2.160: system identification, estimation and learning … · 2.160: system identification,...

2.160: SYSTEM IDENTIFICATION, ESTIMATION AND LEARNING 1

Real-Time Parameter Estimation of BiologicalTissue Using Kalman Filtering

Joshua B. Gafford

Abstract—Soft tissue parameter estimation embodies a widearea of active research in academia with the goal of creatingaccurate finite element models for haptic simulations and needleinsertion modeling. The rise of robotics in surgery has alsoopened up the possibility of robotically palpating tissue and usinglocal mechanical properties of the tissue as biomarkers to localizemetastases in vivo. Up until now, most tissue parameter estima-tion approaches utilize heuristic methodologies or quasi-staticimplementations which can fall victim to the time-dependent,viscoelastic properties that biological tissues exhibit. Approachesthat consider tissue dynamics in parameter fitting typicallyemploy primarily batch-based regressions which are not suitablefor online implementation. In this work, we explore the possibilityof using Kalman filter-based parameter estimation techniques toquantify viscoelastic tissue properties online in real time. In doingso, we develop a discretized model of the nonlinear tool/tissuedynamics and analyze the model for observability. The model isthen linearized and three different Kalman filter methodologies(Extended, Unscented, and Adaptive Fading) are implemented toestimate inherent tissue properties based on a popular model oftissue dynamics (Kelvin-Voigt). Functionality is demonstrated insimulation, and an experimental platform is fabricated to test thefilter properties in hardware. We use this platform to assess theefficacy of the three filters in a variety of different scenarios.Convergence is assessed by analyzing the trace of the errorcovariance. Ultimately, we have demonstrated that it is possible torepeatably estimate the viscoelastic properties of biological tissuewith sufficiently fast (<200 ms) convergence. This work could bea preliminary step in creating robust estimation algorithms thatcould potentially enable low-level robotic control decisions to bemade based on the local mechanical footprint of the biologicaltissue.

Index Terms—Soft Tissue Modeling, Surgical Robotics, Param-eter Estimation, Kalman Filtering

I. INTRODUCTION

THE rise of robotics in minimally-invasive surgery hasequipped surgeons with enhanced dexterity and control

inside the anatomical workspace in ways that are unattainableor unintuitive with manual tools. Advanced robotic controlsystems are able to perform such feats as motion scaling, vir-tual fixturing, and tremor suppression to improve a surgeon’sperformance in vivo. However, the lack of haptic feedbackhas been widely recognized as a barrier to innovation andwidespread application, as surgeons are unable to physicallyfeel the forces being imparted at the tool-tissue interface.

Although force sensing in minimally-invasive surgery is awidely-studied field, most efforts focus on force reflectionand haptics applications, and few groups have explored theopportunity to use these sensing modalities to make low-level, automated decisions at the end-effector level. A po-tential application is to use on-board sensors and actuatorsto characterize the material properties of tissue in order to

β

k

β0

k0

m

BiologicalTissue

Analyzer

u,v x1,x2

Fig. 1: Equivalent mechanical model of tool/tissue interaction assuming aKelvin-Voigt tissue model

make informed control decisions based on the tissue’s localpathology in both a diagnostic and therapeutic scenarios.

Robotic tissue palpation has been identified as a promisingmeans of localizing and diagnosing potentially metastatictissue in vivo without having to acquire a biopsy and performpathological tests. The basic approach is to probe or indentbiological tissue at a known depth and record the axial force,fitting the results to a simple first-order spring model of thetissue. While many groups have focused on quasi-static (<0.1Hz) palpation [1]–[3], the nonlinear and viscoelastic nature ofbiological tissue can lead to widely variable results if testingconditions are not replicated adequately. To get around thisproblem, applying dynamic (>10 Hz) mechanical waveformsto tissue could potentially generate more robust measurementsthat are less vulnerable to errors induced by viscoelasticity(relaxation/creep effects).

The idea of dynamic tissue characterization is not partic-ularly new, and several groups have used piezoceramics [4],[5] and voice coil [6] elements to mechanically excite tissueat high frequencies. However, system identification approachesused in literature have heavily relied on batch-based regressionprocesses (i.e. post-processing/fitting) or heuristics (i.e. ob-serving the empirical Bode plot generated by a chirp signal andfitting a model to the observed data) with no real-time compo-nent. Okamura et. al. implemented a batch-based regression todetermine stiffness and damping parameters involved in needleinsertions [8], and Barbe et. al. performed a similar feat butwith an on-line algorithm based on recursive least-squares [7].In both cases, the authors were able to linearize the system forlinear least squares by ignoring needle dynamics, which maynot be a particularly robust assumption. In [7], the estimationprocess suffered from relatively slow convergence times (>1


seconds), and due to the mechanics of tissue puncture, it wasnot possible for the tissue stiffness estimate to converge as itwas highly dependent on the interaction state.

To the author’s knowledge, no group has explored thepossibility of applying Kalman filtering to a nonlinear modelof tool/tissue interaction to enable real-time, online estimationof the dynamic parameters of an unknown tissue. In addition,most approaches in literature are not particularly concernedwith estimate convergence time, as the intended applicationswere primarily for creating finite element or analytical modelsof tissue behavior. Developing robust estimation algorithmswith fast (<200ms) convergence could lead to the developmentof ‘smart’ instruments and tissue cutting devices that canestimate the tissue parameters in real-time and make low-level control decisions based on the information (i.e. a ‘smart’cutting device that will only cut a certain type of tissue). Thegoal of this work is to apply Kalman filter-based parametricidentification techniques to characterize tissue in real-timebased on a model of the tool/tissue interaction, as shown inFig. 1. Local observability is affirmed via an analysis of thenonlinear system with respect to the Lie derivative. The systemis linearized, and three types of Kalman filter (Extended,Unscented, and Adaptive Fading Extended) are comparedboth in simulation and hardware to determine unknown tissueparameters (stiffness, damping) based on deterministic inputs(displacement) and measured outputs (force). The efficacy ofthe estimation process is quantified in terms of convergencerate, repeatability, and fidelity to baseline measurements.

II. MODELING AND SIMULATION

In designing an optimal filter for parameter estimation, anaccurate model must be developed that characterizes systembehavior. In this section, we consider the material model usedto parametrically characterize viscoelastic tissue behaviors,and develop a coupled system of equations for describing theinterface between the tissue and the tool.

A. Material Model Selection

Several models exist for parametrically modeling viscoelas-tic behavior in biological tissue. These models vary in termsof accuracy with respect to constant-stress (creep) or constant-strain (relaxation) conditions, and complexity in terms ofthe number of parameters required to fully parameterize themodel. Three popular models are listed in Table I, along withthe number of states and parameters required for characteri-zation. Although the Standard Linear Solid and GeneralizedMaxwell models offer the most accurate results, they aretypically difficult to calculate and require many parameters.Due to simplicity, we assume a Kelvin-Voigt model for thebiological tissue characterized by an effective stiffness k anddamping β.

B. Continuous and Discrete Dynamic Model

Optimal parameter estimation requires (1) a model of thereal system (hidden), (2) observations of the system states(obfuscated), and (3) the current best-estimate of the system

TABLE I: Three popular viscoelastic models

Model Nstates NparamsMaxwell 3 2Kelvin-Voigt 2 2Std. Linear Solid 4 3Generalized Maxwell >3 >3

parameters. In this section, we will develop a nonlinear modelof the tool/tissue dynamics (both continuous and discrete).As shown in Fig. 1, the tool is modeled as a mass-spring-damper system with mass m, spring constant k0, and dampingcoefficient β0. The input to the system is a known proximaldisplacement, and the interaction force between the tool andthe tissue is measured by a force sensor. The exogenous inputto the system is a known (measurable) displacement u(t)and velocity u(t) which can vary as a function of time. Themeasured quantity is the reaction force between the tool andthe tissue (given by F (t)). We can construct the continuousdynamics of this system as follows:

mx(t) = k0 (u(t)− x(t))

+ β0 (u(t)− x(t))

− kx(t)− βx(t)

(1)

F (t) = k0(x(t)− u(t)) (2)

Note how we observe the state (i.e., the displacment of proofmass m) through the force measurement.

In order to implement the estimator in hardware, we need toconvert this continuous model into discrete time. To model thedynamics of the tool-tissue interface, we discretize derivativesusing first- and second-order finite difference approximationsgiven a sample rate ∆t. For parameter estimation, it isconvenient to introduce an augmented state matrix X =[x1, x2, k, β]T which contains the tissue parameters to be esti-mated (here we assume parameters are constant). As such, wecan construct a nonlinear system of equations describing thestate and measurement evolution based on nonlinear functionsft(Xt, ut) and ht(Xt, ut).

Xt+1 = ft(Xt, ut) +Gtwt (3)

Yt = ht(Xt, ut) + vt (4)

where wt and vt are the process and measurement noise,respectively, and are characterized by the following:

E[wtws] =

{Qt t = s

0 t 6= s, E[vtvs] =

{Rt t = s

0 t 6= s∀t, s

(5)Expanding ft(Xt, ut) and ht(Xt, ut):

x1,t+1 = x1,t + ∆tx2,t (6)


x2,t+1 = x2,t +∆t

m

[k0 (ut − x1,t)

+β0

(ut − ut−1

∆t− x2,t

)− ktx1,t − βtx2,t

]+ wt

(7)

kt+1 = kt (8)

βt+1 = βt (9)

Our measurement equation is as follows:

yt = k0(x1,t − ut) + vt (10)

Estimating two parameters kt, βt in addition to the statesx1,t, x2,t requires an input u(t) that is persistently exciting tothe system, i.e., an input that satisfies the following condition:

det ΦTΦ =

[Ru(0) Ru(1)Ru(1) Ru(0)

]6= 0 (11)

Ru(τ) = E[u(t)u(t− τ)] (12)

It can easily be shown that u(t) = A sin(ωt) satisfies theP.E. condition to O(2). Therefore, we can estimate the requiredparameters by applying a sinusoidal input which is known tobe persistently exciting to the second order.

C. Observability

Any parameter estimation problem hinges on the systembeing observable. If certain states cannot be accessed throughthe measurement equation, then these states cannot be esti-mated. Whereas, in the case of a linear system, observabilitycan be ascertained through the Cayley-Hamilton theorem, theobservability of a nonlinear system is intimately related tothe Lie derivative. If we can show that a matrix containingthe gradient of n Lie derivatives has rank n, then the systemcan be said to be locally observable. The Lie derivative of ascalar function h : Rn → R (the nonlinear measurement) withrespect to vector field f : Rn → Rn (the nonlinear state) isgiven by the following:

Lf (h) =∂h

∂xf(x) (13)

G =

L0f (h)

L1f (h)

L2f (h)

L3f (h)

(14)

where L0f (h) = h and higher-order derivatives are recur-

sively related by Lnf (h) = ∂∂x

[Ln−1f (h)

]f . We want to show

that the gradient of the Lie derivative vector G ∈ R4×1 hasfull rank:

∂G

∂X=

∂L0

f

∂x1

∂L0f

∂x2

∂L0f

∂k

∂L0f

∂β∂L1

f

∂x1

∂L1f

∂x2

∂L1f

∂k

∂L1f

∂β∂L2

f

∂x1

∂L2f

∂x2

∂L2f

∂k

∂L2f

∂β∂L3

f

∂x1

∂L3f

∂x2

∂L3f

∂k

∂L3f

∂β

(15)

By solving for the determinant of this matrix, we can deriveconditions under which the system is (locally) observable:

det(∂G

∂X

)=

∆t2

m3

[x2

1,t(k0 + kt) + x22,tm

+ x1,tx2,t(β0 + βt)− x1,tutβ0 − k0x1,tut](16)

From this, we can conclude that the system only becomesunobservable when x1,t = x2,t = 0 (i.e. no motion in thesystem). As such, we can proceed with confidence knowingthat our system is locally observable so long as it is beingexcited.

D. Linearization

We can see that the discretized model is nonlinear, as wecannot formulate a state transition matrix that doesn’t containthe states or parameters being estimated. Implementing linearfiltering techniques requires the system to be linearized aboutthe current state estimates Xt. Linearizing the system aboutXt, we formulate the linearized state transition and outputmatrices Ft and Ht, respectively.

∂f

∂X

∣∣∣∣Xt

=

∂f1∂x1

∂f1∂x2

∂f1∂k

∂f1∂β

∂f2∂x1

∂f2∂x2

∂f2∂k

∂f2∂β

∂f3∂x1

∂f3∂x2

∂f3∂k

∂f3∂β

∂f4∂x1

∂f4∂x2

∂f4∂k

∂f4∂β

∣∣∣∣∣∣∣∣∣Xt

=

1 ∆t 0 0

−∆tm (k0 + kt) 1− ∆t

m (β0 + βt) −∆tm x1,t −∆t

m x2,t

0 0 1 00 0 0 1

= Ft ∈ R4×4

(17)

∂h

∂X

∣∣∣∣Xt

=[∂h∂x1

∂h∂x2

∂h∂k

∂h∂β

]∣∣∣Xt

=[k0 0 0 0

]= Ht ∈ R4×1

(18)

III. KALMAN FILTERING

The goal of this section is to present the algorithms forthree different flavors of Kalman filter that will be comparedin simulation and in hardware.

A. Extended Kalman Filter

With the linearized system, we can implement an extendedKalman filter (EKF) to estimate both the states and theparameters involved in the linear system of equations. We


initialize with an a priori error covariance P1|0 = P0 ∈ R4×4

where dim(X) = 4. We compute the Kalman gain Kt ∈ R4×1

as follows:

Kt = Pt|t−1HTt (HtPt|t−1H

Tt +Rt)

−1 (19)

where Rt ∈ R1 is the measurement noise covariance. We usethis to compute our apriori state estimate X−t based on thenonlinear system model h(Xt, ut).

X−t = f(Xt, ut) (20)

We propagate the a priori estimate through the nonlinearoutput model h(Xt, ut):

Xt = X−t +Kt(yt − h(Xt, ut)) (21)

We then update our covariance and propagate it forward(using Joseph’s form for Pt for numerical stability, where Iis a 4× 4 identity matrix):

Pt = (I −KtHt)Pt|t−1(I −KtHt)T +KtRtK

Tt (22)

Pt+1|t = FtPtFTt +GtQtG

Tt (23)

where Qt ∈ R4×4 is the process noise covariance and Gt ∈R4×4 is the process noise state matrix.

B. Adaptive Forgetting

It is well known that when system information is onlypartially known or incorrect, the EKF may diverge or givebiased estimates. This condition could potentially arise whenthe tissue properties abruptly change or go to zero (i.e. theanalyzer dis-engages from the tissue). As properties changein time, the the EKF is not very reactive as past data andrecent data are weighed equally. To mitigate this, an adaptivefading EKF (AFEKF) can be implemented which modifies theExtended algorithm by adding an exponential forgetting factorthat is adapted in real time based on system covariance [9]. Weintroduce three new scalars: Mt ∈ R1, Ct ∈ R1, and Nt ∈ R1.

Mt = HtFtPtFTt H

Tt (24)

Ct = HtPt|t−1HTt +Rt (25)

Nt = Ct −HtGtQtGTq H

Tt −Rt (26)

We use these to calculate the forgetting factor λt which isupdated in real-time based on the covariance.

λt = max(1, αtrace(NtM

−1t ))

(27)

Parameter α is a heuristic that allows us to further adjustthe forgetfulness of the algorithm (where 1 < α < 1.003 hasshown acceptable results). Finally, we modify the a priori statecovariance as follows:

Pt|t+1 = λtFtPtATt +GtQtG

Tt (28)

A notable feature of this algorithm is that the covarianceis typically slower to converge when compared to EKF asexponential forgetting forces the system to be ’less trusting’of old data.

C. The Unscented Transform

It is known that the EKF can perform poorly in the faceof highly nonlinear behavior which exhibit large deviationsfrom the piece-wise linearized dynamics. Unscented KalmanFilters (UKF) attempt to overcome this by estimating thedistribution of the state and propagating it through the stateand measurement equations. The filter relies on the unscentedtransform. Given our state vector X ∈ R4×1, we compute9 sigma points (2n + 1 where n = 4) to characterize thedistribution of X:

X0t−1 = Xt−1 (29)

Xit−1 = Xt−1 +

√n+ κσivi (30)

Xi+nt−1 = Xt−1 −

√n+ κσivi (31)

where σi and vi is the ith eigenvalue and associatedeigenvector in the prior state covariance matrix Px. Each sigmapoint is weighed by:

W0 =κ

n+ κ(32)

Wi = Wi+n =1

2(n+ κ)(33)

We propagate each sigma point through the deterministicpart of the nonlinear state transition equation:

X∗it|t−1 = f(Xt−1, ut−1, t− 1) (34)

and predict the mean of these propagated points:

Xt|t−1,s =

2n∑i=0

WiX∗it|t−1 (35)

The predicted covariance is:

Pt|t−1,s =

2n∑i=0

Wi(X∗it|t−1 − Xt|t−1,s)(X

∗it|t−1 − Xt|t−1,s)

T

+Qt−1

(36)We use the eigenvectors and eigenvalues of covariance

Pt|t−1,s to generate a new set of sigma points, which are fedthrough the observation equation

yit = h(Xit|t−1, t) (37)

and predict the observation by:

yt =

2n∑i=0

Wiyit (38)


Fig. 2: Simulink diagram of system model and three parallel filters

We compute the output estimation covariance using thelinearized observation matrix:

Py = HtPt|t−1HTt +Rt (39)

If we set κ = 2, the sample mean yt and sample covariancePy approach the true mean and variance to the second order.Finally, the Kalman gain can be computed:

Kt = Pt|t−1HTt P−1y (40)

We use the Kalman gain to update the state estimate:

Xt = Xt|t−1 +Kt(yt − yt) (41)

Lastly, we update the state estimate covariance with xt andthe process repeats itself.

D. Simulation

To verify that the filters can successfully estimate bothstates x1, x2 and parameters k, β in the system, the nonlinearmodel in Equation (1) was implemented in Simulink usinga continuous time, variable step solver (ode45). The threeKalman filters were added in parallel and implemented indiscrete form with a sample rate ∆t = 0.5ms (a 2 kHzsampling frequency). The Simulink block diagram is shown inFig. 2. The actual tissue stiffness kact and damping βact wereset initially (to assess filter convergence) and then doubled attime t = 2 (to assess filter robustness to model changes).

The estimated stiffness kt and damping βt are shown inFigs. 3 and 4, where the dotted lines correspond to the actualstiffness and damping values. We see that, in simulation, thefilter estimates are able to converge to the correct parametervalues kact and βact very quickly (<200ms). However, whenthe stiffness changes, only the AFEKF is able to estimate thenew parameter values in a timely manner, as expected. Wewill confirm this result again in hardware.

Time [s]0 0.5 1 1.5 2 2.5 3 3.5 4

Estim

atedStiffnesskt[N

/m]

0

500

1000

1500Estimate (UKF)Estimate(EKF)Estimate (AFEKF)

Fig. 3: Simulink model with a step change in stiffness at t = 2 seconds.The dotted line shows the actual stiffness. TThe AFEKF is quick to convergethanks to exponential forgetting, whereas the EKF and UKF are much slower.

Time [s]0 0.5 1 1.5 2 2.5 3 3.5 4

Estim

atedDampingβt[N

·s/m]

0

5

10

15

20


Fig. 4: Simulink model with a step change in damping at t = 2 seconds.The dotted line shows the actual damping. The AFEKF is quick to convergethanks to exponential forgetting, whereas the EKF and UKF are much slower.The UKF appears to have a biased estimate compared to the other two filters.

IV. EXPERIMENTAL SETUP

Simulations have shown that, assuming the model is cor-rect, the three filter modalities can estimate the stiffness anddamping of an unknown tissue analog, with varying degreesof robustness to model changes. To demonstrate this capabilityin hardware, a system was fabricated to test the Kalmanfilters on actual material analogs. A detailed description of theexperimental setup is presented in the following subsections.

A. Hardware

The experimental setup, shown in Fig. 5 (left), consists ofa high-bandwidth linear voice coil actuator (Moticont LVCM-025-038-01) which provides the input to the system. A linearencoder (Heidenhain AK LIDA 47 TTL) tracks the position ofthe moving coil which is constrained to move unidirectionallyby linear bearings. A beam-type load cell (Omega LCL-005) is attached to the moving coil which provides the forcemeasurement. The stiffness of the load cell provides thesystem stiffness constant k0, and coupled with the proof mass,sets an upper-limit on the system bandwidth based on theresonant frequency (f−3dB = 1

2π

√k0/m). The entire system


Linear Voice Coil Actuator

Dual BeamLoad Cells

SuctionInterface

ThorLabsMicrometer

LinearBearings

Linear Encoder

TissueSpecimen

Fig. 5: (left) Experimental setup. A linear voice coil actuator provides a time-varying excitation input to the tissue analog. A linear encoder measures theposition and beam-type load cells measure the force, (right) Electrical schematic showing high-power dual push-pull amplifier to provide bi-directional currentfrom a single-sided power supply

is mounted on a Thor labs micrometer, angled perpendicularlyto the mounting surface, to allow for fine positioning abovethe tissue surface.

The system was heuristically characterized to obtain thefree-moving stiffness k0 and damping β0. An impulse wasapplied to the system, and a 2nd-order mass spring dampermodel was fitted to the results (shown in Fig. 6). From thisanalysis, k0 = 9700N/m and β0 = 0.8N ·m/s. Given a massof 0.04kg, the resonant frequency of the system is 78 Hz.As such, for our experiments, we want to excite the tissue atsignificantly below this frequency, so that natural oscillationsin the load cell do not contaminate our measurements.

B. Electrical and Software

A National Instruments USB-6002 DAQ provides the exci-tation for the voice coil, and also acquires data from the loadcell and the encoder. The load cell, a thermally-compensatedfull-bridge type with precision-matched resistors, generates adifferential voltage that is amplified by an instrumentationamplifier (AD627, Analog Devices) with a gain of 1000 (andresolution of about 1 mN) to generate a sensitivity of 0.5 V/N.A quadrature encoder IC (Avago HCTL-2032-SC), externallyclocked by a 2 MHz square wave generated by a 555 timer inastable mode, generates a 32-bit output that can be interrogatedby the DAQ at any time without depending on the DAQ’sinternal clock for counting encoder pulses. In order to runthe entire system off of a single-sided 0-10V power supply,a high-power push-pull amplifier creates a virtual ground at5V, which biases the voice coil. Another push-pull amplifiergenerates the bi-directional required current to drive the voicecoil based on the analog output signal from the DAQ. A 5Vregulator provides TTL-level power for the encoder readheadand load cell signal conditioner. A schematic of this systemis shown in Fig. 5 (right).

V. EXPERIMENTAL VALIDATION

Numerous experiments were performed to observe filterperformance in response to a number of different variables.

0 0.05 0.1 0.15 0.2-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2System Response

Model Fit

Time (seconds)

Am

plitu

de [N

]

Fig. 6: Experimental system response to an impulse, and a 2nd-order modelfit showing agreement

Several tissue analogs of varying stiffness were molded outof silicone-based rubbers (EcoFlex 00-10, EcoFlex 00-50,VytaFlex 20, DragonSkin 20, all manufactured by Smooth-On, Easton, PA). Tests were performed under a number ofconditions and data were collected at a rate of 2 kHz. Althoughthe filters are recursive and can be implemented in real-time,for the purpose of demonstrating feasibility, the filters wereapplied in post-processing to generate state and parameterestimates.

A. Baseline Functionality

Preliminary tests were performed to demonstrate that theimplemented filters demonstrated stability and decent con-vergence properties for unknown materials. A sample tissueanalog was molded out of EcoFlex 00-10. The tissue wasprobed at a frequency of 10 Hz for two seconds. Givenan input frequency of 10 Hz, a moving-average filter witha window of 200 samples was applied to the stiffness anddamping estimates to smooth the output. Process noise and


Time [s]0 0.2 0.4 0.6 0.8

Estim

atedStiffnes

kt[N

/m]

-100

0

100

200

300

400

500

600

Estimate (UKF)Estimate(EKF)Estimate (AFEKF)

(a) StiffnessTime [s]

0 0.2 0.4 0.6 0.8

Estim

atedDampingβt[N

·s/m]

0

5

10

15

20

25

30

35


(b) DampingTime [s]

0 0.2 0.4 0.6 0.8

trace[P

t|t−1]

0

0.5

1

1.5

2

2.5

3

3.5

4UKFEKFAFEKF

(c) Covariance Trace

Fig. 7: Baseline experimental results for EcoFlex 00-10. The results show that the filters converge to a steady-state value within 200 ms, and the covariancequickly converges as well. Comparison with a simple first-order contact model demonstrates agreement.

Time [s]0 0.2 0.4 0.6 0.8 1

Force[N

]

-1

-0.5

0

0.5

1Actual ForceEstimate (UKF)Estimate (EKF)Estimate (AFEKF)

Fig. 8: Force estimate vs. actual measured force during baseline experiments.

measurement noise were empirically determined to be wt =[0, 1× 10−8, 0, 0]T and vt = 2.5× 10−5, respectively.

The results of the parameter estimation are shown in Fig. 7.Although it is difficult to verify the accuracy of the results,we can compare the estimated stiffness to that predicted by afirst-order model of the contact mechanics, treating the tool asa cylinder with radius a:

km =2Ea

(1− ν2)= 440N/m (42)

where E is the 100% modulus of the material (specifiedby the manufacturer to be 55 MPa), a is the radius of thesuction probe (3 mm), and ν is the Poisson ratio (ν = 0.5 formost rubbers). As we can see, the model prediction matchesfairly well with the estimated stiffnesses, lending some validityto the estimates. However this model is quasi-static. We willexplore the frequency characteristics of our estimates in thenext section.

In Fig. 7(c), we have plotted the trace of the a prioricovariance Pt|t−1 as a function of time. This provides us witha measure of the ‘uncertainty’ (i.e. variance of the error) in thefilter estimates at any time t. We observe how the covarianceof the EKF and UKF decay almost identically, whereas thecovariance of the AFEKF is a bit slower to decay due toexponential forgetting.

The estimated force (Ft = k0(x1,t − ut)) is compared tothe measured force signal in Fig. 8. All filters perform well

in estimating the actual force, with more noise in the AFEKFestimate due to the forgetting factor.

B. Frequency Dependency

Viscoelastic materials combine properties of linear solidsand Newtonian fluids, and as such, the material behavioris often dependent on the speed (or frequency) of loading.These complex behaviors are encapsulated in storage (E′) andloss (E′′) moduli, which represent elastic and viscous behav-ior, respectively, and combine to form a ‘complex modulus’E∗ = E′ + iE′′ = σ0

ε0eiφ, where σ0 is the applied stress, ε0

is the resulting strain, and φ is the phase angle between thetwo. Typically, for viscoelastic materials, the storage modulusmonotonically increases as a function of frequency, often byfactors of 2 or more, asymptotically approaching some valueE′∞. The loss modulus typically peaks at some frequencyfpeak and decays at higher frequencies, asymptotically ap-proaching some value (E′′∞).

We can draw a parallel between dynamic moduli and ourestimated parameters (E′ → kt and E′′ → βt), as theydescribe similar phenomena in the viscoelastic response.

To understand this behavior further and observe the impli-cations of frequency dependency on our estimated parameters(specifically, the convergence properties), a frequency sweepwas performed on EcoFlex 00-10 silicone rubber. Frequenciesranging from 1Hz to 30Hz were applied to the specimen,resulting in the behavior shown in Fig. 9. As expected, theestimated stiffness tends to increase as a function of frequency,leveling off at around 25 Hz. The damping also behaves asexpected, with an initial decay leveling off at about 25 Hz.Estimates are relatively consistent across all filters with moreerror variance in the AFEKF as expected.

C. Stiffness Dependency

Four different viscoelastic materials with varying stiffnesses(EcoFlex 00-10, EcoFlex 00-50, VytaFlex 20, DragonSkin 20)were parametrically identified with the AFEFK to understandhow filter robustness scales with stiffness. Similar to before,each specimen was probed at a frequency of 10 Hz for aduration of 2 seconds. The results are shown in Fig. 10.An interesting result is that the filter generally takes longerto converge with stiffer materials, and the steady-state errorvariance scales approximately proportionally to the increase


Frequency [Hz]0 5 10 15 20 25 30

Estim

atedStiffnesskt[N

/m]

200

300

400

500

600

700

800

UKFEKFAFEKF

(a) StiffnessFrequency [Hz]

0 5 10 15 20 25 30

Estim

atedDampingβt[N

·s/m]

0

2

4

6

8

10

12

14UKFEKFAFEKF

(b) DampingFrequency [Hz]

0 5 10 15 20 25 30

trace[P

t|t−1]

0

0.05

0.1

0.15

0.2

0.25

0.3UKFEKFAFEKF


Fig. 9: Frequency dependency of estimation results after data is collected for 1 second. As expected, lower-frequency excitation results in the highest estimationuncertainty, perhaps due to a violation of persistence of excitation condition.

Time [s]0 0.5 1 1.5 2

Estim

ated

Stiffnesskt[N

/m]

0

1000

2000

3000

4000BaselineNode 1Node 2Node 3

(a) Stiffness

Time [s]0 0.5 1 1.5 2

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ated

Dam

pingβt[N

·s/m]

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40

60

80

100


(b) Damping

Time [s]0 0.5 1 1.5 2

trace[Pt|t−1]

0

1

2

3

4

5



Fig. 10: Stiffness Dependency: AFEFK estimator performance for four different materials of different specified stiffnesses. Dotted lines in stiffness curvesindicate the manufacturer’s published flexural modulus, normalized and scaled by the estimator results.

Time [s]0 2 4 6 8 10

Estim

atedStiffnes

kt[N

/m]

-500

0

500

1000

1500

2000


(a) StiffnessTime [s]

0 2 4 6 8 10

Estim

atedDampingβt[N

·s/m]

-20

0

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(b) DampingTime [s]

0 2 4 6 8 10

trace[P

t|t−1]

0

0.5

1

1.5

2

2.5

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3.5UKFEKFAFEKF


Fig. 11: Filter responses to sudden changes in stiffness. Here the merits of implementing adaptive fading are made apparent, as the AFEKF is able to convergeto the new stiffness and damping values much more quickly than the EKF and UKF.

in stiffness, as can be seen in Fig. 10 (c). In addition, wesee from Fig. 10 (a) that the algorithm becomes less accurateat higher stiffnesses. Although it is not a 1:1 comparison,the dotted lines show the 100% moduli of each material asspecified by the manufacturer, scaled and normalized by theestimated stiffness values. At higher stiffnesses, the estimatedvalues tend to deviate from what we expect based on a simplelinear model employing the specified material modulus. Thiscould potentially be due to the tissue stiffness approaching thetool stiffness.

D. Model Deviation

As demonstrated in Section III B., EKF and UKF imple-mentations are not particularly robust to model or parameterchanges. The AFEKF attempts to overcome this limitationby introducing an exponential forgetting factor which weighs

recent data more heavily than past data at the cost of morecomputational overhead and potential instability.

To experimentally understand the implications of exponen-tial forgetting, a test was performed wherein the analyzer ispositioned onto a stiff tissue analog, data is collected for 4seconds, and then the analyzer is disengaged from the stifftissue and positioned onto softer tissue, where data is collectedfor another 4 seconds. The results shown in Fig. 11 showthat the AFEKF is very quick to respond to model changes,and we can even see the instance where the analyzer iscompletely disengaged from the tissue (where the stiffnessand damping drop to zero). The EKF and UKF are veryslow to converge, as expected, and provide very inaccuratemeasurements given the duration of contact. Observing thecovariance, we see that the covariance traces for the EKFand UKF monotonically approach zero despite the fact thatmodel changes. The adaptation of the AFEKF is evident in


(a) Stiffness (b) Damping (c) Covariance Trace

Fig. 12: Characterizing unknown tissue stiffness profiles using real-time estimation. Black circles in the x− y plane indicate the locations of the embeddednodules, and the radii of the circles indicate relative depth from the surface (ranging from 1mm, 3mm, and 5mm deep)

(a) Stiffness (b) Damping

Fig. 13: (a) Experimental setup of model deviation/tissue mapping test, (b)experimental setup of porcine liver characterization test

the covariance, as the trace of the covariance converges to anon-zero value.

E. Tissue Mapping

A particularly important potential application of in vivotissue characterization is the localization and marginalizationof metastatic tissue. Tumor localization and marginaalizationis a necessary step in minimally-invasive tumor removal, andit is paramount to provide sufficient margins between healthyand diseased tissue to avoid seeding during resection. Diseasedtissue can be several (2-3) factors stiffer than healthy tissue[10], and this stiffness footprint could potentially be exploitedto locate tumors in vivo without requiring specimen extraction.

To explore the possibility of doing this in real-time, atissue analog (EcoFlex 00-10, Smooth-On) was prepared withembedded nodules at different depths to simulate canceroustissue, as shown in Fig. 13 (left). The sample was analyzedat discrete locations in a uniform 5x5 grid. One second ofdata was collected at each location, and the resulting stiffness,damping, and covariance trace profiles are shown in Fig. 12.The surface curve represents a 2D interpolation of the averageof each filter estimate. The black circles on the x − y planerepresent the locations of the nodules, and the radius of eachcircle represents its relative depth below the surface. We can

see that the estimators were able to detect the location ofeach nodule with relatively high fidelity, purely based on thestiffness characteristics of the tissue. We also observe onceagain that the estimator error variance is higher for stiffertissues, as the trace(Pt|t−1) map indicates greater uncertaintyover the nodules.

F. Tests with Biological TissueCharacterization tests were repeated on an actual biological

tissue analog (porcine liver) to demonstrate that the filterscan reliably and repeatably estimate the viscoelastic propertiesof real biological tissue. The experimental setup is shownin Fig. 13 (b). Fig. 14 shows the results of 5 subsequentcharacterization tests (10 Hz) performed at different locationson the porcine liver, where shaded error bars indicate thestandard deviation amongst tests. We see that the results arefairly repeatable long-term once the transient effects of initialconditions have subsided. The conclusion from this experimentis that it is indeed possible to repeatably characterize unknownbiological tissues using the filtering techniques described.

A frequency sweep test was also performed to see if the liverbehaves as we would expect from a viscoelastic specimen.Again we see that the stiffness (akin to the storage modulusE′) monotonically increases as a function of frequency, wherethe asymptotic frequency (i.e. f where E′ → E′∞) lies beyondtested frequencies. This agrees with our previous conception ofviscoelastic behavior. If we examine the damping behavior, weobserve the initial drop and leveling off as we would expectfrom loss modulus behavior; however at higher frequencies,the EKF and AFEKF estimates begins to blow up. This islikely not a physically meaningful result and could potentiallybe due to linearization error which the UKF was able tosuccessfully overcome. This result speaks to the merits ofusing unscented filtering at higher frequencies to better capturethe nonlinear behavior of the system. Once again, we see inFig. 15 (c) that the covariance begins to increase after about20 Hz, as experienced before during the baseline frequencycharacterization tests. As such, we conclude that biologicalsoft tissue does indeed behave as we would expect froma viscoelasticity standpoint, and this behavior is adequatelycaptured in the model presented herein at lower frequencies,above which, potentially unmodeled nonlinearities tend tocause the EKF and AFEKF to diverge.


Time [s]0 0.5 1 1.5

Estim

atedStiffnesskt[N

/m]

0

500

1000

1500

2000

2500UKFEKFAFEKF

(a) Stiffness (b) DampingTime [s]

0 0.5 1 1.5

trace[P

t|t−1]

0

0.5

1

1.5

2UKFEKFAFEKF


Fig. 14: Repeatability experiment: probing porcine liver with an input frequency of 10 Hz at five different locations. Shaded areas represent the standarddeviation across the five different estimates.

Frequency [Hz]0 5 10 15 20 25 30

Estim

atedStiffnesskt[N

/m]

0

500

1000

1500

2000

2500

3000

3500

4000 UKFEKFAFEKF

(a) StiffnessFrequency [Hz]

0 5 10 15 20 25 30

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atedDampingβt[N

·s/m]

0

2

4

6

8

10UKFEKFAFEKF

(b) DampingFrequency [Hz]

0 5 10 15 20 25 30

trace[P

t|t−1]

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16UKFEKFAFEKF


Fig. 15: Frequency dependency of liver response. The EKF and AFEKF tend to diverge at higher frequencies, potentially due to linearization errors whichthe UKF is able to overcome. As before, the covariance blows up after about 20 Hz.

Time [s]1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045

Force[N

]

-4

-3

-2

-1

0

1

2

3

4 Actual ForceEstimate (UKF)Estimate (EKF)Estimate (AFEKF)

Fig. 16: Force estimate vs. actual measured force.

Fig. 16 shows the actual and estimated forces for an inputfrequency of 30 Hz. The UKF provides a much more accurateestimate at higher frequencies, and we see the EKF andAFEKF begin to overestimate the amplitude and lag in termsof phase. This potentially explains the divergence that we seein Fig. 15, and in fact, the stiffness estimate kt becomesnegative for the EKF and AFEKF as the force estimateis almost 180 degrees out-of-phase with the actual force.Therefore, at higher frequencies were nonlinearities likely playa more substantial role in governing the dynamics, a UKF isthe most robust implementation.

VI. CONCLUSION

In this paper, we demonstrate the feasibility of usingKalman filtering techniques to estimate viscoelastic mechani-cal properties of biological tissue in real-time. A mathematicalmodel of the tool and tissue interaction is derived, and theobservability of the system was ascertained by consideringthe determinant of the gradient of the Lie derivative ofthe nonlinear output equation with respect to the nonlinearstate equations. Three flavors of Kalman filter are derivedand implemented in software to compare their performance.Simulated results proved that the approach was feasible, and anexperimental system was developed to demonstrate function-ality using actual materials and biological tissues. Numerouscase studies are presented and analyzed with respect to speedof convergence and error covariance. Ultimately we found that,at lower input frequencies (1-15 Hz), the AFEKF providessuperior performance and robustness to model changes. How-ever, at higher (>15 Hz) frequencies, linearization errors causethe extended filtering modalities to become unstable, whereasthe UKF is still able to provide reliable state and parameterestimates.

Future work could focus on modifying the UKF algorithmto incorporate adaptive forgetting, thereby maximizing robust-ness to both model changes and system linearity.

REFERENCES

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[2] B. Ahn, Y. Kim, C. Oh and J. Kim, "Robotic palpation and mechanicalproperty characterizattion for abnormal tissue localization," Med Biol EngComput, vol. 50, pp. 961-71, 2012.


[3] J. Kwon, H. Hwang, J. An, G. Yang and D. Hong, "Enhanced TactileSensor for the Minimally Invasive Robotic Palpation," in IEEE/ASMEInternatitonal Conference on Advanced Intelligent Mechatronics, Besan-con, France, 2014.

[4] A. Eklund, A. Bergh and O. Lindahl, "A catheter tactile sensor formeasuring hardness of soft tissue: measurement in a silicone model and inan in vitro human prostate model," Medical and Biological Engineeringand Computing, vol. 37, pp. 618-624, 1999.

[5] A. Menciassi, "Force Sensing Microinstrument for Measuring TissueProperties and Pulse in Microsurgery," IEEE/ASME Transactions onMechatronics, vol. 8, no. 1, pp. 10-17, 2003.

[6] M. Ottensmeyer, "Minimally invasive instrument for in vivo measurementof solid organ mechanical impedance," MIT PhD Thesis, 2001.

[7] L. Barbe, B. Bayle, M. de Mathelin, A. Gangi, "Needle Insertions Model-ing: Identifiability and Limitations." International Journal of BiomedicalSignal Processing and Control, vol. 2, pp 191-198, 2007.

[8] A. Okamura, C. Simone, M. O’Leary, "Force Modeling for Needle In-sertion into Soft Tissue." IEEE Transactions on Biomedical Engineering,vol. 51, no. 10, pp. 1707-16, 2004.

[9] Q. Xia, M. Rao, Y. Ying, and S. Shen, "A New State-EstimationAlgorithm - Adaptive Fading Kalman Filter." Proceedings of the 31stConference on Decision and Control, vol. 31, pp. 1216-1221, 1992

[10] K. Hoyt, B. Castaneda, M. Zhang, P. Nigwekar, P. Anthony diSant’Agnese, J. Joseph, J. Strang, D. Rubens, K. Parker, "Tissue ElasticityProperties as Biomarkers for Prostate Cancer." Cancer Biomark, vol. 4,pp. 213-225, 2008.

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