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- [IEEE 2005 IEEE Conference on Control Applications, 2005. CCA 2005. - Toronto, Canada (Aug. 29-31, 2005)] Proceedings of 2005 IEEE Conference on Control Applications, 2005. CCA 2005. - Friction identification in robotic manipulators: case studies

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<ul><li><p>Friction Identification in Robotic Manipulators: Case Studies</p><p>M.R. Kermani, R.V. Patel, M. Moallemmkermani@uwo.ca, rajni@eng.uwo.ca, mmoallem@engga.uwo.ca</p><p>Department of Electrical and Computer EngineeringUniversity of Western Ontario</p><p>London,Ontario,Canada</p><p>AbstractIn this paper, friction identification in robotic ma-nipulators are studied. A single state dynamic model of thefriction force is utilized for the friction compensation algorithm.A practical method for obtaining parameters of the modelwhich pertains to robotic manipulators is presented. In orderto evaluate the competence of this method, two differentmanipulators with different friction characteristics are exam-ined. A 2-DOF manipulator used for high-speed, micro-meterprecision manipulation and a 4-DOF macro-manipulator usedfor long-reach positioning tasks are considered. It is shown thatdespite the different nature of the two manipulators, the samemethod can effectively improve the speed and performance ofmanipulation in both cases.</p><p>I. INTRODUCTION</p><p>Friction compensation can play an important role in diversecontrol applications. This includes applications that involvehigh-precision and fast motion control as well as thosecontaining heavy and sluggish servomechanisms with slowtracking velocities. In each case, the friction force needs to beadequately compensated for in order to improve the transientperformance and reduce the steady-state tracking errors. Thisalso ensures a smooth control signal without resorting tohigh feedback gains. However, the nature of friction force isnot completely understood and finding a model that capturesall characteristics of friction such as Stribeck, stiction andsliding, is difficult. In addition, friction also includes dynamiceffects which are highly nonlinear and depend on time,position and velocity. In this regard, many researchers havestudied this phenomenon and suggested a number of models.Dahl was the first to present a systematic model for Coulomband sliding friction [5]. This model has been widely usedfor dynamic and adaptive friction compensation by otherresearchers. A dynamic model of friction was proposed byOlsson et al. [10]. The model, also known as LuGre model,incorporates a single continuous state to model pre-slidingdisplacement. Ehrich at al. [7] studied an adaptive model offriction for low-velocity position tracking systems. Dupontat al. [6] introduced a drift-free model of friction for precisemotion control tasks.In order to implement any of the aforementioned models, theparameters of the model need to be identified experimentally.Friction identification is also another challenging part of the</p><p>This research was supported by grants STPGP215729-98, RGPIN1345and RGPIN227612 from the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada.</p><p>friction compensation process. It can be achieved either off-line or online as part of a systems operation. In off-lineidentification, the motion through which the data is gatheredcan be deliberately specified. Online identification must usedata from normal system operations. In this paper, off-lineidentification is studied and a practical method for measuringthe friction force, particularly in robotic manipulators ispresented. It is shown that a dynamic model of frictionforce based on initial measurements results in very accuratemotion control without the necessity of resorting to highfeedback gains. The main motivation for the present workarose after unsuccessful attempts at controlling industrialmanipulators within a certain accuracy. Friction can causemore than 50% error in some heavy industrial manipulatorswhile any changes in control parameters in order to improvethe performance of the systems is usually unsuccessful.Adding a steady-state model of friction, provided that sucha model is obtainable, can alleviate the problem to a certainextent. The final remedy for this problem involves utilizing adynamic model of the friction force. In order to identify theparameters of the friction force, the suggested methods in thecurrent literature are not feasible for robotic manipulators [3],[9]. This is due to the fact that in most robotic manipula-tors with limited workspace, running a joint with constantvelocity may not be possible. In this regard, a new methodfor obtaining the model parameters using a low frequencysinusoidal torque is suggested.The organization of this paper is as follows. In section II, adynamic model of friction force is studied. In section III, ageneralized method for measuring friction force in manipula-tors is discussed. Finally, in section IV, experimental resultsare presented for two different types of manipulators.</p><p>II. MODELING</p><p>Consider the following dynamics of a robot manipulator:</p><p>M(q)q + C(q, q)q +G(q) = F (1)where q = (q1, q2, . . . , qn)T is the joint angle vector, M(q)is the mass matrix, C(q, q)q is the Coriolis and centrifugalterm, G(q) is the gravity term, and and F are vectorsof applied forces/torques and the friction forces/torques,respectively. Henceforth, the term force is used to refer toboth force and torque. Each element of F is the frictionforce between two adjacent joints of the manipulator which</p><p>Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005</p><p>WA1.1</p><p>0-7803-9354-6/05/$20.00 2005 IEEE 1170</p></li><li><p>are moving with relative velocity q with respect to eachother. The friction force at each joint, e.g., joint j of themanipulator, can be dynamically modeled as follows [4]:</p><p>dz</p><p>dt= qj 0|qj |</p><p>s(qj)z (2)</p><p>Fj = 1dz</p><p>dt+ 0z + 2qj</p><p>where z is assumed as an average deflection of bristlesbetween the two surfaces with friction, Fj is the joint jfriction force, 0, 1 and 2 are the stiffness, dampingand viscous coefficients, respectively, and s(qj) is a scalarfunction given by</p><p>s(qj) = Fc + (Fs Fc)e|qj |. (3)In equation (3), Fc is the Coulomb force, Fs is the stictionforce and the determines the variation of s(qj) betweenFs and Fc. As seen, the friction force is characterizedby six parameters, namely Fc, Fs, 0, 1, 2 and , fromwhich Fc, Fs and describe the steady-state part of themodel whereas, 0 and 1 describe dynamic properties ofthe friction force. The values of these parameters, for eachjoint of the manipulator can be estimated experimentally. Ina system with viscous friction, the necessary and sufficientcondition for passivity [1] of the model is given by: 2 1(1 + 21 ), where 1 =</p><p>1s(qj)|qj=0 and 2 =</p><p>1s(qj)|qj .</p><p>III. FRICTION IDENTIFICATION</p><p>The first step in identifying friction parameters of a manipu-lators joint is to obtain a map between the friction force andthe joint velocity. However, obtaining such friction-velocitymap by running the joint at different constant velocities andmeasuring friction force is not always feasible [3], [9]. Thus,an alternative method in which the joint angle is kept withinthe manipulators workspace is desirable. To this end, a lowfrequency sinusoidal torque is applied to each joint of themanipulator while other joints are locked and the correspond-ing velocity is measured. The amplitude and the frequency ofthe applied signal should be deliberately selected so that theeffect of Coriolis and centrifugal forces are negligible, andthe joint angles remain within the manipulators workspace.Nonetheless, if the manipulator is moved too slowly it maynot be possible to capture all friction characteristics as thesystem always remains in pre-sliding state. In the end, findingan appropriate signal is a matter of trial and error. In thesequel, this notion will be exemplified. Provided that theabove conditions are met, the equation of motion for eachjoint of the manipulator (e.g., joint j) can be rewritten asfollows:</p><p>Mjj qj +Gj = j Fj (4)where Mjj is the moment of inertia of joint j around its zaxis, j = Mj sin(jt) + Gj is the applied torque and Fjis the joint friction force. By calculating qj after numericaldifferentiation of joint velocity and assuming perfect gravitycancelation, Fj can be obtained from (4) as follows:</p><p>Fj = Mj sin(jt)Mjj qj (5)</p><p>The value of the friction force (Fj ) is next plotted versusjoint velocity in order to obtain a typical friction-velocitymap as depicted in Fig. 1. All parameters in (3) including 1and 2 can be derived from this graph. Fig. 1 illustrates therelationship between the desired parameters and the graph.In most applications, the exact value of Mjj depends on the</p><p>Fig. 1. Friction force/torque versus velocity.</p><p>manipulator configuration, and using an estimated value ofthis parameter for measuring the friction force is inevitable.Small values of qj can reduce the sensitivity of the measure-ment (friction force) to the exact value of Mjj . However, asmentioned earlier, moving a manipulator too slowly does notreveal all friction characteristics. Thus, finding an appropriateinput signal for an unknown system requires a few trials.To show the effect of Mjj on the friction-velocity graphwe followed the above procedure of applying a sinusoidalsignal to simulate the friction-velocity map of a system givenby (4). The friction force was simulated according to (2) forthe nominal values Fc = 1.2, Fs = 2.0, 0 = 103, 1 =400, 2 = 0.4, = 0.2 and Mjj = 1. To plot the friction-velocity graph, instead of using the simulated values of thefriction force, we calculated the friction force using (5),similar to a practical case. The results are summarized inFig. 2. The solid line shows the friction-velocity graph fornominal values of the parameters and the dashed lines showthe graphs for the perturbed values of Mjj from its nominalvalue. It is observed that the value of Mjj not only affectsthe shape of the graph but it also changes the direction inwhich the graph is followed. A similar effect on the friction-velocity graph is observed for overestimated values of Mjj ,except that the direction of the graph remains unchanged.In the following, the relationship between parameters of themodel, i.e., Fc, Fs, , 2, and the above graph is discussed.First, it should be pointed out that the starting point of themodel (2), assuming that the system is in pre-sliding mode,represents the stress-strain curvature of the average bristledeflections. Thus, the value of 0 can be measured from theslope of the friction force graph versus displacement at theorigin (see for example [2], [10]).For a large value of the joint velocity the average bristledeflections can be assumed constant. This is due to the fact</p><p>1171</p></li><li><p>Fig. 2. The effect of mass/inertia perturbation on the friction-velocitygraph.</p><p>that in lubricated contacts, a fluid layer of lubricant is builtup at high velocities and consequently, friction is determinedby the viscous characteristics of the lubricants [10]. Becauseof the small value of the lubricant viscous coefficient, theaverage deflection of bristles can be considered constant orequivalently dzdt 0. In this case the friction force is givenby,</p><p>0 = qj 0|qj |s(qj)</p><p>z (6)</p><p> z = sgn(qj)s(qj)0</p><p> Fj = sgn(qj)s(qj) + 2qjWithout losing generality, let us assume that qj is positive.Additionally, for the large values of the joint velocity whereqj 1, (3) yields</p><p>s(qj) = Fc + (Fs Fc)e|qj | Fc (7)Now, substituting (7) in (6) renders the friction force as,</p><p>Fj = Fc + 2qj (8)</p><p>Thus, for the large values of the joint velocity, the frictionforce is given by a straight line whose slope and y-interceptare equal to 2 and Fc respectively. This is illustrated inFig. 1.On the other hand, while dzdt 0, small values of the jointvelocity for which qj 1 results in,</p><p>Fj Fs + 2qj Fs (9)The stated conditions (qj 1 and dzdt 0) representthe pre-sliding condition in the system and depending onthe systems direction, i.e., leaving or approaching the pre-sliding condition, two different situations can be observed.Let us first look at the behavior of the system qualitatively.Consider the system in pre-sliding condition. As the inputsignal increases the value of the friction force is increased.At this so called elastic stage the friction behaves as a spring.After rupturing this elastic bond, the system enters the slidingmode. Upon transition of the system from pre-sliding tosliding condition, because of the assumed friction dynamics,</p><p>the value of the friction force may exceed the stiction force(static friction) value. In this case, the friction-velocity graphas depicted Fig. 1 has an overshoot. The amplitude of theovershoot depends on the stiffness and damping coefficientsi.e., 0 and 1, as well as the amplitude of the input signal.Thus, upon leaving the origin of the friction-velocity graph(entering sliding condition), the value of the friction forcedoes not necessarily represent the stiction force. On the otherhand, prior to approaching the origin of the friction-velocitygraph, the system is in sliding condition and, as a result, theaverage bristle deflection is relatively constant and dzdt 0(the stated conditions). Hence, the value of the friction forcerepresents the stiction force. In order to illustrate this factquantitatively, let us consider the linearized model of thesystem in (2) with the dynamics (4) around the equilibriumpoint of the system as follows [8]:</p><p>qj(s)j(s)</p><p>=1</p><p>Mjjs2 + (1 + 2)s+ 0(10)</p><p>Fj = 0qj + (1 + 2)qj , qj = z</p><p>To study the transition of the system from pre-sliding to slid-ing condition, the response of the the linearized model (10)to the input Mj sin(jt) Ajt, where Aj = Mjj , iscalculated. It has been shown that the linearized model,despite the non-smooth behavior of the system at the origin,is valid for both positive and negative velocities [8]. Thus,the complete response of the system is given by,</p><p>qj(t) = wj(t) + hj(t) (11)</p><p>where wj(t) and hj(t) are the zero state and zero inputresponses as follows:</p><p>wj(t) = K1es1t +K2es2t + at+ b (12)hj(t) = K3es1t +K4es2t</p><p>s1,2 = </p><p>2 20 d</p><p>, =1 + 22Mjj</p><p>, 20 =0Mjj</p><p>K1,2 = Aj40</p><p>( d)22d</p><p>, a =Aj20</p><p>, b =2Aj40</p><p>K3,4 = qj(0) + ( d)qj(0)2dIt is clear that the response of the system in pre-slidingcondition is characterized by the values 0, 1, 2 and Mjj .For the large values of ( 0), the roots of the system,s1,2 tend to 0 and , respectively. As a result, qj , qj andFj given by (10), approach their final values at + b, a and0(at+b)+(1+2)a, more gradually. Now, depending onthe values of s1,2 two different situations are recognizable.Let us call the time that the input signal reaches Fs byts = FsAj . Calculating Fj at ts yields,</p><p>Fj = 0qj(ts) + (1 + 2)qj(ts) (13)</p><p>Now, if is such that, at ts the exponential terms in qj andqj expressions are near zero then,</p><p>Fj = 0(ats + b) + (1 + 2)a (14)</p><p>1172</p></li><li><p>Substituting from (12) in (14) yields,</p><p>Fj = Fs</p><p>Therefore, upon transition of the system from pre-sliding tosliding condition the friction force is equal to Fs. Howeverif due to the large values of the exponential terms in qjand qj are nonzero, then</p><p>Fj = Fs + </p><p>where</p><p> 0(K1es1ts+K3es1ts)+(1 + 2)(K1s1es1ts +K3s1es1ts)</p><p>In this case, the friction force is larger than Fs prior toentering the sliding condition.The above discussion indicates that the value of the frictionforce in t...</p></li></ul>