real time identification of the hunt-crossley environment dynamics models

8/13/2019 Real Time Identification of the Hunt-Crossley Environment Dynamics Models

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IEEE TRANSACTIONS ON ROBOTICS, VOL. 28, NO. 3, JUNE 2012 555

Real-Time Identification of HuntCrossley DynamicModels of Contact Environments

Amir Haddadi, Student Member, IEEE, and Keyvan Hashtrudi-Zaad, Senior Member, IEEE

AbstractReal-time estimates of environment dynamics playan important role in the design of controllers for stable interac-tion between robotic manipulators and unknown environments.The HuntCrossley (HC) dynamic contact model has been shownto be more consistent with the physics of contact, compared withthe classical linear models, such as KelvinVoigt (KV). This paperexperimentally evaluates the authors previously proposed single-stage identification method for real-time parameter estimation ofHC nonlinear dynamic models. Experiments areperformed on var-ious dynamically distinct objects, including an elastic rubber ball,a piece of sponge, a polyvinyl chloride (PVC) phantom, and a PVCphantom with a hard inclusion. A set of mild conditions for guar-anteed unbiased estimation of the proposed method is discussed

and experimentally evaluated. Furthermore, this paper rigorouslyevaluatesthe performance of the proposed single-stage method andcompares it with those of a double-stage method for the HC modeland a recursive least squares method for the KV model and itsvariations in terms of convergence rate, the sensitivity to parame-ter initialization, and the sensitivity to the changes in environmentdynamic properties.

Index TermsDynamic model identification, HuntCrossley(HC), KelvinVoigt (KV), online parameter estimation.

I. INTRODUCTION

ROBOTIC tasks often involve continuous or intermittent

contacts between robots and various environments. Theinteraction between a slave robot and body tissues in robot-

assisted minimally invasive surgeries [1], [2], a robotic finger

grasping an object [3], a teleoperated excavator bucket during

remote excavation [4], and the interaction between foot and

ground during the locomotion cycle of a walking machine [5]

are a few examples of contact tasks. Real-time estimates of

contact dynamics have been used for the design of indirect and

model reference adaptive controllers for stable contact in robotic

and telerobotic applications [6][9].

Manuscript received October 18, 2010; revised April 29, 2011 and October25, 2011; accepted December 22, 2011. Date of publication February 7, 2012;date of current version June 1, 2012. This paper was recommended for publica-tion by Associate Editor T. Murphey and Editor B. J. Nelson upon evaluation ofthe reviewers comments. This work was supported in part by Natural Sciencesand Engineering Research Council of Canada.

A. Haddadi was with the Department of Electrical and Computer Engineer-ing, Queens University, Kingston, ON K7L 3N6, Canada. He is now withthe University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail:[email protected]).

K. Hashtrudi-Zaad is with the Department of Electrical and ComputerEngineering, Queens University, Kingston, ON K7L 3N6, Canada (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2012.2183054

Available methods that are widely used to model the envi-

ronment in real-time control applications are mainly limited

to linear KelvinVoigt (KV) models, in which the relationship

between the penetration of contacting bodies and the resulting

force is modeled by a parallel connection of a linear spring

and a linear damper. Linear models have been identified in real

time using estimation methods such as recursive least squares

(RLS) and its variations such as exponentially weighted recur-

sive least squares(EWRLS) [7], [10], [11]. However, the KV

or linear massdamperspring models, in general, are shown to

have physical inconsistencies in terms of power exchange dur-

ing contact. This shortcoming, which results in negative contactforce during rebound, is more visible in dynamic models of soft

environments, such as human body tissues [12]. Therefore, the

estimated forces and dynamic parameters using linear contact

models may not be suitable for control of contact tasks.

In 1975, Hunt and Crossley [13] showed that a position-

dependent environment damping model is more consistent with

physical intuition. It is further shown that the HuntCrossley

(HC)model is consistent with the notion of coefficient of resti-

tution, representing energy loss during impact [14]. Therefore,

such a nonlinear model can potentially improve the estimation

of force and dynamic parameters, which, by itself, will improve

the performance of many robotic, haptic, and telerobotic tasks.However, fast and accurate identification of the HC nonlinear

models remains a challenge and severely limits the use of this

model for real-time applications. Diolaitiet al.proposed a boot-

strapped double-stage method for online identification of the

HC model [12]. However, this method is sensitive to parameter

initial conditions as demonstrated by simulations in [15] and by

experimental results in this paper. Moreover, due to the prop-

agation of error from one stage to the next, the double-stage

method suffers from slow parameter convergence.

Recently, Haddadi and Hashtrudi-Zaad have proposed a novel

single-stage method for online estimation of the HC model [15].

They proved estimation consistency (unbiased estimation) and

provided mild conditions upon which the method is applica-

ble. The single-stage method has been used in [16] to improve

model-mediated teleoperation systems and in [17] for stiffness

modulation in haptic augmented reality applications. Although

the single-stage method has been simulated in [15], it has not

been rigorously evaluated with experiments. In this paper, we

experimentally evaluate the previously proposed single-stage

estimation method for different types of environment contact

materials. We investigate the sensitivity of the method to initial

conditions and model parameter variations and study the esti-

mation convergence rate and force prediction accuracy in com-

parison with the double-stage method applied to an HC model

1552-3098/$31.00 2012 IEEE


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556 IEEE TRANSACTIONS ON ROBOTICS, VOL. 28, NO. 3, JUNE 2012

and an RLS variant applied to a KV model. Then, we provide

intuitive insights on how to design a robotic task for an unknown

environment in order to benefit from efficient identification.

The remainder of this paper is organized as follows. The lin-

ear KV and the nonlinear HC contact models are presented in

Section II. The double-stage and single-stage online identifica-

tion methods are described in Section III. Experimental results

comparing the performance of the two methods are presented

and analyzed under a variety of conditions for three different

contact environments in Section V. The effect of various input

excitation and parameter variation during intermittent contact in

an automated tissue property estimation task are studied in Sec-

tions VI and VII, respectively. Section VIII draws conclusions

and provides suggestions for future work.

II. CONTACTDYNAMICSMODELS

A. KelvinVoigt Linear Contact Model

The most common environment dynamic model for robotic

applications is the linear KV model, which incorporates thedynamics of a linear damperspring system

F(t) =

Kx(t) + Bx(t), x(t) 00, x(t)< 0

(1)

wherex(t)represents the robot penetration inside the environ-ment,F(t) represents the contact force, andK andB denotethe environment stiffness and damping, respectively.

The dynamic parameters of this model can be easily estimated

using linear system identification techniques, such as variations

of least squares. The KV model displays both physical limi-

tations and intuitive inconsistencies [15], [18], which will be

discussed later through experiments.

B. HuntCrossley Nonlinear Contact Model

Nonlinear models have been shown to be in better agree-

ment with the dynamic behavior of physical environments [18].

Specifically, human biological tissues have been reported to

show nonlinear behavior [2]. In order to overcome the problems

of the KV linear model, Hunt and Crossley [13] proposed the

following nonlinear model:

F(t) =

Kc xn (t) + Bc xn (t)x(t), x(t) 00, x(t)< 0

(2)

in which the viscous force depends on contact penetration. Here,Kc xn denotes the nonlinear elastic force, and Bc x

n x denotesthe nonlinear viscous force. The parameter n is typically be-tween 1 and 2, depending on the material and the geometricproperties of contact.

III. LINEARIDENTIFICATION OF THENONLINEAR

HUNTCROSSLEYMODEL

The nonlinear nature of HC models is intuitively consistent

with the physics of contact; however, the resulting computa-

tional complexity of the double-stage identification method has

restricted the use of the HC model in real-time robotic appli-

cations. Therefore, the authors proposed a different approach

Fig. 1. Double-stage identification method for the HC model [12].

to linearize the nonlinear model so that all three parameters of

the HC model can be estimated at the same time in a single

stage. This makes the real-time identification process faster and

computationally more efficient [15].

In this section, we provide a brief overview of the double-

stage parameter estimation method proposed by Diolaiti et al.

[12] and the single-stage identification method proposed by the

authors in [15] and discuss their differences.

A. Double-Stage Identification Method

Fig. 1 illustrates the double-stage parameter estimation

method [12]. In this method, the estimation of the dynamic

parameters (Kc , Bc ) is partially decoupled from the parame-ter n. In stage 1 , assuming a known n, the following lineardynamic equation is used to estimate Kc andBc :

F(t) =Kc [xn(t) ] + Bc [x

n(t) x(t)] + 1 (3)

where1 is the error generated from using n instead ofn. Instage 2 , assuming known Kcand Bcparameters, the parameternis estimated according to

log F(t)

Kc+ Bcx(t)

= n[log x(t)] + 2 (4)

where2 is the error resulting from the estimation of Kc andBc . For the unbiased estimation of all three parameters, both 1and2 must be zero-mean stochastic processes.

Although the convergence of this method has been ana-

lyzed [12], the proof of estimation consistency has been pro-

vided under three conditions that may substantially limit the

applicability of this method.

1) In order for1 to be a zero-mean stochastic process, itis assumed that n = n n is always small such that1 x n log x n = n log x. In other words, theapproximation holds only when the condition x n 1is met. This condition cannot be realized at the begin-

ning of the estimation process, wherex is very small andn is potentially large. For instance, for the initial errorn = 0.2and 1-cm penetration,(0.01)0.2 = 0.398whichis not close to1.

2) In order to have an unbiased estimation in2 , the follow-ing necessary condition must be satisfied:

Kc + Bcx 1xn

(5)


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HADDADI AND HASHTRUDI-ZAAD: REAL-TIME IDENTIFICATION OF HUNTCROSSLEY DYNAMIC MODELS OF CONTACT ENVIRONMENTS 557

where Kc := Kc Kc andB := Bc Bc . This con-dition is met at the initial estimation period where x issmall, as well as after convergence when Kc and Bcbecome small. However, if the parameters of the system

change during operation, condition (5) is no longer met,

and there is no guarantee for2 to be unbiased.

3) The statistical conditionE[n(Kc+ Bcx)x

n ] = 0 (6)

must be satisfied, which is the case if the two estimators

converge independent of each other. However, the conver-

gence of2 is dependent on 1 and vice versa, as eachestimator relies on the resulting estimates from the other

estimator.

B. Single-Stage Identification Method

As discussed in Section III-A, the applicability of the double-

stage parameter estimation method is limited under certain con-

ditions. These limitations may result in inconsistent estimationsdue to the choice of initial conditions that are far from the

actual unknown values. Therefore, the authors provided a dif-

ferent method to linearize the nonlinear HC model so that all

three model parameters can be identified in one stage during a

real-time process.

It has been shown in [15] that by taking the natural logarithm

of both sides of the HC model (2) for x 0, we obtain

ln[F(t)] = ln[Kc xn (t)(1 +

Bcx(t)

Kc+

Kc xn (t))]

= ln(Kc ) + n ln[x(t)] + ln[1 +Bcx(t)

Kc

+

Kc

xn (t)]

(7)

where includes the modeling error and the measured noiseduring the process. As we know for the natural logarithm, ln(1 +) = for || 1. Therefore, assumingBcx(t)Kc +

Kc xn (t)

Bcx(t)Kc

+ Kc xn (t)

1 (8)(7) can be rewritten as

ln[F(t)] =ln(Kc ) +BcKc

x(t) + n ln[x(t)] +

Kc xn (t). (9)

The assumption in (8) implies that |B c x(t)Kc | and | Kcxn (t) | are

negligible. To satisfy |B c x(t)Kc

| 1, as a rule of thumb, we con-sider the condition

x < 0.1Kc

Bc(10)

with the approximation threshold of 0.1. This threshold would

result in a minimum uncertainty of 0.005 in (8), which is small

and acceptable, considering the potential magnitude ofln(Kc )and n ln[x(t)]in (9). Although the effect of the threshold on theidentification process depends on the other terms in (9), such

as the value ofKc , the force, and the amount of measurement

noise, our experiments and simulations have shown that 0.1

is a rather conservative value in many cases. Therefore, being

marginally close to this value, or even slightly violating this

condition, does not necessarily mean that the identification is

not valid. Since the values ofKc andBc are unknown in real-time experiments, their estimates can be used as an alternative

to validate condition (10), as will be seen in Section V. With

regard to the feasibility of condition (10), in many practical

applications, the speed of operation within contact is not high

and condition (10) is often met. However, it should be noted

that the speed of operation is not fully controlled by the user

as it depends on the characteristics of the desired task and the

capabilities of the robot. Therefore, the fact that condition (10) is

not guaranteed can be considered a shortcoming of this method.

With regard to the second term of approximation (9), de-

pending on the power of noise and the type of environment,

i.e.,Kc , a reasonable minimum penetration must be chosen sothat | Kc xn (t) | is small enough to satisfy (8). The identificationprocess must be stopped when the penetration is smaller than

this threshold.

Considering the aforementioned conditions, the linearizedsystem (9) can be identified using the least squares family of

estimation methods. To this purpose, the environment dynamics

(9) is linearly parameterized as

yk =Tkk + k , xk >0 (11)

where Tk is the regressor vector, k is the vector of dynamic

parameters, and k =m k + kKcxnk

represents modeling error

and force measurement noise at the sample time t= k.T, wherek is the iteration number, and Tis the sampling time. In addition

Tk = [1, xk , ln(xk )], = ln(Kc ),Bc

Kc

, nT

, yk = ln(Fk ).

(12)

Among variations of the RLS methods, EWRLS is an estimation

method that is more suitable for environments with variable dy-

namic properties. The EWRLS update equations can be written

as [19]

Lk +1 = Pkk +1

+ Tk + 1Pkk +1

Pk +1 = 1

[Pk Lk +1

Tk + 1Pk ]

k + 1 =k + Lk +1 [Fk + 1 Tk + 1 k ] (13)

where P is the covariance matrix, and is the forgetting factor.When = 1, the RLS method is achieved. At every samplingtime, the estimated parameters of the model are derived accord-

ing to

Kck =ek(1 ) , Bck =e

k(1 ) k (2), nk =k (3).

The algorithm should check for singularities in the logarith-

mic functions that may occur during the operation. As in every

parameter estimation method, the estimation convergence relies

upon the persistency of excitation (P.E.) of the robot end-effector

trajectory. Since there are three parameters to be identified, as a

rule of thumb, a combination of two sine waves would be suffi-

cient [20]. In practical applications, this excitation condition is


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Fig. 2. Experimental setup consisting of a twin pantograph robotic manipula-tor equipped with a Nano-25 force/torque sensor and a poking device.

not guaranteed to be met; however, the nonlinearity of the func-

tion ln() can make the identification signal richer [20]. The P.E.condition will be experimentally investigated in Section VI. The

proof of unbiased estimation for the single-stage identification

method can be found in [15].

In the next sections, we will experimentally evaluate the

single-stage method and demonstrate the advantages of this

method over the double-stage method for the HC model and

the EWRLS method applied to the KV model.

IV. EXPERIMENTALSETUPFORREAL-TIMEIDENTIFICATION

Fig. 2 shows a picture of the experimental setup consisting ofa Quanser planar twin pantograph robotic manipulator equipped

with an ATI Nano-25 force/torque sensor and a poking device.

The twin pantograph is a redundant robot that consists of two

pantographs, each directly driven by two dc motors at the base

joints. The angle of rotation of each motor is measured by a

high-resolution encoder with 20 000 counts/rev. The position of

the end-effector is computed from forward kinematic relations.

The contact force is measured at a resolution of 1/8 N. The robot

is position controlled to follow a desired trajectory within the

contact material. The trajectory, which will be discussed later,

consists of various sinusoidal components. The control system

and the online identification algorithms are implemented usingMATLAB RTW Toolbox and Quanser QuaRC 1.1 real-time

system operating at a sampling rate of 1 kHz on a 2.4-GHz

Quad CPU.

A. Desired Trajectory for the Probe

Different trajectory commands for the robot are used in order

to determine the least possible level of excitation for identifica-

tion. First, the following combination of three sinusoidal signals

is chosen as the desired trajectory in the direction perpendicular

to the contact surface

x(t) = 1 sin(4t) + 1.8 sin(11t) + 1.8 sin(15t) + x0 (mm).

Fig. 3. Desired trajectory perpendicular to the contact surface.

Fig. 4. Three differentcontact materialsidentified through experiments. Fromleft to right: rubber ball, sponge, and PVC phantom.

The frequency and coefficients are chosen to ensure suitable

penetration inside the environment as well as the richness of

the excitation input. The bias x0 is added to push the robotsmoothly inside the environment about 10 mm such that the

robot remains inside the object during the entire contact task. A

sample position trajectory for the first 10 s is shown in Fig. 3. A

2-s delay in the position command is implemented to measure

and remove any bias in the force sensor measurements before

the actual experiment starts.

B. Contact Material

Three different contact materials (environments) are used for

the experiments. They include a rubber ball, a piece of sponge,and a polyvinyl chloride (PVC) phantom. Each environment

displays a specific behavior that results in important conclu-

sions about the identification method and the advantage of the

nonlinear HC model over the linear KV model. Fig. 4 illustrates

the experimental setup with the robot in contact with the three

different environments.

V. EXPERIMENTALPROCEDURE ANDRESULTS

A. Environment: Rubber Ball

An elastic rubber ball is used as the contact material for the

first set of experiments as shown on the left side of Fig. 4. This

type of contact, when compared with the other two, displays adominant elastic behavior, which returns the probe quickly to

its original state once the stress is removed.

1) Initial Conditions: To investigate the sensitivity of the

identification algorithms to parameter initialization, three ex-

periments are conducted for each environment using small and

large initial conditions for the identified parameters. The choice

of small values for Kc0 ,Bc0 , andn0 is trivial. The set of highvalues is established as two or three times the average of the

parameters in the steady state obtained by performing the iden-

tification experiments with the small initial values. To estimate

the parameters of the HC model, we examine the small values

for Bc0 and Kc0 : once with n0 = 1 and once with the maximum


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TABLE IRUBBERBALL: PARAMETERINITIALVALUES FORHC AND KV MODELS

Fig. 5. Rubber Ball. Estimated parameters of the HC model using the single-stage method for three repetitions of the real-time experiment using the first setof initial conditions.

Fig. 6. Convergence condition of the single-stage HC environment contactidentification method for the rubber ball. For reliable estimated parameters, we

should have x< 0 . 1Kc

B c .

expected valuen0 = 2. The two experiments help us to focuson the effect ofn. In the third experiment, we change Bc0 andKc

0

to their larger values to focus on the effect ofBc andKc .The three sets of initial values for Kc0 ,Bc0 , andn0 for the

rubber ball are shown in Table I.1 Since the KV model has only

two parameters, i.e., K andB , two experiments with low andhigh values, as summarized in Table I, are considered.

2) Validity of the Single-Stage Method: The results of the

estimation of the HC model parameters using the single-stage

method are shown in Fig. 5. The figure illustrates the identi-

fied parameters for three repetitions of the real-time experiment

with the first set (set I) of initial conditions for all trials. The

agreement of the estimated parameters verifies the experimental

results and points at the consistency of the single-stage method.

As an alternate method to check the applicability of the single-

stage method and the validity of its results, the approximationcondition (10) is investigated. Since the correct values ofKcandBc are not available, their estimates are used. Fig. 6 com-

pares |x| and0.1KcB c

for a set of collected data and a set of initial

conditions over the entire period of time during which the probe

is in contact with the ball. The figure shows that the condition

x < 0.1Kc

B cis satisfied and the identification results are valid

after approximately 90 ms from the start of the experiment.

3) Effect of Forgetting Factor and Covariance Matrix on

the Single-Stage Method: The identification process for the

1For singularity issues, Kc 0 = 0cannot be usedin our identification process,

and instead,Kc 0 = 0.1is used.

Fig. 7. Effect ofon the identified parameters of the HC model using thesingle-stage method for one experiment with similar initial conditions.

previous three trials starts at t = 2 s with the initial forgettingfactor0.99, which nearly reaches unity in one second according

to the exponential relationship = 1 0.01exp(5(t 2)).The choice of exponential forgetting factor is motivated by the

fact that for constant = 1, as in RLS, the estimation convergesmore slowly, while for a constant below unity, instability

occurs. The forgetting factor is chosen based on the type

of environment under examination. For contacts with varying

parameters, a lower forgetting factor is selected, whereas forcontacts with constant parameters, a value closer to unity is

selected. The choice of the time constant in the exponent also

has significant effect on the convergence rate and convergence

stability.

Fig. 7 shows the slow convergence experienced when was

chosen as unity. Our experiments have shown that using a large

time constant to bring close to unity results in instability.

Using = 1 0.01exp((t 2)), with 5 8, createsa balance between the convergence rate, sensitivity to noise,

and estimation stability. Therefore, in this paper, we use =1 0.01exp(5(t 2)) for the forgetting factor in order to

balance estimation speed and estimation convergence. Fig. 7illustrates the parameters that are estimated in real time after

performing the single-stage identification method on one set of

experimental data with different profiles. It is clear that by

using the RLS method, i.e., = 1, the estimation becomes veryslow. The time constant of 10 s, i.e., = 0.1, results in severefluctuations and instability, whereas = 5 provides the bestperformance.

While the initialization of the covariance matrix Pis expectedto affect the speed of convergence at the beginning of identi-

fication, the experiments show that matrices with norms larger

than 10 result in similar convergence rates. On the other hand,

it has also been approved that the larger the norm of P, the

larger the magnitude of the initial overshoots of the estimatedparameters. It has further been observed that the extent of these

jumps mainly depends on the initial values of the estimated pa-

rameters. The effect of the initial conditions will be discussed

next.

4) Effect of Initial Conditions on the Two HuntCrossley

Identification Methods: Fig. 8 shows the online estimated pa-

rameters and their corresponding force prediction errors for

the various initial conditions listed in Table I. The results in

Fig. 8 are related to the three discussed online identification

methods (HC: single stage, HC: double stage, and KV), which

are all applied to the same set of collected data. The percent

relative root-mean-square error (RMSE%) is also provided in


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Fig. 8. Rubber Ball. Estimated parameters and the force prediction errors for various initial conditions for HC and KV models. The KV model has one fewerparameter to display.

TABLE IIRUBBERBALL: RMSE%OF THEHC AND KV MODELS

Table II.2 Since the KV model has no parameter n to iden-tify, its corresponding RMSE% value for the initial set II is not

available.

It is clear from the results that the single-stage method outper-

forms the double-stage method in terms of convergence rate,3

i.e., speed of convergence at the beginning of the identification

process, force prediction error, and consistency of estimation.

Moreover, the single-stage method is capable of identifying the

environment dynamic parameters for a wide range of initial con-

ditions, whereas the double-stage method lacks this capability.

The reason for such inconsistency using this set of initial con-ditions is that the convergence conditions for the double-stage

method are not satisfied, as discussed in Section III-A. Fig. 9

shows the identification results obtained from the double-stage

method for four different initial conditions that are selected

within the vicinity of the final values. The double-stage method

produces consistent and converging results only for a small

range of initial conditions, which is not desirable for applica-

2In order to exclude the effect of large force prediction error experiencedin the first few milliseconds of contact, the calculation of %RMSE for all themethods are performed after 0.2 s of contact.

3Convergence time is defined as the time required for the estimates of a

parameter for all initial conditions to converge within a 5% difference.

Fig. 9. Rubber Ball. Estimated parameters of the HC model using the double-stage method for different initial conditions.

tions in which limited or no information is available about the

contact material.

5) Comparing the Identified HuntCrossley and Kelvin

Voigt Models: Here, we again focus on the results in Fig. 8

and Table II to compare the HC model obtained from the single-

stage method and the identified KV model. The larger prediction

error for the KV model implies that the HC model better repre-

sents the physical properties of the object. Although parameters

converge for both models, the convergence time is shorter forthe single-stage method. Considering that the same measure-

ments, i.e., position and force, and the same initial conditions

for the EWRLS identification processes have been utilized in

both models, the difference in convergence rate may, then, be re-

lated to the role of the natural logarithms of position and force in

the single-stage method in making the identification excitation

richer and resulting in faster estimation.

As previously mentioned, one potential shortcoming of KV

models is the negative predicted force when the probe reaches

close to the environment surface as it moves in the outward di-

rection at a high velocity. In this case, i.e.,x(t) 0, and since

x(t) is negative and large, a negative force is predicted by the


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Fig. 10. (Left) Position profile and (right) the measured and predicted forceposition hysteresis loop for the KV model.

TABLE IIISPONGE: PARAMETERINITIALCONDITIONS FOR THEHC AND KV MODELS

model. However,in reality, the force exerted on any environmentis always positive during the compression and rebound phases.

Such behavior leads to power transfer from the robot to the en-

vironment during the restitution phase, which is in contrast with

human intuition. This problem primarily occurs in systems with

higher damping at high speeds. In order to demonstrate this be-

havior, a simple experiment is performed on the rubber ball with

a penetration profile, which comes close to zero at certain peri-

ods in time. Fig. 10 illustrates the superimposed forceposition

hysteresis loops obtained from actual measurements and the

estimated model. To avoid the negative force effect in all ex-

periments described throughout the paper, with the exception

of the aforementioned results, we have considered a sufficientlylargex0 . This allows us to focus on comparing the HC and KVmodels to predict contact behavior once the probe is inside the

environment.

B. Environment: Sponge

In this section, a thick piece of sponge, as shown in Fig. 4, is

used as the contact material for dynamic identification. The dy-

namic response of a sponge to contractionsis different compared

with that of the ball. Because of the larger damping property of

the sponge, it requires substantially larger time for restitution.

In this section, we experimentally identify and compare the HC

models, obtained from the single- and double-stage methods,with the KV model for this dynamic property of the sponge.

1) Initial Conditions: Three sets of initial values for Kc ,Bc , n, and two sets of initial values for K and B, as shownin Table III, are considered for the estimation of the dynamic

parameters of the sponge. The selection of the initial values

follows the strategy explained in Section V-A for the rubber

ball. The results of real-time estimations are presented next.

2) Validity of the Single-Stage Method: Fig. 11 shows the

estimated HC parameters of the sponge using the single-stage

identification method for three trials with the Set I initial con-

ditions described in Table III. One property that differentiates

the sponge from the elastic rubber ball is the damping-related

Fig. 11. Sponge. Estimated parameters of theHC modelusing thesingle-stagemethod for three repetitions of real-time experiment with the first set of initialconditions.

Fig. 12. Sponge. Convergence condition of the single stage HC environmentcontact identification method. For reliable estimated parameters, the condition

x < 0 .1 Kc

B c should be met.

parameter Bc of the sponge, which is about ten times higherthan its correspondingKc value.

In order to confirm the applicability of the single-stage

method for the sponge environment, the approximation con-

dition (10) is examined. To this purpose, |x| and 0.1KcB c

are

compared in Fig. 12 for one set of collected data and one set of

initial conditions over the entire period of time that the probe

is in contact with the sponge. The figure shows that condition

(10) is violated for the first 5 s of contact. However, from thatpoint on, condition (10) is met with a small margin. The sponge

is less elastic and more damped than the rubber ball, resulting

in substantially lower value for Kc /Bc or a lower boundaryon the velocity. As a result, for faster operations, i.e., higher

velocity, the identification results lose their accuracy and con-

sistency. Therefore, as previously discussed, one of the main

shortcomings of the single-stage method is that the convergence

conditioners may not be met for a highly damped environment,

especially for the tasks in which the speed of interaction is not

fully controlled by the user. For other tasks for which the speed

of operation can be controlled, such as palpation for physical

examinations, a lower speed of operation is recommended.

3) Effect of Initial Conditions and Comparison Between Dif-ferent Methods: In order to determine the effect of initial condi-

tions on the identification of theHC model using the single-stage

and the double-stage methods, as well as the linear identifica-

tion of the KV model, different initial conditions, as listed in

Table III, are applied to one set of collected data. Fig. 13 shows

the estimated parameters and the prediction error profiles for the

three methods, and Table IV summarizes the RMSE% values.

The results show that the speed of parameter convergence for

the EWRLS applied to the single-stage method for HC model

is the same as that of the KV model and is not sensitive to the

large changes in the initial conditions. In contrast, the double-

stage method shows inconsistent results for such a large range of


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Fig. 13. Sponge. Estimated parameters and the force prediction errors for various initial conditions for the HC and KV models. The KV model has one fewerparameter to identify and display.

TABLE IVSPONGE: RMSE%OF THEHC AND KV MODELS

initial conditions. This highlights the advantage of the single-stage method over the double-stage method in identifying the

contact dynamic parameters for a wide range of initial condi-

tions. Our experiments show that the double-stage method can

only converge when the initial conditions are within 10%oftheir final values. In terms of model force prediction, a larger

force prediction error is observed for the identified KV model of

the sponge than the HC model that uses the single-stage method.

The double-stage method shows larger prediction errors for all

sets of initial conditions.

C. Environment: Polyvinyl Chloride Phantom

Previously, we identified the dynamics of two different con-tact environments: the elastic rubber ball with Kc and Bc ofrelatively the same order ( 2000), and the sponge with a Bcthat was about ten times larger than its Kc value. For bothobjects, the value ofn was estimated to be approximately 1.In this section, we study the distinct dynamic behavior of a

PVC phantom, which is characterized by ann value close to2.Fig. 4 illustrates the PVC phantom used for experiments. PVC

phantoms have been used to mimic tissue properties for various

applications, including experimental analysis and evaluation of

surgical needle insertion methods [21]. The stiffness of the PVC

phantom can be adjusted by changing the ratio of plastic to soft-

ener (or hardener). The PVC phantom used for this experiment

was constructed of five portions of plastic and two portions of

softener.4 For more information on the relationship between the

stiffness of the PVC and the proportion of the plastic and the

softener or hardener, see [22].

In [2], Yamamoto et al. used KV and HC methods, as well

as polynomials of orders two, three, and four, and a second-

order polynomial plus viscous friction to model the dynamic

characteristics of a PVC tissue phantom. The results in [2] have

shown that the second-order polynomial plus viscous friction

predicts the contact force with the same accuracy as that of the

HC model and is more accurate than the KV and other polyno-

mial models. Therefore, in addition to the KV and HC models,

we also report the results of dynamic parameter estimation of

this model, which we call extended KelvinVoigt(EKV) model,

mathematically expressed by

F(t) =

Ke1 x(t) + Ke2 x

2 (t) + Bex(t), x(t) 00, x(t)< 0.

(14)

For identification, we used the same EWRLS estimation

method that has been used for KV models.

1) Initial Conditions: Following the strategy explained inSection V-A, three sets of initial values for the HC model pa-

rameters, two sets of initial values for the KV model parameters,

and two sets of initial conditions for the EKV model parameters

are considered for the estimation of the PVC phantom dynamic

parameters as listed in Table V.

2) Validity of the Single-Stage Method: Fig. 14 shows the

HC estimated parameters for the PVC phantom using the single-

stage parameter estimation method in three repetitions of the

experiment with the first set of the initial conditions (set I).

Although there are differences between the estimated values in

4

Supplier: M-F Manufacturing Co., Inc., Fort Worth, TX.


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TABLE VPVC PHANTOM: PARAMETERINITIALCONDITIONS FORTHREE

ESTIMATION MODELS

Fig. 14. PVC Phantom. Estimated parameters for the HC model using thesingle-stage method for three repetitions of real-time experiment with the firstset of initial conditions.

Fig. 15. PVC Phantom. Convergence condition for the proposed single-stagemethod. For reliable estimated parameters, the conditionx

real time identification of the hunt-crossley environment dynamics models

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