an offset vector to the body frame of

each unicycle. It is shown that stabilizing the desired formation amounts to achieving

consensus of the endpoints of the offset vectors, and simultaneously synchronizing the

unicycles’ heading angles. Extension of formation control to flying robots using strictly

local and distributed feedback is not addressed in this work and remains a challenging

open problem.

iii

Acknowledgements

I would like to sincerely thank my supervisors Manfredi Maggiore and Luca Scardovi

for their continued dedication and availability during the course of my PhD program.

The time they have dedicated to meeting with me on a weekly basis, reviewing my work

and making suggestions has been greatly appreciated. They have been a great source of

wisdom and inspiration. I would also like to thank my family for their continued support

during my PhD program.

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Contents

1 Introduction 1

1.1 Coordination Problems Investigated in This Thesis . . . . . . . . . . . . 7

1.2 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Statement of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Modelling 15

2.1 Attitude Representation as a Rotation Matrix . . . . . . . . . . . . . . . 15

2.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Coordination Problems 31

3.1 Rendezvous of Flying Robots . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Rendezvous of Kinematic Unicycles . . . . . . . . . . . . . . . . . . . . . 32

3.3 Formation Control Problems for Kinematic Unicycles . . . . . . . . . . . 33

3.3.1 Formation Control . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.2 Formation with Final Parallel Collective Motion . . . . . . . . . . 36

3.3.3 Formation with Circular Collective Motion . . . . . . . . . . . . . 39

3.3.4 General Formation Path Following . . . . . . . . . . . . . . . . . 42

3.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Preliminaries 54

4.1 Basic Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

v

4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.2 Reduction Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.3 Lyapunov and Krasovskii-LaSalle Theorems . . . . . . . . . . . . 72

4.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.1 Basic Definitions in Graph Theory . . . . . . . . . . . . . . . . . 79

4.2.2 Classes of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.3 Graph Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Control Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3.1 Control Primitives for Single Integrators . . . . . . . . . . . . . . 87

4.3.2 Control Primitives for Double Integrators . . . . . . . . . . . . . . 91

4.3.3 Control Primitives for Rotational Integrators . . . . . . . . . . . . 92

4.3.4 Control Primitives for Rotating bodies in SO(3) . . . . . . . . . . 96

5 Rend. of Flying Robots with Loc. and Dist. Feedbacks 98

5.1 Solution of the Rendezvous Control Problem (RP− F) . . . . . . . . . . 98

5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 From Rendezvous to Formations . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4.1 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4.2 Proof of Lemma 5.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4.3 Proof of Lemma 5.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 118

6 Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 123

6.1 Solution of the Rendezvous Control Problem (RP− U) . . . . . . . . . . 123

6.1.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2.1 Proof of Theorem 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2.2 Proof of Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . 132

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6.2.3 Proof of Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7 Formations of Kinematic Unicycles 150

7.1 Solution to the Parallel Formation Problem (PP) . . . . . . . . . . . . . 150

7.2 Discussion of the Control Solution . . . . . . . . . . . . . . . . . . . . . . 153

7.3 Special cases: Line formations and full synchronization . . . . . . . . . . 155


7.5 Proof of Theorem 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.5.1 System dynamics in (xi, θi)i∈n coordinates . . . . . . . . . . . . . 159

7.5.2 Lyapunov analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.5.3 Lyapunov analysis in relative coordinates . . . . . . . . . . . . . . 162

7.5.4 Local asymptotic stability of Γp . . . . . . . . . . . . . . . . . . . 165

7.5.5 Almost semiglobal asymptotic stability of Γp . . . . . . . . . . . . 169

8 Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 172

8.1 Solutions to Formation Problems with Final Linear Motion (PFP, PFP-B

and LPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.2 Solutions to Formation Problems with Final Circular Motion (CFP and

CPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.2.1 Formation control of unicycles on a common circle . . . . . . . . . 177


8.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.5 Proof of Theorem 8.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.6 Proof of Theorem 8.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.7 Proof of Theorem 8.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9 General Formation Path Following 195

9.1 Solution to the General Formation Path Following Problem (GPP) . . . . 195


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9.3 Proof of Theorem 9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.3.1 Proof of Theorem 9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 205

10 Unicycle Formation Simulation Trials 212

10.1 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

10.2 Simulation Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.2.1 Study of µf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

10.2.2 Study of high gain parameters α and k . . . . . . . . . . . . . . . 222

10.2.3 Study of state-dependent undirected graphs . . . . . . . . . . . . 224

10.2.4 Study of directed graphs . . . . . . . . . . . . . . . . . . . . . . . 226

10.2.5 Study of input saturation . . . . . . . . . . . . . . . . . . . . . . 228

10.2.6 Study of disturbances and sampling . . . . . . . . . . . . . . . . . 230

10.2.7 Study of dynamic unicycles . . . . . . . . . . . . . . . . . . . . . 232

11 Conclusions and future research 234

11.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Bibliography 239

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Notation and Acronyms

(v1, . . . , vn) in (R)n column vectors v = [v1 · · · vn]⊤ in Rn are identified

with n-tuples (v1, . . . , vn) in (R)n

v · w := v⊤w Euclidean inner product of v, w ∈ Rn

‖v‖ := (v · v)1/2 the Euclidean norm of v

R+ the set of positive real numbers

e1, e2 the natural basis of R2

e1, e2, e3 the natural basis of R3

1 the vector of ones in Rn

⊗ the Kronecker product of matrices

S1 the unit circle, which we identify with the set of

real numbers modulo 2π

Sn the n-dimensional unit sphere

Tn the n-torus Tn := S1 × · · · × S1 (n times)

A\B the set-theoretic difference of the sets A and B

A the closure of the set A

|A| the cardinality of the set A

Bε(Γ) the ε neighborhood of the closed set Γ

N (Γ) a neighborhood of the closed set Γ

n 1, . . . , n

k :n k, . . . , n

ix

(xj)j∈I If I = i1, . . . , in is an index set, the ordered list of

elements (xi1 , . . . , xin) is denoted by (xj)j∈I

Lfϕ(x) If f(x) is a vector field and ϕ(x) is a C1 function

x 7→ Lfϕ(x) := (d/dx)ϕ(x) · f(x) is the Lie derivative

of ϕ along f

Notation for Kinematic Unicycle Model

Notation Explanation

xi ∈ R2 inertial position of unicycle i

x ∈ R2n (xi)i∈n

Ri ∈ SO(2) attitude of unicycle i

θi ∈ S1 heading angle of unicycle i

θ ∈ Tn (θi)i∈n

ωi ∈ R angular velocity of unicycle i

ri = R−1i r coordinate representation of r in frame Bi

xij = xj − xi relative displacement of robot j wrt. robot i

Ni set of neighbors of robot i

yi = (xij)j∈Ni vector of relative positions available to robot i

Notation for Flying Robots


mi, Ji mass and inertia matrix of robot i

xi ∈ R3 inertial position of robot i

x ∈ R3n (xi)i∈n

x

vi ∈ R3 linear velocity of robot i

v ∈ R3n (vi)i∈n

Ri ∈ SO(3) attitude of robot i

Ωi ∈ R3 angular velocity of robot i

Ω ∈ R3n (Ωi)i∈n

qi = −Rie3 thrust direction vector of robot i

Ti = −uiRie3 applied thrust vector of robot i

ri = R−1i r coordinate representation of r in frame Bi

xij = xj − xi relative displacement of robot j wrt. robot i

vij = vj − vi relative velocity of robot j wrt. robot i

Ωi ∈ R3 reference angular velocity of robot i

Ni set of neighbors of robot i

yi = (xij , vij)j∈Ni vector of relative positions and velocities available to robot i

Notation in Chapters 7 to 9


φi ∈ S1 auxiliary state of robot i

σi θi − φi

φi reference angle for φi

C path to follow

c⋆(x) orthogonal projection of x onto the line path C

τ ∈ S1 path parameter for C

(o(τ), r(τ), s(τ)) Frenet-Serret frame on C at τ ∈ S1

π : C → S1 maps a point on C to τ ∈ S1

θr : S1 → S1 angle of the tangent vector r(τ) relative to I

d11i ∈ R2 desired displacement of robot i wrt robot 1

xi

di ∈ R2 desired displacement of robot i wrt the path frame

ψi atan2(di · e2,−di · e1), the angle of di with respect to

the −r(τ) axis

w > 0 desired path following speed

c ∈ R2 circle center for path following

ρi ∈ S1 desired heading offset of robot i wrt robot 1

p ∈ R2 beacon in I

θp ∈ S1 angle of p in I

δi ∈ R2 offset vector attached to frame Biαi component of δi parallel to Rie1

βi component of δi perpendicular to Rie1

xi = xi + δi endpoint of the offset vector δi

τi = π(xi) the path parameter τ corresponding to xi ∈ C

fi((xij)j∈Ni) integrator consensus controller

gi((θij)j∈Ni, η) rotational integrator consensus controller

h(x) integrator path following controller

θh : R2 → S1 angle of vector field h relative to the inertial frame

θ′h : R2 → R derivative of θh as defined in (9.3)

θ′′h : R2 → R second derivative of θh as defined in (9.3)

ξi(σi, xi) σi − θh(xi) + ψi

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Acronyms

Next we present a list of acronyms for each control problem.

Problem Acronym Explanation

Rendezvous RP-F rendezvous problem for flying robots

RP-U rendezvous problem for kinematic unicycles

Γ rendezvous manifold

Formation PP parallel formation problem

Control F formation specification

Γp parallel formation manifold

Parallel PFP parallel formation flocking problem

Formation PFP-B parallel formation flocking problem with a beacon

Flocking PF parallel formation flocking specification

PFB parallel formation flocking specification with a beacon

Γpf parallel formation flocking manifold

Γpfb parallel formation flocking manifold with a beacon

Formation LPP formation line path following problem

Line Path LP formation line path following specification

Following Γlp formation line path following manifold

Circular CFP circular formation flocking problem

Formation CF circular formation flocking specification

Flocking Γcf circular formation flocking manifold

Formation CPP formation circle path following problem

Circle Path CP formation circle path following specification

Following Γcp formation circle path following manifold

General GPP general formation path following problem

Formation GP general formation path following specification

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Path Following Γgp general formation path following manifold

xiv

List of Tables

2.1 Table of Notation for Kinematic Unicycle Model . . . . . . . . . . . . . . 22

2.2 Table of Notation for Flying Robots . . . . . . . . . . . . . . . . . . . . . 28

4.1 Asymptotic and Practical Stability Abbreviations . . . . . . . . . . . . . 60

4.2 Neighbors sets in Figure 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 Table of Control Primitives . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1 Simulation Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2 Control Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.1 Simulation Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 128

10.1 Data Collected from Simulations . . . . . . . . . . . . . . . . . . . . . . . 219

10.2 Base Case Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

10.3 Base Case Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 222

10.4 Simulation Results Varying µf . . . . . . . . . . . . . . . . . . . . . . . . 222

10.5 Simulation Results Varying α . . . . . . . . . . . . . . . . . . . . . . . . 224

10.6 Simulation Results Varying k . . . . . . . . . . . . . . . . . . . . . . . . 225

10.7 Simulation Results Varying the Sensing Radius . . . . . . . . . . . . . . 226

10.8 Simulation Results for the digraph in Figure 10.9 . . . . . . . . . . . . . 228

10.9 Simulation Results with Input Saturation . . . . . . . . . . . . . . . . . . 229

10.10Simulation Parameters in Section 10.2.6 . . . . . . . . . . . . . . . . . . 231

10.11Simulation Results with Perturbations and Sampling . . . . . . . . . . . 231

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10.12Simulation Results for Dynamic Unicycles . . . . . . . . . . . . . . . . . 233

xvi

List of Figures

1.1 Quadrotor fire inspection. The center of the formation moves along the

path in the direction indicated. . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Example of snow removal on a highway. . . . . . . . . . . . . . . . . . . . 3

2.1 Inertial and body frames in two dimensions. . . . . . . . . . . . . . . . . 16

2.2 Inertial and body frames in three dimensions. . . . . . . . . . . . . . . . 16

2.3 Kinematic unicycle class. The right figure shows a differential drive robot

that is modelled as a unicycle. . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Bicycle class. Image from (Francis and Maggiore, 2016). . . . . . . . . . 24

2.5 Flying Robot Class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Illustration of a quadrotor helicopter taken from (Roza, 2012). . . . . . . 29

3.1 Formation in terms of fixed relative displacement vectors d11i, i ∈ 2 :n. . . 34

3.2 formation line path following . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 circular formation flocking with centre of rotation c . . . . . . . . . . . . 40

3.4 The Frenet-Serret frame with origin o(τ) on the smooth Jordan curve C. 43

3.5 Formation path following. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 (left) Diamond formation that rotates rigidly with the curve C and always

points in the path’s tangent direction. (right) Diamond formation defined

with respect to the inertial frame I that does not rotate as it traverses C. 53

xvii

4.1 Illustration on the left shows stability of Γ - solutions with initial conditions

in N (Γ) remain inside Bε(Γ) for all time. Illustration on the right shows

attractivity of Γ with domain of attraction D(Γ) - solutions with initial

conditions in D(Γ) converge to Γ. . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Illustration of almost global attractivity of the set Γ. The set X\D(Γ) has

Lebesgue measure zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Illustration of almost semiglobal attractivity of the set Γ. For compact

sets K1 ⊂ K2 ⊂ X\N , that do not contain the set N of Lebesgue measure

zero, for all k ≥ k⋆1 (left) and k ≥ k⋆2 > k⋆1 (right), solutions with initial

conditions in the sets K1 and K2 respectively, converge to Γ. The domain

of attraction of Γ approaches full measure as the high gain k is increased. 60

4.4 Illustration of global practical stability of the set Γ. For ǫ1 > ǫ2, for any

k ≥ k⋆1 (left) and k ≥ k⋆2 > k⋆1 (right), for any initial conditions in X ,

solutions converge to the neighborhoods Bǫ1(Γ) and Bǫ2(Γ) respectively. . 61

4.5 Illustration of local stability of Γ2 near Γ1 (left) and local attractivity of

Γ2 near Γ1 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6 Example of local stability of Γ2 near Γ1 in Example 4.1.12. Image from (Mag-

giore, 2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Illustration of the Reduction Theorem. The set Γ1 is asymptotically sta-

ble relative to Γ2 and Γ2 is asymptotically stable. By reduction, Γ1 is

asymptotically stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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4.8 Illustration of reduction for almost global asymptotic stability. The figure

on the left illustrates the dynamics on Γ2 which is globally asymptotically

stable relative to X . Γ2 contains three unstable equilibria A = z1, z2, z3

and a compact set K that is almost globally asymptotically stable relative

to Γ2. By the Reduction Thoerem the set K is also AGAS relative to X

as shown in the figure on the right. The set D(A) is Lebesgue measure

zero and contains the initial condition χ0 whose solution converges to the

point z3 ∈ A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.9 Example of a directed (left) and an undirected (right) graph. . . . . . . . 79

4.10 Example of a subgraph G ′ = (V ′, E ′) of the directed graph in Figure 4.9

with V ′ = 1, 2, 3, 5 and E ′ = (5, 1), (5, 2), (2, 3), (3, 5). . . . . . . . . . 80

4.11 Reverse directed spanning tree with root node 1. . . . . . . . . . . . . . . 81

4.12 Directed graph G containing a reverse directed spanning tree and globally

reachable node, node 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.13 Hierarchical sensing graph . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.14 Directed graph G containing a reverse directed spanning tree. The strongly

connected components G0, . . . ,G3 are boxed . . . . . . . . . . . . . . . . . 85

4.15 Condensation digraph C(G) associated with the graph G in Figure 4.14

(left) and reverse directed spanning tree contained in C(G) (right). . . . . 85

4.16 Illustration of properties B1, B2 and B3. . . . . . . . . . . . . . . . . . . 95

5.1 Block diagram of the rendezvous control system for robot i. The outer

loop assigns a desired thrust vector fi(yii). The inner loop thrust control

uses fi(yii) to assign the vehicle input ui while the rotational control uses

fi(yii) to assign the torque input τi. The vector yii contains the relative

displacements and velocities of vehicles that robot i can sense, measured

in the body frame of robot i. . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Illustration of the control input ui and reference angular velocity Ωi in (5.1).102

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5.3 Sensor digraph used in the simulation results. . . . . . . . . . . . . . . . 105

5.4 Rendezvous control simulation without the presence of disturbances. At

the top-left, top-right and bottom-left: positions of the five robots ex-

pressed in the inertial frame I. At the bottom-right: linear speeds ‖vi‖, i ∈

1 : 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5 Rendezvous control simulation with the presence of disturbances. At the

top-left, top-right and bottom-left: positions of the five robots expressed

in the inertial frame I. At the bottom-right: linear speeds ‖vi‖, i ∈ 1 : 5. 107

6.1 Block diagram of the rendezvous control system for robot i. . . . . . . . 125

6.2 Illustration of the control inputs ui = u⋆i (yii) and ωi = ω⋆i (y

ii) in (6.1). . . 127

6.3 Sensor digraph used in the simulation results. . . . . . . . . . . . . . . . 128

6.4 Rendezvous control simulation for: (a) proposed feedback in (6.1), (b)

feedback in (Lin et al., 2005), and (c) feedback in (Zheng et al., 2011) . . 129

6.5 Simulation control inputs for proposed feedback in (6.1) . . . . . . . . . . 129

7.1 Representation of the offset vector δi. . . . . . . . . . . . . . . . . . . . . 151

7.2 Parallel formation where offset vectors xi meet at a common point at a

distance α in front of the formation. . . . . . . . . . . . . . . . . . . . . . 153

7.3 Block diagram of the formation control system for robot i. . . . . . . . . 154

7.4 (a) shows an example of a parallel line formation while (b) shows an ex-

ample of full synchronization, a special case of a parallel line formation. . 156

7.5 Undirected graph G under consideration in the simulation results. . . . . 157

7.6 Triangular formation specified by the offset vectors d112 = (−10, 5), d113 =

(−10,−5), d114 = (−20, 10) and d115 = (−20,−10). . . . . . . . . . . . . . 158

7.7 Simulation for a triangle formation. Initial positions are indicated with

and final positions are indicated with ×. . . . . . . . . . . . . . . . . . . 159

8.1 Undirected graph G under consideration in the simulation results. . . . . 179

xx

8.2 Simulation results for formations with final parallel collective motion: (a)

formation flocking with beacon p = (1, 0) pointing in the direction of

the positive x-axis (b) formation flocking with no beacon where µii =

(w/|Ni|)∑

j∈NiRije1, i ∈ n, (c) formation path following for C(r0, p) =

x ∈ R2 : x = r0 + sp, s ∈ R with r0 = (200, 0), p = (0, 1). Initial

positions are indicated with and positions at the end of the simulation

are indicated with ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.3 Simulation for formations with circular collective motion (a) formation

flocking, (b) formation path following around the point c = (100, 0). Initial

positions are indicated with and positions at the end of the simulation


8.4 Desired formation of unicycles (x, θ) (shaded) is achieved when (z, θ) (not

shaded) lie on a common circle of radius r with the desired spacing θ1i =

ρi(d, β1) for all i ∈ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.1 Illustration of the angle φi and angle σi = θi − φi. . . . . . . . . . . . . . 197

9.2 Illustration of the set Λ in which unicycle i can lie anywhere on the dashed

circle. The desired position of unicycle i is marked by ⋆. . . . . . . . . . 197

9.3 Illustration of angle condition σi = θr(τi)− ψi. . . . . . . . . . . . . . . . 198

9.4 Digraph G under consideration in the simulation results. . . . . . . . . . 202

9.5 Formation specified by the offset vectors d1 = (15, 0), d2 = (5,−8), d3 =

(5, 8), d4 = (−10,−3) and d5 = (−10, 3). . . . . . . . . . . . . . . . . . . 202

9.6 Simulation results: (a) illustrates the curve C corresponding to the stable

limit cycle of the Van der Pol oscillator with µ = 1.5, c0 = 0.04 and shows

the time evolution of x1 illustrated by the dashed line, (b) illustrates the

individual paths traversed by the five unicycles as the formation moves

around C. Initial positions are indicated with and positions at the end

of the simulation are indicated with ×. . . . . . . . . . . . . . . . . . . . 203

xxi

9.7 Conceptual illustration of the reduction sets Γ1, Γ2 and Γ3. The set Γ3 is

asymptotically stable, the compact set Γ2 is asymptotically stable relative

to Γ3 and the point Γ1 is asymptotically stable relative to Γ2. Reduction

implies the set Γ1 is asymptotically stable as illustrated by the solution

starting at χ0 off the set Γ3. . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.1 Illustration of the average of two angles (a) (θ1, θ2) = (0, 3π/2) rad and

(b) (θ1, θ2) = (0, π) rad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

10.2 Illustration of the formation measure. . . . . . . . . . . . . . . . . . . . . 216

10.3 Formation measure. The unicycles at the initial time t0 are not shaded

while the unicycles at time tf are shaded. The average position of the

unicycles are indicated with . In the top figure, x(t0) and x(tf) are the

same and therefore µd = 0. In the bottom figure ‖x(tf )− x(t0)‖ 6= 0 and

therefore drift is present which is roughly twice the size of the formation

itself. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.4 Interaction function f(s) and g(s). . . . . . . . . . . . . . . . . . . . . . 218

10.5 Formation specified by d112 = (−10, 5), d113 = (−5,−5), d114 = (−20, 10)

and d115 = (−15,−10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

10.6 Undirected graph used in the base case. . . . . . . . . . . . . . . . . . . . 221

10.7 Illustration of configurations satisfying different values of µf . The centre of

the circles represent the desired unicycle positions while the actual unicycle

positions are indicated with ×. . . . . . . . . . . . . . . . . . . . . . . . 223

10.8 Illustration of a formation where one agent is disconnected from the rest

of the group. Regardless of this, the rest of the group achieves formation

among themselves. Initial positions are indicated with and final positions


10.9 Directed ring-coupled graph. . . . . . . . . . . . . . . . . . . . . . . . . . 227

10.10Typical input profile during simulation. . . . . . . . . . . . . . . . . . . . 228

xxii

10.11Input profile during simulation for µu,1 = 5 m/s and µω,1 = π/2 rad/s. . 230

10.12Simulation result in the presence of disturbances and sampling rate of 100Hz.232

10.13Simulation result in the presence of disturbances and sampling rate of 10Hz.232

10.14Simulation result for dynamic unicycles. . . . . . . . . . . . . . . . . . . 233

xxiii

Chapter 1

Introduction

The goal of this thesis is to develop control algorithms to coordinate the motion of

autonomous teams of robots in order to achieve some desired collective goal. As an

example, consider a team of autonomous quadrotor helicopters equipped with on-board

cameras given the task of monitoring a forested area during the dry season, searching for

any sign of fire. The control objective would be to make the vehicles organize an efficient

formation that follows a desired path at a desired velocity such that the entire dry area

is scanned. This could be achieved, for example, by flying in a parallel fashion between a

series of straight line segments, circling about each way-point with a desired radius and

for a desired time duration. This scenario is illustrated in Figure 1.1.

Typical coordination problems include attitude synchronization, rendezvous, flock-

ing, and formation control, and find application in many areas such as manufactur-

ing or repair tasks, sensor networks, search and rescue, forest fire control missions, or

surveillance/inspection. Depending on the situation, some tasks may be better suited

for ground-based robot teams, while others may be better suited for flying robots. A

common objective in coordination control is to make the ensemble of robots stabilize

a specific spatial configuration characterized by fixed desired inter-agent displacements.

For example, flocking birds or aircraft formations at an air show are often coordinated

1

Chapter 1. Introduction 2

Figure 1.1: Quadrotor fire inspection. The center of the formation moves along the pathin the direction indicated.

in a characteristic triangular configuration with constant inter-agent displacements. A

special case is called rendezvous when the desired inter-agent displacements all equal

zero, i.e., the robots’ positions coincide.

It is often the case that one would also like the formation to move with a desired

collective motion while maintaining the desired inter-agent spacing at all times. Two

types of collective motions that we consider in this thesis are flocking and path following.

Flocking formations have linear or circular collective motions but the roto-translation

of the line or centre of the circle are not defined a priori. Rather, they depend on the

initial conditions of the agents. The study of parallel and circular flocking for kinematic

unicycles is useful for gaining insight into how coordination seen in nature occurs with

very limited information. In particular, parallel and circular flocking arise in nature with

flocks of birds and schools of fish where no specific path is agreed on ahead of time.

On the other hand, in path following, the formation follows a specific path defined

a priori such as in the forest fire example in Figure 1.1. A key requirement in path

following is that the path be invariant. Namely, if the formation is initialized on the

path, it must stay on the path at all times. This is in contrast with a common trajectory

tracking approach, in which the formation is made to follow a reference point that moves


along the path according to a specific time parametrization. This latter approach is

undesirable in practice because if the formation is initialized on the path but not at the

same location as the reference point, then it could leave the path, and may collide with

off-course objects. There are many practical applications for formation path following.

For example, one may like to have formations of robots follow a series of linear and

circular path segments painted or taped on the floor of a warehouse to retrieve and stock

merchandise. Each robot is only able to measure, using on-board cameras, its relative

position and orientation to other agents as well as its displacement to the closest point

on the path (which may also be measured using on-board cameras). Another example is

a team of automatic snowploughs with the task of snow removal on highways. Typically,

the snowploughs form a diagonal line formation, as illustrated in Figure 1.2, passing snow

accumulated at each plough from one vehicle to the other until it clears to the shoulder

of the highway. In Figure 1.2 snow is being passed from unicycle 1 to unicycle 4 before

clearing the highway. Each plough only observes the vehicle to its left and right in the

process. In nature, formation circle path following is observed, for example, with fish

encircling a food source at a specific location.

Figure 1.2: Example of snow removal on a highway.

The work in this thesis solves a number of coordination problems for two important


classes of robots.

The first class of robots corresponds to ground-based mobile robots modelled as kine-

matic unicycles. Each robot moves in the two-dimensional plane, can rotate freely with a

desired angular speed, but its velocity is restricted to be parallel to its heading direction.

With three degrees-of-freedom and two actuators, each robot is under-actuated with

degree of under-actuation one and its configuration space is SE(2). The second class

of robots corresponds to flying robots. Each robot moves in three-dimensional space

and is endowed with an actuation mechanism producing torques about three orthogo-

nal body axes, but its thrust vector is restricted to be parallel to its thrust axis. With

six degrees-of-freedom and four actuators, each robot is under-actuated with degree of

under-actuation two and its configuration space is SE(3). A quadrotor helicopter is an

example of such a robot. Together, these two classes of robots can be employed in a

broad range of multi-agent coordination applications both on land and in the air.

One of the main difficulties in solving coordination problems for these two classes

of robots arises from nonholonomic constraints which only permit vehicles to move or

accelerate instantaneously along a single body axis.

Another difficulty in solving the coordination problems arises from the constraints

that we impose on sensing and the sharing of information. For kinematic unicycles, each

agent is permitted only to sense its heading angle and position relative to a subset of

neighbors, measured in its own body frame. From a practical standpoint, this infor-

mation can be sensed locally by each agent using a fixed on-board camera. We do not

permit the unicycles to have access to global information like the group centre of mass or

centralized information that is broadcast to the group from a central station. Moreover,

the agents cannot communicate with each other, in the sense of requiring an on-board

communication system using transmitters and receivers to send and receive electromag-

netic signals. This important distinction between sensing and communication will be

used throughout this work. A feedback meeting these sensing requirements is called lo-


cal and distributed, and from a practical standpoint it has the beneficial property that

it can be implemented on each unicycle using information sensed locally from a fixed

on-board camera. In the case of flying robots, each agent typically mounts a camera and

an inertial measurement unit (IMU) that includes a three-axis rate-gyroscope, so that

the robot is able to sense, in the coordinates of its own frame, the relative displacements

and velocities of nearby vehicles, and its own angular velocity. Robots can neither access

their own inertial position and velocity, nor their own attitude.

In this work, as is standard in the literature, the sensing between robots is modelled

as a graph, and is assumed to be static, meaning that the neighbors of each vehicle

remain fixed at all time. In most of the problems considered, unless stated otherwise,

the sensing digraph is assumed to be any digraph containing a globally reachable node.

This is a necessary connectivity condition to solve multi-robot coordination problems for

fixed graphs. The assumption of fixed sensing graphs is questionable in practice, but

is made to render the problem mathematically treatable. In a more realistic situation,

the sensing would be state-dependent, and each unicycle would only sense neighboring

unicycles lying within a given radius of itself. This remains a challenging open problem

for much simpler classes of robot models, such as double-integrators. While it will not be

formally proven that the results in this thesis work for state-dependent graphs, we will

present extensive simulation results to show that our solutions also work considerably

well for state-dependent graphs when the sensing radius is sufficiently large. We will also

run extensive simulation results to illustrate robustness of the approach to unmodelled

effects including sensor noise, input noise and collisions.

Problems will be defined in terms of two specifications. The first specification cor-

responds to the desired formation of the multi-agent system that is stabilized using a

local and distributed control. A second specification enforces a desired final motion of

the ensemble and, in the case of path following, may use cameras or a GPS system to

acquire information about the path. An advantage of this setup is that it is robust to


losses of the GPS signal, in which case the robots will maintain the formation and the

path control can resume once the signal is recovered.

Based on the discussion above, at the most basic level, the aim of this work is

to present solutions to multi-agent coordination problems for classes of nonholonomic

agents in a broad spectrum of ground and air applications. This area has been actively

researched in the last decade. Much of the current research results, however, suffer in

that they apply ad-hoc methodologies and solutions with local domains of operation.

The main contribution and most difficult aspect of this research is its strong emphasis

on enforcing local and distributed feedbacks. Except for problems with a path follow-

ing element, there is no reliance on external landmarks or measurement of a common

inertial frame. The feedbacks are completely independent of time, and do not require a

communication system, thus avoiding the need for synchronized clocks and eliminating

issues surrounding communication delays that ultimately limit performance. This also

makes the solutions presented in this work cost-effective in applications. Moreover, the

feedback expressions that we propose are compact, which makes them very efficient to

compute in real-time on-board each robot. The feedbacks are also highly compartmen-

talized due to the fact that they are constructed out of simpler control primitives such as

consensus controllers for single or double integrators and rotational integrators, making

them easy to interpret. Another important characteristic of the solutions presented in

this thesis is that they are, in most cases, almost global or semi-global, meaning that the

collective motions will be achieved for large sets of initial conditions. Finally, the control

solutions presented in this work are robust against the loss of agents. As long as the

graph maintains a globally reachable node after the loss, the control strategies will still

work. This is in contrast to leader-follower based approaches where the loss of the leader

robot compromises the correct functioning of the control strategies.

Based on these considerations, our solutions very much emulate the distributed co-

ordination behaviour seen in nature by animals, for example, in a herd or flock. These


animals are able to accomplish incredible feats of coordination just by sight and by sensing

the type of information that an IMU unit can provide such as angular velocities. These

formations in nature often arise without requiring much communication beyond sight (al-

though some vocalization may occur) or the sensing of surrounding landmarks (although

it could certainly be influenced by the presence of a food source) and the animals cer-

tainly do not need to measure time. Therefore, there is also an important philosophical

element at the core of this work as it tries to understand and make connections with

coordination phenomena in nature.

Each result in this thesis has had a contribution to the current literature and most

of the results have been published in scholarly journals as discussed in Section 1.3.

1.1 Coordination Problems Investigated in This The-

sis

In this section, we briefly outline the multi-agent coordination problems that will be

studied in this thesis. A common goal in all of the control problems is to stabilize a

pre-specified configuration and, in addition, to optionally enforce a final collective mo-

tion with domain of attraction as large as possible. The difference between rendezvous

and formation control is that with rendezvous, the desired configuration corresponds

to the case in which all inter-agent displacements are zero and relative heading angles

are unconstrained. For the formation problems, on the other hand, the inter-agent dis-

placements are non-zero and relative heading angles will be constrained. The difference

between stopping formations and those with a final collective motion is that, with the

former, the configuration does not translate or rotate in the inertial frame, while with

the latter, it does.

P1. Rendezvous of flying robots (Chapter 5). The objective of the rendezvous

problem for flying robots is to design smooth, local and distributed feedbacks for


each flying robot so as to drive the group to an arbitrarily small neighborhood of

one another from arbitrary initial conditions.

P2. Rendezvous of kinematic unicycles (Chapter 6). The objective of the ren-

dezvous problem for kinematic unicycles is to design smooth, local and distributed

feedbacks for each kinematic unicycle so as to drive the group to a common location

from arbitrary initial conditions.

P3. Formations of kinematic unicycles (Chapter 7). The objective of the for-

mation control problem for kinematic unicycles is to design smooth, local and dis-

tributed feedbacks for each kinematic unicycle so as to drive the ensemble, for

almost all initial conditions in any given compact set, to a parallel formation, i.e.,

one in which the unicycles’ headings are all parallel, and their relative displacement

vectors take on appropriate values corresponding to a desired geometric pattern.

Moreover, it is required that the unicycles come to a stop when they meet the

formation requirement.

The utility of stopping formations is manifest in problems where vehicles are re-

quired to distribute themselves over a terrain in order, for instance, to form an

antenna array or a sensor network. More generally, the problem investigated in

this chapter is a conceptual gateway to the problem of inducing collective motions

in formations.

P4. Formations of kinematic unicycles with final linear and circular col-

lective motions (Chapter 8). The objective of the formation control problem

with final linear and circular collective motions is to design smooth feedbacks for

each kinematic unicycle so as to drive the ensemble, for almost all initial condi-

tions, to a formation in which relative displacement vectors take on appropriate

values corresponding to a desired geometric pattern. Moreover, it is required that

the unicycles move with a final collective parallel or circular motion when they


meet the formation requirement. For formation flocking we require the feedbacks

be local and distributed. For parallel formation flocking with a beacon, unicycles

are permitted also to measure a common beacon in body frame corresponding to

the flocking direction. For formation path following unicycles are permitted also to

compute their perpendicular projection onto the desired path.

P5. General formation path following for kinematic unicycles (Chapter 9).

The objective of the general formation path following problem is to design smooth

feedbacks for each kinematic unicycle so as to asymptotically stabilize a desired

formation defined with respect to the Frenet-Serret frame on a smooth Jordan

curve, possibly having high curvature. Moreover, it is required that the formation

follows the Jordan curve with a desired speed, rotating rigidly with respect to the

curve. We relax the sensor requirements, allowing for some minimal communication

and GPS measurements, due to the complexity of the problem.

1.2 Thesis organization

The main body of this thesis is organized into the following chapters.

• Chapter 2: Modelling

In this chapter we present models for kinematic unicycles and flying robots.

• Chapter 3: Coordination Problems

In this chapter we will introduce the control problems investigated in this thesis

for teams of flying robots and kinematic unicycles.

• Chapter 4: Preliminaries

This chapter contains fundamental definitions and results that will be applied on

a regular basis throughout this thesis.


• Chapter 5: Rendezvous of Flying Robots by means of Local and Distributed Feed-

backs

This chapter presents a solution to P1.

• Chapter 6: Rendezvous of Kinematic Unicycles by means of Local and Distributed

Feedbacks


• Chapter 7: Formations of Kinematic Unicycles


• Chapter 8: Formations of Kinematic Unicycles with Parallel and Circular Collective

Motions

In this chapter, we present almost global solutions for four sub-problems of P4:

parallel formation flocking with and without a beacon (PFP-B, PFP), circular for-

mation flocking (CFP), formation line path following (LPP) and formation circle

path following (CPP).

• Chapter 9: General Formation Path Following of Kinematic Unicycles


• Chapter 10: Unicycle Formation Simulation Trials

This chapter presents extensive simulation trials to study the effectiveness of our

control solution presented in Chapter 7 for formation control of unicycles under

different realistic scenarios not captured by the main theoretical result in that

chapter. This includes performance in the presence of undirected state dependent

sensor graphs, relaxation of high gain requirements and robustness to unmodelled

effects including sensor noise, input noise, sampling and saturated inputs.


1.3 Statement of contributions

What follows is a list of significant original contributions made in this thesis.

1. Chapter 4

• Reduction theorem for almost global asymptotic stability in Theorem 4.1.3

based on lemmas 4.1.15, 4.1.16, 4.1.17.

2. Chapter 5

• The problem of rendezvous of flying robots (P1) using strictly local and dis-

tributed feedbacks is open. Theorem 5.1.1 is the first local and distributed

solution yielding global practical stability.

• This work has been published in the IEEE Transactions on Automatic Control

(TAC) in (Roza et al., 2017).

3. Chapter 6

• The solution presented in Theorem 6.1.1 for rendezvous of kinematic unicy-

cles (P2), is the first one for directed graphs containing a globally reachable

node involving feedbacks that are local and distributed, time-independent,

and continuously differentiable. Brockett’s necessary condition in Theorem

1 of (Brockett, 1983), implies that an equilibrium point cannot be asymp-

totically stabilized for under-actuated driftless systems, including kinematic

unicycles, using a continuously differentiable, time-invariant control law. As

a consequence, one cannot asymptotically stabilize simultaneously a desired

position and angle of a kinematic unicycle using a continuously differentiable,

time-invariant control law. One requires either discontinuous, time-varying, or

dynamic feedback. Our solution to rendezvous of kinematic unicycles (as well

as other problems) does not contradict Brockett’s necessary condition because


the control problem involves asymptotic stabilization of a set as opposed to

a particular equilibrium point in the state space. In particular, we do not

specify, a priori, a final position and angle of the unicycles in the ensemble.

Previous solutions to the rendezvous problem require either time-varying

or discontinuous feedback or are restricted to undirected sensing graphs. We il-

lustrate through simulations in Section 6.1.1 that the proposed time-independent,

continuously differentiable feedback has practical advantages over the time-

varying feedback in (Lin et al., 2005) in that it induces a more natural be-

haviour in the ensemble of unicycles. The feedback in (Lin et al., 2005) makes

the unicycle “wiggle” indefinitely, a behaviour which would be unacceptable in

practice. The feedback in (Zheng et al., 2011) is restricted to undirected sens-

ing graphs and it does not achieve rendezvous for directed graphs containing

a globally reachable node. The feedback in (Dimarogonas and Kyriakopou-

los, 2007) induces instantaneous changes in direction that are impossible to

achieve with realistic implementations.

• This work has been published in the IEEE Transactions on Control of Network

Systems (TCNS) in (Roza et al., 2018).

4. Chapter 7

• The problem of stabilizing static parallel formations (P3) by means of smooth,

local and distributed feedbacks is to date open. Theorem 7.1.1 presents an

almost semiglobal solution to this problem, i.e., a solution making the unicy-

cles achieve the desired objective for almost all initial conditions in any given

compact subset of their collective state space.

• The conditions in Theorem 7.1.1 are relaxed for parallel line formations in

Corollary 7.3.1. A special case of this setup is full synchronization of the

unicycles, a problem of note in its own right. In this case, our solution is to be


compared to the one in (Dimarogonas and Kyriakopoulos, 2007), which relies

on discontinuous control, but considers a more general class of sensor graphs

than ours.

• This work has been conditionally accepted for publication in the IEEE Trans-

actions on Automatic Control (TAC) in 2018.

5. Chapter 8

• For PFP, PFP-B, Theorem 8.1.1 is the first almost-global solution that does

not require communication.

• For CFP, Theorem 8.2.1 is the first almost global solution with local and

distributed feedbacks, not restricted to formations on a common circle.

• We present the first almost global solutions for LPP,CPP in Theorem 8.1.1

and Theorem 8.2.1 respectively for general classes of sensing graphs (digraphs

containing a globally reachable node for line paths and connected undirected

graphs for circle paths) that do not require communication.

6. Chapter 9

• For general formation path following (P5) all approaches in the literature

require at least one of the following assumptions:

– the path to be followed is parametrized by time (formation trajectory

tracking)

– each unicycle needs to compute its own separate path to follow (unrealistic

in practice),

– the digraph has a leader-follower topology,

– all unicycle control inputs require global information, e.g., the group cen-

tre of mass.


Theorem 9.1.1 presents the first solution to P5 along any smooth Jordan curve

for any digraph containing a globally reachable node and not requiring any

of the assumptions above. Due to the complexity of the problem, this solu-

tion possesses some drawbacks compared to the other chapters in this thesis.

Namely, it only has a local domain of attraction and requires communication

of auxiliary states. Resolving these issues can be studied in future research.

Chapter 2

Modelling

In this chapter we present models for kinematic unicycles and flying robots. We start with

a discussion on how the attitude of these vehicles can be represented as rotation matrices

in the special orthogonal groups SO(2) and SO(3). A large potion of the material has

been taken from an analogous discussion in the master’s thesis in (Roza, 2012). Next we

present the system equations for both models. For kinematic unicycles, positions evolve

in Euclidean space R2 while the attitude evolves in SO(2). For flying robots, on the

other hand, positions evolve in R3 and the attitude evolves in SO(3). The combination

of translations in Euclidean space Rk and attitudes in SO(k) for k ∈ 2, 3, known as the

configuration space, also has a group structure and is referred to as the special Euclidean

group and denoted SE(k). Finally, we will discuss how to model the sensing between

neighboring robots.

2.1 Attitude Representation as a Rotation Matrix

Consider a team of n robots and two right-handed orthogonal frames, I and Bi, either

in R2 (Figure 2.1) or R3 (Figure 2.2). Suppose that I is an inertial frame, while Bi is

attached to the i-th agent in an ensemble of robots labelled 1, . . . , n. The attitude of

15

Chapter 2. Modelling 16

Figure 2.1: Inertial and body frames in two dimensions.

I

Bi

ox

oy

oz

bix

biy

biz

Ri

Figure 2.2: Inertial and body frames in three dimensions.

robot i is defined to be the orientation of frame Bi with respect to the inertial frame I.

A rotation matrix R is a k × k matrix in the special orthogonal group SO(k) for

k ∈ 2, 3, defined as

SO(k) = R ∈ Rk×k : R⊤R = Ik = RR⊤, det(R) = 1.

In the above, Ik is the k× k identity matrix. The definition readily implies that rotation

matrices have the property that R−1 = R⊤. One can show, as its name suggests, that

SO(k) constitutes a group under the operation of matrix multiplication. Now consider

the coordinate frames in Figure 2.1 and Figure 2.2, and define the rotation matrix Ri of


frame Bi with respect to frame I as

Ri :=

bix · ox biy · oxbix · oy biy · oy

,

Ri :=

bix · ox biy · ox biz · oxbix · oy biy · oy biz · oybix · oz biy · oz biz · oz

(2.1)

for the cases of R2 and R3 respectively, where “·” denotes the dot product of two geometric

vectors. One can check that Ri as defined in the first equation of (2.1) is in SO(2) while

the second is in SO(3). Vice versa, a rotation matrix Ri uniquely identifies a relative

orientation of frame Bi with respect to frame I, since the columns of Ri are the coordinate

representations of the coordinate axes (bix, biy) and (bix, biy, biz) respectively in the frame

I. In conclusion, SO(k) can be viewed as the set of attitudes of a robot in Rk for

k ∈ 2, 3.

Rotation matrices can be made to act on Rk for k ∈ 2, 3 via multiplication, giving

rise to rotational transformations. That is, if vb in Rk is the coordinate representation of

a geometric vector v in frame Bi, its representation in the coordinates of I is Rivb.

The set SO(k) can be given the structure of a smooth manifold which is compact,

connected, and of dimension k for k ∈ 2, 3. One can show that the tangent space to

SO(k) at the identity element I is the set of skew-symmetric matrices,

so(k) := S ∈ Rk×k : S⊤ = −S.

The sets so(2) and so(3) are vector spaces isomorphic to R and R3 via the isomorphisms


s : R → so(2) and S : R3 → so(3) respectively,

s(ω) =

0 −ω

ω 0

,

S(Ω) =

0 −Ω3 Ω2

Ω3 0 −Ω1

−Ω2 Ω1 0

.

(2.2)

To simplify notation, in this thesis these isomorphisms will be represented with a su-

perscript × as ω× := s(ω) for ω ∈ R and Ω× := S(Ω) for Ω ∈ R3. We represent

the inverse operation with a subscript ×. For example, given a skew-symmetric matrix

M = −M⊤ ∈ so(3), we denote M× := [M32 M13 M21]⊤ where Mij is the (i, j)-th entry

of M . One can show that the set SO(k) constitutes a Lie group with corresponding

Lie algebra so(k) for k ∈ 2, 3. The tangent space at a generic element R ∈ SO(k) is

obtained through left-multiplication by R of matrices in so(k),

TRSO(k) = Rso(k) = RS : S ∈ so(k). (2.3)

This result restricts the structure of the kinematic equations of a rotating body as follows.

If one wants to describe the evolution of a rotation matrix using a vector field on SO(k),

R = f(R), the vector field must have the property that, for all R, f(R) ∈ TRSO(k).

Hence, in light of (2.3), the vector field in two and three dimensions must have the

structure

R = R s(ω), (2.4)

R = RS(Ω), (2.5)

respectively where ω ∈ R and Ω ∈ R3 is an input parameter that may depend on time.

One can show that ω and Ω are precisely the angular velocities of the rotating body in


R2 and R3 respectively. Because of this fact, and in light of the fact that s : R → so(2)

and S : R3 → so(3) are isomorphisms, we can think of elements of so(2) and so(3) as

angular velocities, and equations (2.4) and (2.5) are called the kinematic equation of a

rotating rigid body.

Some important properties of skew-symmetric matrices are given below.

s(ω)⊤ = − s(ω)

(s(ω)d) · d = 0, d ∈ R2

S(Ω)⊤ = −S(Ω)

S(Ω)d = Ω× d, d ∈ R3

S(RΩ) = RS(Ω)R⊤, R ∈ SO(3).

2.2 System model

Having reviewed the rotation matrix parametrization of attitude in two and three di-

mensions, we are ready to introduce the two vehicle classes investigated in this thesis.

The first class corresponds to ground-based mobile robots each modelled as kinematic

unicycles in SE(2). The second class corresponds to flying robots in SE(3).

Kinematic Unicycles

We begin by modelling a group of n kinematic unicycles. We fix an orthogonal frame

I = ox, oy in R2, and attach to unicycle i an orthogonal body frame Bi = bix, biy in

such a way that bix is the heading axis of the unicycle (as in Figure 2.1). We denote by

xi ∈ R2 the position of unicycle i in the coordinates of frame I. The attitude of body

frame Bi relative to I is represented by a rotation matrix Ri ∈ SO(2). Unicycle i’s wheels

point in the direction of its heading axis Rie1 which represents the instantaneous axis of

motion for unicycle i. Since the velocity xi cannot act along the second body axis Rie2,


Figure 2.3: Kinematic unicycle class. The right figure shows a differential drive robotthat is modelled as a unicycle.

this results in the nonholonomic constraint

xi · Rie2 = 0.

Letting θi ∈ S1 be the angle between vectors ox and bix and identifying S1 with the set

of real numbers modulo 2π, one can write the rotation matrix Ri in terms of θi with the

isomorphism

Ri = Ri(θi) =

cos θi − sin θi

sin θi cos θi

,

illustrating the fact that S1 ∼= SO(2). With these conventions and identifying the unit

circle with the set of real numbers modulo 2π, the model of unicycle i is

xi = uiRie1 (2.6)

θi = ωi, i ∈ n, (2.7)


where the pair (ui, ωi), composed of the linear and angular speeds of unicycle i, is the

control input. A kinematic unicycle model is illustrated in Figure 2.3 which includes an

image of an actual robotic unicycle in the lab referred to as a differential-drive robot.

The configuration space of each robot therefore consists of a translation in R2 and an

attitude in SO(2) and can be written as an element in the Special Euclidean Group

SE(2) =

R x

0 1

∈ R

3×3 : R ∈ SO(2), x ∈ R2

which is isomorphic to R2×S1 and just like SO(2) and SO(3), constitutes a group structure

with matrix multiplication as the group operation. The group identity e and inverse are

given respectively by

e =

I 0

0 1

,

R x

0 1

−1

=

R⊤ −R⊤x

0 1

.

(2.8)

We collect the translational and rotational states into the vectors x := (xi)i∈n ∈ R2n and

θ := (θi)i∈n ∈ Tn. The overall system state space is R2n×Tn which also inherits a group

structure through the Cartesian product.

The relative displacement of robot j with respect to robot i is xij := xj−xi while the

relative angles are given by θij = θj − θi. The rotation of robot j with respect to frame

i is defined by Rij := (Ri)

−1Rj, and it is a function of θij . If v ∈ R2 is the coordinate

representation of a vector in frame I, then we denote by vi := R−1i v the coordinate

representation of v in body frame Bi. The notation for the kinematic unicycle is given in

Table 2.1.

Next we will model the sensing between unicycles in the ensemble. In particular,

as the unicycles perform the desired control task, not every unicycle will be able to see


Table 2.1: Table of Notation for Kinematic Unicycle Model

Quantity Description

xi ∈ R2 inertial position of unicycle ix ∈ R2n (xi)i∈nRi ∈ SO(2) attitude of unicycle iθi ∈ S1 heading angle of unicycle iθ ∈ Tn (θi)i∈nωi ∈ R angular velocity of unicycle iri = R−1

i r coord. repr. of r in frame Bixij = xj − xi rel. displacement of robot j wrt robot iNi set of neighbors of robot iyi = (xij)j∈Ni vector of rel. pos. available to robot i

every other unicycle in the ensemble. Instead, each unicycle can only sense a subset of

neighboring unicycles. This sensing convention is naturally represented using the ideas

from graph theory that will be discussed in Section 4.2. We define the sensor graph

G = (V, E), where each node in the node set V represents a robot, and an edge in the

edge set E between node i and node j indicates that robot i can sense robot j. We assume

that G has no self-loops. Given a node i, its set of neighbours Ni represents the set of

vehicles that robot i can sense. For any j ∈ Ni robot i can sense the relative displacement

of robot j in its own body frame, i.e., the quantity xiij , as well as the relative heading

angle θij .

In a realistic scenario, the neighbor set Ni would be the set of robots within the

field of view of robot i. For instance, if each robot mounted an omnidirectional camera,

then one could define Ni to be the collection of robots that are within a given distance

from robot i. With such a definition, the sensor digraph G would be state-dependent,

making the stability analysis too hard at present. Relatively little research has been

done on distributed coordination problems with state-dependent sensor graphs. In this

context, in the simplest case when the robots are modelled as kinematic integrators, it

has been shown in (Lin et al., 2007b) that the circumcentre law of Ando et al. (Ando

et al., 1999) preserves connectivity of the sensor graph and leads to rendezvous if the


sensor graph is initially connected. Despite the simplicity of the robot model, the stability

analysis in (Lin et al., 2007b) is hard, and the control law is continuous but not Lipschitz

continuous.

In light of the above, in this thesis we assume that Ni is static for each i ∈ n (and

hence G is constant as well).

We now define the notion of local and distributed feedback. Define vectors yi :=

(xij)j∈Ni, yii := (xiij)j∈Ni, and ϕi := (θij)j∈Ni. The relative displacements and angles

available to robot i are contained in the vector (yii, ϕi).

Definition 2.2.1. A local and distributed feedback (ui, ωi) for robot i is a locally Lipschitz

function (yii, ϕi) 7→ (ui, ωi).

A local feedback is one in which all quantities are represented in the body frame

of robot i, while a distributed feedback is one in which only relative quantities with

respect to neighboring robots are accessible. In applications, a local and distributed

feedback for robot i can be computed with on-board cameras. No information needs to

be communicated between agents using a communication system or require centralized

information.

Most results in this thesis will require local and distributed feedbacks. However, in

some problems related to higher layer control specifications like formation flocking in

a particular direction and path following, a beacon or access to path information will

be required. Naturally the feedbacks will not be local and distributed in those cases as

information is required from the external environment.

Other Vehicle Models on the Plane

Controllers developed for kinematics unicycles can be extended, with some limitation, to

other models evolving on the plane as discussed below. This illustrates that the study of

kinematic unicycles is fundamental for solving a broad range of mobile robotics problems.


Figure 2.4: Bicycle class. Image from (Francis and Maggiore, 2016).

Bicycles . A model of a bicycle is illustrated in Figure 2.4. The position of bicycle

i’s back wheel in the inertial frame I is denoted xi ∈ R2. The angle of the back wheel is

θi ∈ S1 relative to I and the angle of the front wheel relative to the back wheel is γi ∈ S1

which is called the steering angle. The distance between the center of the two wheels is

B meters. The control inputs are the speed of the back wheel ui and the steering rate

ωi = γi. The equations of motion for the bicycle were derived in (Francis and Maggiore,

2016) as

xi = ui

cos θi

sin θi

,

θi =uiB

tan γi,

γi = ωi.

(2.9)

As stated in (Francis and Maggiore, 2016), this model “turns out to be quite useful

because it captures the essential features of a car with four wheels, only the front two

being steerable”. It is shown in (Francis and Maggiore, 2016) that a controller developed

for a kinematic unicycle can be adapted to the bicycle as long as one shows that the front

wheel of the bicycle is never orthogonal to the back wheel for closed loop solutions.

Dynamic Unicycles . A dynamic model for the unicycle is given in (El-Hawwary


and Maggiore, 2013a) as,

xi = viRi(θi)e1,

vi = ai,

θi = ωi,

ωi = αi, i ∈ n,

(2.10)

where, as for kinematic unicycles, xi ∈ R2 is the position and θi ∈ S1 is the angle.

In the dynamic model vi ∈ R is the speed state and ωi is the angular speed state of

unicycle i. The state of unicycle i can therefore be written as χi = (xi, vi, θi, ωi) ∈

R2 × R2 × S1 × R =: X . The control inputs are the translational acceleration ai and

angular acceleration αi. In (El-Hawwary and Maggiore, 2013a, Proposition V.1.), the

authors show that for uniformly bounded C1 functions ui(χi) and ωi(χi) the controller

ai = ˙ui(χi)− k(vi − ui(χi)),

αi = ˙ωi(χi)− k(ωi − ωi(χi))

(2.11)

globally asymptotically stabilizes the set

Γ = χ ∈ X : vi = ui(χi), ωi = ωi(χi)

for system (2.10). In the set Γ2, the states (xi, θi) in system (2.10) satisfy the equations

of a kinematic unicycle with inputs (ui, ωi)

xi = ui(χi)Ri(θi)e1,

θi = ωi(χi), i ∈ n.

(2.12)

Suppose we design the uniformly bounded C1 functions ui(χi) and ωi(χi) to solve a

particular control problem for kinematic unicycle model in (2.12) (such as rendezvous).

Then under some assumptions, the Reduction Theorem that will be stated formally in


Figure 2.5: Flying Robot Class.

Theorem 4.1.2 tells us that the feedbacks in (2.11) solve the same problem for the dynamic

unicycle model in (2.10). Therefore problems for dynamic unicycles can be reduced to

problems for kinematic unicycles in many cases. However, unicycles must measure their

own speeds which is not local and distributed.

Flying Robots

We now model a group of n flying robots. We fix a right-handed orthonormal inertial

frame I, common to all robots, and attach at the centre of mass of robot i a right-handed

orthonormal body frame Bi = bix, biy, biz (as in Figure 2.2). We denote by (xi, vi) the

inertial position and velocity of robot i. We let g denote the gravity vector in frame I.

The attitude of body frame Bi relative to I is represented by a rotation matrix

Ri ∈ SO(3). The unit vector qi := −Rie3, depicted in Figure 2.5, is referred to as

the thrust direction vector of robot i. We assume that a thrust force uiqi is applied at

the centre of mass of robot i. Notice that uiqi has magnitude ui, is directed opposite

to biz , and has constant direction in body frame Bi. This results in two nonholonomic

constraints on the acceleration when we ignore the effect of gravity

xi · Rie1 = 0

xi · Rie2 = 0.

Robot i is assumed to have an actuation mechanism that induces control torques τix, τiy, τiz


about its body axes. We let τi := (τix, τiy, τiz) be the torque vector, and Ωi denote the

angular velocity of the robot with respect to frame I.

Picking (xi, vi, Ri,Ωii) as the state for robot i, we obtain the equations of motion

xi = vi,

mivi = −uiRie3 +mig = Ti +mig,

(2.13)

Ri = Ri (Ωii)

×,

JiΩii = τi − Ωii × JiΩ

ii.

(2.14)

In the above, mi is the mass of robot i and Ji = J⊤i is its inertia matrix. We define the

(inertial) relative positions and velocities as xij := xj − xi, vij := vj − vi. The model

of a flying robot is illustrated in the Figure 2.5. This model is standard and is widely

used in the literature to model flying vehicles such as quadrotor helicopters. See, for in-

stance, (Hua et al., 2009). Sometimes researchers use alternative attitude representations,

prominently quaternions (Abdessameud and Tayebi, 2011) or Euler angles (Mokhtari

et al., 2006; Castillo et al., 2005). The model (2.13), (2.14) ignores aerodynamic effects

such as drag and wind disturbances (such effects are included in (Hua et al., 2009)). It

also ignores the dynamics of the actuators.

The state vector for each robot is (xi, vi, Ri,Ωi) ∈ R3 × R3 × SO(3) × R3. The

configuration space of each robot consists of its position xi ∈ R3 and an attitude Ri ∈

SO(3) and can be written as an element in the Special Euclidean Group

SE(3) =

R x

0 1

∈ R

4×4 : R ∈ SO(3), x ∈ R3

and analogous to SE(2), constitutes a group. The overall state space of (2.13), (2.14) is

R3n × R3n × SO(3)n × R3n.

In this thesis we adopt the convention that if r ∈ R3 is an inertial vector, the coordi-


Table 2.2: Table of Notation for Flying Robots

Quantity Description

mi, Ji mass and inertia matrix of robot ixi ∈ R3 inertial position of robot ix ∈ R3n (xi)i∈nvi ∈ R3 linear velocity of robot iv ∈ R3n (vi)i∈nRi ∈ SO(3) attitude of robot iΩi ∈ R3 angular velocity of robot iΩ ∈ R3n (Ωi)i∈nqi = −Rie3 thrust direction vector of robot iTi = −uiRie3 applied thrust vector of robot iri = R−1

i r coord. repr. of r in frame Bixij = xj − xi rel. displacement of robot j wrt robot ivij = vj − vi rel. velocity of robot j wrt robot iΩi ∈ R3 reference angular velocity of robot iNi set of neighbors of robot iyi = (xij , vij)j∈Ni vector of rel. pos. and vel. available to robot i

nate representation of r in frame Bi is denoted by ri, that is, ri := R−1i r. In particular,

the angular velocity of robot i in its own body frame is denoted by Ωii. Finally, the

reference angular velocity of vehicle i is denoted Ωi. The notation is summarized in

Table 2.2.

Example 2.2.2. This example has been taken from (Roza, 2012).

A well known vehicle that falls in the class of flying robots is the quadrotor helicopter,

see (Mokhtari et al., 2006) or (Abdessameud et al., 2012). Referring to Figure 2.6, a

quadrotor helicopter consists of four rotors connected to a rigid frame. The distance

from the centre of mass to the rotors is denoted by d. For robot i, each rotor produces

a thrust force fij, j ∈ 1, . . . , 4 parallel to the biz axis, and a reaction torque τrij of the

motor that drives it. To produce a thrust fij in the negative biz (i.e., upward) direction,

the two rotors on the bix axis rotate in the clockwise direction, while the rotors on the

biy axis rotate in the counter-clockwise direction.

The physical inputs are the reaction torques τrij of the motors. Using the development


Figure 2.6: Illustration of a quadrotor helicopter taken from (Roza, 2012).

from (Castillo et al., 2005), the rotor dynamics are given by IrzΩrij = −bΩ2rij+τrij where

Irz is the rotor moment of inertia about the rotor z-axis, Ωrij is the angular speed of

rotor j, and b is a coefficient of friction due to aerodynamic drag on the rotor. There

is also an approximate algebraic relationship between the rotor thrust and rotor speed

given by, fij = γΩ2rij , where γ is a parameter that can be experimentally determined. If

we assume steady-state rotor dynamics such that Ωrij = 0, then fij = (γ/b)τrij = cτrij

where c = γ/b is the algebraic scaling factor between the rotor thrust and the applied

motor torque. Using this fact, it is readily seen that the relationship between the control

inputs and the motor torques is given by

ui

τix

τiy

τiz

=

c c c c

0 −cd 0 cd

cd 0 −cd 0

1 −1 1 −1

τri1

τri2

τri3

τri4

.

In the above, the total thrust is equal to the summation of the four rotor thrusts; the

torque about the bix axis is proportional to the differential thrust of the two rotors on

the biy axis, fi4 − fi2; the torque about the biy axis is proportional to the differential

thrust of the two rotors on the bix axis, fi1 − fi3; and the torque about the biz axis is


equal to the summation of the four reaction torques which are equal and opposite to the

applied motor torques τrij . With the definition of (ui, τi) above, the quadrotor helicopter

is modelled with (2.13), (2.14).

The modelling of the sensor graph G = (V, E) for flying robots is the same as for

kinematic unicycles. If j ∈ Ni is a neighbor of unicycle i, then robot i can sense the

relative displacement and velocity of robot j in its own body frame, i.e., the quantities

xiij , viij. Define the vector yi := (xij , vij)j∈Ni. The relative displacements and velocities

available to robot i are contained in the vector yii := (xiij , viij)j∈Ni. It is also assumed

that robot i can can sense its own angular velocity in its own frame Bi and a local and

distributed feedback is defined below.

Definition 2.2.3. A local and distributed feedback (ui, τi) for robot i is a locally Lipschitz

function (yii,Ωii) 7→ (ui, τi).

In applications, a local and distributed feedback for robot i can be computed with

on-board cameras and rate gyroscopes. Note that in a local and distributed feedback,

we have not allowed unicycle i to measure the attitude of a neighbor j ∈ Ni in its

body frame R−1i Rj or its relative angular velocity represented in its own body frame

R−1i (Ωj − Ωi) = Ri

jΩjj − Ωii as these quantities are more difficult to obtain using sensors

measurements. In this thesis, such measurements are not required to solve the problem

of rendezvous. However, future extension of this result to formation control will most

certainly require these measurements.

Chapter 3

Coordination Problems

In this chapter we will introduce the coordination problems investigated in this thesis

for the kinematic unicycles in (2.6), (2.7) and flying robots in (2.13), (2.14). Instead of

considering the standard control problem of stabilizing equilibrium points, in this work,

each problem will be stated as a set stabilization problem with additional sensing con-

straints. The main motivation behind this is that most of the sets under consideration

simply do not reduce to equilibrium points, even in appropriately chosen error coordi-

nates. Therefore, they are most naturally defined as sets. This chapter uses stability

notions that will be introduced in the preliminaries in Chapter 4.

3.1 Rendezvous of Flying Robots

In this section, we define the rendezvous control problem for flying robots whose solution

is discussed in Chapter 5. The objective of this problem is to design local and distributed

feedbacks in the sense of Definition 2.2.3 to drive a group of n robots to the rendezvous

manifold,

Γ :=

(xi, vi, Ri,Ωii)i∈n ∈ R

3n × R3n × SO(3)n × R

3n : xij = vij = 0,

Ωii = Ωii(yi, Ri), i, j ∈ n

(3.1)

31

Chapter 3. Coordination Problems 32

where Ωii(yi, Ri) is a suitable function to be designed in Chapter 5. The rendezvous

manifold Γ is the subset of the state space in which all positions and velocities are equal.

There is no condition imposed on the relative attitudes and the control input for robot

i is a function of (yii,Ωii) where yii = (xiij , v

iij)j∈Ni are the relative displacements and

velocities of robot i with respect to its neighbors. The Rendezvous Problem for Flying

Robots (RP-F) is stated as follows.

Problem 1 (Rendezvous Problem for Flying Robots (RP-F)). Consider system (2.13), (2.14)

and sensor digraph G = (V, E) containing a globally reachable node. Design local and dis-

tributed feedbacks (u⋆i , τ⋆i ) : (y

ii,Ω

ii) 7→ (ui, τi) for all i ∈ n that globally practically stabilize

Γ under the constraint (u⋆i , τ⋆i )|Γ = (0, 0).

The goal of the rendezvous control problem is to achieve synchronization of the robot

positions and velocities to any desired degree of accuracy from any initial configura-

tion. The requirement (u⋆i , τ⋆i )|Γ = (0, 0) means that the robots are not actuated when

rendezvous is achieved.

3.2 Rendezvous of Kinematic Unicycles

In this section, we define the rendezvous control problem for kinematic unicycles whose

solution is discussed in Chapter 6. The objective of this problem is to design local and

distributed feedbacks in the sense of Definition 2.2.1 to drive a group of n robots to the

rendezvous manifold,

Γ :=

(xi, θi)i∈n ∈ R2n × T

n : xij = 0, i, j ∈ n

. (3.2)

The rendezvous manifold Γ is the subset of the state space in which all positions coincide

with one another. There is no condition placed on the relative heading angles. The

control input for unicycle i must be strictly a function of yii = (xiij)j∈Ni, the relative


displacements between unicycle i and its neighbors. Unicycles are not permitted to

measure any information about their orientation, not even their orientation relative to

their neighbors ϕi = (θij)j∈Ni. The Rendezvous Problem for Kinematic Unicycles (RP-U)

is stated as follows.

Problem 2 (Rendezvous Problem for Kinematic Unicycles (RP-U)). Consider the sys-

tem of kinematic unicycles in (2.6), (2.7) and a sensor digraph G containing a globally

reachable node. Design local and distributed feedbacks (u⋆i , ω⋆i ) : yii 7→ (ui, ωi) for all

i ∈ n that globally asymptotically stabilize the rendezvous manifold Γ under the con-

straint (u⋆i , ω⋆i )|Γ = (0, 0).

The requirement (u⋆i , ω⋆i )|Γ = (0, 0) in the rendezvous problem (RP-U) means that the

unicycles do not move when they achieve rendezvous, i.e., they are stopped and do not

oscillate. This means that the unicycles do not consume any energy when rendezvous is

achieved as one would expect from a good control strategy.

3.3 Formation Control Problems for Kinematic Uni-

cycles

For kinematic unicycles in (2.6), (2.7), a number of formation control problems will be

introduced. A formation of n unicycles is a geometric pattern defined modulo roto-

translations by means of desired inter-agent displacements. Let the fixed vector d11i ∈ R2

denote the desired displacement of unicycle i relative to unicycle 1, measured in the frame

of unicycle 1, i.e., d11i := R−11 (xi − x1). In contrast to the rendezvous problem where the

robots converge to a common position, this specification is usually more useful in practice.

We collect all the fixed relative displacements in a vector d := (d11i)i∈2 :n ∈ R2(n−1) which

specifies the formation.

An example of a formation consisting of four unicycles defined in terms of the offset

vectors d11i, i ∈ 2 :n is illustrated in Figure 3.1. The two configurations of unicycles


depicted in Figure 3.1, labelled 1 and 2, are related to one another through a rigid

roto-translation in the inertial frame I. That is, to arrive at configuration 2 first rotate

configuration 1 by ϕ radians about unicycle 1 followed by a translation of r units along

the dotted line illustrated in Figure 3.1. For both configurations, the offsets d11i ∈ R2

between unicycle 1 and unicycle i as measured in body frame 1, are identical and therefore

the two configurations represent the same formation. We correspondingly say that the

formation is invariant under roto-translations.

Figure 3.1: Formation in terms of fixed relative displacement vectors d11i, i ∈ 2 :n.

The labelling of the unicycles is done solely for the purpose of defining the formation,

and does not imply any attribution of priority to the unicycles. In the problem of stopping

formations discussed in Section 3.3.1 and formations with final parallel collective motion

discussed in Section 3.3.2, it will be assumed, without loss of generality, that unicycle 1

is chosen to be at the front of the formation so that d11i · e1 ≤ 0 for all i ∈ 2 :n. This is

the case in the example in Figure 3.1. For a given formation d, we define the formation

manifold as,

Γ :=

(x, θ) ∈ R2n × T

n : x1i = R1d11i, i ∈ n

. (3.3)

Note that the rendezvous manifold in (3.2) corresponds to the case that d = 0. Now

we present, in detail, the formation control problems for kinematic unicycles studied in


this thesis. Section 3.3.1 discusses formation control, Section 3.3.2 discusses formations

with parallel collective motion, Section 3.3.3 discusses formations with circular collective

motion, and finally Section 3.3.4 discusses formation path following of Jordan curves.

3.3.1 Formation Control

In this section, we define the formation control problem. The corresponding solution to

this problem is presented in Chapter 7.

Let d ∈ R2(n−1) be a desired formation and without loss of generality, choose unicycle

1 to be at the front of the formation so that d11i · e1 ≤ 0 for all i ∈ 2 :n. Let

F := d ∈ R2(n−1) : d11i · e1 ≤ 0, i ∈ 2 :n

be the corresponding set of all formations of n unicycles. For any d ∈ F, the objective

of the formation control problem is to design local and distributed feedbacks to drive a

group of unicycles to the parallel formation manifold,

Γp := (x, θ) ∈ Γ : θi = θ1, i ∈ 2 :n . (3.4)

The parallel formation manifold Γp is the subset of the state space in which the unicycles

have parallel headings, and their relative displacements meet the formation specification.

This type of formation is referred to as a parallel formation, i.e., formations in which the

unicycles’ headings are parallel to each other: θij = 0 for all i, j ∈ n. The formation

illustrated in Figure 3.1 is an example of a parallel formation. The Parallel Formation

Problem (PP) is stated as follows.

Problem 3 (Parallel Formation Problem (PP)). Consider the system of kinematic uni-

cycles in (2.6), (2.7) and a connected, undirected sensor graph G. For any choice of

d ∈ F, design local and distributed feedbacks (u⋆i , ω⋆i ) : (yii, ϕi) 7→ (ui, ωi) for all i ∈ n


to almost semi-globally asymptotically stabilize the formation manifold Γp under the con-

straint (u⋆i , ω⋆i )|Γp

= (0, 0).

The requirement (u⋆i , ω⋆i )|p = (0, 0) means that the unicycles do not move when

they are in formation. The problem of full synchronization in which both positions and

headings synchronize for all unicycles is a special case of PP in which d = 0.

3.3.2 Formation with Final Parallel Collective Motion

Now we introduce two formation control problems with final parallel collective motion:

parallel flocking and line path following. For flocking, the formation moves along a

straight line whose roto-translation with respect to the inertial frame is not defined a

priori. The specific line followed depends on the initial conditions of the agents. On the

other hand, in line following, the formation follows a specific line path defined a priori in

R2. The solutions to these problems are presented in Chapter 8.

Parallel Formation Flocking

Let d ∈ F be a desired formation and w > 0 be a desired flocking speed. The 2-tuple

(d, w) is a parallel formation flocking specification, and

PF := (d, w) ∈ F× R : w > 0

represents the set of all parallel flocking formations of n unicycles. For any (d, w) ∈ PF,

the objective of the formation control problem is to design local and distributed feedbacks

to drive a group of unicycles to a desired formation corresponding to the parallel formation

flocking manifold,

Γpf := Γp (3.5)

which coincides with the parallel formation manifold in Section 3.3.1. However, unlike PP

where the formation is constrained to stop on the set Γp, for parallel formation flocking


they are required to move at the desired steady-state speed w. Formally, the Parallel

Formation Flocking Problem (PFP) is stated as follows.

Problem 4 (Parallel Formation Flocking Problem (PFP)). Consider the system of kine-

matic unicycles in (2.6), (2.7) and sensor digraph G containing a globally reachable node.

For any choice of (d, w) ∈ PF, design local and distributed feedbacks (u⋆i , ω⋆i ) : (y

ii, ϕi) 7→

(ui, ωi) for all i ∈ n to almost globally asymptotically stabilize Γpf , under the constraint

(u⋆i , ω⋆i )|Γpf

= (w, 0).

Alternatively, suppose that each unicycle is permitted to measure, with respect to its

own body frame, a common inertial vector p of unit norm. That is, each unicycle can

measure the vector pi = R−1i p. The vector p is referred to as a beacon and specifies a

desired flocking direction. Let θp ∈ S1 be its angle with respect to the inertial frame.

This is not a local and distributed quantity as the beacon p is defined in the inertial

frame. The 3-tuple (d, p, w), is a parallel formation flocking specification with a beacon,

and

PFB := (d, p, w) ∈ F× R2 × R : ‖p‖ = 1, w > 0

represents the set of all parallel flocking formations of n unicycles with a beacon. The

set Γpf is replaced with the parallel formation flocking manifold with a beacon,

Γpfb := (x, θ) ∈ Γ : θi = θp, i ∈ n (3.6)

in which not only do the unicycles have the same heading angle, therefore constituting

a parallel formation, but the heading coincides with the angle of the beacon θp. The

Parallel Formation Flocking Problem with a Beacon (PFP-B) is stated as follows.

Problem 5 (Parallel Formation Flocking Problem with a Beacon (PFP-B)). Consider

the system of kinematic unicycles in (2.6), (2.7) and sensor digraph G containing a glob-

ally reachable node. For any (d, p, w) ∈ PFB, design control inputs (u⋆i , ω⋆i ) : (y

ii, ϕi, p

i) 7→


(ui, ωi) for all i ∈ n, that almost globally asymptotically stabilize the set Γpfb under the

constraint (u⋆i , ω⋆i )|Γpfb

= (w, 0).

PFP-B will be solved for digraphs containing a globally reachable node, while PFP will

be solved for the specific class of hierarchical digraphs. Simulation results in Section 8.3

suggest that this local and distributed solution may also apply to more general graph

topologies.

Formation Line Path Following

Let d ∈ F be a desired formation and C(r0, p) = x ∈ R2 : x = r0 + sp, s ∈ R ⊂ R2

be a line to be followed by one unicycle in the ensemble (without loss of generalization,

unicycle 1) where r0, p ∈ R2, and p is a unit vector pointing tangent to C(r0, p). Let

w > 0 be a desired path following speed. Assume that unicycle 1 is at the front of the

formation so that d11i · e1 ≤ 0 for all i ∈ 2 :n. The 4-tuple (d, r0, p, w) is a formation line

path following specification, and

LP := (d, r0, p, w) ∈ F× R2 × R

2 × R : ‖p‖ = 1, w > 0

represents the set of all line following formations of n unicycles. The control inputs (ui, ωi)

of unicycle i will be chosen as functions of the local and distributed quantities (yii, ϕi)

and the direction of the path pi measured in body frame. The feedback for unicycle i

will also depend on the displacement vector between xi and its orthogonal projection

c⋆(xi) in C(r0, p), measured in body frame. This quantity is given by (c⋆(xi) − xi)i =

(r0 − xi)i − ((r0 − xi)

i · pi)pi. For any (d, r0, p, w) ∈ LP, the objective of the formation

control problem is to design such feedbacks in order to drive a group of unicycles to the

formation line path following manifold,

Γlp := (x, θ) ∈ Γpfb : x1 ∈ C(r0, p) (3.7)


which coincides with Γpfb with the additional requirement that unicycle 1 lies on the path

C. An example of a configuration of unicycles in Γlp is illustrated in Figure 3.2. The

Formation Line Path Following Problem (LPP) is stated as follows.

Problem 6 (Formation Line Path Following Problem (LPP)). Consider the system of

kinematic unicycles in (2.6), (2.7) and sensor digraph G containing a globally reachable

node. For any (d, r0, p, w) ∈ LP, design feedbacks (u⋆i , ω⋆i ) : (yii, ϕi, p

i, (c⋆(xi) − xi)i) 7→

(ui, ωi) for all i ∈ n to almost globally asymptotically stabilize Γlp under the constraint

(u⋆i , ω⋆i )|Γlp

= (w, 0).

direction ofmotion

Figure 3.2: formation line path following

3.3.3 Formation with Circular Collective Motion

In this section, we state two formation control problems with final circular collective

motion: circular flocking and circle path following. For flocking, the formation moves

around a circle of desired radius whose center is not defined a priori. Rather, it depends

on the initial conditions of the agents. On the other hand, in circle path following, the

formation follows a specific circle path whose center is defined a priori. The solutions


to these problems are presented along with the linear collective motion problems in

Chapter 8.

Circular Formation Flocking

The objective of the circular formation flocking control problem is to design local and

distributed feedbacks (ui, ωi) for all i ∈ n to make a group of unicycles converge to

a desired formation d ∈ R2(n−1) that encircles, with angular speed w > 0, a center

point c ∈ R2 which is not defined a priori and depends on the initial configuration of

the unicycles. For one unicycle in the ensemble (without loss of generality, unicycle 1)

specify, a priori, the radius of rotation β1. The 3-tuple (d, β1, w) is a circular formation

flocking specification, and

CF := (d, β1, w) ∈ R2(n−1) × R× R : β1 > 0, w > 0

represents the set of all circular flocking formations of n unicycles.

Figure 3.3: circular formation flocking with centre of rotation c


When the desired formation d is achieved, each unicycle i ∈ 2 :n must move along a

concentric circular orbit of radius βi =√

(d11i · e1)2 + (β1 − d11i · e2)2 with angular speed

w. The radii βi, i ∈ n are illustrated in Figure 3.3. Unlike in the case of parallel

flocking, where all unicycle headings converge to a common direction, in the case of

circular formation flocking, the heading directions will not align in general. Instead,

the heading of unicycle i is offset relative to the heading of unicycle 1 by the angle

ρi(d, β1) = atan2(d11i · e1, β1 − d11i · e2) illustrated in Figure 3.3 where ρ2 = −ρ4 and

ρ1 = ρ3 = 0.

Inspired by (El-Hawwary and Maggiore, 2013a), define the offset vector δi = βiRie2

for each unicycle i rigidly attached to its body frame and perpendicular to the heading

direction Rie1. The vector xi = xi + δi represents the perceived centre of a circle of

radius βi that unicycle i would trace out if its translational speed were wβi and its

angular speed were w. On the set Λ := (x, θ) ∈ R2n × Tn : x1i = 0, i ∈ n unicycles

lie on concentric circles of desired radii (βi)i∈n about a common centre. Relative to this

set, the desired formation is achieved if, in addition, θ1i = ρi(d, β1) for all i ∈ 2 :n. The

circular formation flocking manifold is therefore given by,

Γcf := (x, θ) ∈ Λ : θ1i = ρi(d, β1), i ∈ 2 :n . (3.8)

The Circular Formation Flocking Problem (CFP) is stated as follows.

Problem 7 (Circular Formation Flocking Problem (CFP)). Consider the system of

kinematic unicycles in (2.6), (2.7) and a connected, undirected sensor graph G. For any

(d, β1, w) ∈ CF, design local and distributed feedbacks (u⋆i , ω⋆i ) : (y

ii, ϕi) 7→ (ui, ωi) for all

i ∈ n to almost globally asymptotically asymptotically stabilize Γcf under the constraint

(u⋆i , ω⋆i )|Γcf

= (wβi, w).


Formation Circle Path Following

For the formation circle path following problem, the desired formation d ∈ R2(n−1) follows

a particular circular orbit with centre c ∈ R2 defined a priori. The 4-tuple (d, β1, c, w) is

a formation circle path following specification, and

CP := (d, β1, c, w) ∈ R2(n−1) × R× R

2 × R : β1 > 0, w > 0

represents the set of all circle following formations of n unicycles. The control inputs

(ui, ωi) of unicycle i are chosen as functions of the local and distributed quantities (yii, ϕi)

and the centre of the desired circle c relative to xi measured in body frame, i.e., (c−xi)i.

The formation circle path following manifold is defined as

Γcp := (x, θ) ∈ Γcf : x1 = c (3.9)

where the only difference from Γcf is that the perceived centers xi = xi + δi not only

coincide, but they coincide at the desired circle center c. The Formation Circle Path

Following Problem (CPP) is stated as follows.

Problem 8 (Formation Circle Path Following Problem (CPP)). Consider the system of

kinematic unicycles in (2.6), (2.7) and a connected, undirected sensor graph G. For any

(d, β1, c, w) ∈ CP, design feedbacks (u⋆i , ω⋆i ) : (y

ii, ϕi, (c− xi)

i) 7→ (ui, ωi) for all i ∈ n that

almost globally asymptotically stabilize the formation circle path following manifold Γcp

under the constraint (u⋆i , ω⋆i )|Γcp

= (wβi, w).

3.3.4 General Formation Path Following

The formation control problems introduced up to this point have considered either forma-

tions that stop or have final parallel or circular collective motions. These basic motions

can be combined to address path following problems where a formation of unicycles


travels between a series of way-points along straight lines and, in turn, stops at each

way-point or encircles each way-point with a desired radius. However, such a scheme

is not appropriate, for example, in a situation where the formation must follow a more

general path with high curvature such as a winding road to deliver packages.

To allow for such situations, the goal of the general formation path following problem

that will be introduced in this section and solved in Chapter 9 is to develop a controller to

make unicycles obtain formation about a desired smooth Jordan curve C ∈ J , where J is

the set of smooth Jordan curves in R2, and follow it at a desired speed w > 0. The set C,

diffeomorphic to S1, has a regular parametrization o : S1 → C, o(τ) where we will refer to

τ as the path parameter. Let π : C → S1 be the inverse of o(τ) that maps a point in C to a

point τ ∈ S1. For every τ ∈ S1, there is a Frenet-Serret frame (o(τ), r(τ), s(τ)) where o(τ)

is the origin of the frame, r(τ) = (d/dτ)o(τ)/‖(d/dτ)o(τ)‖ is the tangent vector to the

curve at τ pointing in the desired direction of motion and s(τ) is the counter clockwise

rotation of r by π/2 rad. Define the rotation matrix with columns corresponding to the

axes of the Frenet-Serret frame at τ ∈ S1 given by R0(τ) = [r(τ) s(τ)]. The Frenet-Serret

frame is illustrated in Figure 3.4.

Figure 3.4: The Frenet-Serret frame with origin o(τ) on the smooth Jordan curve C.

For general formation path following we will no longer define formations with the


collection of fixed offsets (d11i)i∈n relative to the body frame of unicycle 1. Instead the

formation will be defined relative to the Frenet-Serret frame on C. In particular, the

unicycles are said to achieve formation at τ ∈ S1 if each unicycle i ∈ n lies at a fixed

desired displacement di ∈ R2 relative to the Frenet-Serret frame (o(τ), r(τ), s(τ)), i.e.,

R0(τ)−1(xi− o(τ)) = di. The corresponding point about which the formation is achieved

o(τ) = π−1(τ) ∈ C is referred to as the formation origin. This is illustrated in Figure 3.5

for a rectangular formation composed of four unicycles labelled x1, . . . , x4. Denote the

desired, fixed angle of di with respect to the −r(τ) axis by ψi = atan2(di · e2,−di · e1).

We make the assumption that di · e1 6= 0 for all i ∈ n, i.e., in formation, no unicycle lies

on the s(τ) axis corresponding to ψi = ±π/2. The angles ψ1, ψ2 are illustrated for the

example in Figure 3.5.

formationorigin

Figure 3.5: Formation path following.

We collect the above relative displacements in a vector d := (di)i∈n ∈ R2n. A forma-

tion specification d is defined rigidly with respect to the path’s frame of reference and is

not invariant to translations and rotations. The 3-tuple (d, C, w) is a general formation

path following specification, and

GP := (d, C, w) ∈ R2n × J × R : di · e1 6= 0, w > 0, i ∈ n


represents the set of all general path following formations of n unicycles. In the general

formation path following problem, each unicycle stores on-board an additional auxiliary

state φi ∈ S1 with dynamics

φi = ωi ,piv , i ∈ n, (3.10)

where ωi ,piv denotes its angular speed. Let φ := (φi)i∈n ∈ Tn be the collection of the

auxiliary states of all unicycles. A more detailed description of (3.10) will be given in

Chapter 9. For any (d, C, w) ∈ GP, define the general formation path following manifold

as

Γgp :=

(x, θ, φ) ∈ R2n × T

n × Tn : (∃τ ∈ S

1)

R0(τ)−1(xi − o(τ)) = di, φi = φi(xi(xi, θi, φi), θi − φi), i ∈ n

,

(3.11)

where φi(xi(xi, θi, φi), θi − φi) is a desired angle for φi, to be designed in Chapter 9. In

Γgp, there exists τ ∈ S1 such that the unicycles have achieved formation at τ . For any

(d, C, w) ∈ GP, the objective of the General Formation Path Following Problem (GPP)

is to develop feedbacks that asymptotically stabilize a subset of Γgp and such that the

formation origin o(τ), moves at the desired speed w along C in the direction of r(τ).

Problem 9 (General Formation Path Following Problem (GPP)). Consider the system

of kinematic unicycles in (2.6), (2.7) with auxiliary state in (3.10) and sensor digraph G

containing a globally reachable node. For any (d, C, w) ∈ GP, design distributed feedbacks

(u⋆i , ω⋆i , ω

⋆i ,piv) : (xi, θi, φi, (xj, θj , φj)j∈Ni) 7→ (ui, ωi, ωi ,piv)

for all i ∈ n that almost globally asymptotically stabilize a subset of Γgp under the con-

straint that o(τ)|Γgp = wr(τ).

The exact quantities that unicycles need to measure are discussed in detail in Sec-

tion 9.1.


3.4 Literature review

In this section we will discuss the literature related to multi-agent coordination. We will

first discuss the results for single and double integrators. These are the most basic vehicle

classes since they are fully actuated. Results for single and double integrators will be-

come important building blocks for constructing control solutions for the under-actuated

systems in this thesis. Next we discuss results that characterize relative equilibria for

teams of kinematic unicycles using local and distributed feedback. Finally, we review the

literature on rendezvous for kinematic unicycles and flying robots followed by results for

formations of kinematic unicycles that either stop or have final collective motions.

We remark that the control problems introduced in this section cannot be solved using

standard results for output synchronization of nonlinear heterogeneous systems (Wieland

et al., 2011; De Persis and Jayawardhana, 2014; Burger and De Persis, 2015; Liu and

Jiang, 2013; Isidori et al., 2014; Zhu et al., 2016). These results rely heavily on com-

munication of auxiliary states between neighboring agents in the network which is not

permitted in this work. Most importantly, the control feedbacks obtained using these

approaches require agents to measure quantities that are not local and distributed. Con-

sider, for example, the rendezvous problem for kinematic unicycles. In this case, the

outputs to be synchronized need to be chosen as yi = h(xi, θi) = xi for each robot. How-

ever, the final controller for robot i in these results requires measurement of the quantity

yj − yi = xj − xi 6= R−1i (xj − xi) where j is a neighbour of i. This quantity is not local

and distributed.

Coordination problems for single and double integrators . The problem

of rendezvous for networks of single and double integrators is well-established in the

literature. See for example (Ren and Beard, 2005; Moreau, 2004; Olfati-Saber and

Murray, 2004; Ren and Atkins, 2007). The majority of the literature on formation

control considers single and double-integrator models. A dominant approach for single-

integrator formation control is distance-based (Krick et al., 2009; Oh and Ahn, 2014;


Smith et al., 2006), where it is required that the distances between robots take on desired

values. Often in this setting, the feedbacks are deduced from the gradient of a potential

function whose minimum specifies the desired formation modulo roto-translations. This

approach requires the sensing graph to be infinitesimally rigid, which is quite restrictive.

Other approaches define formations in terms of relative angles between neighbouring

robots, instead of distances, (Zhao et al., 2014; Eren, 2012), or in terms of a complex

Laplacian, (Lin et al., 2013; Lin et al., 2016; Lin et al., 2014). In this latter case,

formations are defined modulo scaling and roto-translations. Finally, formation flocking

of double-integrators is considered in (Deghat et al., 2016), where the authors stabilize a

formation and make sure that all robots in the formation achieve a common final velocity.

See also (Tanner et al., 2003).

Relative equilibria for kinematic unicycles . The papers (Justh and Krish-

naprasad, 2004; Sarlette et al., 2010) show that the only possible relative equilibria for

a team of unicycles with local and distributed control laws correspond to either parallel

(including stopping) or circular collective motions. In fact, we will be able to find local

and distributed solutions to the problems studied in this thesis for unicycle rendezvous,

stopping formations and parallel and circular formation flocking where the final collective

motion is either parallel or circular. However, it is not possible to obtain formation flock-

ing with any other type of collective motion, for example, around an oval shaped path

using strictly local and distributed feedbacks. For kinematic unicycles in three dimen-

sions, it is shown in (Justh and Krishnaprasad, 2004; Justh and Krishnaprasad, 2005)

that the only possible relative equilibria correspond to parallel, circular or helical forma-

tions. In (Scardovi et al., 2008), the authors propose distributed controllers to stabilize

relative equilibria but do not specify a particular formation. While kinematic unicycles

in three dimensions are not studied explicitly in this work, we suspect that extension of

the results from two dimensions to three dimensions would not cause any difficulties.

Kinematic unicycle rendezvous . In (Lin et al., 2005), the authors presented


the first solution to this problem. The feedback in (Lin et al., 2005) is local and dis-

tributed, but it requires the use of time-varying feedbacks. In (Zheng et al., 2011) the

authors present a solution using a local and distributed, continuously differentiable, and

time-independent feedback. However the result yields rendezvous only when the sensing

graph is undirected and connected. The feedback in (Zheng et al., 2011) makes the uni-

cycles converge to a circular formation instead of rendezvous for some directed graphs

containing a globally reachable node. In (Zoghlami et al., 2013) the authors also consider

undirected graphs, and present a feedback to achieve rendezvous in finite time. In (Di-

marogonas and Kyriakopoulos, 2007) both positions and attitudes of the unicycles are

synchronized using a time-invariant distributed control. The authors assume an initially

connected sensing graph. The controller that is implemented, however, is discontinuous

and imposes excessive switching. In (Zheng and Sun, 2013) a time-independent, local

and distributed controller is presented. However, the authors make the assumption that

whenever two vehicles get sufficiently close together they merge into a single vehicle,

introducing a discontinuity in the control function. The same merging technique is used

in (Yu et al., 2012) for cyclic and tree graphs where each unicycle keeps its neighboring

vehicle within its windshield’s field of view in order to maintain graph connectivity and

achieve rendezvous. In (Ajorlou et al., 2015; Listmann et al., 2009) distributed solutions

are presented whereby the unicycles move toward the average position of their neighbors.

However, a unicycle’s feedback becomes undefined when it already lies at this average

position which includes the case when the unicycles are at rendezvous. Finally, in (Jafar-

ian, 2015) the authors solve the problem of practical rendezvous using a hybrid controller

in which the unicycles converge to an arbitrarily small neighborhood of one another for

undirected and connected graph topologies. The case of kinematic vehicles in three-space

is investigated in (Nair and Leonard, 2007; Dong and Geng, 2013; Hatanaka et al., 2012).

The authors of (Nair and Leonard, 2007; Dong and Geng, 2013) consider the problem of

full attitude and position synchronization, but assume fully actuated vehicles.


Flying robot attitude synchronization and rendezvous . The problem of atti-

tude synchronization for flying robots is studied in (Ren, 2010; Abdessameud and Tayebi,

2009; Abdessameud and Tayebi, 2013). The proposed controllers do not require mea-

surements of the angular velocity, but they do require absolute attitude measurements.

In (Nair and Leonard, 2007), the authors use the energy shaping approach to design

local and distributed controllers for attitude synchronization. The same approach is

adopted in (Sarlette et al., 2009) to design two attitude synchronization controllers, both

distributed. The first controller achieves almost-global synchronization for directed con-

nected graphs. However, the controller design is based on distributed observers (Scardovi

et al., 2007), and therefore requires auxiliary states to be communicated among neighbor-

ing vehicles. It also employs an angular velocity dissipation term that forces all vehicle

angular velocities to zero in steady-state. The second controller in (Sarlette et al., 2009)

does not restrict the final angular velocities, and does not require communication, but it

requires an undirected sensing graph, and guarantees only local convergence.

For the rendezvous problem of flying robots, in (Wang, 2016) the authors consider

directed graphs containing a globally reachable node and develop an adaptive feedback

that is not local and distributed. In (Lee, 2012; Abdessameud and Tayebi, 2011) the au-

thors consider formation control for flying robots. However, again, the feedbacks are not

local and distributed. In (Abdessameud and Tayebi, 2011) the sensing graph is assumed

to be undirected, and communication among vehicles is required, while in (Lee, 2012)

the graph is balanced, and it is assumed that each vehicle has access to the thrust input

of its neighbors, therefore requiring once again communication between vehicles. Both

approaches in (Lee, 2012; Abdessameud and Tayebi, 2011) use a two-stage backstepping

methodology in which the first stage treats each vehicle as a point-mass system to which a

desired thrust is assigned. A desired thrust direction is then extracted and backstepping

is used to design a rotational control such that vehicle rendezvous or formation control

is achieved.


Formations of kinematic unicycles . A formation controller for single integrator

robots can always be turned into a controller for kinematic unicycles if one considers a

point at a positive distance d in front of each unicycle. These points behave like single

integrators under an appropriate choice of feedback transformation, and can be driven to

a desired formation using the numerous techniques discussed earlier. However, although

the points converge to a formation, the unicycles themselves do not. Choosing a small

value of d reduces this error, but requires large control inputs. This results from the

fact that kinematic unicycles have a nonholonomic constraint which is not present in

the integrator model. For this reason, solutions for integrators do not adapt well into

solutions for kinematic unicycles. We now discuss solutions in the literature designed

specifically for kinematic unicycle formation control.

In (Dimarogonas and Kyriakopoulos, 2008), a discontinuous controller is presented

that stabilizes formations with synchronized heading directions, but unicycles require a

common sense of direction. The papers (Lin et al., 2005; Sepulchre et al., 2007; Tabuada

et al., 2005) discuss the feasibility of achieving various formations using local and dis-

tributed feedback. In (Lin et al., 2005), time-dependent solutions are presented in each

case. For general geometric patterns, unicycles require a common sense of direction.

Similarly, the solution in (Jin and Gans, 2017) is time dependent and requires measure-

ment of a common direction in addition to the velocity input of a neighbouring unicycle,

which can only be obtained if the unicycles can communicate these inputs with each

other. In (Liu and Jiang, 2013), a leader-follower approach is considered. The analysis

transforms the unicycle model into a system of double integrators through dynamic feed-

back linearization. The desired formation is attained for digraphs containing a spanning

tree, but each follower robot requires access to the acceleration of the leader through

communication. In (Oh and Ahn, 2013), each unicycle estimates its own position us-

ing dynamic extension, requiring communication among unicycles. The unicycles use

these estimated states to attain the desired formation globally. The rotational control,


however, is time-dependent and oscillatory.

Kinematic unicycle formations with final collective motions . Results in

the literature for formation flocking remain quite limited. In (Reyes and Tanner, 2015)

it is stated that “although some links between provably convergent formation control

and flocking have been identified, the two behaviors have not been integrated into a

single design.” In (Reyes and Tanner, 2015) a solution is presented for parallel formation

flocking that requires knowledge of a common frame (i.e., beacon) and uses a dynamic

feedback linearization that requires communication of auxiliary states between agents.

In (Sepulchre et al., 2007), a local solution is presented for all-to-all undirected graphs.

However, only stability is shown and not asymptotic stability. For circular formation

flocking, (Sepulchre et al., 2007; Sepulchre et al., 2008) present asymptotically stable

solutions on a common circle with a focus on specific (M,N)-patterns. In (Chen and

Zhang, 2011) the authors present a controller with a repulsion function such that unicycles

achieve equal spacing around a common circle assuming a jointly connected graph, while

in (El-Hawwary and Maggiore, 2013a), the spacing of the unicycles can be freely chosen

beforehand. In these results, the feedbacks are local and distributed, however, the final

formation is restricted to lie on a common circle and the stability results in (Sepulchre

et al., 2007; Sepulchre et al., 2008; El-Hawwary and Maggiore, 2013a) are local.

A far more studied problem in the literature is formation path following. In a common

virtual structure approach (Peng et al., 2015; Loria et al., 2016; Sadowska et al., 2011) a

(possibly virtual) leader moves along the desired path at a desired speed and the other

unicycles converge to their corresponding place-holders with respect to the leader. In

this approach, at least one unicycle must have access to the virtual leader state. This

is trajectory tracking rather than path following and the path is not invariant in this

case. Another approach that is common in the literature (Do and Pan, 2007; Zhang and

Leonard, 2007; Ghabcheloo et al., 2009; Egerstedt and Hu, 2001; Chen and Tian, 2011;

Doosthoseini and Nielsen, 2015), assigns a different path Ci(si) to each unicycle i where


si parametrizes the displacement of unicycle i along its path. Formation path following is

achieved whenever all unicycles lie on their corresponding paths and the path parameters

for all unicycles achieve consensus. To achieve this, each unicycle invokes a path following

controller in order to converge to its own path Ci(si) with a desired speed profile. In the

transient behaviour, the speed control is modified, slowing down or speeding up certain

unicycles, in order to synchronize the si states. While this approach does not require a

virtual leader, it does require communication of the quantities si between neighboring

unicycles and each unicycle needs to compute its own path to be followed. In (Brinon-

Arranz et al., 2014) unicycles converge to a common, compact, time-varying path with

uniform spacing. Each unicycle stores, on-board, the state of an exosystem which must

be communicated with neighboring unicycles. In (Reyes and Tanner, 2015) no virtual

leader is required in the solution, but rather, the formation centre of mass is driven to a

desired path. However, each unicycle must know the location of the formation centre of

mass, requiring all-to-all sensing. Finally, in (Consolini et al., 2012), the authors present

a solution for hierarchical, leader-follower topologies. Unicycles approximately achieve

formation as long as the path followed by the leader has sufficiently small curvature.

However, the stabilizing control of a unicycle depends on the linear speed inputs of its

neighbors.

There are several results in the literature that consider, specifically, formation circle

following. Most consider motion along a common circle (Sepulchre et al., 2007; Sepulchre

et al., 2008; Paley et al., 2008; Yu and Liu, 2016; Yu et al., 2018), while others, more

in line with this thesis, allow unicycles to travel around a common centre with different

radii and correspondingly, with different speeds. This is the problem studied in (Seyboth

et al., 2014) where the undirected sensing graph is assumed to be all-to-all and, by

changing a gradient potential function in the control law, one can achieve either phase

agreement or balancing. In (Zheng et al., 2015), on the other hand, the graph is a

ring and the spacing between neighboring unicycles are equal. Neither of these results,


however, achieve arbitrary formations as in this thesis.

An important distinction among the results in the literature for formation path fol-

lowing is whether or not the formation rotates rigidly with the curve. A formation that

rotates rigidly with the curve is defined with respect to the curve’s frame of reference

rather than with respect to the inertial frame and, as such, will always point tangent to

the path as it follows it. On the other hand, most results define the formation based on

fixed offsets defined in the inertial frame I and the formation maintains a fixed orien-

tation relative to I as it traverses the curve. Each unicycle must be able to sense the

common inertial frame in this case. The distinction between formations defined with

respect to the path frame of reference versus the inertial frame is illustrated in Figure 3.6

for a diamond shaped formation composed of four unicycles.

Directionof motion

Directionof motion

Figure 3.6: (left) Diamond formation that rotates rigidly with the curve C and alwayspoints in the path’s tangent direction. (right) Diamond formation defined with respectto the inertial frame I that does not rotate as it traverses C.

Chapter 4

Preliminaries

This chapter contains fundamental definitions and results that will be applied on a reg-

ular basis throughout this thesis. First we will discuss a number of topics related to the

stability of sets and present corresponding propositions and theorems. Next we discuss

a number of relevant concepts in graph theory and review classes of gradient and homo-

geneous systems. Finally, we discuss a number of results in the literature for multi-agent

coordination in networks where the agents belong to elementary vehicle classes such as

single and double integrators and rotational integrators. These elementary tools will

be referred to as “control primitives”and will serve as building blocks to construct our

control solutions. This way of constructing feedbacks out of simpler building blocks is a

central theme to this thesis that will be seen time and again.

4.1 Basic Stability Theory

The primary objective in each chapter of this thesis will be to design feedbacks for each

agent in a multi-agent team evolving in state-space X , a smooth manifold, in order to

solve a coordination task. The feedbacks will be designed to satisfy a number of sensing

requirements and the control specification for each coordination problem, introduced in

Chapter 3, involves the stabilization of a desired closed subset of the state-space Γ ⊂ X

54

Chapter 4. Preliminaries 55

as opposed to stabilization of an equilibrium point. Solving the coordination task will

amount to “driving” the ensemble of agents to Γ. There are two main components to

this, the first relates to the stability of Γ and the second relates to attractivity. First, we

will discuss each of these concepts and present several formal definitions. Then, we will

present reduction theorems, powerful tools that will be used on numerous occasions in

this work.

4.1.1 Definitions

In this section we will define several notions related to the stability and attractivity

properties of a closed subset Γ ⊂ X . It will be shown how the definitions for stability

and attractivity can be combined to obtain an array of definitions for asymptotic stability

and practical stability. Finally, we will formalize the notion of stability of one set relative

to another. Note that the definitions presented in this section represent one choice and

are not fully inclusive. For a more detailed discussion of stability definitions, the reader

can refer to (Bhatia and Szego, 2002).

Let X ⊂ Rn be open. 1 If d : X × X → [0,∞) is the distance function between two

points in X and Γ ⊂ X is a closed subset of X , then we denote by ‖χ‖Γ := infψ∈Γ d(χ, ψ)

the point-to-set distance of χ ∈ X to Γ. If ε > 0, we let Bε(Γ) := χ ∈ X : ‖χ‖Γ < ε

and by N (Γ) we denote a neighborhood of Γ in X .

In the discussion that follows, consider a smooth dynamical system

Σ : χ = f(χ) (4.1)

with state space X and solutions defined for all time t ≥ 0, and let φ(t, χ0) denote the

solution at time t with initial condition χ(0) = χ0. A closed set Γ ⊂ X is said to be

positively invariant for Σ if for all χ0 ∈ Γ and all t > 0, φ(t, χ0) ∈ Γ.

1One can consider, more generally, a complete Riemannian manifold (X , g) with associated Rieman-nian distance function d : X × X → [0,∞) on X .


Figure 4.1: Illustration on the left shows stability of Γ - solutions with initial conditionsin N (Γ) remain inside Bε(Γ) for all time. Illustration on the right shows attractivity ofΓ with domain of attraction D(Γ) - solutions with initial conditions in D(Γ) converge toΓ.

Set Stability

Roughly, a closed subset Γ ⊂ X for system (4.1) is stable if solutions starting close to Γ re-

main close to Γ for all time. The precise definition for stability is given in Definition 4.1.1

and is illustrated on the left hand side of Figure 4.1.

Definition 4.1.1. The closed set Γ ⊂ X is stable for Σ if for any ε > 0, there exists a

neighborhood N (Γ) ⊂ X such that, for all χ0 ∈ N (Γ), φ(t, χ0) ∈ Bε(Γ), for all t > 0.

Stability is a key factor to ensure robustness in engineering applications. For example,

consider the problem of rendezvous where Γ represents the rendezvous set in which the

positions of all agents coincide. One would expect from a good control solution that

if initial robot positions are close to rendezvous, i.e., if the initial state is in a small

neighborhood of Γ then the corresponding closed-loop solution should not diverge too

much from Γ. If a set is not stable then it is called unstable.

Set Asymptotic Stability

The domain of attraction of a closed set Γ ⊂ X for system (4.1), denoted D(Γ), is the

set of initial conditions in the state space X from which solutions converge to the set


Γ as time approaches infinity. The notion of domain of attraction is formally stated in

Definition 4.1.2.

Definition 4.1.2. The domain of attraction of the closed set Γ ⊂ X for system Σ is the

set D(Γ) = χ0 ∈ X : limt→∞ ‖φ(t, χ0)‖Γ = 0.

Definition 4.1.3. The closed set Γ ⊂ X is (locally) attractive for Σ if D(Γ) is a neigh-

borhood of Γ; Γ is globally attractive for Σ if D(Γ) = X . Moreover, Γ is (locally) asymp-

totically stable for Σ if Γ is stable and locally attractive; Γ is globally asymptotically stable

for Σ if Γ is stable and globally attractive.

While it is desirable to have global results, asking for global asymptotic stability is

often too strong in many applications. This may be because this property is simply

too difficult to prove or, more fundamentally, it may not even be possible to design

continuous time-invariant control inputs to make a desired closed subset Γ ⊂ X globally

asymptotically stable due to a topological obstruction. In fact, this issue is quite common.

Take for example Theorem 1 in (Bhat and Bernstein, 2000), which implies that any

smooth time-invariant vector field on a compact manifold without boundary cannot have

any equilibrium that is globally asymptotically stable. Consequently, it is impossible to

globally asymptotically stabilize an equilibrium point in SO(2) and SO(3) via continuous

time-invariant feedback. For this reason, we will define a slightly weaker form of global

asymptotic stability called almost global asymptotic stability in which the domain of

attraction is not global, but rather, contains almost every point in the state-space.

Before giving a formal definition of almost global asymptotic stability, we need to

understand what a set of Lebesgue measure zero is. We will start by defining Lebesgue

measure zero sets in Rn and then extend this definition to smooth manifolds. Intuitively,

a set of Lebesgue measure zero is negligible in the sense that it occupies no volume in the

state space and the probability of randomly choosing a point in this set is therefore zero.

Define an open cube in Rn as the product of open intervals U = (a1, b1) × (a2, b2) · · · ×


(an, bn) where ai, bi ∈ R and ai < bi for all i ∈ n. Denote the volume of this cube by the

product of the intervals v(U) = (b1−a1)(b2−a2) . . . (bn−an). A set of Lebesgue measure

zero in Rn is one that can be covered by a collection of cubes occupying an arbitrarily

small volume.

Definition 4.1.4. A subset Γ ⊂ Rn has zero measure if for every ǫ > 0, there exist open

cubes U1, U2, . . . such that Γ ⊂ ⋃∞i=1 Ui and

∞∑

i=1

v(Ui) < ǫ.

In the case of a smooth manifold M , a set Γ ⊂M has Lebesgue measure zero in M if

it has zero measure under every smooth coordinate chart on M (Lee, 2013). Moreover,

by Proposition 6.8 in (Lee, 2013) M\Γ is dense in M . This means that the closure of

M\Γ satisfies M\Γ =M .

Definition 4.1.5. If M is a smooth n-dimensional manifold, a subset Γ ⊂ M has zero

measure inM if for every smooth coordinate chart (U, ϕ) forM , the subset ϕ(Γ∩U) ⊂ Rn

has measure zero.

We are now ready to define almost global stability properties.

Definition 4.1.6. The closed set Γ ⊂ X is almost globally attractive for Σ if X\D(Γ)

has Lebesgue measure zero. The set Γ is almost globally asymptotically stable for Σ if Γ

is stable and almost globally attractive.

The notion of almost global attractivity is illustrated in Figure 4.2. Next we give two

final definitions of attractivity and asymptotic stability which require a high gain control

k ∈ R. Consider the dynamical system

Σ(k) : χ = f(χ, k) (4.2)


Figure 4.2: Illustration of almost global attractivity of the set Γ. The set X\D(Γ) hasLebesgue measure zero.

and let φk(t, χ0) denote the solution at time t with initial condition χ(0) = χ0. For

system (4.2) the domain of attraction will depend on the parameter k in general and is

denoted by Dk(Γ).

Definition 4.1.7. The closed set Γ ⊂ X is semiglobally attractive with high-gain param-

eter k for Σ(k) if for each compact set K satisfying Γ ⊂ K ⊂ X , there exists k⋆ > 0 such

that for all k > k⋆, Γ is attractive for Σ(k) and K ⊂ Dk(Γ). The set Γ is semiglobally

asymptotically stable for Σ(k) if Γ is stable and semiglobally attractive.

Definition 4.1.8. A closed subset Γ ⊂ X is almost semiglobally attractive with high-gain

parameter k for Σ(k) if there exists a set N ⊂ X\Γ of Lebesgue measure zero such that

for each compact subset K satisfying Γ ⊂ K ⊂ (X\N), there exists k⋆ > 0 such that

for all k > k⋆, Γ is attractive for Σ(k) and K ⊂ Dk(Γ). The set Γ is almost semiglobally

asymptotically stable for Σ(k) if Γ is stable and almost semiglobally attractive. See

Figure 4.3.

The difference between global asymptotic stability and semiglobal asymptotic stability

of a closed subset Γ ⊂ X is that with the former, solutions converge to the set Γ from all

initial conditions, while with the latter, the domain of attraction can be made arbitrarily

large within the state-space by increasing the control gain k. The difference between

almost global asymptotic stability and almost semiglobal asymptotic stability is that


Figure 4.3: Illustration of almost semiglobal attractivity of the set Γ. For compact setsK1 ⊂ K2 ⊂ X\N , that do not contain the set N of Lebesgue measure zero, for all k ≥ k⋆1(left) and k ≥ k⋆2 > k⋆1 (right), solutions with initial conditions in the sets K1 and K2

respectively, converge to Γ. The domain of attraction of Γ approaches full measure asthe high gain k is increased.

Table 4.1: Asymptotic and Practical Stability Abbreviations

Definition num. Abb. Meaning

4.1.3 LAS or AS local asymptotic stability or asymptotic stability4.1.3 GAS global asymptotic stability4.1.6 AGAS almost-global asymptotic stability4.1.7 SGAS semi-global asymptotic stability4.1.8 ASGAS almost semi-global asymptotic stability4.1.9 GPS global practical stability

with the former, solutions converge to the set Γ with domain of attraction D(Γ) of full

measure, while with the latter, the domain of attraction approaches full measure with

increasing control gain k.

The abbreviations for each type of asymptotic stability are listed in Table 4.1.

In place of asymptotic stability, one can also consider the weaker notion of practical

stability.

Definition 4.1.9. The closed set Γ ⊂ X is globally practically stable for Σ(k) if for any

ε > 0, there exists k⋆ > 0 such that for all k > k⋆, Bε(Γ) has a subset containing Γ which

is globally asymptotically stable for Σ(k).

An illustration of global practical stability of the set Γ is shown in Figure 4.4. Notice

the duality between the concepts of semiglobal stability and global practical stability. If Γ


Figure 4.4: Illustration of global practical stability of the set Γ. For ǫ1 > ǫ2, for anyk ≥ k⋆1 (left) and k ≥ k⋆2 > k⋆1 (right), for any initial conditions in X , solutions convergeto the neighborhoods Bǫ1(Γ) and Bǫ2(Γ) respectively.

is asymptotically stable then it is necessarily practically stable without the requirement

of a high gain parameter k. The purpose of considering practical stability instead of

asymptotic stability is that in most engineering applications, it is sufficient that solutions

converge to an arbitrarily small neighborhood Bε(Γ) of Γ, and not Γ itself. The advantage

of considering notions of practical stability of a subset is that they are often significantly

easier to prove. The downside to practical solutions, however, is that they require a high

gain. If the neighborhood Bε(Γ) is to be very small, then the gain will typically be large.

Now consider an equilibrium p ∈ X for system (4.1). If the linearization of the system

equations in (4.1) at p has all eigenvalues in the open left half complex plane, then p is

exponentially stable. If an equilibrium p is exponentially stable, then for all initial condi-

tions χ0 in a small neighborhood of p, the error ‖φ(t, χ)− p‖ or ‖φk(t, χ)− p‖ converges

to zero at an exponential rate. If at least one eigenvalue at p is positive, the equilibrium

is called exponentially unstable as stated in Definition 4.1.10 taken from (Freeman, 2013).

Definition 4.1.10. An equilibrium p ∈ X is exponentially unstable for vector field f if

the differential dfp at p has at least one eigenvalue in the open right-half complex plane.

Relative and local set stability definitions

In this section we will generalize some of the definitions presented in the previous sections.

The definitions for relative and local set stability, taken from (El-Hawwary and Maggiore,

2013b), are given in Definition 4.1.11.


Definition 4.1.11. Let Γ1 ⊂ Γ2 be two subsets of X that are positively invariant for Σ.

Assume that Γ1 is compact and Γ2 is closed.

• Γ1 is stable relative to Γ2 for Σ if, for any ǫ > 0, there exists a neighborhood N (Γ1)

such that, φ(R+,N (Γ1) ∩ Γ2) ⊂ Bǫ(Γ1). The notions of relative set attractivity,

and asymptotic and practical stability are obtained analogously by restricting initial

conditions to lie in Γ2.

• Γ2 is locally stable near Γ1 if for all x ∈ Γ1, for all c > 0 and all ǫ > 0, there exists

δ > 0 such that for all x0 ∈ Bδ(Γ1) and all t⋆ > 0, if φ([0, t⋆], x0) ⊂ Bc(x) then

φ([0, t⋆], x0) ⊂ Bǫ(Γ2).

• Γ2 is locally attractive near Γ1 if there exists a neighbourhood N (Γ1) such that, for

all x0 ∈ N (Γ1), ‖φ(t, x0)‖Γ2 → 0 as t→ ∞.

The definitions for local stability of Γ2 near Γ1 and local attractivity of Γ2 near Γ1 are

illustrated in Figure 4.5 adapted from (El-Hawwary and Maggiore, 2013b). An explana-

tion of local stability of Γ2 near Γ1 was given in (El-Hawwary and Maggiore, 2013b) as

follows: “Given an arbitrary ball Bc(x) centred at a point x in Γ1, trajectories originating

in Bc(x) sufficiently close to Γ1 cannot travel far away from Γ2 before exiting Bc(x).”

Local attractivity of Γ2 near Γ1, on the other hand, means that all initial conditions

beginning in a neighborhood of Γ1 converge to Γ2 as t→ ∞. It can be seen immediately

that the following implications hold

• Γ2 is stable =⇒ Γ2 is locally stable near Γ1

• Γ1 is stable =⇒ Γ2 is locally stable near Γ1

• Γ2 is attractive =⇒ Γ2 is locally attractive near Γ1

The following illustrative example was taken from course notes (Maggiore, 2015). It

shows that Γ2 may be locally stable near Γ1 even if Γ2 itself is unstable, i.e., Γ2 is locally

stable near Γ1 ; Γ2 is stable.


Figure 4.5: Illustration of local stability of Γ2 near Γ1 (left) and local attractivity of Γ2

near Γ1 (right).

Example 4.1.12. Consider the system

x1 = −x1(1− x22)

x2 = x2,

let Γ1 = 0 be the origin, and let Γ2 be the x2 axis. We claim that Γ2 is unstable, but it

is locally stable near Γ1. Indeed, if x2(0) 6= 0, then x2(t) → ∞, so that eventually x1(t)

will have the same sign as x1(t) and tend to infinity. Thus Γ2 is unstable. On the other

hand, for any ball Bc(0), let ǫ > 0 be arbitrary. We see from the phase portrait that we

can find δ > 0 such that solutions originating in Bδ(Γ1) do not exit Bǫ(Γ2) as long as

they remain in Bc(0). Thus, Γ2 is locally stable near Γ1.

The definitions for exponential stability and instability of an equilibrium point can

similarly be defined relative to a subset Γ ⊂ X as stated in Definition 4.1.13 below.

Definition 4.1.13. Consider the dynamical system Σ, let Γ be a closed subset of X and

suppose Γ is invariant under the flow of the vector field f . Let f |Γ be the restriction of

the vector field f to Γ.

• If p ∈ Γ is an equilibrium of f , we say that p is exponentially stable relative to

Γ if p is exponentially stable for f |Γ, i.e., the differential d(f |Γ)p at p has all its

eigenvalues in the open left-half complex plane.

• If p ∈ Γ is an equilibrium of f , we say that p is exponentially unstable relative to Γ


−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x1

x2

Bǫ(Γ2)

Bc(0)Bδ(Γ1)

Figure 4.6: Example of local stability of Γ2 near Γ1 in Example 4.1.12. Image from (Mag-giore, 2015)

if p is exponentially unstable for f |Γ, i.e., the differential d(f |Γ)p at p has at least

one eigenvalue in the open right-half complex plane.

Theorem 4.1.1 below, taken from (Freeman, 2013), allows us to say when the domain

of attraction of a set of exponentially unstable equilibriaA is Lebesgue measure zero. This

is a desirable property if the states in A represent undesirable equilibrium configurations

of the state-space for the closed loop system that do not correspond with the control

objective.

Theorem 4.1.1 (Proposition 1 in (Freeman, 2013)). Suppose A is a set of equilibria for

system Σ, each equilibrium in A is exponentially unstable, and

D(A) =⋃

z∈A

D(z). (4.3)

Then D(A) is meager and has zero measure.

Condition (4.3) of Theorem 4.1.1 is satisfied if and only if a solution converging to

the set A, necessarily converges to one of the exponentially unstable equilibria in the set


A. A set is meager if it can be expressed as the union of countably many nowhere dense

subsets.

4.1.2 Reduction Theorems

In this section we introduce a powerful tool for analyzing the stability of sets. This tool

is called reduction and allows one to determine the stability and asymptotic stability

properties of a compact positively invariant subset of X , call it Γ1, in a hierarchical

fashion. In particular, if Γ1 satisfies a certain stability or asymptotic stability property

relative to another positively invariant set Γ2 containing it, then the reduction theorem

gives conditions guaranteeing that Γ1 also satisfies this property with respect to the entire

state-space X .

Reduction Theorem for Asymptotic Stability

Theorem 4.1.2 presents the Reduction Theorem for stability, local asymptotic stability,

and global asymptotic stability.

Theorem 4.1.2 (Reduction Theorem (El-Hawwary and Maggiore, 2013b; Seibert and

Florio, 1995)). Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed sets that are positively

invariant for Σ, and suppose Γ1 is compact. Consider the following assumptions:

(i) Γ1 is LAS relative to Γ2;

(i’) Γ1 is GAS relative to Γ2;

(ii) Γ2 is locally stable near Γ1;

(iii) Γ2 is locally attractive near Γ1;

(iii)’ Γ2 is globally attractive;

(iv) all trajectories of Σ are bounded.


Figure 4.7: Illustration of the Reduction Theorem. The set Γ1 is asymptotically stablerelative to Γ2 and Γ2 is asymptotically stable. By reduction, Γ1 is asymptotically stable.

Then, the following implications hold: (i) ∧ (ii) =⇒ Γ1 is stable; (i) ∧ (ii) ∧ (iii) ⇐⇒

Γ1 is LAS; (i)’ ∧ (ii) ∧ (iii)’ ∧ (iv) ⇐⇒ Γ1 is GAS.

Note that one can replace condition (ii) with the stronger condition “Γ2 is stable”

and replace (iii) with the stronger condition “Γ2 is asymptotically stable”. Then it is

clear conditions (ii) and (iii) together are satisfied when Γ2 is asymptotically stable and

(ii) and (iii)’ together are satisfied when Γ2 is globally asymptotically stable. Reduction

is illustrated in Figure 4.7.

To help illustrate how the Reduction Theorem works, we provide a simple proof in

the special case of linear systems using tools learned in an undergraduate control course.

Proposition 4.1.14. Consider the linear system

x = Ax, (4.4)

where x ∈ Rn and A ∈ Rn×n. Suppose the origin Γ1 = 0 is an equilibrium point,

Γ2 is an m-dimensional A-invariant subspace and the following assumptions from the

Reduction Theorem hold

(i)’ Γ1 is (globally) asymptotically stable relative to Γ2

(ii),(iii)’ Γ2 is (globally) asymptotically stable.

Then Γ1 is (globally) asymptotically stable.


Proof. Since Γ2 is A-invariant, there exists a coordinate transformation z = (z1, z2) =

T−1x such that z1 ∈ Rm represents coordinates tangential to Γ2 and z2 ∈ Rn−m repre-

sents coordinates transversal to Γ2. Moreover, let A = T−1AT . By the Representation

Theorem for linear systems, it follows that

z1

z2

=

A11 A12

0 A22

z1

z2

. (4.5)

Restricted to the set Γ2, the system equations in (4.5) are given by

z1 = A11z1

and assumption (i)’ implies that the eigenvalues of A11 belong to the open left half plane.

Moreover, assumptions (ii),(iii)’ imply the origin of the system,

z2 = A22z2

in (4.5) is asymptotically stable and therefore the eigenvalues of A22 belong to the open

left half plane. We conclude that A in (4.5) has all eigenvalues in the open left half plane

and therefore Γ1 is asymptotically stable.

Reduction Theorem: Almost Global Asymptotic Stability

The next result presents a reduction theorem for almost global asymptotic stability. This

is a novel contribution of this thesis.

Theorem 4.1.3. Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed sets that are positively

invariant for Σ, and suppose Γ1 is compact. Consider the following assumptions:

(i) Γ2 is an embedded submanifold of X which is globally asymptotically stable for Σ.

(ii) Γ1 is globally attractive relative to Γ2 and can be decomposed as a disjoint union


Figure 4.8: Illustration of reduction for almost global asymptotic stability. The figureon the left illustrates the dynamics on Γ2 which is globally asymptotically stable relativeto X . Γ2 contains three unstable equilibria A = z1, z2, z3 and a compact set K thatis almost globally asymptotically stable relative to Γ2. By the Reduction Thoerem theset K is also AGAS relative to X as shown in the figure on the right. The set D(A) isLebesgue measure zero and contains the initial condition χ0 whose solution converges tothe point z3 ∈ A.

Γ1 = A⊔K where A is a set of isolated equilibria which are exponentially unstable

relative to Γ2 and K is asymptotically stable relative to Γ2.

(iii) all trajectories of Σ are bounded.

Then K is almost globally asymptotically stable.

The Reduction Theorem for almost global asymptotic stability is illustrated in Fig-

ure 4.8. The proof of Theorem 4.1.3 relies on the three lemmas presented next. The

proof of Theorem 4.1.3 is given after the presentation of the lemmas.

Lemma 4.1.15. Let Γ be a closed embedded submanifold of X which is invariant under

a C1 vector field f : X → TX . Let (U, ϕ) be a smooth coordinate chart centred at p ∈ Γ

and let dfp be the matrix representation of the differential dfp : TpX → Tf(p)(TX ) in

coordinates. If p is an equilibrium of f , then the subspace Tϕ(p)ϕ(Γ ∩ U) of Tϕ(p)ϕ(U) is

(dfp)-invariant.

Proof. Let n = dimX , p ∈ Γ be such that f(p) = 0, and let vp ∈ TpΓ be arbitrary

with vector representation vp ∈ Tϕ(p)ϕ(Γ ∩ U) in coordinates. We need to prove that


dfpvp ∈ Tϕ(p)ϕ(Γ ∩ U). Since Γ is embedded, there exists an open set U ⊂ U containing

p and a smooth submersion h : U → Rn−dimΓ such that Γ∩ U = h−1(0). Let I = (−1, 1)

and σ : I → X be a smooth regular curve whose image is contained in Γ ∩ U , and such

that σ(0) = p and σ(0) = vp. Since Γ is invariant under the flow of f , f(σ(t)) ∈ Tσ(t)Γ

for all t ∈ I or, what is the same, dhσ(t)(f(σ(t))) = 0 for all t ∈ I. Equivalently,

dhσ(t)f(σ(t)) = 0 for all t ∈ I where f(σ(t)) is the vector representation of f(σ(t)) and

dhσ(t) is the matrix representation of the differential dhσ(t) : Tσ(t)U → Th(σ(t))Rn−dimΓ in

coordinates. The derivative of dhσ(t)f(σ(t)) with respect to t must be zero at t = 0,

d

dt

∣

∣

∣

t=0

[

dhσ(t)f(σ(t))]

= 0,

or(

d

dt

∣

∣

∣

t=0dhσ(t)

)

f(σ(0)) + dhσ(0)d

dt

∣

∣

∣

t=0f(σ(t)) = 0.

Since f(σ(0)) = f(p) = 0, using the chain rule we get dhσ(0)dfσ(0)vp = 0, proving that

dfpvp ∈ TpΓ.

Lemma 4.1.16. Let Γ be a closed embedded submanifold of X which is invariant under

a C1 vector field f : X → TX . If p ∈ Γ is an equilibrium of f which is exponentially

unstable relative to Γ, then p is exponentially unstable relative to X .

Proof. Let n = dimX and k = dimΓ. Since Γ is embedded, there exists a smooth

coordinate chart (U, ϕ) centred at p, i.e., ϕ(p) = 0, such that ϕ(Γ ∩ U) = x ∈

Rn : xk+1 = · · · = xn = 0 where x = (x1, . . . , xk, xk+1, . . . , xn) are local coordi-

nates (Lee, 2013). Since Γ is positively invariant, it follows that the restriction of the

vector field f(x) = (fi(x))i∈1,...,n to Γ ∩ U , represented in local coordinates, is given

by f |Γ∩U(x1, . . . , xk) = (fi(x1, . . . , xk, 0, . . . , 0))i∈1,...,k. Since p is exponentially unsta-

ble relative to Γ, d(f |Γ∩U)ϕ(p) has at least one eigenvalue in the open right-half complex

plane. To prove that p is exponentially unstable relative to X , it needs to be shown

that dfϕ(p) has at least one eigenvalue in the open right-half complex plane. It follows


from Lemma 4.1.15 that the tangent space Tϕ(p)ϕ(Γ ∩ U) is (dfϕ(p))-invariant in local

coordinates and therefore dfϕ(p) has the upper triangular form,

dfϕ(p) =

A1 A2

0 A3

where

A1 =

df1(x)dx1

. . . df1(x)dxk

......

dfk(x)dx1

. . . dfk(x)dxk

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

x=0

=

df1(x1,...,xk,0,...,0)dx1

. . . df1(x1,...,xk,0,...,0)dxk

......

dfk(x1,...,xk,0,...,0)dx1

. . . dfk(x1,...,xk,0,...,0)dxk

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(x1,...,xk)=0

= d(f |Γ∩U)ϕ(p)

and therefore dfϕ(p) contains at least one eigenvalue in the open right-half complex plane.

Lemma 4.1.17. For the dynamical system Σ, suppose Γ ⊂ X is a closed embedded

submanifold of X which is globally asymptotically stable. If N ⊂ Γ is a set of Lebesgue

measure zero in Γ, then Γ\N is globally asymptotically stable.

Proof. To show that Γ\N is stable, it needs to be shown that for any ε > 0, there exists a

neighborhood N (Γ\N) such that, for all χ0 ∈ N (Γ\N), φ(t, χ0) ∈ Bε(Γ\N), for all t > 0.

Since Γ is globally asymptotically stable, there exists a neighborhood N1(Γ) such that,

for all χ0 ∈ N1(Γ), φ(t, χ0) ∈ Bε/2(Γ), for all t > 0. Since N is Lebesgue measure zero in

Γ, Γ\N is dense in Γ by Proposition 6.8 in (Lee, 2013) and for all χ ∈ N there exists a

point χ ∈ Γ\N such that χ ∈ Bǫ/2(χ). Stability follows by choosing N (Γ\N) = N1(Γ).

Analogously, global attractivity of Γ\N follows from global attractivity of Γ and

density of Γ\N in Γ.

The proof of Theorem 4.1.3 is presented below.


Proof of Theorem 4.1.3. By Lemma 4.1.16, the isolated equilibria in A, which are ex-

ponentially unstable equilibria relative to Γ2, are also exponentially unstable relative to

X . It holds from Theorem 2.37 in (Rudin, 1976) that since A is compact, if A were an

infinite subset of Γ2, then A would necessarily have a limit point χ ∈ Γ2. Since A is

closed, it contains all its limit points and therefore χ ∈ A. But this contradicts that all

points in A are isolated and therefore A must be finite. Suppose there are m equilibria

in A labelled χ1, . . . , χm. Then for all i ∈ 1, . . . , m, there exists ǫi > 0 such that

Bǫi(χi) ∩ A = χi and the minimum ǫ = minǫ1, . . . , ǫm exists. It follows that for all

i ∈ 1, . . . , m, Bǫ(χi) ∩ A = χi. The condition D(A) =⋃

z∈AD(z) in Theorem 4.1.1

holds if any solution φ(t, χ0) with initial condition χ0 ∈ X that converges to A, necessar-

ily converges to a particular point χ ∈ A. This must be the case since for any solution

φ(t, χ0) converging to A there exists a time T > 0 such that φ(t, χ0) ∈ Bǫ/2(A) for all

t > T . Therefore, there exists a point χ ∈ A such that φ(T, χ0) ∈ Bǫ/2(χ). For t > T it

is impossible for the solution to leave the vicinity of χ and converge to another point in

A located at least a distance ǫ away. Therefore the condition in (4.3) holds and D(A)

has zero measure by Theorem 4.1.1. It holds from analogous arguments that the set

D(A) ∩ Γ2 is Lebesgue measure zero in Γ2.

Denote W := X\D(A) which is a positively invariant domain of full measure and

define Γ2 := W ∩ Γ2, a set of full measure in Γ2. Then Γ2 is GAS relative to X by

Lemma 4.1.17. Since W is positively invariant, it follows that Γ2 is GAS relative to W .

Since K is stable relative to Γ2 and W is positively invariant, K is also stable relative to

Γ2 = W ∩ Γ2 and since Γ1 = A ∪ K is globally attractive relative to Γ2, it immediately

holds that K is globally attractive relative to Γ2. Therefore K is GAS relative to Γ2 and

Γ2 is GAS relative to W implying, by Theorem 4.1.2, that K is globally asymptotically

stable relative toW . This implies that K is almost globally asymptotically stable relative

to X .


4.1.3 Lyapunov and Krasovskii-LaSalle Theorems

In this section we state a number of standard stability results taken from (Maggiore,

2009). These include Lyapunov Stability Theorems and the Krasovskii-LaSalle Invariance

Principle which will be used often in this work. These results are stated here for the

reader’s convenience.

Consider the dynamical system

χ = f(χ), χ ∈ X (4.6)

where X ⊂ Rn is open and f is either locally Lipschitz on X or C1. Suppose that 0 ∈ X .

For a C1 function V (χ), LfV (χ) := (d/dχ)V (χ) · f(χ) is the Lie derivative of V along f .

Theorem 4.1.4 (Lyapunov’s direct method). Let χ = 0 be an equilibrium of (4.6) with

X ⊂ Rn, and suppose there exists a C1 function V : D → R, with D ⊂ X a domain

containing 0, such that

(i) V is positive definite at 0

(ii) The Lie derivative LfV is negative definite at 0

then 0 is an asymptotically stable equilibrium of (4.6).

The global version of the Lyapunov Theorem is given below.

Theorem 4.1.5 (Barbashin-Krasovskii). Let χ = 0 be an equilibrium of (4.6) with

X = Rn, and suppose there exists a C1 function V : Rn → R function such that V is

positive definite at 0, radially unbounded, and LfV is negative definite at 0. Then, χ = 0

is globally asymptotically stable.

Theorem 4.1.6 (Krasovskii-LaSalle Invariance Principle). Let Ω ⊂ D be a compact

positively invariant set. Let V : D → R be a C1 function such that for all χ ∈ Ω, LfV


is negative semi-definite. Let E = χ ∈ Ω : LfV = 0 and let N be the largest invariant

subset of E. Then for all χ0 ∈ Ω, φ(t, χ0) → N as t→ ∞.2

From Theorem 4.1.6, it is also possible to obtain global results. In particular, if

D = Rn, every level set of V is compact, and if for all χ ∈ Rn, LfV is negative semi-

definite, then for all χ0 ∈ Rn, φ(t, χ0) converges to N , the largest invariant set of E =

χ ∈ Rn : LfV = 0. The three theorems just stated have been presented with increasing

generality with the Krasovskii-LaSalle Invariance Principle above being the most general

as it does not require the function V to be positive definite and the attractor set is not

necessarily a point. The following proposition establishes global practical stability of an

equilibrium.

Proposition 4.1.18 (Lyapunov result for global practical stability). Let χ = 0 be an

equilibrium of (4.6) with X = Rn, and suppose there exists a C1 function V : Rn → R

such that V is positive definite at 0, has compact level sets, and LfV is negative outside

the set Vc := χ ∈ Rn : V < c. Then, Vc is globally asymptotically stable.

Proof. Stability follows because LfV is negative outside the sub-level set Vc of the Lya-

punov function V . For attractivity, consider any initial condition χ0 ∈ Rn. There exists

ǫ > c such that χ0 ∈ Vǫ. Since LfV is negative outside of Vc and Vǫ\Vc is a compact set,

d := maxχ∈Vǫ\Vc LfV (χ) < 0 is well-defined. Therefore, φ(t, χ0) ∈ Vc for all t > (ǫ− c)/d

proving attractivity of Vc.

Remark 4.1.7. The three theorems and proposition stated above all require V (χ) to be

a C1 function. The proofs still hold, however, under the milder assumption that V (χ)

and LfV (χ) are both continuous.

2The proof of the Krasovskii-LaSalle Invariance Principle (and in turn, the other theorems in thissection) also apply when the state space is a smooth Riemannian manifold (X , g). See, for example, theproof in (Khalil, 2002, Theorem 4.4).


Application to Gradient and Homogeneous Systems

We now review classes of gradient and homogeneous systems and show how they are well

suited for stability analyses using the Lyapunov and Krasovskii-LaSalle theorems just

discussed. The definition of a gradient system is as follows,

Definition 4.1.19. A gradient system on the state space X is a differential equation of

the form,

χ = −∇V (χ), χ ∈ X (4.7)

where V : X → R is a real valued, twice continuously differentiable function, and ∇V (χ)

is its gradient.

The main feature of a gradient system is that the vector field −∇V (χ) is orthogonal

to the level sets of the corresponding storage function V and point in the direction of

steepest decent of V . The stationary points of the system correspond to the set where

the gradient of V is zero, i.e., the critical points of V . The function V becomes a natural

candidate for a Lyapunov analysis. Taking the time derivative of V along system (4.7)

yields,

V =∂V

∂χχ = −‖∇V (χ)‖2 ≤ 0.

Consider any isolated critical point χ which is a local minimum of V . Then χ is an

equilibrium for (4.7) and it immediately follows from the Lyapunov’s direct method that

χ is asymptotically stable for (4.7). Gradient systems can also be analysed with the

Krasovskii-LaSalle Invariance Principle. In particular, if V is defined globally on the

domain D = Rn and its level sets are compact, then since the time derivative of V is

negative semi-definite and equals zero only in the set E = χ ∈ X : ∇V (χ) = 0 of the

critical points of V , we can conclude by Krasovskii-LaSalle that all solutions converge to

the largest invariant set of critical points of V .

Definition 4.1.20. Let U,W be finite-dimensional vector spaces and let V be a set. A


function f : U → W is homogeneous of degree r ≥ 0 if, for all λ > 0 and for all χ ∈ U ,

f(λχ) = λrf(χ). A function f : U×V →W , (χ,Υ) 7→ f(χ,Υ), is homogeneous of degree

r with respect to χ if for all λ > 0 and for all (χ,Υ) ∈ U × V , f(λχ,Υ) = λrf(χ,Υ).

By this definition, a homogeneous function of degree r is one that scales by powers

of r moving outward along rays going through the origin. Examples of homogeneous

functions are

• The function f(x, y) = atan2(y, x) is homogeneous of degree zero. Notice that f is

undefined at (x, y) = (0, 0);

• Linear functions are homogeneous of degree one directly by the scalar multiplication

property f(λχ) = λχ for any λ > 0;

• The real function f(x, y) = xy is homogeneous of degree two since f(λx, λy) = λ2xy

for any scalar λ > 0;

• The real function f(x, y) = x2y is homogeneous of degree two with respect to x

and one with respect to y.

If the function f(χ) is homogeneous of degree r and g(χ) is homogeneous of degree s

then the product h(χ) is homogeneous of degree r + s since

h(λχ) = f(λχ)g(λχ) = λr+sf(χ)g(χ) = λr+sh(χ).

Similarly, h(χ) = f(χ)/g(χ) is homogeneous of degree r − s. Consider the following

propositions related to homogeneous functions.

Proposition 4.1.21. For a compact set V , let f : Rn × V → R, (χ,Υ) 7→ f(χ,Υ) be a

continuous function, homogeneous of degree zero with respect to χ. Then f achieves a

maximum on Rn × V .


Proof. Since f(χ,Υ) is homogeneous of degree zero with respect to χ, it follows that for all

χ 6= 0, f(χ,Υ) = ‖χ‖0f(χ/‖χ‖,Υ) = f(χ/‖χ‖,Υ). Since f is continuous and (χ/‖χ‖,Υ)

lie on compact sets, f(χ/‖χ‖,Υ) has a maximum value on the domain (Rn\0) × V .

Moreover, f(0,Υ) = 0. Therefore, f achieves a maximum on Rn × V .

Proposition 4.1.22. For a compact set V , let f : Rn × V → R, (χ,Υ) 7→ f(χ,Υ) be

a function, homogeneous of degree r ≥ 1 with respect to χ and continuous on the set

(Rn\0)× V . Then f is continuous on Rn × V .

Proof. It needs to be shown that f is continuous on 0×V . Since f is homogeneous of de-

gree r ≥ 1, f(0,Υ) = 0. For all χ 6= 0, f(χ,Υ) satisfies |f(χ,Υ)| = ‖χ‖r|f(χ/‖χ‖,Υ)| ≤

‖χ‖rmax(χ,Υ)∈(Rn\0)×V |f(χ/‖χ‖,Υ)|. This maximum exists because (χ/‖χ‖,Υ) lie on

compact sets and |f(χ,Υ)| is continuous on (Rn\0)× V . Therefore, for all

δ <

(

ǫ

max(χ,Υ)∈(Rn\0)×Υ |f(χ/‖χ‖,Υ)|

)1/r

,

‖χ‖ < δ implies |f(χ,Υ)| ≤ ǫ and therefore f(χ,Υ) is continuous on the set 0 × V .

Now consider the dynamical system,

Σ : χ = f(χ), χ ∈ X (4.8)

where X is a finite-dimensional vector space. We say that Σ in (4.8) is a homogeneous

system of degree r if f is a homogeneous function of degree r.

Proposition 4.1.23 below shows that if we have a homogeneous system and a positive

definite homogeneous Lyapunov function V , then to prove global asymptotic stability of

the origin 0, it is enough to show negative definiteness of LfV on a compact set rather

than the entire state-space. This is beneficial because compact sets enjoy several useful

properties. For example, the fact that continuous real functions attain a maximum on

compact sets is an important feature that will be used in this thesis.


Proposition 4.1.23. Consider the homogeneous system of degree r > 0 in (4.8), and

assume that X = Rn. Suppose that there exists a continuous positive definite function

V : Rn → R, homogeneous of degree s ≥ 1 and suppose that LfV (χ) is continuous and

satisfies LfV (θ) < 0 for all θ ∈ χ ∈ Rn : ‖χ‖ = 1 ∼= Sn−1. Then χ = 0 is globally

asymptotically stable.

Proof. Since V (χ) is homogeneous of degree s ≥ 1, for all λ > 0 and for all χ ∈ X ,

V (λχ) = λsV (χ). It follows from Euler’s Theorem presented in Section 12.8 in (Allen,

1938) that the derivative DV (χ) := ∂V (χ)/∂χ is homogeneous of degree s−1. Moreover,

V (χ) is radially unbounded because it is both positive definite and homogeneous of degree

s ≥ 1. One can write χ = λθ where λ = ‖χ‖ and θ ∈ Sn−1 is a unit vector. The Lie

derivative LfV (χ) therefore satisfies

LfV (χ) = DV (λθ)f(λθ)

which by the homogeneity properties of DV and f becomes,

LfV (χ) = λr+(s−1)DV (θ)f(θ) = λr+(s−1)LfV (θ).

By assumption, for all θ ∈ Sn−1, LfV (θ) < 0. Therefore LfV (χ) ≤ 0 with equality

if and only if λ = ‖χ‖ = 0 and it follows that LfV (χ) is negative definite. By the

Barbashin-Krasovskii Theorem and Remark 4.1.7, the origin is globally asymptotically

stable.

The proofs in Chapter 5 and, particularly, Chapter 6 use similar arguments to Propo-

sition 4.1.24 below. This proposition is presented primarily as an aid to understanding

the logic in those proofs.

Proposition 4.1.24. Consider the homogeneous system of degree r > 0 in (4.8), and

assume that X = Rn. Let V : X → R be a continuous positive definite function,


homogeneous of degree s ≥ 2. Let g : X → R be a continuous function, homogeneous of

degree s− 1. Finally, let

W (χ) = α√

V (χ) + g(χ)

for α > 0. Then there exists α⋆ > 0 such that for all α > α⋆, W (χ) is positive definite.

Moreover, if LfW (χ) is continuous and LfW (θ) < 0 for all θ ∈ χ ∈ Rn : ‖χ‖ = 1 ∼=

Sn−1 and for all α > α⋆, then χ = 0 is globally asymptotically stable.

Proof. The function W (χ) is continuous and homogeneous of degree s− 1 because both√

V (χ) and g(χ) are such. First it is shown that there exists α⋆ > 0 such that for all

α > α⋆, W (χ) is positive definite. For χ 6= 0, V (χ) > 0 and one can write

W (χ) =√

V (χ)

(

α +g(χ)√

V (χ)

)

.

The function g(χ)/√

V (χ) is homogeneous of degree zero because the numerator and

denominator are both homogeneous of degree s− 1 and therefore, by Proposition 4.1.21,

it attains a maximum over X . Therefore, choosing α⋆ > maxχ∈X ‖g(χ)/√

V (χ)‖ implies

that β := α⋆ + g(χ)/√

V (χ) > 0. Then, for all χ 6= 0, W (χ) >√

V (χ)β > 0 since V (χ)

is positive definite and W (0) = 0 since W (χ) is homogeneous of degree s− 1 > 0. W (χ)

is radially unbounded because it is both positive definite and homogeneous of degree

s − 1 > 0. Moreover, if LfW (θ) < 0 for all θ ∈ χ ∈ Rn : ‖χ‖ = 1 ∼= Sn−1 and for

all α > α⋆, then it follows from Proposition 4.1.23 that χ = 0 is globally asymptotically

stable.

4.2 Graph Theory

Since graph theory is such a broad field, in this background section we will limit the

discussion solely to notions in graph theory that are pertinent to this work. For more in

depth detail on the notions introduced in this section and graph theory in general, we


Table 4.2: Neighbors sets in Figure 4.9

Node i Ni-Directed Graph Ni-Undirected Graph

1 2 2, 52 3 1, 3, 53 4, 5 2, 4, 54 5 3, 55 1, 2, 3 1, 2, 3, 4

refer the reader to (Godsil and Royle, 2001).

4.2.1 Basic Definitions in Graph Theory

We denote a graph by the pair G = (V, E), where V is a set of nodes labelled as 1, . . . , n

and E = (i, j) ∈ V ×V : i connects to j is the set of edges such that (i, j) ∈ E if node i

connects to node j. A directed graph or digraph (undirected graph), is a graph where the

edges are ordered (unordered) pairs in which (i, j) 6= (j, i) ((i, j) = (j, i)). An example of

an undirected and directed graph are illustrated in Figure 4.9. Edges in a directed graph

are indicated with arrows whereas edges in an undirected graph are indicated with lines.

The set of neighbors of node i is the set Ni := j ∈ V : (i, j) ∈ E. The neighbor sets in

Figure 4.9 are listed in Table 4.2:

1

2

3

5

4

1

2

3

5

4

Figure 4.9: Example of a directed (left) and an undirected (right) graph.

Given positive numbers aij > 0, i ∈ n, j ∈ Ni, the associated weighted Laplacian

matrix of G is the matrix L = D − A, where the i-th diagonal entry of D is the sum∑

j∈Niaij , and A is the matrix whose element Aij is aij if j ∈ Ni, and 0 otherwise. The


Laplacian matrix for the directed and undirected graphs in Figure 4.9 for unitary weights,

i.e., aij = 1 for all j ∈ Ni, are given by

L =

1 −1 0 0 0

0 1 −1 0 0

0 0 2 −1 −1

0 0 0 1 −1

−1 −1 −1 0 3

L =

2 −1 0 0 −1

−1 3 −1 0 −1

0 −1 3 −1 −1

0 0 −1 2 −1

−1 −1 −1 −1 4

respectively.

A graph G ′ = (V ′, E ′) is a subgraph of G if V ′ ⊂ V and E ′ ⊂ E . An example of a

subgraph of the directed graph G in Figure 4.9 is shown in Figure 4.10.

1

2

3

5

Figure 4.10: Example of a subgraph G ′ = (V ′, E ′) of the directed graph in Figure 4.9 withV ′ = 1, 2, 3, 5 and E ′ = (5, 1), (5, 2), (2, 3), (3, 5).

A graph G is called static if it is constant for all time and time varying, denoted G(t),

if the set of nodes and/or edges varies with time t.

4.2.2 Classes of Graphs

In this section, we present several important classes of graphs. An undirected graph

G = (V, E) is said to be connected if for any two nodes i, j ∈ V there exists a path from

node i to node j by traversing the edges in the graph. A complete graph is an undirected

graph with an edge between every pair of nodes, that is, for any i, j ∈ V, (i, j) ∈ E . In


the case of directed graphs, a graph is strongly connected if for any two nodes i, j ∈ V

there exists a path from i to j. The undirected graph in Figure (4.9) is connected while

the directed graph is strongly connected. By removing the edge (5, 1), the directed graph

in Figure (4.9) is no longer strongly connected.

A directed spanning tree is a digraph consisting of n− 1 edges such that there exists

a unique directed path from a node, called the root, to every other node. A reverse

directed spanning tree is a graph which becomes a directed spanning tree by reversing the

directions of all its edges. We identify the root of a reverse spanning tree with the root of

its associated spanning tree. Therefore, in a reverse directed spanning tree the root node

can be sensed indirectly by all other nodes in the graph by means of directed paths. In

this sense, the root node can propagate information indirectly through the entire graph.

An example of a reverse directed spanning tree is illustrated in Figure 4.11.

5 6

7 8 9

32

1

4

Figure 4.11: Reverse directed spanning tree with root node 1.

A digraph G contains a reverse directed spanning tree if it has a subgraph G ′ = (V ′, E ′)

satisfying V ′ = V which is a reverse directed spanning tree. A node is globally reachable

if there exists a path from any other node to it. For a digraph G, existence of a globally

reachable node is equivalent to having a directed spanning tree in the reverse graph. An

example of such a digraph is illustrated in Figure 4.12 where node 1 is a globally reachable


node. Strongly connected graphs necessarily contain a globally reachable node. In fact,

every node is globally reachable by definition.

1 2

3 4 5

6

78

9

10

11 12

Figure 4.12: Directed graph G containing a reverse directed spanning tree and globallyreachable node, node 1.

If a digraph does not contain a reverse directed spanning tree, then there does not

exist any node in the graph that can propagate information to the entire network. This

can be interpreted as a lack of connectivity in the graph. In fact, Proposition 4.2.1 relates

the presence of a globally reachable node directly to the eigenvalues of the corresponding

weighted Laplacian.

Proposition 4.2.1 ((Ren and Beard, 2005; Lin et al., 2005)). The following conditions

are equivalent for a digraph G:

(i) G contains a globally reachable node.

(ii) For any set of positive gains aij > 0, i, j ∈ 1, . . . , n the associated weighted

Laplacian matrix L of G is positive semi-definite, has rank n − 1, and KerL =

span1.

A hierarchical digraph, defined in Definition 4.2.2, is a class of digraphs that contains

a globally reachable node.


Definition 4.2.2. A digraph G with n nodes containing a globally reachable node is

called a hierarchical digraph with m ∈ N layers if there exists a partition of V, P =

Lℓℓ∈1,...,m into m nonempty subsets called layers such that for all ℓ ∈ 1, . . . , m and

for all i ∈ Lℓ, if (i, j) ∈ E is an edge from node i to node j then j ∈ ⋃k∈1,...,ℓ−1 Lk.

In a hierarchical digraph G, the nodes are divided into layers in which layer 1 corre-

sponds to a single root. Nodes in layer i have neighbors only in layers j < i as illustrated

in Figure 4.13. As such, the root node is globally reachable and has no neighbors. As

an example, this type of graph can model the behaviour seen in flocking birds that only

observe other birds in front of themselves in a flocking formation.

Figure 4.13: Hierarchical sensing graph

4.2.3 Graph Decomposition

In this section it will be shown how a digraph containing a globally reachable node can

be decomposed into so-called strongly connected components. The strongly connected

components can be treated as nodes in a higher level graph known as a condensation

digraph. We now give a number of formal definitions.

A set of nodes S ⊂ V is an isolated component if it has no outgoing edges, i.e.,

for any edge (i, j) ∈ E , if i ∈ S then j ∈ S. A subgraph G ′ is an induced subgraph


of G if for any two vertices i, j ∈ V ′, (i, j) ∈ E ′ if (i, j) ∈ E . A strongly connected

component G ′ of G is a maximal strongly connected induced subgraph of G. In other

words, there does not exist any other strongly connected induced subgraph of G containing

G ′. Letting G0 = (V0, E0), . . . ,Gr = (Vr, Er) be the strongly connected components of G,

the condensation digraph of G, denoted C(G) = (VC(G), EC(G)), is defined as follows. The

vertex set VC(G) is the set of nodes vii∈0,...,r where the node vi is a contraction of the

vertex set Vi of the i-th strongly connected component Gi. The edge set EC(G) contains

an edge (vi, vj) if there exist vertices i′ ∈ Vi and j′ ∈ Vj such that (i′, j′) ∈ E . The

following properties of the condensation digraph are found in (Hatanaka et al., 2015).

Proposition 4.2.3 ((Hatanaka et al., 2015)). Consider a digraph G containing a globally

reachable node. The condensation C(G) satisfies the following properties:

(i) C(G) is acyclic, i.e., there is no path in C(G) beginning and ending at the same

node.

(ii) C(G) contains a reverse directed spanning tree T with a unique root v0 ∈ VC(G).

(iii) There exists at least one vertex vi ∈ VC(G) such that v0 is the only neighbor of vi.

An example of a digraph G containing a reverse directed spanning tree is shown

in Figure 4.14. The strongly connected components are boxed. The resulting acyclic

condensation digraph C(G) is shown in Figure 4.15. The vertex v0 in the figure is the

unique root of the reverse directed spanning tree in C(G).

As in (Hatanaka et al., 2015), we define the vertex set Lj ⊂ V to be the union

of those vertex sets Vi that correspond to vertices vi in the condensation digraph with

the property that the maximal path length from vi to the root v0 is equal to j. By

this definition, L0 := V0. We let L−1 := ∅. Defining the vertex set Lj := ∪ji=0Li, by

construction, the neighbors of any vertex in Lj are contained in Lj−1. Therefore each


1 2

3 4 5

6

78

9

10

11 12

G0

G3

G1

G2

Figure 4.14: Directed graph G containing a reverse directed spanning tree. The stronglyconnected components G0, . . . ,G3 are boxed

v0

v1

v2

v3

v0

v1

v2

v3

Figure 4.15: Condensation digraph C(G) associated with the graph G in Figure 4.14 (left)and reverse directed spanning tree contained in C(G) (right).

node set Lj is isolated. For the example in Figure 4.15, we have L0 = 1, 2, 3, 4, 5, 6,

L1 = 10 ∪ 11, 12 and L2 = 7, 8, 9.

4.3 Control Primitives

The control solutions presented in each section of this thesis will combine one or more

of the control primitives presented in this section. In the case of kinematic unicycles,

the control primitives are based on two kinematic models in two dimensions: single

integrators in R2 and rotational integrators in SO(2). In the case of flying robots, the

control primitives are based on double integrators in R3 and rotating bodies in SO(3).


The single and double integrators evolve in Euclidean space and represent positions

whereas the rotational integrators in SO(2) and rotating bodies in SO(3) represent angular

quantities. The control primitives considered in this work are listed in Table 4.3.

In the consensus problems, the goal is to globally asymptotically stabilize the set

where states of all agents coincide, be they translational or rotational. In this problem,

each agent can only sense a subset of neighboring agents. Throughout this section, the

sensing convention is defined by a sensor graph G = (V, E) discussed in Section 4.2, where

each node in the set V represents an agent, and an edge in the edge set E between node

i and node j indicates that agent i can sense agent j. Given a node i ∈ V, recall that its

set of neighbours Ni represents the set of agents that agent i can sense.

In the integrator path following problem, the goal is to drive a single integrator to a

smooth path in R2 and follow it with the desired speed and direction.

In the problems of rotational integrator and rotating body equilibrium stabilization,

the goal is to almost globally asymptotically stabilize an equilibrium in SO(2) and SO(3)

respectively.

While single and double integrators have a state-space which is a non-compact man-

ifold, rotational integrators and rotating bodies evolve on compact manifolds without

boundary. This fact leads to a topological obstruction in which global asymptotic con-

sensus or equilibrium stabilization cannot be achieved using continuous, time-invariant

control (Bhat and Bernstein, 2000). In this case, almost global asymptotic stability is

the best one can do with continuous time-invariant control. As a result of this, one may

expect that a control solution built out of a rotational integrator control primitive might

inherit this characteristic.


Table 4.3: Table of Control Primitives

Unicycle

Single integrator consensus (4.10)Uniformly bounded single integrator consensus (4.11)Single integrator path following (4.15)Rotational integrator equilibrium stabilization (4.21)Rotational integrator consensus - Kuramoto (4.23)Rotational integrator consensus (4.24)

Flying Robot

Double integrator consensus (4.19)Rotating body equilibrium stabilization (4.26)

4.3.1 Control Primitives for Single Integrators

Single integrator consensus controllers

Consider n single integrators

xi = vi, i ∈ n, (4.9)

where xi ∈ R2 and vi is the control input of subsystem i. Let xij := xj − xi and

x = (xi)i∈n ∈ R2n.

• The feedback

vi = fi((xij)j∈Ni) :=∑

j∈Ni

aijxij , (4.10)

where aij > 0 for all j ∈ Ni, is an integrator consensus controller. If G contains

a globally reachable node then the set x ∈ R2n : xi = xj , i, j ∈ n is globally

asymptotically stable for system (4.9), (4.10). See (Ren and Beard, 2005; Moreau,

2004; Olfati-Saber and Murray, 2004).

• The feedback

vi = fi((xij)j∈Ni) :=∑

j∈Ni

aijf(‖xij‖)‖xij‖

xij , (4.11)

is a uniformly bounded integrator consensus controller if aij = aji > 0 and f : R →

R, the interaction function, is a locally Lipschitz function satisfying:


A1: sf(s) > 0 for all s 6= 0, f(0) = 0, and there exist c1, c2 > 0 such that

|f(s)| > c1 for all |s| > c2.

A2: sup |f(s)| <∞.

Notice that the conditions on c1 and c2 in A1 imply that f(s) cannot tend to

zero as s tends to infinity. Each element (xij/‖xij‖)f(‖xij‖) of the sum in (4.11)

is continuous at xij = 0 because f(s) is a continuous function and f(0) = 0 by

assumption A1. We will omit the simple proof of the fact that each fi((xij)j∈Ni) is

Lipschitz continuous.

Examples of suitable interaction functions are f(s) = tanh(s) and

f(s) =

s, if |s| ≤ 1

s/|s| if |s| > 1.

(4.12)

The following proposition shows that feedback (4.11) globally asymptotically sta-

bilizes the consensus subspace x ∈ R2n : xi = xj , i, j ∈ n for any connected,

undirected sensor graph G.

Proposition 4.3.1. Consider system (4.9) with feedback (4.11). Assume that

f(s) satisfies assumptions A1 and A2 and the sensor graph G is undirected and

connected. For any parameters aij = aji > 0, the consensus set x ∈ R2n : xi =

xj , i, j ∈ n is globally asymptotically stable.

Proof. Consider system (4.9) with feedback (4.11). The feedback fi in (4.11) for

unicycle i points into the convex hull formed by its neighbours. By Corollary

3.9 in (Lin et al., 2007a) the group of unicycles for system (4.9) achieves global

consensus.

In addition, the following two lemmas will be used later in the thesis.


Lemma 4.3.2. If the sensor graph G is undirected and connected, then for any

parameters aij = aji > 0, system (4.9) with feedback (4.11), where f(s) satisfies

assumptions A1 and A2, is a gradient system, x = −∇Vt(x), with nonnegative

storage function

Vt(x) =1

2

n∑

i=1

∑

j∈Ni

aij

∫ ‖xij‖

0

f(s)ds. (4.13)

Moreover, V −1t (0) = x ∈ R2n : xi = xj , i, j ∈ n.

Proof. Assumption A1 implies that the function xij 7→∫ ‖xij‖

0f(s)ds is nonnegative,

and it attains its global minimum when xij = 0. Since G is connected, Vt attains a

global minimum when xij = 0 for all i, j ∈ n, and therefore Vt is positive definite.

We now show the gradient property, i.e., (∂/∂xi)Vt = −fi((xij)j∈Ni)⊤. We have

∂Vt∂xi

=1

2

∑

j∈Ni

aij∂

∂‖xij‖

(

∫ ‖xij‖

0

f(s)ds

)

∂‖xij‖∂xij

∂xij∂xi

+1

2

∑

j∈Ni

aji∂

∂‖xji‖

(

∫ ‖xji‖

0

f(s)ds

)

∂‖xji‖∂xji

∂xji∂xi

=1

2

∑

j∈Ni

aijf(‖xij‖)(

xij⊤

‖xij‖

)

(−1)

+1

2

∑

j∈Ni

ajif(‖xji‖)(

xji⊤

‖xji‖

)

(1)

= −∑

j∈Ni

aijf(‖xij⊤‖)‖xij‖

xij = −fi((xij)j∈Ni)⊤.

Lemma 4.3.3. Assume G is undirected and connected, and consider system (4.9)

with feedback (4.11), where f(s) satisfies assumptions A1 and A2. For any param-

eters aij = aji > 0, the following three properties hold:

(i) R−1fi((xij)j∈Ni) = fi((R−1xij)j∈Ni) for all i, j ∈ n, R ∈ SO(2).

(ii) x ∈ R2n : fi((xij)j∈Ni) = 0, ∀i ∈ n = x ∈ R2n : xi = xj , i, j ∈ n.


(iii)∑

i fi(·) = 0.

Proof. To show (i), we use the fact that ‖R−1xij‖ = ‖xij‖. Then,

R−1fi(·) =∑

j∈Ni

aijf(‖R−1xij‖)‖R−1xij‖

R−1xij = fi((R−1xij)j∈Ni).

To show (ii), assume fi((xij)j∈Ni) = 0 for all i ∈ n. Then system (4.9) is at a

fixed point. By Proposition 4.3.1, the set x ∈ R2n : xi = xj , i, j ∈ n is globally

asymptotically stable, so it contains all fixed points. Therefore, xij = 0 for all

i, j ∈ n. Conversely, if xij = 0 for all i, j ∈ n, then it follows by definition that

fi = 0 for all i ∈ n. Finally, property (iii) follows by summing over the functions

fi in (4.11) and using the identities aij = aji, xij = −xji.

Path following controllers

Consider a smooth simple (no self intersections) curve C ⊂ R2 and final speed w > 0. If

x = σ(s) is an arc length parametrization of C, then define r(x) = r(σ(s)) = (d/ds)σ(s).

Consider the single integrator,

x = v, (4.14)

where x ∈ R2 and v ∈ R2 is the control input. The feedback,

v = h(x), (4.15)

where h : R2 → R2 is an integrator path following controller for C satisfying the following

properties:

A1: h is globally Lipschitz, i.e., there exists c > 0 such that for all x1, x2 ∈ R2, ‖h(x1)−

h(x2)‖ ≤ c‖x1 − x2‖,

A2: h(x) = wr(x) for all x ∈ C,


A3: the set C is asymptotically stable for (4.14), (4.15).

In this thesis, we will consider paths that are either straight lines of the form C(r0, p) =

x ∈ R2 : x = r0 + sp, s ∈ R ⊂ R2 for r0, p ∈ R2 or smooth Jordan curves. Single inte-

grators are essentially the simplest model that one can design a path following controller

for. For any smooth line or Jordan curve, there always exists a path following controller

satisfying properties A1-A3 above. To obtain such a controller for Jordan curves, one

can use a feedback linearization technique (Nielsen et al., 2010).

For a line C(r0, p) define an integrator line following controller

h(x) = k0(r0 − x)− k0((r0 − x) · p)p+ wp (4.16)

with k0 > 0. The path following controller in (4.16) can be written alternatively as,

h(x) = k0(c⋆(x)− x) + wp (4.17)

where c⋆(x) is the orthogonal projection of the point x onto C(r0, p), i.e., (c⋆(x)−x)·p = 0.

It is easy to verify that h in (4.16) satisfies properties A1-A3. Therefore, this controller

makes x converge to the path C(r0, p) and follow it in the direction of the beacon p with

steady state speed w.

4.3.2 Control Primitives for Double Integrators

Double integrator consensus

Consider a collection of n double-integrators

xi = vi

vi = Ti, i ∈ n,

(4.18)


where xi ∈ R3, vi ∈ R3 and Ti ∈ R3 is the control input of subsystem i. Let xij := xj−xi,

vij := vj − vi, x := (xi)i∈n ∈ R3n and v := (vi)i∈n ∈ R3n.

The feedback

Ti = fi((xij, vij)j∈Ni) :=∑

j∈Ni

aij

(

xij + γvij

)

, i ∈ n, (4.19)

where aij, γ > 0 is a double-integrator consensus controller. The following theorem is

taken from Ren et al. in (Ren and Atkins, 2007, Theorems 4.1, 4.2) and says that for

sufficiently large γ, if the sensing digraph contains a globally reachable node, then the

system of double integrators achieves consensus.

Theorem 4.3.1 ((Ren and Atkins, 2007)). Consider system (4.18) with feedback (4.19)

and sensor digraph G containing a globally reachable node with corresponding weighted

Laplacian matrix L with weights aij > 0. Suppose that γ is chosen to satisfy

γ > maxµi 6=0

√

√

√

√

2

|µi| cos(

π2− tan−1 −Re(µi)

Im(µi)

)

where (µi)i∈n are the eigenvalues of −L. Then the set

(x, v) ∈ R3n × R

3n : xi = xj , vi = vj , i, j ∈ n

is globally asymptotically stable.

4.3.3 Control Primitives for Rotational Integrators

Rotational integrator equilibrium stabilization

Consider a rotational integrator

θ = ω, (4.20)


where θ ∈ S1 and ω is the control input. A feedback

ω = g(θ) := −k0 sin(θ), (4.21)

where k0 > 0, is a rotational integrator equilibrium stabilizer. The following proposition

says that the equilibrium θ = 0 is almost globally asymptotically stable. We do not claim

originality for this result.

Proposition 4.3.4. Consider system (4.20) with feedback (4.21). The union of the two

equilibria θ = 0 and θ = π is globally attractive. The first equilibrium is almost globally

asymptotically stable while the second is exponentially unstable.

Proof. This follows by a standard Lyapunov analysis using the Lyapunov function V =

1 − cos(θ). The equilibrium θ = π is exponentially unstable since computing the lin-

earization at θ = π gives

d

dθ(−k0 sin θ)|θ=π = −k0 cos θ|θ=π = −k0 cosπ = k0 > 0.

Rotational integrator consensus

Consider a collection of rotational integrators,

θi = ωi, i ∈ n (4.22)

where θi ∈ S1, θ := (θi)i∈n ∈ Tn, θij := θj − θi and ωi is the control input of subsystem i.

• A feedback,

ωi = gi((θij)j∈Ni) :=∑

j∈Ni

bij sin(θij) (4.23)


with bij = bji > 0 is a Kuramoto consensus controller. This is the well-known

Kuramoto model for attitude synchronization of angles in S1 (Kuramoto, 1984)

with zero natural frequencies.

Proposition 4.3.5 ((Lin et al., 2007a)). Consider system (4.22) with feedback (4.23)

and connected, undirected sensor graph G. The set θ ∈ Tn : θi = θj , i, j ∈ n is

asymptotically stable with domain of attraction containing

Sπ := θ ∈ Tn : |θij | < π, i, j ∈ n ,

the set where all vehicle angles lie in the same half plane on an open arc of π

radians.

Proof. Without loss of generality, assume θi ∈ (−π/2, π/2) ⊂ S1 for all i ∈ n which

is diffeomorphic to the open segment (−π/2, π/2) ∈ R. If the angle of a unicycle

i is greater than that of all its neighbors then gi will be negative and its angle

will decrease. The opposite is true for a unicycle whose angle is less than all its

neighbors. Therefore, for all unicycles i ∈ n, the input gi points into the convex

hull formed by its neighbours. By (Lin et al., 2007a) the group of unicycles for

system (4.22) achieves consensus with the feedback in (4.23) on (−π/2, π/2).

• A feedback

ωi = gi((θij)j∈Ni, η) := ηi∑

j∈Ni

bijg(θij), (4.24)

is a rotational integrator consensus controller if (i) ηi > 0 where η := (ηi)i∈n, (ii)

bij = bji > 0, and (iii) g : S1 → R is a continuously differentiable interaction

function satisfying the following three assumptions (Mallada et al., 2016):

B1: sg(s) > 0 for all s ∈ (−π, π)\0, g(0) = g(π) = 0.

B2: g(s) is an odd function: g(−s) = −g(s) for s ∈ (−π, π).


B3: g(s) > 0, ∀s ∈ (− πn−1

, πn−1

) and g(s) < 0, ∀s ∈ (−π,− πn−1

) ∪ ( πn−1

, π).

A sample interaction function satisfying B1-B3 is shown in Figure 4.16.

Figure 4.16: Illustration of properties B1, B2 and B3.

The Kuramoto consensus controller corresponds to the choice ηi = 1 for all i ∈ n

and g(s) = sin(s), and satisfies properties B1-B2, but does not satisfy property

B3 for n > 2. In (Mallada et al., 2016), Mallada-Freeman-Tang showed that a

feedback enjoying properties B1-B3 almost globally stabilizes the set θ ∈ Tn :

θi = θj , i, j ∈ n for almost all gains bij = bji > 0. The following result is a special

case of Theorem 2 in (Mallada et al., 2016).

Theorem 4.3.2 ((Mallada and Tang, 2013; Mallada et al., 2016)). Let G be

an undirected and connected sensor graph and consider system (4.22) with feed-

back (4.24) satisfying assumptions B1-B3, expressed in states relative to agent one,

i.e., θ := (θ1i)i∈2 :n,

θ1i = gi((θij)j∈Ni, η)− g1((θ1j)j∈N1, η), i ∈ 2 :n.

There exists a setNb ⊂ (R+)|E| of Lebesgue measure zero such that for any collection

of gains (bij)(i,j)∈E ∈ (R+)|E|\Nb, and for any ηi > 0, i ∈ n, the origin θ = 0

is almost globally asymptotically stable. Moreover, there exists a compact set of


isolated equilibria A such that 0 ∪ A is globally attractive and the equilibria in

A are exponentially unstable.

The above result follows from Theorem 2 in (Mallada et al., 2016). In particu-

lar, system (4.22) with feedback (4.24) satisfies the model in equations (14)-(16)

in (Mallada et al., 2016) by letting χi(s) = ηis, letting ζ be the identity function,

and eliminating the integrator state γi.

The main difference here compared to the solution in (Mallada et al., 2016, Theorem

2) is that, in (Mallada et al., 2016), each system has an additional constant bias.

The integrator state γi is used in (Mallada et al., 2016) to compensate for this

bias. In this thesis, system (4.22) has zero bias, so the integrator state γi is not

needed. Accounting for this small difference, the proof of Theorem 4.3.2 follows

from minimal modifications to the proof of (Mallada et al., 2016, Theorem 2).

4.3.4 Control Primitives for Rotating bodies in SO(3)

Rotating body equilibrium stabilization

Consider the dynamic system for a rotating axis Γ ∈ S2 in three dimensions with sym-

metric inertia matrix J ,

Γ = Γ× ω,

Jω = τ − ω × Jω

(4.25)

where ω ∈ R3 is the body’s angular velocity and τ is the torque control input. A feedback,

τ = τd(Γ, ω) (4.26)

with smooth function τd : R3 × R3 → R3 is a rotating body equilibrium stabilizer if the

closed-loop system (4.25), (4.26) has an AGAS equilibrium point (Γi, ωi) = (e3, 0).

An example of a thrust direction controller is presented in (Chaturvedi et al., 2011)


as,

τd(Γ, ω) = kq (e3 × Γ)−Kωω, (4.27)

where kq is positive and Kω is a symmetric and positive definite matrix.

Proposition 4.3.6 (Theorem 2 in (Chaturvedi et al., 2011)). Feedback (4.27) is a ro-

tating body equilibrium stabilizer.

Chapter 5

Rendezvous of Flying Robots with

Local and Distributed Feedbacks

In this chapter, we present a solution to the rendezvous control problem for flying robots

(RP− F) introduced in Section 3.1. The feedbacks will be constructed out of the double-

integrator consensus control primitive fi((xij , vij)j∈Ni) in (4.19).

5.1 Solution of the Rendezvous Control Problem (RP− F)

We use a consensus controller for double-integrators fi((xij , vij)j∈Ni) = fi(yi) in (4.19) to

construct the feedbacks

ui =u⋆i (y

ii,Ω

ii) = −mifi(y

ii) · e3,

τi =τ⋆i (y

ii,Ω

ii)

=Ωii × JiΩii − k1Ji

(

(Ωii × fi(yii))× e3

)

− k21k2[

Ωii − k1(fi(yii)× e3)

]

, i ∈ n.

(5.1)

The result below states that for sufficiently large k1, k2 > 0, the feedbacks in (5.1)

solve RP-F if the network of robots has a sensor digraph containing a globally reachable

node.

98

Chapter 5. Rend. of Flying Robots with Loc. and Dist. Feedbacks 99

ConsensusControl

RotationalControl

robot

Sensors

ThrustControl

Figure 5.1: Block diagram of the rendezvous control system for robot i. The outer loopassigns a desired thrust vector fi(y

ii). The inner loop thrust control uses fi(y

ii) to assign

the vehicle input ui while the rotational control uses fi(yii) to assign the torque input τi.

The vector yii contains the relative displacements and velocities of vehicles that robot ican sense, measured in the body frame of robot i.

Theorem 5.1.1. RP-F is solvable for system (2.13), (2.14) if and only if the sensor

digraph G contains a globally reachable node, in which case a solution is the following.

Let fi(yi), i ∈ n, be a double-integrator consensus controller in (4.19) satisfying the

conditions in Theorem 4.3.1. The local and distributed feedback in (5.1) where k1, k2 > 0

are control parameters, makes the rendezvous manifold (3.1) globally practically stable,

that is, for any ε > 0, there exist k⋆1 > 1, k⋆2 > 0 such that for all k1 > k⋆1, k2 > k⋆2, the

set Bε(Γ) has a globally asymptotically stable subset containing Γ.

The proof of Theorem 5.1.1 is presented in Section 5.4.1.

Explanation of the proposed controller

Consider the block diagram in Figure 5.1, we now explain in detail the operation of its

two nested loops. We begin with the observation that a double-integrator consensus

controller fi(yi), i ∈ n also makes the system

xi = vi

vi = fi + g, i ∈ n

(5.2)


reach rendezvous, since the addition of the gravity vector g is common to all agents and

does not affect the relative dynamics. Now compare system (5.2) to the translational

dynamics of the flying robots,

xi = vi

vi = − 1

mi

uiRie3 + g, i ∈ n.(5.3)

If it were the case that fi = −(1/mi)uiRie3, systems (5.2) and (5.3) would be identical.

Then, setting −uiRie3 = mifi(yi) in (5.3) would solve RP-F. Inspired by this observation,

the outer loop of the block diagram in Figure 5.1 assumes that −uiRie3 is the control

input of (5.3) and computes a desired double-integrator force mifi(yi) which becomes a

reference signal for the inner loop.

We now explore in more detail the operation of the inner loop. First we observe

that fi(yi) satisfies Rifi(yii) =

∑

j∈Niaij(Rix

iij + γRiv

iij) =

∑

j∈Niaij(xij + γvij) = fi(yi).

Moreover, using the fact that dot products are invariant under rotations, we have

u⋆i (yii,Ω

ii) = −mifi(y

ii) · e3 = mi(Rifi(y

ii)) · (−Rie3) = mifi(yi) · qi,

where qi is the thrust direction vector. Thus, the thrust magnitude is the projection

of the desired thrust mifi(yi) onto the thrust direction vector—see Figure 5.2. Now let

Ωii(yi, Ri) := R⊤

i k1 (fi(yi)× Rie3) = k1 (fi(yii)× e3). Then we have from (5.1) that

τ ⋆i (yii,Ω

ii)=Ωii × JiΩ

ii − k1Ji

(

(Ωii × fi(yii))× e3

)

− k21k2(

Ωii −Ωii(yi, Ri)

)

. (5.4)

For simplicity of notation, we drop the arguments of Ωii(yi, Ri). We will show in the

proof of Theorem 5.1.1 that the torque inputs τi make Ωii converge to an arbitrarily small

neighborhood of Ωii, i ∈ n. Thus, Ωi

i can be seen as a reference angular velocity for the

inner loop. Using the fact that, for all a, b ∈ R3 and all R ∈ SO(3), R(a×b) = (Ra)×(Rb),


we have

Ωi(yi, Ri) = RiΩii = Rik1

(

fi(yii)× e3

)

= k1(

(Rifi(yii))× (Rie3)

)

= k1(fi(yi)×−qi) = k1(qi × fi(yi)).

Thus Ωi is perpendicular to the plane formed by the thrust direction vector qi and the

desired thrust force mifi(yi)—see Figure 5.2. Since the angular velocity vector identifies

an instantaneous axis of rotation, it follows that if Ωi = Ωi, then robot i rotates about

Ωi according to the right-hand rule. Referring to Figure 5.2, we see that such a rotation

closes the gap between uiqi and mifi(yi), and the speed of rotation is proportional to

sinϕ, where ϕ is the angle between uiqi and mifi(yi) marked in the figure. When the gap

is closed, we have ui = ‖mifi(yi)‖, qi = mifi(yi)/‖mifi(yi)‖, and thus uiqi = mifi(yi).

In conclusion, the inner loop assigns (ui, τi) to make Ωi approximately converge to Ωi in

the sense of global practical stability, so that uiqi = −uiRie3 approximately converges to

mifi(yi), which is computed by the outer loop.

While the intuition behind the proposed controller is simple, the proof that the inter-

play between the two nested loop results in global practical stability of the rendezvous

manifold is rather delicate, and it crucially relies on the homogeneity of the functions

fi(yi), i ∈ n.

Remark 5.1.2. Theorem 5.1.1 proves global practical stability of the rendezvous man-

ifold Γ. The reason that the stability is practical and not asymptotic is roughly as

follows. In order to achieve rendezvous of the flying robots, uiqi is driven approximately

tomifi(yi). What’s important is not so much the difference in magnitude of these vectors

but rather the difference in angle between them. In Figure 5.2, one can see that Ωi acts

to reduce this angle with a rate proportional to the magnitude of Ωi. Since Ωi is a linear

function of fi(yi), as the robots approach consensus Ωi converges to zero at the same rate

as fi(yi). This leads to increasing inaccuracy in closing the gap between the vectors uiqi

and mifi(yi) insomuch that in a very small neighborhood of rendezvous, Ωi is so small


Figure 5.2: Illustration of the control input ui and reference angular velocity Ωi in (5.1).

that it fails to make the translational dynamics act as double integrators. More detailed

reasoning is provided in Remark 5.4.1.

Features of the proposed controller

(i) The feedback of Theorem 5.1.1 is static. It does not depend on auxiliary states

that require communication between neighboring robots.

(ii) The feedback of Theorem 5.1.1 is local and distributed in the sense of Defini-

tion 2.2.3. Interestingly, it does not require sensing of relative attitudes, which

can be computed using on-board cameras, but are harder to compute than relative

displacements.

(iii) On the rendezvous manifold Γ there is no prespecified thrust direction qi for robot i

and the robot thrust directions do not need to align at rendezvous. This is desirable

if one wants to employ the proposed controller in a hierarchical control setting to

enforce additional control specifications.

(iv) In (Roza et al., 2014), we developed a solution for almost global asymptotic stability

of the set Γ combining a uniformly bounded double integrator consensus controller


fi(yi) with a rotating body equilibrium stabilizer τd(Γ, ω) in (4.26). The proposed

controller in this chapter has a number of advantages over this previous work. Un-

like (Roza et al., 2014), the inner control loop does not require any derivatives of

the reference thrust force fi(yi). In (Roza et al., 2014), the expressions resulting

from such derivatives are large and may pose difficulty in real-time computation

of the control law. More importantly, the computation of such derivatives requires

communication of the thrust control inputs u⋆i between neighboring robots, a prob-

lem that has been overcome in the present approach. The approach in (Roza et al.,

2014) requires that robots have access to a common inertial vector and its relative

attitude with respect to its neighbors. This requirement is absent in this chapter.

(v) The proposed control law in this chapter does not guarantee hovering of the robots.

While the robots converge to each other, nothing can be said about the motion of

the ensemble. This cannot be otherwise, for it would be impossible to solve RP-F

with hovering without additional sensors. One would need some measurement of

the gravity vector, for example provided by a three-axis accelerometer. The point

of view of these authors is that the proposed solution of RP-F with strictly local

and distributed feedbacks will serve as a layer in a hierarchy of higher-level control

specifications such as hovering and path following. That said, if all agents can

measure gravity, the solution in (Roza et al., 2014) allows the robots to hover in

steady state or have a desired final thrust in the direction of a common inertial

vector sensed by all robots.

5.2 Simulation Results

We consider a group of five robots with the sensor digraph in Figure 5.3. The robot

masses and inertia matrices are: m1 = 3 Kg, m2 = 3 Kg, m3 = 3.4 Kg, m4 = 3.2 Kg,

m5 = 3.2 Kg and J1 := diag (0.13, 0.13, 0.04)Kg·m2, as in (Abdessameud and Tayebi,


Table 5.1: Simulation Initial Conditions

Vehicle i xi(0) (m) vi(0) (m/s) Ri(0)1 (0,−10, 10) (0, 0, 0) side 12 (0, 10, 10) (0, 0, 0) side 23 (0, 0, 0) (0, 0, 0) down4 (−10, 0,−10) (0, 0, 0) up5 (10, 0,−10) (0, 0, 0) up

Table 5.2: Control Effort

Figure 5.4 Figure 5.5maxi supt |ui(t)| (N) 20.4 17.21maxi supt ‖τi(t)‖ (N·m) 15.27 16.47maxi rms(|ui(t)|) (N) 1.72 4.31maxi rms(‖τi(t)‖) (N·m) 1.43 2.24

2011), J2 = J1, J3 = 1.4J1, J4 = 1.2J1, J5 = 1.2J1. We pick aij = 0.3 for all j ∈ Ni

and γ = 30. The control gains k1 and k2 in (5.1) are chosen to be k1 = 2 and k2 = 0.45.

The initial conditions of the robots are shown in Table 5.1. The initial attitudes Ri(0)

of the robots are: up(right), side(ways) 1, side(ways) 2 and (upside)down respectively

given by:

1 0 0

0 1 0

0 0 1

,

1 0 0

0 0 −1

0 1 0

,

1 0 0

0 0 1

0 −1 0

,

1 0 0

0 −1 0

0 0 −1

.

Figure 5.4 shows the simulation without the presence of disturbances while Figure 5.5

shows the simulation when disturbances are present. The disturbances are: an additive

random noise with maximum magnitude of 0.25N on the applied force; an additive

random noise with maximum magnitude of 0.25N·m on the applied torque; an additive

measurement error for the angular velocity, with maximum magnitude of 0.25 rad/s; an

additive random noise on the quantity fi(yii) accounting for errors in measurements of

relative displacements and velocities of the vehicles. The direction of this vector has


3

1

2

4 5

Figure 5.3: Sensor digraph used in the simulation results.

been rotated within 0.25 rad and the magnitude is scaled between 0.75 to 1.25 times

the actual magnitude. The disturbances are updated 10 times per second. In both

cases of Figure 5.4 and Figure 5.5, the vehicles’ positions and velocities converge to a

neighborhood of one another.

In Figure 5.4 the vehicles remain within 0.25m of one another while in Figure 5.5 the

vehicles remain within 1m of one another at steady state. These neighborhoods can be

made even smaller by further increasing the control gains k1 and k2. However, this would

result in having higher control inputs. Metrics related to the thrust and torque inputs are

presented in Table 5.2. The first two rows show peak control norms and the last two show

the root mean square (rms) of the control norms. In these simulations we considered zero

gravity, i.e., g = 0. This was done to improve visibility of the simulation results. In the

presence of gravity, the vehicles would still converge to the same neighborhood of one

another, however at steady state they would accelerate in the direction of gravity since

gravity is not compensated through the control inputs in (5.1).

5.3 From Rendezvous to Formations

A notable omission from this thesis is a solution for formation control of flying robots.

The analogous problem for kinematic unicycles will be presented in Chapter 7 using


0 50 100 150 200−20

−10

0

10

20

time (s)

i x

0 50 100 150 200−10

−5

0

5

10

time (s)

i y0 50 100 150 200

−20

−10

0

10

time (s)

i z

0 50 100 150 2000

0.2

0.4

0.6

0.8

time (s)sp

eed

(m/s

)

Figure 5.4: Rendezvous control simulation without the presence of disturbances. At thetop-left, top-right and bottom-left: positions of the five robots expressed in the inertialframe I. At the bottom-right: linear speeds ‖vi‖, i ∈ 1 : 5.

strictly local and distributed feedbacks. For completeness, in this section we discuss how

one can transform the rendezvous controller for flying robots in (5.1) (or the solution

in (Roza et al., 2014) with hovering) into a formation controller. The final feedback,

however, does not meet the strict local and distributed sensing requirements since each

agent needs to know its own orientation in the inertial frame I.

Consider any desired configuration of n flying robots x = d where d = (d1, . . . , dn) ∈

R3n is centred, without loss of generality, about the origin, i.e.,∑n

i=1 di = 0. A set of

n flying robots is said to be in formation, when they satisfy the desired configuration d

modulo translations. That is, the position of the i-th robot satisfies xi = di + x where

x = (1/n)∑n

i=1 xi is the average position of the robots. Correspondingly, define the

formation manifold as

Γf :=

χ ∈ X : xi = di + x, vij = 0, Ωii = Ωii, i, j ∈ n

. (5.5)


0 50 100 150 200−50

0

50

time (s)

i x

0 50 100 150 200−10

−5

0

5

10

time (s)

i y0 50 100 150 200

−10

0

10

20

time (s)

i z

0 50 100 150 2000

0.5

1

1.5

time (s)sp

eed

(m/s

)

Figure 5.5: Rendezvous control simulation with the presence of disturbances. At thetop-left, top-right and bottom-left: positions of the five robots expressed in the inertialframe I. At the bottom-right: linear speeds ‖vi‖, i ∈ 1 : 5.

For all i ∈ n, define the fixed inertial vector δi := −di attached to robot i with endpoint

xi = xi+ δi. The collection of endpoints is denoted x := (xi)i∈n and xij := xj − xi. Then

the formation manifold in (5.5) can be rewritten as

Γf :=

χ ∈ X : xij = 0, vij = 0, Ωii = Ωii, i, j ∈ n

. (5.6)

This follows because xij = 0 implies that xij = dj − di and therefore, using the fact that∑n

i=1 di = 0,

xi − x =1

n

n∑

j=1

(xi − xj) =1

n

n∑

j=1

(di − dj) = di

as in (5.5). Therefore Γf reduces to the rendezvous manifold in (3.1) in terms of

(xi, vi, Ri,Ωii)i∈n quantities. It can be immediately shown that there is a diffeomorphism

between (xi, vi, Ri,Ωii)i∈n and (xi, vi, Ri,Ω

ii)i∈n where the latter can be treated as new


states where ˙xi satisfies

˙xi = xi + δi = xi = vi.

Since ˙xi is the same as xi, one achieves a formation controller that globally practically

stabilizes Γf by replacing xiij in (5.1) with xiij , satisfying

xiij = xiij + (dj − di)i.

The term (dj − di)i requires measurement of the fixed inertial vector dj − di in body

frame. This, in turn, requires each agent to know its own attitude in the inertial frame.

One could similarly adapt the rendezvous controller in (Roza et al., 2014) to a formation

controller that allows the desired formation to hover, but requires communication of

thrust inputs between neighboring robots.

5.4 Proofs

5.4.1 Proof of Theorem 5.1.1

The feedback in (5.1) is local and distributed because it is a smooth function of yii and

Ωii only. On the rendezvous manifold Γ, yi = 0 for all i ∈ n and it follows that the

requirement u⋆i |Γ = 0 holds and Ωii|Γ = 0. Moreover, since Ωii = Ωi

i|Γ = 0 for all i ∈ n

on Γ, it follows that the requirement τ ⋆i |Γ = 0 also holds. Now we need to show that the

feedback in (5.1) renders the rendezvous manifold Γ in (3.1) globally practically stable.

We begin by expressing the translational portion of the dynamics in coordinates relative


to robot 1, i.e., in terms of the variables (x1j , v1j)j∈2 :n,

x1 = v1,

v1 = − 1

m1

R1e3u1 + g,

x1j = v1j ,

v1j = − 1

mj

Rje3uj +1

m1

R1e3u1, j ∈ 2 :n,

(5.7)

Ri = Ri(Ωii)×,

JiΩii = τi − Ωii × JiΩ

ii, i ∈ n.

(5.8)

Since all relative states (xij , vij) can be expressed in terms of the variables above through

the identity (xij , vij) = (x1j − x1i, v1j − v1i), perfect rendezvous occurs if and only if the

vector (x, v) := (x1j , v1j)j∈2 :n is zero. Denoting,

X := (x, v) ∈ X := R3(n−1) × R

3(n−1),

R := (R1, . . . , Rn) ∈ R := SO(3)n,

Ω := (Ω11, . . . ,Ω

nn) ∈ Ω := R

3n,

the new collective state is (x1, v1, X,R,Ω) ∈ R3 × R3 × X× R × Ω. The meaning of the

new state is this: X contains all translational states (positions and velocities) relative to

robot 1, R contains all the attitudes, and Ω contains all body frame angular velocities.

Due to the identity (xij , vij) = (x1j − x1i, v1j − v1i), the vector yi = (xij , vij)j∈Ni

is a linear function of X which we will denote yi = hi(X). Similarly, the vector yii =

(xiij , viij)j∈Ni is a function of X and R, linear with respect to X . We will denote this

function yii = hii(X,R).

Using the definitions above, we may now express fi(yii) and Ωi

i(yi, Ri) = k1(fi(yii)×e3)

(the latter function was discussed in Section 5.1) in terms of states. Accordingly, we define


gi : X → R3, gii : X× R → R3 and Ω : X× R → Ω as follows:

gi(X) := (fi hi)(X),

gii(X,R) := R−1i gi(X) = (fi hii)(X,R),

Ω(X,R) :=(

Ωii(hi(X), Ri)

)

i∈n.

(5.9)

We remark that gi is linear and gii is linear with respect to its first argument.

Finally, we can rewrite the rendezvous manifold in new coordinates as,

(x1, v1, X,R,Ω) ∈ R3 × R

3 × X× R× Ω : X = 0, Ω = Ω(X,R). (5.10)

Since the feedbacks in (5.1) are local and distributed, it can be seen that the dynamics of

the closed-loop system in (X,R,Ω) coordinates are independent of (x1, v1). The norm of

the right hand side of the translational system in (5.3) grows at most linearly with (x, v)

and therefore (x, v) and, in particular, (x1, v1) have no finite escape times. Moreover, as

we have seen, in these coordinates the set in (5.10) is independent of (x1, v1). In light

of these considerations, for the stability analysis we may drop the variables (x1, v1), and

show that the set

Γ = (X,R,Ω) ∈ X× R× Ω : X = 0, Ω = Ω(X,R). (5.11)

is globally practically stable for the (X,R,Ω) dynamics which will imply that Γ is globally

practically stable as well.


Lyapunov function

Consider n double-integrators with control fi(yi), expressed in X coordinates:

x1j = v1j

v1j = fj(yj)− f1(y1) = gj(X)− g1(X), j ∈ 2 :n.

(5.12)

By Theorem 4.3.1, the origin of this linear time-invariant system is globally asymptot-

ically stable. Thus, there exists a quadratic Lyapunov function V : X → R, V (X) =

X⊤PX , where P is a symmetric positive definite matrix, such that the derivative of V

along the vector field in (5.12) is negative definite.

Let J ∈ R3n×3n be the block-diagonal matrix with the i-th block equal to Ji, and

consider the continuous function W : X× R× Ω → R defined as

W (X,R,Ω) = αWtran(X) +Wrot(X,R,Ω), (5.13)

where α > 0 is a parameter to be assigned later and

Wtran(X) =√

V (X) +1

2V (X),

Wrot(X,R,Ω) =

n∑

i=1

gii(X,R) · e3 +1

2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)).

The function Wtran(X) has been chosen as the summation of two terms:√

V (X) and

12V (X). The first is homogeneous of degree one while the second is homogeneous of

degree two with respect to X . The resulting time derivatives will possess these same

homogeneity properties, a fact that will be crucial in the Lyapunov analysis for proving

that W (X) ≤ 0. Taking the derivative of Wtran(X) yields

dWtran(X)

dX=

1

2√

V (X)

dV (X)

dX+dV (X)

dX


which is continuous everywhere except at V (X) = 0 which is the set X = 0. Therefore

Wtran is C1 everywhere except at X = 0. In light of Remark 4.1.7, this is not an issue

for doing a Lyapunov analysis since we show next that the Lie derivative of Wtran is

continuous. To this end, Wtran satisfies

Wtran = LfWtran(X,R) =dWtran(X)

dX· f(X,R)

where f(X,R) corresponds to the vector field for relative coordinates X = (xij , vij)j∈2 :n

in (5.7) which is homogeneous of degree one with respect to X . LfWtran(X,R) is contin-

uous for X 6= 0 because dWtran(X)/dX and f(X,R) are such. Moreover, LfWtran(X,R)

is the sum of two functions, one (1/2√V )dV (X)/dX · f(X,R) homogeneous of degree

one and the other dV (X)/dX · f(X,R) homogeneous of degree two with respect to X .

Proposition 4.1.22 implies that LfWtran(X,R) is continuous at X = 0.

Lemma 5.4.1. Consider the continuous function W defined in (5.13). Then

α⋆ := sup(X,R)∈(X\0)×R

n∑

i=1

|gii(X/√

V (X), R) · e3| <∞,

and for all α > α⋆, the following properties hold:

(i) W ≥ 0 and W−1(0) = Γ.

(ii) For all c > 0, for all k1 > 0, the sublevel set Wc := (X,R,Ω) :W (X,R,Ω) ≤ c is

compact.

(iii) For all ε > 0, there exists δ > 0 such that Wδ ⊂ Bε(Γ).

The proof is in Section 5.4.2.

From now on we assume α > α⋆. In light of the lemma, if we show that W is

nonincreasing outside a certain compact region of the state space, then all trajectories

of (5.7), (5.8) with feedback (5.1) are bounded, ruling out finite escape times. Moreover,


in light of parts (i) and (iii) of the lemma, to prove that Γ is practically stable it suffices

to prove that for every δ > 0, there exists a gain vector (k1, k2) such that Wδ is globally

asymptotically stable. Therefore, by Proposition 4.1.18 and Remark 4.1.7 we need to

show that W ≥ δ =⇒ W < 0.

Define the quantities

ρ(X) :=√

V (X), µ(X) := X/√

V (X).

Since the numerator and denominator are both homogeneous of degree one, the function

µ(X) is homogeneous of degree zero with respect to X . Therefore, the image satisfies

µ(X\0) = µ(S6(n−1)−1). Since µ(X) is continuous and the set S6(n−1)−1 is compact, the

image, denoted S1 := µ(X\0), is compact.

Dropping the arguments of ρ(X) and µ(X) for notational convenience, the functions

Wtran(X) and Wrot(X,R,Ω) can be expressed as Wtran(X) = ρ+ ρ2

2and

Wrot(X,R,Ω) = ρ

n∑

i=1

gii(µ,R) · e3 +1

2

(

Ω−Ω(ρµ,R))

⊤J(

Ω−Ω(ρµ,R))

.

In writing the above, we used the identity X = ρµ and the fact that the function gii(X,R)

is linear with respect to X , implying that gii(ρµ,R) = ρgii(µ,R).

Stability analysis

Let δ > 0 be arbitrary. We have W ≤ α(ρ+ ρ2/2) + ρ sup(µ,R) |gii(µ,R) · e3|+ (1/2)(Ω−

Ω)⊤J(Ω − Ω). Using the definition of α⋆ in Lemma 5.4.1 and the fact that α > α⋆, we

get

W ≤ α(2ρ+ ρ2/2) + (1/2)(Ω−Ω)⊤J(Ω−Ω).


It readily follows that there exists ∈ (0,min1, δ) such that

Λ := (X,R,Ω) : ρ ∈ (0, ), ‖Ω−Ω(ρµ,R)‖2 < ⊂Wδ.

We will show that there exist α > 0 and a gain vector (k1, k2) such that W < 0 outside

the set Λ. This will imply that W ≥ δ =⇒ W < 0, proving that Wδ is globally


Lemma 5.4.2. Consider the closed-loop system (5.7), (5.8) with feedback (5.1). If

k1 > 1, k2 > 0, then there exist scalars M1, . . . ,M4 > 0 such that the derivatives of ρ

and Wrot along the closed-loop system satisfy the following inequalities:

ρ ≤ρ[

−M2 +M1

n∑

i=1

‖gii(µ,R)× e3‖]

,

Wrot ≤ρ2n∑

i=1

[

−k1‖gii(µ,R)× e3‖2 +M4

k2

]

+ ρM3 −k21k22

‖Ω−Ω(ρµ,R)‖2.(5.14)

The proof is in Section 5.4.3.

From now on we let k1 > 1. Using the inequalities in Lemma 5.4.2, we get

W ≤(ρ+ ρ2)

[

−αM2 + αM1

n∑

i=1

‖gii(µ,R)× e3‖]

+ ρ2n∑

i=1

[

−k1‖gii(µ,R)× e3‖2 +M4

k2

]

+ ρM3 −k21k22

‖Ω−Ω(ρµ,R)‖2.

Denote βi(µ,R) := ‖gii(µ,R) × e3‖, and β(µ,R) := (β1(µ,R), . . . ,βn(µ,R)). For no-

tational convenience, we omit the arguments of the functions β and Ω. With these

definitions, the inequality above may be rewritten as

W ≤ (ρ+ ρ2)(

−αM2 + αM11⊤β)

+ρ2(

−k1‖β‖2+M4n

k2

)

+ ρM3 −k21k22

‖Ω−Ω‖2.


For every k2 > nM4/M3, we have

W ≤ (ρ+ ρ2)(

−αM2 +M3 + αM11⊤β)

− ρ2k1‖β‖2 −k21k22

‖Ω−Ω‖2.

If we further pick α > maxα⋆, 3M3/M2, we have

W ≤ (ρ+ ρ2)(

−2M3 + αM11⊤β)

− ρ2k1‖β‖2 −k21k22

‖Ω−Ω‖2.

Splitting the term −ρ2k1‖β‖2 into two parts and collecting terms for ρ and ρ2, we

obtain

W ≤ρ2(

−2M3 + αM11⊤β − k1

2‖β‖2

)

+ ρ

(

−2M3 + αM11⊤β − ρ

k12‖β‖2

)

− k21k22

‖Ω−Ω‖2.

Consider now the expression

M3 − αM11⊤β +

k1

2‖β‖2=

[

1⊤ β⊤]

M3

nI −αM1

2I

−αM1

2I k1

2I

1

β

.

If k1 > 2n(αM1/2)2/(M3), the above quadratic form is positive definite, implying that

M3 − αM11⊤β +

k1

2‖β‖2 ≥ 0. (5.15)

Since < 1, we also have M3 − αM11⊤β + (k1/2)‖β‖2 ≥ 0. Using the latter inequality,

we get a further upper bound for W ,

W ≤ −ρ2M3 + ρ

(

−2M3 + αM11⊤β − ρ

k12‖β‖2

)

− k21k22

‖Ω−Ω‖2. (5.16)

Using (5.16), we now prove that outside Λ, W < 0. In other words, when either ρ ≥

or ‖Ω−Ω‖2 ≥ , W < 0.


Remark 5.4.1. If the derivative W were negative definite, then the rendezvous manifold

Γ would be globally asymptotically stable. However, this is not guaranteed in (5.16). The

reason is as follows. Suppose ρ is very small and ‖Ω−Ω‖ = 0. Then all terms multiplied

by ρ2 become negligible and what remains in (5.16) is, W ≤ ρ(

−2M3 + αM11⊤β)

. As

we have no control over the value of the constants M1 and M3 in the equation above, W

can be greater than zero if the second term dominates the first.

Suppose first that ρ ≥ . Then from (5.16) we have

W ≤ −ρ2M3 + ρ

(

−2M3 + αM11⊤β − k1

2‖β‖2

)

− k21k22

‖Ω−Ω‖2.

By inequality (5.15) we conclude that

W ≤ −ρ2M3 − ρM3 −k21k22

‖Ω−Ω‖2 < 0.

Next, suppose that ‖Ω−Ω‖2 ≥ . Then from (5.16),

W ≤ −ρ2M3 + ραM11⊤β − k21k2

2

≤ −ρ2M3 + ραM1M5 −k21k22,

where M5 := max(µ,R)∈S1×R1⊤β(µ,R). The maximum exists because β is continuous

and S1 × R is a compact set. If k2 > (αM1M5/k1)2/ then W < 0.

We have therefore proved that, if

• α > maxα⋆, 3M3/M2,

• k1 > max1, 2n(αM1/2)2/(M3), and

• k2 > maxnM4/M3, (αM1M5/k1)2/,

then W > δ implies that W < 0. Therefore, for any initial condition, the solution

of (5.7), (5.8) with feedback (5.1) is bounded and the set Wδ is globally asymptotically


stable.

5.4.2 Proof of Lemma 5.4.1

Recall the definition of W (X,R,Ω), and assume that X 6= 0,

W =α

(

√

V (X) +1

2V (X)

)

+

n∑

i=1

gii(X,R) · e3 +1

2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R))

=√

V (X)

(

α +

∑ni=1 g

ii(X,R) · e3

√

V (X)

)

+α

2V (X) +

1

2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)).

Since gii(X,R) is linear with respect to X , we have

W =√

V (X)

(

α +

n∑

i=1

gii (µ(X), R) · e3)

+α

2V (X)+

1

2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)),

where µ(X) = X/√

V (X) is continuous on X\0 and bounded as follows

‖µ(X)‖ =‖X‖√

V (X)=

‖X‖√X⊤PX

≤ 1√

λmin(P ).

Since gii is continuous, µ(X) is bounded, and R ∈ R, a compact set, it follows that the

function∑n

i=1 |gii (µ(X), R) · e3| has a bounded supremum. Accordingly, let

α⋆ = sup(X,R)∈(X\0)×R

n∑

i=1

∣

∣gii (µ(X), R) · e3∣

∣ .

For all α > α⋆, we have W (X,R,Ω) > W (X,R,Ω),

W (X,R,Ω) :=α

2V (X) +

1

2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)) ≥ 0.

We derived the bound above for X 6= 0, but since gii(0, R) = 0 (by linearity of gii with

respect to X), the bound also holds for X = 0. The above inequality implies that W ≥ 0

and W−1(0) ⊂ W−1(0). But W = 0 if and only if V (X) = 0 (i.e., X = 0) and Ω = Ω.


Thus W−1(0) ⊂ Γ. Moreover, in Γ, (X,Ω −Ω) = 0 and therefore Γ ⊂ W−1(0), proving

part (i) of the lemma.

For part (ii), note that for all c > 0, Wc ⊂ W ≤ c. Since W is a positive definite

quadratic form in the variables (X,Ω −Ω), its sublevel sets are compact in (X,Ω−Ω)

coordinates. Thus if (X,R,Ω) ∈ Wc, X and Ω −Ω(X,R) are bounded. For all k1 > 0,

since Ω(X,R) = k1(gii(X,R)×e3) is continuous and R ∈ R, a compact set, Ω is bounded,

implying that Ω is also bounded. Therefore the set Wc is bounded. Continuity of W

implies that Wc is compact. This concludes the proof of part (ii) of the lemma.

For part (iii), let ε > 0 be arbitrary. Since W is a positive definite quadratic form in

the variables (X,Ω−Ω), there exists δ > 0 such thatW (X,R,Ω) ≤ δ implies ‖X‖ ≤ ε/2,

and ‖Ω − Ω(X,R)‖ ≤ ε/2. This implies that (X,R,Ω) ∈ Bε(Γ). Thus, Wδ ⊂ W ≤

δ ⊂ Bε(Γ), as required. This concludes the proof of Lemma 5.4.1.

5.4.3 Proof of Lemma 5.4.2

Rewrite the dynamics of X in (5.7) as

x1j = v1j

v1j = [gj(X)− g1(X)] +Rj

[

(gjj (X,R) · e3)e3 − gjj (X,R)]

−R1

[

(g11(X,R) · e3)e3 − g11(X,R)]

.

To get the identities above, we added and subtracted in (5.7) the ideal force feedbacks

fj(yj) = gj(X) and f1(y1) = g1(X), and we replaced uj and u1 in (5.7) by the assigned

feedbacks in (5.1). Finally, we used the identity Rigii = gi.


Taking the time derivative of√

V (X) along the above vector field we get

d

dt

√

V (X) =1

2√

V (X)

[

−X⊤QX +

n∑

j=2

∂V

∂v1jRj

(

(gjj(X,R) · e3)e3 − gjj (X,R))

−n∑

j=2

∂V

∂v1jR1

(

(g11(X,R) · e3)e3−g11(X,R))

]

.

The first term in the bracket is the derivative of V (X) along the nominal vector field (5.12),

and Q = Q⊤ is a positive definite matrix. Letting M2 = λmin(Q)/(2λmax(P )) and using

the fact that the Euclidean norm is invariant under rotations, we have

d

dt

√

V (X) ≤ −M2

√

V (X) +1

2√

V (X)·

[

n∑

j=2

∥

∥

∥

∥

∂V

∂v1j

∥

∥

∥

∥

(

‖(gjj(X,R) · e3)e3 − gjj (X,R)‖+ ‖(g11(X,R) · e3)e3 − g11(X,R)‖)

]

.

We claim that ‖(gii(X,R) · e3)e3 − gii(X,R)‖ = ‖gii(X,R) × e3‖. Indeed, writing gii =

(gii · e3)e3 + gii − (gii · e3)e3, we have gii × e3 = (gii − (gii · e3)e3) × e3. Since the vector

gii − (gii · e3)e3 is perpendicular to e3, ‖(gii − (gii · e3)e3)× e3‖ = ‖gii − (gii · e3)e3‖, so that

‖gii × e3‖ = ‖gii − (gii · e3)e3‖. This proves the claim. Using the identity just derived, we

get

d

dt

√

V (X) ≤−M2

√

V (X)

+1

2√

V (X)

[

n∑

j=2

∥

∥

∥

∥

∂V

∂v1j

∥

∥

∥

∥

(

‖gjj (X,R)× e3‖+ ‖g11(X,R)× e3‖)

]

.

Rewriting this in terms of (ρ, µ), we get

ρ ≤ −M2ρ+1

2ρ

[

n∑

j=2

∥

∥

∥

∥

∂V

∂v1j(ρµ)

∥

∥

∥

∥

(

‖gjj(ρµ,R)× e3‖+ ‖g11(ρµ,R)× e3‖)

]

.

Since the functions gii are linear with respect to their first argument, and the partial


derivatives of the quadratic form V are linear functions, by the homogeneity of the norm

we have

ρ ≤ −M2ρ+ρ

2

[

n∑

j=2

∥

∥

∥

∥

∂V

∂v1j(µ)

∥

∥

∥

∥

(

‖gjj (µ,R)× e3‖+ ‖g11(µ,R)× e3‖)

]

.

The functions ‖∂V/∂v1j‖ are continuous. The quantity µ belongs to S1, a compact set.

Therefore ‖∂V/∂v1j‖ has a maximum,

ρ ≤−M2ρ+maxµ∈S1j∈2 :n

∥

∥

∥

∥

∂V

∂v1j(µ)

∥

∥

∥

∥

ρ

2

[

n∑

j=2

(

‖gjj (µ,R)× e3‖)

+ (n− 1)‖g11(µ,R)× e3‖]

≤−M2ρ+maxµ∈S1j∈2 :n

∥

∥

∥

∥

∂V

∂v1j(µ)

∥

∥

∥

∥

ρ

2(n− 1)

n∑

j=1

‖gjj(µ,R)×e3‖.

Letting M1 := max µ∈S1j∈2 :n

‖∂V/∂v1j‖ (n− 1)/2, we get the first inequality in (5.14).

We now turn to the second inequality in (5.14). Recall the definition of Wrot,

Wrot(X,R,Ω) =

n∑

i=1

gii(X,R) · e3 +1

2(Ω−Ω(X,R))⊤J(Ω−Ω(X,R)).

The derivative of Wrot along the vector field in (5.7), (5.8) is

Wrot =n∑

i=1

[

(

d

dtgii

)

· e3 + (Ωii −Ωii(X,R)) ·

(

τi − Ωii × JiΩii − Ji

(

d

dtΩii

))

]

.

To express (d/dt)gii, recall that gii(X,R) = R−1

i fi(hi(X)). Then,

d

dtgii =

(

d

dtR−1i

)

fi(hi(X)) +R−1i

d

dt(fi(hi(X))) .

The function fi(hi(X)) is linear. Its derivative along the vector field (5.7), (5.8) with


feedback (5.1) is given by

d

dtfi(hi(X)) =

∑

j∈Ni

(

aijd

dt(xj1 − xi1) + bij

d

dt(vj1 − vi1)

)

=∑

j∈Ni

(

aij(vj1 − vi1) + bij

(

− 1

mjRje3uj +

1

miRie3ui

))

,

which is linear with respect to X because ui = −gii(X,R) · e3 is such. We will denote

it hi(X,R), hi(X,R) := (d/dt)fi(hi(X)). Consistently with our notational convention

in Table 2.2, we will let hii(X,R) := R−1i hi(X,R). The function hii(X,R) is linear with

respect to X . Returning to the derivative of gii, we have

d

dtgii = −(Ωii)

×R−1i fi(hi(X)) +R−1

i hi(X,R)

= −Ωii × gii(X,R) + hii(X,R).

Similarly, since Ωii(X,R) = k1(g

ii(X,R) × e3), we have d

dtΩii = k1 (−Ωii × gii + hii) × e3.

Substituting the above identities in the expression for Wrot and since τ ⋆i = Ωii × JiΩii −

k1Ji((Ωii × gii)× e3)− k21k2(Ω

ii −Ωi

i), we get

Wrot =

n∑

i=1

[

−(Ωii × gii) · e3 + hii · e3 − k1(Ωii −Ωi

i) · Ji(hii × e3)− k21k2‖Ωii −Ωii‖2]

.

Using the property of the triple product that (Ωii × gii) · e3 = (gii × e3) · Ωii, we obtain

Wrot =

n∑

i=1

[

−(gii × e3) · Ωii + hii · e3 − k1(Ωii −Ωi

i) · Ji(hii × e3)− k21k2‖Ωii −Ωii‖2]

.

Adding and subtracting the term (gii × e3) ·Ωii and collecting the term Ωii −Ωi

i, we have

Wrot =n∑

i=1

[

−(gii×e3) ·Ωii+hii ·e3−((gii×e3)+k1Ji(hii×e3)) ·(Ωii−Ωi

i)−k21k2‖Ωii−Ωii‖2]

.

Substituting in the first term inside the bracket Ωii = k1(g

ii×e3), taking norms, and using


the fact that k1 ≥ 1, we arrive at the inequality

Wrot ≤n∑

i=1

[

− k1‖gii × e3‖2 + ‖hii · e3‖+ k1κi(X,R)‖Ωii −Ωii‖ − k21k2‖Ωii −Ωi

i‖2]

,

where κi(X,R) := ‖gii(X,R)× e3‖+ ‖Ji(hii(X,R)× e3)‖. Note that κi(X,R) is homoge-

neous with respect to X because gii and hi are linear with respect to X and the norm is

a homogeneous function.

Splitting the term −k21k2‖Ωii − Ωii‖2 into two parts and noticing that the function

k1κi(X,R)‖Ωii − Ωii‖ − (k21k2/2)‖Ωii − Ωi

i‖2 is quadratic in the variable ‖Ωii − Ωii‖ with

maximum κ2i (X,R)/(2k2), we get

Wrot ≤n∑

i=1

[

− k1‖gii × e3‖2 + ‖hii · e3‖ −k21k22

‖Ωii −Ωii‖2 +

κ2i (X,R)

2k2

]

.

Now rewriting this in terms of (ρ, µ), we get

Wrot ≤n∑

i=1

[

− k1‖gii(ρµ,R)× e3‖2 + ‖hii(ρµ,R) · e3‖ −k21k22

‖Ωii −Ωii‖2 +

κ2i (ρµ,R)

2k2

]

.

Using the homogeneity with respect to X of ‖gii × e3‖, ‖hii · e3‖, and κi, we get

Wrot ≤n∑

i=1

[

− k1ρ2‖gii(µ,R)× e3‖2 + ρ‖hii(µ,R) · e3‖ −

k21k22

‖Ωii −Ωii‖2 + ρ2

κ2i (µ,R)

2k2

]

.

Since ‖hii(µ,R) · e3‖ and κ2i (µ,R) are continuous functions over the compact set S1 ×

R, they each have a maximum. Letting M3 = n · max(µ,R)∈S1×R

i∈n(‖hii(µ,R) · e3‖), M4 =

max(µ,R)∈S1×R

i∈n

(

κ2i (µ,R)

2

)

, we conclude that

Wrot ≤ ρ2n∑

i=1

[

−k1‖gii(µ,R)× e3‖2 +M4

k2

]

+ ρM3 −k21k22

n∑

i=1

‖Ωii −Ωii‖2,

as required. This concludes the proof of Lemma 5.4.2.

Chapter 6

Rendezvous of Kinematic Unicycles

with Local and Distributed

Feedbacks

In this chapter, we present a solution to the rendezvous control problem for kinematic

unicycles (RP− U) introduced in Section 3.2. The feedbacks will be constructed out of

the integrator consensus control primitive fi((xij)j∈Ni) in (4.10).

6.1 Solution of the Rendezvous Control Problem (RP− U)

Let Fi(G, ρ1, ρ2) denote the set of integrator consensus controllers fi((xij)j∈Ni) = fi(yi)

in (4.10) such that aij > 0 for all j ∈ Ni and 0 < ρ1 < aij < ρ2 with sensor digraph G.

We use f(yi) to construct the feedbacks

ui = u⋆i (yii) = ‖fi(yii)‖fi(yii) · e1,

ωi = ω⋆i (yii) = −kfi(yii) · e2, i ∈ n.

(6.1)

The result below states that for sufficiently large k, the feedbacks in (6.1) solve RP-U

123

Chapter 6. Rend. of Kin. Unicycles with Loc. and Dist. Feedbacks 124

if the network of unicycles has a sensor digraph containing a globally reachable node.

Theorem 6.1.1. RP-U is solvable for system (2.6), (2.7) if and only if the sensor digraph

G contains a globally reachable node, in which case a solution is the following. For each

positive integer n, and each ρ1, ρ2 such that 0 < ρ1 < ρ2, there exists k⋆ > 0 such that

for all k > k⋆, for all sensor digraphs G with n nodes containing a globally reachable

node, and for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n, feedback (6.1) solves RP-U.

Theorem 6.1.1 states that, if the sensor digraph G contains a globally reachable node,

then for a sufficiently large gain k, controller (6.1) solves RP-U. The lower bound on k is

uniform over all sensor digraphs with n nodes, and all consensus controllers fi with gains

aij in a fixed compact interval [ρ1, ρ2]. In practice this implies that one can tune the

controller (6.1) without knowing the sensor digraph G nor the controller parameters aij .

All that is required is to know bounds on these parameters. Moreover, it is clear that

on the rendezvous manifold Γ, yi = 0 for all i ∈ n and it follows that the requirement

(u⋆i , ω⋆i )|Γ = (0, 0) holds.

Remark 6.1.2. Consider a single kinematic unicycle (x1, θ1) ∈ R2 × S1 with inputs

(u1, ω1). As a consequence of Theorem 6.1.1, the controller

u1 = −‖x1‖x1 · R1e1,

ω1 = x1 ·R1e2

globally asymptotically stabilizes the set (x1, θ1) ∈ R2×S1 : x1 = 0. Note that no high

gain is required in this case.

The necessity portion of Theorem 6.1.1 was proved in (Lin et al., 2005). The suffi-

ciency part, namely the fact that the feedback (6.1) solves RP-U, is proved in Section 6.2.


The feedback in (6.1) is very similar in form to the one in (Zheng et al., 2011) given by,

u⋆i (yii) = fi(y

ii) · e1,

ω⋆i (yii) = fi(y

ii) · e2, i ∈ n.

(6.2)

with aij = 1 for all j ∈ Ni. The main difference in (6.1) is the extra multiplicative factor

‖fi(yii)‖ in the control u⋆i (yii) and the control gain k chosen sufficiently large. While the

feedback in (6.2) achieves rendezvous for undirected and connected graphs, the solution

cannot be generalized to the broad class of directed graphs considered in this chapter in

that, as shown in (Zheng et al., 2009), when the sensor digraph is a directed ring, the

feedback (6.2) drives the unicycles to a circular formation instead of achieving rendezvous.

Our solution in (6.1) can be viewed as an adaptation of the controller in (6.2) that allows

for rendezvous with directed graph topologies containing a globally reachable node.

In the presence of link failures in the sensor digraph, as long as the resulting graph

after the last failure maintains a globally reachable node, the presented solution in (6.1)

will still achieve rendezvous. It remains an open problem to solve RP-U with saturated

feedback. If the rotational control gain k > k⋆ results in feedbacks that are too large,

the magnitude of the gains aij can be reduced appropriately to avoid actuator saturation

over any compact set of initial conditions. This, however, would slow the convergence

rate.

ConsensusControl

RotationalControl

robot

Sensors

SpeedControl

Figure 6.1: Block diagram of the rendezvous control system for robot i.


The proposed control architecture is illustrated in the block diagram of Figure 6.1.

There are two nested loops. The outer loop treats each robot as a single-integrator driven

by the linear consensus controller,

xi = fi(yi), i ∈ n. (6.3)

The set (xi)i∈n ∈ R2n : xij = 0, i, j ∈ n is globally asymptotically stable for (6.3) if

the sensing digraph has a globally reachable node (Moreau, 2004). The signal fi(yi(t)) is

computed in the body frame Bi, and used as a reference signal for the inner-loop thrust

and rotational controllers that assign the unicycle control inputs in (6.1). The intuition

behind these controllers is shown in Figure 6.2. The speed input ui is the dot product

ui = u⋆i (yii) = ‖fi(yii)‖fi(yii) · e1. This is the projection of the reference ‖fi(yi)‖fi(yi)

onto the heading axis bix of robot i. The angular speed, on the other hand, is propor-

tional to the dot product between the reference fi(yi) and the second body axis biy. In

Figure 6.2, one can see that ωi = ω⋆i (yii) = −kfi(yii) · e2 = −k‖fi(yi)‖ sin(φi) acts to

reduce the angle φi between bix and fi(yi) with a rate proportional to the magnitude

of fi. Together, these control inputs drive the robot velocity uibix approximately to the

reference ‖fi(yi)‖fi(yi). The convergence is approximate because the control inputs do

not depend on the time derivative of fi. It is the difference in angle between uibix and

‖fi(yi)‖fi(yi) as opposed to the difference in magnitude that is important for obtaining

rendezvous. Since ‖fi(yi)‖fi(yi) is homogeneous of degree two, as the robots approach

consensus, ω⋆i (yii) converges to zero slower than u⋆i (y

ii). This allows ω

⋆i (y

ii) to exert suffi-

cient control authority even as the robots converge to consensus, closing the gap between

the vectors uibix and ‖fi(yi)‖fi(yi).


Figure 6.2: Illustration of the control inputs ui = u⋆i (yii) and ωi = ω⋆i (y

ii) in (6.1).

6.1.1 Simulation Results

We consider a group of five robots with sensor digraph in Figure 6.3. For the feedback

in (6.1), we pick aij = 0.05 for all j ∈ Ni. The control gain k is chosen to be k = 0.2. The

initial conditions of the robots are shown in Table 6.1. The simulation is presented in

Figure 6.4(a) and the control inputs for the five vehicles are plotted in Figure 6.5 showing

reasonable linear and rotational speeds. In Figure 6.4 the trajectories corresponding to

the feedback in (Lin et al., 2005), the feedback in (Zheng et al., 2011), and our solution are

compared. Notice that some curves in the figure have cusps although the actual solution

is C1. These cusps, however, are present only because the state is being projected into

the inertial (ix, iy) plane, i.e., the angular states θ are not illustrated. The simulations

are run with the initial conditions in Table 6.1 and the sensing digraph in Figure 6.3. It

can be seen that the proposed feedback has practical advantages over the time-varying

feedback in (Lin et al., 2005). The proposed feedback induces a more natural behaviour

in the ensemble of unicycles. The feedback in (Lin et al., 2005) makes the unicycle

“wiggle” indefinitely, a behaviour which would be unacceptable in practice. The feedback

in (Zheng et al., 2011) causes the unicycles to converge to a circular formation instead

of rendezvous. Rendezvous is guaranteed only for undirected graphs in this case.


3 1

2

4

5

Figure 6.3: Sensor digraph used in the simulation results.

Table 6.1: Simulation Initial Conditions

Vehicle i xi(0) (m) θi(0) (rad)1 (0, 10) 02 (−10,−10) 2π/53 (−50, 10) 4π/54 (−10, 0) 6π/55 (10, 0) 8π/5

6.2 Proofs

This section presents the sufficiency proof of Theorem 6.1.1. Necessity was proved in (Lin

et al., 2005). This section also presents corresponding propositions, lemmas and claims

and their proofs.

We first observe that the closed loop system with inputs defined in (6.1) has no finite

escape times because ‖xi‖ = ‖uiRie1‖ for all i ∈ n is bounded by a quadratic function

of x and θ is bounded. For any sensor digraph G containing a globally reachable node,

there is a condensation digraph satisfying the properties in Proposition 4.2.3. The key

tool in our proof is this condensation digraph and the isolated node sets Lj defined in

Section 4.2.3. The same tool was employed in (Hatanaka et al., 2015) for pose synchro-

nization (synchronization of positions and attitudes) of fully actuated vehicles.

The dynamics of unicycles associated with an isolated node set Lj are independent of

the nodes outside of this set because, for any robot i ∈ Lj, the feedbacks in (6.1) depend

only on states of robots within Lj. Therefore, the dynamics of the collection of unicycles


−60 −40 −20 0 20−10

−5

0

5

10

15(a)

ix (m)

i y (m

)

−60 −40 −20 0 20−10

−5

0

5

10

15(c)

ix (m)

i y (m

)

−60 −40 −20 0 20−10

−5

0

5

10

15(b)

ix (m)

i y (m

)

Figure 6.4: Rendezvous control simulation for: (a) proposed feedback in (6.1), (b) feed-back in (Lin et al., 2005), and (c) feedback in (Zheng et al., 2011)

0 50 100 150−10

−5

0

5

u i (m

/s)

time (s)

0 50 100 150−0.2

0

0.2

0.4

0.6

omeg

a i (ra

d/s)

time (s)

Figure 6.5: Simulation control inputs for proposed feedback in (6.1)

in Lj,

xi = uiRie1 (6.4)

θi = ωi, i ∈ Lj (6.5)

define an autonomous dynamical system. Henceforth, the dynamics in (6.4), (6.5) are

denoted by ΣLj and we define the reduced rendezvous manifold

ΓLj :=

(xi, θi)i∈Lj : xik = 0, i, k ∈ Lj

.


Recall from Section 4.2.3 that the set L−1 is empty, which implies that the set ΓL−1is

also empty. We adopt the convention that ΓL−1is GAS for ΣL−1

.

The proof of Theorem 6.1.1 relies on an induction argument on the node sets Lj . Key

in the induction argument is the next result stating that if the vehicles in Lj−1 achieve

rendezvous, then so do the vehicles in Lj.

Proposition 6.2.1. Consider system (2.6), (2.7) and assume that the sensor digraph G

contains a globally reachable node and let ρ1 and ρ2 be such that 0 < ρ1 < ρ2. Suppose

that, for some integer j ≥ 0, the set ΓLj−1is globally asymptotically stable for the

dynamics ΣLj−1. Then there exists l⋆ > 0 such that for any k > l⋆ in (6.1) and for all

linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n, feedback (6.1) globally asymptotically stabilizes

ΓLj for the dynamics ΣLj .

In the statement of Proposition 6.2.1, the lower bound l⋆ on k depends on the sensor

digraph G, while in the statement of Theorem 6.1.1, k⋆ does not. In the proof of Theo-

rem 6.1.1 below we show that the lower bound l⋆ can in fact be made uniform over sensor

digraphs G containing a globally reachable node. The same comment holds for Corol-

lary 6.2.1 below. In Section 6.2.1, we use the above proposition to prove Theorem 6.1.1,

and in Section 6.2.2 we prove Proposition 6.2.1.

In the special case when G is strongly connected, we have L0 = V. Since, by definition,

L−1 = ∅, the set ΓL−1is GAS for ΣL−1

, and Proposition 6.2.1 yields the following

corollary.

Corollary 6.2.1. Consider system (2.6), (2.7) and assume that the sensor digraph G

is strongly connected and let ρ1 and ρ2 be such that 0 < ρ1 < ρ2. Then there exists

l⋆ > 0 such that for any k > l⋆ and for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n,

feedback (6.1) solves RP-U.



To begin with, the feedback in (6.1) is local and distributed because it is a smooth function

of yii only. The proof in this section is performed in two steps. First we will show that for

all digraphs G = (V, E) containing a globally reachable node, for all parameter bounds

ρ1, ρ2 such that 0 < ρ1 < ρ2 there exists l⋆ > 0 such that for all k > l⋆, and for all linear

functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n, feedback (6.1) solves RP-U. We then show that the

lower bound on k can be made uniform over all sensor digraphs with n nodes containing

a globally reachable node.

Consider any digraph G = (V, E) containing a globally reachable node, parameter

bounds ρ1 and ρ2 such that 0 < ρ1 < ρ2, and the node sets Lj and Lj defined in

Section 4.2.3. By construction, the node sets Lj are isolated, the subgraph (V0, E0) is

strongly connected, and L0 = L0 = V0.

The proof is by induction. Since the subgraph (L0, E0) is strongly connected, by

Corollary 6.2.1, there exists l⋆0 such that choosing k > l⋆0 makes the set ΓL0globally

asymptotically stable for system ΣL0for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n.

Now consider Lj and suppose the reduced rendezvous manifold ΓLj−1is globally

asymptotically stable for system ΣLj−1. It holds from Proposition 6.2.1 that there exists

l⋆j such that choosing k > l⋆j makes the isolated node set ΓLj globally asymptotically

stable for system ΣLj for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n. By part (ii) of

Proposition 4.2.3, the condensation digraph C(G) contains a globally reachable node, so

there is a path from every node of C(G) to the unique root of C(G). By part (i) of the

same proposition, C(G) is acyclic, which implies that the paths connecting the nodes of

C(G) to the unique root of C(G) have a maximum length, L. Recall that, by definition,

LL =∑L

i=1 Li is the union of those strongly connected components Vi of V that are associ-

ated with nodes vi of the condensation digraph C(G) with the property that the maximum

path length from vi to the root v0 is ≤ L. As we argued earlier, the set of such nodes vi

equals the entire condensation digraph, implying that LL = V. Let l⋆ > maxl⋆0, . . . , l⋆L.


By induction, it must hold that choosing k > l⋆ makes ΓLL = Γ globally asymptotically

stable for system ΣLL = ΣV = Σ for all linear functions fi ∈ Fi(G, ρ1, ρ2), ∈ n. We

conclude that Γ is globally asymptotically stable.

To prove Theorem 6.1.1, it remains to show that the lower bound k can be taken to

be uniform over the set of digraphs with n nodes containing a globally reachable node.

In other words, we need to show that for all n, for all ρ1, ρ2 such that 0 < ρ1 < ρ2,

there exists k⋆ > 0 such that choosing k > k⋆ solves RP-U for all digraphs with n nodes

containing a globally reachable node and for all linear functions fi ∈ Fi(G, ρ1, ρ2), i ∈ n.

Corresponding to n nodes, there is a finite number of directed graphs, m ≥ 1, containing

a globally reachable node. Enumerating these graphs by Gj, j ∈ 1 :m, the result from the

first part of the proof gives values (l⋆)j, such that choosing k > (l⋆)j, (6.1) solves RP-U

for all linear functions fi ∈ Fi(Gj , ρ1, ρ2), i ∈ n. Choosing k⋆ = max(l⋆)1, . . . , (l⋆)m

then solves RP-U for all digraphs Gj , j ∈ 1 :m for all linear functions fi ∈ Fi(Gj , ρ1, ρ2),

i ∈ n.

6.2.2 Proof of Proposition 6.2.1

Consider a digraph G = (V, E) containing a globally reachable node and let ρ1 and ρ2

be such that 0 < ρ1 < ρ2. Recall that the vertex set Lj ⊂ V is the union of those

vertex sets Vi that correspond to vertices vi in the condensation digraph C(G) associated

to strongly connected components with maximal path length j to the root node v0.

Without loss of generality, we assume there is a single vertex set Vi such that Lj = Vi.

In the case that Lj is the union of several vertex sets, one can repeat the argument of

this proof sequentially for each component. We denote A := Lj−1 and B := Lj = Viand therefore Lj = A ∪ B. Denote by r the number of unicycles in the set B. By

assumption, ΓA is globally asymptotically stable for the dynamics ΣA and the digraph

associated to the nodes in B is strongly connected. We need to show that ΓA∪B is

globally asymptotically stable for the dynamics ΣA∪B. The proof relies on the following


coordinate transformation.

Coordinate Transformation

Recall that x = (xi)i∈n ∈ R2n and θ = (θi)i∈n ∈ Tn. For each i ∈ n, define

Xi := fi(yi)/Ai, (6.6)

where Ai :=∑

j∈Niaij , and let X := (X1, . . . , Xn). We may express X as

X = (diag(1/A1, · · · , 1/An)L⊗ I2)x.

In the above, diag(. . .) is the diagonal matrix with diagonal elements inside the parenthe-

sis and L is the weighted Laplacian matrix of the sensor digraph associated with the gains

aij . X lies on the subspace of R2n given by Im(diag(1/A1, · · · , 1/An)L⊗ I2) ⊂ R2n. We

let X := Im(diag(1/A1, · · · , 1/An)L⊗ I2) ⊂ R2n. From now on, we will take into consid-

eration that X is a vector in R2n constrained to lie in the 2(n− 1) dimensional subspace

X representing the values of X which are well-defined. Since the sensor digraph contains

a globally reachable node, by Proposition 4.2.1 the matrix L⊗ I2 has rank 2(n− 1), and

Ker(L ⊗ I2) = span1⊗ e1, 1⊗ e2. Let x := [I2 · · · I2]x =∑

i xi, then the linear map

T : R2n → R2n × R2, x 7→ (X, x) is an isomorphism onto its image. Under the action of

T , the subspace x ∈ R2n : x1 = · · · = xn is mapped isomorphically onto the subspace

(X, x) ∈ ImT : X = 0. Since the feedbacks in (6.1) are local and distributed, it can

be seen that the dynamics of the closed-loop unicycles in (X, x, θ) coordinates are inde-

pendent of x. Moreover, as we have seen, in these coordinates the control specification is

the global stabilization of (X, x, θ) ∈ X × R2 × Tn : X = 0, a set whose description is

independent of x. In light of these considerations, for the stability analysis we may drop

the variable x, and show that the set Γ := (X, θ) ∈ X × Tn : X = 0 is GAS for the

(X, θ) dynamics.


Denote gi(yi) := ‖fi(yi)‖fi(yi). Let a = (aij)(i,j)∈E denote the vector containing the

parameters of the linear consensus controller fi ∈ Fi(Gi, ρ1, ρ2), i ∈ n. Given positive

bounds ρ1 and ρ2, by definition of Fi(G, ρ1, ρ2), the vector of controller gains a lies in the

compact set K(G, ρ1, ρ2) := x ∈ (R+)|E| : ρ1 ≤ xi ≤ ρ2, i ∈ n where R+ is the set of

positive real numbers and |E| is the cardinality of the edge set E . From here on we will use

the hat notation to refer to quantities represented in (X, θ) coordinates. Using (6.6), the

functions fi and gi and their body frame representations are given in (X, θ) coordinates

by

fi(a, Xi) = AiXi, gi(a, Xi) = Ai2‖Xi‖Xi

f ii (a, Xi, θi) = AiR−1i (θi)Xi, g

ii(a, Xi, θi) = Ai

2R−1i (θi)‖Xi‖Xi,

(6.7)

where we have made explicit the dependence of these functions on the parameter vector

a. We can use these functions to rewrite feedback (6.1) in new coordinates as ui =

gii(a, Xi, θi) · e1, ωi = −kf ii (a, Xi, θi) · e2. We remark that fi and fii are homogeneous of

degree one with respect to Xi. Similarly, gi and gii are homogeneous of degree two with

respect to Xi. The closed-loop unicycle dynamics in (X, θ) coordinates are given by

Xi =

∑

j∈Niaij((g

jj · e1)Rje1 − (gii · e1)Rie1)

Ai, (6.8)

θi = −kf ii · e2. (6.9)

We will refer to system (6.8), (6.9) as Σi.

In analogy with what we did earlier, for a set ofm nodes S ⊂ V we letXS := (Xi)i∈S ∈

XS and θS := (θi)i∈S ∈ TnS := Tm. XS lies on the subspace of R2m given by XS := πS(X) in

which πS : R2n → R2m is the projection map whereby πS(X) = (Xi)i∈S ∈ R2m. Moreover,

if S is an isolated node set, the systems Σi, i ∈ S determine an autonomous dynamical

system which we denote by ΣS. We also denote the reduced rendezvous manifold by

ΓS := (XS, θS) ∈ XS × TnS : XS = 0 . In new coordinates, it needs to be shown that the

set ΓA∪B is globally asymptotically stable for the dynamics ΣA∪B under the assumption


that ΓA is globally asymptotically stable for the dynamics ΣA.

Stability analysis

Let

V (γ,XB) =∑

i∈B

γiX⊤i Xi

Wtran(γ,XB) =√

V (γ,XB)

Wrot(a, XB, θB) =∑

i∈B

f ii (a, Xi, Ri) · e1,

(6.10)

where γi(a) > 0 are gains that are continuous functions of a ∈ K(G, ρ1, ρ2) defined in the

proof of Lemma 6.2.3 and let γ(a) := (γ1(a) . . . γr(a)). Consider the continuous function

W : Rr ×K × XB × TnB → R defined as

W (γ, a, XB, θB) = αWtran(γ,XB) +Wrot(a, XB, θB), (6.11)

where α > 0 is a design parameter. Just as in Chapter 5, the Lie derivative ofW (γ, a, XB, θB)

on the vector field in (6.8), (6.9) can be shown to be continuous. Using W (γ, a, XB, θB)

as a Lyapunov function, the proof follows very similar arguments to Proposition 4.1.24.

The next two lemmas are used in the subsequent analysis.

Lemma 6.2.2. Let γ(a) be any positive real-valued continuous function and consider

the continuous function W (γ, a, XB, θB) defined in (6.11). There exists α⋆ > 0 such that,

for all α > 2α⋆ and for all a ∈ K, the following properties hold:

(i) W ≥ 0 and W−1(0) = (XB, θB) : XB = 0.

(ii) For all c > 0, the sublevel set Wc := (XB, θB) : W (γ, a, XB, θB) ≤ c is compact.

(iii) α⋆√

V (γ,XB) < W (γ, a, XB, θB) < 2α√

V (γ,XB).

The proof is in the Section 6.2.3. From now on assume α > 2α⋆.


Lemma 6.2.3. Consider system (6.8), (6.9). There exist positive real-valued functions

γi(a) in (6.10) and l⋆ > 0 such that choosing k > l⋆ implies

d

dtW (γ, a, XB, θB) ≤ −σV (γ,XB) + Φ(a, XA, θ), σ > 0, (6.12)

for all a ∈ K where Φ(a, XA, θ) is continuous with respect to its arguments and Φ(a, 0, θ) =

0.

The proof of Lemma 6.2.3 is presented in Section 6.2.3. Select the positive real-valued

functions γi(a) as in Lemma 6.2.3.

Consider l⋆ in Lemma 6.2.3. We will now show that choosing k > l⋆ implies ΓA∪B

is globally asymptotically stable for ΣA∪B. The proof will make use of the reduction

theorem (Theorem 4.1.2). We will first show that all solutions of the closed-loop sys-

tem are bounded. The rotation matrices live in a compact set, therefore we only need

to show that the states XA∪B = (Xi)i∈A∪B are bounded. Since A is isolated, ΣA is an

autonomous subsystem and by assumption, ΓA = (XA, θA) ∈ XA ×TnA : XA = 0 (com-

pact), is globally asymptotically stable. Therefore, XA is bounded. From the inequality

W (γ, a, XB, θB) ≥ α⋆√

V (γ,XB) in part (iii) of Lemma 6.2.2, to show boundedness of

V (γ,XB), it suffices to show thatW (γ, a, XB, θB) is bounded. Boundedness of V (γ,XB),

in turn, implies boundedness of XB. From the bound on the derivative of W in (6.12),

and by Lemma 6.2.2 we obtain

d

dtW (γ, a, XB, θB) ≤ −σW (γ, a, XB, θB)

2

(2α)2+ Φ(a, XA, θ),

σ > 0. Since XA is bounded and θ ∈ Tn lies on a compact set, it holds that Φ(a, XA, θ)

is bounded and therefore W is bounded, which implies that XB is bounded. Therefore

XA∪B is bounded, as claimed. Now define the set, Λ := (XA∪B, θA∪B) ∈ XA∪B × TnA∪B :

XA = 0. Since the set ΓA is globally asymptotically stable for system ΣA and XA∪B is

bounded, it holds that Λ is globally asymptotically stable for ΣA∪B.


To show that the set ΓA∪B, which is compact, is globally asymptotically stable for

the system ΣA∪B, it suffices to show that ΓA∪B is globally asymptotically stable relative

to Λ. On the set Λ, Φ(a, XA, θ) is equal to zero and the derivative ofW is therefore given

by

d

dtW (γ, a, XB, θB) ≤ −σW (γ, a, XB, θB)

2

(2α)2, σ > 0.

By Lemma 6.2.2, all level sets of W (γ, a, XB, θB) are compact and W−1(0) = (XB, θB) :

XB = 0. This implies ΓA∪B is globally asymptotically stable relative to the set Λ. By

Theorem 4.1.2, ΓA∪B is globally asymptotically stable for ΣA∪B. This completes the

proof.

6.2.3 Proof of Lemmas

Throughout this section we will make use of functions µi and µ defined as follows. Define

the continuous functions µ(γ,XB) := XB/√

V (γ,XB), and µi(γ,XB) := Xi/√

V (γ,XB), i ∈

B. Since the numerator and denominator are both homogeneous of degree one, these

functions are both homogeneous of degree zero with respect to XB. Therefore, the im-

ages satisfy µ(γ,XB\0) = µ(γ, S2r−1 ∩ XB) and µi(γ,XB\0) = µi(γ, S2r−1 ∩ XB), where

S2r−1 is the unit sphere in R2r where r is the number of agents in B. Since µ, µi and

γ(a) are continuous functions and the sets K and S2r−1 ∩ XB are compact, the images

µ(γ(K),XB\0) and µi(γ(K),XB\0) are compact sets.

Proof of Lemma 6.2.2

Recall the definition of W (γ, a, XB, θB),

W = α√

V (γ,XB) +∑

i∈B

f ii (a, Xi, θi) · e1 =√

V (γ,XB)

(

α +

∑

i∈B fii (a, Xi, θi) · e1

√

V (γ,XB)

)

.


Using the fact that f ii (a, Xi, θi) is homogeneous with respect to its second argument, we

have

W =√

V (γ,XB)

(

α +∑

i∈B

f ii (a, µi(γ,XB), θi) · e1)

.

Since f ii is continuous, µi(γ,XB) is bounded, and θB ∈ TnB, a compact set, it follows that

the function∑

i∈B

∣

∣

∣f ii (a, µi(γ,XB), θi) · e1

∣

∣

∣

has a bounded supremum. Accordingly, let

α⋆ = sup(a,XB ,θB)∈K×XB×Tn

B

∑

i∈B

∣

∣

∣f ii (a, µi(γ(a), XB), θi) · e1

∣

∣

∣.

For all α > 2α⋆, we have W (γ, a, XB, θB) ≥ W (γ, a, XB, θB) := α⋆√

V (γ,XB) ≥ 0.

This inequality implies that W ≥ 0 and W−1(0) ⊂ W−1(0). But W = 0 if and only if

V (γ,XB) = 0 (i.e., XB = 0). Thus W−1(0) ⊂ (XB, θB) : XB = 0. Conversely, on the

set (XB, θB) : XB = 0, XB = 0 and hence W = 0, and therefore (XB, θB) : XB =

0 ⊂W−1(0). It follows that W−1(0) = (XB, θB) : XB = 0 proving part (i).

For part (ii), note that for all c > 0, Wc ⊂ W (γ, a, XB, θB) ≤ c. Since the sublevel

sets of W are compact and θB ∈ TnB, a compact set, the set Wc is bounded. Continuity

of W implies that Wc is compact.

For part (iii), it has already been shown that W (γ, a, XB, θB) ≥ α⋆√

V (γ,XB). It

also holds that

W =√

V (γ,XB)

(

α +∑

i∈B

f ii (a, µi(γ,XB), θi) · e1)

≤√

V (γ,XB) (α+ α) ≤ 2α√

V (γ,XB).


Proof of Lemma 6.2.3

We first compute inequalities for Wtran and Wrot for system (6.8) and (6.9). We then

combine them to derive (6.12). Consider unicycle i ∈ B. The dynamics of Xi in (6.8)

are split into two terms, for neighboring robots j ∈ Ni ∩A and j ∈ Ni ∩ B respectively,

Xi =∑

j∈Ni∩A

aij(ujRje1 − uiRie1)

Ai+

∑

j∈Ni∩B

aij(ujRje1 − uiRie1)

Ai. (6.13)

For simplicity of notation, we drop the arguments of gi(a, Xi) and gii(a, Xi, θi). Adding

and subtracting the term,

∑

j∈Ni∩Baij(gj − gi)−

∑

j∈Ni∩Aaij gi

Ai

to (6.13) yields,

Xi =

∑


∑

j∈Ni∩Aaij gi

Ai+

∑

j∈Ni∩Baij(ujRje1 − uiRie1)

Ai

−∑

j∈Ni∩Baij(gj − gi)

Ai+

∑

j∈Ni∩AaijujRje1

Ai+

∑

j∈Ni∩Aaij(gi − uiRie1)

Ai

=

∑


∑

j∈Ni∩Aaij gi

Ai+

∑

j∈Ni∩Baij(ujRje1 − gj)

Ai

−∑

j∈Ni∩Baij(uiRie1 − gi)

Ai+

∑

j∈Ni∩AaijujRje1

Ai+

∑

j∈Ni∩Aaij(gi − uiRie1)

Ai.

Replacing uj and ui by the assigned feedbacks in (6.1) and using the identity Rigii = gi

then,

Xi = ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ) + di(a, XA, θ),


where,

ai(a, XB) :=

∑


∑

j∈Ni∩Aaij gi

Ai

bi(a, XB, θ) :=

∑

j∈Ni∩BaijRj((g

jj · e1)e1 − gjj )

Ai−∑

j∈Ni∩BaijRi((g

ii · e1)e1 − gii)

Ai

ci(a, XB, θ) :=

∑

j∈Ni∩AaijRi(g

ii − (gii · e1)e1)

Ai

di(a, XA, θ) :=

∑

j∈Ni∩Aaij(g

jj · e1)Rje1

Ai.

The time derivative of Wtran =√

V (γ,XB) in (6.10) yields,

Wtran =1

2√V

[

∑

i∈B

∂V (γ,XB)

∂Xi(ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ))

]

+1

2√V

∑

i∈B

∂V (γ,XB)

∂Xidi(a, XA, θ).

(6.14)

The derivative of the first term is considered in Claim 6.2.2.

Claim 6.2.2. There exist gains γi(a) > 0 in (6.10) that are continuous functions of a

and a continuous function κ(γ, a, XB), negative definite and homogeneous of degree three

with respect to XB, such that

∑

i∈B

∂V (γ,XB)

∂Xiai(a, XB) ≤ κ(γ, a, XB).

The proof of Claim 6.2.2 is presented at the end of this chapter. Let the gains γi be

as in Claim 6.2.2. The derivative of the remaining terms in the square brackets of (6.14)


satisfies,

∑

i∈B

∂V (γ,XB)

∂Xi

(bi(a, XB, θ) + ci(a, XB, θ))

≤∑

i∈B

1

Ai

∂V (γ,XB)

∂Xi

[

∑

j∈Ni∩B

aij∥

∥(gjj · e1)e1 − gjj∥

∥

+∑

j∈Ni∩B

aij∥

∥(gii · e1)e1 − gii∥

∥+∑

j∈Ni∩A

aij∥

∥gii − (gii · e1)e1∥

∥

]

≤∑

i∈B

1

Ai

∂V (γ,XB)

∂Xi

[

∑

j∈Ni∩B

aij∥

∥(gjj · e1)e1 − gjj∥

∥+∑

j∈Ni

aij∥

∥(gii · e1)e1 − gii∥

∥

]

.

We claim that ‖(gii(a, Xi, θi) · e1)e1 − gii(a, Xi, θi)‖ = |gii(a, Xi, θi) · e2|. Indeed, writing

gii = (gii · e1)e1 + gii − (gii · e1)e1, we have gii · e2 = (gii − (gii · e1)e1) · e2. Since the

vector gii − (gii · e1)e1 is parallel to e2, |(gii − (gii · e1)e1) · e2| = ‖gii − (gii · e1)e1‖, so that

|gii · e2| = ‖gii − (gii · e1)e1‖. Then,

∑

i∈B

∂V (γ,XB)

∂Xi(bi(a, XB, θ) + ci(a, XB, θ))

≤∑

i∈B

∥

∥

∥

∥

∂V (γ,XB)

∂Xi

∥

∥

∥

∥

(

∑

j∈B

∣

∣gjj · e2∣

∣+ n∣

∣gii · e2∣

∣

)

which is homogeneous of degree three with respect to XB since ∂V (γ,XB)/∂Xi is homo-

geneous of degree one and gii is homogeneous of degree two with respect to XB for all


i ∈ B. The last term in (6.14) satisfies,

1

2√

V (γ,XB)

∑

i∈B

∂V (γ,XB)

∂Xidi(a, XA, θ)

≤ 1

2√

V (γ,XB)

∑

i∈B

1

Ai

∂V (γ,XB)

∂Xi

∑

j∈Ni∩A

aij(gjj · e1)Rje1

≤ 1

2√

V (γ,XB)

∑

i∈B

1

Ai

∥

∥

∥

∥

∂V (γ,XB)

∂Xi

∥

∥

∥

∥

∑

j∈Ni∩A

aij‖gjj‖

≤∑

i∈B

sup(a,XB)∈K×XB

1

2√

V (γ(a), XB)

∥

∥

∥

∥

∂V (γ(a), XB)

∂Xi

∥

∥

∥

∥

∑

j∈Ni∩A

‖gjj‖

:= Φtran(XA, θ).

(6.15)

The bounded supremum of

1√

V (γ(a), XB)

∥

∥

∥

∥

∂V (γ(a), XB)

∂Xi

∥

∥

∥

∥

exists because this term is homogeneous of degree 0 with respect to XB and γ(a) is a

continuous function of a. Moreover, XA = 0 implies that ‖gjj‖ = 0 for all j ∈ A and

hence Φtran(0, θ) = 0. Everything together, (6.14) yields,

Wtran ≤1

2√

V (γ,XB)

[

κ(γ, a, XB) +∑

i∈B

∥

∥

∥

∥

∂V (γ,XB)

∂Xi

∥

∥

∥

∥

(

∑

j∈B

∣

∣gjj · e2∣

∣+ n∣

∣gii · e2∣

∣

)]

+ Φtran(XA, θ).

(6.16)

Since κ(γ, a, XB) is homogeneous of degree three with respect to XB. We can write,

κ(γ, a, XB) =

√

V (γ,XB)V (γ,XB)√

V (γ,XB)V (γ,XB)κ(γ, a, XB)

=√

V (γ,XB)V (γ,XB)κ

(

γ, a,XB

√

V (γ,XB)

)

=√

V (γ,XB)V (γ,XB)κ (γ, a, µ(γ,XB)) .


Analogous operations can be performed with the remaining term in the square bracket

of (6.16) yielding,

Wtran ≤V (γ,XB)

2

[

κ(γ, a, µ(γ,XB)) +∑

i∈B

∥

∥

∥

∥

∂V (γ, µ(γ,XB))

∂Xi

∥

∥

∥

∥

·(

∑

j∈B

∣

∣gjj (a, µj(γ,XB), θj) · e2∣

∣+ n∣

∣gii(a, µi(γ,XB), θi) · e2∣

∣

)]

+ Φtran(XA, θ).

Since κ is continuous and negative definite, a lies on a compact set K and µ(γ(a), XB)

lies on a compact set S1, it follows that κ(γ(a), a, µ(γ(a), XB))/2 has bounded maximum

−M2 < 0. Similarly, the function

∥

∥

∥

∥

∂V (γ(a), µ(γ(a), XB))

∂Xi

∥

∥

∥

∥

has a maximum. Letting

M1 := n max(a,ψ)∈K×S1

i∈B

∥

∥

∥

∥

∂V (γ(a), ψ)

∂Xi

∥

∥

∥

∥

yields,

Wtran ≤V[

−M2 +M1

2n

∑

i∈B

(

∑

j∈B

∣

∣gjj (a, µj(γ,XB), θj) · e2∣

∣+ n∣


∣

)]

+ Φtran(XA, θ)

≤V[

−M2 +M1

2n

∑

i∈B

(

n∣


∣+ n∣


∣

)

]

+ Φtran(XA, θ)

≤V[

−M2 +M1

∑

i∈B

∣


∣

]

+ Φtran(XA, θ)

(6.17)

for all a ∈ K. This proves the first inequality. We now turn to the second. Recall the


definition of Wrot, Wrot(a, XB, θB) =∑

i∈B fii (a, Xi, θi) · e1. The time derivative of Wrot

along the vector field in (6.8), (6.9) is Wrot =∑

i∈B

(

ddtf ii

)

·e1. To express (d/dt)f ii , recall

that f ii (a, Xi, θi) = R−1i fi(a, Xi). Then, d

dtf ii =

(

ddtR−1i

)

fi + R−1i

dfidt. We will denote the

derivative of fi(a, Xi) = AiXi by,

hi(a, X, θ) := (d/dt)fi(a, Xi)

= Ai (ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ) + di(a, XA, θ))

where the first three terms are homogeneous of degree two with respect to XB and the last

term is homogeneous of degree two with respect to XA. Consistently with our notational

convention, we will let hii(a, X, θ) := R−1i hi(a, X, θ). Returning to the derivative of f ii ,

we haved

dtf ii = −(ωi)

×R−1i fi(a, Xi) +R−1

i hi(a, X, θ)

= −

0 −ωiωi 0

f ii (a, Xi, θi) + hii(a, X, θ).

We substitute the above identity in the expression for Wrot,

Wrot =∑

i∈B

−e⊤1

0 −ωiωi 0

f ii (a, Xi, θi) + hii(a, X, θ) · e1

=∑

i∈B

(

(f ii (a, Xi, θi) · e2)ωi + hii(a, X, θ) · e1)

.

Substituting the feedback ωi = −k(f ii (a, Xi, θi) · e2) and taking norms, we arrive at the

inequality

Wrot ≤∑

i∈B

[

− k∣

∣

∣f ii (a, Xi, θi) · e2

∣

∣

∣

2

+ hii(a, X, θ) · e1]

.

This gives,

Wrot ≤[

−k∑

i∈B

∣

∣


∣

∣

∣

2

+ ℓ(a, XB, θ)

]

+ Φrot(a, XA, θ)


where

ℓ(a, XB, θ) :=∑

i∈B

AiR⊤i (ai(a, XB) + bi(a, XB, θ) + ci(a, XB, θ)) · e1

and Φrot(a, XA, θ) :=∑

i∈B AiR⊤i di(a, XA, θ) · e1. Note that

∑

i∈B

∣

∣


∣

∣

∣

2

and

ℓ(a, XB, θ) are homogeneous of degree two with respect toXB. The function Φrot(a, XA, θ)

does not depend on XB and Φrot(a, 0, θ) = 0. This yields,

Wrot ≤V (γ,XB)

[

−k∑

i∈B

∣

∣

∣f ii (a, Xi/

√

V (γ,XB), θi) · e2∣

∣

∣

2

+ ℓ(a, XB/√

V (γ,XB), θ)

]

+ Φrot(a, XA, θ)

≤V (γ,XB)

[

−k∑

i∈B

∣

∣


∣

∣

∣

2

+ ℓ(a, µ(γ,XB), θ)

]

+ Φrot(a, XA, θ).

|ℓ(a, µ(γ(a), XB), θ)| has a bounded supremum. Letting

M3 = sup(a,ψ,θ)∈K×S1×Tn

(|ℓ(a, ψ, θ)|) ,

we conclude that,

Wrot ≤V (γ,XB)

[

−k∑

i∈B

∣

∣


∣

∣

∣

2

+M3

]

+ Φrot(a, XA, θ) (6.18)

for all a ∈ K. By using the inequalities (6.17) and (6.18) we now bound the derivative

of W to derive (6.12). Notice that

W =αWtran + Wrot

≤V (γ,XB)

[

−αM2 + αM1

∑

i∈B

∣


∣

−k∑

i∈B

∣

∣


∣

∣

∣

2

+M3

]

+ Φ(a, XA, θ),


where Φ(a, XA, θ) := αΦtran(XA, θ) + Φrot(a, XA, θ). Choose α > 3M3/M2. This implies,

W ≤ V

[

−2M3 + αM1

∑

i∈B

∣


∣− k∑

i∈B

∣

∣


∣

∣

∣

2]

+ Φ(a, XA, θ).

(6.19)

Since f ii (a, Xi, θi) is homogeneous with respect to Xi, we have,

f ii (a, µi(γ,XB), θi) =

√

‖gii(a, µi(γ,XB), θi)‖‖gii(a, µi(γ,XB), θi)‖

gii(a, µi(γ,XB), θi).

Plugging the last expression into (6.19) yields

W ≤ V

[

−2M3 + αM1

∑

i∈B

∣


∣

−k∑

i∈B

1

‖gii(a, µi(γ,XB), θi)‖∣


∣

2

]

+ Φ(a, XA, θ).

Since gii(a, µi(γ,XB), θi) is a continuous function of its arguments and µi(γ,XB) is com-

pact, |gii(a, µi(γ,XB), θi)| has a maximum M4. This implies,

W ≤V[

−2M2 + αM1

∑

i∈B

∣


∣

−k∑

i∈B

1

M4

∣


∣

2

]

+ Φ(a, XA, θ).

Denote βi(a, µi(γ,XB), θi) := |gii(a, µi(γ,XB), θi) · e2|, and β := (βi(a, µi(γ,XB), θi))i∈B.

Then,

W ≤ V

[

−2M2 + αM11⊤β − k

M4|β|2

]

+ Φ(a, XA, θ)

= V[

1⊤ β⊤]

−2M2

nI αM1

2I

αM1

2I −k

M4I

1

β

+ Φ(a, XA, θ).

There exists l⋆ > 0 such that choosing k > l⋆, the matrix above is negative definite and


therefore the first term satisfies,

V[

1⊤ β⊤]

−2M2

nI αM1

2I

αM1

2I −k

M4I

1

β

≤ −σV (γ,XB), (6.20)

σ > 0 for all a ∈ K. This concludes the proof of Lemma 6.2.3.

Proof of Claim 6.2.2

Recalling that V (γ,XB) =∑

i∈B γiX⊤i Xi with Xi = fi/Ai and defining bij :=

aijAi

2 , it

holds that,

∑

i∈B

∂V (γ,XB)

∂Xiai(a, XB) = 2

∑

i∈B

γifiAi

· ai(a, XB)

≤2∑

i∈B

γifi ·(

∑

j∈Ni∩B

bij(‖fj‖fj − ‖fi‖fi)−∑

j∈Ni∩A

bij‖fi‖fi)

≤2∑

i∈B

γi

(

∑

j∈Ni∩B

bij(−‖fi‖3 + ‖fj‖fj · fi)−∑

j∈Ni∩A

bij‖fi‖3)

≤∑

i∈B

γi∑

j∈Ni∩B

bij

(

−4

3‖fi‖3 +

4

3‖fj‖3

)

+∑

i∈B

γi∑

j∈Ni∩B

bij

(

−2

3‖fi‖3 + 2‖fj‖fj · fi −

4

3‖fj‖3

)

− 2∑

i∈B

γi∑

j∈Ni∩A

bij‖fi‖3.

The first term equals 43γ⊤Mh with h := (‖fi‖3)i∈B. M is the (r× r)-matrix whose (i, j)-

th component is∑

k∈Ni∩Bbik for i = j, bij for j ∈ Ni∩B and zero otherwise for i, j ∈ 1 : r

where it is assumed without loss of generality that B = 1 : r. The components of the

matrix M are continuous functions of a. Since B corresponds to a strongly connected

digraph, the zero eigenvalue of M is unique and all components of left eigenvectors

corresponding to the zero eigenvalue are nonzero and have the same sign (see Proposition

D.5 in (Hatanaka et al., 2015)). Further, it follows from Theorem 3.4.35 in (Abraham

et al., 1988) that there exists a left eigenvector associated to the zero eigenvalue of M ,


γ(a) = (γ1(a), . . . , γr(a)), which is a continuous function of a. Without loss of generality,

we can choose the vector γ(a) with positive components. Therefore,

∑

i∈B

∂V (γ,XB)

∂Xi

ai(a, XB) ≤∑

i∈B

γi∑

j∈Ni∩B

bij

(

−2

3‖fi‖3 + 2‖fj‖fj · fi −

4

3‖fj‖3

)

− 2∑

i∈B

γi∑

j∈Ni∩A

bij‖fi‖3 =: κ(γ, a, XB).

The term

κ1(γ, a, XB) :=∑

i∈B

γi∑

j∈Ni∩B

bij

(

−2

3‖fi‖3 + 2‖fj‖fj · fi −

4

3‖fj‖3

)

≤∑

i∈B

γi∑

j∈Ni∩B

bij

(

−2

3‖fi‖3 + 2‖fi‖‖fj‖2 −

4

3‖fj‖3

)

is less than or equal to zero with equality only when fi = fj for all i, j ∈ B and as such

κ(γ, a, XB) is less than or equal to zero with equality only when fi = fj for all i, j ∈ B.

Now we prove that κ(γ, a, XB) = 0 only if fi = 0 for all robots i ∈ B. In the case

that A is not empty, the inequality κ(γ, a, XB) ≤ −2∑

i∈B γi∑

j∈Ni∩Abij‖fi‖3 implies

κ(γ, a, XB) = 0 only if fi = 0 for any i ∈ B with a neighbor in A. As such, by the

previous arguments, κ(γ, a, XB) = 0 only if fi = 0 for all i ∈ B. On the other hand, if A

is empty, then B is isolated and strongly connected. Therefore κ(γ, a, XB) = κ1(γ, a, XB)

is equal to zero only if κ1(γ, a, XB) = 0 which is the case only if fi = fj for all i, j ∈ B.

For all XB ∈ XB (i.e., all well-defined values of XB ∈ R2r), this implies that there exists

x = (x1, . . . , xr) such that diag(A1, · · · , Ar)XB = (L⊗ I2)x ∈ span1⊗ e1, 1⊗ e2 where

L is the Laplacian matrix for the agents in B (isolated). Since B is strongly connected,

for all a ∈ K(G, ρ1, ρ2) there exists a unique vector γ (with positive entries) such that

γ⊤(L⊗ I2) = 0. Since γ⊤(L⊗ I2)x = γ⊤1⊗ (αe1 + βe2) for some α, β ∈ R, it holds that

γ⊤1 ⊗ (αe1 + βe2) = 0. Since all entries of γ are positive, this implies α = β = 0 and

(L⊗ I2)x = 0. Therefore x ∈ span1⊗ e1, 1⊗ e2 or, equivalently, fi = 0 for all i ∈ B.

Therefore κ(γ, a, XB) = 0 only if Xi = 0 for all i ∈ B and as such κ(γ, a, XB) is


negative definite. Note that κ(γ, a, XB) is homogeneous of degree three with respect to

XB because fi is homogeneous of degree one with respect to XB for all i ∈ B. This

completes the proof of the claim.

Chapter 7

Formations of Kinematic Unicycles

In this chapter we present a control solution solving the Parallel Formation Problem (PP).

The control inputs ui and ωi will be constructed by combining two control primitives dis-

cussed in Section 4.3: a uniformly bounded consensus controller for single-integrators

fi((xij)j∈Ni) in (4.11) and a rotational integrator consensus controller gi((θij)j∈Ni, η)

in (4.24). We will also discuss special cases of parallel line formations and full syn-

chronization of unicycles.

7.1 Solution to the Parallel Formation Problem (PP)

To begin, we define offset vectors rigidly attached to each unicycle. Let α > 0 be a design

parameter, and d = (d1i1)i∈2,...n ∈ F be a desired parallel formation. Define

α1 := α, β1 := 0,

αi := −d11i · e1 + α, βi := −d11i · e2, i ∈ 2 :n.

(7.1)

Referring to Figure 7.1, attach the offset vector δi := αiRie1 + βiRie2 to the body frame

of unicycle i, and let xi := xi+δi be the endpoint of the offset vector in the coordinates of

frame I. Note that the offset vector δi is defined in the body frame of robot i whereas the

150

Chapter 7. Formations of Kinematic Unicycles 151

analogous offset defined in Section 5.3 for formations of flying robots was defined relative

to the inertial frame. Defining the offsets in body frame will permit us to develop a local

and distributed solution for unicycle formation control. Define further

xij := xj − xi,

yi := (xij)j∈Ni, yki := (xkij)j∈Ni.

Figure 7.1: Representation of the offset vector δi.

We now show that PP reduces to synchronizing the unicycles’ heading angles and the

endpoints xi. To this end, suppose that θij = 0 and xij = 0 for all i, j ∈ n. Then,

0 = xi1i = [(xi + δi)− (x1 + δ1)]i

= xi1i + (δi − δ1)i

= xi1i − d11i = x11i − d11i.

The last identity follows from the fact that Ri = R1. We conclude that θij = 0 and

xij = 0 for all i, j ∈ n implies x11i = d11i, so that the unicycles satisfy the parallel formation

requirement. Vice versa, it is clear that if the unicycles form a parallel formation, then

θij = 0 and xij = 0 for all i, j ∈ n.


We have thus shown that PP amounts to the simultaneous synchronization of the

headings θi and the endpoints xi. We now present feedbacks that do just that. Let

fi(·) be a bounded integrator consensus controller as in (4.11), and gi(·) be an attitude

synchronizer as in (4.24). The feedbacks for unicycle i are chosen as follows,

ui =u⋆i (y

ii, ϕi) = fi(y

ii) · e1 + βiω

⋆i (y

ii, ϕi),

ωi =ω⋆i (y

ii, ϕi) =

1

αi

(

fi(yii) · e2 + kgi(ϕi, η)

)

, i ∈ n,(7.2)

where k > 0 is a high-gain parameter, and η = (η1, . . . , ηn), ηi := 1/αi. We are now

ready to present the main result of this chapter.

Theorem 7.1.1. Consider the collection of n unicycles in (2.6), (2.7) with controller (7.2),

where the functions fi(·), gi(·) are defined in (4.11), (4.24) and satisfy properties A1, A2

and B1-B3. Assume that sensor graph G is undirected and connected. For any pa-

rameters aij = aji > 0 in (4.11) and any parameters bij = bji > 0 in (4.24) satisfying

(bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2, there exists α⋆ > 0 such that for any parallel

formation d = (d11i)i∈2 :n ∈ F, choosing α > α⋆maxi∈2 :n (−d11i · e1) in (7.1), the formation

manifold Γp is almost semiglobally asymptotically stable with high-gain parameter k.

The proof is presented in Section 7.5. Roughly speaking, the theorem states that

letting the offset α in (7.1) grow proportionally to the length of the formation (the quan-

tity maxi (−d11i · e1)), and choosing k in (7.2) to be sufficiently large, the controller (7.2)

ensures that almost all initial conditions in any given compact set are contained in the

domain of attraction of the formation manifold Γp . Another property of controller (7.2)

is that (u⋆i , ω⋆i )∣

∣

Γp= 0 for all i ∈ n, and therefore the unicycles come to a stop as Γp

is approached, as required in the statement of PP in Chapter 3. In the next section we

further discuss the controller (7.2).


7.2 Discussion of the Control Solution

As we mentioned earlier, the philosophy behind controller (7.2) is to convert PP into a

synchronization problem in which we make the offset vectors xi and the heading angles

θi converge to one another. This is illustrated in Figure 7.2, where the vectors xi, i ∈ n

all meet at a common point at a distance α in front of the formation, and all heading

directions are aligned.

Figure 7.2: Parallel formation where offset vectors xi meet at a common point at adistance α in front of the formation.

In (7.2), the terms containing fi(yii) aim to achieve consensus on the endpoints of the

offset vectors xi, while the terms containing gi(ϕi, η) aim to achieve consensus on the

unicycle angles. It will be shown in Section 7.5 that the choice of (7.2) achieves both

these, at times competing, objectives simultaneously by making use of gradient properties

of the systems (4.9) and (4.22) with inputs (4.11) and (4.24) respectively.

The block diagram in Figure 7.3 summarizes the design of feedbacks (ui, ωi)i∈n. From

its sensors, unicycle i obtains the vector (yii, ϕi) of its heading and displacement relative

to its neighbours. These quantities can be measured locally in unicycle i’s body frame

using, for example, on-board cameras. The offset extraction block takes as input the

vector (yii, ϕi) and outputs (yii, ϕi), where each component of yii = (xiij)j∈Ni is computed

as,

xiij = xiij + αjRije1 + βjR

ije2 −

[

αi βi]

⊤. (7.3)


PositionConsensus

AttitudeSynchroniztion

OffsetExtraction

Figure 7.3: Block diagram of the formation control system for robot i.

This computation requires that, in addition to (yii, ϕi), unicycle i has access to the

formation parameters (αj , βj)j∈Ni of its neighbours. These quantities must be stored

in memory on-board unicycle i before deployment. Moreover, in order to compute xiij

in (7.3), unicycle i must be able to identify its neighbours so as to use, for each j ∈ Ni,

the appropriate bias constants (αj, βj). Such identification can be achieved, for instance,

by means of visual markers. A consequence of using the constants (αj , βj) is that the

unicycle feedbacks are not identical and the formation is not invariant to a relabelling

of the agents. This is hardly surprising because, in our formulation of PP, we allow for

general, non-symmetric formations.

An important property of the feedback in (7.2) is that it is local and distributed, since

u⋆i and ω⋆i depend on (yii, ϕi). As a consequence of this feature, the asymptotic position

and orientation of the formation with respect to the inertial frame depend only on the

initial configuration of the unicycles.


7.3 Special cases: Line formations and full synchro-

nization

As a by-product of the formation control solution, we present corresponding solutions for

the special cases of parallel line formations and full synchronization.

A parallel line formation is a parallel formation satisfying d11i · e1 = 0 (and hence

αi = α for all i ∈ 2 :n). The set of all such formations will be denoted LF. Clearly,

LF ⊂ F. In the case of full synchronization, the unicycles have the same position and

orientation with respect to the inertial frame, i.e., d11i = 0 for all i ∈ 2 :n (and therefore

αi = α and βi = 0 for all i ∈ n). Full synchronization, therefore, corresponds to the

formation 0 ∈ F. Examples of a parallel line formation and full synchronization are

illustrated in Figure 7.4.

According to Theorem 7.1.1, in both of these cases it suffices that α satisfies the less

strict condition α > 0. This is advantageous, as it will be discussed in Chapter 10 that

large values of α can slow down the rate of convergence of the unicycles to the formation.

Arbitrarily choosing α = 1, the corresponding controller in (7.2) reduces to

u⋆i (yii, ϕi) = fi(y

ii) · e1 + βiω

⋆i (y

ii, ϕi),

ω⋆i (yii, ϕi) = fi(y

ii) · e2 + kgi(ϕi, 1), i ∈ n,

(7.4)

in which, xiij = xiij+Rije1−e1+βjRi

je2−βie2. Since the values αi = α = 1 for all i ∈ n are

equal, unicycle i only needs to store the quantities (βj)j∈Ni of its neighbours on-board.

The next corollary is a specialization of Theorem 7.1.1 to parallel line formations.

Corollary 7.3.1. Consider the collection of n unicycles in (2.6), (2.7) with controller (7.4),

where the functions fi(·), gi(·) are defined as in (4.11), (4.24) and enjoy properties A1,

A2 and B1-B3. Assume that sensor graph G is undirected and connected. For any

parameters aij = aji > 0 in (4.11), any parameters bij = bji > 0 in (4.24) satisfying


Figure 7.4: (a) shows an example of a parallel line formation while (b) shows an exampleof full synchronization, a special case of a parallel line formation.

(bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2, and any parallel line formation d ∈ LF,

the formation manifold Γp is almost semiglobally asymptotically stable with high-gain

parameter k.

In the special case of full synchronization, βi = 0 for all i ∈ n, and the controller

in (7.4) reduces to

u⋆i (yii, ϕi) = fi(y

ii) · e1,

ω⋆i (yii, ϕi) = fi(y

ii) · e2 + kgi(ϕi, 1), i ∈ n,

(7.5)

in which, xiij = xiij +Rije1 − e1. Since the αi and βi parameters are equal for all agents,

unicycle i does not need to store any parameters of its neighbours on-board, the control

inputs are identical for all unicycles, so that in this case the configuration is invariant

to relabelling of agents. This is hardly surprising since the formation is symmetric in

this case. The controller in (7.5) can be viewed as an extension of the result for unicycle

rendezvous in Chapter 6. In fact, in Chapter 6 the controller was defined as

u⋆i (yii, ϕi) = ‖fi(yii)‖fi(yii) · e1,

ω⋆i (yii, ϕi) = −kfi(yii) · e2, i ∈ n,

(7.6)


in which fi(yii) =

∑

j∈Niaijx

iij is a linear single integrator consensus controller.

While the controller in (7.6) guarantees global rendezvous, in which only the unicy-

cle positions are synchronized, the controller in (7.5) guarantees almost semiglobal full

synchronization where both positions and angles of the unicycles are synchronized. The

control inputs in (7.5) and (7.6) are similar in structure. The main difference is that the

full synchronization controller in (7.5) has an additional term kgi(ϕi, η) responsible for

aligning the unicycle heading angles, not required for rendezvous. In fact, for unicycle i,

(7.5) depends on (yii, ϕi) while (7.6) depends only on yii.


This section presents simulations for a group of five unicycles to illustrate our results.

The interaction function f(s) for the bounded integrator consensus control is chosen as

in (4.12) while the interaction function for the attitude synchronizer is chosen satisfying

assumptions B1, B2 and B3 as in Figure 4.16. The undirected sensing graph is cyclic with

connections as shown in Figure 7.5 and the desired triangular formation is specified by

d112 = (−10, 5), d113 = (−10,−5), d112 = (−20, 10) and d112 = (−20,−10), as illustrated in

Figure 7.6. We have chosen random initial unicycle positions on a 40m × 40m area with

1

2

3

5

4

Figure 7.5: Undirected graph G under consideration in the simulation results.

random initial angles. The corresponding plot of a simulation run is shown in Figure 7.7.


Figure 7.6: Triangular formation specified by the offset vectors d112 = (−10, 5), d113 =(−10,−5), d114 = (−20, 10) and d115 = (−20,−10).

Extensive simulation trials will be presented in Chapter 10 to study the effectiveness

of our control solution under different realistic scenarios not captured by the main result

in Theorem 7.1.1 including

• performance in the presence of state dependent sensor graphs in which each unicy-

cle’s neighbors are those that lie within a given radius of itself;

• performance for directed sensing graphs as opposed to undirected sensing graphs;

• performance when the high gain conditions on α and k are ignored;

• robustness of the approach to unmodelled effects including sensor noise, input noise,

sampling and saturated inputs

• extension of the control solution for kinematic unicycles to the dynamic unicycle

model in (2.10).

7.5 Proof of Theorem 7.1.1

We divide the proof of Theorem 7.1.1 in several steps. In Section 7.5.1, we derive the

closed-loop dynamics in (xi, θi)i∈n coordinates. In Sections 7.5.2 and 7.5.3, we propose


-80 -60 -40 -20 0 20 40

x (m)

-20

-10

0

10

20

30

40

50

60

70

80

90

y (m

)

Figure 7.7: Simulation for a triangle formation. Initial positions are indicated with andfinal positions are indicated with ×.

a Lyapunov function V for the closed-loop system, and carry out a Lyapunov analysis

yielding the property V ≤ 0. In Section 7.5.4, we show that, for sufficiently large α > 0,

the zero level set of V coincides with the formation manifold Γp on a neighbourhood

of Γp . This result will imply, via Lyapunov’s direct method, asymptotic stability of Γp .

A further Lyapunov analysis is employed to show that Γp is in fact almost semiglob-

ally asymptotically stable with high-gain parameter k. Each step of the proof will be

presented in its own subsection.

7.5.1 System dynamics in (xi, θi)i∈n coordinates

To simplify the analysis, we consider new coordinates (x, θ) = (xi, θi)i∈n under the dif-

feomorphism F : (x, θ) 7→ (x, θ) given by F ((xi, θi)i∈n) = (xi + δ(θi), θi)i∈n. Computing

the time derivative of xi yields,

˙xi = uiRie1 +Ri

0 −ωiωi 0

(αie1 + βie2)

= uiRie1 + αiωiRie2 − βiωiRie1

= (ui − βiωi)Rie1 + αiωiRie2,


from which we get

˙xi = (ui − βiωi)Rie1 + αiωiRie2

θi = ωi, i ∈ n.

(7.7)

Using Lemma 4.3.3(i) and the fact that the dot product is invariant to rotations, i.e,

R−1i fi(yi) · e1 = fi(yi) · Rie1, the feedbacks in (7.2) can expressed as follows:

u⋆i (yii, ϕi) = fi(yi) ·Rie1 + βiω

⋆i (y

ii, ϕi),

ω⋆i (yii, ϕi) =

1

αi(fi(yi) · Rie2 + kgi(ϕi, η)) .

(7.8)

Substituting ui = u⋆i (yii, ϕi) and ωi = ω⋆i (y

ii, ϕi) from (7.8) into (7.7) and using the

fact that fi(yi) = (f(yi) · Rie1)Rie1 + (f(yi) · Rie2)Rie2 yields the closed-loop system in

(xi, θi)i∈n coordinates,

˙xi = fi(yi) + kgi(ϕi, η)Rie2

θi =1

αi(fi(yi) · Rie2 + kgi(ϕi, η)) , i ∈ n.

(7.9)

Notice that the control inputs are defined precisely in terms of (x, θ) and so the equations

of motion in (7.9) constitute a dynamical system. The closed loop system in (7.9) has no

finite escape times because ‖ ˙x‖ is bounded by a linear function of x and θ is bounded.

The parallel formation manifold Γp in (3.4) in (x, θ) coordinates becomes,

Γp :=

(x, θ) ∈ R2n × T

n : x1i = 0, θ1i = 0, i ∈ n

. (7.10)


7.5.2 Lyapunov analysis

From Lemma 4.3.2, system (4.9) is gradient with nonnegative storage function Vt. In-

spired by (Mallada et al., 2016), define a Lyapunov function Vr(θ) as,

Vr(θ) :=1

2

n∑

i=1

∑

j∈Ni

bij

∫ θij

0

g(s)ds. (7.11)

Since G is connected, we have that Vr ≥ 0 and V −1r (0) = θ ∈ Tn : (∀i, k ∈ n) θi = θj.

Next, combine Vt(x) and Vr(θ) as follows:

V (x, θ) := Vt(x) + kVr(θ). (7.12)

Since Vt and Vr are nonnegative, V is nonnegative and V −1(0) = Γp .

Using (7.9), the time derivative of Vt(x) is given by,

Vt =

n∑

i=1

−fi · (fi + kgiRie2)

=n∑

i=1

(

−‖fi‖2 − (fi · Rie2)kgi)

.

(7.13)

Since g(θij) = −g(θji), we have

∂Vr∂θi

=1

2

∑

j∈Ni

bij∂

∂θij

(∫ θij

0

g(s)ds

)

∂θij∂θi

+1

2

∑

j∈Ni

bji∂

∂θji

(∫ θji

0

g(s)ds

)

∂θji∂θi

= −∑

j∈Ni

bijg(θij).

(7.14)


Using the above, identity (4.24), and the fact that ηi = 1/αi, we obtain

Vr = −n∑

i=1

∑

j∈Ni

bijαig(θij) (fi · Rie2 + kgi)

=

n∑

i=1

−gi (fi · Rie2 + kgi)

=n∑

i=1

(−(fi · Rie2)gi − kg2i ).

(7.15)

Combining (7.13) and (7.15), we get

V = Vt + kVr

=n∑

i=1

(

−‖fi‖2 − 2(fi · Rie2)(kgi)− (kgi)2)

=

n∑

i=1

(

−‖fi · Rie1‖2 − ‖fi · Rie2 + kgi‖2)

≤ 0.

(7.16)

7.5.3 Lyapunov analysis in relative coordinates

To further simplify the stability analysis, we perform another coordinate transformation

with the intention of quotienting out the dynamics of unicycle 1. More precisely, consider

the diffeomorphism

F : R2n × Tn → R

2(n−1) × R2 × T

(n−1) × S1,

F (x, θ) = (x, x11, θ, θ1),

where x := (x11i)i∈2 :n, θ := (θ1i)i∈2 :n. Using Lemma 4.3.3(i) and the fact that fi(yi) ·

Rie2 = fi(y1i ) · R1

i e2, the dynamics in (7.9) can be written in new coordinates as,


˙x11i =[

fi(y1i ) + kgi(ϕi, η)R

1i e2 − f1(y

11)− kg1(ϕ1, η)e2

]

−(

1

α1

(

f1(y11) · e2 + kg1(ϕ1, η)

)

)×

x11i

θ1i =1

αi

(

fi(y1i ) ·R1

i e2 + kgi(ϕi, η))

− 1

α1

(

f1(y11) · e2 + kg1(ϕ1, η)

)

˙x11 =f1(y11) + kg1(ϕ1, η)e2 − ω×

1 x11

θ1 =1

α1

(

f1(y11) · e2 + kg1(ϕ1, η)

)

,

(7.17)

where i ∈ 2 :n.

We remark that y1i = (x1ij)j∈Ni = (x11j − x11i)j∈Ni, ϕi = (θij)j∈Ni = (θ1j − θ1i)j∈Ni

and R1i are functions of the relative quantities (x, θ), and do not depend on the absolute

quantities x11 and θ1. It follows that system (7.17) has a decoupled subsystem with state

(x, θ) ∈ R2(n−1) × Tn−1. Moreover, Γp in new coordinates is given by

(x, x11, θ, θ1) : x11i = 0, θ1i = 0, i ∈ 2 :n

, (7.18)

which is also independent of absolute quantities x11 and θ1.

Based on these considerations, the variables x11 and θ1 may be dropped, yielding a

new dynamical system with state (x, θ) ∈ R2(n−1) × Tn−1. Proving almost semiglobal

asymptotic stability of Γp for system (7.9) is equivalent to proving that the equilibrium

point

Γp :=

(x, θ) = (0, 0) ∈ R2(n−1) × T

(n−1)

. (7.19)

is almost semiglobally asymptotically stable for the (x, θ) subsystem.

We now return to the Lyapunov analysis of Section 7.5.2, expressing V in relative

coordinates (x, θ). Using the fact that ‖xij‖ = ‖R−11 xij‖ = ‖R−1

1 (x1j−x1i)‖ = ‖x11j−x11i‖,


and θij = θ1j − θ1i, we have

Vt(x, θ) := Vt|(x,θ)=F−1(x,x11,θ,θ1)

=1

2

n∑

i=1

∑

j∈Ni

aij

∫ ‖x11j−x11i‖

0

f(s)ds

Vr(x, θ) := Vr|(x,θ)=F−1(x,x11,θ,θ1)

=1

2

n∑

i=1

∑

j∈Ni

bij

∫ θ1j−θ1i

0

g(s)ds.

(7.20)

The identities in (7.20) imply that V can indeed be expressed in terms of relative quan-

tities (x, θ), and in these coordinates it is given by V (x, θ) := Vt(x, θ) + kVr(x, θ).

Since V −1(0) = Γp, it follows that V −1(0) = Γp and therefore V is positive definite

at (x, θ) = (0, 0). For any c > 0, the sublevel set Vc = (x, θ) ∈ R2(n−1) ×T(n−1) : V ≤ c

is closed since V is continuous. Next we show that Vc is bounded, and hence compact.

In the set Vc,

aij

∫ ‖x1ij‖

0

f(s)ds ≤ c,

for all i ∈ n, j ∈ Ni. If ‖x1ij‖ > c2 where c2 is defined in A1, then this implies that

aijc1(‖x1ij‖ − c2) ≤ c where c1 is defined in A1 and therefore ‖x1ij‖ ≤ (c/c1aij) + c2 is

bounded. Since the undirected graph is connected, this proves boundedness of (x, θ).

Moreover, using a standard result in (Lee, 2013, Proposition 8.16), the time derivative

of V satisfies,

˙V =V |(x,θ)=F−1(x,x11,θ,θ1)

=

n∑

i=1

(

−‖fi(y1i ) · R1i e1‖2 − ‖fi(y1i ) · R1

i e2 + kgi(ϕi, η)‖2)

.(7.21)

Once again, since y1i , ϕi and R1i are functions of (x, θ), ˙V (x, θ) is independent of x11

and θ1. In light of (7.21), ˙V ≤ 0, with equality if and only if fi(y1i ) · R1

i e1 = 0 and

fi(y1i ) ·R1

i e2 = −kgi(ϕi, η) for all i ∈ n. Together, these conditions imply that on the set


E := (x, θ) : ˙V (x, θ) = 0 it holds that

fi(y1i ) = −kgi(ϕi, η)R1

i e2, ∀i ∈ n. (7.22)

7.5.4 Local asymptotic stability of Γp

In this section we show that there exists ǫ > 0 such that E ∩ (x, θ) : ‖θ‖ ≤ ǫ =

V −1(0) = Γp, implying that ˙V is negative definite, and Γp is locally asymptotically

stable by Lyapunov’s direct method.

Let (x, θ) ∈ E be arbitrary. By Lemma 4.3.3(iii), we have 0 =∑n

i=1 fi(y1i ) =

R−11

∑ni=1 fi(yi). Using (7.22), we get −∑n

i=1 gi(ϕi, η)R1i e2 = 0, and using (4.24), we

get

−n∑

i=1

ηi∑

j∈Ni

bijg(θij)R1i e2 = 0.

We have R1i e2 =

[

− sin(θ1i) cos(θ1i)

]

⊤, so

−n∑

i=1

ηi∑

j∈Ni

bijg(θij)

− sin(θ1i)

cos(θ1i)

= 0.

The first component of the above identity gives

n∑

i=1

ηi∑

j∈Ni

bijg(θij) sin(θ1i) = 0, (7.23)

which depends solely on relative angles θ. Expanding g(s) and sin(s) about s = 0, we

get

g(s) = g(0)s+ h1(s)s

sin(s) = s+ h2(s)s,

where lims→0 h1(s) = 0 and lims→0 h2(s) = 0. Moreover, g(0) > 0 by B3. Using the


above identities in (7.23) we get

n∑

i=1

ηi∑

j∈Ni

bij [g(0)θijθ1i + g(0)h2(θ1i)θijθ1i

+h1(θij)θijθ1i + h1(θij)h2(θ1i)θijθ1i] = 0.

(7.24)

Dividing by g(0), we have

n∑

i=1

ηi∑

j∈Ni

bijθijθ1i

[

1 + h2(θ1i) +h1(θij)

g(0)+h1(θij)h2(θ1i)

g(0)

]

= 0. (7.25)

Ignoring, for now, higher order terms h1(θij), h2(θ1i) in (7.25) and substituting in ηi =

1/αi, we getn∑

i=1

(θ1i/αi)∑

j∈Ni

bij [θ1j − θ1i] = 0. (7.26)

Define a weighted Laplacian matrix L by L(i, j) = −bij for i 6= j and L(i, i) =∑

j∈Nibij .

Then L is a symmetric Laplacian for the connected undirected graph G, and therefore

kerL = span1. Next, let

λi(α, d) :=maxi αiαi

=α +maxi(−d11i · e1)

α− d11i · e1,

λ = (λ1, . . . , λn),

(7.27)

and

D(λ) := diag(

λi)

i∈n.

The identity in (7.26) can now be rewritten as

− 1

maxi αi

[

0 θ⊤]

D(λ(α, d))L

0

θ

= 0. (7.28)


Letting

L(λ) := P⊤D(λ)LP, P =

01×(n−1)

I(n−1)

,

identity (7.28) implies

θ⊤L(λ(α, d))θ = 0. (7.29)

Denoting by M(λ) := (L(λ)+ L(λ)⊤)/2 the symmetric part of L, identity (7.29) becomes

θ⊤M(λ(α, d))θ = 0. (7.30)

We will show that for large α > 0, M(λ(α, d)) is positive definite. Referring to the

definition of λi in (7.27), note that

λi(α, d) → 1 asα

maxi(−d11i · e1)→ ∞, (7.31)

and λi(α, d) = 1 when −d11i ·e1 = 0 for all i ∈ n. In light of this observation, consider first

the case in which λ = 1, so that D(λ) = D(1) = In, the identity matrix. Then (7.28)

reduces to[

0 θ⊤]

L

0

θ

≥ 0,

with equality if and only if

[

0 θ⊤]

∈ span1 (since kerL = span1), which can occur

only if θ = 0. Owing to the equivalence of (7.28) and (7.30), we have that θ⊤M(1)θ ≥ 0,

with equality holding if and only if θ = 0, and thus M(1) is positive definite and, since

M(λ) is symmetric, all its principal leading minors mi(λ), i ∈ n, have the property that

mi(1) > 0, i ∈ n. Since the functions mi(λ) are continuous, there exists ε > 0 such that

for all λ ∈ Rn such that ‖λ − 1‖ < ε, mi(λ) > 0, i ∈ n. From (7.31), we deduce that


there exists α⋆ > 0 such that

α

maxi(−d11i · e1)> α⋆ =⇒ ‖λ(α, d)− 1‖ < ε

=⇒ mi(λ(α, d)) > 0 i ∈ n.

We have thus established the existence of α⋆ > 0 such that, for all α > α⋆maxi(−d11i ·e1),

the matrix M(λ(α, d)) is positive definite.

Now assuming that α satisfies the above bound so thatM(λ(α, d)) is positive definite,

we return to identity (7.25) including higher-order terms, and rewrite it as

θ⊤M(λ(α, d))θ + r(θ) = 0, (7.32)

where M(·) is as before and

r(θ) =n∑

i=1

ηi∑

j∈Ni

bijθijθ1i

[

h2(θ1i) +h1(θij)


g(0)

]

.

We will show, using similar arguments to (Francis and Maggiore, 2016, Proof of Theo-

rem 6.1), that there exists ǫ > 0 such that in an ǫ-neighborhood of θ = 0, identity (7.32)

holds only if θ = 0.

Condition (7.32) holds only if ‖θ⊤M(·)θ‖ = ‖r(θ)‖. Suppose for a moment that

limθ→0

‖r(θ)‖‖θ⊤M(·)θ‖

= 0. (7.33)

Then for sufficiently small θ, ‖r(θ)‖ ≤ ‖θ⊤M(·)θ‖/2, and the unique solution to (7.32)

is θ = 0, as desired. To show that (7.33) holds, express θ as θ = ‖θ‖φ where φ =

(φ1i)i∈2 :n ∈ Sn−1 is a unit vector. Correspondingly, θ1i = ‖θ‖φ1i for all i ∈ 2 :n. One


can then write,

θ⊤M(·)θ =‖θ‖2φ⊤M(·)φ,

r(θ) =‖θ‖2n∑

i=1

∑

j∈Ni

bijηiφijφ1i

[

h2(θ1i) +h1(θij)


g(0)

]

= ‖θ‖2h(θ, φ),

where h(·, ·) has the property that limθ→0 h(θ, φ) = 0. Then,

limθ→0

‖r(θ)‖‖θ⊤M(·)θ‖

= limθ→0

‖h(θ, φ)‖φ⊤M(·)φ = 0,

since limθ→0 h(θ, φ) = 0 and minφ∈Sn−1(φ⊤M(·)φ) > 0 because M(·) positive definite and

φ is a unit vector.

To summarize, there exists ǫ > 0 such that if ‖θ‖ ≤ ǫ, then (7.25) is zero only if

θ = 0, implying that gi = 0 for all i ∈ n. By (7.22) this implies that fi = 0 for

all i ∈ n and therefore, by Lemma 4.3.3(ii), xi = xj for all i, j ∈ n. It follows that

E ∩ (R2(n−1) × θ : ‖θ‖ ≤ ǫ) = Γp .

To summarize our findings so far, we have shown that V is positive definite, V −1(0) =

(x, θ) = (0, 0) = Γp , and ˙V is negative definite on (R2(n−1) × θ : ‖θ‖ ≤ ǫ), a

neighbourhood of Γp . By Lyapunov’s stability theorem, the equilibrium Γp is locally

asymptotically stable for the (x, θ) subsystem.

7.5.5 Almost semiglobal asymptotic stability of Γp

Having established that for α > 0 sufficiently large, the equilibrium Γp is asymptotically

stable for the (x, θ) subsystem, we now prove that Γp is almost semiglobally asymptoti-

cally stable with high-gain parameter k. The idea is to show that, for sufficiently large

k, for almost all initial conditions in any given compact set the solutions of the (x, θ)

subsystem enter in finite time and remain inside the set (R2(n−1) × θ : ‖θ‖ ≤ ǫ) on

which ˙V is negative definite, which implies that they converge to Γp.


Rewrite the dynamics of the θ subsystem in (7.17) as,

˙θ = kF (θ) + ∆(x, θ), (7.34)

where

Fi(θ) :=

(

1

αigi(ϕi, η)−

1

α1g1(ϕ1, η)

)

∆i(x, θ) :=1

αifi(y

1i ) · R1

i e2 −1

α1f1(y

11) · e2.

After the time scaling τ = kt, system (7.34) reads as

θ′ = F (θ) +1

k∆(x, θ), (7.35)

where prime denotes differentiation with respect to τ . In what follows, we denote by

Σ(0) the nominal system θ′ = F (θ), and by Σ(k) the perturbed system (7.35).

The vector field F coincides with the attitude synchronization dynamics of the col-

lection of rotational integrators in (4.22) with feedback (4.24), expressed relative to inte-

grator 1. Therefore, by Theorem 4.3.2, the equilibrium θ = 0 is almost globally asymp-

totically stable for Σ(0). Let D(0) be the domain of attraction of θ = 0 for Σ(0), a set

of full-measure.

The term (1/k)∆ acts as a perturbation in (7.35). Since, by assumption A2, the

functions fi(y1i ) · R1

i e2 and f1(y11) · e2 are uniformly bounded, the map ∆ is uniformly

bounded, i.e., there exists ∆ > 0 such that sup ‖∆‖ < ∆. The uniform bound on the

perturbation (1/k)∆ tends to zero as k → ∞.

Since θ = 0 is asymptotically stable for Σ(0), there exists r > 0 and a C1 positive

definite Lyapunov function W : Br(0) → R whose derivative along Σ(0), LFW : Br(0) →

R, is negative definite. We may assume, without loss of generality, that r ≤ ǫ. Let c > 0

be such that the sublevel set Wc := θ : W (θ) < c is contained in Br(0) ⊂ Bǫ(0), and let

ǫ′ > 0 be such that Bǫ′(0) ⊂Wc ⊂ Bǫ(0). Since LFW∣

∣

∂Wc< 0, Wc is positively invariant


for Σ(0). Moreover, letting

k0 =max∂Wc

‖∂W/∂θ‖min∂Wc

|LFW | ∆,

we have that for all k > k0 it holds that LF+(1/k)∆W∣

∣

∂Wc< 0, and thus Wc is positively

invariant for the perturbed system Σ(k) in (7.35).

Let θ ∈ D(0) be arbitrary. Then the solution of Σ(0) through θ converges to 0, and

let T > 0 be the first time when the solution enters Bǫ′(0). By continuity of solutions

with respect to initial conditions and bounded perturbations (Khalil, 2002, Theorem 3.4),

there exists µ > 0 and k ≥ k0 such that for all k > k, all solutions of Σ(k) through initial

conditions in Bµ(θ) are contained in Wc at time T . Since Wc is positively invariant for

Σ(k) and contained in Bǫ(0), all solutions of Σ(k) through Bµ(θ) enter in finite time and

remain inside the set Bǫ(0), for all k > k.

Let K ⊂ D(0) be an arbitrary compact set. The arguments above yield an open

cover of K by balls Bµ(θ) and associated gains k, where µ and k depend on θ. Taking a

finite subcover, we obtain points θi, i ∈ k ⊂ Tn−1, associated balls Bµi(θi), and gains

ki, i ∈ k. Letting k⋆ = maxi∈k ki, for all k > k⋆ all solutions of Σ(k) through points in

K enter and remain inside Bǫ(0).

Returning to the (x, θ) dynamics, the x subsystem has no finite escape times because

V is proper and nonincreasing along solutions. Then, from the results just obtained

we have that, for any compact subset K of D(0) (a set of full-measure in Tn−1), there

exists k⋆ > 0 such that, for all k > k⋆, all solutions of the (x, θ) subsystem through

initial conditions in R2(n−1) × K enter in finite time and remain inside the closed set

R2(n−1) × θ : ‖θ‖ ≤ ǫ. For any c > 0, the set Vc ∩ (R2(n−1) × θ : ‖θ‖ ≤ ǫ) is

compact because the sublevel set Vc is compact. Since ˙V is negative definite on this set,

all solutions through R2(n−1)×K converge to Γ. This proves that Γ is almost semiglobally


Chapter 8

Formations of Kinematic Unicycles

with Parallel and Circular Collective

Motions

In this chapter we present control solutions to the following problems introduced in

Chapter 3:

• Parallel FormationFlocking Problem PFP andParallel FormationFlocking Problem

with a Beacon PFP-B

• Formation Line Path Following Problem LPP

• Circular Formation Flocking Problem CFP

• Formation Circle Path Following Problem CPP.

The control inputs ui and ωi for each control problem will be constructed out of one or

more of the following control primitives discussed in Section 4.3:

• a consensus controller for single-integrators fi((xij)j∈Ni) in (4.10),

• a line path following controller for single integrators h(x) in (4.16) and,

172

Chapter 8. Form. of Kin. Unicycles with Par. and Circ. Coll. Mot. 173

• a rotational integrator consensus controller gi((θij)j∈Ni, η) in (4.24).

Control solutions to PFP, PFP-B and LPP are presented in Section 8.1 and control solu-

tions to CFP and CPP are presented in Section 8.2.

8.1 Solutions to Formation Problems with Final Lin-

ear Motion (PFP, PFP-B and LPP)

In this section we present solutions to the formation problems with final linear motion.

Let α > 0 be a design parameter, and let d = (d1i1)i∈2 :n ∈ F be the desired formation.

Define the quantities αi, βi, δi and xi exactly as in Chapter 7 for formations that stop.

Also define yi := (xij)j∈Ni, yki := (xkij)j∈Ni as before. Recall the following set definitions

from Chapter 3

Γ =

(x, θ) ∈ R2n × T

n : x1i = R1d11i, i ∈ n

Γp = (x, θ) ∈ Γ : θi = θ1, i ∈ 2 :n

Γpf = Γp

Γpfb = (x, θ) ∈ Γ : θi = θp, i ∈ n

Γlp = (x, θ) ∈ Γpfb : x1 ∈ C(r0, p) .

By the same arguments as in Chapter 7, stabilizing Γpf = Γp in (3.5) reduces to synchro-

nizing the unicycles’ heading angles and the endpoints xi. For Γpfb there is the additional

requirement that θi = θp for all i ∈ n. Γlp follows by also imposing x1 ∈ C(r0, p). The

latter requirement can be replaced with x1 ∈ C(r0, p) since θ1 = θp and x1 ∈ C(r0, p)

imply x1 = x1 + αRie1 = x1 + αp ∈ C(r0, p). The converse can be shown as well.

For a single integrator consensus controller fi(·) defined in (4.10) choose the feedback

law as,

ui = u⋆i (yii, ϕi, µ

ii) = fi(y

ii) · e1 + βiω

⋆i (y

ii, ϕi, µ

ii) + µii · e1,

ωi = ω⋆i (yii, ϕi, µ

ii) =

1

αi

(

fi(yii) · e2 + µii · e2

)

, i ∈ n.(8.1)


where µi will be defined in the main theorem given below in Theorem 8.1.1 whose proof

is given in Section 8.4.

Theorem 8.1.1 (Solutions to PFP, PFP-B and LPP). Consider system (2.6), (2.7) with

directed sensor graph G containing a globally reachable node and any parameters aij > 0

for i ∈ n, j ∈ Ni.

1. (PFP) Suppose G is a hierarchical digraph. For any (d, w) ∈ PF, the local and dis-

tributed control inputs in (8.1) with α > 0, µ11 = we1, and µ

ii = (w/|Ni|)

∑

j∈NiRije1, i ∈

2 :n almost globally asymptotically stabilize the parallel formation flocking mani-

fold Γpf , thus solving PFP for the class of hierarchical digraphs.

2. (PFP-B) For any (d, p, w) ∈ PFB, the control inputs in (8.1), strictly functions of

(yii, ϕi, pi), with α > 0, and µii = wpi almost globally asymptotically stabilize the

parallel formation flocking manifold Γpfb , thus solving PFP-B.

3. (LPP) For any (d, r0, p, w) ∈ LP the control inputs in (8.1), strictly functions of

(yii, ϕi, pi, (c⋆(xi) − xi)

i), with α > 0, µii = h(xi)i and h(·) a line path following

controller for single integrators defined in (4.16) almost globally asymptotically

stabilize the formation line path following manifold Γlp , thus solving LPP.

The terms in (8.1) containing fi(yii) have the objective of achieving consensus on

endpoints xi while those containing µii have the objective of synchronizing the headings

θi in addition to stabilizing the desired path C(r0, p) in the case of path following. Note

that for path following, the controller in (8.1) will enforce unicycle 1 to follow the line

C(r0, p). If the line is to be followed by unicycle j as opposed to unicycle 1, one simply

changes the value of βi to βi = −d11i · e2 + d11j · e2 for all i ∈ n.

The control inputs in (8.1) for unicycle i are therefore strictly functions of (yii, ϕi),

(αj , βj)j∈Ni and µii where µii is a function of pi in the case of PFP− B. In the case of


LPP, µii (with k0 = 1 for simplicity) satisfies

µii = h(xi)i = (c⋆(xi)− xi)

i + wpi

= (r0 − xi)i − ((r0 − xi)

i · pi)pi + wpi

= (r0 − xi)i − αie1 − βie2 − ((r0 − xi)

i · pi)pi + ((αie1 + βie2) · pi)pi + wpi

= (c⋆(xi)− xi)i − αie1 − βie2 + ((αie1 + βie2) · pi)pi + wpi

which is a function of pi and (c⋆(xi)−xi)i, the displacement between xi and its orthogonal

projection c⋆(xi) on the path represented in body frame.

8.2 Solutions to Formation Problems with Final Cir-

cular Motion (CFP and CPP)

In this section we present solutions to the formation problems with final circular motion.

Recall from Section 3.3.3 that xi = xi+ δi = xi+βiRie2 where βi is the radius of circular

motion of unicycle i. Recall the following set definitions from Chapter 3

Γcf =

(x, θ) ∈ R2n × T

n : x1i = 0, θ1i = ρi(d, β1), i ∈ 2 :n

Γcp = (x, θ) ∈ Γcf : x1 = c

where Γcf corresponds to the set where the perceived centres xi coincide and the relative

headings satisfy θ1i = ρi(d, β1) for i ∈ 2 :n. Γcp follows by also imposing that the

perceived centres xi coincide at the desired circle center c. For gains r,K, k > 0, choose

the feedback law as,

ui = u⋆i (yii, ϕi, νi) = ui + (βi − r)ω⋆i (y

ii, ϕi, νi)

ωi = ω⋆i (yii, ϕi, νi) =

uir+Kϕ(νi)νi, i ∈ n

(8.2)

where


• ui := rw + kgi((θij − ρij(d, β1))j∈Ni, 1) where ρij(d, β1) := ρj(d, β1)− ρi(d, β1)

• gi(·) is an almost global rotational integrator consensus controller in (4.24)

• ϕ(νi) := 1/(1 + ‖νi‖) is a decentralized saturation function, and

• νi will be defined in the main theorem for CFP and CPP given below in Theo-

rem 8.2.1.

Theorem 8.2.1 (Solutions to CFP and CPP). Consider system (2.6), (2.7) with con-

nected, undirected sensor graph G and any parameters aij = aji > 0 in (4.10) and

bij = bji > 0 in (4.24) satisfying (bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2.

1. (CFP) For any (d, β1, w) ∈ CF, there exist K⋆, k⋆ > 0 such that for any K ∈

(0, K⋆), k ∈ (0, k⋆), for any r > 0, the local and distributed control inputs in (8.2)

with νi = −fi(yii) · e1 and fi(·) a single integrator consensus controller defined

in (4.10) almost globally asymptotically stabilize the circular formation flocking

manifold Γcf , thus solving CFP.

2. (CPP) For any (d, β1, c, w) ∈ CP there exist K⋆, k⋆ > 0 such that for any K ∈

(0, K⋆), k ∈ (0, k⋆), for any r > 0, the control inputs in (8.2) with νi = −(c−xi)i·e1,

strictly functions of (yii, ϕi, (c − xi)i), almost globally asymptotically stabilize the

formation circle path following manifold Γcp, thus solving CPP.

The proof of Theorem 8.2.1 is given in Section 8.4. The control inputs in (8.2) for

unicycle i are strictly functions of (yii, ϕi), the formation parameters (βj)j∈Ni and νi. In

the case of flocking, νi is a function of yii and (βj)j∈Ni while in the case of path following,

νi is a function of (c − xi)i = (c − xi)

i − βie2. The feedbacks in (8.2) are based on the

control solution in (El-Hawwary and Maggiore, 2013a), however, unlike (El-Hawwary

and Maggiore, 2013a), they allow for possibly different radii βi of each unicycle so that

formations are not restricted to a common circle. Also, the term gi in ui corresponds

to an almost global rotational integrator consensus controller in (4.24) rather than a


Kuramoto model in (4.23), allowing for an almost global result. An adaptation of the

controller in (El-Hawwary and Maggiore, 2013a) is discussed in Section 8.2.1.

Remark 8.2.2. The solutions presented above for line and circle path following assume

that all unicycles in the ensemble sense the desired path. A more general result can be

obtained for the case where only a non-empty subset of unicycles S ⊂ V sees the path

assuming that at least one unicycle in S is a globally reachable node for G. For undirected

graphs, S can be any non-empty subset of V. For line path following choose the feedback

law as in (8.1) with µii = wpi for i /∈ S and µii = h(xi)i = k0(c

⋆(xi) − xi)i + wpi for

i ∈ S. Measurement of pi is still required by all agents in this case. For circle path

following choose the feedback law as in (8.2) with νi = −f(yii) · e1 for i /∈ S, and

νi = −f(yii) · e1 − (c− xi)i · e1 for i ∈ S.

8.2.1 Formation control of unicycles on a common circle

The solutions in Theorem 8.2.1 are based on an intermediate result presented in this

section that makes unicycles converge almost globally to a common circle of desired radius

r > 0, i.e., βi = r for all i ∈ n, and with desired inter-agent spacings θ1i = ρi(d, β1) ∈ S1

for i ∈ 2 :n. For n kinematic unicycles in (2.6), (2.7) consider the feedbacks,

ui = u⋆i (yii, ϕi, νi) = ui

ωi = ω⋆i (yii, ϕi, νi) =

uir+Kϕ(νi)νi, i ∈ n

(8.3)

where xi = xi + rRie2, ϕ(νi) and ui are defined as in (8.2) and νi is defined in Theo-

rem 8.2.3 below. The feedbacks in (8.3) are a minor modification of those in (El-Hawwary

and Maggiore, 2013a) where the only difference is that the term gi in ui is an almost global

rotational integrator consensus controller in place of the Kuramoto consensus controller

used in (El-Hawwary and Maggiore, 2013a). The result in Theorem 8.2.3 shows that the

set where the n unicycles lie on a common circle of radius r with desired angular spacings


θ1i = ρi(d, β1) for i ∈ 2 :n is almost globally asymptotically stable. In (El-Hawwary and

Maggiore, 2013a), the corresponding result only shows asymptotic stability. Naturally,

the proof of Theorem 8.2.3, presented in Section 8.7, will use the same arguments as

in (El-Hawwary and Maggiore, 2013a) just with the additional requirement of showing

an almost global domain of attraction. Part 1 of Theorem 8.2.3 concerns flocking while

part 2 concerns path following where a specific circle centre c ∈ R2 is specified a priori

which was not considered explicitly in (El-Hawwary and Maggiore, 2013a).

Theorem 8.2.3. Consider the system of unicycles in (2.6), (2.7), radius r > 0, desired

angular speed w > 0, a set of desired angular offsets (ρi(d, β1))i∈2 :n ∈ Tn−1, an integrator

consensus controller fi(·) in (4.10) and a rotational integrator consensus controller gi(·)

in (4.24). Assume that the sensor graph G is connected and undirected and choose

bij = bji > 0 in (4.24) satisfying (bij)(i,j)∈E ∈ (R+)|E|\Nb as in Theorem 4.3.2. Then there

exist K⋆ ∈ (0, (w/2)) and k⋆ > 0 such that for all K ∈ (0, K⋆) and all k ∈ (0, k⋆) the

feedback laws in (8.3) with

1. νi = −f(yii)·e1 almost globally asymptotically stabilize the set Γcf with (u⋆i , ω⋆i )|Γcf

=

(wr, w)

2. νi = −(c − xi)i · e1, almost globally asymptotically stabilize the set Γcp with

(u⋆i , ω⋆i )|Γcp

= (wr, w).


In this section, we present simulation results for formation control with parallel and cir-

cular final collective motions using the feedbacks presented in Section 8.1 and Section 8.2.

We consider a group of five unicycles with undirected graph shown in Figure 8.1 and with

desired triangular formation specified by d112 = (−10, 5), d113 = (−10,−5), d112 = (−20, 10)

and d112 = (−20,−10), illustrated in Figure 7.6.


1

2

3

5

4

Figure 8.1: Undirected graph G under consideration in the simulation results.

We have chosen random initial positions on a 40m × 40m area with random initial

angles. The corresponding simulation results for formations with parallel collective mo-

tions are shown in Figure 8.2 while those for formations with circular collective motions

are shown in Figure 8.3. For parallel formation flocking with no beacon in Figure 8.2(b),

although the sensor graph is not hierarchical, the local and distributed control in (8.1)

with µii = (w/|Ni|)∑

j∈NiRije1, i ∈ n still manages to stabilize Γpf .

8.4 Proofs

In Chapter 3 we introduced the sets Γpf ,Γpfb ,Γlp,Γcf ,Γcp to be almost globally asymp-

totically stabilized. As for stopping formations in Chapter 7, it will be useful to consider

(x, θ) = (xi, θi)i∈n as new coordinates. The parallel formation flocking manifold Γpf

in (3.5), expressed in (x, θ) coordinates, becomes

Γpf :=

(x, θ) ∈ R2n × T

n : x1i = 0, θ1i = 0, i ∈ 2 :n

(8.4)

which is the same as Γp in (7.10). The parallel formation flocking manifold with a beacon

Γpfb in (3.6) becomes,

Γpfb :=

(x, θ) ∈ Γpf : θi = θp, i ∈ n

. (8.5)


Figure 8.2: Simulation results for formations with final parallel collective motion: (a)formation flocking with beacon p = (1, 0) pointing in the direction of the positive x-axis (b) formation flocking with no beacon where µii = (w/|Ni|)

∑

j∈NiRije1, i ∈ n, (c)

formation path following for C(r0, p) = x ∈ R2 : x = r0 + sp, s ∈ R with r0 = (200, 0),p = (0, 1). Initial positions are indicated with and positions at the end of the simulationare indicated with ×.

To stabilize Γlp in (3.7) there is the additional requirement that x1 ∈ C(r0, p). Therefore,

the formation line path following manifold becomes

Γlp :=

(x, θ) ∈ Γpfb : x1 ∈ C(r0, p)

. (8.6)


-60 -40 -20 0 20 40

x (m)

-40

-20

0

20

40

60

y (m

)

(a)

0 50 100

x (m)

-100

-80

-60

-40

-20

0

20

40

60

80

100

y (m

)

(b)

Figure 8.3: Simulation for formations with circular collective motion (a) formation flock-ing, (b) formation path following around the point c = (100, 0). Initial positions areindicated with and positions at the end of the simulation are indicated with ×.

Finally, the sets Γcf ,Γcp in (3.8) and (3.9) expressed in terms of (x, θ) coordinates become

Γcf :=

(x, θ) ∈ R2n × T

n : x1i = 0, θ1i = ρi(d, β1), i ∈ 2 :n

,

Γcp :=

(x, θ) ∈ Γcf : x1 = c

,

respectively. The equations of motion in terms of (x, θ) coordinates is given in (7.7).

Recall that in CFP and CPP, αi = 0 for all i ∈ n. Notice that the control inputs

presented in Section 8.1 and Section 8.2 were defined precisely in terms of (x, θ) and so

the equations of motion in (7.7) constitute a dynamical system.


We will begin by proving the results for PFP-B and LPP corresponding to the sets Γpfb

and Γlp respectively. We will then present the proof for PFP for hierarchical digraphs, cor-

responding to the set Γpf . To this end, consider the closed loop system (2.6), (2.7), (8.1)

with µi = wp for PFP-B and µi = h(xi) for LPP. It needs to be shown that the sets

Γpfb and Γlp in (8.5) and (8.6) respectively are almost globally asymptotically stable.

Since the dot product is invariant to a change of frame and fi(yii) = R−1

i fi(yi), it holds


that fi(yii) · e1 = Rifi(y

ii) · Rie1 = fi(yi) · Rie1, similarly fi(y

ii) · e2 = fi(yi) · Rie2 and

µii · e1 = µi · Rie1. The control inputs in (8.1) represented with respect to the inertial

frame therefore satisfy,

u⋆i (yii, ϕi, µ

ii) = fi(yi) · Rie1 + βiω

⋆i (y

ii, ϕi, µ

ii) + µi ·Rie1,

ω⋆i (yii, ϕi, µ

ii) =

1

αi(fi(yi) · Rie2 + µi · Rie2) , i ∈ n.

(8.7)

Substituting (ui, ωi) = (u⋆i , ω⋆i ) in (8.7) into (7.7) and using the fact that (fi(yi) ·

Rie1)Rie1 + (fi(yi) · Rie2)Rie2 = fi(yi) and (µi · Rie1)Rie1 + (µi · Rie2)Rie2 = µi yields,

˙xi = fi(yi) + µi

θi =1

αi(fi(yi) · Rie2 + µi · Rie2) , i ∈ n.

(8.8)

Since the closed loop system in (8.8) is globally Lipschitz, it has no finite escape times.

The path following controller in (4.16) can be written as in (4.17), i.e.,

h(xi) = k0(c⋆(xi)− xi) + wp, (8.9)

where c⋆(xi) is the orthogonal projection of the point xi onto C(r0, p). Therefore, in the

case of LPP, µi = h(xi) = k0(c⋆(xi) − xi) + wp. In the case of PFP-B, we adopt the

convention that c⋆(xi) = xi such that h(xi) = k0(c⋆(xi)− xi)+wp = wp. Therefore, with

this convention, µi = h(xi) also in this case. Letting p⊥ be a unit vector perpendicular

to p, consider the new states x‖1 := x1 · p, x⊥1 := (x1 − c⋆(x1)) · p⊥ = (r0 − x1) · p⊥,

x := (x1i)i∈2 :n ∈ R2(n−1) and θ = (θi)i∈n ∈ Tn defined under the diffeomorphism

F : R2n × Tn → R× R× R

2(n−1) × Tn

F (x, θ) = (x‖1, x

⊥1 , x, θ)


with smooth inverse yielding

x1 = x‖1p− (x⊥1 − r0 · p⊥)p⊥

(xi)i∈2 :n = (x1i + (x‖1p− (x⊥1 − r0 · p⊥)p⊥))i∈2 :n

θ = (θi)i∈n.

The system equations in (8.8) can be expressed in terms of new coordinates as

˙x‖1 = (f1(y1) + µ1) · p

˙x⊥1 = (f1(y1) + µ1) · p⊥ − c⋆(x1) · p⊥

˙x1i = fi(yi)− f1(y1) + µi − µ1

θi =1

αi(fi(yi) ·Rie2 + µi · Rie2), i ∈ n.

(8.10)

Note that fi(yi) is strictly a function of x, Ri is strictly a function of θ and it can be

shown that

µi = −k0[(x1 − c⋆(x1)) · p⊥]p⊥ − k0x1i + k0(x1i · p)p+ wp

= −k0x⊥1 p⊥ − k0x1i + k0(x1i · p)p+ wp

which is strictly a function of x⊥1 and x. Moreover, for PFP-B, x⊥1 ≡ 0 and therefore

˙x⊥1 = 0. For LPP, since c⋆(x1) lies on the line perpendicular to p⊥ at all times, it follows

that c⋆(x1) · p⊥ = 0. Therefore, the right hand sides of (8.10) are independent of x‖1. The

sets Γpfb and Γlp expressed in new coordinates both can be written as

(x‖1, x⊥1 , x, θ) ∈ R× R× R2(n−1) × T

n : x⊥1 = 0, x = 0, θi = θp, i ∈ n,

where the angle of p in the inertial frame is denoted by θp. This set is also independent

of x‖1. Therefore, x

‖1 can be dropped leaving the remaining states as (x⊥1 , x, θ) ∈ R ×


R2(n−1) × Tn. The sets Γpfb and Γlp reduce to the point

K := (x⊥1 , x, θ) ∈ R× R2(n−1) × T

n : x⊥1 = 0, x = 0, θi = θp, i ∈ n.

For PFP-B, µi = µ1 = wp for all i ∈ 2 :n and system (8.10) reduces to

˙x⊥1 = 0

˙x1i = fi(yi)− f1(y1)

θi =1

αi(fi(yi) · Rie2 + wp · Rie2).

(8.11)

For LPP, since c⋆(xi)− xi is parallel to p⊥, it holds that µi = [k0(c

⋆(xi)− xi) · p⊥]p⊥+wp

for i ∈ n. Moreover, since c⋆(xi) lies on C(r0, p) for all i ∈ n, it follows that (c⋆(xi) −

c⋆(x1)) ·p⊥ = 0 for all i ∈ 2 :n. These two facts imply that µi−µ1 = [k0(c⋆(xi)− c⋆(x1)) ·

p⊥ − k0(xi − x1) · p⊥]p⊥ = (k0xi1 · p⊥)p⊥ in (8.10) for all i ∈ 2 :n. Therefore for LPP,

system (8.10) reduces to

˙x⊥1 = (f1(y1) + µ1) · p⊥

˙x1i · p = (fi(yi)− f1(y1)) · p

˙x1i · p⊥ = (fi(yi)− f1(y1)) · p⊥ + k0xi1 · p⊥

= (fi(yi, xi1)− f1(y1, 0)) · p⊥

θi =1

αi(fi(yi) ·Rie2 + µi ·Rie2), i ∈ n,

(8.12)

where the state x has been split into components parallel and perpendicular to p and

fi(yi, xi1) := fi(yi) + k0xi1 represents the consensus control law fi(yi) with an additional

edge added from unicycle i to unicycle 1. Therefore, both fi(yi) and fi(yi, xi1) are integra-

tor consensus control laws. In (8.12), fi(yi)·p =∑

j∈Niaij(x1j ·p−x1i ·p) is strictly a func-

tion of the elements of x in the direction of p and fi(yi) ·p⊥ =∑

j∈Niaij(x1j ·p⊥− x1i ·p⊥)

is strictly a function of the elements of x in the direction of p⊥. It follows from the


equations of motion for flocking and path following in (8.11), and (8.12) respectively and

the absence of finite escape times that

Γ3 := (x⊥1 , x, θ) ∈ R2 × R

2(n−1) × Tn : x = 0

is globally asymptotically stable in both cases. This implies boundedness of x and, in

turn, boundedness of fi(yi) for all i ∈ n. For PFP-B, x⊥1 ≡ 0 is bounded while for LPP,

˙x⊥1 satisfies

˙x⊥1 = f1(y1) · p⊥ + µ1 · p⊥ ≤ B − k0x⊥1 ,

where B = sup ‖f1(y1)‖ < ∞. This implies boundedness of x⊥1 and boundedness of the

states (x⊥1 , x, θ). On the set Γ3, the equations of motion of (x⊥1 , x, θ) in (8.11), (8.12)

both reduce to,

˙x⊥1 = µ1 · p⊥ = −k0x⊥1˙x1i = 0

θi =1

αi(wp · Rie2) =

w

αisin(θp − θi), i ∈ n.

(8.13)

In the case of flocking, x⊥1 ≡ 0 which implies ˙x⊥1 = 0. For system (8.13), x⊥1 → 0 as

t→ ∞ and the compact set

Γ2 := (x⊥1 , x, θ) ∈ Γ3 : x⊥1 = 0, i ∈ n,

diffeomorphic to Tn, is an embedded submanifold of R2 × R2(n−1) × Tn and is globally

asymptotically stable relative to Γ3. By Theorem 4.1.2, Γ2 is globally asymptotically

stable.

The point K can therefore be written as

K = (x⊥1 , x, θ) ∈ Γ2 : θi = θp, i ∈ n.


To complete the proof it needs to be shown that K is almost globally asymptotically

stable. In K, the control inputs in (8.7) satisfy u⋆i = w and ω⋆i = 0 for all i ∈ n as

desired. Notice that the rotational system in (8.13) is decoupled for each unicycle i ∈ n.

As a consequence of Proposition 4.3.4, the set Γ1 = A ∪ K, where A is the finite set of

isolated equilibria given by

A = (x⊥1 , x, θ) ∈ Γ2 : θi ∈ θp, θp + π, i ∈ n\K,

is globally attractive relative to Γ2. Moreover K is asymptotically stable relative to Γ2

whereas the equilibria in A are exponentially unstable relative to Γ2. In the set A, at

least one unicycle has a heading angle 180 offset from the beacon while the remaining

unicycle headings are aligned with the beacon. It follows from Theorem 4.1.3, setting

Γ2 = Γ2 and Γ1 = Γ1, that K is almost globally asymptotically stable. This concludes

the proof of PFP-B and LPP.

Now we present the proof for PFP assuming hierarchical digraphs. Consider the closed

loop system (2.6), (2.7), (8.1) with µ11 = e1 and µii = (w/|Ni|)

∑

j∈NiRije1, i ∈ 2 :n. The

leader’s control inputs are given by,

u⋆1(y11, ϕ1, µ

11) = w, ω⋆1(y

11, ϕ1, µ

11) = 0 (8.14)

and the control inputs of the follower unicycles are given by,

u⋆i (yii, ϕi, µ

ii) = fi(y

ii) · e1 + βiω

⋆i (y

ii, ϕi, µ

ii) +

w

|Ni|∑

j∈Ni

Rje1 ·Rie1,


ii) =

1

αi

(

fi(yii) · e2 +

w

|Ni|∑

j∈Ni

Rje1 · Rie2

)

,

(8.15)

where∑

j∈NiRje1 · Rie1 =

∑

j∈Nicos(θij) and

∑

j∈NiRje1 · Rie2 =

∑

j∈Nisin(θij). The

feedbacks (8.14), (8.15) for unicycle i are local and distributed since they depend only


on (yii, ϕi).

It needs to be shown that Γpf is almost globally asymptotically stable. The closed

loop equation for the leader is ˙x1 = wR1e1, θ1 = 0. This implies that the leader moves

in a straight line at the desired speed w in the direction of its initial heading. In this

case, since the heading vector p := R1e1 is constant, we can consider the angle θp := θ1

as a parameter rather than a state. The closed loop equations of motion for the follower

unicycles with respect to the inertial frame are given as before in (8.8).

Define the vertex set Lj ⊂ V to be the set of unicycles in layer j of the hierarchical

digraph and Lj := ∪ji=1Li. The equations of motion of unicycles associated with a node

set Lj are independent of the nodes outside of this set because, for any i ∈ Lj, the

feedbacks u⋆i (·) and ω⋆i (·) in (8.14), (8.15) depend only on states of unicycles within Ljitself. Therefore, the dynamics of unicycles in the isolated set Lj in terms of relative

translational and absolute rotational coordinates (x, θ)Lj := ((x1i)i∈Lj\1, (θi)i∈Lj) ∈ Xj ,

given by

˙x1i = fi +w

|Ni|∑

j∈Ni

Rje1 − wp, i ∈ Lj\1

θi = ωi, i ∈ Lj,(8.16)

define an autonomous dynamical system on the reduced state-space Xj . Correspondingly,

one can define the reduced formation flocking manifold for Lj as ΓLj = (x, θ)Lj ∈ Xj :

x1i = 0, θi = θp, i ∈ Lj. The new coordinates are obtained under a differeomorphism

F : R2n × Tn → R2 × R2(n−1) × Tn, F (x, θ) = (x1, x, θ) and the state x1 was dropped

because it does not appear in the dynamics in (8.16) or the reduced flocking manifold

ΓLj . Since fi(yi) = 0 and p = Rie1 for all i ∈ Lj on ΓLj , the control inputs satisfy u⋆i = w

and ω⋆i = 0 for all i ∈ Lj.

Letting m be the total number of layers in the hierarchical digraph, the set Γpf


expressed in (x, θ)Lm ∈ Xm = R2(n−1) × Tn coordinates becomes

Γpf := ΓLm =

(x, θ)Lm ∈ R2(n−1) × T

n : x1i = 0, θi = θp, i ∈ Lm

and u⋆i = w, ω⋆i = 0 for all i ∈ n. In the arguments that follow it will be shown that

Γpf is almost globally asymptotically stable, which solves the formation control problem.

An induction approach based on reduction will be employed. Consider first the isolated

set L2. The leader unicycle with heading vector p = R1e1 is the unique neighbor for all

follower unicycles in L2 and the control inputs in (8.15) reduce to

u⋆i (yii, ϕi, µ

ii) = fi(yi) ·Rie1 + βiω

⋆i (y

ii, ϕi, µ

ii) + wp · Rie1,


ii) =

1

αi(fi(yi) · Rie2 + wp ·Rie2) , i ∈ L2

(8.17)

which has the same form as formation flocking with a beacon and, by the same arguments

as before, the set ΓL2is almost globally asymptotically stable in the state space X2. Now

for k ∈ N, consider the isolated node set Lk−1 and assume the set ΓLk−1is almost globally

asymptotically stable in Xk−1 coordinates with domain of attraction Xk−1\Nk−1 where

Nk−1 is a set of Lebesgue measure zero. Next it will be shown that this implies ΓLk is

almost globally asymptotically stable in Xk coordinates. Inserting ΓLk−1into Xk, implies

Γ3 := (x, θ)Lk ∈ Xk : x1i = 0, θi = θp, i ∈ Lk−1

is almost globally asymptotically stable because the closed loop system in (8.8) is globally

Lipschitz and therefore has no finite escape times. The domain of attraction of Γ3 is given

by Xk\Nk, where

Nk := (x, θ)Lk ∈ Xk : (x, θ)Lk−1∈ Nk−1

is the insertion of Nk−1 into Xk and remains a set of Lebesgue measure zero. All unicycles

in the set Lk have neighbors strictly in Lk−1 and the equation of motion for (x)Lk in (8.16)


can be written as

˙x1i = (fi − f1) +w

|Ni|∑

j∈Ni

Rje1 − wp, i ∈ Lk,

where f1(y1) = 0 and the term (w/|Ni|)∑

j∈NiRje1−wp vanishes in Γ3 because Rje1 = p

for all j ∈ Lk−1. This term is bounded since it is a continuous function of the bounded

quantities (θ)Lk−1. Then, it must hold that (x)Lk is bounded because the origin is expo-

nentially stable for the nominal linear system

˙x1i = (fi − f1), i ∈ Lk.

Therefore, all closed loop solutions remain bounded. In the set Γ3, the equations of

motion for unicycles in Lk in (8.16) reduce to

˙x1i = fi(yi)− f1(y1),

θi =1

αi(fi(yi) · Rie2 + wp · Rie2) , i ∈ Lk,

(8.18)

which has the same form as (8.11) with the redundant state x‖1 ≡ 0 dropped. It follows

that the compact set

Γ2 := (x, θ)Lk ∈ Γ3 : x1i = 0, i ∈ Lk

is globally asymptotically stable relative to Γ3 and reduction in Theorem 4.1.2 implies

that Γ2 is globally asymptotically stable relative to Xk\Nk, a positively invariant set of

full measure. We conclude that Γ2 is almost globally asymptotically stable in Xk. In the

set Γ2, rotational dynamics satisfy θi = (w/αi) sin(θp − θi) for i ∈ Lk and the point

K := ΓLk = (x, θ)Lk ∈ Γ2 : θi = θp, i ∈ Lk


is almost globally asymptotically stable relative to Γ2 with an additional finite number

of exponentially unstable isolated equilibria

A = (x, θ)Lk ∈ Γ2 : θi ∈ θp, θp + π, i ∈ Lk\K.

The set Γ1 = A∪K is globally attractive with respect to Γ2 and applying Theorem 4.1.3,

setting Γ2 = Γ2 and Γ1 = Γ1, the point K is almost globally asymptotically stable relative

to Xk\Nk or, equivalently, almost globally asymptotically stable relative to Xk. It follows

by induction that Γpf is almost globally asymptotically stable. This concludes the proof

of PFP for hierarchical digraphs.


Consider the point zi := xi + γiRie2 offset γi units along unicycle i’s second body axis.

Then under the feedback transformation ui = ui + γiωi, the system (zi, θi) acts as a

unicycle with control inputs (ui, ωi), that is,

zi = uiRie1 − γiωiRie1 = uiRie1

θi = ωi.

Denote the collection of these offsets by z = (zi)i∈n.

In this proof, we convert the problems CFP and CPP in (x, θ) coordinates into

analogous problems for formations on a common circle in (z, θ) coordinates satisfying

the desired angular spacings θ1i = ρi(d, β1) for i ∈ 2 :n. To begin, for any radius

r > 0 let γi = βi − r, define zi := zi + rRie2 for all i ∈ n and denote z = (zi)i∈n.

In the sets Γ1 := (z, θ) ∈ R2n × Tn : zi = z1, θ1i = ρi(d, β1), i ∈ 2 :n and

Γ2 := (z, θ) ∈ Γ1 : z1 = c, the unicycles in (z, θ) coordinates lie on a common cir-

cle of radius r with angular spacing θ1i = ρi(d, β1) for all i ∈ 2 :n. In Γ2, the centre of


Figure 8.4: Desired formation of unicycles (x, θ) (shaded) is achieved when (z, θ) (notshaded) lie on a common circle of radius r with the desired spacing θ1i = ρi(d, β1) for alli ∈ n

the circle is c ∈ R2. First it will be shown that the control inputs in (8.2) almost globally

asymptotically stabilize Γ1 and Γ2 for CFP and CPP respectively. Then, it will be shown

that Γ1 and Γ2 are precisely the representations of Γcf and Γcp in new coordinates, where

(z, θ) represents the radial projection of the unicycle formation in original coordinates

(x, θ) onto a common circle C0 of radius r (see Figure 8.4).

Using the control inputs in (8.2) with ui = u⋆i (yii, ϕi, νi) and ωi = ω⋆i (y

ii, ϕi, νi), (ui, ωi)

become,

ui = ui − γiωi = ui + (βi − r)ω⋆i − γiω⋆i = ui

ωi =uir+Kϕ(νi)νi, i ∈ n.

(8.19)

Using the fact that zi = zi + rRie2 = xi + γiRie2 + rRie2 = xi + βiRie2 = xi, it follows

that νi = −(∑

j∈Nizij) · Rie1 for CFP and νi = −(c − zi) · Rie1 for CPP. Moreover

ui = rw + kgi((θij − ρij(d, β1))j∈Ni, 1). From Theorem 8.2.3, there exist k⋆, K⋆ > 0 such

that for all K ∈ (0, K⋆), k ∈ (0, k⋆), the sets Γ1 and Γ2, in which all unicycles (z, θ)

lie on a common circle of radius r with desired angular spacings, are almost globally

asymptotically stable for CFP and CPP respectively.

Using the fact that zi = xi for all i ∈ n, the sets Γ1 and Γ2 can be written in (x, θ)

coordinates as (x, θ) ∈ R2n×Tn : x1i = 0, θij = ρi(d, β1), i ∈ n and (x, θ) ∈ R2n×Tn :


x1i = 0, θij = ρi(d, β1), x1 = c, i ∈ n respectively which correspond to Γcf and Γcp

respectively. Therefore, Γcf and Γcp are almost globally asymptotically stable for CFP

and CPP respectively. This concludes the proof for CFP and CPP.


This proof makes use of Theorem V.5. in (El-Hawwary and Maggiore, 2013a) where

ui = rw + k∑

j∈Nisin(θij − ρij(d, β1)), a Kuramoto consensus controller, has been re-

placed by an almost global rotational integrator consensus controller ui = rw+kgi((θij−

ρij(d, β1))j∈Ni, 1). Substituting (ui, ωi) = (u⋆i (·), ω⋆i (·)) in (8.3) into (7.7) with αi = 0,

βi = r for all i ∈ n yields,

˙xi = −rKϕ(νi)νiRie1

θi =uir+Kϕ(νi)νi, i ∈ n.

(8.20)

The equations on the right hand side of (8.20) are bounded and therefore the closed

loop system has no finite escape times. In coordinates relative to unicycle 1, (x, θ) where

x := (x1i)i∈2 :n ∈ R2(n−1) and θ := (θ1i)i∈2 :n ∈ Tn−1, define the sets

Γ2 = (x, θ) ∈ R2(n−1) × T

n−1 : x = 0

K = (x, θ) ∈ Γ2 : θ1i = ρi(d, β1), i ∈ 2 :n,

where K corresponds to Γcf in relative coordinates and is a point. Using the same

arguments as in Proposition V.2. in (El-Hawwary and Maggiore, 2013a), there exist

K⋆ ∈ (0, (w/2)), k⋆ > 0 such that for all K ∈ (0, K⋆) and all k ∈ (0, k⋆) the set

Γ2, diffeomorphic to Tn−1 and an embedded submanifold of R2(n−1) × Tn−1, is globally

asymptotically stable relative to R2(n−1) × Tn−1. This implies boundedness of x, while θ

is bounded because it lies on a bounded set. In the set Γ2, νi = 0 for all i ∈ n and the


equations of motion can be written as

˙x1i = 0

θ1i = (k/r)gi((θij − ρij(d, β1))j∈Ni, 1)− (k/r)g1((θ1j − ρ1j(d, β1))j∈N1, 1).

(8.21)

By Theorem 4.3.2, system (8.21) has a globally attractive set of isolated equilibria Γ1 =

A ∪ K relative to Γ2 in which the point K is asymptotically stable and the equilibria

in A are exponentially unstable. Then by Theorem 4.1.3, setting Γ2 = Γ2 and Γ1 =

Γ1, K is almost globally asymptotically stable. Therefore in original coordinates Γcf

corresponding to a formation flocking on a common circle of radius r, is almost globally


Replace νi with νi = −(c − xi) · Rie1 and consider the system dynamics in (8.20).

Define the Lyapunov function V =∑n

i=1 ‖c− xi‖2 with derivative given by

V =n∑

i=1

rKϕ(νi)νi(c− xi) · Rie1

=

n∑

i=1

−rKϕ(νi)‖νi‖2 ≤ 0.

Since

θi =uir+Kϕ(νi)νi,

we have

θi ≥infθ∈Tn ui(θ)

r−K.

There exists k⋆ > 0 such that choosing k ∈ (0, k⋆) implies infθ∈Tn ui > rw/2, supθ∈Tn ui <

∞ for all i ∈ n and K < w/2. Then there exist ζ1, ζ2 > 0 such that 0 < ζ1 < θi < ζ2 for

all i ∈ n, for all time and initial conditions. Stability of Γ2 := (x, θ) ∈ R2n × Tn : xi =

c, i ∈ n holds from the fact that Γ2 corresponds to the minimum of V . Moreover, the

level sets of V are compact in (x, θ) coordinates and since V ≤ 0, the Krasovskii-LaSalle

invariance principle implies that νi → 0 as t → ∞. Using the fact that θi is bounded


away from zero and using ideas from the proof of Proposition V.2. in (El-Hawwary and

Maggiore, 2013a), (xi − c) → 0 as t → ∞. Together with stability, this implies that Γ2

is globally asymptotically stable and, in turn, boundedness of the states (x, θ).

In the set Γ2, νi = 0 for all i ∈ n, the equation of motion for θi reduces to θi =

w + (k/r)gi((θij − ρij(d, β1))j∈Ni, 1), and the set Γcp is almost globally asymptotically

stable relative to Γ2. It can be shown using Theorem 4.1.3, as in the case of Γcf , that

Γcp is almost globally asymptotically stable.

Chapter 9

General Formation Path Following

In this chapter, we present a solution to the general formation path following control

problem (GPP) introduced in Section 3.3.4. The feedbacks will be constructed by com-

bining two control primitives discussed in Section 4.3: an integrator consensus controller

fi((xij)j∈Ni) in (4.10) and a path following controller for single integrators h(x) in (4.15).

In particular, h(x) in this chapter will be chosen as a single integrator path following

controller for smooth Jordan curves.

9.1 Solution to the General Formation Path Follow-

ing Problem (GPP)

In this section we present the solution to GPP. As with the other formation control

problems, we attach an offset vector δi(θi, φi) of fixed length to the body frame of each

unicycle i ∈ n. In particular, we let the auxiliary state −φi in (3.10) be the angle of

δi relative to the heading angle θi of unicycle i and let ‖δi‖ = ‖di‖ as illustrated in

Figure 9.1. Define the endpoint of the offset vector by xi(xi, θi, φi) := xi + δi(θi, φi) and

let x := (xi)i∈n. The endpoint xi(xi, θi, φi) will act as a local estimate of the formation

origin for unicycle i. Also as in the other formation control problems, we decompose

195

Chapter 9. General Formation Path Following 196

the vector δi into a component αi along the heading axis and βi along the perpendicular

axis of unicycle i, i.e., δi := αiRie1 + βiRie2. The offsets αi(φi), βi(φi) are given by

αi(φi) := ‖δi‖ cosφi and βi(φi) := −‖δi‖ sinφi. Unlike in the other formation control

problems, the components αi(φi) and βi(φi) will not be constant as they depend on the

auxiliary state φi which, in turn, depends on the curvature of the path to be followed

which is not constant in general. Define the angle σi := θi−φi as illustrated in Figure 9.1

and denote σ := (σi)i∈n. The angle σi represents the angle of the offset vector δi in inertial

frame.

Our first goal is to make the estimates xi achieve consensus and converge to C so

that all agents possess knowledge of a common formation origin. This configuration

corresponds to the set

Λ =

(x, θ, φ) ∈ R2n × T

n × Tn : x1i = 0, x1 ∈ C, i ∈ 2 :n

.

In the set Λ, the quantity τi := π(xi) is well-defined for all i ∈ n because xi ∈ C

for all i ∈ n, and the endpoint x1 coincides with the origin of a Frenet-Serret frame

(o(τ1), r(τ1), s(τ1)). Relative to this set, we will stabilize the set

Γ′gp :=

(x, θ, φ) ∈ Λ : R0(τ1)−1(xi − o(τ1)) = di, φi = φi(xi, σi), i ∈ n

⊂ Γgp, (9.1)

where Γgp is defined in (3.11). In the set Γ′gp, the formation is achieved at τ1. In the

set Λ, the condition ‖R0(τ1)−1(xi − o(τ1))‖ = ‖xi − o(τ1)‖ = ‖di‖ for Γ′

gp is satisfied

for all i ∈ n. However, the formation is not yet achieved as illustrated in Figure 9.2

where unicycle i can lie anywhere on a circle of radius ‖di‖ about o(τ1) but the condition

R0(τ1)−1(xi−o(τ1)) = −R0(τ1)

−1δi = di for Γ′gp is not satisfied. Or what is the same, the

angle between −r(τ1) and −δi is not necessarily equal to ψi. This requirement can be

written as an additional constraint on Λ as illustrated in Figure 9.3 in which the angle

σi must equal σi := θr(τi) − ψi where θr(τi) is the angle of r(τi) in inertial frame. The


manifold Γ′gp can therefore be expressed alternatively as follows,

Γ′gp := (x, θ, φ) ∈ Λ : σi = θr(τi)− ψi, φi = φi(xi, σi), i ∈ n . (9.2)

Figure 9.1: Illustration of the angle φi and angle σi = θi − φi.

Figure 9.2: Illustration of the set Λ in which unicycle i can lie anywhere on the dashedcircle. The desired position of unicycle i is marked by ⋆.

Let fi(·) be a single integrator consensus controller in (4.10) and h(·) be a path

following controller for C in (4.15) such that r(x) := r π(x) defines the direction of


Figure 9.3: Illustration of angle condition σi = θr(τi)− ψi.

motion for x ∈ C. The angle of the vector field h at the point x ∈ R2 is denoted

θh(x) := atan2(h(x) · e2, h(x) · e1).

The partial derivatives of θh(x) relative to the components of x are well defined as long

as ‖h(x)‖ 6= 0. This is never the case in a neighborhood of C since, by assumption,

‖h(x)‖ = w 6= 0 for all x ∈ C. We can therefore define functions

θ′h(x) :=dθh(x)

dx· h(x)

θ′′h(x) :=dθ′h(x)

dx· h(x).

(9.3)

Letting

ω⋆i (yii, xi) :=

1

αi

(

kfi(yii) · e2 + h(xi)

i · e2)

, (9.4)

choose the feedback law for unicycle i as,

u⋆i (yii, xi) = kfi(y

ii) · e1 + βiω

⋆i (y

ii, xi) + h(xi)

i · e1,

ω⋆i (yii, xi, σi, φi) = ω⋆i ,piv(y

ii, xi, σi, φi) + ω⋆i (y

ii, xi),

ω⋆i ,piv(yii, xi, σi, φi) = −k0 sin(φi − φi(xi, σi)) + φ′

i(yii, xi, σi),

(9.5)


where k, k0 > 0 and the functions φi(xi, σi) and φ′i(y

ii, xi, σi) are defined next. Let

ζi(xi, σi) :=‖δi‖w

−θ′h(xi) + γ sin ξi(xi, σi)

cos(σi − θh(xi))+ tanψi,

ξi(xi, σi) := σi − θh(xi) + ψi,

where γ > 0. Define the functions φi(xi, σi) and φ′i(y

ii, xi, σi) in (9.5) respectively as

φi(xi, σi) =

tan−1 ζi(xi, σi), if di · e1 < 0

tan−1 ζi(xi, σi) + π, if di · e1 > 0,

φ′i(y

ii, xi, σi) =

1

1 + ζi(xi, σi)2‖δi‖w

(−θ′′h(xi) + γ(ω⋆i (yii, xi)− θ′h(xi)) cos ξi(xi, σi)

cos(σi − θh(xi))+

−θ′h(xi) + γ sin ξi(xi, σi)

cos2(σi − θh(xi))sin(σi − θh(xi))(ω

⋆i (y

ii, xi)− θ′h(xi))

)

.

(9.6)

There are two possibilities for φi(xi, σi) in (9.6), depending on the sign of di · e1. The

first entails (xi, σi)φi7−→ (−π/2, π/2), the second (xi, σi)

φi7−→ (π/2, 3π/2). This will ensure

that the unicycles follow the curve moving in the forward direction. The terms in (9.5)

containing fi(yii) have the objective of achieving consensus on the estimates (xi)i∈n of the

formation origin while those containing h(xi)i have the objective of making xi converge

to and follow the path C at the desired speed and direction. Finally, the control input

ω⋆i ,piv(yii, xi, σi, φi) is chosen to achieve the angle condition σi = θr(τi)− ψi. The control

inputs in (9.5), (9.6) are functions of

• relative positions to neighboring unicycles yii = (xiij)j∈Ni,

• the integrator path following controller at xi represented in body frame, h(xi)i

• the angle of δi relative to the angle of h(xi), σi − θh(xi),

• the quantities θ′h(xi) and θ′′h(xi).


The quantity yii can be written as

xiij =R−1i

xj +Rj

αj

βj

− xi −Ri

αi

βi

=R−1i

xij + ‖δj‖RiR

−1i Rj

cosφj

− sinφj

− ‖δi‖Ri

cosφi

− sinφi

=

xiij + ‖δj‖Ri

j

cosφj

− sin φj

− ‖δi‖

cosφi

− sin φi

.

(9.7)

To compute (9.7), unicycle i requires measurement of quantities xiij , Rij , ‖δj‖ and

Rij(cosφj,− sinφj) for all j ∈ Ni. We can alternatively write,

Rij

cosφj

− sinφj

= Ri

j

δjj‖δj‖

=δij

‖δj‖.

To obtain the quantity Rij(cosφj , sinφj), unicycle j must either communicate the angle

φj or have a device on board that indicates the vector δjj visually in its body frame so that

unicycle i can measure δij using cameras. For example, using a string of lights around

the unicycle body. The main theorem for the formation path following problem is given

in Theorem 9.1.1 whose proof is given in Section 9.3.

Theorem 9.1.1. Consider system (2.6), (2.7), (3.10) with sensor digraph G containing a

globally reachable node. Let fi(·) be a single integrator consensus controller in (4.10) with

any parameters aij > 0 for i ∈ n, j ∈ Ni, and h(·) be a path following controller in (4.15)

for the smooth Jordan curve C. For any (d, C, w) ∈ GP, there exist gains k⋆, γ⋆ > 0 such

that choosing k > k⋆ and γ > γ⋆, the control inputs in (9.5), (9.6) asymptotically stabilize

the manifold Γ′gp ⊂ Γgp, thus solving the formation path following control problem.

Remark 9.1.2. The control input for ω⋆i (x, σi, φi) has a singularity when αi = ‖δi‖ cosφi =


0 and the signals φi,φ′i in (9.6) have singularities when cos(σi − θh(xi)) = 0. It will be

shown in the proof of Theorem 9.1.1 that in a neighborhood of Γ′gp neither of these singu-

larities occur and therefore do not affect the asymptotic stability result. To remove the

presence of singularities in the control inputs, one can employ bump functions without

affecting asymptotic stability.

Remark 9.1.3. If one sets φ′i(x, σi) = 0 in (9.5), the result for asymptotic stability in

Theorem 9.1.1 becomes a result for practical stability where instead of solutions converg-

ing exactly to Γ′gp they converge to a neighborhood of Γ′

gp that can be made arbitrarily

small by choosing the control gain k0 in (9.5) sufficiently large. This has the advantage

of significantly simplifying the control input ω⋆i ,piv(yii, xi, σi, φi) in (9.5) which no longer

depends on θ′′h(xi) which may be a large expression. For example, for h(x) defined in (9.9)

for a Van der Pol oscillator, one can compute θ′h(x) as,

θ′h(x) = −c0wµy3z + y2 + 2µyz3 − µyz + z2

((y + µz(y2 − 1))2 + z2)3/2

with y = c0x · e1 and z = c0x · e2. Whereas the expression for θ′′h(x) is roughly five times

this length.


In this section, we present simulation results for formation path following using the

feedbacks presented in (9.5). We consider a group of five unicycles with the directed

sensing graph shown in Figure 9.4 with unitary weights. Each edge is unidirectional

except for the bidirectional edge between unicycles 2 and 4. The desired formation is

specified by d1 = (15, 0), d2 = (5,−8), d3 = (5, 8), d4 = (−10,−3) and d5 = (−10, 3)

about o(t1) as illustrated in Figure 9.5.

We have chosen the path C to be the stable limit cycle of a Van der Pol oscillator


1

2

3

5

4

Figure 9.4: Digraph G under consideration in the simulation results.

Figure 9.5: Formation specified by the offset vectors d1 = (15, 0), d2 = (5,−8), d3 =(5, 8), d4 = (−10,−3) and d5 = (−10, 3).

with equations of motion given by

y = c0z

z = µ(1− (c0y)2)c0z − c0y,

(9.8)

for x = (y, z) ∈ R2, µ = 1.5 and c0 = 0.04. This set is illustrated in Figure 9.6(a) by

the solid line. Correspondingly, we choose the single integrator path following controller

h(x) in (4.15) as,

h(x) =w(c0z, µ(1− (c0y)

2)c0z − c0y)

‖(c0z, µ(1− (c0y)2)c0z − c0y)‖(9.9)

where the vector field in (9.8) is normalized and scaled to the desired speed w. We choose


the initial unicycle positions as x1(0) = (−25,−20)m, x2(0) = (−15,−15)m, x3(0) =

(−27,−25)m, x4(0) = (−20,−20)m, and x5(0) = (−18,−38)m which are indicated by

the circles in Figure 9.6(b). For the control inputs in (9.5), (9.6) choose w = 1, k = 10,

γ = 1. For unicycles i ∈ 1, 2, 3, for which di · e1 > 0, let φi(xi, σi) = tan−1 ζi + π while

for unicycles i ∈ 4, 5, for which di · e1 < 0, let φi(xi, σi) = tan−1 ζi. The corresponding

simulation results are shown in Figure 9.6. Figure 9.6(b) illustrates the individual paths

traversed by the five unicycles as the formation converges to and moves around C. Clearly

the paths followed by each unicycle are significantly different from one another in order

to maintain the desired formation. One advantage of this approach is that unicycles do

not need to compute their individual paths, but rather, only require knowledge of the

common curve C. The path traversed by x1, unicycle 1’s estimate of the formation origin,

is illustrated by the dashed line in Figure 9.6(a) which converges to the curve C as desired

and moves along it with the speed w.

-50 0 50

x (m)

-80

-60

-40

-20

0

20

40

60

80

y (m

)

(b)

-60 -40 -20 0 20 40 60

x (m)

-80

-60

-40

-20

0

20

40

60

80

y (m

)

(a)

Figure 9.6: Simulation results: (a) illustrates the curve C corresponding to the stable limitcycle of the Van der Pol oscillator with µ = 1.5, c0 = 0.04 and shows the time evolutionof x1 illustrated by the dashed line, (b) illustrates the individual paths traversed by thefive unicycles as the formation moves around C. Initial positions are indicated with and positions at the end of the simulation are indicated with ×.



In this section we will drop the arguments for a number of functions for simplicity of

notation: φi(xi, σi) → φi, φ′i(y

ii, xi, σi) → φ′

i, ζi(xi, σi) → ζi, ξi(xi, σi) → ξi. Before

presenting the proof for Theorem 9.1.1, the formation path following problem will be

reformulated in terms of new states (x, σ, φ) related to the original coordinates under the

diffeomorphism F : R2n × Tn × Tn → R2n × Tn × Tn given by

F (x, θ, φ) = ((xi + δ(θi, φi))i∈n, θ − φ, φ) = (x, σ, φ)

and with inverse

F−1(x, σ, φ) = ((xi − δ(σi + φi, φi))i∈n, σ + φ, φ) = (x, θ, φ).

The time derivative of σi is given by σi = θi−φi which, after setting ωi = ω⋆i ,piv(yii, xi, σi, φi)+

ω⋆i (yii, xi) and ωi ,piv = ω⋆i ,piv(y

ii, xi, σi, φi), becomes σi = ω⋆i (y

ii, xi). The time derivative of

xi = xi + αiRie1 + βiRie2 is given by

˙xi = uiRie1 + αiRi

0 −θiθi 0

e1 + βiRi

0 −θiθi 0

e2 + αiRie1 + βiRie2

= uiRie1 + αiθiRie2 − βiθiRie1 + αiRie1 + βiRie2.

Substituting θi = ω⋆i ,piv(yii, xi, σi, φi) + ω⋆i (y

ii, xi) yields

˙xi = u⋆iRie1 + αiω⋆i ,pivRie2 − βiω

⋆i ,pivRie1 + αiω

⋆iRie2 − βiω

⋆iRie1 + αiRie1 + βiRie2.


Substituting the quantities αi = ‖δi‖ cosφi, βi = −‖δi‖ sinφi, α = −‖δi‖ sinφiω⋆i ,piv and

β = −‖δi‖ cosφiω⋆i ,piv , one computes

˙xi = u⋆iRie1 + αiω⋆iRie2 − βiω

⋆iRie1,

which is independent of ω⋆i ,piv and the closed-loop dynamics for the (x, σ, φ) system are

given by

˙xi = (u⋆i (yii, xi)− βiω

⋆i (y

ii, xi))Rie1 + αiω

⋆i (y

ii, xi)Rie2,

σi = ω⋆i (yii, xi),

φi = ω⋆i ,piv(yii, xi, σi, φi).

(9.10)

Notice that the control inputs presented in (9.5) are defined precisely in terms of (yii, xi, σ, φ)

and so the equations of motion in (9.10) constitute a dynamical system.

The manifold Γ′gp represented in (x, σ, φ) coordinates becomes

Γgp :=

(x, σ, φ) ∈ R2n × T

n × Tn : x1i = 0, x1 ∈ C,

σi = θr(π(xi))− ψi, φi = φi(xi, σi), i ∈ n ,(9.11)

where φi : R2 × S1 → (−π/2, π/2) ∪ (π/2, 3π/2) is a desired angle of φi in (9.6). It

needs to be shown that Γgp is asymptotically stable for system (9.10). Moreover, the

state x1 should move at the desired speed w > 0 along C in the direction of r(τ1).


Consider the closed loop system (2.6), (2.7), (3.10), (9.5), (9.6). It needs to be shown

that the set Γgp is asymptotically stable. Since the dot product is invariant to a change

of frame and fi(yii) = R−1

i fi(yi), it holds that fi(yii) · e1 = Rifi(y

ii) · Rie1 = fi(yi) · Rie1,

similarly fi(yii) ·e2 = fi(yi) ·Rie2, h(xi)

i ·e1 = h(xi) ·Rie1 and h(xi)i ·e2 = h(xi) ·Rie2. The

control inputs (u⋆i , ω⋆i ) in (9.5) represented with respect to the inertial frame therefore


satisfy,

u⋆i = kfi(yi) ·Rie1 + βiω⋆i + h(xi) ·Rie1,

ω⋆i =1

αi(kfi(yi) · Rie2 + h(xi) · Rie2) , i ∈ n

(9.12)

where k > 0, which will be chosen later, scales the primitive fi(yi) in the controller. Sub-

stituting (9.12) into (9.10) and using the fact that (fi(yi)·Rie1)Rie1+(fi(yi)·Rie2)Rie2 =

fi(yi) and (h(xi) · Rie1)Rie1 + (h(xi) ·Rie2)Rie2 = h(xi) yields

˙xi = kfi(yi) + h(xi)

σi =1

αi(kfi(yi) · Rie2 + h(xi) · Rie2)

φi = ω⋆i ,piv , i ∈ n.

(9.13)

The system equations in (9.13) can be expressed in terms of new states: x1 ∈ R2,

x := (x1i)i∈2 :n ∈ R2(n−1), σ := (σi)i∈n ∈ Tn and φ := (φi)i∈n ∈ Tn as

˙x1 = kf1(y1) + h(x1)

˙x1i = kfi(yi)− kf1(y1) + h(xi)− h(x1)

(9.14)

σi =1

αi(kfi(yi) · Rie2 + h(xi) · Rie2)

φi = ω⋆i ,piv ,

(9.15)

where xi = x1i + x1. The set Γgp expressed in (x1, x, σ, φ) coordinates becomes

Γgp :=

(x1, x, σ, φ) ∈ R× R2(n−1) × T

n × Tn : x1i = 0,

x1 ∈ C, σi = θr(π(x1))− ψi, φi = φi(x1i + x1, σi), i ∈ n .

It needs to be shown that Γgp is asymptotically stable which is sufficient to solve the

formation path following control problem. The x dynamics in (9.14) can be written as

˙x = −kLx+ h(x1, x), (9.16)


where h(x1, x) = (h(xi) − h(x1))i∈2 :n acts as a vanishing perturbation as x → 0 and L

is the positive definite reduced Laplacian matrix of L where L(i, j) = −aij for i 6= j

and L(i, i) =∑

j∈Niaij . Moreover, using the fact that h(xi) is globally Lipschitz with

Lipschitz constant c one can write

‖h(x1, x)‖ = ‖(h(xi)− h(x1))i∈2 :n‖

≤n∑

i=2

‖h(xi)− h(x1)‖ ≤n∑

i=2

c‖x1i‖ ≤ c(n− 1)‖x‖.(9.17)

By (Khalil, 2002, Lemma 9.1), there exists k⋆ > 0 such that choosing k > k⋆ implies

that the set

Γ3 := (x1, x, σ, φ) ∈ R× R2(n−1) × T

n × Tn : x = 0

is globally exponentially stable provided that x1 has no finite escape times. System (9.14),

(9.15) is globally Lipschitz (since fi(·) and h(·) are globally Lipschitz) and therefore has

no finite escape times.

In the set Γ3, it holds that h(x1, x) = 0, fi(yi) = 0 for all i ∈ n and the equations

in (9.14) reduce to,

˙x1 = h(x1),

˙x1i = 0,

σi =1

αi(h(xi) · Rie2),

φi = ω⋆i ,piv .

(9.18)

Two important things take place on the set Γ3 (i) xi → C and (ii) φ → φi. Property (i)

follows directly from the fact that on Γ3, ˙xi = h(xi) for all i ∈ n, where h(xi) is a single

integrator path following controller. To show property (ii), consider the derivatives of


θh(xi) and θ′h(xi) which on Γ3 satisfy,

θh(xi) =∂θh(xi)

∂xi· h(xi) = θ′h(xi)

θh(xi) = θ′h(xi) =∂θ′h(xi)

∂xi· h(xi) = θ′′h(xi).

(9.19)

Moreover, recalling that

ζi =‖δi‖w

−θ′h(xi) + γ sin ξicos(σi − θh(xi))

+ tanψi,

in Γ3 the function φi in (9.6) has the time derivative φi given by

φi =1

1 + ζ2iζi

=1

1 + ζ2i

‖δi‖w

(

−θ′h(xi) + γ cos ξiξicos(σi − θh(xi))

− −θ′h(xi) + γ sin ξicos2(σi − θh(xi))

d

dtcos(σi − θh(xi))

)

=1

1 + ζ2i

‖δi‖w

(−θ′′h(xi) + γ(ω⋆i − θ′h(xi)) cos ξicos(σi − θh(xi))

+−θ′h(xi) + γ sin ξicos2(σi − θh(xi))

sin(σi − θh(xi))(ω⋆i − θ′h(xi))

)

=φ′i,

(9.20)

where φ′i was defined in (9.6). We have used the fact that the time derivative of ψi is zero

and the derivative of ξi is given by ξi = σi − θh(xi) + ψi = ω⋆i − θ′h(xi). Then, from (9.5)

the equation of motion for φi is given by

φi = −k0 sin(φi − φi) + φi,

for which the set φ ∈ S1 : φ = φ is asymptotically stable. Therefore, the compact set

Γ2 = (x1, x, σ, φ) ∈ Γ3 : x1 ∈ C, φi = φi, i ∈ n


is asymptotically stable relative to Γ3. It holds from Theorem 4.1.2 that the set Γ2 is

asymptotically stable. In the set Γ2, xi = x1 ∈ C for all i ∈ n and x1, traverses C

with speed w. It follows that π(xi) is well-defined and h(xi) = wr(xi) = wr(π(xi))

by property A2 for integrator path following controllers in (4.15). Therefore θh(xi) =

θh(x1) ≡ θr(π(x1)) =: θr (correspondingly θh(xi) = θr) for all i ∈ n. Therefore ξi =

σi − θh(xi) + ψi = σi − θr + ψi,

φi = φi =

tan−1(

‖δi‖w

−θr+γ sin ξicos(σi−θr)

+ tanψi

)

, if di · e1 < 0

tan−1(

‖δi‖w

−θr+γ sin ξicos(σi−θr)

+ tanψi

)

+ π, if di · e1 > 0

(9.21)

and ω⋆i is given by

ω⋆i =1

αi(h(xi) · Rie2) =

‖h(xi)‖αi

sin(θh(xi)− θi) =w

αisin(θr − θi).

Now we write out the dynamics of ξi = σi− θr +ψi in Γ2 which we’d like to converge

to zero. Taking the derivative of ξi yields

ξi = σi − θr = ω⋆i − θr =w

αisin(θr − θi)− θr

= − w

‖δi‖ cosφisin((θi − φi − θr + ψi) + φi − ψi)− θr

= − w

‖δi‖ cosφisin(ξi + φi − ψi)− θr

= − w

‖δi‖ cosφi

sin(ξi + φi − ψi)− θr,

where we used the fact that the time derivative of ψi is zero, φi = φi and αi = ‖δi‖ cosφi =

‖δi‖ cosφi. Using the identity sin(ξi + φi − ψi) = sin ξi cos(φi − ψi) + cos ξi sin(φi − ψi)

implies,

ξi = − w

‖δi‖ cosφi

[sin ξi cos(φi − ψi) + cos ξi sin(φi − ψi)]− θr

= − w

‖δi‖

[

sin ξicos(φi − ψi)

cosφi

+ cos ξisin(φi − ψi)

cosφi

]

− θr.

(9.22)


The following identities can be derived using standard trigonometric identities,

cos(φi − ψi)

cosφi

=cosφi cosψi + sinφi sinψi

cosφi

= cosψi + tanφi sinψi

sin(φi − ψi)

cosφi

=sinφi cosψi − cosφi sinψi

cosφi

= tanφi cosψi − sinψi,

which after substitution into (9.22) yields

ξi = − w

‖δi‖[sin ξi cosψi + tanφi(sin ξi sinψi + cos ξi cosψi)− cos ξi sinψi]− θr

= − w

‖δi‖[sin ξi cosψi + tanφi cos(σi − θr)− cos ξi sinψi]− θr,

(9.23)

where we have used the identity sin ξi sinψi+cos ξi cosψi = cos(σi− θr). Substituting φi

from (9.21) into (9.23) yields

ξi = − w

‖δi‖[sin ξi cosψi + tanψi cos(σi − θr)− cos ξi sinψi] + θr − γ sin ξi − θr

= − w

‖δi‖

[

sin ξi cosψi +sinψicosψi

(sin ξi sinψi + cos ξi cosψi)− cos ξi sinψi

]

− γ sin ξi

= − w

‖δi‖

[

sin ξi cosψi +sin2 ψicosψi

sin ξi + cos ξi sinψi − cos ξi sinψi

]

− γ sin ξi

= −(

γ +w

‖δi‖cosψi +

w

‖δi‖sin2 ψicosψi

)

sin ξi

(9.24)

and choosing

γ > maxi∈n

(

w

‖δi‖

∣

∣

∣

∣

cosψi +sin2 ψicosψi

∣

∣

∣

∣

)

=: γ⋆

implies that the set

Γ1 := Γgp

is asymptotically stable relative to Γ2. Figure 9.7 illustrates the sets Γ1, Γ2 and Γ3 just

discussed. Based on reduction in Theorem 4.1.2, Γgp is asymptotically stable as long as

two assumptions hold in a neighborhood of Γgp: (i) cos(σi − θh(xi)) 6= 0 for all i ∈ n so

that φi and φ′i in (9.6) are well defined and (ii) αi = ‖δi‖ cosφi is bounded away from


zero for all i ∈ n (equivalently φi is bounded away from ±π/2) ensuring that ω⋆i in (9.4)

is well defined.

Figure 9.7: Conceptual illustration of the reduction sets Γ1, Γ2 and Γ3. The set Γ3 isasymptotically stable, the compact set Γ2 is asymptotically stable relative to Γ3 andthe point Γ1 is asymptotically stable relative to Γ2. Reduction implies the set Γ1 isasymptotically stable as illustrated by the solution starting at χ0 off the set Γ3.

Starting with (i), since ψi 6= ±π/2 there exists ρ > 0 such that in the set Γgp,

cos(σi − θh(xi)) = cos(σi − θr) = cosψi is bounded away from 0 by ρ. Since Γgp is

compact, there exists an ǫ1 > 0 neighborhood Bǫ1(Γgp) in which cos(σi − θh(xi)) is

bounded away from 0 by ρ/2, i.e., | cos(σi − θh(xi))| > ρ/2.

To show (ii), in the set Γgp, φi in (9.21) reduces to

φi = tan−1

[

−‖δi‖w

θrcosψi

+ tanψi

]

∈ (−π/2, π/2).

Since ψi 6= ±π/2 and θr is bounded (since C is compact), there exists ρ > 0 such that

φi is bounded away from ±π/2 by ρ in Γgp, i.e., |φi ± π/2| > ρ. Since Γgp is compact

there exists an ǫ2 < ρ/4 neighborhood of Γgp, Bǫ2(Γgp), in which φi is bounded away

from ±π/2 by ρ/2, i.e., |φi ± π/2| > ρ/2. In addition, in Bǫ2(Γgp), |φi−φi| < ρ/4. This

implies that |φi ± π/2| > ρ/4 in Bǫ2(Γgp).

Therefore choosing ǫ = min(ǫ1, ǫ2), conditions (i) and (ii) hold in the set Bǫ(Γgp).

Therefore Γgp is asymptotically stable.

Chapter 10

Unicycle Formation Simulation

Trials

In this chapter, we present extensive simulation trials to study the effectiveness of our

control solution presented in Chapter 7 for formation control of unicycles under different

realistic scenarios not captured by the main theoretical result in Theorem 7.1.1. The

simulations will study the following items:

• performance in the presence of state dependent sensor graphs in which each unicy-

cle’s neighbors are those that lie within a given radius of itself

• performance for directed sensing graphs as opposed to undirected sensing graphs

• performance when the high gain conditions on α and k are ignored

• robustness of the approach to unmodelled effects including sensor noise, input noise,

sampling, and saturated inputs

• extension of the control solution for kinematic unicycles to the dynamic unicycle

model in (2.10)

We will not present extensive simulations for formations with final collective motion

discussed in Chapter 8 and Chapter 9. However, we predict similar outcomes since the

212

Chapter 10. Unicycle Formation Simulation Trials 213

solutions in these chapters are based on a similar methodology. For the simulation results

in this chapter, unless stated otherwise, we consider the five-unicycle formation illustrated

in Figure 7.6 with initial angles (θi)i∈n chosen randomly in Tn and the positions (xi)i∈n

chosen randomly in a 60× 60 meter grid.

10.1 Performance Measures

In this section we introduce a number of performance measures that will be used to

evaluate and compare the effectiveness of the control solution in Chapter 7 under various

scenarios considered in the simulation results. These include a formation measure, drift

measure, collision measure and input measures and are described below.

• Formation Measure. The most important measure of performance for the forma-

tion control problem is whether or not the unicycles achieve the desired formation.

Any test trial that does not meet this condition is considered a failure. This aspect

will be captured by the formation measure. The formation measure µf : X → R is

a function of the state χ ∈ X that satisfies two conditions

1. it is positive semi-definite

2. it equals zero if and only if the formation is achieved, i.e., µ−1f (0) = Γp,

where Γp is the parallel formation manifold defined in (3.4). These conditions would

suggest that a natural candidate for µf(χ) is the Lyapunov function V (χ) defined

in (7.12). However, recall that V was designed to meet the two conditions above

under the assumption of fixed connected undirected graphs. This assumption will

not necessarily hold when we consider the scenario of state-dependent graphs. For

example, if the initial positions of the robots are too sparsely separated to sense one

another, i.e., the edge set E is empty, then all the neighbor sets (Ni)i∈n are empty

and V = 0 even though the formation is not achieved. This violates Condition


2. The formation measure should therefore be defined independent of the graph,

control gains or any other factors that don’t intrinsically specify the spatial config-

uration of the robots. Moreover, a Lyapunov function has the requirement that its

derivative be negative definite along system trajectories which is not necessary for

choosing the formation measure. It follows that designing µf will allow for a lot

more flexibility compared to designing the Lyapunov function.

We proceed to the design of µf . We have seen in Chapter 7 that achieving formation

amounts to the simultaneous synchronization of the heading angles (θi)i∈n and

the endpoints (xi)i∈n. One choice of formation measure could therefore simply

be µf(χ) =∑n

i=1

∑nj=1(‖xij‖ + ‖ sin(θij)‖). However, we opt for a more intuitive

choice. Consider a collection of n unicycles. The average endpoint position ¯x ∈ R2

and angle θ ∈ S1 are given by,

¯x :=1

n

n∑

i=1

xi,

θ := atan2

(

n∑

i=1

sin θj ,n∑

i=1

cos θj

)

.

This choice of average for angular quantities on S1 is standard and has a singu-

larity when (∑n

i=1 sin θj ,∑n

i=1 cos θj) = (0, 0). For example, in Figure 10.1 (a) the

average angle is θ = −π/4 while in (b) the average angle is undefined. Note that

(1/n)∑n

i=1 θi does not correspond with the average angle and leads to erroneous

results. For example, in Figure 10.1 this calculation yields (a) 3π/4 rad and (b)

π/2 rad.

Consider Figure 10.2 where the set of endpoints (xi)i∈n are indicated by the black

points drawn within the dotted circle and their average position ¯x is indicated in

the figure by . The angle θ represents the average heading angle of the unicycles

with respect to the inertial frame (only unicycle i is illustrated in the figure). The


Figure 10.1: Illustration of the average of two angles (a) (θ1, θ2) = (0, 3π/2) rad and (b)(θ1, θ2) = (0, π) rad.

actual position of unicycle i has been illustrated at position xi with heading angle

θi − θ with respect to the average. The desired formation is achieved in the set

χ ∈ X : xi = ¯x, θi = θi, i ∈ n

,

where all offset vectors and heading angles coincide with the average. The desired

position of unicycle i in this set is denoted by xi ,des in Figure 10.2. The error

between xi and xi ,des is bounded by

‖xi − xi ,des‖ < ‖xi − ¯x‖ + 2√

α2i + β2

i

∥

∥

∥

∥

sin

(

θi − θ

2

)∥

∥

∥

∥

=: ei(χ).

Correspondingly, define the formation measure as the maximum distance maxi∈n ei,

normalized by the size of the formation maxi∈2 :n ‖d11i‖, that is,

µf(χ) :=maxi∈n ei(χ)

maxi∈2 :n ‖d11i‖.

Note that normalizing by the size of the formation allows us to use the same mea-


sure to compare the performance of the control strategy for arbitrary formations.

We say that a formation is achieved if there exists a time tf after which the for-

mation measure satisfies µf < µf where µf > 0 is called the formation threshold.

Otherwise, the test is considered a failure. In the simulation trials, we have chosen

µf = 0.05.

Figure 10.2: Illustration of the formation measure.

• Drift Measure. In an ideal scenario, a formation should be achieved without

translating or drifting too much from its initial configuration. For example, if the

unicycles are constrained to move within an enclosed building, drifting too much

might mean that some unicycles collide with the walls before the formation is

achieved. We capture the aspect of drift with a drift measure which compares the

average position of unicycles at the initial time x(t0) = (1/n)∑

i∈n x(t0) to the

average position of unicycles at a final time x(tf ) = (1/n)∑

i∈n x(tf ) where tf is

the time when the formation is achieved, i.e., when the formation measure satisfies

µf < µf . The drift measure is therefore defined as

µd := ‖x(tf )− x(t0)‖


and is illustrated in Figure 10.3.

Figure 10.3: Formation measure. The unicycles at the initial time t0 are not shaded whilethe unicycles at time tf are shaded. The average position of the unicycles are indicatedwith . In the top figure, x(t0) and x(tf ) are the same and therefore µd = 0. In thebottom figure ‖x(tf ) − x(t0)‖ 6= 0 and therefore drift is present which is roughly twicethe size of the formation itself.

• Collision Measure. The collision measure µc is the minimum distance attained

between any two unicycles between the initial time t0 and the final time tf when

the formation is achieved and is defined as

µc := mint∈[t0,tf ],i,j∈n

‖xij(t)‖.

Assuming robots are 0.5m in radius or less, we say that a collision occurs if µc < 0.5.

• Input Measures. Finally, there are four input measures that correspond to the

supremum and mean of the control input magnitudes from the initial time t0 to the

formation time tf , maximized over i ∈ n. These are given by

µu,1 := maxi∈n supt∈[t0,tf ] |ui(t)| (m/s)

µω,1 := maxi∈n supt∈[t0,tf ] |ωi(t)| (rad/s)


µu,2 := maxi∈n meant∈[t0,tf ](|ui(t)|) (m/s)

µω,2 := maxi∈n meant∈[t0,tf ](|ωi(t)|) (rad/s).

10.2 Simulation Trials

In this section, we present numerous simulation trials to understand the effectiveness of

our control solution for stopping formations under different parameter choices. Each trial

is composed of N = 500 simulations. We consider the five-unicycle formation illustrated

in Figure 7.6 as the formation under study and for each simulation, initial angles (θi)i∈n

are chosen randomly in Tn and the positions (xi)i∈n are chosen randomly in a 60 × 60

meter grid. We choose the interaction function f(s) in (4.11) and rotational interaction

function g(s) in (4.24) as shown in Figure 10.4. Let aij = 30 and bij = αi + αj for all

j ∈ Ni and ηi = 1/αi. For each test, we compute the quantities in Table 10.1 and let

t0 = 0.

-5 -4 -3 -2 -1 0 1 2 3 4 5

s

-1

-0.5

0

0.5

1

f(s)

-4 -3 -2 -1 0 1 2 3 4

s

-1

-0.5

0

0.5

1

g(s)

X: 0.7884Y: 0.9987

Figure 10.4: Interaction function f(s) and g(s).

We begin by studying a base case that will serve as a reference of comparison between

test trials. The control parameters, i.e., (µf , k, α) used for the base case are presented

in Table 10.2. In the subsequent sections, we will vary each of the parameters one at a


Table 10.1: Data Collected from Simulations

Quantity Meaning

C # of tests that begin with a connected graphD # of tests that begin with a disconnected graphF # of tests that achieve formationCF # of tests that begin with a connected graph

and end in formation (µf < µf)DF # of tests that begin with a disconnected graph

and end in formation (µf < µf)tf formation timeµd drift measureµc collision measureNc # of tests in which a collision occursµu,1 , µω,1 , µu,2 , µω,2 input measures

time and study the effect this has. The undirected graph that will be used for the base

case simulations is shown in Figure 10.6.

Now we present the simulation results for the base case in Table 10.3. The column

called F/N (%) represents the percentage of simulations in which formation is achieved,

i.e., there exists a time tf after which µf < 0.05. The column called Nc/N (%) represents

the percentage of simulations in which a collision occurs. The remaining columns cor-

respond to the performance measures. The quantity tfµu,2 upper bounds the distance

travelled by any single robot in the ensemble and tfµω,2 upper bounds the number of

radians that any robot rotates before formation is achieved. The quantity tfµω,2/2π

therefore upper bounds the number of revolutions spun by any robot. The values pre-

sented in Table 10.3 are averages taken over all N iterations and we draw the following

conclusions:

• F/N (%)= 100 indicates that all simulations achieve formation.

• the average drift of the formation is 76m, i.e., 3.4 times the formation size.

• there is a collision 68 percent of the time which is high. To avoid this, one would

have to design a high level collision avoidance layer. It turns out that the colliding


agents are precisely those sharing common αi values. If one considers, instead, the

formation in Figure 10.5, where no two unicycles share a common αi value then

there is a collision only 11 percent of the time.

Figure 10.5: Formation specified by d112 = (−10, 5), d113 = (−5,−5), d114 = (−20, 10) andd115 = (−15,−10).

• the average time to achieve formation is 220 seconds.

• the maximum speed input is 76.4238 m/s and the maximum angular speed input is

17.1043 rad/s. These are both large but can be resolved by using input saturation

as we discuss later.

• the maximum displacement of any single unicycle in the ensemble is 129.7534m and

the maximum number of revolutions of any single unicycle is 4.1278/2π = 0.66.

Therefore the solution has very little oscillation which is a desirable property.


Table 10.2: Base Case Parameters

Quantity Base Value

µf (formation threshold) 0.05k 15α 5fixed or state-dependent graphs fixedundirected or directed graph undirectedinput saturation nosampling and disturbances no

1

2 3

4 5

Figure 10.6: Undirected graph used in the base case.

10.2.1 Study of µf

In this section, we will show how changing the formation threshold µf affects the per-

formance measures µd and tf . The average results for N = 500 simulations are shown

in Table 10.4 where the bold values correspond to the base case. Both µd and tf are

inversely proportional to µf . We can see that the smaller the threshold values of µf , the

further the formation needs to drift. Also the time to achieve smaller formation threshold

values grows very rapidly. To speed up convergence in a neighborhood of the formation

(where the rate of convergence is slow), one can always add a gain K to the controller

in (7.2) as follows,

u⋆i (yii, ϕi) = Kfi(y

ii) · e1 + βiω

⋆i (y

ii, ϕi),

ω⋆i (yii, ϕi) =

K

αi

(

fi(yii) · e2 + kgi(ϕi, η)

)

, i ∈ n.(10.1)

This will increase the rate of convergence by a factor of K. However, this will also

increase the magnitude of the control inputs by a factor of K and one may therefore


Table 10.3: Base Case Simulation Results

F/N (%) µd (m) Nc/N (%) tf (s)

100 76.4330 68.20 220.3635

µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)76.4238 17.1043 0.5888 0.0187 129.7534 4.1278

Table 10.4: Simulation Results Varying µf

µf µd (m) tf (s)

0.01 115.0104 5794.50.02 99.0958 1403.70.05 76.4330 220.36350.1 60.0024 54.07230.15 51.3388 24.10760.2 43.0169 13.15820.25 39.2520 8.9690

need to add input saturation. We illustrate formations in Figure 10.7, relative to the

frame of unicycle 1, corresponding to different values of µf .

10.2.2 Study of high gain parameters α and k

Based on Theorem 7.1.1, we need to choose α and k as high gain parameters. In this

section we show that these gains do not actually need to be chosen too large to still

achieve formation. The simulation results varying α are presented in Table 10.5 and the

results varying k are presented in Table 10.6. The values of the base case are in bold.

The main conclusions from the simulation results in Table 10.5 for varying α are:

• For α < 5 (the base value), some simulations fail. Only 2.2 percent of simulations

fail when α is chosen as low as 0.1.

• drift is proportional to α.

• collisions always remain above 60 percent.


-10 0 10 20 30-20

-10

0

10

20µ

f = 0.02

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.01

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.05

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.1

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.15

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.2

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.25

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.3

-10 0 10 20 30-20

-10

0

10

20µ

f = 0.35

Figure 10.7: Illustration of configurations satisfying different values of µf . The centre ofthe circles represent the desired unicycle positions while the actual unicycle positions areindicated with ×.

• The time to reach formation tf increases with α.

• All four input measures are inversely proportional to α. The maximum angular

velocity is as high as 994 rad/s when α = 0.1. This would certainly require satura-

tion.

• The lower the value of α, the more oscillatory the response. The most revolutions

that a unicycle makes is 1.58 when α = 0.1.

The main conclusions from the simulation results in Table 10.6 for varying k are:

• For k < 15 (the base value), some simulations fail. Only 0.4 percent of simulations

fail when k is chosen as low as 1.

• drift is proportional to k.


Table 10.5: Simulation Results Varying α

α F/N (%) µd (m) Nc/N (%) tf (s)

0.1 97.80 65.8306 61.76 196.41900.5 98.00 69.6150 66.33 192.79161 99.40 68.6228 64.59 187.50875 100 76.4330 68.20 220.363510 100 89.1039 70.00 315.933725 100 138.9444 69.20 853.3169

α µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)0.1 85.4778 994.4596 0.7337 0.0508 144.1049 9.97320.5 84.3687 660.0438 0.7010 0.0312 135.1392 6.00961 81.4894 197.9675 0.6934 0.0290 130.0245 5.43435 76.4238 17.1043 0.5888 0.0187 129.7534 4.127810 74.8656 7.6092 0.4822 0.0120 152.3422 3.776425 73.5076 3.2741 0.2501 0.0034 213.4184 2.9089

• collisions attain a minimum of 45 percent when k = 50.

• The time to reach formation attains a minimum of 92.59s when k = 50.

• All four input measures are proportional to k.

• The most revolutions that a unicycle makes is 1.07 when k = 100.

The simulations reveal that we cannot choose α and k much less than the base values

of α = 5 and k = 15 without running a risk of failure (i.e., formation may not be

achieved). It is also important not to choose these values too large, especially α, as this

can increase drift and/or tf .

10.2.3 Study of state-dependent undirected graphs

In this section, we study our control solution under state-dependent undirected graphs

where each unicycle’s neighbors are those that lie within a given radius of itself. We vary

the sensing radius from 20m to 45m. The results are presented in Table 10.7. The column

called C/N (%) represents the percentage of simulations that begin connected, CF/C

(%) is the percentage of initially connected graphs that achieve formation, and DF/D


Table 10.6: Simulation Results Varying k

k F/N (%) µd (m) Nc/N (%) tf (s)

1 99.60 73.4022 67.07 2994.85 99.80 73.8609 74.75 609.732915 100 76.4330 68.20 220.363550 100 125.2349 45.80 92.594075 100 168.9605 50.60 99.6497100 100 217.0383 53.60 129.4235

k µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)1 73.2869 9.0937 0.0423 0.0016 126.7413 4.79635 73.7068 10.1675 0.2139 0.0081 130.4128 4.931015 76.4238 17.1043 0.5888 0.0187 129.7534 4.127850 103.6473 46.4832 2.0498 0.0531 189.7969 4.921175 127.2770 65.3029 2.2921 0.0555 228.4063 5.5290100 156.9870 91.0888 2.1674 0.0521 280.5187 6.7436

(%) is the percentage of initially disconnected graphs that achieve formation. Naturally,

the sensing radius has to be large enough to maintain connection when unicycles lie in

formation.

The main conclusions from the simulation results in Table 10.7 for state-dependent

undirected graphs are:

• the proportion of initially connected undirected graphs goes from 5 percent to 95

percent as the sensing radius goes from 20m to 45m.

• the proportion of initially connected undirected graphs that achieve formation goes

from 59 percent to 98 percent as the sensing radius goes from 20m to 45m.

• the proportion of initially disconnected undirected graphs that achieve formation

goes from 11 percent to 70 percent as the sensing radius goes from 20m to 45m.

• the sensing radius does not have a large effect on the drift.

• tf is inversely proportional to the sensing radius due to increased connectivity as

the sensing radius increases.


Table 10.7: Simulation Results Varying the Sensing Radius

rad.(m) C/N (%) CF/C (%) DF/D (%) µd (m) Nc/N (%) tf (s)

20 5.80 58.62 11.25 74.5940 81.43 227.979825 22.20 68.47 24.68 72.4965 72.67 163.533630 45.40 79.30 44.69 70.5644 72.19 112.720835 72.20 89.20 45.32 71.9680 73.25 112.302240 86.20 94.90 52.17 70.2627 71.46 100.828145 95.40 97.90 69.57 69.6455 65.42 101.9338

rad. µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)20 65.9074 15.7912 0.7856 0.0312 179.0938 7.122325 76.2224 16.2460 1.0513 0.0402 171.9238 6.579330 85.0675 19.1963 1.2967 0.0482 146.1645 5.434535 94.0355 19.9240 1.2721 0.0442 142.8599 4.961540 101.7096 21.7516 1.3289 0.0436 133.9915 4.398145 107.6055 23.6756 1.3108 0.0427 133.6198 4.3515

• the input measures are typically proportional to the sensing radius.

• The maximum number of revolutions that a unicycle makes is inversely proportional

to the sensing radius.

One benefit of the control solution is that it is robust to the disconnection of agents.

This is illustrated in Figure 10.8 where a single agent begins too far from the rest of the

group to sense anyone. Despite this, the formation is still attained among the remaining

agents.

10.2.4 Study of directed graphs

We present simulation results for the directed ring-coupled graph in Figure 10.9 as op-

posed to the undirected graph in Figure 10.6. The main conclusions from the simulation

results in Table 10.8 are:

• 99 percent of the simulations achieved formation.

• The drift and collision measures are not changed significantly compared to the base

case.


-80 -60 -40 -20 0 20 40 60 80

x (m)

-100

-80

-60

-40

-20

0

20

y (m

)

Figure 10.8: Illustration of a formation where one agent is disconnected from the rest ofthe group. Regardless of this, the rest of the group achieves formation among themselves.Initial positions are indicated with and final positions are indicated with ×.

• For this particular digraph, tf is 2.23 times the value of the base case. This makes

sense because the undirected graph in the base case has 2.4 times the number of

edge connections. The input measures are also higher in the base case for the same

reason.

1

2 3

4 5

Figure 10.9: Directed ring-coupled graph.


Table 10.8: Simulation Results for the digraph in Figure 10.9

Graph F/N (%) µd (m) Nc/N (%) tf (s)

Fig 10.6 100 76.4330 68.20 220.3635Fig 10.9 99.00 79.7923 63.03 493.0773

Graph µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)Fig 10.6 76.4238 17.1043 0.5888 0.0187 129.7534 4.1278Fig 10.9 38.0586 11.6481 0.2805 0.0102 138.2853 5.0263

10.2.5 Study of input saturation

We have seen in the base case that the input measures, µu,1 = 76.4238 m/s and µω,1 =

17.1043 rad/s, are unacceptably high for most applications. Typical profiles of the inputs

ui and ωi are plotted in Figure 10.10. The large control inputs are only present at the

start of the simulation and decrease rapidly to more reasonable values as the formation

is reached. This suggests that we may be able to saturate the inputs without affecting

performance too much. In this section, we saturate the inputs so that µu,1 and µω,1

are bounded by different values corresponding to the first two columns in Table 10.9.

Table 10.9 presents the simulation results where the bold quantities correspond to the

base case.

0 20 40 60 80 100

time (s)

0

10

20

30

40

|ui| (

m/s

)

0 20 40 60 80 100

time (s)

0

0.5

1

|ωi| (

rad/

s)

Figure 10.10: Typical input profile during simulation.


Table 10.9: Simulation Results with Input Saturation

µu,1 (m/s) µω,1 (rad/s) F/N (%) µd (m) Nc/N (%) tf (s)

0.2 π/8 100 80.5316 64.60 624.76211 π/4 100 78.3436 65.60 275.70205 π/2 100 76.2906 67.60 216.513010 π 100 78.2598 72.60 213.968315 3π/2 99.80 77.5945 69.34 216.593120 2π 100 75.3784 72.00 211.3089

76.4238 17.1043 100 76.4330 68.20 220.3635

µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s) tfµu,2 (m) tfµω,2 (rad)0.2 π/8 0.1687 0.01 105.4219 6.21861 π/4 0.4080 0.0199 112.4854 5.48355 π/2 0.5836 0.0230 126.3490 4.987910 π 0.5896 0.0209 126.1483 4.468015 3π/2 0.5959 0.0210 129.0616 4.555420 2π 0.6095 0.0208 128.7945 4.4015

76.4238 17.1043 0.5888 0.0187 129.7534 4.1278

The profiles of the inputs ui and ωi for µu,1 = 5 m/s and µω,1 = π/2 rad/s are

plotted in Figure 10.11. The main conclusions from the simulation results in Table 10.9

for varying input saturation values are:

• the saturation did not lead to any significant formation failures (however, there was

a single failure out of 500 tests for the case where µu,1 = 15 and µω,1 = 3π/2).

This suggests that the solution is very robust to input saturation.

• the saturation did not lead to any significant change in drift and collisions.

• There was no significant increase in the formation time tf unless µu,1 < 1 m/s and

µω,1 < π/4 rad/s.

• There was no significant change in the mean control magnitudes µu,2 and µω,2

unless µu,1 < 1 m/s and µω,1 < π/4 rad/s where the values decrease.


0 10 20 30 40 50 60

time (s)

0

2

4

6

|ui| (

m/s

)

0 10 20 30 40 50 60

time (s)

0

0.5

1

1.5

2

|ωi| (

rad/

s)

Figure 10.11: Input profile during simulation for µu,1 = 5 m/s and µω,1 = π/2 rad/s.

10.2.6 Study of disturbances and sampling

In this section, we present a simulation result including a number of disturbances as well

as input sampling not included in the theoretical results. Due to slow simulation speed,

only a single test is performed instead of 500 in the other trials. Moreover, the inputs will

be saturated in which µu,1 ≤ 5m/s and µω,1 ≤ π/2 rad/s. The simulation parameters

are listed in Table 10.10. The disturbances are:

• an additive random noise with maximum magnitude of 0.25m/s on the input ui;

• an additive random noise with maximum magnitude of 0.25 rad/s on the input ωi;

• an additive random noise with maximum magnitude of 0.25 rad on the quantity

gi((θij)j∈Ni, η) accounting for errors in measurements of relative headings;

• an additive random noise on the quantity fi(yii) accounting for errors in measure-

ments of relative displacements of the vehicles. The direction of this vector has been

rotated within 0.25 rad and the magnitude is scaled between 0.75 to 1.25 times the

actual magnitude.


Table 10.10: Simulation Parameters in Section 10.2.6

Quantity Base Value

k 15α 5fixed or state-dependent graphs fixedundirected or directed graph undirectedinput saturation yessampling and disturbances yes

Table 10.11: Simulation Results with Perturbations and Sampling

µu,1 (m/s) µω,1 (rad/s) µu,2 (m/s) µω,2 (rad/s)5 π/2 3.9174 1.2296

Each unicycle samples its input 100 times per second. The simulation results are pre-

sented in Figure 10.12. The simulation results were run for 750s and the unicycles

achieved the formation measure µf = 0.1123 and no collisions. This shows that the

system still works well with perturbations and a sampling rate of 100Hz. In this time,

the formation drifted 74m and the input measures satisfied the values in Table 10.11.

In particular, the average input values µu,2 and µω,2 are significantly higher than in the

base case due to the disturbances. Reducing the sampling rate to 10Hz no longer has

acceptable performance as illustrated in Figure 10.13 where the unicycles achieve the for-

mation measure µf = 0.6455 after 750s and the unicycles move with noisy trajectories.


Figure 10.12: Simulation result in the presence of disturbances and sampling rate of100Hz.

Figure 10.13: Simulation result in the presence of disturbances and sampling rate of10Hz.

10.2.7 Study of dynamic unicycles

In this section, we present simulation results for the dynamic unicycle model presented

in (2.10). We select the control inputs as

ai = ˙ui(χi)− k(vi − ui(χi)),

αi = ˙ωi(χi)− k(ωi − ωi(χi))

where ai is the translational acceleration input, αi is the rotational acceleration input

and (ui, ωi) are the feedbacks for formation control of kinematic unicycles with the same

parameters as the base case. Moreover, we select k = 1. Denote


Table 10.12: Simulation Results for Dynamic Unicycles

F/N (%) µd (m) Nc/N (%) tf (s)

98 90.7796 58.33 175.2707

µa,1 (m/s2) µα,1 (rad/s2) µa,2 (m/s2) µα,2 (rad/s2) tfµu,2 (m) tfµω,2 (rad)2 π/2 0.6354 0.1268 334.4075 12.4730

µa,1 := maxi∈n supt∈[t0,tf ] |ai(t)| (m/s2)

µα,1 := maxi∈n supt∈[t0,tf ] |αi(t)| (rad/s2)

µa,2 := maxi∈n meant∈[t0,tf ](|ai(t)|) (m/s2)

µα,2 := maxi∈n meant∈[t0,tf ](|αi(t)|) (rad/s2).

The acceleration inputs are saturated such that µa,1 ≤ 2m/s2 and µα,1 ≤ π/2 rad/s2. The

simulation results are presented in Figure 10.14 and Table 10.12. Compared to the base

case, the drift increases, the collisions decrease to 58 %, the formation time decreases

unexpectedly as a result of input saturation and the maximum number of revolutions

made by any agent has increased to 12.4730/2π = 1.96.

-60 -40 -20 0 20 40 60 80

x (m)

-20

0

20

40

60

80

100

y (m

)

Figure 10.14: Simulation result for dynamic unicycles.

Chapter 11

Conclusions and future research

The work discussed in this thesis has been motivated by applications in multi-agent co-

ordination both for land and aerial based applications. We have studied fundamental

coordination problems including rendezvous, full synchronization, and formation con-

trol with and without final collective motions. We have focussed on two robot models:

ground-based kinematic unicycles in SE(2), and flying robots in SE(3). The main source

of difficulty in controlling these systems arises due to nonholonomic constraints that re-

strict the instantaneous velocities or accelerations of robots to a single direction defined

in their body frame.

The coordination problems have two goals. The first goal is to stabilize, using lo-

cal and distributed feedback, a fixed pre-defined configuration of agents modulo roto-

translations. In addition, there may be a higher-level control specification that enforces

a desired final motion of the ensemble. In the case of path following, achieving such

motion will require cameras or a GPS system allowing the robots to detect the desired

path. All the control problems addressed in this thesis have been formulated as set sta-

bilization problems with additional constraints on the inputs to ensure the final desired

collective motion.

In order to prove the results throughout this thesis, we have relied on numerous sta-

234

Chapter 11. Conclusions and future research 235

bility theorems stated in Section 4. Of particular importance are the reduction theorems

which have been employed consistently throughout the thesis. Many of the results would

have been far too difficult to prove without reduction, especially to the degree of globality

sought out in this work. All the control solutions have been constructed out of a number

of control solutions, or building blocks, solving analogous multi-agent coordination tasks

for much simpler integrator models. This framework has led to control solutions that

are quite intuitive. The control solutions presented also enjoy the following desirable

properties, often absent from results in the literature

• The feedbacks are completely independent of time thus avoiding the need for syn-

chronized clocks

• The feedbacks do not require a communication system eliminating issues surround-

ing communication delays

• Feedback expressions solving all the control problems are short and efficient

• Solutions are typically almost global or semi-global

• Solutions work for graphs containing a globally reachable node (undirected or di-

rected graphs depending on the problem)

• The control solutions presented in this work are robust to the loss of agents.

11.1 Future Research

Below is a list of possible future research directions.

• The reduction theorem for almost global asymptotic stability in Theorem 4.1.3

assumes that the set of equilibria in A, exponentially unstable relative to Γ2, are

isolated. It would be interesting to study whether the result still holds in cases

when the equilibria in A are not necessarily isolated.


• The result presented for rendezvous of flying robots in Chapter 5 shows global

practical stability of the rendezvous manifold. As in the case of kinematic unicycles

in Chapter 6, it may be possible to improve this result to global asymptotic stability.

In a preliminary conference paper (Roza et al., 2014) we presented an almost global

solution, however the feedback was not local and distributed and therefore was

excluded from this thesis.

• The proposed control law in Chapter 5 does not guarantee hovering of the robots.

While the robots converge to each other, nothing can be said about the motion of

the ensemble. The point of view of these authors is that the proposed solution of

the rendezvous problem with strictly local and distribution will serve as a layer in a

hierarchy of higher-level control specifications such as hovering and path following.

In a preliminary conference paper (Roza et al., 2014) we did provide a control

solution, combining a uniformly bounded double integrator consensus controller

and a rotating body equilibrium stabilizer in (4.26), for rendezvous of flying robots

where all agents did hover in steady state. However, this result was excluded from

this thesis because it required communication of thrust inputs between neighboring

agents as well as measurement of relative attitudes and angular velocities between

neighboring agents, none of which was required in Chapter 5.

• A common drawback of all the solutions that we have presented in this thesis

is the assumption of fixed, undirected graphs. This assumption is questionable

in practice and a future research direction might be extension of the developed

approach to state-dependent sensor graphs. For instance, if each robot mounted

an omnidirectional camera, then one could define Ni to be the collection of robots

that are within a given distance from robot i. If the camera is not omni-directional,

then the sensing will also depend on the robot heading angles. Stability is hard to

analyse for state dependent graphs. It needs to be shown any formation starting


with in a connected configuration are guaranteed to stay connected for all time

and achieve the goals. We believe that solutions in the literature for consensus of

double-integrators with time-dependent sensor digraphs could be extended to rigid

bodies using the framework in this thesis. However the Lyapunov function used in

the analysis would need to be modified extensively. Another common drawback is

the absence of collision avoidance in the control scheme or system perturbations.

• In Chapter 7, the solution requires two high gain parameters k and α. We have seen

in simulation that increasing these gain too large will have negative consequence

on the rate of convergence of the formation. In extensive simulations, we have seen

that these gains do not typically need to be very large for the controller to work

well. While this is encouraging, a future direction would be to characterize exactly

how large these gains need to be, or find another control solution that does not

require them altogether.

• In Chapter 8, we presented a solution to PFP for the class of hierarchical digraphs.

Based on the simulation results, this controller may also work for any sensor di-

graph containing a globally reachable node. Also, the solutions to PP, CFP and

CPP assume undirected sensor graphs but may also work for any sensor digraph

containing a globally reachable node.

• The result for formation path following for Jordan curves in Chapter 9 has a number

of drawbacks compared to the results for formation line and circle path following in

Chapter 8 that might be improved in the future. The asymptotic stability is local

as opposed to almost global, it requires communication of pivot angles and every

agent is required to see the common path. These assumptions weren’t necessary in

Chapter 8.

• A notable omission from this work is a discussion of formation control of flying

robots using local and distributed feedbacks, analogous to the work in Chapter 7


for kinematic unicycles. As we discussed in Section 5.3, one could easily modify

the solution for rendezvous in Chapter 5 (or the preliminary result in (Roza et al.,

2014)) to a solution for formation control by making the robots converge to fixed

offsets from one another defined in inertial frame instead of converging exactly to

one another. However, an approach like this assumes that each agent can measure

their own attitude in the inertial frame which is not a local and distributed quantity.

Instead, inspired by the work on formation control of kinematic unicycles in Chap-

ter 7, one might try to attach fixed body referenced offset vectors to each robot,

xi = xi + αiRie1 + βiRie2 + γiRie3, such that the formation is achieved when the

endpoints xi achieve consensus and all robot attitudes synchronize. One might

stabilize this formation by appropriately combining a double integrator consensus

controller with a consensus controller for rotating bodies in SO(3). A special case

of this solution would be full synchronization of flying robots. We have attempted

to do this but without success. One difficulty of adapting the approach in 7 to the

dynamic case of flying robots is its reliance on control primitives that are gradient

systems.

After the formation control problem for flying robots is addressed, one could then

look at including high level specifications, some of which may require GPS or other

sensors, like hovering (requiring each agent to measure gravity), formation path

following and flocking, analogous to the work in Chapter 8 for kinematic unicycles.

• The results in this thesis could be extended in the future to other classes of robots

not considered in this work. For example, fixed-wing aircraft or boats. They could

also be extended to allow for formations that are not necessarily fixed, but change

with time. Such dynamic formations are seen in swarms of fish, for example.

• Experiments using the control theory in this thesis will be performed as part of

future undergraduate student projects.

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