Dual-Channel Haptic Synthesis of Viscoelastic Tissue ... · Dual-Channel Haptic Synthesis of Viscoelastic Tissue Properties Using Programmable Eddy Current Brakes Andrew H. C. Gosline

Authors’ version. International Journal of Robotics Reseach, 28(10):1387-1399, 2009.

Dual-Channel Haptic Synthesis of Viscoelastic Tissue Properties

Using Programmable Eddy Current Brakes

Andrew H. C. Gosline and Vincent Hayward

Haptics Laboratory, Centre of Intelligent Machines, McGill University, Canada

Abstract

We describe a novel method for haptic synthesis ofviscoelastic responses which employs a dual-channelhaptic interface. It has motors that generate torqueindependently from velocity and brakes that gener-ate viscous torque independently from position. Thisway, twice as many states are directly accessible,which reduces reliance on observation and feedback.Torque-generating actuators, e.g. dc motors, arewell known. For the viscous actuators, we use eddycurrent brakes as programmable, linear, non-contact,physical dampers. By decomposing a mechanicalimpedance to be realized into viscous and elasticcomponents, we can dedicate each actuator to thatcomponent it is ideally suited to synthesize, dampersfor the viscous component, and motors for the elas-tic component. The decomposition is in general notunique so it possible to select the option that takesthe best advantage of the hardware. Experimentalresults show that this technique can render a vari-ety of viscoelastic models without the artifacts thatcan occur when synthesizing viscous components onconventional haptic interfaces. The synthesized me-chanical impedances have guaranteed passivity, andcan have arbitrarily high or low viscous and elasticcomponents.

1 Introduction

Medical virtual environments, analogues of flightsimulators for pilots, provide an accurately moni-tored and safe method for medical training. Studentsand practitioners alike can learn new skills and re-hearse difficult and complicated surgical procedures.Haptic rendering (also referred to as force feedback)is a vital component of these interactive simulationsbecause in a number of cases, much of the infor-mation necessary to perform surgical procedures isextracted through touch. Examples include simu-lators for procedures such as laparoscopy (Basdo-gan et al., 2001) and brachytheraphy (Goksel et al.,2006). Other contemporary and emerging applica-tions are described in recent surveys (Harders, 2008;Basdogan et al., 2007).

Most tissues exhibit viscoelastic behavior, whichcomplicates the simulations above purely elastic de-formations because the rate of deformation must en-ter in the formulation. Creep, relaxation and hys-teresis are examples of deformation-rate-dependentphenomena that are germane to viscoelastic tis-sues (Fung, 1993). While simulating rate-dependentphenomena can have a great impact on the visualrealism of a simulation, the same is true for hapticrealism. Inserting and retracting a needle in an elas-tic medium does not feel the same as doing the samein a viscoelastic medium. The role of realism in sim-ulators may be discussed, nevertheless, it is beyonddebate that surgical simulators that are capable ofsynthesizing a wide range of tissue behaviors offerfar more pedagogical options than those that don’t.Haptic rendering is possible only if complex mechan-ical impedances can be precisely synthesized.

A variety of methods have been proposed to dis-play viscoelasticity for surgery simulators, most ofwhich use networks of discrete springs and dampers


to model viscoelastic effects, or the Finite Ele-ment Method (fem), as used by Astley and Hay-ward (1998), Sedef et al. (2006), or Brouwer et al.(2007). Alternatively, Discrete Green’s Functionshave been used by Schoner et al. (2004). Recently, anexplicit simulation and interpolation approach usingradial basis functions were described by Hover et al.(2009). Regardless of what numerical method au-thors have used to compute a viscoelastic response,the result yields a force relationship that is depen-dent on both the velocity and the position of thevirtual instrument. Velocity measurement or esti-mation is therefore an integral part of the problemof realizing a haptic simulation of viscoelastic me-chanical behavior.

2 Options For The MeasurementAnd Estimation Of Velocity

With impedance-type devices—the causality ofchoice for most surgical simulators which, whenneeded, must be able to present very low impedancesto the user—closing the loop through a velocity mea-surement or a velocity estimate is subject to trade-offs that are similar to those needed when closing theloop through a position sensor: one must cope withnoise, sampling, delay, mechanical modes, as wellas time-varying and configuration-dependent systemdynamics.

To give a sense of the requirements, it is usefulto consider that inserting a needle is an act thatnormally occurs at low velocity, viz. 10−2 m·s−1.Taking the Phantom 1.0 as an example of a hard-ware platform (the highest performing model in therange), this device has an advertised resolution of30× 10−6 m. This number is arrived at by dividingthe encoder resolution by the mechanical transmis-sion ratio, which indicates that its practical resolu-tion is necessarily lower. For the sake of example, wecan retain this theoretical resolution, given that it isa resolution guaranteed to be never achieved in prat-ice. If we chose to update forces at a rate of 1 kHz,the basic quantum of velocity achieved by a singlebackwards difference estimation is 3 × 10−2 m·s−1,or approximately three times larger than the veloc-ity we wish to measure.

2.1 Inverse-time methods

One option is to use the so-called inverse-time ve-locity estimation method where a velocity estimateis obtained by measuring the time elapsed betweentwo encoder pulses rather than the number of pulsesbetween two clock interrupts (Wallingford and Wil-son, 1977; Cavusoglu et al., 2002). For medical sim-ulations, however, this method suffers from criticalshortcomings. Estimates are obtained at a rate dic-tated by the pulses of the encoder and not by thesystem clock. Causal interpolation schemes to rec-oncile sampling rates mismatch will inevitably intro-duce a variable delay, destabilizing in closed-loop.Delay-compensating, non-causal predictive interpo-lation schemes could be contemplated, but even then,estimates are necessarily erroneous around each ve-locity reversal and have the wrong sign until a newpulse is sensed.

2.2 Velocity smoothing

When sampling an encoder counter on a regular clocksignal, the standard approach is to low-pass the ve-locity estimates to filter out the noise. The tradeoffsinvolving types of filters and filter orders are muchtoo numerous to be discussed here, but in generalphase delay and noise attenuation always create op-posing requirements. The implementer can also ap-ply various types of tracking estimators to obtain abetter velocity signal but invariably, tradeoffs mustbe made (Su et al., 2006), and/or a system modelmust be assumed (Belanger et al., 98). Tradeoff-free adaptive fir filters can be helpful (Janabi-Sharifiet al., 2000). In our experience those filters permitthe level of simulated viscosity to be raised by an or-der of magnitude, roughly, but there is always a pointwhere limit cycles occur and their onset is difficultto predict.

2.3 Velocity sensing

Current haptic interface hardware typically does notprovide velocity sensing, but one could consider de-signing or retrofitting devices with velocity sensors.The mechatronic method of choice for sensing veloc-ity is the tachometer. It could be an option to pro-vide it together with the indispensable position sen-sor. However, by principle, they do not operate wellat low velocities. A high-quality tachometer (Model


dct 22, Maxon Motors ag, Sachseln, Switzerland)mounted on the motor shaft of a Phantom 1.0 wouldgive less than 1 mV (with 6% of ripple) for the ex-ample given above, which poses an instrumentationchallenge without eliminating filter design tradeoffs.

3 Options for Transducing Viscos-ity Directly

All these options, velocity measurement, model-freeestimation, or state reconstruction yield artifactssuch as delayed response or oscillations due to the ap-proximations that each method must contend with.It is therefore natural to examine the possibility ofproducing viscosity directly in a programmable fash-ion, thereby bypassing the necessity to estimate ormeasure velocity.

3.1 Fluid viscosity

The most common industrial approach is to take ad-vantage of the viscosity of fluids. Such systems in-volve a hydrostatic pump or a piston pushing fluidthrough an electromagnetic (or electrorheological flu-idic) valve. There is a number of reasons whythis approach is not directly suitable to create pro-grammable viscosity for medical simulations. Chieflyamong them is the necessity to establish turbulentflow in the valve orifice, giving a quadratic relation-ship between velocity and force that has zero slope atthe origin. The resulting strongly nonlinear dampingrelationship would not fit conveniently into roboticcontrol frameworks (Blackburn et al., 1960).

3.2 Electromagnetic motors

Another approach is to leverage electromotive forces.These forces are present in any electrical machine.Conductors moving in magnetic fields always inducean electromotive force. If a path is provided forcurrent to flow, the current interacts with the mag-netic field to oppose motion. Nominally, because ofthe linear nature of Lorentz’ law, the torque is pro-portional to the angular velocity, creating a viscous-like torque that opposes motion. Dc motors whichare frequently used to make haptic devices, whetherof the wound or coreless types, exhibit this effect.For example, shorting the terminals of the motorsof a Phantom device (model re25, Maxon Motors

ag, Sachseln, Switzerland) creates a viscous coeffi-cient at the tip which evaluates to approximatively0.6 N·s·m−1. At 10−2 m·s−1, the viscous force is only6 mN which too small for a realistic simulation.

Of course, it is possible to consider the design ofa feedback circuit to modulate the effective internalresistance of the motor, so that the induced currentis higher than that permitted by the natural resis-tance of the windings, but such a design is delicateto achieve. If the motor is brushed, its terminal re-sistance varies considerably as the motor turns (1:10ratios are not uncommon) and when the brushes slipfrom segment to segment on the commutator. More-over, each time a new winding is connected and an-other is disconnected, the circuit would have to dealwith a new inductor circuit. Another problem is thedrift of resistance when the windings heat and cool.

Commutation-free electromagnetic actuators canbe considered to alleviate these problems, i.e. lim-ited angle torquers or galvanometric motors. Never-theless, the feedback design would have to be able tomodify the electrical behavior of a motor such that,once converted to the mechanical domain, the vis-cous component and the elastic components of the re-sulting mechanical impedance can be independentlyspecified. Undoubtably, such a feedback controllerwould have to be realized with mixed signal technol-ogy, partly digital, partly analog, to be truly pro-grammable. Recently, a circuit able to add mod-est programmable electrical damping over a limitedrange of motion with a brushless motor was pre-sented (Weir et al., 2008). If such circuits can berealized to achieve wide-range modulation of damp-ing independently from torque, then the methodsdescribed in Section 5 would apply directly to thisapproach.

3.3 Taking the bull by the horns

It is possible to take advantage of Lorentz’ law togenerate programmable viscosity directly. Devicesthat are quite appropriate to realize this functionare eddy current brakes or ecb’s. With this ap-proach, velocity dependent phenomena can be syn-thesized, keeping the velocity signal out of the feed-back loop (Gosline et al., 2006).

Ecb’s are simple magnetic devices consisting ofa conductor that moves through a magnetic field.Eddy currents are induced as a result of the motionin the conductor itself, and the interaction between


these currents and the applied magnetic field gener-ates a resistive force that is proportional to the rel-ative velocity, according to the Lorentz’ Force Law.Although the phenomena are difficult to analyze forcomplex geometries (Wiederick et al., 1987), the un-derlying relationship between velocity and force canbe illustrated with several simplifying assumptions.Following Heald, 1988, the induced current density inthe conductor, J

[A·m−2

], is a function of the veloc-

ity, v[m·s−1

], the specific resistivity of the material,

ρ [Ω·m], the electrostatic field of Coulomb charge in-duced within the conductor, E [V], and the magneticfield, B [T], that is

J =1ρ

(E + v ×B). (1)

Computing the current paths given the motion andapplied field is complex, and is beyond the scope ofthis paper. Closed form solutions exist for simplegeometries, such as an infinite conductor. Regard-less of the eddy current flow regime that ensues, thebraking force, F [N] can be computed by integratingover the pole projection volume, τ

[m3]

F =∫J ×B dτ. (2)

Assuming a rectangular pole with width w [m] andlength l [m], neglecting the air gap, and assuming aconstant thickness of the conductor, d [m], the eddycurrent brake force is then

F = −αdlwρB2v, (3)

where α is a correction factor related to the eddycurrent distribution through the pole projection vol-ume. For further details, refer to Heald, 1988. Ac-cording to (3), the drag force varies linearly withvelocity, and prior work has found this to be true forlow speeds (Heald, 1988; Anwar, 2002; Lee and Park,2002). As the relative speed increases, however, theinduced magnetic field from the eddy currents be-comes too large to neglect, and the braking forcebegins to diminish. Because haptic interfaces are de-signed for interaction with humans, the velocity thatan interface is subjected to is typically low, of order200 mm·s−1 (Lederman et al., 1999), and prior workhas shown that eddy current brakes do behave lin-early at speeds typical of haptic interaction (Goslineet al., 2006).

Fig. 1 shows an experimental eddy current brakeused to perform experimental validation of the meth-ods described in this article. The figure shows atoroidal electromagnet capable of producing a fieldof about 1.0 T inside a 3.18 mm air-gap. A torquemotor drives a rotating arm directly so that a usercan interact with a manipuladum moving on the arcof a circle. The rotating arm supports a 1.59 mmthick, 50 mm radius aluminum annulus that movesconcentrically with the motor. When the electro-magnet is activated, eddy currents — also called Fou-cault currents — induced in the annulus create a vis-cous torque that adds to the torque created by thetorque motor. The preferred geometry of such vis-cous brakes is more fully described by Gosline andHayward (2008) .

Figure 1: Experimental eddy current brake. Thissystem comprises a blade in the shape of an annulussection of average radius 50 mm moving in the airgap of a toroidal electromagnet. The brake is coupleddirectly to a coreless dc motor.

Ecb’s synthesize viscosity without any mechanicalcontact nor any electrical contact. By their physics,for all practical purposes, they create perfectly ac-curate viscosity. This has two consequences. Whensynthesizing the viscoelastic response of a virtual en-vironment, the more they contribute, the more accu-rate is the response, without any tradeoff. Secondly,since the passivity of a system can only increase withthe addition of any combination of dissipative ele-ments, there is no tradeoff either in attempting tomaximize their use. The optimal use of their prop-erties is explored in the next sections.


4 Modeling Viscoelasticity

Classical models of viscoelasticity usually consist ofa network of discrete springs, (Fig. 2a) and dampers(Fig. 2b) to reproduce the relaxation, creep and hys-teresis characteristics of common materials. Figs. 2cand 2d show the two simplest viscoelastic models,the Kelvin-Voight model and Maxwell model respec-tively. While the Kelvin and Maxwell models ex-hibit rate dependence, creep, and relaxation, authorsgenerally prefer more complex models for fitting realdata. Fig. 2e shows a 2nd order Maxwell model thatis a good example of a more complex viscoelasticconstitutive law. This model was shown to providea good fit for biological tissue behavior, and is de-scribed as a “general viscoelastic solid” (Christensen,1971). Recently, Brouwer et al. (2007) used a 2nd

order model to fit experimental data from a porcinebrain deformation. With a similar model, Wang andHayward (2007) obtained a good fit for the in-vivobehavior of fingertip skin.

a b c d e

η2η1ημ

μ1 μ2

μ3

Figure 2: Single Spring (a). Damper (b). Kelvin (c).Maxwell (d). 2nd order model (e).

Our immediate aim is to investigate the simulationand the solution of spring-damper networks so thatthey may be physical re-synthesized with a hapticdisplay for a user to experience them as if she/hewas interacting with real tissues.

Because of its practical importance we focus onthe 2nd order generalized Maxwell model. If thesprings and dampers can be assumed to be linear,the Laplace transform method can be used to solvefor the deformation response or for the force re-sponse (Findley et al., 1976). In the case of the 2nd

order model, the total stress in the element, σ [Pa],is the summation of the stresses in each of the loadpaths,

σ = σ1 + σ2 + σ3. (4)

For each element, the stress is related to strain, ε [%],by,

ε1 =σ1

µ1+σ1

η1, ε2 =

σ2

µ2+σ2

η2, σ3 = µ3 ε3. (5)

With the assumption of time independence, each dif-ferential equation can be solved independently. Itfollows that, given identical initial conditions, eachsolution can be combined in the time domain, suchthat the stress to a ramp input of strain (constantstrain rate) at t = 0 is:

σ = ε η1 − ε η1e−(µ1/η1)t + εη2 − εη2e

−(µ2/η2)t + µ3 ε t.(6)

Note that (6) contains two exponential decay func-tions that correspond to the separate Maxwell ele-ments, hence the nomenclature that it is a 2nd or-der model. The time-domain solution to a givenviscoelastic constitutive law is useful to verify theaccuracy of a simulation update law using the samemodel. In Section 5, we use these results to verifythat that the discrete realization of viscoelastic func-tions agree with the closed form solutions.

5 Realization of ViscoelasticFunctions for Haptic Synthesis

In a haptic synthesis framework, a force update lawis required given inputs of position and velocity,

fk = f(uk,k−1,..., uk,k−1,...), (7)

where fk is the force to be returned by the device atstep k, uk is the kth position measurement from thedevice, and uk is the kth velocity estimation or mea-surement. Input and output could be assumed to oc-cur in synchrony if the sampling period is sufficientlylong compared to the conversion delays. However, inmost practical realizations, it is often preferred toincrease the sampling as high as it is feasible. Theupdate law is then

fk = f(uk−1,k−2,..., uk−1,k−2,...). (8)

We now examine the realization of viscoelastic con-stitutive laws into an update law in the form of (8).Earlier, we discussed different approaches for real-izing devices that would be more capable than sys-tems based on measuring position only and produc-ing force only.

With the haptic simulation of viscoelastic behav-iors in mind, we examined two possibilities. The firstwas to improve the measurement or estimation ofvelocity at low speeds. The second was to considerprogrammable viscous actuators in addition to stan-dard motors (or perhaps the electronic modification


of the electromechanical behavior of standard mo-tors). We concluded that the later approach wouldbe more straightforward to realize. This is the as-sumption we adopt for the remainder of this paper.It should be noted however that similar manipula-tions could be carried out to account for the avail-ability of high quality velocity signals, that is downto 10−3 or 10−4 m·s−1.

In the next subsections we examine basic cases andlook at the realization of a 2nd order model that cantake best advantage of the physical properties of vis-cous brakes in parallel with torque generating mo-tors. It is also worthwhile noting that the series con-nection of a torque-generating motor with a viscousbrake, in essence a viscous clutch, could also havecertain applications but wouldn’t be directly appli-cable to simulating tissue behavior.

At this point, since the interaction with a tissuecan be assumed to be effected through a tool, weno longer need to use the notion of stress-strain inthe tissue or at its boundary. We can lump thesedistributed quantities into a force applied to a toolin response to a displacement.

5.1 Realization of a Kelvin Element

The realization of a Kelvin element is straightfor-ward due its parallel nature. The interaction with a“virtual wall” in the haptics literature is most oftenmodeled as a contact with a Kelvin element, suchthat

fk = Kk uk−1 +Bk uk−1, (9)

where uk [m] is the deflection of the spring andKk

[N·m−1

]and Bk

[N·s·m−1

]are the desired val-

ues of stiffness and damping at a particular samplingperiod. The quantities Kk and Bk can be allowed tovary provided that they do so slowly compared tou and u, lest they also participate in the variationsof f . For the purpose of this paper focused on vis-coelastic simulations of tissues, this assumption canbe assumed to hold.

A single 1-dof kelvin element, as described by (9)maps directly to a programmable parallel-connecteddamper-motor pair. Instead of providing damping byvelocity feedback (virtual damping), as would be re-quired for interfaces without programmable physicaldampers, two command signals can be sent in paral-lel. One to the amplifier-motor channel and the otherto the amplifier-damper channel. Provided that the

brake responds sufficiently fast, the system (and thehaptic experience) will remain passive for any valueof K and B provided that a minimal damping is com-manded (Colgate and Schenkel, 1994). Estimates ofthis minimum damping can easily be computed inopen loop from each sample Kk and from the sam-pling period (Hayward, 2007), or determined incre-mentally from cycle to cycle (Gosline and Hayward,2008). A single Kelvin element is the optimal vis-coelastic function for our dual-channel haptic inter-face, as it is a mirror image of the hardware.

Following Findley et al. (1976), the Kelvin model,like the Maxwell model, can be generalized to repre-sent a rich set of behaviors by connecting additionalelements, provided that they have different relax-ation times. Just like the 2nd order model mentionedin Section 4 is an example of a generalized Maxwellmodel with two time constants, the 2nd order gen-eralized Kelvin model has two elements in series torepresent two retardation times. It is shown in Fig. 3.

xu

B2B1

K1 K2

Figure 3: Generalized Kelvin model.

Because the Kelvin elements are in series, theyshare the same force. For simulation it is necessaryto compute the state, x [m], that represents the dis-placement of the elements relatively to the total dis-placement. The force balance is

K1(u− x) +B1(u− x) = K2x+B2x. (10)

Time discretization of (10) leads to an update lawfor x,

xk =(K1k uk−1 +B1k uk−1 +

B1k +B2k

∆txk−1

)(K1k +K2k +

B1k +B2k

∆t

)−1

. (11)

The force output can be computed using the lefthand side of (10), such that the static spring compo-nent K1(u−x) plus a term due to the slow migrationof the hidden state −B1x can be output by the mo-tors, while the dampers can handle the output of B1uwithout measuring velocity.


As an additional benefit, it is possible to choosewhich Kelvin element is used as an observer for theforce output. Numerical tests have shown that it isbest to select the Kelvin element that is responsiblefor the largest amount of dissipation within the as-sembly to maximize the passivity advantage of dualchannel rendering. This way, a user forcefully ma-nipulating the simulation will constantly be resistedby the damper—which comes “free”—and not by themotor. This realization generalizes to any number ofKelvin elements in series.

5.2 Realization of a Maxwell Element

The realization of a Maxwell element is less straight-foward than that of a Kelvin element because of theadditional state which must be computed. To furthercomplicate the matter, there are two possible config-urations which are mechanically equivalent but com-putationally distinct, yielding four different methodsto compute the force output of a Maxwell element.Referring to Fig. 4a or 4b, the force output can eitherbe computed by the difference of velocity across thedamper, or difference in position across the spring.

xuxu

a b

BK B K

Figure 4: Generalized Maxwell model. The two pos-sible realizations.

In the realization of Fig. 4a, the force balance is:

K(u− x) = Bx. (12)

Time discretization of (12) leads to an update lawfor x of the form

xk =(Kuk−1 +

B

∆txk−1

)(K +

B

∆t

)−1

. (13)

Once the state x is known, the output force can eitherbe computed by

fk = Bxk−1, (14)

or byfk = K(uk−1 − xk−1). (15)

It is clear from (14) and (15) that this method doesnot allow to take direct advantage of the motor-damper tandem. In the case of (15), there are no

velocity dependent terms. Equation (14) is veloc-ity dependent, and would work for an initial contactwith a Maxwell element. However, when the velocitydecreases, goes to zero, or changes sign, dampers areunable to generate the required transient forces. Analternate method for realizing a Maxwell element isas in Fig. 4b. The force balance is

Kx = B(u− x). (16)

In this realization, the contribution of Bu can be di-rectly provided by the physical dampers, leaving themotors to responsible for small and transient elasticcorrection forces according to −Bx. In this formula-tion, the update of x is computed by

xk =(Bk uk−1 +

Bk∆t

xk−1

)(Kk +

Bk∆t

)−1

. (17)

Once the state x is known, for the configuration asin Fig. 4b, the force output can either be computedby

fk = Kkxk, (18)

or byfk = Bk(uk−1 − xk). (19)

The only hardware-realizable force observation lawis (19), since the physical dampers can be used todirectly produce the term Bkuk by physics, and themotor be left to handle the term −Bkxk. This isrequired for transient elastic components only.

Velocity estimation is still required to computethe state x in (17), risking noisy velocity estimationto make its way into the force output law. How-ever, (17) is an integration process which is robustto noisy inputs. Moreover, in the force observationexpression (19), the correction term −Bx appearsonly during transients, and acts when the physicaldampers also contribute the Bu term completely ina physically exact manner. As a result, the physicaldampers contribute the greatest part of the outputforce, and the effect of a noisy and/or delayed veloc-ity estimation is minimized.

It is also worthwhile noting that this formulationcorresponds exactly to an early suggestion by Col-gate and Schenkel (1994) to maintain passivity of ahaptic interface by adding excessive physical damp-ing and removing it with negative computationaldamping. Such approach is guaranteed to err on theside to too much passivity rather than on the side oftoo little of it.


Generalized Maxwell models have a plurality ofMaxwell elements in parallel. For this arrangement,see Fig. 2e, an intermediate state, x1,2,···n, must becomputed for each of the n Maxwell elements con-nected in parallel. Due to linearity, the contributionfrom each discrete damper will sum, thus the totaldamping coefficient is the summation of each discretedamper, and the total motor contribution is the sumof each motor contributions.

5.3 Closed Form Solution Vs. Realiza-tion of a 2ndOrder Maxwell Model

Fig. 5 shows a comparison between the closed-formsolution of a 2nd order Maxwell model (Section 4,Fig. 2e) and the time-discretized realization as de-scribed above for contact with a constant strain rate.The upper left panel illustrates that the simulationagrees well with the closed form solution, even usinga fixed time step of 200 ms (K1= 12 N·m−1; K2 =10 N·m−1; K3= 5 N·m−1; B1 = 5 N·s·m−1; B2 =6 N·s·m−1). The same simulation with a time stepof 1 ms would be graphically undistinguishable fromthe closed-form solution. The lower left panel showsthe strain applied, a constant ramp of strain for 4seconds, followed by a constant strain after 4 sec-onds. The upper and lower right panels show thecontributions of the damper and the motor respec-tively, see (19). This figure shows the transients nec-essary to compensate for the physical damping when-ever the manipulandum changes velocity. The sharptransients in the motor’s contribution are needed dueto the sharp changes in velocity that are possible insimulation. In practice, humans accelerate at a finiterate, so under normal operating conditions, the mo-tor’s contribution always remains well conditioned.

5.4 Simulation of a Maxwell Element forPassivity Analysis

Using the dual-channel realization methods for a sin-gle Kelvin element can completely remove the veloc-ity estimation signal from the feedback loop. How-ever, for a Maxwell element, simulation of the in-termediate state x using (17) requires that the ve-locity of the interaction point, u, be known. Fig. 6shows results from a numerical comparison betweenthe closed form solution (grey line), a simulation ofconventional rendering using a motor for elastic andviscous components according to (19) (thick black

damper contribution

motor contribution

0 2 4 6 8 10

0

1

2

3

0 2 4 6 8 10

0 2 4 6 8 10

-1

0

1

2

-1

0

1

2

3

time [s]time [s]

time [s]

strain applied

force

0 2 4 6 8 10

0

0.1

0.2

0.3

.04

time [s]

solution

closed form

forc

e [N

]

forc

e [N

]ds

ipla

cem

ent [

m]

disp

lace

men

t [m

]

Figure 5: Realized simulation versus closed-form So-lution for 2nd order model. This example was com-puted with a time step of 200 ms to exaggerate theerror, which still remains small.

line), and a simulation of dual-channel rendering us-ing a motor for elastic and a physical damper for vis-cous components according to (19) (thin black line).The difference between the conventional and dual-channel cases is that the viscous component, Bu, iscomputed with a delayed velocity in the conventionalcase, while with the delay-free velocity in the dual-channel case. In both simulated cases, conventionaland dual-channel, Bx is computed with a delayedvelocity by (17). In the simulation, B = 5, K = 50,and ∆t = 0.001 s. The input, u, is modeled as 0.5 Hzsine wave of velocity, and the velocity estimation ismodeled as a 5 ms delay in velocity.

Note that in the output force plot, the single-motortrace (thick black line) exhibits considerable delay,but the dual-channel output (thin black line) ex-hibits a reduced force, and a smaller delay due tothe correction term −Bx and the delay-free physi-cal dampers. The energy trace in Fig. 6 is computedusing a passivity observer (po), as described by Han-naford and Ryu (2002):

Eobsv(k) = f(k)v(k)∆t. (20)

It is clear from the plots that using a single motor tooutput viscous components causes the energy traceto return too much energy due to delay in the velocityestimate as well as in the elastic term. By the con-vention used by Hannadord and Ryu (2002), energy


4.8

4.7

4.6

4.5 4.6 4.7time [s] time [s]

forc

e [N

]

-0.12

-0.10

-0.08

-0.06

5.02 5.04 5.06 5.08

ener

gy [m

J]

closed form

dual-channel

single motor

Figure 6: Output force and discrete energy of a singleMaxwell element using various velocity estimationmodels. Thick grey trace is closed form, thin blacktrace is dual-channel, thick black trace is motor only.It is clear that the single-motor synthesis yields non-passive energetic behavior whereas the dual channelsolution tracks the closed form solution nearly per-fectly.

dissipation is positive, so energy traces in the nega-tive regions indicates active behavior. In the energyplot of Fig 6 the energy trace goes lower into the neg-ative energy region and stays there longer than in theother cases. The dual energy behavior of the dual-channel technique indicates nearly perfect behaviorbut, as predicted, errs toward a slightly over-dampedbehavior. This finding agrees with prior work, no-tably that of Colgate and Schenkel (1994). Com-pensation for physical dissipation removes too muchenergy, and is thereby guaranteed to be passive.

6 Experimental Validation

As this work is primarily aimed at surgery simula-tion, multidimensional simulation problems shouldeventually be studied. But for the purpose of thisarticle, we restrict validation to the one-dimensionalcase since its success is a prerequisite to the feasi-bility of multidimensional cases. Moreover, it canbe observed that numerous medically relevant tool-tissue interaction cases occur along one single di-mension. Such is the case of needle insertion andcatheterization, both of which have dedicated com-mercially available one-dof simulators.1

As indicated earlier, when employing ecb’s forhaptic synthesis, fidelity and passivity can only in-crease in proportion to their overall contribution tothe final mechanical impedance experienced by theuser, without tradeoff. This is due to the reduction

1see www.immersion.com/medical

of reliance on velocity estimation suffering from delayand quantization effects.

6.1 Hardware

The prototype dual-channel haptic interface hard-ware comprises the Pantograph haptic interfacewhich is retrofitted with concentric annular alu-minum sections on each base arm. As shown inFig. 1, only one axis is used. The 1.59 mm thick,50 mm radius “damper blade” moves through theair gap of a powerful electromagnet driven by aswitching amplifiers set in current mode (Model amc20a20 Advanced Motion Controls, Camarillo, CA,usa) with a 150 vdc power rail. The motor is a core-less dc motor (Model re25, 118751, Maxon Motorsag) driven by a linear current amplifier (model lcam,Quanser Inc., Markham, on, Canada) with a 27 vdcpower rail. The position sensor is a 65k counts-per-revolution quadrature optical encoder (model r119,Gurley Precision Instruments, Troy, ny, usa). Sig-nal sampling and reconstruction is done with a isabus io card (model stgii-8, Servo To Go Inc., Indi-anapolis, in, usa) on a 2.8 GHz pc running LinuxKernel 2.6 and the Xenomai Realtime Framework.

6.2 Velocity Estimation

It is important to gain an appreciation for the mag-nitude of the noise and delay that is commonplacein haptic simulations with optical encoders. Despitethe reliable, drift -free position sensing ability of en-coders, raw velocity estimation by backward differ-ence is very noisy, and requires considerable filter-ing before it can be used in a haptic simulation. Asmentioned in the introduction, filtering techniquesare required to allow a velocity estimation to be fedback into the synthesis algorithm.

Fig. 7a shows the level of noise that is presentwith a single backward difference velocity estima-tion using our high precision encoders at low speed.Each step corresponds to a velocity quantum of 0.95radians-per-second with many switches. Fig. 7bshows three signals. The first is the quantized po-sition signal. The second is the output of non-causaldelay-free 100 samples averaging filter. The third isa filtered signal using a 120 Hz fourth-order Butter-worth fir low-pass filter. This figure exemplifies thebasic tradeoff that every velocity dependent hapticsimulation system faces. A smooth signal can be ob-


tained only at the cost of delay, here, the filteredsignal estimates cross zero with approximately 5 msof delay. The 120 Hz filter is necessary to generatevirtual damping of the same magnitude than thatcan be be produced by the ecbs prototypes, approx-imately 7 mN·m·s.

1.2 1.4 1.6 1.8 0.002 0.006 0.010 0.012time [s] time [s]

a b

position

zero delay filtered velocity

FIR filtered velocity

singlebackwarddifference

Figure 7: Noise and/or delay are associated with ap-proximating velocity from a discretized position sig-nal. Plots are drawn with arbitrary units.

6.3 Examples of Viscoelastic BehaviorHaptic Synthesis

Experiments were performed to demonstrate the useof physical damping for three different examples ofviscolastic behavior. The first case is the synthesisof a Kelvin model that exemplifies the difficulties as-sociated with the use of computational viscosity andthe second demonstrates the synthesis of a Maxwellmodel. Finally we show the realization of a visco-elasto-plastic model which can be used, for example,to simulate the feel of a needle insertion procedure.

6.3.1 Kelvin Model

It is apparent from Fig. 8 and caption that theaddition of physical dampers removes the delay-related artifacts that occur on contact and releasewith a Kelvin object. Force spikes are clearlypresent in the traces when the manipulandum crossesx = 0, and spikes results in the object feeling“sticky”. This effect was previously noted by severalresearchers (Rosenberg and Adelstein, 1993). Fol-lowing the initial contact with the Kelvin element,the manipulandum is released to show the dampedrecovery. With both physical and virtual damping,the recovery states look similar, and resemble theexponential function that represents the tissue re-covery. The position traces are different because theexperiment was performed manually.

0 1 2 3

0 1 2 3

time [s]

time [s]

position [cm]

0

0.5

forc

e [n

]

0

0.5

forc

e [n

]

0

8

position [cm]0

8

force

force

position

position

Figure 8: Kelvin viscolelasticity synthesized as per(9). The top panel shows the simulation realizedcomputationally. The bottom panel shows the com-manded torque to the motor. The viscous compo-nent is produced physically by the non-contact eddy-current viscous dampers.

6.3.2 Maxwell Model

Fig. 9 shows synthesis results from a 2nd order gen-eralized Maxwell viscoelastic model as in Fig. 2e. Inthis experiment, a solid metallic obstacle was placedon the path of the manipulandum, inside the vis-coelastic object, so it could be stopped sharply toshow the relaxation characteristic of the model. Boththe physical and virtually damped methods show analmost identical relaxation period. They are bothphysically realized by the motor. Since velocitychanges abruptly, a discontinuity must appear some-where in the simulation. The difference between thephysical and virtually damped methods is apparentat the time of contact and under active compres-sion of the virtual object. In the case of the virtuallydamped object, the shape of the curve resembles thatshown in Fig. 5 (upper left panel). In the case of thephysically damped object, the shape of the curve re-sembles that shown in Fig. 5 (lower right panel), asonly the motor contribution can be monitored in real-time using the hardware setup. From these plots itis apparent that a dual-channel haptic interface doesnot have as clear reduction of artifacts compared toa single motor. Nevertheless, the physical viscousdamping ensures passivity, and allows for stiff inter-actions with rigid objects, such as a bone. While thiscannot be shown by plots, these sharp transients areactually not felt because their occurrence scoincidewith the periods where the dampers are activated.


0 1 2 3

0 1 2 3

time [s]

time [s]

position [cm]

0

0.5fo

rce

[n]

0

0.5

forc

e [n

]

0

8

position [cm]0

8

force

position

Figure 9: Maxwell viscolelasticity synthesized as per(19).The top panel shows the simulation realizedcomputationally. The bottom panel shows the com-manded torque to the motor. The rest is producedphysically by the non-contact eddy-current viscousdampers.

6.3.3 Visco-elasto-plastic model

Percutaneous procedures are very common in mod-ern medicine to deliver therapeutic material or sam-ple tissues (Abolhassani et al., 2007). Proceduressuch as needle insertion (Alterovitz et al., 2003),catheter insertion (Gobbetti, 2000), or lumbar punc-ture (Gorman, 2000) are examples of medical simu-lations that could benefit from dual-channel haptics.Several authors have measured the forces of inter-action of needles as they slide in biological tissues,typically in porcine liver (Barbe et al., 2007; Oka-mura et al., 2004; Hing et al., 2007). These studiesshow that a model which can describe the behav-ior of a needle moving inside a tissue must includea plastic component describing the sliding of the in-strument in the tissues (irrecoverable displacement),an elastic component describing the deformation ofsurrounding tissues that can recover, and a viscouscomponent that changes according to whether theinstrument slides or is stuck. This general behav-ior can be modeled by the combination of elementsshown in Fig. 10 which can represent visco-elasto-plastic needle-tissue behavior.

The model shown in Fig. 10 corresponds to a vis-coelastic, stip-slick friction model where the squarebox represents a plastic slip element. In the stuckstate, it behaves like a Kelvin element with stiffnessK and damping B1. During sliding, the nonlinearplastic slip element slides, making x = u. Thus, in

xu

K B2

B1

Figure 10: Visco-elasto-plastic model.

sliding, the output is the summation of a constantspring force, Kδmax, and the two dampers in paral-lel, u(B1 +B2). The update law for this model is:

fk+1 =

Kδmax + uk−1(B1 +B2),xk+1 = uk − |uk−xk−1|

uk−xk−1δmax,

if |uk − xk−1| > δmax

K(uk − xk) +B1uk−1, otherwise,(21)

where δmax is the spring deflection at the transitionto sliding. Since our hardware does not have a forcesensor, it has only a position sensor, and hence isof the impedance display type, δmax represents thepresliding displacement in a manner similar to thefriction synthesis algorithm described by Haywardand Armstrong (2000). Update law (21) can be sep-arated into a elastic component to be realized by amotor and a viscous component to be realized bydamper.

Fig. 11 shows the results of the synthesis of a visco-elasto-plastic model. In the computational version(upper panel), transient spikes occur with a delayafter each velocity reversals. As expected from ourfindings where realizing a Kelvin element, these ar-tifacts are eliminated when using the dual-channelhaptic synthesis hardware. The haptic simulationusing the viscous damper and motor combinationimproves the fidelity considerably and also guaran-tees passivity of a needle insertion simulation havingvisco-elasto-plastic components.

The elasto-visco-plastic element, in essence, mod-els the local deformation of tissues interacting withan instrument. It could combined with other el-ements, specifically of the generalized Kelvin type(Section 5.1), to account for the global deformationdeformation of an organ.

Due to the parallel nature of both the viscoelasticpresliding and the viscoplastic sliding states of themodel, a parallel attached motor/damper is ideallysuited for display of such models.


0 1 2 3

0 1 2 3

time [s]

time [s]

position [cm]

0

0.5

-0.5

forc

e [n

]

0

5

-5

position [cm]

0

5

-5

force

force

position

position

0

0.5

-0.5

forc

e [n

]

Figure 11: Visco-elasto-plastic model synthesized asper (21). The top pannel shows the torque com-mand from the visco-elasto-plastic simulation real-ized computationally. The bottom pannel shows thedual-channel realization torque command.

6.4 Energetic Behavior of Maxwell Ele-ments

Simulation results from Section 5.4 showed that adelayed velocity estimation yielded extra energy atvelocity reversals in a Maxwell element. In this sec-tion, we present experimental results that supportthis finding. Note that only Maxwell elements areanalysed in detail, because the analysis of a Kelvinelement has been studied in detail by previous re-searchers, for example by Colgate and Brown (1994).

We turn our attention to the energetic behaviorof a Maxwell element. Of particular importanceis the behavior at velocity reversals because this iswhere observation-based methods generate eroneousresults. Fig. 12 shows experimental results of aMaxwell element at a velocity reversal, comparableto the simulation results described in Section 5.4.Shortly following the velocity reversal at approxi-mately 3.12 s, the Maxwell element changes fromcompression to elongation. It is clear from the figurethat the virtually damped Maxwell element returnsconsiderably more energy than the ideal case, whilethe dual-channel Maxwell element returns too little,which agrees with the previously discussed simula-tion results. For the experimental energy plots, adelay-free velocity signal was generated using first anon-causal 20 sample center-weighted averaging fil-ter, followed by a 200 Hz Butterworth zero-delaydigital filter implemented with the matlab R© Filter-

ing Toolbox, zero phase function filtfilt(). Thisdelay-free velocity signal was used to infer the con-tribution of the physical dampers necessary that isnecessary to generate the dual-channel trace. It isimportant to note that the energy traces were com-puted offline, as it is not feasible to use non-causal,delay-free filters online.

1.5

1.0

0.5

time [s]3.0 3.1 3.2 3.3

2

-1

-2

time [s]3.0 3.1 3.2 3.3

posi

tion

[rd]

ener

gy [J

]

0

1 ideal

dual-channel

singlemotor

Figure 12: Energetic comparison at a velocity rever-sal of Maxwell element realizations. Thick grey lineis ideal, thin black line is dual-channel, thick blackline is single-motor actuation. As predicted, the sin-gle motor implementation displays active energeticbehavior whereas the dual channel implementationis clearly passive.

From a separate test, Fig. 13 shows the resultsfrom a drag and release experiment done with a sin-gle Maxwell element. Upon release, it is clear thatthe dual-channel realization exhibits a more dampedresponse, which yields fewer oscillations and a fastersettling time compared to the more oscillatory be-havior of the single motor realization. Because theexperiments were performed manually, attention waspaid to the initial conditions. At 4.2 s in the single-motor, feedback-damped case, and 11.2 s in dual-channel, physically-damped plot, the release velocitywas 1.8 rd·s−1. Notice how the virtually damped ele-ment oscillates down to −3.8 rd·s−1, while the phys-ically damped element reaches only −2 rd·s−1. Thedifference between these two results is entirely at-tributed to the delayed velocity estimation necessaryfor vibration-free display of a virtual damping coef-ficient.

7 Conclusions And Future Work

We introduced a method for the haptic synthesis ofviscoelastic media using a dual-channel haptic in-terface. The prototype uses a dc motor as a pro-grammable torquer in parallel with an ecb as a pro-grammable viscous damper. This design is aimed


posi

tion

[rd]

2.0

1.5

-4.03.8 4.0 4.2 4.4 4.6 4.8 10.8 11.0 11.2 11.811.611.4

1.0

0.5

2.0

1.5

1.0

0.5

-2.0

0

2.0

-4.0

-2.0

0

2.0

time [s] time[s]

velo

city

[rd/

s]

single-motorrealization of aMaxwell element

dual-channelrealization of aMaxwell element

Figure 13: Drag and Release of a single Maxwell el-ement. The left pannels show a virtually damped,single-motor realization of a Maxwell element. Theright pannels show the physically damped, dual-channel realization of the same Maxwell element.The small circle indicates release of the manipulan-dum. In each case, the release velocity was nearlyidentical, approximately 1.8 rd·s−1.

at avoiding or minimizing the dependence on a noisyand/or delayed velocity estimation signal. Due to theparallel nature of the device, viscoelastic functionsthat are connected in parallel, such as the KelvinModel, or a visco-elasto-plastic model, can be sim-ulated without velocity estimation in the feedbackloop of the torquer. This approach minimizes arti-facts that result from a delayed signal and actuatorsaturation.

For viscoelastic functions that are serial in nature,such as the Maxwell Model, the benefits of using thehybrid device are not as clear cut. A state must besimulated between each serially connected damper-spring pair, and due to the purely dissipative na-ture of the ecb damper, the torquer must be used tocompensate for the damper during transients. Thesetransient torque corrections are velocity based, thusvelocity estimation is required in the torquer feed-back loop. The torquer corrections, however, areoutput when the physical viscous dampers are en-gaged, which minimizes the destabilizing effects thata delayed velocity signal can have on a haptic ren-dering and does not translate into tangible artifacts.Simulation and experimental results have shown thatthe dual-channel method improves the stability and

passivity of a Maxwell element.The examples we have given were produced with

arbitrary viscoelastic parameters which may notmatch the medical reality. They were selected ac-cording to the capabilities of our available hardwareand to exemplify the various properties of the dual-channel haptic synthesis approach. Medical proce-dures are numerous and varied, from orthopedic op-erations to retinal scrapping. It is clear that no sin-gle piece of haptic hardware will be capable to beadequately applied to the simulation of all possiblecases. But starting from insertion procedures, to pal-pation, to more complex surgical gestures, one, twoand higher numbers of actuated degrees of freedomin haptic devices could benefit from dual-channel ac-tuation.

Our present efforts are directed at extending thehardware haptic realization theory and techniquesto multi-dimensional cases, as per the recent workby Hover et al. (2009). This includes the specifi-cation of multidimensional visco-elasto-plastic func-tions, not at one single point of interaction, but any-where on the surface or inside a virtual organ andthe methods by which discontinuities can be han-dled without violations of hardware constraints suchas actuator saturation and amplifier slew rates. Animportant aspect of a multidimensional synthesis ofa force field is the generation of an arbitrary viscousfield. While the theory behind the generation of suchfields is not yet fully developed, preliminary resultswere shown in (Gosline et al., 2006), using motor cor-rections to eliminate the parasitic effects from cou-pled dampers. This approach requires further devel-opment and testing, and a method for online inter-polation of dynamics response functions is requiredas well. There are also interesting new avenues toexplore such as the design of compact and efficientecbs. Preliminary engineering data indicates thatsuch devices could be realized with form factors andmasses comparable to the dc motors to which theycould be optimally coupled.

Acknowledgment

This work was funded by a Collaborative Researchand Development Grant “High Fidelity Surgical Sim-ulation” from nserc, the Natural Sciences and Engi-neering Council of Canada and by Immersion Corp.,and by a Discovery Grant also from nserc.


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Dual-Channel Haptic Synthesis of Viscoelastic Tissue ... · Dual-Channel Haptic Synthesis of Viscoelastic Tissue Properties Using Programmable Eddy Current Brakes Andrew H. C. Gosline

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