The original version of this paper was presented at the IFACWorldCongress which was held in San Francisco, USA during 1996. Thispaper was recommended for publication in revised form by AssociateEditor J.Z. Sasiadek under the direction of Editor Mituhiko Araki.

*Corresponding author. Tel.: #1-405-744-6579; fax: #1-405-744-7873.E-mail address: pagilla@ceat.okstate.edu (P. R. Pagilla).

Automatica 37 (2001) 983}995

An adaptive output feedback controller for robot arms:stability and experiments

Prabhakar R. Pagilla*, Masayoshi TomizukaSchool of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078-5016, USA

Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1970, USA

Received 8 May 1998; revised 24 December 1999; received in "nal form 3 January 2001

An adaptive output feedback controller for robot arms is developed in this paper. A nonlinear observerbased on desired joint velocities and bounded joint position error is used to estimate joint velocities.Experimental results validate the ewectiveness of the proposed adaptive output feedback controller.

Abstract

An adaptive output feedback controller for robot arms is developed in this paper. To estimate the joint velocities, a simple nonlinearobserver based on the desired velocity and bounded position tracking error is proposed. The closed-loop system formed by theadaptive controller, observer and the robot system is shown to be semi-global asymptotically stable. Extensive experiments conductedon a two link robot manipulator con"rm the e!ectiveness of the proposed controller}observer structure. To highlight the performanceof the proposed scheme, it is compared via experiments with a well-known passivity based control algorithm. 2001 ElsevierScience Ltd. All rights reserved.

Keywords: Adaptive control; Robot control; Observers; Lyapunov stability

1. Introduction

Adaptive control of robot arms based on completestate measurements has been dealt in great detail in theliterature. The feed-forward and passivity based algo-rithms for robot arms proposed in Sadegh and Horowitz(1990), Slotine and Li (1991) and Ortega and Spong(1989), and the references therein, have been extensivelyused. A comparative experimental study of the standardand new algorithms has been done in Whitcomb, Rizzi,and Koditschek (1993). Most of these algorithms needcomplete state measurements. A major drawback of suchschemes is that both joint position and joint velocitymeasurements of the robot are required for feedback

control. Sensors for measuring robot joint velocities areexpensive. Further, measurements from these sensors areoften contaminated by noise. Velocity estimated feed-back control of robot arms can be used instead and therequirement of robots to be equipped with velocity sen-sors can be eliminated. Most of the robot adaptiveschemes use velocity errors or modi"ed velocity errors todrive the parameter adaptation algorithms. When theactual velocities are not available, estimated velocitiesand position errors have to be used to drive the para-meter adaptation algorithms. This leads to an addeddi$culty in proving the stability of these algorithms.Considerable research is being conducted in the area of

output feedback control of nonlinear systems. Outputfeedback control of robot arms has been studied by manyresearchers. In Berghuis and Nijmeijer (1993), theauthors consider passivity based controller}observer de-sign for robots. A linear observer is designed to estimatethe velocities. It is shown that the closed-loop systemformed by the controller}observer and the robot is lo-cally exponentially stable. A linear velocity observer isdesigned assuming complete knowledge of the structuralparameters of the robot. A robust variable structure

0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 0 4 8 - 6

controller and a nonlinear observer is designed in Zhu,Chen, and Zhang (1992). Berghuis and Nijmeijer (1994)proposes a linear controller and a linear observer forrobust control in the presence of parameter uncertainties.In Canudas de Wit and Fixot (1992), tracking control ofrobot manipulators is proposed by combining a passivitybased controller and a nonlinear sliding observer. Localasymptotic convergence of the position tracking errorsand the velocity estimation errors was shown.A nonlinear observer based on the robot error dynam-

ics was designed in Nicosia and Tomei (1990), and a con-trol design that uses joint position measurements andestimated velocity is proposed. Repetitive and adaptivecontrol of robot manipulators with velocity estimation ispresented in Kaneko and Horowitz (1997). In the case ofrepetitive control, the robot achieves tracking of thedesired periodic trajectory through repeated learningtrials. An adaptive controller is also designed. A linearobserver is designed to estimate the velocities. Localasymptotic stability is shown for both the repetitive andthe adaptive cases.In this work, an adaptive feedback controller for robot

arms is designed using partial state feedback, i.e., onlyjoint position measurements are used to design the adap-tive controller. A simple nonlinear observer is designed toestimate the robot joint velocities. The closed-loop sys-tem formed by the adaptive controller, observer and therobot system is shown to be semi-global asymptoticallystable, i.e., the region of attraction can be increasedarbitrarily by increasing the controller and the observergains.Convergence of the estimated parameters to the true

parameters depends on whether the regressor matrixsatis"es the persistence of excitation condition. In theproposed adaptive controller the regressor matrix entire-ly depends on the desired trajectory. Hence, the persist-ence of excitation condition is satis"ed by choosinga persistently exciting desired trajectory. Experimentswere conducted on a two link planar arm for the pro-posed controller}observer. Successful experimental re-sults show the validity of the proposed controller andobserver. The proposed scheme is compared, via experi-ments, with a well-known passivity based controller. Thepassivity based controller used for this comparisonassumes that the parameters are exactly known anda "rst-order numerical di!erentiation of joint positionmeasurements is used to estimate velocities.The remainder of this paper is organized as follows. In

Section 2, robot dynamics and problem formulation isgiven. Section 3 gives the proposed adaptive controllerand observer. Closed-loop error dynamics is also derivedin Section 3. Stability of the closed-loop system is shownin Section 4. Section 5 discusses the experimentalplatform and the experimental results. Some concludingremarks with a summary of this paper are given inSection 6.

2. Robot dynamics and problem formulation

Consider the dynamics of an n degree of freedom robotarm

x"x

,

M(x)x

#C(x

,x

)x

#g(x

)", (1)

where x3, x

3 are the generalized position and

velocity, respectively, M(x)3 is the inertia matrix,

C(x,x

)3 is the matrix composed of Coriolis and

centrifugal terms, g(x)3 is the gravity vector, and

3 is the vector composed of joint torques. The struc-ture of the robot dynamics satis"es the properties givenin Appendix A.Given a desired trajectory of the robot, the objective is

to design a stable tracking controller that only requiresjoint position measurements for feedback. To achievethis objective, an adaptive controller together witha simple nonlinear observer to estimate joint velocities isproposed. Let x

(t) and x

(t) be the desired position and

velocity, respectively. It is assumed that the desired statetrajectory is twice continuously di!erentiable. Let x(

(t)

and x((t) denote the estimated position and estimated

velocity, respectively. Let 3 denote the actual para-meter vector as given by property (iv) in Appendix A. LetK (t) denote the estimate of . De"ne the tracking and theestimation errors by

e(t) :"x

(t)!x

(t), e(

(t) :"x(

(t)!x

(t),

e(t) :"x

(t)!x

(t), e(

(t) :"x(

(t)!x

(t),

e(t) :"e

(t)#e

(t), I (t) :"K (t)!,

e(t) :"

(e

)e

(t),

where e(t) and e

(t) are the position and velocity track-

ing errors, respectively, e((t) and e(

(t) are the estimated

position and estimated velocity errors, respectively, e(t)

is the reference velocity error, I (t) is the parameter es-timation error, e

(t) is an auxiliary bounded position

tracking error, and(e

(t)) is a positive de"nite diagonal

matrix given by

(e

(t))"diag

1#e(t),2,

1#e(t), (2)

where e(t),2, e(t) are the components of the position

error vector e(t) and

is a positive gain. Notice that this

choice of (e

) renders e

(t) to be bounded by

. In the

remainder of the paper, whenever it is clear from thecontext, explicit dependence of variables on time isnot shown. Also, throughout the paper A denotes the2-norm of A. The following section gives the adaptivecontroller, observer, and the closed-loop error dynamics.

984 P. R. Pagilla, M. Tomizuka / Automatica 37 (2001) 983}995

3. Adaptive controller and observer

The following control scheme is proposed:

"Y(x

,x

,x

)K !K

(e

#e(

)!K

e, (3)

where K, K

are positive de"nite gain matrices and K is

the estimated parameter vector of the robot. Note thatthe second term in the control law is a function ofestimated velocity, desired velocity, and actual positionerror, i.e., e

#e(

"x(

!x

#e

. The desired regressor

matrix, Y(x

, x

, x

), is given by

Y(x

, x

, x

)K "MK (x

)x

#CK (x

, x

)x

#g( (x

),

where MK (x), CK (x

,x

), and g( (x

) are the estimates of

M(x), C(x

,x

), and g(x

), respectively. The desired re-

gressor matrix depends only on the desired trajectoryand can be pre-computed. The parameter adaptation lawis chosen as follows:

K (t)"K (0)!Ye (t)!

YQ e() d, (4)

where K (0) is the initial estimate of the unknownparameter vector, is a positive de"nite gain matrix, andeis given by

e(t)"e

(t)!e(

(t)#

e() d!

e(() d.

The following observer is proposed to estimate the states:

x("!

e(#x(

, (5)

x("x

!

e(#e

, (6)

where and

are positive gains. Thus, by rearranging

terms, the observer error dynamics is given by

e("!

e(#e(

,

e("!

e(#

e(#2e

!e

.

(7)

The closed-loop error dynamics is derived in the follow-ing section.

3.1. Error dynamics

Noting that e"e

#e

and !e

#e

#e

"0,

M(x)e

can be expressed as

M(x)e

"!C(x

,x

)e

#M(x

)e

#C(x

, x

)e

!g(x)#M(x

)e

#C(x

,x

)e

. (8)

From (1) and (8), we obtain

M(x)e

"!C(x

,x

)e

#!M(x

)x

#C(x,x

)x

#M(x

)e

#C(x

, x

)e

. (9)

Using the control law and noting that e"E

e, where

E"

, the error equation is

M(x)e

"!C(x

,x

)e

#YdI !W!K(e#e( )

!Ke#M(x

)E

e!M(x

)E

e, (10)

where W is given by

W"[M(x)!M(x

)]x

#g(x

)!g(x

)

# [C(x,x

)(x

!e

)!C(x

, x

)x

].

Using Eq. (7), the observer error equation can be derivedas follows:

M(x)e(

"!

M(x

)e(

#

M(x

)e(

# 2M(x)e

!M(x

)e

"!C(x, x

)e(

!M(x

)e

#C(x

,x

)e

! M(x

)e(

#C(x

, x

)(e(

#e

)

# 2M(x)e

#

M(x

)e(

. (11)

On substitution of the robot error dynamics (10) andusing x

"e

!e

#x

, we obtain

M(x)e(

"!C(x,x

)e(

!Y

I #W#K

(e

#e(

)#K

e

#M(x)E

e!M(x

)E

e!

M(x

)e(

# M(x

)e(

#C(x

, e

)!C(x

, e

)(e(

#e

)

#C(x, x

)(e(

#e

). (12)

4. Stability

First, de"ne an extended vector z given byz :"[e

, e

, e(

, e(

, I ]. The following theorem gives the

stability of the closed-loop system with the proposedcontroller}observer structure.

Theorem 1. For the robot dynamics given in (1), using theadaptive controller (3) together with the update law (4) andthe observer (6), it is always possible to choose feedbackgains K

, K

and

and the observer gains

and

such

that z"0 is locally uniformly stable, e, e

, e(

and

e(

locally asymptotically converge to zero. Further, theclosed-loop system is semi-globally asymptotically stable,i.e., the region of attraction can be arbitrarily increased byincreasing the controller and the observer gains.

Proof. Consider the following Lyapunov function candi-dates,