机器人摩擦补偿综述

Friction Compensation in Robotics:an Overview

Basilio Bona and Marina Indri

Abstract—Friction e?ects are particularly critical for industrial robots,since they can induce large positioning errors,stick-slip motions,and limit cycles. This paper o?ers 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 a?ected 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 e?ects are particularly critical for industrial robots:it has been observed[1]that“friction can cause50%error in some heavy industrial manipulators”.A poor(or absent)fric-tion compensation action in the control scheme may lead to signi?cant 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 di?erent 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 de?nitely considered more e?ective than the others, since many factors can signi?cantly a?ect 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 o?er a guideline to the reader among the various solutions that have been developed in literature.A classi?cation of the various control schemes can be made following di?erent 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?nd previous results among their references) and the control schemes that have been experimentally tested.

The paper is organized as follows.In Section II,the most common friction models are brie?y 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 24,10129Torino,Italy basilio.bona@polito.it,ma-rina.indri@polito.it while Section III deals with the di?erent 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 e?ects 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 di?erent 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 models.

The simplest static friction model that can be adopted is the Coulomb+viscous one:

F(v)=F c sgn(v)+βv(1) where F is the friction force,v is the relative velocity of the contact surfaces,βis the viscous friction coe?cient, and F c is the Coulomb friction.

Model(1)can be easily upgraded taking into account the presence of stiction,i.e.a higher friction force F s 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 F s[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, di?erent mathematical models have been formulated to represent friction vs.velocity by continuous functions that account for the so-called Stribeck e?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 type:

F(v)=

F c+(F s?F c)e?|v v s|δs

sgn(v)+βv(2) where v s is the Stribeck velocity that de?nes the region in which such an e?ect is present,andδs can 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

44th IEEE Conference on Decision and Control, and

the European Control Conference 2005

Seville, Spain, December 12-15, 2005

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viscous e?ects,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 e?ects 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 de?ections 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 de?ection at the contact points of the moving surfaces) and de?ning the friction force accordingly as:

˙z=v?|v|

F c

σz(3a)

F=σz(3b) whereσrepresents the bristle sti?ness parameter.The Dahl model neither includes stiction,nor the Stribeck e?ect,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?rst time in[5],is complete from this point of view, since it includes also these e?ects,describing the behavior of the friction force as:

˙z=v?

|v|

g(v)

σ0z(4a)

F=σ0z+σ1˙z+f(v)(4b) where z is an internal state variable as in(3),σ0and σ1are model parameters assumed to be constant,and functions g(v)and f(v)model the Stribeck e?ect and the viscous friction,respectively.For constant velocity, the steady-state friction force is then given by:

F ss=g(v)sgn(v)+f(v)(5) Di?erent 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 identi?cation pro-cedure for the LuGre model can be found in[15].It must be underlined that the identi?cation of a friction dynamic model like the LuGre one always presents some peculiar di?culties,related to di?erent 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.σ0andσ1for 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?nal solution from the modelling point of view,since it does not take into account possible hysteresis e?ects.Alternative friction models,including also hysteresis e?ects 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

compensation

Several solutions have been proposed to compensate for friction in robotics,considering di?erent kinds of fric-tion phenomena,and hence di?erent models to represent it,and various strategies to eliminate or satisfactorily reduce its undesirable e?ects on the robot motion.In practice the choice of a particular solution is strongly in?uenced 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‘?xed’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 o?-line,on the basis of an assigned(more or less complicated) friction model,following proper,ad hoc identi?ca-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 o?-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 signi?cant variations are present.

D)Strategies that are not based on a particular friction

model can be applied to counteract the friction e?ects,by properly choosing the control gain param-eters or by using non model-based observers[40]-

[43].

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 identi?cation phase(possibly with high-precision sensors),and they cannot,obviously, account for friction variations.Solutions B)could o?er some more robustness features,but a correct tuning of the compensation action is crucial to avoid limit cycles, and to obtain an accurate?nal positioning.Solutions C) often require a signi?cant 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 e?ects are often estimated and compensated only together with other disturbances acting on the robot motion,so that the actual control performances could signi?cantly vary in practice for di?erent 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 classi?cation,are discussed in the following,to analyze with more details their main characteristics.

A.Fixed friction compensation techniques Satisfactory results can be obtained by adding a ‘?xed’friction compensation term to standard control algorithms,if a good o?-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 de?ne the compensation term must be su?ciently 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 identi?cation 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 o?-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,de?ned by using the identi?ed 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 di?erent 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:

τf

i

=

3

k=1

f i,k

1?

2

e2w i,k˙q i+1

+b i˙q i,i=1,2,3(6)

whereτf

i

is the friction torque acting on the i-th joint,˙q i is the i-th joint velocity,and f i,k,w i,k and b i are the parameters that de?ne the friction model.It is important to note that model(6)does not capture the static friction

component,sinceτf

i

=0at˙q i=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 good.

If the dynamic friction e?ects in the presliding phase cannot be neglected,the compensation term must be de-?ned 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 su?ciently accurate friction description to be used for a pure?xed compensation,some kind of on-line correction action is added to the compensation provided by the o?-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 o?-line,whereas the nonlinear,Stribeck e?ects 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 e?ects 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.identi?ed o?-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 ?F

as:

˙?z =v ?|v |g (v )

σ0?z ?Ke (7a)

?F =σ0?z +σ1˙?z +βv (7b)where e is the tracking error,K is 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 identi?cation problems for a 2dof micro-manipulator and a 4dof 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 characteristics

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 speci?c 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 τf acting 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 =

?

??τ+s

0<τ+

s <τc τc τ?s ≤τc ≤τ+

s τ?s τc <τ?s

<0

(8a)τslip (˙q )=τ+d (˙q )μ(˙q )+τ?

d (˙q )μ(˙q )

(8b)

where τc is the command torque,τ+s and τ?

s are the positive and negative limits of the static friction torque,τ+d (˙q )and τ?d (˙q )are the slipping torque functions for positive and negative velocities,respectively,supposed to be bounded within the ?rst 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 q L ≤q ≤q H ,

with q L =?τ+s /K P <0,q H =?τ?

s /K P >0,K P being the position control gain,and assuming q r =0as reference position,for the sake of simplicity.In the nonlinear solution proposed in [29],the steady-state position error is eliminated by de?ning the command torque as τc =?K D ˙q ?τn ,with

τn =???????τ?

s 0

??τ

+

s ?q L ≤q ≤0K P q otherwise (9a)where

?q H =q H +ε,?q L =q L ?ε,ε>0?τ+s =?K P ?q L =τ+s +K P ε?τ?s =?K P ?q H =τ?s ?K P ε

(9b)

The nonlinear compensation torque τn is active only

when the joint position is between the augmented stick-ing limits ?q L ,?q H ,while the controller is essentially a PD one outside such a region.

Some modi?cations 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,δL and δH ,denot-ing the bounds of the velocity-dependent dead zone (see [30]for details).Their choice is crucial to eliminate the destabilizing e?ect of the time delay:greater values im-prove the stability margin,but enlarge the error bounds,since the steady-state position will lie between δL and δH .Slight modi?cations to the method,e.g.in [31],have been proposed to try to obtain a smaller ?nal 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 in?uences the choice of δL ,δH and 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 )part of the 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 di?erent 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 brie?y recalled in Section III-A,are developed to cope with structured friction variations in two cases.In the?rst one, variations of the normal forces exchanged between the contact surfaces are assumed to mainly a?ect the friction static parameters,whereas the dynamic parameters are considered as invariant due to the unchanged lubricant characteristics,and possible variations of the viscous coe?cientβare directly dealt with the linear controller. In the second case,temperature variations are assumed to uniformly a?ect both static and dynamic friction pa-rameters.The resulting parameter uncertainties in both cases are captured by a unique variable parameterθ,by rewriting model(4)with f(v)=βv as follows:

Case1):?

?

?

˙z=v?θ

|v|

g(v)

σ0z

F=σ0z+σ1˙z+βv

(10)

Case2):?

?

?

˙z=v?

|v|

σ0z

F=θ(σ0z+σ1˙z+βv)

(11)

where parametersσ0,σ1,βand function g(v)are sup-posed to be known,i.e.a priori identi?ed,andθis 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 di?erent 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 di?erent control loops.On the basis of such observers,two adaptive controllers are proposed:the ?rst one compensates for all the mechanical parameters variations,except for the parameters associated to the Stribeck e?ect(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 e?ect function,following the same approach of case1)in[15],but utilizing a nonlinear?lter 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 di?erent 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 di?erent 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 de?ning function g(v)in(4a) are a priori identi?ed,and an observer is inserted for the estimation of z and˙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 e?ect 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 e?ect parameters,as in[36],where friction is modelled as:

F(v)=

F c+F s e(?Fτv2)

sat(v)(12) where the sat(·)function can be de?ned 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 F s and Fτare updated by laws of the following types:

˙ F

s

=?γ0v sat(v)e(? Fτv2)(13a)

˙ F

τ=γ1v

3sat(v)e(? Fτv2)(13b) whereγ0andγ1are 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) withδs=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 F L(v)and a nonlinear one F N(v)as F(v)=F L(v)+F N(v),with

F L(v)=F c sgn(v)+βv(14a)

F N(v)=(F s?F c)e?(v2/v2s)sgn(v)(14b) where F N(v)can be rewritten as the product of two Lipschitzian functions in the parameter vectorθ= [θ1θ2]T:=

(F s?F c)1/v2s

T

as:

F N(v)=g(v,θ)h(v,θ)(15) with g(v,θ)=[10]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 di?erent 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 s and 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 o?-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 coe?cients,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 the90’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 di?erent control schemes, including also some classic integral-based techniques(a rate-varying integral algorithm and a reset-o?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 di?erent PI or PID controllers(with di?erent 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?nal 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 u Irev de?ned as:

u Irev=K I sgn(v)

t

0e(τ)sgn(v(τ))dτ(16)

where K I is the integral control gain and e(t)is the

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 v changes,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 e?ects 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 speci?c 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 e?ects 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 in?uence

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?nal 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 speci?c task to be performed;(ii )signi?cantly di?erent perfor-mances can be achieved by a certain friction compen-sation approach when di?erent tasks or trajectories are executed;in these cases,an average performance method might preferable to those providing an high accuracy only in speci?c 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 8·10?5rad (more details can be found in [44]).Its dynamic model can be expressed as:

D d (q,˙q,¨q )θd +τf (˙q )=τc

(17)

where q,˙q ,and ¨q are the vectors of joint angles,an-gular velocities and angular accelerations,respectively,

τf (˙q )is the friction torque vector,τc is the command torque vector,while the contributions of the inertial,centrifugal and Coriolis torques are regrouped in the term D d (q,˙q,¨q )θd that is linear with respect to the vector of the identi?able inertial parameters θd [45].

The following inverse dynamics control law has been applied:

τc =D d (q,˙q,¨q r ?v c )?θd +?τf (˙q ),(18)where ¨q r is the reference acceleration vector,?θ

d and ?τf (˙q )ar

e estimates o

f the inertial parameter vector θd and of the friction torques τf (˙q ),respectively.The term v c represents the command vector of the outer loop,obtained by a standard PID control algorithm,identical in all the tests.In the ?rst implemented solution (denoted as C.I),the available nominal values of the inertial

parameters have been used to de?ne ?θd ,while ?τf (˙q )has

been de?ned according to the static,steady-state LuGre friction model (5),with:

g i (˙q i )= α0i +α1i e ?

˙

q i

ωs,i 2 sgn(˙q i )(19)

f (˙q i )=α2i ˙q

i (20)

The friction parameters have been identi?ed o?-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 ?tting of these simpler functions g i (˙q i )and f (˙q i )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 coe?cients have been identi?ed o?-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 di?erences between the so-identi?ed inertial parameters and their nominal values are extremely small [16],so that the di?erent control performances of C.I and C.II are mainly due to the di?erent friction compensation solutions.

The same circular trajectory has been tracked twice for each of the two control solutions:in the ?rst case (LV)with low joint velocities (less than 2rad/s),and in the second case (HV)with high joint velocities (values greater than 4rad/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-

https://www.sodocs.net/doc/6816655231.html,parison 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 ?τf (˙q )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 1.8mm in the LV test and 2.9mm in the HV one (while the maximum error values are 3.2mm and 4.9mm,respectively),whereas with the C.I solution the rms value passes from 0.5mm to 3.5mm,and the maximum one from 1.0mm to 5.9mm.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 signi?cantly reduced on-line computational burden.

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机器人控制器的现状及展望概要

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