adjusting output-limiter for stable haptic rendering in virtual environments

768 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 4, JULY 2009

Adjusting Output-Limiter for Stable HapticRendering in Virtual Environments

Kyungno Lee and Doo Yong Lee, Senior Member, IEEE

Abstract—This paper presents a control method using anadjusting output-limiter for stable haptic rendering in virtualenvironments. In a simulation of force-reflecting interaction withdeformable objects in the virtual environment, a quick compu-tation of the accurate impedance of deformable objects is rare.This is particularly true when physics-based models, such astensor-mass models or mass-spring models, are used. The problemis aggravated if the simulation involves changes in the geometryand/or impedance of the deformation model, such as cutting orsuturing. The proposed control method guarantees stable hapticinteraction with deformable objects of unknown and/or varyingimpedance. The method is based on the time-domain passivitytheorem and the two-port network model. The controller adjuststhe maximum permissible force to guarantee the passivity of thehaptic system at every sampling instant. The controller notes onlythe magnitude of the reflective force, and does not depend onproperties of the employed force model. This allows the proposedcontrol method applicable to haptic systems involving deformableobjects with unknown, nonlinear, and/or time-varying impedance.Designs of the controllers are presented for impedance-type andadmittance-type haptic systems. The method is also extended formultiple degrees-of-freedom.

Index Terms—Haptic control, haptic interface, haptic rendering,passivity.

I. INTRODUCTION

M ULTIMODAL interaction involving haptic and visualfeedback enhances the fidelity and experience of the

simulation in a virtual environment. It is important in haptic ren-dering to generate a realistic reflective force at more than 300Hz for soft objects and 1 kHz for rigid ones to guarantee goodhaptic sensation and stability of the haptic system [1]. Med-ical simulation requires realistic physics-based rendering of de-formation of the organs and tissues including cutting and su-turing. These physics-based deformation models such as finite-element models [2] and mass-spring models [3] are used to dis-play realistic deformation and reflective force. However, thesemodels incur heavy computational burden. Estimation methods[1], [4]–[6] exploiting previous positions and forces are used for

Manuscript received October 23, 2007; revised August 03, 2008. Manuscriptreceived in final form December 10, 2008. First published April 21, 2009; cur-rent version published June 24, 2009. Recommended by Associate Editor M.de Mathelin. This work was supported in part by the Korea Science and Engi-neering Foundation (KOSEF) grant funded by the Korean government (MOST)(R01-2007-000-11353-0) and by the Brain Korea 21 Project in 2008.

K. Lee is with the Samsung Electro-Mechanics, Suwon 443-743, Republic ofKorea (e-mail: [email protected]).

D. Y. Lee is with the Department of Mechanical Engineering, Korea Ad-vanced Institute of Science and Technology (KAIST), Daejeon 305-701, Re-public of Korea (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCST.2008.2011281

real-time computation of a reflective force, due to the low up-date rate that results from the computational complexity of thedeformation model.

Estimation error can jeopardize the stability of the hapticsystem. Moreover, the impedance of the deformation modelis comparatively small, nonlinear, time-varying and often un-known. When a finite-element model or a mass-spring modelis used, it is difficult to compute accurate impedance valuesfor deformable objects in real time because it is a kind ofnetwork model that consists of many nodes. This problem ismagnified if the simulation involves changes in the geometryand/or impedance of the deformation model, such as cuttingor suturing. Hence, it is difficult to guarantee stable hapticinteractions with the deformation model.

Barbagli et al. [7] described an adaptive local haptic modelthat allowed users to interact haptically with a deformable objectsimulation featuring computational delays and low servo rates.Asymptotic stability is guaranteed when the stiffness of the localhaptic model is smaller than that of the deformable object at acontact point; however, it is necessary for the real stiffness ofdeformable objects to be estimated. Mahvash and Hayward [8]propose a passivity condition considering the combined effectsof zero-order-hold and computational delay. The passive-forceresponse is synthesized from the local force fields and the pas-sivity condition. It can be applied when the impedance of thedeformable object is known beforehand.

The large computational load of the deformation model in-curs a computational time-delay and a low update rate. However,control methods based on virtual coupling and on time-domainpassivity are developed based on high servo rates, which areoften incompatible when interacting with deformable objects.

Colgate et al. [9] introduced a virtual coupling comprised ofan artificial spring and damper model that connects the hapticdisplay. The passivity condition [10] is derived using a general-ized two-port linear virtual coupling under the assumption thatthe human operator is an arbitrary linear time-invariant passiveimpedance. Adams and Hannaford [11], [12] proposed a methodof designing a two-port network-based virtual coupling for ab-solute stability. Miller and Colgate [13] derived a sufficient con-dition for stable haptic interactions with nonlinear and time-de-layed virtual environments. Gil et al. [14] analyzed the sta-bility of a 1-degree-of-freedom (DOF) haptic interface using theRouth-Hurwitz criterion and presented the parameter conditionsto guarantee the stability. Diolaiti et al. [15] presented how thesampling rate, quantization, computational delay, and amplifierdynamics interact with the inertia, natural viscous, and Coulombfriction of the haptic device. The electrical dynamics of DC mo-tors is exploited to implement maximum achievable object stiff-ness [16]. Lee and Lee [17] studied a stability condition for a

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LEE AND LEE: ADJUSTING OUTPUT-LIMITER FOR STABLE HAPTIC RENDERING IN VIRTUAL ENVIRONMENTS 769

nonlinear virtual coupling with 1-DOF, which takes into accountthe response of a human arm. Haptic control methods based onthe virtual coupling and a high update rate can guarantee stablehaptic interaction with a rigid body regardless of its impedance.The human operator feels parallel summation of the virtual en-vironment and the virtual coupling impedances. For example,the human operator feels a half of the impedance of the virtualenvironment when the impedance of the virtual environment isas large as that of the virtual coupling. The control methodsbased on the virtual coupling, while certainly applicable to in-teract with a low impedance virtual environment, may degradefidelity of the reflected force. In addition, the impedance of thedeformable objects is unknown and may be time-varying be-cause of suturing or cutting frequently used in medical simu-lation. The previous control methods using virtual coupling donot deal with the time-varying impedance of the virtual environ-ment.

Hannaford and Ryu [18] employ a time-domain passivitystrategy. When the time-domain passivity controller [18] isapplied, the haptic system might become active for a short time.The time-domain passivity controller, then, promptly makesthe haptic system passive again. In this process, the originalcontroller [18] has a potential to generate a large impulse thatmay incur transiently unstable behavior. Time-domain passivitycontrol with reference-energy following [19] is proposed toresolve this transient unstable behavior. The controller attemptsto make the output of the passivity controller smooth, that is, toalleviate the negative effect of the impulse. However, it is nec-essary to compute the correct reference-energy at a high rate.The update rate of the deformation model is relatively low andthe correct reflective force is unknown during the inter-sampleperiod. It is difficult to estimate correctly the reference-energyin this case. In addition, the maximum available stiffness of thetime-domain passivity controller can be obtained only throughexperiments. If the stiffness of deformable objects becomeslarger than the maximum available stiffness, the haptic systembecomes unstable. The stiffness of the deformable objectsshould be smaller than the maximum stiffness for stable hapticinteraction. Hence, the controller may not guarantee stableinteraction with deformation models of unknown impedance.

Stramigioli et al. [20] proposed a port-Hamiltonian ap-proach to track and dissipate energy excess. This methodhas not been applied to nonlinear multidimensional virtualenvironments. The energy bounding algorithm [21], [22]makes a haptic system stable regardless of the range of theimpedance and the sampling frequency. The energy boundingalgorithm is a method to consume the energy generated by thesample-and-hold operator by using energy dissipation elementsin the physical haptic device. Namely, the energy boundingalgorithm controls the energy generated by the reflective forcecomputed from a deformation model that is always smaller thanthe energy dissipated by the device damping. The actuator forceis controlled at every sampling instant. And the actuator forcemay be different from the force computed from the deformationmodel at some sampling instants, which deteriorates the fidelityof the reflective force.

An adjusting output-limiter (AOL) is presented in this paperfor stable force-reflection in virtual environments. The AOL is

Fig. 1. Adjusting output-limiter. (a) Reflective force versus maximum permis-sible force. (b) Force controlled by AOL.

developed based on the passivity theorem and on a two-portnetwork model. It monitors the maximum permissible force toguarantee the passivity of the haptic system at every samplinginstant. In addition, the AOL adjusts the reflective force for thepassivity while sacrificing the fidelity of the reflective force onlyif the force computed from the deformation model is larger thanthe maximum permissible force. The adjusted reflective force istransferred to a human operator through a haptic device. Oth-erwise, the high fidelity of the reflective force is maintained.The AOL can maintain the stability of the haptic system regard-less of the unknown, nonlinear, and/or time-varying impedanceof a deformable object. It creates a stable haptic system evenwhen large impulses or discontinuous reflective forces that canarise from estimation errors are generated. The adjusting output-limiter for an impedance or admittance-type haptic system ispresented here. It is extended straightforwardly to haptic sys-tems with multiple DOFs and it can be applied to fully multidi-mensional virtual environment. The efficacy of the controller isshown through experiments.

II. ADJUSTING OUTPUT-LIMITER

The AOL monitors the maximum permissible force to guar-antee passivity of the haptic system at each sampling period,as shown in Fig. 1(a). As shown in Fig. 1(b), the AOL adjuststhe reflective force while sacrificing the fidelity of the reflectiveforce, only if the force computed from the deformation model islarger than the maximum permissible force. Otherwise, the highfidelity of the reflective force is maintained, while guaranteeingthe passivity of the system.

A human operator can feel realistic haptic interaction whena force response from a virtual environment is realistic enoughto reproduce the response of the actual interactions and is trans-ferred faithfully to the operator. The method to reproduce real-istic response of the virtual environment is explained in Mah-vash and Hayward [23]. The force response from a virtual en-vironment is assumed to be realistic enough to reproduce theresponses of the actual interactions. It is important how faith-fully the force from the virtual environment is transferred to theoperator. The fidelity of the reflective force means how closethe actuator force is to the force computed from the deforma-tion model under the same displacement. In other words, highfidelity of the reflective force is obtained when the force com-puted from the virtual environment is transferred to the operatorwithout modification.


Fig. 2. Two-port network model of the impedance-type haptic system and theadjusting output-limiter. (a) Human operator. (b) Haptic device. (c) Sample andhold. (d) Virtual environment and adjusting output-limiter.

The AOL is based on the time-domain passivity theories. Ittakes the sample-and-hold (ZOH) effect into account. The en-ergy generated by the ZOH and computational time-delay, aswell as energy dissipated by viscous and Coulomb friction ofthe haptic device, are all observed at every sampling instant,while only the energy of the virtual environment is observed inthe time-domain passivity controllers [18], [19]. The maximumenergy generated at the next sampling instant is predicted, andthe AOL controls the maximum permissible force to maintainstable haptic rendering in the virtual environment.

The bilateral reflective force in a two-port network model ofthe impedance-type haptic system is considered, as shown inFig. 2. It is difficult to guarantee stability when it depends onhuman operator behavior because the biomechanical impedanceproperties of the human operator are unpredictable. We wouldlike to maintain a stable system for any level of human interac-tion, the only restriction being that the human operator is passive[11], [21].

Energy in the haptic device at sampling instant , ,is derived as shown in (1), using[21]. Energies in the sample-and-hold system and in the virtualenvironment with AOL are shown in (2) and (3), respectively.The sum of the energy in the sample-and-hold and the virtual en-vironment and the total energy of the haptic system

at the sampling instant are shown in (4) derived by[21] and (5), respectively. , , and are the sam-pling time, Coulomb friction, viscous friction and velocity ofthe haptic device, respectively. The Coulomb and viscous fric-tion of the haptic device are nonlinear and depend on the posi-tion of the device. and are defined as the minimumof the Coulomb and the viscous friction at all positions, respec-tively, which are assumed to be constant

(1)

since

(2)

(3)

(4)

(5)

We have to estimate the amount of energy dissipation to applythe AOL. We can make a haptic system passive, only if an es-timate of the dissipated energy is smaller than the amount ofactual energy dissipation. Hence, the AOL is developed basedon the minimum values of frictions empirically measured in var-ious positions and moving directions.

The is the th reflective force computed from the con-troller, which should be adjusted to maintain the passivity ofthe system at the sampling instant . The total energyof the haptic system at the sampling instant becomes(6), where is unknown because the velocity is deter-mined by the interaction between the human operator and thevirtual environment. Hence, the condition that is independentof velocity is necessary to guarantee passivity of the system atall the sampling instances.

The reflective force is controlled to satisfy the condi-tion for . The velocity to minimize

, is derived from differentiating(6) this is a function of the sign of and the range of


TABLE IFOUR CASES TO COMPUTE THE RANGE OF � �� THAT SATISFIES

��

the reflective force as shown in (7). Four cases are consid-ered according to the sign of and the range of ,as shown in Table I

(6)

where sgn

(7)

Case I: When andbecomes negative. The ve-

locity to minimize in (6) under the condition, becomes zero, as

is a second-order function of and thecoefficient of is positive. When is zero,

becomes nonnegative, that is, the system is passiveregardless of the value of . Hence, the system becomespassive at the instant if .

Case II: When andbecomes non-positive.

becomes . Thefollowing equation is obtained from (6), andis defined as the predicted minimum of

where .

Hence, the haptic system becomes passive at if

Case III: When andbecomes nonnegative.

becomes .The following equation is obtained from (6)

where

Hence, the haptic system becomes passive at if

Case IV: When and , thesystem is passive at the instant under the condition of

, as was similarly derived in Case I.Combining Cases I to IV, the system becomes passive at the

instant if the reflective force, , is limited to sat-isfy the following condition regardless of :

If the reflective force computed from the deformation modelis , the controller can then be designed to make the systempassive at every sampling instant, as follows.

Adjusting Output-Limiter:

If then

(8)

Otherwise, , where

The controller notes only the magnitude of the reflective forceregardless of properties of the reflective force model. Hence, thecontroller guarantees stable haptic interactions even when theimpedance of the virtual environment is unknown, time-varying,and/or nonlinear. The controller can be also used when the sam-pling rate is low and/or a computational delay exists.

The magnitude of the impedance of a virtual object is lim-ited for the stable haptic interaction in the time-domain passivitycontrollers [18], [19]. The impedance of the virtual object mustbe designed, with taking into account the maximum allowableimpedance for the stability. Otherwise, the haptic system be-comes unstable. However, the AOL allows the design of force-deflection models without considering the stability, and guaran-tees the passivity of the system.


Fig. 3. Two-port network model for the admittance-type haptic system andthe adjusting output-limiter. The network models of the human operator, thehaptic device, and the sample-and-hold are identical to those shown in Fig. 1.(a) Human. (b) Haptic device. (c) Sample and hold. (d) Virtual environment andadjusting output-limiter.

The clipping in the AOL might incur the oscillation of thehaptic system. In the haptic system, a human operator interactswith a virtual environment, and physical damping of the humanoperator somewhat alleviates the negative effect of the clippingin the AOL. As the reflective force that human operator feelsincreases, the physical damping of the human operator also in-creases [24]. Hence, the negative effect of the clipping may notbe so devastating. Actually, the negative effect of the clipping isnot generated during the experiments.

Theorem 1: The haptic system becomes passive if the reflec-tive force is always chosen so that it satisfies the (9)

(9)

Proof: The proof is explained in detail in Appendix B.

A. Adjusting Output-Limiter for Admittance-TypeHaptic Systems

In an admittance-type interaction, the displacement is gener-ated in response to the measured force, while the force is gener-ated in response to the measured displacement in an impedance-type haptic system. Adams and Hannaford [11] derived such anadmittance-type haptic display by adding a proportional-inte-gral (PI) velocity control loop, , to theimpedance-type interaction model, in whichare the force of actuation, the velocity of the haptic device, thecommanded velocity, and the PI-controller, respectively. Thesymbol “*” means discrete-time variable.

A two-port network model for an admittance-type hapticsystem is designed as shown in Fig. 3. The force applied bythe human operator, , and the velocity of the haptic device,

, can be measured at a high rate. The desired velocityin response to input is computed from the admit-

tance-type deformation model. The commanded force, isdetermined by the PI-velocity controller, and the AOL adjuststhe commanded force to make the system passive. Hence,the two-port network model for the admittance-type hapticsystem is equivalent to the model for the impedance-type hapticsystem. The energy for each sub-system in the admittance-type

Fig. 4. Workspaces and motions to measure the frictions of the PHANToM. (a)Workspace. (b) Motions in �-axis.

haptic system can be computed using procedures identical tothose of the impedance-type haptic system.

The AOL in (8) can be applied to the admittance-type hapticsystem by substituting the commanded force for thereflective force, . While the AOL adjusts the reflectiveforce from the deformation model, , in the impedance-type haptic system, it adjusts the commanded force, , inthe admittance-type haptic system.

B. Adjusting Output-Limiter for a Haptic System With MultipleDOF

The total energy of the haptic system can be computed from(10) when the system is -DOF [25]. The AOL can be extendedstraightforwardly to a haptic system with multiple DOF using(10). , and are the total en-ergy, minimum viscous and Coulomb friction, velocity, and thereflective force at the sampling instant kT for the -directionalmotion, respectively. The total energy of the system becomesnonnegative if the energy for each DOF is nonnegative; that is,if . Each controller for the -directional motion makesthe total energy nonnegative

(10)

where

III. EXPERIMENTS AND RESULTS

A. Friction Measurement of a Haptic Device

The AOL is experimented using a 6-DOF haptic device,PHANToM and it is necessary to measure the frictions ofPHANToM to apply the AOL. Coulomb and viscous frictionsof the PHANToM can be measured using the friction measure-ment procedure presented by Kelly et al. [26]. The procedure


Fig. 5. Workspaces and motions to measure the frictions of the PHANToM. (a) Workspace. (b) Pose of PHANToM. (c) Motions in �-axis. (d) Motions in �-axis.(e) Motions in �-axis.

is explained in the following. It is assumed that the frictionis a memoryless mapping of the relative velocity between thecontacting bodies. When a ramp force with slope (N/s) isapplied to the PHANToM, the velocity response at time ismeasured. By using this result, the slope of the velocity curveover time, the parameter , and the intersection with the timeaxis, the parameter , are computed as shown in Fig. 4(a).Fig. 4(b) shows one of measurement results in -axis. ThenCoulomb and viscous frictions are computed from (11) and(12) that are derived by [26], where the inertia mass of thePHANToM is set at 0.075 Kg

Coulomb friction: (11)

Viscous friction: (12)

Since the coefficients of friction of PHANToM varies ac-cording to the position, we set the workspace of the PHANToMas shown in Fig. 5(a). The -axis and the -axis represent hor-izontal and vertical direction, respectively. The -axis is or-thogonal direction to the -plane. The frictions for motions inthe -axis are measured at five different positions, as shown inFig. 5(c). In this case, only motions in -axis are available be-cause motions in - and -axis are constrained using a virtualspring with large stiffness. The velocity responses to a rampforce input with slope (N/s) are measured. The pa-rameters and are computed. The procedure is repeatedover five times at each position to average out any stochastic in-fluence. The Coulomb and viscous frictions are, then, computedfrom (11) and (12), and the average frictions at each position areshown in Table II. The Coulomb and viscous frictions for mo-tions in - and -axis are measured at five positions as shownin Fig. 5(d) and (e) using the same procedure. The friction inTable III is the minimum of measured frictions at each axis inTable II. The AOL is designed based on the minimums of fric-tions in Table III.

B. Experiments and Results

The AOL is proposed for stable haptic interaction with de-formable objects such as organs and tissues, not rigid objectssuch as a virtual wall. The exact impedance of the deformableobject designed with a physics-based deformation model such

TABLE IICOULOMB AND VISCOUS FRICTIONS MEASURED AT EACH POSITION

TABLE IIIMEASURED MINIMUM FRICTIONS OF PHANTOM

as a finite element model or a mass-spring model is unknown,time-varying, and/or nonlinear. Five different experiments aredesigned to show the efficacy of the AOL. The AOL is appliedto a 1-DOF motion. The frictions at the -axis in Table III areused. These experiments include when the impedance of the de-formation model is unknown, very large, time-varying, and/ornonlinear, as shown in Table IV. The effects of a low samplingrate and a large time-delay are also investigated. The samplingrate of the experiments is set to 1 kHz with the exception of ex-periment D the sampling rate of which is set to 25 Hz. The delaytime in experiment C is set to 300 ms. The PHANToM deviceand the AOL for an impedance-type haptic system are used forall the experiments in this study.

The deformable object can be pushed or pulled with bilateralconstraints. The contact node is pulled and pushed repeatedly,as shown in Fig. 6(a), and the reflective force from the designedforce model is generated. Fig. 6(b) shows the input motion.The magnitudes of the impedance of the five force modelsin Table IV are not the maximum allowable impedance. Theproposed control method makes the system stable even ifthe impedance of the virtual model becomes larger than inthe five experiments. The operational stiffness is defined as


TABLE IVEXPERIMENTAL CONDITIONS OF THE EXPERIMENTS (� � force��

displacement (MM), � � time� �� sampling frequency)

Fig. 6. Force model and motion used in the experiments. (a) Virtual model. (b)Operator’s input motion (position versus time).

Fig. 7. Results of the experiment A; � � ��N/m � s� � �� . (a)Unstable behavior without control. (b) Stablized position profile by AOL. (c)Force from deformation model and force adjusted by AOL. (d) Energy of thehaptic system.

in which , andare the initial and the kth measured positions, along with the

corresponding forces, respectively.The haptic system became unstable without the developed

control in all the five experiments. Experiment results for thevirtual environment with time-varying impedance, nonlinearimpedance, large time delay, low sampling rate, and largestiffness are shown in Figs. 7–11, respectively. The unstablebehavior without control is shown in (a) of Figs. 7–11. Theposition profiles stabilized by AOL is presented in (b) ofFigs. 7–11. The operator can move as shown in Fig. 6(b)and interacts stably with the virtual environments, which isa benefit of the AOL. Namely, the force computed from theemployed force model is adjusted. Forces from the force modeland the AOL are shown in (c) of Figs. 7–11. Solid graph is the

Fig. 8. Results of the experiment B: � � �� (N/mm ) �� . (a) Unstablebehavior without control. (b) Stablized position profile by AOL. (c) Force fromdeformation model and force adjusted by AOL. (d) Energy of the haptic system.

Fig. 9. Results of the experiment C: � � 300 ms. (a) Unstable behaviorwithout control. (b) Stablized position profile by AOL. (c) Force from deforma-tion model and force adjusted by AOL. (d) Energy of the haptic system.

force computed from the employed force model and solid linemarked with “pentagram” is the adjusted force generated fromthe adjusting output-limiter in (c) of Figs. 7–11. The -axison the left is for the reflective force computed from each forcemodel and that on the right is for the adjusted reflective force.This figure shows the difference between the force adjusted bythe proposed controller and the force computed from the forcemodel. The proposed output-limiter adjusts the magnitude ofthe reflective force to maintain the passivity of the systemand the adjusted reflective force is transferred to the humanoperator. The adjusted output force, transferred to the humanoperator through the PHANToM device, is smaller than 8.5Newton which is the maximum exertable force. The hapticsystem energy over time is shown in (d) of Figs. 7–11, which isnonnegative all the time. The haptic system is always passiveby the virtue of the AOL.


Fig. 10. Results of the experiment D: � � 5000 (N/m) �� 25 Hz. (a)Unstable behavior without control. (b) Stablized position profile by AOL. (c)Force from deformation model and force adjusted by AOL. (d) Energy of thehaptic system.

Fig. 11. Results of the experiment E: � � �� N/m� � ��. (a) Unstablebehavior without control. (b) Stablized position profile by AOL. (c) Force fromdeformation model and force adjusted by AOL. (d) Energy of the haptic system.

Result for experiment A is shown in Fig. 7. As time goes,the stiffness becomes very large and haptic interaction becomesunstable. High-frequency oscillation is generated as shown inthe circles A of Fig. 7(a). The force computed from the forcemodel, represented in solid line in Fig. 7(c), is adjusted to makethe system passive. Energy over time in Fig. 7(d) is always non-negative, which means the system is passive all the time. The op-erator’s input is stably transferred to the virtual environment andthe operator feels the corresponding response. In this case, theoperational stiffness measurement is approximately 530 N/mwhich is large enough to simulate the interaction with the de-formable objects.

Result for experiment B is shown in Fig. 8. As the depthincreases, the stiffness becomes very large and haptic interac-tion becomes unstable. Circle B in Fig. 8(a) represents high-fre-quency oscillation. The AOL stabilizes the system, as shown in

Fig. 8(b). The operational stiffness is approximately 530 N/mwhich is as large as that of experiment A.

The AOL depends only on the magnitude of the current re-flective force. Hence, the proposed control method makes thesystem stable even when the time-delay is quite large or whenthe sampling rate is low as shown in Figs. 9 and 10. As shown inFig. 9(a), oscillation is generated and amplified when large timedelay exists although the stiffness of the force model is small,that is, 100 N/m. On the other hand, the AOL makes the systemstable as shown in Fig. 9(b) although the stiffness is twenty timeslarger than that of Fig. 9(a). The stiffness of the force model isnot the maximum allowable impedance and the AOL can stabi-lize the system even with larger stiffness. The operational stiff-ness measurement is approximately 180 N/m which is one thirdof the experiment A.

Oscillation is generated and its amplitude increases gradu-ally because of the low sampling rate as shown in Fig. 10(a).Fig. 10(b) shows stabilization by the AOL. The accumulated en-ergy is small because the sampling rate is low. Maximum per-missible force is the smallest among the five experiments. Theoperational stiffness is about 16 N/m which is approximatelyone-thirtieth of the experiment A. This implies that the opera-tional stiffness of the AOL becomes larger as the update ratebecomes higher.

High-frequency oscillation is generated because of the largestiffness as shown in Fig. 11(a), which can be prevented bythe AOL. The operational stiffness is approximately 500–600N/m which is large enough to simulate the impedance of the de-formable objects while the stiffness of the virtual model is 4000N/m. This difference between the operational stiffness and thedesigned stiffness is because the AOL limits magnitude of thereflective force to maintain the passivity.

These experiments show that the AOL makes the systemstable even when the impedance of the virtual model is un-known and also time-varying, nonlinear, or very large.

Another force model with time-varying impedance,N/m s , is introduced to

show how the AOL maintains the fidelity of the reflective force.Once the impedance of the force model is increased, however,it is decreased again. This is done repeatedly. The positionand total energy of the haptic system for time are presented inFigs. 12(a) and (b). A large reflective force may be generatedas the impedance of the force model and the penetrated depthincrease; however, the negative effect of a large reflectiveforce such as oscillation is alleviated immediately, as shownin Fig. 12(a). The total energy in Fig. 12(b) is always positive,which shows that the haptic system is always passive.

The impedance of the employed force model changes and thecorresponding reflective force varies, as shown in Fig. 12(c).The solid line indicates the force computed from the force modeland the line with “pentagram” is the force controlled by the AOLin Fig. 12(c). The AOL controls the force only if the force com-puted from the deformation model is larger than the maximumpermissible force, as shown in Fig. 12(c). The adjusted reflectiveforce is transferred to a human operator, however, if the adjustedforce is larger than the maximum exertable force of the hapticdevice, PHANToM, the maximum exertable force is transferredto the human operator. Otherwise, a reflective force with high


Fig. 12. Efficacy of the AOL for the time-varying impedance. (a) Positionversus time. (b) Energy versus time. (c) Force versus time. (d) Force from 2.8to 3.6 s.

fidelity is generated. The reflective force from 3.054 to 3.190s is generated without the adjustment of the AOL, as shown inFig. 12(d).

The effects of a large estimation error on the stability andperformance of the AOL are investigated. The single-input,single-output (SISO) output estimator with multirate sam-pling, as developed by Lee and Lee [4], [5], is used. Theoutput estimator with P-matrix resetting [5] is proposed forreal-time haptic rendering even when a deformation modelis refreshed at low rate. The estimator is developed basedon the least-squares algorithm and output-error estimationmodel. The estimation parameters continuously trace thecorrect parameters of the input-output relationship of thedeformation model. Inter-sample output force is estimatedat a high rate by the output estimator, and traces the cor-rect output force computed from the deformation model,even when the parameters of the input-output relationshipof the deformation model change. Here, The reflective forcemodel is designed as .

is selected for the esti-mation model [5], where and are estimation parameters.The input displacement is measured at 1 kHz and the outputforce is computed at 25 Hz from the deformation model. Thecomputational time-delay is assumed to be 40 ms. The P-matrixis reset whenever .

The contact node is pulled and pushed repeatedly, as shown inFig. 6. When the estimation error is small, the interaction withthe deformation model is stable. However, a large estimationerror is often generated, as shown in circle A and B of Fig. 13(a).The solid line represents the force computed from the deforma-tion model, and the line marked with the ‘pentagram’ indicatesthe estimated force in Fig. 13(a). The correct output force com-puted from the deformation model is two thirds of the estimatedoutput force. The estimation error is larger than 4 N.

Fig. 13. Unstable behavior due to a large estimation error. (a) Force versustime. (b) Position versus time.

Fig. 14. Stabilized behavior when the AOL is used. (a) Force controlled byAOL. (b) Position versus time.

When a large reflective force due to the estimation error isgenerated, as shown in circle A and B of Fig. 13(a), the un-desirable oscillation and the motion contrary to the intentionsof the human operator appear, as shown in circle A and B ofFig. 13(b). It is necessary to prevent the unexpected large re-flective force that incurs the undesirable behavior and threatensthe safety of the haptic system. The AOL limits the large reflec-tive force due to the estimation error and alleviates the negativeeffect, as shown in Fig. 14. The solid line is the output forcefrom the controller and the line with the “pentagram” is the es-timated force in Fig. 14(a). The operator can move as the inputmotion as shown in Fig. 14(b). The stable haptic interaction isrealized by the AOL.

IV. CONCLUSION

This paper presents a control method that uses an adjustingoutput-limiter for stable haptic simulation with deformableobjects in a virtual environment. The adjusting output-limiter(AOL) monitors the maximum permissible force to guaranteethe passivity of the haptic system at every sampling instant.The AOL adjusts the reflective force only if the force com-puted from the deformation model is larger than the maximumpermissible force. Otherwise, the high fidelity of the reflectiveforce is maintained and the reflective force with high fidelity istransferred to the human operator. The passivity of the systemis always guaranteed. Hence, this enables us to feel realisticreflective forces computed from deflection-force curves ofphysical matters such as human organs and tissues, whilemaintaining the stability of the simulation.

Only the magnitude of the reflective force is used in theproposed control method regardless of the properties of theemployed reflective force models. Hence, the AOL guaranteesstable haptic interaction with deformable objects, even when the


impedance of the deformable objects is unknown, time-varying,and/or non-linear. Thus the control method presented in thispaper allows the design of force-deflection models withouttaking into account the stability of the models. In addition,the AOL alleviates the negative effects of large estimationerrors. The operational stiffness achieved by the AOL at 1 kHzupdate rate is approximately 500 N/m, which is large enoughfor realistic medical simulations with deformable objects. Theoperational stiffness is reduced as the time delay increases orthe sampling rate is low. The operational stiffness of the virtualenvironment with 25 Hz rate is approximately one thirtiethof that of the time-varying model or nonlinear model with 1kHz sampling rate. It is important to increase the sampling rateor compensate the time-delay for increasing the operationalstiffness, which can be resolved by using the multirate outputestimation algorithm [5].

The presented AOL can be used for both the impedance andadmittance-type haptic systems. It can also handle multipleDOF systems.

APPENDIX I

A. Intermediate Steps for (1)

Equation is proved bythe following procedure [21]:

Hence

End of proof.

B. Proof of the Theorem 1

Theorem 1: The haptic system becomes passive if the reflec-tive force is always chosen so that it satisfies the (9)

(9)

Proof: The energy of the haptic system at ,is as that shown in (5), and (9) can be equivalently represented,as shown in (B-1). It is shown that the system always becomespassive through a mathematical induction. The initial energy ofthe haptic system is set at zero; that is,

(5)

(B-1)

(B-2)

(B-3)

(B-4)

ifif

(B-5)


Step 1) When

where .Hence the system is passive at . The reflectiveforce is chosen to satisfy (B-2), shown atthe bottom of the previous page. When , thetotal energy of the haptic system, , becomes(B-3). isalways nonnegative regardless of the sign ofand . Hence, is always nonnegative ac-cording to (B-2), (B-4), and (B-5). Thus, the hapticsystem becomes passive at .

Step 2) The haptic system is assumed to be passive at ;that is, the following condition is satisfied

(B-6)The reflective force is chosen that satisfies (B-7). The

energy becomes (B-8). Hence the haptic system atis passive

(B-7)

(B-8)

(B-9)

if

if

(B-10)

is always non-negative regardless of the sign of and . Hence,

is always nonnegative according to (B-8), (B-9), and(B-10). The haptic system then becomes passive at .Finally, the haptic system becomes passive if the reflective force

is always chosen to satisfy (9).This completes the Proof of the Theorem 1.

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Kyungno Lee received the B.S. degree from theDepartment of Mechanical Engineering, YonseiUniversity, Seoul, Korea, in 1996 and the M.S. andPh.D. degrees from the Department of MechanicalEngineering, Korea Advanced Institute of Scienceand Technology (KAIST), Daejeon, Korea, in 1998and 2007.

He is currently a Senior Researcher with SamsungElectro-Mechanics Co. Ltd., Suwon, Korea. He wasa postdoctoral Research Associate with BK21 Valu-facture Institute of Mechanical Engineering, KAIST

(2007–2008). His research interests include tactile sensors, haptic device, hapticcontrol and rendering, and its application to robotics and virtual-reality-basedsimulation.

Doo Yong Lee (S’89–M’93–SM’99) receivedthe B.S. degree from the Department of Controland Instrumentation Engineering, Seoul NationalUniversity, Seoul, Korea, and the M.S. and Ph.D. de-grees from the Department of Electrical, Computer,and Systems Engineering, Rensselaer PolytechnicInstitute, Troy, NY.

He has been with the faculty of the Departmentof Mechanical Engineering, Korea Advanced Insti-tute of Science and Technology (KAIST), Daejeon,Korea, since 1994, and is now a Professor. His cur-

rent research interests include robotics and medical simulation.Prof. Lee was a recipient of the Charles M. Close Doctoral Prize from Rensse-

laer Polytechnic Institute (1993), the Baekam Paper Award from the Korean So-ciety of Mechanical Engineers (1999), the Young Researcher Paper Award fromthe Institute of Control, Robotics, and Systems (2002), the Franklin V. TaylorMemorial Award from the IEEE Systems, Man, and Cybernetics Society (2004),and the Outstanding Paper Award of the 2007 International Conference on Con-trol, Automation, and Systems (ICCAS 2007). He serves as an Associate Editorof the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B:CYBERNETICS and the Journal of Mechanical Science and Technology. He wasthe Program Chair of the 2008 International Conference on Control, Automa-tion, and Systems (ICCAS 2008). He is a Member of the Society for Simulationin Healthcare (SSH), Senior Member of the Society of Manufacturing Engineers(SME), Member of the Korean Society of Mechanical Engineers (KSME), andManaging Director of the Institute of Control, Robotics, and Systems (ICROS).

adjusting output-limiter for stable haptic rendering in virtual environments

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