Isotropic force control for haptic interfaces

Control Engineering Practice 12 (2004) 1423–1436

ARTICLE IN PRESS

$$This work

Foundation (IIS

Research (N000

*Correspondi

7881.

E-mail addre

colorado.edu (D

0967-0661/$ - see

doi:10.1016/j.con

Isotropic force control for haptic interfaces$

Christopher D. Leea, Dale A. Lawrenceb,*, Lucy Y. Paoc

aElectrical Engineering, University of Colorado, Boulder, CO 80309-0526, USAbAerospace Engineering, University of Colorado, Boulder, CO 80309-0429, USAcElectrical Engineering, University of Colorado, Boulder, CO 80309-0425, USA

Received 12 April 2003; accepted 15 December 2003

Abstract

This paper describes a multivariable modeling and control strategy that increases the bandwidth of isotropic force transmission in

multi-degree of freedom haptic interfaces. The controller structure leads to a straightforward model identification procedure and

yields a simple control law that can be easily implemented. Experimental results show that the multivariable closed-loop control

significantly improves performance on a five degree-of-freedom in-parallel haptic interface.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Multivariable force control; Haptic interface; Impedance modeling; Isotropic bandwidth

1. Introduction

Haptic interfaces provide forces on a user’s hand inresponse to hand motion. These forces can be actualforces transmitted from a remote location, as inteleoperation applications, or forces produced by amodel in a computer, as in virtual reality applications.An ideal haptic interface would have a large range ofmotion in multiple degrees of freedom, and accuratetransmission of desired forces from the remote or virtualenvironment to the user.Forces due to interface dynamics, such as mass,

friction, backlash, and cogging, mask transmittedforces, reducing the transparency of the interface inrendering desired forces. Requirements for a transparenthaptic interface depend on human haptic sensitivity,since forces below sensory thresholds do not contributeto loss of transparency. Sensitivity to these dynamicforces is a complex issue, and little psychophysical datais available (Lawrence et al., 1998). The existing datadoes suggest that a truly transparent interface would beextremely difficult to achieve. The work of Westling and

was supported in part by the National Science

-9711936 and HRD-0095944) and the Office of Naval

14-97-1-0354).

ng author. Tel.: +1-303-492-3025; fax: +1-303-492-

sses: [email protected] (C.D. Lee), dale.lawrence@

.A. Lawrence), [email protected] (L.Y. Pao).

front matter r 2004 Elsevier Ltd. All rights reserved.

engprac.2003.12.020

Johansson (1987) and Bolanowski, Gescheider, Verrillo,and Checkosky (1998) shows that extremely smallvibrations can be detected, with sensitivity increasingto a maximum at about 300 Hz: This suggests that atransparent interface must produce accurate forces onthe fingers up to a bandwidth exceeding 300 Hz; withlarge range of motion, and in multiple degrees offreedom. Such performance from a mechanical system isdaunting, even in one degree of freedom (Lee, Lawrence,& Pao, 2000) or with a small range of motion (Berkel-man & Hollis, 1998). This is due to low stiffness loadpaths from actuator to fingers inherent in movinglinkages. Together with the linkage mass, this resultsin low-frequency structural dynamics which make highgain (high bandwidth) feedback control difficult toachieve (Eppinger & Seering, 1987). However, there areseveral factors mitigating the high bandwidth require-ment: force direction sensing may have lower sensitivityat high frequencies (Srinivasan, Whitehouse, &LaMotte, 1990; Kontarinis & Howe, 1995); the handgrip (e.g., a stylus) filters rendered forces (Lederman,Klatzky, Hamilton, & Grindley, 2000); visual cues candominate haptic sensing (Wu, Basdogan, & Srinivasan,1999); and the application may not require high-frequency force rendering (e.g., no hard virtual surfaces,as occurs in scientific visualization (Lawrence, Lee, Pao,& Novoselov, 2000a)).The authors’ experience in the development of a

five degree of freedom (DOF), in-parallel interface(see Fig. 1) has indicated that force tracking bandwidths

ARTICLE IN PRESSC.D. Lee et al. / Control Engineering Practice 12 (2004) 1423–14361424

on the order of 10 Hz are sufficient for many tasks,provided this tracking is isotropic, i.e., tracking accu-racy is similar in all degrees of freedom. Otherwise, theinterface has perceptible directional characteristics, andvirtual surfaces do not feel ‘‘real’’ due to unexpectedreaction force directions upon surface contact. Theperceived hardness of the surface is a strong function ofthe force rate of change upon contact (Rosenberg &Adelstein, 1993; Salcudean & Vlaar, 1994; Lawrence,Pao, Dougherty, Salada, & Pavlou, 2000b), whichrequires higher-frequency force transmission. Althoughnot supported by quantitative experiments, it appearsthat significant errors in tracking the magnitude ordirection of these higher-frequency components can betolerated with little perceptual confusion. This suggeststhat isotropic force tracking is only important up tosome mid-frequency range (e.g., about 10 Hz), andconsiderable latitude is permitted in higher-frequencyforce tracking, i.e., high-frequency forces are needed insome (but not all) directions.Force tracking accuracy can be improved by closing a

force loop which includes as much as possible of thehaptic interface, i.e., with force sensors close to theuser’s grasp (Lawrence & Chapel, 1994). This reducesthe loss of force due to moving the mechanism (i.e., dueto mass and damping in the interface), and also reducesthe corruption of force due to cross-coupling in themechanism dynamics (i.e., non-diagonal mass anddamping matrices). Usually, static coupling is removedby proper use of kinematic transformations betweenCartesian and actuator-space force commands. Theforce loop can also attenuate non-linear disturbance

Fig. 1. CU parallel haptic interface with five prismatic actuators

connected to a hand-held stylus.

forces arising from actuator/transmission friction (Leeet al., 2000), cogging, etc.Haptic interfaces can be viewed as robotic manip-

ulators providing forces to the user, and manipulatorforce control has a long history. Conventionally, forcecontrol is applied in Cartesian directions where motionsare constrained by the workpiece, which is usuallyassumed to be rigid. Position control is used in Cartesiandirections where the workpiece does not constrainmotion, leading to the so-called hybrid position-forcemethod (Raibert & Craig, 1981), and its impedancecontrol generalizations, beginning with (Hogan, 1985).In a haptic interface, virtual surfaces are created in asimilar, but complementary way, applying positioncontrol to create surfaces which constrain motion, and(less often) applying force control in directions tangentto surfaces to improve the sense of uninhibited motion.In both application areas, this hybrid effect is oftenaccomplished by an inner/outer loop structure. Anisometric, position-based, or admittance control struc-ture uses a high-gain position loop to make themechanism a position source, insensitive to appliedforces. The outer loop senses forces at the interactionpoint, and is used to increase the mobility or ‘‘soften’’the mechanism only in selected directions. An isotonic,force-based, or impedance control structure provides aninner loop which approximates a force source, and theouter loop provides position control constraints to‘‘stiffen’’ the mechanism only in desired directions. Seee.g., Lawrence (1988), Hayward and Astley (1996),Carignan and Cleary (2000)) for a more detaileddiscussion. We follow the isotonic approach here, usingthe inner force loop to improve the quality of theinterface as a force source. An important distinction,however, is that in a haptic interface the ‘‘workpiece’’ isthe user’s hand, which is neither rigid nor free, andwhose dynamics are a significant factor in stability andperformance of any control loop. Force control forhaptic interfaces has been studied in one DOF(Lawrence & Chapel, 1994; Hayward & Cruz-Hernan-dez, 1998; Avizzano, Barbagli, Frisoli, & Bergamasco,2000; Lee et al., 2000), and in multi-DOF usingconstant, diagonal, high-gain control (Bergamascoet al., 1994; Eom, Suh, & Yi, 2000; Kwon & Woo,2000; Ando, Szemes, Korondi, & Hashimoto, 2002).Here the focus is on non-constant, non-diagonal

multivariable force control to improve the bandwidth ofisotropic force transmission over that which can beobtained with simpler (constant, diagonal) controllers.A relatively simple modeling and control design methodis described, which is well suited to haptic interfaceapplications because it explicitly includes the dynamicsof the user’s hand (in contrast to work in robot forcecontrol), and it results in a control structure which canbe easily gain scheduled (if necessary) to compensatefor kinematic and dynamic changes throughout the

ARTICLE IN PRESSC.D. Lee et al. / Control Engineering Practice 12 (2004) 1423–1436 1425

workspace. This method can be used to improve themid-frequency transparency of any haptic interfacewhich incorporates force sensing.This paper is organized as follows. The haptic

interface used is described in more detail in Section 2.Section 3 outlines the performance measures and controlstrategy. Section 4 discusses a parameter identificationtechnique that yields a model that can be used forinverse model control. Section 5 addresses digitalcontrol implementation issues and presents closed-loopcontrol results.

2. Haptic interface description

The University of Colorado (CU) haptic interface(Figs. 1 and 2) was designed to promote high fidelityforce sensing and transmission to the user in 3 DOF intranslation and 2 DOF in rotation (Lee et al., 2000). Itwas designed to enable accurate force transmission tothe user’s fingers, yet provide adequate range of motionto conduct studies of human haptic perception andhaptic rendering modes, and serve applications inscientific visualization. The user can move the stylusthroughout a spherical translational workspace approxi-mately 40 cm in diameter, and rotate the stylus in aconical workspace of71 radian about the nominal posepictured. This range of motion is compatible with thecomfortable range of a user’s wrist and arm motion witha fixed elbow rest. Each link consists of a thin rodcontrolled by a prismatic friction drive actuator,providing a stiff, no-backlash transmission. Actuatorsare attached to a hexagonal base via two-DOF gimbals.The hand grip is a pen-shaped stylus attached to threeactuator rods on one end and two rods on the other.Each rod attaches to the stylus through a three axisgimbal. Force sensors are located at the tip of each rod,

6

5

P QActuator

Actuator

Actuator

Actuator

3

2

4Actuator

Rod 5Rod 3

Rod 4

Rod 2Rod 6

Stylus

Actuator 1 (not used)

Fig. 2. Diagram of the haptic interface, indicating actuator, rod, and

stylus connections.

near the stylus, to provide feedback of measured axialrod forces to the force control loop. Accurate transmis-sion of sensed forces to the user’s fingers is provided bylight, stiff connections between the stylus and thesesensors. Optical encoders embedded in the actuatorsmeasure rod displacement with a Cartesian resolution of7 mm: A DSP computer running between 1 and 2 kHzloop rates (depending on the application) implementsthe force control law, in addition to performing therequired kinematic transformations and data interpola-tion to render desired effects as the user investigatesdata, e.g., for scientific visualization of vector fields(Lawrence et al., 2000a).

2.1. Force control approach

There is a natural separation of the haptic interfacedynamics into a component representing the interfacemechanism itself, and a component representing theuser’s hand. Fig. 3 shows this partition for a single rodcase, as well as the location and sign convention of rodactuator forces RA; sensed (mechanism/hand interac-tion) rod forces RI ; voluntary user forces RV ; andmeasured rod length L: Voluntary forces result from theuser’s active positioning of the stylus, which areindependent inputs to the overall system. ZM is themechanical impedance looking into the mechanism, andZH is the impedance looking into the hand at theinteraction point where forces are sensed.In the multiple-rod case, the hand interacts with the

measured forces through a stylus and rod tip gimbals,which induce an unusual partition of mechanism andhand dynamics that is expedient for model identifica-tion. Here, the multivariable Mechanism impedance,ZM ; includes all hardware up to the force sensors,whereas the Hand impedance, ZH ; includes all dynamicsoutboard of the force sensors, i.e., the stylus, rod tipgimbals, and user’s hand lumped together.Models of these impedances are developed by

parameterizing effective mass, stiffness, and dampingin each component. Although the actual impedances areundoubtedly more complex than this, it is shown thatthese models can capture the dominant multivariable

ZM

ZH

RVRA

Force sensor

RodActuator

RI

Hand

L

Fig. 3. Hand/mechanism impedance partition, actuator force RA;measured rod axial force RI ; voluntary force RV ; and rod length L for

a one-rod case.

ARTICLE IN PRESS

G Z 1M ZH

R D R E L R H

R V

R IR A

++

−

− −

−−

Fig. 4. Overall force control structure with separate Hand and

Mechanism impedances.

C.D. Lee et al. / Control Engineering Practice 12 (2004) 1423–14361426

dynamics, and this leads to improved control ofrendered forces compared to separate single-input-single-output force control on each rod.Fig. 4 shows the 5-DOF force control loop resulting

from the attachment of the five rods to the stylus andhand. Forces and motions are described not in Cartesiancoordinates, but along the five moving rod axes. G is theforce controller, whose design is the focus of this paper.This force loop is designed to cause the five-tuple of

measured rod interaction forces, RI ; to track the fiveforce loop commands (desired forces) RD; using themultivariable control law G; which outputs five actuatorforces, RA: The RI are measurements of interactionforces along each rod’s long axis. Motion of theMechanism causes changes in the distance between eachactuator and the end of the rod. The 5-tuple of these rodlengths is denoted L: Changes in L cause a reaction forcefrom the Hand, denoted RH ; due to the impedance ofthe Hand, ZH : This reaction force and the voluntaryuser force RV combine to form the measured interactionforce RI : In turn, the motion L results from the sum ofactuator forces RA and interaction forces RI acting onthe Mechanism admittance Z�1

M : The Mechanismdynamics consist primarily of mass and friction ele-ments, which cause significant differences between theapplied actuator forces RA and the forces RI applied tothe Hand. Friction dominates at low frequencies, and isprimarily Coulombic. At high frequencies, much of theactuator force is diverted into accelerating the Mechan-ism mass. Provided the force loop is stable, high loopgains over suitable bandwidths improve the tracking ofdesired forces RD by measured forces RI ; reducing theperceived friction and mass of the interface.Although the Coulomb friction in the interface is non-

linear, Laplace transform representations of signals inFig. 4 are employed. The implication is that the resultinglinear model impedance ZM relating measured motionsand applied forces is an approximation to the non-lineardynamics, and this approximation may vary withoperating conditions. For example, effective dampingin the interface will be a function of motion amplitudeused to obtain modeling data. Furthermore, ZM willvary (slowly) with location and pose in the workspace,due to kinematic non-linearities. In other words, ZM

should be thought of as belonging to a set of transferfunctions, which cover the range of variations encoun-

tered in operation. Actually, ZH must be considered aset of functions also, due to variations among user handsizes, in stylus grip force, and in location/pose of thehand in the workspace. Control design based on thesemodels must therefore be robust to these variations.Guaranteed robustness is difficult to provide, since itrequires bounds on model uncertainty (which areunknown), or bounds on model variation (which oftenrequire extensive test data). Practically, this problem isoften solved by designs which provide sufficient stabilitymargins, using representative members of the model set,and by thorough testing of the resulting design undertypical operating conditions. This practical approach isfollowed here.Under the condition that the user does not supply any

voluntary forces (i.e., RV ¼ 0), and with the definition ofthe Total interface impedance

ZT ðsÞ ¼ ZMðsÞ þ ZH ðsÞ ð1Þ

the following multivariable Laplace transform relationsare obtained between signals in the block diagram ofFig. 4:

RI ðsÞ ¼ �ZH ðsÞLðsÞ;

RI ðsÞ � RAðsÞ ¼ ZM ðsÞLðsÞ;

RAðsÞ ¼ �ZT ðsÞLðsÞ ¼ ZT ðsÞZ�1H ðsÞRI ðsÞ: ð2Þ

These definitions lead to a 5� 5 ‘‘plant’’ transferfunction matrix PðsÞ relating actuator forces RA andmeasured interaction forces RI :

RI ðsÞ ¼ PðsÞRAðsÞ;

PðsÞ ¼ ZH ðsÞZ�1T ðsÞ: ð3Þ

The decomposition into a Mechanism and Handimpedance, and the corresponding modeling and controlmethods discussed here, are applicable to any hapticinterface, whether parallel, serial, or serial-parallel indesign. Ideally, the force sensing should be located asclose to the user’s fingers as possible, so that themeasured forces closely represent those felt by the user,and so that the force loop causes these to track thedesired forces.

3. Performance measures and control design strategy

3.1. Control performance measures

The goal of the force controller is to increase thebandwidth of isotropic force tracking. Here multi-variable, or multiple-input multiple-output (MIMO)control is investigated, since independent single-inputsingle-output (SISO) control on each rod is extremelylimited in its ability to affect isotropy. The forcetransmission frequency response TðjoÞ relating desiredforces RD to measured interaction forces RI can be

ARTICLE IN PRESS

100

101

102

0

0.5

1

1.5

2

2.5

Frequency, Hz

Mag

nitu

de

Fig. 5. Singular values of ðTðjoÞ � IÞ under an SISO control design

(Lee et al., 2000).

C.D. Lee et al. / Control Engineering Practice 12 (2004) 1423–1436 1427

derived from the block diagram in Fig. 4 and Eq. (3):

RI ðjoÞ ¼ TðjoÞRDðjoÞ;

TðjoÞ ¼ ðIþ PðjoÞGðjoÞÞ�1PðjoÞGðjoÞ:

Ideally, TðjoÞ ¼ I; so that the user feels the desired forceRD in all degrees of freedom and at all frequencies.While studying the magnitude of the scalar open-loop

frequency response jPðjoÞGðjoÞj is sufficient for asses-sing tracking performance in SISO systems, it isnecessary in MIMO systems to consider the singularvalue decomposition (SVD) of the loop gain matrixPðjoÞGðjoÞ:

PðjoÞGðjoÞ ¼ URV� ¼ U

s1 0 ? ? 0

0 s2 & ^

^ & & & ^

^ & & 0

0 ? ? 0 sn

26666664

37777775V�;

where U and V are unitary matrices, � denotes theconjugate transpose, and R is a diagonal matrix of thesingular values si; with s1Xs2X?XsnX0 (Strang,1988). At frequencies o where the loop gain is large, i.e.,when all sib1;

TðjoÞEðPðjoÞGðjoÞÞ�1PðjoÞGðjoÞ ¼ I ð4Þ

and the actual interaction force vector RI will closelytrack the desired force vector RD in both magnitude anddirection. At frequencies o where the loop gain is small,i.e., all si51;

TðjoÞEPðjoÞGðjoÞ ð5Þ

and the actual interaction forces RI will not track thedesired forces RD; being smaller in magnitude and(often) in other directions.TðjoÞEI can sometimes be achieved (typically only at

low frequencies) by designing the controller GðjoÞ suchthat the minimum singular value

%sðPðjoÞGðjoÞÞ � snb1

so that TðjoÞEI: At higher frequencies, the maximumsingular value %sðPðjoÞGðjoÞÞ necessarily becomes smal-ler to ensure stability, hence so does

%sðPðjoÞGðjoÞÞ:

When the condition number %sðPðjoÞGðjoÞÞ=

%sðPðjoÞGðjoÞÞ is large, the minimum and maximumsingular values are widely separated, resulting in a widevariation in loop gain in various directions. This cancause either relatively poor force tracking in somedirections or poor stability margins, or both.Therefore, the first step toward isotropic force

tracking is to reduce the condition number ofPðjoÞGðjoÞ; particularly in the neighborhood of the0-dB magnitude crossover frequency. This extends thebandwidth of isotropic force tracking, and simulta-neously promotes better stability margins. However, thecondition number criterion is not sufficient: even if thecondition number of T is 1 (a unitary matrix), T maystill cause vector rotation if the eigenvalues of T are

complex. Instead it will be specified that %sðTðjoÞ � IÞ issuitably small, since this bounds the magnitude of theerror between a desired force vector RD and its renderedvalue RI at each frequency. To be specific, thebandwidth of isotropic force tracking is defined hereas the maximum frequency where %sðTðjoÞ � IÞ is lessthan 1� 1=

ffiffiffi2

pE0:293 (analogous to the 3 dB band-

width definition for scalar systems). The quantity T� Iis called the sensitivity of the closed loop, and itsminimization is often thought of as the primary goal inoptimization-based MIMO control approaches (see e.g.,Vidyasagar, 1985; Chiang & Safonov, 1992). Hereoptimization is avoided in the interest of maintaining asimple controller, and in retaining some insight into itsparameters.This performance criterion is novel in the field of

haptic interfaces, and is rather stringent. Consider itsapplication to a previously designed control system (Leeet al., 2000) which used an SISO approach. In thisexample, a force loop was designed for each actuatorrod independently, providing a closed loop trackingbandwidth of about 100 Hz in each case. Fig. 5 showsthe result of applying each of these control systemssimultaneously to the haptic interface, and measuringperformance according to the isotropic criterion. Notethat the vertical axis is magnitude (not in dB). Theisotropic force transmission bandwidth (as definedabove) is not visible on this plot, since tracking in somedirections is poor over all frequencies shown. At thesame time, tracking in other directions is significantlybetter, with approximately 10 Hz force tracking band-width, according to the 3 dB criterion, and less accuratetracking up to about 100 Hz: This corresponds with thefeel of the interface which is better in some directionsthan others, e.g., virtual surfaces in some orientationsfeel harder, with less-directional distortion in contact


forces. Clearly, SISO performance measures can fail tocapture the weakening of loop gain in some Cartesiandirections caused by dynamic coupling. As an aside, theextreme peaking in some singular values in Fig. 5 near100 Hz was not noticed by users, presumably due to therather poor human force direction sensing capability athigh frequencies, combined with the poor force render-ing in this case at low frequencies. One could alsoquestion how voluntary motion could elicit forcecommands at the higher frequencies (e.g., above100 Hz). This is difficult indeed in linear renderedmedia, but high-frequency forces are an essential partof the ‘‘crisp’’ feeling of good virtual surfaces, whereforces must rise rapidly as the surface is penetrated(Rosenberg & Adelstein, 1993; Salcudean & Vlaar, 1994;Lawrence et al., 2000b).As applied here, the isotropic performance measure

also relies on force measurements close to the user’shand, providing a more accurate estimate of the forcesperceived by the user than a criterion based on actuatorforce commands would provide. This paper uses theisotropic criterion to pursue improved control designsfor haptic interfaces. A particular hybrid MIMO/SISOapproach is taken to design multivariable controllers,first ‘‘compressing’’ loop gain singular values togetherby approximately inverting the MIMO plant, thendesigning scalar compensation according to a ‘‘classi-cal’’ approach applied to loop transfer function eigen-values. Tracking performance is improved by increasingeigenvalue magnitudes at low frequencies. Stability ofthe force loop is assessed using the multivariableNyquist criterion (MacFarlane & Postlethwaite, 1977),which uses the loop eigenvalues instead of scalar loopgain, counting encirclements of the critical point in asimilar way as for scalar systems. Gain and phasemargins can be similarly inferred, provided that theeigenvectors are approximately orthogonal.It is important to mention that a stable force loop

may not be sufficient when the system contains an outerimpedance loop, since the overall system may bedestabilized by this outer loop. Fig. 6 shows a typicalsituation where the desired force RD is a function ofhaptic interface motion L: Here the interface isattempting to render a desired impedance ZD; and theforce loop produces a closed (inner) loop impedanceZCLðjoÞ ¼ Z�1

H ðjoÞTðjoÞ (see Fig. 4). Note that in theideal case where TðjoÞ ¼ I; RI ¼ RD ¼ ZDðLD � LÞ; and

LD RD

ZD Z −1CL

L

+

−

Fig. 6. An isotonic impedance loop can be built around the inner force

loop, in which Z�1CL ¼ Z�1

H T:

the user feels only ZD: That is, the Mechanismimpedance ZM is suppressed by the force control loop,resulting in a perfectly transparent interface. Short ofthis ideal, improvements in isotropic force trackingincrease the fidelity of the rendered ZD; provided theouter loop is stable, with adequate margins. Stability ofthis loop can also be assessed using the multivariableNyquist criterion. Other criteria, e.g., passivity (Adams& Hannaford, 1999) could also be applied. However, thefocus is on the inner force loop in the remainder of thispaper.

3.2. Control design strategy

Given the objective of ‘‘compressing’’ the singularvalues of the loop frequency response, an inverse modelcontrol strategy is pursued. If #ZH ðsÞ and #ZT ðsÞ aremodels of the Hand and Total impedances, respectively,then an MIMO controller can take the form

RAðsÞ ¼ GðsÞREðsÞ

with

GðsÞ ¼ gðsÞ #ZT ðsÞ #Z�1H ðsÞ ¼ gðsÞ #P�1ðsÞ; ð6Þ

where #P�1ðsÞ is the (inverse) plant model and gðsÞ is ascalar compensator used to further shape the frequencyresponse by increasing low-frequency gain (to improveforce tracking) and providing roll-off at higher frequen-cies (to provide stability robustness). To the extentthat the plant model matches the plant, the singularvalue magnitudes of PðsÞ #P�1ðsÞ will be compressed intoa narrow band around 0 dB; allowing force trackingto be similarly effective in all directions. The second-order polynomial forms used for the impedances in#P (see below) are well suited to model inversion,since the resulting model has relative degree zero, andthe inverse is realizable without large high-frequencygains.

4. Impedance modeling

The goal of this section is to obtain a parametricmodel of the plant transfer function of the form

#PðsÞ ¼ #ZH ðsÞ #Z�1T ðsÞ: ð7Þ

To this end, separate Mechanism and Hand impedancemodels will be sought first. Previous work (Lee,Lawrence, & Pao, 2002) suggests that a second-orderlinear model of the Mechanism provides considerableaccuracy in modeling the non-Coulomb friction portionof the Mechanism dynamics at low frequencies. Themodel is of the form

#ZM sð Þ ¼ #MMs2 þ #BMs þ #KM :

ARTICLE IN PRESS

2The superscript in RmIc; m ¼ 2;y; 6; is used to index rod number


The feasibility and performance of Hand and Totalimpedance models of the same form will be investigated:

#ZH ðsÞ ¼ #MHs2 þ #BHs þ #KH ;

#ZT ðsÞ ¼ #ZH ðsÞ þ #ZM ðsÞ ¼ #MT s2 þ #BT s þ #KT : ð8Þ

The mass matrices #MH and #MT ; damping matrices #BH

and #BT ; and stiffness matrices #KH and #KT will beobtained in Section 4.2 using a least-squares fitting offrequency domain data and Eq. (2).

4.1. Hand and total impedance frequency response data

The method for obtaining frequency response data issimilar for the Hand and Total impedances. For brevity,only the Hand will be discussed. In the data collectionexperiment described below, the user was instructed tomaintain a comfortable grip on the stylus, with a relaxedhand and arm pose. Since the intent is to obtain modelsof the natural impedance of the hand, user efforts(voluntary forces) to actively position the stylus or reactto interface motions were to be avoided. Control designwas based on data taken from a single user, at onelocation and pose near the center of the haptic interfaceworkspace.The relationship at a frequency o between measured

rod forces RI and measured rod displacements L in theabsence of voluntary user force RV is the Handimpedance at that frequency

RI ðjoÞ ¼ �ZH ðjoÞLðjoÞ:

Let ZH ðjoÞ ¼ ½zijðjoÞ:1 To obtain the scalar frequency

responses of the zij at a set of frequencies½o1;o2;y;oN ; the system must be excited at thosefrequencies. Let RAðtÞ denote the 5-tuple of actuatorinput forces at time t; and consider the following five setsof actuator inputs over a time period t; to be used in aset of five data collection experiments

RA1 ðtÞ ¼ rf ðtÞ½�1 1 1 1 1T;

RA2 ðtÞ ¼ rf ðtÞ½1 � 1 1 1 1T;

RA3 ðtÞ ¼ rf ðtÞ½1 1 � 1 1 1T;

RA4 ðtÞ ¼ rf ðtÞ½1 1 1 � 1 1T;

RA5 ðtÞ ¼ rf ðtÞ½1 1 1 1 � 1T; ð9Þ

where rf ðtÞ is a pseudo-random signal

rf ðtÞ ¼XN

k¼1

bk cosðokt þ akÞ

and the ak are uniformly distributed random phases inthe interval 0; 2p½ ; and the component amplitudes bk areselected to produce an adequate signal to noise ratio inthe measured data. If N is the number of time samplesduring time t; i.e., t ¼ Nts with ts being the sample

1The subscript index j should not be confused with the imaginary j:

period, then the frequencies ok must be integer multiplesof 2p=t to avoid frequency sampling errors in the fastfourier transform (FFT). For each experiment (forc ¼ 1;y; 5), by taking the FFT of the measured rodforces Rm

IcðjoÞ and rod2 lengths Lm

c ðjoÞ; m ¼ 2;y; 6; oneobtains, at each frequency o

RIcðjoÞ ¼

R2IcðjoÞ

R3IcðjoÞ

R4IcðjoÞ

R5IcðjoÞ

R6IcðjoÞ

266666664

377777775¼ �ZH ðjoÞ

L2cðjoÞ

L3cðjoÞ

L4cðjoÞ

L5cðjoÞ

L6cðjoÞ

266666664

377777775

) RIcðjoÞ ¼ �ZH ðjoÞLcðjoÞ:

By concatenating these results by column into matrixform, one obtains

RI ðjoÞ ¼ �ZH ðjoÞLðjoÞ; ð10Þ

where

RI ðjoÞ ¼ ½RI1ðjoÞ RI2ðjoÞ ? RI5ðjoÞ;

LðjoÞ ¼ ½L1ðjoÞ L2ðjoÞ ? L5ðjoÞ

are both 5� 5 matrices at each frequency o: Hence, theempirical (measured) impedance ZH ðjoÞ can be found as

ZH ðjoÞ ¼ �RI ðjoÞLðjoÞ�1

provided the columns of L are linearly independent. Inexperiments, choice (9) provided sufficiently indepen-dent columns in L: Other choices are possible, but theparticular choice of RAcðtÞ affects the quality of thefrequency response: if one of the rods is not sufficientlyactuated, data will be corrupted by quantization noise,friction, etc. This is most likely to occur at highfrequencies when the amplitude of motion is reducedand friction may dominate. This deficiency was ob-served, for example, when only one rod was excited at atime. This caused so little motion in some rods that theempirical impedance was highly inaccurate.The model for ZT can be constructed in a similar

manner

ZT ðjoÞ ¼ �RAðjoÞLðjoÞ�1;

where RAðjoÞ ¼ ½RA1ðjoÞ RA2 ðjoÞ ? RA5ðjoÞ is thecommanded actuator force vector. Although notdirectly needed in this work, the Mechanism impedancecan also be found from

ZMðjoÞ ¼ ðRI ðjoÞ �RAðjoÞÞLðjoÞ�1:

These frequency domain matrices are now parameter-ized to extract mass, spring, and damping matrixdescriptions of the Hand and Total impedances.

and is not a power exponent. In this work, actuator 1 (and hence

m ¼ 1) is not used—see Fig. 2


4.2. M;B;K Model of ZH and ZT

The identification of the mass, spring, and dampingmatrices for the Hand and Total impedances eachfollow the same method. Only the Hand will bediscussed, but results will be shown for both the Handand Total impedances.For each element zijðjoÞ of the Hand impedance ZH ; a

parameterized model of the form

#zijðjoÞ ¼ ½ðjoÞ2 jo 1

#mij

#bij

#kij

2664

3775

is sought. Stacking these equations by frequency, oneobtains

#Zij ¼

#zijðo1Þ

#zijðo2Þ

^

#zijðoNÞ

26664

37775 ¼

ðjo1Þ2 jo1 1

ðjo2Þ2 jo2 1

^ ^ ^

ðjoN Þ2 joN 1

266664

377775

#mij

#bij

#kij

2664

3775;

which can be written as

#Zij ¼ /�hij ;

where ‘‘�’’ denotes the conjugate transpose. The realparameter 3-tuple hij ¼ ½ #mij ; #bij ; #kijT can be found bysolving the least-squares problem

#hij ¼ arg minðZij � /�hijÞ�ðZij � /�hijÞ

where Zij is the N-tuple of measured impedances zijðjoÞ:Although the magnitude of the frequency response ofeach Zij is relatively constant between 0 and 7 Hz; itthen follows an upward trend at a rate of roughly40 dB/decade (see Fig. 7). The plain least-squaresminimization thus weights errors at the higher frequen-cies heavily, since the magnitude is larger at the higherfrequencies. This causes greater modeling error at low

100

101

102

−50

−40

−30

−20

−10

0

10

20

30

40

50

60

Frequency, Hz

Mag

nitu

de, d

B

Fig. 7. Singular values of ZH (solid) and #ZH (dashed).

frequencies. In order for the model to be equallyaccurate at all frequencies of identification, the mea-sured data and the information matrix / will beweighted with a simple model of the inverse of thefrequency response of the data:

hðjoÞ ¼1

ðjoþ 7 � 2pÞ2:

The weighted data is

*Zij ¼

hðo1Þzijðo1Þ

hðo2Þzijðo2Þ

^

hðoN ÞzijðoN Þ

26664

37775

*/� ¼

hðo1Þðjo1Þ2 hðo1Þjo1 hðo1Þ1

hðo2Þðjo2Þ2 hðo2Þjo2 hðo2Þ1

^ ^ ^

hðoN ÞðjoN Þ2 hðoN ÞjoN hðoN Þ1

266664

377775:

The modified minimization problem can be written as

*hij ¼ arg minð *Zij � */� *hijÞ�ð *Zij � */� *hijÞ: ð11Þ

The least-squares solution to this modified problem is(see e.g., Ljung, 1987)

*hij ¼ ðRealð */ */�ÞÞ�1Realð */ *ZijÞ:

Parameter 3-vectors *hij are found in this way for each ofthe 25 elements of the 5� 5 ZH matrix. To obtain amodel for the total impedance ZT ; this process isrepeated, with Zij derived from the elements of theempirical ZT data matrix at each frequency. Althoughthere are a large number of parameters, the process iseasily automated and involves only a 3� 3 matrixinverse for each impedance element.

4.3. Modeling results

ZH and ZT were modeled using the method abovewith excitation at the 28 frequencies 1; 2; 3;y; 9;10; 15; 20;y; 100 Hz: The singular values of ZH andits model #ZH are shown in Fig. 7. Those for ZT and itsresulting model #ZT are shown in Fig. 8. It can be seenfrom these plots that the impedances are not those ofstrict second-order matrix polynomials, since the modelfits cannot be equally good at both low and highfrequencies. In the interest of capturing a good low-frequency model for control design, the models are fitclosely to the data up to about 50 Hz; but are allowed todeviate significantly at higher frequencies. Another viewof model accuracy is given by the singular values ofPðjoÞ � #P�1ðjoÞ; where PðjoÞ is the plant measured at the28 frequencies above. These singular values ideallywould all be 1: Fig. 9 shows that these singular valuesoccupy a magnitude band within 710 dB of 0 forfrequencies up to 50 Hz; and within 77 dB for

ARTICLE IN PRESS

100

101

102

−40

−20

0

20

40

60

80

Frequency, Hz

Mag

nitu

de, d

B

Fig. 8. Singular values of ZT (solid) and #ZT (dashed).

100

101

102

−50

−40

−30

−20

−10

0

10

20

Frequency, Hz

Mag

nitu

de, d

B

Fig. 9. Singular values of P � #P�1:


frequencies between 2 and 20 Hz: The plant PðjoÞ alonehas a singular value span ranging from 25 to 50 dB forfrequencies up to 30 Hz; hence this model inversioncomponent of the controller has reduced anisotropy bya factor ranging from 5 to 36 dB over this frequencyrange.

5. Control implementation

The controller takes the form given in Eq. (6). Beforeaddressing the scalar compensator design gðsÞ; twoimplementation issues related to the inverse modelcomponent #P�1ðsÞ will be discussed. First, model right-half-plane poles and zeros may appear from the leastsquares parameter matching; second, the continuousmultivariable inverse model must be discretized forimplementation.

5.1. Removing right-half-plane poles and zeros

Since the Hand and Total impedances are realphysical systems, their impedance realizations #ZH ðsÞand #ZT ðsÞ should have zeros in the complex open lefthalf-plane (OLHP). Unfortunately, due to noise in thedata measurements, the least-squares algorithm mayreturn impedance models with zeros in the complexclosed right half plane (CRHP). From (6), CRHP zerosof #ZH ðsÞ cause GðsÞ to be unstable. This is undesirablesince it makes the controller implementation difficult todebug: a frequency response of the controller cannot beobtained before the loop is closed. CRHP roots of #ZT ðsÞcause GðsÞ to be non-minimum phase and are alsoundesirable since they induce phase loss with an increasein magnitude, limiting closed-loop bandwidth (Skoges-tad & Postlethwaite, 1996). These two issues can oftenbe remedied by ‘‘positizing’’ the mass, spring, anddamping matrices of #ZH ðsÞ and #ZT ðsÞ as follows. IfVDV�1 is the eigenvector/eigenvalue decomposition of amatrixM; thenMP is defined to be the positized versionof M as follows:

MP ¼ VjDjV�1;

where j � j is the absolute value of matrix elements. Notethe eigenvectors and eigenvalue magnitudes of theparameter matrices are left unchanged.Once the terms of the impedance model have been

positized, the roots of the impedance models #ZHðsÞ and#ZT ðsÞ are checked for the presence of remaining CRHProots. Positizing normally is sufficient, but if CRHProots are found, one can return to the original matrices,symmetrize them, and then positize them. This willresult in positive definite mass, spring, and dampingmatrices, ensuring OLHP roots (Bellman, 1970). Un-fortunately, it was found that symmetrization cansignificantly reduce the accuracy of the model #P byaltering the eigenvectors and the magnitudes of theeigenvalues, and should be avoided if possible.

5.2. Discrete-time implementation

To implement the multivariable analog controller inEq. (6), define

yðsÞ ¼ #Z�1T ðsÞRAðsÞ ¼ #Z�1

H ðsÞgðsÞREðsÞ:

Then, using the plant models from Eq. (8),

gðsÞREðsÞ ¼ #MHs2yðsÞ þ #BHsyðsÞ þ #KHyðsÞ

) s2yðsÞ ¼ gðsÞ #M�1H REðsÞ � #M�1

H#BHsyðsÞ

� #M�1H

#KHyðsÞ

and

RAðsÞ ¼ #MT s2yðsÞ þ #BT syðsÞ þ #KTyðsÞ:

Fig. 10 shows the continuous-time version of thecontroller.

ARTICLE IN PRESS

g(s)RE

RA

ysys2y

M̂ 1H

M̂ 1H B̂ H

M̂ −

−

−

1H K̂ H

K̂T

B̂T

M̂ T

+

++

+

−−

1s

1s

Fig. 10. Continuous-time implementation of the plant inverse model

controller.


A simple way to find a discrete-time equivalent is toreplace the integrators 1=s with the forward differencetsz=ðz � 1Þ; as shown in Fig. 11. One could also use thebackward difference ts=ðz � 1Þ; but this would induceextra delay, reducing the closed-loop stability margin.Unfortunately, the use of undelayed integrators pro-duces algebraic loops in Fig. 11. These can be avoidedby computing the closed loop relationship between y1and RA:

y5ðkÞ ¼ y5ðk � 1Þ þ tsy4ðkÞ

) y5ðkÞ � tsy4ðkÞ ¼ y5ðk � 1Þ;

y4ðkÞ ¼ y4ðk � 1Þ þ ts#M�1

H ½y1ðkÞ

� #BHy4ðkÞ � #KHy5ðkÞ

) y4ðkÞ þ ts#M�1

H#BHy4ðkÞ þ ts

#M�1H

#KHy5ðkÞ

¼ ts#M�1

H y1ðkÞ þ y4ðk � 1Þ:

It is expedient to place these equations into matrix form

y5ðk � 1Þ

ts#M�1

H y1ðkÞ þ y4ðk � 1Þ

" #

¼I �tsI

ts#M�1

H#KH Iþ ts

#M�1H

#BH

" #y5ðkÞ

y4ðkÞ

" #:

Defining the (constant) 10� 10 matrix A as

A ¼I �tsI

ts#M�1

H#KH Iþ ts

#M�1H

#BH

" #�1

allows one to write

y5ðkÞ

y4ðkÞ

" #¼ A

y5ðk � 1Þ

y4ðk � 1Þ

" #þ A

0

ts#M�1

H

" #y1ðkÞ:

From Fig. 11, the actuator force is then computed from

RAðkÞ ¼ #MTy3ðkÞ þ #BTy4ðkÞ þ #KTy5ðkÞ;

where y3ðkÞ is given by

y3ðkÞ ¼y4ðkÞ � y4ðk � 1Þ

ts

:

The design and discretization of gðsÞ to obtain gðzÞ isdiscussed in Section 5.3.

Note that any model found with the method abovehas the same structure. Thus, if Hand or Mechanismdynamics vary sufficiently throughout the workspace, itmay be necessary to develop a suite of models andcorresponding controllers, and gain schedule these as afunction of position in the workspace. Having identicalstructure, the suite of controllers can be gain scheduledby simple parameter interpolation of the A; #M�1

H ; #MT ;#BT ; and #KT matrices. Stability of the gain schedulingscheme is not a trivial issue, since the system has anadditional non-linear dependence on location in theworkspace. However, gain scheduling often works wellwhen the parameter variations are slow enough. This isan area of ongoing research, e.g., Apkarian and Gahinet(1995). In the present case, a single controller usingvalues at the center of the workspace yielded goodresults in all but the edges of the workspace. In aninterface which has a larger range of motion, or moresevere kinematic variations, gain scheduling may berequired.

5.3. Scalar compensator design

Once the goal of singular value ‘‘compressing’’ isachieved with the inverse model controller, it remains toshape the open-loop frequency response to provide highgain at low frequencies to improve force tracking, androll-off at the higher frequencies to subdue unmodeleddynamics. A ‘‘classical’’ design approach is employed,which examines the loop gain eigenvalues in Bode andNyquist plot formats, applying compensating filters toshape the eigenvalues over frequency in a similarmanner as is done for the scalar loop gain in SISOsystems.The scalar gðsÞ factor was chosen to include a gain,

three notch filters, a lag filter, and two first-order low-pass filters according to the following design strategy.The overall goal is to increase the loop gain as much aspossible, subject to stability constraints and uncertaintycaused by imperfect plant inversion. This improves forcetracking fidelity at frequencies where the loop gainsingular values are larger than 1. To this end, a lag filterwith a zero at 30 Hz and a pole at 1 Hz is used to boostthe low-frequency loop gain.The plant contains high-frequency dynamics which

are not well modeled (and hence not inverted) by theMIMO controller, which can be seen in the widesingular value spread in P #P�1 above 50 Hz in Fig. 9.These dynamics produce large excursions in loop gaineigenvalue magnitude and phase, and can lead toinstability (encirclements of the �1 point in the multi-variable Nyquist plot). A standard approach is to ‘‘‘gainstabilize’’ unmodeled dynamics above the desired closedloop bandwidth by rolling off the loop gain at these highfrequencies. Low-pass filters are introduced at 100 and200 Hz for this purpose.

ARTICLE IN PRESS

g(z)RE

RA

y1 y2 y3 y4y5

M̂ −

−

−

1H

M̂ 1H B̂ H

M̂ 1H K̂ H

K̂ T

B̂ T

M̂ T

+

++

+

− −

t s zz − −1

t s zz 1

Fig. 11. Direct discrete-time implementation of the plant inverse model controller.


Similarly, the MIMO controller only approximatelyinverts the plant at lower frequencies, resulting inundesirable peaking in the loop gain eigenvalues atintermediate frequencies. While this situation could beimproved by using a more complex MIMO plant model#P�1; additional compensation can also be provided bythe scalar gðsÞ: Attempting to keep controller complexityat a minimum, the latter approach was followed,introducing three shallow notch filters in gðsÞ at 17, 55,and 100 Hz to reduce corresponding peaks in the loopgain eigenvalues.The scalar controller’s transfer function is thus

gðsÞ ¼ ks2 þ 2z1o1s þ o21s2 þ 2z2o1s þ o21

� s2 þ 2z1o2s þ o22s2 þ 2z2o2s þ o22

�

�s2 þ 2z1o3s þ o23s2 þ 2z2o3s þ o23

� s þ z1

s þ p1

�

�p2

s þ p2

� p3

s þ p3

� ;

where

k ¼ 0:85;

o1 ¼ 100 � 2p rad=s;

o2 ¼ 55 � 2p rad=s;

o3 ¼ 17 � 2p rad=s;

z1 ¼ 0:3;

z2 ¼ 1:0;

z1 ¼ 30 � 2p rad=s;

p1 ¼ 1 � 2p rad=s;

p2 ¼ 100 � 2p rad=s;

p3 ¼ 200 � 2p rad=s:

This continuous scalar compensator is discretized bypole-zero matching using z ¼ ests and matching DCgains (see e.g., Franklin, Powell, & Workman, 1990).Note that a zero-order hold has not been added to eitherof the discrete filters making up the controller, becausethe plant model already contains the effects of the D/Aconverter in its measured data.

5.4. Design results

When the controller GðjoÞ is applied to the measuredplant frequency response PðjoÞ; the predicted loopfrequency response (see Figs. 12 and 13) shows a gainmargin of about 6 dB and a phase margin of about 35degrees. Note the tight grouping of the loop gaineigenvalues in magnitude up through the 0 dB cross-over, and the general 20 dB/decade reduction of loopgain, which provides large loop gain at low frequencies,and suppression of unmodeled dynamics at highfrequencies. The phase plot is well grouped up to30 Hz; but 360 degree phase ‘‘wrap-arounds’’ make thisdifficult to assess above 30 Hz: These design results areonly predictors of performance, since non-linearities inthe plant may produce a different plant model whenloop gain is identified in the presence of this particularcontroller. Measured behavior is discussed in the nextsection.

5.5. Measured results

This section presents measured results after imple-menting and applying the above controller design on the5 DOF haptic interface (shown in Fig. 1). Figs. 14 and15 show the Bode and Nyquist plots for the open-loopfrequency response, measured as the relation betweenforce errors RE to measured forces RI in the hardwareimplementation. The achieved gain margin is about10 dB; and the phase margin is about 40 : The maindifference from the predicted loop gain (Figs. 12 and 13)is the somewhat larger eigenvalue magnitude spread,and what appears to be a slight improvement in stabilitymargins. Apparently, non-linearities do not have asignificant effect on the linearized models, since thedesign and measured loop gains are quite similar.Fig. 16 shows the measured singular values of

ðTðjoÞ � IÞ for the MIMO controller. These singularvalues are ideally 0. The isotropic bandwidth for theMIMO controller is about 2:5 Hz; with only slightlyworse tracking up to 5 Hz: It can be seen that the

ARTICLE IN PRESS

100

101

102

−40

−20

0

20

40

Frequency, [Hz]

Mag

nitu

de(e

ig(P

G))

, [dB

]

100

101

102

−200

−100

0

100

200

Frequency, [Hz]

Pha

se(e

ig(P

G))

, [de

g]

Fig. 14. Eigenvalues of measured PðsÞGðsÞ in Bode plot format.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real(eig(PG))

Imag

(eig

(PG

))

Fig. 15. Eigenvalues of measured PðsÞGðsÞ in Nyquist plot format.Dotted line indicates the envelope used to estimate gain and phase

margin.

100

101

102

0

0.5

1

1.5

2

2.5

Frequency, Hz

Mag

nitu

de

Fig. 16. Singular values of measured TðsÞ � I under MIMO control.

100

101

102

−40

−20

0

20

40

Frequency, [Hz]

Mag

nitu

de(E

ig(P

G))

, [d

B]

100

101

102

−200

−100

0

100

200

Frequency, [Hz]

Pha

se(E

ig(P

G))

, [d

eg]

Fig. 12. Eigenvalues of designed PðsÞGðsÞ in Bode plot format.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real(eig(PG))

Imag

(eig

(PG

))

Fig. 13. Eigenvalues of designed PðsÞGðsÞ in Nyquist plot format.Dotted line indicates the envelope used to estimate gain and phase

margin.


singular values of ðTðjoÞ � IÞ are much smaller for theMIMO control case than for the independent SISOcontrol case (see Fig. 5) for frequencies up to 9 Hz: Thelarge singular value of ðTðjoÞ � IÞ at 15 Hz reflects apeak in TðjoÞ of about 6 dB; which results from therelatively small phase margin seen in Figs. 13 and 15.Despite this peaking, the MIMO results are a very

large improvement over independent SISO designresults. The ‘‘feel’’ of the interface under the newcontrol scheme is noticeably better due to improvedforce tracking in the directions which had relatively lowloop gain under SISO control. Specifically, free spacefeels freer, and virtual walls are stiffer with lessdirectional distortion. The interface does have a slight‘‘alive’’ feel to it due to the underdamped closed loopmode at 15 Hz:

ARTICLE IN PRESSC.D. Lee et al. / Control Engineering Practice 12 (2004) 1423–1436 1435

Improvements in the plant inversion process near15 Hz would reduce the observed peaking, by ensuringthat the eigenvectors of the loop gain are nearlyorthogonal in this regime. This makes the eigenvalueplots a more accurate predictor of closed-loop singularvalue performance. Improvements to the plant modelcould be made by changing the weighting function h inthe least-squares model fitting procedure, or conductinga more focused identification with larger signals in the15 Hz range.Although robustness has not been fully quantified, the

following preliminary remarks can be made. The systemis highly robust to increases in grip tightness, but is lessrobust for very loose grips. However, this is not aproblem, since users tend to use a firm-enough grip tomaintain a confident grasp of the stylus. Although theidentification experiments were taken from only oneuser, performance is not particularly user-sensitive,based on the lack of any observed anomalies amongseveral other users. Performance is also robust toposition changes up to the outer 20% of the workspace,where the actuator rods approach full extension. Giventhe relatively mild variations in behavior with work-space location, gain scheduling in 324 regimes may besufficient to overcome this problem.

6. Conclusion

Transparency of haptic interfaces can be improved byforce control loops, which cause rendered forces to moreaccurately track desired forces. Isotropic tracking at lowfrequencies is important in haptic interfaces, so thatrendered objects produce reaction forces not only withappropriate magnitudes, but also in appropriate direc-tions. The modeling and control methods presented herewere shown to lead to effective isotropic force control ina particular parallel haptic interface. These methods areapplicable to all haptic interfaces equipped with forcesensing.Simplicity of the models derives from the second-

order polynomial matrix models for constituent im-pedances. These capture the dynamics of the hapticinterface and human hand with sufficient accuracy toenable significant improvement in isotropic bandwidthover independent SISO control designs. The multi-variable controller greatly improves the feel of renderedobjects on the CU haptic interface.Force control may be sufficient for haptic interfaces

which directly render data values as forces (as in someapplications of scientific visualization). Other applica-tions may render forces which are a function of handposition, which results in an impedance loop enclosingthe inner force loop. Here, the force loop can provide animprovement in rendered impedance accuracy.

The structure of the controller lends itself to gainscheduling, when dynamics of the hand or interface arestrong functions of position or pose. This is because allcontrollers generated from linear models at variouslocations have the same structure, allowing a straight-forward interpolation of control parameters as afunction of location.

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