a unified geometric framework for kinematics, dynamics and ... · cially, i would like to mention...

A Unified Geometric Framework forKinematics, Dynamics and Concurrent Control ofFree-base, Open-chain Multi-body Systems with

Holonomic and Nonholonomic Constraints

by

Robin Chhabra

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

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

c© Copyright 2014 by Robin Chhabra

Abstract

A Unified Geometric Framework for

Kinematics, Dynamics and Concurrent Control of

Free-base, Open-chain Multi-body Systems with

Holonomic and Nonholonomic Constraints

Robin Chhabra

Doctor of Philosophy

Graduate Department of Aerospace Science and Engineering

University of Toronto

2014

This thesis presents a geometric approach to studying kinematics, dynamics and

controls of open-chain multi-body systems with non-zero momentum and multi-degree-

of-freedom joints subject to holonomic and nonholonomic constraints. Some examples

of such systems appear in space robotics, where mobile and free-base manipulators are

developed. The proposed approach introduces a unified framework for considering holo-

nomic and nonholonomic, multi-degree-of-freedom joints through: (i) generalization of

the product of exponentials formula for kinematics, and (ii) aggregation of the dynamical

reduction theories, using differential geometry. Further, this framework paves the ground

for the input-output linearization and controller design for concurrent trajectory tracking

of base-manipulator(s).

In terms of kinematics, displacement subgroups are introduced, whose relative config-

uration manifolds are Lie groups and they are parametrized using the exponential map.

Consequently, the product of exponentials formula for forward and differential kinematics

is generalized to include multi-degree-of-freedom joints and nonholonomic constraints in

open-chain multi-body systems.

As for dynamics, it is observed that the action of the relative configuration manifold

corresponding to the first joint of an open-chain multi-body system leaves Hamilton’s

equation invariant. Using the symplectic reduction theorem, the dynamical equations

ii

of such systems with constant momentum (not necessarily zero) are formulated in the

reduced phase space, which present the system dynamics based on the internal parameters

of the system.

In the nonholonomic case, a three-step reduction process is presented for nonholo-

nomic Hamiltonian mechanical systems. The Chaplygin reduction theorem eliminates the

nonholonomic constraints in the first step, and an almost symplectic reduction procedure

in the unconstrained phase space further reduces the dynamical equations. Consequently,

the proposed approach is used to reduce the dynamical equations of nonholonomic open-

chain multi-body systems.

Regarding the controls, it is shown that a generic free-base, holonomic or nonholo-

nomic open-chain multi-body system is input-output linearizable in the reduced phase

space. As a result, a feed-forward servo control law is proposed to concurrently control

the base and the extremities of such systems. It is shown that the closed-loop system is

exponentially stable, using a proper Lyapunov function. In each chapter of the thesis,

the developed concepts are illustrated through various case studies.

iii

To my love, Fahimeh

iv

Acknowledgements

First of all, I would like to thank my supervisors, M. Reza Emami and Yael Karshon.

Reza showed me how to define practical problems and approach them in a scientific

manner. He was the one who introduced me to the field of robotics, starting from the

basics. Throughout my graduate studies, he was always inspiring and supportive, and

he familiarized me with ethics in research. During the last four years of my Ph.D., Yael

helped me to understand differential geometry and use it towards the final goals of my

research. She was always patient to hear me and advise me in the theoretical aspects of

my Ph.D. dissertation. She always encouraged me and reminded me that my research

was a valuable piece of work.

During my studies at the University of Toronto, I had the opportunity of knowing

great professors who gave me constructive pieces of advice about my research. Amongst

them, I particularly would like to thank Gabriele D’Eleuterio and Christopher J. Damaren,

the members of my Doctorla Examination Committee.

Further, I want to sincerely thank my friends in the Space Mechatronics group who

made a very friendly and comfortable environment for me to perform my research. Spe-

cially, I would like to mention my amazing friends, Sina, Peter, Victor, Jason, Michael

Anthony and Adrian.

Finally, I would like to take a moment and appreciate my best friends and family who

accompanied me in this journey. Special thanks go to Payman and Ali, my best friends,

whose friendship and help has been endless. My parents and my brother Arvind have

been always supportive in different perspectives of life. Without their help and support,

I was not able to complete my Ph.D. degree. Thank you mama, thank you papa, and

thank you Arvind!

Last but not least, my sincere thanks go to Fahimeh and her beautiful smile. Since

the first day we met, she has been encouraging and supporting me, as a friend and as

my wife. She has been emotionally and technically supportive, and filled my life with

happiness and joy. While I was writing this dissertation, she was the only one who was

with me at all the moments, happy and sad. Thank you Fahimeh, and please keep smiling

in the rest of our lives!

v

Contents

1 Introduction 1

1.1 Kinematics of Open-chain Multi-body Systems . . . . . . . . . . . . . . . 1

1.2 Dynamical Reduction of Holonomic and Nonholonomic Hamiltonian Sys-

tems with Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Control of Free-base Multi-body Systems . . . . . . . . . . . . . . . . . . 7

1.4 Statement of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.3 Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.4 Produced Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 A Generalized Exponential Formula for Kinematics 15

2.1 Holonomic and Nonholonomic Joints . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Displacement Subgroups . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 Nonholonomic Displacement Subgroups . . . . . . . . . . . . . . . 21

2.2 Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Differential Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Coordinate Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.1 Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.2 Differential Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Reduction of Holonomic Multi-body Systems 38

3.1 Hamilton-Pontryagin Principle and Hamilton’s Equation . . . . . . . . . 38

3.2 Hamiltonian Mechanical Systems with Symmetry . . . . . . . . . . . . . 45

3.3 Symplectic Reduction of Holonomic Open-chain Multi-body Systems with

Displacement Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

vi

3.3.1 Indexing and Some Kinematics . . . . . . . . . . . . . . . . . . . 54

3.3.2 Lagrangian and Hamiltonian of an Open-chain Multi-body System 58

3.3.3 Reduction of Holonomic Open-chain Multi-body Systems . . . . 59

3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Reduction of Nonholonomic Multi-body Systems 82

4.1 Nonholonomic Hamilton’s Equation and

Lagrange-d’Alembert-Pontryagin principle . . . . . . . . . . . . . . . . . 83

4.2 Nonholonomic Hamiltonian Mechanical Systems with Symmetry . . . . . 87

4.3 Reduction of Nonholonomic Open-chain Multi-body Systems with Dis-

placement Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.4 An Investigation on Further Symmetries of Open-chain Multi-body Systems113

4.4.1 Identifying Symmetry Groups using AP1 . . . . . . . . . . . . . . 114

4.4.2 Identifying Symmetry Groups using AP2 . . . . . . . . . . . . . . 115

4.5 Further Reduction of Nonholonomic Open-chain Multi-body Systems . . 118

4.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.6.1 Further Reduction of the System . . . . . . . . . . . . . . . . . . 141

5 Concurrent Control of Multi-body Systems 144

5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.1.1 Mathematical Formalization and Assumptions . . . . . . . . . . . 145

5.1.2 Reduced Hamilton’s Equation and Reconstruction . . . . . . . . . 149

5.2 End-effector Pose and Velocity Error . . . . . . . . . . . . . . . . . . . . 152

5.2.1 Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.2.2 Velocity Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3 Input-output Linearization and Inverse Dynamics in the Reduced Phase

Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.4 An Output-tracking Feed-forward

Servo Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6 Conclusions 176

6.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.2.3 Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

vii

List of Tables

2.1 Categories of displacement subgroups [38, 71] . . . . . . . . . . . . . . . 19

3.1 Displacement subgroups and their corresponding isotropy groups . . . . . 69

viii

List of Figures

2.1 A mobile manipulator on a six d.o.f. moving base . . . . . . . . . . . . . 33

2.2 Coordinate frames assigned to A0, ..., A6 at the initial configuration . . . 34

3.1 A six-d.o.f. manipulator mounted on a spacecraft . . . . . . . . . . . . . 70

3.2 The coordinate frames attached to the bodies of the robot . . . . . . . . 71

4.1 An example of a mobile manipulator . . . . . . . . . . . . . . . . . . . . 122

4.2 The coordinate frames attached to the bodies of the mobile manipulator

(Note that, the Zi-axis (i = 0, · · · , 6) is normal to the plane) . . . . . . . 123

4.3 An example of a crane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.4 The coordinate frames attached to the bodies of the crane . . . . . . . . 132

5.1 Feed-forward servo control for a generic free-base, open-chain multi-body

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.2 Servo controller for concurrent control of a three-d.o.f. manipulator mounted

on a two-wheeled rover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

ix

Notation

Lr Left composition/translation by a Lie group element r

Rr Right composition/translation by a Lie group element r

Kr Conjugation by a Lie group element r

Adr Adjoint operator corresponding to a Lie group element r

adξ adjoint operator corresponding to a Lie algebra element ξ

[ξ, η] Lie bracket of two Lie algebra elements or matrix commutator of

two matrices

diag(A1, ..., An) Block diagonal matrix of the matrices A1, ..., An

v Skew-symmetric matrix corresponding to the vector v in R3

R(θ) 2× 2 rotation matrix for the angle θ

R(θ, v) 3× 3 rotation matrix of a rotation for the angle θ, about the vector

v ∈ R3

ω The 3× 3 anti-symmetric matrix corresponding to the vector ω in R3

Tmf Tangent map corresponding to the map f at m, an element of the

domain manifold

T ∗mf Cotangent map corresponding to the map f at m, an element of the

target manifold

TmM Tangent space of the manifold M at the element m

TM Tangent bundle of the manifold M

T ∗mM Cotangent space of the manifold M at the element m

T ∗M Cotangent bundle of the manifold M

exp(ξ) Group/matrix exponential of a Lie algebra element ξ

Lie(G) Lie algebra of the Lie group G

Lie∗(G) Dual of the Lie algebra of the Lie group G

Gµ Coadjoint isotropy group for µ ∈ Lie∗(G)

n Semi-direct product of groups

·, · Euclidean metric

‖v‖h Norm of the vector v with respect to the metric h

〈·, ·〉 Canonical pairing of the elements of tangent and cotangent space

LX Lie derivative with respect to the vector field X

ξM Vector field on the manifold M induced by the infinitesimal action of

ξ ∈ Lie(G)

ιXΩ Interior product of the differential form Ω by the vector field X

X(M) Space of all vector fields on the manifold M

x

Ω2(M) Space of all differential 2-forms on the manifold M

dΩ Exterior derivative of the differential form Ω

dH Exterior derivative of the function H

M/G Quotient manifold corresponding to a free and proper action of

the Lie group G

xi

Chapter 1

Introduction

Holonomic and nonholonomic open-chain multi-body systems appear in the field of

robotics. In the context of geometric mechanics, these systems can be considered as

Hamiltonian mechanical systems. In this thesis, we have a geometric approach towards

studying the kinematics, dynamics and controls of generic open-chain multi-body sys-

tems with holonomic and nonholonomic constraints. This study includes: revisiting the

notion of lower kinematic pairs and generalizing it to define displacement subgroups,

studying and unifying the reduction of Hamiltonian mechanical systems for holonomic

and nonholonomic open-chain multi-body systems with symmetry, and deriving an out-

put tracking, feed-forward servo controller for such systems. In the following, we first

report the existing literature for different topics appearing in this thesis. Then, we list

the main contributions of the thesis, and finally we give the outline of the thesis.

1.1 Kinematics of Open-chain Multi-body Systems

The product of exponentials formula for Forward Kinematics of serial-link multi-body sys-

tems with revolute and/or prismatic joints was first introduced by Brockett in 1984 [11].

This formulation was further developed and its roots in Lie group and screw theory were

illustrated by Murray et al. in 1994 [57]. One of the most important contributions of

this method of multi-body system modeling is the elimination of intermediate coordinate

frames in the kinematic analysis of serial-link manipulators. Since then, a number of

researchers have investigated the computational efficiency of this formulation [62], and

have applied it to different robotic problems [64, 24, 37, 67, 68]. In 1995, Park et al.

used this formulation to reformulate the dynamical equations of serial-link multi-body

systems [63], and later in 2003 Muller et al. attempted to unify the kinematics and dy-

namics of open-chain multi-body systems with one degree-of-freedom (d.o.f.) joints [56].

1

Chapter 1. Introduction 2

The exponential map used in the product of exponentials formula is the exponential

map of Lie groups, which maps an element of the corresponding Lie algebra to an element

of the Lie group. For a rigid body the configuration manifold is the Lie group SE(3), and

the elements of its Lie algebra se(3) are the screws associated with the possible motions

of a rigid body in 3-dimensional space [57]. Screw theory, which was first introduced by

Ball in 1900 [4] and also appeared in the work of Clifford [22, 23], has been extensively

investigated as a powerful means for the kinematic modeling of mechanisms [47, 45, 32, 33,

38, 10] and robotic systems [24, 80, 92, 31], by defining the notion of screw systems [71].

Moreover, the relationship between screw theory, Lie groups and projective geometry in

the study of rigid body motion was elaborated in a paper by Stramigioli in 2002 [82]. He

subsequently defined the notions of relative configuration manifold and relative screw to

study multi-body systems [81]. In 1999 Mladenova also applied Lie group theory to the

modeling and control of multi-body systems [54]. As opposed to the geometric nature of

most of the above-mentioned works, her approach was mainly algebraic.

Based on a well-known theorem in the theory of Lie groups, any element of a connected

Lie group can be written as product of exponentials of some elements of its Lie algebra.

Accordingly, Wei and Norman introduced a product of exponentials representation for

the elements of a connected Lie group [91], which was adopted by Liu [46] and Leonard et

al. [44] to reformulate Kane’s equations for multi-body systems and solve nonholonomic

control problems on Lie groups, respectively. On the other hand, surjectivity of the

exponential map of SE(3) that is a direct consequence of Chasles’ Theorem [57] implies

that any element of SE(3) can be written as the exponential of an element of se(3).

However, not much work has been done on the exponential parametrization of the

Lie subgroups of SE(3). Only for the one-parameter subgroups of SE(3), which corre-

spond to one-d.o.f. joints, the exponential map has been used to parametrize the relative

configuration manifold that leads to the standard product of exponentials formula. In

fact, we will show that the Lie subgroups of SE(3) correspond to the relative configura-

tion manifolds of displacement subgroups [38, 36]. These joints are generally multi-d.o.f.

holonomic joints. For generic multi-d.o.f. joints, Stramigioli in [81] briefly mentions that

at each point the exponential map can be used as a local diffeomorphism between the

relative configuration manifold and its tangent space. He later used this local diffeomor-

phism to introduce singularity-free dynamic equations of a generic open-chain multi-body

system with holonomic and nonholonomic joints [29]. In Chapter 2, we give the necessary

and sufficient conditions for surjectivity of the exponential map of the relative configu-

ration manifolds of displacement subgroups. Under those conditions the corresponding

Lie subgroups are locally parametrized using the elements of their Lie algebras.


1.2 Dynamical Reduction of Holonomic and Non-

holonomic Hamiltonian Systems with Symmetry

A symplectic manifold is a pair (M,Ω), where M is an even dimensional smooth manifold

and Ω is a nondegenerate, closed 2-form. Such a 2-form is called a symplectic form.

Consider the action of a Lie group G on M ; the G-action is called symplectic if it

preserves the symplectic form Ω, i.e., ∀g ∈ G, Φ∗gΩ = Ω, where Φg : M →M is the action

map. Now consider an Ad∗-equivariant map M : M → Lie∗(G) such that ∀ξ ∈ Lie(G) it

satisfies the identity

ιξMΩ = d〈M, ξ〉, (1.2.1)

where ξM is the vector field on M induced by the infinitesimal action of G in the direction

of ξ. Such a map is called the momentum map. The symplectic reduction theorem states

that in the presence of a free and proper G-action and an (Ad∗-equivariant) momentum

map, for any value µ ∈ Lie∗(G) of the momentum map the quotient manifold Mµ :=

M−1(µ)/Gµ inherits a symplectic form Ωµ, where Gµ is the coadjoint isotropy group for

µ, Ωµ is identified by the equality i∗µΩ = π∗µΩµ, and where the maps iµ : M−1(µ) → M

and πµ : M−1(µ) → M−1(µ)/Gµ are the canonical inclusion and quotient maps [53].

The pair (Mµ,Ωµ) is called the symplectic reduced manifold. This theorem by Marsden

and Weinstein made a huge impact on unifying the reduction methods that had been

previously developed for holonomic dynamical systems, such as classical Routh method

and the reduction of Lagrangian systems by cyclic parameters [70].

For mechanical systems, the space of momenta, or phase space, i.e., the cotangent

bundle of the configuration manifold T ∗Q, admits a canonical symplectic 2-form, which

is the exterior derivative of the tautological 1-form Θcan defined by (Θcan)pq(Zpq) :=

〈pq, TpqπQ(Zpq)〉, ∀pq ∈ T ∗qQ and ∀Zpq ∈ TpqT∗Q and where πQ : T ∗Q → Q is the

cotangent bundle projection. That is, (T ∗Q,Ωcan := −dΘcan) is a symplectic manifold.

Let H : T ∗Q → R be the Hamiltonian of a mechanical system that is defined by a

Riemannian metric and a function on Q. The solution curves of this system satisfy

Hamilton’s equation

ιXΩcan = dH,

where X ∈ X(T ∗Q) is everywhere tangent to the solution curves. In general, for any

function f ∈ C∞(T ∗Q), the vector field Xf ∈ X(T ∗Q) that satisfies Hamilton’s equation

is called the Hamiltonian vector field of f . Let G be a group acting properly on the

configuration manifold Q. The cotangent lifted action on the phase space is symplectic.

In this case, if the Hamiltonian of the system is also invariant under the cotangent lift


of the G-action, the group G is called the symmetry group of the mechanical system,

and the system is called a mechanical system with symmetry [48, 50]. In the reduction

process of mechanical systems with symmetry, we start with a Riemannian metric on

Q, a symplectic structure on T ∗Q, the Hamiltonian H, and a Lie group whose action

preserves the above structures, and after the reduction, we have a mechanical system on

the reduced phase space, which is a symplectic manifold, with the induced Riemannian

metric and Hamiltonian.

A Poisson manifold is a pair (P, ·, ·), where P is a smooth manifold and ·, · :

C∞(P )×C∞(P )→ C∞(P ), called the Poisson bracket, satisfies the following properties:

∀f, g, h ∈ C∞(P ) and ∀λ ∈ R,

i) f, g = −g, f (antisymmetry property)

ii) f + λh, g = f, g+ λ h, g (linearity property)

iii) hf, g = h f, g+ h, g f (Leibniz property)

iv) f, g , h+ h, f , g+ g, h , f = 0. (Jacobi identity)

For a mechanical system, the phase space T ∗Q admits a canonical Poisson structure using

the canonical symplectic form, given by f, h := −Ωcan(Xf , Xh), ∀f, h ∈ C∞(T ∗Q),

where Xf and Xh satisfy the identities ιXfΩcan = df and ιXhΩcan = dh. Based on this

definition of the Poisson bracket, one has f, h = LXfh, where LXf is the Lie derivative

in the direction of the vector field Xf . For a mechanical system with symmetry, suppose

that the symmetry group G acts freely and properly on Q, and hence on T ∗Q. The

Poisson bracket is invariant under the cotangent lifted action, i.e., the action is a Poisson

action on (T ∗Q, ·, ·). The Poisson bracket on T ∗Q descends to a Poisson bracket on

the quotient manifold (T ∗Q)/G, defined by

f, h(T ∗Q/G) π = f π, h π ,

where f and h are smooth functions on (T ∗Q)/G, and π : T ∗Q → (T ∗Q)/G is the

quotient map. This bracket is well-defined since f π, h π and ·, · are G invariant.

This process, which has been introduced in [50, 8], is called Poisson reduction. The major

difference between the Poisson reduction and the symplectic reduction is the concept of

momentum map, which is not necessary for Poisson reduction, and as a result the induced

Hamilton’s equation on the quotient phase space evolves in a bigger space. This approach

unifies the Euler-Poincare and Lagrange-Poincare equations for mechanical systems with

symmetry [50].


Both of the abovementioned reduction theories were developed and extended to La-

grangian systems, in the 1990s [15, 52, 51]. Since the trivial behaviour of a mechanical

system due to symmetry are eliminated during a reduction process, the behaviour of

the system is more explicit in the reduced space. The reduction procedures are help-

ful for extracting coordinate-independent control laws for the mechanical systems with

symmetry [8, 13], which is the subject of Chapter 5.

A nonholonomic mechanical system with symmetry is a mechanical system with sym-

metry together with a G-invariant distribution D, i.e., a distribution D such that ∀g ∈ Gand ∀q ∈ Q, TqΦg(D(q)) = D(Φg(q)). The distribution D is a linear sub-bundle of TQwhere the velocities of the physical trajectories of the system should lie. Generally, this

distribution is non-involutive, and it is the result of kinematic nonholonomic constraints

such as rolling without slipping. If D is involutive, we say that the constraints are holo-

nomic. Although in general the physical constraints can be nonlinear, time dependant

or affine, we only restrict our attention to the constraints that are linear in velocity. The

distinguishing characteristics of nonholonomic systems from the holonomic ones are that

i) they satisfy the Lagrange-d’Alembert principle instead of the Hamilton principle [9],

and

ii) the momentum is not generally conserved for them.

A Chaplygin system is a nonholonomic mechanical system with symmetry such that

the space of directions of the infinitesimal G-action is complementary to the distribution

D. On the Lagrangian side, in [16] Chaplygin reduces such systems considering only

abelian symmetry groups . Afterwards, Koiller generalizes his result to non-abelian

symmetry groups [42]. He considers two cases:

i) Nonholonomic systems whose configuration manifold is a total space of aG-principal

bundle and the constraints are given by a connection, and

ii) Nonholonomic systems whose configuration manifold is G itself with left invariant

constraints and left invariant metric, which defines the Lagrangian.

A more general reduction procedure for the tangent lifted symmetries of a nonholo-

nomic system that results in Lagrange-d’Alembert-Poincare equations [8, 14] is reported

in [9]. This method is centred on defining a nonholonomic connection as the sum of an

Ehresmann connection and the mechanical connection and introducing a nonholonomic

momentum map. The analogue of this approach in Poisson formalism is also explained

in [8] that is originated in a paper by van der Schaft and Maschke [86]. In this paper,


the authors use an Ehresmann connection to project the canonical Poisson bracket of

T ∗Q to the image of the nonholonomic distribution under the Legendre transformation,

and they show that the resulting bracket satisfies the Jacobi identity if and only if the

original distribution is involutive.

On the Hamiltonian side, Bates and Sniatycki first show that the vector field repre-

senting the dynamics of a nonholonomic system, which is the solution of Hamilton’s equa-

tion for nonholonomic systems, indeed lies in the distribution T (FL(D))∩v ∈ T (T ∗Q)|TπQv ∈ D ⊆ T (T ∗Q). Here, the fibre-wise linear map FL : TQ → T ∗Q is the Legen-

dre transformation. Then under the symmetry hypotheses, after restricting Hamilton’s

equation to this distribution, they show that the flow of the vector field, which is the so-

lution of Hamilton’s equation, descends to the quotient manifold FL(D)/G [6, 25, 26, 27].

Later on, based on this method of reduction, which is called distributional Hamiltonian

approach [27], the Noether theorem is extended to nonholonomic systems and accordingly

a two-stage reduction procedure is introduced. In the first stage, the symplectic reduc-

tion theorem is applied to reduce Hamilton’s equation by a normal subgroup G0 ⊆ G,

whose momentum is conserved, and yields another distributional Hamiltonian system.

For the second stage, the method in [6] is used to reduce the equations by G/G0 [76].

This method is further extended to singular reduction of nonholonomic systems, and it

is reformulated for almost Poisson manifolds in [77]. Here, an almost Poisson manifold

is a manifold equipped with a bracket that satisfies the properties of the Poisson bracket

except the Jacobi identity.

An extension of reduction of Chaplygin systems is also reported in the concept of

nonholonomic Hamilton-Jacobi theory [59, 39], which uses the symplectic reduction the-

orem in the presence of further symmetries of the system to reduce a Chaplygin system

in two steps. The first step is equivalent to the Chaplygin reduction in [42], which results

in an almost symplectic 2-form to describe Hamilton’s equation in the reduced space. An

almost symplectic 2-form is a non-degenerate differential 2-form (which is not necessarily

closed). In the second step, under some assumptions an almost symplectic reduction [69]

is performed. Based on this idea, a three-step reduction procedure for nonholonomic me-

chanical systems with symmetry is presented in Chapter 4 that generalizes the two-step

reduction in [59] by trying to find constants of motion that are not necessarily correspond

to the action of abelian Lie groups.


1.3 Control of Free-base Multi-body Systems

An example of a mechanical system with symmetry is a free-base multi-body system,

which is mostly studied in the field of robotics and aerospace. Vafa and Dubowsky

introduce the notion of Virtual Manipulator [85] (for a free-floating manipulator with

zero total momentum), and they show that this approach decouples the system centre of

mass translation and rotation. Dubowsky and Papadopoulos in [28] use this notion to

solve for the inverse dynamics problem that yields to designing linear controllers in joint

and task space. Since the trivial behaviour of a multi-body system due to momentum

conservation is eliminated during a reduction process, the behaviour of the system is

more explicit in the reduced space. The reduction procedures have been helpful for

extracting control laws for space manipulators by restricting the dynamical equations to

the submanifold of the phase space where the momentum of the system is constant (and

usually equal to zero). Yoshida et al. investigate the kinematics of free-floating multi-

body systems utilizing the momentum conservation law. They derive a new Jacobian

matrix in generalized form and develop a control method based on the resolved motion

rate control concept [84, 58].

McClamroch et al. propose an articulated-body dynamical model for free-floating

robots based on Hamilton’s equation, and implement it to derive an adaptive motion

control law [90]. Based on the concept of Virtual manipulator, Parlaktuna and Ozkan

also develop an adaptive controller for free-floating space manipulators [65]. Wang and

Xie introduce an adaptive control law for position/force tracking of free-flying manipula-

tors [87, 88], and later they use recursive Newton-Euler equations to derive a novel adap-

tive controller for position tracking of free-floating manipulators in their task space [89].

In this controller, they estimate the inertia tensor of the spacecraft (base body) by a

parameter projection algorithm. As an application, Pazelli et al. investigate different

nonlinear H∞ control schemes implemented to a free floating space manipulator, subject

to parameter uncertainty and external disturbances [66].

In the case of underactuated space manipulators, Mukherjee and Chen in [55] show

that even if the unactuated joints do not possess brakes, the manipulator can be brought

to a complete rest provided that the system maintains zero momentum. In [83] an alterna-

tive path planning methodology is developed for underactuated manipulators using high

order polynomials as arguments in cosine functions to specify the desired path directly

in joint space. Note that all of the above mentioned control strategies were developed

for holonomic multi-body systems with one-d.o.f. joints and for zero momentum of the

system.


Geometric methods have also been used to reduce the dynamical model of free-base

multi-body systems and introduce effective control laws. For example, in [78, 79] Sreenath

reduces Hamilton’s equation by SO(2) for free-base planar multi-body systems with non-

zero angular momentum. He uses the symplectic reduction theory to first reduce the

dynamical equations and then derive a control law for reorienting the free-base system.

Chen in his Ph.D. thesis [17] extends Sreenath’s approach to spatial multi-body systems

with zero angular momentum. Duindam and Stramigioly derive the Boltzmann-Hamel

equations for multi-body systems with generalized multi-d.o.f. holonomic and nonholo-

nomic joints by restricting the dynamical equations to the nonholonomic distribution [29].

This is the first attempt to reduce the dynamical equations of a generic open-chain multi-

body systems with generalized holonomic and nonholonomic joints. Furthermore, Shen

proposes a novel trajectory planning in shape space for nonlinear control of multi-body

systems with symmetry [74, 72, 73]. In his work he performs symplectic reduction for zero

momentum and assumes multi-body systems on trivial bundles. Then, in [75] he extends

his results to include nonholonomic constraints. Hussein and Bloch study optimal control

of nonholonomic mechanical systems, using an affine connection formulation [40]. Sliding

mode control of underactuated multi-body systems is also studied in [3]. In the control

community, Olfati-Saber in his Ph.D. thesis [60] studies the reduction of underactuated

holonomic and nonholonomic Lagrangian mechanical systems with symmetry and its ap-

plication to nonlinear control of such systems. He uses a feedback linearization method

in the reduced phase space to extract control laws for such systems [61]. However, he

only considers abelian symmetry groups, and he does not take into account non-zero

momentum of the system in his approach. As a continuation of Olfati-Saber’s work,

Grizzle et al. in [34] show that a planar mechanism with a cyclic unactuated parameter

is always locally feedback linearizable, and they derive a nonlinear control law for such

systems. Further, Bloch and Bullo extract coordinate-independent nonlinear control laws

for holonomic and nonholonomic mechanical systems with symmetry [8, 12, 13].

1.4 Statement of Contributions

This section presents the contributions of this dissertation in different aspects of study-

ing open-chain multi-body systems. In this work, we consider nonholonomic constraints

as linear constraints on the joint velocities. Normally, systems with nonholonomic con-

straints are treated separately in the literature. This thesis is an attempt to use geo-

metric tools to unify and extend the existing approaches for analyzing the kinematics

and dynamics of open-chain multi-body systems with non-zero momentum and holo-


nomic/nonholonomic constraints.

As a result, based on the developments in Chapters 2 to 4, we are able to derive a

nonlinear control scheme in Chapter 5 for concurrent trajectory tracking of a generic free-

base, open-chain multi-body system with multi-d.o.f. holonomic and/or nonholonomic

joints. In the following sections, we elaborate on the contributions of this thesis in

kinematics, dynamics and controls.

1.4.1 Kinematics

The main contributions of the thesis in kinematics can be listed as:

i) group theoretic classification of multi-d.o.f. joints, and

ii) development of a generalized exponential formula for forward and differential kine-

matics of open-chain multi-body systems with multi-d.o.f. holonomic and/or non-

holonomic joints.

In the following, we detail different steps of this phase of the research, which is the content

of Chapter 2.

Displacement Subgroups

We start with the definition of joint as a distribution on the relative configuration mani-

fold of a body with respect to another body. This configuration manifold is diffeomorphic

to the Lie group SE(3). We observe that for a left invariant distribution (corresponding

to a joint) the involutivity of the distribution and closedness of the Lie bracket (of the

Lie algebra) coincide. Based on this observation, we show that the relative configuration

manifolds of lower kinematic pairs are indeed Lie subgroups of SE(3), and in Section 2.1

we generalize this class of multi-d.o.f. holonomic joints by introducing the notion of dis-

placement subgroups. In Table 2.1 we list different categories of displacement subgroups.

Accordingly, we use the exponential map for Lie subgroups of SE(3) to introduce a new

joint parametrization, called screw joint parameters. This joint parametrization is used

in Chapter 3 and 4 to embed an open subset of a quotient manifold in the relative con-

figuration manifold and in Chapter 5 to define the error function for the controller. We

study the relationship between the screw and classic joint parameters in Theorem 2.1.5.

We then define the nonholonomic constraints for a multi-d.o.f. joint in section 2.1.2. The

contribution of this part of the thesis is stated in Proposition 2.1.3, in which we prove the

surjectivity of the exponential map for all categories of displacement subgroups except

for the 2-d.o.f. prismatic + helical category of joints.


Forward and Differential Kinematics

The main contribution of this chapter is generalizing the existing product of exponential

formula [57] for forward and differential kinematics of open-chain multi-body systems to

include displacement subgroups, in Theorem 2.2.3 and 2.3.1. We accordingly derive a

modified Jacobian for the screw joint parameters in (2.3.13), by considering the nonholo-

nomic constraints. Finally in Section 2.4, we study different operators appearing in the

developed differential kinematics formulation using the standard basis for the Lie algebra

of SE(3). The results of this section are summarized in Proposition 2.4.5. To illustrate

the contents of Chapter 2, we present a detailed example in Section 2.5.

1.4.2 Dynamics

The main contributions of this phase of research are:

i) symplectic reduction of holonomic open-chain multi-body systems with multi-d.o.f.

joints and non-zero momentum as a generalization of the existing reduction methods

for free-base manipulators, which are for single-d.o.f. joints and zero momentum,

ii) generalization of the existing approaches to the reduction of nonholonomic Hamil-

tonian mechanical systems and its application to dynamical reduction of nonholo-

nomic open-chain multi-body systems with multi-d.o.f. joints, and

iii) unification of the developed reduction methods for holonomic and nonholonomic

cases.

In addition, a new approach to the derivation of Hamilton’s equation for holonomic

and nonholonomic Lagrangian systems is developed, using Lagrange-d’Alembert-Pontryagin

principle on Pontryagin bundle. (See Section 3.1 and 4.1.)

The study of the dynamical reduction of open-chain multi-body systems is the subject

of Chapters 3 and 4. Different steps of this part of the research are detailed in the

following.

Holonomic

Chapter 3 focuses on the case of holonomic Hamiltonian mechanical systems with symme-

try. We denote the symmetry group by G and its coadjoint isotropy group corresponding

to an element µ ∈ Lie∗(G) by Gµ. The Hamiltonian function H of a Hamiltonian mechan-

ical system consists of a quadratic term coming from the kinetic energy metric on the

configuration manifold Q plus the potential energy function. We revisit the dynamical


reduction of Hamiltonian mechanical systems with symmetry, using the symplectic re-

duction theorem. We also use the mechanical connection, which is a principal connection

compatible with the kinetic energy metric, to identify the symplectic reduced space with

a vector sub-bundle of T ∗(Q/Gµ).

One of the contributions of this chapter is identifying the relative configuration man-

ifold of the first joint of a holonomic open-chain multi-body system with displacement

subgroups as a symmetry group for the system (see Theorem 3.3.3). We then define

the notion of a holonomic open-chain multi-body system with symmetry. Consequently,

we apply the symplectic reduction procedure for Hamiltonian mechanical systems to

holonomic open-chain multi-body systems with symmetry. The main contribution of

this chapter is summarized in Theorem 3.3.6. In this theorem, we reduce the Hamil-

ton’s equation for a holonomic open-chain multi-body system with symmetry in T ∗Qto a Hamilton’s equation in the reduced phase space, which is a vector sub-bundle of

T ∗(Q/Gµ). This theorem generalizes the existing reduction methods for holonomic open-

chain multi-body systems at zero momentum, e.g., used in [17, 28, 90, 72, 74].

Nonholonomic

In Section 4.2, we consider nonholonomic Hamiltonian mechanical systems with sym-

metry, where the linear distribution corresponding to the nonholonomic constraints is

denoted by D. In this section we restrict our attention to the nonholonomic systems

with symmetry whose symmetry group has a Lie subgroup G that satisfies the Chaply-

gin assumption in (4.2.10). One of the contributions of this section is the proof of the

Chaplygin reduction theorem [42]. In Theorem 4.2.4, we state the Chaplygin reduction

theorem for the systems on T ∗Q. And, we give a proof that is independent of the choice

of local coordinate charts, and it illustrates the geometry behind the Chaplygin reduction

theorem. Using this proof, we geometrically show the similarities and distinctions be-

tween this reduction procedure and the symplectic reduction of holonomic Hamiltonian

mechanical systems with symmetry. The main difference between these two reduction

methods is that in the holonomic case the reduced phase space is a symplectic mani-

fold, as opposed to the almost symplectic manifold for the case of a Chaplygin system.

This proof can be used to unify the reduction processes developed for holonomic and

nonholonomic Hamiltonian mechanical systems with symmetry. Accordingly, we give a

nonholonomic version of Noether’s theorem for reduced Chaplygin systems in Proposi-

tion 4.2.12, which is equivalent to the theorem presented in Section 3 of [76]. Another

contribution of this section is using this proposition along with the almost symplectic

reduction presented in [69] to perform a three-step reduction of nonholonomic Hamilto-


nian mechanical systems with symmetry. The main results of this section are presented

in Proposition 4.2.14 and Theorem 4.2.18. Note that the three-step reduction process in

this section is a generalization of the 2-step reduction of Chaplygin systems presented in

[59]. To illustrate the contents of Chapter 3 and 4, we include three detailed case studies

in Sections 3.4 and 4.6.

In Section 4.3, we apply the developed reduction process to nonholonomic open-

chain multi-body systems with symmetry. We report the result of the first step of the

reduction process in Theorem 4.3.1, which is one of the main contributions of Chapter

4. Before performing the next steps of the reduction process, in Section 4.4 we present

a number of sufficient conditions, under which a nonholonomic open-chain multi-body

system admits a symmetry group bigger than G = Q1, which is one of the contributions

of this dissertation. Then, Theorem 4.5.2 finalizes Chapter 4 by performing the second

step of the reduction presented in Section 4.2 for nonholonomic open-chain multi-body

systems with symmetry. This theorem is one of the main contribution of this dissertation.

1.4.3 Controls

The main contributions of this research in controls can be listed as:

i) solving the input-output linearization problem in the reduced phase space of a free-

base, holonomic (with non-zero momentum) or nonholonomic controlled open-chain

multi-body system and multi-d.o.f. joints, and

ii) deriving a coordinate-independent, trajectory tracking, feed-forward servo control

law for concurrent control of the base and other extremities of a generic open-chain

multi-body system with multi-d.o.f. joints, and proving the exponential stability

of the closed-loop system by introducing a proper Lyapunov function.

In the following, we detail different steps of this phase of research.

Chapter 5 is devoted to the concurrent control of underactuated holonomic and non-

holonomic open-chain multi-body systems with displacement subgroups. We only restrict

our attention to systems in which there is no actuation in the directions of the group

action and nonholonomic constraints. We call such systems free-base, open-chain multi-

body systems. The control problem considered in this thesis is a trajectory tracking

problem for the extremities of an open-chain multi-body system. In order to formally

define this problem, we need to make sense of pose and velocity error on the output

manifold of a holonomic or nonholonomic open-chain multi-body system (see Section

5.2). For technical reasons we assume that the output manifold of the system can be


identified by a Lie subgroup of a Cartesian product of copies of SE(3). As a result,

we use the exponential map of Lie groups and right trivialization of the tangent bundle

of Lie groups to define an error function and connection on the output manifold. In

order to control the pose of the extremities in the inertial coordinate frame, we need not

only the reduced Hamilton’s equation but also the reconstruction equations. In Section

5.1.2, we derive the reconstruction equations for holonomic and nonholonomic open-chain

multi-body systems.

As mentioned above, one of the contributions of this dissertation is unification of

the reduction of holonomic and nonholonomic open-chain multi-body systems. This

enables us to develop a unified framework to derive control laws for both categories

of multi-body systems. In Section 5.3, we first show that a controlled holonomic or

nonholonomic open-chain multi-body system with symmetry is input-output linearizable

in the reduced phase space. This result generalizes the existing linearization methods

for underactuated, holonomic and nonholonomic mechanical systems presented, e.g., in

[2, 5, 28, 34, 60, 61], to include non-abelian symmetry groups, non-zero momentum

(of holonomic systems) and nonholonomic constraints. In addition, under a dimensional

assumption and feasibility of the desired trajectory we solve the inverse dynamics problem

for a generic holonomic or nonholonomic open-chain multi-body system with symmetry

in the reduced phase space.

Finally, in Theorem 5.4.2 (Section 5.4) we present a coordinate-independent, output

tracking, feed-forward servo control law for a holonomic or nonholonomic open-chain

multi-body system. And, using an appropriate Lyapunov function we prove that this

controller exponentially stabilizes the closed-loop system for any feasible trajectory of

the extremities. This control law depends only on the elements of the reduced phase

space and the symmetry group, and it is independent of the velocity of the system in the

directions of the group action.

1.4.4 Produced Manuscripts

Four manuscripts [20, 19, 18, 21] have been produced for publication (one is accepted for

publication), as listed in the following:

i) R. Chhabra and M.R. Emami. A Generalized Exponential Formula for Forward

and Differential Kinematics of Open-chain Multi-body Systems. Accepted in Mech-

anism and Machine Theory, September 2013.

ii) R. Chhabra and M.R. Emami. Symplectic Reduction of Holonomic Open-chain

Multi-body Systems with Constant Momentum. Submitted to Multibody System


Dynamics, September 2013.

iii) R. Chhabra and M.R. Emami. A Geometric Approach to Dynamical Reduction of

Open-chain Multi-body Systems with Nonholonomic Constraints. Submission to

Mechanism and Machine Theory, October 2013.

iv) R. Chhabra and M.R. Emami. A Unified Approach to Input-output Linearization

and Concurrent Control of Underactuated Holonomic and Nonholonomic Open-

chain Multi-body Systems. Submission to Journal of Dynamical and Control Sys-

tems, October 2013.

1.5 Outline of the Thesis

A brief outline of the content of different chapters of this dissertation is as follows:

Chapter 2: In this chapter we study the kinematics of holonomic and nonholonomic open-chain

multi-body systems with multi-d.o.f. joints.

Chapter 3: This chapter is devoted to the study of the symplectic reduction of holonomic

Hamiltonian mechanical systems with symmetry and its application to holonomic

open-chain multi-body systems.

Chapter 4: This chapter presents a three-step reduction method for nonholonomic Hamiltonian

mechanical systems with symmetry and its application to nonholonomic open-chain

multi-body systems.

Chapter 5: An output tracking, feed-forward servo control law in the reduced phase space of

a generic free-base holonomic or nonholonomic open-chain multi-body system is

developed in this chapter, and exponential stability of the closed-loop system is

proven.

Chapter 6: This chapter includes some concluding remarks and states some future directions

of the research presented in this dissertation.

Chapter 2

A Generalized Exponential Formula

for Forward and Differential

Kinematics of Open-chain

Multi-body Systems

This chapter presents a generalized exponential formula for Forward and Differential

Kinematics of open-chain multi-body systems with multi-degree-of-freedom, holonomic

and nonholonomic joints. We revisit the notion of displacement subgroup, and show

that the relative configuration manifolds of such joints are Lie groups. Accordingly, we

categorize displacement subgroups, and prove that except for one class of displacement

subgroups the exponential map is surjective. Screw joint parameters are defined to

parametrize the relative configuration manifolds of displacement subgroups using the

exponential map of Lie groups. For nonholonomic constraints, the admissible screw

joint speeds are introduced, and the Jacobian of the open-chain multi-body system is

modified accordingly. Then by assigning coordinate frames to the initial configuration

of the multi-body system, employing the matrix representation of SE(3) and choosing

a basis for se(3), we explore the computational aspects of the developed formulation

for Forward and Differential Kinematics of open-chain multi-body systems. Finally, we

study the developed formulation for an example of a mobile manipulator mounted on a

spacecraft, i.e., on a six-degree-of-freedom moving base.

15

Chapter 2. A Generalized Exponential Formula for Kinematics 16

2.1 Holonomic and Nonholonomic Joints

A physical 3-dimensional (3D) space can be mathematically modelled as a 3D affine

space, denoted by A, which is modelled on a vector space V , and a rigid body B is the

closure of a bounded open subset of A. Let us fix a coordinate frame in the physical

space. Considering a multi-body system MS(N) = (Ai, Bi)|Bi ⊂ Ai, i = 0, ..., N and

a body Bi in it, the space of all absolute poses (position and orientation) of Bi with

respect to the fixed coordinate frame is then Gi = SE(3), Special Euclidean group. One

can also introduce the relative pose of two bodies of a multi-body system. Let Bi and

Bj be two bodies in MS(N), the space of all relative poses of Bi with respect to Bj

forms a smooth manifold P ji :=

g−1j · gi

∣∣ gi ∈ Gi = SE(3), gj ∈ Gj = SE(3) ∼= SE(3).

When i = j this manifold, which is the space of all possible coordinate transformations

of Ai, inherits Lie group structure isomorphic to Gi = SE(3) with the identity element

ei and the Lie algebra denoted by Lie(P ii ). In the case of i = j, to simplify the notation

only the lower index is used, e.g., Pi := P ii . A relative motion of Bi with respect to

Bj is a smooth curve rji : [0, 1] → P ji , and the relative velocity at time t is the vector

vji (t) = (drji /dt)(t) ∈ Trji (t)Pji , where Trji (t)

P ji is the tangent space of P j

i at the element

rji (t). At each instant t, one can show that this vector induces a vector field Xt on Aj

corresponding to the relative motion of Bi with respect to Bj such that ∀a ∈ Aj,

Xt(a) = limδ→0

exp(δ(Trji (t)

Rrij(t)

)vji (t)

)(a)− (a)

δ; (2.1.1)

where Rrij(t): P j

i → Pj denotes the right composition map by rij(t). If we identify the

manifolds P ji and Pj by the Lie group SE(3), the right composition map becomes just

the right translation map by rij(t) =(rij(t)

)−1 ∈ SE(3). For a relative motion, if this

vector field is independent of time, the relative motion is called relative screw motion. In

other words, a relative screw motion is the curve on P ji corresponding to the flow of a

left-invariant Killing vector field [82] on Pj. An interpretation of the Chasles’ Theorem

indicates that from any initial relative pose, any relative pose of Bi with respect to Bj can

be reached by a relative screw motion. Therefore, the exponential map of SE(3) ∼= P ji

is onto [57].

Given two rigid bodies of a multi-body system, Bi and Bj, a joint is a mechanism

that restricts the relative motion of Bi with respect to Bj, and specifies a subset Dji

of TP ji . A joint may be time dependant, called rheonomic joint, or time independent,

which is called scleronomic joint. A special type of scleronomic joints, which is mostly

considered in the literature, is when we have Dji ⊆ TP j

i being a distribution on P ji that


corresponds to admissible directions of the relative velocity of Bi with respect to Bj.

We only consider this category of joints in this thesis. In particular, we assume that Dji

has constant rank. If Dji is involutive, i.e. its space of sections is closed under the Lie

bracket of vector fields, the joint is called holonomic; otherwise, it is a nonholonomic

joint. For any non-involutive distribution Dji , under the existence assumption, let Dj

i be

the involutive closure of Dji . The involutive closure of a distribution Dj

i is the smallest

vector sub-bundle of TP ji containing Dj

i that is closed under the Lie bracket of vector

fields. Based on the global Frobenius Theorem [43], either Dji or Dj

i (for a holonomic

or nonholonomic joint) gives a foliation on P ji . The leaf Qj

i ⊆ P ji that contains the

initial relative pose of Bi with respect to Bj, rji,0, is called the relative configuration

manifold. The manifold Qji is the space of all admissible relative poses considering the

joint constraints. The dimension of this manifold, k, is called the number of d.o.f. of

a joint, which is greater than or equal to the dimension of the joint distribution for a

nonholonomic or holonomic joint, respectively.

One can define the submanifold Qi ⊆ Pi as the left composition of Qji by rij,0, i.e.,

Qi = Lrij,0(Qji ), where rji,0 rij,0 = ej and rij,0 r

ji,0 = ei. This submanifold contains

the identity element of Pi, which corresponds to rji,0 ∈ Qji . A local coordinate chart

for a neighbourhood W ⊂ Qi of ei is a diffeomorphism ϕ : Rk ⊃ U → W such that

ϕ([0, ..., 0]T ) = ei. Therefore, any element rji ∈ Lrji,0(W ) ⊆ Qj

i can be parametrized

by a q ∈ U , which is called the classic joint parameter, through the diffeomorphism

Lrji,0 ϕ. A velocity vector vji ∈ TrjiQ

ji can also be identified with a k-dimensional vector

q ∈ TqU ∼= Rk by the linear isomorphism (Tϕ(q)Lrji,0)(Tqϕ). Note that the coordinate

chart ϕ induces a basis ( ∂∂qb

)|q|b = 1, ..., k for Tϕ(q)W , where qb is the bth element of q,

and in this basis Tqϕ is the identity matrix, idk.

2.1.1 Displacement Subgroups

In this subsection, displacement subgroups are defined as a class of holonomic joints,

and it is shown that their relative configuration manifolds are connected Lie groups.

In Proposition 2.1.3, the necessary and sufficient conditions for the surjectivity of the

exponential map of these relative configuration manifolds are given. Based on this iden-

tification of displacement subgroups, a set of new joint parameters, called screw joint

parameters, is introduced. These joint parameters can be physically interpreted as the

initial classic joint speeds for a screw motion on the corresponding relative configuration

manifold. Finally, the relationship between the screw joint parameters and the classic

joint parameters is formalized in Theorem 2.1.5.


For a holonomic joint, define the distribution Dj := TrjiRrij,0

(Dji ) ⊆ TPj. Based on the

definition of a holonomic joint, Dj is involutive, i.e., its space of sections is closed under

the Lie bracket of vector fields on Pj. This bracket coincides with the definition of the

Lie bracket [41] on Lie(Pj) if Dj is left-invariant, i.e., Dj(rj) = TejLrj(Dj(ej)),∀rj ∈ Pj.We denote the integral manifold of Dj containing ej by Qj ⊆ Pj. Particularly, Dj(ej),

which is a linear subspace of Lie(Pj), is closed under the Lie bracket of Lie(Pj); hence

TejQj = Dj(ej) is a Lie sub-algebra of Lie(Pj).

Proposition 2.1.1. For a holonomic joint, if Dj (defined above) is left-invariant, its

integral manifold containing ej, i.e., Qj ⊆ Pj, is the unique k-dimensional connected Lie

subgroup of Pj with the Lie algebra Lie(Qj) = Dj(ej).

Note that conversely, for any Lie subgroup Q′j ⊆ Pj, there exists a unique involutive

distribution corresponding to a holonomic joint, by left translating Lie(Q′j) over Pj and

right composing it with rji,0.

Definition 2.1.2. A holonomic joint is called displacement subgroup if the corresponding

distribution Dj (defined above) on Pj is left-invariant.

Therefore, based on Proposition 2.1.1 and since Pj ∼= SE(3), different types of dis-

placement subgroups are identified by the connected Lie subgroups of SE(3), up to

conjugation, which are tabulated in Table 2.1 [38, 71]. In this table, Hp is the Lie sub-

group of SE(3) corresponding to a simultaneous rotation about and translation along a

vector in R3, where the ratio of translation to rotation is equal to the constant p. From

this table, one can observe that the displacement subgroups consist of the six lower kine-

matic pairs, i.e., revolute, prismatic, helical, cylindrical, planar and spherical joints, and

combinations of them. Therefore, in this joint categorization, the relative configuration

manifolds of lower kinematic pairs are indeed subgroups of SE(3). There also exist other

types of holonomic joints, e.g., universal joint and higher kinematic pairs, which are not

included in the category of displacement subgroups. However, the relative configuration

manifolds of these joints are not subgroups of SE(3). To parametrize the relative config-

uration manifolds of these joints one needs a product of exponentials of some elements

of a basis for the tangent space of the relative configuration manifold at the identity

element.

Proposition 2.1.3. The group exponential map exp: Lie(Qj)→ Qj is surjective for all

categories of displacement subgroups, except for a three-d.o.f. joint where a helical joint is

combined with a two-d.o.f. prismatic joint such that the helical joint axis is perpendicular

to the plane of the prismatic joint. This case is considered as two separate joints in this

thesis.


Table 2.1: Categories of displacement subgroups [38, 71]

Dim. Subgroups of SE(3)/displacement subgroups

6 SE(3) = SO(3)nR3

freea

4 SE(2)× Rplanar+prismaticb

3 SE(2) = SO(2)nR2

planarSO(3)ball (spherical)

R3

3-d.o.f. prismaticHp nR2

helical + 2-d.o.f. prismaticc

2 SO(2)× Rcylindricald

R2

2-d.o.f. prismatic1 SO(2)

revoluteRprismatic

Hp

helical0 e

fixeda

a These two subgroups are the trivial subgroups of SE(3).b The axis of the prismatic joint is always perpendicular to the plane of the planar joint.c The axis of the helical joint is always perpendicular to the plane of the 2-d.o.f. prismatic joint.d The axis of the revolute and prismatic joints are always aligned.

Since this proposition is proved by coordinate chart assignment, its proof is presented

in Section 2.4.

Definition 2.1.4. Let ϕ be a coordinate chart for a neighbourhood of ei. By Propo-

sition 2.1.3 any relative configuration manifold Qji of a displacement subgroup can be

parametrized by vectors s ∈ Rk, called screw joint parameters, such that every rji ∈Qji ⊆ P j

i can be expressed as

rji = exp(τ ji s) rji,0 := exp

((Adrji,0

)(Teiι)(T0ϕ)s) rji,0, (2.1.2)

where ι : Qi → Pi is the inclusion map.

Therefore, for a relative motion rji : [0, 1]→ Qji the relationship between (s, s), which

are the screw joint parameters and their speeds, and (q, q), which are the classic joint

parameters and their speeds, can be summarized in the following theorem. In this the-

orem, ∀η ∈ Lie(Qj) adη : Lie(Qj) → Lie(Qj) is the endomorphism of Lie(Qj) such

that ∀ξ ∈ Lie(Qj) we have adη(ξ) := [η, ξ] [41]. The linear map Z(s) (defined in The-

orem 2.1.5) is an isomorphism between T0Rk and TqRk if and only if adT0ϕ(s) has no

eigenvalue in 2πiZ, where i =√−1.

Theorem 2.1.5. For a displacement subgroup, consider a coordinate chart for Qi, ϕ : Rk ⊃U → W such that ϕ([0, ..., 0]T ) = ei, and a relative motion rji : [0, 1] → Qj

i in the neigh-

bourhood of rji,0, denoted by W ′ := Lrji,0(W ) ⊆ Qj

i . Then, rji (t) = exp(τ ji s(t)) rji,0 where


s(0) = 0, and

q(s) = ϕ−1 exp(T0ϕ s), (2.1.3a)

q(s, s) = Z(s)s

:= (Tq(s)ϕ)−1TejLexp(T0ϕs)

(∫ 1

0

exp(−x adT0ϕs) dx

)T0ϕ s. (2.1.3b)

Proof. For the relative motion rji ⊂ W ′, let ri = Lrij,0 rji ⊂ W be the corresponding

curve on Qi. This curve on Pi is ι ϕ(q) = Lrij,0 Rrji,0 exp(τ ji s) = Krij,0

exp(τ ji s).

Based on (2.1.2) and the fact that exponential map is compatible with the Lie group

homomorphisms [41], in this case conjugation and inclusion map, ιϕ(q) = Krij,0Krji,0

ι exp(T0ϕs) = ι exp(T0ϕs). Therefore, (2.1.3a) is true since the inclusion map ι is an

embedding, and ϕ is a diffeomorphism.

Differentiating (2.1.3a) with respect to the curve parameter results in

q =(Texp(T0ϕs)ϕ

−1)

(TT0ϕs exp)T0ϕs = (Tqϕ)−1 (TT0ϕs exp)T0ϕs.

For a Lie group G, it can be shown that the differential of the exponential map at

ξ ∈ Lie(G) is [30]

Tξ exp = TeLexp(ξ)

∫ 1

0

exp(−x adξ)dx. (2.1.4)

Hence, substituting (2.1.4) and (2.1.3a) in the above equation completes the proof for

(2.1.3b).

In (2.1.3b), Z(s) is defined as the composition of several linear operators, and it is

invertible if and only if all of the linear operators are invertible. Since left translation is

a global diffeomorphism and ϕ is a coordinate chart, it suffices to check the conditions

under which Θ :=∫ 1

0exp(−x adT0ϕs) dx is invertible. For z ∈ C, consider the solution of∫ 1

0exp(−x z) dx that is equal to the entire holomorphic function f(z) = 1−exp(−z)

zsuch

that f(0) = 1. Thus, the eigenvalues of Θ are equal to 1−exp(−λi)λi

, where λi’s are the

eigenvalues of adT0ϕs. The Lie algebra endomorphism Θ is invertible if and only if it has

no eigenvalues equal to zero, i.e., λi 6∈ 2πiZ where i =√−1.

This theorem gives a condition for the size of the image of the coordinate chart

associated with the screw joint parametrization. On Pj ∼= SE(3) this condition dictates

that the coordinate chart cannot include elements of Pj corresponding to 2π radian

rotation about an axis in Aj. Also, note that the integral term in (2.1.4) is equal to

the identity map for abelian Lie groups, and in general this term corresponds to the

non-commutativity of ξ, ξ ∈ Lie(Qj) with respect to the Lie bracket.


2.1.2 Nonholonomic Displacement Subgroups

A nonholonomic displacement subgroup is a displacement subgroup together with k lin-

early independent constraints in the space of the speeds of the classic joint parameters

that are not integrable, i.e., C(q)q = 0, where C(q) ∈ Rk×k, and C(q) is assumed to be

a differentiable linear operator on Qi. In other words, for the neighbourhood W of the

initial relative pose rji,0, ∀q ∈ U ⊂ Rk q ∈ TqRk should lie in the ker(C(q)) ∼= Rk−k that

can be considered as the range of another linear operator C(q), i.e., C(q)C(q) = 0. The

C(q) ∈ Rk×(k−k) is a differentiable linear operator on Qi of constant rank k − k. This

linear operator identifies a smooth non-involutive distribution on Qji corresponding to the

space of all admissible instantaneous relative velocities of the joint. Therefore, an admis-

sible joint speed has the form q = C(q) ˙q ∀ ˙q ∈ Rk−k. Note that the representation of C(q)

in the local coordinates is not unique, and it could be chosen such that the admissible

classic joint speeds are collocated with the joint control forces and torque to simplify the

dynamic analysis. Based on (2.1.3b) in Theorem 2.1.5 and considering the screw joint

parameters, the space of all admissible screw joint speeds at s can be identified by

s = Σ(s) ˙s := Z−1(s)C(q(s)) ˙s. ∀ ˙s ∈ Rk−k (2.1.5)

2.2 Forward Kinematics

Definition 2.2.1. An open-chain multi-body system is a multi-body system MS(N)

together with N − 1 joints between the bodies, such that there exists a unique path

between any two bodies of the multi-body system. In an open-chain multi-body system,

bodies with only one neighbouring body are called extremities.

In robotics, the relative pose and velocity of the extremities with respect to a base

body, labeled as B0 in MS(N), is usually of interest. The base body is possibly an inertial

observer.

Definition 2.2.2. A branch of an open-chain multi-body system is a chain of m+1 ≤ N

bodies together with m joints that connects B0 to an extremity.

In this chapter, an open-chain multi-body system is assumed to have n branches with

both holonomic and nonholonomic multi-d.o.f. joints. In the branch i, joint j connects

body Bj−1 to Bj. The branch configuration ri is defined as the collection of the relative

poses of rigid bodies, i.e., ri :=(r0

1, ..., rmi−1mi

)∈ Q0

1 × ...×Qmi−1mi

.

Index the jth body of the branch i by ji. Let kji be the number of d.o.f. of the joint j

in the ith branch, for an initial branch configuration, the set of all screw joint parameters


of the branch is denoted by Gi :=

is =[isT1 , ...,

isTmi

]T |isj ∈ Rkji , j = 1, ...,mi

. For-

ward Kinematics of the ith branch of an open-chain multi-body system is a smooth map

FKi from the set of screw joint parameters of the branch to P 0mi

for an initial branch

configuration that indicates the relative pose of the body Bmiwith respect to B0, i.e.,

FKi : Gi → P 0mi

such that FKi(is) := r0

1 ... rmi−1mi

.

Theorem 2.2.3. For an open-chain multi-body system MS(N) along with N holonomic

and nonholonomic displacement subgroups, the generalized exponential formula for the

Forward Kinematics map corresponding to the ith branch can be formulated as

FKi(is) = exp

(0τ 0

1is1

) ... exp

(0τmi−1mi

ismi

) r0

mi, (2.2.6)

where 0τ j−1j = (Adr0

j,0)(Tejιj)(T0ϕj), ιj : Qj → Pj is the inclusion map, and ϕj is a coordinate

chart for a neighbourhood of ej ∈ Pj ∀j = 1, ...,mi.

Proof. Using the screw joint parameters and the definition of the Forward Kinematics

map,

FKi(is) =

(exp(τ 0

1is1) r0

1,0

) ...

(exp(τmi−1

mi

ismi) rmi−1

mi,0

).

Due to the fact that rj−1j,0 = rj−1

0,0 r0j,0, associativity of the composition operator, and

compatibility of the exponential map with the conjugation map,

FKi(is) = exp(τ 0

1is1)

(r0

1,0 exp(τ 12

is2) r10,0

) ...

(r0mi−1,0 exp(τmi−1

mi

ismi) rmi−1

0,0

) r0

mi,0

= exp(τ 01

is1) exp(Adr01,0

(τ 12

is2)) ... exp(Adr0mi−1,0

( τmi−1mi

ismi)) r0

mi,0.

Substituting the definition of τ j−1j , ∀j = 1, ...,mi, from (2.1.2) completes the proof.

Note that since Forward Kinematics is only a function of the relative poses, non-

holonomic constraints do not appear in (2.2.6). Forward Kinematics of an open chain

multi-body system, FK, is defined as the collection of the relative poses of the extrem-

ities with respect to the base body B0, i.e., FK : G1 × ... × Gn → P 0m1× ... × P 0

mn such

that

FK(s) :=

FK1(1s)

...

FKn(ns)

,where s = [1sT , ..., nsT ]T .

For a serial-link multi-body system MS(N) with one-d.o.f. revolute and/or prismatic

joints, sj(t) ∀j = 1, ..., N is a real number function, instead of a vector function. Based


on the interpretation of the screw joint parameters given in the beginning of Subsec-

tion 2.1.1, sj(t) is the constant speed of a classic joint parameter during a screw motion

from 0 to qj(t), in the interval of [0,1]. Therefore, its number is equal to the corresponding

classic joint parameter. Moreover, since the joint has only one d.o.f., the linear operator0τ j−1

j reduces to the joint screw at the initial configuration, which corresponds to the axis

of rotation for a revolute joint or the direction of translation for a prismatic joint [57, 71].

Consequently, it can be shown that in this special case the formulation for Forward Kine-

matics of an open-chain multi-body system is equivalent to the product of exponentials

formula suggested by Brockett [11]. This relationship is further illustrated in the case

study in Section 2.5.

2.3 Differential Kinematics

For the ith branch of an open-chain multi-body system, Differential Kinematics is a

linear map that relates the speed of the screw joint parameters of the branch to the

instantaneous relative twist of Bmiwith respect to B0 and observed in A0, i.e., expressed

in the vector space associated with A0, V0. The corresponding linear operator 0J0mi

(is),

called the Jacobian, for an initial branch configuration is 0J0mi

(is) : TisGi → Lie(P0) such

that 0J0mi

(is) :=(TFKi(is)R(FKi(is))−1

)TisFKi.

Theorem 2.3.1. For an open-chain multi-body system MS(N) along with N holonomic

displacement subgroups, the generalized exponential formula for the Jacobian of the branch

i can be formulated as

0J0mi

(is) =[(

∆10τ 0

1

) (exp

(ad0τ0

1is1

)∆2

0τ 12

)· · ·(

exp(

ad0τ01

is1

)... exp

(ad0τ

mi−2mi−1

ismi−1

)∆mi

0τmi−1mi

)], (2.3.7)

where ∆j :=∫ 1

0exp(x ad0τ j−1

j (isj))dx is an endomorphism of Lie(P0).

Proof. Consider a curve is : [0, 1] → Gi, such that t 7→i s(t), in the set of screw joint

parameters of the branch i. Let γj(t) := exp(0τ j−1j

isj(t)) ∀j = 1, ...,mi. Using (2.2.6) and

the product rule for Lie groups,

d

dtFKi(

is(t)) = Tis(t)FKiis(t) =

(Tγ1Rγ2...γmi

r0mi,0

)γ1

+(Tγ2...γmi

r0mi,0

Lγ1

)(Tγ2Rγ3...γmi

r0mi,0

)γ2 + ...

+(Tγmi

r0mi,0

Lγ1...γmi−1

)(Tγmi

Rr0mi,0

)γmi

.


By the definition of the Differential Kinematics map and rearranging the differential of

the right and left composition maps,

0J0mi

(is) is =(Tγ1Rγ−1

1

)γ1 +

(Tγ1γ2R(γ1γ2)−1

)(Tγ2Lγ1) γ2 + ...

+(Tγ1...γmi

R(γ1...γmi)−1

)(Tγmi

Lγ1...γmi−1

)γmi

. (2.3.8)

Now, use (2.1.4) for the exponential map exp: Lie(P0)→ P0, and the equality of opera-

tors [41]

Adexp(ξ) = exp(adξ), ∀ξ ∈ Lie(P0) (2.3.9)

to calculate γj(t) = (Te0Lγj)(∫ 1

0Adexp(−x 0τ j−1

jisj)dx)

0τ j−1j

isj(t). Substitute γj and use

the identity Adr := TrRr−1 Te0Lr ∀r ∈ P0 in (2.3.8) to achieve

0J0mi

(is) is = Adγ1

(∫ 1

0

Adexp(−x 0τ01

is1)dx

)0τ 0

1is1 + ...

+ Adγ1...γmi

(∫ 1

0

Adexp(−x 0τ

mi−1mi

ismi)dx

)0τmi−1mi

ismi. (2.3.10)

Define ∆j ∀j = 1, ...,mi as

∆j := Adγj

(∫ 1

0

Adexp(−x 0τ j−1j

isj)dx

)=

∫ 1

0

Adexp((1−x) 0τ j−1j

isj)dx

=

∫ 1

0

exp(x ad0τ j−1

jisj

)dx,

where the first equality holds since [x 0τ j−1j

isj,0 τ j−1

jisj] = 0, and the second equality is

the consequence of a change of variable and using (2.3.9). Finally, by substituting ∆j

in (2.3.10) and employing the equality of operators in (2.3.9) one can show the desired

expression for the Jacobian in (2.3.7) .

For a serial-link multi-body system with one-d.o.f. revolute and/or prismatic joints,

since sj(t) is a real number function, 0τ j−1j sj(t) ∈ Lie(P0) and 0τ j−1

j sj(t) ∈ Lie(P0) com-

mute, i.e., [0τ j−1j sj,

0 τ j−1j sj] = 0, and hence ∆j becomes the identity map. In this case,

the developed formulation simplifies to the existing product of exponentials formula for

Differential Kinematics [57, 71].

Based on the definition of the Differential Kinematics map, 0J0mi

(is)is is the twist of

Bmiwith respect to B0 and expressed in A0. This twist can be viewed in the affine space


attached to the body j of the branch i, Aji , using the Adjoint operator, i.e.,

jiJ0mi

(is) = Adrji0 (is)

0J0mi

(is), (2.3.11)

where according to (2.2.6) r0ji(is) = exp

(0τ 0

1is1

) ... exp

(0τ j−1

jisj) r0

ji,0. In addition,

following the same calculations performed in the proof of Theorem 2.3.1, the Jacobian

for the instantaneous relative twist of the body Bj with respect to Bl in the ith branch of

MS(N) and observed in A0, i.e., 0J lj (

is) j > l > 0, can be determined to be the truncated

version of the Jacobian in (2.3.7):

0J lj (

is) =[exp

(ad0τ0

1is1

)... exp

(ad0τ l−1

l

isl

)∆l+1

0τ ll+1 · · ·

exp(

ad0τ01

is1

)... exp

(ad0τ j−2

j−1isj−1

)∆j

0τ j−1j

]. (2.3.12)

In order to include the nonholonomic constraints in the Jacobian of the ith branch of

MS(N), one can define admissible screw joint speeds according to (2.1.5). Therefore,

the Jacobian in (2.3.7) can be modified to introduce the modified Jacobian for the ith

branch of a multi-body system consisting of both holonomic and nonholonomic joints.

0J0mi

(is) := 0J0mi

(is)diag(Σ1(is1), · · · ,Σmi

(ismi))

; (2.3.13)

where diag(Σ1(is1), · · · ,Σmi

(ismi))

is the block diagonal matrix of its entries, and Σj =

idkji for a holonomic joint. The modified Jacobian is a linear operator from the space of all

admissible screw joint speeds, i.e., Gi :=

i ˙s =[i ˙sT1 , ...,

i ˙sTmi

]T |i ˙sj ∈ Rkji−kji , j = 1, ...,mi

,

to Lie(P0). For an open-chain multi-body system MS(N), the modified Jacobian is

defined as the collection of the modified Jacobians of the extremities with respect to

the base body and observed in A0, i.e., J(s) ˙s := diag(

0J0m1

(1s), ...,0 J0mn(ns)

)˙s, where

˙s =[

1 ˙sT , ...,n ˙sT]T

.

2.4 Coordinate Assignment

At the computational level, consider a base point Oi for the affine space Ai in a multi-

body system MS(N). Every point in this affine space can now be realized by a vector

in Vi ∼= R3 through the action of (Vi,+) on Ai [7]. Therefore, any relative pose rji ∈P ji can be represented by an orientation preserving isometry, Hj

i : Vi → Vj such that

Hji := σOj rji (σOi)

−1 ∈ SE(3), where σOl : Vl → Al for l = i, j is the map induced by

the vector space action of Vl on Al. A matrix representation of SE(3) is the group of


orientation preserving linear isometries of R4 that preserve the plane x4 = 1 [7], i.e.,

SE(3) ∼=

Hji =

[Rji pji

01×3 1

]|Rj

i ∈ SO(3), pji ∈ R3

,

where Rji is the rotation matrix whose columns are the elements of a basis for Vi expressed

in terms of a basis for Vj and pji is the position of the point rji (Oi) from Oj and expressed

in Vj. In this representation, the Lie algebra of SE(3) is denoted by

se(3) ∼=

T ji =

[ωji wji

01×3 0

]|ωji ∈ so(3), wji ∈ R3

,

where wji is the relative velocity of the point rij(Oj) with respect to Oj and expressed

in Vj. The element ωji ∈ so(3) corresponds to the relative angular velocity of Bi with

respect to Bj and expressed in Vj, and it can be identified with the column vector

ωji = [ω1 ω2 ω3]T ∈ R3. This identification is through the following equality:

ωji =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

.

By choosing a basis for se(3) asE1 :=

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

, E2 :=

0 0 0 0

0 0 0 1

0 0 0 0

0 0 0 0

, E3 :=

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

,

E4 :=

0 0 0 0

0 0 −1 0

0 1 0 0

0 0 0 0

, E5 :=

0 0 1 0

0 0 0 0

−1 0 0 0

0 0 0 0

, E6 :=

0 −1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

,

and using the propositions presented in the sequel, one can perform the computations

for Forward and Differential Kinematics in the matrix representation of SE(3).

Proposition 2.4.1. For any element ξ = [wT , ωT ]T ∈ se(3), where ω,w ∈ R3, ω 6= 0,


expressed in the basis E1, ..., E6,

exp(ξ) =

[exp(ω) (id3 − exp(ω)) ωw

‖ω‖2 + ωωTw‖ω‖2

01×3 1

], (2.4.14)

where‖ · ‖ is the Euclidean norm of R3 and exp(ω) is evaluated using the Rodrigues’

formula for the exponential of skew-symmetric matrices,

exp(ω) = id3 +ω

‖ω‖sin(‖ω‖) +

ω2

‖ω‖2(1− cos(‖ω‖)). (2.4.15)

When ω = 0, exp(ξ) =

[id3 w

01×3 1

].

Proof. See Appendix A in [57].

Now, using the matrix representation of SE(3) and the above proposition, the proof

for Proposition 2.1.3 is presented.

Proof. (Proposition 2.1.3) In the matrix representation, the exponential map for a

connected Lie subgroup of SE(3) coincides with the restriction of the matrix exponential

to the Lie sub-algebra corresponding to the subgroup. Up to conjugation, all of the

connected Lie subgroups of SE(3) are listed in Table 2.1. Hence, to prove this proposition,

it suffices to check the surjectivity of the exponential map for the matrix representation

of each connected Lie subgroup, individually. Consider the following proposition and two

lemmas.

Proposition 2.4.2 (Chasles’ Theorem [57]). Every relative pose of a rigid body can be

realized by a rotation about an axis combined with a translation parallel to that axis. In

other words, the exponential map of the Lie group SE(3) is surjective.

Lemma 2.4.3. The exponential map of a compact, connected Lie group is surjective [30].

Lemma 2.4.4. For a vector space V, Lie(V) = V with zero Lie bracket, and the expo-

nential map is the identity map, i.e., exp(v) = v, ∀v ∈ V.

Based on the Chasles’ Theorem and the above lemmas, the exponential maps of the

subgroups SE(3), and SO(2), SO(3), R, R2 and R3 are surjective. In addition, since

SO(2) × R is the direct product of two subgroups with surjective exponential maps, its

own exponential map is also surjective.


In the following, we check the surjectivity of the exponential map for the remaining

four non-trivial Lie subgroups of SE(3), i.e., Hp, SE(2), SE(2)×R and HpnR2, respec-

tively. The subgroup Hp with p 6= 0 is a one dimensional subgroup of SE(3) that can be

represented as

Hp∼=

cos(θ) − sin(θ) 0 0

sin(θ) cos(θ) 0 0

0 0 1 pθ

0 0 0 1

|θ ∈ R

. (2.4.16)

It is easy to check that the Lie algebra of Hp is

Lie(Hp) = TidHp = spanR

Ep :=

0 −1 0 0

1 0 0 0

0 0 0 p

0 0 0 0

. (2.4.17)

Therefore, based on (2.4.14),

∀H =

h11 h12 0 0

h21 h22 0 0

0 0 1 h34

0 0 0 1

∈ Hp

there exists θ = h34/p such that exp(θEp) = H. For

SE(2) = SO(2) nR2 ∼=

cos(θ) − sin(θ) 0 x

sin(θ) cos(θ) 0 y

0 0 1 0

0 0 0 1

|θ ∈ S1, x, y ∈ R

, (2.4.18)

the corresponding Lie algebra is spanRE1, E2, E6. Based on Lemma 2.4.4,

∀H =

1 0 0 h14

0 1 0 h24

0 0 1 0

0 0 0 1

∈ SE(2),


exp(h14E1 + h24E2) = H, and otherwise for a general element of SE(2),

H =

h11 h12 0 h14

h21 h22 0 h24

0 0 1 0

0 0 0 1

∈ SE(2),

there exists θ = atan2(h21, h11), where, based on (2.4.14), one has

exp

(θE6 +

(θh24

2+θh14

2cot(

θ

2)

)E1 +

(θh24

2cot(

θ

2)− θh14

2

)E2

)

=

cos(θ) − sin(θ) 0

sin(θ) cos(θ) 0

0 0 1

x′

y′

z′

[0 0 0

]1

, (2.4.19)

where x′

y′

z′

=

1− cos(θ) sin(θ) 0

− sin(θ) 1− cos(θ) 0

0 0 0

0 −1

θ0

1θ

0 0

0 0 0

θh24

2+ θh14

2cot( θ

2)

θh24

2cot( θ

2)− θh14

2

0

=

h14

h24

0′

. (2.4.20)

Hence, the exponential map of SE(2) is surjective, and since SE(2) × R is the direct

product of two subgroups with surjective exponential maps, its own exponential map is

also surjective.

In the case of

Hp nR2 ∼=


sin(θ) cos(θ) 0 y

0 0 1 pθ

0 0 0 1

|θ, x, y ∈ R

, (2.4.21)


the Lie algebra is equal to spanREp, E1, E2. If θ ∈ 2πZ \ 0, then

H =

1 0 0 x

0 1 0 y

0 0 1 pθ

0 0 0 1

∈ Hp nR2,

and there does not exist any τ ∈ spanREp, E1, E2 such that exp(τ) = H. Therefore,

for Hp nR2 the exponential map is not surjective.

The following proposition presents closed form formulae for exp(adxi), for any ξ ∈se(3), and its integral that are used in the Differential Kinematics of open-chain multi-

body systems with displacement subgroups.

Proposition 2.4.5. For any element ξ = [wT , ωT ]T ∈ se(3), where ω,w ∈ R3 and ω 6= 0,

expressed in the basis E1, ..., E6,

adξ =

[ω w

03×3 ω

],

exp(adξ) =

[exp(ω) 1

‖ω‖2 [[ω, w], exp(ω)] + ωωTw‖ω‖2 exp

(ωωTw‖ω‖2

)03×3 exp(ω)

], (2.4.22)

where [·, ·] is the matrix commutator, exp(ω) is evaluated using (2.4.15) and,

∫ 1

0

exp(x adξ) dx =

[M1 M2

03×3 M1

], (2.4.23)

where,

M1 = id3 + ω‖ω‖2 (1− cos(‖ω‖)) + ω2

‖ω‖2

(1− 1

‖ω‖ sin(‖ω‖))

, and

M2 = 1‖ω‖2 [[ω, w],M1]− ω

ωTw+(

ωωTw− ω2

‖ω‖2

)cos(ωTw‖ω‖

)+(

ω‖ω‖ + ω2

‖ω‖ωTw

)sin(ωTw‖ω‖

). For

the case ω = 0,

exp(adξ) =

[id3 w

03×3 id3

],

and ∫ 1

0

exp(x adξ) dx =

[id3 w/2

03×3 id3

].


Proof. Case 1) When ω = 0,

adξ =

[03×3 w

03×3 03×3

].

Using the Taylor expansion of the matrix exponential, exp(adξ) =∑∞

i=0

(adiξ/i!

), and

the fact that adξ is nilpotent of degree two, i.e., adiξ = 0 for i ≥ 2, it is easy to show the

result.

Case 2) To prove the result for ω 6= 0, the following lemma is required.

Lemma 2.4.6. ∀ω,w ∈ R3 and ω ∈ so(3),

(i) ω2 = ωωT − ‖ω‖2id3 [57],

(ii) ω3 = −‖ω‖2ω [57],

(iii) ωw = −wω = ω × w,

(iv) ˜ωw = [ω, w].

The proof for the above lemma is a straight forward computation. Now, consider the

Adjoint operator corresponding to the element H, AdH , for

H =

[id3

−ωw‖ω‖2

01×3 1

]∈ SE(3),

and its action on ξ ∈ se(3). Based on Lemma 2.4.6,

ξ′ : = AdHξ =

[id3 − [ω,w]

‖ω‖2

03×3 id3

][w

ω

]=

[w − ωw ω

‖ω‖2 + wω ω‖ω‖2

ω

]

=

[w +

(ωωT − ‖ω‖2id3

)w‖ω‖2

ω

]=

[(ωTw)ω‖ω‖2

ω

]=:

[hω

ω

].

Hence,

exp(adξ′) =∞∑i=0

adiξ′

i!=

∞∑i=0

1

i!

[ωi i(hω)i

03×3 ωi

]=

[exp(ω)

∑∞i=1

(hω)i

(i−1)!

03×3 exp(ω)

]

=

[exp(ω) ∂

∂µ|µ=1 exp(hωµ)

03×3 exp(ω)

]=

[exp(ω) hω exp(hω)

03×3 exp(ω)

].


According to the definition of the adjoint operator, one has the following:

exp(adξ) = exp(adAdH−1 (ξ′)) = Adexp(AdH−1 (ξ′)) = Ad(H−1 exp(ξ′)H) = AdH−1 exp(adξ′)AdH .

A straightforward calculation proves the first part of the proposition. For the second

part of the proposition,∫ 1

0

exp(x adξ) dx

=

∫ 1

0

[exp(xω) 1

x2‖ω‖2 [[xω, xw], exp(xω)] + xhω exp(xhω)

03×3 exp(xω)

]dx.

Since the matrix commutator is a bilinear operator, and the integral operator and partial

derivative can commute,∫ 1

0

exp(x adξ) dx

=

[∫ 1

0exp(xω) dx 1

‖ω‖2

[[ω, w],

∫ 1

0exp(xω) dx

]+∫ 1

0xhω exp(xhω) dx

03×3

∫ 1

0exp(xω) dx

].

Using (2.4.15) and substituting h = ωTw‖ω‖2 , one can show the second part of the proposition.

2.5 Case Study

In this section, the kinematic analysis of a mobile manipulator moving on a spacecraft

is performed to elaborate the computational aspects of the proposed formulation for

Forward and Differential Kinematics of open-chain multi-body systems. The spacecraft

can be considered as a six-d.o.f. moving base for the mobile manipulator that is shown

in Figure 2.1. The multi-body system MS(6) = (Bi, Ai)|i = 0, ..., 6, Bi ⊂ Ai consists

of two branches and six joints. The first branch consists of B0 to B5. The second branch

contains B6 and joint six is its last joint. Joint one is a free joint, the second joint is

a nonholonomic three-d.o.f. planar joint, the next joint is a three-d.o.f. spherical joint

and the rest of the joints are one-d.o.f. revolute joints. The coordinate frames assigned

to A0, ..., A6 at the initial configuration are shown in Figure 2.2. In the sequel, the joint

parameters are specified, and Forward and Differential Kinematics maps of MS(6) are

determined. Note that in the following, a basis for Vj at the initial configuration is denoted


Figure 2.1: A mobile manipulator on a six d.o.f. moving base

by Xj, Yj, Zj, and the linear operator 0τ j−1j in the chosen coordinates is represented by

the matrix 0T j−1j .

2.5.1 Forward Kinematics

The first joint is a six-d.o.f. holonomic joint between B0 and B1. The classic joint pa-

rameters are q1 = [x1, y1, z1, θ1,x, θ1,y, θ1,z]T , where [x1, y1, z1]T is the position of H0

1 (t)(O1)

with respect to H01,0(O1) and expressed in V0, and [θ1,x, θ1,y, θ1,z]

T is the rotation angles

of V1 with respect to the axes of V1 at the initial configuration. Therefore, the local

coordinate chart ϕ1 for Q1 is

ϕ1(q1) =

[R(θ1,x, X1)R(θ1,y, Y1)R(θ1,z, Z1) [x1, y1, z1]T

01×3 1

],

where R(θ, W ) is the 3 × 3 rotation matrix corresponding to θ radian rotation about

the vector W . For this coordinate chart, any element of Lie(P0) corresponding to the

relative pose of B1 with respect to B0 is parametrized with the screw joint parameters

s1 = [s1,1, ..., s1,6]T , such that

0T 01 s1 =

(AdH0

1,0

)(Tid6ι1) (T0ϕ1) s1.


Figure 2.2: Coordinate frames assigned to A0, ..., A6 at the initial configuration

With some basic calculations one can show that

∂ϕ1

∂x1

|0 = E1,∂ϕ1

∂y1

|0 = E2,∂ϕ1

∂z1

|0 = E3,∂ϕ1

∂θ1,x

|0 = E4,∂ϕ1

∂θ1,y

|0 = E5, and∂ϕ1

∂θ1,z

|0 = E6,

which coincides with the basis selected for se(3) ∼= Lie(P1). For this joint since Q1 = P1,

Tid6ι1 and T0ϕ1 are equal to the identity matrix. In the basis E1, ..., E6,

∀Hji,0 =

[Rji,0 pji,0

01×3 1

]

the Adjoint operator can be represented by the matrix [81]

AdHji,0

=

[Rji,0 pji,0R

ji,0

03×3 Rji,0

].

Therefore, 0T 01 s1 = AdH0

1,0s1.

Joint number two is a three-d.o.f. nonholonomic joint between B1 and B2. The classic

joint parameters can be chosen as q2 = [x2, y2, θ2,z]T , where [x2, y2, 0]T is the position of

H12 (t)(O2) with respect to H1

2,0(O2) and expressed in V2, and θ2,z is the rotation angle of

V2 about Z2. Hence, the local coordinate chart ϕ2 for Q2 is

ϕ2(q2) =

[R(θ2,z) R(θ2,z)[x2, y2]T

01×2 1

],

where R(θ2,z) is the 2× 2 rotation matrix for θ2,z. For this coordinate chart, any element


of Lie(P0) corresponding to the relative pose of B2 with respect to B1 is parametrized

by the screw joint parameters s2 = [s2,1, s2,2, s2,3]T , such that

0T 12 s2 =

(AdH1

2,0

)(Tid3ι2) (T0ϕ2) s2,

where

Tid3ι2∂ϕ2

∂x2

|0 = E1, Tid3ι2∂ϕ2

∂y2

|0 = E2, and Tid3ι2∂ϕ2

∂θ2,z

|0 = E6.

Thus,

0T 12 s2 = AdH0

2,0

1 0 · · · 0

0 1 · · · 0

0 0 · · · 1

T

s2.

The third joint is a three-d.o.f. holonomic joint between B2 and B3. The classic joint

parameters are q3 = [θ3,x, θ3,y, θ3,z]T , and the local coordinate chart for Q3 is ϕ3(q3) =

R(θ3,x, X3)R(θ3,y, Y3)R(θ3,z, Z3). The elements of Lie(P0) corresponding to the relative

poses of B3 with respect to B2 are parametrized by the screw joint parameters s3 =

[s3,1, s3,2, s3,3]T , such that

0T 23 s3 = AdH0

3,0

[03×3

id3

]s3.

Joint 4 is a one-d.o.f. revolute joint, its classic joint parameter is q4 = θ4,z, and the

local coordinate chart for Q4 is ϕ4(q4) = R(θ4,z).The line in Lie(P0) corresponding to the

relative pose of B4 with respect to B5 is parametrized by the screw joint parameter s4,

such that0T 3

4 s4 = AdH04,0

[0, ..., 1]T s4.

By a simple calculation

0T 34 =

[p0

4,0 ×0 Z4

0Z4

],

where 0Z4 is the joint screw axis expressed in V0. Hence, 0T 34 s4 coincides with the

argument of the exponential map in the existing product of exponentials formula for a

revolute joint [11, 57, 71]. Similarly, for the fifth and sixth joints

0T 45 s5 = AdH0

5,0[0, ..., 1]T s5,

0T 46 s6 = AdH0

6,0[0, ..., 1]T s6,

respectively.

Therefore, based on (2.2.6), the Forward Kinematics map corresponding to MS(6) is


FK(s) =

[exp(0T 0

1 s1)... exp(0T 45 s5)H0

5,0

exp(0T 01 s1)... exp(0T 4

6 s6)H06,0

],

where exp is the matrix exponential for SE(3) that can be evaluated by (2.4.14) and

s = [sT1 , ..., vT6 ]T .

According to the calculation performed in the case of joint four, for a serial-link

multi-body system with revolute and/or prismatic joints, where the multi-body system

consists of one branch, the above formulation for FK reduces to the existing product of

exponentials formula.

2.5.2 Differential Kinematics

Based on Proposition 2.4.1 and 2.4.5, the Jacobian maps of B5 and B6 with respect to B0

and expressed in V0, i.e., 0J05 (s) and 0J0

6 (s), can be determined as 6× 14 matrices. The

nonholonomic constraints at the second joint can be expressed in terms of the classical

joint parameters as

C2(q2)q2 = [0, 1, 0]q2 = 0,

which indicates that the mobile base cannot drift side way. The annihilator of C2 can be

selected to be

C2(q2) =

[1 0 0

0 0 1

]T,

and therefore using (2.1.3b) and (2.1.5)

Σ2(s2) =

sin(s2,3)

s2,3s2,2

(cos(s2,3)+s2,3 sin(s2,3)−1)

s22,3+ s2,1

(cos(s2,3)+sin(s2,3)/s2,3)

s2,3(cos(s2,3)−1)

s2,3s2,1

(1−cos(s2,3)−s2,3 sin(s2,3))

s22,3+ s2,2

(cos(s2,3)−sin(s2,3)/s2,3)

s2,3

0 1

.Note that when s2,3 = 0,

Σ2(s2) =

[1 0 0

s2,2/2 −s2,1/2 1

]T.

Finally, according to (2.3.13) the modified Jacobian of the multi-body system MS(6)

becomes

J(s) =

[0J0

5 (s) 06×13

06×130J0

6 (s)

],


which can be calculated as a 12× 26 matrix using Proposition 2.4.1 and 2.4.5.

Chapter 3

Symplectic Reduction of Holonomic

Open-chain Multi-body Systems

with Displacement Subgroups

This Chapter presents a symplectic geometric approach to the reduction of Hamilton’s

equation for holonomic open-chain multi-body systems with multi-degree-of-freedom dis-

placement subgroups.

First in Section 3.1, we revisit Hamilton’s principle for Lagrangian systems, and we

use the Hamilton-Pontryagin principle to study the geometry of Hamiltonian systems. In

Section 3.2 we use the symplectic reduction theorem to express Hamilton’s equation in

the symplectic reduced manifold, for holonomic Hamiltonian mechanical systems. Then

by identifying the symplectic reduced manifold with a cotangent bundle, we express the

reduced Hamilton’s equation in that cotangent bundle. Consequently, in Section 3.3

we apply this procedure to open-chain multi-body systems with multi-degree-of-freedom

displacement subgroups, for which the symmetry group is identified with the configura-

tion manifold corresponding to the first joint. Then we derive their reduced dynamical

equations in local coordinates, in Theorem 3.3.6.

3.1 Hamilton-Pontryagin Principle and Hamilton’s

Equation

In this section we first explore the geometry of Hamilton’s principle for Lagrangian sys-

tems. Then we show how this principle leads to the Hamilton-Pontryagin principle on

the Pontryagin bundle TQ ⊕ T ∗Q. The Lagrangian systems that satisfy the Hamilton-

38

Chapter 3. Reduction of Holonomic Multi-body Systems 39

Pontryagin principle are called implicit Lagrangian systems, and the resulting equation of

motion is called the implicit Euler-Lagrange equation [93, 94]. In addition, we show that

for hyper-regular Lagrangian systems the implicit Euler-Lagrange equation is equivalent

to Hamilton’s equation. In the next chapter, we use an analogous method to derive the

equations of motion for nonholonomic systems, using Lagrange-d’Alembert-Pontryagin

principle.

Let TQ be the tangent bundle of the configuration manifold Q, and let L : TQ → Rbe a smooth function; we call L the Lagrangian. Let t 7→ vq(t)(t) ∈ Tq(t)Q be a smooth

curve in TQ. This curve corresponds to a tangent lift of a curve in Q if vq(t)(t) = dqdt

(t) =:

qq(t)(t), ∀t. For a time interval [ts, tf ], let (t, ε) 7→ q(t, ε) ∈ Q, for ε ∈ R, be a variation

of a smooth curve t 7→ q(t) ∈ Q with fixed end points qs, qf ∈ Q, i.e., q(ts, ε) = qs and

q(tf , ε) = qf , along with the condition that q(t, 0) = q(t). Hamilton’s principle states that

a Lagrangian system evolves on a curve t 7→ vq(t)(t) that is the tangent lift of the curve

t 7→ q(t) and that makes the action functional stationary for any arbitrary variation of

the curve t 7→ q(t) with fixed end points. That is,

∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

L(vq(t,ε)(t, ε))dt = 0 (3.1.1)

for any variation as described above. This holds if and only if the curve t 7→ vq(t)(t)

satisfies the Euler-Lagrange equation, which is written in coordinates as

d

dt(∂L

∂q(qq(t)(t)))−

∂L

∂q(qq(t)(t)) = 0. (3.1.2)

We present the Euler-Lagrange equation (and upcoming dynamical equations) in the

form of paired elements of cotangent and tangent bundles, for the sake of generalizing

them to nonholonomic systems in the next chapter:⟨(d

dt(∂L

∂q(qq(t)(t)))−

∂L

∂q(qq(t)(t))

)dq, wq(t)

⟩= 0,

∀t ∈ (ts, tf ) and ∀wq(t) ∈ Tq(t)Q,

As was mentioned above, in Hamilton’s principle the variational problem deals only

with tangent lifted curves in TQ. One may implicitly impose this kinematic constraint

in the variational problem, and form a variational problem in the Pontryagin bundle

PQ := TQ ⊕ T ∗Q. The Pontryagin bundle is a vector bundle over the configuration

manifold Q with the canonical projection ΠQ : PQ → Q such that ∀vq ∈ TqQ and

∀pq ∈ T ∗qQ we write (vq, pq) ∈ PqQ and we have ΠQ(vq, pq) = q. The resulting equivalent


principle is called Hamilton-Pontryagin principle. This principle states that an implicit

Lagrangian system evolves on a curve t 7→ (vq(t)(t), pq(t)(t)) ∈ Pq(t)Q that makes the

following functional stationary for any arbitrary variation of the curve in PQ with fixed

end points in Q, i.e., q(ts, ε) = qs and q(tf , ε) = qf :

∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

(L(vq(t,ε)(t, ε)) +

⟨pq(t,ε)(t, ε), qq(t,ε)(t, ε)− vq(t,ε)(t, ε)

⟩)dt = 0. (3.1.3)

For the time interval [ts, tf ], we denote any variation of the curve t 7→ (vq(t)(t), pq(t)(t)) ∈Pq(t)Q by a function γ : [ts, tf ]× R→ PQ:

γ(t, ε) = (vq(t,ε)(t, ε), pq(t,ε)(t, ε)).

For γ to be a variation with fixed end points in Q we assume that for all ε ∈ R,

ΠQ(γ(ts, ε)) = qs and ΠQ(γ(tf , ε)) = qf . We denote the induced map by ΠQ on the

tangent bundles by TΠQ : TPQ → TQ and the projection map that projects the Pon-

tryagin bundle onto T ∗Q by ΠT ∗Q : PQ → T ∗Q. Let Θcan and Ωcan := −dΘcan be the

tautological 1-form and the canonical 2-form on T ∗Q, and let γ := ∂γ∂t∈ Tγ(t,ε)(PQ) and

δγ := ∂γ∂ε

∣∣ε=0∈ Tγ(t,0)(PQ). We can write the left hand side of (3.1.3) as

∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

(L(vq) + 〈pq, TΠQ(γ)− vq〉) γ dt

=

∫ tf

ts

(〈(dL− d〈pq, vq〉) γ(t, 0), δγ(t)〉+

∂

∂ε

∣∣∣∣ε=0

〈(T ∗ΠT ∗Q(Θcan)) γ, γ〉)dt

=

∫ tf

ts

∂

∂t〈(T ∗ΠT ∗QΘcan) γ(t, 0), δγ(t)〉 dt

+

∫ tf

ts

⟨(dL− d〈pq, vq〉) γ(t, 0) + ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0)) , δγ(t)

⟩dt

(by Lemma 3.1.1 bellow)

= [〈(T ∗ΠT ∗QΘcan) γ(t, 0), δγ(t)〉]tfts

+

∫ tf

ts


⟩dt

=

∫ tf

ts


⟩dt

(since the variation in Q at the end points is zero)

In the above calculation, the first equality follows from the definition of the tautological


1-form Θcan ∈ Ω1(T ∗Q) and from the following diagram:

PQΠT∗Q //

ΠQ

T ∗Q

πQ

Q Q

where πQ : T ∗Q → Q is the canonical projection of the cotangent bundle. Since δγ(t) ∈Tγ(t,0)PQ is arbitrary, we can write (3.1.3) as, ∀Wγ(t,0) ∈ Tγ(t,0)(PQ),

⟨d(L− 〈pq, vq〉) γ(t, 0) + ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0)) ,Wγ(t,0)

⟩= 0,

or equivalently,

d(L− 〈pq, vq〉) γ(t, 0) + ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0)) = 0. (3.1.4)

The 2-form T ∗ΠT ∗QΩcan is a closed degenerate 2-form on PQ. It is degenerate only

in the direction of vq, that is, ∀(vq, pq) ∈ PqQ and ∀W(vq ,pq) ∈ T(vq ,pq)(PQ), we have

ιW(T ∗ΠT ∗QΩcan) = 0 if and only if T(vq ,pq)ΠT ∗Q(W(vq ,pq)) = 0.

Lemma 3.1.1. For any variation γ as described above and for all α ∈ Ω1(PQ), we have

the equality

∂

∂ε

∣∣∣∣ε=0

〈α γ(t, ε), γ(t, ε)〉 =∂

∂t〈α γ(t, 0), δγ(t)〉+ 〈ιγ(t,0)(−dα γ(t, 0)), δγ(t)〉.

Proof. The proof is based on a straightforward calculation.

∂

∂ε〈α γ(t, ε), γ(t, ε)〉 =

∂

∂ε

⟨α γ, Tγ(

∂

∂t)

⟩= L∂/∂ε

⟨T ∗γ(α),

∂

∂t

⟩=

⟨L∂/∂ε(T ∗γ(α)),

∂

∂t

⟩+

⟨T ∗γ(α),L∂/∂ε(

∂

∂t)

⟩=

⟨L∂/∂ε(T ∗γ(α)),

∂

∂t

⟩(t and ε are two independent variables)

=

⟨ι∂/∂ε(T

∗γ(dα)) + d

⟨T ∗γ(α),

∂

∂ε

⟩,∂

∂t

⟩(by Cartan’s formula)

=

⟨−ι∂/∂t(T ∗γ(dα)),

∂

∂ε

⟩+∂

∂t

⟨T ∗γ(α),

∂

∂ε

⟩


=

⟨T ∗γ (ιγ(−dα γ)) ,

∂

∂ε

⟩+∂

∂t

⟨T ∗γ(α),

∂

∂ε

⟩=

⟨ιγ(−dα γ), Tγ(

∂

∂ε)

⟩+∂

∂t

⟨α γ, Tγ(

∂

∂ε)

⟩Based on the definition of δγ(t), at ε = 0 we have the desired equality.

We define the function E : PQ → R by E(vq, pq) := 〈pq, vq〉 − L(vq); it is called

the energy function. We call the triple (PQ, T ∗ΠT ∗QΩcan ∈ Ω2(PQ), E) an implicit

Lagrangian system.

In a coordinate chart, we have (γ(t, 0), γ(t, 0)) = (q(t), v(t), p(t), q(t), v(t), p(t)) and

T ∗ΠT ∗QΩcan = −dp ∧ dq. We can write (3.1.4) as

∂L

∂q(q, v)dq +

∂L

∂v(q, v)dv + qdp− pdq − vdp− pdv = 0,

or equivalently,

p =∂L

∂q(q, v), p =

∂L

∂v(q, v), q = v. (3.1.5)

This gives a bijection between the tangent lift of the curves in Q that satisfy the Euler-

Lagrange equation (3.1.2) and the curves in PQ that satisfy the implicit Euler-Lagrange

equation (3.1.5).

The fibre derivative of the Lagrangian L induces a fibre preserving map FL : TQ →T ∗Q, called Legendre transformation,

〈FLq(vq), wq〉 :=d

dε

∣∣∣∣ε=0

L(vq + εwq) =

⟨∂L

∂v(vq), wq

⟩. ∀wq ∈ TqQ (3.1.6)

For all vq ∈ TqQ we can define the embedding grph : TQ → PQ by grphq(vq) :=

(vq,FLq(vq)) ∈ PqQ. By (3.1.5), we have that the solution curve of an implicit La-

grangian system is always in this submanifold.

The Lagrangian L is called hyper-regular if FL is a diffeomorphism. Under the

assumption that L is hyper-regular, we also have the embedding grph : T ∗Q → PQthat maps any element pq ∈ T ∗Q to (FL−1(pq), pq) ∈ PqQ. In this case, we have

grph(T ∗Q) = grph(TQ); hence in (3.1.4) the curve t 7→ γ(t, 0) is in the image of grph,

and it has a unique pre-image t 7→ λ(t) = pq(t)(t) ∈ T ∗q(t)Q, such that λ(t) = ΠT ∗Q(γ(t, 0)),

for all t. Assuming that L is hyper-regular, we can now rewrite (3.1.4) in T ∗Q as

T ∗grph(ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0))− dE γ(t, 0)

)= 0,


⇐⇒ ιλ(t) (Ωcan λ(t))− dE grph(λ(t)) = 0,

since we have the following diagram:

T ∗Q

T ∗Q grph // PQ

ΠT∗Q

OO

Here, λ(t) := dλdt

(t). We define the Hamiltonian function H : T ∗Q → R on the cotangent

bundle by

H(pq) := E grph(pq) = 〈pq,FL−1(pq)〉 − L(FL−1(pq)). (3.1.7)

For a hyper-regular Lagrangian, the solution curve of an implicit Lagrangian system, i.e.,

t 7→ γ(t, 0), satisfies (3.1.4) if and only if the curve t 7→ λ(t) satisfies Hamilton’s equation,

defined by

ιλ(t) (Ωcan λ(t)) = dH λ(t). (3.1.8)

Let πQ : T ∗Q → Q be the canonical projection map for the cotangent bundle, and

(with some abuse of notation) denote a variation of the curve t 7→ λ(t) ∈ T ∗Q by the

function (t, ε) 7→ λ(t, ε) ∈ T ∗Q. Under the assumptions considered to derive (3.1.4),

Hamilton’s equation in T ∗Q can also be derived from the Hamilton-Pontryagin principle,

once we restrict the variational problem to the image of the embedding grph. That is, we

only consider the variations (t, ε) 7→ γ(t, ε) ∈ grph(T ∗Q) such that λ(t, ε) = ΠT ∗Q(γ(t, ε)):

∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

(L(vq) + 〈pq, TΠQ(γ)− vq〉) γ dt = 0

⇐⇒ ∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

(〈λ(t, ε), TπQ(λ(t, ε))〉 −H λ(t, ε))

)dt = 0

⇐⇒⟨ιλ(t,0) (Ωcan λ(t, 0))− dH λ(t, 0), δλ(t)

⟩= 0, ∀δλ(t) ∈ Tλ(t,0)(T

∗Q)

⇐⇒ ιλ(t) (Ωcan λ(t)) = dH λ(t).

Here, λ(t, ε) := ∂∂tλ(t, ε) and δλ(t) := ∂

∂ε

∣∣ε=0

λ(t, ε). Note that the details are omitted

here, since the derivation presented above is similar to the derivation of (3.1.4).

Using any coordinate chart for T ∗Q, we have (λ(t), λ(t)) = (q(t), p(t), q(t), p(t)), and


we can write (3.1.8) as

q =∂H

∂p, p = −∂H

∂q.

Now, let X be a vector field on the cotangent bundle T ∗Q. It induces a vector field

on grph(T ∗Q) whose smooth extension to PQ is denoted by X . Note that X is not a

unique vector field on PQ. In other words, ∀pq ∈ T ∗Q we have Tpq grph(Xpq) = Xgrph(pq).

If the curve t 7→ γ(t) ∈ PQ is an integral curve of the vector field X and it satisfies

(3.1.4), then ∀Wγ(t) ∈ Tγ(t)(PQ) we have

⟨(−dE + ιX (T ∗ΠT ∗QΩcan)) γ(t),Wγ(t)

⟩= 0,

⇐⇒ (−dE + ιX (T ∗ΠT ∗QΩcan)) γ(t) = 0. (3.1.9)

It is easy to show that the pull back of the 1-form ιX (T ∗ΠT ∗QΩcan) − dE ∈ Ω1(PQ)

(restricted to the image of grph) by the embedding grph is equal to

ιXΩcan − dH ∈ Ω1(T ∗Q).

Consequently, any curve t 7→ γ(t) ∈ grph(T ∗Q) such that Xγ(t) = dγdt

(t) satisfies (3.1.9) if

and only if the curve t 7→ λ(t) = ΠT ∗Q(γ(t)) ∈ T ∗Q, which is the integral curve of the

vector field X, satisfies

⟨(ιXΩcan − dH) λ(t),Yλ(t)

⟩= 0, ∀Yλ(t) ∈ Tλ(t)(T

∗Q)

⇐⇒ (ιXΩcan − dH) λ(t) = 0. (3.1.10)

If (3.1.10) holds for any integral curve of X ∈ X(T ∗Q), we can define Hamilton’s

equation as

ιXΩcan = dH. (3.1.11)

In general, one can have a system satisfying Hamilton’s equation (3.1.11) on T ∗Q for a

Hamiltonian H ∈ C∞(T ∗Q) that does not necessarily come from a Lagrangian. Such

system is called a Hamiltonian system. The unique vector field that satisfies Hamilton’s

equation for a Hamiltonian H is called the Hamiltonian vector field for the Hamiltonian

H. We define a Hamiltonian system to be a triple (T ∗Q,Ωcan, H), as above.


3.2 Hamiltonian Mechanical Systems with Symme-

try

For a mechanical system, the Lagrangian is defined by L := 12Kq(vq, vq) − V (q), where

Kq : TqQ×TqQ → R is a Riemannian metric, called the kinetic energy metric, and where

V : Q → R is a smooth function, called the potential energy function. This Lagrangian is

hyper-regular, and the corresponding Legendre transformation is equal to the fibre-wise

linear isomorphism that is induced by the metric K:

〈FLq(vq), wq〉 := Kq(vq, wq). ∀vq, wq ∈ TqQ (3.2.12)

Likewise, the Hamiltonian of the system is

H(pq) =1

2Kq(FL−1

q (pq),FL−1q (pq)) + V (q), (3.2.13)

which is the total energy of the mechanical system. A Hamiltonian mechanical system is

such a quadruple (T ∗Q,Ωcan, H,K).

Let G be a Lie group with the Lie algebra Lie(G). Consider an action of G on Q, and

denote the action by Φg : Q → Q, ∀g ∈ G. This action induces an action of G on T ∗Qby the cotangent lift of Φg, which is denoted by T ∗Φg : T ∗Q → T ∗Q.

Lemma 3.2.1. For every g ∈ G, the map T ∗Φg is a symplectomorphism, i.e., it preserves

Ωcan [50].

Proof. The proof relies on the fact that T ∗Φg preserves the tautological 1-form Θcan:

⟨(T ∗T ∗Φg(Θcan))pq ,Ypq

⟩=⟨T ∗Φg(pq), (TπQ)(TT ∗Φg)(Ypq)

⟩= ∀Ypq ∈ Tpq(T ∗Q)⟨

T ∗Φg(pq), T (πQ T ∗Φg)(Ypq)⟩

=⟨T ∗Φg(pq), T (Φg−1 πQ)(Ypq)

⟩=⟨

pq, TπQ(Ypq)⟩

=⟨(Θcan)pq ,Ypq

⟩.

The third equality holds, since the following diagram commutes:

T ∗Q

πQ

T ∗Φg // T ∗Q

πQ

Q

Φg−1// Q


Finally, we have

T ∗T ∗Φg(Ωcan) = T ∗T ∗Φg(−dΘcan) = −d (T ∗T ∗Φg(Θcan)) = −dΘcan = Ωcan.

Consider the infinitesimal action of Lie(G) on Q. For any ξ ∈ Lie(G), this action

induces a vector field ξQ ∈ X(Q) such that ∀q ∈ Q,

ξQ(q) =∂

∂ε

∣∣∣∣ε=0

(Φexp(εξ)(q)

). (3.2.14)

Denote the fibre-wise linear map corresponding to the infinitesimal action of Lie(G) by

φq : Lie(G) → TqQ, where φq(ξ) = ξQ(q). Likewise, we define ξT ∗Q ∈ X(T ∗Q) such that

∀pq ∈ T ∗qQ,

ξT ∗Q(pq) =∂

∂ε

∣∣∣∣ε=0

(T ∗Φexp(εξ)(q)

Φexp(−εξ)(pq)). (3.2.15)

Now, consider the fibre-wise linear map M : T ∗Q → Lie∗(G), defined by

〈Mq(pq), ξ〉 := 〈φ∗q(pq), ξ〉 = 〈pq, ξQ(q)〉. (3.2.16)

Lemma 3.2.2. The map M is an Ad∗-equivariant momentum map corresponding to the

cotangent lifted action T ∗Φg.

Proof. To prove that M is a momentum map, it suffices to show that M satisfies the

momentum equation (1.2.1) for the G-action on T ∗Q,

ιξT∗QΩcan = d〈M, ξ〉.

Therefore, we have

d〈M(pq), ξ〉 = d〈pq, ξQ〉 = d(ιξT∗QΘcan

)= LξT∗QΘcan − ιξT∗QdΘcan = ιξT∗QΩcan.

The forth equality is true, since the cotangent lifted action preserves the tautological

1-form.

To prove that M is Ad∗-equivariant, we have to show

M(T ∗Φg(pq)) = Ad∗gM(pq).


Using the definition of action and the map M, ∀ξ ∈ Lie(G) one has

〈M(T ∗Φg(pq)), ξ〉 = 〈T ∗Φg(pq), ξQ(Φg−1(q))〉 = 〈pq,∂

∂ε

∣∣∣∣ε=0

(Φg exp(εξ)g−1(q)

)〉 =

〈pq, (Adgξ)Q(q))〉 = 〈Ad∗gM(pq), ξ〉.

Proposition 3.2.3 (Noether’s Theorem). Let H : T ∗Q → R be the Hamiltonian of a

mechanical system. If H is invariant under the cotangent lifted group action, the mo-

mentum map M is constant along the flow of the Hamiltonian vector field X for the

Hamiltonian H. That is, ∀ξ ∈ Lie(G) we have LX(〈M, ξ〉) = 0.

Proof.

LX(〈M, ξ〉) = 〈d〈M, ξ〉, X〉 =⟨ιξT∗QΩcan, X

⟩= −Ωcan(X, ξT ∗Q) = −〈ιXΩcan, ξT ∗Q〉 =

− 〈dH, ξT ∗Q〉 = −LξT∗QH = 0.

The forth equality is true, since X is a Hamiltonian vector field for the function H, and

the last equality is the consequence of the hypothesis that H is G-invariant.

We define a Hamiltonian system with symmetry to be a quadruple (T ∗Q,Ωcan, H,G),

as above, where the Hamiltonian H is invariant under the cotangent lifted action of G. A

Hamiltonian mechanical system with symmetry is defined by a quintuple (T ∗Q,Ωcan, H,K,G),

where K is the kinetic energy metric on Q, and in addition to H, K is invariant under

the G-action.

Theorem 3.2.4 (Symplectic Reduction Theorem [53]). Assume that the action of G on

Q is free and proper, and let µ ∈ Lie∗(G) be a regular value of its momentum map M.

Also, let Gµ = g ∈ G|Ad∗gµ = µ be the coadjoint isotropy group for µ ∈ Lie∗(G).

Then the quotient manifold (T ∗Q)µ := M−1(µ)/Gµ is a symplectic manifold, called

the symplectic reduced space, with the unique symplectic form Ωµ that is identified by

the equality T ∗πµ(Ωµ) = T ∗iµ(Ωcan). Here, the maps πµ : M−1(µ) → M−1(µ)/Gµ and

iµ : M−1(µ) → T ∗Q are the projection map and inclusion map, respectively.

This theorem was first stated and proved in a paper by Marsden and Weinstein in

1974 [53], and since then this result has been extended to non-free actions [27] and almost

symplectic manifolds [39]. An almost symplectic manifold is a manifold equipped with

a nondegenerate 2-form, which may not be closed. Based on the symplectic reduction


theorem, in the presence of a group action that preserves the symplectic structure and

an Ad∗-equivariant momentum map (corresponding to the symmetry group) we say that

the phase space of a Hamiltonian system along with its symplectic 2-form can be reduced

to the symplectic reduced space ((T ∗Q)µ,Ωµ). In order to have a well-defined projection

of Hamilton’s equation onto the symplectic reduced space, the Hamiltonian of the system

should be invariant under the group action, as well. Under these hypotheses, Hamilton’s

equation can be written on (T ∗Q)µ as

ιXµΩµ = dHµ, (3.2.17)

where Hµ is defined by H iµ = Hµ πµ and Xµ πµ = Tπµ(X iµ).

We say that the Hamiltonian system with symmetry (T ∗Q,Ωcan, H,G) has been re-

duced to the Hamiltonian system ((T ∗Q)µ,Ωµ, Hµ).

In the theory of cotangent bundle reduction, there exist two equivalent ways to de-

scribe the symplectic reduced space in terms cotangent bundles and coadjoint orbits [49]:

i) Embedding version: in which the symplectic reduced space is identified with a

vector sub-bundle of the cotangent bundle of Q := Q/Gµ, called µ-shape space of a

Hamiltonian system.

ii) Bundle version: in which the symplectic reduced space is identified by a (locally

trivial) fibre bundle over T ∗Q, whereQ := Q/G, and where the fibre is the coadjoint

orbit through µ. The manifold Q is called the shape space of the Hamiltonian

system.

In this section, the embedding version of the cotangent bundle reduction is used to

write Hamilton’s equation (3.2.17) in a sub-bundle of the cotangent bundle of the µ-shape

space, i.e., a sub-bundle of T ∗Q. Prior to reporting the final result, we introduce a number

of necessary objects. Note that since we consider multi-body systems for the application,

from now on we only focus on Hamiltonian mechanical systems with symmetry, unless

otherwise stated.

Consider a Hamiltonian mechanical system with symmetry (T ∗Q,Ωcan, K,H,G), and

∀g ∈ G denote the action map by Φg : Q → Q. Assume that the action is free and proper.

The quotient manifold Q := Q/G gives rise to the principal bundle π : Q → Q with the

base space Q, and the fibres of the bundle are isomorphic to the group G. A principal

connection on the principle bundle π : Q → Q is a fibre-wise linear mapA : TQ → Lie(G),

such that A(ξQ(q)) = ξ (∀ξ ∈ Lie(G) and ∀q ∈ Q), and it is Ad-equivariant, i.e.,

A(TqΦg(vq)) = AdgA(vq) (∀vq ∈ TqQ). Accordingly, for any base element q ∈ Q the


tangent space of Q can be written as the following direct sum

TqQ = ker(Tqπ)⊕ ker(Aq). (3.2.18)

Note that V := ker(Tπ) = ξQ = φ(ξ)| ξ ∈ Lie(G) is called the vertical vector sub-

bundle of TQ, and H := ker(A) is called the horizontal vector sub-bundle of TQ. As

a result, any vq ∈ TqQ can be decomposed into the horizontal and vertical components

such that vq = hor(vq)+ver(vq), where ver(vq) := φq Aq(vq) and hor(vq) := vq−ver(vq).

For any q ∈ Q and q := π(q) ∈ Q the restriction of the tangent map Tqπ : TqQ → TqQto the horizontal subspace of TqQ, namely Hq, is a linear isomorphism between Hq and

TqQ. Therefore, for any vq ∈ TqQ it defines a horizontal lift map by

hlq(vq) := (Tqπ|Hq)−1(vq). (3.2.19)

The choice of the principal connection A is arbitrary; however, for a Hamiltonian

mechanical system, we can use the Legendre transformation, which is induced by the

kinetic energy metric K, to define an appropriate principal connection.

For any q ∈ Q consider the linear map Iq : Lie(G)→ Lie∗(G), defined by

Iq := φ∗q FLq φq, (3.2.20)

such that the following diagram commutes:

Lie(G)

Iq

φq // TqQ

FLq

Lie∗(G) T ∗qQφ∗q

oo

This map is a linear isomorphism for any q ∈ Q, and it is called the locked inertia

tensor. For a Hamiltonian mechanical system with symmetry ∀ξ, η ∈ Lie(G) we have

〈Iq(ξ), η〉 = Kq(ξQ(q), ηQ(q)). The principal connection A can now be chosen to be the

mechanical connection AMech, which can be interpreted as the orthogonal projection with

respect to the kinetic energy metric K, and defined by the following commuting diagram:


TqQ

AMechq

FLq // T ∗qQ

Mq

Lie(G) Lie∗(G)

I−1q

oo

Therefore, ∀q ∈ Q we have

Aq = AMechq := I−1

q Mq FLq. (3.2.21)

For any µ ∈ Lie∗(G), let the action of G restricted to the subgroup Gµ = g ∈ G|Ad∗gµ =

µ ⊆ G be denoted by Φµh : Q → Q (∀h ∈ Gµ). Similarly, for this action we have a prin-

cipal bundle π : Q → Q := Q/Gµ. Using the same procedure detailed above, the locked

inertia tensor Iµq : Lie(Gµ) → Lie∗(Gµ) and the (mechanical) connection Aµq : TqQ →Lie(Gµ) (∀q ∈ Q) for the Gµ-action are defined by

Iµq := (φµq )∗ FLq φµq , (3.2.22)

and

Aµq := (Iµq )−1 Mµq FLq, (3.2.23)

respectively. Here, the map φµq : Lie(Gµ) → TQ corresponds to the infinitesimal Gµ-

action, and Mµ : T ∗Q → Lie∗(Gµ) is the Ad∗-equivariant momentum map for the cotan-

gent lifted Gµ-action, which are defined based on (3.2.14) and (3.2.16). Let the map

iµ : Gµ → G be the canonical inclusion map. Denote the induced map in the Lie alge-

bras by iµ∗ : Lie(Gµ) → Lie(G) and in the dual of the Lie algebras by (iµ)∗ : Lie∗(G) →Lie∗(Gµ). The following diagrams commute:

Lie(G)

φq

""Lie(Gµ)?

iµ∗

OO

φµq // TqQ

Lie∗(G)

(iµ)∗

Lie∗(Gµ) T ∗qQ(φµq )∗

oo

φ∗q

bb


Based on these commuting diagrams, we have the following relations:

Iµq = (iµ)∗ φ∗q FLq φq iµ∗ = (iµ)∗ Iq iµ∗ ,

Mµq = (iµ)∗ Mq,

Aµq = (Iµq )−1 (iµ)∗ Mq FLq = (Iµq )−1 (iµ)∗ Iq Aq.

For the principal bundle π : Q → Q with the principal connection Aµ, the horizontal

and vertical sub-bundles are Hµ := ker(Aµ) and Vµ := ker(π) = ηQ = φµ(η)| η ∈Lie(Gµ), respectively. It is easy to check that Vµ ⊆ V and H ⊆ Hµ as vector sub-

bundles. The horizontal lift map corresponding to the connection Aµ can be defined as

hlq(vq) := (Tqπ|Hµq )−1(vq),

where q := π(q) and vq ∈ TqQ.

Now, consider the 1-form αµ := A∗µ ∈ Ω1(Q).

Lemma 3.2.5. The 1-form αµ takes values in M−1(µ), and it is invariant under Gµ-

action.

Proof. Using the definition of the momentum map and principal connection, we have

∀ξ ∈ Lie(G)

〈M(αµ), ξ〉 = 〈αµ, ξQ〉 = 〈A∗qµ, φq(ξ)〉 = 〈µ, (Aq φq)(ξ)〉 = 〈µ, ξ〉.

As a result, αµ ∈M−1(µ).

Finally, consider the action of an arbitrary element h ∈ Gµ, and denote the action

simply by h · q := Φh(q) and h · vq := TΦh(vq). Based on the Ad∗-equivariance of A and

the definition of Gµ, one can show that αµ is Gµ invariant. For all vq ∈ TqQ,

〈αµ(h · q), h · vq〉 = 〈A∗h·qµ, h · vq〉 = 〈µ,Ah·q(h · vq)〉

= 〈µ,Adh−1Aq(vq)〉 = 〈Ad∗h−1µ,Aq(vq)〉 = 〈µ,Aq(vq)〉.

According to the Cartan Structure Equation derived in [49, Theorem 2.1.9] for prin-

cipal connections, ∀Z, Y ∈ X(Q) the exterior derivative of αµ evaluated on Y and Z is

equal to

dαµ(Z, Y ) = 〈µ, dA(Z, Y )〉 = 〈µ,B(Z, Y ) + [A(Z),A(Y )]〉, (3.2.24)


where Bq(Zq, Yq) := (dA)q(horq(Zq), horq(Yq)) = −Aq([hor(Z), hor(Y )]q) is the curvature

of the connection A, and [·, ·] in (3.2.24) corresponds to the Lie bracket in Lie(G).

Lemma 3.2.6. For all η ∈ Lie(Gµ), the interior product of the 2-form dαµ with ηQ is

zero, i.e., ιηQdαµ = 0.

Proof.

ιηQdαµ = LηQ(αµ)− d(ιηQαµ).

The Lie derivative term is zero since αµ is invariant under the Gµ-action (see Lemma

3.2.5), and the exterior derivative term is zero since

ιηQαµ = 〈αµ, ηQ〉 = 〈µ,A φµ(η)〉 = 〈µ, η〉

is a constant function on Q, since A φµ(η) = η, for all η ∈ Lie(Gµ).

By this lemma and Lemma 3.2.5 the 2-form dαµ is basic; hence, a closed 2-form

βµ ∈ Ω2(Q) can be uniquely defined by the relation T ∗π(βµ) = dαµ, and its pullback Ξµ

by the cotangent bundle projection πQ : T ∗Q → Q will be a closed 2-form on T ∗Q,

Ξµ := T ∗πQ(βµ).

Theorem 3.2.7. There is a symplectic embedding ϕµ : ((T ∗Q)µ,Ωµ) → (T ∗Q, Ωcan−Ξµ)

onto [T π(V)]0 ⊂ T ∗Q that covers the base Q, where Ωcan is the canonical 2-form on T ∗Qand 0 indicates the annihilator with respect to the natural pairing between tangent and

cotangent bundle. The map ϕµ is identified by

〈ϕµ([γq]µ), Tqπ(vq)〉 = 〈γq − αµ(q), vq〉, (3.2.25)

∀γq ∈ M−1q (µ) and ∀vq ∈ TqQ, where [·]µ refers to a class of elements in the quotient

manifold M−1(µ)/Gµ [49].

Based on the above theorem, the inverse of the map ϕµ exists only on [T π(V)]0 ⊂ T ∗Q,

and it is a diffeomorphism on this vector sub-bundle. Hence, one may rewrite the reduced

Hamilton’s equation (3.2.17) in [T π(V)]0 ⊂ T ∗Q as

ιX(Ωcan − Ξµ) = dH, (3.2.26)

where H := Hµ ϕ−1µ for ϕ−1

µ : [T π(V)]0 → (T ∗Q)µ being the inverse of ϕµ, X ϕµ =

TϕµXµ, and Ξµ can be calculated as follows. Consider two vector fields Z,Y ∈ X(T ∗Q),


denote an element of Q by q := π(q), and ∀αq ∈ T ∗Q define Zq := TπQZ(αq), Yq :=

TπQY(αq):

(Ξµ)αq(Z(αq),Y(αq)) =⟨µ,−Aq([hor(hl(Z)), hor(hl(Y ))]q) + [Aq(hlq(Zq)),Aq(hlq(Yq))]

⟩.

(3.2.27)

For all h ∈ Gµ, we show the action of h at any q ∈ Q by h · q := Φµh (q). The 2-form

Ξµ ∈ Ω(T ∗Q) is well-defined, since we have⟨µ,−Ah·q([hor(hl(Z)), hor(hl(Y ))]h·q) + [Ah·q(hlh·q(Zq)),Ah·q(hlh·q(Yq))]

⟩=⟨µ,−Ah·q([TΦµ

h (hor(hl(Z))), TΦµh (hor(hl(Y )))]h·q)

+[Ah·q(TΦµh (hlq(Zq))),Ah·q(TΦµ

h (hlq(Yq)))]⟩

=⟨µ,−Ah·q(TΦµ

h [hor(hl(Z)), hor(hl(Y ))]q) + [AdhAq(hlq(Zq)),AdhAq(hlq(Yq))]⟩

=⟨µ,−AdhAq([hor(hl(Z)), hor(hl(Y ))]q) + Adh[Aq(hlq(Zq)),Aq(hlq(Yq))]

⟩=⟨µ,−Aq([hor(hl(Z)), hor(hl(Y ))]q) + [Aq(hlq(Zq)),Aq(hlq(Yq))]

⟩.

The first equality is the result of the definition of the bundle maps hl and hor. The

second and third equalities are true since the principal connection A is Ad-equivariant.

And the last equality holds because h ∈ Gµ.

If in Theorem 3.2.7 we assume Gµ = G, whose special examples are when G is Abelian

or µ = 0, then the map ϕµ becomes a symplectomorphism. Under this assumption, since

hl = hl and A hl = 0, Ξµ can be calculated by a simpler formulation

(Ξµ)αq(Z(αq),Y(αq)) =⟨µ,−Aq([hl(Z), hl(Y )]q)

⟩. (3.2.28)

3.3 Symplectic Reduction of Holonomic Open-chain

Multi-body Systems with Displacement Subgroups

In this section, we show that holonomic open-chain multi-body systems can be considered

as Hamiltonian mechanical systems with symmetry. In Theorem 3.3.3 we prove that the

configuration manifold of the first joint is a symmetry group, and the corresponding

group action is left translation. We identify the spaces and maps introduced in the

previous section. Accordingly, we apply the reduction theory for Hamiltonian mechanical

systems with symmetry to holonomic open-chain multi-body systems with displacement

subgroups.


3.3.1 Indexing and Some Kinematics

Based on Definition 2.2.1 in the previous chapter, a holonomic open-chain multi-body

system is a multi-body system MS(N) together with N holonomic displacement sub-

groups between the bodies, such that there exists a unique path between any two bodies

of the multi-body system, where B0 (a fixed body) corresponds to the ground (inertial

coordinate frame). In a holonomic open-chain multi-body system, bodies with only one

neighbouring body are called extremities.

We label the bodies starting from the inertial coordinate frame (ground), B0, out-

wards. That is, we label the bodies connected to B0 by joints successively as B1, · · · , BN0

(N0 ≤ N), and we repeat the same procedure for all N0 bodies starting from B1, e.g.,

all of the bodies connected to B1 by joints are labeled as BN0+1, · · · , BN0+N1 and so on.

Thus, one has∑

l=0Nl = N . Joints are numbered using the larger body label, e.g., the

joint between Bi and Bj, where i > j, is labeled as Ji. Considering the bodies and joints

in an open-chain multi-body system as vertices and edges of a graph, respectively, we can

encode the structure of the system in an N × (N + 1) matrix. We label this matrix by

GM. The N rows of this matrix correspond to the joints, J1, · · · , JN , and the columns

represent the bodies, B0, · · · , BN . Each row of this matrix consists of only two non-zero

elements and the rest is equal to 0. The non-zero elements in the row i correspond to the

two bodies that Ji connects. We put the element corresponding to the body with lower

index equal to −1 and the one with the higher index is equal to 1. With the choice of

numbering that was explained above, we have

GMij =

−1 if Ji connects Bj to Bi

1 if i = j

0 otherwise

,

which is a lower triangular matrix. We have the following properties of the matrix GM.

Corollary 3.3.1. Let GMj denote the jth column of the matrix GM.

i) The summation of the columns of the matrix GM is equal to zero, i.e.,

N+1∑j=1

GMj =

J1 0...

...

JN 0

.ii) The summation of the rows corresponding to the edges (joints) that are connecting


the vortex (body) Bj to Bi for i > j, has the following form

[ B0 ··· Bj−1 Bj Bj+1 ··· Bi−1 Bi Bi+1 ··· BN

0 · · · 0 −1 0 · · · 0 1 0 · · · 0].

Denote the transpose of GM by GMT . For all i, j = 1, · · · , (N + 1)

iii) ((GM)T (GM))ii= the number of neighbouring vortices (bodies) connected to Bi−1.

iv) if ((GM)T (GM))ij = −1 for i 6= j, then the vortex (body) Bi−1 is connected to Bj−1,

either with the edge (joint) Ji−1 if i > j, or with the edge (joint) Jj−1 if j > i.

Note that for any i = 2, · · · , (N + 1), if ((GM)T (GM))ii = 1 then the body Bi−1 is an

extremity. The body corresponding to the kth 1 is called the kth extremity. Accordingly,

the path between B0 and the kth extremity is called the kth branch.

Corollary 3.3.2. Let the row matrix Phi represent the path between the vertex (body) Bi

(∀i = 1, · · · , N) and B0. The jth element of Phi is equal to 1 if the path crosses the edge

(joint) Jj. Then we have

Phi×GM =[ B0 B1 ··· Bi−1 Bi Bi+1 ··· BN

−1 0 · · · 0 1 0 · · · 0].

Hence, the matrix of all paths, i.e.,

Ph =

Ph1

...

PhN

is equal to GM

−1, where GM is the matrix GM when the first column is removed.

For example, consider the following structure of an open-chain multi-body system

B0J1

B1J3

J2

B3J4

B4

B2

(3.3.29)


We have

GM =

B0 B1 B2 B3 B4

J1 −1 1 0 0 0

J2 0 −1 1 0 0

J3 0 −1 0 1 0

J4 0 0 0 −1 1

,

(GM)T (GM) =

B0 B1 B2 B3 B4

B0 1 −1 0 0 0

B1 −1 3 −1 −1 0

B2 0 −1 1 0 0

B3 0 −1 0 2 −1

B4 0 0 0 −1 1

,

Ph =

J1 J2 J3 J4

Ph1 1 0 0 0

Ph2 1 1 0 0

Ph3 1 0 1 0

Ph4 1 0 1 1

.

From the matrix (GM)T (GM) one can see that the open-chain multi-body system rep-

resented by the above graph has two extremities, B2 and B4. The body B2 is the first

extremity and B4 is the second one.

Since only displacement subgroups are considered, the relative configuration manifold

corresponding to the joint Ji is diffeomorphic to the Lie group Qi := Lr0i,0Rri0,0

Qi, where

Qi is defined in Section 2.1 and r0i,0 ∈ P 0

i is the initial pose of Bi with respect to B0,

for i = 1, ..., N . Note that every Qi is a di dimensional Lie subgroup of P0∼= SE(3),

where di is the number of degrees of freedom of Ji, and D :=∑N

i=1 di is the total number

of degrees of freedom of the holonomic open-chain multi-body system. Accordingly, any

state of the system can be realized by q := (q1, · · · , qN) ∈ Q := Q1 × · · · ×QN , where Qis the configuration manifold. The manifold Q along with the group structure induced

by Qi’s is also a Lie group. Let rcm,i ∈ SE(3) be the initial pose of the centre of mass

of Bi with respect to the inertial coordinate frame. Considering Qi’s as subgroups of

SE(3), we define the map F : Q → SE(3)× · · · × SE(3) =: P by

F (q) := (q1rcm,1, q1q2rcm,2, · · · , q1 · · · qNrcm,N). (3.3.30)


This map determines the position of the centre of mass of all bodies with respect to

the inertial coordinate frame. Note that the ith component of this map consists of the

joint parameters of all joints that connect B0 to Bi in the open-chain multi-body system.

Also, in this thesis, we consider open-chain multi-body systems with only one joint (J1)

attached to B0, i.e., N0 = 1. Because, for any N0 > 1 we can split the open-chain multi-

body system to N0 decoupled open-chain multi-body systems and study each of them

separately.

For any motion of the open-chain multi-body system, i.e., a curve t 7→ q(t) ∈ Q, the

velocity of the centre of mass of the bodies with respect to the inertial coordinate frame

(absolute velocity) is calculated by p := ddtF (q(t)) = TqF (q). Based on Corollary 3.3.2,

we can explicitly write the tangent map TqF using the matrix Ph. First, we substitute

the zero elements in the matrix Ph by 6× 6 block matrices of zero. Then, ∀i = 1, · · · , Nwe substitute all the elements equal to 1 in Phi by the linear maps that look like

T (Rrcm,i)T (R∏r qr

)T (L∏l ql

),

where the maps L• : SE(3) → SE(3) and R• : SE(3) → SE(3) are the left and right

translation maps on SE(3), respectively. Here,∏

l ql and∏

r qr are the product of some

elements of the relative configuration manifoldsQi ⊆ P0∼= SE(3), considered as elements

of SE(3). In order to specify which joints contribute to the left or right translation

maps, in Phi we look at the 1s that are on the left or right of the corresponding element,

respectively. If there does not exist any element equal to 1 on left (right), then we put

the argument of the left (right) translation map equal to the identity element of SE(3).

Finally, TqF is the right multiplication of the resulting matrix by

Tq1ι1 ⊕ · · · ⊕ TqN ιN =

Tq1ι1 · · · 0

.... . .

...

0 · · · TqN ιN

,where for all i = 1, · · · , N , ιi : Qi → SE(3) is the canonical inclusion map and Tιi : TQi →TSE(3) is the induced map on the tangent bundles.

This simple procedure becomes clear in an example. Consider the structure of the


system in (3.3.29), we calculate TqF to be the following matrixTRrcm,1 06×6 06×6 06×6

TRrcm,2TRq2 TRrcm,2TLq1 06×6 06×6

TRrcm,3TRq3 06×6 TRrcm,3TLq1 06×6

TRrcm,4TRq3q4 06×6 TRrcm,4TRq4TLq1 TRrcm,4TLq1q3

Tq1ι1 · · · 0

.... . .

...

0 · · · TqN ιN

.

3.3.2 Lagrangian and Hamiltonian of an Open-chain Multi-body

System

As mentioned in Section 3.2, the Lagrangian of an Open-chain Multi-body System

L : TQ → R is L(vq) = 12Kq(vq, vq) − V (q). In this section, we describe how the La-

grangian L and subsequently the Hamiltonian H of an Open-chain Multi-body System

is calculated.

Let hi for i = 1, · · · , N be the left-invariant kinetic energy metric for the rigid body Bi

in the open-chain multi-body system. They induce h := h1 ⊕ · · · ⊕ hN as a left-invariant

metric on P . For the open-chain multi-body system, the metric K := T ∗F (h), where

T ∗F (h) refers to the pull back of the metric h by the map F . That is, ∀q ∈ Q and

∀vq, wq ∈ TqQ we have

Kq(vq, wq) = hF (q) (TqF (vq), TqF (wq))

= he(TF (q)LF (q)−1(TqF (vq)), TF (q)LF (q)−1(TqF (wq))

), (3.3.31)

where e is the identity element of the Lie group P and Lp is the left translation map by

any element p ∈ P .

In this thesis, wherever we consider a non-zero potential energy function it is in-

duced by a constant gravitational field g in A0, which is defined in Section 2.1 as the

3-dimensional affine space corresponding to the inertial coordinate frame. Using the Eu-

clidean inner product of R3, which is denoted by ·, · , the potential energy function

for an open-chain multi-body system is defined as

V (q) :=N∑i=1

mig,O0 − Fi(q)(Oi), (3.3.32)

where mi is the mass of the rigid body Bi, and Fi(q) : Ai → A0 is the ith component of

the map F that can be considered as an isometry between Ai and A0 with respect to the

Euclidean norm of R3. The points O0 ∈ A0 and Oi ∈ Ai are the base points for the affine


spaces A0 and Ai, where Oi is located at the centre of mass of Bi.

Subsequently, using the Legendre transformation one can define the Hamiltonian

H : T ∗Q → R for an open-chain multi-body system by

H(pq) := 〈pq,FL−1q (pq)〉 − L(FL−1

q (pq)). (3.3.33)

Here, we remind the reader that FL : TQ → T ∗Q is the fibre-wise invertible Legendre

transformation induced by the kinetic energy metric, i.e., ∀vq, wq ∈ TqQ, 〈FLq(vq), wq〉 =

Kq(vq, wq).

Accordingly, a holonomic open-chain multi-body system can be considered as a Hamil-

tonian mechanical system described by the quadruple (T ∗Q,Ωcan, H,K). Here, the metric

K and the Hamiltonian H are defined by (3.3.31) and (3.3.33), respectively.

3.3.3 Reduction of Holonomic Open-chain Multi-body Systems

Based on the definition of the kinetic energy metric K for a holonomic open-chain multi-

body system, we immediately find the following symmetry for K.

Theorem 3.3.3. For a holonomic open-chain multi-body system, the action of G = Q1 on

Q by left translation on the first component leaves the kinetic energy metric K invariant.

Here,for any g ∈ G the action map Φg : Q → Q is given by Φg(q) = (gq1, q2, · · · , qN),

where q = (q1, · · · , qN) ∈ Q.

Proof. For any g ∈ G, let TΦg : TQ → TQ be the induced action of G on the tangent

bundle. For simplicity, ∀q ∈ Q and ∀vq ∈ TqQ we respectively write Φg(q) and TqΦg(vq)

as g · q and g · vq. Then, ∀wq ∈ TqQ we have

Kg·q(g · vq, g · wq) = he((TF (g·q)LF (g·q)−1)(Tg·qF )(g · vq), (TF (g·q)LF (g·q)−1)(Tg·qF )(g · wq)

)= he

((TF (g·q)LF (g·q)−1)(Tq(F Φg))(vq), (TF (g·q)LF (g·q)−1)(Tq(F Φg))(wq)

)= he

((TF (g·q)LF (g·q)−1)(Tq(∆g F ))(vq), (TF (g·q)LF (g·q)−1)(Tq(∆g F ))(wq)

)= he

((T∆gF (q)(LF (q)−1 ∆g−1))(Tq(∆g F ))(vq)

, (T∆gF (q)(LF (q)−1 ∆g−1))(Tq(∆g F ))(wq))

= he(Tq(LF (q)−1 F )(vq), Tq(LF (q)−1 F )(wq)

)= he

((TF (q)LF (q)−1)(TqF )(vq), (TF (q)LF (q)−1)(TqF )(wq)

)= Kq(vq, wq).

The first equality is based on the definition of the metric K, and the third and fourth

equalities are true since the following diagram commutes. Note that ∆g = L(g,...,g) is the


diagonal action of G on P .

Q F //

Φg

P

∆g

Q F // P

For the special case of open-chain multi-body systems in space where the potential

energy function is equal to zero, this theorem indicates that the Hamiltonian of the

system is also invariant under the cotangent lifted action of G. In general, there exist

joints for which the potential energy function V defined by (3.3.32) is also invariant

under the G-action, e.g., if Q1 corresponds to a planar joint with the direction of the

gravitational field g being perpendicular to the plane of the joint. For such first joints,

the Hamiltonian of the system H becomes invariant under the cotangent lifted action

of G. From here on, we always assume that V is also invariant under the G-action,

unless otherwise stated. Accordingly, the quintuple (T ∗Q,Ωcan, H,K,G) with the group

action defined in Theorem 3.3.3 is called a holonomic open-chain multi-body system with

symmetry. It is a mechanical system with symmetry.

For a holonomic open-chain multi-body system with symmetry, the G-action is ba-

sically the left translation on Q1. Therefore, the quotient manifolds Q = Q/G and

Q = Q/Gµ are equal to (Q2 × · · · × QN) and (Q1/Gµ × Q2 × · · · × QN), respectively.

We remind the reader that ∀µ ∈ Lie∗(G) the subgroup Gµ ⊆ G is the coadjoint isotropy

group corresponding to G. For any q1 ∈ Q1, let q1 ∈ Q1/Gµ denote the equivalence class

corresponding to q1. Indeed, ∀q = (q1, · · · , qN) ∈ Q the quotient maps π : Q → Q and

π : Q → Q are defined by q := π(q) = (q2, · · · , qN) and q := π(q) = (q1, q2, · · · , qN),

respectively.

For an open-chain multi-body system with symmetry, we then calculate the infinites-

imal action of ξ ∈ Lie(G) on Q at q = (q1, ..., qN) by

ξQ(q) =∂

∂ε

∣∣∣∣ε=0

(exp(εξ)q1, q2, · · · , qN) = (ξq1, 0, · · · , 0),

where ξq1 corresponds to the right translation of ξ by q1 ∈ Q1. This relation indicates


that the map φ is the right translation of a Lie algebra element on Q1, i.e.,

φq :=

Te1Rq1

0...

0

. (3.3.34)

Accordingly, based on (3.2.16) ∀pq := (p1, · · · , pN) ∈ T ∗Q the momentum map M : T ∗Q →Lie∗(G) for a holonomic open-chain multi-body system can be determined by the follow-

ing calculation,

〈Mq(pq), ξ〉 = 〈(p1, · · · , pN), (ξq1, 0, · · · , 0)〉 = 〈p1, ξq1〉 = 〈T ∗e1Rq1p1, ξ〉.

As a result,

Mq = φ∗q =[T ∗e1Rq1 0 · · · 0

]. (3.3.35)

Denote the block components of the kinetic energy tensor, which is equal to the Legendre

transformation in the case of Hamiltonian mechanical systems, by Kij(q)dqi ⊗ dqj for

i, j = 1, · · · , N . Hence, we have FLq =∑N

i,j=1Kij(q)dqi ⊗ dqj or equivalently

FLq =

K11(q) · · · K1N(q)

.... . .

...

KN1(q) · · · KNN(q)

.

Lemma 3.3.4. For all q ∈ Q we have the following equality:

FLq =

(T ∗q1Lq−1

1)(K11(q))(Tq1Lq−1

1) (T ∗q1Lq−1

1)(K12(q)) · · · (T ∗q1Lq−1

1)(K1N(q))

(K21(q))(Tq1Lq−11

) K22(q) · · · K2N(q)...

.... . .

...

(KN1(q))(Tq1Lq−11

) KN2(q) · · · KNN(q)

,

where q = π(q) and Kij(q) = Kij((e1, q)).

Proof. By Theorem 3.3.3, ∀vq, wq ∈ TqQ and q = π(q) ∈ Q we have

Kq(vq, wq) = K(e1,q)(TqΦq−11vq, TqΦq−1

1wq).


By the definition of Legendre transformation in (3.2.12), we can rewrite this equation as

〈FLq(vq), wq〉 =⟨FL(e1,q)(TqΦq−1

1)(vq), TqΦq−1

1(wq)

⟩=⟨

(T ∗q Φq−11

)FL(e1,q)(TqΦq−11

)(vq), wq

⟩.

We prove the equality in the lemma, since we have

TqΦq−11

= Tq1Lq−11⊕ idTqQ =

[Tq1Lq−1

10

0 idTqQ

],

where idTqQ is the identity map on TqQ.

Based on this lemma we calculate the locked inertia tensor Iq = φ∗q FLq φq for a

holonomic open-chain multi-body system by

Iq = (T ∗e1Rq1)K11(q)(Te1Rq1) = Ad∗q−11K11(q)Adq−1

1. (3.3.36)

Consequently, using (3.2.21) we determine the (mechanical) connection A corresponding

to the G-action, for a holonomic open-chain multi-body system:

Aq = I−1q Mq FLq

= (Adq1)K11(q)−1(Ad∗q1)[T ∗e1Rq1 0 · · · 0

]K11 · · · K1N

.... . .

...

KN1 · · · KNN

= Adq1

[Tq1Lq−1

1K11(q)−1K12(q) · · · K11(q)−1K1N(q)

]=: Adq1

[Tq1Lq−1

1Aq

],

(3.3.37)

where the last line of (3.3.37) is the consequence of Lemma 3.3.4, and the fibre-wise linear

map A : TQ → Lie(G) is defined by the last equality.

According to (3.2.19), ∀q ∈ Q and ∀vq ∈ TqQ the horizontal lift map hlq : TqQ → TqQbecomes

hlq =

[−(Te1Lq1)Aq

idTqQ

],

where q = (q1, q).

Using the decomposition TQ = H ⊕ V introduced in the previous section, we then

show that ∀q ∈ Q the map horq : TqQ → Hq, which maps any vector in the tangent space


TqQ to its horizontal component, is

horq = idTqQ − verq = idTqQ − φq Aq = idTqQ −

Te1Rq1

0...

0

Adq1

[Tq1Lq−1

1Aq

]

=

0 · · · 0 −Te1Lq1Aq...

...

0 · · · 0 idTqQ

. (3.3.38)

We consider the principal bundle π1 : Q1 → Q1/Gµ to locally trivialize the Lie group

Q1. Let Uµ ⊆ Q1/Gµ be an open neighbourhood of e1, where e1 is the equivalence class

corresponding to the identity element e1 ∈ Q1. We denote the map corresponding to

a local trivialization of the principal bundle π1 by χ : Gµ × Uµ → Q1. This map can

be defined by embedding Uµ in Q1, for example by using the exponential map of Lie

groups. We denote this embedding by χµ : Uµ → Q1 such that ∀q1 ∈ Q1/Gµ we have

χµ(q1) = exp(ζ) for some ζ ∈ C, where C ⊂ Lie(Q1) is a complementary subspace to

Lie(Gµ) ⊂ Lie(G). Accordingly, ∀h ∈ Gµ we define the map χ by the equality χ((h, q1)) :=

hχµ(q1). It is easy to show that the map χ is a diffeomorphism onto its image [35].

Using this diffeomorphism, any element q1 ∈ π−11 (Uµ) ⊆ Q1 can be uniquely identified

by an element (h, q1) ∈ Gµ × Uµ. As a result, we have q = (q1, q) = (χ((h, q1)), q).

From now on, for brevity we write q = (h, q1, q). Accordingly, by Lemma 3.3.4, for all

q = (h, q1, q) ∈ Gµ × Uµ ×Q we can calculate Aµ as

Aµq = Adh

[ThLh−1 Aq

],

where q = π(q) = (q1, q) ∈ Uµ ×Q and Aq : Tq(Uµ ×Q)→ Lie(Gµ) is calculated by

Aq :=[KGµ1 (q)−1K

Q1/Gµ1 (q) K

Gµ1 (q)−1Kµ

12(q) · · · KGµ11 (q)−1Kµ

1N(q)].

Here, according to the local trivialization that we chose we have the following form for


the tensor FLq

FLq =

KGµ1 (q) K

Q1/Gµ1 (q) Kµ

12(q) · · · Kµ1N(q)

KGµ2 (q) K

Q1/Gµ2 (q)

Kµ21(q)

.... . .

......

.... . .

...

KµN1(q) · · · · · · · · · Kµ

NN(q)

.

And, we have KGµ1 (q) = K

Gµ1 ((eµ, q)), K

Q1/Gµ1 (q) = K

Q1/Gµ1 ((eµ, q)), and Kµ

1i(q) = Kµ1i((eµ, q))

for all i = 2, · · · , N . Here, eµ ∈ Gµ is the identity element of the Lie group Gµ ⊆ G = Q1.

Now, for any h ∈ Gµ and ∀q = (h, q1, q) ∈ Gµ × Uµ × Q, we calculate the horizontal

lift map hlq : Tq(Uµ ×Q)→ TqQ for the principal bundle π : Q → Q by

hlq =

[−(TeµLh)Aq

idTq1Uµ ⊕ idTqQ

], (3.3.39)

where idTq1Uµ is the identity map on the tangent space Tq1Uµ. Let µ ∈ Lie∗(G) be a

regular value of the momentum map M. For a holonomic open-chain multi-body system

with symmetry, the level set of the momentum map M at µ becomes

M−1(µ) = pq = (p1, · · · , pN) ∈ T ∗Q| p1 = T ∗q1Rq−11µ ⊂ T ∗Q.

Furthermore, we determine αµ = A∗µ ∈ Ω1(Q) in the local trivialization by

αµ(q) =

[T ∗(h,q1)L(h,q1)−1

A∗q

]Ad∗(h,q1)µ =

[T ∗(h,q1)L(h,q1)−1

A∗q

]Ad∗(eµ,q1)µ, (3.3.40)

where (h, q1)−1 = χ−1 ((χ(h, q1))−1), by definition. The second equality is true by the

definition of the map χ, and because h ∈ Gµ.

Lemma 3.3.5. Based on Theorem 3.2.7, the inverse of the map ϕµ : M−1/Gµ → T ∗Q is

defined on [T π(V)]0 and in the local trivialization ∀pq = (p1, p) ∈ T ∗q (Uµ ×Q),

ϕ−1µ (pq) =

[T ∗(h,q1)R(h,q1)−1(µ)

p+ A∗q(Ad∗(eµ,q1)µ)

]µ

. (3.3.41)

Proof. First we show that p ∈ [T π(V)]0 if and only if p1 = 0. For any p ∈ [T π(V)]0 and


∀ξ ∈ Lie(G) = Lie(Q1) we have

〈(p1, p), T π(ξQ)〉 =⟨φ∗q(0, p1, p), ξ

⟩=⟨T ∗e1Rq1(0, p1), ξ

⟩= 0.

The first equality is true based on the definition of ξQ and the local trivialization that is

chosen. The second equality is the consequence of the definition of the map φ in (3.3.34).

Since the above equality should hold for every ξ ∈ Lie(G) and the right translation map

is a diffeomorphism ∀q1 ∈ Q1, we have p1 = 0. Now, based on (3.3.40) and the definition

of the map ϕµ in Theorem 3.2.7 we have the desired equation in the lemma.

Based on the definition of H(pq) := Hµ ϕ−1µ (pq) and the above lemma, we calculate

H on [T π(V)]0 using the local trivialization:

H(pq) =1

2

⟨(Ad∗(eµ,q1)µ, p+ A∗q(Ad∗(eµ,q1)µ)),

,FL−1(e1,q)

(Ad∗(eµ,q1)µ, p+ A∗q(Ad∗(eµ,q1)µ))⟩

+ V (eµ, q1, q). (3.3.42)

Now we are ready to state the main result of this section in the following theorem.

Theorem 3.3.6. Let µ ∈ Lie∗(G) be a regular value of the momentum map M. A

holonomic open-chain multi-body system with symmetry (T ∗Q,Ωcan, H,K,G) is reduced

to a Hamiltonian mechanical system ([T π(V)]0 ⊆ T ∗Q, Ωcan|[T π(V)]0 − Ξµ, H, K), where

Ωcan is the canonical 2-form on T ∗Q, H is defined by (3.3.42) and K is a metric on Qsuch that ∀uq, wq ∈ TqQ we have

Kq(uq, wq) = Kq(hlq(uq), hlq(wq)).

Here, in the local coordinates Ξµ is calculated as follows. Let πQ : T ∗Q → Q be the

canonical projection map of the cotangent bundle and let TπQ : T (T ∗Q) → T Q be its

tangent map. For every αq ∈ T ∗Q and ∀U , W ∈ X(T ∗Q) we introduce uq = TαqπQ(Uαq)and wq = TαqπQ(Wαq). In the local trivialization, we have q = (q1, q) ∈ Uµ × Q, uq =

(u1, u) and wq = (w1, w):

(Ξµ)αq(Uαq , Wαq) =

⟨µ,−Adχµ(q1)

([Aqu,Aqw] + (

∂Aq∂q

w)u− (∂Aq∂q

u)w

)+[(−Aqu+ (Tχµ(q1)Rχµ(q1)−1)(Tq1χµ)(u1) + Adχµ(q1)Aqu

),(

−Aqw + (Tχµ(q1)Rχµ(q1)−1)(Tq1χµ)(w1) + Adχµ(q1)Aqw)]⟩

, (3.3.43)

where χµ : Uµ → Q1 is the embedding that is used to define the local trivialization map


χ, using the exponential map of Q1.

Finally, in local coordinates we have X = (˙q1, q, p) as a vector field on [T π(V)]0.

Hamilton’s equation in the vector sub-bundle [T π(V)]0 of the cotangent bundle of µ-shape

space reads

ι( ˙q1,q,p)(−dp ∧ dq − Ξµ) =

∂H

∂pdp+

∂H

∂q1

dq1 +∂H

∂qdq, (3.3.44)

where Ξµ is calculated by (3.3.43).

Proof. In order to prove (3.3.43), we start with (3.2.27):

(Ξµ)αq(Uαq , Wαq) =⟨µ,−Aq([hor(hl(u)), hor(hl(w))]q) + [Aq(hlq(uq)),Aq(hlq(wq))]

⟩.

Using the local trivialization, we write q = (h, q1, q) ∈ Gµ × Uµ × Q, and accordingly

u = (u1, u) and w = (w1, w). By (3.3.39), the horizontal lift of u and w can be calculated

as

hlq(uq) = (−(TeµLh)Aqu, u1, u), hlq(wq) = (−(TeµLh)Aqw, w1, w),

and using (3.3.38), the terms hor(hl(u)) and hor(hl(w)) are

horq(hlq(uq)) = (−(T(eµ,e1)L(h,q1))Aqu, u), horq(hlq(wq)) = (−(T(eµ,e1)L(h,q1))Aqw,w).

Now, by (3.3.37) we have

Aq(hlq(uq)) = Ad(h,q1)

((T(h,q1)L(h,q1)−1)

(−(TeµLh)Aqu, u1

)+ Aqu

). (3.3.45)

Using the definition of the local trivialization map χ we have

T(h,q1)L(h,q1)−1

(−(TeµLh)Aqu, u1

)= Thχµ(q1)Lχµ(q1)−1h−1

(ThRχµ(q1)(−(Te1Lh)Aqu) + (Tχµ(q1)Lh)(Tq1χµ)(u1)

)= Adχµ(q1)−1(−Aqu) + (Tχµ(q1)Lχµ(q1)−1)(Tq1χµ)(u1),

where χµ : Uµ → Q1 is the embedding map that is defined using the exponential map.

Therefore, we have

Aq(hlq(uq)) = Adh

(−Aqu+ (Tχµ(q1)Rχµ(q1)−1)(Tq1χµ)(u1) + Adχµ(q1)Aqu

).


Similarly,

Aq(hlq(wq)) = Adh

(−Aqw + (Tχµ(q1)Rχµ(q1)−1)(Tq1χµ)(w1) + Adχµ(q1)Aqw

).

Since for all g ∈ G and ξ, η ∈ Lie(G) we have the equality Adg[ξ, η] = [Adgξ,Adgη]:

[Aq(hlq(uq)),Aq(hlq(wq))] = Adh

[(−Aqu+ (Tχµ(q1)Rχµ(q1)−1)(Tq1χµ)(u1) + Adχµ(q1)Aqu

),(

−Aqw + (Tχµ(q1)Rχµ(q1)−1)(Tq1χµ)(w1) + Adχµ(q1)Aqw)].

For all q ∈ Q, to calculate the Lie bracket [hor(hl(u)), hor(hl(w))]q, we express the vector

fields hor(hl(u)) and hor(hl(w)) in coordinates:

horq(hlq(uq)) =(−(T(eµ,e1)L(h,q1))Aqu

) ∂

∂(h, q1)+ u

∂

∂q

horq(hlq(wq)) =(−(T(eµ,e1)L(h,q1))Aqw

) ∂

∂(h, q1)+ w

∂

∂q.

In any coordinates chosen for Qi (i = 2, · · · , N), Gµ and Q1/Gµ we have

[hor(hl(u)), hor(hl(w))] = [((T(eµ,e1)L(h,q1))Aqu

) ∂

∂(h, q1),((T(eµ,e1)L(h,q1))Aqw

) ∂

∂(h, q1)]

+ [u∂

∂q, w

∂

∂q] + [

((T(eµ,e1)L(h,q1))Aqw

) ∂

∂(h, q1), u

∂

∂q]

− [((T(eµ,e1)L(h,q1))Aqu

) ∂

∂(h, q1), w

∂

∂q]

Based on the definition of the Lie bracket for Lie groups, ∀q ∈ Q the first bracket on the

right hand side can be written as

[((T(eµ,e1)L(h,q1))Aqu

) ∂

∂(h, q1),((T(eµ,e1)L(h,q1))Aqw

) ∂

∂(h, q1)]

=((T(eµ,e1)L(h,q1))[Aqu,Aqw]

) ∂

∂(h, q1)

+

((T(eµ,e1)L(h,q1))Aq

∂w

∂(h, q1)

((T(eµ,e1)L(h,q1))Aqu

)) ∂

∂(h, q1)

−(

(T(eµ,e1)L(h,q1))Aq∂u

∂(h, q1)

((T(eµ,e1)L(h,q1))Aqw

)) ∂

∂(h, q1).


The second bracket is equal to

[u∂

∂q, w

∂

∂q] =

(∂w

∂qu

)∂

∂q−(∂u

∂qw

)∂

∂q.

We calculate the third bracket as

[((T(eµ,e1)L(h,q1))Aqw

) ∂

∂(h, q1), u

∂

∂q] =

(∂u

∂(h, q1)(T(eµ,e1)L(h,q1))Aqw

)∂

∂q

−(

(T(eµ,e1)L(h,q1))

(∂Aq∂q

u

)w + (T(eµ,e1)L(h,q1))Aq

∂w

∂qu

)∂

∂(h, q1).

Similarly, the last bracket can be calculated. Accordingly, using (3.3.37),

Aq([hor(hl(u)), hor(hl(w))]q) = Ad(h,q1)

([Aqu,Aqw] +

(∂Aq∂q

w

)u−

(∂Aq∂q

u

)w

).

Finally, knowing that h ∈ Gµ, we have the equation for Ξµ in the theorem.

Regarding Hamilton’s equation, we should note that based on Lemma 3.3.5 the re-

striction of Ωcan to [T π(V)]0 is equal to −dp ∧ dq, in coordinates.

Corollary 3.3.7. Let us assume that Gµ = G, in the above theorem. A holonomic open-

chain multi-body system with symmetry (T ∗Q,Ωcan, H,K,G) is reduced to a Hamiltonian

mechanical system (T ∗Q,Ωcan − Ξµ, H,K), where Ωcan is the canonical 2-form on T ∗Q,

H(pq) :=1

2

⟨(µ, p+ A∗qµ),FL−1

(e1,q)(µ, p+ A∗qµ)

⟩+ V (e1, q), (3.3.46)

and K is a metric on Q such that ∀uq, wq ∈ TqQ we have

Kq(uq, uq) = Kq(hlq(uq), hlq(wq)).

Here, in the local coordinates Ξµ is calculated by a simpler formulation. Let πQ : T ∗Q →Q be the canonical projection map of the cotangent bundle and let TπQ : T (T ∗Q)→ TQbe its tangent map. For every αq ∈ T ∗Q and ∀U ,W ∈ X(T ∗Q) we introduce uq =

TαqπQ(Uαq) and wq = TαqπQ(Wαq). We have

(Ξµ)αq(Uαq ,Wαq) =

⟨µ,−[Aqu,Aqw]− (

∂Aq∂q

w)u+ (∂Aq∂q

u)w

⟩. (3.3.47)

Finally, in local coordinates we have X = (q, p) as a vector field on T ∗Q. Hamilton’s


equation in the cotangent bundle of shape space reads

ι(q,p)(−dp ∧ dq − Ξµ) =∂H

∂pdp+

∂H

∂qdq,

where Ξµ is calculated by (3.3.47).

We show the isotropy groups for different types of displacement subgroups in Ta-

ble 3.1. Note that for different values of µ ∈ Lie∗(G), the isotropy groups are isomorphic

to the groups listed in the table, and the isomorphism map is conjugation by an element

of SE(3). In this table we consider the configuration manifold of the first joint as a Lie

sub-group of SE(3) whose Lie algebra is a vector space isomorphic to so(3)⊕R3, where

so(3) is the Lie algebra of SO(3). For any element ξ ∈ se(3), we call its component in

R3 the linear and the one in so(3) the angular component of ξ, where se(3) denotes the

Lie algebra of SE(3).

Table 3.1: Displacement subgroups and their corresponding isotropy groups

DisplacementSubgroups

Gµ (µ = (µv, µω)a)

Q1∼= G µv 6= 0, µω 6= 0 µv = 0, µω 6= 0 µv 6= 0, µω = 0 µv = µω = 0

SE(3) SO(2)× R SE(2)× R SO(2)× R SE(3)SE(2)× R R2 (SE(2)× R)b SE(2)× R R2 (SE(2)× R)b SE(2)× RSE(2) R SE(2) R SE(2)SO(3) SO(2) SO(3)R3 R3 R3

Hp nR2 R Hp nR2 R Hp nR2

SO(2)× R SO(2)× R SO(2)× R SO(2)× R SO(2)× RR2 R2 R2

SO(2) SO(2) SO(2)R R RHp Hp Hp

a µv is the linear component and µω is the angular component of the momentum.b If the linear momentum is in the direction of the allowed direction of rotation in the space.

3.4 Case Study

In this section we study the dynamics of an example of a holonomic open-chain multi-

body system. We derive the reduced dynamical equations of a six-d.o.f. manipulator

mounted on top of a spacecraft whose initial configuration is shown in Figure 3.1.

Using the indexing introduced in the previous section and starting with the spacecraft

as B1, we first number the bodies and joints. The following graph shows the structure of


Figure 3.1: A six-d.o.f. manipulator mounted on a spacecraft

the holonomic open-chain multi-body system.

B4

B0J1

B1J2

B2J3

B3

J5

J4

B5

We then identify the relative configuration manifolds corresponding to the joints of

the robotic system. The relative pose of B1 with respect to the inertial coordinate frame

is identified by the elements of the Special Euclidean group SE(3). We identify the

elements of the relative configuration manifold corresponding to the first joint, which is

a six-d.o.f. free joint, by

Q01 =

r01 =

RY (θY )RX(θX)RZ(θZ)

xyz

[0 0 0

]1

∣∣∣∣∣∣∣∣∣∣x, y, z ∈ R, θX , θY , θZ ∈ S1

,


Figure 3.2: The coordinate frames attached to the bodies of the robot

where we have

RX(θX) =

1 0 0

0 cos(θX) − sin(θX)

0 sin(θX) cos(θX)

,

RY (θY ) =

cos(θY ) 0 sin(θY )

0 1 0

− sin(θY ) 0 cos(θY )

,

RZ(θZ) =

cos(θZ) − sin(θZ) 0

sin(θZ) cos(θZ) 0

0 0 1

.The second joint is a three-d.o.f. spherical joint betweenB2 andB1, and its corresponding

relative configuration manifold is given by

Q12 =

r12 =

RX(ψX)RY (ψY )RZ(ψZ)

0

l1

0

[0 0 0

]1

∣∣∣∣∣∣∣∣∣∣ψX , ψY , ψZ ∈ S1

.

The third joint is a one-d.o.f. revolute joint between B3 and B2, and its relative config-


uration manifolds is

Q23 =

r23 =

1 0 0 0

0 cos(ψ1) − sin(ψ1) l2

0 sin(ψ1) cos(ψ1) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ1 ∈ S1

.

The forth and fifth joints are one-d.o.f. revolute joints whose axes of revolution are

assumed to be the Xi-axis (i = 4, 5). The relative configuration manifolds of these joints

are identified by

Q34 =

r34 =

1 0 0 c

0 cos(ψ2) − sin(ψ2) l3


0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ2 ∈ S1

,

Q35 =

r35 =

1 0 0 −c0 cos(ψ3) − sin(ψ3) l3


0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ3 ∈ S1

.

Here, we denote the distance between J4 and J5 by 2c, i.e., the origins of the coordinate

frames V4 and V5 are located at c and −c in the x direction of V3, respectively. Further,

l1, · · · , l5 are defined in Figure 3.2.

We assume that the initial pose of B1 with respect to the inertial coordinate frame

r01,0 is the identity element of SE(3). We have located the coordinate frame attached to

B1 on its centre of mass. Hence, in matrix form we have r01,0 = rcm,1 = id4, where id4 is

the 4 × 4 identity matrix. For the second body, the initial relative pose with respect to

B1 is

r12,0 =

1 0 0 0

0 1 0 l1

0 0 1 0

0 0 0 1

,and we have

rcm,2 =

1 0 0 0

0 1 0 l1 + l2/2

0 0 1 0

0 0 0 1

.


The initial relative pose of B3 with respect to B2 is

r23,0 =

1 0 0 0

0 1 0 l2

0 0 1 0

0 0 0 1

,

and the relative pose of the centre of mass of B3 with respect to the inertial coordinate

frame is

rcm,3 =

1 0 0 0

0 1 0 l1 + l2 + l3/2

0 0 1 0

0 0 0 1

.Here we have assumed that the centre of mass of B2 and B3 are in the middle of the

links. For the forth and fifth bodies we have (i = 4, 5)

r3i,0 =

1 0 0 ±c0 1 0 l3

0 0 1 0

0 0 0 1

,

rcm,4 =

1 0 0 c

0 1 0 l1 + l2 + l3 + l4

0 0 1 0

0 0 0 1

, rcm,5 =

1 0 0 −c0 1 0 l1 + l2 + l3 + l5

0 0 1 0

0 0 0 1

,where the plus and minus signs correspond to the body B4 and B5, respectively.

With the above specifications of the system we identify the configuration manifold of

the holonomic open-chain multi-body system in this case study by Q = Q1 × · · · × Q5,

where

Q1 =

q1 =


xyz

[0 0 0

]1

∈ SE(3)

,


Q2 =

q2 =

R

0

l1

0

−R0

l1

0

[0 0 0

]1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣R = RX(ψX)RY (ψY )RZ(ψZ)

,

Q3 =

q3 =

1 0 0 0

0 cos(ψ1) − sin(ψ1) 2(l1 + l2) sin2(ψ1/2)

0 sin(ψ1) cos(ψ1) −(l1 + l2) sin(ψ1)

0 0 0 1

∈ SE(3)

,

Q4 =

q4 =

1 0 0 0

0 cos(ψ2) − sin(ψ2) 2(l1 + l2 + l3) sin2(ψ2/2)

0 sin(ψ2) cos(ψ2) −(l1 + l2 + l3) sin(ψ2)

0 0 0 1

∈ SE(3)

,

Q5 =

q5 =

1 0 0 0

0 cos(ψ3) − sin(ψ3) 2(l1 + l2 + l3) sin2(ψ3/2)

0 sin(ψ3) cos(ψ3) −(l1 + l2 + l3) sin(ψ3)

0 0 0 1

∈ SE(3)

.

In order to calculate the kinetic energy for the system under study, we need to first

form the function F : Q → P =

5−times︷︸︸︷SE(3)× · · · × SE(3), which determines the pose of

the coordinate frames attached to the centres of mass of the bodies with respect to the

inertial coordinate frame.

F (q1, · · · , q5) = (q1rcm,1, q1q2rcm,2, q1q2q3rcm,3, q1q2q3q4rcm,4, q1q2q3q5rcm,5)

Using (3.3.31), we can calculate the kinetic energy metric for the open-chain multi-

body system. In matrix form we have the following equation for the tangent map

Tq(LF (q)−1F ) : TqQ → Lie(P)

Tq(LF (q)−1F ) =

Adr−1

cm,1· · · 0

.... . .

...

0 · · · Adr−1cm,5

JqTq1(Lq−1

1 ι1) · · · 0

.... . .

...

0 · · · Tq5(Lq−15 ι5)

,


where we have

Jq =

id6 06×6 06×6 06×6 06×6

Adq−12

id6 06×6 06×6 06×6

Ad(q2q3)−1 Adq−13

id6 06×6 06×6

Ad(q2q3q4)−1 Ad(q3q4)−1 Adq−14

id6 06×6

Ad(q2q3q5)−1 Ad(q3q5)−1 Adq−15

06×6 id6

,

and where id6 is the 6× 6 identity matrix. Let us denote the standard basis for se(3) by

E1, · · · , E6, such that

E1 =

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

, E2 =

0 0 0 0

0 0 0 1

0 0 0 0

0 0 0 0

, E3 =

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

E4 =

0 0 0 0

0 0 −1 0

0 1 0 0

0 0 0 0

, E5 =

0 0 1 0

0 0 0 0

−1 0 0 0

0 0 0 0

, E6 =

0 −1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

Using the introduced joint parameters, we have the following equalities:

Tq1(Lq−11 ι1) =

R−1Z (θZ)R−1

X (θX)R−1Y (θY ) 03×3

03×3

cos(θZ) cos(θX) sin(θZ) 0

− sin(θZ) cos(θX) cos(θZ) 0

0 − sin(θX) 1

,

Tq2(Lq−12 ι2) =

−l1 sin(ψY ) 0 −l10 0 0

l1 cos(ψY ) cos(ψZ) −l1 sin(ψZ) 0

− cos(ψY ) cos(ψZ) sin(ψZ) 0

cos(ψY ) sin(ψZ) cos(ψZ) 0

− sin(ψY ) 0 1

,

Tq3(Lq−13 ι3) =

[0 0 l1 + l2 1 0 0

]T,

Tq4(Lq−14 ι4) =

[0 0 l1 + l2 + l3 1 0 0

]TTq5(Lq−1

5 ι5) =

[0 0 l1 + l2 + l3 1 0 0

]T.


Note that ∀r0 ∈ SE(3) that is in the following form (R0 ∈ SO(3) and p0 = [p0,1, p0,2, p0,3]T ∈R3)

r0 =

[R0 p0

01×3 1

],

we calculate the Adr0 operator by

Adr0 =

[R0 p0R0

03×3 R0

],

where

p0 =

0 −p0,3 p0,2

p0,3 0 −p0,1

−p0,2 p0,1 0

is a skew-symmetric matrix. We choose the standard basis E1, · · · , E6 for se(3). For

this case study, the left-invariant metric h = h1⊕· · ·⊕h6 on P is identified, in the above

basis, by the following metrics on the Lie algebras of copies of SE(3) corresponding to

the bodies:

he,i =

miid3 03×3

03×3

jx,i 0 0

0 jy,i 0

0 0 jz,i

,

where i = 1, · · · , 5, id3 and 03×3 are the 3×3 identity and zero matrices, respectively, mi

is the mass of Bi, and (jx,i, jy,i, jz,i) are the moments of inertia of Bi about the X, Y and

Z axes of the coordinate frame attached to the centre of mass of Bi. Note that we chose

this coordinate frame such that its axes coincide with the principal axes of the body Bi.

For the body Bi (i = 2, · · · , 5), since we assume a symmetric shapes with Yi-axis being

the axis of symmetry, we have jx,i = jz,i. Finally, in the coordinates chosen to identify

the configuration manifold (joint parameters), we have the following matrix form for FLq

FLq = T ∗q (LF (q)−1F )

he,1 · · · 0

.... . .

...

0 · · · he,5

Tq(LF (q)−1F ) =

K11(q) · · · K15(q)

.... . .

...

K51(q) · · · K55(q)

,and the kinetic energy is calculated by

Kq(q, q) =1

2qTFLq q,


where, with an abuse of notation, q is the vector corresponding to the speed of the joint

parameters.

We assume zero potential energy for this holonomic open-chain multi-body system,

Hence, we have the Hamiltonian of the system as

H(q, p) =1

2pTFL−1

q p,

where p is the vector of generalized momenta corresponding to the joint parameters.

In the following, we derive the reduced Hamilton’s equation for this system, with

the initial total momentum µ =[0 µ1 0 µ2 0 0

]T∈ se∗(3) represented in the dual

of the standard basis for se(3). That is, the system has a constant linear momentum

in the direction of Y0, equal to µ1, and a constant angular momentum in the direction

of X0, equal to µ2. The kinetic energy (and hence the Hamiltonian) of the this multi-

body system is invariant under the action of G = Q1 = SE(3). The isotropy group

corresponding to µ is

Gµ =

h =

cos(θY ) 0 sin(θY ) µ2

µ1sin(θY )

0 1 0 y

− sin(θY ) 0 cos(θY ) −2µ2

µ1sin2(θY /2)

0 0 0 1

∈ SE(3)

,

which is a Lie subgroup of G, and it is isomorphic to SO(2)×R. Now, consider the action

of G = SE(3) by left translation onQ1. Using the joint parameters, ∀(x0, y0, z0, θX,0, θY,0, θZ,0) ∈G we have

Φ(x0,y0,z0,θX,0,θY,0,θZ,0)(q) = (RY (θY,0)RX(θX,0)RZ(θZ,0)[x y z

]T+[x0 y0 z0

]T, RY (θY,0)RX(θX,0)RZ(θZ,0)RY (θY )RX(θX)RZ(θZ), q)

where q = (ψX , ψY , ψZ , ψ1, ψ2, ψ3). We have the principal G-bundle π : Q → Q = Q2 ×· · ·×Q5, and using the joint parameters its corresponding principal connection A : TQ →se(3) is defined by (3.3.37)

Aq =


xyz


03×3 RY (θY )RX(θX)RZ(θZ)

[Tq1Lq−1

1Aq

],


where we have xyz

=

0 −z y

z 0 −x−y x 0

,

Tq1Lq−11

=

R−1Z (θZ)R−1

X (θX)R−1Y (θY ) 03×3

03×3

cos(θZ) cos(θX) sin(θZ) 0

− sin(θZ) cos(θX) cos(θZ) 0

0 − sin(θX) 1

,

Aq =[K11(q)−1K12(q) · · · K11(q)−1K1N(q)

],

where K1i(q) = K1i(e1, q) for i = 1, · · · , N , and consequently, the horizontal lift map

hlq : TqQ → TqQ is

hlq =

−

RY (θY )RX(θX)RZ(θZ) 03×3

03×3

cos(θZ) − sin(θZ) 0

sin(θZ)/ cos(θX) cos(θZ)/ cos(θX) 0

sin(θZ) tan(θX) cos(θZ) tan(θX) 1

Aq

id6

,

where id6 is the 6× 6 identity matrix. Then, we use the principal bundle π : Q → Q/Gµto introduce the local trivialization of G = Q1. The Lie algebra of Gµ as a vector subspace

of se(3) is spanned byE2,

µ2

µ1E1 + E5

, and a complementary subspace to this subspace

is spanned by E1, E3, E4, E6. Now, ∀q1 ∈ Uµ ⊂ Q1/Gµ we introduce the embedding

χµ : Uµ → Q1

χµ(q1) =

RX(θX)RZ(θZ)

x0z

01×3 1

,which identifies the elements of Q1/Gµ by elements of an embedded submanifold of Q1,


and in the local coordinates its induced map on the tangent bundles is

Tq1χµ =

1 0 0 0

0 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

0 0 0 1

.

Subsequently, we define the local trivialization of the principal bundle π : Q → Q/Gµ by

χ : Gµ × Uµ → Q1

χ((h, q1)) = hχµ(q1),

and its induced map on the tangent bundles (in the local coordinates) is calculated as

T(h,q1)χ =

0 (µ2

µ1+ z) cos(θY )− x sin(θY ) cos(θY ) sin(θY ) 0 0

1 0 0 0 0 0

0 −(µ2

µ1+ z) sin(θY )− x cos(θY ) − sin(θY ) cos(θY ) 0 0

0 0 0 0 1 0

0 1 0 0 0 0

0 0 0 0 0 1

,

where we use (y, θY ), (x, z, θX , θZ), and (x, y, z, θX , θY , θZ) as the local coordinates for

the manifolds Gµ, Q1/Gµ, and Q1, respectively. Accordingly, we can calculate the map

Aµq : T(q1,q)(Uµ ×Q)→ Lie(Gµ) using the following equalities:

Aµq :=[KGµ1 (q)−1K

Q1/Gµ1 (q) K

Gµ1 (q)−1K

Gµ12 (q) · · · K

Gµ1 (q)−1K

Gµ1N(q)

],

[KGµ1 ((h, q)) K

Q1/Gµ1 ((h, q))

KGµ2 ((h, q)) K

Q1/Gµ2 ((h, q))

]= T ∗(h,q1)χ (K11(χ(h, q)))T(h,q1)χ,[

KGµ12 ((h, q)) · · · K

Gµ1N((h, q))

KQ1/Gµ12 ((h, q)) · · · K

Q1/Gµ1N ((h, q))

]= T ∗(h,q1)χ

[K12(χ(h, q)) · · · K1N(χ(h, q))

].

And, we have KGµ1 (q) = K

Gµ1 ((eµ, q)), K

Q1/Gµ1 (q) = K

Q1/Gµ1 ((eµ, q)), and K

Gµ1i (q) =


KGµ1i ((eµ, q)) for all i = 2, · · · , N . We also have the reduced Hamiltonian on [T π(V)]0:

H(pq) =1

2

[AdT(eµ,q1)µ

p+ ATq AdT(eµ,q1)µ

]TFL−1

(e1,q)

[AdT(eµ,q1)µ

p+ ATq AdT(eµ,q1)µ

], (3.4.48)

where

AdT(eµ,q1)µ =

RTZ(θZ)RT

X(θX) 03×3

−RTZ(θZ)RT

X(θX)

x0z

RTZ(θZ)RT

X(θX)

0

µ1

0

µ2

0

0

.

In order to calculate the 2-form Ξµ, we compute the following matrices in the local

coordinates:

Tχµ(q1)Rχµ(q1)−1(Tq1χµ) =

1 0 0 z sin(θX)

0 0 z −x cos(θX)

0 1 0 −x sin(θX)

0 0 1 0

0 0 0 − sin(θX)

0 0 0 cos(θX)

,

Adχµ(q1) =

RX(θX)RZ(θZ)

x0z

RX(θX)RZ(θZ)

03×3 RX(θX)RZ(θZ)

,Dq : = −Aµq +

[Tχµ(q1)Rχµ(q1)−1(Tq1χµ) Adχµ(q1)Aq

],

Fq1 :=

0

µ1

0

µ2

0

0

T

Adχµ(q1) =

µ1 cos(θX) sin(θZ)

µ1 cos(θX) cos(θZ)

−µ1 sin(θX)

µ1(z cos(θZ)− x sin(θX) sin(θZ)) + µ2 cos(θZ)

−µ1(z sin(θZ) + x cos(θZ) sin(θX))− µ2 sin(θZ)

−µ1x cos(θX)

T

.


Finally, we have the following expression for the 2-form Ξµ:

Ξµ =∑i<j

6∑a=1

Fa

((∂Aaj∂qi− ∂Aai∂qj

)−∑l<k

Ealk(AliAkj − AljAki )

)(dqi ∧ dqj)

+∑i′<j′

∑l<k

((µ1E2

lk + µ2E4lk)(Dli′Dkj′ −Dlj′Dki′)

)(dqi′ ∧ dqj′)

=:∑i′<j′

Υi′j′(q)dqi′ ∧ dqj′ ,

where a, l, k, i, j ∈ 1, · · · , 6 and i′, j′ ∈ 1, · · · , 10. Here, in the local coordinates q =

(x, z, θX , θZ , ψX , ψY , ψZ , ψ1, ψ2, ψ3), q = (ψX , ψY , ψZ , ψ1, ψ2, ψ3), and for the standard

basis for se(3), i.e., E1, · · · , E6, we have

[El, Ek] =6∑

a=1

EalkEa,

Fq1 =6∑

a=1

Fa(q1)Ea,

Aq =

A1

1(q) · · · A16(q)

.... . .

...

A61(q) · · · A6

6(q)

,

Dq =

D1

1(q) · · · D110(q)

.... . .

...

D61(q) · · · D6

10(q)

.As a result, in matrix form we have the following reduced equations of motion for the

holonomic multi-body system under study:

˙q1

q

p

=

0 −Υ12(q) · · · · · · −Υ110(q)

Υ12(q) 0 −Υ23(q) · · · −Υ210(q)... · · · . . . · · · ...

Υ19(q) · · · Υ89(q) 0 −Υ910(q)

Υ110(q) · · · · · · Υ910(q) 0

[

04×6

−id6

]

[06×4 id6

]06×6

−1 ∂H∂q1∂H∂q∂H∂p

,

where H is calculated by (3.4.48).

Chapter 4

Reduction of Nonholonomic


with Displacement Subgroups

This Chapter presents a two-step geometric approach to the reduction of Hamilton’s

equation for nonholonomic open-chain multi-body systems with multi-degree-of-freedom

displacement subgroups.

In Section 4.2 we consider open-chain multi-body systems whose first joint is non-

holonomic. we reduce Hamilton’s equation for these systems in two steps. In the first

step, we consider a subgroup of the symmetry group whose orbits are complementary

to the nonholonomic distribution. Using the reduction theory of Chaplygin systems, we

express Hamilton’s equation of these systems in the cotangent bundle of the quotient

configuration manifold. In this space, the 2-form representing the dynamics is almost

symplectic. In the second step, under some assumptions, we employ a generalization of

the symplectic reduction theorem to almost symplectic manifolds to express the resulting

Hamilton’s equation from the first step in a smaller space.

In Section 4.4 we give some conditions under which K, the kinetic energy metric of a

nonholonomic open-chain multi-body system, admits further symmetries corresponding

to the joints other than the first joint. The metric K is induced by a left-invariant metric

h on a bigger space SE(3) × · · · × SE(3). We follow two approaches to find the bigger

symmetry groups. First, we study the scenario in which the left invariance property of h

induces further symmetries for K. Then we consider the metric K on the configuration

manifold and find the conditions under which the action of a group corresponding to the

joints other than the first joint leaves K unchanged.

82

Chapter 4. Reduction of Nonholonomic Multi-body Systems 83

4.1 Nonholonomic Hamilton’s Equation and

Lagrange-d’Alembert-Pontryagin principle

In this section we first state the Lagrange-d’Alembert principle for nonholonomic La-

grangian systems, following our approach in Section 3.1. Then we relate this principle

to the Lagrange-d’Alembert-Pontryagin principle [93, 94] on the Pontryagin bundle PQ,

and introduce nonholonomic implicit Lagrangian systems. Finally, for hyper-regular La-

grangian systems we show that the resulting equation of motion for nonholonomic implicit

Lagrangian systems is equivalent to Hamilton’s equation for nonholonomic systems in the

phase space, T ∗Q.

A nonholonomic Lagrangian system is a Lagrangian system described with a config-

uration manifold Q and a Lagrangian L ∈ C∞(TQ), along with a regular non-involutive

linear distribution D ⊂ TQ that is bracket-generating. A distribution D is bracket

generating if after a finite number of iterations we have [D,D] ⊆ [D, [D,D]] ⊆ · · · ⊆[D, ..., [D, [D,D]]...] = TQ. The distribution D indicates the submanifold of TQ that

contains any solution curve of the nonholonomic Lagrangian system. A nonholonomic

Lagrangian system evolves on a curve that satisfies the Lagrange-d’Alembert [8] princi-

ple. The Lagrange-d’Alembert principle states that the solution curve of a nonholonomic

Lagrangian system, t 7→ vq(t)(t) ∈ Tq(t)Q, satisfies Hamilton’s principle for arbitrary

variations of the curve t 7→ q(t) ∈ Q with fixed endpoints such that

∂

∂εq(t, ε)

∣∣∣∣ε=0

∈ D(q(t)),

along with the constraint that qq(t)(t) ∈ D(q(t)). As a result, Based on (3.1.2), in

coordinates we have the Euler-Lagrange equation for nonholonomic systems, known as

the Lagrange-d’Alembert equation for nonholonomic Lagrangian systems.⟨(d

dt(∂L

∂q(qq(t)(t)))−

∂L

∂q(qq(t)(t))

)dq, wq(t)

⟩= 0,

∀t ∈ [ts, tf ] and ∀wq(t) ∈ D(q(t)) and qq(t)(t) ∈ D(q(t)),

⇐⇒(d

dt(∂L

∂q(qq(t)(t)))−

∂L

∂q(qq(t)(t))

)dq ∈ D0(q(t)), qq(t)(t) ∈ D(q(t)), (4.1.1)

where D0 ⊂ T ∗Q indicates the annihilator of the distribution D. For the details of

Hamilton’s principle including the definition of variations, see Section 3.1.

As defined in Section 3.1, an implicit Lagrangian system is a system that satis-

fies Hamilton-Pontryagin principle on the Pontryagin bundle PQ, and it is denoted


by the triple (PQ, T ∗ΠT ∗QΩcan, E). A nonholonomic implicit Lagrangian system is

an implicit Lagrangian system along with a regular non-involutive linear distribution

D ⊂ TQ. We denote a nonholonomic implicit Lagrangian system with the quadruple

(PQ, T ∗ΠT ∗QΩcan, E,D). Note that, since in this thesis we only consider nonholonomic

constraints that come from the kinematics, we only consider a distribution on Q. A non-

holonomic implicit Lagrangian system evolves on a curve t 7→ (vq(t)(t), pq(t)(t)) ∈ Pq(t)Qthat satisfies the Lagrange-d’Alembert-Pontryagin principle [94]. Let ∆ := (TΠQ)−1(D) ⊂TPQ be the distribution on the Pontryagin bundle induced by D. Here, ΠQ : PQ → Qis the canonical projection map for the Pontryagin bundle and TΠQ : TPQ → TQ is

its induced map on the tangent bundles. The Lagrange-d’Alembert-Pontryagin princi-

ple states that the solution curve of a nonholonomic implicit Lagrangian system, t 7→(vq(t)(t), pq(t)(t)) = γ(t, 0), satisfies the Hamilton-Pontryagin principle for arbitrary vari-

ations of the solution curve with fixed endpoints in Q, such that

δγ(t) =∂

∂εγ(t, ε)

∣∣∣∣ε=0

∈ ∆(γ(t, 0)),

along with the constraint γ(t, 0) = ∂γ∂t

(t, 0) ∈ ∆(γ(t, 0)). As a result, based on (3.1.4) we

have the implicit Euler-Lagrange equation for nonholonomic systems as

⟨dE γ(t, 0) + ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0)) ,Wγ(t,0)

⟩= 0,

∀Wγ(t,0) ∈ ∆(γ(t, 0)) and γ(t, 0) ∈ ∆(γ(t, 0))

⇐⇒ dE γ(t, 0) + ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0)) ∈ ∆0(γ(t, 0)), γ(t, 0) ∈ ∆(γ(t, 0)),

(4.1.2)

where ∆0 ⊂ T ∗PQ indicates the annihilator of the distribution ∆. For the details of the

Hamilton-Pontryagin principle including the definition of variations, see Section 3.1.

Using any coordinate chart and based on (3.1.5), we can write the implicit Euler-

Lagrange equation for nonholonomic systems, known as Lagrange-d’Alembert-Pontryagin

equation [94], in coordinates as

∂L

∂q(q, v)dq +

∂L

∂v(q, v)dv + qdp− pdq − vdp− pdv ∈ ∆0(q, v, p), (q, v, p) ∈ ∆(q, v, p)

⇐⇒(p− ∂L

∂q(q, v)

)dq ∈ D0(q), p =

∂L

∂v(q, v), q = v, q ∈ D(q), (4.1.3)

where we did not change the symbols when we wrote the distributions D and ∆ and

their annihilators in coordinates. This gives a bijection between the curves in TQ that


satisfy the Lagrange-d’Alembert equation (4.1.1) and the curves in PQ that satisfy the

Lagrange-d’Alembert-Pontryagin equation (4.1.3).

By (4.1.3), for a hyper-regular Lagrangian the curve t 7→ γ(t, 0) is in the image of the

embedding grph : T ∗Q → PQ restricted to FL(D) ⊂ T ∗Q. Note that, in Section 3.1 we

defined grph by the equation grphq(pq) = (FL−1q (pq), pq). For a hyper-regular Lagrangian,

the curve t 7→ γ(t, 0) has a unique pre-image t 7→ λ(t) = pq(t)(t) ∈ FLq(t)(D(q(t)), such

that λ(t) = ΠT ∗Q(γ(t, 0)), for all t. We can now rewrite (4.1.2) in T ∗Q as

T ∗grph(ιγ(t,0) ((T ∗ΠT ∗QΩcan) γ(t, 0))− dE γ(t, 0)

)∈ T ∗grph(∆0(γ(t, 0))),

γ(t, 0) ∈ ∆(γ(t, 0))

⇐⇒ ιλ(t) (Ωcan λ(t))− dE grph(λ(t)) ∈(T ∗πQ(D0)

)(λ(t)),

Tλ(t)πQ(λ(t)) ∈ D(πQ(λ(t)))

since we have the following two diagrams:

T ∗Q

T ∗Q grph // PQ

ΠT∗Q

OO

PQΠT∗Q //

ΠQ

T ∗Q

πQ

Q Q

Here, λ(t) := dλdt

(t), and πQ : T ∗Q → Q is the canonical projection map for the cotangent

bundle. For a hyper-regular Lagrangian, we define the Hamiltonian function H : T ∗Q →R on the cotangent bundle by (3.1.7). The solution curve of an implicit Lagrangian

system, i.e., t 7→ γ(t, 0), satisfies (4.1.2) if and only if the curve t 7→ λ(t) satisfies

nonholonomic Hamilton’s equation, defined by

ιλ(t) (Ωcan λ(t))−dH λ(t) ∈(T ∗πQ(D0)

)(λ(t)). Tλ(t)πQ(λ(t)) ∈ D(πQ(λ(t))) (4.1.4)

With some abuse of notation, denote a variation of the curve t 7→ λ(t) ∈ T ∗Q


by the function (t, ε) 7→ λ(t, ε) ∈ T ∗Q. Under the assumptions considered to derive

(4.1.2), nonholonomic Hamilton’s equation in T ∗Q can also be derived from the Lagrange-

d’Alembert-Pontryagin principle, once we restrict the variational problem to the image of

the embedding grph. That is, we only consider the variations (t, ε) 7→ γ(t, ε) ∈ grph(T ∗Q)

such that λ(t, ε) = ΠT ∗Q(γ(t, ε)):

∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

(L(vq) + 〈pq, TΠQ(γ)− vq〉) γ dt = 0, γ(t, 0) ∈ ∆(γ(t, 0))

⇐⇒ ∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

(〈λ(t, ε), TπQ(λ(t, ε))〉 −H λ(t, ε))

)dt = 0,


⇐⇒⟨ιλ(t,0) (Ωcan λ(t, 0))− dH λ(t, 0), δλ(t)

⟩= 0, ∀δλ(t) ∈ (TπQ)−1(D)(λ(t, 0))


⇐⇒ ιλ(t) (Ωcan λ(t))− dH λ(t) ∈(T ∗πQ(D0)

)(λ(t, 0)).


Here, λ(t, ε) := ∂∂tλ(t, ε) and δλ(t) := ∂

∂ε

∣∣ε=0

λ(t, ε). For the details of the derivation

presented above, see Section 3.1.

Using any coordinate chart for T ∗Q, we have (λ(t), λ(t)) = (q(t), p(t), q(t), p(t)), and

we can write (4.1.4) as

q =∂H

∂p(q, p),

(p+

∂H

∂q(q, p)

)dq ∈ D0(q), q ∈ D(q)

As we presented in Section 3.1, we extend our result to the systems whose solution

curve is the integral curve of a vector field. Let X be a vector field on the cotangent

bundle T ∗Q. It induces a vector field on grph(T ∗Q) whose smooth extension to PQis denoted by X . Note that X is not a unique vector field on PQ. In other words,

∀pq ∈ T ∗Q we have Tpq grph(Xpq) = Xgrph(pq). If the curve t 7→ γ(t) ∈ PQ is an integral

curve of the vector field X and it satisfies (4.1.2), then ∀Wγ(t) ∈ ∆(γ(t)) we have

⟨(−dE + ιX (T ∗ΠT ∗QΩcan)) γ(t),Wγ(t)

⟩= 0, X (γ(t)) ∈ ∆(γ(t))

⇐⇒ (−dE + ιX (T ∗ΠT ∗QΩcan)) γ(t) ∈ ∆0(γ(t)). X (γ(t)) ∈ ∆(γ(t)) (4.1.5)

By pulling back (4.1.5) by the embedding grph we can write the dynamic equation in


T ∗Q as

(ιXΩcan − dH) λ(t) ∈(T ∗πQ(D0)

)(λ(t)). X ⊂ (TπQ)−1(D) (4.1.6)

Based on (4.1.3), not only X is a section of the distribution (TπQ)−1(D) but also it is a

subset of T (FL(D)), i.e., X ⊂ (TπQ)−1(D) ∩ T (FL(D)).

Consequently, any curve t 7→ γ(t) ∈ grph(T ∗Q), such that X (γ(t)) = dγdt

(t), satisfies

(4.1.5) if and only if the curve t 7→ λ(t) = ΠT ∗Q(γ(t)) ∈ T ∗Q, which is the integral

curve of the vector field X, satisfies (4.1.6). If (4.1.6) holds for any integral curve of

X ∈ X(T ∗Q), we can define nonholonomic Hamilton’s equation as

ιXΩcan − dH ⊂ T ∗πQ(D0). X ⊂ (TπQ)−1(D) (4.1.7)

In general, one can have a system satisfying nonholonomic Hamilton’s equation (4.1.7) on

T ∗Q for a Hamiltonian H ∈ C∞(T ∗Q) that does not necessarily come from a Lagrangian.

Such system is called a nonholonomic Hamiltonian system. We define a nonholonomic

Hamiltonian system to be the quadruple (T ∗Q,Ωcan, H,D), as above.

4.2 Nonholonomic Hamiltonian Mechanical Systems

with Symmetry

In Section 3.2, a Hamiltonian mechanical system is defined by a quadruple (T ∗Q,Ωcan,

H,K), where Q is the configuration manifold and the pair (T ∗Q,Ωcan) is the cotangent

bundle of Q along with its symplectic structure, the smooth function H : T ∗Q → R is the

Hamiltonian, and the Riemannian metric K is the kinetic energy metric. A nonholonomic

Hamiltonian mechanical system is defined by a quintuple (T ∗Q,Ωcan, H,K,D), where

D ⊂ TQ is a regular non-involutive linear distribution that is bracket generating, and

the rest of the objects are defined as above. Suppose that the distribution D can be

defined using a set of (constraint) 1-forms ωs ⊂ T ∗Q| s = 1, · · · , f on Q such that

D(q) = vq ∈ TqQ|ωs(q)(vq) = 0, s = 1, · · · , f, (4.2.8)

where f is the number of nonholonomic constraints and it is less than the dimension of Q.

The kinetic energy metric K and Hamiltonian H for a nonholonomic Hamiltonian me-

chanical system are defined similar to the holonomic case (see (3.2.13)). As shown in the

previous section, a nonholonomic mechanical system satisfies nonholonomic Hamilton’s


equation (4.1.7), or equivalently we have

ιXΩcan − dH = −f∑s=1

κsT∗πQωs, ωs(TπQ(X)) = 0, ∀s = 1, · · · , f (4.2.9)

where πQ : T ∗Q → Q is the canonical projection map for the cotangent bundle.

Let G be a Lie group with the Lie algebra Lie(G). Consider a free and proper action

of G on Q, and denote the action by Φg : Q → Q, ∀g ∈ G. The cotangent lift of this

action T ∗Φg : T ∗Q → T ∗Q is the action of G on T ∗Q that preserves Ωcan on T ∗Q (see

Lemma 3.2.1). The momentum map M for the action T ∗Φg on (T ∗Q,Ωcan) is defined by

(3.2.16). By Lemma 3.2.2 this momentum map is Ad∗-equivariant.

Definition 4.2.1. A nonholonomic Hamiltonian mechanical system (T ∗Q,Ωcan, H,K,D)

is called a nonholonomic Hamiltonian mechanical system with symmetry, if the distribu-

tion D and kinetic energy metric K are invariant under the tangent lifted action of G,

and the Hamiltonian H is invariant under the cotangent lift of the G-action. We denote

such a system by a sextuple (T ∗Q,Ωcan, H,K,D,G), as defined above.

Definition 4.2.2. (Chaplygin System) A nonholonomic Hamiltonian mechanical system

with symmetry (T ∗Q,Ωcan, H,K,D,G) is called a Chaplygin system if ∀q ∈ Q we also

have

TqQ = D(q)⊕ TqOq(G), (4.2.10)

where Oq(G) := Φg(q)| g ∈ G is the orbit of the G-action through q.

In this section, we restrict our attention to nonholonomic Hamiltonian mechanical

systems with symmetry whose symmetry group G possesses a Lie subgroup G ⊆ G that

satisfies the definition of a Chaplygin system. Under this assumption, we can perform the

Chaplygin reduction that was presented by Koiller in [42]. Before stating Koiller’s result,

let us remind the reader of some preliminary mathematical objects. For a Chaplygin

system (T ∗Q,Ωcan, H,K,D, G), we have a G-principal bundle π : Q → Q := Q/G, whose

corresponding connection A : TQ → Lie(G) may be defined by

A :=

f∑s=1

ωsεs, (4.2.11)

where εs| s = 1, · · · , f is a basis for Lie(G). As a result, at each point q ∈ Q,

Hq := ker(Aq) = D(q) is the horizontal subspace of the G-principal bundle, and Vq :=

TqOq(G) = ker(Tqπ) = ηQ(q)| η ∈ Lie(G) is the vertical subspace. Denote the map


corresponding to the action of G on Q by Φh : Q → Q, for all h ∈ G, and its infinitesimal

action by φ : Lie(G) → TQ. Accordingly, for the cotangent lifted G-action we have

the momentum map M : T ∗Q → Lie∗(G) corresponding to Ωcan that is defined by the

equality 〈M(pq), η〉 = 〈pq, ηQ〉 for all pq ∈ T ∗Q and η ∈ Lie(G), i.e., we have M = φ∗.

Then the vertical and horizontal projection maps ver : TQ → TQ and hor : TQ → TQcan be defined by ver(vq) := φq Aq(vq) and hor(vq) := vq − ver(vq), respectively, ∀vq ∈TqQ. Also, the horizontal lift corresponding to the G-principal bundle can be defined by

(3.2.19), i.e., hlq := (Tqπ|H)−1. Let K be the metric on Q induced by the kinetic energy

metric K, that is, ∀uq, wq ∈ TqQ we have Kq(uq, wq) = Kq(hlq(uq), hlq(wq)), which is well-

defined since ∀h ∈ G we have hlΦh(q) = TqΦh hlq and since K is G-invariant. Then, we

can define Legendre transformation on Q by 〈FLq(uq), wq〉 := Kq(uq, wq), where q := π(q)

and uq, wq ∈ TqQ. Let M = FL(D) be the vector sub-bundle of T ∗Q corresponding to

the nonholonomic distribution. Since D and K are invariant with respect to the G-action,

the vector sub-bundle M is also invariant under the cotangent lifted G-action. We may

also define the horizontal lift map hlMq : T ∗q Q →M(q) toM by hl

Mq := FLq hlq FL−1

q ,

where q = π(q).

Lemma 4.2.3. The map hlMq : T ∗q Q →M(q) is G-equivariant under the action of G on

its base point.

Proof. We have to show that ∀h ∈ G and ∀q ∈ Q,

hlMΦh(q) = T ∗

Φh(q)Φh−1 hl

Mq .

By the definition of hlM

and the G-invariance of K, for all q := π(q) we have

hlMΦh(q) = FLΦh(q) hlΦh(q) FL

−1q = T ∗

Φh(q)Φh−1 FLq TΦh(q)Φh−1 TqΦh hlq FL−1

q

= T ∗Φh(q)

Φh−1 FLq hlq FL−1q = T ∗

Φh(q)Φh−1 hl

Mq .

Based on this lemma and the fact that the Hamiltonian H is invariant under the

cotangent lifted G-action, we can define the reduced Hamiltonian H : T ∗Q→ R by

H := H iM hlM, (4.2.12)

where iM : M → T ∗Q is the canonical inclusion map.


Theorem 4.2.4 (Chaplygin Reduction [42]). A Chaplygin system (T ∗Q,Ωcan, H,K,D, G),

whose solution curves satisfy the nonholonomic Hamilton’s equation (4.2.9), can be re-

duced to a system (T ∗Q, Ωcan − Ξ, H, K), where Ωcan is the canonical 2-form on the

cotangent bundle of the quotient manifold Q = Q/G, H : T ∗Q → R is the reduced Hamil-

tonian defined by (4.2.12), K is the induced metric on Q, and Ξ is a non-closed (possibly

degenerate) 2-form that is defined in the sequel. The reduced system satisfies Hamilton’s

equation for the reduced Hamiltonian H with the almost symplectic 2-form Ωcan− Ξ. An

almost symplectic 2-form is a non-closed non-degenerate 2-form. That is

ιX(Ωcan − Ξ) = dH, (4.2.13)

where X is a vector field on T ∗Q.

Consider two vector fields Z,Y ∈ X(T ∗Q), denote an element of Q by q := π(q),

and ∀αq ∈ T ∗Q define Zq := TπQZ(αq), Yq := TπQY(αq), where πQ : T ∗Q → Q is the

cotangent bundle projection. Then, we have

Ξαq(Z(αq),Y(αq)) :=⟨Mq iM hl

Mq (αq),−Aq([hl(Z), hl(Y )]q)

⟩. (4.2.14)

Proof. We start with the nonholonomic Hamilton’s equation

ιXΩcan +

f∑s=1

κsT∗πQωs = dH. ωs(TπQ(X)) = 0, ∀s = 1, · · · , f

We first use the invariance of the Hamiltonian H and the definition of Chaplygin systems

(specifically the dimension condition (4.2.10)) to determine the Lagrange multipliers κs,

for s = 1, · · · , f . Let εl| l = 1, · · · , f be a basis for Lie(G).

ι(εl)T∗QιXΩcan + ι(εl)T∗Q

f∑s=1

κsT∗πQωs = 0, ∀l = 1, · · · , f

− ιXd〈M, εl〉+

f∑s=1

κs 〈T ∗πQωs, (εl)T ∗Q〉 = 0, ∀l = 1, · · · , f

− ιXd〈M, εl〉+

f∑s=1

κs 〈ωs, (εl)Q〉 = 0, ∀l = 1, · · · , f

κl = ιXd〈M, εl〉 = LX〈M, εl〉. ∀l = 1, · · · , f

In the above calculation, first line is true since H is G-invariant. In the second line we

use the definition of the momentum map, and the last line is true because of the choice


of ωs (s = 1, · · · , f) such that we have 〈ωs, (εl)Q〉 = δls, where δls is the Kronecker delta

function.

As a result, the nonholonomic Hamilton’s equation can be written as

ιXΩcan +

f∑s=1

((ιXd〈M, εs〉)T ∗πQωs

)= dH. TπQ(X) ⊂ D (4.2.15)

In the following we show that a Chaplygin system can be considered as a nonholo-

nomic system that satisfies Hamilton’s equation for the non-closed degenerate 2-form

Ωnhl := Ωcan + d〈M, A〉 − 〈M, B〉 along with the nonholonomic constraints, where B is

the curvature of the connection A. The 2-form d〈M, A〉 is the exterior derivative of the

one form 〈M, A〉 that is evaluated at each point αq ∈ T ∗qQ and ∀Uαq ∈ Tαq(T ∗Q) by

〈Mq(αq), Aq(TαqπQ(Uαq))〉 = 〈αq, φq Aq(TαqπQ(Uαq))〉 = 〈αq, verq(TαqπQ(Uαq))〉.

And ∀Uαq ,Wαq ∈ Tαq(T ∗Q) the non-closed 2-form 〈M, B〉 is evaluated by

〈M(αq), Bq(TαqπQ(Uαq), TαqπQ(Wαq))〉 = 〈αq, φq Bq(TαqπQ(Uαq), TαqπQ(Wαq))〉.

If the vector field X ∈ X(T ∗Q) satisfies (4.2.15) then it also satisfies the equations

bellow:

ιX

(Ωcan +

f∑s=1

(d〈M, εs〉 ∧ T ∗πQωs

))= dH, TπQ(X)) ⊂ D

ιX

(Ωcan +

f∑s=1

(d(〈M, εs〉T ∗πQωs)− 〈M, εs〉T ∗πQdωs

))= dH, TπQ(X)) ⊂ D

ιX

(Ωcan + d〈M, A〉 − 〈M, dA〉

)= dH, TπQ(X)) ⊂ D

ιX

(Ωcan + d〈M, A〉 − 〈M, B〉 − 〈M, [A, A]〉

)= dH, TπQ(X)) ⊂ D

ιX

(Ωcan + d〈M, A〉 − 〈M, B〉

)= dH, TπQ(X)) ⊂ D

ιXΩnhl = dH. TπQ(X)) ⊂ D

In the above, the first equation is valid since the vector field X should satisfy the nonholo-

nomic constraints, i.e., ωs(TπQ(X)) = 0 for all s = 1, · · · , f . Third and fourth equations

are the consequences of the definition of the principal connection A and the Cartan Struc-

ture Equation, respectively. In the fifth equation, we use the fact that A(TπQ(X)) = 0,

since the distribution D is the kernel of the principal connection A.


Lemma 4.2.5. The 2-form Ωnhl is invariant under the cotangent lifted G-action and it

also vanishes in the directions of the infinitesimal action.

Proof. In the first part of the lemma we have to show that the 2-form Ωnhl = Ωcan +

d〈M, A〉 − 〈M, B〉 is invariant under the cotangent lifted G-action, which is the conse-

quence of invariance of each term of this 2-form.

The canonical 2-form Ωcan is invariant under the cotangent lifted G-action based on

Lemma 3.2.1. Also the 1-form 〈M, A〉 and the 2-form 〈M, B〉 are invariant under the

cotangent lifted G-action due to Ad∗-equivariance of M and Ad-equivariance of A and

B. For all h ∈ G we have

T ∗T ∗Φh〈M, A〉 = 〈M T ∗Φh, A T (πQ T ∗Φh)〉 = 〈M T ∗Φh, A T (Φh−1 πQ)〉

= 〈Ad∗hM,Adh−1A TπQ〉 = 〈M, A〉,

and also

T ∗T ∗Φh〈M, B〉 = 〈M T ∗Φh, B(T (πQ T ∗Φh)(·), T (πQ T ∗Φh)(·))〉

= 〈M T ∗Φh, B(T (Φh−1 πQ)(·), T (Φh−1 πQ)(·))〉

= 〈Ad∗hM,Adh−1B(TπQ(·), TπQ(·))〉 = 〈M, B〉.

For the second part, we have to show that ιηT∗QΩnhl = 0 for all η ∈ Lie(G):

ιηT∗QΩnhl = ιηT∗QΩcan + ιηT∗Qd〈M, A〉 − ιηT∗Q〈M, B〉

= d〈M, η〉+ LηT∗Q〈M, A〉 − dιηT∗Q〈M, A〉

= d〈M, η〉 − d〈M, η〉 = 0.

The second equality is the consequence of the definition of curvature, i.e., ∀uq, vq ∈ TqQwe have Bq(uq, vq) := (dA)q(horq(uq), horq(vq)), and the fact that TπQ(ηT ∗Q) = ηQ,

which is a section of the vertical vector sub-bundle V . And the third equality is true

based on the definition of the 1-form 〈M, A〉 and its invariance under the cotangent

lifted G-action.

Now let us define the map Γ: T ∗Q → T ∗Q by

Γ(αq) := αq − A∗q Mq(αq),


for all αq ∈ T ∗qQ. The image of this map is M−1

(0) ⊂ T ∗Q, since we have

Mq(αq − A∗q Mq(αq)) = Mq(αq)− φ∗q A∗q Mq(αq) = Mq(αq)− (A φ)∗q Mq(αq)

= Mq(αq)− Mq(αq) = 0.

Note that in the above calculation we use the definition of the momentum map for the

cotangent lifted G-action and the property of the principal bundle that Aq φq = idLie(G).

As a result, the map Γ is a projection map that projects the cotangent bundle to the

vector sub-bundle M−1

(0). The restriction of the map Γ to the vector sub-bundle Mcan be considered as the shear translation of M along spanRωs| s = 1, · · · , f, which

is orthogonal to M with respect to the induced metric on T ∗Q by K. We also have

that ∀q ∈ Q the subspace spanRωs(q)| s = 1, · · · , f is transverse to M−1

q (0), since we

have chosen ωs such that 〈ωs(q), (εs)Q(q)〉 = 〈Mq(ωs(q)), εs〉 = 1 for all s = 1, · · · , f .

Therefore, considering the fact that dim(M) = dim(M−1

(0)), the map ψ := ΓiM : M→M−1

(0) is a diffeomorphism, where iM : M → T ∗Q is the inclusion map.

Both vector sub-bundles M and M−1

(0) are invariant under the cotangent lifted G-

action. Also, the map ψ is G-equivariant due to Ad∗-equivariance and Ad-equivariance

of M and A, respectively. That is, ∀αq ∈M and ∀h ∈ G we have

ψ(T ∗Φh(αq)) = T ∗Φh(αq)− A∗ M(T ∗Φh(αq)) = T ∗Φh(αq)− A∗ Ad∗h M(αq)

= T ∗Φh(αq)− T ∗Φh A∗ M(αq) = T ∗Φh(ψ(αq)).

Hence, the map ψ can descend to a map ψ : M/G→ M−1

(0)/G such that the following

diagram commutes

M ψ //

πM

M−1

(0)

π0

M/Gψ

// M−1

(0)/G

where π0 and πM are the projection maps to the quotient manifolds.

On the other hand, based on the theory of cotangent bundle reduction at zero mo-

mentum [49], we have the map ϕ0 : M−1

(0)→ T ∗Q that is defined by

〈ϕ0(αq), Tqπ(vq)〉 = 〈αq, vq〉 ,


for all αq ∈ M−1

(0) and vq ∈ TqQ, where π : Q → Q is the projection map. Let

π0 : M−1

(0)→ M−1

(0)/G be the projection map to the quotient. Then a symplectomor-

phism ϕ0 : M−1

(0)/G→ T ∗Q is uniquely characterized by the relation

ϕ0 π0 = ϕ0,

such that T ∗ϕ0Ωcan = Ω0, where Ωcan is the canonical 2-form on T ∗Q and Ω0 is the

induced symplectic 2-form on M−1

(0)/G that satisfies the relation T ∗i0Ωcan = T ∗π0Ω0.

Here the map i0 : M−1

(0) → T ∗Q is the inclusion map. Finally, a diffeomorphism ϕM :=

ϕ0ψ : M/G→ T ∗Q can be defined. The following diagram summarizes the construction

of ϕM.

T ∗Q Γ // M−1

(0) i0 //

π0

ϕ0

""

T ∗Q

M?

iM

OO

ψ

;;

πM

M−1

(0)/Gϕ0

// T ∗Q

M/G

ψ

;;

ϕM

55

(4.2.16)

The next step is to show that the 2-form T ∗iMΩnhl is a basic 2-form under the induced

G-action on M and hence it can descend to a 2-form Ωnhl on M/G that satisfies the

relation T ∗iMΩnhl = T ∗πMΩnhl.

Lemma 4.2.6. The 2-form T ∗iMΩnhl is a basic 2-form under the induced G-action on

M.

Proof. In order to prove that T ∗iMΩnhl is basic, we have to show that T ∗iMΩnhl is invari-

ant under the induced G-action on M and it vanishes in the directions of infinitesimal

action.

Based on Lemma 4.2.5, we have that Ωnhl is invariant under the G-action, i.e., ∀h ∈ Gwe have T ∗T ∗ΦhΩnhl = Ωnhl. Also, since M is invariant under this action, ∀h ∈ G the

induced action ΦMh : M → M can be uniquely defined by the relation iM ΦM =


T ∗Φh iM. Therefore, we have

T ∗ΦMh (T ∗iMΩnhl) = T ∗(iM ΦM)Ωnhl = T ∗(T ∗Φh iM)Ωnhl

= T ∗iM(T ∗T ∗ΦhΩnhl) = T ∗iMΩnhl,

which shows that T ∗iMΩnhl is invariant under the induced G-action.

Next, ∀η ∈ Lie(G) and ∀αq ∈ M we define the infinitesimal action of G on M by

ηM(αq) := ∂∂ε

∣∣ε=0

ΦMexp(εη)(αq). Then, we have

ιηM(αq)(T∗iMΩnhl) = ιT iM(ηM(αq))(Ωnhl)iM(αq) = ιηT∗Q(iM(αq))(Ωnhl)iM(αq) = 0.

The first equality is basically the definition of the pull-back of a 2-form. The second

equality is the result of the following calculation

TiM(ηM(αq)) = TiM

(∂

∂ε

∣∣∣∣ε=0

ΦMexp(εη)(αq)

)=

∂

∂ε

∣∣∣∣ε=0

(iM ΦMexp(εη)(αq)

)=

∂

∂ε

∣∣∣∣ε=0

(T ∗Φexp(εη) iM(αq)

)= ηT ∗Q(iM(αq)).

And the last equality follows from Lemma 4.2.5.

It remains to show that T ∗(ϕM πM)(Ωcan − Ξ) = T ∗iMΩnhl. We first show that

the 2-form Ξ ∈ Ω2(T ∗Q) in the statement of the theorem is in fact equal to the 2-form

Λ ∈ Ω2(T ∗Q) that is characterized by the relation T ∗(ϕM πM)Λ = T ∗iM〈M, B〉.

Lemma 4.2.7. For all αq ∈ T ∗Q the 2-form Λαq = 〈M(αq), B〉 such that q = π(q),

αq ∈ M and ϕM πM(αq) = αq. Based on (4.2.14), we also have that the 2-form

Ξαq = 〈M(α′q′), B〉 such that q = π(q′) and α′q′ = hlMq′ (αq). Then, there is a h ∈ G such

that q′ = Φh(q) and we have α′q′ = T ∗Φh(q)

Φh−1(αq).

Proof. Based on the definition of the map hlM and Lemma 4.2.3, it suffices to show that

the following diagram commutes.

M πM //

FL−1

M/GϕM // T ∗Q

D T π // T Q

FL

==


In other words, ∀αq ∈M, we have

FL T π FL−1(αq) = ϕM πM(αq). (4.2.17)

For the right hand side of (4.2.17), ∀vq ∈ TQ we have

〈ϕM πM(αq), T π(vq)〉 = 〈ϕ0 ψ(αq), T π(vq)〉 = 〈ψ(αq), vq〉 =⟨αq − A∗q Mq(αq), vq

⟩= 〈αq, vq〉 −

⟨αq, φq Aq(vq)

⟩= 〈αq, vq〉 − 〈αq, verq(vq)〉

=⟨αq, horq(vq)

⟩.

The first equality is the consequence to the commuting diagram (4.2.16), and the second

and third equalities are true based on the definition of the maps ϕ0 and ψ, respectively.

As for the left hand side of (4.2.17), ∀vq ∈ TQ we have

〈FLq Tqπ FL−1q (αq), Tqπ(vq)〉 =

⟨hl∗Φh(q) FLΦh(q) hlΦh(q) Tqπ FL

−1q (αq), Tqπ(vq)

⟩=⟨

hl∗q T ∗q Φh FLΦh(q) TqΦh hlq Tqπ FL−1

q (αq), Tqπ(vq)⟩

=⟨

hl∗q FLq hlq Tqπ FL−1

q (αq), Tqπ(vq)⟩

=⟨

hl∗q(αq), Tqπ(vq)

⟩=⟨αq, hlq Tqπ(vq)

⟩=⟨αq, horq(vq)

⟩.

The first equality is correct based on the definition of FL. The third equality is the

consequence of invariance of the metric K under the G-action, and the forth equality is

true because of the fact that αq ∈M and hlq Tqπ∣∣∣D(q)

= idD(q).

Note that, ∀αq ∈ T ∗Q the 2-form Ξαq sees the vector hlq TαqπQ(Yαq) for any vector

Yαq ∈ Tαq(T∗Q), and the 2-form Λαq sees the vector horq TαqπQ(Yαq) for any vector

Yαq ∈ TαqM that satisfies Tαq(ϕM πM)(Yαq) = Tαq(ϕ0 ψ)(Yαq) = Yαq . Also, since the

map ψ is identity on the base point and the map ϕ0 acts by π on the base point we have

the equality πQ ϕ0 ψ = π πQ. Therefore, we have

hlq TαqπQ(Yαq) = hlq TαqπQ Tαq(ϕ0 ψ)(Yαq)

= hlq Tqπ TαqπQ(Yαq) = horq TαqπQ(Yαq).

Based on the above calculation and Lemma 4.2.7 we have that Λ = Ξ.


Lemma 4.2.8. We have the following equality:

T ∗(ϕM πM)Ωcan = T ∗iM(Ωcan + d〈M, A〉).

Proof. Based on the diagram 4.2.16 and the equality T ∗i0Ωcan = T ∗π0Ω0, we have

T ∗(ϕM πM)Ωcan = T ∗(ϕ0 ψ πM)Ωcan = T ∗(ϕ0 π0 ψ)Ωcan

= T ∗(π0 ψ)Ω0 = T ∗(i0 ψ)Ωcan = T ∗(i0 Γ iM)Ωcan

= T ∗iMT∗(i0 Γ)Ωcan = T ∗iM(Ωcan + d〈M, A〉).

The last equality is true based on the definition of the map Γ and the fact that T ∗(i0 Γ)Θcan = T ∗iM(Θcan − 〈M, A〉), where Θcan is the tautological 1-form on T ∗Q.

Therefore, if X ∈ X(T ∗Q) satisfies (4.2.9) for a Chaplygin system, then there exists

a vector field X ∈ X(T ∗Q) that can be characterized by X = T (ϕM πM)(X iM) that

satisfies Hamilton’s equation for the Hamiltonian H : T ∗Q → R such that H ϕMπM =

H iM or H = H iM hlM,

ιX(Ωcan − Ξ) = dH.

This completes the proof of Theorem 4.2.4.

Remark 4.2.9. Note that for Hamiltonian mechanical systems, where the Hamiltonian

is in the form of kinetic plus potential energy, one may also work in the Lagrangian side to

perform the reduction. Specially, we refer the reader to the counterexample 3.4.32 in [27],

where the Lagrangian approach is helpful to show that the inverse of the statement “If

D is involutive then Ξ = 0” is not correct.

Remark 4.2.10. The reduced Hamilton’s equation in the case of reduction of holonomic

Hamiltonian mechanical systems with symmetry at non-zero momentum involves the

closed 2-form Ξµ = 〈µ, dA(hl(·), hl(·))〉. If G = Gµ, this 2-form is simplified to Ξµ =

〈µ,B(hl(·), hl(·))〉, which is analogous to the Chaplygin case where the non-closed 2-form

Ξ = 〈M iM hlM, B(hl(·), hl(·))〉 is involved. Note that, since the momentum is not

conserved along the flow of the vector field X ∈ X(T ∗Q) for nonholonomic systems, Ξ

sees non-constant momentum comparing to the constant µ that appears in Ξµ.

Remark 4.2.11. The proof presented above for Chaplygin reduction theorem of Koiller

is almost analogous to the proof of embedding version of symplectic reduction of cotan-

gent bundles [49]. There are two major distinguishing points:


i) The 2-form Ωnhl ∈ Ω2(T ∗Q) that is defined in the case of Chaplygin systems is a

non-closed degenerate 2-form, as opposed to the canonical 2-form Ωcan ∈ Ω2(T ∗Q)

in the cotangent bundle reduction case.

ii) The map Γ: T ∗Q → M−1

(0), such that Γ(αq) = αq − A∗ M(αq), is a projection

onto M−1

(0) as opposed to the diffeomorphism Shiftµ : T ∗Q → T ∗Q, such that

Shiftµ(αq) = αq −A∗µ, which is defined in the case of cotangent bundle reduction

(see [49]).

Now we state a nonholonomic version of Noether’s theorem for reduced Chaplygin

systems.

Proposition 4.2.12. For a reduced Chaplygin system (T ∗Q, Ωcan− Ξ, H, K), a function

h : T ∗Q → R is constant of motion if and only if its Hamiltonian vector field Xh ∈X(T ∗Q) corresponding to the almost symplectic 2-form Ωcan−Ξ preserves the Hamiltonian

H.

Proof. Suppose that there exists a function h : T ∗Q → R that is constant of motion. Its

Hamiltonian vector field Xh corresponding to Ωcan − Ξ is defined by

ιXh(Ωcan − Ξ) = dh.

Then we have

0 = LX h =⟨dh, X

⟩=⟨ιX

h(Ωcan − Ξ), X

⟩= −

⟨ιX(Ωcan − Ξ), Xh

⟩= −

⟨dH, Xh

⟩= −LX

hH.

Conversely, based on the above calculation, if there exists a Hamiltonian vector field Xh

for the function h that preserves the Hamiltonian H, then h is a constant of motion.

If a function h : T ∗Q → R is a constant of motion, then based on the diagram (4.2.16)

the function h : T ∗Q → R defined by h ϕM πM = h iM is a G-invariant function,

which is constant on the trajectories of the vector field X ∈ X(T ∗Q), i.e., LXh = 0.

If a Chaplygin system has a bigger symmetry group, as we assumed at the beginning

of this section, we may find constants of motion for the reduced Chaplygin system by

using the directions of symmetry in D. We investigate this possibility in two steps. In

the first step, we assume that a Chaplygin system after reduction (T ∗Q, Ωcan− Ξ, H, K)

is still invariant under a group action in the following sense. Let N ⊂ G be a Lie group

with a free and proper action denoted by Φn : Q → Q, for all n ∈ N , such that


i) The Hamiltonian H is invariant under the cotangent lifted N -action, i.e., ∀n ∈ N ,

we have H T ∗Φn = H, where T ∗Φn : T ∗Q → T ∗Q denotes the cotangent lift of the

N -action.

ii) The infinitesimal generator for the cotangent lifted action ζT ∗Q, ∀ζ ∈ Lie(N ),

satisfies the condition

ιζT∗Q

Ξ = 0.

Let M : T ∗Q → Lie∗(N ) be the Ad∗-equivariant momentum map for the cotangent

lifted N -action corresponding to the canonical 2-form Ωcan, which is defined by (3.2.16).

That is, ∀ζ ∈ Lie(N ) we have

ιζT∗Q

Ωcan = d〈M, ζ〉.

Under the assumption (ii), we have that the map M is also the momentum map corre-

sponding to the almost symplectic 2-form Ωcan − Ξ. As a result, we have the following

corollary of Proposition 4.2.12.

Corollary 4.2.13. Under the above assumptions (i) and (ii), the momentum map M is

conserved along the trajectories of the vector field X.

Let ϑ ∈ Lie∗(N ) be a regular value for the momentum M. Since M is a momentum

map with respect to the almost symplectic 2-form Ωcan − Ξ, we can perform the almost

symplectic reduction presented in [69] at ϑ. And consequently, drop the dynamics to

M−1

(ϑ)/Nϑ, where Nϑ is the coadjoint isotropy group of ϑ ∈ Lie∗(N ).

Proposition 4.2.14. Under the assumptions (i) and (ii) stated above, we have

i) There exists an almost symplectic 2-form Ωϑ − Ξϑ ∈ Ω2(M−1

(ϑ)/Nϑ) that is

uniquely characterized by

T ∗πϑΩϑ = T ∗iϑΩcan,

and

T ∗πϑΞϑ = T ∗iϑΞ,

where iϑ : M−1

(ϑ) → T ∗Q and πϑ : M−1

(ϑ) → M−1

(ϑ)/Nϑ are the canonical in-

clusion and projection maps, respectively.

ii) The reduced Hamilton’s equation (4.2.13) can be further reduced to

ιXϑ(Ωϑ − Ξϑ) = dHϑ, (4.2.18)


where we have

Tπϑ(X iϑ) = Xϑ πϑ,

and Hϑ is uniquely defined by

Hϑ πϑ = H iϑ.

For all q ∈ Q, we denote the infinitesimal action of Lie(N ) by φq : Lie(N ) → TqQand define the locked inertia tensor Iq : Lie(N )→ Lie∗(N ) by

Iq = φ∗q FLq φq.

Then the mechanical connection corresponding to the N -action Aq : TqQ → Lie(N ) is

Aq = I−1q Mq FLq.

The 1-form αϑ ∈ Ω1(Q) is then defined by αϑ := A∗(ϑ), which introduces a closed 2-form

βϑ ∈ Ω2(Q) by the equality T ∗πβϑ = dαϑ, where Q := Q/Nϑ is the quotient manifold

with the canonical projection map π : Q → Q. The pullback of βϑ by the cotangent

bundle projection map πQ : T ∗Q → Q will be a closed 2-form on T ∗Q that is denoted by

Ξϑ := T ∗πQ(βϑ). This 2-form can be calculated based on (3.2.27).

We can use the symplectic embedding map ϕϑ : M−1

(ϑ)/Nϑ → T ∗Q, defined by

(3.2.25) in Theorem 3.2.7, and write the reduced Hamilton’s equation (4.2.18) in the

cotangent bundle of Q as

ιXϑ(Ωcan − Ξϑ − Ξϑ) = dHϑ. (4.2.19)

Here, Ωcan is the canonical symplectic form on T ∗Q, and Ξϑ := T ∗ϕ−1ϑ (Ξϑ). The map

ϕ−1ϑ is only defined on a vector sub-bundle of T ∗Q, i.e., ϕ−1

ϑ : [T πV ]0 → M−1

(ϑ)/Nϑ,

where V ⊂ T Q is the vertical sub-bundle corresponding to the N -action. The vector

field Xϑ and the Hamiltonian Hϑ are defined by the relations Xϑ ϕϑ = T ϕϑ(Xϑ) and

Hϑ := Hϑ ϕ−1ϑ .

In the special case ofNϑ = N , the map ϕϑ is a symplectomorphism between M−1

(ϑ)/Nand T ∗Q, and one can use (3.2.28) to evaluate the closed 2-form Ξϑ. Based on the

above paragraph, we say that a reduced Chaplygin system with symmetry (T ∗Q, Ωcan −Ξ, H, K,N ⊂ G) is further reduced to the system (T ∗Q, Ωcan−Ξϑ−Ξϑ, Hϑ, K), as defined

above. Here, K is the induced metric on Q by K.

The two-step reduction of the Hamiltonian mechanical system with symmetry (T ∗Q,


Ωcan, H,K,D, G×N ) can be summarized in the following diagram:

T ∗Q Γ // M−1

(0)

ϕ0

M−1

(ϑ)L l

iϑ

zz

πϑ

M−1

(0)shift−1

ϑ

oo

M?

iM

OO

ψ

<<

πM

T ∗QhlMoo M−1

(ϑ)/Nϑ

ϕϑ

M/G

ϕM

<<

[T πV ]0 ⊂ T ∗Q

hlM

CC(4.2.20)

Here, hlM

: [T πV ]0 ⊂ T ∗Q → M−1

(0) is defined by the induced metric K on Q and

the horizontal lift map hlq := (Tqπ|Hq)−1, where H := ker(A) is the horizontal vector

sub-bundle and q := π(q),

hlMq = FLq hlq FL−1

q .

Also, the map shiftϑ : M−1

(ϑ) → M−1

(0) is the shifting map that ∀αq ∈ M−1

(ϑ) is

defined by

shiftϑ(αq) := αq − αϑ(q).

In the second step, we study the case where there still left directions of the symmetry

group G in D that may result in constants of motion that do not necessarily come from

a cotangent lifted group action in T ∗Q.

Proposition 4.2.15. For a nonholonomic Hamiltonian mechanical system with symme-

try (T ∗Q,Ωcan, H,K,D,G), assume that

i) The configuration manifold Q can be globally trivialized, i.e., Q ∼= G × Q, where

Q = Q/G.

ii) We have the condition D(q)∩ TqOq(G) = S(q) 6= 0, where the dimension of S(q)

is constant for all q ∈ Q.

For all q ∈ Q, by Kq ⊆ Lie(G) we denote the subspace of Lie(G) for which we have

S(g, q) = (gς q, 0)| ς q ∈ Kq for all g ∈ G. Then for a ς q ∈ Kq the function

h(g, q, pg) = 〈T ∗e Lg(pg), ςq〉 ,


where pg is the component of FL(g,q)(g, ˙q) in T ∗G, is a constant of motion if and only if⟨T ∗e Lg(pg), ad(g−1g)(ς

q) +∂ς q

∂q˙q

⟩= 0,

along the trajectories of X = (g, ˙q, pg, pq). Note that, h is a G-invariant function since the

metric K, which defines the map FL for Hamiltonian mechanical systems, is invariant

under the group action.

Proof. The proof relies on the Lagrange-d’Alembert principle and the invariance of the

Lagrangian L with respect to the tangent lift of the G-action. Note that, since the

distribution D is G-invariant, the subspace Kq is independent of the group parameters.

For a time interval [ts, tf ], let (t, ε) 7→ q(t, ε) = (g(t) exp(ρ(t, ε)ς q(t)), q(t)) ∈ G×Q (ε ∈ R),

where ρ : R2 → R is a function that satisfies ρ(t, 0) = ρ(ts, ε) = ρ(tf , ε) = 0, be a variation

of a smooth curve t 7→ q(t) = (g(t), q(t)) with fixed end points. Based on Hamilton’s

principle we have∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

L(q(t, ε), q(t, ε))dt = 0,

for any variation of the curve t 7→ q(t), as defined above, along with the constraint

q(t) ∈ D(q(t)). Note that, by construction we have δq(t) := ∂∂ε

∣∣ε=0

q(t, ε) ∈ D(q(t)). Let

ρ′(t) := ∂∂ε

∣∣ε=0

ρ(t, ε). Then we have∫ tf

ts

(⟨∂L

∂g, ρ′gς q

⟩+

⟨∂L

∂g, ρ′gς q + ρ′gς q + ρ′g

∂ς q

∂q˙q

⟩)dt = 0.

On the other hand, the Lagrangian L, hence its integral, is invariant under the group

action. Consider the action of exp(ρ(t, ε)Adg(t)(ςq(t))) ∈ G:

∂

∂ε

∣∣∣∣ε=0

∫ tf

ts

L((exp(ρAdg(ςq))g, q), (exp(ρAdg(ς

q))g, ˙q))dt

=

∫ tf

ts

(⟨∂L

∂g, ρ′Adg(ς

q)g

⟩+

⟨∂L

∂g, ρ′Adg(ς

q)g

⟩)dt = 0.

By subtracting the above two equations and using integration by parts, we have the

following equation:

d

dt

⟨T ∗e Lg(

∂L

∂g), ς q

⟩=

⟨T ∗e Lg(

∂L

∂g), ad(g−1g)(ς

q) +∂ς q

∂q˙q

⟩,

where we denote the left translation map on the group G by Lg : G → G, and its induced

map on the cotangent bundle by T ∗Lg : T ∗G → T ∗G. Then the function⟨T ∗e Lg(

∂L∂g

), ς q⟩

,


which is a function of g−1g ∈ Lie(G) and q ∈ Q, is a constant of motion if the right hand

side of the above equation is zero along the trajectories of X = (g, ˙q, pg, pq).

From now, we restrict our attention to the case where Q ∼= G × Q, which is the case

for nonholonomic open-chain multi-body systems. If there exists a constant of motion

h : T ∗Q → R, as identified above, which is invariant under the cotangent lifted G-action

on T ∗Q, then the function h : [T πV ]0 ⊂ T ∗Q → R, which is uniquely defined by the

equality h ϕϑ πϑ = h iM hlM iϑ, is well-defined and remains constant on the

trajectories of Xϑ on [T πV ]0 ⊂ T ∗Q. Please see the diagram (4.2.20).

Proposition 4.2.16. For a reduced Chaplygin system (T ∗Q, Ωcan − Ξϑ − Ξϑ, Hϑ, K),

assume that there exists a constant of motion h1, whose Hamiltonian vector field corre-

sponding to the almost symplectic 2-form Ωcan − Ξϑ − Ξϑ comes from an infinitesimal

action of a one-parameter Lie subgroup G1 ⊂ G, i.e., for all η1 ∈ Lie(G1) we have

ι(η1)T∗Q(Ωcan − Ξϑ − Ξϑ) = η1dh1,

where (η1)T ∗Q ∈ X(T ∗Q) is the vector field generated by the infinitesimal action of

Lie(G1) corresponding to η1. Then for a regular value ν1 ∈ R of the function h1 the

level set h−11 (ν1) is invariant under the G1-action. Also, under the assumption that

T ∗iν1

(ι(η1)T∗Q

dΞϑ

)= 0,

the 2-form T ∗iν1(Ωcan−Ξϑ−Ξϑ) is basic with respect to the G1 action, where iν1 : h−11 (ν1) →

T ∗Q is the canonical inclusion map.

Proof. In order to show that the level set h−11 (ν1) is G1-invariant, it suffices to show that

the function h1 is invariant under the action of G1. In other words, ∀η1 ∈ Lie(G1)

L(η1)T∗Q(h1) =

⟨dh1, (η1)T ∗Q

⟩=

1

η1

⟨ι(η1)T∗Q

(Ωcan − Ξϑ − Ξϑ), (η1)T ∗Q

⟩= 0.

In order to show that the 2-form T ∗iν1(Ωcan − Ξϑ − Ξϑ) is basic with respect to G1

action, we need to show that it vanishes in the direction of the infinitesimal action of

G1 and it is invariant under the G1-action. Let η1 be an element of Lie(G1), and denote

the vector field generated by the infinitesimal action of Lie(G1) in the direction of η1 by


(η1)T ∗Q. First, we show that

ι(η1)T∗QT ∗iν1(Ωcan − Ξϑ − Ξϑ) = T ∗iν1

(ι(η1)T∗Q

(Ωcan − Ξϑ − Ξϑ))

= T ∗iν1dh1 = d(h1 iν1) = 0,

which is the consequence of the fact that h−11 (ν1) is invariant under the G1-action, and

it proves that T ∗iν1(Ωcan − Ξϑ − Ξϑ) vanishes in the direction of the infinitesimal action

of G1. Secondly, we show that

L(η1)T∗Q

(T ∗iν1(Ωcan − Ξϑ − Ξϑ)

)= dι(η1)T∗Q

(T ∗iν1(Ωcan − Ξϑ − Ξϑ)

)+ ι(η1)T∗Q

d(T ∗iν1(Ωcan − Ξϑ − Ξϑ)

)= T ∗iν1d

(ι(η1)T∗Q

(Ωcan − Ξϑ − Ξϑ))− ι(η1)T∗Q

(T ∗iν1dΞϑ

)= T ∗iν1

(ddh1

)− T ∗iν1

(ι(η1)T∗Q

dΞϑ

)= 0,

which proves that T ∗iν1(Ωcan− Ξϑ−Ξϑ) is invariant under the group action. Hence, the

2-form T ∗iν1(Ωcan − Ξϑ − Ξϑ) is basic with respect to the G1-action.

Following the same lines of proof, we can extend the above proposition to the following

theorem.

Theorem 4.2.17. For a reduced Chaplygin system (T ∗Q, Ωcan − Ξϑ − Ξϑ, Hϑ, K), as-

sume that there exist m constants of motion h1, · · · , hm, whose Hamiltonian vector fields

corresponding to the almost symplectic 2-form Ωcan − Ξϑ − Ξϑ comes from infinitesimal

actions of one-parameter Lie subgroups G1, · · · , Gm ⊂ G, i.e., for all ηi ∈ Lie(Gi), where

i = 1, · · · ,m, we have

ι(ηi)T∗Q(Ωcan − Ξϑ − Ξϑ) = ηidhi,

where (ηi)T ∗Q ∈ X(T ∗Q) is the vector field generated by the infinitesimal action of Lie(Gi)

corresponding to ηi. Then for a regular value ν = (ν1, · · · , νm) ∈ Rm of the function

h := (h1, · · · , hm) the level set h−1(ν) is invariant under the (G1 × · · · × Gm)-action.

Also, ∀η := (η1, · · · , ηm) ∈ Lie(G1)× · · · × Lie(Gm) under the assumption that

T ∗iν

(ι(η)T∗Q

dΞϑ

)= 0,

the 2-form T ∗iν(Ωcan− Ξϑ−Ξϑ) is basic with respect to the (G1×· · ·×Gm)-action. Here,

(η)T ∗Q ∈ X(T ∗Q) is the vector field generated by the infinitesimal action of G1×· · ·×Gm

corresponding to η, and iν : h−1(ν) → T ∗Q is the canonical inclusion map.


Under the hypotheses of the above theorem, we can even further reduce the system

(T ∗Q, Ωcan − Ξϑ − Ξϑ, Hϑ, K) using the m constants of motion.

Theorem 4.2.18. Under the hypotheses of the above theorem, we have

i) There exists an almost symplectic 2-form Ωnhl ∈ Ω2(h−1(ν)/(G1 × · · · ×Gm)) that

is uniquely characterized by

T ∗πνΩnhl = T ∗iν(Ωcan − Ξϑ − Ξϑ),

where πν : h−1(ν)→ h−1(ν)/(G1 × · · · ×Gm) is the canonical projection map.

ii) The reduced Hamilton’s equation (4.2.19) can be further reduced to

ιXν Ωnhl = dHν , (4.2.21)

where we have

Tπν(Xϑ iν) = Xν πν ,

and Hν is uniquely defined by

Hν πν = Hϑ iν .

The three-step reduction process can be summarized in the following diagram

T ∗Q Γ // M−1

(0)

ϕ0

M−1

(ϑ)K k

iϑ

yy

πϑ

M−1

(0)shift−1

ϑ

oo

M?

iM

OO

ψ

<<

πM

T ∗QhlMoo M−1

(ϑ)/Nϑ

ϕϑ

h−1(ν)I i

iν

vv

πν

M/G

ϕM

<<

[T πV ]0 ⊂ T ∗Q

hlM

>>

h−1(ν)/(G1 × · · · ×Gm)

(4.2.22)


4.3 Reduction of Nonholonomic Open-chain Multi-

body Systems with Displacement Subgroups

In this section, we show that nonholonomic open-chain multi-body systems can be con-

sidered as nonholonomic Hamiltonian mechanical systems with symmetry. As we showed

in Theorem 3.3.3, the configuration manifold of the first joint is indeed the symmetry

group for holonomic multi-body systems. If the distribution corresponding to the non-

holonomic joints of a multi-body system is also invariant under the action of this group or

a Lie subgroup of that, then we can apply the reduction theory developed in the previous

section to holonomic open-chain multi-body systems with displacement subgroups.

A nonholonomic open-chain multi-body system with displacement subgroups is ba-

sically a multi-body system with at least one nonholonomic joint that is denoted by a

quintuple (T ∗Q,Ωcan, H,K,D). Here, Q = Q1 × · · · × QN is the configuration mani-

fold, H : T ∗Q → R is the Hamiltonian of the system, which is defined similar to the

holonomic case by (3.3.33), K is the kinetic energy metric defined by (3.3.31), and

a distribution D ⊂ TQ. We can write the distribution D as a Cartesian product

D = D1 × · · · × DN ⊂ TQ1 × · · · × TQN of distributions Di’s over Qi’s, which cor-

respond to the nonholonomic joints. The joint distributions Di’s may be defined by the

constraint one-forms ωsi ⊂ T ∗Qi| s = 1, · · · , fi such that ∀qi ∈ TqiQi

Di(qi) = vqi ∈ TqiQi|ωsi (qi)(vqi) = 0, s = 1, · · · , fi, (4.3.23)

where fi < di = dim(Qi) is the number of linear constraints on the relative velocities

at the joint Ji. Note that for a holonomic joint Ji0 , Di0 = TQi0 . We use the indexing

and consequently the forward kinematics and Jacobian maps introduced for open-chain

multi-body systems in Section 3.3.1.

Similar to the holonomic case, Theorem 3.3.3 may be also applied to the systems with

nonholonomic joints to show that the Hamiltonian of such systems is invariant under the

cotangent lift of the action of G = Q1, as defined in Theorem 3.3.3. Note that, it is under

the assumption that the potential energy function is also G-invariant. Since in the field of

robotics, nonholonomic joints usually appear in the form of wheeled mobile platforms, in

this thesis we restrict our attention to the case where only the first joint (corresponding

to the mobile platform) of the open-chain multi-body system is nonholonomic. Also, we

assume that D1 is invariant under the G-action, and there is a Lie subgroup G ⊂ G for

which we have the Chaplygin assumption (4.2.10). We summarize the above-mentioned

assumptions as


NHR1) Only the first joint is nonholonomic, i.e., Di = TQi for i ∈ 2, · · · , N.

NHR2) The distribution D1 ⊂ TQ1 is invariant under the G-action, i.e., ∀q ∈ Q and ∀g ∈ Gwe have D1 Φg(q) = TqΦg(D1(q)).

NHR3) There exists a Lie subgroup of G, namely G, such that ∀q1 ∈ Q1 we have

Tq1Q1 = D1(q1)⊕ TqOq(G). (4.3.24)

We call a nonholonomic multi-body system that satisfies the assumptions stated above,

a nonholonomic open-chain multi-body system with symmetry and denote it by the sex-

tuple (T ∗Q,Ωcan, H,K,D1,G), as defined above. The nonholonomic Hamilton’s equation

for a nonholonomic open-chain multi-body system is written on T ∗Q as:

ιXΩcan = dH −f1∑s=1

κsT∗πQω

s1. ωs1(TπQ(X)) = 0 ∀s = 1, · · · , f1

where πQ : T ∗Q → Q is the cotangent bundle projection map, and κs’s are the Lagrange

multipliers. Under the assumption NHR2, one has a G-principal bundle π : Q → Q :=

Q/G = Q1/G×Q2×· · ·×QN , and the corresponding connection A : TQ → Lie(G) may

be defined by

A :=

f1∑s=1

ωs1εs,

where εs for s ∈ 1, · · · , f1 are elements of a basis for Lie(G). We represent any

element of Q/G by q = (q1, q) ∈ Q1/G × Q, where q1 ∈ Q1/G is the equivalence class

corresponding to q1 ∈ Q1 and q ∈ Q = Q2 × · · · × QN . We consider the principal

bundle π1 : Q1 → Q1/G to locally trivialize the Lie group Q1. Let U ⊆ Q1/G be an

open neighbourhood of e1, where e1 is the equivalence class corresponding to the identity

element e1 ∈ Q1. We denote the map corresponding to a local trivialization of the

principal bundle π1 by χ : G × U → Q1. This map can be defined by embedding U in

Q1, for example by using the exponential map of Lie groups. We denote this embedding

by χ : U → Q1 such that ∀q1 ∈ Q1/G we have χ(q1) = exp(ζ) for some ζ ∈ C, where

C ⊂ Lie(Q1) is a complementary subspace to Lie(G) ⊂ Lie(G). Accordingly, ∀h ∈ G we

define the map χ by the equality χ((h, q1)) := hχ(q1). The map χ is a diffeomorphism onto

its image. Using this diffeomorphism, any element q1 ∈ π−11 (U) ⊆ Q1 can be uniquely

identified by an element (h, q1) ∈ G×U . As a result, we have q = (q1, q) = (χ((h, q1)), q).

Note that, from now on, for brevity we write q = (h, q1, q).


The map corresponding to the infinitesimal action ofG onQ is denoted by φq : Lie(G)→TqQ. Based on the above local trivialization, ∀(h, q1, q) ∈ G×U×Q this map is calculated

by

φq =

Te1Rh

0...

0

,where with an abuse of notation we show the identity element of G by e1. Accordingly,

we calculate the momentum map M : T ∗Q → Lie∗(G) by

Mq = φ∗q =[T ∗e1Rh 0 . . . 0

].

Then by defining the fibre wise linear map Aq1 : Tq1U → Lie(G) according to the non-

holonomic constraint 1-forms ωs1’s and based on the properties of principal connections,

for all q = (h, q1, q) ∈ G× U ×Q we can write A as

Aq =: Adh

[ThLh−1 Aq1 0

], (4.3.25)

As a result, at each point q = (h, q1, q), we have the horizontal lift map hlq : TqQ → TqQ,

which is determined by

hlq =

[[−(Te1Lh)Aq1 0

]idTq1U ⊕ idTqQ

],

where idTq1U and idTqQ are the identity maps on the tangent spaces Tq1U and TqQ,

respectively.

Similar to Section 3.3, we denote the block components of the kinetic energy tensor,

which is equal to the Legendre transformation for Hamiltonian mechanical systems, by

Kij(q)dqi ⊗ dqj (i, j = 1, · · · , N). Hence, we have the matrix form for FLq, and then

using the local trivialization we can rewrite this matrix as follows:

FLq =

K11(q) · · · K1N(q)

.... . .

...

KN1(q) · · · KNN(q)

,


FL(h,q) =

KG1 ((h, q)) K

Q1/G1 ((h, q)) KG

12((h, q)) · · · KG1N((h, q))

KG2 ((h, q)) K

Q1/G2 ((h, q)) K

Q1/G12 ((h, q)) · · · K

Q1/G1N ((h, q))

KG21((h, q)) K

Q1/G21 ((h, q)) K22((h, q)) · · · K2N((h, q))

......

.... . .

...

KGN1((h, q)) K

Q1/GN1 ((h, q)) KN2((h, q)) · · · KNN((h, q))

,

where q = (q1, q), q1 = χ(h, q1), and we have[KG

1 ((h, q)) KQ1/G1 ((h, q))

KG2 ((h, q)) K

Q1/G2 ((h, q))

]= T ∗(h,q1)χ (K11(χ(h, q)))T(h,q1)χ,[

KG12((h, q)) · · · KG

1N((h, q))

KQ1/G12 ((h, q)) · · · K

Q1/G1N ((h, q))

]= T ∗(h,q1)χ

[K12(χ(h, q)) · · · K1N(χ(h, q))

],

KG21((h, q)) K

Q1/G21 ((h, q))

......

KGN1((h, q)) K

Q1/GN1 ((h, q))

=

K21(χ(h, q))

...

KN1(χ(h, q))

T(h,q1)χ.

Now, based on Lemma 3.3.4, since K is invariant under the G-action, we have

FL(h,q) =

(T ∗hLh−1)(KG1 (q))(ThLh−1) (T ∗hLh−1)(K

Q1/G1 (q)) (T ∗hLh−1)(KG

12(q)) · · ·(KG

2 (q))(ThLh−1) KQ1/G2 (q) K

Q1/G12 (q) · · ·

(KG21(q))(ThLh−1) K

Q1/G21 (q) K22(q) · · ·

......

.... . .

(KGN1(q))(ThLh−1) K

Q1/GN1 (q) KN2(q) · · ·

(T ∗hLh−1)(KG1N(q))

KQ1/G1N (q)

K2N(q)...

KNN(q)

,

where we introduce the new block components by

KG1 (q) = KG

1 ((e1, q)),

KQ1/G1 (q) = K

Q1/G1 ((e1, q)),

KG2 (q) = KG

2 ((e1, q)),

KQ1/G2 (q) = K

Q1/G2 ((e1, q)),

KG1j (q) = KG

1j ((e1, q)), ∀j = 2, · · · , N


KQ1/G1j (q) = K

Q1/G1j ((e1, q)), ∀j = 2, · · · , N

KGi1 (q) = KG

i1 ((e1, q)), ∀i = 2, · · · , N

KQ1/Gi1 (q) = K

Q1/Gi1 ((e1, q)), ∀i = 2, · · · , N

Kij(q) = Kij((e1, q)). ∀i, j = 2, · · · , N

Since K is G-invariant, it induces a metric on Q, namely K, which defines the Leg-

endre transformation on Q by

〈FLq(uq), wq〉 : = Kq(uq, vq) = Kq(hlq(uq), hlq(wq))

= 〈FLq hlq(uq), hlq(wq)〉 = 〈hl∗q FLq hlq(uq), wq〉,

where q = (q1, q) and ∀uq, wq ∈ TqQ. Therefore,

FLq =

K11(q) K12(q) · · · K1N(q)

K21(q) K22(q) · · · K2N(q)...

.... . .

...

KN1(q) KN2(q) · · · KNN(q)

,

with the following equalities:

K11(q) = (A∗q1)(KG1 (q))(Aq1)− (A∗q1)(K

Q1/G1 (q))− (KG

2 (q))(Aq1) + KQ1/G2 (q),

K1j(q) = −(A∗q1)(KG1j (q)) + K

Q1/G1j (q), ∀j = 2, · · · , N

Ki1(q) = −(KGi1 (q))(Aq1) + K

Q1/Gi1 (q). ∀i = 2, · · · , N

LetM = FL(D1×TQ2×· · ·×TQN) be the vector sub-bundle of T ∗Q corresponding

to the nonholonomic distribution. We then define the horizontal lift map hlM(h,q) : T ∗q Q →

M((h, q)) on the cotangent bundle of the reduced space by

hlM(h,q) := FL(h,q) hl(h,q) FL−1

q

=

[T ∗hLh−1 0

0 idTqQ

]

−(KG1 (q))(Aq1) + K

Q1/G1 (q) KG

12(q) · · · KG1N(q)

−(KG2 (q))(Aq1) + K

Q1/G2 (q) K

Q1/G12 (q) · · · K

Q1/G1N (q)

−(KG21(q))(Aq1) + K

Q1/G21 (q) K22(q) · · · K2N(q)

......

. . ....

−(KGN1(q))(Aq1) + K

Q1/GN1 (q) KN2(q) · · · KNN(q)

FL−1

q

where idTqQ is the identity map on TqQ. Based on the definition of H(pq) := H hlMq (pq),


where pq ∈ T ∗Q and q = π(q), we calculate H on T ∗Q using the local trivialization and

the definition of the map hlM

:

H(pq) =1

2

⟨hlM(h,q)(pq),FL−1

(h,q) hlM(h,q)(pq)

⟩+ V (e1, q) =

1

2

⟨pq,FL−1

q (pq)⟩

+ V (q),

(4.3.26)

where the function V (q) := V (e1, q).

Now performing the Chaplygin reduction in Theorem 4.2.4 we can write the reduced

dynamical equations for nonholonomic multi-body systems on T ∗Q.

Theorem 4.3.1. A nonholonomic open-chain multi-body system with symmetry

(T ∗Q,Ωcan, H,K,D, G) is reduced to a system (T ∗Q, Ωcan − Ξ, H, K), where Ωcan is the

canonical 2-form on T ∗Q, H is defined by (4.3.26) and K is the induced metric on

Q. Here, in the local coordinates Ξ is calculated as follows. Let πQ : T ∗Q → Q be the

canonical projection map of the cotangent bundle and let TπQ : T (T ∗Q) → T Q be its

induced map on the tangent bundles. For every αq ∈ T ∗q Q and ∀U , W ∈ X(T ∗Q) we

introduce uq = TαqπQ(U(αq)) and wq = TαqπQ(W(αq)). In the local trivialization, we

have q = (q1, q) ∈ U ×Q, uq = (u1, u) and wq = (w1, w):

Ξαq(U(αq), W(αq)) =⟨[−(KG

1 (q))(Aq1) + KQ1/G1 (q) KG

12(q) · · · KG1N(q)

]FL−1

q (αq),

−[Aq1u1, Aq1w1]− (∂Aq1∂q1

w1)u1 + (∂Aq1∂q1

u1)w1

⟩. (4.3.27)

Finally, in local coordinates we have X = (˙q, ˙p) as a vector field on T ∗Q, and Hamilton’s

equation in this space reads

ι( ˙q, ˙p)(−dp ∧ dq − Ξ) =∂H

∂pdp+

∂H

∂qdq,

where Ξ is calculated by (4.3.27).

Proof. In order to prove (4.3.27), we start with (4.2.14):

Ξαq(U(αq), W(αq)) =⟨Mq iM hl

Mq (αq),−Aq([hl(u), hl(w)]q)

⟩.

Using the local trivialization, we write q = (h, q1, q) ∈ G × U × Q, and accordingly


u = (u1, u) and w = (w1, w). The horizontal lift of u and w can be calculated as

hlq(uq) = (−(Te1Lh)Aq1u1, u1, u), hlq(wq) = (−(Te1Lh)Aq1w1, w1, w).

For all q ∈ Q, to calculate the Lie bracket [hl(u), hl(w)]q, we express the vector fields

hl(u) and hl(w) in coordinates:

hl(u) =(−(Te1Lh)Aq1u1

) ∂

∂h+ u

∂

∂q

hl(w) =(−(Te1Lh)Aq1w1

) ∂

∂h+ w

∂

∂q.

In any coordinates chosen for Qi (i = 2, · · · , N), G and Q1/G we have

[hl(u), hl(w)

]=

[((Te1Lh)Aq1u1

) ∂

∂h,(

(Te1Lh)Aq1w1

) ∂

∂h

]−[(

(Te1Lh)Aq1u1

) ∂

∂h, w

∂

∂q

]−[u∂

∂q,(

(Te1Lh)Aq1w1

) ∂

∂h

]+

[u∂

∂q, w

∂

∂q

]Based on the definition of the Lie bracket for Lie groups, the first bracket on the right

hand side can be written as[((Te1Lh)Aq1u1

) ∂

∂h,(

(Te1Lh)Aq1w1

) ∂

∂h

]=(

(Te1Lh)[Aq1u1, Aqw1

]) ∂

∂h

+

((Te1Lh)Aq1

∂w1

∂h

((Te1Lh)Aq1u1

)) ∂

∂h

−(

(Te1Lh)Aq1∂u1

∂h

((Te1Lh)Aq1w1

)) ∂

∂h.

We calculate the second bracket as[((Te1Lh)Aq1u1

) ∂

∂h, w

∂

∂q

]=∂w

∂h

((Te1Lh)Aq1u1

) ∂

∂q

−

((Te1Lh)

(∂Aq1∂q1

w1

)u1 + (Te1Lh)Aq1

∂u1

∂qw

)∂

∂h.

Similarly, the third bracket can be calculated. The last bracket is equal to

[u∂

∂q, w

∂

∂q] =

(∂w

∂qu

)∂

∂q−(∂u

∂qw

)∂

∂q.


Accordingly, using the definition of A for nonholonomic multi-body systems,

Aq([hl(u), hl(w)]q) = Adh

([Aqu,Aqw]−

(∂Aq∂q

w

)u+

(∂Aq∂q

u

)w

).

As for the term Mq iM hlMq (αq), based on the definition of the maps M and hl

Mq

we have

MqiM hlMq (αq) =

[T ∗e1Rh 0 · · · 0

] [T ∗hLh−1 0

0 idTqQ

]

−(KG1 (q))(Aq1) + K

Q1/G1 (q) KG

12(q) · · · KG1N(q)

−(KG2 (q))(Aq1) + K

Q1/G2 (q) K

Q1/G12 (q) · · · K

Q1/G1N (q)

−(KG21(q))(Aq1) + K

Q1/G21 (q) K22(q) · · · K2N(q)

......

. . ....

−(KGN1(q))(Aq1) + K

Q1/GN1 (q) KN2(q) · · · KNN(q)

FL−1

q (αq)

= Ad∗h−1

[−(KG

1 (q))(Aq1) + KQ1/G1 (q) KG

12(q) · · · KG1N(q)

]FL−1

q (αq).

As a result, we have the equation for Ξ in the theorem.

The hypotheses of the second stage of reduction for nonholonomic open-chain multi-

body systems with symmetry are satisfied if we have a symmetry group of the original

system that is bigger than G1 := Q1. This might happen when we have the original

system being invariant under the action of a group in the form of G := G1×G2×· · ·×GN ,

where Gi ⊆ Qi (i = 2, · · · , N) is a Lie subgroup of Qi corresponding to the joint Ji.

In the following section we investigate the possibility that a nonholonomic open-chain

multi-body system being invariant under such a group action.

Remark 4.3.2. For a holonomic open-chain multi-body system (T ∗Q,Ωcan, H,K,G),

we may also have that the symmetry group G is in the form of G1 × G2 × · · · × GN , as

defined above. In this case, we can reduce the holonomic system following the same steps

presented in Section 3.3 for the action of this group. See Theorem 3.3.6.

4.4 An Investigation on Further Symmetries of Open-

chain Multi-body Systems

In this section we introduce a number of sufficient conditions under which the kinetic

energy metric of a nonholonomic open-chain multi-body system admits further symme-


tries. That is, the system is invariant (in the sense that was presented in the previous

section) under the action of other groups in addition to the one presented in Theorem

3.3.3. We investigate two approaches:

AP1) Identifying symmetry groups due to left invariance of the kinetic energy metric h

on P = SE(3)× · · · × SE(3). See Section 3.3 for the definition of the metric h.

AP2) Identifying symmetry groups by studying the metric K on Q.

4.4.1 Identifying Symmetry Groups using AP1

As for the approach AP1, we consider the embedding F : Q → P , defined by (3.3.30),

which determines the position of the centre of mass of all bodies with respect to the


F (q) = (q1rcm,1, q1q2rcm,2, · · · , q1 · · · qNrcm,N),

where rcm,i (i = 1, · · · , N) is the initial pose of a coordinate frame attached to the centre

of mass of body Bi with respect to the inertial coordinate frame, i.e., B0.

For any element (a1, · · · , aN) ∈ P we define the group action ΘN(a1,··· ,aN ) : P → P by

ΘN(a1,··· ,aN )(p) := (a1p1, (a1a2)p2, · · · , (a1 · · · aN)pN),

where p = (p1, · · · , pN) ∈ P . Since the metric h on P is left-invariant, it is also invariant

under this action. That is, we have T ∗ΘN(a1,··· ,aN (h) = h. This action induces an action

on Q by the embedding F , if and only if the image of the map F , i.e., F (Q), is invariant

under the action ΘN for a Lie subgroup of P . We denote this Lie subgroup by G1×· · · GN ,

where Gi ⊆ SE(3) (i = 1, · · · , N) is a Lie subgroup of SE(3). Then the induced action

on Q, denoted by ΦN(a1,··· ,aN ) : Q → Q, is defined by ΦN(a1,··· ,aN ) := F−1 ΘN(a1,··· ,aN ) F ,

where (a1, · · · , aN) ∈ G1× · · · GN . Here, F−1 : F (Q)→ Q is only defined on the image of

the map F . In order to identify the group G1 × · · · × GN , we impose the condition that

F (Q) is invariant under the action of this group. By the definition of the map F and

ΘN(a1,··· ,aN ), we have

ΘN(a1,··· ,aN ) F (q) = (a1q1rcm,1, (a1a2)q1q2rcm,2, · · · , (a1 · · · aN)q1 · · · qNrcm,N)

The image of F is invariant under the group action if and only if we have the following


conditions:

a1 ∈ Q1

q−11 a2q1 ∈ Q2 ∀q1 ∈ Q1,

...

(q1 · · · qN−1)−1aN(q1 · · · qN−1) ∈ QN ∀q1 ∈ Q1 and · · · and ∀qN−1 ∈ QN−1.

Hence, the biggest symmetry group G1 × · · · GN that leaves the kinetic energy metric K

invariant under the induced action ΦN is equal to

G1 × · · · GN =(a1, · · · , aN)| a1 ∈ Q1, a2 ∈⋂

q1∈Q1

(q1Q2q−11 ), · · ·

, aN ∈⋂

q1∈Q1···

qN−1∈QN−1

((q1 · · · qN−1)QN(q−1N−1 · · · q

−11 )) ⊆ Q1 × · · · × QN .

Noteworthy examples of open-chain multi-body systems whose kinetic energy metric K

is invariant under the action of this group include but not limited to the systems with

identical multi-degree-of-freedom joints and systems with commutative joints. In general,

this symmetry group may be as small as G1 = Q1, specially when most of the joints are

actuated, since the actuation force breaks the symmetry.

4.4.2 Identifying Symmetry Groups using AP2

For any velocity vector q ∈ TqQ, we denote the left translation of q to Lie(Q) by

τ = (τ1, · · · , τN) := q−1q = (q−11 q1, · · · , q−1

N qN) ∈ Lie(Q)

Now let iτ ji (i, j = 1, · · · , N) be the relative twist of the body Bi with respect to Bj

and expressed in the coordinate frame attached to Bi. In order to determine the kinetic

energy of an open-chain multi-body system we need to have the relative twist of each

body Bi with respect to B0 and expressed in a coordinate frame attached to the centre

of mass of Bi, i.e.,

iτ 0i = Adr−1

cm,i

(Ad(q2···qi)−1(τ1) + · · ·+ Adq−1

i(τi−1) + τi

)


for a sequence of bodies from B0 to Bi [20]. Then the kinetic energy of a multi-body

system can be calculated by

1

2Kq(q, q) =

1

2

N∑i=1

‖ iτ 0i ‖2

hi, (4.4.28)

where hi denotes the left invariant metric corresponding to the body Bi on se(3), and

‖ · ‖hi refers to its induced norm on se(3). In the second approach AP2, first the case of

a multi-body system with only two bodies is investigated in the sequel, and the result is

generalized for the case of N bodies.

Let G1 = Q1 and G2 ⊆ Q2 be a Lie subgroup of Q2, and consider the action of G1×G2

by left translation on the configuration manifold Q = Q1 ×Q2, i.e., ∀(a1, a2) ∈ G1 × G2

we have (q1, q2) 7→ (a1q1, a2q2) for all q = (q1, q2) ∈ Q. It is easy to show that under this

action the kinetic energy of the system becomes

1

2K(a1q1,a2q2)(a1q1, a2q2) =

1

2

(‖ Adr−1

cm,1τ1 ‖2

h1+ ‖ Adr−1

cm,2

(Ad(a2q2)−1τ1 + τ2

)‖2h2

),

where (a1q1, a2q2) denotes the left translation of the velocity vector (q1, q2) to (a1q1, a2, q2).

As it was expected, the kinetic energy remains invariant under the G1-action. We define

the metric h′2 := Ad∗r−1cm,2

(h2)e on the Lie algebra of SE(3) corresponding to the body

B2. Note that, here e ∈ SE(3) denotes the identity element of SE(3). Kinetic energy is

invariant under the action of G1 × G2 if and only if it is invariant under the infinitesimal

action of all elements $ ∈ Lie(G2) at the identity element e2. Hence, we have the

following necessary and sufficient condition for the metric K being invariant under the

action of G1 × G2 by left translation:

∂

∂ε

∣∣∣∣ε=0

(1

2‖ Ad(exp(−ε$)q2)−1τ1 + τ2 ‖2

h′2

)= h′2(Adq−1

2ad$(τ1),Adq−1

2τ1 + τ2) = 0.

(4.4.29)

∀q2 ∈ Q2, ∀τ1 ∈ Lie(G1) and ∀τ2 ∈ Lie(G2)

The largest Lie sub-algebra of Lie(Q2) whose elements satisfy the above condition is the

Lie algebra of G2, and G2 can be identified by integrating this Lie sub-algebra on Q2.

Noteworthy examples of the systems that admit such a symmetry group are any two

commutative joints, a planar rover with a rotary joint orthogonal to it, and a planar

rover moving on a rotating disc. With similar calculations, we can extend this result to

the case of open-chain multi-body systems with N bodies, and write the necessary and


sufficient condition (4.4.29) as

N∑i=2

h′i(Ad(q2···qi)−1ad$(τ1),Ad(q2···qi)−1(τ1 + · · ·+ Ad(q2···qi)τi)) = 0. (4.4.30)

∀qi ∈ Qi (i = 2, · · · , N) and ∀τi ∈ Lie(Gi) (i = 1, · · · , N)

where h′i := Ad∗r−1cm,i

(hi)e. Note that, the expression in the parentheses in the second

argument of h′i is the relative twist of Bi with respect to B0 and expressed in a coordinate

frame attached to B1. Based on this condition, we may derive a sufficient condition for

the metric K being invariant under the action of G1 × G2 by left translation.

Proposition 4.4.1. For an open-chain multi-body system, the metric K is invariant

under the action of G1 × G2, as defined above, by left translation, if ∀$ ∈ Lie(G2) and

∀τ1 ∈ Lie(Q1) we have

ad$(τ1) = 0.

Similarly, we can derive sufficient conditions for the metric K being invariant under

the action of a group in the form of G1 × · · · × GN by left translation. Here Gi ⊆ Qi is

a Lie subgroup of Qi for i = 2, · · · , N . However, since it is very unlikely that we have

the invariance of K under the action of such a big group, we do not go through the

calculations for this most general case.

Finally, suppose that Bi0 is an extremity of the open-chain multi-body system. Con-

sider the action of Gi0 as a Lie subgroup of Qi0 by right translation. The kinetic energy

of the system after the action of an element ai0 ∈ Gi0 becomes

1

2Kqai0

(qai0 , qai0) =1

2

N∑i=1i 6=i0

‖ iτ 0i ‖2

hi+

1

2‖ Ada−1

i0

Adrcm,i0i0τ 0

i0‖2h′i0

. (4.4.31)

The kinetic energy metric is invariant under this action if and only if it is invariant under

the infinitesimal action of any element % ∈ Lie(Gi0) at the identity element.

∂

∂ε

∣∣∣∣ε=0

(1

2‖ Ad(exp(−ε%))−1(Adrcm,i0

i0τ 0i0

) ‖2h′i0

)= h′i0(ad%(Adrcm,i0

i0τ 0i0

),Adrcm,i0i0τ 0

i0) = 0,

(4.4.32)

for all i0τ 0i0

, i.e., all admissible relative twists of Bi0 with respect to the inertial coordinate

frame and expressed in the same frame. The largest Lie sub-algebra of Lie(Qi0) that

satisfies the above condition is Lie(Gi0), and Gi0 ⊆ Qi0 is identified by integrating this


Lie sub-algebra on Qi0 . Therefore, the kinetic energy K is invariant under the Gi0-action

by right translation on Qi0 if and only if we have the above condition.

4.5 Further Reduction of Nonholonomic Open-chain

Multi-body Systems

Let N be a Lie subgroup of Q. We define the action of N on Q, i.e., Φn : Q → Q, by

left translation on Q. For any element n ∈ N we have

Φn(q1, q) = (q1, nq).

Hence, the tangent and cotangent lift of the N -action are

TqΦn(vq) =

[idTq1Q1

0

0 TqLn

][v1

v

]

T ∗Φn(q)

Φn−1(pq) =

[idTq1Q1

0

0 T ∗nqLn−1

][p1

p

].

Let us assume that the Hamiltonian H and the metric K of the reduced nonholonomic

open-chain multi-body system (T ∗Q, Ωcan − Ξ, H, K) are invariant under the cotangent

and tangent lift of the N -action, respectively. We locally trivialize Q such that we have

q = (q1, n, q′) ∈ U × N × U ′, where U ′ ⊆ Q′ := Q/N is an open subset of Q′. In

this trivialization, the map corresponding to the infinitesimal N -action φq : Lie(N ) ⊂Lie(Q)→ T Q is calculated by

φq =

0

TeRn

0

,where e ∈ N ⊆ Q is the identity element. Since the cotangent lift of the N -action leaves

p1 invariant, its infinitesimal generator, ζT ∗Q for all ζ ∈ Lie(N ), satisfies the condition

ιζT∗Q

Ξ = 0.

We also define the momentum map Mq : T ∗q Q → Lie∗(N ) by

Mq = φ∗q =[0 T ∗eRn 0

].


Accordingly, the locked inertia tensor Iq : Lie(N ) → Lie∗(N ) and the principal connec-

tion Aq : TqQ → Lie(N ) for N -action are calculated as

Iq = φ∗q FLq φq = Ad∗n−1(KN1 (q1, q′))Adn−1

Aq = I−1q Mq FLq

= Adn

[(KN1 (q1, q

′))−1KN12(q1, q′) TnLn−1 (KN1 (q1, q

′))−1KQ/N1 (q1, q

′)]

=:[A(q1,q′) TnLn−1 B(q1,q′)

], (4.5.33)

where we define the linear maps Aq : TU → Lie(N ) and Bq : TU ′ → Lie(N ) by the last

equality, and we have

FLq =

[K ′11(q) K ′12(q)

K ′21(q) K ′22(q)

]

=:

K ′11(q1, e, q

′) (KN12(q1, q′))TnLn−1 K

Q/N12 (q1, q

′)

T ∗nLn−1(KN21(q1, q′)) T ∗nLn−1(KN1 (q1, q

′))TnLn−1 T ∗nLn−1(KQ/N1 (q1, q

′))

KQ/N21 (q1, q

′) (KN2 (q1, q′))TnLn−1 K

Q/N2 (q1, q

′)

.As a result we can calculate the map hor in the local trivialization by

horq = idTqQ − φq Aq =

idTq1U 0 0

−TeLn(A(q1,q′)) 0 −TeLn(B(q1,q′))

0 0 idTq′U ′

, (4.5.34)

where idTq1U and idTq′U ′ are the identity maps on the tangent spaces Tq1U and Tq′U′,

respectively.

We also locally trivialize the principal bundle N → N /Nϑ, and similarly we calculate

the (mechanical) principal connection Aϑ corresponding to the principal bundle Q →Q = Q/Nϑ. We use this connection to calculate the horizontal lift map hl. Let us

assume that the principal connection Aϑ in the local trivialization is written as:

Aϑq :=[Aϑq TkLk−1 Bϑ

q

],

for all q ∈ U × Nϑ × Uϑ × U ′, where Uϑ ⊆ N /Nϑ is an open subset of N /Nϑ, k ∈ Nϑand q ∈ U × Uϑ × U ′ ⊆ Q = Q/Nϑ. Here, the linear maps Aϑq : TU → Lie(Nϑ) and

Bϑq : T (Uϑ×U ′)→ Lie(Nϑ) are defined based on the Legendre transformation FLq in the

local trivialization of the principal bundle N → N /Nϑ. Consequently, the horizontal lift


map hlq : Tq(U × Uϑ × U ′)→ Tq(U ×Nϑ × Uϑ × U ′) is calculated by

hlq =

[−TeLk

[Aϑq Bϑ

q

]idTq(U×Uϑ×U ′)

], (4.5.35)

where idTq(U×Uϑ×U ′) is the identity map on the tangent space Tq(U × Uϑ × U ′). Now, we

use (4.5.33), (4.5.34) and (4.5.35) to calculate the 2-form Ξϑ in (3.2.27) for a reduced

nonholonomic open-chain multi-body system. Furthermore, based on Theorem 3.2.7, for

a reduced multi-body system we have

Lemma 4.5.1. Based on Theorem 3.2.7, the inverse of the map ϕϑ : M−1

(ϑ)/Nϑ → T ∗Qis defined on [T π(V)]0 and in the local trivialization ∀pq = (p1, pϑ, p

′) ∈ T ∗q (U ×Uϑ×U ′),

ϕ−1ϑ (p1, pϑ, p) =

p+ A∗(q1,q′)(Ad∗(e,n)ϑ)

T ∗(k,n)R(k,n)−1(ϑ)

p′ + B∗(q1,q′)(Ad∗(e,n)ϑ)

ϑ

, (4.5.36)

where in local trivialization we have n = (k, n) ∈ N .

It is easy to show that on the vector sub-bundle [T π(V)]0, pϑ = 0. Hence, using this

lemma we can determine the reduced Hamiltonian Hϑ : [T π(V)]0 → R by

Hϑ(q, p1, 0, p) := Hϑ(ϕϑ(p1, 0, p)). (4.5.37)

Theorem 4.5.2. We say that a reduced nonholonomic open-chain multi-body system with

symmetry (T ∗Q, Ωcan − Ξ, H, K,N ), and whose solution curves satisfy the reduced non-

holonomic Hamilton’s equation (4.3.27), can be further reduced to the system ([T π(V)]0 ⊂T ∗Q, Ωcan− Ξϑ−Ξϑ, Hϑ, K), where Ωcan is the canonical 2-form on the cotangent bundle

of the quotient manifold Q = Q/Nϑ. The 2-form Ξϑ := T ∗ϕ−1ϑ (Ξϑ) is calculated based on

Lemma 4.5.1. The Hamiltonian Hϑ : [T π(V)]0 → R is the further reduced Hamiltonian

in (4.5.37), and K is the induced metric on Q

K(vq, wq) := K(hlq(vq), hlq(wq)). ∀vq, wq ∈ TqQ

Also the closed 2-form Ξϑ is defined by (3.2.27), using (4.5.33), (4.5.34) and (4.5.35).

The further reduced system satisfies Hamilton’s equation (4.2.19) for the Hamiltonian

Hϑ with the almost 2-form Ωcan − Ξϑ − Ξϑ. That is, in the local coordinates, where we

have Xϑ = (˙q1,˙n, q′, ˙p1, p

′) as a vector field on [T π(V)]0, further reduced nonholonomic


Hamilton’s equation reads

ι( ˙q1,˙n,q′, ˙p1,p

′)(−dp1 ∧ dq1 − dp′ ∧ dq′ − Ξϑ − Ξϑ)

=∂Hϑ

∂p1

dp1 +∂Hϑ

∂p′dp′ +

∂Hϑ

∂q1

dq1 +∂Hϑ

∂ndn +

∂Hϑ

∂q′dq′, (4.5.38)

where Ξϑ and Ξϑ are calculated as explained above.

The third stage of reduction can also be done similar to the second stage using the

theory developed in the previous section.

4.6 Case Study

In this section we study the dynamics of two examples of nonholonomic open-chain

multi-body systems. In the first example, we derive the reduced dynamical equations of

a three-d.o.f. manipulator mounted on top of a two-wheeled, differential rover whose top

view in the initial configuration is shown in Figure 4.1. In the second example, we study

the two-step reduction of the dynamical equations of a two-d.o.f. crane on a four-wheel

car.

Example 4.6.1. Using the indexing introduced in the previous section and starting

with the rover without wheels and the manipulator as B1, we first number the bodies

and joints. The following graph shows the structure of the nonholonomic open-chain

multi-body system.

B2

B0J1

B1J4

J3

J2

B4J5

B5J6

B6

B3

We then identify the relative configuration manifolds corresponding to the joints of

the robotic system. The relative pose of B1 with respect to the inertial coordinate

frame is identified by the elements of the Special Euclidean group of plane SE(2). For

simplicity of calculations, we take the middle point of the line connecting the wheels C

as the reference point and identify the elements of the relative configuration manifold


Figure 4.1: An example of a mobile manipulator

corresponding to the first joint, which is a three-d.o.f. planar joint, by

Q01 =

H21 =


sin(θ) cos(θ) 0 y

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣x, y ∈ R, θ ∈ S1

,

as a submanifold (Lie subgroup) of SE(3). Here, (x, y) is the position of C with respect

to the inertial coordinate frame and θ is the angle between the X1-axis and X0-axis (see

Figure 4.2). The second joint is a one-d.o.f. revolute joint between B2 and B1, and its

corresponding relative configuration manifold is given by

Q12 =

H12 =

cos(ψ1) 0 sin(ψ1) 0

0 1 0 c

− sin(ψ1) 0 cos(ψ1) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ1 ∈ S1

,

where c is the distance between the point C and the wheels. Similarly, for the third joint

we have

Q13 =

H13 =


0 1 0 −c− sin(ψ2) 0 cos(ψ2) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ2 ∈ S1

.


Figure 4.2: The coordinate frames attached to the bodies of the mobile manipulator(Note that, the Zi-axis (i = 0, · · · , 6) is normal to the plane)

The forth, fifth and sixth joints are one-d.o.f. revolute joint whose axes of revolution are

the Z4, X5 and X6 axes, respectively. The relative configuration manifolds of these joints

are identified by

Q14 =

H14 =

cos(ϕ1) − sin(ϕ1) 0 l0 + l1

sin(ϕ1) cos(ϕ1) 0 0

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ1 ∈ S1

,

Q45 =

H45 =

1 0 0 0

0 cos(ϕ2) − sin(ϕ2) l2

0 sin(ϕ2) cos(ϕ2) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ2 ∈ S1

,

Q56 =

H56 =

1 0 0 0

0 cos(ϕ3) − sin(ϕ3) l3

0 sin(ϕ3) cos(ϕ3) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ3 ∈ S1

.



H01,0 is the identity element of SE(3). In matrix form we have H0

1,0 = id4, where id4 is

the 4 × 4 identity matrix. As a result, the initial pose of the centre of mass of B1 with

respect to B0 is

rcm,1 =

1 0 0 l0

0 1 0 0

0 0 1 0

0 0 0 1

.For the second and third body, the initial relative pose with respect to B1 is

H1i,0 =

1 0 0 0

0 1 0 ±c0 0 1 0

0 0 0 1

,

and we have

rcm,i =

1 0 0 0

0 1 0 ±c0 0 1 0

0 0 0 1

,where plus and minus signs correspond to i = 2 and i = 3, respectively.

The initial relative pose of B4 with respect to B1 is

H14,0 =

1 0 0 l0 + l1

0 1 0 0

0 0 1 0

0 0 0 1

,

and the relative pose of the centre of mass of B4 with respect to the inertial coordinate

frame is

rcm,4 =

1 0 0 l0 + l1

0 1 0 l2/2

0 0 1 0

0 0 0 1

.Here we assume that the centre of mass of B4 and B5 are in the middle of the links. For


the fifth and sixth bodies we respectively have

H45,0 =

1 0 0 0

0 1 0 l2

0 0 1 0

0 0 0 1

,

rcm,5 =

1 0 0 l0 + l1

0 1 0 l2 + l3/2

0 0 1 0

0 0 0 1

,and

H56,0 =

1 0 0 0

0 1 0 l3

0 0 1 0

0 0 0 1

,

rcm,6 =

1 0 0 l0 + l1

0 1 0 l2 + l3 + l4

0 0 1 0

0 0 0 1

.With the above specifications of the system we identify the configuration manifold of

the nonholonomic open-chain multi-body system in this case study by Q = Q1×· · ·×Q6,

where

Q1 =

q1 =


sin(θ) cos(θ) 0 y

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣x, y ∈ R, θ ∈ S1

,

Q2 =

q2 =


0 1 0 0

− sin(ψ1) 0 cos(ψ1) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ1 ∈ S1

,

Q3 =

q3 =


0 1 0 0

− sin(ψ2) 0 cos(ψ2) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ2 ∈ S1

,


Q4 =

q4 =

cos(ϕ1) − sin(ϕ1) 0 2(l0 + l1) sin2(ϕ1/2)

sin(ϕ1) cos(ϕ1) 0 −(l0 + l1) sin(ϕ1)

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ1 ∈ S1

,

Q5 =

q5 =

1 0 0 0

0 cos(ϕ2) − sin(ϕ2) 2l2 sin2(ϕ2/2)

0 sin(ϕ2) cos(ϕ2) −l2 sin(ϕ2)

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ2 ∈ S1

,

Q6 =

q6 =

1 0 0 0

0 cos(ϕ3) − sin(ϕ3) 2(l2 + l3) sin2(ϕ3/2)

0 sin(ϕ3) cos(ϕ3) −(l2 + l3) sin(ϕ3)

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ3 ∈ S1

.






F (q1, · · · , q6) = (q1rcm,1, q1q2rcm,2, q1q3rcm,3, q1q4rcm,4, q1q4q5rcm,5, q1q4q5q6rcm,6)




Tq(LF (q)−1F ) =

Adr−1

cm,1· · · 0

.... . .

...

0 · · · Adr−1cm,6

id6 06×6 06×6 06×6 06×6 06×6

Adq−12

id6 06×6 06×6 06×6 06×6

Adq−13

06×6 id6 06×6 06×6 06×6

Adq−14

06×6 06×6 id6 06×6 06×6

Ad(q4q5)−1 06×6 06×6 Adq−15

id6 06×6

Ad(q4q5q6)−1 06×6 06×6 Ad(q5q6)−1 Adq−16

id6

Tq1(Lq−1

1 ι1) · · · 0

.... . .

...

0 · · · Tq6(Lq−16 ι6)

,where id6 is the 6 × 6 identity matrix, and we have the following equalities, using the


introduced joint parameters:

Tq1(Lq−11 ι1) =

cos(θ) − sin(θ) 0 0 0 0

sin(θ) cos(θ) 0 0 0 0

0 0 0 0 0 1

T

,

Tq2(Lq−12 ι2) =

[0 0 0 0 1 0

]T,

Tq3(Lq−13 ι3) =

[0 0 0 0 1 0

]T,

Tq4(Lq−14 ι4) =

[0 −l0 − l1 0 0 0 1

]T,

Tq5(Lq−15 ι5) =

[0 0 −l2 1 0 0

]T,

Tq6(Lq−16 ι6) =

[0 0 −l2 − l3 1 0 0

]T.

Note that, ∀H0 ∈ SE(3) that is in the following form (R0 is a rotation matrix and

p0 ∈ R3)

H0 =

[R0 p0

01×3 1

],

we calculate the AdH0 operator by

AdH0 =

[R0 p0R0

03×3 R0

].

We choose E1, · · · , E6, defined in Section 2.4, as a basis for se(3). For this case

study, the left-invariant metric h = h1⊕· · ·⊕h6 on P is identified, in the above basis, by

the following metrics on the Lie algebras of copies of SE(3) corresponding to the bodies:

(hi)e =

miid3 03×3

03×3

jx,i 0 0

0 jy,i 0

0 0 jz,i

,

where i = 1, · · · , 6, id3 and 03×3 are the 3 × 3 identity and zero matrices, respectively,

mi is the mass of Bi, and (jx,i, jy,i, jz,i) are the moments of inertia of Bi about the X,

Y and Z axes of the coordinate frame attached to the centre of mass of Bi. Note that,

we chose this coordinate frame such that its axes coincide with the principal axes of the

body Bi. For the body Bi (i = 2, · · · , 5), since we assume a symmetric cylindrical shape

whose axis is aligned with the Yi-axis, we have jx,i = jz,i. Therefore, in the coordinates


chosen to identify the configuration manifold (joint parameters), we have the following

matrix form for FLq

FLq = T ∗q (LF (q)−1F )

(h1)e · · · 0

.... . .

...

0 · · · (h6)e

Tq(LF (q)−1F ) =

K11(q) · · · K16(q)

.... . .

...

K61(q) · · · K66(q)

,and the kinetic energy is calculated by

Kq(q, q) =1

2qTFLq q,


parameters.

The potential energy of the nonholonomic open-chain multi-body system is calculated

by (3.3.32). We assume a constant potential field [0 0 g]T in the inertial coordinate

frame. Therefore, using the joint parameters, we have

V (q) = g(l4m6 sin(ϕ2 + ϕ3) + l3(m5

2+m6) sin(ϕ2)).

As a result, the Hamiltonian of the nonholonomic open-chain multi-body system is cal-

culated by

H(q, p) =1

2pTFL−1

q p+ V (q),


The nonholonomic constraints for the multi-body system under study are the non-

slipping conditions on the wheels, i.e., B2 and B3. The linearly independent 1-forms

corresponding to the constraints are

ω11 = − sin(θ)dx+ cos(θ)dy,

ω21 = cos(θ)dx+ sin(θ)dy − cdθ − bdψ1,

ω31 = cos(θ)dx+ sin(θ)dy + cdθ − bdψ2,

where b is the radius of each wheel. The distribution D ⊂ TQ is the annihilator of these

constraint 1-forms, and it is the span of the following vector fields:∂

∂ψ1

+b

2

(cos(θ)

∂

∂x+ sin(θ)

∂

∂y− 1

c

∂

∂θ

)


,∂

∂ψ2

+b

2

(cos(θ)

∂

∂x+ sin(θ)

∂

∂y+

1

c

∂

∂θ

),∂

∂ϕ1

,∂

∂ϕ2

,∂

∂ϕ3

.

Here in this example, the base of the multi-body system consists of three bodies, B1,

B2 and B3, and its configuration manifold Q1 ×Q2 ×Q3 is isomorphic to G = SE(2)×SO(2)×SO(2), as a group. The kinetic and potential energy of the system are invariant

under the action of G by left translation on Q1 × Q2 × Q3. Also, the distribution D is

invariant under this action. Now, consider the action of G = SE(2) ⊂ G as a subgroup

of G, which satisfies the dimensional assumption (4.2.10) for Chaplygin systems. Using

the joint parameters, ∀(x0, y0, θ0) ∈ G we have

Φ(x0,y0,θ0)(q) = (x cos(θ0)− y sin(θ0) + x0, x sin(θ0) + y cos(θ0) + y0, θ + θ0, q1, q),

where q1 = (ψ1, ψ2) and q = (ϕ1, ϕ2, ϕ3). We have the principal G-bundle π : Q → Q =

Q2 × · · · × Q6, and using the joint parameters its corresponding principal connection

A : TQ → se(2) is defined by

Aq =

Adh︷︸︸︷cos(θ) − sin(θ) y

sin(θ) cos(θ) −x0 0 1

ThLh−1︷︸︸︷ cos(θ) sin(θ) 0

− sin(θ) cos(θ) 0

0 0 1

Aq1︷︸︸︷−b/2 −b/2

0 0

b/(2c) −b/(2c)

03×3

,

where h = (x, y, θ) is an element of Q1. And consequently, the horizontal lift map

hlq : TqQ → TqQ is

hlq =

b cos(θ)/2 b cos(θ)/2

b sin(θ)/2 b sin(θ)/2

−b/(2c) b/(2c)

03×3

id5

,where id5 is the 5× 5 identity matrix. Then, we have

FLq = hlT

q FLqhlq =

K11(q) · · · K14(q)

.... . .

...

K41(q) · · · K44(q)

,


where the following equalities hold:

K11(q) = ATq1K11((e1, q))Aq1 − ATq1

[K21((e1, q))

K31((e1, q))

]T−

[K21((e1, q))

K31((e1, q))

]Aq1

+

[K22((e1, q)) K23((e1, q))

K32((e1, q)) K33((e1, q))

],

K1j(q) = −ATq1K1(j+2)((e1, q)) +

[K2(j+2)((e1, q))

K3(j+2)((e1, q))

], ∀j = 2, 3, 4

Kj1(q) = K1j(q)T , ∀j = 2, 3, 4

Kij(q) = Kij((e1, q)). ∀i, j = 2, 3, 4

Here,

Aq1 =

−b/2 −b/20 0

b/(2c) −b/(2c)

.As a result, we can calculate the 2-form Ξ by (4.3.27)

Ξpq = pTFL−1q

−ATq1K11((e1, q)) +

[K21((e1, q))

K31((e1, q))

]K41((e1, q))

...

K61((e1, q))

0

b2/(2c)

0

dψ1∧dψ2 = Υ(q, p)dψ1∧dψ2,

where p is the vector of generalized momenta in the reduced space. Finally, in matrix

form we have the following reduced equations of motion for the nonholonomic multi-body

system under study:

[˙q˙p

]=

05×5 id5

−id5

0 Υ(q, p) 0 0 0

−Υ(q, p) 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

∂H∂q∂H∂p

,

where H is calculated by (4.3.26), with V (q) = V ((e1, q)).

Example 4.6.2. In this example, we study the two-step reduction of the dynamical

equations of a two-d.o.f. crane on a four-wheel car whose top and side view in the


Figure 4.3: An example of a crane

initial configuration is shown in Figure 4.3. Using the indexing introduced in the pre-

vious section and starting with the car without the rear wheels and the crane as B1,

we first number the bodies and joints. The following graph shows the structure of the

nonholonomic open-chain multi-body system.

B4 B2J5J4

B5

B0J1

B1

J2

J3B3

J6B6

Similar to the previous example, we then identify the relative configuration manifolds

corresponding to the joints of the robotic system. We identify the elements of the relative

configuration manifold corresponding to the first joint, which is a three-d.o.f. planar joint,

by

Q01 =

H21 =


sin(θ) cos(θ) 0 y

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣x, y ∈ R, θ ∈ S1

.

Here, (x, y) is the position of C with respect to the inertial coordinate frame and θ is the

angle between the X1-axis and X0-axis (see Figure 4.4). The second joint is a one-d.o.f.

revolute joint between B2 and B1, and its corresponding relative configuration manifold


Figure 4.4: The coordinate frames attached to the bodies of the crane

is given by

Q12 =

H12 =

cos(ψ1) − sin(ψ1) 0 l

sin(ψ1) cos(ψ1) 0 0

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ1 ∈ S1

,

where l is the distance between the front and rear wheels. Similarly, for the third joint

we have

Q13 =

H13 =

cos(ϕ1) − sin(ϕ1) 0 l1

sin(ϕ1) cos(ϕ1) 0 0

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ1 ∈ S1

.

The forth and fifth joints are one-d.o.f. revolute joints whose axes of revolution are the

Yi-axis (i = 4, 5). The relative configuration manifolds of these joints are identified by

Q24 =

H24 =


0 1 0 c

− sin(ψ2) 0 cos(ψ2) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ2 ∈ S1

,


Q25 =

H25 =


0 1 0 −c− sin(ψ3) 0 cos(ψ3) 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ3 ∈ S1

,

where c is the distance between the steering point and the front wheels. Note that, if

we assume that the front wheels are rotating together, then we can substitute the front

wheels with a cylinder. Finally, the sixth joint is a one-d.o.f. revolute joint with the

Y6-axis being its axis of revolution. So, we have

Q56 =

H56 =

cos(ϕ2) 0 sin(ϕ2) 0

0 1 0 0

− sin(ϕ2) 0 cos(ϕ2) l2

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ2 ∈ S1

.


H01,0 is the identity element of SE(3). As a result, the initial pose of the centre of mass

of B1 with respect to B0 is

rcm,1 =

1 0 0 l0

0 1 0 0

0 0 1 0

0 0 0 1

.For the second and third body, the initial relative pose with respect to B1 is

H12,0 =

1 0 0 l

0 1 0 0

0 0 1 0

0 0 0 1

,

H13,0 =

1 0 0 l1

0 1 0 0

0 0 1 0

0 0 0 1

,and we have

rcm,2 =

1 0 0 l

0 1 0 0

0 0 1 0

0 0 0 1

,


rcm,3 =

1 0 0 l1

0 1 0 l2/2

0 0 1 0

0 0 0 1

,where we assume that the centre of mass of B3 is located in the middle of the body. The

initial relative pose of B4 and B5 with respect to B2 is

H2i,0 =

1 0 0 0

0 1 0 ±c0 0 1 0

0 0 0 1

,

and the relative pose of the centre of mass of B4 and B5 with respect to the inertial

coordinate frame is

rcm,i =

1 0 0 l

0 1 0 ±c0 0 1 0

0 0 0 1

,where i = 4, 5 and plus and minus signs refer to B4 and B5, respectively. For the sixth

body we have

H36,0 =

1 0 0 0

0 1 0 l2

0 0 1 0

0 0 0 1

,

rcm,6 =

1 0 0 l1

0 1 0 0

0 0 1 l2

0 0 0 1

,where we assume that the centre of mass of this body is at the sixth joint J6.

Knowing the above specifications of the system, we identify the configuration manifold

of the nonholonomic open-chain multi-body system in this case study by Q = Q1×· · ·×


Q6, where

Q1 =

q1 =


sin(θ) cos(θ) 0 y

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣x, y ∈ R, θ ∈ S1

,

Q2 =

q2 =

cos(ψ1) − sin(ψ1) 0 2l sin2(ψ1/2)

sin(ψ1) cos(ψ1) 0 −l sin(ψ1)

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ1 ∈ S1

,

Q3 =

q3 =

cos(ϕ1) − sin(ϕ1) 0 2l1 sin2(ϕ1/2)

sin(ϕ1) cos(ϕ1) 0 −l1 sin(ϕ1)

0 0 1 0

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ1 ∈ S1

,

Q4 =

q4 =

cos(ψ2) 0 sin(ψ2) 2l sin2(ψ2/2)

0 1 0 0

− sin(ψ2) 0 cos(ψ2) l sin(ψ2)

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ2 ∈ S1

,

Q5 =

q5 =

cos(ψ3) 0 sin(ψ3) 2l sin2(ψ3/2)

0 1 0 0

− sin(ψ3) 0 cos(ψ3) l sin(ψ3)

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ψ3 ∈ S1

,

Q6 =

q6 =

cos(ϕ2) 0 sin(ϕ2) 2l1 sin2(ϕ2/2)− l2 sin(ϕ2)

0 1 0 0

− sin(ϕ2) 0 cos(ϕ2) l1 sin(ϕ2) + 2l2 sin2(ϕ2/2)

0 0 0 1

∈ SE(3)

∣∣∣∣∣∣∣∣∣∣ϕ2 ∈ S1

.






F (q1, · · · , q5) = (q1rcm,1, q1q2rcm,2, q1q3rcm,3, q1q2q4rcm,4, q1q2q5rcm,5, q1q3q6rcm,6)





Tq(LF (q)−1F ) =

Adr−1

cm,1· · · 0

.... . .

...

0 · · · Adr−1cm,6

id6 06×6 06×6 06×6 06×6 06×6

Adq−12

id6 06×6 06×6 06×6 06×6

Adq−13

06×6 id6 06×6 06×6 06×6


06×6 id6 06×6 06×6


06×6 06×6 id6 06×6

Ad(q3q6)−1 06×6 Adq−16

06×6 06×6 id6

Tq1(Lq−1

1 ι1) · · · 0

.... . .

...

0 · · · Tq6(Lq−16 ι6)

.where we have the following equalities, using the introduced joint parameters:

Tq1(Lq−11 ι1) =

cos(θ) − sin(θ) 0 0 0 0

sin(θ) cos(θ) 0 0 0 0

0 0 0 0 0 1

T

,

Tq2(Lq−12 ι2) =

[0 −l 0 0 0 1

]T,

Tq3(Lq−13 ι3) =

[0 −l1 0 0 0 1

]T,

Tq4(Lq−14 ι4) =

[0 0 l 0 1 0

]T,

Tq5(Lq−15 ι5) =

[0 0 l 0 1 0

]T,

Tq6(Lq−16 ι6) =

[−l2 0 l1 0 1 0

]T.

We choose E1, · · · , E6 as a basis for se(3) and define the following metrics on the

Lie algebras of copies of SE(3) corresponding to bodies:

(hi)e =

miid3 03×3

03×3

jx,i 0 0

0 jy,i 0

0 0 jz,i

,

where i = 1, · · · , 6, mi is the mass of Bi, and (jx,i, jy,i, jz,i) are the moments of inertia of

Bi about the Xi, Yi and Zi axes of the coordinate frame attached to the centre of mass


of Bi. Note that, we chose this coordinate frame such that its axes coincide with the

principal axes of the body Bi. For the body Bi (i = 2, · · · , 6), we assume a symmetric

cylindrical shape. The cylinder axis is aligned with the Yi-axis for i = 2, 4, 5, so we have

jx,i = jz,i. Similarly, for the bodies B3 and B6, the cylinder axes are aligned with Z3 and

X6 axes, and we have the equalities jx,3 = jy,3 and jy,6 = jz,6. Also, since the wheels

are assumed identical, only the dynamic parameters of B4 is going to appear in the

calculations. Therefore, in the coordinates chosen to identify the configuration manifold

(joint parameters), we have the following matrix form for FLq

FLq = T ∗q (LF (q)−1F )

(h1)e · · · 0

.... . .

...

0 · · · (h6)e

Tq(LF (q)−1F ) =

K11(q) · · · K16(q)

.... . .

...

K61(q) · · · K66(q)

.Here, we have

K11(q) =

mtot 0 − sin(θ)(lm2 + 2lm4 + l0m1 + l1m3 + l1m6)

? mtot cos(θ)(lm2 + 2lm4 + l0m1 + l1m3 + l1m6)

? ? jz,tot

K21(q) = K12(q)T =

[0 0 2m4c

2 + jx,2 + 2Jx,4

],

K31(q) = K13(q)T =[0 0 jz,3 + jx,6 sin2(ϕ2) + jy,6 cos2(ϕ2)

],

Ki1(q) = K1i(q)T =

[0 0 0

],∀i = 4, 5, 6

K22(q) = 2m4c2 + jx,2 + 2jx,4, Ki2 = K2i = 0, ∀i = 3, · · · , 6

K33(q) = jz,3 + jx,6 sin2(ϕ2) + jy,6 cos2(ϕ2), Ki3 = K3i = 0,∀i = 4, 5, 6

K44(q) = jy,4, Ki4 = K4i = 0,∀i = 5, 6

K55(q) = jy,4, K65 = K56 = 0, K66(q) = jy,6,

where

mtot = m1 +m2 +m3 + 2m4 +m6,

jz,tot = jz,1 + jx,2 + jz,3 + 2jx,4 + jx,6 + l2m2 + 2l2m4 + l20m1 + l21m3

+ l21m6 + 2c2m4 − jx,6 cos2(ϕ2) + jy,6 cos2(ϕ2).

For K11(q), we did not include the lower diagonal elements, since the matrix is symmetric.


The kinetic energy is calculated by

Kq(q, q) =1

2qTFLq q,


parameters.

For this case study, the potential energy of the nonholonomic open-chain multi-body

system is constant, and it does not enter the dynamical equation. As a result, the

Hamiltonian of the nonholonomic open-chain multi-body system is calculated by

H(q, p) =1

2pTFL−1

q p,


The nonholonomic constraints for the multi-body system under study are the non-

slipping conditions on the wheels, i.e., B4 and B5. The linearly independent 1-forms

corresponding to the constraints are

ω11 = − sin(θ)dx+ cos(θ)dy,

ω21 = − sin(θ + ψ1)dx+ cos(θ + ψ1)dy + l cos(ψ1)dθ,

ω31 = cos(θ + ψ1)dx+ sin(θ + ψ1)dy + (l sin(ψ1)− c)dθ − cdψ1 − bdψ2,

ω41 = cos(θ + ψ1)dx+ sin(θ + ψ1)dy + (l sin(ψ1) + c)dθ + cdψ1 − bdψ3,

where b is the radius of each wheel. The distribution D ⊂ TQ is the annihilator of these

constraint 1-forms, and it is the span of the following vector fields:∂

∂ψ1

+cl cos(ψ1)

l − c sin(ψ1)

(cos(θ)

∂

∂x+ sin(θ)

∂

∂y+

tan(ψ1)

l

∂

∂θ+

2

b cos(ψ1)

∂

∂ψ3

),∂

∂ψ2

+bl cos(ψ1)

l − c sin(ψ1)

(cos(θ)

∂

∂x+ sin(θ)

∂

∂y+

tan(ψ1)

l

∂

∂θ+l + c sin(ψ1)

bl cos(ψ1)

∂

∂ψ3

),∂

∂ϕ1

,∂

∂ϕ2

.

Here in this example, the base of the multi-body system consists of four bodies, B1, B2,

B4 and B5, and its configuration manifold is Q1×Q2×Q4×Q5. The Hamiltonian of the

system H and the distribution D are invariant under the action of G = Q1×Q3×Q4×Q5,

which is isomorphic to SE(2)× SO(2)× SO(2)× SO(2) as a group, by left translation.

Now, consider the action of G = Q1 × Q5 ⊂ G as a subgroup of G, which satisfies

the dimensional assumption (4.2.10) for Chaplygin systems. Using the joint parameters,


∀(x0, y0, θ0, ψ3,0) ∈ G we have

Φ(x0,y0,θ0,ψ3,0)(q) = (x cos(θ0)−y sin(θ0)+x0, x sin(θ0)+y cos(θ0)+y0, θ+θ0, ψ3+ψ3,0, q1, q),

where q1 = (ψ1, ψ2) and q = (ϕ1, ϕ2). We have the principal G-bundle π : Q → Q =

Q2×Q4×Q3×Q6, and using the joint parameters its corresponding principal connection

A : TQ → Lie(G) is defined by

Aq =

Adh︷︸︸︷cos(θ) − sin(θ) y 0

sin(θ) cos(θ) −x 0

0 0 1 0

0 0 0 1

ThLh−1︷︸︸︷cos(θ) sin(θ) 0 0

− sin(θ) cos(θ) 0 0

0 0 1 0

0 0 0 1

Aq1︷︸︸︷1

l − c sin(ψ1)

−lc cos(ψ1) −lb cos(ψ1)

0 0

−c sin(ψ1) −b sin(ψ1)

−2lc/b −(l + c sin(ψ1))

04×2

,

where h = (x, y, θ, ψ3) is an element of Q1 × Q5. And consequently, the horizontal lift

map hlq : TqQ → TqQ is

hlq =

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1

1

l−c sin(ψ1)

lc cos(ψ1) cos(θ) lb cos(ψ1) cos(θ)

lc cos(ψ1) sin(θ) lb cos(ψ1) sin(θ)

c sin(ψ1) b sin(ψ1)

2lc/b l + c sin(ψ1)

04×2

id4

,

where in the above formulation, the first matrix in the multiplication is necessary only

to match the order of parameters.


Then, we have

FLq = hlT

q FLqhlq =

K11(q) · · · K14(q)

.... . .

...

K41(q) · · · K44(q)

,where the following equalities hold:

K11(q) = ATq1

[K11((e1, q)) K15((e1, q))

K51((e1, q)) K55((e1, q))

]Aq1 − ATq1

[K21((e1, q)) K25((e1, q))

K41((e1, q)) K45((e1, q))

]T

−

[K21((e1, q)) K25((e1, q))

K41((e1, q)) K45((e1, q))

]Aq1 +

[K22((e1, q)) K24((e1, q))

K42((e1, q)) K44((e1, q))

],

K12(q) = −ATq1

[K13((e1, q))

K53((e1, q))

]+

[K23((e1, q))

K43((e1, q))

]= K21(q)T ,

K13(q) = −ATq1

[K16((e1, q))

K56((e1, q))

]+

[K26((e1, q))

K46((e1, q))

]= K31(q)T ,[

K22(q) K23(q)

K32(q) K33(q)

]=

[K33((e1, q)) K36((e1, q))

K63((e1, q)) K66((e1, q))

].

Here,

Aq1 =1

l − c sin(ψ1)

−lc cos(ψ1) −lb cos(ψ1)

0 0

−c sin(ψ1) −b sin(ψ1)

−2lc/b −(l + c sin(ψ1))

.As a result, we can calculate the 2-form Ξ by (4.3.27)

Ξpq = pTFL−1q

−ATq1

[K11((e1, q)) K15((e1, q))

K51((e1, q)) K55((e1, q))

]+

[K21((e1, q)) K25((e1, q))

K41((e1, q)) K45((e1, q))

][K31((e1, q)) K35((e1, q))

K61((e1, q)) K65((e1, q))

]

−b(c− l sin(ψ1))

0

−b cos(ψ1)

−2c cos(ψ1)

l

(l − c sin(ψ1))2dψ1 ∧ dψ2 = Υ(q, p)dψ1 ∧ dψ2,

where p is the vector of generalized momenta in the reduced space. Finally, in matrix


form we have the following reduced equations of motion for the nonholonomic multi-body

system under study:

[˙q˙p

]=

04×4 id4

−id4

0 Υ(q, p) 0 0

−Υ(q, p) 0 0 0

0 0 0 0

0 0 0 0

∂H∂q∂H∂p

,

where H is calculated by (4.3.26), with V (q) = 0.

4.6.1 Further Reduction of the System

In this subsection we investigate if the system under study demonstrates any conserved

quantity due to the action of a bigger symmetry group (bigger than Q1 × Q5). In this

case study, since originally FLq is independent of ϕ1, the Hamiltonian H is invariant

under the cotangent lift of the action of N = Q3 by left translation. Using the joint

parameters, for any ϕ1,0 we have the action of N on T ∗Q defined as

T ∗Φϕ1,0(ψ1, ψ2, ϕ1, ϕ2, pψ1 , pψ2 , pϕ1 , pϕ2) = (ψ1, ψ2, ϕ1 + ϕ1,0, ϕ2, pψ1 , pψ2 , pϕ1 , pϕ2),

where (ψ1, ψ2, ϕ1, ϕ2, pψ1 , pψ2 , pϕ1 , pϕ2) is a coordinate for T ∗Q, which can be considered

as the reduced space of joint parameters and their corresponding momenta. Also, it is

easy to check that ∀ζ ∈ Lie(N ),

ιζT∗Q

Ξ = 0,

where based on the definition of the cotangent lifted action of N , defined above, we have

φq =[0 0 1 0

]T,

ζT ∗Q =

[φq(ζ)

04×1

]=[0 0 ζ 0 0 0 0 0

]T.

As a result, we have that the momentum map Mq : T ∗q Q → Lie∗(N ) for the N -action and

corresponding to the 2-form Ωcan is conserved along the solution curves of the reduced

system. Here, the momentum map is defined by

Mq =[0 0 1 0

].


We have a principal bundle π : Q → Q = Q2 ×Q4 ×Q6 with the (mechanical) principal

connection Aq : TqQ → Lie(N )

Aq =1

l − c sin(ψ1)

[c sin(ψ1) b sin(ψ1) l − c sin(ψ1) 0

].

For a regular value of the momentum map ϑ ∈ Lie∗(N ), the coadjoint isotropy group

Nϑ = N , and the level set of the momentum map is

M−1

(ϑ) =

(ψ1, ψ2, ϕ1, ϕ2, pψ1 , pψ2 , pϕ1 , pϕ2) ∈ T ∗Q∣∣∣ pϕ1 = ϑ

⊂ T ∗Q.

Also, we have

αϑ =ϑ

l − c sin(ψ1)(c sin(ψ1)dψ1 + b sin(ψ1)dψ2 + (l − c sin(ψ1))dϕ1) ∈ Ω1(Q),

and hence,

Ξϑ =ϑbl cos(ψ1)

(l − c sin(ψ1))2dψ1 ∧ dψ2 ∈ Ω2(T ∗Q).

We then can calculate the map ϕ−1ϑ : T ∗Q → M

−1(ϑ)/N by

ϕ−1ϑ (ψ1, ψ2, ϕ2, pψ1 , pψ2 , pϕ2) = (ψ1, ψ2, ϕ2, pψ1 +

ϑc sin(ψ1)

l − c sin(ψ1), pψ2 +

ϑb sin(ψ1)

l − c sin(ψ1), pϕ2).

As a result, we determine the 2-forms Ξϑ ∈ Ω2(T ∗Q):

Ξϑ = Υ(q, p)dψ1 ∧ dψ2,

where

Υ(q, p) := Υ(ψ1, ψ2, 0, ϕ2, pψ1 +ϑc sin(ψ1)


ϑb sin(ψ1)

l − c sin(ψ1), ϑ, pϕ2).

Finally, we have the reduced equations of motion in T ∗Q as:

[˙q˙p

]=

03×3 id3

−id3

0 Υ(q, p) + ϑbl cos(ψ1)

(l−c sin(ψ1))2 0

−Υ(q, p)− ϑbl cos(ψ1)(l−c sin(ψ1))2 0 0

0 0 0

∂Hϑ∂q∂Hϑ∂p

,


where

Hϑ(q, p) = H(ψ1, ψ2, 0, ϕ2, pψ1 +ϑc sin(ψ1)


ϑb sin(ψ1)

l − c sin(ψ1), ϑ, pϕ2).

Chapter 5

Concurrent Control of Free-base,


This chapter presents a unified geometric approach to output tracking control of under-

actuated holonomic and nonholonomic open-chain multi-body systems with multi-d.o.f.

joints, and constant (non necessarily zero) momentum. We focus our attention on the

case of holonomic and nonholonomic open-chain multi-body systems with symmetry,

whose joints corresponding to the symmetry group are unactuated. The immediate ap-

plications of this case are free-base space manipulators with non-zero total momentum

and mobile manipulators. We first formalize the control problem, and rigorously state

an output tracking problem for such systems in Problem 5.1.10 in Section 5.1.2. Then,

in Section 5.2 we introduce a geometrical definition of the end-effector pose and velocity

error. The main contribution of this chapter is brought in Section 5.3, where we solve for

the input-output linearization of the highly nonlinear problem of coupled manipulator

and base dynamics with non-zero momentum subject to holonomic and nonholonomic

constraints. This enables us to design a coordinate-independent linear controller, such

as a proportional-derivative with feed-forward, for concurrently controlling a free-base,

open-chain multi-body system with non-zero momentum and under holonomic and/or

nonholonomic constraints, which is discussed in Section 5.4. Finally, by defining a Lya-

punov function we prove in Theorem 5.4.2 that the closed-loop system is exponentially

stable. A detailed case study in Section 5.5 concludes this chapter.

5.1 Problem Statement

In this section we formally state an output trajectory tracking control problem for free-

base, open-chain multi-body systems with multi-d.o.f. holonomic (non-zero momentum)

144

Chapter 5. Concurrent Control of Multi-body Systems 145

and nonholonomic joints. The output of such systems is usually the pose of the extremi-

ties and the base. More generally, one may be interested in controlling only parts of the

motion of the extremities and the base. For example, in the case of a free-floating space

manipulator, one may want to control the position of the end-effector and the orientation

of the base. In this case, the output manifold of the open-chain multi-body system is a

quotient manifold, which is locally identified by a submanifold of the space of all possible

poses of the end-effector and the base. In this chapter, by output manifold we mean a

submanifold of the smooth manifold that consists of all possible poses of the extremities

and the base of a free-base, open-chain multi-body system.

5.1.1 Mathematical Formalization and Assumptions

Let a holonomic or nonholonomic open-chain multi-body system with symmetry be de-

noted by (T ∗Q,Ωcan, H,K,G) or (T ∗Q,Ωcan,D, H,K,G), where in the holonomic case

G = Q1 and G = G × N for a nonholonomic system. Also, we denote the number of

independent control directions by nc; it is equal to dim(Q)− dim(G).

CON1) We assume that the multi-body system is invariant under the action of G, in the

sense introduced in Chapter 3 and 4.

This assumption implies that in the holonomic case the open-chain multi-body system

has a conserved momentum. Note that in this case the constant momentum does not

have to be zero. As for the nonholonomic open-chain multi-body system, we assume that

it is a Chaplygin system, and it possibly has a conserved quantity. We recall that control

directions are modelled by 1-forms on Q.

CON2) We also assume that there is no control input collocated with the G orbits for a

holonomic or nonholonomic multi-body system with symmetry. That is, for a set of

linearly independent differential 1-forms Ti ∈ Ω1(Q)| i = 1, · · · , nc corresponding

to the directions of the (available) control inputs (in the form of control force or

torque), ∀ξ ∈ Lie(G) we have

ιξQTi = 0. i = 1, · · · , nc (5.1.1)

This condition guarantees that there is no actuator at the first joint for a holonomic

open-chain multi-body system, and also the actuators of a nonholonomic open-chain

multi-body system do not overlap with the directions of the nonholonomic constraints.


Definition 5.1.1. A holonomic or nonholonomic open-chain multi-body system with

symmetry is called free-base, if we have the assumption CON2, i.e., the joints corre-

sponding to the symmetry group G are unactuated.

Corollary 5.1.2. Let the codistribution F := spanC∞(Q) Ti ∈ Ω1(Q)| i =

1, · · · , nc annihilate the distribution spanC∞(Q) ξQ ∈ X(Q)| ξ ∈ Lie(G). This distri-

bution is involutive and it specifies a global foliation of Q whose leaves are the orbits G(embedded submanifolds of Q), since the G-action is free and proper. Based on Frobenius

theorem [43], for any point there exists an open neighbourhood U ⊂ Q there exist nc exact

1-forms Ui|U = dhi (i = 1, · · · , nc), where hi ∈ C∞(U), such that we have

F|U = spanC∞(U)

Ui ∈ Ω1(U)

∣∣Ui = dhi, i = 1, · · · , nc. (5.1.2)

We say that the codistribution F is integrable, if its corresponding distribution on TQ is

integrable.

Proposition 5.1.3. The 1-forms Ui ∈ Ω1(U) for all i = 1, · · · , nc are locally G invariant.

Proof. This result follows immediately from

LξQUi = d(ιξQUi) + ιξQdUi = ιξQddhi = 0, i = 1, · · · , nc

since based on the previous corollary we have Ui|U = dhi.

CON3) From now on, we assume that (globally) there exist nc closed 1-forms Ui’s whose

span is the codistribution F , and we use them as a basis for F . As a result, the

Ui’s are basic 1-forms with respect to the G action.

In a case of free-base manipulator, CON3 means that there exist sufficient number of

control directions that are independent of the full (holonomic) or partial (nonholonomic)

pose of the base. We specify a set of n2c smooth functions yij ∈ C∞(Q)| i, j = 1, · · · , nc

by Ui =∑nc

j=1 yijTj.

Definition 5.1.4. Under the assumption CON3, for a free-base holonomic or nonholo-

nomic open-chain multi-body system with symmetry, we call (T ∗Q,Ωcan, H,K,G, Uinci=1)

or (T ∗Q,Ωcan, H,K,D,G, Uinci=1), respectively, a controlled multi-body system with sym-

metry.

Let ui ∈ C2(T ∗Q× R)| i = 1, · · · , nc be a set of twice differentiable functions on

the extended phase space (by the time direction). We define the control input for a


controlled multi-body system with symmetry by

nc∑i=1

uiUi. (5.1.3)

We write the control Hamilton’s equation for a controlled multi-body system with

symmetry as

ιXCΩcan = dH −f∑s=1

κsT∗πQ(ωs) +

nc∑i=1

uiT∗πQ(Ui), (5.1.4)

ωs(TπQ(XC)) = 0 s = 1, · · · , f

where T ∗πQ : T ∗Q → T ∗(T ∗Q) is the induced map on the cotangent bundles by the

canonical projection map of the cotangent bundle πQ : T ∗Q → Q. Here, ωs’s are the

constraint 1-forms defining the nonholonomic distribution D and κs’s are the Lagrange

multipliers.

Remark 5.1.5. Note that this equation is a generic Hamilton’s equation, which applies

to both holonomic and nonholonomic multi-body systems, and in the holonomic case

ωs ≡ 0.

Remark 5.1.6. Also, note that the vector field XC in this equation is a time-dependent

vector field on T ∗Q, since the ui’s are functions of time.

We are interested in controlling the motion of the extremities (the base of the multi-

body system is also considered as one of the extremities) of a multi-body system with

symmetry, in certain directions. Let ne be the number of extremities of a controlled multi-

body system with symmetry, and let FKi : Q1 × · · · × Qi0 → SE(3) (for i = 1, · · · , ne)be the forward kinematics maps for the extremities, defined by

FKi(q1, · · · , qi0) = q1 · · · qi0r0i0,0, i = 1, · · · , ne (5.1.5)

where Bi0 is the ith extremity (base body is considered as the first extremity, i.e., 10 = 1),

r0i0,0∈ P0

∼= SE(3) is the initial pose of the ith extremity with respect to the inertial

coordinate frame. Note that only the elements of the relative configuration manifolds

of the joints that are in the path of bodies connecting B0 to Bi0 are involved in the

above equation (see Section 3.3.1). For the ith extremity, we identify it by the embedded

submanifold Ri ⊆ SE(3), which corresponds the directions of motion of Bi0 that we

are interested in. By FK : Q → R := R1 × · · · × Rne we denote the collection of


FKi composed with the projection maps ri : SE(3) → Ri, such that we have ri ιRi =

idRi . Here, ιRi : Ri → SE(3) is the canonical inclusion map, and idRi indicates the

identity map on Ri. We also denote the induced projection map that projects Pe :=ne−times︷︸︸︷

SE(3)× · · · × SE(3) to R by r : Pe → R. The manifold R is called the output manifold,

and no := dim(R) indicates the dimension of the output.

Note that in general the projection map r is not defined globally. In the cases where

this projection does not make sense globally, we define it in a tubular neighbourhood in

Pe around the submanifold R. When we are working in an appropriate local coordinates

for Pe, the projection map r can be considered as a projection to an affine subspace of

the Euclidean space.

Consider a curve in the output manifold γ : R → R corresponding to the desired

motion of the extremities of a controlled multi-body system with symmetry.

CON4) It is always assumed that the curve t 7→ γ(t) is a feasible trajectory for a controlled

holonomic or nonholonomic open-chain multi-body system. That is, it respects the

nonholonomic constraints and the momentum conservation, and also it is in the

image of the forward kinematics map FK with a full rank Jacobian.

CON5) We also assume that the number of control inputs nc is greater than or equal to

the dimension of the output manifold, i.e., D = dim(Q) ≥ nc ≥ no.

Remark 5.1.7. Condition CON5 together with the fact that the control directions are

linearly independent guarantee local controllability of a controlled holonomic or nonholo-

nomic open-chain multi-body system with symmetry at the configurations away from the

singularities of the Jacobian introduced in Section 5.3 [72]. As a result, in the following

we assume that a controlled holonomic or nonholonomic open-chain multi-body system

with symmetry is always away from singular configurations.

Problem 5.1.8 (Control Problem). Let (T ∗Q,Ωcan, H,K,D,G, Uinci=1) be a controlled

multi-body system with symmetry, and let γ : R→ R be a desired motion of its extremi-

ties. Find a set of twice differentiable functions ui ∈ C2(T ∗Q× R)| i = 1, · · · , nc, such

that the output FK(q(t)) tracks the curve γ with an exponentially decreasing error. We

can formulate the controlled system as

Controlled System: ιXCΩcan = dH −f∑s=1

κsT∗πQ(ωs) +

nc∑i=1

ui

nc∑j=1

yijT∗πQ(Tj),

Nonholonomic Constraints: ωs(TπQ(XC)) = 0, s = 1, · · · , f (5.1.6)

Output: FK(q(t)) = (r1(q1(t)), r2(q1(t) · · · q20(t)), · · · , rne(q1(t) · · · qne0(t))).


Note that we defined the problem for a nonholonomic open-chain multi-body system with

symmetry. For the holonomic case, we have no constraint 1-form ωs in (5.1.6).

This problem is not precise yet, since we have not defined the error, nor the expo-

nential stability on manifolds. After reformulating the problem in the following section,

we rigorously define the error in Section 5.2 and the exponential stability in Definition

5.4.1.

5.1.2 Reduced Hamilton’s Equation and Reconstruction

In this section, we use the reduction theories developed in the previous chapters and

their corresponding reconstruction equations to reformulate Problem 5.1.8, in the re-

duced phase space. One of the premises of this section is to introduce a notation to treat

holonomic and nonholonomic cases at the same time. We denote a reduced holonomic

(nonholonomic) open-chain multi-body system by (S ⊆ T ∗Q, Ω, H, K), where S is the

reduced phase space, which is a vector sub-bundle of T ∗Q, Ω ∈ Ω2(S) is the (almost)

symplectic 2-form on the reduced phase space, H : S → R is the reduced Hamiltonian,

and ∀q ∈ Q we have the induced metric Kq : TqQ × TqQ → R on the reduced config-

uration manifold Q. The reduced Hamilton’s equation for the reduced holonomic or

nonholonomic multi-body system (S, Ω, H, K) reads

ιXΩ = dH, (5.1.7)

where for the holonomic case this equation is equivalent to (3.2.26) or (3.3.44), and

for the nonholonomic case it is (4.2.19) or (4.5.38). In order to control the extremities

of a controlled multi-body with symmetry in the inertial coordinate frame, not only

we need the reduced Hamilton’s equation but also the equations corresponding to the

reduced parameters of the system. The process of recovering these equations is called

reconstruction.

For a holonomic open-chain multi-body system with symmetry, where G = Q1, the

reconstruction yields the velocity of B1 (base) with respect to the inertial coordinate

frame and expressed in the coordinate frame attached to B1 (body velocity):

Tq1Lq−11

(q1) = K−1

11 (q)Ad∗χµ(q1)(µ)− Aq q, (5.1.8)

where µ ∈ Lie∗(G) is the constant momentum of the system, K11(q) is defined in Lemma

3.3.4, χµ : Uµ → Q1 is the embedding map corresponding to the local trivialization of

the principal bundle Q1 → Q1/Gµ introduced in Section 3.3, and q1 ∈ Uµ ⊆ Q1/Gµ.


For a nonholonomic open-chain multi-body system with symmetry, where G = G×Nas defined in Section 4.3, the reconstruction leads to the body velocities corresponding

to the joints involved in the symmetry group N , and also body velocity of the base.

TnLn−1(n) = (KN1 (q1, q′))−1Ad∗(e,n)(ϑ)−

[A(q1,q′) B(q1,q′)

] [ ˙q1

q′

], (5.1.9)

ThLh−1(h) = −Aq1 ˙q1, (5.1.10)

where ϑ ∈ Lie∗(N ) is the constant momentum corresponding to the joints involved in

the symmetry group N , e ∈ Nϑ ⊆ Q is the identity element, KN1 (q, q′), A(q,q′) and B(q,q′)

are defined in Section 4.5. In the local trivialization, we have n = (k, n) ∈ Nϑ × Uϑ ⊆Nϑ × N /Nϑ, and Aq1 is defined in (4.3.25). For the detailed description of the maps

and variables appear in (5.1.8), (5.1.9) and (5.1.10), we refer the reader to the previous

chapters.

Remark 5.1.9. Note that there is an overlap between the reduced Hamilton’s equation

in µ-shape space and ϑ-shape space for holonomic and nonholonomic multi-body systems,

respectively, and the reconstruction equations. In the design of the controller for either

systems, we will ignore the dynamical equations corresponding the speed of the elements

of Q1/Gµ or N /Nϑ. One can use this overlap to design an estimator for some uncertain

parameters of the system, which is out of the scope of this thesis.

Based on the assumption CON3 and Proposition 5.1.3, Ui’s are basic with respect

to the G action, and they can drop to the reduced configurations space Q/G. The

reduced 1-forms Ui ∈ Ω1(Q/G) are uniquely identified by T ∗π(Ui) = Ui for i = 1, · · · , nc.Here, π : Q → Q/G is the canonical projection map corresponding to the G action, and

T ∗π : T ∗(Q/G) → T ∗Q is the induced map on the cotangent bundles. We call the five

tuple (S, Ω, H, K,Uinci=1

) the reduced controlled multi-body system. As a result, we can

redefine Problem 5.1.8.

Problem 5.1.10. Let (S, Ω, H, K,Uinci=1

) be the reduced controlled multi-body sys-

tem, and let γ : R → R be a desired motion of the extremities of the original con-

trolled multi-body system with symmetry. Find a set of twice differentiable functionsui ∈ C2(G × S × R)

∣∣ i = 1, · · · , nc

, such that the output FK(q(t)) tracks the curve γ

with an exponentially decreasing error. We can reformulate the controlled multi-body


system as

Controlled System: ιXC Ω = dH +nc∑i=1

uiT∗πQ(Ui),

Reconstruction Equation: (5.1.8) or (5.1.9) and (5.1.10)

Output: FK(q(t)) = (r1(q1(t)), r2(q1(t) · · · q20(t)), · · · , rne(q1(t) · · · qne0(t))).

Here, πQ : T ∗Q → Q is the canonical projection map for the cotangent bundle T ∗Q, and

XC ∈ X(S) is a time dependent vector field. Now, let [Ω]] : T S → T ∗S be the vector

bundle map naturally associated to the non-degenerate 2-form Ω ∈ Ω2(S), such that

∀vpq , wpq ∈ Tpq(S) we have

Ω(vpq , wpq) = [Ω](pq)(vpq , wpq) =⟨[Ω]](pq)(vpq), wpq

⟩.

Then, we can write the above equations in a coordinate chart for a controlled holonomic

multi-body system as

Controlled System: [Ω]]

˙q1

q

p

=

∂H∂q1∂H∂q∂H∂p

+

0

u

0

,Reconstruction Equation: Adχµ(q1)−1ThLh−1(h) + Tq1(Lχµ(q1)−1 χµ)( ˙q1)

= K−1

11 (q)Ad∗χµ(q1)(µ)− Aq q, (5.1.11)

Output: FK(q(t)) = (r1(q1(t)), r2(q1(t) · · · q20(t)), · · · , rne(q1(t) · · · qne0(t))),

where u is the control input in the given coordinate chart, and h ∈ Gµ ⊆ G is the element

of the isotropy group corresponding to µ ∈ Lie∗(G), such that q1 = hχµ(q1).

And, for a controlled nonholonomic multi-body system we have

Controlled System: [Ω]]

˙q1

˙n

q′

˙p1

p′

=

∂Hϑ∂q1∂Hϑ∂n∂Hϑ∂q′

∂Hϑ∂p1

∂Hϑ∂p′

+

u1

0

u′

0

0

,

Reconstruction Equations: Adχϑ(n)−1TkLk−1(k) + Tn(Lχϑ(n)−1 χϑ)( ˙n)

= (KN1 (q1, q′))−1Ad∗χϑ(n)(ϑ)−

[A(q1,q′) B(q1,q′)

] [ ˙q1

q′

],


ThLh−1(h) = −Aq1 ˙q1, (5.1.12)

Output: FK(q(t)) = (r1(q1(t)), r2(q1(t) · · · q20(t)), · · · , rne(q1(t) · · · qne0(t))),

where u1 and u′ are the control input in the given coordinate chart, and h ∈ G ⊆ Q1

is an element of the subgroup of Q1, the embedding χϑ : Uϑ ⊆ N /Nϑ → N corresponds

to the local trivialization such that n = kχϑ(n) for the element k ∈ Nϑ, and Nϑ ⊆ N is

the isotropy group corresponding to ϑ ∈ Lie∗(N ). The equations (5.1.11) and (5.1.12)

formally define the control problem in the reduced phase space.

5.2 End-effector Pose and Velocity Error

5.2.1 Error Function

In this section we introduce a quadratic error function, based on an induced metric on

the output manifold R from a left-invariant metric on Pe. This error function represents

the distance between the actual output and the desired output of the system in the am-

bient manifold Pe. Different methods of defining error functions and their corresponding

gradients are discussed in Bullo’s thesis [12]. The definition of the error function adopted

in this thesis is due to its geometrical interpretation. But, the following development can

readily be applied to other definitions of the error function.

Definition 5.2.1. A smooth two variable function Er : R × R → R≥0 is a symmetric

error function on R, if ∀r1, r2 ∈ R we have Er(r1, r2) ≥ 0 and we have equality if and

only if r1 = r2.

Let ιR : R → Pe be the inclusion map, and let Ki (for i = 1, · · · , ne) be an arbitrary

left invariant Riemannian metric on SE(3) corresponding to the ith extremity. These

metrics induce a left invariant metric K := K1⊕ · · · ⊕Kne on Pe. Consider two elements

r1, r2 ∈ R and the one-parameter subgroup σ : R → Pe in Pe that connects ιR(r1) to

ιR(r2). We define the distance between r1 ∈ R and r2 ∈ R by the length of the portion

of σ that connects ιR(r1) ∈ Pe to ιR(r2) ∈ Pe in the ambient manifold. That is,

dis(r1, r2) =

∫ 1

0

√Kσ(s)

(dσ(s)

ds,dσ(s)

ds

)ds σ(0) = ιR(r1), σ(1) = ιR(r2)

=

∫ 1

0

√KeP

(σ−1(s)

dσ(s)

ds, σ−1(s)

dσ(s)

ds

)ds,


where s ∈ R is the curve parameter for σ, and eP is the identity element of Pe. Since

one-parameter subgroups are the integral curves of left invariant vector fields, we can

uniquely write the curve as

σ(s) = ιR(r1) exp(s exp−1(re)

),

where re := ιR(r1)−1ιR(r2) that is called output pose error. Consequently, we have

σ−1(s)dσ(s)

ds= exp−1(re),

which is a constant vector in Lie(Pe). As a result, we can simplify the above equation

dis(r1, r2) =√KeP (exp−1(re), exp−1(re)) =‖ exp−1(re) ‖KeP ,

where ‖ · ‖KeP is the induced norm on Lie(Pe) by the left invariant metric K. This length

is also equal to the length of the one-parameter subgroup that connects eP ∈ Pe to re.

It is easy to show that the error function defined by

Er(r1, r2) =1

2dis(r1, r2)2 =

1

2‖ exp−1(re) ‖2

KeP(5.2.13)

is a quadratic, smooth, symmetric error function on R.

Remark 5.2.2. Although it would have been natural to define the length of the geodesic

corresponding to the induced metric on R by K as the error function, the error function

defined by 5.2.13 is more efficient computationally.

We denote the exterior derivative of the error function with respect to the first and

second input of the function by d1 and d2, respectively. Also, let us use KP for the self-

adjoint positive definite map between Lie(Pe) and Lie∗(Pe) corresponding to the induced

metric on the Lie algebra. Then, for all vr1 ∈ Tr1R and wr2 ∈ Tr2R we have the following

equations:

〈d1Er(r1, r2), vr1〉 =1

2

⟨d1KeP

(exp−1(re), exp−1(re)

), vr1

⟩= −KeP

(exp−1(re),Θ(re)

−1AdιR(r2)−1(TιR(r1)RιR(r1)−1)(Tr1ιR)(vr1))

=⟨−KP exp−1(re),Θ(re)

−1AdιR(r2)−1(TιR(r1)RιR(r1)−1)(Tr1ιR)(vr1)⟩

=⟨−(T ∗r1ιR)(T ∗ιR(r1)RιR(r1)−1)Ad∗ιR(r2)−1(Θ(re)

−1)∗KP exp−1(re), vr1⟩,

(5.2.14)

〈d2Er(r1, r2), wr2〉 =1

2

⟨d2KeP

(exp−1(re), exp−1(re)

), wr2

⟩


= KeP(exp−1(re),Θ(re)

−1(TιR(r2)LιR(r2)−1)(Tr2ιR)(wr2))

=⟨KP exp−1(re),Θ(re)

−1(TιR(r2)LιR(r2)−1)(Tr2ιR)(wr2)⟩

=⟨(T ∗r2ιR)(T ∗ιR(r2)LιR(r2)−1)(Θ(re)

−1)∗KP exp−1(re), wr2⟩, (5.2.15)

where d1Er(r1, r2) ∈ T ∗r1R and d2Er(r1, r2) ∈ T ∗r2R. The map Θ(re) : Lie(P) → Lie(P)

defined by

Θ(re) =

∫ 1

0

Adexp(s exp−1(re))ds

is the linear map that appears in the tangent map of the exponential of Lie groups, which

was defined in the proof of Theorem 2.1.5 and calculated by (2.4.23) in Proposition 2.4.5

for SE(3). This map is invertible in an open neighbourhood of the identity element.

From now on, we always assume that re = ιR(r1)−1ιR(r2) belongs to a symmetric neigh-

bourhood of identity such that Θ(re) is invertible.

5.2.2 Velocity Error

The notion of connection on manifolds is usually used to linearly identify the tangent

spaces of the manifold at different base points, along a path. It is also used to define

the derivative of a vector field along a curve on the manifold, and consequently defining

the notion of parallel transport along a curve. A well-known example of connections on

manifolds is the affine connection on Riemannian manifolds. For Lie groups, one can

use the tangent to left or right translation maps to globally define a well-defined (trivial)

connection on Lie groups. In this way, we identify the tangent space at each element of

the Lie group with the Lie algebra of the Lie group. In this section we use the induced

connection on R by the right translation map of the Lie group Pe to define the output

velocity error of a controlled multi-body system.

For any r′1, r′2 ∈ Pe, let Tr′2R(r′2)−1r′1

: Tr′2Pe → Tr′1Pe be the tangent map of the right

translation by (r′2)−1r′1 that is a linear isomorphism between the tangent spaces Tr′1Pe and

Tr′2Pe. Recall that the canonical inclusion map and the projection map for the output

manifold R are denoted by ιR : R → Pe and r : Pe → R, respectively. We define the

linear isomorphism <(r1,r2) : Tr2R → Tr1R by

<(r1,r2) = (TιR(r1)r)(TιR(r2)RιR(r2)−1ιR(r1))(Tr2ιR) = (TιR(r1)r)(TιR(r2)Rr−1e

)(Tr2ιR),

(5.2.16)

which is a well-defined connection for any r2 ∈ R in a neighbourhood of r1 ∈ R. Note

that unlike the tangent map of the right translation on a Lie group, which globally defines

the trivial connection on a Lie group, the connection that is defined in (5.2.16) can only


make sense, locally. The size of the neighbourhood of r1, in which the above map is a

linear isomorphism, depends on the given projection map r.

Definition 5.2.3. Let γ1 : R → R and γ2 : R → R be two curves, and t ∈ R be their

curve parameter. We call

Ve(t) =d

dtγ1(t)−<(γ1(t),γ2(t))(

d

dtγ2(t)) =: γ1(t)−<(γ1(t),γ2(t))(γ2(t)) (5.2.17)

the output velocity error of a system.

Note that unlike the case of systems on linear spaces, where the velocity error can be

simply defined by subtracting the velocity of curves, in the case of systems on manifolds,

we need the notion of connection to define the velocity error. In the next section, we use

this notion to design control laws for controlled multi-body systems with symmetry.

Definition 5.2.4. A connection is called compatible with the error function, if ∀r1, r2 ∈ Rthe following equality holds [12]:

d2Er(r1, r2) = −<∗(r1,r2)d1Er(r1, r2). (5.2.18)

Corollary 5.2.5 ([12]). Let γ1 : R→ R and γ2 : R→ R be two curves. If the connection

is compatible with the error function, then

d

dtEr(γ1(t), γ2(t)) = 〈d1Er(γ1(t), γ2(t)), Ve(t)〉 . (5.2.19)

Proposition 5.2.6. If R is the right translation of a Lie subgroup of Pe by an element

r′0 ∈ Pe, then the connection in (5.2.16) is compatible with the error function in (5.2.13).

Proof. It is easy to check that for all r1, r2 ∈ R the connection <(r1,r2) is compatible with

Er(r1, r2), if ∀vr2 ∈ Tr2R we have

(TιR(r2)RιR(r2)−1ιR(r1)

)Tr2ιR(vr2) ∈ Tr1ιR(Tr1R),

since the tangent of the projection map T r is identity on Tr1ιR(Tr1R).

We claim that the above condition is satisfied if R = Gr′0, where G is a Lie subgroup

of SE(3). Let g1, g2 ∈ G such that r1 = g1r′0 and r2 = g2r

′0. Then, we have

Tr1R = (Tg1G)r′0 = Lie(G)g1r′0 = Lie(G)r1.

Similarly, it is easy to show that Tr2R = Lie(G)r2. As a result, ∀vr2 ∈ Tr2R there exists

a ξ ∈ Lie(G), such that vr2 = ξr2. If we apply the connection in (5.2.16) to this vector,


we get

vr2r−12 r1 = ξr1 ∈ Lie(G)r1 = Tr1R,

which proves our claim.

CON6) From now on, we assume that R ⊆ Pe is a Lie subgroup of Pe. Note that any

statement in the rest of this chapter also holds for any right translation of Lie

subgroups of Pe.

To simplify our notation, from now on, we do not use the inclusion map ιR to show

elements of the Lie subgroup R in Pe, whenever it does not result any confusion.

5.3 Input-output Linearization and Inverse Dynam-

ics in the Reduced Phase Space

In this section, we present an input-output linearization process for the reduced dynamics

of controlled open-chain multi-body systems with symmetry, based on left trivialization

of the tangent bundles of Q and SE(3). This process is useful for deriving an output

tracking feed-forward PD (proportional-derivative) controller for such systems, which is

the subject of the next section.

Consider the Jacobian maps for the extremities that map the joint velocities to the

twist of the extremities with respect to the inertial coordinate frame and expressed in

the body coordinate frames (attached to the extremities). We may left trivialize the

tangent bundle of Q, and denote the resulting Jacobian maps by J0i : Q×Lie(Q)→ se(3)

(i = 1, · · · , ne), which are calculated by

J01 : =

(TFK1(q)LFK1(q)−1

)TqFK1 (Te1Lq1) (Te1ι1) = Ad(r0

1,0)−1

[Te1ι1 0

],

J0i : =

(TFKi(q)LFKi(q)−1

)TqFKi

(Te1Lq1 ⊕ · · · ⊕ Tei0Lqi0

)(Te1ι1 ⊕ · · · ⊕ Tei0 ιi0)

= Ad(r0i0,0

)−1

[Ad(q2···qi0 )−1Te1ι1 · · · Tei0 ιi0

],

where ιi : Qi → SE(3) for i = 1, · · · , N are the canonical inclusion maps. These maps

are fibre-wise linear maps that relate the relative twist of the joints to the twist of the

extremities. We denote the collection of J0i ’s by

Jq :=

(J0

1 )q...

(J0ne)q

: Lie(Q)→ Lie(Pe).


Note that the Jacobian maps q 7→ (J0i )q’s and consequently q 7→ Jq are Q1 invariant,

as was detailed in Chapter 1. We then consider the Jacobian maps whose images are

projected to the Lie algebra of the output manifold:

0i := TeP(Lri(FKi(q))−1 ri LFKi(q)

)J0i =: Ei J0

i , i = 1, · · · , ne (5.3.20)

where the fibre-wise linear maps (Ei)q : se(3) → Lie(Ri) are obtained from the dif-

ferentiations of ri : SE(3) → Ri (see page 148) by the left trivialization, and Eq :=

(E1)q ⊕ · · · ⊕ (Ene)q : Lie(Pe) → Lie(R) denotes the collection of them. As a result, we

introduce the fibre-wise linear (Jacobian) map q : Lie(Q)→ Lie(R) by

q :=

(01)q

...

(0ne)q

= Eq Jq.

Consider an initial phase for the controlled multi-body system with symmetry (q0, p0) ∈T ∗Q that specifies an initial phase for the reduced controlled multi-body system (q0, p0) ∈S.

CON7) We assume that these initial conditions respect a pre-chosen constant (non-zero)

momentum of the system, and respect the nonholonomic constraints.

We use the local coordinates to denote the integral curves of the vector fields XC ∈X(T ∗Q) and XC ∈ X(S) (with the above-mentioned initial conditions) by t 7→ (q(t), p(t))

and t 7→ (q1(t), q(t), p(t)), respectively.

CON8) For the nonholonomic case, the integral curve of XC is t 7→ (q1(t), n(t), q′(t), p1(t),

p′(t)), in the local coordinates. However, in order to unify our approach, from

now on, by q1, q and p we mean n, (q1, q′) and (p1, p

′), respectively, in the case of

nonholonomic multi-body systems. We also denote the control input (u1, u′) by u,

in the local coordinates.

In the next step, we restrict the map : Q×Lie(Q)→ Lie(R) to the curve (q(t), τ(t) =

Tq(t)Lq(t)−1 q(t) = Tq(t)Lq(t)−1FL−1q(t)(p(t))

)to get a curve in the Lie algebra of the output

manifold corresponding to the evolution of the relative twists of the extremities. That

is,

t 7→ q(t)(τ(t)) = Eq(t) Jq(t)(τ(t)) =: Eq(t) Jq(t)((τ1(t), · · · , τN(t)) ∈ Lie(R) (5.3.21)


In the holonomic case the first entry of this map τ1(t) = Tq1(t)Lq(t)−1 q1(t) is the relative

twist of the first body with respect to the inertial coordinate frame and expressed in

the body coordinate frame, which is the outcome of the reconstruction equation (5.1.8).

That is,

τ1(t) = K−1

11 (q(t))Ad∗χµ(q1(t))(µ)− Aq(t)q(t). (5.3.22)

Based on (5.1.10), for a nonholonomic open-chain multi-body system with symmetry

we have

τ1(t) = (h(t)χ(q1(t)))−1(h(t)χ(q1(t)) + h(t)Tχ( ˙q1(t))

)= Adχ(q1(t))−1

(h−1(t)h(t)

)+ χ(q1(t))−1Tχ( ˙q1(t))

= −Adχ(q1(t))−1

(Aq1(t)

˙q1(t))

+ χ(q1(t))−1Tχ( ˙q1(t)). (5.3.23)

Also, if the relative configuration manifold of a joint appears in the symmetry group N ,

we have the same equation as in (5.3.22) for the relative twist of the corresponding joint.

Let [Ω]] : T S → T ∗S be the vector bundle map corresponding to the 2-form Ω ∈Ω2(S), in the dynamical equations of a reduced controlled holonomic or nonholonomic

open-chain multi-body system. Consider rearranging of the coordinate variables of non-

holonomic systems based on CON8, then for both the holonomic and nonholonomic case

the momenta in the reduced phase space S are denoted by the coordinates pi. According

to the notation of (3.2.27) and (Eq.Ham.Non.Coor) we have

ι∂/∂piΞµ = 0,

ι∂/∂pi(Ξϑ − Ξϑ) = 0.

As a result, it is easy to check that ∀(q, p) ∈ Sq in the reduced phase space [Ω]]((q, p))

has the following form:

[Ω]](q, p) =

[Ω]]11(q, p) [Ω]]12(q, p) 0

−([Ω]]12(q, p))T [Ω]]22(q, p) −id0 id 0

,and the inverse of it ([Ω]])−1 : T ∗S → T S has the following form:

([Ω]](q, p))−1 =

([Ω]]11)−1 0 −([Ω]]11)−1[Ω]]12

0 0 id

−([Ω]]12)T ([Ω]]11)−1 −id [Ω]]22 + ([Ω]]12)T ([Ω]]11)−1[Ω]]12

.


Therefore, we can write the speed of the integral curve of the controlled holonomic or

nonholonomic open-chain multi-body system as: ˙q1

q

p

= ([Ω]](q(t), p(t)))−1

∂H∂q1

∂H∂q

+ u∂H∂p

=

([Ω]]11)−1 0 −([Ω]]11)−1[Ω]]12

0 0 id

−([Ω]]12)T ([Ω]]11)−1 −id [Ω]]22 + ([Ω]]12)T ([Ω]]11)−1[Ω]]12

∂H∂q1

∂H∂q

+ u∂H∂p

=

([Ω]]11)−1 ∂H

∂q1−(

([Ω]]11)−1[Ω]]12

)∂H∂p

∂H∂p

−(

([Ω]]12)T ([Ω]]11)−1)∂H∂q1− ∂H

∂q− u+

([Ω]]22 + ([Ω]]12)T ([Ω]]11)−1[Ω]]12

)∂H∂p

.(5.3.24)

Based on the first two equations in (5.3.24) we can write

˙q1(t) = ([Ω]]11)−1∂H

∂q1

−(

([Ω]]11)−1[Ω]]12

) ∂H∂p

=: q(q(t), p(t)), (5.3.25)

q(t) =∂H

∂p=: q(q(t), p(t)), (5.3.26)

By substituting these equations in (5.3.22) and (5.3.23) (and any reconstruction equation

due to the second step of the reduction of nonholonomic systems) we obtain the following

relations for holonomic and nonholonomic cases, respectively:

τ1(t) = K−1

11 (q(t))Ad∗χµ(q1(t))(µ)− Aq(t)q(q(t), p(t)).

and

τ1(t) = −Adχ(q1(t))−1

(Aq1(t)q1((q(t), p(t)))

)+ χ(q1(t))−1Tχ(q1(q(t), p(t))),

where q1 : S → T (Q1/G) specifies the components of XC ∈ X(S) in T (Q1/G) as a

portion of the components of q. In general, we can write τ(t) =: τ(q(t), p(t)), where

the function τ : S → Lie(Q) is defined based on (5.3.25), (5.3.26) and reconstruction

equations. Therefore, the curve in (5.3.21) can be rewritten as

t 7→ q(t) τ(q(t), p(t))) ∈ Lie(R),


where we have τ : G × S → Lie(R). Taking the derivative of this curve with respect

to time, we obtain a curve in TLie(R) ∼= Lie(R):

t 7→ d

dt

(q(t) τ(q(t), p(t))

)=

(∂q(t)∂q

q(t)

)τ(q(t), p(t)) + q(t)

(∂τ

∂q˙q(t) +

∂τ

∂p˙p(t)

)=

(∂q(t)∂q

(TeLq(t)τ(q(t), p(t))

))τ(q(t), p(t)) + q(t)

(∂τ

∂q˙q(t) +

∂τ

∂p˙p(t)

)=

(∂q(t)∂q


))τ(q(t), p(t))

+ q(t)

(∂τ

∂q

[q(q(t), p(t))

q(q(t), p(t))

]+∂τ

∂p(p(q(t), p(t))− u)

)∈ Lie(R),

(5.3.27)

where the last line is the consequence of substituting (5.3.25) and (5.3.26) in the equation,

and the map p is defined based on the last equation of (5.3.24):

p(q(t), p(t)) = −(

([Ω]]12)T ([Ω]]11)−1) ∂H∂q1

− ∂H

∂q+(

[Ω]]22 + ([Ω]]12)T ([Ω]]11)−1[Ω]]12

) ∂H∂p

.

(5.3.28)

Equation (5.3.27) is the input-output linearized form in the reduced phase space of a

generic holonomic or nonholonomic open-chain multi-body system with multi-d.o.f. joints

and non-zero momentum. The input-output linearization method presented in this sec-

tion generalizes different approaches to the input-output linearization of underactuated,

holonomic and nonholonomic multi-body systems used, e.g., in [2, 5, 28, 34, 60, 61],

to derive nonlinear control laws. Equation (5.3.27) holds for any holonomic open-chain

multi-body system with non-abelian symmetry group and non-zero momentum, and also

it holds for Chaplygin systems with underactuated joints.

In (5.3.27), u is going to be designed such that the output of the controlled holonomic

or nonholonomic open-chain multi-body system follows the desired trajectory t 7→ γ(t).

As a result, we solve the inverse dynamics problem for a free-base, open-chain multi-body

system with symmetry by equating the curve in (5.3.27) and ddt

(γ−1(t)γ(t)):

d

dt(r−1(t)r(t)) =

d

dt

(q(t) τ(q(t), p(t))

)=

d

dt

(γ−1(t)γ(t)

).

Now, we use the last line of (5.3.27) to find the solution for the inverse dynamics problem,

by solving for u in the reduced phase space.

u(q(t), p(t), γ(t), γ(t), γ(t)) =

(q(t)

∂τ

∂p

)−1((∂q(t)∂q


))τ(q(t), p(t))


+q(t)∂τ

∂q

[q(q(t), p(t))

q(q(t), p(t))

]− d

dt

(γ−1(t)γ(t)

))+ p(q(t), p(t)).

(5.3.29)

Note that (5.3.29) matches with equation (17) in [28] in a special case where the total

momentum of the system is equal to zero and there is no nonholonomic constraints at

the base. Also, the formulation in [28] is based on a specific parametrization of the

output manifold of the system. Therefore, (5.3.29) can be considered as a coordinate-

independent generalization of the inverse dynamics solution in the reduced phase space

for non-zero momentum subject to holonomic or nonholonomic constraints.

Remark 5.3.1. The matrix q(t)∂τ∂p

is square if the dimension of the control codistribution

F is equal to the dimension of the output manifold, i.e., no = nc. In case no < nc, we

can either choose u amongst all possible solutions by, for example, optimizing a function

along the trajectories of the system, or use a pseudo inverse matrix in the above equation.

Remark 5.3.2. Note that since the Legendre transformation is invertible for the re-

duced open-chain multi-body system with symmetry, the matrix ∂τ∂p

is always full rank.

Furthermore, for a feasible desired trajectory t 7→ γ(t) the Jacobian q(t) is also full rank.

Therefore, the inverse dynamics problem in the reduced phase space has a unique solu-

tion u (or the matrix q(t)∂τ∂p

is invertible), if no = nc and the desired trajectory t 7→ γ(t)

is feasible.

CON9) In the next section, we assume that the dimension of the output manifold is equal to

the number of control inputs of a controlled holonomic or nonholonomic open-chain

multi-body system, i.e., no = nc.

5.4 An Output-tracking Feed-forward

Servo Controller

In this section, under the dimensional assumption CON9 and the feasibility of the desired

trajectory t 7→ γ(t), we develop an output tracking feed-forward servo controller for a

generic open-chain multi-body system with symmetry. The system can include multi-

d.o.f. joints and can be subject to holonomic or nonholonomic constraints. Also, for the

holonomic case the total momentum of the system can be non-zero. In this process, we use

the definition of the error function and velocity error of the system output introduced in

Section 5.2. Consequently, we show that the developed controller exponentially stabilizes

the closed-loop system using Lyapunov function t 7→ VL(r(t), r(t), γ(t), γ(t)) ∈ R.


Definition 5.4.1 ([12]). Let t 7→ r(t) := FK(q(t)) ∈ R denote the output of a controlled

holonomic or nonholonomic, open-chain multi-body system, and let t 7→ γ(t) ∈ R be a

feasible desired output trajectory. The desired trajectory γ

i) is Lyapunov stable with Lyapunov function t 7→ VL(r(t), r(t), γ(t), γ(t)) ∈ R, if

VL(t) ≤ VL(0) from all initial conditions (r(0), r(0)).

ii) is exponentially stable with Lyapunov function t 7→ VL(r(t), r(t), γ(t), γ(t)) ∈ R, if

there exist two positive constants δ1 and δ2 such that VL(t) ≤ δ1VL(0)eδ2t from all

initial conditions (r(0), r(0)).

Theorem 5.4.2. Consider the controlled holonomic or nonholonomic, open-chain multi-

body system in (5.1.11) or (5.1.12), and let the curve t 7→ γ(t) ∈ R be a twice differ-

entiable feasible trajectory in the output manifold that satisfies the assumption CON4.

Also, let Er : R × R → R≥0 be the error function in (5.2.13) and < be its compat-

ible connection (assuming CON6), defined by (5.2.16). Let KP : Lie(R) → Lie∗(R),

KD : Lie(R) → Lie∗(R) and I : Lie(R) → Lie∗(R) be self-adjoint positive-definite ten-

sors, such that I induces a norm on Lie(R) denoted by ‖ · ‖I. Under the condition

CON9 the control input in the reduced phase space is

u(q, p, γ, γ, γ) =

(q∂τ

∂p

)−1((

∂q∂q

(TeLq τ(q, p))

)τ(q, p) + q

∂τ

∂q

[q(q, p)

q(q, p)

]− ν

)+ p(q, p),

(5.4.30)

where we have the control law as:

ν = νPD + νFF , (5.4.31)

νPD = −I−1T ∗ePLr(t) (d1Er(r, γ))− I−1KDve(r, γ, r, γ), (5.4.32)

νFF = ad(r−1r)Ad(r−1γ)

(γ−1γ

)+ Ad(r−1γ)

(d

dt

(γ−1γ

))(5.4.33)

where r(t) := FK(q(t)) ∈ R is the output of the system, and

ve(r, γ, r, γ) := r−1r − r−1<(r,γ)(γ) = r−1r − Ad(r−1γ)(γ−1γ) (5.4.34)

is the left translated output velocity error to Lie(R). Then, the desired trajectory t 7→ γ(t)

is Lyapunov stable with the Lyapunov function VL : R→ R≥0:

VL(t) = Er(r, γ) +1

2〈Ive(r, γ, r, γ), ve(r, γ, r, γ)〉 . (5.4.35)


Further, the desired trajectory t 7→ γ(t) is exponentially stable with Lyapunov function

VL from all initial conditions, such that we have VL(0) < W 2R. Here, WR is the length

of the radius of an open ball in Lie(R) with respect to the norm induced by I, where the

exponential map is a diffeomorphism.

Proof. In order to show Lyapunov stability of the desired trajectory, we have to show

that the time derivative of a candidate Lyapunov function is always less than or equal

to zero. We choose the Lyapunov function to be (5.4.35), and we start with the time

derivative of the error function. Based on CON6 and Corollary 5.2.5, we have

d

dtEr(r, γ) = 〈d1Er(r, γ), Ve〉 =

⟨T ∗ePLr(t) (d1Er(r, γ)) , ve

⟩. (5.4.36)

The time derivative of the second term in (5.4.35) is also calculated as follows:

d

dt

(1

2〈Ive, ve〉

)=

⟨Ive,

d

dtve

⟩=

⟨Ive,

d

dt

(r−1r − r−1<(r,γ)(γ)

)⟩=

⟨Ive,

d

dt(q τ)− d

dt

(r−1<(r,γ)(γ)

)⟩=

⟨Ive, ν −

d

dt

(r−1(γ)γ−1r

)⟩=

⟨Ive, νPD + νFF −

d

dt

(Ad(r−1γ)(γ

−1γ))⟩

=

⟨Ive, νPD + νFF − ad(r−1r)Ad(r−1γ)

(γ−1γ

)+ Ad(r−1γ)

(d

dt

(γ−1γ

))⟩= 〈Ive, νPD〉

=⟨Ive,−I−1T ∗ePLr(t) (d1Er)− I−1KDve

⟩=⟨−T ∗ePLr(t) (d1Er)−KDve, ve

⟩. (5.4.37)

By adding (5.4.36) and (5.4.37), we calculate the time derivative of VL as:

dVLdt

(t) = −〈KDve, ve〉 ,

which is always less than or equal to zero due to the fact that KD is positive definite.

This proves the Lyapunov stability of the feasible desired trajectory.

In order to show the exponential stability, we need to add a term to VL and define a


new Lyapunov function VL : R→ R≥0:

VL(t) := Er(r, γ) +1

2〈Ive(r, γ, r, γ), ve(r, γ, r, γ)〉+ δ

d

dtEr(r, γ),

where we have to find δ > 0 such that VL is positive definite, i.e., VL is greater than

or equal to zero and it is zero if and only if Er(r, γ) = 0 and ve(r, γ, r, γ) = 0. Let

re(t) = r(t)−1γ(t):

VL(t) = Er +1

2〈Ive, ve〉+ δ

⟨T ∗ePLr (d1Er) , ve

⟩=

1

2

⟨KP exp−1(re), exp−1(re)

⟩+

1

2〈Ive, ve〉+ δ


⟩=

1

2

⟨KP exp−1(re), exp−1(re)

⟩+

1

2〈Ive, ve〉 − δ

⟨Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re), ve

⟩.

From the proof of Lyapunov stability of the closed loop t 7→ γ(t), we have

VL(t) ≤ VL(0) =: W0 =⇒ Er(r, γ) ≤ W0, ‖ ve(r, γ, r, γ) ‖I≤ W0.

We consider the induced norm by KeP and I on the space of all automorphisms of Lie(R),

as a vector space. For any linear map = : Lie(R)→ Lie(R), this induced norm is defined

by

‖ = ‖I := max ‖ =ξ ‖I | ξ ∈ Lie(R), ‖ ξ ‖I= 1 ,

for I, and similarly we can define the norm, which is induced by KeP . Since the er-

ror function is bounded by the Lyapunov stability and re ∈ R is assumed to be in a

neighbourhood of the identity where Θ(re) is invertible, ∀t ∈ R we have the following

bounds:

‖ Adr−1e‖I ≤ sup

‖ Adr−1

e‖I∣∣ re ∈ R, ‖ exp−1(re) ‖KeP≤

√W0

= W1,

‖ Θ(re)−1 ‖I ≤ sup

‖ Θ(re)

−1 ‖I∣∣ re ∈ R, ‖ exp−1(re) ‖KeP≤

√W0

= W2.

As the result of these bounds,

VL(t) ≥ 1

2‖ exp−1(re) ‖2

KeP+

1

2‖ ve ‖2

I −δ ‖ exp−1(re) ‖KeP ‖ Θ(re)−1Adr−1

eve ‖I

≥ 1

2‖ exp−1(re) ‖2

KeP+

1

2‖ ve ‖2

I

− δ ‖ Adr−1e‖I‖ Θ(re)

−1 ‖I‖ exp−1(re) ‖KeP ‖ ve ‖I

≥ 1

2‖ exp−1(re) ‖2

KeP+

1

2‖ ve ‖2

I −δW1W2 ‖ exp−1(re) ‖KeP ‖ ve ‖I


=1

2

[‖ exp−1(re) ‖KeP

‖ ve ‖I

]T [1 −δW1W2

−δW1W2 1

][‖ exp−1(re) ‖KeP

‖ ve ‖I

]

=:1

2


‖ ve ‖I

]TW


‖ ve ‖I

],

where, W is a positive definite matrix if 0 < δ < 1√W1W2

. Therefore, for any δ in this

range, VL is a well-defined Lyapunov function.

Now, to calculate the time derivative of VL(t) we only need to take the derivative of

the term ddtEr(r, γ):

d

dt

(d

dtEr(r, γ)

)=

d

dt


⟩= − d

dt

⟨Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re), ve

⟩= − d

dt

⟨KP exp−1(re),Θ(re)

−1Adr−1eve⟩

= −⟨KP

d

dt

(exp−1(re)

),Θ(re)

−1Adr−1eve

⟩−⟨KP exp−1(re),

d

dt

(Θ(re)

−1Adr−1eve)⟩

.

In the following, we calculate the terms appeared in the above equation.⟨KP

d

dt

(exp−1(re)

),Θ(re)

−1Adr−1eve

⟩=⟨KPΘ(re)

−1TePLr−1ere,Θ(re)

−1Adr−1eve⟩

= −⟨KPΘ(re)

−1Adr−1eve,Θ(re)

−1Adr−1eve⟩

= − ‖ Θ(re)−1Adr−1

eve ‖2

KeP, (5.4.38)

since we have

re =d

dt

(r−1γ)

)= −r−1rr−1γ + r−1γ = −

(r−1r − r−1γγ−1r

)re

= −TePRre

(r−1r − r−1<(r,γ)(γ)

)= −TePRre(ve).

And,

d

dt

(Θ(re)

−1Adr−1eve)

=

(∂Θ−1

∂rere

)Adr−1

e(ve)−Θ−1Adr−1

eadrer−1

e(ve) + Θ−1Adr−1

e(ve)

=

(∂Θ−1

∂re(−vere)

)Adr−1


eadve(ve) + Θ−1Adr−1

e(νPD)

= −(∂Θ−1

∂re(vere)

)Adr−1

e(ve) + Θ−1Adr−1

e(νPD)

= −(∂Θ−1

∂re(vere)

)Adr−1


e

(I−1T ∗ePLr (d1Er) + I−1KDve

)


= −(∂Θ−1

∂re(vere)

)Adr−1

e(ve)

+ Θ−1Adr−1e

(I−1Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re)− I−1KDve

),

(5.4.39)

which is the result of the following equalities:

re = −vere,

ve = νPD,

d

dtAdr−1

e(η) =

d

dt

(r−1e ηre

)= −r−1

e rer−1e ηre + r−1

e ηre = r−1e

(ηrer

−1e − rer−1

e η)re

= −Adr−1e

adrer−1e

(η),

where in the last equality η is an element of Lie(R). Therefore, by (5.4.38) and (5.4.39)

we have

d

dt

(d

dtEr(r, γ)

)= ‖ Θ(re)

−1Adr−1eve ‖2

KeP

+

⟨KP exp−1(re),

(∂Θ−1

∂re(vere)

)Adr−1

e(ve)

⟩−⟨

Ad∗r−1e

(Θ(re)−1)∗KP exp−1(re), I−1Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re)

⟩+⟨KP exp−1(re),Θ

−1Adr−1eI−1KDve

⟩.

Based on the norm equivalence inequality and since we have the bounds ‖ Adr−1e‖I≤ W1

and ‖ Θ(re)−1 ‖I≤ W2, there exists W3 > 0 such that⟨

Ad∗r−1e

(Θ(re)−1)∗KP exp−1(re), I−1Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re)

⟩=‖ Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re) ‖2

I

≥ W3 ‖ KP exp−1(re) ‖2KeP

= W3 ‖ exp−1(re) ‖2KeP

,

(5.4.40)

wherever we have an element of Lie∗(R) inside the norm we mean the naturally induced

norm by a metric on Lie∗(R). Also, we have the following inequalities:

‖ Θ(re)−1Adr−1

eve ‖2

KeP≤ W4 ‖ ve ‖2

I , (5.4.41)⟨KP exp−1(re),

(∂Θ−1

∂re(vere)

)Adr−1

e(ve)

⟩≤ W5 ‖ ve ‖I‖ exp−1(re) ‖KeP , (5.4.42)


⟨KP exp−1(re),Θ

−1Adr−1eK−1P KDve

⟩≤ W6 ‖ ve ‖I‖ exp−1(re) ‖KeP , (5.4.43)

where W4,W5,W6 > 0 are three positive real numbers. The inequalities in (5.4.40),

(5.4.41), (5.4.42) and (5.4.43) yields to

d

dt

(d

dtEr(r, γ)

)≤

− 1

2


‖ ve ‖I

]T [2W3 −(W5 +W6)

−(W5 +W6) −2W4


‖ ve ‖I

]

=: −1

2


‖ ve ‖I

]TW


‖ ve ‖I

],

where W is a symmetric matrix. As a result, we have

d

dtVL(t) ≤

− 1

2


‖ ve ‖I

]T [2δW3 −δ(W5 +W6)

−δ(W5 +W6) −2(δW4 −W6)


‖ ve ‖I

]

=: −1

2


‖ ve ‖I

]TW


‖ ve ‖I

]. (5.4.44)

It is easy to check that if

0 < δ <4W3W6

(W5 +W6)2 + 4W3W4

,

then W is a symmetric positive-definite matrix. Therefore, for any positive δ less than

the smaller of 1√W1W2

and 4W3W6

(W5+W6)2+4W3W4VL is a Lyapunov function and (5.4.44) holds.

Until this step, we have proved the asymptotic stability of the desired feasible trajectory

t 7→ γ(t) ∈ R.

In the final step, we show that in fact this trajectory is exponential stable. Consider

the Lyapunov function VL for an appropriate δ. We have

VL(t) ≤ 1

2‖ exp−1(re) ‖2

KeP+

1

2‖ ve ‖2

I +δW1W2 ‖ exp−1(re) ‖KeP ‖ ve ‖I

=1

2


‖ ve ‖I

]T [1 δW1W2

δW1W2 1


‖ ve ‖I

]


=:1

2


‖ ve ‖I

]TW ′[‖ exp−1(re) ‖KeP

‖ ve ‖I

],

whereW ′ is a symmetric positive definite matrix. Using the norm equivalence inequality,

there exists a positive number δ such that

d

dtVL(t) ≤ −1

2


‖ ve ‖I

]TW


‖ ve ‖I

]

≤ −δ2


‖ ve ‖I

]TW ′[‖ exp−1(re) ‖KeP

‖ ve ‖I

]≤ −δ VL(t).

Based on this inequality, VL(t) ≤ VL(0)e−δt. Consequently, there exists a positive number

δ such that

VL(t) =1

2


‖ ve ‖I

]T [1 0

0 1


‖ ve ‖I

]

≤ δ

2


‖ ve ‖I

]TW


‖ ve ‖I

]≤ δVL(t) ≤ δVL(0)e−δt ≤ 2δVL(0)e−δt,

where the last inequality holds due to the fact that VL(0) ≤ 2VL(0).

Remark 5.4.3. Theorem 5.4.2 presents an output tracking, feed-forward PD controller

in the reduced phase space that exponentially stabilizes the closed-loop generic holonomic

or nonholonomic multi-body system. The control input is a function of the joint displace-

ments, including those of the unactuated joints eliminated in the reduction process, and

the velocities of the actuated joints.

Remark 5.4.4. The controller input in (5.4.30) is in the reduced space. In order to

find the control input for the original system in (5.1.6), we have to first express u in the

reduced basis for the control directions, i.e.Uinci=1

, then we have to lift the result to

the original phase space of the system. That is,

u =:nc∑i=1

uiUi =nc∑i=1

uiUi,

where Ui = T ∗π(Ui).

Figure 5.4 depicts the block diagram of the proposed control scheme. In this diagram,


g is an element of the symmetry group G, and r′, r′ correspond to the actual motion of

the robot. Also, we have

νP = −I−1T ∗ePLr(t) (d1Er(r, γ)) = I−1Ad∗r−1e

(Θ(re)−1)∗KP exp−1(re),

νD = −I−1KDve(r, γ, r, γ) = −I−1KD(r−1r − Adre(γ

−1γ)),

νFF = ad(r−1r)Adre(γ−1γ

)+ Adre

(d

dt

(γ−1γ

))= ad(r−1r)Adre

(γ−1γ

)+ Adre

(γ−1γ − (γ−1γ)(γ−1γ)

)

In Theorem 5.4.2, we introduce a feed-forward PD controller at the output of a generic

controlled open-chain multi-body system. We used the group structure of the output

manifold to define the pose and velocity error for the extremities, and consequently, to

construct this controller. As a result, the controller, i.e., ν, is dependent on the group

structure of the output manifold. For this controller, the behaviour of the closed-loop

system can be presented in the form of the following set of coupled differential equations:

d

dt(r−1r) = νFF + νP + νD

γ−1γ − (γ−1γ)(γ−1γ)

γ−1γ

re = r−1γ

νFF

νD

νP

∑u(g, q, p, ν) ROBOT

FK(g, q)

(g, q, p)

(5.1.8)OR

(5.1.9) & (5.1.10)

p(q1, q, q)

q1

γ

γ

γ

g

q q

r

r′

r′

r−1r

Figure 5.1: Feed-forward servo control for a generic free-base, open-chain multi-bodysystem


=d

dt(Adre(γ

−1γ))− Ad∗r−1e

(Θ(re)−1)∗KP exp−1(re)−KD

(r−1r − Adre(γ

−1γ)),

where we assume that the linear map I : Lie(R) → Lie∗(R), which is used to define

Lyapunov function, is the identity matrix in a basis for the Lie algebra of R and its dual.

This simplification helps illustrating the behaviour of the closed-loop system. By some

manipulation, we get the following set of coupled differential equations for the output

error (re = r−1γ):

d

dt(rer

−1e ) +KD(rer

−1e ) + Ad∗

r−1e

(Θ(re)−1)∗KP exp−1(re) = 0.

Now by appropriately choosing the self-adjoint tensors KP and KD, we can achieve a

desired performance of the closed-loop system.

If the output manifold of the system is an abelian subgroup of Pe, the above differential

equation can be simplified to the familiar second order linear differential equation. Then

by choosing diagonal matrices for KP and KD we can decouple the differential equations

representing the behaviour of the closed-loop system, and we can explicitly design the

controller to achieve any desired performance of the system. In this case, the output

error is simply the difference between the desired and actual output of the system, and

the maps Adre , exp−1 and Θ(re) are the identity maps. Therefore, we have

d2

dt2(r − γ) +KD

d

dt(r − γ) +KP (r − γ) = 0.

This situation occurs, for example, when we want to control the position of the extremities

without considering their orientation. This idea is illustrated in the next section, where we

derive the developed controller for the example of a three-d.o.f. manipulator mounted on

top of a two-wheeled differential rover. Note that in case we are interested in controlling

the orientation of the extremities, e.g., orientation of the base body of a free-floating

manipulator, then the controller design is not as simple as the presented case study.

5.5 Case Study

In this section, we derive the control law presented in Theorem 5.4.2 for the example

of a three-d.o.f. manipulator mounted on top of a two-wheeled, differential-drive rover

(Example 4.6.1 in Section 4.6). In this example, we assume that the two wheels of

the rover and the three joints of the manipulator are actuated. We consider an output

manifold R = R2 × R3 ⊂ SE(3)× SE(3) that corresponds to the position of the centre


of mass of the rover and the centre of mass of the end-effector with respect to the

inertial coordinate frame and expressed in the same coordinate frame. Note that R is

a subgroup of Pe = SE(3) × SE(3). Using the local coordinates defined in Example

4.6.1, the forward kinematics maps for the extremities of the controlled nonholonomic,

open-chain multi-body system are

FK1(x, y, θ) = q1rcm,1 =

cos(θ) − sin(θ) 0 x+ l0 cos(θ)

sin(θ) cos(θ) 0 y + l0 sin(θ)

0 0 1 0

0 0 0 1

,

FK2(x, y, θ, ϕ1, ϕ2, ϕ3) = q1q4q5q6rcm,6 =:

[RE pE

01×3 1

],

where RE(x, y, θ, ϕ1, ϕ2, ϕ3) ∈ SO(3) and pE(x, y, θ, ϕ1, ϕ2, ϕ3) ∈ R3 specify the pose

of the end-effector with respect to the inertial coordinate frame and expressed in the

same coordinate frame. In this case study, the projection maps r1 : SE(3) → R2 and

r2 : SE(3) → R3 are simply projection to the position components of the poses of the

rover and the end-effector, respectively. These projection maps are defined globally;

accordingly the output of the system is calculated in the local coordinates as

FK(x, y, θ, ϕ1, ϕ2, ϕ3) = (r1 FK1(q), r2 FK2(q))

= (x+ l0 cos(θ), y + l0 sin(θ), pE(x, y, θ, ϕ1, ϕ2, ϕ3)).

Denote the output of the controlled open-chain multi-body system by r(t) = FK(x(t),

y(t), θ(t), ϕ1(t), ϕ2(t), ϕ3(t)), and consider a desired feasible trajectory t 7→ γ(t) ∈ R.

Since the output manifold is an abelian subgroup of Pe, the output pose error is just

re(t) = γ(t) − r(t), and considering KP = diag(K1P , · · · ,K5

P ) (the proportional gain) as

the self-adjoint positive-definite map between Lie(R) ∼= R and Lie∗(R) ∼= R, the error

function is defined by Er(r(t), γ(t)) = 〈KP re(t), re(t)〉. Note that the exponential map

restricted to the abelian subgroup R ⊂ Pe is the identity map, and it is everywhere

invertible. The compatible connection with this error function is the identity map, and

the output velocity error is simply calculated by Ve = ve = r(t) − γ(t). We denote the

derivative gain by KD = diag(K1D, · · · ,K5

D), which is a diagonal matrix with positive

diagonal elements.


Next, we calculate the Jacobian maps:

J01 =

id3 −

l000

03 id3

1 0 0 0 0 0

0 1 0 0 0 0

0 0 0 0 0 1

T

=

1 0 0 0 0 0

0 1 0 0 0 0

0 l0 0 0 0 1

T

J02 = Adr−1

cm,6

[Ad(q4q5q6)−1Te1ι1 Ad(q5q6)−1Te4ι4 Adq−1

6Te5ι5 Te6ι6

],

where

Te1ι1 =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 0 0 0 1

T

,

Te2ι2 =[0 0 0 0 1 0

]T,

Te3ι3 =[0 0 0 0 1 0

]T,

Te4ι4 =[0 −l0 − l1 0 0 0 1

]T,

Te5ι5 =[0 0 −l2 1 0 0

]T,

Te6ι6 =[0 0 −l2 − l3 1 0 0

]T.

And accordingly we have

01 =

[cos(θ) − sin(θ) −l0 sin(θ)

sin(θ) cos(θ) l0 cos(θ)

]

02 =

RE −RE

˜ l0 + l1

l2 + l3 + l4

0

[Ad(q4q5q6)−1Te1ι1 Ad(q5q6)−1Te4ι4 Adq−1

6Te5ι5 Te6ι6

],

q =

[[(01)q 02×3

](02)q

].

Consider an initial phase (q(0), p(0)) for the controlled nonholonomic open-chain multi-

body system under study that satisfies the condition CON7. It induces an initial phase

(q(0), p(0)) in the reduced phase space T ∗Q. We denote the integral curve for the reduced

system by t 7→ (q(t), p(t)). As a result, the controlled Hamilton’s equation in the reduced


phase space is:

[˙q˙p

]=

05×5 id5

−id5

[

0 Υ(q, p)

−Υ(q, p) 0

]02×3

03×2 03×3

[∂H∂q

+ u∂H∂p

].

From this equation and the reconstruction equation we have

˙q =: q(q, p) =:

[q1(q, p)

q(q, p)

]=

[∂H∂p1

∂H∂p

]= FL−1

q p,

p(q, p) : =

[

0 Υ(q, p)

−Υ(q, p) 0

]02×3

03×2 03×3

FL−1q p− ∂H

∂q,

τ1 = −Aq1 q1(q, p),

where p = (p1, p) = ((pψ1 , pψ2), (pϕ1 , pϕ2 , pϕ3)) are the momenta corresponding to q =

(q1, q) = ((ψ1, ψ2), (ϕ1, ϕ2, ϕ3)) in the reduced phase space, and

Aq1 =

−b/2 −b/20 0

b/(2c) −b/(2c)

.In the following, we calculate the components of the nonlinear control law stated in

(5.4.30). Since Q is an abelian group, TqLq−1 is the identity map, and we have

τ(t) =: τ(q(t), p(t)) =

[−Aq1(t)q1(q(t), p(t))

q(q(t), p(t))

]=

[−Aq1(t) 03×3

03×2 id3

]FL−1

q(t)p,

∂τ

∂p=

[−Aq1(t) 03×3

03×2 id3

]FL−1

q ,

∂q∂q

(TeLq τ(q, p)) = −∂q∂x

[cos(θ) − sin(θ) 0

]Aq1 q1(q, p)

− ∂q∂y

[sin(θ) cos(θ) 0

]Aq1 q1(q, p)− ∂q

∂θ

[0 0 1

]Aq1 q1(q, p)

+∂q∂ϕ1

∂H

∂pϕ1

+∂q∂ϕ2

∂H

∂pϕ2

+∂q∂ϕ3

∂H

∂pϕ3

.


As a result, the following control law exponentially stabilizes the output of the system

for any feasible desired trajectory t 7→ γ(t):

u(q, p, γ, γ, γ) = FLq(t)

(q

[−Aq1(t) 03×3

03×2 id3

])−1((∂q∂q

(TeLq τ)

)[−Aq1(t) 03×3

03×2 id3

]FL−1

q p

+q∂τ

∂qFL−1

q p− ν)

+ p,

such that for this case study we have

ν = νPD + νFF ,

νPD = νP + νD = −KP (r(t)− γ(t))−KD(r(t)− γ(t)),

νFF = γ(t)

where r(t) := FK(q(t)) ∈ R is the output of the system. Here, we have chosen I to

be an identity matrix for the standard basis of Lie(R) ∼= R5 and Lie∗(R) ∼= R5, and

KD = diag(K1D, · · · ,K5

D) is a diagonal matrix with positive diagonal elements, as defined

above. In this case study, we can explicitly write the differential equation that governs

the behaviour of the closed-loop system:

d2

dt2(r − γ) +KD

d

dt(r − γ) +KP (r − γ) = 0,

∑u(q1, q, p, ν) ROBOT

p = FLq( ˙q)

(q1, q, p)

FK(q1, q)

q1 = RZ(θ)Aq1˙q1KD

KP

q1

q1

˙q q

r

r

γ

γ-+

γ-+

r′

r′

Figure 5.2: Servo controller for concurrent control of a three-d.o.f. manipulator mountedon a two-wheeled rover


where γ, r ∈ R5, and by choosing KP and KD diagonal we decouple this differential

equation. As a result, we can choose the diagonal elements of KP and KD such that

the closed-loop system, in this case, becomes decoupled with a desired performance.

Consequently, the gains KiP ’s and KiD’s can be design so that the system error dynamics

will have a desired behaviour in each channel. Also, note that the feed-forward function,

in this case, becomes gain one. The complete block diagram of the closed-loop system is

shown in Figure 5.5. In this figure, RZ(θ) ∈ SO(3) corresponds to the principal rotation

about the Z axis for θ radian.

Chapter 6

Conclusions

In this thesis we studied different aspects of open-chain multi-body systems from a geo-

metrical point of view. We started with the kinematic modelling of such systems, where

we mostly focused on the exponential parametrization of the configuration manifold.

Then, by using relevant tools in geometric mechanics, such as principal connections, we

introduced reduction methods for the dynamical equations of holonomic and nonholo-

nomic open-chain multi-body systems. The asset of our treatment is to unify and extend

the existing dynamical reduction methods for such systems, and especially to include the

dynamical reduction of holonomic multi-body systems with non-zero momentum. Fi-

nally, the input-output linearization problem was solved in the reduced phase space for a

generic holonomic or nonholonomic open-chain multi-body system. As a result, we pro-

posed a coordinate-independent, feed-forward servo control law for concurrent trajectory

tracking of the base and other extremities of such systems, which makes the closed-loop

system exponentially stable. We summarize the main contributions of this thesis in the

forthcoming section, and state some future directions of this research afterwards.

6.1 Summary of Contributions

An extension of the product of exponentials formula for Forward and Differential Kine-

matics of generic open-chain multi-body systems with multi-d.o.f., holonomic and non-

holonomic joints was formalized in Chapter 2. Towards this goal, we classified multi-

d.o.f. joints, and introduced the notion of displacement subgroups. It was shown that

the relative configuration manifolds of such joints are Lie subgroups of SE(3), and the ex-

ponential map is surjective for all types of displacement subgroups except for one type.

Accordingly, we defined the screw joint parameters, and formalized their relationship

with the classic joint parameters. We then considered the nonholonomic constraints in

176

Chapter 6. Conclusions 177

the Pfaffian form on displacement subgroups, and by introducing admissible screw joint

speeds we modified the Jacobian of an open-chain multi-body system. The proposed gen-

eralized exponential formulation for forward and differential kinematics is independent of

the intermediate coordinate assignment to the bodies, the choice of the joint parametriza-

tion and the choice of the basis for the Lie algebra of the configuration manifold. We

explored the computational aspects of the developed formulation in Section 2.5 through

an example, where forward and differential kinematics of a mobile manipulator mounted

on a spacecraft were calculated.

Chapter 3 presents an extension of the reduction procedures for free-base multi-body

systems to more general cases with non-zero momentum and multi-d.o.f. holonomic

joints. We used the symplectic reduction theorem in geometric mechanics to express

Hamilton’s equation in the symplectic reduced manifold, for holonomic Hamiltonian me-

chanical systems. We then identified the symplectic reduced manifold with the cotangent

bundle of a quotient manifold. Accordingly, we developed a reduction process for the

dynamical equations of open-chain multi-body systems with non-zero momentum and

multi-d.o.f. holonomic joints, for which one symmetry group is the relative configuration

manifold corresponding to the first joint. Finally, we derived the reduced dynamical

equations in the local coordinates for an example of a six d.o.f. manipulator mounted on

a spacecraft in Section 3.4 to illustrate the results of this chapter.

In Chapter 4, we developed a reduction method to reduce the dynamical equations of

a nonholonomic open-chain multi-body system with symmetry. Through this process, we

considered more general cases of multi-body systems, where there exist holonomic and

nonholonomic displacement subgroups. We used Chaplygin reduction theorem to express

Hamilton’s equation in the cotangent bundle of a quotient manifold. Then, we found some

sufficient conditions, under which the kinetic energy metric is invariant under the action

of a subgroup of the configuration manifold. Accordingly, we extended the Chaplygin

reduction theorem to a three-step reduction process for the dynamical equations of open-

chain multi-body systems with holonomic and nonholonomic displacement subgroups.

To illustrate the results of this chapter, we derived the reduced dynamical equations in

the local coordinates for two examples in Section 4.6.

In Chapter 5, a generic output tracking, feed-forward servo control law was derived

for open-chain multi-body systems with constant (non-zero) momentum and holonomic

and/or nonholonomic multi-d.o.f. joints. We focused on the cases, where there is no

actuation in the directions of the group action and in the direction of nonholonomic con-

straints. The control problem considered in this chapter was to concurrently track the

trajectories of the base and other extremities of an open-chain multi-body system in the


inertial coordinate frame. We used the exponential map of Lie groups and right trivial-

ization of the tangent bundle of Lie groups to define an error function and a connection

on the output manifold. One of the main contributions of this dissertation is unification

of the reduction of holonomic and nonholonomic open-chain multi-body systems. As a

result, we were able to show that the generic class of controlled open-chain multi-body

systems with both holonomic and nonholonomic constraints is input-output linearizable

in the reduced phase space. Then, we solved for the inverse dynamics problem of such

systems, in the reduced phase space. Accordingly, we proposed a unified coordinate-

independent control law. Finally, we proved the exponential stability of the closed-loop

system for a feasible desired trajectory of the extremities (including the base) of a free-

base holonomic or nonholonomic open-chain multi-body system using an appropriate

Lyapunov function, in Theorem 5.4.2.

6.2 Future Work

The future directions of this research can be divided into three main streams, as listed

bellow:

6.2.1 Kinematics

In this thesis, we used Lie group theory to develop a kinematic model of open-chain multi-

body systems with holonomic and nonholonomic multi-d.o.f. joints. In this modelling

process, we identified the relative configuration manifolds of the joints by Lie subgroups

of SE(3). Obviously, a natural way of constructing this kinematic model would have

been using Lie groupoids.

A relevant problem in the study of multi-body systems is inverse kinematics problem.

This problem solves for the trajectory(ies) in the configuration space corresponding to a

desired trajectory in the output manifold. In the case of redundant systems, this problem

has infinitely many solutions, where we can, for example, conduct an optimization to

choose a trajectory in the configuration space. In this thesis, since we designed the

controller in the work-space of a free-base holonomic or nonholonomic open-chin multi-

body system, we did not need to deal with this problem. In the case of designing the

controller in the joint space, we will need to study inverse kinematics problem. Another

future direction of this thesis could be studying this problem specially for redundant

systems.

We have studied the forward and differential kinematics of open-chain multi-body


systems. However, we did not consider the singularities of the resulting Jacobian for

these system. These singularities correspond to the configurations of the system, where

the Jacobian is not full rank. In general, forward kinematics can be a mapping, which

is neither surjective nor injective and not even a submersion or immersion. Studying

the properties of the image of such mapping can help us understand and deal with

singularities in the mapping.

6.2.2 Dynamics

In this thesis, we unified the derivation of Hamilton’s equation for holonomic and nonholo-

nomic Lagrangian systems with hyper-regular Lagrangian, using Hamilton-Pontryagin

and Lagrange-d’Alembert-Pontryagin principles. This method can be generalized to sin-

gular Lagrangian systems, which may appear in robotics where, for example, we have

springs and dampers in the system, using Dirac structure [93, 94].

In addition, we tried to unify the reduction of holonomic and nonholonomic Hamil-

tonian mechanical systems. We have listed the similarities and distinctions between the

existing approaches; however, this phase of research has not been completed yet. This

unification is useful for studying integrable nonholonomic Hamiltonian mechanical sys-

tems, and affine and nonlinear nonholonomic distributions.

Another future direction of this research is to study closed-chain multi-body systems,

where in addition to nonholonomic constraints there are holonomic constraints on the

system due to the existence of kinematical loops in the topology of the system (see Section

3.3.1). Conventionally, the dynamics of closed-chain multi-body systems is studied on a

configuration manifold constructed by integrating these holonomic constraints. Another

possible approach is to consider the additional holonomic constraints as integrable distri-

butions and treat them similar to the nonholonomic distributions. The advantage of this

method is that we can, for example, use the group structure of the configuration mani-

fold of open-chain multi-body systems or the left(right)-invariance of the nonholonomic

distributions.

6.2.3 Controls

As for controls, there exist many practical control problems (other than trajectory track-

ing) for controlled holonomic or nonholonomic open-chain multi-body systems with sym-

metry. For example, studying the relative stability of these systems and using nonholo-

nomic constraints to change the momentum of the system. Furthermore, the trajectory

planing of such systems is challenging. Although in [72] Shen attempts to solve the (lo-


cal) trajectory planning problem for controlled holonomic multi-body systems with zero

momentum, there is no global approach to the trajectory planning problem for generic

controlled holonomic or nonholonomic open-chain multi-body systems with symmetry.

Shen in his thesis uses a series expansion for the integral curves of the controlled vector

fields of the system.

The control law derived in chapter 5 is dependent on the kinematic and dynamic

model of the system. Regarding the dynamic model, we may use the reconstruction

equations to design an estimator for an adaptive control law. To estimate the kinematic

model of the system we cannot use usual adaptive control laws, since the system model is

not linear with respect to the kinematic parameters of the system. In this case, we should

work with the space of diffeomorphisms of the configuration manifold of the system. That

is, we can associate a diffeomorphism of the configuration manifold to a set of kinematic

parameters of the systems. Consequently, we can attempt to derive an estimator for an

adaptive control law that can estimate both the dynamic and kinematic parameters of

the system.

In addition, in this thesis we assumed perfect sensors and environment for an open-

chain multi-body system. As another future direction of this research, we can, for exam-

ple, study the sensitivity of the proposed control law in the presence of noise in sensory

data and disturbance from the environment. As a result, we can find the conditions

under which the proposed controller is robust with respect to noise and disturbance.

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a unified geometric framework for kinematics, dynamics and ... · cially, i would like to mention...

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