IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.5,OCTOBER 2005 925

Contact Impedance Estimation for Robotic Systems

Nicola Diolaiti,Student Member,IEEE,Claudio Melchiorri,Senior Member,IEEE,and

Stefano Stramigioli,Senior Member,IEEE

Abstract—In this paper,the problem of online estimation of

the mechanical impedance during the contact of a robotic system

with an unknown environment is considered.This problem is of

great interest when controlling a robot in an unstructured and

unknown environment,such as in telemanipulation tasks,since it

can be easily shown that the exploitation of the knowledge of the

mechanical properties of the environment can greatly improve the

performance of the robotic system.In particular,a single-point

contact is considered,and the (nonlinear) Hunt–Crossley model is

taken into account,instead of the classical (linear) Kelvin–Voigt

model.Indeed,the former achieves a better physical consistency

and also allows describing the behavior of soft materials.Finally,

the online estimation algorithm is described and experimental

results are presented and discussed.

Index Terms—Compliant contact,Hunt–Crossley model,online

estimation.

I.I

NTRODUCTION

T

HE knowledge of contact dynamics plays a fundamental

role in several complex robotics tasks,such as the grasping

or the manipulation of soft and delicate objects.Indeed,the

stability and the performances of interaction controllers,such

as impedance controllers [2] or intrinsically passive controllers

(IPCs) [3],are directly affected by the mechanical properties of

the touched environment [4].Therefore,the availability of an

accurate description of contact dynamics could help to handle

these problems in a more effective way,allowing adapting robot

controllers to current working conditions,i.e.,the interaction

with compliant or stiff environments.

This concept has been already considered in the literature,

and in [5] it has been shown,in a linear setting,that the stability

region of an impedance controller is enlarged by means of an

adaptation lawbased on the real-time estimation of environment

parameters.

Asimilar work is presented in [6],where different estimation

algorithms for the contact stiffness and damping parameters of a

linear environment are presented and discussed,with particular

attention to the online simulation of space operations;for this

Manuscript received June 25,2004;revised January 27,2005.This paper was

recommended for publication by Associate Editor P.Dupont and Editor H.Arai

upon evaluation of the reviewers’ comments.This work was supported by Geo-

plex under EU Project N.IST-2001-34166.This paper was presented in part

at the IEEE/RSJ International Conference on Intelligent Robots and Systems,

2004.

N.Diolaiti and C.Melchiorri are with DEIS,Department of Electronics,

Computer Science,and Systems,University of Bologna,Bologna 40136,Italy

(e-mail:ndiolaiti@deis.unibo.it;cmelchiorri@deis.unibo.it).

S.Stramigioli is with the Centre for Telematics and Information Technology

(CTIT) and Drebbel Institute for Mechatronics,University of Twente,7500AE

Enschede,The Netherlands (e-mail:S.Stramigioli@ieee.org).

Digital Object Identiﬁer 10.1109/TRO.2005.852261

kind of task,an extensive discussion can be found in [7].Anal-

ogous concepts can be applied to master/slave bilateral teleop-

eration systems,designed to enable a human operator to interact

with remote environments.In fact,user perception is affected by

the inherent dynamics of robotic devices and by possible time

delays in data transmission,thus leading to poor performances

in terms of transparency of the overall apparatus [8].On the

other hand,the ﬁdelity of the force feedback is extremely im-

portant,e.g.,in surgical applications,where the surgeon should

be able to recognize human tissues by means of their mechan-

ical properties (i.e.,by perceiving their impedance).The use of

an online estimation algorithm to adapt the controller’s param-

eters has been suggested in [9] as a method to improve user per-

ception in micro-macro manipulation tasks.More recently [10],

the estimation of the of the contact dynamics has been proﬁtably

used in order to improve the transparency of both rate and posi-

tion controllers for telemanipulation systems.In this context,the

identiﬁcation of the dynamics haptic display also plays a role,

and an efﬁcient algorithmbased on the support vector machines

(SVM) theory has been presented in [11].

Fromthis perspective,a crucial aspect is the choice of a suit-

able model to describe the dynamics of the contact between a

robotic device and its surrounding environment.In the literature,

see [12] for an extensive review,the behavior of viscoelastic

materials has been characterized in several ways.Focusing on

1-D contacts,the simplest model,known as the

Kelvin–Voigt

model,to describe the relation between the relative penetra-

tion (approach) of contacting bodies and the resulting force,

is represented by the parallel of a linear spring and a viscous

damper.This model has been further extendedin [13] and in [14]

by accounting for the dynamics of the contact surface.How-

ever,these models are linear,and with particular reference to

the Kelvin–Voigt model,it is shown that they are not suitable

to describe the behavior of soft materials,such as human tis-

sues,where viscous effects are substantial.Therefore,a non-

linear model has to be considered to characterize the dynamics

of both stiff and compliant materials.Hunt and Crossley [15]

showed that it is possible to obtain a behavior that is in better

agreement with the physical intuition if the damping coefﬁcient

is made dependent on the body’s relative penetration.Moreover,

the Hunt–Crossley model is consistent with the notion of coef-

ﬁcient of restitution used to characterize energy loss during im-

pacts [15],[16] and,even if nonlinear,retains a certain computa-

tional simplicity.Beside these properties,it is important to note

that the physical consistency of the model can be preserved by a

proper generalization to the full geometrical contact description

[six degrees of freedom(DOF)],as discussed in [17] and [18].

This paper shows that the Hunt–Crossley model is suitable

to describe the contact with both stiff and compliant objects,

1552-3098/$20.00 © 2005 IEEE

926 IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.5,OCTOBER 2005

and presents an online estimation algorithm to identify char-

acteristics parameters of different materials.In particular,

Section II provides a brief overview on contact models,dis-

cussing the drawbacks affecting the Kelvin–Voigt linear model

and the properties of the Hunt–Crossley model.Then,in Sec-

tion III,an online recursive algorithm for the parameters of the

Hunt–Crossley model is presented,discussing its convergence

properties.This algorithm has been applied to different mate-

rials in order to provide experimental conﬁrmation (Section IV)

to the theoretical considerations.Conclusions and future work

plans are ﬁnally addressed in Section V.

II.C

OMPLIANT

C

ONTACT

M

ODELS

As well discussed in [12],the characterization of contact

dynamics is usually carried out following two different ap-

proaches.On one side,the use of coefﬁcients,as suggested

by Newton,Poisson,and more recently,Brach [19],allows

describing variations of certain dynamical or kinematic param-

eters,such as relative velocities,due to impacts between two or

more bodies.On the other hand,the Kelvin–Voigt [20] (linear),

further extended in [13] and [14],or the Hunt–Crossley [15]

(nonlinear) models have been developed in order to achieve

a continuous description of the relation between the body’s

penetration and the consequent reaction force.Therefore,these

models can be applied to describe and estimate the contact

dynamics between the end-effector of a robotic manipulator and

surrounding objects.The main advantage of the Hunt–Crossley

nonlinear model over the Kelvin–Voigt formulation is its

consistency with the coefﬁcient of restitution,and for this

reason,it is more suitable to describe energy dissipation effects,

especially for soft materials.

A.Coefﬁcient of Restitution

The contact is a complex phenomenon whose dynamics de-

pends on many properties of contacting bodies,such as material,

geometry,and relative velocity.In particular,the phase starting

with the contact at time

and ending at time

,when

the maximum deformation

is reached,is called compres-

sion,and is followed by the restitution phase,taking place from

to the instant

when bodies separate [21],[22].Be-

cause of energy dissipation,the total kinetic energy of colliding

bodies diminishes after impact,and the coefﬁcient of restitu-

tion

,empirically determined for several materials,is widely

used to characterize energy loss due to impacts.According to

the Newton model [12],

relates the relative velocity of bodies

along the normal direction after impact

to the initial rela-

tive velocity

(1)

The coefﬁcient of restitution is unitary only in perfectly re-

versible impacts,where energy is completely conserved.For

low impact velocities within the elastic range of materials,ex-

periments showed that

depends on the initial velocity

,and

that with reasonable accuracy,the following relation holds:

(2)

Fig.1.Behavior of a material described by the Kelvin–Voigt model (pictures

are in different scales).(a) Hyteresis loop.(b) Power exchange.

where

,usually between 0.08 and 0.32 s/m,depends on the

materials of colliding bodies.According to (2),the energy

,

dissipated during an impact by a point mass

with initial mo-

mentum

,can be expressed as

(3)

Since the quadratic term

is relatively small,the following

approximation is usually considered:

(4)

B.Linear Contact Model

The coefﬁcient of restitution is an accurate way to experimen-

tally characterize impacts andsummarize energy dissipationdue

to complex phenomena,such as wave propagation,plasticity,

friction,and heat.However,in most applications,information

about contact force and dynamics is also needed,and different

models have been proposed in order to provide a continuous

representation of the collision phenomenon at a macroscopic

level.The simplest way to describe the mechanical impedance

of an object is the linear Kelvin–Voigt [20] contact model.It has

been obtained considering that an ideal viscoelastic material is

represented by the mechanical parallel of a linear spring and

a damper.If

is the force exerted by the material on a probe

during contact,the linear model is expressed by

(5)

where

is the penetration velocity of the probe,and

and

are the elastic and viscous parameters of the contact.The

hysteresis loop obtained simulating impact of mass

with a

material modeled by (5) is shown in Fig.1(a);the compression

phase corresponds to the upper arc,connecting point

to the

maximum penetration

,while the lower arc represents the

restitution phase.Physical inconsistencies of (5) are related to

the unnatural shock forces at the moment of impact (point

)

and of tensile or “sticky” forces at the moment of load removal

(point

).Moreover,the dissipated energy

is represented

by the area enclosed by the hysteresis loop,and can be computed

as the algebraic sumof the energies

,

,

.These energies

are also plotted in Fig.1(b),showing the power ﬂowobtained by

multiplying the reaction force

by the penetration velocity

.Since positive areas represent energy

ﬂowing from the mass to the touched material and vice versa,

DIOLAITI et al.:CONTACT IMPEDANCE ESTIMATION FOR ROBOTIC SYSTEMS 927

Fig.2.Behavior of a material described by the Hunt–Crossley model (pictures

are in different scales).(a) Hyteresis loop.(b) Power exchange.

represents power that is extracted from the mass,even in

the restitution phase,where

,and this behavior is in

contrast both with physical intuition and experimental results.

These physical inconsistencies happen because when the

penetration is small,the force

in (5) mainly depends on

the damping term,which is assumed to be independent on

.

As well discussed in [12],[15],and [16],this condition leads

to energetic inconsistencies,and the coefﬁcient of restitution

obtained from (5) does not depend on the impact velocity,in

contrast with (2).

C.The Hunt–Crossley Model

These problems can be overcome if the viscous force is made

dependent on the penetration depth,as ﬁrst proposed by Hunt

and Crossley [15]

(6)

where the exponent

is a real number,usually close to unity,

that takes into account the geometry of contact surfaces.Indeed,

since the contact surface increases as the penetration depth

increases,the exponent

allows taking intoaccount the stiffness

variation due to a larger contact area.Notice that when

,

the elastic term of (6) exactly matches the force resulting from

the Hertzian theory for spheres contacting in static conditions

[22].The newhysteresis loop,shown in Fig.2(a),shows that (6)

produces a behavior that is more consistent with experimental

observations.

Moreover,in [15],and more formally in [16],it has been

shown that for low impact velocities

,the Hunt–Crossley

model is consistent with (2).Indeed,the energy balance

equation during impact with a material whose impedance is

described by (6) is

(7)

where

is the initial momentum,

is the stored

elastic energy,and

represents the energy dissipated be-

cause of the damping term

(8)

At ﬁrst,it is convenient to analyze the properties of (6) in the

case

,and then to extend the results to generic materials.

Therefore,for perfectly elastic materials

,the hysteresis

Fig.3.Online parameter estimator for the Hunt–Crossley model.

loop degenerates to a line,and it is possible to relate initial ki-

netic energy to the maximum penetration

(9)

and ﬁnally,the relation between penetration

and velocity

is

given by

(10)

This expression can be considered approximately valid

also for materials having small viscous losses,compared

with the elastic energy storage.Then

(11)

and the substitution of (10) into (11) ﬁnally leads to

(12)

On the other hand,

can also be computed by substituting

(10) into (4).This allows relating,through (2),the coefﬁcients

of the Hunt–Crossley model to the coefﬁcient of restitution

(13)

In this way,

turns out to depend on the impact velocity

,con-

sistent with the previous discussion.Moreover,

does not affect

(13),so we can conclude that the coefﬁcient of restitution does

not depend on the shape/dimension of contact surfaces repre-

sented by this parameter.

III.E

STIMATION

A

LGORITHM

Because of its simplicity and its advantages with respect to

the linear model,the Hunt–Crossley model has been chosen to

represent the viscoelastic contact dynamics.

A linear regression algorithm for estimating the (unknown)

parameters

,

,and

can be implemented if force,position,

and velocity measures are available.In order to handle the non-

linearity of (6) with respect to the exponent

,the main idea is to

separate the estimation of

and

fromthe estimation of

,by

taking advantage of the partial decoupling of parameters noted

in (13).In this way,we can write two recursive least-squares

(RLS) estimators

and

[23] interconnected via feedback,

as shown in Fig.3.

With respect to Kalman ﬁltering techniques [24],the pro-

posed algorithmdoes not require performing local linearizations

of the process (6),and by taking advantage of the particular

928 IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.5,OCTOBER 2005

structure of the Hunt–Crossley model,handles the nonlinearity

by means of the interconnection of two linear estimators.

The block

estimates

and

minimizing the cost function

(14)

(15)

where

represents the discrete time variable

,

being the sample time,and

is the number of mea-

sures acquired from the beginning of the estimation process.

It is considered here,as commonly assumed in recursive

identiﬁcation algorithms,that

is a zero-mean stochastic

process,and summarizes model errors and measurement noise.

Let

be the vector of estimates at time

,

the vector of input

signals,and

the system output.The estimator

is implemented by the following recursive equations [23]:

(16)

where

represents the forgetting factor limiting the estimation

to more recent measures,and

the identity matrix,whose di-

mensions correspond to the number of parameters to be esti-

mated (two,in this case).

On the other hand,an expression of (15) that is linear with

respect to the parameter

can be obtained by means of algebraic

manipulation

(17)

with

(18)

If

is small with respect to the force computed according to the

Hunt–Crossley model (6),it is possible to write the following

series expansion of (17) (the index

is omitted for notational

simplicity):

(19)

Therefore,if the previous assumption holds,and

is inde-

pendent on the force computed according to (6),

can

be considered a zero-mean stochastic process.It is then possible

to estimate

by means of an RLS procedure that minimizes the

cost function

(20)

and the implementation of

is analogous to (16),with

,

and

.

A.Convergence Analysis

Because of the estimator structure (see Fig.3),the values of

,and

and

used by

and

,respectively,are not the “true”

parameters,but their estimates.For this reason,the convergence

of the overall estimation procedure depends on the additional

uncertainty introduced by the feedback interconnection of

and

.Let the stochastic processes

,

represent

the residuals between measured and estimated forces for (

,

)

and

,respectively.Previous considerations lead to rewriting

estimation errors (15) and (17) as

(21)

(22)

where the effect introduced by the use of estimates instead of

true parameter values is explicitly taken into account.There-

fore,provided that elementary estimators would converge,the

convergence of their feedback interconnection is obtained if ad-

ditional disturbances do not bias

,

.

In particular,(21) can be rewritten as

(23)

and,if the estimation error

is small,the following

approximation holds:

(24)

Therefore,

is a zero-mean stochastic process if

(25)

and this condition is satisﬁed if the estimator

converges in-

dependently on

.

By letting

and

,the relation between

and

is given by

(26)

that can be approximated by means of its series expansion

(27)

where (19) has been used.

Hence,the additional noise due to feedback interconnection

has to be dominated by

,so that

can be still consid-

ered a zero-mean process

(28)

Clearly,the most delicate part for the convergence of the al-

gorithm is the initialization,where initial values (

,

) have

to be provided to

and,since the real parameter values are un-

known,the initial estimation errors (

,

) can be substan-

tial.However,at the beginning of the estimation process,the

penetration

is small,thus satisfying (28).Moreover,since the

impact velocity required by the Hunt–Crossley model is limited,

a sufﬁcient number of samples with small

can be acquired,so

that (

,

) are quickly reduced.Therefore,the use of (

,

)

within the

estimator does not alter its convergence properties,

DIOLAITI et al.:CONTACT IMPEDANCE ESTIMATION FOR ROBOTIC SYSTEMS 929

Fig.4.Forgetting-factor adaptation depending on error.Parameters

,

,

and

can be adjusted on the basis of the noise level.

and the overall feedback estimator provides unbiased estimates

,

,

of parameters of the Hunt–Crossley model.

Notice that the derivation of the proposed estimator relies on

two main properties of the Hunt–Crossley model.First of all,

it is possible to obtain linearly parameterized equations for dif-

ferent subsets of parameters (

,

) and

,respectively.More-

over,since the estimation starts at

and the exponent

be-

longs,because of its physical meaning,to the interval [1,2],the

convergence of the estimator

can be shown to be independent

on the block

.Therefore,online estimation procedures,po-

tentially simpler than fully nonlinear min-max algorithms [25],

can be designed whenever similar conditions occur.

B.Performance Tuning

The need to identify modiﬁcations in touched materials sug-

gests computing the forgetting factor

dynamically,according

to the magnitude of the error

between the esti-

mated and the measured force [23].In particular,when the error

is small,the forgetting factor should be close to one,while it has

to be decreased when the error is large,so that the weight related

to older samples decays.In particular,as shown in Fig.4,

is

computed as

(29)

where

is the forgetting-factor value for large errors,

has the meaning of a threshold between the small error and large

error conditions,while the amplitude of the transition region is

governed by

.

C.Persisting Excitation Condition

Performances of recursive estimation are inﬂuenced mainly

by two factors:the correctness of the Hunt–Crossley model in

describing the behavior of compliant material,and the trajectory

followed by the probe device that should provide a sufﬁcient

level of excitation to the system.Indeed,the minimization of

(14) and (20) is aimed at minimizing the error on the estimated

force,but does not guarantee that parameter estimates converge

to the “true” values.

Since

and

are simple recursive estimators,the usual

notion of persistent excitation [23] can be used to formalize re-

quirements on the input signals

,

in order for the

parameter estimates

,

,and

to converge to the “true” values,

avoiding local minima.

and

being nonlinear func-

tions of the position and of the velocity,a fairly generic motion

proﬁle is sufﬁcient to obtain the convergence.Indeed,

re-

quires

to span

,and therefore,

that the position changes sufﬁciently over time;on the other

hand,

is not singular whenever

.

Note that these conditions are certainly satisﬁed by the random

motion of a human operator (e.g.,the palpation procedure per-

formed by the surgeon’s ﬁngers in order to detect the properties

of touched tissues).However,much simpler input signals,i.e.,

a sinusoidal motion proﬁle,are also capable of producing sufﬁ-

cient excitation for the identiﬁcation algorithm.

Simulation results obtained by initializing the estimation al-

gorithmwith a relatively wide range of randomly distributed ini-

tial conditions conﬁrm the convergence properties (see Fig.5).

The simulated material was characterized by the following pa-

rameters:

,

,and

,while the initial

estimates were assumed to be

,

,

.The sampling time of the simulation was

ms

and,according to previous discussion,an input signal obtained

by the superposition of two sinusoids of frequency 1 and 3 Hz

has been used.Notice also the effect of the forgetting factor,

ranging in the interval

,that causes small disconti-

nuities in the plotted diagrams.

IV.E

XPERIMENTAL

R

ESULTS

Alaboratory setup,shown in Fig.6,based on a linear electric

motor equipped by a load cell,similar to what was described

in [26],has been used to experimentally validate both the the

suitability of the Hunt–Crossley model to describe contact dy-

namics with stiff and soft objects,and the convergence proper-

ties of the estimation algorithm.The linear motor is a LinMot

P01-23x80,controlled by a LinMot E-110 unit,that implements

a position sensor based on a magnetic ﬁeld whose equivalent

resolution is

mm at the center of the workspace.The

tip of the mobile slider is equipped by a customload cell of mass

11 g and stiffness

N/mm;the surface used to

probe material is a half sphere made of brass.The device is in-

terfaced to a standard PCrunning real-time position control and

estimation tasks in a mixed MATLAB/RTLinux environment.

In the experimental activity,the sampling time has been set to

ms,and the resolution for the load cell was of

N,corresponding to 12 b with the saturation occurring at 30 N.

Signal conditioning includes temperature compensation in order

to avoid drifts during experiments;in these conditions,the av-

erage measurement error is smaller than 0.2 N.

The measures required by the estimation algorithm have

been obtained by imposing a motion proﬁle on the linear motor,

touching different materials.As mentioned in Section III-C,

both sinusoidal and random proﬁles have been used in order

to identify the parameters in “ideal” and more realistic con-

ditions,respectively.This allows verifying the convergence

properties of the estimation algorithm and to detect possible

frequency-dependent behavior of different materials.In partic-

ular,the randommotions have been obtained by a user handling

the slider of the linear motor,and their bandwidth is,therefore,

approximately 20 Hz.Moreover,the penetration velocity has

930 IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.5,OCTOBER 2005

Fig.5.Simulations results about the convergence to the “real” parameter values independently on initial estimates.(a) Convergence of

.(b) Convergence of

.

(c) Convergence of

.

Fig.6.Experimental setup.

been kept limited in order to achieve a stable contact,both in

the compression and the restitution phase,and to satisfy the

hypothesis required by the Hunt–Crossley model.

The penetration

is measured by comparing the current

motor position with the position measured at the time of impact.

Finally,the penetration velocity

is obtained from

by means

of a discrete-time adaptive windowing ﬁlter [27] that optimizes

the signal-to-noise ratio while minimizing the phase delay

with respect to the true velocity signal.

1

The contact force

is

measured by means of the load cell,whose intrinsic dynamics is

faster (

0.4 ms) than the sampling rate,and,more importantly,

than the bandwidth of the input signals.Therefore,the effect of

its inertial load on the measured force has been lumped in the

sensor noise,and did not need to be handled explicitly.

Finally,the parameters related to the forgetting-factor adap-

tation law (29) have been tuned to

,

,and

for all the experiments.

Several materials have been used,and here,in particular,

we present the results obtained with a thin layer (3-mm thick)

of plastic material (polycarbonate),characterized by a stiff

behavior,and a soft silicone gel used in [28],and whose energy

dissipation is substantial.

A.Stiff Material

Experimental results related to the thin layer of polycarbonate

probed with a sinusoidal motion proﬁle of frequency 2.5 Hz

1

The adaptive windowing ﬁlter is based on the concept that a more reliable

velocity estimation can be obtained by taking into account a larger number of

position measurements if the velocity is small,while a tighter windowhas to be

considered when the velocity increases;in this way,noise at low velocities is

minimized,as well as the delay at high velocities.

Fig.7.Polycarbonate layer.Sinusoidal input (top:

dashed and

solid)

and contact forces (bottom:

dashed,

solid).

(Fig.7) are presented in Figs.8 and 9.In particular,the hys-

teresis loop reported in Fig.8(a) shows that energy dissipation

is low,and the behavior of the material depends essentially on

the elastic coefﬁcient

.

Moreover,both the hysteresis loop and the related power

exchange,reported in Fig.8(b),show good correspondence

between the experimental curves and the estimated ones.Note

that this class of materials could be adequately modeled also

by means of the Kelvin–Voigt model,since points

and

of

Fig.1(a) are,in this case,very close to the origin,and therefore,

a damping coefﬁcient independent of

does not signiﬁcantly

alter the force estimation

with respect to

.Finally,Fig.9

shows the parameter estimates.In particular,the following

values are obtained:

,

,

.

To evaluate the quality of these estimates,the Matlab optimiza-

tion toolbox has been used a posteriori to compute the best-ﬁt

parameters on the overall set of measures,obtaining the fol-

lowing values:

,

,

,that

conﬁrmthe convergence properties of the estimation algorithm.

According to what previously discussed about properties of the

Hunt–Crossley model,the exponent

,that takes into account

geometry of contact surfaces,is about one.We notice that the

settling time for

about its ﬁnal value is a little longer than the

settling time of

,and this depends mainly on the numerical

differentiation used to compute

,which makes this parameter

more sensitive to measurement noise.

DIOLAITI et al.:CONTACT IMPEDANCE ESTIMATION FOR ROBOTIC SYSTEMS 931

Fig.8.Hunt–Crossley model.Measured (dashed) and estimated (solid) hysteresis loop and power exchange for a thin layer of polycarbonate.

Fig.9.Hunt–Crossley model.Parameter estimation for a thin layer of

polycarbonate material.

B.Soft Material

For compliant materials,the advantages of the Hunt–Crossley

model with respect to the linear model are more evident.Fig.10

illustrates the input sinusoidal proﬁle of frequency 1 Hz,the

measured contact force,and the force estimated according to

the Hunt–Crossley model.The application of a linear regression

algorithmto estimate parameters

and

of the Kelvin–Voigt

model for a soft gel provides the results of Fig.11,with

N/m and

Ns/m.In particular,drawbacks

related to nonzero estimated force when

are evident,as

well as inconsistencies in power exchange between the probe

device and the silicone gel.

On the contrary,hysteresis loop estimated by means of the

Hunt–Crossley model is more similar to the measured one,

shown in Fig.12(a),as well as the estimated power exchange,

shown in Fig.12(b),that does not exhibit positive “spikes,” as

in Fig.11(b).

Finally,Fig.13 shows the convergence of parameters to their

ﬁnal values

,

,and

.Notice

that in this case,the value of

is greater than that obtained for

Fig.10.Silicone gel.Sinusoidal input (top:

dashed and

solid) and

contact forces (bottom:

dashed,

solid).

the stiff material,since the contact surface between the probe

device and the silicone gel is slightly different.For comparison,

the values obtained from the a posteriori Matlab curve ﬁtting

are

,

,and

.

C.Change of Material

As mentioned in Section III,the use of a forgetting factor

allows improving the detection of material changes.In partic-

ular,the case of a switching from the silicone gel to the layer

of plastic materials has been considered (see Fig.14).The

hysteresis loop and the power exchange diagram are shown

in Fig.15,which conﬁrm the adequacy of the Hunt–Crossley

model to describe both stiff and soft materials.

The ability of the estimation algorithm to detect a change in

the touched material is shown in Fig.16.Indeed,after a short

transient,the convergence to parameters of the new material is

achieved,and previous estimates do not affect ﬁnal values.In

particular,at the time when the material was changed,the pa-

rameter estimates were

,

,and

,almost identical to those detected in Section IV-B,

while at the end of the experiment,we had

,

,and

,according to the ﬁndings of Sec-

tion IV-A.

932 IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.5,OCTOBER 2005

Fig.11.Linear model.Measured (dash) and estimated (solid) hysteresis loop and power exchange for silicone gel.

Fig.12.Hunt–Crossley model.Measured (dashed) and estimated (solid) hysteresis loop and power exchange for silicone gel.

Fig.13.Hunt–Crossley model.Parameter estimation for silicone gel.

D.Random Motion

Finally,the proposed estimation algorithmhas been tested by

imposing a random motion proﬁle on the linear motor slider.

In particular,a human operator grasped the shaft of the linear

motor and manually produced the contact between the probe tip

and the test surface.

Fig.17 shows the motion proﬁle (top),followed by the motor

slider during the interaction with the silicone gel,while the

Fig.14.Change of material.Sinusoidal input (top:

dashed and

solid)

and contact forces (bottom:

dashed,

solid).

good correspondence between measured and estimated forces

is shown in the bottomplot.Notice that both amplitude and fre-

quency of vibrations due to impact are very limited because of

the limited penetration velocity,and therefore,they do not affect

the acquisition and estimation processes.

Parameter estimates are plotted in Fig.18.The ﬁnal values are

very close to those obtained in the experiment with sinusoidal

motion:

,

,

,thus con-

ﬁrming the adequacy of the Hunt–Crossley model to describe

the contact dynamics and the convergence properties of the es-

timation algorithm (see Fig.13).

DIOLAITI et al.:CONTACT IMPEDANCE ESTIMATION FOR ROBOTIC SYSTEMS 933

Fig.15.Measured (dashed) and estimated (solid) hysteresis loop and power exchange when silicone gel is substituted by the thin plastic layer.(a) Hy

teresis loop.

(b) Power exchange.

Fig.16.Estimated parameters when material is changed.

Fig.17.Silicone gel.Randommotion proﬁle (top:

dashed and

solid)

and contact forces (bottom:

dashed and

solid).

Similar results,obtained with the stiff plastic layer,are pre-

sented in Figs.19 and 20.Notice that in this case,it has been

Fig.18.Silicone gel.Parameter estimates with randommotion.

Fig.19.Polycarbonate layer.Random motion proﬁle (top:

dashed and

solid) and contact forces (bottom:

dashed and

solid).

more difﬁcult to achieve a stable contact with the material,and

the effect of measurement noise,especially on the force and the

velocity,is somehow ampliﬁed.

Indeed,when the user is slowly penetrating the material,the

contact force can become larger than user’s grip,which is usu-

934 IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.5,OCTOBER 2005

Fig.20.Polycarbonate layer.Parameter estimates with randommotion.

ally quite loose during this kind of task,and the motor shaft

quickly tends to return to its rest position,see,e.g.,

.

However,as more samples are collected,the estimates converge

to values close to those obtained in the sinusoidal case,namely,

,

,and

.

By comparing the estimation outputs with randommotion to

previous experiments with sinusoidal motions,we can conclude,

both for the silicone gel and the polycarbonate layer,that the pa-

rameters

,

,and

do not exhibit frequency-dependent behav-

iors,and that the estimation algorithmconverges in both cases.

V.C

ONCLUSIONS AND

F

UTURE

W

ORK

The need to accurately represent the contact dynamics be-

tween a robotic system and a compliant object requires the

choice of a suitable model and of an online estimation algo-

rithm.In this paper,the properties of the Hunt–Crossley model

have been discussed and compared with those of the traditional

linear model.However,the nonlinearity of the former model

requires designing an online recursive estimator that combines

efﬁciency and good convergence properties.The proposed

algorithm has been used to experimentally identify parameters

characterizing different materials,and the results relative to two

cases of practical interest have been reported and discussed.In

the case of a stiff material,the advantages of the Hunt–Crossley

model are more limited,while for the silicone gel,it has been

shown that the classical Kelvin–Voigt model does not provide

satisfactory results.

Future work,besides a more formal proof of the convergence

of the estimation algorithm,will be aimed at the generalization

of the described models and algorithms to the full geomet-

rical contact (6 DOF),following the approach outlined in [17]

and [18],aimed,as is [29],at representing the physical and

geometrical properties of the contact surfaces.Moreover,the

application of this estimation technique for the improvement

of performance of robotic systems interacting with unknown

environments,such as in telemanipulation systems,will be

investigated.

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DIOLAITI et al.:CONTACT IMPEDANCE ESTIMATION FOR ROBOTIC SYSTEMS 935

Nicola Diolaiti (S’02) received the M.Sc.degree

(cum laude) in electrical engineering in July 2001

from the University of Bologna,Bologna,Italy,

where he is currently working toward the Ph.D.

degree.

In 2003,he visited the Drebbel Insitute,Univer-

sity of Twente,Enschede,The Netherlands,and in

2004 and 2005,he was appointed Visiting Scholar at

the Stanford AI-Robotics Lab,Stanford University,

Stanford,CA.His research activity is focused on the

modeling and control aspects of interactive robotic

systems,and in particular,on bilateral teleoperation and haptic interfaces.

Claudio Melchiorri (M’92–SM’00) was born on

October 23,1959.He received the Laurea degree

in electrical engineering in 1985 and the Ph.D.

degree in 1990,both fromthe University of Bologna,

Bologna,Italy.

He was appointed Adjunct Associate in Engi-

neering in the Department of Electrical Engineering,

University of Florida,Gainesville,in 1988,and

Visiting Scientist in the Artiﬁcial Intelligence

Laboratory,Massachusetts Institute of Technology,

Cambridge,for periods in 1990 and 1991.Since

1985,he has been with DEIS,the Department of Electrical Engineering,

Computer Science and Systems,University of Bologna,working in the ﬁeld of

robotics and automatic control.He currently holds the position of Associate

Professor in Robotics at the University of Bologna.His research interests

include dexterous robotic manipulation,haptic interfaces,telemanipulation

systems,advanced sensors,and nonlinear control.He is the author or coauthor

of about 150 scientiﬁc papers presented at conferences or published in journals,

of three books on digital control,and is coeditor of three books on robotics.

Stefano Stramigioli (SM’00) received the M.Sc.de-

gree with honors (cum laude) fromthe University of

Bologna,Bologna,Italy,in 1992,and the Ph.D.de-

gree with honors (cum laude) fromthe Delft Univer-

sity of Technology,Delft,The Netherlands,in 1998.

Since 1998,he was ﬁrst an Assistant Professor

with the Delft University of Technology,and then

Associate Professor with the University of Twente,

Enschede,The Netherlands.He has more than 50

publications,including a book.He is involved in

different projects related to control,robotics,MEMS,

and intelligent transportation systems,and is Coordinator of the European

Project Geoplex (http://www.geoplex.cc).He has been teaching modeling,

control,and robotics for under- and post-graduates,and has received teaching

nominations and an award.

Dr.Stramigioli is currently the Vice President for Technical Activities of the

Intelligent Transportation Systems Society,and the IEEE Representative of the

IEEE Robotics and Automation Society (RAS) for ITSC.He chairs the Tech-

nical Committee on Intelligent Transportation Systems for the IEEE RAS.He

is a member of the ESA Topical Team on Dynamics of Prehension in Micro-

gravity and its application to Robotics and Prosthetics.He is Editor-in-Chief of

the IEEE ITSC Newsletter and the IEEE Robotics and Automation Magazine,

and has been Guest Editor for others.

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