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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
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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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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