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Friction Compensation in Robotics: an Overview


Abstract and Figures

Friction effects are particularly critical for industrial robots, since they can induce large positioning errors, stick-slip motions, and limit cycles. This paper offers a reasoned overview of the main friction compensation techniques that have been developed in the last years, regrouping them according to the adopted kind of control strategy. Some experimental results are reported, to show how the control performances can be affected not only by the chosen method, but also by the characteristics of the available robotic architecture and of the executed task.
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Friction Compensation in Robotics: an Overview
Basilio Bona and Marina Indri
Abst ract — Friction effects are particularly critical
for industrial robots, since they can induce large
positioning errors, stick-slip motions, and limit cycles.
This paper offers a reasoned overview of the main
friction compensation techniques that have been de-
veloped in the last years, regrouping them according
to the adopted kind of control strategy. Some experi-
mental results are reported, to show how the control
performances can be affected not only by the chosen
method, but also by the characteristics of the available
robotic architecture and of the executed task.
I. Introduction
Each time precise motion control must be achieved
by a robotic system, friction compensation represents
a crucial step for the designer, who must solve various
theoretical and practical problems. Friction effects are
particularly critical for industrial robots: it has been
observed [1] that “friction can cause 50% error in some
heavy industrial manipulators”. A poor (or absent) fric-
tion compensation action in the control scheme may lead
to significant tracking errors (especially at low velocities),
stick-slip motions, hunting in the stopping phase of the
robot movement, and limit cycles when velocity reversals
occur in the assigned trajectory.
Several friction models have been proposed [2]-[12],
having different levels of accuracy, and a wide variety
of control solutions can be found in literature [1], [3]-
[6], [11], [13]-[15], [19]-[25], [27]-[43], but no strategy can
be definitely considered more effective than the others,
since many factors can significantly affect the practical
implementation and the performances of each scheme.
This paper proposes an overview of the most used
and recent control strategies for friction compensation
in robotics, with the purpose to offer a guideline to
the reader among the various solutions that have been
developed in literature. A classification of the various
control schemes can be made following different criteria:
in this paper the control strategy is considered instead
of the assumed friction model (as it is often done), to
help the controller designer to choose the most suitable
solution for the task to be executed, given the available
hardware and software control architecture. We have
considered only the most recent papers (as the interested
reader can find previous results among their references)
and the control schemes that have been experimentally
The paper is organized as follows. In Section II, the
most common friction models are briefly pointed out,
This work was supported by ASI and MIUR.
B. Bona and M. Indri are with Dipartimento di Automatica
e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi
24, 10129 Torino, Italy, ma-
while Section III deals with the different control strate-
gies developed for friction compensation, distinguishing
four main types of solutions, which are discussed in
the four corresponding subsections. Section IV draws
some conclusions concerning the reviewed methods, and
discusses some additional issues, on the basis of exper-
imental tests carried out in our laboratory for a planar
manipulator with standard-resolution resolvers.
II. Friction models
Friction forces between two surfaces in contact arise
as a consequence of the irregularities and asperities at
microscopical level, and their effects depend on many
factors, such as displacement and relative velocity of the
bodies, properties of the surface materials, presence of
lubrication, temperature, etc. The experimental obser-
vation of friction phenomena has led to various, deeply
different models, which capture the friction components
in a more or less accurate way. Interesting reviews of the
main friction characteristics and classical models can be
found in [3], [4], [6], starting from the basic concept of
friction as a force that opposes motion, captured by the
pure Coulomb model, up to complex static and dynamic
The simplest static friction model that can be adopted
is the Coulomb + viscous one:
F(v)=Fcsgn(v)+βv (1)
where Fis the friction force, vis the relative velocity of
the contact surfaces, βis the viscous friction coefficient,
and Fcis the Coulomb friction.
Model (1) can be easily upgraded taking into account
the presence of stiction, i.e. a higher friction force Fs
acting while the system is at rest: motion can start
only when the applied external forces (i.e. the command
torque applied to the joint in the robotic case) are greater
than Fs[6]. From experimental observation it is evident
that the passage from stiction to friction during motion
is not discontinuous: on the basis of such a consideration,
different mathematical models have been formulated to
represent friction vs. velocity by continuous functions
that account for the so-called Str ibec k eff ect, i.e. the
decrease of the friction amount as velocity increases
within the low-velocity Stribeck region. The most used
nonlinear expression leads to a model of the following
F(v)=Fc+(FsFc)e−| v
vs|δssgn(v)+βv (2)
where vsis the Stribeck velocity that defines the region
in which such an effect is present, and δscan be equal to
2 (but not necessarily).
Other models have been used for control purposes
to represent friction, including Stribeck, Coulomb and
Proceedings of the
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viscous effects, such as a simple polynomial function of
a proper order as in [13], [16], or sigmoid-functions as
in [7], [17]. All the models analyzed up to this point,
and many others that have been developed from the
Karnopp model (see Section III-B and references [2],
[6]) are static models, as friction depends only on the
current velocity value. The friction phenomena related
to non-stationary velocities, variations in the break-away
force, small displacements occurring during the stiction
phase, and hysteretic effects can be captured only by
dynamic models, which describe the pre-sliding behavior
representing the microscopic asperities of the contact sur-
faces by means of elastic bristles, whose deflections give
rise to the friction forces. The well-known Dahl model
(illustrated and discussed in many papers, such as [4], [6],
[14]) is the simplest possible description of the dynamic
friction behavior, and it can be expressed introducing an
internal state variable z(representing the average bristle
deflection at the contact points of the moving surfaces)
and defining the friction force accordingly as:
σz (3a)
F=σz (3b)
where σrepresents the bristle stiffness parameter. The
Dahl model neither includes stiction, nor the Stribeck
effect, since it simply adds a lag in the changes of the
friction forces as velocity changes.
The widely used LuGre friction model, proposed for
the first time in [5], is complete from this point of view,
since it includes also these effects, describing the behavior
of the friction force as:
where zis an internal state variable as in (3), σ0and
σ1are model parameters assumed to be constant, and
functions g(v)andf(v) model the Stribeck effect and
the viscous friction, respectively. For constant velocity,
the steady-state friction force is then given by:
Fss =g(v)sgn(v)+f(v)(5)
Different parameterizations are possible for g(v)and
f(v): the most used choices are those leading to an
expression for the steady-state friction similar to (2).
Details about them and about the identification pro-
cedure for the LuGre model can be found in [15]. It
must be underlined that the identification of a friction
dynamic model like the LuGre one always presents some
peculiar difficulties, related to different aspects, such as
the impossibility to directly measure the internal state z,
the high sensitivity of the stick-slip motions to the values
of the dynamic parameters (i.e. σ0and σ1for the LuGre
model), and the necessity of high precision sensors to
correctly capture the phenomena of the pre-sliding phase.
Even if the LuGre model is often used for control, it
does not represent the final solution from the modelling
point of view, since it does not take into account possible
hysteresis effects. Alternative friction models, including
also hysteresis effects at low velocity, have been proposed
and discussed e.g. in [8], [9], and [11], whereas updates
to the LuGre model have been proposed in [10] to avoid
nonphysical drift phenomena in the presliding phase.
More recently, single and multistate integral friction
models have been proposed in [12], to account for the
hysteresis behavior with non local memory.
III. Control strategies for friction
Several solutions have been proposed to compensate
for friction in robotics, considering different kinds of fric-
tion phenomena, and hence different models to represent
it, and various strategies to eliminate or satisfactorily
reduce its undesirable effects on the robot motion. In
practice the choice of a particular solution is strongly
influenced by factors like the available actuators, sensors,
and hardware/software control architecture, as well as
by constraints on the real-time computational burden.
It can be useful then to analyze the various friction
compensation techniques classifying them according to
the adopted kind of control strategy, thus recognizing the
following four main types of solutions.
A) A ‘fixed’ friction compensation term is added to a
more general control scheme, like a joint indepen-
dent one or an inverse dynamics control scheme [18],
by estimating the friction parameters off-line, on
the basis of an assigned (more or less complicated)
friction model, following proper, ad hoc identifica-
tion procedures [17], [22], [23]. If a dynamic friction
model is considered, an observer is inserted to esti-
mate the friction internal state [24], [25]. A further
correction or a robust control action can be possibly
added to the a priori estimated compensation term
[1], [5], [14], [27], [28].
B) Only some main characteristics of the friction phe-
nomena (e.g. the maximum stick component) are
taken into account for compensation within the con-
trol scheme. The estimates of the required friction
characteristics are determined off-line, whereas the
on-line compensation action is tuned on the basis of
such estimates [29]-[31].
C) Model-based adaptive algorithms are applied for
on-line friction compensation [15], [34]-[39]. The
adaptive schemes are based on a particular, static
or dynamic, friction model, whose parameters are
tuned on-line to obtain a satisfying compensation
action also when significant variations are present.
D) Strategies that are not based on a particular friction
model can be applied to counteract the friction
effects, by properly choosing the control gain param-
eters or by using non model-based observers [40]-
Another group of friction compensation techniques
takes advantage of the so-called soft computing approach,
using fuzzy, neural, and genetic algorithms to reconstruct
the friction torques to be compensated or for a suitable
self-tuning of the controller gains (see e.g. [19]-[21]).
These approaches will not be reviewed in detail in the
present paper for space reasons.
Each type of control solution has advantages and
drawbacks that cannot be ignored. Solutions of type
A), especially if based on static friction models, are
the ‘cheapest’ to be implemented in practice, since the
required on-line computational burden is limited (a feed-
forward compensation term could be possibly used, if
necessary, in the static case using the reference posi-
tion/velocity values instead of the current ones), but they
require an accurate friction identification phase (possibly
with high-precision sensors), and they cannot, obviously,
account for friction variations. Solutions B) could offer
some more robustness features, but a correct tuning of
the compensation action is crucial to avoid limit cycles,
and to obtain an accurate final positioning. Solutions C)
often require a significant on-line computational burden,
and high-precision sensors when dynamic friction models
are considered. Besides, only some of the friction param-
eters can be easily updated on-line, as it will be discussed
in Section III-C. In the control solutions of type D),
the friction effects are often estimated and compensated
only together with other disturbances acting on the robot
motion, so that the actual control performances could
significantly vary in practice for different trajectories
and/or be satisfying only if high control values can be
sustained by the considered robotic system.
It is then evident that none of these types of control
solutions is superior to the others, in every case. Some
control schemes proposed in literature, and regrouped
according to the above classification, are discussed in
the following, to analyze with more details their main
A. Fixed friction compensation techniques
Satisfactory results can be obtained by adding a
‘fixed’ friction compensation term to standard control
algorithms, if a good off-line estimate of the friction
parameters has been performed, provided that friction
variations with time, temperature, etc. are negligible.
It is quite obvious that the friction model used to
define the compensation term must be sufficiently accu-
rate: compensation terms based on the pure Coulomb
+ viscous friction model (1) cannot provide accurate
tracking and positioning results, as shown e.g. in [22],
where this kind of solution is compared with friction
compensation performed on the basis of the nonlinear,
static model (2) in the case of a simple servomechanism.
As expected, the best results are obtained when the
second kind of compensation term is added to a model-
based control algorithm, also thanks to a scrupulous
identification of the friction model parameters, aimed at
the avoidance of limit cycles generation.
More recently, another compensation solution based
on the same kind of discontinuous, static friction model
has been proposed in [23], where an observer-based
compensation scheme is developed by estimating again
the parameters of the friction model off-line. A lin-
ear observer reconstructs the system states (i.e., joint
position and velocity in the robotic case), whereas a
nonlinear friction compensation term, defined by using
the identified friction model and the estimated velocity,
is added to a standard observer-based state feedback
control. Good performances can be obtained by a proper
tuning of the controller and the observer gains, if the
presliding friction dynamics is actually negligible.
A different kind of static, a priori estimated friction
compensation solution is proposed in [17] for a three
revolute joints robot, on the basis of the following three-
sigmoid-function friction model:
fi,k 12
e2wi,k ˙qi+1+bi˙qi,i=1,2,3(6)
where τf
iis the friction torque acting on the i-th joint,
˙qiis the i-th joint velocity, and fi,k ,wi,k and biare the
parameters that define the friction model. It is important
to note that model (6) does not capture the static friction
component, since τf
i=0at ˙qi= 0: its compensation
then cannot be performed by the model-based term but
only via an integral control action. The results obtained
in a hand-writing task by a complete inverse dynamics
control scheme with friction compensation are extremely
If the dynamic friction effects in the presliding phase
cannot be neglected, the compensation term must be de-
fined on the basis of a dynamic model, with the insertion
of an observer for the friction internal state estimation, as
in [24], [25], where the Dahl model (3) is considered, with
the addition of a viscous friction term, to compensate
friction within velocity control schemes. In particular, in
[24] an inverse dynamics control scheme and a PD-like
algorithm (given by a nonadaptive version of the motion
controller proposed in [26]) are applied to a two dof
direct-drive robot arm, comparing the results obtained
by a pure Coulomb + viscous friction compensation with
those provided by the Dahl model-based compensation.
As expected, the second friction compensation solution
gives the best results with both controllers, while the
PD algorithm provides better performances with respect
to the inverse dynamics scheme, when the same friction
compensation term is inserted, perhaps because of a
higher sensitivity of the last controller to model and
parameter estimation errors.
Since in practice it is not always possible to obtain a
sufficiently accurate friction description to be used for a
pure fixed compensation, some kind of on-line correction
action is added to the compensation provided by the
off-line estimated friction model to obtain better control
performances, thus getting to schemes that lie between
types A) and C). In [27], the Coulomb and viscous
friction parameters are estimated off-line, whereas the
nonlinear, Stribeck effects are compensated within a
sliding model control scheme by means of a disturbance
observer; only an upper bound of the Stribeck term is
required for the implementation of such a controller.
It is important to note that the estimated disturbance
term is not necessarily given by friction only, even if
its effects are certainly dominant in the low velocity
region. A nonlinear H-controller is proposed in [14],
where friction is supposed to be described by the Dahl
friction model, assuming that its parameters are known
(i.e. identified off-line), while the discrepancies between
the actual friction and the estimated one are overcome
by the robustness properties of the approach.
The control solution proposed in [5] is constituted
by a linear position or velocity controller + a friction
compensation term, estimated according to the LuGre
friction dynamic model (4) with the insertion of an
‘observer’ term, proportional to the tracking error in
(4a), thus obtaining an estimate of the friction force ˆ
g(v)σ0ˆzKe (7a)
ˆz+βv (7b)
where eis the tracking error, Kis the observer gain,
and f(v)=βv has been considered in (4b). A similar
approach for friction compensation on robot joints has
been considered in [1], together with a PD algorithm
with gravity compensation; an interesting discussion is
developed by the authors about the friction identification
problems for a 2 dof micro-manipulator and a 4 dof
macro-manipulator, showing the importance of such a
phase for the control performances, together with the
inability of a pure static, steady-state model to describe
friction beyond a certain accuracy, at least for the consid-
ered manipulators. A similar compensation term is added
also in [28] to an output feedback controller, by intro-
ducing in (7a) a properly designed output function K(y)
instead of Ke, but no experimental results are reported
to test the actual performances of such a scheme.
B. Techniques based on a partial knowledge of the friction
These techniques are based on the addition of a friction
compensation term to standard control algorithms, like
in the previous cases, but without requiring the knowl-
edge of the exact friction model.
In particular, this kind of approach, originally devel-
oped in [29], introduces a specific nonlinear compensation
term that supplements a standard PD control algorithm;
asymptotic stability is guaranteed for a stick-slip friction
system, provided that the upper bounds of the static
friction levels are known; robustness is also assured with
respect to the characteristics of the slipping force, as-
sumed to lie within a piecewise linear band. The friction
torque τfacting on each joint of a robot is described
according to the Karnopp model [2], as the sum of the
static torque τstick and the slipping torque τslip, with:
τstick =
where τcis the command torque, τ+
sand τ
sare the
positive and negative limits of the static friction torque,
dq) are the slipping torque functions for
positive and negative velocities, respectively, supposed
to be bounded within the first and third quadrants, and
µ(·) is the right-continuous Heaviside step function.
The application of a traditional PD control leads to a
steady-state position error, since all the trajectories end
up within an equilibrium region in which qLqqH,
with qL=τ+
s/KP<0, qH=τ
s/KP>0, KP
being the position control gain, and assuming qr=0
as reference position, for the sake of simplicity. In the
nonlinear solution proposed in [29], the steady-state
position error is eliminated by defining the command
torque as τc=KD˙qτn, with
˜qH=qH+ε, ˜qL=qLε, ε > 0
The nonlinear compensation torque τnis active only
when the joint position is between the augmented stick-
ing limits ˜qLqH, while the controller is essentially a PD
one outside such a region.
Some modifications and upgrades have been proposed
more recently to this technique, extending in particular
its application to digital control systems, too, for which
a stable limit cycle response would be induced otherwise,
due to the time delay introduced by the sample-and-hold
operations. In particular, in [30] an hysteresis expression
has been proposed for the nonlinear compensation term
τn, introducing two nonzero constants, δLand δH,denot-
ing the bounds of the velocity-dependent dead zone (see
[30] for details). Their choice is crucial to eliminate the
destabilizing effect of the time delay: greater values im-
prove the stability margin, but enlarge the error bounds,
since the steady-state position will lie between δLand
δH. Slight modifications to the method, e.g. in [31], have
been proposed to try to obtain a smaller final error,
but the actual possibility to achieve good performances
seems to be strongly related to the characteristics of the
considered robotic systems and of the hardware control
architecture, which influences the choice of δL,δHand
of the sampling time.
C. Adaptive compensation schemes
As friction characteristics vary with time, temperature
and system operating conditions, the adaptive compen-
sation approach seems to be the natural solution to
maintain satisfying and constant control performances
in the various situations. Even if the researchers interest
has been devoting to this kind of approach since the
beginning of the 90’s (see e.g. [32], [33]), two main
problems emerge each time a complete, dynamic friction
model is considered: (i) some friction parameters enter
in a nonlinear way in the model, and (ii)partofthe
system dynamics (i.e. the internal friction state z), that
in turn depends on the unknown parameters, is not
measurable. There is no global solution to a problem like
this, in which system state variables and parameters of
a nonlinear model should be simultaneously estimated.
Leaving to the interested reader more details about each
particular adaptive algorithm that has been proposed in
the last years, it is interesting to compare the different
approaches that has been followed to overcome the above
two problems.
In [15] adaptive versions of the observer-based fric-
tion compensation scheme proposed in [5], and briefly
recalled in Section III-A, are developed to cope with
structured friction variations in two cases. In the first one,
variations of the normal forces exchanged between the
contact surfaces are assumed to mainly affect the friction
static parameters, whereas the dynamic parameters are
considered as invariant due to the unchanged lubricant
characteristics, and possible variations of the viscous
coefficient βare directly dealt with the linear controller.
In the second case, temperature variations are assumed
to uniformly affect both static and dynamic friction pa-
rameters. The resulting parameter uncertainties in both
cases are captured by a unique variable parameter θ,by
rewriting model (4) with f(v)=βv as follows:
Case 1):
Case 2):
where parameters σ0,σ1,βand function g(v) are sup-
posed to be known, i.e. a priori identified, and θis
updated on-line, according to a proper adaptation law
(see [15] for details and for some experimental results).
This kind of approach has been upgraded in [34], where
three different observers are developed for the estimation
of the unmeasurable friction state, in order to relax
the conditions required in [5], and hence to facilitate
the use of different control loops. On the basis of such
observers, two adaptive controllers are proposed: the
first one compensates for all the mechanical parameters
variations, except for the parameters associated to the
Stribeck effect (i.e. the parameters of function g(v)in
(4a)); the second one compensates for only a single
parameter associated with normal force variations in the
Stribeck effect function, following the same approach of
case 1) in [15], but utilizing a nonlinear filter structure
that allows an active compensation for the observer
transient. Experimental results are available only in a
1-dof case, given by a switched reluctance motor, with a
metal disk attached to the rotor.
An interesting comparison of different static and dy-
namic friction compensation techniques, including adap-
tive algorithms, is presented in [35], in the case of a
two-dof planar manipulator, showing in particular the
different performances that can be obtained by using
(i) a pure computed torque scheme, (ii) the adaptive
controller developed in [26] without friction compensa-
tion, the same kind of adaptive control scheme with (iii)
static or (iv) dynamic friction compensation. In the static
friction case, only the Coulomb and viscous components
are considered and directly included in the adaptive law,
since they enter linearly in the model, and similarly, in
the dynamic friction case, only the parameters that lin-
early enter in equation (4b) are included in the adaption
law, while the parameters defining function g(v) in (4a)
are a priori identified, and an observer is inserted for the
estimation of zand ˙z. This last solution gives the best
results, as expected, thanks also to the available high-
precision resolver.
All the considered adaptive algorithms, based on dy-
namic friction models, result in compromises between
the necessity of estimating the unmeasurable part of the
friction dynamics and of updating on-line the friction
parameters, typically disregarding the nonlinear param-
eters related to the Stribeck effect in the adaptation
law. If a static model is considered to represent friction,
nonlinear adaptive control schemes can be developed,
taking into account also the variations of the Stribeck
effect parameters, as in [36], where friction is modelled
se(Fτv2)sat(v) (12)
where the sat(·) function can be defined as the standard,
discontinuous signum function (see [36] for details). In
particular, one of the two control schemes that are
proposed is given by an adaptive set-point controller (i.e.
performing only regulation tasks), which compensates for
the uncertainty associated with all the friction parame-
ters. In particular, the stiction and Stribeck parameters
sand Fτare updated by laws of the following types:
where γ0and γ1are positive, constant gains. Experimen-
tal results are shown for the same setup used in [34].
The tracking problem has been addressed in [37],
where a general framework of adaptive control is pro-
posed to compensate for uncertain nonlinear parame-
ters appearing in robot dynamic model, assuming that
friction is described by the static, nonlinear model (2)
with δs= 2. The resulting adaptive controller, applicable
under Lipschitzian conditions, incorporates observers of
minimum dimension, independently of the dimension
of the unknown parameter vector. In particular, the
applicability of the proposed controller is guaranteed
by the possibility of decomposing the friction force (2)
into a linear part FL(v) and a nonlinear one FN(v)as
F(v)=FL(v)+FN(v), with
FL(v)=Fcsgn(v)+βv (14a)
s)sgn(v) (14b)
where FN(v) can be rewritten as the product of two
Lipschitzian functions in the parameter vector θ=
[θ1θ2]T:= (FsFc)1/v2
FN(v)=g(v, θ)h(v, θ) (15)
with g(v, θ)=[1 0]θand h(v, θ)=e
(v2θ2)sgn(v). De-
tails about the adaptive controller can be found directly
in [37], together with some experimental results for a
two-dof planar manipulator.
In [38], [39], a decomposition-based friction compensa-
tion method is proposed, designing a separate compen-
sator for each type of friction, utilizing different control
techniques. Friction is described by the static model (12)
with the addition of the viscous component and a further
term that takes into account the position dependency of
friction and other modelling errors. The Stribeck term is
linearized at the nominal parameter values F
sand Fτ,
so that all the friction parameters appear linearly in the
linearized model, and adaptive control and robust control
techniques can be easily applied. In particular, while
the nominal friction is compensated by feedforward (on
the basis of off-line estimates), an adaptive compensator
is designed to compensate for parametric unmodelled
friction with unknown but constant parameters, and a
robust compensator is used to deal with friction model
parameter variations, as well as non-parametric unmod-
elled friction.
Finally, it is worth to be noted how an adaptive
friction compensation law has been successfully utilized
in [13] in some hybrid force/velocity contour tracking
tests, by using a polynomial friction model of properly
high degree: the advantage of this solution is given by
the fact that the friction parameters, i.e. the polynomial
coefficients, enter linearly in the model, and hence update
laws can be easily designed for their on-line adaptation.
D. Non model-based compensation schemes and neural-
fuzzy techniques
The friction compensation action of non-model based
control schemes is generally accomplished by proper
choices of the gains of standard control algorithms. Since
the beginning of the 90’s, the properness of using in-
tegral control actions has been repetitively discussed,
since although a conventional integral term eliminates
the steady-state positioning error, it could produce limit
cycles about a set-point for stick-slip systems. Varying
integral actions must then be applied. An interesting
experimental comparison of different control schemes,
including also some classic integral-based techniques (a
rate-varying integral algorithm and a reset-off integral
law) can be found in [40], showing that satisfying results
can be actually obtained in practice. More recently, the
importance of the integral action has been put in evi-
dence in [11], where the precision-limit positioning (PLP)
is experimentally obtained for a direct-drive DC motor
by using different PI or PID controllers (with different
control gains) in the stick and slip phases; in particular
a large integral action, that could not be applied in the
slip phase (otherwise the system would become unstable)
is used in the final positioning stage to achieve PLP.
An integral-based solution has been developed also in
[41] for a parallel manipulator with unknown Coulomb
friction; the proposed control law is composed of a
position PD controller and a reversed position error
integral controller, given by a nonlinear control input
uIrev defined as:
uIrev =KIsgn(v)t
where KIis the integral control gain and e(t)isthe
tracking error. A correct compensation action is based on
the fact that the sign of the integrated output is reversed
each time the sign of the velocity vchanges, and the
integral controller consequently restarts.
A nonlinear proportional-integral-derivative (NPID)
control has been designed and experimentally imple-
mented in [42], showing the possibility to compensate
friction effects and improve tracking accuracy by apply-
ing a state feedback NPID control law with time-varying
state feedback gains, properly switching between higher
and lower values according to the system conditions.
All these solutions, which are not based on a particular
friction model, obviously lead to compensation actions
that include not only friction, but also all the other
disturbances acting on the system. This consideration
is at the basis of the control scheme proposed in [43],
where the problem of friction compensation is solved by
means of a nonlinear disturbance observer for robotic
manipulators, where friction is considered as a distur-
bance on the control torque, similar to other unknown
torques, without using a specific friction model. Even
if the stability properties of the controlled system are
analytically proven under the assumption that the dis-
turbance term varies slowly with respect to the observer
dynamics (thus assuming it as practically constant), the
reported simulated and experimental results show that
also some fast varying disturbances can be tracked by
the observer.
Finally, various control schemes (that are not discussed
in detail in this paper) have been proposed in literature
to compensate for friction effects by using genetic, neural
and fuzzy techniques. Such methods are used to ensure
stability properties of the controlled system by means
of a fuzzy self-tuning of the controller gains as in [19],
or to generate the shapes of the applied torque pulses,
to achieve a high positioning accuracy for stick-slip sys-
tems as in [21], or again to approximate the unknown
dynamics by fuzzy logic systems, thus obtaining a signal
to compensate for both structured and unstructured
uncertainties as in [20].
IV. Some experimental results and conclusions
In our opinion, no method among the reviewed ones
can be considered as intrinsically superior to the oth-
ers. The choice of a particular friction compensation
technique must be made taking into account the char-
acteristics of the considered robotic systems and of its
hardware/software control architecture, since practical
implementation issues, as sensors accuracy, actuators
characteristics, and real-time constraints, can discourage
the application of a certain method, or deeply influence
the achieved results. Hence, an experimental comparison
of the main friction techniques in the case of a particular
robot would be interesting, but it would not lead anyway
to a final judgement.
Some experimental results, obtained for a two-dof
planar manipulator with standard resolution resolvers,
are reported to underline two further important issues,
which are not strictly related to a particular friction
compensation method: (i) model-based techniques are
suitable only if there is a good correspondence between
the assumed and the actual friction model for the specific
task to be performed; (ii) significantly different perfor-
mances can be achieved by a certain friction compen-
sation approach when different tasks or trajectories are
executed; in these cases, an average performance method
might preferable to those providing an high accuracy
only in specific conditions.
The considered two revolute-joint planar manipulator,
moving in an horizontal plane, is actuated by direct-drive
(i.e. without reduction gears) brushless motors, and it
is equipped by resolvers, having a standard resolution
of 8 ·105rad (more details can be found in [44]). Its
dynamic model can be expressed as:
Dd(q, ˙q, ¨q)θd+τfq)=τc(17)
where q, ˙q,anqare the vectors of joint angles, an-
gular velocities and angular accelerations, respectively,
τfq) is the friction torque vector, τcis the command
torque vector, while the contributions of the inertial,
centrifugal and Coriolis torques are regrouped in the
term Dd(q, ˙q, ¨q)θdthat is linear with respect to the vector
of the identifiable inertial parameters θd[45].
The following inverse dynamics control law has been
τc=Dd(q, ˙q, ¨qrvc)ˆ
where ¨qris the reference acceleration vector, ˆ
ˆτfq) are estimates of the inertial parameter vector θd
and of the friction torques τfq), respectively. The term
vcrepresents the command vector of the outer loop,
obtained by a standard PID control algorithm, identical
in all the tests. In the first implemented solution (denoted
as C.I), the available nominal values of the inertial
parameters have been used to define ˆ
θd, while ˆτfq)has
been defined according to the static, steady-state LuGre
friction model (5), with:
ωs,i 2sgn( ˙qi) (19)
The friction parameters have been identified off-line,
moving the joints at constant velocity values by means
of a PD joint-independent control law; a new improved
prototyping architecture has allowed a better fitting
of these simpler functions giqi)andfqi) than that
obtained in previous papers [16] and [46] with more
complex expressions. In the second implemented solu-
tion (denoted as C.II), friction on each joint has been
described by a third-order polynomial function, whose
coefficients have been identified off-line together with the
robot inertial parameters by a Least-Squares algorithm,
collecting data during the execution of an “optimal”
trajectory, according to the method developed in [45]
(more details can be found in [16]). The differences
between the so-identified inertial parameters and their
nominal values are extremely small [16], so that the
different control performances of C.I and C.II are mainly
due to the different friction compensation solutions.
The same circular trajectory has been tracked twice
for each of the two control solutions: in the first case
(LV) with low joint velocities (less than 2 rad/s), and in
the second case (HV) with high joint velocities (values
greater than 4 rad/s). As shown in Figure 1, while in the
LV test the LuGre static compensation gives the best
tracking accuracy, in the HV test the best results are
achieved by the polynomial friction function. The accu-
0.3 0.4 0.5
a) LV cartesian trajectory with C.I
x (m)
y (m)
0.3 0.4 0.5
b) LV cartesian trajectory with C.II
x (m)
y (m)
0.3 0.4 0.5
c) HV cartesian trajectory with C.I
x (m)
y (m)
0.3 0.4 0.5
d) HV cartesian trajectory with C.II
x (m)
y (m)
Fig. 1. Comparison of the experimental results: reference (dashed
line) and tracked (solid line) trajectory with C.I and C.II in the LV
and HV tests.
rate friction model used in C.I has led to the best results
at the low velocities range of the LV test, i.e. where
friction is the main disturbance acting on the robot, and
it is well described by such a model. On the contrary, in
the high-velocity range of the HV test, the polynomial
model seems to better capture the behavior of the ˆτfq)
term, considered as friction but that can actually contain
also other dynamic disturbances. From the control point
of view then, the polynomial friction model could be
preferable, since the quality of the achieved tracking
results are comparable in the two tests: the rms value
of the Cartesian tracking error with the polynomial C.II
solution is 1.8 mm in the LV test and 2.9 mm in the
HV one (while the maximum error values are 3.2 mm
and 4.9 mm, respectively), whereas with the C.I solution
the rms value passes from 0.5 mm to 3.5 mm, and the
maximum one from 1.0 mm to 5.9 mm. Similar results
have been obtained also by a feedforward compensation,
i.e., when the reference joint position and velocity values
have been used, instead of the current ones, in the inner
loop of the inverse dynamics control scheme and in the
friction compensation term, with a significantly reduced
on-line computational burden.
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... This kind of simplifying approach to friction compensation may yield unsatisfactory results if high-precision control, e.g., at velocity reversals, is required [14]. Friction compensation with precise models can considerably improve the overall performance of a manipulator in terms of accuracy and stability, as reported by Olsson et al. [15]; Bona and Indri [16]; Guo et al. [17]; Susanto et al. [18]; Kim et al. [19]. de Wit et al. [20] and Olsson [21] stated that the LuGre model is capable of representing most of the nonlinear effects inherent to friction, such as varying break-away forces, Stribeck effects, and hysteresis. ...
... Given that the compensation of dynamic friction plays an important role in precise motion control, model-based friction compensation is widely used where model identification plays an important role for reliability. A detailed survey on modeling, analysis, and compensation methods for the control of machines with friction can be found in Ref. [33] and an overview in Ref. [16]. ...
This paper presents an experimental study on friction compensation for the high-precision tracking control of parallel manipulators. A Luenberger-like observer (LLO) and an extended state observer (ESO) are designed and implemented in real-time control of a 6-DoF heavy-duty Stewart-Gough platform (SGP). The dynamic Lu-Gre model is used in the identification of friction. Performances of the proposed observer-based friction compensators are compared to those of a model-based compensator in computed torque control. Experimental results show that the observer-based compensators significantly improve the tracking performances in high speed motions. Among the investigated observers, the ESO results in minimum RMS error in position tracking. Improvement in position tracking at velocity reversals of the individual leg motions is also observed with the contribution of observer-based compensation. The observer error dynamics is exponentially stable, and the convergence rate can be arbitrarily increased by tuning the observer gain.
... Friction modelling and identification has a long history and this topic is so vast and rich that it is impossible to cite all the relevant works. The interested reader can refer to the following survey [16] and the references therein. Friction models can be divided into two classes: static models that depend only on the current velocity and dynamic models that are more general. ...
... This approach is common in robotics: see e.g. [10], [16]- [18]. When only the link j is moving, while the other links stay at their steady-states, its IDM is given by ...
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This paper addresses the topic of robot model identification. The standard identification method is based on the use of the inverse dynamic model (IDM) and the application of Least Squares (LS) estimation while the robot is tracking trajectories. Although this approach has been applied successfully to several industrial robots, the standard friction model is assumed to be described by a linear relationship in the viscous and Coulomb coefficients and the consistency of LS estimates is not secured when the system is identified in closed loop. In this paper, a Separable Instrumental Variable (SIV) approach is proposed that corrects these limitations. This SIV method consists of two-steps. In the first one, a Nonlinear LS estimation is combined with the IDM to identify the nature of friction effect, which is potentially nonlinear. In a second step, an iterative linear IV method is performed in order to obtain the estimates of the inertia and gravity parameters. The experimental results obtained on the 6 degrees-of-freedom TX40 robot show the effectiveness of this new approach.
... Friction is the natural resistance to the relative motion between two contacting bodies and it is a detrimental factor that can result in steady-state error in position/force tracking, oscillations, and even cause instability of the controlled systems. [26][27][28] The experimental observations lead to the definition of several friction models, which can be divided into static and dynamic models. The former include models such as Coulomb, viscous, and Stribeck ones; while examples of the latter are Dahl, LuGre, and generalized Maxwell-Slip models, which are difficult to completely and accurately define due to the presence of unknown or immeasurable parameters. ...
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High‐performance robot control is one of the most investigated topics in both research and industry. Being able to compensate for robot dynamics is indeed one major challenge. Joint friction is commonly the main issue, especially in sensorless (i.e., no availability of torque/force sensors) compliance‐controlled robots for interaction application purposes. The presented paper aims to propose a methodology for sensorless Cartesian impedance controlled robots to learn local friction compensation controllers. Exploiting a 6D virtual sensor to quantify the joint friction effects, a Bayesian optimization (BO)‐based algorithm is proposed to minimize the estimated external interaction in free‐motion tasks (related to friction effects). The BO algorithm enhances the impedance control performance by tuning the model‐based friction compensator parameters. To validate the proposed approach, experimental tests have been executed on a Franka EMIKA panda robot, highlighting the suitability of the proposed 6D virtual sensor and BO‐based algorithm to minimize the estimated external interaction for joint friction compensation purposes.
... Therefore, the generalized force is described τ = u + τ . Within the literature many friction models have been proposed (Olsson et al. 1998;Albu-Schäffer 2002;Bona and Indri 2005;Wahrburg et al. 2018). These models assume that the motor friction only depends on the joint velocity q of the th-joint and is independent of the other joints. ...
Deep learning has been widely used within learning algorithms for robotics. One disadvantage of deep networks is that these networks are black-box representations. Therefore, the learned approximations ignore the existing knowledge of physics or robotics. Especially for learning dynamics models, these black-box models are not desirable as the underlying principles are well understood and the standard deep networks can learn dynamics that violate these principles. To learn dynamics models with deep networks that guarantee physically plausible dynamics, we introduce physics-inspired deep networks that combine first principles from physics with deep learning. We incorporate Lagrangian mechanics within the model learning such that all approximated models adhere to the laws of physics and conserve energy. Deep Lagrangian Networks (DeLaN) parametrize the system energy using two networks. The parameters are obtained by minimizing the squared residual of the Euler-Lagrange differential equation. Therefore, the resulting model does not require specific knowledge of the individual system, is interpretable, and can be used as a forward, inverse, and energy model. Previously these properties were only obtained when using system identification techniques that require knowledge of the kinematic structure. We apply DeLaN to learning dynamics models and apply these models to control simulated and physical rigid body systems. The results show that the proposed approach obtains dynamics models that can be applied to physical systems for real-time control. Compared to standard deep networks, the physics-inspired models learn better models and capture the underlying structure of the dynamics.
... F RICTION is a common nonlinear effect in mechanical systems that restricts their motion. To design an effective friction compensator, a good knowledge of the friction model along with a precise identification scheme is required [1]. The identified friction model can be applied either in a feedback or feedforward configuration [2], [3]. ...
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This paper proposes a tracking controller for servomechanisms with a continuous friction model. The parameters of this model are estimated through a proposed two-step off-line identification methodology that uses a Linear Extended State Observer (LESO) based on the integral of position measurements, termed as ILESO. In the first step, the viscous and Coulomb coefficients of this friction model are estimated by the ordinary Least Squares (LS) method. The second step employs these estimates to obtain the initial guess for the Nonlinear Least Squares (NLLS) method that identifies all the parameters of the nonlinearly parameterized friction model. Finally, a backstepping controller is designed for reference tracking, which uses the estimated off-line friction model and on-line estimations of the servo position, velocity, and system uncertainty. These on-line estimations are provided by a modified ILESO, termed as FILESO, that has a similar structure to that of the ILESO but incorporates the identified off-line friction model into its dynamics. The convergence and stability of the proposed observer and controller are established using contraction analysis. Experimental results show that the proposed approach exhibits satisfactory performance.
Obtaining dynamics models is essential for robotics to achieve accurate model-based controllers and simulators for planning. The dynamics models are typically obtained using model specification of the manufacturer or simple numerical methods such as linear regression. However, this approach does not guarantee physically plausible parameters and can only be applied to kinematic chains consisting of rigid bodies. In this article, we describe a differentiable simulator that can be used to identify the system parameters of real-world mechanical systems with complex friction models, holonomic as well as non-holonomic constraints. To guarantee physically consistent parameters, we utilize virtual parameters and gradient-based optimization. The described Differentiable Newton-Euler Algorithm (DiffNEA) can be applied to a class of dynamical systems and guarantees physically plausible predictions. The extensive experimental evaluation shows, that the proposed model learning approach learns accurate dynamics models of systems with complex friction and non-holonomic constraints. Especially in the offline reinforcement learning experiments, the identified DiffNEA models excel. For the challenging ball in a cup task, these models solve the task using model-based offline reinforcement learning on the physical system. The black-box baselines fail on this task in simulation and on the physical system despite using more data for learning the model.
The standard robot identification method is based on the use of the inverse dynamic model (IDM) and the application of Least Squares (LS) estimation while the robot is tracking trajectories. Although this approach has been successfully applied to several industrial robots, the standard friction model is assumed to be linear. In this paper, a two-step LS approach is proposed that corrects this limitation. In the first step, State-Dependent-Parameter (SDP) estimation is combined with the IDM to identify the nature of the friction effect. In a second step, the standard LS method is performed in order to obtain the estimates of the inertia and gravity parameters. The experimental results obtained on the 6 degrees-of-freedom TX40 robot show the effectiveness of this new approach.
We present a novel approach for discrete-time state feedback control implementation which reduces the deteriorating effects of sampling on stability and performance in digitally controlled nonlinear mechanical systems. We translate the argument of energy shaping to discrete time by using the symplectic implicit midpoint rule. The method is motivated by recent results for linear systems, where feedback imposes closed-loop behavior that exactly represents the symplectic discretization of a desired target system. For the nonlinear case, the sampled system and the target dynamics are approximated with second order accuracy using the implicit midpoint rule. The implicit nature of the resulting state feedback requires the numerical solution of an in general nonlinear system of algebraic equations in every sampling interval. For an implementation with pure position feedback, the velocities/momenta have to be approximated in the sampling instants, which gives a clear interpretation of our approach in terms of the Störmer–Verlet integration scheme on a staggered grid. Both the Hamiltonian and the Lagrangian perspective are adopted. We present discrete-time versions of impedance or energy shaping plus damping injection control as well as computed torque tracking control in the simulation examples to illustrate the performance and stability gain compared to the quasi-continuous implementation. We discuss computational aspects and show the structural advantages of the implicit midpoint rule compared to other integration schemes in the appendix.
The independent joint control approach with motor-side velocity and position feedback still dominates most industrial robot applications. Although it is known, that the elasticity of the drivetrains has a significant influence on the achievable dynamic path accuracy. As a result, an increasing number of industrial robots are available that use secondary encoders to measure the joint position. This paper addresses the robust design of independent joint control of industrial robots with secondary encoders under explicit consideration of the drivetrain elasticity. For this purpose, the nonlinear robot dynamics are linearized and the configuration dependency of the mass matrix is treated as structured uncertainty. The trade-off between the velocity control bandwidth and the gain margin of the position control loop is solved by means of a nonsmooth optimization-based H_inf synthesis. The design method is validated for the three base joints of a KUKA KR210-2 industrial robot. Experimental results for linear Cartesian paths and a milling experiment are presented to confirm the approach.
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A nonlinear compensation force for stick-slip friction is developed to supplement a proportional + derivative control law applied to a one- degree-of-freedom mechanical system. Inertial control objects acted on by stick-slip friction are common mechanical components in mechanical servo systems and the conceptual model chosen for this investigation is a mass sliding on a rough surface. The choice of a discontinuous compensation force is motivated by the requirement that the desired reference be a unique equilibrium point of the system. The stick-slip friction force, modelled with a sticking force term and a slipping force term, generates discontinuous state derivatives. A Lyapunov function is introduced to prove global asymptotic stability of the desired reference using a modification of the direct method for discontinuous systems. Stability is verified numerically as well as experimentally. The nonlinear compensation force is robust with respect to the character of the slipping force which is assumed to lie within a piecewise linear band. Exact knowledge of the static friction force levels is not required, only upper bounds for these levels. Stability and control effectiveness is verified analytically, numerically and experimentally on a laboratory test stand.
The paper suggests a method for compensating the impact of friction for the control nonlinear systems, where the original controller (derived without taking into account friction) results from the passivity relation. The illustrative example is given
Even though the nonlinear compensator proposed by Southward, et al. can guarantee the closed-loop stability for stick-slip friction systems in continuous-time control, applying this to digital control systems may often generate a limit cycle response in the vicinity of a reference position because of the inherent time delay of sample-and-hold operation. In this paper, a new robust digital friction compensator (RDFC) is proposed, which consists of a digital P controller+hysteresis friction compensator part (position feedback loop) and an analog D controller part (velocity feedback loop). The hysteresis compensator in the position feedback loop increases the phase lead in comparison with Southward’s technique such that the phase lag coming from the time delay of digital control can be compensated for. A modified form of Lyapunov’s stability theorem is employed to verify the asymptotic stability of the RDFC. Stability and control effectiveness are verified analytically and experimentally on a single-axis robot system.
Grey-box modeling covers the domain where we want to use a balanced amount of white box modeling based on first principles and black-box modeling based on empiricism. The two grey-box models presented combine a white-box model with a black-box model, i.e., a neural network model and a polytopic model that are capable of identifying friction characteristics that are left unexplained by first principles modeling. In an experimental case-study, both grey-box models are applied to identify a rotating arm subjected to friction. An augmented state extended Kalman filter is used iteratively and off-line for the estimation of unknown parameters. For the studied example and defined blackbox topologies, little difference is observed between the two models. In addition, the applicability of the identified models is illustrated in a model basedfriction compensation control scheme with the objective to linearize the system.
In this paper, we design two adaptive controllers for a second-ordermechanicalsystem which incorporates frictional effects such as Coulomb, static, Stribeck, andviscousfriction. First, we design a modular position tracking controller that can accommodate avarietyof adaptive update laws. The proposed controller is shown to compensate foruncertaintyassociated with the friction parameters which appear linearly in the model. In thesecond controlscheme, we show how a Lyapunov-based adaptive position setpoint controller canbe designed tocompensate for parametric uncertainty throughout the mechanical systemincluding the Stribeckeffect related constant which does not appear linearly in the model.Experimental results areprovided to illustrate the performance of the proposed controllers.
Stick-slip friction is present to some degree in almost all actuators and mechanisms and is often responsible for performance limitations. Simulation of stick-slip friction is difficult because of strongly nonlinear behavior in the vicinity of zero velocity. A straightforward method for representing and simulating friction effects is presented. True zero velocity sticking is represented without equation reformulation or the introduction of numerical stiffness problems. 3 refs.
Precise low speed motion control of a robot manipulator calls for precise position and velocity measurement and joint friction compensation, as well as robustness and adaptability of the control scheme. However, precise velocity measurement and friction compensation remain challenging research tasks, especially for very slow motions. In the present work, a simple and efficient method is proposed to estimate velocity from a sampled incremental encoder pulse train, which is then utilized in the experimental investigation on a proposed robust decomposition-based friction compensation method. The experimental results on a direct drive robot arm have demonstrated precise motion control at very slow speeds and in the presence of significant joint friction.
In this paper, a linear parametric friction model is formulated by linearizing a nonlinear empirical friction model in parameters. A proposed decomposition-based control design framework is applied to synthesize the friction compensation scheme. A separate compensator is designed for each type of friction utilizing the most suitable control technique. The nominal friction is compensated by feed forward. An adaptive compensator is derived to compensate for parametric unmodeled friction with unknown but constant parameters, and a robust compensator is used to deal with friction model parameter variations, as well as non-parametric unmodeled friction. The combination of the compensators yields the overall compensation scheme. Adaptive and robust compensators complement each other in compensating the effects of model uncertainties. The analytical and simulation studies have confirmed the efficiency of the proposed friction compensation method.