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Friction modeling and identification for industrial manipulators
Marina IndriIvan Lazzero
Alessandro AntoniazzaDipartimento di Automatica e Informatica
Politecnico di Torino, Torino, Italy{marina.indri, ivan.lazzero}@polito.it
Aldo Maria BotteroComau S.p.A.
Grugliasco (Torino), [email protected]
Abstract
The paper is focused on the development of an ade-quate model of the friction acting on the joints of an in-dustrial manipulator, suitable to be used for simulationand control purposes. The experimental tests required forthe identification are executable via the standard interfacefor the robot programming, without any change in the pathplanning procedure or in the the robot control. The pro-posed friction model is developed and validated for thesix-dof Comau SMART NS12 manipulator.
1. Introduction
It is well known that undesirable friction effects can becritical for industrial robots [3], each time precise motioncontrol is required, as well as in advanced robotic appli-cations involving visual servoing for constrained robots orin force-feedback telerobotic systems.
Several friction models, with different levels of accu-racy and mathematical properties, can be found in liter-ature, together with numerous control solutions [1]-[3],[5]-[12], [14]-[18], but the various characteristics that fric-tion may present, in conjunction with the possible tech-nological limitations and specific features of the controlarchitecture of the industrial robots, do not allow consid-ering a particular solution as definitely the best one forfriction modeling and compensation. In particular, severalof the friction models proposed in literature could be wellidentified only through specific tests, using high precisionsensors, or working on single-dof systems not subject toother dynamic effects apart from friction. The most com-mon HW/SW architectures of the industrial robots imposea series of limitations to the actual execution of such tests,e.g., in the motion planning procedure, or from the con-trol point of view, since it is not possible to apply anycommand input to the motor of each joint for obvious safereasons. On the other side the accuracy errors due to fric-tion in industrial manipulators can be often effectively re-duced only with a model based approach to the problem.
This paper is focused on the development of an ad-equate mathematical model of the friction acting on thejoints of an industrial manipulator, suitable to be used towell simulate the actual behavior of the robot, correctlyestimating the torques (and consequently the motor cur-
rents) required for the execution of a given trajectory, aswell as for control purposes, as the basis for the defini-tion of a proper compensation action, especially at lowvelocity regimes. The friction model is built up taking intoaccount some general, but fundamental issues for practi-cal implementation: (i) the experimental tests required forfriction identification must be executable via the standardinterface for the robot programming, without any changein the path planning procedure or in the the robot con-trol; (ii) even if a static description of friction could beprobably sufficient for the definition of a proper frictioncompensation action suitable for the most common robottasks (even at low velocity), a complete (static + dynamic)friction model must be developed, to have the possibilityto well describe the pre-sliding effects, when they resultto be significant for some joint and/or for some particularmotion; (iii) the model must be experimentally validatednot only by using particular robot motions, enhancing spe-cific friction peculiarities, but it must be suitable for itsinclusion in a simulator of the complete robot behavior,so to effectively contribute to the estimate of the requiredtorques (and hence of the motor currents) for the execu-tion of a generic working trajectory. Moreover, the avoid-ance of discontinuous operators could facilitate, from thetheoretical point of view, the analysis of the properties ofa friction compensation scheme designed on the basis ofsuch a model. Friction modeling and identification is de-veloped in the paper following these guidelines with ref-erence to a Comau SMART NS12 manipulator, i.e., a six-dof industrial robot, with anthropomorphic structure andspherical wrist, actuated by brushless motors through geartrains.
The paper is organized as follows: Section 2 briefly re-calls the most used friction models, while Section 3 offersa detailed description of the experimental tests performedto acquire data for friction identification. Section 4 is de-voted to the construction and identification of a suitablefriction model, starting from some existing models in liter-ature. Section 5 reports and discusses the results obtainedin the tests carried out to experimentally validate the de-veloped friction model, while Section 6 proposes a furtherextension to such a model, and briefly outlines current andfuture works.
2. Classical friction models
Friction phenomenon is the result of complex inter-actions between surfaces in nanoscale perspective. At amicroscopic scale, surfaces are affected by irregularities,whose properties depend on several parameters like thenature of the solid, the machinery method used for the sur-face finishing, the interaction with the environment andthe efforts supported. The resulting friction phenomenadepend on a lot of physical parameters such as tempera-ture, velocity, lubrication and wear of the contact surfaces.
Due to the friction complexity, several researchers havefocused their attention on this topic during the last years,attempting to find an accurate mathematical model of thephenomenon. In order to devise proper models for simu-lation and control purposes, a good compromise betweenthe model complexity and the represented phenomenaneeds to be found. For this purpose, such models attemptto capture the largest possible number of phenomena us-ing the smallest number of parameters, and that is whythey are called integrated models.
Reviews of the main friction characteristics and clas-sical models can be found in [2], [3], and [15]. A gen-eral classification of the most common friction modelscan be made distinguishing static and dynamic models.In the first ones friction depends only on the current ve-locity value, while the friction phenomena related to non-stationery velocities, break-away forces, and small dis-placements during the stiction phase can be captured onlyby dynamic models, including the description of the pre-sliding behavior.
The well-known Dahl model [3], [16], is the simplestdynamic model of the friction force F , given by:
(1)dz
dt= v − σ
Fc|v| z
(2)F = σ z
where v is the relative velocity of the contact surfaces,Fc is the Coulomb friction, σ is the stiffness of the bris-tles representing the microscopic asperities of the sur-faces, and the internal state variable z represents the av-erage bristles deflection. The Dahl model reduces to pureCoulomb friction in steady-state, i.e., it does not includestiction and the Stribeck effect, so that it is suitable fornon lubricated contacts only.
The so-called LuGre model [6] extends somehow theDahl model, capturing also several dynamic effects suchas hysteresis, frictional lag and varying breakaway force.It is based, too, on the average deflection of elastic bris-tles between the contact surfaces: when a tangential forceis applied, the bristles deflect like springs, generating areaction force proportional to the displacement until theystart to slip.
The LuGre model has the following form:
(3)dz
dt= v − σ0
|v|g(v)
z
(4)F = σ0z + σ1z + f(v)
(5)g(v) = α0 + α1 e(−v/v0)2
The model is fully characterized by seven parameters.Some important physical information is obtained from themodel parameters. The αi and v0 parameters are relatedto the static friction behavior. The sum α0 + α1 corre-sponds to stiction force and α0 itself to Coulomb force.The parameter v0 represents the Stribeck velocity and de-scribes how fast the friction force drops until it reachesthe minimum. The f(v) term represents the viscous fric-tion and is typically replaced with α2v where α2 standsfor the viscous friction parameter. Finally, σ0 and σ1 arethe so-called dynamic parameters, corresponding to bris-tle stiffness and damping, respectively.
Even though the LuGre model is effective to describethe most relevant aspects of friction, it may fail when rep-resenting the friction force during the presliding regime.Experiments show that friction force in relation to dis-placement is affected by a hysteretic behavior with a non-local memory. Over the last decades, a lot of authors havesuggested extensions of the LuGre model or devised newmodels (i.e. [7], [8], [9], [12], [18]), trying to capturethe non-local memory hysteretic relation between frictionforce and displacement during the presliding regime. Un-fortunately, some of these models are very complex, dis-couraging their use for control purposes in the industrialfield. Friction modeling and identification techniques,specifically developed for control purposes of robots andservomechanisms, are proposed in [11], [14].
3 The experimental tests for identification
Several of the friction models proposed in literaturewould require the executions of specific tests, with theavailability of high precision sensors, to well identify theirparameters, or the possibility to work on a single-dof sys-tem, not subject to other dynamic effects apart from fric-tion. Such conditions cannot be fully satisfied by the stan-dard HW/SW architecture of an industrial robot, e.g., dueto limitations in the motion planning procedure, to the ac-tual availability of sensors information, or to technologicallimitations.
This section describes how experimental tests were car-ried out to collect data for joint friction identification inthe case of a Comau SMART NS12 manipulator, i.e., asix-dof industrial robot, with anthropomorphic structureand spherical wrist, actuated by brushless motors throughgear trains. The tests were executed via the standard inter-face for the robot programming (in joint space), withoutany change in the path planning procedure or in the robotcontrol. No specific friction model is addressed in this ini-tial phase, whose goal is the collection of friction torquedata for all the joints, at different constant velocity values,moving one joint at a time while the others are locked. Themotion planning procedure of the considered manipulatoris based on a trapezoidal velocity profile: data must becollected during the central constant velocity phase, tak-ing into account that the joint stroke ends limit the trajec-tory length, and that as the maximum velocity achievedincreases, the duration of the acceleration phase (as wellas that of the deceleration one) increases, until the profiledegenerates into a bang–bang one, in which no constantvelocity phase is present.
Several experiments were carried out moving eachjoint at various constant velocities of its motor from 0.8
rad/s to approximately 400 rad/s, while current and shaftposition were acquired (motor shaft positions are mea-sured by means of position encoders with a resolution of21 bits, while currents are coded on a signed 16 bit stringgiving a resolution of 1.3mA for the first three joints and396 µA for the wrist joints). Special attention was paidto the low velocity regime in order to capture the possi-ble Stribeck effect, by increasing velocity logarithmicallyinstead of linearly in such a regime (so to acquire morevelocity-friction torques couples). To reduce the measure-ment noise, the position and the current measurementswere filtered by a fourth order Butterworth filter with acut off frequency of 10Hz. For the j-th joint the commandtorque was then determined from the motor armature cur-rent Ij using the well known DC motor equation:
(6)τj = Kτ,j Ij
whereKτ,j is derived from motor datasheet. Velocity (andacceleration) values were obtained from position mea-surements by differentiation. For each movement of thejoint from one stroke end to the other, the velocity and cur-rent samples acquired during the constant velocity phasewere isolated, and their average values were computed soto obtain a single velocity-current couple, and hence a sin-gle velocity-friction torque couple for each test.
The applied joint torque is due to several overlappingphenomena such as inertia, gravity and friction. To isolatethe friction torque, components like inertia and gravitymust be removed somehow. A smart choice of the jointstrajectories may reduce or even cancel the influence ofthese components. When this is not possible, these com-ponents must be specifically computed. Therefore a goodtest design can minimize the overlapping of different ef-fects on the measurements.
Let the dynamic model of the manipulator be expressedas [4]:
M(q)q + C(q, q)q + τ f (q, q) + g(q) = τ (7)
where q is the joint position vector, M(q) is the iner-tia matrix, C(q, q)q includes the centrifugal and Corio-lis torques, g(q) the gravity torques, τ f (q, q) is the jointfriction torque vector, and τ is the vector of the appliedjoint torques. Friction is considered generically as a func-tion of joint position and velocity.
Moving one joint at a time, the coupling effects in-cluded in the term C(q, q)q can be considered as neg-ligible, as well as the inertia term M(q)q, if data arecollected during the constant velocity phase of the jointmotion. A deeper discussion must be made about gravity,distinguishing in particular joints driving a link subject ornot to gravity effects during its motion.
3.1 Joints not subject to gravityJoints not subject to gravity are those whose motion
axis is perpendicular to the ground. Only the first jointbelongs to this category of joints.
To obtain a friction velocity map, the first joint wasmoved at different constant velocities keeping the manip-ulator in a vertical posture, so to minimize the inertia mo-ment. Since g1(q) = 0 for this joint, the dynamic equationfor the first joint in the constant velocity phase is simplygiven by:
(8)τf,1(q1, q1) = τ1
The friction torque samples so acquired for the firstjoint are shown in Figure 2, in Section 5.
3.2 Joints always subject to gravityJoints always subject to gravity are those whose mo-
tion axis is parallel to the ground; the second and the thirdjoint fulfill this requirement for every robot configuration.For such joints the gravity term gj(q) cannot be neglectedand must be directly compensated to isolate the frictiontorques from the joint dynamic equation, which is givenin this case by:
τf,j(qj , qj) + gj(q) = τj (9)
3.2.1 Gravity compensation
There are two main ways to “eliminate” the gravity con-tribution in (9) and isolate the friction torque:
1. evaluating the gravity torque on the basis of a sim-plified model of the robot manipulator, when a singlejoint is moved;
2. dynamically compensating the gravity torquethrough forward-backward movements of the joint.
According to the first method the manipulator is assumedto behave like an inverted pendulum with a fixed centerof mass, when a single joint is moved at constant velocity,and all the others are locked. The possibility of using sucha simplified model to evaluate the gravity torque relies onthe knowledge of the system physical parameters (mass,position of the center of mass, inertia). Their values canbe obtained:
a) from the CAD model of the robot (but unfortunatelythese data are often affected by a high degree of un-certainty);
b) through experimental identification: the linearity ofthe inverted pendulum model with respect to the vec-tor of the dynamic parameters allows using the sim-ple, least-squares algorithm for the estimation. Itmust be underlined that inaccuracies in the estima-tion of the gravity torque are unavoidable in thiscase, too, due to different reasons, as discrepanciesin the robotic arm structure with respect to the sim-plified model (i.e., the center of mass is not exactlylocated on the rod main axis, like in the pendulummodel), the presence of friction itself, which affectsthe identification of the parameters , and the presenceof noise in the measurements.
The second, alternative solution to isolate the frictiontorque from the gravity one (successfully used in [1]) isbased on the execution of forward-backward movementsof the joint. The torques required to move the joint for-ward (τ+) and backward (τ−) at a given constant velocityqj , according to (9), are given by:
(10)τf,j(qj , qj) + gj(q) = τ+
(11)τf,j(qj ,−qj) + gj(q) = τ−
Considering τf,j(qj ,−qj) ' −τf,j(qj , qj), it directly fol-lows that:
(12)τf,j(qj , qj) =τ+ − τ−
2
In this way the gravity contribution is automatically can-celed and the friction torque can be determined from thedifference between the applied torques in the two oppositemotions.
Both methods have been experimentally tested. For thefirst one both solutions a) and b) have been implemented,considering a simplified Coulomb-viscous model for fric-tion in the experimental identification of the dynamic pa-rameters of the simplified inverted pendulum model. Theobtained results (not reported for space reasons, but avail-able on request) have shown that the last method, basedon the forward-backward motion of the joint, is more ro-bust and offers a higher degree of repeatability, so that thislast solution has been adopted for all the tests performedin this work.
The friction torque samples so acquired for the thirdjoint are shown in Figure 3, in Section 5.
3.3 Joints subject to gravity in some specific jointconfigurations.
Joints subject to gravity in some specific joint config-urations are those having a motion axis whose directionvaries during the robot motion. Using an appropriate jointconfiguration the motion axis of these joints can be placedperpendicular to the ground, thus suppressing the gravityinduced torque. All the wrist joints belong to this cate-gory.
The friction torque samples so acquired for the fifthjoint are shown in Figure 4, in Section 5.
3.4 Tests for dynamic friction identificationThe experimental tests described so far were aimed at
acquiring data for the identification of the static behav-ior of friction. Several approaches have been proposedin literature to capture the dynamic characteristics of fric-tion in the pre-sliding regime. A widely used method con-sists in applying a low frequency sinusoidal torque to eachjoint, while the others are locked, and then record the cor-responding position measurements. The controller of theconsidered manipulator actuates the motors using a posi-tion control algorithm. In order to directly apply a lowfrequency sinusoidal torque signal to a joint, the controlalgorithm should be bypassed, but this way gravity com-pensation would be no more performed and the robot armwould “collapse”.
An alternative solution has been developed by usingthe “FLY Motion” option provided by the Comau C4Gcontroller. Applying such option, the joint motion issmoothed along a sinusoidal trajectory obtaining a nearlysinusoidal velocity profile, and consequently a nearly si-nusoidal profile for acceleration and torque, too. Figure 1shows the dynamic and hysteresis characteristics of fric-tion at very low velocities, captured for the first joint byusing the standard encoder.
4 Friction modeling and identification
A complete (static + dynamic) friction model is devel-oped in this section, starting from existing models in liter-
Figure 1. Joint 1: friction at very low veloci-ties
ature, like the LuGre (3)-(5), with the purpose of captur-ing nonlinear effects of the static friction at low and highvelocities, and avoiding discontinuous operators, so to ob-tain a model suitable for control.
4.1 Static model definitionStatic friction models are usually defined by the sum of
proper mathematical functions, each one representing oneor more aspects of friction. For example, the linear-in-parameter (LIP) model proposed in [5] uses three math-ematical functions to describe the stiction, the Coulombfriction, the viscous friction and the Stribeck effect.
In order to capture all the nonlinear effects of friction,both at low and high velocities, avoiding discontinuousoperators (like the sign function), the following frictionmodel has been built up, including four terms:
(13)τf (v) =
2 τsπ
arctan(v Kv) + τsc arctan(v)
+ τv v + (τnlv v2)
2
πarctan(v Kv)
where the operator2
πarctan(v Kv) has been used to re-
place the sign function used in most of the friction models(Kv is a compression factor, which must be sufficientlyhigh to achieve a good approximation of the sign func-tion) [17]. The Stribeck effect is captured by functionarctan(v) in the second term, while the last squared termfits the bends of the friction curve at high velocity. Param-eter τsc is given by the difference between the Coulombfriction and the stiction torque; τv is the viscous frictioncoefficient.
Summarizing, the first two terms of model (13) capturestiction, Coulomb friction and Stribeck effect, while thelast two take into account the linear, viscous friction andthe nonlinear effects present at high velocity.
Experimental evidences have shown that the shape ofthe friction curve in the Stribeck region depends on vari-ous factors, like the characteristics of the surfaces in con-tact, the applied load, temperature and lubricant proper-ties. A small modification can be made to model (13)
to take into account the dependence on such factors, byinserting a proper, positive compression factor δ in thearctan(v) function in the second term, thus obtaining:
(14)τf (v) =
2 τsπ
arctan(v Kv) + τsc arctan(v δ)
+ τv v + (τnlv v2)
2
πarctan(v Kv)
Although the introduction of the δ parameter improvesthe accuracy of the model, it introduces a nonlinearity,which must be properly managed in the model identifi-cation process, as discussed in the next subsection. Theδ parameter describes how fast the friction force dropsbefore reaching its minimum value: the greater δ is, thefaster the friction torque decreases.
4.1.1 Static model identification
The developed friction model (14) non-linearly dependson parameter δ, while linearity holds with respect to theother parameters, once the compression factor Kv hasbeen suitably chosen. Restoring the sign(v) function in
(14) instead of2
πarctan(v Kv), τf (v) can be differenti-
ated with respect to v for positive values of v, obtaining:
(15)d τf (v)
d v= τsc δ
1
1 + δ2 v2+ τv + 2 τnlv v
Assuming that τv is very small, and thus negligiblewith respect to the first term in (15), for v → 0+ it holds:
(16)d τf (v)
d v' τsc δ
Therefore, δ can be determined by estimating thederivative of τf (v) with respect to v and τsc.
The derivative of τf (v) can be estimating by fittingthe initial few friction torque-velocity map samples witha polynomial function (a third order function is gener-ally sufficient), then differentiating the polynomial func-tion with respect to velocity, and finally evaluating thefriction curve slope as closest to zero as possible. Arough estimation of τsc (that will be properly identifiedtogether with the other parameters of the friction model)can be determined as the difference between the minimumfriction torque, measured at the end of the initial phase,in which friction monotonically decreases as velocity in-creases (i.e., the friction value at the Stribeck velocity v0in (5)), and the greatest friction torque, acquired at thesmallest velocity at which the joint has been moved.
Once estimated δ, the other friction parameters can beeasily identified, thanks to the linearity of model (14),rewriting it as:
(17)τf (v) = φ(v)θ
with:
(18)φ(v)
= [2
πarctan(v Kv), arctan(v δ), v, v
2 2
πarctan(v Kv)]
(19)θ = [τs, τsc, τv, τnlv]T
τs τsc τv τnlv δ
J1 1.5237 -0.6757 0.0027 -2.1 · 10−6 10.0972J2 1.0419 -0.4026 0.0120 -1.3 · 10−5 3.5088J3 1.1257 -0.5719 0.0057 -5.4 · 10−6 3.7748J4 0.1623 -0.0581 0.0012 -1 · 10−6 7.3695J5 0.5144 -0.1903 0.0022 -7.7 · 10−7 3.4900J6 0.6855 -0.3718 0.0023 -1.6 · 10−6 5.4420
Table 1. Estimated parameters of the staticfriction model.
and collecting the values of τf (v) and φ(v) for an ade-quate number nm of velocity values.
By defining
Φ :=
φ(v1)...
φ(vnm)
(20)
and T := [τf (v1) · · · τf (vnm)]T , the friction parameters
are then estimated by applying the least squares algorithmas:
(21)θ = (Φ ΦT )−1 ΦT
The estimated values of the parameters of the staticfriction model (14) for the six joints of the considered Co-mau SMART NS12 manipulator are reported in Table 1 inthe proper SI units.
4.2 Dynamic model definitionThe pre-sliding behavior of friction, detected in the ex-
perimental tests discussed in Section 3.4, can be capturedonly by extending the static friction model defined in Sec-tion 4.1 to account for the dynamic effects.
The classical LuGre model (3)-(5) can be modified toinclude the developed static friction function, and elimi-nate the discontinuous operators present in it. The com-plete (static + dynamic) friction model is then defined as:
(22)dz
dt= v − σ0
|v|g(v)
z
(23)F = σ0z + σ1z + f(v)
(24)g(v) = τs + τsc arctan(v δ)
(25)f(v) = τv v + (τnlv v2)
2
πarctan(v Kv)
to maintain all the static properties discussed in the previ-ous section. The discontinuous operators can be replacedby continuous approximation functions as proposed in[17], modifying the dynamic equation (22) in:
(26)dz
dt= S1(v) v − σ0
S2(v)
g(v)z
σ0 σ1
J1 765500 1456J2 795920 989J3 398901 1512J4 31129 395J5 297090 911J6 8973 232
Table 2. Estimated parameters of thedynamic friction model.
where S1(v) and S2(v) are defined as:
(27)S1(v) = (S0(v))2
(28)S2(v) = S0(v) v
with S0(v) =2
πarctan(v Kv).
A further modification can be introduced to let thedamping term in (23) decrease as velocity increases (assuggested in [10]), thus redefining F as:
(29)F = σ0z + (1− 2
πarctan(v δ))σ1z + f(v)
It must be noted that function (1 − 2π arctan(v δ))
varies from 1 to zero as velocities increases.
4.2.1 Dynamic model identification
The dynamic behavior of this model is thoroughly char-acterized by σ0 and σ1 parameters, corresponding respec-tively to the fictitious bristle stiffness and damping. Non-linear identification methods must be applied to identifysuch parameters. Table 2 reports the estimated values (inthe proper SI units) of the dynamic friction parametersfor the six joints of the NS12 robot. Such values havebeen obtained by iteratively applying the “ODE parame-ter estimation” function [13] of the “System IdentificationToolbox” of MATLAB to different sets of sampled data,acquired along various trajectories like the one discussedin Section 4.1, each time considering the last parameterestimates as initial values, and maintaining for the staticparameters the values identified in Section 4.1.1.
5 Experimental validation of the frictionmodel
Both the static and the dynamic friction models havebeen experimentally validated. In particular, a new set ofexperiments have been carried out, moving each joint atvarious constant velocities and keeping all the other jointslocked, so to isolate the friction torque on each joint (fol-lowing a procedure similar to the one applied to collectdata for the identification). The experimental friction sam-ples so collected have been compared with the values ob-tained by the identified friction model (14). Figures 2, 3,and 4 compare the friction behavior defined by the devel-oped model (solid black line), the acquired friction sam-ples (red circles), and the friction curve (dashed blue line)
corresponding to the Coulomb plus viscous (CV) modelimplemented in the C4G controller, for the first, the thirdand the fifth joint, respectively. As shown, the proposedstatic model well fits the real friction behavior; similar sat-isfying results have been obtained for all the joints.
Figure 2. Static model validation: frictiontorque on joint 1
Figure 3. Static model validation: frictiontorque on joint 3
In order to obtain a quantitative evaluation of the modelperformances, the root-mean-square (RMS) errors are re-ported in Table 3 for all the joints, comparing the re-sults obtained with the proposed static friction model withthose provided by the CV model.
The RMS errors are significantly lower for the pro-posed static friction model, suggesting its possible use tosimulate the robot behavior (or for control purposes), forall the manipulator motions in which the pre-sliding fric-tion effects do not represent a fundamental issue.
The complete, dynamic friction model described inSection 4.2 has been validated by using the simulator (de-veloped by Comau), which simulates the entire robot dy-namics. This tool computes the motor currents required
Figure 4. Static model validation: frictiontorque on joint 5
CV model Static model
Joint 1 0.21679272 0.00761328Joint 2 0.30840432 0.05687789Joint 3 0.20677194 0.02160639Joint 4 0.06630509 0.01278170Joint 5 0.60089415 0.01720321Joint 6 0.51918911 0.00915342
Table 3. Friction model RMS erros:comparison between CV and static model.
to track the assigned reference trajectory, on the basis ofthe complete manipulator dynamic model. In the origi-nal version of the simulator, the simple CV model wasused to represent friction on the joints. A new release hasbeen implemented, inserting instead the complete frictionmodel previously discussed. The same planned trajectoryhas been applied to both the real robot and the simulator,to compare the motor currents directly measured on thereal manipulator with the simulated ones (with the twofriction models): the smaller is the difference between thereal measured current and the simulated one, the higheris the quality of the model. A nearly sinusoidal velocityprofile (obtained by using the “FLY Motion” option, asdiscussed in Section 3.4) has been considered just to em-phasize the pre-sliding dynamic behavior. The obtainedresults for the first joint are reported in Figure 5, whichcompares the motor current measured on the real robotwith the estimates provided by the simulator with the twofriction models.
Despite the noise affecting the measured current, it ispossible to see that significant improvements are achievedin its estimate when the developed dynamic friction modelis used. The RMS errors between the real current and thesimulated one, computed for both the proposed and theCV friction model for all the joints, are reported in Table4.
A trajectory mimicking a typical operating procedure,
Figure 5. Comparison between the mea-sured motor current and the simulated onesfor joint 1 for a nearly sinusoidal profile
CV model Proposed model
Joint 1 0.29989036 0.21385734Joint 2 0.35687425 0.26258745Joint 3 0.22458652 0.15698745Joint 4 0.40256359 0.31885426Joint 5 0.32458654 0.24758865Joint 6 0.28456245 0.20124577
Table 4. Current RMS errors: comparisonbetween CV and dynamic friction model.
in which a mechanical part is picked up from a dispenserand placed in a specific location, has been applied to theNS12 manipulator, to evaluate if the proposed frictionmodel is able to significantly improve the estimate of therequired motor currents for a generic robot motion. Fig-ure 6 shows the results obtained for the second joint, con-firming that the proposed friction model considerably im-proves the simulation accuracy, especially at low veloci-ties.
The general effectiveness of the proposed frictionmodel has been finally tested for a different robot, i.e.,a Comau SMART5 NJ60, having an anthropomorphicstructure with six degrees of freedom and spherical wrist,too, but a bigger and heavier mechanical structure, sinceit supports payloads up to 60 kg instead of 12. Despite thedifferent friction characteristics of the two manipulators,satisfying results have been obtained for the NJ60 robot,too (the results are not reported for space reasons, but theyare available on request).
6 Possible extensions of the friction modeland future works
Some further factors could contribute to let frictionvary, like the temperature, the robot configuration, and thepayload.
Figure 6. Comparison between the mea-sured motor current and the simulated onesfor joint 2 for a pick-and-place trajectory
The repetition of the experimental tests carried out forfriction identification in two opposite situations, i.e., a)after a long period of inactivity of the robot, and b) aftersome hours of continuous activity, has shown that the fric-tion variations due to the different internal temperaturesof the joints are negligible, as well as those due to dif-ferent robot configurations (as verified by other specifictests). On the contrary, since friction depends on the nor-mal force applied on the contact surfaces, variations of theapplied manipulation torque, due to the effects of gravityon the robot payload, can actually induce significant vari-ations of the friction torque acting on a joint. In particular,it must be stressed that variations of the applied manipula-tion torque on a joint can be determined as a consequenceof the variations of the gravity-induced torque, when thedirection of the joint rotation axis changes during the robotmotion.
Some preliminary tests have been carried out, mount-ing a calibration payload tool on the wrist flange, just toenhance the variations of the gravity induced torque. Theobtained results have shown that the relation between theidentified values of parameters τs and τsc and the manipu-lation torque is approximately linear. The friction depen-dence on τm can then be included in the proposed staticmodel (14) by substituting τs and τsc with two linear func-tions of τm, thus obtaining:
τf (v) =2 τs,τmπ
arctan(v Kv)− τsc,τm arctan(v δ)
+ τv v + (τnlv v2)
2
πarctan(v Kv)
(30)
with:(31)τs,τm = τs0 + τs,mτm
(32)τsc,τm = τsc0 + τsc,mτm
Current works are devoted to analyze more deeply theeffectiveness of such an extension of the friction model
in the estimation of the required motor currents for theexecution of a given reference trajectory.
Goal of the future works will be the insertion of a fric-tion compensation term in the robot control scheme, deter-mined on the basis of the developed friction model. Partic-ular attention will be devoted to specific corrective actionsat low and very low velocity regimes.
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