An approach to adaptive control of robotic manipulators

0005 1098/85$3.00+ 0.00 PergamonPressLtd. t'. 1985InternationalFederationof AutomaticControl

Amomatica. Vol. 21, No. 6, pp. 639 647, 1985. Printed in Great Britain.

An Approach to Adaptive Control of Robotic Manipulators* M. VUKOBRATOVIt2t and N. KIRt~ANSKIt Industrial robots manipulating various payloads represent non-linear and time-varying systems which should be controlled by robust adaptive controllers with self-adjustable parameters, suitable for microcomputer implementation. Key Words--Decentralized control; adaptive control; estimation; multivariable systems; non-linear systems; stability criteria; self-adjusting systems; servomechanisms; robots; manipulation; microprocessors; actuators.

problem of the decentralized control synthesis for large-scale mechanical systems with uncertain and variable parameters. In particular, we consider open-chain active mechanisms (OCAM), i.e. mechanical systems with several degrees of freedom powered by actuators. The modelling of OCAM, based on Newtonian mechanics, results in very complex multivariable, non-linear, stationary dynamic models (Luh, Walker and Paul, 1980; Vukobratovi6 and Potkonjak, 1982). In contrast to most physical or chemical processes where the parameters are partially unknown, the parameters of the actuators and the mechanism itself can be considered as completely known or directly measurable. In this regard, the decompositionaggregation approach is very convenient to be applied to OCAM-control synthesis. Nevertheless, the problem becomes more complex when one tries to consider an OCAM operating in real, environmental conditions. For example, consider a mechanical manipulator handling objects with variable masses (inertias) from one place to another. Under the supposition that object parameters are not always a priori known, the complete system (OCAM and the object) becomes a system with partially unknown and highly variable parameters. Similar examples can be found in biomechanics (hand or leg orthoses), aircraft, etc. Obviously, the parameter estimation problem now becomes of fundamental importance. However, papers devoted to the parameter estimation in large-scale systems (Guinzy and Sage, 1973; Hassan and Singh, 1977) cannot be directly applied to the large-scale mechanical systems because of the following facts. Variable parameters within some classes of largescale mechanical systems (e.g. robotic systems, etc.) do not represent the parameters of isolated subsystems (free of coupling). Variable parameters in this case represent in fact some of the parameters

Abstract The control synthesis for the robotic systems in which parameters are partially unknown is considered. We propose synthesis of robust, non-adaptive, decentralized control which has to stabilize robots for all allowable variations of the parameters. If the robust non-adaptive control cannot withstand all expected variations of parameters, we propose synthesis of indirect adaptive control, i.e. the estimation of the robot parameters is performed first and then used for adjusting the decentralized control gains. The non-adaptive and adaptive control syntheses are illustrated by simulation of an industrial robot with unknown payload mass. INTRODUCTION

CONTROL synthesis in multivariable mechanical systems, such as aircraft, hydraulic systems, industrial robots etc., imposes certain special requirements when applying general approaches to the control synthesis of complex dynamic systems. As opposed to the approach via linear regulator theory resulting in complex control with a great number of feedback loops, the approach via the decomposition-aggregation method enables the synthesis of a reliable and simple control scheme, justifiable from an engineering standpoint (Sandell, Varaiya, Athans and Safonov, 1978; Siljak, 1978; Weissenberger, 1973). The problem of decoupling complex mechanical systems into subsystems, the design of local controllers and the introduction of the global feedback loop have been intensively investigated during the past few years (Vukobratovi6 and Stoki6, 1980, 1981, 1982). Relying on these results, this paper considers the

* Received 8 August 1983; revised 20 February 1985; revised 15 April 1985. The original version of this paper was presented at the Third IFAC/IFORS Symposium on Large Scale Systems: Theory and Applications which was held in Warsaw, Poland during 11-15 July. The Published Proceedings of this IFAC Meeting may be ordered from Pergamon Press Limited, Headington Hill Hall, Oxford OX30BW, U.K. This paper was recommended for publication in revised form by Associate Editor M. Jamshidi under the direction of Editor A. P. Sage. t Mihailo Pupin Institute, Beograd, Yugoslavia. 639

640

M. VUKOBRATOVIC and N. KIRd~ANSK1

of the coupling function (load). The equations which describe the coupling among subsystems represent the complete dynamic model of the mechanism itself. On the other hand, the algorithms for parameter estimation (like maximum a posteriori parameter estimation and others) use the system model (the models of the subsystems and the coupling~ as a part of the control procedure (Chemouil, Katebi, Sastry and Singh, 1981 ). Within the large-scale mechanical systems, this would seriously burden the computation eflorts and make the real-time application difficult. F'or these reasons, tollowing one specific (indirect) adaptive control approach, we have accepted an engineering two-stage procedure. First, the estimates of plant parameters are generated and then used for adjusting the coefficients of local regulators. The plant parameters are estimated using the input output variables of the mechanical system's part and by using a fast, real-time algorithm for sensitivity model construction. Such an approach to the estimation problem was initially suggested by Feldbaum (Feldbaum, 1960; Timofeev and Ekalo, 1976). These algorithms are used as a basis for the development of a parameter estimator, because of the simplicity of the sensitivity model of an OCAM. Namely, this model is much simpler than the complete dynamic model due to the linearity of generalized forces with respect to the dynamic parameters (masses and moments of inertial of the mechanism. In the second stage, taking into account the estimated parameters and following the decomposition-aggregation approach the local regulators are synthesized. The coefficients of the regulators are adjusted according to the estimates of unknown parameters. The complete system stability is investigated using the Lyapunov functions of subsystems (Sandell et al., 1978: Vukobratovic and Stokic, 1980). In this paper variable parameters are assumed to be the dynamic parameters of the payload, although the position of its mass center and the orientation of its principal inertial axes are also variable parameters. The influence of inertial parameters compared with the influence of payload mass was analyzed by Vukobratovic and Kircanski (1984). It was shown that the variations of inertial parameters influence the system behaviour less than the variation of payload mass by an order of magnitude. In the text to follow we shall suppose that the mentioned "'geometric" parameters influence the system behaviour much less than the payload mass. These variations as well as variations in actuator parameters (for example, friction in reducers and joints) should be compensated by the non-adaptive part of the control system. This means that the local regulators should be designed to be robust with respect to these variations, and adaptive with

respect to mass variation. Otherwise, the adaptive regulators would be very complicated and difficult for implementation. In closing, a number of prior works on adaptive robot control should be mentioned. For example, Timofeev and Ekalo (1976) presented an algorithm which utilizes complete dynamic manipulator model in "inverse control structure". The basic deficiency of this method is the usage of the complete dynamic model as a part of control algorithm. Dubowsky and DesForges (1979) suggested a decentralized control structure based on "model reference" approach. Here, the reference model of the manipulator is considered as a set of linear, time-invariant subsystems. The model of the manipulator itself is considered as a linear timevarying system. This approach resulted in a simple control structure applicable to industrial robot systems. The limitations of this algorithm might appear in high speed manipulator applications. Horowitz and Tomizuku 11980) presented an adaptive scheme combining an inner loop model reference adaptive system and an outer loop fixed PID control. Here, the coordinate coupling and the system non-linearity are taken into account. The gravity effects and the actuator dynamics are neglected. Koivo and Guo (1981) considered an adaptive control scheme based on self-tuning regulators. The manipulator was modeled by a set of subsystems with time-varying parameters. These parameters are estimated by an on-line recursive algorithm based on the least squares errors criterion. This algorithm does not use any knowledge of the system dynamics which can be partially or, in some cases, completely calculated on-line. In this paper as well as in most other papers, practical stability of the complete non-linear system including the actuators is not considered. Also, the real necessity to introduce adaptive control, the implementation of which might be complex and expensive, is not analyzed. The procedure presented in this paper differs considerably from the above-mentioned. Our aim is to synthesize, at first, simple non-adaptive and robust decentralized control which can withstand all expected variations of parameters. Here we consider a manipulator system as a set of subsystems each associated to one degree of freedom Id.o.f.t of the mechanism. For each subsystem we synthesize a local controller which stabilizes decoupled (free) subsystem (Vukobratovic and Stoki6, 1982). Then we directly examine practical stability of the global (coupled) system for the given set of allowable parameter values. However, it might appear that we cannot find unique local regulators which accommodate the system to all values of parameters. In that case we have to introduce the adaptive control.

Adaptive control of robotic manipulators The efficiency of the proposed technique is illustrated by an industrial robot powered by hydraulic actuators. MATHEMATICAL M O D E L A N D C O N T R O L TASK

Let us consider the robotic system S consisting of the mechanical part of the system SM and actuators Si. The system has n degrees of freedom each powered by one actuator Si. The model of the actuator is given by linear time-invariant system: si: 2 i = Aix i + biN(u i) + fipi,

i e I = {i:i = 1,2 ..... n}

(1)

where x i e R"' is the vector of state coordinates of S/, ni is the order of the model S i, A i e R "'×"' is system matrix, b~s R"' is input distribution vector, f i e R"' is the load distribution vector, ui e R 1 is the input to actuator S i, N(u i) is the non-linearity of the amplitude saturation type, [uq ~< U~,x, P~ is the driving torque (load) acting upon the ith d.o.f. (actuator). This torque can be computed from

Pi = ~ Ito(q,O)glj + hi(q,(1,0),

i~l

(2)

j=l

where ~ j are scalar functions describing inertial coupling between degrees of freedom i and j, q = [ql... q, ]r ~ R" is vector of joint coordinates, 0 E R"~ is the vector of mechanism parameters and hi is a scalar function describing gravitational, centrifugal and Coriolis effects. Equations (2) form the model of the mechanical part of the system and can be presented in matrix form S"" P = H(q, O)gl + h(q, (1, O)

(3)

where P = [,°1... p , ] r e R" is the vector of driving torques (forces), H(q, 0) = [Nj]e R" ×" is a full-rank positive definite inertial matrix and h(q,(1,0) = [hl...h.]reR" is the vector of gravitational centrifugal and Coriolis effects. We shall assume that two coordinates of the state vector x i are joint coordinate qi and its velocity (1( Therefore, there exists a simple relation between x i and qi

q i = 7~xi

(4)

641

actuators S ~, i ~ I, and mechanical part of the system SM, is x = [xlT... x"r] w. Its order is N = ~ hi. i=1

Let us define the control task which will be considered in the text to follow. The nominal trajectory of the state coordinates of the system S is given x ° ( t ) = Ix°IT(t), x°2T(t) . . . . . x°nT(t)]T, V t ¢ T , T = {t: t¢ (0, r)}, r--given time period. The control should be synthesized ensuring practical stability of the system about the nominal trajectory, so that Vx(0)~X I and V0~® implies x(t,x(O))~X'(t), Vt~T, where X I ~ R N is given finite region of allowable initial conditions, and Xt(t) ~ R N, Vt ~ T, is given finite region of allowable system states at arbitrary moment t~ T. Here, it is assumed that x°(O)~X ~ and x°(t)~X'(t), Vt~T. Here, ®~R"~ denotes the set of allowable values of parameters 0. In the scope of decentralized control synthesis it is necessary to perform the distribution of the complete system model into the local subsystems and the coupling. The distribution may be done in various ways. Since the coupling P~ is a function of the state coordinates x i of subsystem Si, it is possible to extract some terms from the model of the mechanical system part SM and associate them to free subsystems. Diagonal elements of matrix H(q, O) can be presented in the form Hii(q , O) = Hii(O) -4- g~(q, O)

where Hii(O) > O, gin : ttii - Hii(O). The ith element of vector h(q, (1, O) can be written as

hi(q, (1, O) = hi(O)qi + g~h(q, (1, O)

Pi = HiiiJi + hiqi + Pi

ffi = ~ I-Iij(q, O)qj + ein(q, O);~i + eih(q, (t, 0). j=l

(j¢i)

Taking into account relations (4) and (5), equation (8) becomes

Oi = Tia~i

S~: ;¢i = 3,(O)x i + bi(O)N(u i) + fi(0)/~

AUTO 21:6-B

(8)

where

Substituting this equation into (1), we obtain

where Tia~ R 1 x hi. The state vector of the overall system S including

(7)

where hi(O)q~ originates from gravity forces acting upon ith actuator and el, = hi - hiqi. Substituting (6) and (7) into (2), we obtain

where T~i~R l ×,,. For example, if x i = [qi (1i]r, then T'c = [1 0]. A similar relation holds for acceleration (5)

(6)

e, =&VaS¢' + h,r x, + P,.

(9)

where .~i= (7,i(0) (Ai _f_fihi(O)~), ~i = ~i(O)bi ' fi = Ci(O)fi with Ci(O) = (I,i - fi~,(O)T~,)- '. The

642

M. VUKOBRATOVIC and N. KIRCANSKI

introduction of ~i(0) in the subsystem model S i is very common: each actuator is considered with inertiality (moment of inertia or equivalent mass) which includes not only actuator's own inertiality, but also the inertiality of the mechanical system part, which is moved by the considered actuator. The introduction of hi(O)qi in the subsystem model S~means that the actuator model is considered with gravitational load, which is a function of q~. N O N - A D A P T I V E C O N T R O L SYNTHESIS

In order to synthesize decentralized control for practical stabilization of the robotic system S, we shall observe an approximate decentralized model of the system, i.e. we shall observe system S as a set of free subsystems

where H ~> 0 is prescribed degree of exponential stability of free subsystem (12). Here, we observe criterion over the infinite time interval although the practical stability on the finite time interval is desired. The control minimizing the criterion (131 under the constraint (12) is given by A~(t) = - r [ 1DcrKiAxi(t),

Vt~ T,

where K i e R "'×"' is the positive definite matrix, which is the solution of the corresponding algebraic Ricatti equation. Now, the control ui(t) = u°~(O +

aft(t)

= u°i(t) - ri-lbirKiAxi(t),

Si: )Ci = Ai(O)xi + bi(O)N(tl i)

0 = Oa: (11)

where/~i = ~i(Od) and bi = bi(Oa). The synthesis of local controllers which have to stabilize (11 ) can be performed in the following way. Let us assume that there exists a nominal control u°i(t) and a nominal trajectory x°i(t) satisfying (11). Obviously, since the model (t 1) is quite simple, u°i(t) can be generated on-line. The model of deviation of free subsystem (11) from nominal trajectory x°~(t) and nominal programmed control u°~(t) is given by

Si: Vt~ T,

Vi~l

(121

where Axi(t) = xi(t) - x°i(t), Aui(t) = ui(t) - u°i(t). The stabilization of the subsystem (12) about nominal trajectory x°~(t) and nominal control u°~(t) can be achieved by minimization of the performance index: Ji(Axi(0)) =

(AuiriAu i + A x i T Q i A x i )

exp(2Ilt)dt

Viel

(15)

(10)

which are coupled through .f~(O)~. The control task will be solved as follows. First the local controllers are synthesized for free (decoupled) subsystems and then practical stability of the overall systems (1), (3) is checked. However, matrices of subsystems (10) depend on the unknown parameters 0 ~ ®. First, we shall synthesize simple linear control for one chosen value of the parameter vector 0 and then we shall examine whether such control is robust enough to withstand all allowable parameter variations, i.e. whether such control can stabilize system S for all assumed values 0~®. Let us consider free subsystems (10) for the chosen parameters values si: 2 i = Aixi -~- f f N ( R i)

(14)

Vi~l

should stabilize free subsystem (l l) about the nominal trajectory x°~(t). As the next step in control synthesis we have to examine whether the global system S (1), (3) (when coupling and parameter variations are taken into account) is practically stable. This analysis can be carried on using aggregation-decomposition method as has been presented in (Weisenberger, 1973; Siljak, 1978; Vukobratovic and Stokic, 1980). This analysis can be used to establish iterative procedure for the choice of weighting matrices Qi and r~ in (13) and the choice of parameter values Oa in (11 ), so that the synthesized local controller (15) can ensure practical stability about nominal trajectory x°(t} and for any given 0 e ®. The short summary of the stability analysis is given in the Appendix. If the influence of coupling among subsystems and influence of parameter variations are too strong it might happen that it is impossible to find out unique local controllers (15) which are robust enough to withstand non-linear coupling and all parameter variations. In this case it is necessary to introduce either global control which should compensate destabilizing influence of coupling and parameter variations, or adaptive control which should change the local gains in accordance to parameter variations. As we have already recognized (Vukobratovid and Stokid, 1982), coupling among d.b.f, of the robotic systems is represented by load (force) P~ which can be directly measured or calculated using the complete or simplified model of the mechanical system part S M. The main advantage of the force feedback is its simplicity, while the main disadvantage is the sensitivity to transducers noise. On the other hand the on-line calculation of SM might require powerful microcomputers. ADAPTIVE C O N T R O L SYNTHESIS

(13)

In order to accommodate the local regulators to unpredictable parameter variations, it is necessary

Adaptive control of robotic manipulators to introduce an estimation technique which can successfully furnish real-time parameter estimates to the regulators. As explained in the Introduction, we shall estimate only the dynamic parameters (mass or moments of inertia) of the last link (endeffect®r) with a payload. Denoting by n the index of the end-effector link and by np the number of dynamic parameters that we estimate, we obtain (Vukobratovi6 and Kir6anski, 1984)

643

where T

=

~ R 2n is the state vector of S M,

s.(()= [S.~(~)...S...p(~)]TeR "×". with S,,(~) = H.,(~)ij ° + h,t((), A0 = [A01 ... A0,p ]T E R rip, and

tip

Aplq=qo,) = ~ stit(q°,(1°,ij°)AO,

(16)

1-1

with

sMI: s.~ = H.l(q°)q ° + h.t(q °, (1o), (17)

l = 1..... tip where q°(t) represents the nominal trajectory, A0t the dynamic parameter variation. Equations (16) and (17) permit the calculation of the deviation of the input torque/force vector which enables the system to follow the nominal trajectory in the

e(t) > 0 is a given scalar function. It should be pointed out that S.~ differs from s.~, since S.t is the sensitivity function calculated for real state vector ~ and nominal acceleration qo, while s.t corresponds to nominal state vector and nominal acceleration. Expression (18) represents an infinite set of inequalities as it can be written for ¥tk ~ T. It is therefore necessary to discretize the time interval T: T = {tk}, with k-- 1.... , [z/At], At--timeinterval of discretization. The estimation A0e of the real value of A0 can be calculated by the recursive algorithm

~AOe(tk), (18)--fulfilled AOe(tk+ 1) = ~AOe(tk) q- Pro I-IIS.(G)I[- 2 S~.(~k)(APk -- S.(¢k)AOe(tk)) ]

presence of finite parameter variations. The function s,~ represents a "part" of system dynamics which relates the input vector AP to the deviation of a given dynamic parameter 0~. It does not depend on 0~, i.e. models S M~are valid for both small and large parameter deviations. Matrix H.t(q): R" ~ R" ×" is analogous to the matrix H(q, 0), which corresponds to the inertial effects, and the vector h,~(q,(1): R" x R " ~ R" is analogous to the vector h(q, (1, O) related to the effects of gravity, centrifugal and Coriolis forces. As shown in Vukobratovi6 and Kir6anski (1984), the number of floating-point multiplications/additions for the calculation of H,t(q) and h,l(q, (1) is nearly n times less than that for dynamic model matrices H (q, O) and h(q, (t, 0). This is of fundamental importance for real-time implementation. The basic ideas for the presented estimation technique can be found in Feldbaum (1960). These results are also used in Timofeev and Ekalo (1976) employing the centralized manipulator model which is too complex for real-time application. In order to pose the estimation algorithm, consider the inequality

]AP(tk) -- S,(~(tk))AOJ < e(tk)

(18)

(19)

where Pro ['] is the operation of projection to the parameter space ®. In the simplest case the operator P~0 can be substituted by a scalar multiplicative constant a.. Obviously, the rate of change of the estimate A0e can be decreased by decreasing the value of a.. On the other hand, that increases the adaptation time. A compromise value should be chosen by the designer. As described in the previous section, the synthesis of local regulators (12)-(15) is exactly valid when 0 is constant. When 0 is variable over time t ~ T, the procedure can be regarded as correct only when the rate of change AOe(t) is much less than that corresponding to the system dynamics. This is the quasi-stationarity condition which can be fulfilled by decreasing the constant a, of the estimation algorithm (19). In that case the synthesis of local regulators (12)-(15) should be repeated for all 0 ~ ®. In the case of robotic systems, only a few parameters could be considered as variable (for example, payload mass). Now, it is easy to form the function Di(O) (Di matrix consisting of local feedback coefficients of the ith subsystem) by repeating the optimization procedure for several 0 e O. The general block scheme of the algorithm is given in Fig. 1. The efficiency of the dynamic

644

M. VUKOBRATOVIC and N. KIRCANSKI P,

1 u,

system part ( manipulator with a payload )

- i th s u b s y s t e m ~ i th Local controller

I

'i

IMechonicol

/~Oe

11

. . .

Dynamic parameter estimator Fl(i. 1. General block-scheme.

estimator and the complete adaptive algorithm may best be recognized through the numerical example presented below. Sensors required for the recursive algorithm (19) are: position, velocity and force sensors. Combining these informations and the presented recursive algorithm, one obtains extremely short adaptation time and high precision.

remaining joints are powered by revolute vane actuators. In the second case, the subsystems S i ( i = 4, 5 and 6) state vector is taken as x i = [ql, q/, pi ] r where qi, (h are the corresponding angle and angular velocity. The actuator inputs are servovalve coil currents which are constrained in amplitude. As the first two d.o.f, are revolute, but driven by linear actuators, the connection between the actuator model state coordinates x i and the state coordinates of the mechanical system part model (i.e. the corresponding angles and angular .velocities) is not linear. These non-linear relations are shown in Fig. 1 by vectorial function j(.). The nominal trajectory for the particular manipulation task is given in the following way: the tip of the manipulator should move along straight line from the point A [coordinates of which with respect to the reference coordinate system are (0, 1.14, 1.342)r(m)] to the point B [with coordinates ( -0.31, 1.14, 1.65)r(m)] while the angles of joints 4, 5 and 6 should be kept constant (q4= 0.0,

EXAMPLE

The synthesis of the proposed non-adaptive and adaptive control will be illustrated by a hydraulic 6 d.o.f, manipulator, presented in Fig. 2. Manipulator parameters (link lengths, masses, moments of inertia) are given in Table 1. Hydraulic cylinders, controlled by hydraulic servovalves, are modeled by models of order ni = 3. The state vector of the hydraulic actuator is Xi --_ [1i, ii, i f ] r , where Ii is the piston travel, iz is the piston velocity, ff is the oil pressure in the cylinder. The first three manipulator degrees of freedom are driven by means of hydraulic cylinders while the z3

L6

L~L4

i ',, ,

I

l

/|

in-h-h~l

/o

I z2

!L~ q~"

/

%,,

L

/~

--

y, , L.

lo

,""

, ~ ' ~ 7 - - ' - - ' Y ' ~ 2 2:J

n

a

I

\

,

II

/

/

I

I

L, -Link i L, - l e n g t h of link i q, - j o i n t coordinate i xHy,~z , -coordinate frame of link i FIG. 2. Manipulator.

Adaptive control of robotic manipulators

645

TABLE 1. PARAMETERSOF THE MECHANISMSHOWN IN FIG. 2 1

MEMBER MASS mi(kg )

LENGTH Oi)(kgm2) Zi(m)

1.2

J<~(kgm 2) i^

Jiz(kgm2)

0.322

2

3

27.8

22.34

4,

5

OOS

6

2.33

0.33 0.26

/

1.142

0.14

2.98

2.98

0.004

.0007

0.004

.0009

0.004

.0009

3.701

I .21

r~

qO

02 04 .............. . ..... e,o, ..... ;o

0

"~{,L-~"

=

-008

'I

%=5kg

posit.gain vel.gain

ed:lOkg

posit.gain

vel.gain

--r.r-/

004

------

mp=Okg 8~=lOkg

......

mp: IOkg ed= 5kg

.....

mp=lOkg 8~=lOkg

......

mp=5kg 8~=5kg

- -

mp:lOkg 8 d - e s t i m a t e d

Is]

~,,

•

oo2

/"

~

7"

~.

<3

-

o

1 " J o ,z

/

.

.

o

2

"~"

I

.

.

~

•. J "

"',..Z

~

~

t

/

-004

FIG. 3. Simulation with fixed and adaptive local regulators.

IOO

I x.

75

J

5o!

I I I

25, I

I

--~

Estimation Reo(

moss

I I

o2

o!4

ol6 t Is]

FIG. 4. Payload mass estimation.

2

50.81 161 .77 10.86: 25.40 89.41 21 2.89 19,17 44,03 130.89 28.08

~ . . ~ : ...~,

~,"

TABLE 2. LOCALGAINS (FOR rl = 5)

8d=0

/ 06. /

~w~'~')'~'"~~

l

First we shall try to synthesize non-adaptive control which could withstand all variations of the payload, in the assumed range. We have synthesized local controllers (14), (15) for the maximal allowed mass of the payload in (11), i.e. for 0e = 10kg. The local gains are given in Table 2. We have simulated the dynamics of the manipulation system S (from Fig. 2) while tracking nominal trajectory with the synthesized local controllers. If the payload is mp = 10kg the tracking is satisfactory, but in the case when mp = 0 the tracking is poor, since local gains are too high for the manipulation system without payload. The results of simulation are shown in Fig. 3, where Aq~(t) denotes deviations of the joint coordinate i from its nominal value. We have tried to stabilize the system with local controllers synthesized for the mass of the payload in (11) 0d = 5 kg. The results of tracking the nominal trajectory (Fig. 3) with these local gains (Table 2) show good tracking for rap= 5kg and for mp = 0 kg, but for mp = 10kg these gains are too low. This means that we cannot find unique local controllers (15) which are able to withstand all assumed parameter variations. In order to stabilize manipulation system S for all possible payloads we have synthesized adaptive control algorithm as it has been presented before. The local gains in (14) have been synthesized for various payload masses (Table2). Figure 4 shows the simulation result of parameter estimation technique (19). The difference between the real mass

posit.gain vel.gain

./

l • ,1~ "~" , . . . . . . " " ~' o . " - . . . I . ~ .. . . 1:...:.,,~" ....... t - .

• ~~; .

{mp:m p e (0, 10kg)}.

SUBSYSTEM

/

/

<~

q5 = _ 1.57 rad, q6 ~- 0.0). The triangular velocity profile of the manipulator tip is accepted and the time duration of the movement is T = 0.8 s. The only unknow, n parameter is the payload mass 0 - - m p which can have values belonging to the region 0

/

//../

004 -

0.142

-

/

349.16 74.40

3

4

5

6

154.35 2.30 154.38 2.65

35,11 35.09 35.00 0.54 0.52: 0.35 36.12 36 .I0 35.28 2.21 2.18 0.87

154.42 3.01

37.86 4.03

37.83 4.00

35.57 I .37

i 08

646

M. VUKOBRATOVIC and N. KIRCANSKI

step disturbance and the estimated value is quite small. This difference, obviously, depends on the prescribed degree of exponential stability H, operator Pro of the dynamic estimator (19), and the nominal trajectory (velocities and accelerations) of mechanical segments. By increasing the degree of exponential stability H, the estimation error and the adaptation time decreases. On the other hand, that increases the local gains and the energy consumption. In this example the operator Pro is substituted by a constant (a, = l). The influence of the nominal trajectory itself to the estimated value is rather low due to (18). For example at t -~ 0.4s the end-effector acceleration changes from 3.8 to - 3 . 8 m / s 2 influencing the estimated value only about 10~o. Figure 3 shows the tracking of the desired nominal trajectory with adaptive (as well as fixed) control. It can be seen that we now achieve satisfactory tracking of nominal trajectory for arbitrary payload mass in a given range. However, such control is more complex for implementation than the simple robust non-adaptive control law presented above. The number of floating-point additions and multiplications that have to be performed at each sampling period for non-adaptive control in this particular example is hADD = 114

nMu,, = 96

while for identification of payload mass equals

APPENDIX: STABILITY ANALYSIS Consider the set of free subsystems S~ (12) with the performance indices J~ (13). Under the assumption that the pair (A ~,b~) is completely controllable and when input constraints N(u ~) are not taken into account, the optimal control minimizing (13) with the constraint (12) is given by (14). Assuming that Q~ = C~Cr, where C~ ~ R"' ×"', and that the pair (/i ~,Ci) is completely observable, subsystem (12) with the closed feedback loop is exponentially stable around x°(t) with the stability degree ll. If the amplitude constraint on the input N(ff} is taken into account, it can be shown that the exponential stability of subsystem (12) can be guaranteed for the initial conditions belonging to the finite region of the subsystem state space, i.e. for x~(O)~Sl, it follows that Ilxi(t)ll _< IIx~(0)ll~n' for Vte T. Let us now examine whether the global system S is asymptotically stable around x°(t). For this purpose, we shall use well-known composite system method to examine Liapunov stability of the system. In order to estimate the finite region in which the composite system S is asymptotically stable we shall apply Weissenberger's method (1973). For this purpose we introduce Liapunov functions for subsystems r, = (x ir K S ) I/2, ¥i~ 1, where ci: R"' x R"'--, R. The finite region X~ can be estimated by Liapunov function: X i - [xi:vAx ') -< voiAx'e X,I, Viel, where voi > (I should be chosen so that X~ be the best possible estimation of X~ (Siljak, 1978). Since lim AP,. - , 0 for Vie I, the numbers ~j < ~- ~. can be x~0

determined satisfying inequalities

{gradt:il~.fiAPdt, x) < i J

(t,x)eTxX(O)

28

n~u, = 64.

Using the INTEL 8086 microprocessor with 8087 coprocessor, we can achieve a sampling period of about 10ms for non-adaptive and of 16ms for adaptive control. CONCLUSION

The limitations of the admissible range of payload masses and inertials are mainly imposed by robot control strategy due to difficult compensation of non-linear and time-varying effects. These effects can, however, be evaluated as functions of states, or estimated combining measurements and knowledge of the model structure. A method for parameter estimation end control synthesis based on decentralized control strategy is presented in this study. The mathematical models of actuators and robot arm, as well as a sensitivity robot model are described. The results are illustrated by an example of a 6 d.o.f, manipulator powered by hydraulic actuators. Acknowledgement--This work was partially supported by the Institute of Mathematics, Belgrade, Yugoslavia.

ViEI

where .~(0) = )(1 x )(2 x ... x )~, and (ii > 0 for i e j, gi, j e l . Now, we want to examine whether the composite system S is asymptotically stable in the region X(0) which estimates the finite region in which decoupled system is exponentially stable. According to Weissenberger (1973), if the following condition is satisfied: Gu 0 < O,

n~D D =

and

~ijt:j" 1

Gij = - ( l q + p/2)dij + ~:ii

where vo = [vol . . . . . to,] T, G = [Gi;;eR "×", 6q is Kronecker symbol, p = 2,,(Wj/2M(Ki), 14,~= Qi + Kibir[- lb'r Ki, 2,. and ,~M denote minimal and maximal eigenvalue of the corresponding matrix, region )((0) is an estimate of the region of the global system asymptotic stability. The practical stability of the system is achieved if X t ~ . ~ ( O ) and X ( t ) ~ X ' ( t ) , t e 7 i If these conditions are not fulfilled, the weighting matrices Q~ and r~ should be modified. An iterative procedure for choice of weighting matrices Q~ and r~, so that the obtained local controllers can ensure practical stability around the nominal trajectory, is presented by Vukobratovi6 and Stoki~ (1982). REFERENCES Chemouil, P., M. R. Katebi, D. Sastry and M. G. Singh (1981). Maximum a posteriori parameter estimation in large-scale systems. Automatica, 17, 845. Dubowsky, S. and T. D. DesForges (1979). The application of model-references adaptive control to robotic manipulators. Trans. ASME, J. Dynam. Syst., Meas. Control, 101, 193. Feldbaum, A. A. (1960), Theory of dual control. Aut. Remote Control, 9, 1240. Guinzy, N. J. and A. P. Sage (1973). System identification in large-scale systems with hierarchical structures. Comp. Elec. Engng, 1, 1. Hassan, M. F. and M. G. Singh (1977). A two-level co-state prediction algorithm for nonlinear systems. Automatica, 6. Horowitz, R. and M. Tomizuku (1980). An adaptive control scheme for mechanical manipulators-compensation of nonlinearity and decoupling control. A S M E Paper No. 80Wal/DSC-6. Koivo, J. A. and H. T. Guo (1981). Control of robotic manipulator with adaptive controller. Proc. Decision Control, pp. 271 276. San Diego.

A d a p t i v e c o n t r o l of r o b o t i c m a n i p u l a t o r s Luh, J. Y. S., M. W. Walker and R. P. C. Paul (1980). On-line computational scheme for mechanical manipulators. Trans. ASME J. Dynam. Syst., Meas. Control, 102, 69. Sandell, N. R., P. Varaiya, M. Athans and M. G. Safonov (1978). Survey of decentralized control methods for large-scale .. systems. IEEE Trans. Control, AC-23, 108. Siljak, D. D. (1978). Large-scale Dynamics Systems: Stability and Structure. North-Holland, New York. Timofeev, A. V. and Yu. V. Ekalo (1976). Stability and stabilization of programmed movements of robotic manipulators. Aut. Remote Control, 10, 148. Vukobratovi6, M. and N. Kir/zanski (1984). Computer-assisted sensitivity model generation in manipulation robots dynamics. Mechanism and Machine Theor., 2~ 223.

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Vukobratovi6, M. and V. Potkonjak (1982). Dynamics of Manipulation Robots: Theory and Application. SpringerVerlag, Berlin. Vukobratovi6, M. and D. Stoki6 (1980). Contribution to the decoupled control of large-scale mechanical systems. Automatica, 16, 9. Vukobratovi6, M. and D. Stoki6 (1981). One engineeringconcept of dynamic control of manipulators. Trans. ASME, J. Dynam. Syst., Meas. Control, 103, 108. Vukobratovi6, M. and D. Stoki6 (1982). Control of Manipulation Robots: Theory and Application. Springer-Verlag, Berlin. Weissenberger, S. (1973). Stability regions of large-scale systems. Automatica, 9.