An Extended Friction Model to Capture Load and Temperature Effects in Robot Joints

Technical report from Automatic Control at Linköpings universitet

An Extended Friction Model to capture Load and Temperature eﬀects in Robot Joints

André Carvalho Bittencourt, Erik Wernholt, Shiva Sander-Tavallaey, Torgny Brogårdh Division of Automatic Control E-mail: [email protected], [email protected], shiva [email protected], [email protected]

7th April 2010

Report no.: LiTH-ISY-R-2948 Accepted for publication in the 2010 IEEE/RSJ International Confer- ence on Intelligent Robots and Systems (IROS 2010) Taipei International Convention Center, Taipei, Taiwan October 18-22, 2010

Address: Department of Electrical Engineering Linköpings universitet SE-581 83 Linköping, Sweden

WWW: http://www.control.isy.liu.se

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications. Abstract Friction is the result of complex interactions between contacting surfaces in a nanoscale perspective. Depending on the application, the dierent models available are more or less suitable. Available static friction models are typically considered to be dependent only on relative speed of interacting surfaces. However, it is known that friction can be aected by other factors than speed. In this work, static friction in robot joints is studied with respect to changes in joint angle, load torque and temperature. The eects of these variables are analyzed by means of experiments on a standard industrial robot. Jus- tied by their signicance, load torque and temperature are included in an extended static friction model. The proposed model is validated in a wide operating range, reducing the average error a factor of 6 when compared to a standard static friction model.

Keywords: friction, modeling, identication, industrial robots An Extended Friction Model to capture Load and Temperature effects in Robot Joints

Andre´ Carvalho Bittencourt, Erik Wernholt, Shiva Sander-Tavallaey and Torgny Brogardh˚

Abstract— Friction is the result of complex interactions be- found in [17], where the author designs joint friction models tween contacting surfaces in a nanoscale perspective. Depending based on physical models of elementary joint components as on the application, the different models available are more helical gear pairs and pre-stressed roller bearings. or less suitable. Available static friction models are typically considered to be dependent only on relative speed of interacting Empirically motivated friction models have been successfully surfaces. However, it is known that friction can be affected by used in many applications, including robotics [5], [18]– other factors than speed. In this work, static friction in robot joints is studied with [20]. This category of models was developed through time respect to changes in joint angle, load torque and temperature. according to empirical observations of the phenomenon [2]. The effects of these variables are analyzed by means of Considering a set of states, X , and parameters, θ, these experiments on a standard industrial robot. Justified by their models can be described as the sum of N functions that significance, load torque and temperature are included in an describe the behavior of friction, F, extended static friction model. The proposed model is validated N in a wide operating range, reducing the average error a factor X of 6 when compared to a standard static friction model. F(X , θ) = fi(X , θ). (M) I.INTRODUCTION i=1 X = [z, q,˙ q] gives the set of Generalized empirical Friction Friction exists in all mechanisms to some extent. It can be Model structures (GFM) [1], where z is an internal state defined as the tangential reaction force between two surfaces related to the dynamic behavior of friction and q is a in contact. It is a nonlinear phenomenon which is physically generalized coordinate and q˙ = dq/dt. dependent on contact geometry, topology, properties of the Among the GFM model structures, the LuGre model [5], materials, relative velocity, lubricant, etc [1]. Friction has [19] is a common choice in the robotics community. For a been constantly investigated by researchers due to its impor- revolute joint, it can be described as tance in several fields [2]. In this paper, friction has been studied based on experiments on an industrial robot. τf = σ0z + σ1z˙ + h(ϕ ˙ m) (ML) One reason for the interest in friction of manipulator joints |ϕ˙ m| is the need to model friction for control purposes [3]–[7], z˙ =ϕ ˙ m − σ0 z, g(ϕ ˙ ) where a precise friction model can considerably improve m the overall performance of a manipulator with respect to where τf is the friction torque and ϕm is the joint motor accuracy and control stability. Since friction can relate to angle. The state z is related to the dynamic behavior of the wear down process of mechanical systems [8], including asperities in the interacting surfaces and can be interpreted as robot joints [9], there is also interest in friction modeling for their average deflection, with stiffness σ0 and damping σ1. robot condition monitoring and fault detection [9]–[16]. The function h(ϕ ˙ m) represents the velocity strengthening A friction model consistent with real experiments is nec- (viscous) friction, typically taken as h(ϕ ˙ m) = Fvϕ˙ m, and essary for successful simulation, design and evaluation. Due g(ϕ ˙ m) captures the velocity weakening of friction. Motivated to the complexity of friction, it is however often difficult to by the observations of Stribeck [18], [21], g(ϕ ˙ m) is usually obtain models that can describe all the empirical observations modeled as α | ϕ˙ m | (see [1] for a comprehensive discussion on friction physics g(ϕ ˙) = Fc + Fse− ϕ˙ s . and first principle friction modeling). In a robot joint, the Where F is the Coulomb friction, F is, in this paper, de- complex interaction of components such as gears, bearings c s fined as the standstill friction parameter∗, ϕ˙ s is the Stribeck and shafts which are rotating/sliding at different velocities, velocity and α is the exponent of the Stribeck nonlinearity. makes physical modeling difficult. An example of an ap- The model structure M is a GFM with X = [z, ϕ˙ ] and proach to model friction of complex transmissions can be L m θ = [σ0, σ1,Fc,Fs,Fv, ϕ˙ s, α]. According to [19] it can This work was supported by ABB and the Vinnova Industry Excellence successfully describe many of the friction characteristics. Center LINK-SIC at Linkoping¨ University. Since z is not measurable, a difficulty with ML is the A. C. Bittencourt and E. Wernholt are with the Division of Auto- estimation of the dynamic parameters [σ , σ ]. In [5], these matic Control, Department of Electrical Engineering, Linkoping¨ University, 0 1 Linkoping,¨ Sweden [andrecb,erikw]@isy.liu.se parameters are estimated in a robot joint by means of open S. Sander-Tavallaey is with ABB Corporate Research, Vaster¨ as,˚ Sweden ∗ [email protected] Fs is commonly called static friction. An alternative nomenclature T. Brogardh˚ is with ABB Robotics, Vaster¨ as,˚ Sweden was adopted to make a distinction between the dynamic/static friction [email protected] phenomena. loop experiments and by use of high resolution encoders. Open-loop experiments are not always possible, and it is common to accept only a static description of ML. For constant velocities, ML is equivalent to the static model MS:

τf (ϕ ˙) = g(ϕ ˙ m)sign(ϕ ˙ m) + h(ϕ ˙ m) (MS) which is fully described by the g- and h functions. In fact, ML simply adds dynamics to MS. The typical choice for (a) ABB IRB 6620 robot with (b) Schematics of the 3 first g and h as defined for ML yields the static model structure 150 kg payload and 2.2 m joints including the torque reach. definitions for joint 2. M0:

h ϕ˙ m α i Fig. 1. The experiments were made on joint 2 of the ABB robot IRB | ϕ˙ | τf (ϕ ˙ m) = Fc + Fse− s sign(ϕ ˙ m) + Fvϕ˙ m. (M0) 6620. ϕa is the joint angle, T the joint temperature, τm the manipulation torque and τp the perpendicular torque.

M0 requires a total of 4† parameters to describe the velocity weakening regime g(ϕ ˙ m) and 1 parameter to capture viscous where T is the joint (more precisely, lubricant) temperature friction h(ϕ ˙ m). See Figure 3 for an interpretation of the parameters. and ϕa the joint angle at arm side. Ideally, the chosen model should be coherent with the From empirical observations, it is known that friction can be empirical observations and, simultaneously, with the lowest affected by several factors, dimension of θ, the parameter vector, and with the lowest temperature, velocity, number of describing functions (minimum N). For practical • • force/torque levels, acceleration, purposes, the choice of fi should also be suitable for a useful • • position, lubricant/grease properties. identification procedure. • • A shortcoming of the LuGre model structure, as with any The document is organized as follows. Section II presents GFM, is the dependence only of the states X = [z, q,˙ q]. In the method used to estimate static friction in a robot joint, more demanding applications, the effects of the remaining together with the guidelines used during the experiments. variables can not be neglected. For instance in [17], the Section III contains the major contribution of this work, with author observes a strong temperature dependence, while in the empirical analysis, modeling and validation. Conclusions [5] joint load torque and temperature are considered as and future work are presented in Section IV. disturbances and estimated in an adaptive framework. In [9], the influence of both joint load torque and temperature II. STATIC FRICTION ESTIMATION AND are observed. However, more work is needed in order to EXPERIMENTATION understand the influence of different factors on the friction A manipulator is a multivariable, nonlinear system that properties. A more comprehensive friction model is needed can be described in a general manner through the rigid body to improve the performance of control and diagnosis of dynamic model systems including friction phenomena. The objective of this contribution is to analyze and model M(ϕa)ϕ ä + C(ϕa, ϕ˙ a) + τg(ϕa) + τf = u (1) the effects in static friction related to joint angle, load torques where ϕ and ϕ are the vectors of robot angles at arm and and temperature. The phenomena are observed in joint 2 of a m‡ motor side of the joint gearbox, M(ϕ ) is the inertia matrix, an ABB IRB 6620 industrial robot, see Figure 1(a). Two load a C(ϕ , ϕ˙ ) relates to speed dependent terms (e.g. Coriolis torque components are examined, the torque aligned to the a a and centrifugal), τ (ϕ ) are the gravity-induced torques and joint DoF (degree of freedom) and the torque perpendicular g a τ contain the joint friction components. The system is to the joint DoF . These torques are in the paper named f controlled through the input torque, u, applied to the joint manipulation torque and perpendicular torque , see τm τp motor (in the experiments the torque reference from the servo Figure 1(b). was measured§). By means of experiments, these variables are analyzed and modeled based on the empirical observations. The task For single joint movements (C(ϕa, ϕ˙ a) = 0) at constant of modeling is to find a suitable model structure according speed (ϕ ä ≈ 0), Equation (1) simplifies to to: τg(ϕa) + τf = u. (2) N X X X M ‡ τf ( ∗, θ) = fi( ∗, θ) ( ∗) Notice that for the rigid model (1) follows the equivalence ϕa = r·ϕm, i=1 where r is the gearbox ratio. Both nomenclatures are kept to emphasize friction as a joint phenomenon. X ∗ = [ϕ ˙ m, ϕa, τp, τm,T ] , §It is known that this abstraction might not always hold, for instance under high temperatures. The deviations are however expected to be small †Many times α is considered a constant between 0.5 and 2 [19]. and therefore neglected during the experiments.

The applied torque u drives only friction and gravity-induced 0.14 0.13 f torques. If realistic estimates of τg(ϕa) are available, it is τ 0.12 easy to isolate the friction component in Equation (2). If such 0.11 0.1 estimate is not possible (e.g. not all masses are completely 0.09

0.08 known), τf can still be estimated as follows. Fv F 0.07 s vel-weakening, g(ϕ ˙ ) + 0.06 m The required torques to drive a joint in forward, u , and 0.05 vel-strengthening, h(ϕ ˙ m) Fc 0.04 ¯ reverse, u−, directions at constant speed ϕ˙ m and at a joint 0 50 100 150 200 250 ϕ˙ s ϕ˙ (rad/s) angle ϕ¯a (so that τg(ϕa) is equal in both directions), are m,2 + τf (ϕ¯˙ m) + τg(ϕ ¯a) = u Fig. 3. Static friction values calculated from experiments shown together with M0 parameters obtained by best ﬁt. The curve is divided into two τf (−ϕ¯˙ m) + τg(ϕ ¯a) = u−. regions according to its velocity dependent functions, g and h. Subtracting the equations yields + are required to achieve their estimate. Considering the static τf (ϕ¯˙ m) − τf (−ϕ¯˙ m) = u − u− friction curve in the ﬁrst quadrant, M0 can be written as the and supposing a direction independent friction, regression i.e. τ (−ϕ¯˙ ) = −τ (ϕ¯˙ ), the resulting direction f m f m T independent friction is: τˆf (ϕ ˙ m) = f(ϕ ˙ m)θ (4a) h | ϕ˙ m |α i + 1 ϕ˙ s u − u− f(ϕ ˙ m) = , e− , ϕ˙ m (4b) τf (ϕ¯˙ m) = . (3) 2 θ = [Fc, F s, Fv] (4c)

Due to nonlinearities of friction, it is important to define where f(ϕ ˙ m) is a regressor vector. The chosen identi- an excitation signal including several different (constant) fication method combines linear regression with extensive velocities. The signal used moves one axis at a time at 12 search (grid search over a predetermined range) for the speed levels in both directions, taking 2:15 min and sampled nonlinear parameters ϕ˙ s and α. For the curve in Fig- at 2 KHz¶. Figure 2 shows the motor speed- and torquek ure 3, the identified parameters are [Fc,Fs,Fv, ϕ˙ s, α] = signals in the experiments. [3.40 10−2, 4.63 10−2, 3.68 10−4, 10.70, 1.95]. Notice that, as seen in Figure 3, the model structure M0 can describe static

1 300 friction dependence on speed fairly well. In fact, the sum of

200 P P 0.5 absolute prediction errors, |ε| = |τf − τˆf |, in Figure 3 100 0 is no more than 0.03. 0 (rad/s) u 2 −0.5

m, −100 ˙ ϕ B. Guidelines for the Experiments −200 −1 −300 In order to be able to build a friction model including more 0 2 4 6 8 10 12 14 4 0 2 4 6 8 10 12 14 x 10 sample sample 4 x 10 variables than the velocity, it is important to separate their (a) Motor speed (b) Motor applied torque influences. The situation is particularly critical regarding Fig. 2. Excitation signal used for the static friction curve estimation. temperature as it is difficult to control it inside a joint. Moreover, due to the complex structure of an industrial robot, changes in joint angle might move the mass center of the The data was segmented at the different constant speeds and, robot arm system, causing variations of joint load torques. To using Equation (3), the friction torque was computed for each avoid undesired effects, the guidelines below were followed speed. The result of the estimation can then be presented during the experiments. in a static friction curve, sometimes referred to as Stribeck 1) Isolating joint load torque dependency from joint angle curve, see Figure 3. Notice that, since it is assumed that dependency: Using an accurate dynamic robot model∗∗, it is friction is independent of the joint direction of movement, possible to predict the joint torques for any given robot con- the friction torques for negative velocities would have the figuration (a set of all joints angles). For example, Figure 4 same amplitude as in Figure 3 but with opposite sign. shows the resulting τm and τp at joint 2, related to variations A. Parametric Description and Identification of joint 2 and 4 angles (ϕa,2 and ϕa,4) throughout their workrange. Using this information, a set of configurations The solid line in Figure 3 is obtained by model-based can be selected a priori in which it is possible to estimate estimates of the friction curve with an instance of the static parameters in an efficient way. model structure M0. Since the parameters ϕ˙ s and α enter 2) Isolating temperature effects: Some of the experiments M0 in a nonlinear fashion, nonlinear identification methods require that the temperature of the joint is under control. ¶Similar results have been experienced with sampling rates down to Using joint lubricant temperature measurements, the joint 220 Hz. thermal decay constant κ was estimated to 3.04 h (see kThroughout the paper all torques are normalized to the maximum manipulation torque at low speed. ∗∗An ABB internal tool was used for simulation purposes.

0.14 0.14 20 30 40 50 60 70 0.05 0.06 0.07 0.08 0.09 0.1 0.13 0.13

0.12 0.12

0.11 0.11

0.1 0.1 f f τ τ 0.09 0.09

0.08 0.08

0.07 0.07

0.06 (a) Simulated τm (b) Simulated τp 0.06 0.05 0.05

Fig. 4. Simulated joint load torques at joint 2. Notice the larger absolute 50 100 150 200 250 50 100 150 200 250 ϕ˙ (rad/s) ϕ˙ (rad/s) values for τm when compared τp. m,2 m,2

(a) Effects of ϕa at τm = −0.39, (b) Effects of τp at τm = −0.39, T = 34◦ C. T = 36◦ C. [22] for more details). Executing the static friction curve Fig. 5. Static friction curves for experiments related to ϕa and τp. identification experiment periodically, for longer time than 2κ (i.e. > 6.08 h), the joint temperature is expected to have reached an equilibrium. Only data related to the expected thermal equilibrium was considered for the analysis. Because of the mechanical construction of the robot, only small variations of the perpendicular load torque, τp, are III. EMPIRICALLY MOTIVATED MODELING possible to achieve for joint 2 (see Figure 4(b)). A total of 20 Using the described static friction curve estimation experiments at constant temperature were performed for joint method, it is possible to design a set of experiments to 2, in the range τp = [0.04, 0.10]. As Figure 5(b) shows, τp analyze how the states X ∗ affect static friction. As shown in values in the obtained range did not play a significant role for Section II-A, the model structure M0 can represent static the static friction curve. No extra terms are therefore needed friction dependence on ϕ˙ m fairly well. M0 is therefore for joint 2 and M0 is considered valid. The observation is taken as a primary choice, with α considered constant at 1.3 true at least over a narrow τp interval. In [22], a similar (the value is motivated from [22]). Whenever M0 can not experiment is presented for joint 1, for which a larger range describe the observed friction behavior, extra terms fi(X ∗, θ) of τp is possible. In that case a small change in the velocity- are proposed and included in M0 to achieve a satisfactory weakening regime could be noticed. model structure M∗. As seen in Figure 4(a), large variations of the manipulation A. Joint angles torque τm are possible by simply varying the arm config- Due to asymmetries in the contact surfaces, it has been uration. A total of 50 static friction curves were estimated observed that the friction of rotating machines depends on over the range τm = [−0.73, 0.44]. As seen in Figure 6, the the angular position [1]. It is therefore expected that this effects appear clearly. Obviously, a single M0 instance can dependency occurs also in a robot joint. Following the not describe the observed phenomena. A careful analysis of experiment guidelines from the previous section, a total of the effects reveals that the main changes occur in the velocity 50 static friction curves are estimated in the joint angle range weakening part of the curve. From Figure 6(c), it is possible ϕa = [8.40, 59.00]deg. As seen in Figure 5(a), little effects to observe a (linear) bias-like (Fc) increase and a (linear) can be observed. The subtle deviations are comparable to the increase of the standstill friction (Fs) with |τm|. Furthermore, errors of the friction curve identified under constant values of as seen in Figure 6(b), the Stribeck velocity ϕ˙ m is maintained [ϕa, τp, τm,T ]. In fact, even a constant instance of M0 can fairly constant. The observations support an extension of M0 describe the friction curves satisfactorily, no extra fi terms to are thus required. B. Joint load torque τf (ϕ ˙ m, τm) = {Fc,0 + Fc,τm |τm|}+ 1.3 Since friction is related to the interaction between contact- ϕ˙ m + {F + F |τ |}e− ϕ˙ s,τm + F ϕ˙ . (M ) ing surfaces, one of the first phenomena observed was that s,0 s,τm m v m 1 friction varies according to the applied normal force. The observation is thought to be caused by the increase of the In the above equation the parameters are written with sub- true contact area between the surfaces under larger normal script 0 or τm in order to clarify its origin related M0 or to forces. A similar reasoning can be extended to joint torques the effects of τm. Assuming that any phenomenon not related in a robot revolute joint. Due to the elaborated joint gear- and to τm is constant and such that the 0 terms can capture bearing design it is also expected that torques in different them, good estimates of the τm-dependent parameters can be directions will have different effects on the static friction achieved. Using an identification procedure similar to the one curve . †† presented in Section II-A, the model M1 is identified with ††In fact, a full joint load description would require 3 torque and 3 force the data set from Figure 6. The resulting model parameters components. describing the dependence on τm are shown in Table I. (a) Estimated friction curves for different values of τm. (a) Estimated friction curves for different values of T .

0.16 0.16

0.2 0.4 0.6 0.8 1 50 100 150 200 250 300 0.14 79.1 0.14 0.7 40 50 60 70 80 0.14 50 100 150 200 250 300 0.14 250 0.13 0.13 250 5 250 0.1273.3 0.12 0.12 0.10.1 200 0.30.3 0.12 0.5 200 0.5 0.1167.5 0.11 5 0.7 200 150 0.1 38.5 61.7 f 0.1 0.1 10 τ f 0.1 0.3 150 τ 150 f 44.3 0.09 f 20 τ 55.9 0.09 10 τ 0.3 100 50.1 0.08 0.08 30 0.10.1 0.08 50.1 55.9 0.08 100 250 0.30.3 61.7 100 44.3 50 0.1 0.07 67.5 0.07 200 0.1 0.5 73.3 0.06 5020 79.1 70 0.7 0.06 38.5 70 150 2530 5 0.06 0.06 100 50 5 10 50 0.05 0.05 10 0.04 203025 0.04 Stribeck 50 100 150 200 250 0.04 0.04 30 velocity −0.6 −0.4 −0.2 0 0.2 0.4 20 50 100 150 200 250 ϕ˙ m,2 (rad/s) τm 40 50 60 70 80 ϕ˙ m,2 (rad/s) T (◦ C) (b) Friction surface cuts for differ- (c) Friction surface cuts for differ- (b) Friction surface cuts for differ- (c) Friction surface cuts for dif- ent values of τm. ent values of ϕ˙ m,2 (rad/s). ent values of T at τm = −0.02. ferent values of ϕ˙ m,2 (rad/s) at τ = −0.02 Fig. 6. The dependence of the static friction curves on the manipulation m . torque, τ , at T = 34◦ C. m Fig. 7. The temperature dependence of the static friction curve. TABLE I DENTIFIED τ DEPENDENTMODELPARAMETERS I m- . exponential-like) decrease of the velocity-dependent slope, as seen in Figures 7(b) and 7(c). Fc,τm Fs,τm ϕ˙ s,τm 2.32 10−2 1.28 10−1 9.07 It is also interesting to study combined effects of τm and T . To better see these effects, the friction surfaces in Figure 7(a) are subtracted from each other, yielding τ˜ . As it can be C. Temperature f seen from the resulting surface in Figure 8(a), the difference The friction temperature dependence is related to the between the surfaces is fairly temperature independent. This change of properties of both lubricant and contacting sur- is an indication of independence between effects caused by faces. In lubricated mechanisms, both the thickness of the T and τm. lubricant layer and its viscosity play an important role Given that the effects of T and τm are independent, it is for the resulting friction properties. In newtonian fluids, possible to subtract the τm-effects from the surfaces in Figure the shear forces are directly proportional to the viscosity 7(a) and solely obtain temperature related phenomena. The which, in turn, varies with temperature [23]. Dedicated previously proposed terms to describe the τm-effects in M1 experiments were made to analyze temperature effects. The were: 1.3 ϕ˙ m joint was at first warmed up to 81.2◦ C by running the ϕ˙ s,τm τˆf (τm) = Fc,τm |τm| + Fs,τm |τm|e− . (5) joint continuously back and forth. Then, while the robot cooled, 50 static friction curves were estimated over the With the parameters values given from Table I, the manipulation torque effects were subtracted from the friction curves range T = [38.00, 81.20] ◦ C. In order to resolve combined of the two surfaces in Figure 7(a), that is, the quantities effects of T and τm, two manipulation torque levels were τ − τˆ (τ ) were computed. The resulting surfaces are used, τm = −0.02, and τm = −0.72. As it can be seen in f f m Figure 7, the effects of T are significant. shown in Figure 8(b). As expected, the surfaces become Temperature has an influence on both velocity regions quite similar. The result can also be interpreted as an of the static friction curves. In the velocity-weakening re- evidence on the fact that the model structure used for the τm-dependent terms and the identified parameter values are gion, a (linear) increase of the standstill friction (Fs) with temperature can be observed according to Figure 7(b). In correct. Obviously, the original model structure M0 can not Figure 7(c) it can moreover be seen that the Stribeck velocity characterize all observed phenomena, even after discounting the τm-dependent terms. (ϕ˙ s) increases (linearly) with temperature. The effects in the velocity-strengthening region appear as a (nonlinear, A proposal for M∗. From the characteristics of the T - D. Validation A separate data set is used for the validation of the proposed model structure M∗. It consists of several static friction curves measured at different τm- and T values, as seen in Figure 9. With an instance of M∗ given by

(a) Difference τ˜f between the two static friction surfaces in Figure 7(a).

(a) Static friction curves

1 40

0.5 35 ) C

m 0 30 ◦ τ ( T −0.5 25 T (◦C)

τm −1 20 0 50 100 150 200 250 (b) Static friction surfaces in Figure 7(a) after subtraction experiment # of the τm-dependent terms. (b) τm- and T conditions

Fig. 8. Indication of independence between effects caused by T and τm. Fig. 9. Validation data set. Notice the large variations of T - and τm values in Figure (b) when registering the static friction curves in (a). related effects and the already discussed τm-effects, M1 is the parameter values from Tables I and II, the resulting extended to: prediction errors for the validation data set are shown in Figure 10. As a comparison, the errors related to τf (ϕ ˙ m, τm,T ) = a single instance of M0, with [Fc,Fs,Fv, ϕ˙ s, α] = 1.3 ϕ˙ m [4.90 10−2, 8.44 10−2, 5.00 10−4, 6.50, 1.30], are also shown − ϕ˙ s,τm {Fc,0 + Fc,τm |τm|} + Fs,τm |τm|e + (Mg∗ ) τm in the ﬁgure. As it can be seen, M∗ performs visibly better 1.3 ϕ˙ m { } − {ϕ˙ s,0+ϕ ˙ s,T T } M + Fs,0 + Fs,T T e + ( g∗T ) −T { TVo } (M ) + Fv,0 + Fv,T e ϕ˙ m. h∗T

The model describes the effects of τm and T for the investigated robot joint. The ﬁrst Mg∗ expressions relate to the velocity-weakening friction while Mh∗ relates to the velocity-strengthening regime. τm only affects the velocity- weakening regime and requires a total of 3 parameters,

[Fc,τm ,Fs,τm , ϕ˙ s,τm ]. T affects both regimes and requires

4 parameters, [Fs,T , ϕ˙ s,T ,Fv,τm ,TVo]. The 4 remaining parameters, [Fc,0,Fs,0, ϕ˙ s,0,Fv,0] , relate to the original friction model structure M0. Notice that under the assump- tion that τm- and T effects are independent, their respective expressions appear as separated sums in M∗. Fig. 10. Models absolute prediction error. Notice the considerable better The term T /TVo in M is motivated by the performance of M∗. Fv,T e− h∗T exponential-like behavior of viscous friction (recall Figure when compared to M , with only speed dependence. The 7(c)). In fact, the parameter TVo is a reference to the Vogel- 0 −2 −3 Fulcher-Tamman exponential description of viscosity and maximum and mean errors for M∗ are [1.86 10 , 3.39 10 ], −2 −2 temperature [23]. compared to [7.09 10 , 2.07 10 ] for M0. IV. CONCLUSIONS AND FURTHER RESEARCH Given the already identified τm-dependent parameters in Table I, the remaining parameters from M∗ are identified The main contribution of this paper is the empirically from the measurement results presented in Figure 8(b), after derived model of static friction as a function of the variables the subtraction of the τm-terms. The values are shown in X ∗ = [ϕ ˙ m, ϕa, τp, τm,T ]. While no significant influences Table II. of joint angle and perpendicular torque could be found by TABLE II

IDENTIFIED T -DEPENDENTAND M0-RELATED MODEL PARAMETERS.

Fc,0 Fs,0 Fs,T Fv,0 Fv,T ϕ˙ s,0 ϕ˙ s,T TVo 3.04 10−2 −2.44 10−2 1.69 10−3 1.29 10−4 1.31 10−3 −25.00 1.00 21.00

the experiments, the effects of manipulation torque (τm) REFERENCES and temperature (T ) were significant. The effects of τm and [1] F. Al-Bender and J. Swevers, “Characterization of friction force T were included in the proposed model structure M∗, an dynamics,” IEEE Control Systems Magazine, vol. 28, no. 6, pp. 64–81, extended version of the model structure M dependent only 2008. 0 [2] D. Dowson, History of Tribology. Professional Engineering Publish- on velocity. As shown in Figure 10, the proposed model M∗ ing, London., 1998. is needed in applications where the manipulation torque and [3] H. M. Kim, S. H. Park, and S. I. Han, “Precise friction control for the the temperature play significant roles. nonlinear friction system using the friction state observer and sliding mode control with recurrent fuzzy neural networks,” Mechatronics, The description of the velocity-weakening regime, g, in vol. 19, no. 6, pp. 805 – 815, 2009. [4] Y. B. Yi Guo, Zhihua Qu, Z. Zhang, and J. Barhen, “Nanotribology M∗ with an exponential temperature-dependent function was and nanoscale friction,” Control Systems Magazine, IEEE, vol. 28, based on the observed phenomena at a (large but) limited no. 6, Dec. 2008. temperature range. To capture the static friction behavior at [5] H. Olsson, K. J. Astr˚ om,¨ C. C. de Wit, M. Gafvert, and P. Lischinsky, “Friction models and friction compensation,” Eur. J. Control, vol. 4, even larger temperature ranges, more complex expressions no. 3, pp. 176–195, 1998. may be needed [23]. [6] B. Bona and M. Indri, “Friction compensation in robotics: an overview,” in Decision and Control, 2005. Proceedings., 44th IEEE The model M∗ has a total of 7 terms and 3 parameters International Conference on, Dec 2005. which enter the model in a nonlinear fashion. The identifica- [7] F. L. Witono Susanto, Robert Babuska and T. van der Weiden, “Adaptive friction compensation: application to a robotic manipulator,” tion of such a model is computationally costly and requires in The International Federation of Automatic Control, 2008. Proceed- data from several different operating conditions. Studies ings., 17th World Congress, Dec 2008. on defining sound identification excitation and estimation [8] P. J. Blau, “Embedding wear models into friction models,” Tribology Letters, vol. 34, no. 1, Apr. 2009. routines are therefore important. It will also be important to [9] A. C. Bittencourt, “Friction change detection in industrial robot arms,” validate M∗ on other robot joints and on other robot types MSc. thesis, The Royal Institute of Technology, 2007. and even on other types of rotating mechanisms. [10] F. Caccavale, P. Cilibrizzi, F. Pierri, and L. Villani, “Actuators fault diagnosis for robot manipulators with uncertain model,” Control Only static friction (measured when transients caused by Engineering Practice, vol. 17, no. 1, pp. 146 – 157, 2009. velocity changes have disappeared) was considered in the [11] M. Namvar and F. Aghili, “Failure detection and isolation in robotic manipulators using joint torque sensors,” Robotica, 2009. studies. It would be interesting to investigate if a dynamic [12] M. McIntyre, W. Dixon, D. Dawson, and I. Walker, “Fault identifica- model, for instance given by the LuGre model structure ML, tion for robot manipulators,” Robotics, IEEE Transactions on, vol. 21, could be used to describe dynamic friction with extensions no. 5, pp. 1028–1034, Oct. 2005. [13] A. T. Vemuri and M. M. Polycarpou, “A methodology for fault from the proposed M∗. However, to make experiments on a diagnosis in robotic systems using neural networks,” Robotica, vol. 22, robot joint in order to obtain a dynamic friction model is a no. 04, pp. 419–438, 2004. big challenge. Probably, such experiments must be made on [14] D. Brambilla, L. Capisani, A. Ferrara, and P. Pisu, “Fault detection for robot manipulators via second-order sliding modes,” Industrial a robot joint mounted in a test bench instead of on a robot Electronics, IEEE Transactions on, vol. 55, no. 11, pp. 3954–3963, arm system, which has very complex dynamics. Nov. 2008. [15] R. Mattone and A. D. Luca, “Relaxed fault detection and isolation: A practical limitation of M∗ is the requirement on avail- An application to a nonlinear case study,” Automatica, vol. 42, no. 1, ability of τ and T . Up to date, torque- and joint temperature pp. 109 – 116, 2009. m [16] B. Freyermuth, “An approach to model based fault diagnosis of sensors are not available in standard industrial robots. The industrial robots,” in Robotics and Automation, 1991. Proceedings., gears of the robot used in the studies was lubricated with oil, 1991 IEEE International Conference on, vol. 2, Apr 1991, pp. 1350– not grease, this gave an opportunity to obtain well defined 1356. [17] R. Waiboer, “Dynamic modelling, identification and simulation of temperature readings by having a temperature sensor in the industrial robots,” Ph.D. dissertation, University of Twente, 2007. circulating lubricant oil. Moreover, as mentioned in Section [18] B. Armstrong-Helouvry,´ Control of Machines with Friction. Kluwer II-B, the joint torque components can still be estimated Academic Publishers, 1991. [19] K. J. Astr˚ om¨ and C. Canudas-de Wit, “Revisiting the lugre friction from the torque reference to the drive system by means of model,” Control Systems Magazine, IEEE, vol. 28, no. 6, pp. 101– an accurate robot model. In this situation, it is of course 114, Dec. 2008. important to have correct load parameters in the model [20] B. F. S. C. Avraham Harnoy, “Modeling and measuring friction effects,” Control Systems Magazine, IEEE, vol. 28, no. 6, Dec. 2008. in order to calculate the manipulation- and perpendicular [21] R. Stribeck, “Die wesentlichen eigenschaften der gleit-und rollenlager torques. – the key qualities of sliding in roller bearings,” Zeitschrift Des Vereines Deutcher Ingenieure, vol. 46, no. 36–38, pp. 1342–1348, Regardless these experimental challenges, there is a great 1432–1437, 1902. [22] A. C. Bittencourt, “Friction in a manipulator joint – an empirical potential for the use of M∗ for simulation- and evaluation model,” Tech. Report, Linkoping¨ University, to appear. purposes. The designer of control algorithms, the diagnosis [23] C. J. Seeton, “Viscosity-temperature correlation for liquids,” Tribology engineer, the gearbox manufacturer, etc. would very likely Letters, vol. 22, no. 1, pp. 67–78, Mar. 2006. see benefits in using a more realistic friction model.