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Copyright © t 996 IF AC 13th Triennial World Congrcs;, San Francisco, USA

OPTIMIZATION-BASED TELEOPERATION CONTROLLER DESIGN Zbongzbi Hu', S. E. Sakudean' and P_ D. Loewen 1 Departmenl of Electrical Engineering' and Institute of Applied Marhematics' University of British Columbia Vancouver, B. C. Canada V6T lZ4 E-mail: [email protected]

Abstract: This paper addresses issues of performance and stability robustness specifications and trade-otts, and computational techniques in optimization-based teleoperation controller design. With Youla's Q-parametrization of stabilizing controllers, a transparency measure, defined as the H 00 distance to the ideal teleoperator model, and a robust stability constraint, defined as positive realness of the transmitted admittance to the environment, are convex in the free design parameters. Therefore.. the controller design problem can be formulated as a convex optimization problem. The limit of performance acbievabl" with the designed controller, and thus the exact form of the trade-otts between performance and robust stability, can be c.)mputed numerically. The solution procedure is illustrated by a design example for a motion-scaling teleoperation system. Key words: Robust control, Multiobjective optimizations, Numerical methods, Robotic manipulators, Thleoperation.

1. INTRODUCTION Teleoperation has found wide applications in space exploration, waste management and undersea exploration, by extending an operator's sensing and manipulation capabilities to remote and hazardous locations. RecenUy, teleoperation has started to encompass the extension of such capabilities through barriers of scale, allowing human involvement at scales much smaller or much larger than possible directly. One of the major issue; in teleoperation control is to design a controller to achieve transparency while maintaining stability of the \eleoperation system. Several controller design methods have been proposed for this problem. The scaling concepts of impedance and power as a function of position and force scaling based on specifying two-port hybrid parameters are discussed in (Hannaford, 1989); a four-channel control structure has been suggested to achieve transparency in (Lawrence, 1993) and the ideal response ofteleoperation in (Yokokohji et al .. 1994); Hoo-optimization theory has been used to best shape the closed-loop responses of interest in (Yan and Salcudean. 1996) and to sbape the relationships between forces and positions at both ends of the teleoperator in (Kazerooni et al., 1993); a control algorithm was proposed based on a semi-autonomous task-oriented virtual tool (Kosuge et al., 1995); and a robust stability criterion for a power scaling \eleoperation was obtained by using the structured singular values of the scattering matrix (Col gate, 1993). However, none of the above work has explicitly in-

corporated robust stability into the controller design. The goal of this paper is to design robust controllers with good transparency for scaled teleoperalion systems. First, the ideal teleoperator is defined by using a two-pon admittance matrix that cllaracterizes the ideal situation for scaled teleoperalion. Then, by using a four-channel control structure, the controller is designed to be robustly stable for a given human operator impedance and unknown passive environments, and to match the ideal one as closely as possible. The teleoperation controller design problem is formulated as a constrained multiple objective optimization problem, which is convex and numericaJly solvable. In the following section. we review some basic passivity concepts and stability conditions for teleoperation systems, and present an ideal teleoperator. In section 3, the controller design problem is fonnulated as an optimization problem. In Section 4, a numerical solution procedure is described and illustrated by a design example of a controller for a motion-scaling system. Some concluding remarks are included in the final section.

2. AN IDEAL TELEOPERATOR 2.1. Passivity and stability Definition 1: For a linear time-invariant (LTI) n-port network as shown in Fig. I, the impedance matrix Z is defined

1844

=

as the map from v to f by f Z v; the admittance malrix Y as the map from J to v by v = Y f; and the scattering malrix S as the map from the input wave a

~ (f + v)/2 to

the output wave 6 ~ (f - v)/2 , i.e., satisfying the equation 6 = S • . These malri(:es are interrelated by

S

slave, and the operator hand and environment impedances are represented, respectively. by Zh and Z,.

= (l- Y)(I + y)-' = (Z -/)(Z + f)-' .

t- - - - - - - - -- - - --·

-----------~

"I ... v. :v. -=--:-_, IOJ-.Of 1= ::1......... 1: :ICoi"'.....~ EnvlroM.enI I 1.

(I)

1

f. : t. L_________ _______ _!~~_ J '"

Figure 3: General tcleoperation system

Figure I: An n-p<>n network

Theorem I (Desoer and Vidyasagar, 1975): (a) An LTI n-port as shown in Fig. I is strictly passive if and only if the malrix criterion ""Iow holds for some 6 > 0 : Y(jw) -I Y·Uw)

~

61 , 11", E R.

=

Theorem 1 (COl gate and Hogan, 1988): Consider the network in Fig. 2, in which an LTI one-pan with admittance Y is coupled to an impedance Z. This network will be stable for every strictly pas

wEn

Figure 4 : 2n1'on representation of a teleoperation system The following robustness criterion for the 2n-pon teleoperation system is based on lbe finding lbat the system can be transformed into a structured uncertainty problem by using the scatlering malrix (Colgate, 1993), which can be addressed by considering the structured singular value (Doyle, 1981).

(3)

V.Ooitioo Z (Wen, 1988): The v-index, also referred to as passivily distance, is defined as the distance of a stable LTI system to strict passivity. Let the system transfer function be T( s), then

v ~ - inf {Re[TUwl]).

MCS

(2)

Notice that if n I , condition (2) reduces to Re[Y(jw)] ~ 6/2. (b) An LTI n-pen is passive if and only if criterion (2) holds for 6 = O.

Re[Y(jw)1 ~ O,lIw E R.

Teleoperator

(4)

Theorem 3 (Colgate, 1993): Consider the bilateral system shown in Fig. 4. This teleoperation system will be stable for every pair of strictly rassive Zh and Z. if and only if •

•

the scaltenng matrIX S, operator MCS has DO and moreover satisfies

= 811 " s f:j.

sup {1',,(S,(jw))} =sup

Iln S~m L ._.__-.F::=r-= A typical teleoperation system consists of five interacting subsystems: human operator, master manipulator, controller, slave manipulator and environment as shown in Fig. 3. To a ftrSt approximation, this can be modeled as an LTI 2n-p<>rt network illustrated in Fig. 4, in which the master, controller and slave are grouped into one block called the te1eoperator MCS . Here, Vm is the master velocity, 11, the slave velocity, fh the fonce that the operator applies to the master, f, the force that the environment applies to the

]

ps"] ) :s: 1. .22

(5) Here, denotes the maximum Singular value, and 1''' denotes the structured singular value against the block struc-

1T

ture /':,. Figure 2: A representation of an LTI I-pan network, Y, coupled to Z.

12

inf1T([ 8 :,'/'P

w '>0

w

8

22 of the 2n-pen tele• poles in lbe closed right-half-plane,

= [~h ~.], where 3 .

and S, are the scattering

matrices of strictly passive Z. and Z" respectively. When the hand impedance Zh is fixed, Theorem 3 reduces to Theorem 2 and therefore the stability criterion in (5) is less conservative than the stability criterion in (3). In practice, the variation of the operator hand impedance is relatively small if compared with drastic changes of the environment impedance, Therefore, to design a less conservative controller for teleoperation, the stability criterion in (3) could be used. The advantage of using criterion in (3) is that it is convex in design parameters, as we will show in Section 4 .

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2.2. An ideal scaled l

An ideal teleoperator is one that provides complete transparency of the man-machine interface sucb that tile operator bas the perception of working directly on the task environment. In order to achieve this, for non-scaled teleaperation, the effort and impedance of the operator port should be identical to the effort and impedance of the environment and vice versa (Hannaford, 1989; Lawrence. 1993), i.e., Vm V.. " = f ,. This can be represented by an infinitely stiff and weightless mechanical connection between the master and the slave (Dudragne, 1989), which cannot be physically realized. For scaled teleoperation, an ideal teleaperator is proposed here and is modeled by a two-port network as illustrated in Fig, 5. Instead of eliminating the dynamics of the teleoperator and realizing the ideal response defined by Vrn == fl pV" , fh = nlle ! where np and nJ are constant scaling ratio! of position and force respectively, a passive tool represented by impedance Zd Is used by the operator, as similarly done in (Kosuge et al .. 1995). The force is multiplied by n J and directly fed forward to tile operator's hand so that the operator can get a feel of tile task, and the motion command from the operator is divided by np' For micro-manipulations, n J np > 1. There are instances in which n J , " p = I (passive tool) or nt, np < 1, i.e., manipulation is at a large scale. Here Z1h denotes the transmilled impedance to the hand, and Z" the transmitted impedance to the enviroomem .

=

I

Theot-em 4: (I). Z,. = Zd + !!.LZ. ", . and Z" = !!LZd ", + ~ Z •. (2). If Yd is passive, then Yd Is passive if and only if ndiw)n;Uw) = I, Vw E R (3). (3a). If Yd is passive, then sup {1'",(SdUW»)} = I for any con-

n,

w

stan~

positive and real scalings

and

"p '

(1 + ;;;)Vd is passive, then sup {1'",(SdUW»}

(3b).

=

If

1 for

w

any frequency dependent scalings n,(jw) and np(jw) sat-

R.e{ ;;;(j..,)},Vw E R . Here, Sd is the

Isfying 1;;;(jw)1 = scattering matrix of the

structure is given by 6. and

IIS,ll oo <

Sd _ ---Ln/+n,.

tele~rator

=

1. (4). If

[np - nJ 2

('t

MCS, and the block

~.]

IYd(jw)1

~

IIS.II", < I 00, Vw E R, then with

2nJnp],

7l, -

np

3. CONTROLLER DESIGN A general four channel structure in "admiltance" form is

used for controller design as shown in Fig. 6. Laplace transforms and transfer function notation are assumed throughout. For simplicity, we consider only the one-degree-offreedom case. In this figure, ;l:m = vm/s is the master poSition, z:, = VJ / s the slave pOSition, Pm the master plant. and p. the slave plant. The hand force f. is deeoupled into

an active component fh and a passive component -Hx m. and similarly. the environment force f. is deeoupled into f~o. and -Ex, where H sZ" and E = sZI! ' ' ft..

Figure 5: 'TWo-pOri representation of ideal scaled teleaperation

=

.,.

+fm

'M Figure 6: A four channel control strucrure.

The dynamics of this MCS telcoperator block are given by: (6)

(7)

where Yd ;; 1/2d. Therefore, the ideal MCS teleaperator block can also be represented by the following admittance

maUix

The controller K, which is a 2 by 4 matrix of real-rational transfer functions, takes (orces and positions from both mas· ter and slave, and generates the actuator driving forces Im on the master and f. on the slave. Our objective here is to design the controller h ' to realize scaled teleoperation with good transparency while maintaining stability. This controller design problem can be put into the standard H oc -framework (FranciS, 1988; Boyd and Barrat~ 1991) shown in Fig. 7, where we define the signals w = [h ]T, u = [fm I.]T, y = [f. f. Xm and z = [v m v.] .

I'

".f,

(8)

= =

Wben nJ n. 1, this corresponds to the case in wbich tile operator manipulates the task environment direclly with the assistance o( a tool with impedance Zd. The important properties of this MCS teleoperator block are summarized in the following theorem.

y

Figure 7: General control system

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The generalized plant is described by

[yz]

=

[C,w C,"] [w], G G

(9)

U

yu

lIW

where

to match the matrix Yd,H as in the model in the ideal teleoperation. To maintain stability against any arbitrary but strictly passive environment, the admittance seen from the environment, Yte : It ----j. v" should be passive according to the stability criterion in (3). Therefore the robust controller design problem for this simpler case is

(10)

and

~],

C,"

= [}m

P,

( 18)

s.t. : inf {Re[Y,,(K)(jw)]}

~ ],

w

(11 )

p.

0

(P') :

The master and slave transfer matrices Pm and P, are realrational and strictly proper, and therefore C will be proper and H oo-optimization design techniques can be applied, The teleoperator can be described by the admittance matrix Y. : w -> z, Clearly, to get good transparency, it should be designed to match the model Yd for the ideal teJeoperator MCS proposed in the previOUS section as closely as possible, Therefore. a transparency measure can be defined as (12)

where W is a weighting matrix that re!leets the frequency bands of interest To guarantee stability against any strictly passive environment or operator impedances, the scattering matrix 5. ~ (Y. - I)(~; + I)-I should satisfy the stability criterion in (5), Therefore the problem of designing a robustly stable controller for the scaled teleoperation system is a constrained optimization problem as follows:

~ -11,

(19)

where W H is a weighting matrIx penalizing the frequencies of interest. and the positive parameter v is used to ensure a given distance to passivity defined as in (4) and determines the degree of conservatism of the controlter design.

4. NUMERICAL SOLUTION

4.1. Convex optimization The Youla parametrization of the stabilizing controlters K (Francis, 1988) makes the transfer matrices Y., Y"H and Yte affine functions of Q E RH!X4, However the scattering operator 5. is not affine in Q, so the scalar quantity sup {!,,,(S,(jw))} :0; I is not a convex function of Q. w

Therefore, the general controller design for scaled teleoperation is a non.convex optimization problem:

s.t,: sup!,,,(S',(Q)(jw)):S I,

(21)

w

and only in the simpler case, does it reduce into a convex

optimization problem:

(pI) :

( 13)

.IIW[Y.! K) - Ydlll oo

min

~1abtktln9

1\

(14)

The H 00 theory can also be applied in the simpler case, where the hand impedance is assumed to be known and fixed and the environment impedance is arbitrary but strictly

passive. Here the si,gnals

u, y,

and

those defined before except now w the generalized plant beeomes

[yz]

=

z

are the same as

= [ha f, f.

Then,

[C,w C,"] [w], G G yw

yu

S.t. :

inf{Re[Y,,(Q)(jw)]} w

2:

-I/.

(23)

In this paper, we only solve convex problem (P'). This problem is infinite-dimensional. To produce a finitedimensional approximation, Q E RH!X4 can be approximated as a linear combination of fixed scalar stable basis functions Q, E RH ~. as in N

(15)

U

Q(X\,X" ... ,XN)

= I:X,Q"

X,

ERH',

(24)

i=l

where

s.], I - ~PH [

PH

o

(16)

~],

where the N real-valued matrices X,(i= 1,2, ... ,N) are the design parameters. For example, the basis functions can be chosen as all-pass function,

Q;=

(17)

S

p.

and PH ~ Pm/(I + l'mH). To get transparency, we have to shape the closed-loop transfer matrix Yt,H : w - z

(~)N-' +p

(25)

for some fixed p with positive real part. The linear approximation in (24) reduces problem (p4) to a finite-dimensional convex program.

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There are several numerical solvers that can be used to solve this program (e.g., Bayd et al., 1988, Polak and Salcudean, 1989; Boyd et al., 1994). Of those, the following were tried: • Quadratic program based solver: This solver was developed based on functions provided by the MaUab optintization loolbox. Even though it worked but was very slow. • LMI based solver: Problem (p4) could also be expressed in terms of state·space linear mamx inequalities (LMI) (Chen and Wen, 1994), wbich can be solved by interior point methods (Boyd et al., 1994). In our design examples, these method, only worked for low-order approximation of Q in (24). With an increase of the order of the approximated Q in (24), the resulting LMrs for (p4) became ilI-conditioned and therefore failed to produce the desired results. • Cutting-plane based solver: We found that the cuttingplane algorithm with rules of dropping old constraints developed by (Elzinga and Moore, 1975) is most effective for solving problem (p4). The cutting-pl.ne method is an iterative algorithm. It is attractive since the subproblem to be solved at each iteration is a simple linear program that changes only sightly from one iteration to the next and no line searches are required. The original cutting-plane algorithm developed by (Kelley, 1960) suffers the size problem, I.e., the number of constraints in the linear program that must be solved at each iteration grows rapidly with the total number of elapsed iterations. The cutting-plane algorithm by (Elzinga and Moor, 1975) drops old constraints, so that the linear programs solved at each iteration does not grow rapidly in size as the algorithm proceeds. Therefore this algorithm was used in the development of the cutting-plane-based solver. To apply the cutting-plane algorithm, we simply approximate the semi-inifinite constraint and objective by discretization, e.g., replacing a H," nonn objective by some number of single frequency objectives log·spaced in • specified frequency range. For our design examples, the algorithm by Elzinga and Moor reduced the computation time by an order of magnitude compared with the Kelley's algorithm.

~, and specified the tool's Impedance as Zd = 0.628 + 3 + 1:0. In this example, we chose the force scaling ratio 10 and n. 5, and the motion scaling ratio as n! respectively. The weighting mamx was selected as

=

(

WH 5)

=

w. = -s+wlf - - [ 2 x ] , ""11 = 511" rad/sec,

which reflects the frequency bandwidth of transparency, and is a IOW-pass filter. This controller design problem was formulated in the fonn of (P'), which was approximated by (P 4 ) with N = 10 and p 1, and was solved by the cutting-plane-based O. The obtained solver. First we solve (p4) when v controller is robustly stable for any smctly passive environments. To look at transparency of the designed teleoperator, we compute its transntitted impedances to the hand, Z'h = fh /v m , corresponding to three different environment imped0 (free motion), (b) Z, 58 + 10 + ;lQ ances: (a) Z, (soft environment), and (c) Z, = 10s + 100 + 10,00 (StiiT environment) and compare with Zd+ !!L.Ze. where !!.1... np np = 2 was obtained from the proposed ideal teleoperator. The Bode plots of the transntitted impedances Z'h and Zd+2Z, are presented in Fig. 8 to Fig. 10, from which we can see that the designed teleoperator sbows very good transparency in the specified frequency range « 5" rad/sec). In order to display the trade-offs between perfonnance and stabi!ity robustness, the convex problem (p 4 ) was solved WIth different values of v. The performance vs robust stability trade-off curve can be plotted as shown in Figure 11. As expected, the perfonnance gets better as the passivity dis-

=

=

=

=

tance increases. Note the big increase in perfonnance index imposed by the passi vity requirement. S. CONCLUDING REMARKS An optimization method has been used to design a controller for the teleoperation system that has not only good transparency but also robust stability. Fonnulation of the controller design problem in optintization form was described, the solution procedure was presented, and some computational issues were discussed. A design example demonstrates the effectiveness of the approach.

4.2. Design example The design problem considered here is that of controlling the force-scaling and motion-scaling system, • prototype telerobotic system for use in nticrosurgery experiments (Yan and Salcudean, 1996). Both force and position need to be scaled down from the operator's hand to the task. The transfer functions mapping force to position for the master Pm(8) and P,(s) are, respectively,

Pm(s) =

I

0.628'

+ 38 + 150'

and

(26)

I

P,(8)

= 0.0358' + 0.178 + 8.6'

(28)

(27)

We assumed that the operator's hand is a constant mass0.58 + 2.5 + spring-damper system with impedance Zh

Figure 8: Transntitted impedance Z'h (solid line) O. and Zd + 2Z, (dotted line) : with Z,

=

=

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REFERENCES

Frequency(rad'•• c)

I :::W. :t l:l:: Itn:::, :: ll::,

1111 . .

"""

1:::1:;: : 11:::;: : :::: :~l : 11: '" : I:::::: : :n: '''''". : :: :11\: : :11 1::11 : 11 ,H: : I::::: : :1::: -~-HUl~--~- ~ i.u iii--+~ :11:::: .. : :' ,::: .. U pm:__ ~_p W'

:::::1:: I , 1,1".

!i:::!~ :!~f:Tl,:~[lf!nrm '""~ " I "...

"Il,,,. I, ,11,,,, ,

:'!ii!!~

t ll::::: Ill::::: I::::::': 10'

10'

H'

1cf

!imm I I I Ill"

!11!!!rl1 11!1!!!

",

Frequency(radlsec)

Figure 9: Transmitted impedance Z" (solid line) and Z. + 2Z, (dotted line) : willl Z, 5. + 10 +

'f.

=

"'1;;;TI1En:::m;nTfmm,TnmnT/[Illliif=i=fi,' " : ' _-+ W:'~'-j .. Hijjjj. _i.j.l, !! 1:1 :: :: I I

Z;::LJ.llllill,-"-llJlillll,,-l-ll III

Figure 10: Transmilted impedance Z" (solid line) and

Z. + 2Z, (do t ted line) : willl Z,

= 10. + 100 +

10,0 0.

t

I:

~

'" ••• OD1 .

"U

G.M

c....

~

.'

o.u

.. H

DII&Inc. to P ......1y

e...

v...

U, l

Figure 11 : Trade-offs between transparency and robuslness

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