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Adaptive motion control design of robot manipulators:an input-output approachRAFAEL KELLY a , RICARDO CARELLI a & ROMEO ORTEGA aa Universidad Nacional Aulonoma de Mexico, Apdo, Postal 70-256, 04510, Mexico, D.F.,Mexico

Version of record first published: 27 Apr 2007.

To cite this article: RAFAEL KELLY, RICARDO CARELLI & ROMEO ORTEGA (1989): Adaptive motion control design of robotmanipulators: an input-output approach, International Journal of Control, 50:6, 2563-2581

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INT. J. CONTROL, 1989, VOL. 50, No.6, 2563-2581

Adaptive motion control design of robot manipulators:

an input-output approach

RAFAEL KELLYt, RICARDO CARELLIt and ROMEO ORTEGAt

An input-output approach to adaptive motion control design of robot manipu­lators is presented.The main technicaldevicein our approach is the passivitytheory.This formulation provides a framework suitable for the design of new control andadaptation laws.A newcontrol law which consists of a computed torque part and afeedforward compensation part is analysed using this approach.

l. IntroductionAdaptive control is a design technique suitable for applications where high

performance is demanded and the system uncertainty can be captured with a fewunknown constant parameters, but the structural information is otherwise consi­derable. The problem of robot motion control belongs to this class, since robotdynamics models are described by well-structured non-linear differential equationswith some uncertain parameters, i.e. payload, link inertias, etc. It is now well known(Spong and Vidyasagar 1988), that robot dynamics are feedback linearizable viainverse dynamics (also called computed torque) control. To attain high performancein non-adaptive designs, the inverse dynamics controller must ensure exact cancel­lation of some non-linear dynamics, and is therefore non-robust. This situationmotivates our interest in adaptive motion control of robot manipulators.

In this work, we are concerned with the problem of establishing a framework forthe analysis and design of adaptive controllers for rigid robot motion control. Thiscontrol problem can be stated as follows. Given the robot dynamic model with someuncertain constant parameters and a desired motion trajectory, find a parametrizedcontroller structure and parameter update law to compute the input signals such thatasymptotic position tracking is attained. If this objective is achieved for all initialconditions and desired trajectories then we will say that the scheme is 'globallyconvergent'.

The first globally convergent adaptive controller for robot manipulators with rigidlinks was presented by Craig et al. (1986). A key point in that work was the use of aparametrization of the robot equations that yields a linear regression in terms of asuitably selected set of robot and load parameters, as pointed out by Khosla andKanade (1985). Based on this parametrization an adaptive computed torquecontroller that required acceleration measurement and the inversion of a matrix ofestimated parameters was proposed. A scheme that does not need the matrix inversionwas later given by Amestegui et al. ( 1987), where an outerloop adaptive compensationscheme is used. The need of acceleration measurement was obviated via filtering byMiddleton and Goodwin (1986). Using the relationship between the inertia and

Received 15 June 1988. Revised 15 November 1988.t Universidad Nacional Autonoma de Mexico, Apdo, Postal 70-256, 04510 Mexico, D.F.,

Mexico.

0020·7179/89 $3.00 © 1989Taylor & Francis Ltd.

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2564 R. Kelly et al.

Coriolis torques matrices, pointed out by Arimoto and Miyazaki (1984) andKoditschek (1984), a globally stable scheme has been devised by Siotine and Li (1986)and Sadegh and Horowitz (1987). A particularly simple and elegant derivation of thecontroller has been given by Slotine and Li (1987 a) using an energy interpretationthat has strongly inspired our work. (For further references see the work by Ortegaand Spong 1988.)

For the stability analysis of the controllers that do not require accelerationmeasurement, the well known Lyapunov analysis cannot be applied directly. Instead,a problem specific mathematical machinery has been developed, which roughlyspeaking proceeds as follows. First, a quadratic function of (not necessarily all) thestates of the system is used to establish some properties (boundedness and squareintegrability) of some signals. Then, chasing the signals through the loop, the globalboundedness and asymptotic error tracking is established. The same procedure is usedin stability proofs of adaptive controllers for linear systems with relative degree largerthan one. It is worth remarking that, in general, this procedure does not allow us toestablish Lyapunov stability of the dynamic system, a situation that may cast doubtson the robustness properties of the algorithms.

The purpose of this work is to present an alternative framework for the problem ofthe analysis and design of adaptive motion controllers for rigid robots. It is based onthe system theoretic formulation of the input-output stability theory (see, e.g.,Willems 1971 and Desoer and Vidyasagar 1975). Input-output theory, in contrastwith Lyapunov analysis, is much coarser; it deliberately discards information so as toproduce general results that are easy to use. The input-output approach helps us tothink in terms of the structure of the system and to realize that the pattern of theinterconnections is more important than the detailed behaviour of the components. Itis our belief that the well-established body of analytic results available ininput-output theory provides an adequate framework to study the stability ofadaptive controllers of robot manipulators. Some previous work along this line hasbeen reported by Kelly and Ortega (1988) and Landau and Horowitz (1988). Theusefulness of this approach is illustrated with a new control law, which consists of acomputed torque plus a feedforward compensation. It is shown that this scheme,together with either a proportional plus integral or a composite (Slotine and Li1987 b) estimator, is globally convergent.

The layout of this work is as follows. First some input-output preliminaries aregiven in § 2. Section 3 contains the problem formulation and main results, which arederived using the input-output formalism. In § 4 we analyse the new adaptive robotcontrol system illustrating our main results. Some concluding remarks are given in § 5.

2. Mathematical preliminariesThe input-output formulation of the adaptive control problem is the principal

view taken throughout this work (see, e.g., Willems 1971, Desoer and Vidyasagar1975, for further details on input-output stability analysis). We will denote IR+ the setof non-negative real numbers, and by IR" the n-dimensional vector space over IRendowed with the euclidean (II . Ill, Villi . 112) or L~ (II . II ",) norm. It is shown that allsignals are imbedded in the normed function spaces

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Adaptive motion control design of robot manipulators

or its extensions

L2•= {x: IR+ -+ 1R"lllxlltT ~ LT Ilxf dt < 00, If T E IR+}

L"oo. = {x: IR+ -+ 1R·llIxll~.T~ sup IIxl1 2 < 00, If T E IR+}I~T

Here Vie is an inner product space, with inner product defined by

2565

The definitions below follow those by Kosut and Friedlander (1985). Let G : L2e-+ L2eand let u, p, fJ, be constants with p > O. Then If u E L2e s If T E IR +: G is passive if,

(u, GU)T;;' p

G is input strictly passive if,

(U, GU)T;;' p + pllulI~.T

G is output strictly passive if,

(U, GU)T;;' p + JlIIGulltTG has finite gain if,

(p and p are not the same throughout).The main ingredient in the stability proofs is a simplified version of the passivity

theorem suitably tailored for the analysis of adaptive robot controllers. In order tomake the work self-contained, the proof is also given.

Theorem I

Consider the feedback system of Fig. I with u, = U 2 = 0 so that

e1 = -H2 e2

e2 = HIe!

Figure 1. Feedback system.

(2.1 a)

(2.1 b)

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2566 R. Kelly et al.

(2.2)

(2.3)

Here HI and Hz map L'2e into L'2e. We assume that there are solutions e l , ez in L'2e.Suppose that there are positive constants PI' Pz, and Jlz such that

(II, HIII)r~ -PI(II, H zll)r ~ - pz + JlzlIHzlllltr

for all II E L'2e> T E [0, CX»), then e I E L'2.

ProofFor the proof see Appendix A.

RemarkTheorem I is a particular case of more general passivity theorems (Desoer and

Vidyasagar 1975). Inequalities (2.2) and (2.3) mean that HI is passive and Hz is outputstrictly passive, respectively. The theorem holds for any finite constants PI and P2provided fll + P2 > o.

We will also need the following result due to Desoer and Vidyasagar (1975, p. 59).

Theorem 2Let the transfer function H(s) E IR" x "(s) be exponentially stable and strictly proper.

Let II and y be its input and output, respectively. If II E L'2, then y E L'2 n L';", yE L'2and y(/) --t 0 as I ..... CX).

The key idea in the adaptive motion robot control analysis via the input-outputapproach is to describe the closed-loop system as a feed back interconnection of twodynamic operators, say H I and H 2 as in Fig. 1. Here H I and H 2 are non-linear time­varying operators defined by the parameter estimator and the controller structure,respectively. This procedure allows us to effectively separate the task of designing theestimator from that of choosing the controller structure. This is the main advantage ofthe input-output formulation. Throughout the work we will assume, that alloperators of interest map L2e into L 2e and their feedback interconnection is well posed(see, e.g., Willems 1971).

3. Adaptive motion robot control designWe consider the adaptive control of rigid link n degrees of freedom robot

manipulators described by (see, e.g., Spong and Vidyasagar 1988),

H(q)ij + C(q, 4)4 + C(q) = r (3.1)

where q E IR" is the vector of joint displacements, r E IR" is the vector of applied jointtorques (or forces), H(q) E IR" x" is the symmetric manipulator inertia matrix,C(q, q)4 E IR" is the vector of centripetal and Coriolis torques, and G(q) E IR" is thevector of gravitational torques.

3.1. Some propertiesSome fundamental properties of the motion equation (3.1) follow (see also Ortega

and Spong 1988).

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Adaptive motion control design of robot manipulators 2567

Property 1 (Arimoto and Miyazaki 1984, Siotine and Li 1987 a)

Using a proper definition of matrix qq, 4), H(q) and qq,4) in (3.1) satisfy

xT(H - 2C)x = 0, VX E IRe

Property 2Consider the robot dynamic equation (3.1). Then the operator

HR : L'2e-+ L'2e

is passive.

Proof

For the proof see Appendix B.

Remark 2This fundamental property of rigid robots was established by Kelly and Ortega

(1988) and Landau and Horowitz (1988) for the particular case when G(q) =0. Theproof for the general case is given by Ortega and Spong (1988) using a hamiltonianformulation. It has also been derived, without explicit reference to passivity, byKhorrami and Ozguner (1988) to prove that PI regulation is possible withoutcancellation of gravity. It is important to remark that the same property holds even inthe presence of dissipative friction forces.

The corollary below follows immediately from Property 2.

Corollary 1

Consider the differential equation

H(q)v + qq, 4)v = ({Jo

where H(q), qq, 4) are as in (3.1). Then, for all v, ({J E L'2e the operator

HE: L'2e -+ L'2e

: ({Jo-+ v

is passive. Furthermore, if ({Jo is generated as (see Fig. 2)

({Jo = ({J - HFv

f1I

'----------;vr-----~I

H2

Figure 2. Feedback system of corollary I.

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2568

then, the closed-loop operator

R. Kelly et al.

H 2: L2e--+ L2e:cp--+v

is output strictly passive for all input strictly passive operators HF: L2e.... L2e.

Property 3 (Khosla and Kanade 1985, Craig et al. 1986, Slotine and Li 1986)

The dynamic structure (3.1) is linear in terms of a suitably selected set of robot andload parameters, i.e.

H(q)q + C(q, eM + G(q) =Q(q, q, it)fJ

where Q(q, cj, it) E IR" x m is a matrix and fJ E IRm is a vector containing the unknownmanipulator and load parameters.

Property 4 (Craig 1988)

There exists a positive constant <X such that

<Xl ,;; H( q), . If q E IR"

where I is the n x n identity matrix. Matrix H-\ (q) exists and is positive and bounded.

3.2. A general control law structureWe will consider a general controller law structure given by

r = H(q)II + C(q, cj)I2 + G(q) - K oI 3

where

11 = Fdq, cj, qd, qd, itd) E IR"

12 = F2(q, cj, qd, qd, itd) E IR"

13 = F3(q, cj, qd, cjd, itd) E IR"

(3.2)

(3.3)

(3.4)

(3.5)

are functionals to be chosen by the designer and H(q), C(q, cj), and G(q) have the samefunctional form as H(q), C(q, cj), and G(q), respectively, with estimated parameters 0.Here Ko E IR" X", qd E IR" is the vector of desired reference trajectories.

Remark 3The control law proposed by Siotine and Li (1987 a) can be obtained from (3.2) by

choosing

1\ = itd - Aq = t.12 = cjd - Aq

13= q+ Aq = cj - 12

Ko>O

where A E IR" X" is a matrix whose eigenvalues are strictly in the right complex half­plane and q~ q - qd' The control law proposed by Sadegh and Horowitz (1987) is

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Adaptive motion control design oj robot manipulators 2569

given by (3.2) with

J, = ijr «,I ij(a) da - Kij - Kd 4=t.

J2 = f>l (a) d«

J3= IJI(a)da- q

KD>O

(3.6)

where K" K p , Kd are n x n positive definite matrices.Using Property 3, (3.2) can be written in terms of the estimated selected set of

robot and load parameters as

where

</J(q, q, qd, 4d' ijd)8= H(q)Jl + C(q, 4)J2 + G(q)

</J(q, 4, qd' 4d' ijd) E IR" x m and 11 E IRm is a vector of estimated parameters.

(3.7)

(3.8)

3.3. Error equation

As usual in adaptive control theory, we will find it convenient to write thesystem equations in terms of the parameter error (see, e.g., Anderson et al. 1986). Tothis end, we denote Of!, 11- e, H(q) & H(q) - H(q), C(q, 4)!'; C(q, 4) - C(q, 4), andG!'; G(q) - G(q). From (3.8) we have

(3.9)

and (3.7) yields

r = [H(q) + H(q)]J, + [C(q, 4) + C(q, 4)]J2 + [G(q) + G(q)] - KDJ3 (3.10)

The closed-loop system equations are obtained from (3.1) and (3.10), thus

H(q)(ij - J,) + C(q, 4)(4 - J2) = H(q)JI + C(q, 4)J2+ G(q) - K DJ3

which using (3.9) yields

H(q)(ij - J,) + C(q, 4)(4 - J2) = </J(q, 4· qd' 4d' ijdJO - KDJ3 (3.11)

From the closed-loop equation (3.11) we will define an operator denoted by H2 withinput </J(q, 4. qd' 4d' ijd)O and output v

H 2 : L2e L2e: </JO v

(3.12)

where v = v(q, 4, qd' 4d' ijd) is a measurable signal chosen by the designer.Operator H 2 defines a non-linear time-varying map which depends only on the

robot model (3.1) and control law (3.2) but is independent of the parameter updatelaw.

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2570 R. Kelly et al.

Remark 4The interest of defining such an operator stems from the fact that it is possible to

establish for it 'nice' passivity properties. For instance, from Corollary I we can easilyestablish that H2 is an output strictly passive operator for the choices ofII J2' and 13used by Siotine and Li (1987 a) with

V=/3In this case, H F is a constant gain operator K D •

3.4. A general update law

The control law (3.2), alternatively written in an explicit parametrized form in(3.7), uses an estimated parameter vector tJ to compute the applied torques T. Theparameter estimates tJ are adjusted on-line by an update law. We will consider ageneral parameter update law structure given by

(3.13)

where the functional 0( . ) is chosen by the designer.Most of the adaptive laws used in continuous-time adaptive systems such as

gradient-type, proportional plus integral, least-squares, etc., can be written in the formof (3.13).

Using the parameter error vector {J = tJ - e, (3.13) becomes

(3.14)

(3.15)

From (3.14) we will define an operator HI which maps -y into rjJ(q, ti, qd' tid' iidW,

HI :L~e L~e

:-v rjJ{J

The input-output properties of the non-linear time-varying map HI are defined solelyby the functional 0( . ) and its relationship with v,

3.5. Feedback interconnection

The closed-loop behaviour of the adaptive motion robot control system as statedbefore can be analysed by the feedback interconnection of operators H I and H2 asshown in Fig. 3.

Figure 3. Error model.

To carry out the stability analysis via passivity theory we need to prove someinput-output properties for the maps HI and H 2 for some suitable selected functions}; (i = 1,2,3) and v. We present our main results in the following.

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Adaptive motion control design of robot manipulators 2571

Theorem 3Consider the adaptive motion robot control problem of the robot model (3.1)

using the control law structure (3.2) given in parametrized form by (3.7) and theupdate law (3.13). Let the operators HI and H2 be as defined in (3.15) and (3.12),respectively. Assume that v and r/J{J are in L'2e. If;; and v are chosen such that

<- v, r/J{J)T ~ - PI- 2<r/JO, v)T ~ -P2 + 112 II v112.T

for TE [0, (0) and positive constants PI' P2, and 1l2, then the following hold:

(3.16)

(3.17)

(a) v E L'2If in addition, v is defined such that there exists a stable, linear time-invariant, strictlyproper, operator F(p), p ~ dtdt so that qd- q = F(p)v then

(b)

(c)

(d)

ProofConclusion

Theorem 2.

(qr q) E L'2(qr q) E L'2nL':"

qd(t) - q(t) ->0 as t-> <X)

(a) follows directly from Theorem I, and (b)-(d) follow fromD

Remark 5

It is worth mentioning that, with the theorem formulation above, it is possible alsoto establish asymptotic zeroing of the velocity error and boundedness of all internalsignals when qd, qd' iid are bounded. The proof requires Lyapunov-type argumentswhich are beyond the scope of the present work. For further details see the work byOrtega and Spong (1988).

Remark 6To analyse within this framework the controller given by Siotine and Li (1987 a) we

chooseF(p) = -(pI + 1\)-1

with 1\ E IR" x" a Hurwitz matrix. For the scheme of Sadegh and Horowitz (1987) let

(I I )-1

F(p) = - pI + Kd+ pKp + p2 K I

where K d , K p , K, E IRnx " are chosen to ensure stability of F(p).

4. New controller structureIn this section, we present the input-output analysis of an adaptive motion robot

control system. The adaptive controller consists of a computed torque plus compens­ation control law structure. To show the generality of the formulation we consider twodifferent update laws as follows: proportional plus integral and the new compositeestimator given by Slotine and Li (1987 b).

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2572 R. Kelly et al.

(4.4)

(4.1 b)

4.1. Control law

We propose the controller structure

r= H(q)[ijd+ k)i+ kpif] + C(q,q)q+ G(q) - C(q, q)v (4.1 a)

where Ii = qd - q, H(q), C(q, q), and G(q) are the estimates of H(q), C(q, q), and G(q),respectively, and v is given by

1 .. .v = - p + ), [Ii + K,1i + kpq] (4.2)

with), E IR+ and K" K; are n x n positive definite matrices. Note that

H).v= -(q+K,q+Kpii'j (4.3)

and also

p, 1 ~v = ---q - --[K,q + K if]

p+), p+). p

Thus measurement of acceleration is not needed to compute the control law (4.1).

Remark 7The proposed control law (4.1) can be expressed in terms of the general control

Jaw structure (3.2) with

!,=ijd+K,q+Kpli, !2=q-V, !3=0, Kv=O

Note that, from Property 4 and (3.8) we can write the control as

t = H(q)!, + C(q, q)!2 + G(q) = ¢(q, qJI J2)0

Remark 8It is worth pointing out that the controller above can be related to the controller

given by Siotine and Li (1987 a), which for convenience is rewritten here as

r = H(q)j2 + C(q, q)!2 + G(q) - K D!3, K D > 0

with!2 and!3 as in Remark 3. On the other hand, the new controller may be expressedas

r = H(q)j2 + C(q, q)!2 + G(q) - ).H(q)v

with!2 and I' given by Remark 7. Even though both controllers share the samestructure, the restriction that K D be positive definite imposed in the first controllerdoes not necessarily hold for the new controller, since H(q) may be non-positivedefinite.

4.2. Adaptation laws

To update the parameter vector (j we consider first the so-called integral plusproportional adaptation law

6(1)= -K, I ¢T(q,qJ,J2)vdr-K¢T(q,qJIJ2)v+6(0) (4.5)

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Adaptive motion control design of robot manipulators 2573

(4.6 a)

(4.6 b)

(4.6 c)

where Ki and K are m x m positive definite matrices (see, e.g., Landau 1979, andLandau and Horowitz 1988). Note that for K = 0, this adaptation law corresponds tothe gradient-type used by Siotine and Li (1986) and Sadegh and Horowitz (1987). Wewill also consider the composite update law of Slotine and Li (1987 b)

Ii = - F[¢T (q, q,fl Js)» + ¢}(q. q,fI,f2)8]

r> = ¢HJ(q, q,fl ,12), F(O) = F(W > 0

8 = - ¢J( q, q,fl ,f2)(j + TJ

where we have used the notationw

( .) ,@, --( .), ill> 0J p+w

and T is defined by (4.1 b).

Remark 9

The interesting features of the laws above are the following. Parameter updatelaws with a proportional term have been in the literature for a long time (Landau1979). However, its role, e.g. for robustness enhancement, has not been fullyinvestigated. Tomizuka (1988) has shown that its inclusion is essential for establishingsome interesting robustness results of an adaptive system. On the other hand, thecomposite update law uses both prediction and tracking errors and the descentdirection is given by a least squares metric. Thus, at least locally, better convergenceproperties are expected. For further comments see Remark 10.

4.3. Stability analysis

To carry out the stability analysis we will establish first the input output propertiesof the map HI (3.15) and H 2 (3.12) for this choice of controller structure and updatelaws, respectively. Then, invoking the general stability Theorem 3 for the feedbackinterconnection, we prove the global convergence of the adaptive scheme.

4.3.1. Properties ofH2 . ReplacingflJ2' and j, in (JII) we get the error equation

H(q)[q + K"q + Kpt[] - C(q, q)v = -¢(q, q,JI,f2)fJ

which on substituting (4.3) yields

H(q)(i' + ).v) + C(q, q)v = ¢(q, q,fl ,12)8

Equation (4.8) defines a mapH2 : L'2e -> L'2e

whose properties are summarized in the proposition below.

(4.7)

(4.8)

(4.9)

Proposition 5

Consider the map H 2 as defined in (4.9). Then H 2 is output strictly passive, i.e.

<U,H2U>T~ -{J2+1l21IH2ullt T' VUEL'2" VTEIR+ (4.10)

where 112 = ).a > 0 and {J2 =1[vT(O)H(q(O))v(O)].

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2574 R. Kelly et al.

ProofThe proof follows immediately from Corollary I choosing H F = },H(q) and noting

that, in view of Property 5, it defines an output strictly passive operator.An alternative proof is as follows. Let the non-negative function of time

V(t) =![vTHv];;> 0

whose time derivative along the trajectories of (4.8) is

V = !vT[l1 - 2C]v + vT<pO - AVTv

and from Property 1 we have

V=vT<pO-}]Hv

Integrating from 0 to T we get

V(Ti - V(O) = -IT JcvTHv de + foT vT<pOdi

and noting that

- IT T -(u,H 2u)T=(V,<pO)T= 0 V <pOd,

(u,H 2u)T;;> -V(O)+},LT

vTHvd,

However, by Property 3 there exists IX> 0 such that IXllxl1 2,,; xT Hx for all t;;> 0, thus

(u, H2u)T;;> - V(O) + },IXLT vTv di

As

IIH2uIIL·=lIvll~.T=(v,v)T= LT

vTvdt

the proposition is proven. o

4.3.2. Properties of H" The update laws (4.5) or (4.6) together with the vectorparameter error 0 define an operator

H, : Lie-> Lie

: -v-><p O

whose properties are summarized in the proposition below.

(4.11)

Proposition 6Consider the map HI as defined in (4.5) and (4.11). Then, H, is passive, i.e.

(u,H lu)T;;>-{33' VueLie> VTelR+

where

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Adaptive motion control design of robot manipulators

Proof

For the proof see the work by Landau and Horowitz (1988).

2575

Proposition 7

Consider the map HI as defined by (4.6) and (4.11). Then HI is passive, i.e.

<u, HI U>T ~ - P3, 1/uE L1e> 1/T E IR+where

Proof

Letv= tiJTF- 1 0

whose derivative along the trajectories of (4.6) is

Ii = _OTeth - OT.pie + tiliiT.p112 (4.12)

Now, combining (4.1 b) and (4.6 c) we get

e= -.pfO (4.13)

The proof is completed by replacing (4.13) in (4.12) and integrating to yield.- 1 - 2<.pO, -v>T~211.pfOIl2.T-P3~ -P3 (4.14)

Remark 10A block diagram for the error model of the adaptive system with the composite

update law is given in Fig. 4. It is interesting to remark that the map - v -->.pO has a

Hr-----------------------------, 1

III II IL ~

'------------------1 H2I--------------J

v

Figure 4. Error model for the composite update law.

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2576 R. Kelly et al.

property which we conjecture might be useful, e.g. for robustness enhancement. To seethis, note that

4>fO = [~4>0J(t) + ~4>ftTp+w w

which, replaced in (4.13) gives a sort of sectoricity property.

(4.15)

We are in a position to present the main stability result of the adaptive motion controlsystem studied in this section.

Corollary 2Consider the control law (4.1) with the adaption laws (4.5) or (4.6) in closed-loop

with the manipulator model (3.1). Assume that v and 4>0 are in L~e' Then the followinghold:

(a)

(b)

(c)

(d)

v E L~

tiE L~

ij E L~nL"oo

ij( t) --+0 as t --+ co

Proof

The proof follows immediately from Theorem 3 considering Propositions 5-7.

o

4.4. Simulation results

Computer simulations have been carried out to demonstrate the stability andperformance of the new adaptive controller. The manipulator used for the simulationsis the two degree of freedom arm moving in the horizontal plane presented by Siotineand Li (1987 a). The corresponding dynamic model can be written as,

()I iiI + (()3 C21 +.~4S21 ):'2 - ()3S21 ~~ + ()4C21 ~~ = f l}

(()3 C21 + ()4S2tlql + ()2q2 + ()3S21ql - ()4 C21ql = f2(4.16)

with Ql' Q2' <11' <12'iiI' ii2 the corresponding two axes angular positions, velocities andaccelerations; fl' f2 the joint torques; ()1' ()2' ()3' ()4' are constants depending onthe geometrical, mass, and inertia parameters of the manipulator and the load;C21 = cos (Q2 - Qtl, S21 = sin (Q2 - Qtl·

Numerical values of the parameters are,

()I = 0'15, ()2 = 0·04

()3 = 0'03, ()4 = 0·025

In the controller design, matrices K p and K; are chosen to be diagonal,

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Adaptive motion control design of robot manipulators 2577

(4.17)

(4.18)

The control law, = HCiid + Kv 4+ K.ifl + C[q - v] + Gcan be written explicitly interms of the parameter vector 0 as.

'I = <P1l01 + <P13 0, + <P14 04}

'2 = <P22 02+ <P2'0, + rP24 04

where

rPll=fl" <PI,=C2J12- S2Iq2f22. rPI4=S2JI2+ C2Iq2f22

<P22=f22' <P2,=C2JIl+S2IqJ21' rP24=S2JII-CZ,qJ21

with

fll = lidl + kvI4I + kpi iii

fl2 = lid2 + kv242+ kp2ii2

f21 = ql - VI

f22 = q2 - V2

The unknown parameters 81.82.8,,84 are updated using a pure integral adaptive lawwith l=diag(y):

61= - Y<P l IVI

62 = - Y<P22 V2

63 = -Y(<PI,VI + <P2J V2 )

64 = -Y(<P14 VI + <P24 V2)

The desired trajectories last for two seconds, the first half second covering trackingfrom the initial to final reference positions, the remaining time for regulation. Desiredtrajectories for joint I are shown in Fig. 5. Those for joint 2 are constantqd2 = 1·745 rad, qd2 = 0, lid2 = O.

Si "~ i \I' . .,i \ ,q'dII \ /~/:;:::>, -L.- ~ _

8~' .", .__

i

"\

Figure5. Desired trajectories.

Simulations show the performance of the adaptive control algorithm of (4.17) and(4.18). A typical performance is presented in Figs 6-9 using the following controlparameters:

K" = diag (2k"

2k2 ) , K; = diag (ki, kn

with k, = 20, k2 = 30, ;, = 30, and Y = 0·2.

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2578 R. Kelly et al.

8.1

/\/ \, -! V q

,

1,1 \8.851 i \II - \L cL' \,,---

i 8.5 ---7L'"I- ----z

Figure 6. Position errors.

ii\8jL. I'v'"'-~--:;;:=-.---------I \ \~ /" ..

"8.5/ \ 'II \ I

..Ijl \ 1--.."-\ ! q,

. I

-L5~__ ":"\1..-8T---~I------'-fS'---"--Z

Figure 7. Velocity errors.

"i 1\ t

Zii I \-- ,

I \I I \1 ( \

! 1/,.....,,\ei 1.1 \ t 2-I ~.,/r'?

" ....._"" "j \ ...I ',r

-z] \ ,.L '-i_~----r_____--.---------.- ..- ..--8 8.5 1 1.5 I

Figure 8. Joint torques.

O.Z' __--.-/0---- -------- - -- .. -_..-.......- "'" -- - 9,(

,Ii I!

8.1/ iI i

!! .r-~ _ /~,i !({~ -<, ~,

el)k./.. ..__....----.-- ..__.._. __~ .. __..__ 8,8 8.5 1 1.5 Z

Figure 9. Parameter estimates.

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Adaptive motion control design of robot manipulators 2579

Figures 6 and 7 show the zero convergent position and velocity servo errors inboth joints. Joint torques are plotted in Fig. 8. Finally Fig. 9 shows the parameterestimates from an initial guess of zero values.

5. ConclusionsAn input-output framework for the analysis and design of adaptive motion

controllers of robot manipulators has been presented. The controller design reducesto finding a suitable controller structure and adaptation laws such that two associatednon-linear maps preserve some input-output properties. The closed-loop systemanalysis is carried out via the passivity theory. This input-output formalism was usedto analyse an adaptive motion controller which consists of a new control law togetherwith PI and composite adaptation laws.

It is worth emphasizing that we do not claim that with the input-output approachwe are able to prove stronger stability results than the ones reported elsewhere, at leastnot in its present stage. Also, it is important to note that, similarly to Lyapunovanalysis, to establish the input-output properties of the operators, suitable energyfunctions should be found, and this task requires ingenuity from the designer. Ourcontention is that the functional analytic framework allows us to treat in a unifiedfashion a broad class of robot models and most importantly, the approach itself ismore design-oriented than other techniques.

ACKNOWLEDGMENT

The work by R. Carelli was supported by the CONACYT-Mexico andCONICET-Argentina.

Appendix A

Proof of Theorem I

Using (2.1), then (2.2), (2.3) can be written as

(e"e2>T~ -fJ,

<e2' -e, >T ~ -fJ2 + Jl211-e,llh

for all e" e2 E Lie and TE [0, (0).Combining (A I) and (A 2)

_fJ"'-.-1_+_fJ..'-2 ~ II-e l IltTJl2

(A I)

(A 2)

where (fJI + fJ2)/Jl2 is a finite positive constant so that e1 belongs to Li. 0

Appendix BProof of Property 2

Let

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2580

Note that

R. Kelly et al.

where the term in square brackets is the robot kinetic energy

K ~+?H(q)ti~ 0

Also, the potential energy V~ 0 satisfies

dV = G(q)dq

thus

tiT G(q) dt = dV

Replacing the previous equation in (B 1) and using Property 1 we get

<ti, r)T = K(T) - K(O) + V(T) - V(O) ~ - K(O) - V(O)

which completes the proof. o

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