1-s2.0-S0957415808000688-main

Mechatronics 18 (2008) 641–652

Contents lists available at ScienceDirect

Mechatronics

journal homepage: www.elsevier .com/ locate/mechatronics

Technical note

Guaranteed robustness/performance adaptive controlwith limited torque for robot manipulators q

C.Q. Huang a,*, X.F. Peng a, C.Z. Jia b, J.D. Huang b

a Department of Automation, Xiamen University, Xiamen 361000, PR Chinab Department of Mechanical Engineering, Shijiazhuang College of Mechanical Engineering, Shijiazhuang 050030, PR China

a r t i c l e i n f o

Article history:Received 13 November 2006Accepted 6 May 2008

Keywords:Robust-adaptiveRobot manipulatorTracking controlRobustnessUniformly ultimately boundedLp input/output stableLimited torque

0957-4158/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.mechatronics.2008.05.001

q This work is supported by the National Natural Scie60704043) and Research Fund for the Doctoral Progr20070384031).

* Corresponding author.E-mail address: [email protected] (C.Q. Hu

a b s t r a c t

A novel guaranteed robustness/performance adaptive algorithm with limited torque is proposed fortracking control of robot manipulators in this paper. In framework of the adaptive controller, a continu-ous differentiable increasing function is introduced into the parameter estimator of controller so as torestrain from parameter drifting. Furthermore, both robustness and performance are guaranteed withoutany persistent excitation (PE) condition: (i) asymptotic stability is achieved in absence of disturbancewhen the estimation of parameter-range is available; (ii) the close-loop system is Lp input/output stableand its solution is uniformly ultimately bounded regardless whether the true parameters are locatedwithin the estimated parameter-range or not. Finally, taking some practical complexity into account,such as disturbances, dead-zone as well as dynamics of brushed direct-current motor, simulation resultsvia three-dof robot manipulator show the significant improvement in terms of robustness and perfor-mance in comparison with the conventional adaptive and projection algorithm.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Since robot manipulator is a highly non-linear, dynamicallycoupled electro-mechanical system, its control design poses a chal-lenge for the robotics and control community, and has attractedmany research interests. Different algorithms, such as robust con-trol, adaptive control [3,26], variable structure control [14,27] andneural network control [10,19], etc., have been introduced in theliterature. Adaptive control is yet an effective strategy to deal withrobot uncertainties.

Indirect adaptive control of robot manipulators has evolved dueto the property of parameter linearization of robot dynamics. Oneof major problems of applications is its poor robustness. Unmod-eled dynamics or even a small bounded disturbance could causemost of the adaptive control to go unstable since it results inparameter drifting in absence of excitation. A number of redesignsand modifications with respect to adaptive control were proposedand analyzed to improve its robustness, e.g., applying normaliza-tion techniques, projection algorithm, dead zone modification, e-modification, and r-modification (see [9] and references therein).Although these techniques ensure the stability of the system, some

ll rights reserved.

nce Foundation of China (No.am of Higher Education (No.

ang).

performances such as transient performance or asymptotic stabil-ity are degraded (see [7,17,18]).

Another major problem is about the unsatisfactory transientperformance. In spite of the development of adaptive control, tran-sient performance of the closed-loop systems is not clear, which isactually more important in practical applications. The design ofadaptive controller with improved transient performance is a cur-rent research topic. Research reveals that a trade-off between per-formance and robustness has to be made for adaptive controlsystems. Performance improvement has been considered in a num-ber of ways, including the use of dominant rich reference input[12], fast adaptation [32], multiple models [15,21–23], adaptivebackstepping [16], and the use of compensators [28]. Unfortu-nately, either persistent excitation (PE) condition is required [2]or tuning these controllers gains too large to improve the transientperformance results in some actuator saturation [30,31]. Switchedadaptive (i.e. multiple-model adaptive control) seems to be apromising and attractive approach for the improvement of tran-sient performance [15]. Its basic idea is to choose the best modelfor the plant from a priori known set of models at every instant,and apply the output of the corresponding controller to the plant.It is however not clear how these controllers behave during slowplant drift and occasional plant jumps [8]. Furthermore, this algo-rithm has great calculation and great time consumption and hencebecomes of significant difficulty for real-time execution when ro-bot degree of freedom increases.

Although many adaptive schemes have been proposed to im-prove robustness and transient performance, they unfortunately

mailto:[email protected]

http://www.sciencedirect.com/science/journal/09574158

http://www.elsevier.com/locate/mechatronics

642 C.Q. Huang et al. / Mechatronics 18 (2008) 641–652

demand highly skilled and educated personnel for tuning andmaintenance. Therefore, bridging the theory/practice gap is still achallenge problem. In many practical problems, we may have somea priori knowledge of parameter of a physical plant. This knowl-edge usually comes from in terms of upper or lower bounds forevery parameter or a well-defined subset. Such a priori informationis likely to be employed to design adaptive controllers that are con-strained to search for these parameter estimates in the set wherethe parameter is located. In this case, projection algorithm[6,7,13,24,30,33] is an alternative. But its robustness is still a prob-lem in some cases (see the example showed in following simula-tion section).

This paper continues to explore new robust adaptive controlalgorithm for tracking control of robot manipulators. This is moti-vated by the underlying ideas in mind:

(i) to guarantee transient performance of the resulting adaptivesystem;

(ii) to make best of some knowledge of the physical plant aswell as its parameters;

(iii) to lay particular emphasis on simple but robust adaptivealgorithm with easy synthesis;

(iv) to remove the requirement of PE condition for adaptiverobustness;

(v) to take limited torque into account in some sense. Therobust-adaptive algorithm developed in this paper solvesthe aforementioned problems.

2. Preliminaries

Throughout this paper, km{�}(kM{�}) is the minimal (maximal)eigenvalue of matrix; and for matrix M(q),

kmfMg ¼minq2RnfkmfMðqÞgg;

�kMfMg ¼maxq2RnfkMfMðqÞgg:

The norm of vector x is defined by kxk ¼ffiffiffiffiffiffiffiffixTxp

and norm of matrix M

is defined by the corresponding induced norm kMk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikMfMTMg

q.

j�j stands for the absolute value.For a Lebesgue measurable function x(t) : R+ ? Rn, the Lp norm

p 2 [1,1) for is defined to be

kxð�Þkp ¼Z 1

0kxðsÞkpds

� �1=p

:

For p =1, the norm is defined as

kxðtÞk1 ¼ suptP0fkxðtÞkg

almost everywhere.

2.1. Continuous piecewise-differentiable increasing function

A set of all continuous increasing functions introduced by [11] isdeveloped and described as follows,

Definition. Fðl; q; e; xÞ denotes the set of all continuous piecewise-differentiable increasing function vectors

sðxÞ ¼ ½s1ðx1Þs2ðx2Þ � � � snðxnÞ�T

with

l ¼ ½l1l2 � � �ln�; li P 0;q ¼ ½q1q2 � � �qn�; qi > 0;e ¼ ½e1e2 � � � en�; ei > 0; i ¼ 1; . . . ;n

such that (Notation si(xi) can be denoted as si (x))

� qijxijP jsi(x)jP lijxij, "xi 2 R:jxij < ei,� qiei P jsi(x)jP liei, "xi 2 R:jxijP ei,� 0 6 dsi(x)/dxi 6 qi, "xi 2 R,

where x 2 Rn. Some examples, such as saturated function, sinefunction etc., are given by [1] or [11].

Define a matrix Js(x) = os(x)/oxT, it shows that Js(x) is diagonal.According to the following inequalities:

ksðxÞk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

s2i ðxiÞ

vuut and jsiðxiÞj 6 qe;

we have

ksðxÞk 6 qeffiffiffinp

; kJsðxÞk 6 q; ð1Þwhere q = maxi{qi}, e = maxi{ei}.

2.2. Dynamic properties of Robot manipulator

In the absence of friction, the dynamics of rigid serial n-link ro-bot manipulator is given in joint space as follows:

MðqÞ€qþ Cðq; _qÞ _qþ gðqÞ ¼ sþ sd; ð2Þ

where q is n � 1 vector of joint angles, M(q) 2 Rn�n is the robot iner-tial matrix; Cðq; _qÞ _q, g(q), s 2 Rn denote the centrifugal Coriolisforce, the gravitational force and control inputs respectively. sd issome bounded disturbance.

In this paper, only robot manipulator with all revolute joints isconsidered. A list of useful properties of its dynamic model is givenas follows [3,25]:

P1. A suitable definition of Cðq; _qÞ makes the matrix ( _M � 2CÞskew-symmetric, i.e.

_MðqÞ ¼ Cðq; _qÞ þ CTðq; _qÞ ð3Þ

P2. M(q) is a symmetric positive-definite matrix; for anyq1,q2 2 Rn and some constant aM, it verifies

kmfMgI 6MðqÞ 6 kMfMgI; ð4ÞkMðq2Þ �Mðq1Þk 6 aMkq2 � q1k: ð5Þ

P3. For all vector v,w,x,y,z 2 Rn, there exist positive constants c1,c2 such that

Cðx; y þ zÞw ¼ Cðx; yÞwþ Cðx; zÞw; ð6ÞkCðx; yÞzk 6 c1kyk � kzk; ð7ÞkCðx; zÞw� Cðy; vÞwk 6 c1kz � vk � kwk þ c2kx� yk � kzk � kwk:

ð8Þ

P4. For x,y 2 Rn, there exist constants kg, �g such that

kgðxÞ � gðyÞk 6 kgkx� yk; kgðxÞk 6 �g: ð9Þ

P5. There exist some sið�Þ 2Fðli; qi; ei; �Þ for i = M,C,g and all~q ¼ q� qd 2 Rn such that

k½MðqÞ �MðqdÞ�€qdk 6 k€qdk � ksMð~qÞk; ð10Þk½Cðq; _qÞ � Cðqd; _qdÞ� _qdk 6 c1k _qdk � k _~qk þ k _qdk2ksCð~qÞk; ð11ÞkgðqÞ � gðqdÞk 6 ksgð~qÞk: ð12Þ

P6. For any vector v,w 2 Rn, we have

MðqÞv þ Cðq;wÞwþ gðqÞ ¼ Uðv;w;qÞhþ Kðv;w; q; h0Þ;ð13Þ

where K(v,w,q) is the known dynamic counterpart in which con-cludes the known parameters h0 2 Rm0 ; U(v,w,q) 2 Rn�m is regres-

C.Q. Huang et al. / Mechatronics 18 (2008) 641–652 643

sor matrix; h 2 Rm is a parameter vector containing only the interestor unknown robot parameters. Robot parameters consist of h0 2 Rm0

plus h 2 Rm.

3. The proposed adaptive controller design

3.1. The proposed adaptive control law

The control problem is subjected to the following assumptions:

A.1 The desired reference positions qd, velocities _qd and acceler-ations €qd are all bounded.

A.2 We assume that each joint actuator has a maximum torque�si, namely, jsij 6 �si.

A.3 Throughout the operating procedure, _~q always stays withinsome region X for some constant �c _~q > 0 such that

j _~qij 6 �c _~q; i ¼ 1; . . . ;n: ð14Þ
It implies that
js v;ið _~qÞjP c _~qj _~qij or _~qTsvð _~qÞP c _~q � k _~qk2; ð15Þ

where c _~q ¼miniflv;iev;i=�c _~qg.
A.4 Suppose that the estimated parameter-range H is available:
H ¼ f- 2 Rmjhi 6 -i 6�hi; i ¼ 1;2; . . . ;mg; ð16Þ

where hi, �hi are the lower and upper estimated bound ofmanipulator parameter hi (i = 1,2, . . . ,m).

0

iπ

))(ˆ(ˆ ts iθθ ))(ˆ(ˆ ts iθθ

i,θπ

)(ˆ tiθ

i)( θε

i)( θε−

ii t ,)(ˆθπθ −

Fig. 1. Element shðhiÞ of function vector.

Remark 1. Assumption A.1 is always satisfied in the practicalsituation. A.4 is often available in practice since we may have somea priori knowledge of parameters. From point of the practical view,actuator torque is always bounded. In this paper, the boundedtorque of joint actuator is taken into account by A.2.

In A.3, (14) implies finite region X with constant radius consid-ered in this paper. Actually, (14) does not impose any conditions onparameter estimates directly: according to the proof of Theorem 1,(14) is only related to magnitude of Lp stable region but not param-eter update law.

As for robot manipulator with unlimited actuator torque, wehave the following nominal adaptive control law [26]:

s ¼ Kdðh0Þ � Kp~q� Kv_~qþUdh; ð17Þ

hðtÞ ¼ hð0Þ � 1c

Z t

0UT

d � ½ _~qðfÞ þ asð~qðfÞÞ�df; ð18Þ

where a and c are positive constants, Kp, Kv are all positive definitediagonal matrices. hðtÞ (hð0ÞÞ is the estimate (the initial estimate) ofunknown robot parameter h. Kdðh0Þ :¼ Kð€qd; _qd;qd; h0Þ,Ud :¼ Uð€qd; _qd; qdÞ and ~q :¼ q� qd denote the known dynamic, thedesired regressor matrix and the tracking error, respectively. Inthe case of non-adaptive, hðtÞ is constant, i.e. hðtÞ ¼ hð0Þ.

Since some aforementioned problems involve in the adaptivesystems, the aim of this paper is to find a robust adaptive algorithmsuch that the following performance of the resulting system issatisfied:

(a) Asymptotic stability if the true robot parameter is within theestimated parameter-range H, i.e., h 2H;

(b) Lp input/output stable and uniformly ultimately bounded inpresence of disturbance or/and h R H.

From intuition, we can make a conjecture that drawback of adap-tive control, parameter drifting, is overcome when output of theestimator is restricted within some limits, and thus better robust-ness and better transient performance are expected.

Made use of the property P6., the following tracking control lawwith the parameter update law (18) is proposed in this paper

s ¼ Kdðh0Þ � Kpsð~qÞ � Kvsvð _~qÞ þUd � hcðtÞ; ð19aÞhcðtÞ ¼ shðhðtÞÞ; ð19bÞshðhÞ ¼ �shðhÞ þ p; ð19cÞ�shð� � phÞ 2Fðlh; qh; eh; � � phÞ; ð19dÞ

where hcðtÞ is called the applied parameter-estimate, p is a constantvector, each entry of shð�Þ belongs to the area between the two sha-dow that is shown in Fig. 1, and

sð~qÞ 2Fðl;q; e; ~qÞ; ð20Þsvð~qÞ 2Fðlv;qv; ev; ~qÞ: ð21Þ

Remark 2. According to (18) and (19), it shows that the proposedcontroller employs the applied parameter-estimate hcðtÞ instead ofhðtÞ as a robot parameter estimate. That is the main feature of thisproposed adaptive scheme, which distinguishes itself from otheradaptive techniques. It can been seen that the applied parameter-estimate hcðtÞ is always bounded since �shð�Þ in (19d) and hence shð�Þis bounded. As a result, parameter drifting is restrained in theresulting adaptive system because hcðtÞ is always bounded.

3.2. Preparatory parameter design of controller

In general, some prior knowledge of robot such as the estimatedparameter-range H is available in practice. Or rather, the range ofthe unknown parameter hi is known.

According to (19c), function shð�Þ defines the so-called controllerparameter-range N, that is

N ¼ f 2 Rmj infh2Rmfs h;iðhÞg 6 fi 6 sup

h2Rmfsh;iðhÞg

( ): ð22Þ

As for function shð�Þ, we are trying to design its parameters such thatthe controller parameter-range N is coincident with the range of H,or in other words, H is used to design function shð�Þ.

The parameter design of shð�Þ should guarantee that the range ofeach Ni involves that of Hi and is as closed to it as possible. Usually,we believe that the true parameter hi is located at the center of theestimated parameter-range Hi (i = 1, . . . ,m). Consequently, param-eters of shð�Þ are chosen as follows for every i = 1, . . . ,m:

pi ¼12ð�hi þ hiÞ; ð23Þ

ph;i ¼ 2pi=ðqh;i þ lh;iÞ; ð24Þ

ðlhÞiðehÞi ¼12ð�hi � hiÞ: ð25Þ


Meanwhile, deviation qh,i � lh,i should be chosen as small as possi-ble so as to make deviation of N from H smaller.

Proposition 1. According to (16), (19c), (19d) and (23)–(25), itverifies that

H � N ð26Þ

and hence

h 2 N: ð27Þ

if the true parameter h locates within the estimated parameter-rangeH, namely, h 2H.

Remark 3. From (19c), the value of pi and (lh)i (eh)i (with (qh)i (eh)i)decide the position and the volume of the range of sh;iðhÞ. On theother hand, the basic idea in the above parameter design is to makethe estimated-range H and that of N coincident with each other.Therefore, (23) and (25) is the key to parameter design for functionshð�Þ.

For the given parameters of function shð�Þ viz. qh, lh and eh,different function choice results in little change of the character-istics of shð�Þ and hence leads to small performance difference. Inother words, performance stemmed from shð�Þ depends primarilyon its parameters, instead of its types. Due to multiple parameterchoice of function shð�Þ, robust-adaptive algorithm (18) and (19)provides some flexibility for adaptive controller design and hencemake more robustness and better transient performance possibly.

If value of lh,ieh,i is large enough, condition (27) is alwayssatisfied. When lh,i = qh,i = 1 and eh,i ? +1, it means that theproposed robust-adaptive algorithm is identical with the conven-tional one. Namely, in the conventional case, it implies thathcðtÞ ¼ hðtÞ.

For the sake of convenience of presentation, suppose that

Kp ¼ kpI; Kv ¼ kvI: ð28Þ

Furthermore, on the basis of (10)–(12), function s(�) can be chosento satisfy the following inequality:

ksð�ÞkP maxfksMð�Þk; ksCð�Þk; ksgð�Þkg: ð29Þ

4. Asymptotic stability of the resulting system

Since function shð�Þ is bounded (see (19c)), we can find someconstant vector h* satisfying equation

shðh�Þ � h ¼ 0 ð30Þ

if and only if value of robot parameter hi does not go beyond therange of function shð�Þ for every i = 1, . . . ,m, namely, (27) is valid.

After that, we define a scalar function U~hð~hÞ with vector Wð~hÞsuch that

r~hU~hð~hÞ ¼ Wð~hÞ; ð31aÞwhere the new parameter h* is called the mapped parameter,

~h ¼ h� h� and Wð~hÞ ¼ shðhÞ � h: ð31bÞAccording to Remark 4 below, we have

Wð~hÞ ¼ shðhÞ � h ¼ 0() ~h ¼ 0 or h ¼ h�; ð32Þ

then the following result is obtained.

Proposition 2. Suppose that (27) is satisfied, then equality (30) issolvable with respect to h*, and the following results are valid:

U~hð~hÞ > 0 if ~h–0;U~hð~hÞ ¼ 0 iff ~h ¼ 0;

(ð33Þ

U~hð~hÞ ! þ1 when ~h! 1: ð34Þ

Similarly, we define another scalar function by

r~qU~qð~qÞ ¼ ½Kp þ aqvKv� � sð~qÞ: ð35Þ

Obviously

U~qð~qÞ > 0 if ~q–0;U~qð~qÞ ¼ 0 iff ~q ¼ 0;

(ð36Þ

U~qð~qÞ ! þ1 when ~q! 1: ð37Þ

Remark 4. Note that function vector Wð~hÞ in (31a) satisfies all con-ditions presented in Definition, although its entry usually is notcentro-symmetric with respect to the origin. In fact, each entryof Wð~hÞ is still of a continuous piecewise-differentiable increasingfunction when condition (27) is valid since so does the functionvector shð�Þ in (19c) and we can always find some parameters viz.(l,q,e) so as to make all conditions of Definition satisfied. Thus, itis easily to verify (33) and (34) in Proposition 2. Similarly, it verifies(36) and (37) since

U~qð~qÞP kmfKp þ aqvKvg �Z

sð~qÞd~q; ð38Þ

where Kp and Kv is diagonal.

Using property P6, the control law can also be rewritten as

s ¼ �Kpsð~qÞ � Kvsvð _~qÞ þUð€qd; _qd;qdÞ½shðhÞ � h� þ Kd: ð39Þ

Then the closed-loop system dynamics is obtained as

MðqÞ€~qþ Cðq; _qÞ _~q� Hð _~q; ~qÞ ¼ �Kpsð~qÞ � Kvsvð _~qÞ þUd½shðhÞ � h�;ð40Þ

where

Hð _~q; ~qÞ ¼ ½MðqdÞ �MðqÞ�€qd þ gðqdÞ � gðqÞ þ ½Cðqd; _qdÞ � Cðq; _qÞ� _qd:

Theorem 1. Consider robot manipulator (2) with sd = 0, the robust-adaptive control law (18) and (19) is chosen such that (27) and (29)and the following conditions are all satisfied:

ðiÞ ðkp þ aqvkvÞI > a2kMfMg � Js; ð41Þ

ðiiÞ kp > gþ 12½g=aþ qvkv þ ð1þ c1Þk _qdk1�; ð42Þ

ðiiiÞ kvc _~q > aqðkMfMg þ c1effiffiffinpÞ þ c1k _qdk1

þ 12½gþ aqvkv þ að1þ c1Þk _qdk1�; ð43Þ

where �c _~q, a are defined in (14) and (18), respectively, and

g ¼ k€qdk1 þ c2k _qdk21 þ 1:

Then the resulting closed-loop system is asymptotically stable.Moreover, under assumption A.1, the torque constraints in A.2 issatisfied provided that

�si P qieikp þ qv;iev;ikv þ jKd;iðh0Þj þmaxh2NfðUdhÞig: ð44Þ

Proof. Consider the Lyapunov function candidate

Vð _~q; ~q; ~hÞ ¼ 12

_~qTM _~qþ a _~qTMsð~qÞ þ U~qð~qÞ þ cU~hð~hÞ; ð45Þ

where U~hð~hÞ, U~qð~qÞ are defined in (31a) and (35).

Step 1. The globally positive definition of V.From (33), (34) and (36) and (37), the Lyapunov functioncandidate (45) is radially unbounded positive definitionprovided that Vað _~q; ~qÞP 0; where


Va ¼12

_~qTMðqÞ _~qþ a _~qTMðqÞsð~qÞ þ U~qð~qÞ: ð46Þ

The Hessian matrix Ha of Va is given as follows:

Ha ¼M aMJs

aJTs M ðKp þ aqvKvÞJs

� �: ð47Þ

According to the result in Appendix C,

Ha > 0() ðKp þ aqvKvÞJs � a2JTs MJs > 0: ð48Þ

And that matrix Js is a diagonal,

ð41Þ ) ðkp þ aqvkvÞJs P a2kMfMgJTs Js

) ðkp þ aqvkvÞJs P a2JTs MJs;

with the help of convex function condition, it is easily to verify thatVað _~q; ~qÞP 0.

Step 2. To prove the negative definition of _V . The time derivativeof (45) is given as follows:

_V ¼ ½ _~qþ asð~qÞ�TM€~qþ a _~qT½ _Msð~qÞ þM _sð~qÞ� þ 12

_~qT _M _~q

þ _~qTr~qU~qð~qÞ þ c _~hTr~hU~hð~hÞ: ð49ÞSubstituting (40) and parameter update law (18) with (3) in prop-erty P1 into the above equation

_V ¼ ½ _~qþ asð~qÞ�T½H � Cðq; _qÞ _~q� Kpsð~qÞ � Kvsvð _~qÞ

�Uð€qd; _qd;qdÞWð~hÞ� þ a _~qT½ _Msð~qÞ þM _sð~qÞ� þ 12

_~qT _M _~q

þ _~qTr~qU~qð~qÞ þ c _~hTr~hU~hð~hÞ

¼ a _~qT½CTðq; _qÞsð~qÞ þM _sð~hÞ� þ _~qTr~qU~qð~qÞ þ ½ _~qþ asð~qÞ�T½H

� Kpsð~qÞ � Kvsvð _~qÞ�:

For the sake of clarity, we carry out the following manipulations be-fore next procedure,

[1] sTð~qÞCðq; _qÞ _~q ¼ sTð~qÞ½Cðq; _~qÞþCðq; _qdÞ� _~q 6 qec1

ffiffiffinp� k _~qk2 þ k _qdk � k _~qk � ksð~qÞk,

[2] _~qTMðqÞ _sð~qÞ 6 qkM � k _~qk2,[3] sð~qÞKv½qv

_~q� svð _~qÞ� 6 qvkvksð~qÞk � k _~qk,[4] ½ _~qþ asð~qÞ�THð _~q; ~qÞ 6 ½k _~qk þ aksð~qÞk� � fgksð~qÞk þ c1k _qdk � k _~qkg,

where (10)–(12) in property P5 and (19c) are applied. Then

_V 6 �kv_~qTsvð _~qÞ � akpksð~qÞk2 þ aqvkvksð~qÞk � k _~qk þ aq½c1e

ffiffiffinpþ kM�

� k _~qk2 þ ak _qdk � ksð~qÞk � k _~qk þ ½k _~qk þ aksð~qÞk� � ½gksð~qÞk

þ c1k _qdk � k _~qk�

¼ �kv_~qTsvð _~qÞ þ ½aqvkv þ gþ að1þ c1Þk _qdk�k _~qk � ksð~qÞk þ ½c1k _qdk

þ aqðkM þ c1effiffiffinpÞ�k _~qk2 � aðkp � gÞksð~qÞk2

:

Note that (15) and the fact xy 6 12 ðx2 þ y2Þ for all x,y 2 R, hence

_V 6 �k1k _~qk2 � k2ksð~qÞk2; ð50Þ

where (15) is applied, k1 and k2 are given as follows:

k1 ¼ kvc _~q � aqðkMfMg þ c1effiffiffinpÞ � c1k _qdk1

� 12½gþ aqvkv þ að1þ c1Þk _qdk1�; ð51Þ

k2 ¼ aðkp � gÞ � 12½gþ aqvkv þ að1þ c1Þk _qdk1�: ð52Þ

Therefore, (42) and (43) implies _Vð _~q; ~q; ~hÞ 6 0.On the basis of the aforementioned fact, the conclusion

concerning stability follows by invoking Appendix A.

Because of assumption A.1, i.e., the boundedness of the desiredreference trajectory, the applied control torque si of the algorithm(19) can be bounded as follows:

jsij 6 qieikp þ qv;iev;ikv þ jKd;iðh0Þj þmaxh2NfðUdhÞig ð53Þ

then the proof is completed. h

Remark 5. Theorem 1 shows that in the absence of disturbances,the robust-adaptive law (18) and (19) renders the asymptotictracking of the closed-loop system under actuator constraintswhen the true robot parameter locates within the estimatedparameter-range H viz. (27) is valid.

Remark 6. As for robotic manipulator with joint servo-motor, themaximal torque �si is depended on the corresponding angle veloc-ity. It can be denoted as �sið _qiÞ that is easy to get according to themotor manual instruction. In other words, the maximal torque �si

of joint actuator is not constant in real robot. Actually, the lastthree terms of (44) are also depended upon _q or _qd ¼ _q� _~q. Its lessconservative description is given as follows:

�sið _qÞP qieikp þ kvsv;ið _qÞ þ f ð _q� _~qÞ; ð54Þor �sið _qd þ _~qÞP qieikp þ kvsv;ið _qd þ _~qÞ þ f ð _qdÞ; ð55Þ

where f ð�Þ ¼ jKið€qd; �;qd; h0Þj þmaxh2NfðUð€qd; �;qdÞhÞig.Therefore, it is enough to check (54) or rather (55) through the

desired reference trajectory ð€qd; _qd;qdÞ with rough estimate of thecorresponding _~q. By virtue of modern computer, it would not be ofhard job. An easier and more conservative way is to use�s0i ¼min _qf�sið _qÞg to check (54) or (55) within the working space.

Remark 7

(i) From the controller structure in (19), the estimated parame-ters are shaped by function shð�Þ and hence outputs of esti-mator hð�Þ are no longer regarded as the estimatedparameters for the controller. In other words, robust-adap-tive algorithm (18) and (19) is an extensive conception ofadaptive since controllers of the conventional adaptive makeuse of hð�Þ directly; the latter can be conceived as a specialcase of the former.

(ii) When every element of vector �shð�Þ is chosen as some satu-rated functions, the proposed algorithm (18) and (19)becomes much more simple. Furthermore, the upper andlower bound of each element of function shð�Þ are the counter-parts of the estimated parameter-range H respectively. It issaid that function shð�Þ can be chosen as follows (i = 1, . . . ,m),

sh;iðhÞ ¼�hi if hi > �hi;

hi if �hi P hi P hi;

hi if hi < hi:

8><>: ð56Þ

This is something similar to Projection Algorithm, but the algorithm(18) and (19) has more explicit physical meaning [13]: if the upperand lower bound of the unknown parameter h are both available,we can easily choose the form and parameters of function shð�Þ like(56). Therefore, it shows the easier and simpler synthesis and calcu-lations of the proposed algorithm.

5. Robustness/performance of the resulting system

For a given adaptive algorithm, it is more interest in robustnessto disturbance from the point of practical view. Since the proposedrobust algorithm (18) and (19) requires the estimated parameter-range H, so it is necessary to deal with stability analysis in the

0 5 10 15 20 25 30-10

0

10

20

30

40

50

60

Ang

le tr

acki

ng e

rror

(gr

ad)

t (s)(a) Angle tracking error of RA


presence of improper estimation of parameter-range viz. h R H(h R H) h R N).

When h R N, i.e. (27) is not satisfied, the closed-loop systemwithout disturbance can be rewritten as follows:

MðqÞ€~qþ Cðq; _qÞ _~q ¼ Hð _~q; ~qÞ � Kp~q� Kv_~qþ sh; ð57Þ

where sh ¼ Uð€qd; _qd;qdÞ½shðhÞ � h�: ð58Þ

It is said the parameter error can be received as a disturbance sh,boundedness of function vector shðhÞ and the bounded trajectory(assumption A.1) implies bounded sh.

Before the detailed presentation, we give a necessary result asfollows:

Lemma 1. Suppose that the manipulator parameters are all exactlyknown, position error ~qðtÞ always stays within some region j~qij 6 �c~q

for constant �c~q > 0 and every i = 1, . . . ,n, the control law for manip-ulator (2) is given as follows:

s ¼ �Kpsð~qÞ � Kvsvð _~qÞ þ KdðhÞ ð59Þ

and besides, (41)–(43) in Theorem 1 are all satisfied, then

(i) The disturbed closed-loop system (2) is Lp input/output stablewith respect to pairs ( _~q; sdÞð~q; sdÞ for all p 2 [1,1], namely

-50

0

50

l sig

nal (

Nm

)

k~qkp 61x½V

12að0Þ þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=kmfMg

p� ksdkp� ð60Þ

k _~qkp 61ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12 kMfMg

q þ aqx

0B@

1CA V

12að0Þ þ

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 kmfMg

q � ksdkp

264

375;ð61Þ

where V12að0Þ :¼ V

12að _~qð0Þ; ~qð0ÞÞ.

-100

al c

ontr

o
(ii) Solution ½~qðtÞ _~qðtÞ� of the system is uniformly ultimatelybounded and converges to the residual set, respectively, asfollows:
0 5 10 15 20 25 30-200

-150

9

10

Act

u

t (s)(b) Actual control signal of RA

R~q ¼ ~qjk~qk 6 1ffiffiffiffiffiffiffiffiffiffiffi2=km

p� ksdkmax=x

n o; ð62Þ

R _~q ¼ _~qjk _~qk 6 1ð1þ aqffiffiffiffiffiffiffiffiffiffiffi2=km

p=xÞ � ksdkmax

n o; ð63Þ

where k1,k2 are defined in (51) and (52) and, respectively, and

c~q ¼minifliei=�c~qg;

1 ¼ 12

kMfHag=minfk1; k2gðHa in ð47ÞÞ;

x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12c~qðkp þ aqvkvÞ � a2kMfMg � kmfJsg

r:

8

te (

kg)

Proof. See Appendix B. h

2m2l

2q

1q

3q

1l

3m

1m

Fig. 2. The 3-dof robot manipulator.

When disturbance sd is consider for system (2), with the aid ofLemma 1, the we have the following result:

0 5 10 15 20 25 300

1

2

3

4

5

6

7

App

lied

para

met

er e

stim

a

t (s)

(c) Applied parameter estimate qc

Fig. 3. Asymptotic stability of RA scheme.


Theorem 2. When the true robot parameters h go beyond theestimated parameter-range H, the robust-adaptive control law (18)and (19) for the disturbed system (2) makes all the properties ofLemma 1 valid except that sd(t) is replaced by sd(t) + sh(t), wheresd(t), sh(t) are defined in (2) and (57), respectively.

Remark 8. In the presence of external disturbances and/orunmodeled dynamics, some existing adaptation algorithms mayresult in parameter drifting when PE condition is not valid [2].The result in [2] shows that exponential tracking error convergenceis guaranteed but the prediction error is employed in adaptive con-troller and hence a parameter-dependent PE condition is required.In the robust-adaptive algorithm (18) and (19), bounded functionshð�Þ is introduced into controller (17). As a result, bounded param-eter estimate, instead of unlimited one in some adaptive scheme, isactually employed in the controller, and thus, it restrains the adap-tive system from parameter drifting. From the above results, itshows that the proposed adaptive algorithm (18) and (19) dealswith this problem effectively; furthermore, even the true robotparameter h exceeds the estimated parameter-range H, robustnessis guaranteed without the PE condition. Compared with the exist-ing adaptive algorithms and their modifications, the algorithm (18)and (19) is relatively simpler and more effective.

Remark 9. Note that projection algorithm guarantees boundednessof parameter-estimate on the condition that the true parametersare located within the known parameter-rangeH. So some robust-ness is also guaranteed accordingly because parameter drift isrestrained by this algorithm. However, in the case of h R H, itsrobustness and transient performance is not clear, for example,what does it happen when the true parameter h goes beyond theestimated parameter-rangeH. Furthermore, simulations in nextsection show that projection algorithm is not robust enough tosome real-world robots e.g. robot systems actuated by brusheddirect-current motors. With the help of Theorems 1 and 2 in thispaper, these above problems are solved by robust-adaptive (18)and (19) accordingly.

Fig. 5. Disturbance in joint space.

6. Numeric example

As the sine–cosine–sine (amplitude 1 rad, frequency 1 Hz) trajec-tory-tracking task in joint-space for a weighting-lifting operation isconsidered for 3-dof manipulator described in Fig. 2 [20]. Theparameters are given as follows:

0 5 10 15 20 25 30-10

0

10

20

30

40

50

60

t (s)

Ang

le tr

acki

ng e

rror

(gr

ad)

Fig. 4. Robustness of RA to improper

l1 ¼ 1 ðmÞ; l2 ¼ 1 ðmÞ; m1 ¼ 10 ðkgÞ; m2 ¼ 7 ðkgÞ;m3 ¼ 6 ðkgÞ;

Only parameter m3 is unknown, its initial estimation is 7 (kg). Theinitial configuration is

qð0Þ ¼ ½111�T ðradÞ; _qð0Þ ¼ ½000�T ðrad=sÞ:

In order to execute the above tracking task of the manipulator, it re-quires that the maximal control signals of joint-2 exceed the value189.2.

Disturbance sd ¼ ½sd1 sd2sd3 �T, where sd1 ¼ 4ðradÞ, sd2 is square-

wave (amplitude 2 rad, frequency 1 Hz), sd3 is a chirp signal.Suppose that the estimated parameter-range is available, i.e.,

H ¼ N ¼ ½m3 �m3�. The real-world complexity such as dead-zone,external disturbances, measurement noises and motor dynamicsis taken into account to demonstrate the effectiveness of the pro-posed algorithm (18) and (19). The parameters of dead-zone ofthree join-motors are 10, 10 and 5, respectively.

Dynamics of brushed direct-current motor [4] is described as

L_I þ RI þ Kb _q ¼ v þ d; s ¼ KtI;

where R is the resistance of the armature circuit; L is the induc-tance; Kb is the back emf constant of the motor; I is the armature

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

t (s)

Act

ual c

ontr

ol s

igna

l (N

m)

estimation of parameter-range.


current, and v is the armature input voltage. d denotes external dis-turbance. Here R = diag{[2 0.8 1]} (X), L = diag{[0.025 0.1 0.05]} (H),

0 5 10 15 20 25 30-10

0

10

20

30

40

50

60

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l (N

m)

App

lied

para

met

er e

stim

ate

(kg)

t (s)

t (s)

)(st

(a) Angle tracking error of RA

(b) Actual control signal of RA

(c) Applied parameter estimate qc

Fig. 6. Result of RA (without motor dynamics) in the presence of dead-zone anddisturbance.

Kb = diag{[0.9 1.2 0.9]} (N m/A), Kt = 0.9 (N m/A). d includes whitenoise and square-wave like sd2. The maximal signals of threejoint-actuators are 80(v), 189.3(v), 80(v), respectively.

The controller parameters are chosen as follows:

a ¼ 0:3; c ¼ 56; Kp ¼ diagf½7 7:5 9�g;

Kv ¼ diagf½4 3 3�g; sð~qÞ : l ¼ q ¼ ½48 57:6 48�;e ¼ ½0:19 0:16 0:12�svð~qÞ : lv ¼ qv ¼ ½36 48 36�;

ev ¼23

12732

1� �

:

In robust adaptive controller (19), Shð�Þ is chosen as follows:

shðm3Þ ¼12ð �m3 þm3Þ þ

12ð �m3 �m3Þ � satðm3Þ;

where sat(�) is a saturated function.Using SIMULINKTM in MATLAB, this simulation is to

demonstrate:

(i) the effectiveness of the proposed robust adaptive algorithm(denotes RA);

(ii) significant enhancement of robustness and transient perfor-mance of RA in comparison with the conventional adaptive(denotes CA) and the smooth projection algorithm (denotesPA).

Note that controller of the conventional adaptive is the case ofhcðtÞ ¼ hðtÞ. The smooth Projection Algorithm [30] is given asfollows:_h ¼ ProjðyÞ=c; y ¼ UTð€qd; _qd;qdÞ½ _~qþ asð~qÞ�;

ProjðyÞ ¼I � pðhÞpT

hðhÞphðhÞ

kphðhÞk2

� �y if pðhÞ > 0 & phðhÞy > 0;

y otherwise

8<:

with pðhÞ ¼ ðhTh� h2MÞ=ðu2 þ 2uhMÞ, phðhÞ ¼ dpðhÞ=dh, u = 0.1 and

satisfies jhðtÞj 6 hM þu.Throughout all the simulation results, history of joint-1, joint-2

and joint-3 are denoted by ‘‘solid”, ‘‘dot” and ‘‘dash” lines, respec-tively. For the sake of clarity, measurement noise and motor noiseis omitted since it does not affect the validity of simulation results.

(i) Demonstrate effectiveness of RA.When m3 2H = N = [49], Fig. 3 shows asymptotic stability ofRA in absence of dead-zone, disturbance and noise. In theabsence of motor dynamics, its robustness to improper esti-mate of parameter-range is shown in Fig. 4. Meanwhile, per-formance of the resulting system, in the presence ofdisturbance (Fig. 5) and dead-zone, demonstrates in Fig. 6.Where parameters of start (end) of dead-zone for every jointare �10(10), � 10(10), �5(5) respectively. The above resultsverify the effectiveness of the proposed robust-adaptive con-trol algorithm.

(ii) Comparison between RA and CA/PA.In the presence of bounded disturbances, simulations for RA,CA, PA are shown in Fig. 7, which demonstrates more effec-tiveness of RA over CA & PA. When motor dynamics is takeninto account in robot dynamics, comparison results in Fig. 8show poor robustness of CA and PA, their performance isdeteriorated greatly. On the contrary, robustness and tran-sient performance of RA are guaranteed.

7. Conclusion

This paper develops a novel guaranteed robustness/perfor-mance adaptive control with limited torque for tracking control

0 5 10 15 20 25 30-1500

-1000

-500

0

500

1000

1500

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30-10

0

10

20

30

40

50

60

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

0 5 10 15 20 25 30-10

0

10

20

30

40

50

60

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l (N

m)

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l (N

m)

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l (N

m)

(a) Simulation results of RA

(b) Simulation results of CA

(c) Simulation results of PA

Fig. 7. Comparison between RA and CA and PA without motor dynamics and dead-zone in presence of disturbances.


of robot manipulators. It provides a simple alternative of adap-tive controller, whose robustness and performance are guaran-teed without any PE conditions. Asymptotic stability is

obtained when the true robot parameters are contained withinthe estimated parameter-range. In the presence of boundeddisturbances or noises, the resulting system is Lp input/output

0 5 10 15 20 25 30-20

-10

0

10

20

30

40

50

60

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

0 5 10 15 20 25 30-3000

-2000

-1000

0

1000

2000

3000

4000

5000

6000

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30-50

0

50

100

150

0 5 10 15 20 25 30-200

-150

-100

-50

0

50

100

150

200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l

Ang

le tr

acki

ng e

rror

(gr

ad)

Act

ual c

ontr

ol s

igna

l

(a) Simulation results of RA

(b) Simulation results of CA

(c) Simulation results of PA

Fig. 8. Comparison between RA and CA & PA for robot with motor dynamics.


stable and its solution is uniformly ultimately bounded eventhough the true parameters go beyond the estimated parame-ter-range. Compared with the conventional adaptive and projec-

tion algorithm, this proposed algorithm is a simpler and moreeffective scheme for improvement of robustness and perfor-mance for adaptive systems.


Appendix A

Lemma 2 (Lyapunov-like stability theorem). For a give system withstate vector X with some Y � X, if there exists a Lyapunov candidateV(X), such that

(i) V(X) P 0 for any X 2 Rn;(ii) _VðXÞ 6 �a1YTY , for some constant a1 > 0;

(iii) suptP0_YTðtÞ _YðtÞ 6 a2, for some constant a2; then

limt?1Y(t) = 0.

Proof. From (i) and (ii), we have 0 6 V1 6 V0 andZ 1

0YTðtÞYðtÞ ¼ ðV0 � V1Þ=a1 6 a3 ð64Þ

for some constant a3 > 0. It implies that Y(t) is square integrable.According to a simple alternative to Barbalat Lemma presented by[29] with (iii), the proof is completed. h

Remark 10. Asymptotic stability of Y(t) in Lemma 2 requiresboundedness of suptP0

_YTðtÞ _YðtÞ instead of uniform continuity of_V , and the later is derived directly from Barbalat Lemma [26]. It isa weaker condition that would be satisfied when Y(t) is a vectorof continuous piecewise-differentiable functions defined in Section2.1.

Appendix B

Proof of Lemma 1. Consider the same Lyapunov function candidateVa like (46), Va is radially unbounded positive definition when (41)is satisfied.

With the help of manipulations in the proof of Theorem 1, wehave

_Va 6 �k1k _~qk2 � k2ksð~qÞk2 þ k _~qþ asð~qÞk � ksdk: ð65Þ

On the one hand, according to convex function condition, it is easilyto verify

Va 612

kMfHag � ðk _~qk2 þ ksð~qÞk2Þ ) k1k _~qk2 þ k2ksð~qÞk2 P Va=1:

ð66Þ

On the other hand,

Va ¼12½ _~qþ asð~qÞ�TMðqÞ½ _~qþ asð~qÞ� þ Vb;

where

Vb ¼ U~qð~qÞ �12a2sð~qÞTMðqÞsð~qÞ

P ðkp þ aqvkvÞZ

sTð~qÞd~q� a2kMfMgZ

sTð~qÞdsð~qÞ

¼Z

sTð~qÞ � ½ðkp þ aqvkvÞI � a2kMfMg � Js� � d~q ð67Þ

then (41)

) Va P 0

) Va P12

kmfMg � ½ _~qþ asð~qÞ�T½ _~qþ asð~qÞ�

) k _~qþ asð~qÞk 6 V12a=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

kmfMgr

:

ð68Þ

With the combination of (66) and (68), then

_Va 6 �11� Va þ


p� V

12a � ksdk )

ddt

V12a þ

121� V

12a

612


p� ksdk: ð69Þ

Multiplying both sides of inequality (69) by et/21 and integrating,we obtain

V12að _~qðtÞ; sð~qðtÞÞÞ 6 e�

121t � V

12að0Þ þ Vd; ð70Þ

where

Vd ¼12


p Z t

0e�ðt�sÞ=21 � ksdkds:

The term Vd is a convolution operator on ksdk, and from well-knownresult in linear system theory [5], we obtain

kVdkp 6 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=kmfMg

p� ksdkp ð71Þ

for all p 2 [1,1]. Based on (70), it verifies

j _~qþ asð~qÞkp 61ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12 kMfMg

q V12að0Þ þ


q ksdkp

264

375: ð72Þ

According to, j~qij 6 �c~qj it means that

jsið~qiÞjP c~qj~qij; i ¼ 1; . . . ;n: ð73Þ

Since matrix Js is diagonal, from (67), we have

Vb P kmfðkp þ aqvkvÞI � a2kMfMg � Jsg �Z

sTð~qÞd~q

P c~qkmfðkp þ aqvkvÞI � a2kMfMg � Jsg �Z

~qT d~q

¼ 12c~qkmfðkp þ aqvkvÞI � a2kMfMg � Jsg � k~qk

2

) V12að _~qðtÞ; sð~qðtÞÞÞP x � k~qk ð74Þ

Combined (74) with (70), it verifies that

k~qkp 61x

V12að0Þ þ


q � ksdkp

264

375 ð75Þ

and hence

k _~qkp 61ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12 kMfMg

q þ aqx

0B@

1CA V

12að0Þ þ


q � ksdkp

264

375 ð76Þ

with the help of the following inequality:

k _~qkp 6 k _~qþ asð~qÞkp þ aksð~qÞkp 6 k _~qþ asð~qÞkp þ aqk~qkp:

When t ?1, from (70) and (74), we obtain

x � k~qkt¼1 6 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=kmfMg

p� ksdkmax: ð77Þ

Because k _~qk 6 k _~qþ asð~qÞk þ aqk~qk, according to (70) and (77), ityields

k _~qkt¼1 6 1ð1þ aqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=kmfMg

p=xÞ � ksdkmax: ð78Þ

The proof is completed. h

Appendix C

Suppose that a symmetric matrix is, P ¼ A BBT C

� �where A and

C are square. This matrix is positive definite if and only if


A > 0 and C � BA�1BT > 0; or

C > 0 and A� BTC�1B > 0:

References

[1] Arimoto S. Fundamental problems of robot control: Part 1. Innovations in therealm of robot servo-loop. Robotica 1995;13:9–27.

[2] Arteaga MA, Tang Y. Adaptive control of robots with an improved transientperformance. IEEE Trans Automat Contr 2002;47(7):1198–202.

[3] Canudas de Wit C, Siciliano B, Bastin G, editors. Thoery of robotcontrol. London: Springer-Verlag; 1997. p. 61–3.

[4] Chang Y-C. Adaptive tracking control for electrically-driven robots withoutoverparametrization. Int J Adapt Contr Signal Process 2002;16:123–50.

[5] Desoer CA, Vidyasagar M. Feedback systems: input/output properties. NewYork: Academic Press; 1975.

[6] Dixon WE, de Queiroz MS, Zhang F, Dawson DM. Tracking control of robotmanipulators with bounded torque inputs. Robotica 1999;17:121–9.

[7] Feng G. Analysis of a new algorithm for continuous-time robust adaptivecontrol. IEEE Trans Automat Contr 1999;44:1764–8.

[8] Gutman P-O. Robust and adaptive control—fidelity or a free relationship? In:Moheimani SO Reza, editor. Perspectives in robust control. Great Britain,London: Springer-Verlag; 2001. p. 85–101.

[9] Ge SS, Hang CC, Lee TH, Zhang T. Stable adaptive neural networkcontrol. Boston: Kluwer Academic Publishers; 2002.

[10] Ge SS, Lee TH, Harris CJ. Adaptive neural network control of roboticmanipulators. London: World Scientific; 1998.

[11] Huang CQ, Wang XG, Wang ZG. A class of transpose Jacobian-based NPIDregulators for robot manipulator with uncertain kinematics. J Robot Syst2002;19(11):527–39.

[12] Ioannou PA, Tsakalis K. A robust direct adaptive controller. IEEE Trans AutomatContr 1986;31:1033–4.

[13] Ioannou PA, Sun J. Robust adaptive control. New Jersey, USA: Prentice HallPTR; 1996. p. 203–8.

[14] Jiang L, Wu QH. Nonlinear adaptive control via sliding model state andperturbation observer. IEE Proc Control Theory Appl 2002;149(4):269–77.

[15] Karimi A, Landau ID. Robust adaptive control of a flexible transmission systemusing multiple models. IEEE Trans Control Syst Technol 2000;8(2):321–31.

[16] Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive controldesign. New York: John Wiley and Sons; 1995.

[17] Landau ID, Lozano R, M’Saad M. Adaptive control. London, UK: Springer-Verlag; 1997.

[18] Landau ID. Adaptive control: a perspective, IEEE Adaptive Systems for SignalProcessing, Communications, and Control Symposium, AS-SPCC(2000). AltaCanada. 2000; 181-186.

[19] Lewis FL, Jagannathan S, Yesildirek A. Neural network control of robotmanipulators and nonlinear systems. London: Taylor and Francis; 1999.

[20] Murray RM, Li Z, Sastry SS. A mathematical introduction to roboticmanipulation. Florida: CRC Press; 1994. p. 172–5.

[21] Narendra KS, Balakrishnan J. Improving transient response of adaptive controlsystems using multiple models and switching. In: Proc. 32nd IEEE Conf.Decision and Control, San Antonio, Texas; 1993. p. 1067–72.

[22] Narendra KS, Mukhopadhyay S. Adaptive control using neural networks andapproximate models. IEEE Trans Neural Networks 1997;8(3):475–85.

[23] Narendra KS, Driollet OA, Feiler M, George K. Adaptive control using multiplemodels, switching and tuning. Int J Adapt Control Signal Process2003;17:87–102.

[24] Pomet JB, Praly L. Adaptive nonlinear regulation: estimation from theLyapunov equation. IEEE Trans Automat Contr 1992;37(6):729–40.

[25] Santibanez V, Kelly R. Global convergence of the adaptive PD controllerwith computed feedback for robot manipulators. In: Proceedings of theIEEE International Conference on Robot Automation, Belgium; 1999. p.1831–6.

[26] Slotine J-JE, Li W. Applied nonlinear control. USA: Pentice-Hall, Inc.; 1991.[27] Su CY, Stepanenko Y. Adaptive variable structure set-point control of

underactuated robots. IEEE Trans Automat Contr 1999;44(11):2090–3.[28] Sun J. A modified model reference adaptive control scheme for improved

transient performance. IEEE Trans Automat Contr 1994;34(3):630–5.[29] Tao G. A simple alternative to the Barbalat Lemma. IEEE Trans Automat Contr

1997;42(5):698.[30] Tomei P. Robust adaptive control of robots with arbitrary transient

performance and disturbance attenuation. IEEE Trans Automat Contr1999;44(3):654–8.

[31] Yao A, Tomizuka M. Robust desired compensation adaptive control of robotmanipulators with guaranteed transient performance. In: Proceedings ofthe IEEE International Conference on Robotics and Automation; 1994. p.1830–6.

[32] Ydstie BE. Transient performance and robustness of direct adaptive control.IEEE Trans Automat Contr 1992;37(8):1091–105.

[33] Zergeroglu E, Dixon W, Behal A, Dawson D. Adaptive set-point control ofrobotic manipulators with amplitude-limited control inputs. Robotica2000;18:171–81.

1-s2.0-S0957415808000688-main

Documents

Transcript of 1-s2.0-S0957415808000688-main