2008 Bilateral Teleoperation Experiments Scattering Transformation and Passive Output Synchronization Revisited

8/3/2019 2008 Bilateral Teleoperation Experiments Scattering Transformation and Passive Output Synchronization Revisited

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Bilateral Teleoperation Experiments:

Scattering Transformation and Passive

Output Synchronization Revisited

Emmanuel Nuno Luis Basanez Erick Rodrguez-Seda

Mark W. Spong

Institute of Industrial and Control Engineering, Technical Universityof Catalonia. Barcelona 08028, Spain.

(e-mail: [email protected]; [email protected]). Coordinated Science Laboratory, University of Illinois at Urbana

Champaign. 1308 W. Main St., Urbana, IL 61801, USA.(e-mail: [email protected]; [email protected]).

Abstract: It is well known that the scattering variables transform the transmission delays into

a passive virtual transmission line, hence, its interconnection with passive subsystems preservespassivity of the overall system. However, wave reflections may occur. Using a symmetric velocitycontroller, on the master and the slave, and by matching the impedances, the scatteringtransformation reduces to a passive output synchronization scheme. In this paper we revisitthis relation and perform some experiments on this line.

Keywords: Telerobotics, Time-delay, Robot control, Lyapunov functions.

1. INTRODUCTION

Most bilateral teleoperators use a communication channelthat imposes a time-delay between data transfers. It iswell known that this time-delay affects the overall stability

of the teleoperator. The control of these systems hasbecome an highly active research field amongst engineeringscientists. The groundbreaking work of Anderson andSpong [1989] has ever since dominated this field. Theyproposed to send the scattering signals to transform thetransmission delays into a passive virtual transmissionline. The transmission line is then interconnected withthe master and slave robots, which define passive forceto velocity operators, while the human operator and thecontact environment constitute the terminations to thetransmission line. Since powerpreserving interconnectionof passive systems is again passive L2stability of theoverall system is ensured under the reasonable assumption

that the human operator and the environment definepassive (force to velocity) maps. Since then, the use ofscattering theory has been widely extended. The readeris referred to Hokayem and Spong [2006] and Arcara andMelchiorri [2002] for two detailed surveys regarding thecontrol of teleoperators.

Recently, Chopra and Spong have presented an interestingcontrol architecture, based on the passive output synchro-nization of n-agents. In their work they have achieveddelay-independent output synchronization of the agents,for any constant time-delay (Chopra and Spong [2007],Chopra and Spong [2006]). They have shown that using a

This work has been partially supported by the spanish CICYT

projects: DPI2005-00112 and DPI2007-63665, the FPI program withreference BES-2006-13393, and also by the mexican CONACyTgrant-169003.

symmetric controller on the teleoperator and by matchingtheir impedances with the virtual transmission line, thescheme reduces to the one used for the passive outputsynchronization. The stability for both schemes is ana-lyzed using the passivity property of the transmission line,

for the scattering transformation, and with a Liapunov-Krasovskii functional, respectively. In this work we revisitthis analysis and present some teleoperation simulationsand experiments that show the stable behavior of theoverall system.

The paper is arranged as follows: modeling the n-DOFteleoperator is shown in Section 2; Section 3 and Section 4analyze the stability of the scattering transformation andthe passive output synchronization schemes, respectively;some simulations are shown in Section 5 with the experi-ments in Section 6; finally, the conclusions and future workare outlined in Section 7.

2. MODELING THE NDOF TELEOPERATOR

Before continuing, let us introduce the following notation.R := (, ), R+ := (0, ), R+0 := [0, ). m{A} andM{A} represent the minimum and maximum eigenvalueof matrix A, respectively. || stands for the Euclidean normand 2 for the L2 norm. In order to keep equationsas clear as possible, the argument of all time dependentsignals will be omitted (e.g. q(t) q), except for thosewhich are time-delayed (e.g. q(t T)). Following thisreasoning, the argument of the signals inside the integrals

will be omitted, and it is supposed that is equal to thevariable on the differential, unless otherwise noted (e.g.t0

x()d t0

xd).

Proceedings of the 17th World CongressThe International Federation of Automatic ControlSeoul, Korea, July 6-11, 2008

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The master and the slave are modeled as a pair ofndegreeof freedom (DOF) serial links with revolute joints. Theircorresponding nonlinear dynamics are described by

Mm(qm)qm + Cm(qm, qm)qm + gm(qm) =

m hMs(qs)qs + Cs(qs, qs)qs + gs(qs) = e

s, (1)

where qi, qi, qi Rn are the acceleration, velocity andjoint position, respectively. Mi(qi) Rnn are the inertiamatrices, Ci(qi, qi) Rnn the coriolis and centrifugaleffects, gi(qi) Rn represent the vectors of gravitationalforces, i Rn are the control signals and h Rn,e Rn are the forces exerted by the human operator andthe environment interaction, respectively. i = m representsthe master and i = s the slave.

These robot dynamic models have some important prop-erties 1 :

P1. Due to the fact that all joints are revolute thenMi(qi) is lower and upper bounded. i.e.

0 < m(Mi(qi))I Mi(qi) M(Mi(qi))I < P2. The Coriolis matrix Ci(qi, qi) is given by

Cjki (qi, qi) =

nl=1

1

2

M

jki

q li+

Mjli

qki M

kli

qji

ijkl(qi)

qli

where ijkl(qi) are the Christoffel symbols of the first

kind with the symmetric property that ijkl(qi) =ijlk(qi), hence, the matrix Mi(qi) 2Ci(qi, qi) isskew-symmetric. i.e.

Mi(qi) = Ci(qi, qi) + CTi (qi, qi) (2)

2.1 General Assumptions

A1. Following standard considerations, we assume thehuman operator and the environment define passive(force to velocity) maps, that is, there exists i R+0s.t.t

0

qmhd m, t0

qs ed s, (3)for all t 0.

A2. In order to simplify some calculations and to focuson the main idea of this article, we will assume thatthe gravitational forces are precompensated by the

controllers m, s (i.e. m = m + gm(qm) ands = s gs(qs)). Hence, the dynamical models (1)

are reduced toMm(qm)qm + Cm(qm, qm)qm = m h

Ms(qs)qs + Cs(qs, qs)qs = e s (4)A3. We assume that the time-delay imposed by the com-

munication channel is constant on each direction, butit may differ from one to another. The total roundtrip time-delay is equal to Tm + Ts 0.

3. SCATTERING TRANSFORMATION

The scattering transformation is given by

1 The reader may refer to Kelly et al. [2005] and Spong et al. [2005]for a complete guide on advanced robot modeling.

um =12b

[md bqmd] us = 12b

[sd bqsd]

vm =12b

[md + bqmd] vs =12b

[sd + bqsd](5)

where b is the virtual impedance of the transmission line.The master and the slave are interconnected, for constant

time-delays in the forward and the backward paths (Tmand Ts, respectively), as

us = um(t Tm) vm = vs(t Ts) (6)Proposition 1. Consider the teleoperator given by (4) con-trolled by

m = md Bmqms = sd + Bsqs

(7)

wheremd = Kdm[qm qmd]sd = Kds[qs qsd], (8)

with the communications governed by (5) and (6). Thecontrol gains 2 b, Bi, Kdi R+. Under the assumptionsA1 and A3

i. The system velocities asymptotically converge to theorigin for any time-delay Ti 0.

ii. By matching the impedances (i.e. Kdm = Kds = b),wave reflections do not occur, and the system velocityerror, defined by e = qm qs(t Ts), asymptoticallyconverges to the origin for any time-delay and anyb > 0.

Proof. Let us propose the following Liapunov functionV(qi, qi) given by

V =1

2qmMm(qm)qm +

1

2qs Ms(qs)qs + m + s + (9)

+ t0

[qmh qs e]d + t0

[qsdsd qmdmd]d,relying on the assumption A1, the property P1 of robotmanipulators, and the well known fact that

t0

[qsdsd qmdmd]d =1

2

ttTm

|um|2d +

+1

2

ttTs

|vs|2d

0which is the main contribution of the scattering trans-formation, we show that the Lyapunov candidate (9) ispositive definite and radially unbounded.

Using the property P2, the resulting time derivative of Vis given by

V = qmm qs s + qsdsd qmdmd (10)Substituting the expressions (7) and (8) we get

V = Bm|qm|2 Bs|qs|2 Kdm|qm qmd|2 Kds|qs qsd|2 (11)

02 The use of scalar gains is made for simplicity. When these gainsare positive definite diagonal matrices can b e treated with slightmodifications to the proof.

17th IFAC World Congress (IFAC'08)

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From which can conclude that the system velocities qmand qs asymptotically converge to the origin (part i, ofthe proposition). Moreover, invoking LaSalle Theorem fortime-delayed systems (Hale [2006]), all solutions of (4),(7) and (6) converge to M, the largest invariant set in

S. Where S = {qi Rn, i = {m, s} : V = 0} s.t.{

qm =

qmd

, qs =

qsd}.

In order to prove part ii, from (7) and (8) we can find that

qmd =Kds

b + Kdmqs(t Ts) + Kdm

b + Kdmqm +

+b Kdsb + Kdm

qsd(t Ts)

qsd =Kdm

b + Kdsqm(t Tm) + Kds

b + Kdsqs +

+b Kdmb + Kds

qmd(t Tm) (12)note that, by matching the impedances, the terms qid(t

Ti) on (12) disappear, and so do wave reflections. Usingthese new expressions the controllers become

m =b

2[qs(t Ts) qm] Bmqm

s =b

2[qs qm(t Tm)] + Bsqs

(13)

and the time derivative of V given by (11) transforms into

V = Bm|qm|2 Bs|qs|2 b4|qm qs(t Ts)|2

b4|qs qm(t Tm)|2 (14)

Similarly by LaSalle, qm

qs(t

Ts)

0 as t

. 2

4. OUTPUT SYNCHRONIZATION

The control laws for the teleoperator are given by

m = K[qs(t Ts) qm] Bmqms = K[qs qm(t Tm)] + Bsqs (15)

where K > 0 is the controller gain. Ti 0, i {m, s}, is the time delay in the forward and backwardpaths, respectively. It is assumed that the initial conditionsqi() Cn,r for [Ti, 0].Proposition 2. (Chopra and Spong [2006]). Consider theteleoperator system (4) controlled by (15). Assume that

h and e verify (3). Then all signals are bounded andthe system velocity error, defined by e = qm qs(t Ts),asymptotically converges to the origin, for any K > 0 andall Ti 0.Proof. Consider the following Lyapunov-Krasovskii func-tional, V(t, qi, qi(t Ti)), given by 3

V = qmMm(qm)qm + q

s Ms(qs)qs +

+

t0

[qmh qs e]d + m + s +

+ Kt

tTm |qm()

|2d + K

t

tTs |qs()

|2d (16)

3 For a complete description of Liapunov-based analysis for time-delay systems, the reader should refer to Niculescu [2001].

Taking the time derivative ofV and using the property P2of robot manipulators, we obtain

V = 2qmm 2qs s + K|qm|2 K|qm(t Tm)|2 ++ K|qs|2 K|qs(t Ts)|2 (17)

The last four terms come from the Krasovskii-functional,and can be written as:

K[qm qs(t Ts)][qm + qs(t Ts)] ++K[qs qm(t Tm)][qs + qm(t Tm)] (18)

using the expressions (15) and some algebraic manipula-tions, it is easily seen that (17) becomes

V = 2Bm|qm|2 2Bs|qs|2 K|qm qs(t Ts)|2 K|qs qm(t Tm)|2 (19)

0

Towards this end, the Liapunov-Krasovskii functionalonly ensures uniform stability, but, using an extensionof LaSalles Invariance principle for FDE, we can obtainuniform asymptotic stability of the V signals.

Note that, the function V(t, qi, qi(t Ti)) is positivedefinite, and V 0 any arbitrary initial level set ispositively invariant, hence, all signals in (4) are bounded.Now, consider the set S = {qi Rn, i = {m, s} :V = 0} characterized by all trajectories s.t. {qs(t Ts) = qm, qm(t Tm) = qs}. Let M S be the largestinvariant set in S. Using LaSalle Theorem (Hale [2006]),all solutions of (4) and (15) converge to M as t . Thisimplies that the velocity error e asymptotically convergesto the origin for all Ti 0 and all positive definite gainK. 2

5. SIMULATIONS

This section presents a simulation of the teleoperator sys-tem. The master and slave are two link manipulators withrevolute joints. Their corresponding nonlinear dynamicsfollow (1). The inertia matrix Mi(qi) is given by

Mi(qi) =

i + 2i cos(q2i) i + i cos(q2i)i + i cos(q2i) i

qki , k {1, 2} is the articular position of each link,i = l

22im2i + l

21i(m1i + m2i), i = l1i l2im2i and i =

l22im2i . The lengths for both links l1i and l2i , in eachmanipulator, are 0.38m. The mass of each link correspondto m1m = 3.9473kg, m2m = 0.6232kg, m1s = 3.2409kg andm2s = 0.3185kg, respectively. These values are the sameof those used by in Lee and Spong [2006]. Coriolis andcentrifugal forces are modeled as the vector Ci(qi, qi)qiwhich are

Ci(qi, qi)qi =

i sin(q2i)q22i i sin(q2i)q1i q2ii sin(q2i)q

21i

q1i and q2i are the respective revolute velocities of the twolinks. The gravity effects (gi(qi)) for each manipulator arerepresented by

gi(qi) = 1

l2igi cos(q1i + q2i) + 1l1i

(i i) cos(q1i)1l2i

gi cos(q1i + q2i)



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Note that this vector is precompensated following Assump-tion 2. h and e are the operator and environmentaltorques. At this point, it should be addressed that thehuman exerts a force fh on the master manipulators tip,and the slave interaction force with the environment fe isalso measured at the manipulators tip. Hence, for the sim-ulations the following expressions are used h = J

m

(qm)fhand e = Js (qs)fe, where Ji (qi) is the Jacobian trans-posed of the robot manipulator. The controllers for thesesimulations are given by (15), with K = 5 and Bi = 0.5.The time delay is Tm = 2.5s and Ts = 3.5s. The simulationhas been carried out using MatLab SimuLink TM.

Two simulations were carried out: on the first, the humanexerts a force on the master and the slave moves freelyon the environment (Fig. 1); on the second, the humanmoves the master and the slave touches a high-stiffnessvirtual wall on the environment (Fig. 2). This virtual wallhas been located in cartesian coordinates at y = 0.25m.The stiffness of the wall was set to 20000 N

mwith 10 Ns

mof damping. We assume that the system is frictionless. OnFig. 1 the system moves freely, velocity error converges tothe origin and it is clearly seen that the teleoperator isstable. Moreover, if touching a high-stiff wall, around 6son Fig. 2, the system remains stable, but, position driftarises.

6. EXPERIMENTS

The experimental test-bed mainly consists of two direct-drive two DOF nonlinear manipulators. These manipu-lators are made of aluminium and are actuated by twopairs of Compumotor DM1015-B brushless DC motors.Optical encoders are used to measure the joint position,

the joint velocity is digitally estimated and filtered. TwoJR3 force-torque sensors, located at the manipulators end-effectors, are used to measure the force interaction with thehuman operator and environment, respectively. The con-trollers are implemented using WinCom 3.3 that enablesSimulinkTM models to interact with external hardware inreal time. The sampling time is set to 4ms. An aluminiumwall is located at one side of the slave in order to testthe stability while interacting with an stiff environment.This setup is located at the CSL, UIUC, and is depictedin figure 3.

There are two experiments, the teleoperator moves freelyon Fig. 4 and touches an aluminium wall around 12s on

Fig. 5. The static friction has caused the teleoperator toexhibit a position error, although moving freely (q1, q2 onFig. 4). However, the system is stable despite touching thealuminium wall. In order to set the control gain K it can beused a model-based tuning method such as the one on pp.213 of Kelly et al. [2005]. The master and slave controllersfollow (15) and the gains were set to K = 10, Bi = 2, forboth experiments. The two 2 DOF manipulators move ona parallel plane tangential to the earth surface, hence, thegravity vector is zero.

7. CONCLUSIONS AND FUTURE WORK

Chopra and Spong [2006] introduced the passive outputsynchronization, which has been the base platform for thispaper. Using a symmetric controller with the scattering

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Fig. 1. Simulation of the teleoperator system with Tm =2.5s and Ts = 3.5s.



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Fig. 2. Simulation of the teleoperator system touching astiff wall with Tm = 2.5s and Ts = 3.5s.

MasterSlave

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Fig. 3. Experimental teleoperator at CSL, UIUC.

transformation and matching the impedances, one canreduce the system to a passive output synchronizationscheme. Both schemes ensure stability for any arbitraryconstant time-delay. However, under velocity control, bothschemes do not provide position synchronization. Futuresteps on this line are the study of the effects of variabletime-delays and the analysis of position drift free schemes.

ACKNOWLEDGEMENTS

The first author gratefully acknowledges the hospitality ofthe people at the CSL at the UIUC, especially to Prof.Mark W. Spong, he also thanks the comments of theanonymous reviewers, that helped to improve the contentsof this work.

REFERENCES

R.J. Anderson and M.W. Spong. Bilateral control ofteleoperators with time delay. IEEE Transactions onAutomatic Control, 34(5):494501, May 1989.

P. Arcara and C. Melchiorri. Control schemes for teleop-eration with time delay: A comparative study. Roboticsand Autonomous Systems, 38:4964, 2002.

N. Chopra and M.W. Spong. Passivity-Based Controlof Multi-Agent Systems, pages 107134. Advances inRobot Control From Everyday Physics to Human-LikeMovements. Springer, 2006.

N. Chopra and M.W. Spong. Adaptive Synchronization ofBilateral Teleoperators with Time Delay, pages 257270.Advances in Telerrobotics. Springer, 2007.

J.K. Hale. History of Delay Equations, chapter 1, pages128. Number 205 in Delay Differential Equations andApplications. Springer-Verlag, 2006.

P.F. Hokayem and M.W. Spong. Bilateral teleoperation:An historical survey. Automatica, 42:20352057, 2006.R. Kelly, V.Santibanez, and A. Lora. Control of robot

manipulators in joint space. Advanced textbooks incontrol and signal processing. Springer-Verlag, 2005.

D. Lee and M.W. Spong. Passive bilateral teleopera-tion with constant time delay. IEEE Transactions onRobotics, 22(2):269281, April 2006.

S.I. Niculescu. Delay Effects on Stability: A Robust ControlApproach. Number 269 in Lecture Notes in Control andInformation Sciences. Springer-Verlag, 2001.

M.W. Spong, S. Hutchinson, and M. Vidyasagar. RobotModeling and Control. Wiley, 2005.



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Fig. 5. Experiments touching the aluminium wall, withTm = 2.5s and Ts = 3.5s.



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2008 Bilateral Teleoperation Experiments Scattering Transformation and Passive Output Synchronization Revisited