A Gain-switching Control Scheme for

Position-error-based Force-reflecting Teleoperation

Liya Ni David W. L. Wang

Dept. of Electrical & Computer Engineering

University of Waterloo

Waterloo, Ontario, Canada

email: lni@kingcong.uwaterloo.ca, dwang@kingcong.uwaterloo.ca

Abstract

To lower the cost of a force-reflecting teleoperation sys-tem, position errors instead of direct measurementsfrom force/torque sensors can be used for force feed-back. However, the position-position architecture withfixed PD controllers provides poor transparency. Again-switching control scheme is developed in this pa-per to solve this problem. Based on the detection ofthe impedance change at the slave site, the master andslave PD controller gains are switched accordingly. Ex-perimental results are shown to demonstrate the effec-tiveness of this method.

1 Introduction

Over the past few decades, teleoperation has been em-ployed in a large variety of applications including spaceand underwater exploration, nuclear operation, micro-surgery, etc. When there is a contact with the environ-ment at the remote site, force feedback is importantto achieve the goal of “telepresence” — to feel as ifphysically present at the remote site.

The strategy of adjusting the master and slave con-trollers in a dual manner has been applied in someresearch for bilateral teleoperation. Reboulet et. alproposed the dual hybrid control scheme in [1]. Thetask cartesian space is separated into two orthogonalsubspaces. In the unconstrained subspace, the masteris zero force controlled while the slave is position con-trolled; in the constrained subspace, the master is posi-tion controlled while the slave is force controlled. Thiswas extended to the “matched impedance” methodin [2], which uses the relative sizes of contact forcesand velocities to adjust the master and slave targetimpedances to match high or low impedance environ-ments. The variable damping impedance control pro-posed in [3] presented a similar idea, but the modifi-cation of the desired impedance is based on the com-

manded slave velocity, the contact force and the anglebetween them.

The above control schemes on bilateral teleoper-ation were designed under the assumption that thecontact force is measured by force/torque sensors di-rectly. However, force/torque sensors are expensiveand noise-prone. An alternate method is the position-error-based bilateral teleoperation [4], in which bothof the master and slave manipulators are controlledusing the position measurements from each other asreference. No force sensors are utilized. Lawrence[5] showed that with linear time-invariant controllers,the position-position architecture provides poor trans-parency. To improve transparency, the position con-trollers should be adjusted when the dynamics at theslave site changes, such as hitting a hard object. An in-direct adaptive bilateral control scheme was proposedin [6], which uses master and slave position, velocityand acceleration measurements and requires no forcesensing. The environment estimation was based on the“composite adaptive control” scheme which is aimedat position tracking. To eliminate the need for forcesensors, a sliding mode type robust controller was de-veloped in [7], which is also designed for the conver-gence of position error to zero. In some applications,however, accurate position tracking at the slave site isnot desirable or even harmful when the slave is in con-tact with a stiff environment. Shared Compliant Con-trol (SCC) [8] was developed to soften collisions. WithSCC, the telerobot has a compliant hand implementedwith feedback of the low-pass filtered force/torque mea-surements.

In this paper, the philosophy of dual adjustmentof master and slave controller is interpreted in thecontext of position-error-based teleoperation. A gain-switching control scheme is proposed. The changesof slave site dynamics between free motion and hardcontact are detected based on impedance estimation,and the master and slave controller gains are switched

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

accordingly. In Section 2, a two port model is usedfor the transparency analysis of a position-error-basedteleoperation. The on-line slave site impedance esti-mation and design of the gain-switching rules are pre-sented in Section 3. The experimental results and com-ments from a group of users are shown in Section 4.The contact stability is discussed briefly in Section 5,followed by conclusions in Section 6.

2 Transparency Analysis

The proposed position-error-based bilateral teleoper-ation system has a symmetric structure as shown inFig. 1. Gm(s) and Gs(s) are the transfer functions

e u y

f

-s s

e

C (s) G (s)s+ -

+s

s

channelcommunication

K2

1 KC (s) G (s)e u

f

+-

m

h

m m+

+ymm

local site remote site

Figure 1: Structure of the position-error-basedbilateral teleoperation system

for the master and slave dynamics. PD controllersare applied at both sides as Cm and Cs. fh repre-sents the force exerted by the operator on the masterwhile fe represents the force exerted by the slave onthe environment. The other variables ym, ys, em, es,um, us denote the position signals, position errors andcontroller outputs at the master and slave sites respec-tively. K1 and K2 are the scaling factors which satisfyK1 × K2 = 1 to make the two workspaces virtuallyoverlap each other. The time delays in the commu-nication channel are assumed to be negligible in thispaper.

Under the assumption that each degree of free-dom (DOF) has linear dynamics decoupled from theother DOF, the analysis and design will be focused ona one DOF linear system hereafter. The mechanicalimpedances of master and slave are normally charac-terized by their inertia J and viscous damping ratioB:

Zm(s) =1

sGm(s)= Jms+Bm; (1)

Zs(s) =1

sGs(s)= Jss+Bs. (2)

For convenience, we define the “impedance form” ofthe controllers as:

Cm(s) =Cm(s)

s= Kdm +

Kpm

s; (3)

Cs(s) =Cs(s)

s= Kds +

Kps

s. (4)

With the analogy to an electrical network, the tele-operation system can be represented by a two-portmodel [9], as shown in Fig. 2. fh and fe correspond to

Teleoperator

-

++

- f f

v v sm

eh Z

Z

Local operator Remote Task

e

th

Figure 2: Two-port model of a bilateral teleop-eration system

the voltages applied at the two ports, and the veloci-ties of the master and slave, where vm = ym, vs = ys,correspond to the currents. Ze = fe/vs representsthe mechanical impedance of the environment, andZth = fh/vm represents the impedance transmittedto the human operator. The dynamics of the humanoperator is not included here because it’s unnecessaryfor our analysis and design.

In a position-error-based teleoperation, the two-port model can be represented by an impedance matrixZ 1:

[

fh

fe

]

= Z

[

vm

−vs

]

. (5)

The impedance matrix for the system shown in Fig. 1can be derived as:

Z(s) =

[

Zm(s) + Cm(s) K2Cm(s)

K1Cs(s) Zs(s) + Cs(s)

]

. (6)

Complete transparency is defined in [5] as: (i) theslave follows the master (xs = xm or vs = vm); (ii)for any environment impedance, Zth = Ze. In theposition-error-based teleoperation system, the relation-ship between Zth and Ze can be obtained from theimpedance matrix as:

Zth = Zm + Cm −CmCs

Ze + Zs + Cs

. (7)

We only look at complete transparency in two extremecases: (i) the slave is in free motion, i.e. Zth = Ze → 0;(ii) the slave is in hard contact, i.e. Zth = Ze → ∞.When Ze → 0, Zth becomes:

Zth = Zm +CmZs

Zs + Cs

. (8)

1The sign of vs is reversed here since it is leaving rather than

entering the two-port system.

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

Transparency in free motion requires (8) to be Zth =Ze = 0, which is obviously unachievable due to thefact that Zth > Zm with positive parameters in thetransfer functions Zm, Zs, Cm, Cs. By allowing an“intervening impedance” [10], which can be taken tobe the master impedance Zm, to be added on the envi-ronment impedance and transmitted to the operator’shand, the transparency condition for free motion canbe modified to:

CmZs

Zs + Cs

→ 0. (9)

This requires that, for a given Zs, the master PD con-troller should have very low gains, while the slave con-troller should have very high gains. When Ze → ∞,Zth becomes:

Zth = Zm + Cm.

This shows that the operator only feels the dynamicsof the master and its control system, which is muchsmaller than Ze with our assumption. However, ac-cording to (7), if the master controller has very highgains, while the slave controller has low gains, it’s pos-sible to make the impedance transmitted to the humanoperator be high enough to let the operator feel as ifthe master device is in hard contact.

Intuitively, there should be no force feedback tothe operator when the slave manipulator is in free mo-tion, which indicates that the master controller shouldhave gains as low as possible; otherwise, the systemwould feel “sluggish” due to the position error. Theslave controller should have high gains for good posi-tion tracking performance in free motion. When theslave manipulator has a hard contact, the master con-troller should have high gains to make the operatorsensitive to the collision happening at the remote site;while the slave controller should have low gains in or-der to soften the collision. These rules are consistentwith the above transparency analysis.

3 Design of Gain-switching Con-

trollers

The transparency in position-error-based teleoperationcan be improved by on-line adjustment of the con-troller gains according to the changes in environmentimpedance. The scenario studied in this paper is thatthe slave is either in free motion or in hard contact.According to the analysis in Section 2, the master andslave PD controller gains should be switched between“high” and “low” values. Fig. 3 shows the block dia-gram of this control scheme.

1s

1smC (s) (s)mZ C (s) Z (s)

Z (s)

s s

e

-1-1K

K

Impedance

Gain SwitchingRules

+

+ ++

-

-

-e

f

1

2

m y ym su m

+e s

h

u s vs

fe

Estimator

Figure 3: Block diagram of the proposed gain-switching control

3.1 Slave Site Impedance Estimation

The environment is modelled by a mass-damper-springsystem with the impedance:

Ze = Jes+Be +Ke

s. (10)

If the slave is in free motion, we have Je = Be = Ke =0. However, only the slave position ys and controlleroutput us are available for estimation. The transferfunction from Us to Ys is:

Ys(s)

Us(s)=

1

Jts2 +Bts+Kt

. (11)

This describes the entire dynamics at the slave sitewith Jt = Js + Je, Bt = Bs +Be, Kt = Ke.

Because of noise consideration, we want to avoidusing acceleration calculated as the second order dif-ferentiation of ys. The filtering method in [11] is ap-plied. Rewrite (11) as:

A(s)Ys(s) = B(s)Us(s), (12)

where

A(s) = s2 +Bt

Jt

s+Kt

Jt

, B(s) =1

Jt

.

Divide both sides of (12) by a known monic secondorder polynomial A0(s) = s2 + a1s+ a0, leading to

Ys(s) =A0(s)−A(s)

A0(s)Ys(s) +

B(s)

A0(s)Us(s)

=[

a0 −Kt

Jta1 −

Bt

Jt

1Jt

]

Ys(s)A0(s)sYs(s)A0(s)Us(s)A0(s)

(13)

By filtering with A0(s), a DC gain of 1/a0 is intro-duced. This effect can be eliminated by multiplyingboth sides of (13) by a0. The continuous-time modelfor identification becomes:

a0Ys(s) = θTΦ(s), (14)

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

where

θ =[

a0 −Kt

Jta1 −

Bt

Jt

1Jt

]T

,

Φ =[

a0Ys(s)A0(s)

a0sYs(s)A0(s)

a0Us(s)A0(s)

]T

.

The on-line impedance estimation will be imple-mented in a sampled-data control system. Therefore adiscrete-time version of (14) has to be constructed asfollows:

y(k) = θTφ(k), (15)

where y(k) = a0ys(k), φ(k) =[

y(k) v(k) u(k)]T

.y(k) and u(k) are obtained by applying the discretiza-tion of the filter a0/A0(s) on the output sample ys(k)and input sample us(k), while v(k) is obtained by ap-plying the discretization of the filtered differentiationa0s/A0(s) on ys(k).

The Recursive Least Square (RLS) algorithm withexponential data weighting [12] is employed for theimpedance estimation. The update laws for the pa-rameter estimate vector θ, gain vector K and covari-ance matrix P are as follows:

θ(k) = θ(k − 1) +K(k)[y(k)− θT (k − 1)φ(k)];

K(k) = P (k − 1)φ(k)/[λ+ φT (k)P (k − 1)φ(k)];

P (k) =1

λ[P (k − 1)−K(k)φT (k)P (k − 1)]. (16)

The choice of λ involves compromises: too fast dis-counting (small λ) will cause the covariance matrix towindup, too slow discounting will make it difficult totrack fast parameter variations.

3.2 Gain Switching Control

According to the analysis in Section 2, the master andslave PD controller gains should be switched betweenhigh and low values. Since we only consider two casesof the slave site dynamics (free motion and hard con-tact), we don’t have to wait until the estimation ofimpedance converges, which is too slow for practicaluse. The gain-switching can be triggered by the de-tection of impedance changes between free motion andhard contact. A linear combination of the estimatedslave site impedance parameters δ ≡ αJt + βBt + γKt

is used as an indicator or metric for the gain switching(note that δ is not itself an impedance). The followinglogic can be applied as the gain switching rules:I. If Kpm, Kdm are low, Kps, Kds are high and δ >δcontact, then switch Kpm, Kdm to high, and switchKps, Kds to low.II. If Kpm, Kdm are high, Kps, Kds are low and δ <δfree, then switch Kpm, Kdm to low, and switch Kps,Kds to high.

The values of α, β, γ, δcontact and δfree are tunedexperimentally. Under the assumption that the con-tact surface is perpendicular to the free motion justprior to contact and the slave force is applied orthog-onal to the contact surface, an alternate and probablymore efficient way to detect when the slave is leav-ing the environment is to check the position of masterrelative to the “contact point” (scaled to the masterworkspace). The master position must have passed the“contact point” when the master controller gains areswitched to high. If the position shows that it’s pass-ing the “contact point” again in the opposite direction,then the master intends to leave, so the controller gainsshould be switched to the free motion settings. The“low” and “high” controller gains are selected sepa-rately for the master and slave devices. Controlleroutput saturation should be considered for choosinghigh gains. When the slave is in hard contact, theslave controller gains need to be low enough to avoidexerting too much force to the environment, but can-not be too low as the slave should remain in contactwith the environment and keep the impedance estima-tion process persistently excited. These parametersare obtained by trial and error in the experiment.

4 Experimental Evaluation

4.1 Experimental Setup

The experimental setup, as shown in Fig. 4, consists ofa five-bar linkage manipulator as master and a VirtualReality Mouse (VRM) as slave. They are connectedto a single computer, so there is no time delay in thecommunication channel.

Figure 4: Experimental setup

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

The five-bar linkage manipulator was designed andbuilt at the University of Waterloo. It is gravity bal-anced and dynamically decoupled using the counter-balancing scheme developed by Ching and Wang [13].Furthermore, the robot uses direct-drive motors sothat non-linear effects, such as gear backlash and fric-tion, are not present or are minimal. The VRM is acommercial product from Betacom Inc. 2 It can movealong two orthogonal directions. The motion can becaused by the two motors, by the user or by a com-bination of both. The dynamics of the VRM in thetwo orthogonal directions are decoupled. Hence, thefive-bar linkage robot and the VRM are ideally suitedfor use as haptic interfaces.

The mapping of robot motion versus VRM motionin the teleoperation is illustrated in Fig. 5. The extra

0 x

y

x y

five-bar linkage manipulator

link 3

link 2

link 1

link 4Virtual Reality Mouse

Figure 5: Mapping of the five-bar linkage robotmotion versus VRM motion

DOF of the five-bar robot — the base joint doesn’tmove during the experiments. For safety reasons, themotion of the five-bar robot is only allowed to be withina limited range of its workspace. Therefore, the VRMworkspace is actually mapped to the “safe region” ofthe robot workspace. An iron object is clamped ontothe table and located within the VRM workspace.

Both of the robot and VRM positions are measuredby encoders mounted on the motors, and the controlsignals are sent to D/A converters with PWM voltageoutputs. There is no direct force/torque measurementavailable.

4.2 Friction Compensation on the VRM

It was noticed that the estimated impedance parame-ters sometimes increase as a result of the considerablestatic friction between the VRM handle and the tablesurface. Therefore, friction compensation is necessaryfor implementing correct and prompt gain switching.

Static friction occurs when the VRM is starting tomove from rest, and it acts in the opposite directionof pending motion. Kinetic friction applies when the

2www.betacom.com

VRM is moving on the table surface. A friction com-pensation scheme for the VRM is proposed in [14], asshown in Fig. 6. However, the static friction was notcompletely compensated for due to the difficulty ofdeciding the direction of pending motion without thebenefit of a force sensor. In our particular case, theVRM acts as the slave in teleoperation. Therefore,the direction of the motor torque indicates the direc-tion of pending motion when the VRM is at rest. Amodified version of the friction compensation is shownin Fig. 6 as well.

+

+

++

filterd( )/dtVRMu vyfriction compensation in [.

additional static friction compensation

-1

1

14]

Figure 6: Block diagram of the friction compen-sation

4.3 Results and Comparison

The proposed gain-switching control scheme was testedon the experimental setup described in Section 4.1.One of the implementation issues is that the force gen-erated by the spring part of the environment impedanceis actually a function of the penetration, i.e. Ke(ys −

ye), which depends on the unknown contact point ye.Keys is used instead in our modelling and this is trueonly when the origin of the measurement axis y0 equalsye. To solve this problem, we introduced offsets to theys measurements by using several different locationswithin the workspace as the origins, and then per-formed impedance estimation in parallel for ys withdifferent offsets. The average of the results from theseimpedance estimators are used for gain-switching.

The Butterworth filter F (s) =ω2

c

s2+√

2ωcs+ω2c

with a

cutoff frequency wc = 20πrad/sec is applied as a0/A0(s)in the impedance estimation. The exponential forget-ting rate λ is selected as 0.92. The sampling rate is10 milliseconds. By trial and error, α, β, γ are se-lected to be 1, 1, 2. With the friction compensation,δcontact = 0.05 is big enough to detect the hard con-tact. It takes about 100 milliseconds for δ to reach0.05 from the moment when the VRM just hits theenvironment. However, the change in δ seems sluggishwhen the VRM is leaving the environment. The time

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

for δ to drop below δfree = 0.02 is almost 300 millisec-onds from the moment when the VRM just starts toleave the environment. Therefore, the robot positionrelative to the scaled “contact point” is used as an in-dicator of the leaving stage. The gain switch flips whenthe “contact” or “leaving” behavior of the VRM is de-tected. When the gain switch is 1, the robot controllergains become Kh

pm = 20Klpm, Kh

dm = 5Kldm, and the

VRM controller gains become K lps, K

lds. When the

gain switch is 0, the robot controller gains are K lpm,

Kldm, while the VRM controller gains are Kh

ps = 4Klps,

Khds = 2Kl

ds. The superscripts l and h denotes low andhigh gains respectively. The ratios are tuned by trialand error.

As both of the five-bar linkage robot and VRMhave decoupled dynamics on the two DOF of teleop-eration, we only show the experimental results on oneDOF in Fig. 7. Because the position control is in-volved mapping between linear motion and rotation,the master and slave positions are shown in the unitof encoder counts (scaled to the same range).

0 5 10 15 20 25 30−4000

−2000

0

2000

4000

y m &

ys (

enc.

cts

.)

(a)

0 5 10 15 20 25 30−1

0

1

2

time(sec.)

gain

sw

itch

(c)0 5 10 15 20 25 30

0

0.05

0.1

δ

(b)

δcontact

Figure 7: Experimental results (ym : —, ys : · · ·in (a))

Since no force sensor is available, we show the im-provement of transparency from the observation of po-sition responses and users’ comments, instead of directforce measurements. It can be observed from Fig. 7that once the VRM hits the iron object (indicated inFig. 7 by the flattening of the ys response), the userholding the robot arm will experience an impact shownas the “overshoot” in ym and then cannot move for-ward as if there were a virtual barrier in its workspace.In the experiment, telepresence was achieved not onlyby the force feedback to the user, but also by the “col-

lision” sound from the robot motor when its controllergains suddenly switch to high values.

For comparison, a set of experiments were testedby six users. Fixed PD controllers were tested in Ex-periment 1-3. Low gains of Cm and high gains of Cs

were applied in Experiment 1 while high gains of Cm

and low gains of Cs in Experiment 2. The averageof high and low gains were applied in Experiment 3.In Experiment 4, the estimated δ was used for gainswitching in both of the “contact” and “leaving” cases,which exactly follows the rules I and II in Section 3.2.In Experiment 5, it was modified by using the robotposition relative to the “contact point” to indicate the“leaving” case. The users were asked to move aroundthe five-bar linkage manipulator and describe how theyfeel when the VRM is in free motion and in hard con-tact. Their comments are summarized in Table 1.

The comparison of the experimental results is alsoshown in Fig. 8. Fig. 8(a)-(c) were obtained from Ex-periment 1-3 respectively, and Fig. 8(d) corresponds tothe proposed method as in Experiment 5. When low

0 5 10 15−4000

−2000

0

2000

4000

y m &

ys (

enc.

cts

.)

time(sec.)

(a)

0 5 10 15−4000

−2000

0

2000

4000

y m &

ys (

enc.

cts

.)

time(sec.)

(b)

0 5 10 15−4000

−2000

0

2000

4000

y m &

ys (

enc.

cts

.)

time(sec.)

(c)

0 5 10 15−4000

−2000

0

2000

4000

y m &

ys (

enc.

cts

.)

time(sec.)

(d)

Figure 8: Comparison of experimental results

master gains and high slave gains are applied (Fig. 8(a)),we have good free motion response, but the contact inthe slave site almost has no effect at the master site.While high master gains and low slave gains (Fig. 8(b))result in sluggish free motion with large tracking er-rors, although the master cannot go further due to thecontact at the slave site. As shown in Fig. 8(c), byusing the average of high and low gains, we couldn’tget much improvement. Only with the proposed gain-switching scheme, we obtained good position trackingat slave site in free motion and duplication of the hardcontact at the master site, as shown in Fig. 8 (d).

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

Table 1. Comments from a group of users.Exp. slave in free motion slave in hard contact1 Very light/smooth, as if the power is off (6)a. Cannot feel anything(6).2 Very heavy/sluggish(6). Can feel the contact(3).

It’s hard to move beyond some point,but the change is not drastic(2).Can hardly feel the contact(1).

3 Lighter than Exp. 2 but heavier than Exp. 1(6). Feel as if in contact with a spring(3).Feel contact, but not as stiff as Exp. 2(3).

4 Very light/smooth, as if the power is off (6). Feel as if in contact with a “sticky” wall(6).Feel the impact like hitting a wall(3).

5 Very light/smooth, as if the power is off (6). Can feel the hard contact, and leavethe “wall” without feeling “sticky”(6).

aThe number after every comment is the number of users who have that comment.

5 Discussion on Contact Stabil-

ity

Most previous stability studies on teleoperation sys-tems [4, 5, 15, 16, 17] are based on the passivity theory.Under the assumption that the human operator andenvironment can be modelled as passive terminations,passivity of the teleoperation system is a sufficient con-dition for stability in either unconstrained motion orconstrained motion. However, when the slave robotcollides with a stiff surface, the teleoperation systemmay still exhibit a severe chattering instability. Thiswas noticed in simulation and experiments with di-rect feedback of force sensor measurements [18], butwe’ve never observed chattering in our gain-switchingposition-error-based teleoperation experiment.

The rigorous contact stability analysis for the gain-switching teleoperation system is currently in progress.In [19], we proved the contact transition stability fora position-error-based bilateral teleoperation systemwith constant PD controller gains, which forms thebase of the stability proof for the gain-switching sys-tem. The results presented in [19] are summarized inthe following theorem.

Theorem 1. Consider the system shown in Fig. 1with constant gains in Cm(s) and Cs(s), the environ-ment impedance isZe(s) = 0 if ys < ye;Ze(s) = Be +

Ke

sif ys ≥ ye,

where ys = ye is the contact surface. Under the fol-lowing conditions:

1.Kpm

Kps= Kdm

Kds;

2. the dynamics of the human operator during thetransition can be modelled by a mass-damper-spring system, i.e.

−fh = Jhym +Bhym +Kh(ym − yd),

where yd, which satisfiesK1yd > ye, is a constantdenoting the desired position;

3. the positions and velocities of the master andslave are continuous at the switching instances;

4. within any finite closed time interval, there areonly finite number of switchings,

the system will remain in the constrained space ys ≥ ye

after finite numbers of switchings and asymptoticallyconverge to the equilibrium point defined by ym = y1,ys = y2, vm = 0, vs = 0, where y1 and y2 are constantssatisfying K1yd > K1y1 > y2 > ye.

To simplify the stability analysis, we assumed thatthe environment is stationary and infinitely massiveand modelled it as a simple spring and damper sys-tem.

The outline of the proof for Theorem 1 is as follows:According to the Llewellyn’s Absolute Stability Crite-ria [20], we derived condition 1 as the sufficient con-dition for passivity of the unconstrained system andconstrained system. Under condition 2, we provedthat both of the unconstrained and constrained sys-tems are asymptotically stable and their total energyis non-increasing and only flattens at certain isolatedtime instances. Furthermore, there exists an energythreshold below which the slave robot cannot switchfrom the constrained space to the unconstrained space.Under condition 3 and 4, the total energy will be belowthat threshold after finite numbers of switchings, thenthe system will settle down in the constrained spaceand approach the equilibrium point asymptotically.

This theorem is now being extended to the case ofa gain switching controller as presented in this paper.There are several difficulties which arise due to thediscontinuities which are introduced as well as the dif-ficulty in finding a Lyapunov function. However, it is

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE

expected that Theorem 1 will become the foundationfor such a proof.

6 Conclusions

A gain-switching control scheme is proposed in thispaper. Experimental results show that this schemeis successful in improving transparency of a position-error-based teleoperation system. The advantages ofthis method are multi-fold: it doesn’t require directforce/torque measurements; transparency is improvedsignificantly over fixed PD controller strategy; goodposition tracking in free motion and collision soften-ing are obtained at the slave site; the use of accelera-tion is avoided in the impedance estimation. Frictioncompensation on the slave device was implemented toimprove the performance. Future research will be con-ducted on how to extend this method to handle variousenvironment impedances. A rigorous stability analysison the gain-switching teleoperation system has to bestudied and to be verified with simulations and exper-iments.

References[1] C. Reboulet, Y. Plihon and Y. Briere. “Interest of the

Dual Hybrid Control Scheme for Teleoperation withTime Delays”. Experimental Robotics IV: The 4th Int.

Symp., Stanford, CA., 1995, Springer Verlag, LNCISvol. 223, 1997, pp. 498–506.

[2] S. E. Salcudean, K. Hashtrudi-Zaad, S.Tafazoli, S.P. DiMaio and C. Reboulet. “Bilateral Matched-impedance Teleoperation with Application to Exca-vator Control”. IEEE Control System Magazine, Dec.1999, pp. 29–37.

[3] R. V. Dubey, T. F. Chan and S. E. Everett. “Vari-able Damping Impedance Control of a Bilateral Teler-obotic System”. IEEE Control System Magazine,February, 1997, pp. 37–45.

[4] G. J. Raju, G. C. Verghese and T. B. Sheridan. “De-sign Issues in 2-port Network Models of Bilateral Re-mote Manipulation”. Proc. IEEE. Int. Conf. Robot.

Automat., 1989, pp. 1316–1321.

[5] D. A. Lawrence. “Stability and Transparency in Bi-lateral Teleoperation”. IEEE Trans. Robot. Automat.,

Vol.9, No.5, 1993, pp. 624–637.

[6] K. Hashtrudi-Zaad and S. E. Salcudean. “Adap-tive Transparent Impedance Reflecting Teleopera-tion”. Proc. IEEE Int. Conf. Robotics and Automa-

tion, Minneapolis, Minnesota, April, 1996, pp. 1369–1374.

[7] D. Lee and P. Y. Li. “Passive Control of Bilateral Tele-operated Manipulators: Robust Control and Experi-ments”. Proceedings of the American Control Confer-

ence, Arlington, VA, June, 2001, pp. 4612–4618.

[8] W. S. Kim, B. Hannaford, and A. K. Bejczy. “Force-reflection and Shared Compliant Control in Operat-ing Telemanipulators with Time Delay”. IEEE Trans.

Robot. Automat., Vol.8, No.2, 1992, pp. 176–185.

[9] B.Hannaford. “A Design Framework for Teleopera-tors with Kinesthetic Feedback”. IEEE Trans. Robot.

Automat., Vol.5, No.4, 1989, pp. 426–434.

[10] Y. Yokokohji and T. Yoshikawa. “Bilateral Controlof Master-slave Manipulators for Ideal KinestheticCoupling—Formulation and Experiment”. IEEE

Trans. Robot. Automat., Vol.10, No.5, 1994, pp. 605–620.

[11] Slotine, J-J. E. and W. Li. Applied Nonlinear Control.Prentice-Hall, Englewood Cliffs, NJ, 1991.

[12] Warwick, K. Implementation of Self-tuning Con-

trollers. Peter Peregrinus Ltd., London, United King-dom, 1988.

[13] M. Ching and D. W. L. Wang. “A Five-bar-linkageForce Reflecting Interface for a Virtual Reality Sys-tem”. Proc. IEEE. Int. Conf. Robot. Automat., Albu-

querque, NM, 1997, pp. 3012–3017.

[14] Turner, C. F. and D. W. L. Wang. A Virtual Re-

ality Mouse Friction Model. Technical Report UW-E&CE#2001-13, Dept. of Electrical & Computer En-gineering, The University of Waterloo, Waterloo, On-tario, Canada, December, 2001.

[15] R. J. Anderson and M. W. Spong. “Bilateral Con-trol of Teleoperators with Time Delay”. IEEE Trans.

Automat. Contr., Vol.34, No.5, 1989, pp. 494–501.

[16] G. Niemeyer and J. J. E. Slotine. “Stable AdaptiveTeleoperation”. IEEE Journal of Oceanic Engineer-

ing, Vol.16, No.1, 1991, pp. 152–162.

[17] K. Hashtrudi-Zaad and S. E. Salcudean. “Analysisand Evaluation of Stability and Performance Robust-ness for Teleoperation Control Architectures”. Proc.

IEEE. Int. Conf. Robot. Automat., 2000, pp. 3107–3113.

[18] B. Hannaford and R. J. Anderson. “Experimental andSimulation Studies of Hard Contact in Force Reflect-ing Teleoperation”. Proc. IEEE Int. Conf. Robot. Au-

tomat., 1988, pp. 584–589.

[19] L. Ni and D. W. L. Wang. “Contact Transition Sta-bility Analysis for a Bilateral Teleoperation System”,submitted to the IEEE International Conference of

Robotics and Automation. May, 2002.

[20] Haykin S. S. Active Network Theory. Addison-Wesley,1970.

Proceedings of the 10th Symp. On Haptic Interfaces For Virtual Envir. & Teleoperator Systs. (HAPTICS�02) 0-7695-1489-8/02 $17.00 © 2002 IEEE