IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 1

Nonlinear Control of Hydraulic RobotsMohammad Reza Sirouspour, Septimiu E. Salcudean

Abstract| This paper addresses the control problem ofhydraulic robot manipulators. The backstepping designmethodology is adopted to develop a novel nonlinear po-sition tracking controller. The tracking errors are shownto be exponentially stable under the proposed control law.The controller is further augmentedwith adaptation laws tocompensate for parametric uncertainties in the system dy-namics. Acceleration feedback is avoided by using two newadaptive and robust sliding type observers. The adaptivecontrollers are proven to be asymptotically stable via Lya-punov analysis. Simulation and experimental results per-formed with a hydraulic Stewart platform demonstrate thee�ectiveness of the approach.

Keywords|Hydraulic robots, control of robot manipula-tors, adaptive/nonlinear control, backstepping.

I. Introduction

HY draulic robots and machinery are widely used inthe construction and mining industries, as well as

in motion simulators. They have rapid responses andhigh power-to-weight ratios suitable for many applications.High performance controllers can have a signi�cant impacton the e�ectiveness of hydraulic robots. Furthermore, thepotential complexity of such controllers is becoming lessand less of an implementation issue due to the inexpen-sive and powerful processors available today for real-timecontrol.In general, the control of hydraulic manipulators is more

challenging than that of their electrical counterparts. Itmight seem that a potentially e�ective way of increasingthe performance of hydraulic robots is to consider controlmethods that neglect actuator dynamics but incorporatethe manipulator rigid body dynamics, such as computedtorque [1], passivity-based [2], [3], [4], adaptive [2], [5] androbust [6], [7], [8] control methods. However, this is not thecase in general. Unlike their electrical counterparts that re-semble force sources, hydraulic actuators resemble velocitysources. They also exhibit signi�cant nonlinear character-istics. Therefore, the above control methods cannot beapplied e�ectively to hydraulic manipulators as hydraulicactuators cannot accurately apply forces or torques over asigni�cant dynamic range.Actuator dynamic models have been successfully incor-

porated in the controller design for rigid link electricallydriven (RLED) robots to improve position tracking per-formance. The complete dynamics of the manipulators,including their actuators, are third order nonlinear di�er-ential equations. [9] used feedback linearization to linearizeand decouple these dynamics. [10] developed an adaptivecontroller for RLED manipulators that does not requireacceleration feedback. [11] and [12] and other papers also

The authors are with the Robotics and Control Laboratory,Depart-ment of Electrical and Computer Engineering, University of BritishColumbia, Canada. Emails: shahins@ece.ubc.ca, tims@ece.ubc.ca.

considered the adaptive control of RLED robots based onmodels that include actuator dynamics.

While actuator dynamics are generally linear in RLEDrobots and can be ignored in many cases due to their fasttime constants, they are highly nonlinear and dominantin hydraulic manipulators. Therefore, the incorporationof these dynamics in the design of controllers is of criti-cal importance in hydraulic robots. Research in the areaof hydraulic systems has mainly focused on the control ofsingle-rod hydraulic actuators, e.g. see [13], [14], [15]. Inparticular, [13] developed a nonlinear position tracking con-troller for hydraulic servo-systems following the backstep-ping approach. There are only a few papers that addressthe control of robot manipulators driven by hydraulic ac-tuators. In [16], the authors established a simpli�ed modelin a standard form suitable for the application of singularperturbation methods. No experimental or numerical re-sults are presented in this work. A decentralized adaptivecontroller was proposed to control a hydraulic manipulatorin [17], [18]. The use of pressure feedback in the controlof a Stewart type hydraulic manipulator was proposed in[19]. However, these approaches lack stability proofs thatare important from both theoretical and implementationpoints of view. Only recently, simultaneous to this work,[20] proposed a Lyapunov-based adaptive controller for hy-draulic robots.

The backstepping design methodology [21], [22] has be-come increasingly popular in the control community. Forsome recent applications of this method see [23], [24]. Inthis paper, backstepping is adopted to develop a novel non-linear controller for hydraulic manipulators. Both rigidbody and actuator dynamics are incorporated into the de-sign. The controller is also extended to compensate forparametric uncertainties in the system dynamics, includinghydraulic and rigid body dynamics. Two types of observersare developed to avoid the use of acceleration feedback inthe proposed adaptive control laws. The �rst observer is anextension of the passivity-based observers proposed by [3],to the case in which the system parameters are unknown.The concept of sliding observers [25] is also adopted to de-velop a robust acceleration observer. The tracking errorsare proven to converge to zero asymptotically using Lya-punov analysis. It can be shown that these errors remainbounded in the presence of Coulomb friction in the actua-tors. The bounds on the tracking errors are adjustable bythe controller gains.

The main di�erences between this work and the adaptivecontroller introduced in [20] are the following: (i) the adap-tive controller / adaptive observer proposed here uses thesame set of estimated rigid body parameters in the observerand controller, as opposed to the use of two distinct setsof parameter estimates and adaptation laws in [20]; and

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 2

(ii) the introduction of an adaptive control method witha robust observer that is simpler to implement because ithas reduced computational complexity. The form of thecontrol laws and the observers are di�erent from those of[20].

Position, velocity and hydraulic pressure measurementsare required for the implementation of the proposed con-trollers. Simulation and experimental results for a hy-draulic Stewart platform are presented to show the e�ec-tiveness of the approach.The paper is organized as follows. System dynamics, in-

cluding rigid body and hydraulic dynamics are presented inSection II. In Section III a nonlinear controller is proposedassuming that the dynamics are known exactly. The adap-tive control of hydraulic robots is addressed in Section IVfor the cases in which the robot dynamics are subject toparametric uncertainty. In Section V simulation results arepresented. The experimental evaluation of the controllersis discussed in Section VI. Finally, conclusions are drawnin Section VII.

II. Manipulator/Actuators Dynamics

The dynamics of an n-link robot with rigid links are gov-erned by a second-order nonlinear di�erential equation

D(q)�q + C(q; _q) _q +G(q) = � (1)

where q 2 Rn is a vector of generalized joint positionsand � 2 Rn is a vector of generalized joint torques. D(q) 2Rn�n is the manipulatormass matrix, C(q; _q) 2 Rn�n con-tains coriolis and centripetal terms and G(q) 2 Rn repre-sents gravitational e�ects. Unlike electrically driven ma-nipulators, hydraulic robots exhibit signi�cant nonlinearactuator dynamics. Assuming a three-way valve con�gura-tion, these dynamics can be written in the following form,

_� = f(q; _q) + g(q; �; u) (2)

where u is the control command vector and f , g are non-linear functions of q, _q and � . The detailed expressions forf and g are given in Appendix A.

The matrices describing the rigid body dynamics in (1)satisfy the following properties [3]:

(i) xT�_D(q)� 2C(q; _q)

�x = 0 8x 2 Rn

(ii) C(q; x)y = C(q; y)x 8q; x; y 2 Rn

(iii) 9Dm; DM s:t: 0 < Dm � kD(q)k � DM <18q 2 Rn

(iv) 9CM s:t: kC(q; x)k � CMkxk 8q; x 2 Rn

(v) 9GM s:t: kG(q)k � GM 8q 2 Rn

(3)

which are exploited in deriving the proposed control laws.According to (1) and (2), the overall actuator/manipulatordynamics are governed by a set of third-order nonlineardi�erential equations.

III. Nonadaptive Controller

In this section, the backstepping design methodology [21]is adopted to derive a nonlinear position tracking controllerfor hydraulic manipulators in the case in which the systemparameters are known.

Theorem 1: Consider the system described by (1), (2) withthe control law given by the solution u of the followingalgebraic equation:

g(q; �; u) = �f(q; _q)� ��1s + _�d �K� ~� (4)

and

�d = D(q)�qr + C(q; _q) _qr + G(q)�Kpe�Kds (5)

with

e = q � qd; _qr = _qd � �e;

s = _q � _qr = _e +�e; ~� = � � �d(6)

where Kp, Kd, K� , � and � are positive de�nite diagonalmatrices, and qd 2 Rn and _qd 2 Rn are the desired jointposition and velocity trajectories, respectively.Then 0 is an exponentially stable equilibrium point for

the state ~x =�eT sT ~�T

�Tof (1),(2),(4),(5).

Remark: From the expression of g from Appendix A, it canbe seen that (4) can be easily solved for u.

Proof: Substituting (5) into (1) yields the following errordynamics,

D(q) _s +C(q; _q)s +Kds +Kpe = ~� (7)

Note that the e�ect of actuator dynamics emerges as anon-zero ~� , as the controller reduces to a passivity-basedcontroller [26] in the absence of actuator dynamics. Let V1be de�ned as

V1 =1

2sTD(q)s +

1

2eTKpe (8)

It can be shown that the derivative of V1 along trajectoriesof the closed loop system becomes

_V1 = �sTKds � eT�Kpe+ sT ~� (9)

where (7) and the properties given in (3) have been used.Following the backstepping methodology, V2, which is aLyapunov function for the closed-loop system, is de�ned as

V2 = V1 +1

2~�T�~� (10)

with � > 0 diagonal. Note that:

�mk~xk2 < V2 < �Mk~xk2; �m; �M > 0 (11)

By taking the derivative of (10) and using the control law(4), one can write

_V2 = �sTKds� eT�Kpe� ~�T�K� ~� < ��k~xk2 (12)

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 3

with � > 0. Therefore, the system is exponentially stable inthe Lyapunov sense. This means that the position trackingerror converges to zero exponentially. Furthermore, sinces = _e+�e, the velocity tracking error is also exponentiallystable. Note that in the realization of (4) one needs tocompute _�d which is equal to

_�d = _D(q)�qr +D(q)...q r +C(q; _q)�qr + _C(q; _q; �q) _qr

+ _G(q) �Kp _e�Kd _s(13)

Since...q r =

...q d � ��e, _s = �e + �_e and _C are functions of

�q, link accelerations appear in the proposed control law.However, if � is measured through pressure sensors, thelink accelerations �q can be obtained from position, velocityand pressure measurements using

�q = D(q)�1 [� � C(q; _q) _q � G(q)] (14)

Thus q, _q and � are required to implement the proposedcontrol law that leads to exponentially stable tracking er-rors.

Remark: Since the system dynamics are fully known andthe states are assumed to be measured, feedback lineariza-tion could also be used to derive a stabilizing controller.[9] adopted this approach to develop a controller for electri-cally driven manipulators in the presence of linear actuatordynamics.

IV. Adaptive Controller

The control law derived in the previous section requiresfull knowledge of the system parameters. However, themanipulator rigid body dynamics are uncertain and sub-ject to changes, e.g. due to an unknown variable payload.It is also di�cult to measure some of the manipulator's pa-rameters. Moreover, the hydraulic parameters are usuallyunknown and time varying. In this section, the nonlinearcontroller proposed in Section III is extended to compen-sate for parametric uncertainties in the system dynamics.To deal with uncertainties in rigid body dynamics, the lin-ear parameterization of manipulator dynamics is used [26]:

D(q)�q + C(q; _q) _q + G(q) = Y (q; _q; �q)� (15)

where Y (q; _q; �q) is a regressor matrix and � 2 Rm is thevector of unknown parameters. Similarly, as shown in Ap-pendix A, the hydraulic dynamics (2) can be written as

_� = f0(q; _q) 1 + g0(q; �; u) 2 (16)

where 1 =� 11 � � � n1

�T, 2 =

� 12 � � � n2

�Tare two

sets of hydraulic parameters and f0, g0 are de�ned as

f0(q; _q) = diagff i0(qi; _qi)gg0(q; �; u) = diagfgi0(qi; � i; ui)g

(17)

In the non-adaptive controller, (14) was used to computejoint accelerations from joint positions and velocities andhydraulic pressure measurements. This can not be done

if D(q), C(q; _q) and G(q) are not known. To deal withthis problem, novel adaptive and robust observers are in-troduced. The following Lemma [27] will be used in thestability proofs.

Lemma 1: Consider the scalar function � = (�� �)T (�� _�),

with �; �; � 2 Rn and ai � �i � bi.

If _� = �(a; b; �)�, where �(a; b; �) is a diagonal matrix

with entries

�i(a; b; �) =

8><>:0 if �i � ai, �i � 0

0 if �i � bi, �i � 0

1 otherwise

(18)

then � � 0.

A. Adaptive Controller/ Adaptive Observer

The �rst solution is an adaptive controller using an adap-tive passivity-based observer. Before stating the result, thefollowing notation must be de�ned:

_qr = _qd � �1(q � qd) = _qd � �1(e � ~q)

_qo = _q � �2(q � q) = _q � �2~q

s1 = _q � _qr = _e+ �1(e � ~q)

s2 = _q � _qo = _~q +�2~q

(19)

where q 2 Rn is the estimated value of q, e = q � qd,and ~q = q� q are position tracking and observation errors,respectively. �1;�2 > 0 are diagonal. Note that in thede�nitions of _qr and _qo, _q has been replaced by _qd and _q.This will be shown to eliminate the need for accelerationfeedback.

Theorem 2: Consider the system described by (1), (2), theobserver dynamics

_q = z + �2~q

_z = D(q)�1[� � C(q; _q) _qo � G(q) + Lp~q +Kds1

+K0

ds2]

(20)

and the controller obtained by solving the following alge-braic equation

g0(q; �; u) 2 = _�d � f0(q; _q) 1 � ���1s1 �K� ~� (21)

where

�d = D(q)�qr + C(q; _qr) _qr + G(q) �Kd(s1 � s2)�Kpe

= Y1(q; _qr; �qr)� �Kd(s1 � s2)�Kpe

(22)

with unknown rigid body parameter adaptation law

_� = �����1

�Y T1 (q; _qr; �qr)s1 + Y T

2 (q; _q; _qo; �qo)s2�

(23)

where

Y1(q; _qr; �qr)� = D(q)�qr + C(q; _qr) _qr + G(q)

Y2(q; _q; _qo; �qo)� = D(q)�qo + C(q; _q) _qo + G(q)(24)

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 4

and with hydraulic parameter adaptation laws

_ 1 = � 1��1 1��f0(q; _q)~�

_ 2 = � 2��1 2���(

~�

2)�_�d � f0(q; _q) 1 � ��

�1s1 �K� ~��

(25)

where �( ~� 2) = diagf ~�i

2i g. Then, if the condi-

tions given below in (26) are satis�ed, 0 is an asymp-totically stable equilibrium point of the state ~x =�eT ~qT sT1 sT2 ~�T

�T.

(a) �(Kp)�(Lp)�(�1)�(�2) >1

4��2(Kp)��

2(�1)

(b) k~x(0)k �s

�m

3�M

��(Kd) �CM _qdm

CM ��(�1)

�2� VpM � Vpm

�M

(26)

where

�m =1

2minfDm; �(Kp); �(Lp); �(�� )g

�M =1

2maxfDM ; ��(Kp); ��(Lp); ��(�� )g

Vpm � 1

2

�~�T�~� + ~ T1 � 1 ~ 1 + ~ T2 � 2~ 2

�� VpM

(27)

Here, ��(:) and �(:) denote the maximum and minimumsingular values of their matrix argument, respectively, and_qdm is an upper bound on the norm of the desired veloc-ity. The projection gains ��, � 1 and � 2 are de�ned asin (18). All of the gains used in the controller and ob-server, i.e. Kp;Kd;K

0

d; Lp;�;� 1;� 2 ;�� ;K� are constantpositive de�nite diagonal matrices.

Remark 1: The inequality given in (26.b) speci�es theboundary of the attraction region that can be enlarged byadjusting the controller gains, i.e. Kd and Kp. There-fore, the closed loop system is semiglobally asymptoticallystable. While the controller is guaranteed to be stable,in practice the parameters should be tuned to achieve thedesired performance.

Remark 2: In the above formulation, D, C and G are the es-timated dynamical matrices corresponding to �. Note thatthe controller and observer use the same set of estimatedparameters which compares favorably to the approach pro-posed in [20], in which di�erent parameter estimates areemployed in the controller and observer.

Remark 3: The use of projection gains �, in the adap-tation laws guarantees that the estimate of each parame-ter remains in a prede�ned interval [a; b]. In particular, if 2

i becomes zero the control law u in (21) is unde�ned.This can be avoided by using ai > 0 for the estimationof i2. Furthermore, the parameter estimates can not driftbecause of the upper and lower bounds on their values.Therefore, parameter adaptation is robust against unmod-eled disturbances [27].

Proof : By substituting (22) into (1) the following errordynamics are obtained

D(q) _s1 +C(q; _q)s1 +Kds1 +Kpe =

Kds2 � C(q; s1)( _q � s1) � Y1(q; _qr; �qr)~� + ~� (28)

The observer closed-loop dynamics can also be written as

D(q) _s2 +C(q; _q)s2 +K0

ds2 + Lp~q =

�Kds1 � Y2(q; _q; _qo; �qo)~� (29)

where (1) and (20) have been used in deriving (29).Now, let the Lyapunov-like function V1 be de�ned as:

V1 =1

2s1

TD(q)s1 +1

2eTKpe+

1

2sT2D(q)s2

+1

2~qTLp~q +

1

2~�T�~�

(30)

It can be shown that the derivative of V1 along the trajec-tory of the closed loop system is then given by

_V1 = �s1TKds1 � s2TK0

ds2 � eTKp�1e � ~qTLp�2~q

+ eTKp�1~q + s1T ~� + ~�T [�� _� � Y T

1 (q; _qr; �qr)s1

� Y T2 (q; _q; _qo; �qo)s2]� s1

TC(q; s1)( _q � s1)

(31)

With the adaptation law given in (23) and using Lemma1, we have that

_V1 � �s1TKds1 � s2TK0

ds2 � eTKp�1e� ~qTLp�2~q

+ eTKp�1~q + sT1 ~� � sT1 C(q; s1)( _qd � �1e+ �1~q)

� � (�(Kd)� CM( _qdm + ��(�1)kek + ��(�1)k~qk)) ks1k2

� �(K0

d)ks2k2 � �(Kp)�(�1)kek2 � �(Lp)�(�2)k~qk2

+ ��(Kp)��(�1)kekk~qk+ s1T ~�

= H(kek; k~qk; ks1k; ks2k) + sT1 ~�

(32)

Note that since

��(Kp)�(�1)kek2��(Lp)�(�2)k~qk2+��(Kp)��(�1)kekk~qk

= � � kek k~qk � � �(Kp)�(�1) �12��(Kp)��(�1)

�12 ��(Kp)��(�1) �(Lp)�(�2)

�� kekk~qk

�(33)

the condition given in (26.a) and the inequality

kek+ k~qk < �(Kd)� CM _qdmCM��(�1)

(34)

guarantee that

H(kek; k~qk; ks1k; ks2k) � ��(kek2 + k~qk2 + ks1k2 + ks2k2)(35)

with � > 0. It is not di�cult to show that if (26.b) holdsthen (34) is also satis�ed.

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 5

Following the backstepping approach, V2, which is a Lya-punov function for the system dynamics, is de�ned as

V2 = V1 +1

2~�T�� ~� +

1

2~ T1 � 1~ 1 +

1

2~ T2 � 2 ~ 2 (36)

where ~ 1 =�~ 11 : : : ~ n1

�Tand ~ 2 =

�~ 12 : : : ~ n2

�Tare the vectors of hydraulic parameter errors. By takingthe derivative of (36) and employing the control law givenin (21), one can show after some manipulation that

_V2 � H(kek; k~qk; ks1k; ks2k)� ~�T��K� ~�

+ ~ T1

hf0(q; _q)�� ~� � � 1

_ 1

i+ ~ T2 [���(

~�

2)

( _�d � f0(q; _q) 1 � ���1s1 �K� ~� )� � 2 _ 2]

(37)

Using the adaptation laws given in (25) and Lemma 1, thederivative of V2 becomes

_V2 � H(kek; k~qk; ks1k; ks2k)� ~�T��K� ~�

� � (kek2 + k~qk2 + ks1k2 + ks2k2 + k~�k2) (38)

Thus the position and velocity tracking errors convergeto zero asymptotically.

Remark 1: For parameter convergence the condition of per-sistency of excitation must be satis�ed [26].

Remark 2: Inspection of (22) reveals that �d does not con-tain _q. This means that acceleration term �q does not ap-pear in _�d and hence in the control law. This is achievedby the particular de�nition of _qr in (19) and also by usings1 � s2 = _qo � _qr instead of s1 in (22). In summary, theproposed controller requires q, _q and � to be measured.

Remark 3: In order to implement the observer proposed in(20), D�1(q) must exist. This can be guaranteed by choos-ing bounds on the estimates of the rigid body parameters.

B. Adaptive Controller/ Robust Observer

In this subsection, an adaptive/nonlinear controller uti-lizing a sliding type observer [25] is proposed that yieldsglobally asymptotically stable tracking errors.Before stating the result the following variables should

be de�ned:

_qr = _qd � �e; s = _q � _qr = _e +�e (39)

and � > 0 is diagonal.

Theorem 3: Consider the system described by (1),(2) andthe following observer:

_z = �o _~q +�osgn(_~q) �WT (q; _qr; �)s + ��q (40)

with

W (q; _qr; �) = �D(q)� + C(q; _qr)�Kd (41)

��q = �D�1�� � �C _q � �G

�(42)

where z = _q is the observed velocity. �D, �C and �G are con-stant matrices (rough estimates of dynamical matrices).Let the control law be given by the solution u of the fol-lowing algebraic equation

g0(q; �; u) 2 = _�d � f0(q; _q) 1 � ���1s�K� ~� (43)

where

�d = D(q)(�qr +�_~q) + C(q; _q) _qr + G(q)�Kd(s� _~q)�Kpe

(44)

and let the adaptation laws be given by

_� = �����1� Y T (q; _q; _qr; �qr)s (45)

and

_ 1 = � 1��11 ��f0(q; _q)~�

_ 2 = � 2��12 ���(

~�

2)�_�d � f0(q; _q) 1 � ��

�1s �K� ~��(46)

for the rigid body and hydraulic parameters, respectively.Then, 0 is an asymptotically stable equilibrium point of the

state ~x =heT sT _~q

T~�TiT. In the above equations,

Kp;Kd;�o;��;�1;�2;�� ;K� are positive de�nite diagonalmatrices.

Remark : �d does not contain any velocity terms. This canbe seen from:

�qr + �_~q = �qd � �( _q � _qd) + �( _q � _q) = �qd � �(_q � _qd)

s � _~q = _q � _qr � _q + _q = _q � _qr(47)

Proof: By substituting (44) into (1) the following closedloop dynamics are obtained

D(q) _s +C(q; _q)s +Kds +Kpe =

� Y (q; _q; _qr; �qr)~� �W (q; _qr; �)_~q + ~�(48)

De�ne the Lyapunov-like function V1 to be

V1 =1

2eTKpe +

1

2sTD(q)s +

1

2_~qT _~q +

1

2~�T��~� (49)

The derivative of V1 becomes

_V1 = sT [�Y (q; _q; _qr; �qr)~� �W (q; _qr; �)_~q �C(q; _q)s

�Kds �Kpe+ ~� ] +1

2sT _D(q)s + eTKp(s � �e)

+ _~qTh�q � �o _~q � �osgn(_~q) +WT (q; _qr; �)s � ��q

i+ ~�T��

_~�

(50)

which can be written in the following form:

_V1 = �sTKds� eTKp�e� _~qT�o _~q � _~q

T[��q � �q

+ �osgn(_~q)] + ~�T [��_~� � Y T (q; _q; _qr; �qr)s] + sT ~�

(51)

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(a) (b)

Fig. 1. Controller implementation block diagrams. (a) Controller with adaptive observer. (b) Controller with robust observer.

With the adaptation law given in (45), _V1 becomes

_V1 = �sTKds � eTKp�e � _~qT�o _~q +� + sT ~� (52)

where � = � _~qT[��q� �q+�osgn(_~q)]. It is not di�cult to show

that

k��q � �qk � �0 + �1k _qk2 + �2k�k+ �3k _qk: (53)

with �i > 0. The properties given in (3) have been ex-ploited in deriving (53). The following choice of �o makes� < 0:

�o = diagf�iog (54)

�io = �i0 + �i1k _qk2 + �i2k�k+ �i3k _qk (55)

and �ik > �k for i = 1; � � � ; n and k = 0; � � � ; 3. The rest ofthe proof is the same as before and will not be presentedhere.Note that there are no limitations on the norms of the ini-

tial state tracking errors in this approach. However, chat-tering phenomena, which are inherent in sliding mode sys-tems, can a�ect stability. For example, if high frequencydynamics (e.g. valve dynamics) are excited, instabilitycould result. The problem could be solved by using a piece-wise linear approximation to sgn(:).

The E�ect of FrictionIn the controllers proposed in this paper, friction in the

hydraulic actuators has been neglected. It is easy to han-dle viscous friction since it acts as additional damping inthe system. It can also be shown that in the presence ofCoulomb friction, the tracking errors do not converge tozero but remain bounded. The error bounds can be re-duced by increasing the gains. The proof will be omittedhere.

V. Simulation Results

Simulations have been performed to investigate the e�ec-tiveness of the proposed controllers and to obtain guidelines

for experimentation. For this purpose, a realistic model ofthe experimental setup, a hydraulic Stewart-type platform,has been used (see Appendices A and B). The system pa-rameters were selected based upon their actual values andare given in Table I.

In the simulations and experiments conducted for thispaper, a task-space control strategy has been followed. Theadvantage of this approach is that the dynamical matriceshave simpler forms in these coordinates for parallel manip-ulators such as the Stewart platform. However, the forwardkinematics problem must be solved on-line to convert themeasured link positions to robot positions in task-space co-ordinates. Newton's method was utilized for this purpose.The control algorithms and the robot dynamics were allimplemented using the Matlab SimulinkTM toolbox. Theimplementation block diagrams of the controllers are shownin Figures 1(a) and 1(b).

Extensive simulations showed similar performance forboth of the proposed adaptive control methods, thus onlythe results obtained for the controller using the robust ob-server are presented here. The system parameters were ini-tially set to values di�erent from those used in the model toinvestigate the ability of the controllers to cope with para-metric uncertainties. The reference trajectory was chosento be xd = 0:02 sin(2�t)+0:01 sin(4�t)+0:01sin(6�t), yd =0, zd = 0:02 sin(2�t)+0:01 sin(4�t), d = 0:0873 sin(2�t)+0:0349 sin(4�t), �d = 0:0524 sin(2�t) + 0:0175 sin(4�t),�d = 0:0524 sin(2�t) + 0:0175 sin(4�t). Positions and an-gles are expressed in meters and radians, respectively. Thetracking errors clearly converge to zero in all coordinatesas shown in Figure 2. The pro�les of the parameter es-timates are given in Figure 3. The parameter adaptationlaws were activated after t = 0:5 s. Both rigid body andhydraulic parameters converge to their actual values, eventhough the parameter convergence is not guaranteed in the-ory. The estimates of Ix; Iy reach their boundaries duringsome periods of the simulation as seen in Figure 3.

In summary, the controller with the robust observer com-

IEEETRANSACTIONSONROBOTICSANDAUTOMATION,VOL.XX,NO.Y,APRIL

2001

7

02

46

8

−2 0 2

x 10−

3

02

46

8−

0.02

−0.01 0

0.01

0.02

02

46

8−

1

−0.5 0

0.5 1x 10

−3

Position Tracking Errors (m)

02

46

8−

0.02

−0.01 0

0.01

0.02

Position Tracking Errors (rad)

02

46

8

−2 0 2

x 10−

3

Tim

e (s)0

24

68

−0.01

−0.005 0

0.005

0.01

Tim

e (s)

x−axis

y−axis

z−axis

ψ−axis

θ−axis

φ−axis

Fig.2.Positio

ntra

ckingerro

rs(sim

ulatio

n).

pares

favorab

lyto

theonewith

adaptiv

eobserv

er,sin

ceit

requires

fewer

computatio

nsandperfo

rmssim

ilarly.

VI.

ExperimentalResults

Theprop

osedcontro

lmeth

odswere

experim

entally

eval-uated

usin

gthemotion

simulator

attheUniversity

ofBritish

Columbia

[28](see

Figure

4).This

simulator

isdriven

bysix

hydraulic

cylin

ders.

Each

cylin

der

iscap

able

ofexertin

gforces

inexcess

of400

0N

at1m/s,

andover

8000Natzero

rodspeed

.Thehydrau

licactu

atio

nsystem

isequipped

with

Rexro

th4WRDEthree-stage

proportion

alvalves

connected

inathree-w

aycon

�guration

.Low

fric-tio

nTe on

sealsare

used

inthehydrau

liccylin

ders.

The

installed

sensors

measure

theactu

ator

lengths,

thevalve

spool

position

sandthepressu

resboth

inthecon

troland

supply

sides

ofthecylin

ders.

High

bandwidth

valveswith

abandwidth

ofarou

nd80Hzhavebeen

used

inthesetu

pso

thatthedynamics

ofthevalves

may

beignored

.The

actuato

rvelo

citiesthatare

required

inthecon

trollaw

sare

estimated

from

themeasu

redactu

atorlen

gthsusin

g�xed

gain

Kalm

an�lters.

O�-lin

eexperim

ents

were

perform

edto

identify

theinitia

lvalu

esof

theparam

eterestim

ates.

Thecomputatio

nalsetu

pwas

aPCrunningVxWorksTM

5.4

andaSparc

1eboard

runningVxWorksTM

5.2

(seeFigu

re4).TheSparc

1eperform

stheI/Oandsafety

mon-

itorin

gfunctio

nsandthecon

troller

runson

thePC.The

contro

llerwas

implem

ented

usin

gtheMatla

bRealTim

eWorkshopTM

toolbox

targetin

gTornadoTM

2.0.

Data

betw

eenthePC

andtheVME

board

are

communicated

throu

ghacustom

para

llelI/O

communica

tionproto

col.Usin

gthissetu

pacon

trolfreq

uency

of512

Hzwas

suc-

cessfully

achieved

.Thesamecontroller

block

used

inthe

simulation

studies

wasutilized

tocon

troltheplatform

.

Only

theresu

ltsoftheexperim

ents

with

adaptiv

econ

-tro

ller/robust

observ

erare

presen

tedhere,

while

similar

perform

ancewasobserv

edfortheoth

ercon

troller.

Figu

re5

02

46

8−

60

−40

−20 0 20

02

46

8−

50 0 50

100

02

46

8

0 50

100

Parameter Estimation Errors (%)

02

46

8

0 50

100

Parameter Estimation Errors (%)

02

46

8−

60

−40

−20 0 20

Tim

e (s)0

24

68

−20 0 20 40 60

Tim

e (s)

Mp

Ix

γ1 γ2

Iy Iz

Fig.3.Parameter

estimatio

nerro

rs(sim

ulatio

n).

show

sthetrack

ingbehavior

ofthenonlin

earcon

trollercom

pared

with

that

ofawell-tu

ned

Pcon

trollerintrack

ing

thereferen

cetra

jectoryzd=�2:34

+0:05

sin(2�t)m

(the

bias

isnot

show

n).

Themaxim

umtrack

ingerrors

are4%

and43%

forthenonlin

earandPcon

troller,resp

ectively.Theresp

onse

ofthesystem

toa2H

zreferen

cetra

jectorywas

alsoexam

ined

andispresen

tedinFigu

re6.

Inthiscase

zd=�2:34

+0:02

sin(4�t)mandthemaximumtrack

inger-

rorsare

14%and69%

.Sim

ilarresu

ltswere

obtain

edinthe

other

coord

inates.

For

exam

ple,

Figu

re7show

sthetrack

-ingresu

ltsalon

gthe axiswhere

d=

0:09

sin(2�t)

radwith

4%and41%

maximumtrack

ingerror

forthenonlin

-ear

andPcon

troller,resp

ectively.In

allof

these

casesthe

prop

osedadaptiv

enonlin

earcon

trollerclearly

outperform

sthewell-tu

ned

Pcon

trollerandexhibits

excellen

ttrack

ing

perform

ance.

Note

that

dueto

thefriction

intheactu

a-tors,

trackingerrors

donot

converge

tozero

butrem

ainbounded

asclaim

edin

thepaper.

Durin

gtheexperim

ents,

theestim

atedparam

etersdid

not

converge

to�xed

values,

contrary

towhat

was

ob-

servedin

thesim

ulation

s.Friction

isan

importan

tfactor

that

intro

duces

trackingerrors

andpreven

tstheparam

-eters

fromcon

verging.

Theprop

osedcon

trollersmay

be

interp

retedas

cascadecom

bination

sof

passiv

ity-based

po-

sitioncon

trollersandactu

atorforce

controllers.

Thevery

sti�dynam

icsof

hydrau

licactu

atorsmake

theforce

(pres-

sure)

control

loop

sensitive

tovelo

cityestim

ationerrors

(orvelo

citymeasu

rementnoise)

andpressu

remeasu

rement

noise.

Thislim

itsthelevel

ofthepressu

refeed

back

gains

andmay

deteriorate

forcetrack

ingandsubseq

uently

pa-

rameter

estimation

,esp

eciallyfor

thehydrau

licparam

e-ters.

Other

factorssuch

asinsu�cien

texcitation

andun-

modeled

dynam

ics,e.g.

valveandleg

dynam

ics,cou

ldalso

preven

ttheparam

etersfrom

convergin

g.Moreov

er,it

should

bestressed

that

theparam

etercon

vergence

isnot

evenguaran

teedin

theory,

therefore

theexperim

ental

results

donot

contrad

ictthetheoretical

arguments.

The

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 8

Fig. 4. The experimental setup.

0 0.5 1 1.5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time (s)

Pos

ition

(m

)

P

Non.

Ref.

P Error

Non. Error

Fig. 5. Position tracking (1Hz) along z coordinate (experiment).

adaptation was found to be quite helpful in improving thetracking performance. The projection gains used in theadaptation laws proved e�ective in preventing the largeparameter swings that can occur especially during start-up transients. The step response of the controller alongthe z axis is also compared with that of the P controllerin Figure 8. As it can be seen, the nonlinear controllerexhibits a much faster response with some overshoot.

VII. Conclusions

This paper addresses the control problem for hydrauli-cally driven manipulators. The highly nonlinear dominantactuator dynamics prevents the use of standard robot con-trol methods. In fact, inclusion of actuator dynamics in thedesign is of critical importance in hydraulic robots. Whilemost of the reported work in the literature consider the con-trol of single-rod hydraulic actuators, this paper proposednovel nonlinear controllers for hydraulic manipulators us-ing backstepping. A realistic model of the system was uti-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time (s)

Pos

ition

(m

)

P

Non. Ref.

P Error

Non. Error

Fig. 6. Position tracking (2Hz) along z coordinate (experiment).

lized in developing these Lyapunov-based controllers. Todeal with parameter uncertainties, the controllers were aug-mented with adaptation laws. Acceleration feedback wasavoided by proposing adaptive and sliding-type observers.Simulations and experiments were carried out with a hy-draulic Stewart platform to investigate the e�ectivenessof these approaches. The results demonstrated excellenttracking position tracking behavior and satisfactory tran-sient responses for these new controllers.

Acknowledgments

The authors would like to thank Wen-Hong Zhu for histechnical comments, Ruediger Six for his contribution tothe real-time implementation of the controllers, and SimonBachmann for his assistance in maintenance and modi�-cation of the hydraulic system. This work was supportedby the Canadian IRIS/PRECARN Network of Centers ofExcellence.

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 9

0 0.5 1 1.5−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (s)

Pos

ition

(ra

dian

)

P

Non.

Ref.

P Error

Non. Error

Fig. 7. Position tracking (1Hz) along coordinate (experiment).

Appendix A

The dynamics of a typical hydraulic actuator are pre-sented in this Appendix. A three-way valve con�guration isassumed to be used in the actuators, as shown in Figure 9.For such a con�guration, the control pressure dynamics aregoverned by [29]

Vt

�_pc = ql + cl(ps � pc) � _Vt (56)

where Vt is the trapped uid volume in the control side,� is the e�ective bulk modulus, pc is the control pressureacting on the control side, ps is the supply pressure actingon the rod side, ql is the load ow, and cl is the coe�cientof total leakage. The load ow, ql, is a nonlinear functionof the control pressure and the valve spool position and isgiven by

ql =

8><>:c(u� d)

ppc u < �d

c(u+ d)pps � pc + c(u� d)

ppc �d � u � d

c(u+ d)pps � pc u > d

(57)

and c = cdwq

2�, where cd is the e�ective discharge coe�-

cient, w is the port width of the valve, � is the density ofthe uid, d is the valve underlap length and u is the valvespool position which is the control command. Note that theactuator output force is � = pcA � psa. Therefore, using(56) and (57), the dynamics of the i'th hydraulic actuatorcan be written in the following form (assuming cl t 0)

_� i = �A�i _qi

qi � li+

�i

qi � liqil (�

i; ui) = f i(qi; _qi) + gi(qi; � i; ui)

(58)

where l is the actuator stroke length. For a Stewart plat-form, there are six actuators driving the system. The ac-tuator subsystem dynamics can be represented in matrixform as in (2).

24.9 25 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8−2.342

−2.34

−2.338

−2.336

−2.334

−2.332

−2.33

−2.328

−2.326

Time (s)

Pos

ition

(m

)

Non.

P

Ref.

Fig. 8. Step response along z coordinate (experiment).

Note that (58) can be rewritten in the following formwhich is suitable for adaptive control:

_� i = i1fi0(q

i; _qi) + i2gi0(q

i; � i; ui) (59)

where i =��i �ici

�T, f i0 = � Ai _qi

qi�li, and gi0 =

qil

ci(qi�li)

(does not depend on csi , see (57)). These equations can bewritten in matrix form as in (16).

ps

p =0t XV

qlPc

A

a

d

CylinderValve

l

q-l

Fig. 9. A typical three-way valve con�guration.

Appendix B

The Stewart platform is a parallel manipulator widelyused in conventional motion simulators. The dynamicsof an inverted, ceiling-mounted Stewart platform (Figure(10)) are presented here [19].In task-space coordinates, the dynamics of the platform

are governed by:

D(q)�q +C(q; _q) _q +G = (JL)T � (60)

where q =�x y z � �

�Tand �, � and are roll-

pitch-yaw angles, respectively. Furthermore, J is the ma-nipulator Jacobian matrix and L is de�ned as

L =

�I3�3 00 T

�(61)

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. XX, NO. Y, APRIL 2001 10

TABLE I

The system parameters used in the simulations and experiments.

Hydraulic ParametersParameter A (m2) a (m2) L (m) Ps (psi) d (m) c � (Mpa)Value 1:14� 10�3 6:33� 10�4 1:37 m 1500 55:4� 10�6 1:5� 10�4 700

Rigid Body ParametersParameter Mp (kg) Ix (kg.m2) Iy (kg.m2) Iz (kg.m

2) {Value 250 45 45 43 {

Base

A

BCD

E

F

Platform

Actuator

ryp

p

pγ

z

y

z

y

p

pP

B

B

B

B

BB

P PP

P P

P AE

CD

BF hp

bb i

bdp

ba i

ppi

xp

120 degy

γb

rb

Base

x

120 deg

Platform

P

B

BEB

DB

FB AB

CB

BB

PB

PA

PE

PF

PD PC

Fig. 10. The schematic of the Stewart platform.

with

T =

24cos(�) cos(�) � sin(�) 0cos(�) sin(�) cos(�) 0� sin(�) 0 1

35 (62)

Finally, D(q), C(q; _q) and G have the following forms:

D(q) =

�MpI3�3 0

0 TT bIpT

�(63)

C(q; _q) =

�0 00 c22

�c22 = TTS(!)bIpT + T T bIp _T

(64)

G =�0 0 Mpg 0 0 0

�T(65)

where ! is the angular velocity vector of the platform and

S(!) =

24 0 �!z !y!z 0 �!x�!y !x 0

35 (66)

In the above equations, bIp is the platform inertia matrixwith respect to the base frame and is given by

bIp = R pIpRT (67)

where

pIp =

24Ix 0 00 Iy 00 0 Iz

35 (68)

and R is a rotation matrix giving the coordinates of theplatform-attached basis vectors in a base frame. Note that(60) is not exactly as (1). However, since J is a functionof platform position and is known, the controllers can beeasily modi�ed to be used in this case. Moreover, the rigidbody dynamics may be written in so-called linear in pa-rameters form.

D(q)�q + C(q; _q) _q +G = Y6�4(q; _q; �q)� (69)

where � =�Mp Ix Iy Iz

�Tis the vector of unknown

parameters. The detailed expressions of the elements ofY are long and fairly straightforward and will not be pre-sented here.

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[1] J.Y.S. Luh, M.W. Walker, and R.P.C. Paul, \Resolved accelera-tion control of mechanicalmanipulators," IEEE Tran. Automat.Cont., vol. 25, pp. 468{474, 1980.

[2] R. Ortega and M.W. Spong, \Adaptive motion control of rigidrobots: A tutorial," Automatica, vol. 25, pp. 877{888, 1989.

[3] H. Berhuis and H. Nijmeijer, \A passivity approach to controller-observer design for robots," IEEE Tran. Robot. Automat., vol.9, no. 6, pp. 740{754, 1993.

[4] B. Paden and R. Panja, \Globally asymptotically stable 'PD+'controller for robot manipulators," Int. J. Control, vol. 47, pp.1697{1712, 1988.

[5] J. J.-E Slotine and W. Li, \On the adaptive control of robotmanipulators," Int. J. Robot. Research, vol. 6, pp. 49{59, 1987.

[6] M. W. Spong, \On the robust control of robot manipulators,"IEEE Tran. Automat. Cont., vol. 37, pp. 1782{1786, 1992.

[7] S. Zenieh and M. Corless, \Simple robust tracking controllersfor robotic manipulators," in SYROCO'94, September 1994, pp.193{198.

[8] C. Abdallah, D. Dawson, P. Dorato, and M. Jamshidi, \Surveyof robust control for rigid robots," IEEE Control Systems, pp.24{30, 1991.

[9] T. J. Tarn, A. K. Bejczy, X. Yun, and Z. Li, \E�ect of motordynamics on nonlinear feedback robot arm control," IEEE Tran.

Robot. Automat., vol. 7, no. 1, pp. 114{122, February 1991.[10] J. Yuan, \Adaptive control of robotic manipulators including

motor dynamics," IEEE Tran. Robot. Automat., vol. 11, no. 4,pp. 612{617, 1995.

[11] T. Burg, D. Dawson, J. Hu, , and M. de Queiroz, \An adaptivepartial state-feedback controller for RLED robot manipulators,"

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IEEE Tran. Automat. Contr., vol. 41, no. 7, pp. 1024{1030, July1996.

[12] C. Y. Su and Y. Stepanenko, \Redesign of hybrid adap-tive/robust motion control of rigid-link electrically-driven robotmanipulator," IEEE Tran. Robot. Automat., vol. 14, no. 4, pp.651{655, 1998.

[13] M. R. Sirouspour and S. E. Salcudean, \On the nonlinear con-trol of hydraulic servo-systems," in Int. Conf. Robot. Automat.,April 2000, pp. 1276{1282.

[14] G. A. Sohl and J. E. Boborow, \Experiments and simulations onthe nonlinear control of a hydraulic servosystem," IEEE Tran.

Cont. Syst. Tech., vol. 7, no. 2, pp. 238{247, March 1999.[15] G. Vossoughi and M. Donath, \Dynamic feedback linearization

for electrohydraulically actuated control systems," J. Dyn. Sys.,Meas. and Contr., vol. 117, pp. 468{477, Dec. 1995.

[16] B. Novel, M.A. Garnero, and A. Abichou, \Nonlinear control ofa hydraulic robot using singular perturbations," in Int. Conf.

Syst. Man and Cyber., 1994, pp. 1932{1937.[17] K. A. Edge and F. Gomes de Almeida, \Decentralized adap-

tive control of a directly driven hydraulic manipulator Part I:theory," in Instn. Mech. Engr., 1995, vol. 209, pp. 191{196.

[18] K. A Edge and F. Gomes de Almeida, \Decentralized adaptivecontrol of a directly driven hydraulic manipulator Part II: ex-periments," in Instn. Mech. Engr., 1995, vol. 209, pp. 197{202.

[19] D. Li and S. E. Salcudean, \Modeling, simulation and controlof a hydraulic Stewart platform," in IEEE Int. Conf. Robot.

Automat., April 1997, pp. 3360{3366.[20] F. Bu and B. Yao, \Observer based coordinated adaptive robust

control of robot manipulators driven by single-rod hydraulic ac-tuators," in IEEE Int. Conf. Robot. Automat., April 2000, pp.3034{3039.

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Mohammad Reza Sirouspour was bornin Tehran, Iran, in 1973. He received theB.Sc. and M.S. degrees in Electrical Engi-neering from Sharif University of Technology,Tehran, Iran in 1995 and 1997, respectively.During1996-1998 he was a research engineer atthe Sharif Electronic Research Center, Tehran,Iran. Since September 1998, he has joinedthe Robotics and Control Laboratory at theUniversity of British Columbia, Vancouver,Canada, where he is pursuing a doctoral de-

gree in Electrical and Computer Engineering. His research interestsinclude control of robotmanipulators, nonlinear control, tele-robotics,haptics, and estimation theory.

Septimiu E. Salcudean received the B. EngandM.Eng degress fromMcGill University andthe Ph.D. degree from U.C. Berkeley, all inElectrical Engineering. From 1986 to 1989, hewas a Research Sta� Member in the roboticsgroup at the IBM T.J. Watson Research Cen-ter. He then joined the Department of Electri-cal and Computer Engineering at the Univer-sity of British Columbia, Vancouver, Canada,where he is now a Professor and holds a CanadaResearch Chair. He spent one year at ONERA

in Toulouse, France, in 1996-1997, where he held a Killam ResearchFellowship. Dr. Salcudean is interested in haptic interfaces, tele-operation and virtual environments. He is pursuing applications tomedical diagnosis and intervention and to the control of heavy-dutyhydraulic machines such as excavators.