Adaptive Hybrid Control for a Robot Manipulator Interacting with Uncertain Flexible Environments

Jb-07 5

Copyright © 1996 IFAC . 13th Triennial World Congress, San FrancIsco. USA

ADAPTIVE HYBRID CONTROL FOR A ROBOT MANIPULATOR INTERACTING WITH UNCERTAIN FLEXIBLE ENVIRONMENTS

Jianqing Wu", Zhiwei Luo", Masaki Yamakita 2 ),3) and Koji Ito"'"

1 }.Def'artment (lln.t(1rInalion Clnd Conl/JUter Sciences, Toyohashi University of Technology Tempaku~clw. Toyohashi 441, JlWan. E~mail: ~o@sy.\.tem.tutic.v.tttt.ac.jp 2).Bio·Mimetic Control Research Center. The Institute (~l Physical and Chemical Research H~31, Rokuhaf/ 3-chome, At'\"uta~ktl, Nap,oya 456. Japlll1. E-mail: 'uo@nllp,oya.riken.[?oJp 3). FllCU/ty (~l EIl~illeerillg. Tokyo Institute (~r Technology 2-I2~J Oh~Okaml1ia. Mef!,uro-ku, Tokyo 152, Japan, E~maif: [email protected].}1'

Abstract: This paper presents an adaptive position/force hybrid control approach for a robot manipulator to interact with its uncertain flexible environments. Because of its flexibility, the environment dynamics influences the robot's control system, and since it is a distributed parameter system, the environment dynamics as seen from the robot chilllges when the robot moves. The problem becomes further complicated such that it is difficult to decompose the robot's position and force control loops. In Ihis paper, the environment's distributed parameter dynamics is approximated into a lumped "position state~varying" model. The robot's control space is decomposed into a position control subspace and an environment torque control subspace. Optimal state feedback is designed for the position control loop, and model-reference simple adaptive control is applied to the force control loop. Experiments show the effectiveness of this control approach. Keywords: Robotic manipulator. Uncertain flexible environment. Position/force hybrid control. Adaptive control.

I. INTRODlICTION Because of its lightweight. less energy consumption and fast motion abilities, flexible objects have received increasing attention and have been used in many important applications especially in space industry. During the last few years, significant efforts have been made which considered the motion control of robots with flexible arms

(Siciliano and Book. 1988). However. there is no complete research to consider a robot's contact tasks which involve the mechanical interactions with flexible environments. This k.ind of tasks may appear when we want to manufacture and/or to maintain a flexible object such as a space stmcture.

Consider how to cover a sheet on a cantilever beam as

235

shown in Fig.1. To perform this task successfully. the robot is required to move on the beam while maintaining a constant contact force along the normal direction of the beam's surface. If the contact force is too s mall. the robot is nOI able to stick the sheet ont o the beam; ilIld conversely, if the contact force is roo large, the robot may destroy both the beam and the sheet. Clearly. because of its fle xibililY, the beam dynamic s wi ll intluence the robot's control system, and since it is us unlly a di stributed parameter sys tem, the beam dyni.lmics as seen from the robot will change when the robot moves I..1n its differe nt positions. Furthermore, as will be shown in the next section , the problem becomes more complicated as it is dift1cult to decompose the robot' s position and force control loops.

Fig. I. Robot manipul ator's contact task o n a flexible beam For a robot manipul ator to perform the contact tasks, impedance control (Hogan, 1985; Luo and Ito. 1993) . .m hybrid position/force control (Raibert and Craig, 1985; Yoshikawa et al" 1988) have heen proposed . Impedance control considers the robot's d ynamic interaction wi th ils environment. It uses the force feedback [() adjus t the robot's impedance as seen fro m the environment. However, this approach does not conrrol the robo!'s contact force . directly. The contact force is con lrnlled implicitly through turning the robot's equilibrium pos ition. Therefor~ , the force is onl y controlled in the steady stall! without any environment uncertaimies. On the orher hand. hybrid control divides (he robot's motion space into free motion s uhs pace a-k.I constraint subspace. It specifies the position control along the free motion subspace and controls the contac t force explicitly along th e constraint subspace. However, few studies considered the environment dynami cs (Ynshikawa et al.. 1988 ). Recently. Yo,hikawa and Ume no (1993) proposed an approach ( 0 consider the o bject dynamics in designing a robot's dynamic hybrid control . However. they considered the environment as a lumped time~invariant system, and did not discussed on the environm~nt variation as well as the decomposi tion difficulties. Because the se two problems are the most fundamental difficulties arise when wc specify the hybrid cont rol on a flexible env ironment, they will mainly be snldied in this paper.

In thi s paper, the environment's distributed parameter dynamics is simplified into a lumped "position statevarying" model. The robot's control space is decomposed into the position control subspace and the environmental torque co ntrol subspace usi ng non linear feedback co mpensation . Optimal stale feedhack is designed for the position control loop. and the robot's contact force is contro lled through contro ll ing Ihe resultant torque of the environment. Since the torque control loop is influenced by the environment dynamics , and the environment dynamics varies with respect to the robot's contact position, modelreference adaptive control strategy is applied for Ihis uncen ain "state-varying" torque control loop. Adaptive control theori es us uall y consider the linear lumped timeinvariant system. Only betw(!en in recent years, theoretical studie s have been proposed for the adaptive control of a time-varying system. One of these studies is sImple adaptive control (Middleton and Goodwin. 1988 ; Kaufman el "I.. 1994: Bar-kana and Kaufman, 1988). In our study, the torque control loop is "state-varying" with respect to the robo r's posilio n. Ho wever. if the position control loop is predetermined , the lorque loop Ihen becomes a typical time-varying syste m. Simple adaptive control will be applied for Ihis loop. The problem on how to select a reasonable reference model is also studied. Experiments of a PUMA robot interacting wilh an aluminum beam show lhe effectiveness of thi s approach.

2. SYSTEM FORMULATION AND APPROXIMATION

2.1. Dynamic equation

(~tjle.xihl('

environment

In thi s section, it is assumed that the flexible environment onl y makes a small deflection when the robot impose a contact force along ils normal direc tion. the environment's di stri buled parameter dynamics is then approximated 10 a lumped parameter model. In Fig.2(al, :Exy is the world Cartesia n coordinate. When the robot is pushing the beam OA at the point B with the interaction force F'''i , line ab is a tangent of the beam at B which crosses the axis X with angle e. Shifting ab to cross the origin 0 of :Exy. we get a line OA'. OA' is defined as the approximated o bjecl .m point S'is defined as the virtual contact point. :Ex~, y,. is the cons tra int coo rdinat(! a nd a ll variations of the beam dynamics sl!en from the robot is finally approximated into the variations of the approximated beam's inertia moment, viscosi ty and elasticity. respecti vely. Therefore, the robot's contact task is simplified as Fig .2( b). and the envi ronment dynamics as scen from the robot becomes:

236

y

Differentiating eq.(3) with respect to time t, then

x=JijXI J. = [

and cos 9

x' =illXI + Jexl

sin 9

xccos ()

-x,sin

(4)

6]

(5)

where JH is the Jacobian matrix from rxcYc to I.xy. By specifying the nonlinear feedback compensation

la)

F = A(.lie', + p(x,;<) + F,,,, + AIXlleM:,'I( u,) _( \) U2

)k~n(y~

)1

(6)

to eq.(2), the robot's control space is then simplified as: mlx.,(t)

= UI

m:?9(t) =

where [u j • u1 P: is a new control vector,

Ib) Fig. 2. The approximated lumped parameter model of the flexible environment

.

..

I(xd9(t) + dlxc)9(t) + k(x,j9(t) = xdt)f,,,,y, where 6(t) is the angle, fintYt is the direction, xJt) is the I(X), d(X) and ktx)

(I)

approximated environment's rotation contact force acting on its normal robot's end-effector position on ~x"y~,. are the environment's inertia moment,

2.2. Dynamic equation (~f" manipulator The robot manipulator's dynamic equation is given in the world coordinate LX y as

where x

=[x. y]T is

x) = F -

(8) Md

= diag(m 1• m 2 )

is a desired inertia matrix. Eqs.(7). (8) together with eq.(1) represent the simplified robot-environment dynamics. Although the position xc's control loop is independent from the contact force finIY': in eq.(7), the force fintyt·'S control loop is influenced by the contact position xc in eq.(8), the relation from the control input u 2 to the contact force fmtyc is nonlinear, this system is called a lumped "position statevarying" system. This is the second difficulty in the robot's hybrid control.

viscosity and elasticity parameters, respectively. They are unknown and varying with respect to the robot's position x,.(t). this is the tirst difficulty for hybrid control.

A(x)x + p(x.

(7)

U2 - x~(t)fintyc

3. HYBRID CONTROL DESIGN The objective is now to design the control [u\, u1 F for the system eq.1 I) and (7), (8), such that the robot makes the desired motion x~ on the object smface while maintaining a constant force fint)'c 3./. Po.\·ition contra/loop

Finl

the robot's end-effector position. A(x)

x)

In eq.(7). the position x, is only controlled by u, and is

is the coriolis aOO is the robot's inertia matrix. p(x. centrifugal force. F = [f,... t~,F is the robot's driving force

independent from the contact force fimyc, It is straightforward to design this s"bloop first. In detail, set m, = I, aod

and Filii = [fintx ' fillly]T is the -contact force from the environment as seen on the world coordinate LXY.

rewrite the eq. (7) into the following state-space form

"dO = Ax..;x,,(t) + yell) = c ."x,(t)

The geometric relationship between x = Ix. y]T and x, = [x,. 9]1' illustrated in Fig.2 is .(t) =

x,(I)COS [

bx"UI(t)

where the state vector is x~· = [xc.\]T and

ell)·1

x,(t)sin 9(t)

A" (3)

237

=1:::,1, b" =1':I,c" =[1,0]

(9)

(10)

x.(t) = A.(!) •• It) + b.(!)u,(t) yolt) = c .(t)x.(t) + d.(t)u,(!)

The optimal state feedback which optimize the following performance index

(11 )

where X!:I(t) = A.(t) =

is designed as

(J

[

(16)

(17)

[e,er. ya(t)=x.

:fiJiljl'-'.

I botl) [ " . c =

I

m2k tn2d

J

np+l j -,

- -

nl'H-l nn+L

(L)

[m"I'

= m,.1

H

m..lI

delL)

rn':", -

=[:] m2+'

(12)

The original problem of designing the control u, such

where matrix P is the solution of 'he Riccati equation T

-I

T

PA x( + Ah·P - Pbx~·R b~cP + Q =

t;nty~,(t) ~ til~('K

o.

(13)

becomes to make

-

Iim [yml!)· y.(t)]= lim x,(t)ertt) = 0 t-, ...

d (d .d)T and Xc = XC,Xc is the desired end-effector state.

$

(18)

Set the matrices Q = I and R = I. then For this control problem, a reference model (14)

The transfer function G lI , from the desired position x~

(0

Xmlt) = Amxm(t) + bmlt)u" Ym(t) = Cm(t)xrn(t)

the

(19) (20)

robot's real end-effector position x~.o.l is:

(15)

The position control block diagram is shown in Fig. 3.

L

_ _ _ _ _ _ lilT

-I R' h"px ,I t)

I

can be selected and model reference adaptive control can be used, such that the torque control system follows the output of the model. perfectly. In this reference model, Urn

= Xc" 11'inly~' is the desired output

of

the model and Yrn is

the model's real output. The problem in practice is how to select a proper refereoce model. Compare eqs.(19), (20) with eq.(I5), it is found that, the best reference model of the system eqs.(I6) and (17) satisfies the relation that, its transfer function equals that of the robot's position control loop. That is

Gm = Ym(s) = cm(sl '- Amr1b m= Gx~' umls)

(21)

Fig. 3. Optimal state feedback for position control loop

3.2. Force contra/lool' The main problem of this paper is how to design the controlu, for the force control loop of eqs.1 I) and (8). such that the robot maintains a constant contact force on the uncertain flexible environment. Since the position control has already been designed above, XI: is then a detennined time function. The force control loop of eqs. (1) and (8) is then a typical time-varying system_ Moreover. although the relation from u 2 to finlyc(t) is non linear. the relation between xc(t)finlyc(t) and u]: is linear. Since xdt) is already defined, it is then natural to control tinly~·(t) implicitly through controlling (he new variable X~·(t)finlyc(t) --- that is a torque imposed on the object. Rewriting the eqs. ( I ) and (8) into the following state-space formulation:

Adaptive control algorithms usually considered a lumped time invariant system with parameter uncertainties, they generally require the slow adaptation rates. Recently, a simple adaptive control algorithm has been presented which also considered the control of a timevarying system. and it guarantees the system's robust stability at any rate of adaptation. The higher the adaptation rate, the faster the transient response and the smaller the steady-state error. Motivated by these advantages. tbe following control design will apply the simple adaptive control algorithm. The control ", is designed such that the output y.(l) of the system eqs. (16) and (17) perfectly tracking the output of the reference model ym(l). Defining the output error as e y = Y6 - Ym, a error system is derived from eqs.(I6). (17) and eqs.(I9), (20) as follows:

238

= A"e,(t) + b",(t)$J.{J)(t)

e,(t) e,(t)

T

= elke,(!) + d,dt)$",{Jl(t)

(22) (23)

(See Appendix A forthe detailed derivation.). The t()fce loop's control input is then designed as lie

= k T(t)(t)(O

(24)

where kit) is the adaptive gains. which is adjusted according to the rule kit) = -re,·ffi(t)

125)

f = diag[r\,f,.f,.f,J > 0 IS a posltlve definite scaling matrix. This parameter adjustment rule eq.(25) together with the simple adaptive control eq.(24) guarantee the output error ey(t) to approach zero, asymptotically. Therefore, w(t) ----70 0, the robot's contact force is controlled stably. The over all model matching simple adaptive control of the robot's contact force is shown in Fig. 4.

Reference Model G,,(s)

Ym= x,(l) f "illly(

In the experiments, a rotary is equipped at the end-effecter of

the robot to prevent the friction. A six-degree of freedom force sensor is used to measure the contact force. and to calculate the tangent angle e of the beam at the contact point. The sampling time in all experiments is 2[msJ. The desired position and contact force are given as x..~ = 0.2 rml and = 20.0 rNl.

e,,",

Comparative experiments on the conventional hybrid control and the adaptive hybrid control is made. The force control input in the conventional hybrid control is

Figs. 6 and 7 show the experiment results. Fig. 6(a) shows the robot's position responses along the normal of the aluminum beam, and (b) is the contact force responses. The solid lines are our results ;md the dotted lines are the results of the conventional hybrid control. From Fig.6(b). it is understood that. because the conventional hybrid control has not considered the influence of the beam dynamics, the force response has large oscillation.

However, in our approach, the force response reaches to its

Xm Fig. 4. Simple adaptive control of contact force

desired value. smoothly. Comparing Fig.6(a) and (b). it is also find that, in our approach, the contact force responses faster than the relative pOSition. which means that the force control loop is fast adjusted to adapt to the slow changes of the beam dynamics with respect to the robot's contact position. Fig.? shows the parameter convergence of our control approach. The control parameter matrix r in eq.(25) are selected as f=diag[150.50. 100.IOJ in the experiments.

4. EXPERIMENTS -J""I"".rt"",[h"d

A PUMA robot is used to perform various contact tasks on the surface of an aluminum beam shown in Fig.5 to verify the effectiveness of above adaptive hybrid control approach.

•••

C,"I1'"~i,,,al ""_1I1L~1

Ti"" [lICe'

(b)

(a)

Fig. 6. Position responses· of a robot's end-effector and contact force responses between robot and object '.0»

~.~--~---

,

-Ko

:.;r=~~1 -K"'"

I.JlI

1

.- ... K,",[ !

"

'I'~",i"",

Fig.5. Contact tasks of a PUMA robot on a beam

Fig.7. Convergence of the verified input gains

239

i. CONCLlISIONS This paper presented an adaptive hybrid control approach for a robot manipulator to perform the contact tasks on its flexible environment. The environmental distributed parameter dynamics was approximated into a lumped "statevarying" system. By using a nonlinear teedback compensation, the robot's control space was decomposed into the position control subs pace and the environmental torque control subspace. The optimal state feedback was designed for the position control loop, and the robot's contact force was controlled through controlling a new control variable -~ the resultant torque on the environment. Since the torque control loop is intluenced by the environmental dynamics. and the environmental dynamics is varied with respect to the robot's contact position, a model-reference adaptive control approach --- simple adaptive control --- was Lhen applied for this conLrol loop. The effectiveness of this control approach was shown through a PUMA robot's experiments.

ACKNOWLEDGMENTS This work was supported through the scientific research fund from the Ministry of Education. Science and Culture of Japan (#06750454, #0645225.1). REFERENCES Bar-Kana, I. and H.Kallfmanl 1988). Simple adaptive control of uncertain systems. Int. 1. on Adaptive Control (Inll Sixnal Processin;!. 2, 133-143. Hogan, N. (1985). Impedance control: an approach to manipulation, Part I - Ill. ASME J. Dm. Sys. Mea,\'. Cont., 107, 1-24. Kaufman, H., I.Bar-kana and K.Sobeli I 994).Dim·1 adllplive control algorithms. New York: Springer-Verlag. Luo, Z.w., and M.Ito 11993). Control design of robot for compliant manipulation on dynamic environments. IEEE Tr(ws. 011 Roh. mui Auto .. 9, 2g6- 296. Middleton, R. H. and G. C.Goodwin 11988). Adaptive control of time-varying linear system. IEEE Tram. AC, 33. liO-155. Raibert, M.H. andJ.J.Craig (1985). Hybrid PositionlForce Control of Manipulators. ASME 1. Drn. S1'S. Meas. ConI., 102, 126-133. Siciliano, B. and W. J.Book I 1988). A singular perturbation approach to control of lightweight flexible manipulators. Int. J. Rohotic Research, 7. 79--90.

Yoshikawa, T, TSllgie and M.Tanaka (1988). Dynamic hybrid position/force control of robot Manipulators controller design and experiment. IEEE J. of Rob. arrl Aato., 4. 699-705. Yoshikawa, T and A.Umeno (1993). Dynamic hybrid control of manipulators considering object dynamics. 1. Rob. Soc. Jtq'Ul, 11, 122Y-1235.

APPENDIX Consider model-reference adaptive control of a time-varying system x.(t) ; A,(t)x.(t) + b.(t)u,(!)

(AI)

y.(t); co(t)x.lt) + d,(!)u,(!)

(A2l

to follow the output of a reference model Xm(t) = Amxm(t)

+ hm(t)um

Ym(t) == C 1II(t)Xm(t)

(A3) (A4)

Assume that the system (A I) and (A2) are "almost strictly passivc"(ASP). When perfect following is realized, the "ideal" trajectories ~:It) and the "ideal" control input u;1t) are defined, which satisfy

x~(t) ; A.(t)x;(t) + b.(l)u;(t)

(A5)

y;(t); c.(t)x;1t) + d,(t)u;1t)

(A6)

Represent x~(t) and u:m as a linear combinations of Xm(t) and Unt as x~(I); S(t)x,,(1) + lI(t)u" (A7) ~

ue(t) = Kxmxm(t)

~

+ Kumum

(A8)

using the time-varying matrices S(1), U(1), and constant matrices kXIll and kum. By defining the state error as

.,It); "It) - x;(t),

and using (A I), following error state equation

(A5) and (A8), the

(A9)

is obtained. Also from (A2) and IA6), the output error ey(t); Y' - Ym ; c"e,(1)

T + d",(!)$mCll(t)

(AIO)

is obtained. In our problem of the system eqs.(I6) and (17),

240