formulation of generalized block backstepping control law of underactuated mechanical systems

Design of Control Law for Underactuated Mechanical Systems: A Modif ied Backstepping Approach

byShubhobrata Rudra

Inspire Research FellowElectrical Engineering Department

Jadavpur University

1

Contents

• A few words on the Underactuated Mechanical Systems [UMS]

• Motivations and Contributions of the Present Work

• Generalized Formulation of the Proposed Block-Backstepping

Control Law for Generic Underactuated Mechanical System

• Comparative Analysis with well known Olfati Saber’s Method

• Applications of the Proposed Control Law on different

Underactuated Mechanical Systems

• Conclusions2 2

Definition: underactuated mechanical systems are a special class

of mechanical systems that have fewer actuating

inputs (control inputs) than configuration variables

(outputs from the system).

3

Advantages of Underactuated Mechanical System

• Cost Effective Design

• Backup control algorithm for fully-actuated system

Underactuated Mechanical Systems

• Use of fewer control inputs

• Complicated nonlinear structure

• Lack of Controllability

• Most of the system fails to satisfy Brockett’s condition

of feedback linearization

4

Control problems of the UMS

Important Contributions in the Field of Underactuated Control System

Year of

PublicationTitle of the Publication Main Contribution

1976Control of unstable mechanical system Control of

pendulum [35]

Proposed a control algorithm for swing up and stabilization of

Inverted Pendulum

1990Nonlinear controllers for non-integrable systems: the

Acrobot example [30]

Developed a basic control algorithm for nonholonomic

underactuated system

1991

Control of mechanical systems with second-order

nonholonomic constraints: Underactuated

manipulators [97]

First paper that identify second order nonholonomic constraints

and discussed the control aspects of such system

1993An analysis of the kinematics and dynamics of under-

actuated manipulators [8]

Provided a detail analysis of forward and inverse dynamics of

underactuated manipulators

1994Partial feedback linearization of underactuated

mechanical systems [32]

Devised a feedback control law that partially linearizes any

arbitrary underactuated system

1995Control of vehicles with second-order nonholonomic

constraints: Underactuated vehicles [84]

Proposed control algorithm for higher order underactuated

system with second order nonholonomic constraints

5


1996

Energy based control of a class of underactuated


First Proposed a novel approach of controlling the flow of

energy in an UMS

Hybrid feedback laws for

a class of cascade nonlinear control systems [24]

Proposed the concept of a logic based switching controller

to select the suitable control law between various time-

periodic control functions at discrete-time instants.

Control of underactuated mechanical systems using

switching and saturation [125]

Developed a saturation control algorithm for stabilization of

UMS

1997

Non-smooth stabilization of an underactuated

unstable two degrees of freedom mechanical

system [112]

Proposed the formulation of a nonsmooth control law to

address the control problem of nonholonomic system

1998

Controller design for a class of underactuated

nonlinear systems [20]

Proposed a novel state transformation to convert the state

model of 2 DOF systems into normal form and then apply

the concept of integral backstepping on that.

1999

Dynamics and control of a class of underactuated


Provided a detailed analysis of controllability and

stabilizability of the underactuated mechanical system

Adaptive Variable Structure Set-Point Control of

Underactuated Robots [130]

Devised a model based adaptive control scheme where

the uncertainty bounds only depends on the inertial

parameter of the system 6


2000

Nonlinear control of underactuated

mechanical systems with application to

robotics and aerospace vehicles [14]

Developed a novel coordinate transform that converts the state

model of the underactuated system into Byrens-Isidori normal

form (for both 2-DOF and n-DOF system).

Discontinuous feedback control of a

special class of underactuated mechanical

systems [17]

Devised a generalized discontinuous control law for class of

nonholonomic system with 2nd order motion constraints.

Controlled Lagrangians and the

stabilization of mechanical systems I:

Potential shaping [123]

Devised a kinematic symmetry based stabilization algorithm for

UMS. Modification to the Lagrangian of uncontrolled system is

achieved by means of the energy shaping technique.

2001

Controlled Lagrangians and the

stabilization of mechanical systems II:

Potential shaping [124]

Extended version of the previous approach to address the control

problem of the system without symmetry.

Stabilization of underactuated mechanical

systems: A nonregular backstepping

approach [19]

At first the state model of the underactuated system transformed

into a chained system. And then the authors applied cascade

backstepping approach on the chained system.

2002

Stabilization of a class of underactuated

mechanical systems via interconnection

and damping assignment [57]

This is basically a passivity based method. The proposed

algorithm generates a smooth asymptotic stabilizing control law to

ensure the local stability of the UMS. 7


2003Position tracking of underactuated

vehicles [118]

Proposed a novel controller that ensures exponential convergence of

Underactuated vehicles to any arbitrary trajectory.

2004

Design of a stable sliding-mode

controller for a class of second-order

underactuated systems [140]

Proposed the concept of hierarchical sliding mode control to address

the control problem of second order underactuated system

2005Global time-varying stabilization of

underactuated surface vessel [85]

Proposed three different time varying control laws to address the

stabilization control problem of USV

2006Optimal sliding mode control for


Based on nonlinear predictive control the authors proposed a novel

design of optimal linear surfaces for sliding model control of

underactuated nonlinear systems.

2007

Trajectory-tracking and path-following

of underactuated autonomous vehicles

with parametric modeling uncertainty

[91]

Proposed an adaptive supervisory control combined with

backstepping control law to address the control problem of UAV with

uncertain parameter.

2008Sliding mode control of a class of


The authors proposed a sliding mode control approach to stabilize a

class of underactuated systems that could be represented in

cascaded form.8

Important Contributions in the Field of Underactuated Control System-

2009Control of a class of underactuated mechanical

systems using sliding modes [144]

Developed a higher order sliding mode control algorithm

to address the stabilization control problem of the

underactuated system

2010

Output feedback control of a quadrotor UAV using

neural networks

[148]

Neural network is introduced to learn the uncertain

dynamics of UAV and then apply a output feedback

control algorithm to stabilize the UAV.

2011

Inverse dynamics of underactuated mechanical

systems: A simple case study and experimental

verification [149]

Studied the dynamic inversion problem for underactuated

mechanical systems

2012Controller design for a class of underactuated


A backstepping-like adaptive controller based on function

approximation technique (FAT) is designed to address the

control problem of UMS

2013Output-feedback stabilization control for a class

of under-actuated mechanical system [150]

An output feedback based backstepping controller is

designed to address the control problem of UMS

(actuated shape variable)

2014

Nonlinear State Feedback Controller design for

Underactuated Mechanical System: a Modified

Block Backstepping Approach [29]

Proposed a novel block-backstepping design approach to

address the control problem of generalized underactuated

systems9

Solutions to the Control Problems of UMS

• By means of some state transformation state model of the

systems could be converted into a new state model that

conforms application of well-known control laws.

• Design a control law for the system in new state space using

advanced control law that will also ensure proper functioning

of the physical system.

• A few popular control algorithms for UMS are feedback

linearization, backstepping, sliding mode control, etc.

10

Important applications of Backstepping Control on Underactuated System

Year of

PublicationTitle of the Publication Main Contributions

1996Nonlinear tracking of underactuated surface vessels

[86]

Proposed a backstepping based output decoupling

controller design for USV

1998

Controller design for a class of underactuated

nonlinear systems [20]

Proposed a novel state transformation to convert the

underactuated system state model into normal form and

then apply the concept of integral backstepping on that.

Global practical stabilization and tracking for an

underactuated ship—A combined averaging and

backstepping approach [88]

Developed a combined averaging and backstepping control

approach to address the tracking and stabilization for USV.

2000Trajectory tracking control design for autonomous

helicopters using a backstepping [135]

Devised a backstepping based controller on the

approximate dynamic model of autonomous helicopter.

2001Stabilization of underactuated mechanical systems:

A non-regular backstepping approach [19]

At first the state model of the underactuated system

transformed into a chained system. And then the authors

have applied cascade backstepping approach on the

chained system.

2002Global tracking control of underactuated ships by

Lyapunov's direct method [111]

Proposed a cascade backstepping approach to address the

control problem of the USV 11


2003 Position tracking of underactuated vehicles [118]

Proposed a novel controller that ensures exponential

convergence of Underactuated vehicles to any arbitrary

trajectory.

2004

Global uniform asymptotic stabilization of an

underactuated surface vessel: Experimental

results [83]

Proposed a smooth time varying state feedback to achieve

the global asymptotic stabilization of USV.

2005Global time-varying stabilization of


Proposed three different time varying control laws to address

the stabilization control problem of USV

2006Asymptotic backstepping stabilization of an


Proposed a natural coordinate transformation that converts

the USV state model into a third order chained system and

then implement a discontinuous feedback control on it.

2007

Trajectory-tracking and path-following of

underactuated autonomous vehicles with

parametric modeling uncertainty [91]

Proposed an adaptive supervisory control combined with

backstepping control law to address the control problem of

UAV with uncertain parameter.

2008

Modeling and Backstepping-based Nonlinear

Control Strategy for a 6 DOF Quadrotor

Helicopter [151]

Proposed a backstepping based pd controller design

approach for 6 DOF helicopter

12


2010Global stabilisation and tracking control of

underactuated surface vessels [90]

Proposed an unified backstepping control design approach

to address the stabilization and exponential tracking control

of USV

2011

Backstepping design for cascade systems with

relaxed assumption on Lyapunov functions

[59]

Use of the feedforoward control with backstepping approach

has been proposed

2012Controller design for a class of underactuated


A backstepping-like adaptive controller based on function

approximation technique (FAT) is designed to address the

control problem of UMS

2013

Output-feedback stabilization control for a

class of under-actuated mechanical system

[150]

An output feedback based backstepping controller is

designed to address the control problem of UMS (actuated

shape variable)

13

A Few Words on Backstepping

• Simplifies the control problem of an n-dimensional

system by treating it as a cascade of n number 1st

order systems.

• Does not cancel useful nonlinearities.

• It requires system representation in controllable

canonical form.

14

Motivations & Contributions of the Present Work

• Motivations:

• An effective way of solving the UMS control problems is to convert the

state model of the UMS into a new state model using some state

transformation.

• However, most of the previously proposed control laws of UMS have

employed complicated state transformation that in turn makes the

control law inapt for practical applications.

• Conversely, some of the application oriented approaches have

resulted in a handy controller for real-time implementation, but have

failed to ensure global asymptotic stability.

• As a matter of fact, scientist and practice engineers are still looking

for a plausible solution to the control problem of the UMS.15

16


Main Contributions of the Present Work

• A simple algebraic state transformation technique has been proposed to

convert the state model of the UMS into a convenient form that is suitable

for backstepping control law design.

• A thorough analysis has been made to find out the condition of global

diffeomorphism for the proposed algebraic state transformation.

• Block-backstepping based control law has been designed to address the

control problem of the generic underactuated mechanical systems.

• Stability of the internal dynamics has thoroughly been analyzed to ensure

the global asymptotic stability of the control law.

17


Main Contributions of the Present Work

• Integral action has been incorporated in the stabilizing function to make

the system partially insensitive with respect to parameter variation and

model uncertainty.

• Proposed control law has been applied on several underactuated systems

to corroborate the theoretical claims (Seven 2-DOF systems, and three 3-

DOF systems).

• Performances of the proposed control law during application on different

UMS have been compared with the performances of the Hierarchical

Adaptive Backstepping Sliding Mode Control.

Formulation of the Block Backstepping Control Law

11 12 1

21 22 2

01 2

1 2

q q q q h q,p

q q q q h q,p q

m m

m m B

Lagrangian model of n-DOF underactuated system

Feedback Law by Spong

1 1 1 1

21 11 1 22 21 11 12, ,2

q h q p q q h q p q q q q q uB m m B m m m m

,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

Where , , ,

, , ,

, , ,

1

1q

n 2

2q

n ,

1 2q q q

ncol

1 1

11 qn n

m

1 2

12 qn n

m

1 2

21 qn n

m

2 2

22 qn n

m

q p

1,1

h q pn

2

2 ,h q pn

2 2n nB q

2n

18

1

11 1

1

11 12

, ,f q p q h q

q q q

m p

g m m

0X

State Model of n-DOF underactuated system

State Vector: 21 1 2X q p q pT

Desired Configuration Vector:

Control Objective:2

as t E 0

State model is

not in Controllable

Canonical form

19

,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

0 E X XError:


• Define new state variable z1

• Derivative of state variable z1

• Define Stabilizing Function for z1

• Define 2nd State variable z2

• Derivative of z1

1 2 1 2z p p f pK D g

1 1 1c K D g 1 1 1 2

α z p f p 1

nij

ij k

k k

gD g p

q

2 2 1z p α

1 1c 1 2 1 1z z z χ

Where , is a constant matrix such

that only when i=j or otherwise.

2 1n nK

0ijk 0ijk

20

Integral action

1 2 1 1 2z q q p pK g ……(i)

……(ii)

..(iii)

…...(iv)

…...(v)


Hence,

1

1 1 2

1 1 1 1

2 1 2

2 2 2 2

1 2 2 2

1 1 1 1

q q p p

p p p

c c

K g Df Df Df g Df

D g D g D g D g g D g

c c

2 1 2 1 1

1 2

2

1 2 1 1

z u z z χ z

f u p p f u u

u p f u u

u z z χ z φ

1 2

2 2

1 2 2 2p p p pI K g Df g Df D g D g g D g

2 2

1 2 1 1 2 1q q p pK Df Df Df D g D 1 2 2φ f p p f p f

where

21

…...(vi)


• Desired Dynamics of the system

• Desired expression of the time derivative of z2

• Actual time derivative of z2

• Expression of the control input:

1 1

2

c

c

1 2 1 1

2 1 2

z z χ z

z z z

22 1 2z z zc

1 1 1 1 2c c c 1 2 1 1 1 2

u z z χ z Φ z z 1 2

1 1 1 1 1 21 c c c 1 1 2u c χ z z φ

1 1 1 1c c 2 1 2 1 1

z u z z χ z φ

22

…...(vii)

…...(viii)

…(ix)

..(x)


• Remark 1: The proposed control law (u) ensures global asymptotic

stability of the reduced order system in (z1, z2 ) coordinate.

• Remark 2: The proposed control law relies on the fact that ψ is invertible.

• Remark 3: The Proposed control law only ensures the global asymptotic stability of reduced order system (order =2n2).

1 1

2

c

c

1 2 1 1

2 1 2

z z z χ

z z z

23


• The proposed control law transforms original system (order 2n) into

reduced order system (2n2).

• Now, 2n = 2n2 + 2n1

• Therefore, the order of the Internal dynamics of the system is 2n1.

Original State Model

,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

Reduced order Model

1 1 1

2

c

c

1 2 1

2 1 2

z z z

z z z

24


• Stability of the internal dynamics can be analyzed using the concept of

zero dynamics stability.

• Two state variables q1 and p1 are selected to represent the internal

dynamics of the underactuated system.

• z1 identically equal to zero implies , z1 , first derivative of z1 , second

derivative of z1 are equal to zero. That is

Solution of these three

Equations will give the

expressions of q2 and p2

in terms of q1 and p1`.

Zero Dynamics Analysis

25

01 2 1 1 2

z q q p pK g

1 02 1 2z = p p f pK D g

1z u φ 0

..(xi.a)

..(xi.c)

..(xi.b)

• Zero dynamics:

• Stability of the zero dynamics depends on the choice of controller

parameter K.

• A Lyapunov function Vz may be constructed to analyze the stability of the

zero dynamics system.

• The controller parameter K should be selected in a manner to ensure the

negative definiteness of the

1

1,

1 1

1 1

q p

p f u f Φ F q pg g

1 1

2 2

T T

1 1 1 1q q p pzV

1

1

T

zV T

1 1q p p f g φ

26

Zero Dynamics Analysis

..(xii)

..(xiii)

..(xiv)

zV

27

Pictorial Representation of Proposed Control Law

Fig: 1. Pictorial representation of the proposed control law

• proposed Transformation

28

Condition of Global Diffeomorphism

1 2

1

1

2 1 1 21

2 2 1 1 1

3

4

q q p pz

z p z p fZ X

z

z

K g

c K D g pT

q

p

..(xv)

1 1 1 1

12

2

1

1 1

1 0 0 0

0 1 0 0

2

2

1 2

12 2

1 2 2

q

fp

p qf

fqX p p

p q q

gK K K Kg

gK c c K c Kc g KD g

TD gK c

D g D g Kpp

..(xvi)

• Simplification yields

• Therefore, it can be concluded that invertibility of ψ implies

that the proposed algebraic transformation is a global

diffeomorphism.

• In turn, it can be inferred that invertibility of ψ together with

zero dynamics stability ensures that the proposed control law

can be used to stabilize an underactuated system at a desired

equilibrium. 29

Condition of Global Diffeomorphism

….(xvii)detX

T

Comparative Analysis with Olfati’s Method

• Olfati’s method in brief:

11 12 1

21 22 2

, 0

,

1 2

1 2

q q q q h q p

q q q q h q p q

m m

m m B

Lagrangian model of n-DOF underactuated system

Spong’s Feedback Law

1 1 1 1

21 11 1 22 21 11 12, ,2

q h q p q q h q p q q q q q uB m m B m m m m

,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

State Model of n-DOF underactuated system

Define change of coordinate

2

2

1 2

2 2

1 1

1

y q γ q

yp

ξ q

ξ p

L

12

11

2q

2

0

γ q dsm s

m s

1

2p p q

TL M V

30

State model in new coordinate

1

11

,

1 1 2

2 1 1 1

1 2

2

y ξ y

y y γ ξ ξ

ξ ξ

ξ u

m

g

1

,1 2q qq

Vg

Define change of coordinate

2

2

1 2

2 2

1 1

1

y q γ q

yp

ξ q

ξ p

L

12

11

2q

2

0

sγ q ds

s

m

m

1

2p p q

TL M V

31



• A short demonstration on Inertia Wheel Pendulum

32

1 1

2 2

1 1

2 2

q

q

p

p

1 1

21 2 0 12

1 1

11 11

2 2

2

sin

q p

m m g mp q u

m m

q p

p u

2 2

11 1 1 2 1 1 2 12 21 22 2

0 1 1 2 1 0

, m m l m L I I m m m I

m m l m L g

where

33


0

1 1

21 2 . 12

1 1

11 11

2 2

2

sin

q p

m m g mp q u

m m

q p

p u

State model of IWP

Olfati’s method Proposed method

12

1 1 2

11

2 11 1 12 2

1

1 2

2 2

my q q

m

Ly m p m p

p

q

p

12

1 2 1 1 2

11

21 2 0

1 2 1 1

11

21 2 0

1 1 1 1 1 1 1

11

2 2 1

sin

sin

mz q k q p p

m

m m gz p k p q

m

m m gc z k p q

m

z p

New set of state variable Definition of error variables

State Model State Model

1

1 11 2

12

2 12 2 0 1 1

11

1 2

2

sin

y m y

my m m g y

m

u

1 2 1 1 1 1

2 1 1 1 2 1 1 1 1

11

12

21 0 0 21 0 0

1 1 1

11 11

1

sin cos

z z c z

z u z c z c z

mk

m

m m g m m gk q q p

m m

34


Olfati’s method Proposed method

State Model State Model

1

1 11 2

12

2 12 2 0 1 1

11

1 2

2

sin

y m y

my m m g y

m

u

1 2 1 1 1 1

2 1 2 1 1

11

12

21 0 0 21 0 0

1 1 1

11 11

1

sin cos

z z c z

z u c z c z

mk

m

m m g m m gk q q p

m m

Expression of Control Law Expression of Control Law

011 12

2 1 3 2 1 1 1

12 11 11

1 0 1 1

212

1 0 1 0 1 1 1 2

11

2 12 1 12 2

1 0 1 0 1 2 1 1

11 11

12

1 0 1 1 1

11

sin

tanh

sin 1 tanh

1 tanh cos

2 tanh sin

mm mu c c y k

m m m

k c c z

mk c c m y c y

m

m y mk c c m c y y

m m

mc m c y y

m

2

1

12 2 12 2

1 1 1 1 2 1

11 11

; m y m

y k km m

1

211

1 1 1 1 2 2

12

21 0 0 21 0 0

1 1 1 1 1 1

11 11

1 1

sin cos

mu k c z c c z

m

m m g m m gc k q q p

m m

35

Application of the Proposed Control Algorithm

• Acrobot (nonholonomic system with actuated shape variable)

• Pendubot (nonholonomic system with un actuated shape variable)

• TORA (Holonomic system with actuated shape variable)

• Furuta pendulum (nonholonomic system with un actuated shape variable)

• Inertia Wheel Pendulum (Flat underactuated system)

• Inverted Pendulum (Holonomic system with unactuated shape variable)

• Overhead crane (Holonomic system with unactuated shape variable)

• USV (nonholonomic system with actuated shape variable)

• VTOL ( Nonminimum phase Flat underactuated system)

• 3DOF redundant manipulator (nonholonomic system, interacting input) 36

Conclusions

37

• Proposed Control law relies on a simple algebraic state transformation.

• Global diffeomorphism of the state transformation ensures controllability.

• It converts the system into a block-strict feedback form that consists of a

reduced order system, and internal dynamic model.

• Stability of the zero dynamics has thoroughly been analyzed.

• Integral actions has been incorporated in the control law.

• Proposed algorithm is quite generalized.

• It performs well in both the simulation as well as in the real-time environments.

• Only drawback is the tedious calculation of the zero dynamics.

References• S.Rudra, R. K. Barai, and M. Maitra, “Nonlinear State Feedback Controller design for Underactuated

Mechanical System: a Modified Block Backstepping Approach,” ISA Transactions, Vol. 53, No. 2, pp. 317-

326, Mar. 2014.

• Y. Liu and H. Yu, “A survey of underactuated mechanical systems,” IET Control Theory & Applications,

Vol. 7, No. 7, pp. 921-935, 2013.

• M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, New York:

Wiley Interscience, 1995.

• M. Bloch, M. Reyhanoglu, and N. H. McClamroch, “Control and stabilization of nonholonomic dynamic

systems,” IEEE Trans. on Autom. Control, Vol. 37, No. 11, pp.1746–1757, Nov. 1992.

• A. Astolfi, “Discontinuous control of nonholonomic system,” Systems and Control Letters, Vol. 27, No. 1,

pp. 37–45, Jan. 1996.

• M. W. Spong, “Underactuated Mechanical Systems,” Control Problems in Robotics and Automation, Vol.

230, Springer-Verlag, 1998, pp.135-150.

• I. Fantoni, R. Lozano, Nonlinear control for underactuated mechanical systems, London: Springer-Verlag,

2001.

• A. Jain and G. Rodriguez, “An analysis of the kinematics and dynamics of under-actuated manipulators,”

IEEE Trans. on Robotics and Automation, Vol. 9, No. 4, pp. 411-422, Aug. 1993.

• H. Arai, S. Tachi, “Position control of a manipulator with passive joints using dynamic coupling,” IEEE

Trans. on Robotics and Automation, Vol. 7, No. 4, pp. 528–534, Aug. 1991.

• M. Reyhanoglu, A. van der Schaft, N. H. McClamroch, and I. Kolmanovsky, “Dynamics and control of a

class of underactuated mechanical systems,” IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1663–1671,

1999.

• contd. 38






1 2 1 1 2z q q p pK g

1 2 1 2z p p f pK D g

11 1 1 2α z p f pc K D g

1

nij

ij k

k k

gD g p

q

2 2 1z p α

11 2 1z z zc

Where , is a constant matrix such

that only when i=j or otherwise.

2 1n nK

0ijk 0ijk

1

2

3

0 0

0 0

0 0

k

K k

k

Detailed Analysis

40


1 1

1 2 1

2

2

2 2 1

1 2

2

z u z z

f u p p f u

u u p

q q p

p

c c

K g Df Df Df g

Df D g D g

2 2 2

1 2D g D g D g

2

2

1

1 1

n nij

ij k l

l k k l

gD g p p

q q

2

2

1

nij

ij k

k k

gD g p

q

Detailed Analysis

• But derivative of p vector will generate components of u vector

1 1

2

p f uP

p u

g

• Structure of matrix 2

2D g

2

1 1 2

111

1 1

2

2

1

1 1

n nn

k k

k kk k

n nn n n

k k

k kk k

ggp p

q q

D g

g gp p

q q

41

Detailed Analysis

2

1 1 2 2

111

1 1 21

2

2

1 2

1 1

2p

n nn

k k

k kk k

n nn n n n

k k

k kk k

ggp p

q q p

D g

g g pp p

q q

Instead of , let us consider 2

2D g 2

2 2pD g

Or,

2

2

1 1 2

2

111

21 2

1 1

2

2

1

21 2

1 1

2p

n nn

k n k

k kk k

n nn n n

k n k

k kk k

ggp p p p

q q

D g

g gp p p p

q q

2 2

2

1 1 1 2 1 2

2

1 111 11

21 2

1 1

2

2

1 1

21 2

1 1

2

p+ + p

p

p+ + p

n n

n

n n

n n n n n n

n

n n

g gg gp p

q q q q

D g

g g g gp p

q q q q

2 2

2

1 1 1 2 1 2

2

1 111 11

21 2

1 1

2

2

1 1

21 2

1 1

2

+ +

p p

+ +

n n

n

n n

n n n n n n

n

n n

g gg gp p

q q q q

D g

g g g gp p

q q q q

42

Detailed Analysis

2 2

2

1 1 1 2 1 2

2

1 111 11

21 2

1 1

2

2

1 1

21 2

1 1

1

2

2

+ +

pp

p

+ +

n n

n

n n

n n n n n n

n

n n

g gg gp p

q q q q

D g

g g g gp p

q q q q

Or,

2 2 2 2

1 1

2 2 2 2

1 1 1 1

1 1

1 1 1 1

2 2 2 2

1 1 1 11 1

2

2 2

2 2 2 2

1 1 1 11 1

1 2p p

n n n n

r r r r

r r r r

r r r rn n n

n n n nn r n r n r n r

r r r r

r r r rn n n

g g g gp p p p

q q q q

D g p

g g g gp p p p

q q q q

2 2

2 2

1 1

1 1

2 2

1 11

12

2 2

2

2 2

1 11

n n

r r

r r

r r n

n nn r n r

r r

r r n

g gp p

q qp

D g pp

g gp p

q q

43

Detailed Analysis

1p f ug

2p u

Since

2 2 2 2

1 1

2 2 2 2

1 1 1 1

1 1

1 1 1 1

2 2 2 2

1 1 1 11 1

2

2

2 2 2 2

1 1 1 11 1

2p f u u

n n n n

r r r r

r r r r

r r r rn n n


r r r r

r r r rn n n

g g g gp p p p

q q q q

D g g

g g g gp p p p

q q q q

2 2 2 2

1 1

2 2 2 2

1 1 1 1

1 1

1 1 1 1

2 2 2 2

1 1 1 11 1

2

2 2

2 2 2 2

1 1 1 11 1

1 2p p

n n n n

r r r r

r r r r

r r r rn n n


r r r r

r r r rn n n

g g g gp p p p

q q q q

D g p

g g g gp p p p

q q q q

Or,

2 2 2 2

2 2 2 21 1 22 p p pp f u uD g D g D g g D g

44

Detailed Analysis Hence,

1

1 1 2

1 1

2 1 2

2 2 2 2

1 2 2 2

1 1

2 2 1

1 2

2

2 1

z u z z

f u p p f u u

u p f u u

u z z Φ

q q p p

p p p

c c

K g Df Df Df g Df


c c

1 2

2 2


2 2

1 2 1 1 2 11 2 2f p p f p f q q p pK Df Df Df D g D

where

45

• Desired Dynamics of the second order system




1

2

1 2 1

2 1 2

z z z

z z z

c

c

22 1 2z z zc

1 1 21 2 1 1 1 2u z z z χ Φ z zc c c 1 2

1 1 21 1 2u z z Φc c c

Detailed Analysis

1 12 2 1z u z z Φc c

46

Stepwise Development of the

Proposed Control Algorithm-2 DOF

0 E X X

1 1

2 2

1 1 1

2 2 2

q,p q

q,p q

q p

q p

p f g u

p f g u

State Model of 2 DOF underactuated system

State Vector: 1 2 1 2

Tq q p pX

State Error:

Control Objective: E 0 as t

State model is

not in Controllable

Canonical form

48 c






1 2 1 2 1 1 2z q k q g p g p

1 2 1 2 1 1 1 2 22 1dg p dg p z p k p g f p g f p

1 1 1 1 2 1 1 1 2 22 1dg p dg pc z k p g f p g f p

2 2 1z p

1 2 1 1z z c z

Contd.

49 c

• Derivative of z2 :

• Expressions of ψ, and ϕ

2 1 2 1 1z u c z c z

21 1 2 2

2 1 1 2 2 1 1 1 1 2

1 2 1 2

2 2 2 1 11 1 2 2 2 1 2 2

1 2 1 2

2

1

dg p

dg p

f f g gg k g g g g g p g p g

p p q q

f f g gg g g g p g p g

p p q q

2 2 2

2 22 2 2 2 2

2 1 1 1 1 2 1 1 2 22 2

1 2 1 21 2

1 1 1 1

1 2 1 2 1 2

1 2 1 2

2

2 2

2

2

1

dg p

dg

g g g g gf k f f p f f p p p p

q q q qq q

f f f fg g p p f f

q q p p

f

2 2 2

2 21 1 2 2 2

2 1 2 1 1 2 22 2

1 2 1 21 2

2pg g g g g

p f f p p p pq q q qq q

Contd.

50 c

• Desired Dynamics of the second order system




1 2 1 1

2 1 2 2

z z c z

z z c z

2 1 2 2z z c z

2 1 2 1 1z u c z c z

1 1 2 1 1 1 1 2 2u z c z c z z c z 1 2

1 1 1 2 21u c z c c z

Contd.

51 c

• Remark 1: The proposed control law (u) ensures global asymptotic

stability of the reduced order system in (z1, z2 ) coordinate.

• Remark 2: The proposed control law relies on the fact that ψ is invertible.

• Remark 3: The Proposed control law only ensures the global asymptotic stability of reduced order system (2nd order system).

1 2 1 1

2 1 2 2

z z c z

z z c z

Contd.

52 c

1 1

2

2 2 1 2 121

11 11

2 2

2

sin 2l

q p

M q p p p mp u

m m

q p

p u

Application on Acrobot

2

2 2 1 2 1 1 2 1 0 1 2 2 0 1 2

2 2 2

1 1 2 1 2 1 2 2 1 2

2 2 2

1 1 2 1 2 1 2 2 1 2

2 2 2

1 1 2 1 2 1 2 2 1 2

sin 2 cos cos

2 cos

2 cos

2 cos

l c c

c c c

c c c

c c c

M q p p p m l m l g q m l g q qf

m l m l l l l q I I

m l m l l l l q I Ig

m l m l l l l q I I

State Model of the Acrobot

Standard 2-DOF state model

1 1

2 2

1

2

,q p q

q p

q p

p f g u

p u

2

1 1 2 1 0 1 2 2 0 1 2 2 1 2 2 2 2 2cos cos , , c c l c cm l m l g q m l g q q M m l l M m l I

2 2 2

1 1 1 2 1 2 1 2 11 1 2 12 2 2, 2 cos , cos c c l lM m l m l l I I m M M q m M M q55

• Definition of z1:

• Dynamics of z1:

• Stabilizing Function:

• Definition of z2 :

• Control Input Required to realize the desired dynamics for z2:

where

and

1 2 1 1 2z q k q p gp

1 1 1 1 1 1 2c z k p f p dg p

2 2 1z p

1 2 1 2z p k p f p dg p

1 2

1 1 1 1 2 2 1 1 11u c z c c z c

2

1 2 2

1 2f f g

k g g pp p q

23

1 2 22

1 2 1 2

f f f gk f p p f p

q q p q

1 1 2

1 11

sin sinq q qf

q m

2 2

1 11

2 sinlM q pf

p m

2

2 2 2 1 1 2 2

2 11

cos 2 sin 2 sinl lM q p p p q q M q ff

q m

2 1 2

2 11

2 sinlM q p pf

p m

1

0 g

q

2

2 11

1 2 sinlg M qg

q m

2 222 11 2

2 22 11

2 1 2 sin 1 2 cosl lg M q m g M qg

q m

56

Application on Acrobot

Results obtained from simulation

Fig: Acro_1. Angular displacement of the base link (q1)

57

1 2 1 23, 4, 0 and 0q q p p Initial Conditions:

58

Fig: Acro_2. Angular Velocity of the base link (p1)


59


Fig: Acro_3. Angular displacement of the upper link (q2)

60


Fig: Acro_4. Angular Velocity of the upper link (p2)

61


Fig: Acro_5. Control Input (u)

Hierarchical Adaptive Backstepping Sliding

Mode Control [HABSMC]

• 2 DOF Systems

1 1

1

2 2

2

q p

p f X g X u

q p

p u

Algorithm

i) Design backstepping sliding mode control

law for two subsystems. These two sliding

surfaces are known as first layer sliding

surface.

ii) Similar to MRAC Identify the disturbance

bound for each subsystems.

iii) Design a second layer sliding surface to

establish the coupling between two of the

first layer sliding surface.

iv) Second layer sliding surface yields the

complete expression of input

Comparison with HABSMC

63Fig: Acro_6. Comparison of Angular displacement of the base link (q1)



64Fig: Acro_7. Comparison of Angular velocity of the base link (p1)


Comparison of q2

65

Fig: Acro_8. Comparison of Angular displacement of the upper link (q2)


Comparison of p2

66Fig: Acro_9. Comparison of Angular Velocity of the upper link (p2)


Comparison of input u

67Fig: Acro_10. Comparison of the input (u)

1 1

1

2 2

2

2 2 1 2 12

2

11 11

sin 2l

q p

p u

q p

M q p p p mp u

m m

Application on Pendubot

2

2 2 1 2 1 1 2 1 0 1 2 2 0 1 2

2 2 2

1 1 2 1 2 1 2 2 1 2

2 2 2

1 1 2 1 2 1 2 2 1 2

2 2 2

1 1 2 1 2 1 2 2 1 2

sin 2 cos cos

2 cos

2 cos

2 cos

l c c

c c c

c c c

c c c

M q p p p m l m l g q m l g q qf

m l m l l l l q I I

m l m l l l l q I Ig

m l m l l l l q I I

State Model of the Pendubot


1 1

2 2

1

2 q,p q

q p

q p

p u

p f g u

2

1 1 2 1 1 2 2 1 2 2 1 2 2 2 2 2cos cos , , c c l c cm l m l g q m l g q q M m l l M m l I

2 2 2

1 1 1 2 1 2 1 2 11 1 2 12 2 2, 2 cos , cos c c l lM m l m l l I I m M M q m M M q69


• Dynamics of z1:




where

and

2

2 11

1 2 sinlg M qg

q m

2

2 2 2 1 1 2 2

2 11

cos 2 sin 2 sinl lM q p p p q q M q ff

q m

1 1 2 2 1z q k q p gp

1 1 1 1 1 2 1c z k p f p dg p

2 1 1z p

1 1 2 1z p k p f p dg p

1 2

1 1 1 1 2 2 1 1 11u c z c c z c

2

1 2 2

1 2f f g

k g g pp p q

23

1 2 22

1 2 1 2

f f f gk f p p f p

q q p q

1 1 2

1 11

sin sinq q qf

q m

2 2

1 11

2 sinlM q pf

p m

2 1 2

2 11

2 sinlM q p pf

p m

1

0 g

q

2 222 11 2

2 22 11

2 1 2 sin 1 2 cosl lg M q m g M qg

q m

70

Application on Pendubot


Fig: Pendu_1. Angular displacement of the base link (q1)

71

Initial Conditions:1 2 1 24, 3, 0 and 0q q p p

72

Fig: Pendu_2. Angular Velocity of the base link (p1)


73


Fig: Pendu_3. Angular displacement of the upper link (q2)

74


Fig: Pendu_4. Angular Velocity of the upper link (p2)

75


Fig: Pendu_5. Control Input (u)



• 2 DOF SystemsAlgorithm




surface.








1 1

1

2 2

2

q p

p f X g X u

q p

p u


77

Fig: Pendu_6. Comparison of Angular displacement of the base link (q1)



78

Fig: Pendu_7. Comparison of Angular velocity of the base link (p1)


Comparison of q2

79

Fig: Pendu_8. Comparison of Angular displacement of the upper link (q2)


Comparison of p2

80

Fig: Pendu_9. Comparison of Angular Velocity of the upper link (p2)



81

Fig: Pendu_10. Comparison of the input (u)

1 1

1

2 2

2 1

2 2 2 1 3 2

2

tan sincos

q p

p u

q p

kp k q p k q u

q

Application on Furuta Pendulum

21 2

2 1 2

1 2 1

2

2 2 2

2 1 2 2

tan sin

cos

gl lf q p q

L l L

J m lg

m l l q

State Model of the Furuta Pendulum

Standard UMS state model

1 1

2 2

1

2 q,p q

q p

q p

p u

p f g u

2

2 2 2 1 2

1 2 3

2 1 2 1 2 1

, ,

J m l gl l

k k km l l L l L 83

1

2

1

2

q

q

p

p

Actuated Configuration

Variable: φ

Unactuated Configuration

Variable: θ


• Dynamics of z1:




where

and

1 1 2 2 2 1 z q k q p g p

1 1 1 1 1 2 2 1 2c z k p f p dg p

2 1 1 z p

1 1 2 2 1 2 z p k p f p dg p

1 2

1 1 1 1 2 2 1 1 11u c z c c z c

2

1 2

1 2f g

k g g pp q

23

2 22

2 1 2

f f gk f p f p

q p q

1

0f

q

3 2 1

1

2 sinf

k q pp

1

0 g

q

1 2 2

2

sec tang

k q qq

2

2 21 2 2 22

2

sec tan secg

k q q qq

2 2

2 2 3 2 1

2

sec cosf

k q k q pq

2

0f

p

84

Application on Furuta Pendulum


Fig: FP_1. Angular displacement of the base (q1)

85


86

Fig: FP_2. Angular Velocity of the base (p1)


87


Fig: FP_3. Angular displacement of the upper link (q2)

88


Fig: FP_4. Angular Velocity of the upper link (p2)

89


Fig: FP_5. Control Input (u)







surface.








1 1

1

2 2

2

q p

p f X g X u

q p

p u


91

Fig: FP_6. Comparison of the Angular displacement of base (q1)



92

Fig: FP_7. Comparison of the Angular velocity of the base (p1)


Comparison of q2

93

Fig: FP_8. Comparison of the Angular displacement of Pendulum (q2)


Comparison of p2

94

Fig: FP_9. Comparison of the Angular Velocity of Pendulum(p2)



95

Fig: FP_10. Comparison of the input (u)

1 1

2

1 3 1 2 2 1 2 2

2 2

2

sin cos

q p

p k q k q p k q u

q p

p u

Application on TORA system

2

3 1 2 2 1

2 2

sin

cos

f k q k q p

g k q

State Model of the TORA System


1 1

2 2

1

2

q,p q

q p

q p

p f g u

p u

2 2 1 2 3 1 1 2, k m r m m k k m m

where

97

1

2

1

2

x

q x

q

p v

p


Variable: θ


Variable: x


• Dynamics of z1:




where

and

1 2 1 1 2z q k q p gp

1 1 1 1 1 1 2c z k p f p dg p

2 2 1z p

1 2 1 2z p k p f p dg p

1 2

1 1 1 1 2 2 1 1 11u c z c c z c

2

1 2

1 2f g

k g g pp q

23

1 2 22

1 1 1 2

f f g gk f p f p f p

q p q q

2 2 1

1

2 sinf

k q pp

1

0 g

q

2 2

2

cosg

k qq

2

2 222

sing

k qq

2

0f

p

3

1

fk

q

2

2 2 1

2

cosf

k q pq

98

Application on TORA system


Fig: TORA_1. Displacement of the cart (q1)

99


q1

[m]

100

Fig: TORA_2. Velocity of the cart (p1)


101


Fig: TORA_3. Angular displacement of the rotor(q2)

102


Fig: TORA_4. Angular Velocity of the rotor(p2)

103


Fig: TORA_5. Control Input (u)







surface.








1 1

1

2 2

2

q p

p f X g X u

q p

p u


105

Fig: TORA_6. Comparison of Cart displacement (q1)

1 2 1 20.5, 4, 0 and 0q q p p Initial Conditions:


106

Fig: TORA_7. Comparison of the base velocity (p1)


Comparison of q2

107

Fig: TORA_8. Comparison of the rotor Angular displacement (q2)


Comparison of p2

108

Fig: TORA_9. Comparison of the rotor angular velocity (p2)



109

Fig: TORA_10. Comparison of the input (u)

1 1

21 2 0 12

1 1

11 11

2 2

2

sin

q p

m m g mp q u

m m

q p

p u

Application on IWP

21 2 0

1

11

2

2 2

1 1 2 1 1 2

sinm m g

f qm

Ig

m l m L I I

State Model of the IWP


1 1

2 2

1

2

q,p q

q p

q p

p f g u

p u

2 2

11 1 1 2 1 1 2 12 21 22 2, m m l m L I I m m m I

where

111

1 1

2 2

1 1

2 2

q

q

p

p


Variable: θ2


Variable: θ1


• Dynamics of z1:




where

and

1 2 1 1 2z q k q p gp

1 1 1 1 1 1 2c z k p f p dg p

2 2 1z p

1 2 1 2z p k p f p dg p

1 2

1 1 1 1 2 2 1 1 11u c z c c z c

1 kg

1

1

fk f p

q

1

0f

p

1

0 g

q

2

0g

q

2

0f

p

1

1

cosf

qq

2

2 2 1

2

cosf

k q pq

112

Application on IWP


Fig: IWP_1. Angular displacement of the pendulum (q1)113


114

Fig: IWP_2. Angular Velocity of the base (p1)


115


Fig: IWP_3. Angular displacement of the Wheel (q2)

116


Fig: IWP_4. Angular Velocity of the Wheel (p2)

117


Fig: IWP_5. Control Input (u)







surface.








1 1

1

2 2

2

q p

p f X g X u

q p

p u


119Fig: IWP_6. Comparison of the angular displacement of pendulum (q1)



120

Fig: IWP_7. Comparison of the angular velocity of the pendulum (p1)


Comparison of q2

121

Fig: IWP_8. Comparison of the angular displacement of wheel (q2)


Comparison of p2

122

Fig: IWP_9. Comparison of the angular velocity of wheel (p2)



123

Fig: IWP_10. Comparison of the input (u)

1 1

2 2

2

2 2 2 1

1

2

2 2 2 1

2

sin 2 sin

2

sin 2 sin

2

q p

q p

gl q p q bp aup

d d d d

g q p q bp lup

d d d d

2

2 2 2 1

1

2

2 2 2 1

2

1

2

sin 2 sin

2

sin 2 sin

2

gl q p q bpf

d d d

g q p q bpf

d d d

ag

d

lg

d

125

Applications on Inverted Pendulum

μ=l M+m 2

2sind J l q 2 J

a lM m

State Model of Inverted Pendulum


1 1

2 2

1 1 1

2 2 2

q,p q

q,p q

q p

q p

p f g u

p f g u

1

2

1

2

x

q x

q

p v

p


• Dynamics of z1:




where

and

1 2 1 1 2z q k q ldp adp

1 2 1 1 2 1 2z p k p d lp ap d lf af

1 1 1 1 3 3 4 1 2c z k x d lx ax d lf af

2 2 1z p

1 2

1 1 1 2 2 1 11u c z c c z c

2 2

2 2 2 2 2sin 2 2 sin 2 sinl q l q p alp ql a bak l a

d d d d d d

2 2

2 1 2 2 1 2 2 1 2 2 22

3

2 2 2 2 2 1 2 2 2 2 2

3

2 2 1 2 2

2 cos 2 2 sin 2 cos 2

2 sin cos cos 2 sin

+ cos sin 2

f k f lp q lp ap lp q lf af gl p q

lp q f lp q blf agp q ap q f

ap q baf lf q

126

Applications on Inverted Pendulum

Results obtained from Real-time experiment

Fig: IP_1. Displacement of the cart (q1)

127

128

Fig: IP_2. Velocity of the cart (p1)


129


Fig: IP_3. Angular displacement of the pendulum(q2)

130


Fig: IP_4. Angular Velocity of the pendulum(p2)

131


Fig: IP_5. Control Input (u)







surface.








1 1

1

2 2

2

q p

p f X g X u

q p

p u


133

Fig: IP_6. Comparison of Cart displacement (q1)


134

Fig: IP_7. Comparison of the cart velocity (p1)


Comparison of q2

135

Fig: IP_8. Comparison of the pendulum angular displacement (q2)


Comparison of p2

136

Fig: IP_9. Comparison of the pendulum angular velocity (p2)

Application on Granty crane

2

2 2 2 1

1

2

2 2 2 1

2

1

2

sin 2 sin

2

sin 2 sin

2

gl q p q bpf

d d d

g q p q bpf

d d d

ag

d

lg

d

μ=l M+m 2

2sind J l q 2 J

a lM m

State Model of Granty Crane


1 1

2 2

1 1 1

2 2 2

q,p q

q,p q

q p

q p

p f g u

p f g u

1 1

2 2

2

2 2 2 1

1

2

2 2 2 1

2

sin 2 sin

2 2

sin 2 sin

2

q p

q p

gl q p q bp aup

d d d

g q p q bp lup

d d d d

138

1

2

1

2

x

q x

q

p v

p

Application on Granty crane


• Dynamics of z1:




where

and

1 2 1 1 2z q k q ldp adp

1 2 1 1 2 1 2z p k p d lp ap d lf af

1 1 1 1 3 3 4 1 2c z k x d lx ax d lf af

2 2 1z p

1 2

1 1 1 2 2 1 11u c z c c z c

2 2

2 2 2 2 2sin 2 2 sin 2 sinl q l q p alp ql a bak l a

d d d d d d

2 2

2 1 2 2 1 2 2 1 2 2 22

3

2 2 2 2 2 1 2 2 2 2 2

3

2 2 1 2 2

2 cos 2 2 sin 2 cos 2

2 sin cos cos 2 sin

+ cos sin 2

f k f lp q lp ap lp q lf af gl p q

lp q f lp q blf agp q ap q f

ap q baf lf q

139


Fig: OC_1. Displacement of the cart (q1)

140

141

Fig: OC_2. Velocity of the cart (p1)


142


Fig: OC_3. Angular displacement of the payload (q2)

143


Fig: OC_4. Angular Velocity of the payload (p2)

144


Fig: OC_5. Control Input (u)







surface.








1 1

1

2 2

2

q p

p f X g X u

q p

p u


146

Fig: OC_6. Comparison of Cart displacement (q1)


147

Fig: OC_7. Comparison of the cart velocity (p1)


Comparison of q2

148

Fig: OC_8. Comparison of the payload angular displacement (q2)


Comparison of q2

149

Fig: OC_9. Comparison of the payload angular velocity (p2)


Comparison of q2

150

Fig: OC_10. Comparison of the input signal(u)

1 3

1 6

2 1 2

2 4 5

4 2 6 2

tan & tan 0

T

T

y

q x

p x

q x x

p x x

cf x x x g x

m

1 4

2 5

3 6

4 1

5 2

6 1 2 4 2 6tan tan

y

x x

x x

x x

x u

x u

cx u x x x x

m

Application on USV

State Model of the USV


,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

152

1

2

3

4

5

6

x

y

x x

x

x y

x v

x

x v


Variable: x,θ


Variable: y


• Dynamics of z1:



Application on USV

K g1 2 1 1 2

z q q p p

K D g1 2 1 2

z p p f p

0K k

1 1c K D g 1 1 1 2

α χ z p f p

3

2 2 2 24 5 6

1 1 2 3

tan tan tan tan0 0k

k k

x x x xD g p x x x

q x x x

2 2z p α

2

2 5sec 0D g x x

153

Dynamics of z2:

1

1 1 2

1 1

2 1 2

2 2 2 2

1 2 2 2

1 1

2 2 1

1 2

2

2 1

z u z z

f u p p f u u

u p f u u

u z z Φ

q q p p

p p p

c c

K g Df Df Df g Df


c c

Application on USV

Dynamics of z2:

1 1

1 2 1

2

2

2 2

2 1

1 2

2

z p α

u z z

f u p p f u

u u p

q q p

p

c c

K g Df Df Df g

Df D g D g

(5.9)

23 3

2

1

1 1

ij

ij k l

l k k l

gD g p p

q q

32

2

1

ij

ij k

k k

gD g p

q

2 2

2 1 2D g D g D g

1 1 2

2 2 2 2

2 2 2 22p f u up p pD g D g D g g D g

154

2

1

2

2 2 1

1 1

1, ,n

mi

p mr i

i r

gD g p m n

q

2 22 2

2 2 4 5 4 5 2 4

1 1 2 2

0 0tan tan0 secp

x xD g x x x x x x

x x x x

2 22 1 4 5

3 3

tan 00p

xD g x x

x x

1 4 2 6

6

tany y

p

c cDf x x x

m x m

2 4 2 6 4 2 6 2

4 5

tan tan tan 0y y

p

c cDf x x x x x x x

m x x m

2

2

2

2 2 1

1 2

1, ,n

mi

p mr i

i r

gD g p m n

q

4 2 6 2tan & tan 0yc

f x x x g xm

Application on USV

Control Input 1 2

1 1 1 2 1 11 c c c c 1 2 1u z z χ Φ

1 2

2 2


155

23 3

2 2 22

1 2 2 5

1 1

tan0 2 tan sec 0ij k l

l k k l

xD g p p x x x

q q

1 4 2 6

3

tan 0y

q

cDf x x x

m x

2

2 4 2 6 4 2 6 4 2

1 2

tan tan 0 secy y

q

c cDf x x x x x x x x

m x x m

Application on USV

2 2


1 4 2 6

6

tany y

p

c cDf x x x

m x m

2

2 1 0pD g

2

2

1

1 1

n nij

ij k l

l k k l

gD g p p

q q

156


1 2 1 20.8, 0.19 , 0.1 and 1.62 2

q q p p

Initial Condition

Fig: USV_1. Longitudinal Displacement

157


Fig: USV_ 2. Longitudinal Velocity

158

Fig: USV_3. Angular Orientation of the surface vessel

159


Fig: USV_4. Angular Velocity of the surface vessel160


Fig: USV_ 5.Lateral Displacement

161


Fig: USV_6. Lateral Velocity

162


163


Fig: USV_7. Control Input u1

164


Fig: USV_8. Control Input u2



• 3 DOF Systems

1 1

1 1

21 21

22 22

21 1

22 2

q p

p f X g X u

q p

q p

p u

p u

Algorithm

(i) Follow the same principle of 2-DOF

hierarchical architecture for controller

design for first input (u1)

(ii) Repeat the adaptive backstepping

sliding mode design for input u2 too.


1 2 1 21, 0.3 , 0.1 and 2.43 3

q q p p

Initial Condition

Fig: USV_9. Performance Comparison of the Longitudinal Displacement

166


Fig: USV_10. Performance Comparison of the Longitudinal Velocity

167


Fig: USV_11. Performance Comparison of the Angular Orientation of the USV168


Fig: USV_12. Performance Comparison of the Angular Velocity of the USV

169


Fig: USV_13. Performance Comparison of the Lateral Displacement

170


Fig: USV_14. Performance Comparison of the of Lateral Velocity

171


Fig: USV_15. Performance Comparison of the Control Input u1

172


Fig: USV_16. Performance Comparison of the Control Input u2

173

1 5

1 6

2 1 3

2 2 4

5 5 5

sin & cos sin

T

T

q x

p x

q x x

p x x

f g x g x x

1 2

2 1

3 4

4 2

5 6

6 1 5 2 5 5cos sin sin

x x

x u

x x

x u

x x

x u x u x g x

Application on VTOL

State Model of the VTOL


,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

175

1

2

3

4

5

6

y

x

x y

x v

x

x

x x

x v


Variable: y,θ


Variable: x


• Dynamics of z1:



Contd.

K g1 2 1 1 2

z q q p p

K D g1 2 1 2

z p p f p

0K k

1 1c K D g 1 1 1 2

α z χ p f p

3 3

3 33 6 3 6

1 1

cos sinsin cosk k

k kk k

x xD g p p x x x x

q q

2 2z p α

176

Application on VTOL

Dynamics of z2:

1 1

1 2 1

2

2

2 2

2 1

1 2

2

z p α

u z z

f u p p f u

u u p

q q p

p

c c

K g Df Df Df g

Df D g D g

(5.9)

23 3

2

1

1 1

ij

ij k l

l k k l

gD g p p

q q

32

2

1

ij

ij k

k k

gD g p

q

2 2

2 1 2D g D g D g

1 1 2

2 2 2 2


Dynamics of z2:

1

1 1 2

1 1

2 1 2

2 2 2 2

1 2 2 2

1 1

2 2 1

1 2

2

2 1

z u z z

f u p p f u u

u p f u u

u z z Φ

q q p p

p p p

c c

K g Df Df Df g Df


c c

177

2

1

2

2 2 1

1 1

1, ,n

mi

p mr i

i r

gD g p m n

q

5 5 5 52

2 2 4 2 4

1 1 2 2

cos sin cos sin0 0p

x x x xD g x x x x

x x x x

2 5 52 1 2 4 5 2 5 4

5 5

cos sinsin cosp

x xD g x x x x x x

x x

1 5

6

9.81 sin 0pDf xx

2 5 5

2 4

9.81 sin sin 0 0pDf x xx x

2

2

2

2 2 1

1 2

1, ,n

mi

p mr i

i r

gD g p m n

q

5 5 59.81 sin & cos sinf x g x x

Application on VTOL

Control Input 1 2

1 1 1 2 1 11 c c c c 1 2 1u z z χ Φ

1 2

2 2


5 6 5 6sin cosD g x x x x

178

2 23 3 3 3

2 25 5

1 5 5 6

1 1 1 1

cos sincos sinij k l k l

l k l kk l k l

x xD g p p p p x x x

q q q q

5

1 5

5

sin9.81 9.81 cosq

xDf x

x

2 5 5

1 3

9.81 sin sin 0 0qDf x xx x

Application on VTOL

2 2


1 5

6

9.81 sin 0pDf xx

2

2

1

1 1

n nij

ij k l

l k k l

gD g p p

q q

2

2 1 5 2 5 4sin cospD g x x x x

179


1 2 1 22, 4 , 3 and 1 03

q q p p

Initial Condition

Fig: VTOL_1. X axis Displacement of VTOL

180


Fig: VTOL_ 2. X axis Velocity of VTOL

181

Fig: VTOL_3. Y axis displacement of VTOL

182


Fig: VTOL_4. Y axis Velocity of the VTOL

183


Fig: VTOL_ 5. Angular Displacement of VTOL

184


Fig: VTOL_6. Angular Velocity of the VTOL

185


186


Fig: VTOL_7. Control Input v1

187


Fig: VTOL_8. Control Input v2









1 1

1 1

21 21

22 22

21 1

22 2

q p

p f X g X u

q p

q p

p u

p u


1 2 1 23, 3 , 2.5 and 1.5 04

q q p p

Initial Condition

Fig: VTOL_9. Performance Comparison of the x-axis Displacement

189


Fig: VTOL_10. Performance Comparison of the x-axis Velocity

190


Fig: VTOL_11. Performance Comparison of the Y axis displacement of the VTOL

191


Fig: VTOL_12. Performance Comparison of the Y axis Velocity of the VTOL

192


Fig: VTOL_13. Performance Comparison of the angular orientation of VTOL

193


Fig: VTOL_14. Performance Comparison of the of angular velocity of VTOL

194


Fig: VTOL_15. Performance Comparison of the Control Input v1

195


Fig: VTOL_16. Performance Comparison of the Control Input v2

196

1 3

1 6

2 1 2

2 4 5

2

0 & tan 0

T

T

q x

p x

q x x

p x x

f g x

1 4

2 5

3 6

4 1

5 2

6 1 2

1 2 3

tan

, ,

x x

x x

x x

x u

x u

x u x

x x x x y

Application on 3-DOF Redundant Manipulator

State Model of the Manipulator

Standard n-DOF state model

,

1 1

2 2

1

2

q p

q p

p f q p q u

p u

g

198


• Dynamics of z1:




K g1 2 1 1 2

z q q p p

K D g1 2 1 2

z p p f p

0K k

1 1 1c K D g 1 1 2

α z p f p

3

222 5

1

tan0 sec 0k

k k

xD g p x x

q

2 2z p α

199


Dynamics of z2:

1 1 1 1

1 2 1

2

2

q q p

p

c c

K g Df Df Df g

Df D g D g

2 2

2 1 1 1

1 2

2

z p α

u z z χ z

f u p p f u

u u p

(5.9)

23 3

2

1

1 1

ij

ij k l

l k k l

gD g p p

q q

32

2

1

ij

ij k

k k

gD g p

q

2 2

2 1 2D g D g D g

1 1 2

2 2 2 2


Dynamics of z2:

1

1 1 2

1 1

2 1 2

2 2 2 2

1 2 2 2

1 1

q q p p

p p p

c c

K g Df Df Df g Df


c c

2 2 1

1 2

2

2 1

z u z z

f u p p f u u

u p f u u

u z z Φ

200

2

1

2

2 2 1

1 1

1, ,n

mi

p mr i

i r

gD g p m n

q

2 22 2

2 4 5 4 5 2 4

1 1 2 2

0 0tan tan0 secp

x xD g x x x x x x

x x x x

2 22 1 4 5

3

tan0p

xD g x x

x

1 0pDf 2 0 0pDf

2

2

2

2 2 1

1 2

1, ,n

mi

p mr i

i r

gD g p m n

q

20 & tan 0f g x


Control Input 1 2

1 1 1 2 1 11 c c c c 1 2 1u z z χ Φ

1 2

2 2


3

222 5

1

tan0 sec 0k

k k

xD g p x x

q

201

23 3

2 2 22

1 2 2 5

1 1

tan0 2 tan sec 0ij k l

l k k l

xD g p p x x x

q q

1 0qDf

2 0 0qDf


2 2


1 0pDf

2

2

1

1 1

n nij

ij k l

l k k l

gD g p p

q q

2 21 4 5

3

tan0p

xD g x x

x

202


1 2 1 20.755, 1 , 0.5 and 0.5 04

q q p p

Initial Condition

Fig: 3-DOF_1. X axis Displacement of 3-DOF

203


Fig: 3-DOF_ 2. X axis Velocity of 3-DOF

204

Fig: 3-DOF_3. Y axis displacement of 3-DOF

205


Fig: 3-DOF_4. Y axis Velocity of the 3-DOF

206


Fig: 3-DOF_ 5. Angular Displacement of 3-DOF

207


Fig: 3-DOF_6. Angular Velocity of the 3-DOF

208


209


Fig: 3-DOF_7. Control Input u1

210


Fig: 3-DOF_8. Control Input u2









1 1

1 1

21 21

22 22

21 1

22 2

q p

p f X g X u

q p

q p

p u

p u


1 2 1 20.5, 0.755 , 0.5 and 0.45 06

q q p p

Initial Condition

Fig: 3-DOF_9. Performance Comparison of the x-axis Displacement

212


Fig: 3-DOF_10. Performance Comparison of the x-axis Velocity

213


Fig: 3-DOF_11. Performance Comparison of the Y axis displacement of the 3-DOF

214


Fig: 3-DOF_12. Performance Comparison of the Y axis Velocity of the 3-DOF

215


Fig: 3-DOF_13. Performance Comparison of the angular orientation of 3-DOF

216


Fig: 3-DOF_14. Performance Comparison of the of angular velocity of 3-DOF

217


Fig: 3-DOF_15. Performance Comparison of the Control Input u1

218


Fig: 3-DOF_16. Performance Comparison of the Control Input u2

219

What is Backstepping?

Stabilization Problem of Dynamical System

Design objective is to construct a control input u which ensures the

regulation of the state variables x(t) and z(t), for all x(0) and z(0).

Equilibrium point: x=0, z=0

Design objective can be achieved by making the above mentioned

equilibrium a GAS.

221

Contd.

Block Diagram of the system:

222

Contd.

First step of the design is to construct a control input for the scalar subsystem

z can be considered as a control input to the scalar subsystem

Construction of CLF for the scalar subsystem

Control Law:

But z is only a state variable, it is not the control input.

223

Contd.

Only one can conclude the desired value of z as

Definition of Error variable e:

z is termed as the Virtual Control

Desired Value of z, αs(x) is termed as stabilizing function.

System Dynamics in ( x, e) Coordinate:

224

Modified Block Diagram

Contd.

Feedback Control Law αsBackstepping

Signal -αs225

So the signal αs(x) serve the purpose of feedback control law inside the block and “backstep” -αs(x) through an integrator.

Contd.

Feedback loop with + αs(x) Backstepping of Signal -αs(x)

Through integrator 226

Construction of CLF for the overall 2nd order system:

Derivative of Va

A simple choice of Control Input u is:

With this control input derivative of CLF becomes:

Contd.

227

Consider the scalar nonlinear system

Control Law( using Feedback Linearization):

Resultant System:

Edurado D. Sontag Proposed a formula to avoid the Cancellation of these

useful nonlinearities.

Why Backstepping?

is it essential to cancel out the

term ?

Not at

all!!!!

This is an Useful Nonlinearity, it has an Stabilizing

effect on the system.

228

Sontag's Formula:

Control Law (Sontag’s Formula):

Control Law (using Backstepping):

Contd.

For large values of x, the

control law becomes

u≈sinx

So this control law avoids the cancellation of

useful nonlinearities!

For higher values of x

But this formula leads a

complicated control input

for intermediate

values of x

0 0

0

42

gx

Vfor

gx

Vfor

gx

V

gx

Vf

x

Vf

x

V

u

back229

Basic Concept of Zero Dynamics

• Let us consider a simple 3rd order system transfer function

• State Model Representation

• Three consecutive differentiation of y yields explicit relation

between input and output

• Relative Degree of the system is THREE.231

3 2

1

6 11 6

Y s

U s s s s

1 1

2 2

3 3

1

0 1 0 0

0 0 1 0

6 11 6 1

x x

x x u

x x

y x

1 2 36 11 6y x x x u

• Relative degree indicates the excess number of poles over the

number of zeros.

• Order of the internal dynamics 3-3=0.

• Now append one zero to that same transfer function

• State Model

232


3 2

4

6 11 6

Y s s

U s s s s

1 1

2 2

3 3

1 2

0 1 0 0

0 0 1 0

6 11 6 1

4

x x

x x u

x x

y x x

• Two consecutive differentiation of y yields explicit relation

between input and output

• Relative degree of the system is 2.

• Order of the internal dynamics is 1.

• Now use of the following feedback law

yields

233


1 2 36 11 2y x x x u

1 2 36 11 2u x x x v

y v

• Therefore, if we consider the output y as the state of the system

and signal v as the input to the system, then

• Basically this is a second order state model, so we can conclude

that application input , converts the

original system into a 2nd order state model!!!!!

• Actually, it converts the system into a cascade combination of

reduced order system and internal dynamics.

234


1 2

1 2

2

, z y z y

z z

z v

1 2 36 11 2u x x x v

• Internal dynamics can be represented as

• Stability of internal dynamics can be assessed by the location

of the zero at s plane!!!!

• Consider a nonlinear system

• First order differentiation of y yields

235


1 14x x y

1 1 2 1

2 1 2 2

2

2 3

x x x x

x x x x u

y x

1 2 22 3y x x x u

• Now application of input yields

• Internal dynamics

• Zero Dynamics can be found using the concept of output

zeroing input.

• Output identically equal to zero implies

• Now that implies

• Again

• Therefore, zero dynamics equation is

236


1 1

2

x x

x u

1 2 22 3u x x x v y v

1 1 1x x x y

0, 0 y and y

20 0 y x

20 0 0y x u

1 1x x

237

Contd.

238

Contd.

239

Contd.

• P. Bars, P. Colaneri, C. Souza, L. Dugard, F. Allgower, A. Kleimenov, C. Scherer, “Theory, algorithms and

technology in the design of control systems,” Annual Reviews in Control, Vol. 30, No. 1, pp. 19–30, 2006.

• A. Ollero, S. Boverie, D. Cho, H. Hashimoto, M. Tomizuka, W. Wang, D. Zuhlke, “Mechatronics, robotics and

components for automation and control,” IFAC milestone report, Annual Reviews in Control, Vol. 30, No. 1, pp. 41–

54, 2006.

• P.V. Kokotovic and M. Arcak, “Constructive nonlinear control: a historical perspective,” Automatica, Vol. 37, No. 5,

pp. 637-662, May. 2001.

• R. Olfati-Saber, “Nonlinear control of underactuated mechanical systems with application to robotics and

aerospace vehicles,” Ph.D. thesis, Dept. of Electrical Engineering and Computer, Massachusetts Institute of

Technology, 2001.

• N. P. I. Aneke, “Control of underactuated mechanical systems,” Ph.D. thesis, Mechanical Engineering, Eindhoven

University of Technology, 2003.

• A.de Luca, R. Mattone, G. Oriolo, “Control of underactuated mechanical systems: Application to the planar 2R

robot,” Proc. of Decision and Control Conf., Kobe, Japan, 1996, pp. 1455–1460.

• M. Reyhanoglu, S. Cho, and N. M. McClamroch, “Discontinuous feedback control of a special class of

underactuated mechanical systems,” International Journal of Robust and Nonlinear Control, vol. 24, July 2000, pp.

265-281.

• M. Nikkhah, H. Ashrafiuon, K. R. Muske, “Optimal sliding mode control for underactuated systems,” Proc. of the

American Control Conf., Minneapolis, Minnesota, USA, 2006, pp. 4688–4693.

• Z. Sun, S. S. Ge, and T. G. Lee, “Stabilization of underactuated mechanical systems: A non-regular backstepping

approach,” Int. J. Control, vol. 74, no. 11, pp. 1045–1051, Jun. 2001.

• R. Olfati-Saber and A. Megretski, “Controller design for a class of underactuated nonlinear systems,” in Proc. 37th

Conf. Decision and Control, vol. 4, Tampa, FL, Dec. 1998, pp. 4182–4187.

240

Contd.

• M. W. Spong, “The swing-up control problem for the Acrobot,” IEEE Control System Magazine, Vol. 47, No. 1, pp.

49–55, Feb. 1995.

• I. Fantoni, R. Lozano, M. W. Spong, “Energy based control of the Pendubot,” IEEE Trans. Autom. Control, Vol. 45,

No. 4, pp. 725–729, Apr. 2000.

• M. W. Spong, “Energy based control of a class of underactuated mechanical systems,” Proc. of the IFAC World

Congress, San Francisco, CA, USA, 1996, pp. 431–435.

• I. Kolmanovsky and N. H. McClamroch, “Hybrid feedback laws for a class of cascade nonlinear control systems,”

IEEE Trans. Automatic Control, vol. 41, No. 9, pp. 1271-1282, 1996.

• K. J. Åstorm, K. Furuta, “Swinging up a pendulum by energy control,” Automatica, Vol. 36, No. 2, pp. 287–295,

Feb. 2000.

• M. Zhang, T. Tarn, “A hybrid switching control strategy for nonlinear and underactuated mechanical systems,”

IEEE Trans. on Automatic Control, Vol. 48, No. 10,pp. 1777–1782, Oct. 2003.

• X. Z. Lai, J. H. She, S. X. Yang, M. Wu, “Comprehensive unified control strategy for underactuated two-link

manipulators,” IEEE Trans. on Syst. Man Cybern. B, Vol. 39, No. 2, pp. 389–398, Apr. 2009.

• T. Albahkali, R. Mukherjee, T. Das, “Swing-up control of the Pendubot: an impulse momentum approach,” IEEE

Trans. on Robotics, Vol. 25, No. 4, pp. 975–982, Aug. 2009.

• R. Jafari, F. B. Mathis, R. Mukherjee, “Swing-up control of the Acrobot: An impulse momentum approach,” Proc. of

the American Control Conf., San Francisco, CA, USA, 2011, pp. 262–267.

• John Hauser and Richard M. Murray, “Nonlinear controllers for non-integrable systems: The acrobot example”, in

Proceedings of the 1990 American Control Conference, vol. 1, , San Diego, CA, 1990, pp. 669–671.

241

Contd.

• H. Yabuno, T. Matsuda, N. Aoshima, “Reachable and stabilizable area of an underactuated manipulator without

state feedback control,” IEEE/ASME Trans Mechatronics, Vol. 10, No. 4, pp. 397–403, Aug. 2005.

• M. W. Spong, “Partial feedback linearization of underactuated mechanical systems,” Proc. of the IEEE/RSJ/GI

International Conference on Intelligent Robots and Systems, Munich, Germany, 1994, pp. 314–321.

• J. Zhao and M. W. Spong, “Hybrid control for global stabilization of the cart-pendulum system,” Automatica, Vol.

37, No. 12, pp. 1941–1951, Dec. 2001.

• D. Chatterjee, A. Patra, H. K. Joglekar, “Swing-up and stabilization of a cart pendulum system under restricted cart

track length,” System & Control Letter, Vol. 47, No. 4, pp. 355–364, Nov. 2002.

• S. Mori, H. Nishiharaa & K. Furuta, “Control of unstable mechanical system Control of pendulum,” International

Journal of Control, Vol. 23, No. 5, pp. 673-692, 1976.

• N. Muskinja, B. Tovornik, “Swinging up and stabilization of a real inverted pendulum,” IEEE Trans. on Industrial

Electronics, Vol. 53, No. 2, pp. 631–639, Apr. 2006.

• R. Fierro, F. L. Lewis, A. Lowe, “Hybrid control for a class of underactuated mechanical systems,” IEEE Trans. on

Syst. Man and Cyberen. A, Vol. 29, No. 6, pp. 649–654, Nov. 1999.

• K. Pathak, J. Franch, S. K. Agrawal, “Velocity and position control of a wheeled inverted pendulum by partial

feedback linearization,” IEEE Trans. on Robotics, Vol. 21, No. 3, pp. 505–513, June 2005.

• Y. Kim, S. H. Kim, Y. K. Kwak, “Dynamic analysis of a nonholonomic two-wheeled inverted pendulum robot,” J.

Intelligent and Robotic Systems, Vol. 44, No. 1, pp. 25–46, Sep. 2005.

• J. Huang, Z. H. Guan, T. Matsuno, T. Fukuda, K. Sekiyama, “Sliding-mode velocity control of mobile-wheeled

inverted-pendulum systems,” IEEE Trans. on Robotics, Vol. 26, No. 4, pp. 750–758, Aug. 2010.

242

Contd.

• X. Xin, M. Kaneda, “Analysis of the energy-based control for swinging up two pendulums,” IEEE Trans. on Autom.

Control, Vol. 50, No. 5, pp. 679–684, May. 2005.

• W. Zhong, H. Rock, “Energy and passivity based control of the double inverted pendulum on a cart,” Proc. of the

IEEE Int. Conf. on Control Applications, Mexico City, Mexico, 2001, pp. 896–901.

• N. A. Chaturvedi, N. H. McClamroch, D. S. Bernstein, “Asymptotic smooth stabilization of the inverted 3-D

pendulum,” IEEE Trans Automatic Control, Vol. 54, No. 6, pp. 1204–1215, June 2009.

• H. Li, K. Furuta, F. L. Chernousko, “A pendulum-driven cart via internal force and static friction,” Proc. of IEEE Int.

Conf. on Physics and Control, St. Petersburg, Russia, 2005, pp. 15–17.

• H. Yu, Y. Liu, T. C. Yang, “Closed-loop tracking control of a pendulum-driven cart-pole underactuated system,”

Proc. of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 222, No.

2, pp. 109–125, Mar. 2008.

• Y. Liu, H. Yu, S. Cang, “Modelling and motion control of a double-pendulum driven cart,” Proc. of the Institution of

Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 226, No. 2, pp. 175–187, Feb.

2012.

• Y. Fang, W. Dixon, D. Dawson, E. Zergeroglu, “Nonlinear coupling control laws for an underactuated overhead

crane system,” IEEE/ASME Transactions on Mechatronics, Vol. 8, No. 3, pp. 418–423, Sept. 2003.

• B. Ma, Y. Fang, Y. Zhang, “Switching-based emergency braking control for an overhead crane system,” IET

Control Theory and Applications, Vol. 4, No. 9, pp. 1739–1747, Sept. 2010.

• Y. Fang, B. Ma, P. Wang, X. Zhang, “A motion planning-based adaptive control method for an underactuated crane

system,” IEEE Trans. on Control Syst. Technol., Vol. 20, No. 1, pp. 241–248, Jan. 2012.

• N. Sun, Y. Fang, Y. Zhang, B. Ma, “A novel kinematic coupling-based trajectory planning method for overhead

cranes,” IEEE/ASME Trans Mechatronics, Vol. 17, No. 1, pp. 166–173, Feb. 2012.243

Contd.

• I. Fantoni, R. Lozano, “Stabilization of the Furuta pendulum around its homoclinic orbit,” Int. J. Control, Vol. 75, No. 6, pp. 390–398, 2002.

• G. Viola, R. Ortega, R. Banavar, J. A. Acosta, A. Astolfi, “Total energy shaping control of mechanical systems: Simplifying the matching equations via coordinate changes,” IEEE Trans. Autom. Control, Vol. 52, No. 6, pp. 1093–1099, June 2007.

• M. S. Park, D. K. Chwa, “Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method,” IEEE Transaction on Industrial Electronics, Vol. 56, No. 9, pp. 3541–3555, Sept. 2009.

• D. E. Chang, “Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation by the energy-shaping method,” IEEE Trans Autom. Control, Vol. 55, No. 8, pp. 1888–1893, Aug. 2010.

• A. S. Shiriaev, L. B. Freidovich, A. Robertsson, R. Johansson, A. Sandberg, Virtual holonomic-constraints-based design of stable oscillations of Furuta pendulum: Theory and experiments, IEEE Trans. on Robotics, Vol. 23, No. 4, pp. 827–832, Aug. 2007.

• L. Freidovich, A. Shiriaev, F. Gordillo, F. G´omez-Estern, J. Aracil, “Partial-energy shaping control for orbital stabilization of high-frequency oscillations of the Furuta pendulum,” IEEE Trans. on Control Syst. Technol., Vol. 17, No. 4, pp. 853–858, July 2009.

• R. Ortega, M. W. Spong, F. G´omez-Estern, G. Blankenstein, “Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment,” IEEE Trans. on Autom. Control, Vol. 47, No. 8, pp. 1218–1232, Aug. 2002.

• H. Ye, H. Wang, H. Wang, “Stabilization of a PVTOL aircraft and an inertia wheel pendulum using saturation technique,” IEEE Trans. Control Syst. Technol., Vol. 15, No. 6, pp. 1143–1150, Nov. 2007.

• H. Ye, W. Gui, Z. P. Jiang, “Backstepping design for cascade systems with relaxed assumption on Lyapunov functions,” IET Control Theory and Applications, Vol. 5, No, 5 pp. 700–712, Mar. 2011.

• M. Lopez-Martinez, J. A. Acosta, J. M. Cano, “Non-linear sliding mode surfaces for a class of underactuated mechanical systems,” IET Control Theory and Applications, Vol. 4, No. 10,pp. 2195–2204, Oct. 2010.

244

Contd.

• J. Wan, D. S. Bernstein, V. T. Coppola, “Global stabilization of the oscillating eccentric rotor,” Proc. of the Decision and Control Conf., Orlando, USA, 1994, pp. 4024–4029.

• M. Jankovic, D. Fontaine, P. V. Kokotovic, “TORA example: cascade and passivity based control designs,” IEEE Trans. on Control Syst. Technol., Vol. 4, No.3, pp. 292–297, May. 1996.

• Z. P. Jiang, D. J. Hill, Y. Guo, “Stabilization and tracking via output feedback for the nonlinear benchmark system,” Automatica, Vol. 34, No. 7, pp. 907–915, July 1998.

• Z. P. Jiang, I. Kanellakopoulos, “Global output-feedback tracking for a benchmark nonlinear system,” IEEE Trans. on Autom. Control, Vol. 45, No. 5, pp. 1023–1027, May 2000.

• A. Pavlov, B. Janssen, N. van de Wouw, H. Nijmeijer, “Experimental output regulation for a nonlinear benchmark system,” IEEE Trans. Control Syst. Technol., Vol.15, No.4, pp. 786–793, July 2007.

• Y. F. Chen, A. C. Huang, “Controller design for a class of underactuated mechanical systems,” IET Control Theory and Applications, Vol. 6, No. 1, pp. 103–110, Jan. 2012.

• J. Hauser, S. Sastry, G. Meyer, “Nonlinear control design for slightly non-minimum phase systems,” Automatica, Vol. 28, No. 4, pp. 665–679, July 1992.

• R. Olfati-Saber, “Global configuration stabilization for the VTOL aircraft with strong input coupling,” IEEE Trans. Autom. Control, Vol. 47, No. 11, pp. 1949–1952, Nov. 2002.

• P. Setlur, D. Dawson, Y. Fang, B. Costic, “Nonlinear tracking control of the VTOL aircraft,” Proc. of the Decision and Control Conf., Orlando, USA, 2001, pp. 4592–4597.

• A. Zavala-Rio, I. Fantoni, R. Lozano, “Global stabilization of a PVTOL aircraft model with bounded inputs,” Int. J. Control, Vol. 76, No. 18, pp. 1833–1844, Dec. 2003.

• R. Lozano, P. Castillo, A. Dzul, “Global stabilization of the PVTOL: real-time application to a mini-aircraft,” Int. J. Control, Vol. 77, No. 8, pp. 735–740, May 2004.

• R. Wood, B. Cazzolato, “An alternative nonlinear control law for the global stabilization of the PVTOL vehicle,” IEEE Trans. Autom. Control, Vol. 52, No. 7, pp. 1282–1287, July 2007.

245

Contd.

• R. Xu, U. Ozguner, “Sliding mode control of a class of underactuated systems,” Automatica, Vol. 44, No. 1, pp. 233–241, Jan. 2008.

• P. Martin, S. Devasia, B. Paden, “A different look at output tracking: control of a VTOL aircraft,” Automatica, Vol. 32, No. 1, pp. 101–107, Jan. 1996.

• F. Lin, W. Zhang, R. D. Brandt, “Robust hovering control of a PVTOL aircraft,” IEEE Trans. on Control Syst. Technol., Vol. 7, No. 3, pp. 343–351, May 1999.

• S. A. Al-Hiddabi, N. H. McClamroch, “Tracking and maneuver regulation control for nonlinear non-minimum phase systems: application to flight control,” IEEE Trans. on Control Syst. Technol., Vol. 10, No. 6, pp. 78-0–792, Nov. 2002.

• L. Marconi, A. Isidori, A. Serrani, “Autonomous vertical landing on an oscillating platform: an internal-model based approach,” Automatica, Vol. 38, No. 1, pp. 21–32, Jan. 2002.

• K. D. Do, Z. P. Jiang, J. Pan, “On global tracking control of a VTOL aircraft without velocity measurements,” IEEE Trans. Autom. Control, Vol. 48, No. 12, pp. 2212–2217, Dec. 2003.

• G. Notarstefano, J. Hauser, R. Frezza, “Trajectory manifold exploration for the PVTOL aircraft,” Proc. of IEEE conference on decision and control and European control conference, Seville, Spain, 2005, pp. 5848–5853.

• L. Consolini, M. Tosques, “On the VTOL exact tracking with bounded internal dynamics via a Poincar´emap approach,” IEEE Trans. Autom. Control, Vol. 52, No. 9, pp. 1757–1762, Sept. 2007.

• L. Consolini, M. Maggiore, C. Nielsen, M. Tosques, “Path following for the PVTOL aircraft,” Automatica, Vol. 46, No. 8, pp. 1284–1296, Aug. 2010.

• A. Behal, D. M. Dawson, W. E. Dixon, Y. Fang, “Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics,” IEEE Trans. Autom. Control, Vol. 47, No. 3, pp. 495–500, Mar. 2002.

• K. Y. Pettersen, F. Mazenc, H. Nijmeijer, “Global uniform asymptotic stabilization of an underactuated surface vessel: Experimental results,” IEEE Trans. Control Syst. Technol., Vol. 12, No. 6, pp. 891–903, Nov. 2004.

• K. Y. Wichlund, O. J. Sordalen, and O. Egeland, "Control of vehicles with second-order nonholonomic constraints: Underactuated vehicles," Proc. European Control Conf., 1995, pp. 3086 -3091.

246

Contd.

• W. Dong, Y. Guo, “Global time-varying stabilization of underactuated surface vessel,” IEEE Trans. Autom. Control,

Vol. 50, No. 6, pp. 859–864, Jun. 2005.

• J. M. Godhavn, “Nonlinear tracking of underactuated surface vessels,” in Proc. 35th Conf. Decision Control, Kobe,

Japan, Dec. 1996, pp. 975–980.

• J. Ghommam, F. Mnif, A. Benali, N. Derbel, “Asymptotic backstepping stabilization of an underactuated surface

vessel,” IEEE Trans. Control Syst. Technol., Vol. 14, No.6, pp. 1150–1157, Nov. 2006.

• K. Y. Pettersen and H. Nijmeijer, “Global practical stabilization and tracking for an underactuated ship—A

combined averaging and backstepping approach,” in Proc. IFAC Conf. Syst. Structure Control, Nantes, France,

Jul. 1998, pp. 59–64.

• W. Dong, J. A. Farrell, “Formation control of multiple underactuated surface vessels,” IET Control Theory and

Applications, Vol. 2, No. 12, pp. 1077–1085, Dec. 2008.

• J. Ghommam, F. Mnif, N. Derbel, “Global stabilization and tracking control of underactuated surface vessels,” IET

Control Theory Applications, Vol. 4, No. 1, pp. 71–88, Jan. 2010.

• A. P. Aguiar, J. P. Hespanha, “Trajectory-tracking and path-following of underactuated autonomous vehicles with

parametric modeling uncertainty,” IEEE Trans. Autom. Control, Vol. 52, No. 8, pp. 1362–1379, Aug. 2007.

• F. Fahimi, “Sliding-mode formation control for underactuated surface vessels,” IEEE Trans. Robotics, Vol. 23, No.

3, pp. 617–622, Jun. 2007.

• R. Yu, Q. Zhu, G. Xia, Z. Liu, “Sliding mode tracking control of an underactuated surface vessel,” IET Control

Theory and Applications, Vol. 6, No. 3, pp. 461–466, Feb. 2012.

• O. Egeland, M. Dalsmo, O. J. Sordalen, “Feedback control of a nonholonomic underwater vehicle with constant

desired configuration,” Int. J. of Robotics Research, Vol. 15, No. 1, pp. 24–35, Feb. 1996.

• F. Bullo, “Nonlinear control of mechanical systems: A Riemannian geometry approach,” Ph.D. thesis, Department

of Control and Dynamical Systems, California Institute of Technology, 1998.

• A. de Luca, B. Siciliano, “Trajectory control of a non-linear one-link flexible arm,” Int. J. of Control, Vol. 50, No, 5,

pp. 1699–1715, Nov. 1989.

247

Contd.

• W. Dong, Y. Guo, “Global time-varying stabilization of underactuated surface vessel,” IEEE Trans. Autom. Control,

Vol. 50, No. 6, pp. 859–864, Jun. 2005.

• J. M. Godhavn, “Nonlinear tracking of underactuated surface vessels,” in Proc. 35th Conf. Decision Control, Kobe,

Japan, Dec. 1996, pp. 975–980.

• J. Ghommam, F. Mnif, A. Benali, N. Derbel, “Asymptotic backstepping stabilization of an underactuated surface

vessel,” IEEE Trans. Control Syst. Technol., Vol. 14, No.6, pp. 1150–1157, Nov. 2006.

• K. Y. Pettersen and H. Nijmeijer, “Global practical stabilization and tracking for an underactuated ship—A

combined averaging and backstepping approach,” in Proc. IFAC Conf. Syst. Structure Control, Nantes, France,

Jul. 1998, pp. 59–64.

• W. Dong, J. A. Farrell, “Formation control of multiple underactuated surface vessels,” IET Control Theory and

Applications, Vol. 2, No. 12, pp. 1077–1085, Dec. 2008.

• J. Ghommam, F. Mnif, N. Derbel, “Global stabilization and tracking control of underactuated surface vessels,” IET

Control Theory Applications, Vol. 4, No. 1, pp. 71–88, Jan. 2010.

• A. P. Aguiar, J. P. Hespanha, “Trajectory-tracking and path-following of underactuated autonomous vehicles with

parametric modeling uncertainty,” IEEE Trans. Autom. Control, Vol. 52, No. 8, pp. 1362–1379, Aug. 2007.

• F. Fahimi, “Sliding-mode formation control for underactuated surface vessels,” IEEE Trans. Robotics, Vol. 23, No.

3, pp. 617–622, Jun. 2007.

• R. Yu, Q. Zhu, G. Xia, Z. Liu, “Sliding mode tracking control of an underactuated surface vessel,” IET Control

Theory and Applications, Vol. 6, No. 3, pp. 461–466, Feb. 2012.

• O. Egeland, M. Dalsmo, O. J. Sordalen, “Feedback control of a nonholonomic underwater vehicle with constant

desired configuration,” Int. J. of Robotics Research, Vol. 15, No. 1, pp. 24–35, Feb. 1996.

• F. Bullo, “Nonlinear control of mechanical systems: A Riemannian geometry approach,” Ph.D. thesis, Department

of Control and Dynamical Systems, California Institute of Technology, 1998.

• A. de Luca, B. Siciliano, “Trajectory control of a non-linear one-link flexible arm,” Int. J. of Control, Vol. 50, No, 5,

pp. 1699–1715, Nov. 1989.

248

Contd.

• G. Oriolo, T. Nakamura, “Control of mechanical systems with second-order nonholonomic constraints:

Underactuated manipulators,” Proc. of the Decision and Control Conf., Brighton, UK, 1991, pp. 2398–2403.

• R. Martinez, J. Alvarez, Y. Orlov, “Hybrid sliding-mode-based control of underactuated systems with dry frictions,”

IEEE Trans. on Industrial Electronics, Vol. 55, No. 11, pp. 3998–4003, Nov 2008.

• R. M. A. de Luca, S. Iannitti, G. Oriolo, “Control problems in underactuated manipulators,” Proc. of the IEEE/ASME

Int. Conf. on Advanced Intelligent Mechatronics, Como, Italy, 2001, pp. 8–12.

• Y. Liu, H. Yu, T. C. Yang, “Analysis and control of a Capsubot,” Proc. of IFAC World Congress, Seoul, Korea, 2008,

pp. 756–761.

• R. Olfati-Saber, “Normal forms for underactuated mechanical systems with symmetry,” IEEE Tran. Autom. Control,

Vol. 47, No. 2, pp. 305–308, Feb. 2002.

• M. Zhang, T. Tarn, “Hybrid control of the Pendubot,” IEEE/ASME Trans. on Mechatronics, Vol. 7, No. 1, pp. 79–86,

Mar. 2002.

• R. Olfati-Saber, “Trajectory tracking for a flexible one-link robot using a nonlinear non-collocated output,” Proc. of

the Decision and Control Conf., Sydney, Australia, 2000, pp. 4024–4029.

• K. Y. Pettersen, H. Nijmeijer, “Tracking control of an underactuated surface vessel,” Proc. of the Decision and

Control Conf., Tampa, Florida, USA, 1998, pp. 4561–4566.

• L. C. McNich, H. Ashrafiuon, K. R. Muske, “Sliding mode set-point control of an underactuated surface vessel:

simulation and experiment,” Proc. of the American Control Conf., Baltimore, MD, USA, 2010, pp. 5212–5217.

• A. Shiriaev, H. Ludvigsen, O. Egeland, A. Pogromsky, “On global properties of passivity based control of the

inverted pendulum,” Proc. Decision and Control Conf., Phoenix, Arizona, USA, 1999, pp. 2513–2518.

• A. Gao, X. Zhang, H. Chen, J. Zhao, “Energy-based control design of an underactuated 2-dimensional TORA

system,” Proc. of the IEEE/RSJ International Conf. on Intelligent Robots and Systems, St. Louis, USA, 2009, pp.

1296–1301.

• M. W. Spong, J. K. Holm, D. Lee, Passivity-based control of bipedal locomotion, IEEE Robotics and Automation

Magazine 14, (2007), 30–40.

249

Contd.

• A. vander Schaft, “L2-gain analysis of nonlinear systems and nonlinear state feedback H∞ control,” IEEE Trans

Automatic Control, vol. 37, no. 6, pp. 770–784, Jun. 1992.

• A. van der Schaft, L2-Gain and Passivity in Nonlinear Control, New York: Springer-Verlag, 1999.

• Z. P. Jiang, “Global tracking control of underactuated ships by Lyapunov direct method,” Automatica, vol. 38, no. 2,

pp. 301–309, Feb. 2002.

• C. Rui, M. Reyhanoglu, I. Kolmanovsky, S. Cho, and N. H. McClamroch, "Non-smooth stabilization of an

underactuated unstable two degrees of freedom mechanical system," Proc. 36th IEEE Conf. Decision Control,

1997, pp. 3998 -4003.

• W. N. White, M. Foss, X. Guo, “A direct Lyapunov approach for a class of underactuated mechanical systems,” in

Proc. of the American Control Conf., Minneapolis, Minnesota, USA, 2006, pp. 103–110.

• P. V. Kokotovic, M. Krstic, I. Kmellakopoulos, “Backstepping to passivity: Recursive design of adaptive systems,” in

Proc. of the 31st Conference on Decision and Control, Tucson, Arizona, USA, 1992, pp. 3276–3280.

• K. D. Do, Z. P. Jiang, J. Pan, “Underactuated ship global tracking under relaxed conditions,” IEEE Trans. on

Autom. Control, Vol. 47, No. 9, pp. 1529–1536, 2002.

• J. Ghommam, F. Mnif, N. Derbel, "Global stabilisation and tracking control of underactuated surface vessels," IET

Control Theory & Applications, Volume 4, Issue 1, pp. 71 – 88, January 2010.

• F. Y. Bi, Y. J. Wei, J. Z. Zhang, W. Cao, “Position-tracking control of underactuated autonomous underwater

vehicles in the presence of unknown ocean currents,” IET Control Theory Applications, Vol. 4, No. 11, pp. 2369–

2380, 2010.

• A. P. Aguiar and J. P. Hespanha, “Position tracking of underactuated vehicles,” in Proc. 2003 Amer. Contr. Conf.,

vol. 3, Denver, CO, Jun. 2003, pp. 1988-1993.

• A. Abdessameuda, A. Tayebi, “Global trajectory tracking control of VTOL-UAVs without linear velocity

measurements,” Automatica, vol. 46, no. 6, pp. 1053–1059, 2010.

• A. Gruszka, M. Malisoff, F. Mazenc, “On tracking for the PVTOL model with bounded feedbacks,” in: Proc. of the

American Control Conf., San Francisco, CA, USA, 2011, pp. 1428–1433.

250

Contd.

Can’t reach any position Outside the track

Holonomic System

`Nonholonomic systemWith velocity constraint

No velocity componentBut the position is reachable

`Nonholonomic systemWith acceleration

constraint

Position is reachable

May have a sidewise velocity component too

But no sidewiseacceleration

251