tracking control of second-order chained form systems by cascaded

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2003; 13:95–115 (DOI: 10.1002/rnc.709)

Tracking control of second-order chained form systems bycascaded backstepping

N. P. I. Aneke*,y, H. Nijmeijerz and A. G. de Jager}

Dynamics and Control Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600 MB Eindhoven, The Netherlands

SUMMARY

A design methodology is presented for tracking control of second-order chained form systems. Themethodology separates the tracking-error dynamics, which are in cascade form, into two parts: a linearsubsystem and a linear time-varying subsystem. The linear time-varying subsystem, after the firstsubsystem has converged, can be treated as a chain of integrators for the purposes of a backsteppingcontroller. The two controllers are designed separately and the proof of stability is given by using a resultfor cascade systems. The method consists of three steps. In the first step we apply a stabilizing linear statefeedback to the linear subsystem. In the second step the second subsystem is exponentially stabilized byapplying a backstepping procedure. In the final step it is shown that the closed-loop tracking dynamics ofthe second-order chained form system are globally exponentially stable under a persistence of excitationcondition on the reference trajectory. The control design methodology is illustrated by application to asecond-order non-holonomic system. This planar manipulator with two translational and one rotationaljoint (PPR) is a special case of a second-order non-holonomic system. The simulation results show theeffectiveness of our approach. Copyright # 2002 John Wiley & Sons, Ltd.

KEY WORDS: chained form; underactuated systems; cascaded systems; tracking, backstepping

1. INTRODUCTION

In recent years, control of non-holonomic systems has received increasing interest in the controlliterature [1]. However, the studies are primarily limited to the stabilization and tracking of first-order non-holonomic systems, or systems satisfying non-integrable kinematic or velocityconstraints. In Reference [2] it is shown that certain second-order non-holonomic systems have astructural obstruction to the existence of smooth time-invariant stabilizing feedback laws; theydo not meet Brockett’s well-known necessary condition for smooth time-invariant feed-back stabilization [3]. For these systems the feedback stabilization problem can only be solvedby smooth time-varying feedback, non-smooth time-varying homogeneous feedback or

Received 2 October 2000Published online 30 September 2002 Revised 18 February 2002Copyright # 2002 John Wiley & Sons, Ltd. Accepted 1 March 2002

*Correspondence to: N. P. I. Aneke, Eindhoven University of Technology, Department of Mechanical Engineering,Dynamics and Control Group, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

yE-mail: [email protected]: [email protected]}E-mail: [email protected]

discontinuous feedback. It should be noted that there are examples of second-order non-holonomic systems that can be stabilized by smooth time-invariant feedback. The planarvertical take-off and landing vehicle (PVTOL) problem [4] is such an example. Similarly, itfollows that for smooth feedback tracking additional constraints on the desired trajectory arerequired, see e.g. [5].

In this contribution we restrict ourselves to second-order non-holonomic systems, orsystems satisfying non-integrable dynamic or acceleration constraints [6]. The tracking controlproblem for second-order non-holonomic systems has hardly received any attention. Weconsider the tracking control problem for a specific class of second-order non-holonomicsystems. This class consists of second-order non-holonomic systems that can be transformedby co-ordinate and feedback transformation into the second-order chained form. Second-order non-holonomic systems can arise by imposition of design constraints on the allowablemotions of redundant manipulators. Such systems can also arise as models of underactuatedsystems, or systems having more degrees of freedom than actuators. For example, inunderactuated manipulators the unactuated joints give rise to second-order non-holonomicconstraints.

It is known that certain mechanical systems can be transformed into second-orderchained form. For example, planar underactuated manipulators such as the planar horizontalPPR [7], the RRR [8] manipulators and the 3-dof redundant manipulator [9] can be transformedinto second-order chained form. Also, the PVTOL aircraft [10] can be transformed into a systemthat is equal to the second-order chained form, up to some constant drift term resulting fromgravity. We expect this to be true for other types of underactuated manipulators orunderactuated mechanical systems as well. By transforming the system into second-orderchained form the dynamics are considerably simplified, and thus facilitates control design.Systems that can be written in terms of the second-order chained form include the idealizedmodels of mechanical systems such as underactuated surface vessels [11] and underwatervehicles [12].

In particular, the underactuated manipulators are interesting from a control pointof view because they allow the control of more joints than actuators and thereforereduce weight, energy consumption and cost of manipulators. They are also interesting forapplications that involve hitting or hammering an object, because the impact may causeno damage to the joint actuators. In addition, the dynamics of an underactuated mani-pulator is equivalent to those of dynamic manipulation [13], where the object is loosely graspedor simply pushed by the robot. Control methodologies for underactuated manipulators also adda fail-safe protection or fault tolerance to robotics applications in, for example, space robotics.In the case of actuator failure it may still be possible to perform certain tasks with the givenmanipulator by using control methodologies that are developed for underactuated manip-ulators.

To our knowledge the few existing results for tracking control of underactuated manipulatorsare given in References [13–16]. In Reference [14] time-varying controllers were developed tolocally exponentially stabilize the system to a desired trajectory. The system is linearized alongthe desired trajectory and thus results in a linear time-varying system. This linear time-varyingsystem can be exponentially stabilized if it is uniformly completely controllable on intervals offixed length. If the system is not uniformly completely controllable on these intervals, it mightbecome so if the interval length is increased. Similarly, controllability can be lost if the intervallength is decreased. For simple trajectories, such as straight lines, the exponentially stabilizing

Copyright # 2002 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2003; 13:95–115

N. P. I. ANEKE, H. NIJMEIJER AND A. G. DE JAGER96

controller can be obtained in closed-form, but for more general trajectories the controller has tobe computed numerically. The main drawbacks of this approach is the fact that the controllerscompletely depend on the trajectory to be tracked, and have to be recalculated when thetrajectory changes. Moreover, the parameters have to be chosen such that the system linearizedalong the desired trajectory is uniformly completely controllable on intervals of a specifiedlength.

In Reference [13] smooth time-varying feedback controllers were given for tracking of simpletranslational and rotational movements of an underactuated robot manipulator. Thesetranslational and rotational motions allow the execution of rest-to-rest motions. In Reference[15] a discontinuous and flatness-based tracking controller could be derived for a class oftrajectories of the second-order chained form system. Finally, in Reference [16] trackingcontrollers were designed by using dynamic feedback linearization.

The previously mentioned controllers achieve either: (1) local exponential stability of thetracking error dynamics [14], (2) exponential convergence towards the desired trajectory[15] and therefore are no stabilizers, or (3) only stabilize certain trajectories [13] or trajectoriesthat avoid certain singularities. In this contribution we try to design tracking controllersthat can track trajectories that pass to singularities and also achieve more than just alocal stability result. The trajectory tracking problem for the second-order chained formsystem will be solved by using a combined cascade and backstepping approach. Thesecond-order chained form is an extension of the chained form [17] and exhibitssecond-order, instead of first-order, derivatives. Cascaded systems are defined inReference [18] and the backstepping approach [19] followed in this contribution has somesimilarities with the approach in Reference [5]. Our approach distinguishesitself from flatness-based approaches such as in References [15,16] because we considerarbitrary trajectories that might be passing through points where the endogenous transforma-tion generated by the flat output is singular. In these singularities, which turn outto be interesting points of operation, the state of the system can not be obtained byinversion of the endogenous transformation and flatness-based approaches fail. Moreover,the resulting linear time-varying feedback controllers globally asymptotically stabilizethe second-order chained form system to the desired trajectory with exponentialconvergence.

The approach followed in this contribution consists of three steps and results in a linear time-varying tracking controller. The second-order chained form system being in cascade form, westart with applying a linear stabilizing feedback to the first subsystem. In the second step thesecond subsystem is exponentially stabilized by applying a backstepping procedure. In the finalstep we make conclusions on the exponential stability of the closed-loop tracking dynamics ofthe second-order chained form system by using a result for cascaded systems from Reference[20], which was based on the result in Reference [18].

The paper is organized as follows. In Section 2 the trajectory tracking problem for second-order chained form is formulated. Section 3 presents the control design methodology and inSection 4 it is shown that the closed-loop system satisfies our design objective. In Sections 5 and6 a planar horizontal underactuated PPR manipulator example is used as a benchmark to testthe effectiveness of our control methodology. Section 5 introduces a 3-link planar underactuatedmanipulator which is used in the simulation study of section. Conclusions are drawn andrecommendations for further research are given in Section 7. A preliminary version of this paperwas presented in Reference [21].


TRACKING CONTROL BY BACKSTEPPING 97

2. PROBLEM FORMULATION

Consider the second-order chained form system given by

.xx1 ¼ u1

.xx2 ¼ u2

.xx3 ¼ x2u1 ð1Þ

Define x ¼ ðx1; x2; x3Þ: Suppose that we want the system to follow a predefined trajectory, i.e. wewant the state ðx; ’xxÞ to follow a prescribed path ðxd ; ’xxd Þ: This reference trajectory xd thus satisfiesa system of equations given by

.xx1d ¼ u1d

.xx2d ¼ u2d

.xx3d ¼ x2du1d ð2Þ

The tracking-error x 2 R6 is given by x ¼ ½x11; x12; x21; x22; x31; x32�T where

xi1 ¼ xi � xid ; xi2 ¼ ’xxi � ’xxid ð3Þ

The tracking-error dynamics in state-space form are derived from (1), (2) and are given by

’xx11 ¼ x12 ’xx12 ¼ u1 � u1d

’xx21 ¼ x22 ’xx22 ¼ u2 � u2d

’xx31 ¼ x32 ’xx32 ¼ x21u1d þ x2ðu1 � u1d Þ ð4Þ

Problem 1 (State feedback tracking control problem)The tracking control problem is solvable if we can design appropriate continuous time-varyingstate feedback controllers of the form

u1 ¼ u1ðt; x; %uud Þ; u2 ¼ u2ðt; x; %uudÞ ð5Þ

such that the closed-loop system (4), (5) is globally uniformly asymptotically stable. Thevector %uud contains ud ¼ ½u1d ; u2d � and higher-order derivatives up to some order k; i.e. %uud ¼ðud ; ’uud ; . . . ; u

ðkÞd Þ:

Remark 1The second-order chained form system (1) is differentially flat [22] and, therefore, any systemthat can be written in terms of the second-order chained form is necessarily flat. Possible flatoutputs are ðx1; x3Þ; but a singularity in the endogenous transformation generated by the flatoutputs occurs at .x1x1 ¼ 0: This makes it difficult or even impossible to solve the stabilization ortracking problem near points where .xx1d ¼ 0: However, in the case of first-order non-holonomicsystems the flatness property has been successfully used for motion planning [22]. For a car withn-trailers, a trajectory and its corresponding inputs connecting two points could be generated byusing a geometric interpretation of the ‘rolling without slipping’ condition and the use of the



Fr!eenet formula, see also References [23,24]. It is still unclear if these results can be extended tounderactuated manipulators or the second-order chained form system.

In the case of second-order non-holonomic systems, it still remains an open problem how toavoid the singularity in the endogenous transformation (generated by the flat outputs). InReference [16] a flatness-based trajectory tracking controller has been developed for trajectoriesof a second-order non-holonomic system that do not cross the singularity. Therefore the initialconditions have to be chosen carefully in order to avoid singularities during the motion.

In this paper, an alternative to flatness-based approaches is presented. In this approach it ispossible to track trajectories that pass through singularities in the endogenous transformationgenerated by the flat outputs. The approach uses the fact that the system can be transformedinto the second-order chained form. This transformation of the dynamic equations, however,may have configuration singularities where the transformation is not valid. These configurationsingularities must be avoided by the trajectory to be tracked. The results in this paper aretherefore only valid on a subspace where the co-ordinate transformation is well-defined. In thefollowing section we investigate whether the tracking-error dynamics can be asymptoticallystabilized.

3. CASCADED BACKSTEPPING CONTROL

In this section we apply a cascade design to stabilize the equilibrium x ¼ 0 of the error dynamics(4). We start by rewriting the tracking dynamics into a more convenient form.

D1

’xx31 ¼ x32;

’xx32 ¼ x21u1d þ x2ðu1 � u1d Þ;

(D2

’xx21 ¼ x22

’xx22 ¼ u2 � u2d

(

D3

’xx11 ¼ x12

’xx12 ¼ u1 � u1d

(ð6Þ

Suppose that the subsystem D3 is stabilized by a controller u1ðu1d ; x11; x12Þ: Then x12 � 0 andtherefore u1 � u1d � 0: We design the remaining input u2 such that the remaining subsystemðD1;D2Þ is stabilized for u1 � u1d � 0: In order to make conclusions on the exponential stabilityof the complete closed-loop system we use a result for cascaded systems from References [20,25]that was based on Reference [18].

Consider the cascaded system with equilibrium ðz1; z2Þ ¼ ð0; 0Þ given by

’z1z1 ¼ f1ðt; z1Þ þ gðt; z1; z2Þz2

’z2z2 ¼ f2ðt; z2Þ ð7Þ

where z1 2 Rn; z2 2 Rm; f1ðt; z1Þ is continuously differentiable in ðt; z1Þ and f2ðt; z2Þ; gðt; z1; z2Þ arecontinuous in their arguments and locally Lipschitz in z2 and ðz1; z2Þ; respectively. The totalsystem (7) is a system S1 with state z1 that is perturbed by the state z2 of the system S2; where

S1: ’z1z1 ¼ f1ðt; z1Þ; S2: ’z2z2 ¼ f2ðt; z2Þ ð8Þ

and the perturbation term is given by gðt; z1; z2Þz2: If S2 is asymptotically stable, z2 tends to zeroand the dynamics of z1 reduces to S1: If S1 is also asymptotically stable we may investigate



whether this implies asymptotic stability of the cascaded system (7). We state the followingresult from Reference [20].

Theorem 1 (Lefeber et al. [20])The cascaded system (7) is globally uniform asymptotically stable (GUAS) if the following threeassumptions hold:

(1) S1 Subsystem: the subsystem ’z1z1 ¼ f1ðt; z1Þ is GUAS and there exists a continuouslydifferentiable function V ðt; z1Þ :Rþ � Rn ! R and positive definite proper functionsW1ðz1Þ and W2ðz1Þ such that

ðiÞ W1ðz1Þ4V ðt; z1Þ4W2ðz1Þ; 8t5t0;8z1 2 Rn

ðiiÞ@V@t

þ@V@z1

f1ðt; z1Þ40; 8jjz1jj5Z

ðiiiÞ@V@z1

��

�� jjz1jj4zV ðt; z1Þ; 8jjz1jj5Z ð9Þ

where z > 0 and Z > 0 are constants.(2) Interconnection: the function gðt; z1; z2Þ satisfies

jjgðt; z1; z2Þjj4k1ðjjz2jjÞ þ k2ðjjz2jjÞjjz1jj; 8t5t0 ð10Þ

where k1;k2 :Rþ ! Rþ are continuous functions.(3) S2 Subsystem: the subsystem ’z2z2 ¼ f2ðt; z2Þ is GUAS and satisfiesZ 1

t0

jjz2ðt0; t; z2ðt0ÞÞjjdt4zðjjz2ðt0ÞjjÞ; 8t050 ð11Þ

where the function zð�Þ is a class K function.

Lemma 2 (cf. Panteley et al. [26])If in addition to the assumptions in Theorem 1 both S1 and S2 are globally K-exponentiallystable, then the cascaded system (7) is globally K-exponentially stable.

Remark 2In (6) the perturbation term gðt; z1; z2Þz2 can be written as ðx21 þ x2d Þðu1 � u1d Þ; and depends onthe, to be designed, feedback u1ðt; xÞ: When considering D3 as the unperturbed subsystem S2; inorder to satisfy condition (2) in Theorem 1 the resulting perturbation matrix gðt; z1; z2Þ has to belinear with respect to the variable z1 ¼ ðx21; x22; x31; x32Þ: This is the case when choosing thefeedback u1 ¼ u1d þ kðx11; x12Þ with k:R2 ! R a linear function in ðx11; x12Þ:

3.1. Stabilization of subsystem D1

Suppose that the D3 subsystem has been stabilized by choosing

u1 ¼ u1d � k1x11 � k2x12; k1 > 0; k2 > 0 ð12Þ



where the polynomial pðlÞ ¼ l2 þ k1lþ k2 is Hurwitz. The time-varying subsystem D1 withu1 � u1d � 0 can be written as

’xx31 ¼ x32

’xx32 ¼ x21u1d ð13Þ

We aim at designing a stabilizing feedback x21 for subsystem (13). This stabilizing feedback isdesigned using a backstepping procedure in which x21 is a virtual input. First we need to makesome assumptions on the reference input signal u1d :

Assumption 1Assume that the function u1d ðtÞ :Rþ ! R is uniformly bounded in t and smooth ðC1Þ:Moreover, assume that u1d ðtÞ is persistently exciting, i.e. there exist an integer r50 such thatthere exist d > 0; e1 > 0 and e2 > 0 for which

e14Z tþd

tu2rþ21d ðtÞ dt4e2; 8t50 ð14Þ

Consider the first equation ’xx31 ¼ x32 and assume that x32 is the virtual input. A stabilizingfunction x32 ¼ a1ðx31Þ for the x31-subsystem is

a1ðu1dðtÞ; x31Þ ¼ �c1u2d1þ21d x31; c1 > 0; d1 2 N

Define %xx32 ¼ x32 � a1ðx31Þ ¼ x32 þ c1u2d1þ21d x31 and consider the %xx32-subsystem

’%xx%xx32 ¼ x21u1d þ c1u2d1þ21d x32 þ c1ð2d1 þ 2Þu2d1þ1

1d ’uu1dx31

Suppose that x21 is the virtual input, then a stabilizing function x21 ¼ a2ð %uu1d ; x31; x32Þ for the %xx32-subsystem is given by

a2ð %uu1d ; x31; x32Þ ¼ � ðc21u4d1þ31d %xx31 � c1ð2d1 þ 2Þu2d11d ’uu1d Þ %xx31

� ðc1u2d1þ11d þ c2u

2d2þ11d Þ %xx32

¼ � ðc1c2u2d1þ2d2þ31d þ c1ð2d1 þ 2Þu2d11d ’uu1dÞx31

� ðc1u2d1þ11d þ c2u

2d2þ11d Þx32; c2 > 0; d2 2 N ð15Þ

where we have substituted %xx32 ¼ x32 þ c1u2d1þ21d x31: Define %xx31 ¼ x31 and %xx21 ¼ x21 � a2ð %uu1d ; x31;

x32Þ: The closed-loop system becomes

’%xx%xx31 ¼ �c1u2d1þ21d %xx31 þ %xx32

’%xx%xx32 ¼ �c2u2d2þ21d %xx32 þ %xx21u1d ð16Þ

Under Assumption 1 we can prove that the closed-loop system given by

’%xx%xx31 ¼ �c1u2d1þ21d %xx31 þ %xx32

’%xx%xx32 ¼ �c2u2d2þ21d %xx32 ð17Þ

is globally exponentially stable (GES). This is shown in Proposition 4, by applying the followinglemma and some basic theory for linear time-varying systems [27]. The influence of the term%xx21u1d on the stability of system (16) will be considered in Section 4.



Lemma 3Suppose that Assumption 1 holds for some d > 0 and e1; e2 > 0: There exist %mm1; %mm2 2 R

such that

t � t0d

e1 þ %mm14Z t

t0

u2rþ21d ðtÞ dt4

t � t0d

e2 þ %mm2; 8t5t0 ð18Þ

ProofFor t ¼ t0 the result is trivial. Consider the situation t > t0 and divide the interval ½t0; t� into Nequal subintervals of length d and a subinterval of length 04m15d: For any t > t0; N is thenearest integer towards zero of the positive real number ðt � t0Þd

�1; i.e.

N ¼t � t0d

j k; m1 ¼

t � t0d

� N

This division into N subintervals ½t0 þ id; t0 þ ðiþ 1Þd�; i ¼ 0; . . . ;N � 1 with length d and oneinterval ½t0 þ Nd; t� of length m1d yieldsZ t

t0

u2rþ21d ðtÞ dt ¼

XN�1

i¼0

Z t0þðiþ1Þd

t0þidu2rþ21d ðtÞ dtþ m2 ð19Þ

where the difference 04m24e2 between the integral over ½t0; t� and the integral over ½t0; t0 þ Nd�is given by

m2 ¼Z t

t0þNdu2rþ21d ðtÞ dt

Substitution of (14) into the summation term in (19) gives

Ne1 þ m24Z t

t0

u2rþ21d ðtÞ dt4Ne2 þ m2

Define %mmi ¼ m2 � m1ei; i ¼ 1; 2: Then substitution of N ¼ ðt � t0Þd�1 � m1 into the previous

equation yields the desired result (18). &

Proposition 4Consider system (17). Suppose that the reference input u1dðtÞ satisfies the persistence ofexcitation condition (14) for some r50: Suppose that minðd1; d2Þ5r holds. Then the equilibriumx ¼ 0 is globally exponentially stable (GES).

ProofOne can easily verify that the general solution of the linear time-varying system (17) with initialcondition %xxðt0Þ ¼ ½ %xx31ðt0Þ; %xx32ðt0Þ� is given by

%xx31ðtÞ ¼ %xx31ðt0Þ exp �c1

Z t

t0

u2d1þ21d ðtÞ dt

� �þ GðtÞ

%xx32ðtÞ ¼ %xx32ðt0Þ exp �c2

Z t

t0

u2d2þ21d ðtÞ dt

� �ð20Þ



where the additional term GðtÞ is given as

GðtÞ ¼Z t

t0

%xx32ðt0Þ exp �c1

Z t

su2d1þ21d ðtÞ dt� c2

Z s

t0

u2d2þ21d ðtÞ dt

� �ds ð21Þ

We can use Lemma 3 to prove the exponential stability of the solution ð %xx31; %xx32Þ: Using (18) insolution (20) yields

j %xx31ðtÞj4j %xx31ðt0Þj expð�c1 %mm1Þ exp �c1e1d

ðt � t0Þ� �

þ jGðtÞj

j %xx32ðtÞj4j %xx32ðt0Þj expð�c2 %mm1Þ exp �c2e1d

ðt � t0Þ� �

Define the coefficients g1 ¼ e1c1=d; g2 ¼ e1c2=d and j1 ¼ expð�c1 %mm1Þ; j2 ¼ expð�c2 %mm1Þ: Using(18) in (21) gives

jGðtÞj4j1j2j %xx32ðt0Þj expð�ðg1t � g2t0ÞÞZ t

t0

expð�ðg2 � g1ÞsÞ ds; 8t5t0

We distinguish two cases; c1 ¼ c2 and c1=c2: (1) In the case c1 ¼ c2 we have g1 ¼ g2 and theperturbation term GðtÞ satisfies

jGðtÞj4j1j2j %xx32ðt0Þj expð�g1ðt � t0ÞÞðt � t0Þ; 8t5t0

The general solution %xx ¼ ð %xx31; %xx32Þ then satisfies the inequality

j %xx31ðtÞj4j1j %xx31ðt0Þj expð�g1ðt � t0ÞÞ þ j1j2j %xx32ðt0Þjðt � t0Þ expð�g1ðt � t0ÞÞ4y1ðtÞ

j %xx32ðtÞj4j2j %xx32ðt0Þj expð�g1ðt � t0ÞÞ4y2ðtÞ

where yðtÞ ¼ ðy1ðtÞ; y2ðtÞÞ is the solution of the linear time-invariant system

’yy ¼�g1 1

0 �g2

" #y; yðt0Þ ¼ ðj1j %xx31ðt0Þj;maxðj1j2j %xx32ðt0Þj;j2j %xx32ðt0ÞjÞÞ

Since j %xxðtÞj4jyðtÞj; 8t5t0 and the solution yðtÞ is exponentially stable, we conclude that thelinear time-varying system (13) is also exponentially stable. (2) In the case c1=c2 we have thatg1=g2 and the perturbation term GðtÞ satisfies

jGðtÞj4j1j2jx32ðt0Þj

g1 � g2ðexpð�g2ðt � t0ÞÞ � expð�g1ðt � t0ÞÞÞ

The general solution %xx ¼ ð %xx31; %xx32Þ then satisfies the inequality

j %xx31ðtÞj4z1 expð�g1ðt � t0ÞÞ þ z2 expð�g2ðt � t0ÞÞ

j %xx32ðtÞj4z3 expð�g2ðt � t0ÞÞ

where we defined

z1 ¼ j1j %xx31ðt0Þj � j1j2

j %xx32ðt0Þjg1 � g2

� �; z2 ¼ j1j2

j %xx32ðt0Þjg1 � g2

; z3 ¼ j2j %xx32ðt0Þj

The equilibrium %xx ¼ 0 is thus globally exponentially stable, i.e.

jjxðtÞjj4kjjxðt0Þjj expð�gðt � t0ÞÞ; k ¼j1

r1r1g1�g2

0 j2

24

35

��

��



with g ¼ minfg1; g2g: Concluding, we have shown that the linear time-varying system (13) isexponentially stable in both the cases c1=c2 and c1 ¼ c2: This concludes the proof. &

Concluding, Proposition 4 states that the subsystem D1 in closed-loop with thelinear time-varying feedback (15) is GES when the D3 subsystem has been stabilized, i.e.u1 � u1d � 0: In other words, system (13) can be stabilized by the linear time-varyingfeedback (15).

3.2. Stabilization of subsystem ðD2Þ

In Section 3.1 the D1-subsystem with u1 ¼ u1d has been exponentially stabilized by thevirtual input x21 ¼ a2ð %uu1d ; x31; x32Þ given by (15). The virtual input x21 is a state of the system D2

and we continue the backstepping procedure for the D2-subsystem as follows. Denote the stateof the D1-subsystem by *xx ¼ ½x31; x32�: Define %xx21 ¼ x21 � a2ð %uu1d ; *xxÞ and consider the %xx21-subsystem

’%xx%xx21 ¼ x22 �d

dt½a2ð %uu1d ; *xxÞ�

where x22 denotes a new virtual input. We define a new variable %xx22 ¼ x22 � a3ð %uu1d ; %xx21; *xxÞwhere

a3ð %uu1d ; %xx21; *xxÞ ¼ �c3 %xx21 þd

dt½a2ð %uu1d ; *xxÞ�

The %xx21-subsystem is then given by ’%xx%xx21 ¼ �c3 %xx21 þ %xx22: Consider the %xx22-subsystem

’%xx%xx22 ¼ ðu2 � u2dÞ �d

dt½a3ð %uu1d ; %xx21; *xxÞ�

This subsystem can be stabilized by choosing the input u2 as

u2 � u2d ¼ � c4 %xx22 þd

dt½a3ð %uu1d ; %xx21; *xxÞ�

¼ � c3c4x21 � ðc3 þ c4Þx22 þ c3c4a2ð %uu1d ; *xxÞ þ ðc3 þ c4Þd

dt½a2ð %uu1d ; *xxÞ�Þ

þd2

dt2½a2ð %uu1d ; *xxÞ� ð22Þ

The closed-loop D2 subsystem then becomes

’%xx%xx21 ¼ �c3 %xx21 þ %xx22

’%xx%xx22 ¼ �c4 %xx22 ð23Þ

4. STABILITY OF TRACKING-ERROR DYNAMICS

In this section we show that the complete tracking dynamics are globally exponentially stable. Inthe previous sections we have stabilized the ðD1;D2Þ-subsystem when u1 ¼ u1d and the D3

subsystem in (6). We can now use Theorem 1 to investigate the stability properties of thecomplete system. The result is stated in the following proposition.



Proposition 5Consider the system (6) and the controllers u1 given by

u1 ¼ u1d � k1x11 � k2x12; pðsÞ ¼ s2 þ k2sþ k1 is Hurwitz ð24Þ

and u2 given by (22). Suppose that the reference input u1dðtÞ satisfies Assumption 1and the parameters d1 and d2 satisfy minðd1; d2Þ5r: If the signal xd2ðtÞ and the derivative ’uu1din (2) are uniformly bounded in t; then the closed-loop system is globally K-exponentiallystable.

ProofThe closed-loop system ((6), (24), (22)), using (16) and (23), is given by

’%xx%xx31 ¼ � c1u2d1þ21d %xx31 þ %xx32

’%xx%xx32 ¼ � c2u2d2þ21d %xx32 þ %xx21u1d � x2ðk1x11 þ k2x12Þ

’%xx%xx21 ¼ � c3 %xx21 þ %xx22

’%xx%xx22 ¼ � c4 %xx22

’xx11 ¼ x12

’xx12 ¼ � k1x11 � k2x12

The closed-loop system can be written in form (7) with

z1 ¼ ½ %xx31; %xx32�T; f1ðt; z1Þ ¼ A1ðtÞz1

z2 ¼ ½ %xx21; %xx22; x11; x12�T; f2ðt; z2Þ ¼ A2z2 ð25Þ

and the matrices A1ðtÞ;A2 are given by

A1ðtÞ ¼�c1u

2d1þ21d ðtÞ 1

0 �c2u2d2þ21d ðtÞ

" #; A2 ¼

�c3 1 0 0

0 �c4 0 0

0 0 0 1

0 0 �k1 �k2

2666664

3777775

The perturbation matrix gðt; z1; z2Þ is given by

gðt; z1; z2Þ ¼ �ðx21 þ x2dÞ0 0 0 0

0 0 k1 k2

" #þ

0 0 0 0

u1dðtÞ 0 0 0

" #ð26Þ

In order to apply Theorem 1 we verify the three assumptions.

(1) Due to Assumption 1 and Proposition 4 the S1 subsystem, i.e. (17), is GES. By converseLyapunov theory, i.e. Theorem 3.12 in Reference [28], a suitable Lyapunov functionV ðt; z1Þ is guaranteed to exist when the matrix A1ðtÞ is uniformly bounded in t: Byassumption the reference input u1d is uniformly bounded and therefore also the matrixA1ðtÞ; which gives the desired result.



(2) By assumption the signals u1d ; ’uu1d and x2d are bounded, i.e. ju1d ðtÞj4M1; j ’uu1dðtÞj4M2; andjx2d ðtÞj4M3 8t: Therefore, we have

jjgðt; z1; z2Þjj4jjkjjðjx21j þ jxd2 jÞ þ ju1d j4jjkjjðjx21j þM3Þ þM1

where k ¼ ½k1; k2�: Furthermore, using the states %xx21 ¼ x21 � a2ðu1d ; x31; x32Þ; %xx31 ¼ x31and %xx32 ¼ x32 þ c1u

2d1þ21d x31 from the backstepping procedure in Sections 3.1 and 3.2 yields

jx21j ¼ j %xx21 � ðc21u4d1þ31d � c1ð2d1 þ 2Þu2d11d ’uu1d Þ %xx31 � ðc1u

2d1þ11d þ c2u

2d2þ11d Þ %xx32j

Using the boundedness of u1d and ’uu1dðtÞ yields the inequality

jx21j4 j %xx21j þ ðc21M4d1þ31 þ c1ð2d1 þ 2ÞM2d1

1 M2Þj %xx31j

þ ðc1M2d1þ11 þ c2M

2d2þ11 Þj %xx32j

4 jjz2jj þ ðc21M4d1þ31 þ c1ð2d1 þ 2ÞM2d1

1 M2 þ c1M2d1þ11 þ c2M

2d2þ11 Þjjz1jj

Introducing the continuous function k1ðjjz2jjÞ ¼ jjkjjðjz2j þM3Þ þM1 and the parameterk2 ¼ jjkjjðc21M

4d1þ31 þ c1ð2d1 þ 2ÞM2d1

1 M2 þ c1M2d1þ11 þ c2M

2d2þ11 Þ; this finally gives the

desired result

jjgðt; z1; z2Þjj4k1ðjjz2jjÞ þ k2jjz1jj

(3) The characteristic polynomial of the S2 subsystem is given by wðsÞ ¼ ðsþ c1Þðsþ c2ÞpðsÞwhere pðsÞ is given in (24). Because the polynomial pðsÞ is Hurwitz and the ci’s are positive,the S2 subsystem is GES. The existence of a class K function zð�Þ satisfying condition (11)follows directly from the GES of the S2 subsystem.

By Theorem 1 and Lemma 2 we conclude K-exponentially stability of the completeclosed-loop system. &

Summarizing, we have exponentially stabilized the ðD1;D2Þ and D3 subsystems separately. Wethen concluded by Theorem 1 and Lemma 2 that the combined system isK-exponentially stablewhen the reference input u1d satisfies Assumption 1 and its derivative ’uu1d is uniformly boundedin t:

5. AN EXAMPLE

Consider a three-link underactuated planar manipulator where the first two prismatic ortranslational joints are actuated and the third revolute or rotational joint is not actuated. Weassume that both prismatic joints and the revolute joint are frictionless. Figure 1 shows the PPRplanar manipulator. This system is a special case of an underactuated manipulator, i.e. amanipulator having more degrees of freedom than actuators. The unactuated degrees offreedom give rise to non-integrable dynamic or acceleration constraints that are also known assecond-order non-holonomic constraints. Examples of underactuated mechanical systems are



underactuated robot manipulators [7], the 3-dof redundant manipulator [9], underactuatedsurface vessels [29], underwater vehicles [30], the planar vertical takeoff and landing aircraft [4],the Acrobot [31] and the inverted pendulum [32].

Underactuated manipulators are interesting from a control point of view because they allowthe control of more joints than actuators and therefore reduce weight, energy consumption andcost of manipulators. They are also interesting for applications that involve hitting orhammering an object, because the impact may cause no damage to the joint actuators. Inaddition, the dynamics of an underactuated manipulator is equivalent to those of dynamicmanipulation [13], where the object is loosely grasped or simply pushed by the robot. Controlmethodologies for underactuated manipulators also add a fail-safe protection or fault toleranceto robotics applications in, for example, space robotics. In the case of actuator failure it may stillbe possible to perform certain tasks with the given manipulator by using control methodologiesthat are developed for underactuated manipulators.

In References [2,33] it is shown that planar underactuated manipulators have a structuralobstruction to the existence of smooth time-invariant stabilizing feedback laws; they do notmeet Brockett’s well-known necessary condition for smooth time-invariant feedback stabiliza-tion [3]. Moreover, due to the absence of gravity the linear approximation is not controllableand equilibria cannot be locally stabilized by smooth time-invariant state feedback. It can alsobe shown that there is no continuous time-invariant state feedback that solves the globalstabilization problem [32].

By Theorem 1 in Reference [2], it can be deduced that underactuated manipulators arestrongly accessible. In some cases it is even possible to prove a stronger controllability propertysuch as small time local controllability (STLC). In Reference [15] it is shown that the n linkplanar underactuated manipulator is completely controllable if the first joint is actuated and notcompletely controllable if the first joint is unactuated. In Reference [34] the underactuatedplanar PPR manipulator is shown to be controllable by constructing inputs to generate atrajectory from any given initial state to any given final state in finite time. The trajectoriesconsist of simple translations and rotations together with combined translational and rotational

ry

rx

y

l�

CG

x

Figure 1. A planar PPR manipulator.



movements. Also, feedback controllers [35] were presented for the translational and rotationalmovements, based on exact linearization and time-scaling, respectively.

The underactuated PPR manipulator example will be used as a benchmark to test theeffectiveness of our control methodology. The dynamics are significantly simplified by assumingthat there are no un-modelled dynamics, in particular all joints are assumed to be frictionless,and no parameter uncertainties are present. This underactuated PPR manipulator bears aresemblance to the ‘planar skater’ example in Reference [36], which is a two-dimensionalrestriction of the three-body robot satellite considered in that reference.

5.1. Equations of motion

Define q ¼ ½rx; ry ; y�; where ðrx; ryÞ denotes the displacement of the third joint from the originðx; yÞ ¼ 0 and y denotes the orientation of the third link with respect to the positive x-axis, seeFigure 1. Let tx; ty denote the inputs to the prismatic actuated joints. The mass and centralmoments of inertia of the links are denoted by mi and Ii; respectively, and l denotes the distancebetween the centre of mass of the third link and the third joint. The dynamic equations ofmotion can be written as

mx .rrx � m3l sinðyÞ.yy� m3l cosðyÞ’yy2¼ tx

my .rry þ m3l cosðyÞ.yy� m3l sinðyÞ’yy2¼ ty

I .yy� m3l sinðyÞ.rrx þ m3l cosðyÞ.rry ¼ 0 ð27Þ

where the configuration variables q lie in a, to be defined, configuration-space C � R3: We alsointroduced the parameters mx ¼ ðm1 þ m2 þ m3Þ; my ¼ m2 þ m3 and I ¼ I3 þ m3l2; see Refer-ence [7]. The non-holonomic constraint of the system (27) can be written as

l.yyðtÞ � .rrxðtÞ sin yðtÞ þ .rryðtÞ cos yðtÞ ¼ 0 ð28Þ

where l ¼ I=ðm3lÞ equals the effective pendulum length when the third link is treated as a rigid-body pendulum suspended under the passive joint. The first-order linear approximation of (27)is not controllable since the dynamics are not influenced by gravity [13].

5.2. Transformation into second-order chained form

In Reference [7] a co-ordinate and feedback transformation ðx; ’xxÞ ¼ Fðq; ’qqÞ with x ¼ ðx1; x2; x3Þand t ¼ Oðq; ’qqÞ with t ¼ ðtx; tyÞ are proposed to transform the system (27) into the second-orderchained form. Let x ¼ 0 correspond to the equilibrium q ¼ 0: The mapping F : ðq; ’qqÞ 2C� R3 ! ðx; ’xxÞ 2 R6 then follows from the relations

x1 ¼ rx þ lðcos y� 1Þ

x2 ¼ tanðyÞ

x3 ¼ ry þ l sin y ð29Þ



The feedback transformation O : ðq; ’qqÞ 2 C� R3 ! t 2 R2 is given by

tx

ty

" #¼

�m3l cos y’yy2þ mx �

I

l2sin2 y

� �vx þ

I

l2sin y cos y

� �vy

�m3l sin y’yy2þ

I

l2sin y cos y

� �vx þ my �

I

l2cos2 y

� �vy

26664

37775 ð30Þ

where vx and vy are new inputs given by

vx

vy

" #¼

cos y sin y

sin y �cos y

" # u1cosðyÞ

þ l’yy2

lðu2 cos2 y� 2’yy2tan yÞ

264

375

When y ¼ p=2� kp; k 2 N this co-ordinate transformation is not well-defined. The configura-tion-space C of the configuration co-ordinates q ¼ ðrx; ry ; yÞ is thus given by

C ¼ fðrx; ry ; yÞ 2 R3 j y 2 ð�p=2þ kp;p=2þ kpÞg ð31Þ

The co-ordinate transformation (29) from local co-ordinates q 2 C to local co-ordinates x 2 R3 isa diffeomorphism. The dynamics of the underactuated manipulator are transformed into thesecond-order chained form system

.xx1 ¼ u1

.xx2 ¼ u2

.xx3 ¼ x2u1 ð32Þ

Applying the co-ordinate transformation to the non-holonomic constraint (28), directly showsthat it is transformed into the constraint

.xx3 ¼ x2 .xx1 ð33Þ

The non-holonomic constraint (28) is thus preserved under the co-ordinate and feedbacktransformation.

6. SIMULATION

Our control objective is to move the end-point of the third link along a prescribed path, i.e. thestate ðqðtÞ; ’qqðtÞÞ should follow a prescribed trajectory ðqdðtÞ; ’qqd ðtÞÞ where qdðtÞ 2 C 8t: Considerthe reference second-order chained form system (2) subject to the sinusoidal inputs

u1dðtÞ ¼ �r1a2 sinðatÞ; u2d ðtÞ ¼ �r2a2 cosðatÞ ð34Þ

With initial conditions given by

ðxd ð0Þ; ’xxdð0ÞÞ ¼ ½x1d ð0Þ; r2; x3dð0Þ; r1a; 0; ar1r2=4� ð35Þ

where x1d ð0Þ and x3d ð0Þ still have to be chosen, the closed-form solution xd is given by

x1dðtÞ ¼ x1dð0Þ þ r1 sinðatÞ; x2dðtÞ ¼ r2 cosðatÞ

x3dðtÞ ¼ x3dð0Þ þr1r28

sinð2atÞ ð36Þ



The reference input u1d is continuously differentiable and its derivative ’uu1d and xd2 are uniformlybounded in t: Moreover, the persistence of excitation condition (14) is satisfied with r ¼ 0: Wetherefore conclude by Proposition 5 that the tracking controller (22.24) uniformly exponentiallystabilizes the tracking-error dynamics (4) for any positive values of d1 and d2:

Remark 3Note that the reference trajectory (36) crosses singularities .xx1d ðtiÞ ¼ 0 of the endogenoustransformation generated by the flat outputs, cf. ðx1; x3Þ; at time instances ti ¼ kp=a: Moreover,the second-order chained form system (32) is not controllable and the persistence of excitationcondition (32) is not satisfied when .xx1d ðtÞ ¼ 0; 8t50:

The resulting reference trajectory qd for the mechanical system (27) is given by

rxdðtÞ ¼ x1d ð0Þ þ r1 sinðatÞ � lðcosðarctanðr2 cosðatÞÞÞ � 1Þ

ryd ðtÞ ¼ x3dð0Þ þr1r28

sinð2atÞ � l sinðarctanðr2 cosðatÞÞÞ

ydðtÞ ¼ arctanðr2 cosðatÞÞ ð37Þ

We consider a trajectory around the origin ðrx; ryÞ ¼ 0 and select x1dð0Þ ¼ 0; x3d ð0Þ ¼ 0 andr1 ¼ r2 ¼ a ¼ 1 in the reference trajectory (37). The controller u1 is given by (24). We choosed1 ¼ d2 ¼ 0 in the virtual input x21 (15) and d3 ¼ d4 ¼ 0 in the controller u2 (22). The controlparameters are chosen as

k1 ¼ 4; k2 ¼ 2ffiffiffi2

p; c1 ¼ 2; c2 ¼ 2; c3 ¼ 4; c4 ¼ 4

The perturbation term in the D1-subsystem of (6) depends on u1 � u1d : Therefore, inorder to obtain fast convergence to the reference trajectory u1 must converge to the referencevalue u1d fast, which motivates our choice for k1; k2: The parameters c1; c2 determinethe convergence of the chained state x3 and its corresponding mechanical state ry : Theparameters c3; c4 mainly determine the convergence of the state x2 and its correspondingmechanical state y:

The result of tracking the trajectory with initial condition qð0Þ ¼ 0 is shown in Figure 2. Theconfiguration variables of the mechanical system are shown in Figure 3. From the simulation itis clear that it takes about 5 s before the trajectory is successfully tracked.

In Figure 4 we have shown the variables of the second-order chained form system. Initiallyat time t ¼ 0 only a tracking-error exists in the variable x2 that corresponds to the orientationy of the link. The co-ordinates ðx1 þ l; x3Þ denote the position in the base frame of thecentre of percussion of the third link. This centre of percussion can be characterized asbeing the point on the third link that stays at rest in the base frame when the third jointrotates around it at any velocity or acceleration [13]. This point is situated at a distance l fromthe third joint and lies between the centre of gravity and the end-point of the third link, seeFigure 1.

Between approximately t ¼ 0 s and 0:5 s the tracking-error ðx11; x31Þ of the position of thecentre of percussion increases because the reference trajectory has non-zero initial velocity.Between t ¼ 0:5 and 2s the feedback loop decreases this tracking-error to zero. In the meantime,and while the tracking-error for the centre of percussion is zero, the tracking-error x21 of the



variable x2 is being reduced to zero until the reference trajectory xd is perfectly tracked at timet ¼ 5 s: There still remains some freedom of choice in the control parameters and whenconsidering more complex trajectories this freedom of choice can be exploited to improve thetransient response of the underactuated manipulator. Note that the closed-loop tracking-errordynamics of the mechanical system before co-ordinate and feedback transformation are onlyglobally exponentially stable on a subspace where the co-ordinate transformation is well-defined.

7. CONCLUSIONS

We have presented a control design method for exponential tracking of a second-order non-holonomic system. The main contribution of this method is that it provides a solution to thetracking control problem that allows trajectories to be tracked that cross singularities of theflatness-based approach. To our knowledge only tracking controllers have been developed thatachieve local exponential stability [14], or only achieve exponential convergence [15]. InReference [14] the system is linearized along the desired trajectory and the tracking controllerhas to be re-computed when the trajectory changes. For simple trajectories the controller can beobtained in closed form, but for more complicated trajectories, the controller has to becomputed numerically. Then parameters a and d have to be chosen such that the system,linearized along the desired trajectory, is uniformly completely controllable over intervals oflength d: The tracking controllers in Reference [15] can not be used to track trajectories passingto singularities.

t = 0.0 _4.2 s

t = 15.0 _19.2 s t = 19.2 _24.5 s t = 24.5 _30.0 s

t = 4.2 _9.4 s t = 9.42 _15.0 s

Figure 2. Motion of the third joint and link in simulation of Figure 3 ðDt ¼ 0:25 sÞ: The circle ð8Þ representsthe position of the third joint.



The presented linear time-varying state feedback globally exponentially stabilizes thetracking-error dynamics of the second-order chained form system. A necessary condition forglobally exponentially stability is that the reference trajectory is persistently exciting. Thepresented controller can stabilize the system to trajectories passing to singularities of flatness-based approaches and can is given in a closed-form. In the simulations, we considered thetracking control problem of a planar underactuated robot manipulator [7]. This benchmarkexample consists of three links with two actuated translational joints and an unactuated revolutejoint. The simulations show the effectiveness of our approach. The closed-loop tracking-errordynamics of the mechanical system before co-ordinate and feedback transformation are onlyglobally exponentially stable on a subspace where the co-ordinate transformation is well-defined. Moreover, in practice, there will be disturbances and mechanical friction in theunactuated joint that can considerably deteriorate the performance of our controller.

Figure 3. Tracking of the trajectory (37); co-ordinates of the mechanical system (27) with respect to time,rx (solid), ry (dashed), y (dash–dotted), inputs tx (solid), ty (dashed).



In this contribution, the trajectory tracking problem has only been solved for the second-order chained form system with 6 states, i.e. (1). The result can be extended to higher-orderchained form systems, or systems with more than three states or more than two inputs byadapting the backstepping procedure in Section 3.1. This will be shown in a forthcoming paper.Our goal for future research is to extend our results with some experiments.

REFERENCES

1. Kolmanovsky I, McClamroch NH. Developments in non-holonomic control problems. IEEE Control SystemsMagazine 1995; 15:20–36.

2. Reyhanoglu M, van der Schaft AJ, McClamroch NH. Non-linear control of a class of underactuated systems. InProceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996; 1682–1687.

3. Brockett RW. Asymptotic stability and feedback stabilization. In Differential Geometric Control Theory. BrockettRW, Milman RS, Sussmann HJ (eds). Birkhauser: Boston, 1983; 181–191.

Figure 4. Tracking of the trajectory (36); co-ordinates of the second-order chained form system (32), x1(solid), x2 (dashed), x3 (dash–dotted), inputs u1 (solid) u2 (dashed).



4. McClamroch NH, Kolmanovsky I. A hybrid switched mode control for v/stol flight control problems. InProceedings 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996; 2648–2653.

5. Jiang ZP, Nijmeijer H. A recursive technique for tracking control of non-holonomic systems in chained form. IEEETransactions on Automatic Control 1999; 44:265–279.

6. Reyhanoglu M, van der Schaft AJ, McClamroch NH, Kolmanovsky I. Dynamics and control of a class ofunderactuated mechanical systems. In IEEE Transactions on Automatic Control 1999; 44:1663–1671.

7. Imura J, Kobayashi K, Yoshikawa T. Non-holonomic control of a 3 link planar manipulator with a free joint. InProceedings of the 35th IEEE Conference on Decision and Control, vol. 2, Kobe, Japan, 1996; pp. 1435–1436.

8. Yoshikawa T, Kobayashi K, Watanabe T. Design of desirable trajectory with convergent control for 3-d.o.fmanipulator with a non-holonomic constraint. In Proceedings of the 2000 IEEE International Conference on Robotics& Automation, vol. 2, San Fransisco, CA, 2000; 1805–1810.

9. Baillieul J. Kinematically redundant robots with flexible components. IEEE Control Systems Magazine 1993;13(1):15–21.

10. Sastry S. Non-linear Systems, Analysis, Stability and Control. Springer: Berlin, 1999.11. Reyhanoglu M. Control and stabilization of an underactuated surface vessel. In Proceedings 35th IEEE Conference

on Decision and Control, Kobe, Japan, December 1996; 2371–2376.12. Egeland O, Berglund E. Control of an underwater vehicle with non-holonomic acceleration constraints. In

Proceedings of the IFAC Conference on Robot Control, Capri, Italy, 1994; 845–850.13. Arai H, Tanie K, Shiroma N. Non-holonomic control of a three-dof planar underactuated manipulator. IEEE

Transactions on Robotics and Automation 1998; 14:681–695.14. Walsh G, Tilbury D, Sastry S, Murray R, Laumond J-P. Stabilization of trajectories for systems with non-

holonomic constraints. IEEE Transactions on Automatic Control 1994; 39(1):216–222.15. Kobayashi K. Controllability analysis and control design of non-holonomic systems. Ph.D. Thesis, Department of

Mechanical Engineering, Kyoto University, Japan, 1999.16. De Luca A, Oriolo G. Motion planning and trajectory control of an underactuated three-link robot via dynamic

feedback linearization. In Proceedings of the 2000 IEEE International Conference on Robotics & Automation SanFransisco, CA, 2000; 2789–2795.

17. Murray RM, Sastry SS. Non-holonomic motion planning: Steering using sinusoids. IEEE Transactions on AutomaticControl 1993; 38(5):700–716.

18. Panteley E, Lor!ııa A. On global uniform asymptotic stability of non-linear time-varying systems in cascade. Systemsand Control Letters 1998; 33(2):131–138.

19. Krsti!cc M, Kanellakopoulos I, Kokotovi!cc P. Non-linear and Adaptive Control Design. Adaptive and Learning Systemsfor Signal Processing, Communications and Control. Wiley Interscience, Wiley Inc.: New York, 1995.

20. Lefeber E, Robertsson A, Nijmeijer H. Linear controllers for exponential tracking of systems in chained-form.International Journal of Robust and Nonlinear Control, 2000.

21. Aneke NPI, Nijmeijer H, de Jager AG. Trajectory tracking by cascaded backstepping control for a second-ordernon-holonomic mechanical system. In Nonlinear Control in the Year 2000. Lecture Notes in Control and InformationSciences, Isidori A, Lamnabhi-Lagarrigue F, Respondek W (eds). vol. 258, Springer: Paris, 2000; 35–49.

22. Fliess M, L!eevine J, Martin P, Rouchon P. Flatness and defect of non-linear systems: Introductory theory andexamples. International Journal of Control 1995; 61(6):1327–1361.

23. Fliess M, L!eevine J, Martin P, Rouchon P. Non-linear control and lie-b.aacklund transformations: Towards a newdifferential geometric standpoint. In Proceedings of the 33rd Conference on Decision and Control, vol. 1, Lake BuenaVista, FL, December 1994; 339–344.

24. Fliess M, L!eevine J, Martin P, Rouchon P. Design of trajectory stabilizing feedback for driftless flat systems. InProceeding of the 3rd European Control Conference, vol. 3, Rome, Italy, 1995; 1882–1887.

25. Lefeber E, Robertsson A, Nijmeijer H. Linear controllers for tracking chained-form systems. In Stability andStabilization of Non-linear Systems. Lecture Notes in Control and Information Sciences. Aeyels D, Lamnabhi-Lagarrigue F, van der Schaft AJ (eds). vol. 246, Springer: Berlin, 1999; 183–199.

26. Panteley E, Lefeber E, Lor!ııa A, Nijmeijer H. Exponential tracking control of a mobile car using a cascade approach.In Proceedings of the IFAC Workshop on Motion Control, Grenoble, September 1998; 221–226.

27. Rugh WJ. Linear System Theory (2nd edn). Prentice-Hall: Englewood Cliffs, NJ, 1993.28. Khalil HK. Nonlinear systems (2nd edn). Prentice-Hall: Upper Saddle River, New York, 1996.29. Pettersen KY. Exponential stabilization of underactuated vehicles. Ph.D. Thesis, Norwegian University of Science and

Technology, Department of Engineering Cybernetics, 1996.30. Egeland O, Berglund E, S�rdalen OJ. Exponential stabilization of a non-holonomic underwater vehicle with

constant desired configuration. In Proceedings IEEE International Conference on Robotics and Automation, vol. 1,San Diego CA, U.S., May 1994; 20–25.

31. Spong MW. The swing up control problem for the acrobot. IEEE Control Systems Magazine 1995; 15(1):49–55.32. Shiriaev A, Pogromsky A, Ludvigsen H, Egeland O. On global properties of passivity-based control of an inverted

pendulum. International Journal of Robust and Nonlinear Control 2000; 10:283–300.



33. Oriolo G, Nakamura Y. Free-joint manipulators: Motion control under second-order non-holonomic constraints.In Proceedings of the IEEE/RSJ International Workshop on Intelligent Robots and Systems, Intelligence forMechanical Systems, vol. 3, IROS ’91, 1991; 1248–1253.

34. Arai H. Controllability of a 3-dof manipulator with a passive joint under a non-holonomic constraint. InProceedings of the IEEE International Conference on Robotics and Automation 1996; 3707–3713.

35. Arai H, Tanie K, Shiroma N. Feedback control of a 3-dof planar underactuated manipulator. In Proceedings of theIEEE International Conference on Robotics and Automation 1997; 703–709.

36. Walsh G, Sastry S. On reorienting linked rigid bodies using internal motions. IEEE Transactions on Robotics andAutomation 1995; 11(1):139–146.



tracking control of second-order chained form systems by cascaded

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