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 singlepoint
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 realtime 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.IST200134166.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
(email: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 (email: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 micromacro 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
1D 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,
15523098/$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 endeffector 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 leastsquares
(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 zeromean 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 zeromean 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 zeromean 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 zeromean 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.Forgettingfactor 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 minmax 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 forgettingfactor 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
P0123x80,controlled by a LinMot E110 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 realtime 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 IIIC,
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
frequencydependent 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 discretetime adaptive windowing ﬁlter [27] that optimizes
the signaltonoise 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 forgettingfactor 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 (3mm 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 bestﬁt
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 IVB,
while at the end of the experiment,we had
,
,and
,according to the ﬁndings of Sec
tion IVA.
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 frequencydependent 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 AIRobotics 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 postgraduates,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 EditorinChief of
the IEEE ITSC Newsletter and the IEEE Robotics and Automation Magazine,
and has been Guest Editor for others.
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