FRICTION MODELING WITH TEMPERATURE EFFECTS FOR INDUSTRIAL ROBOT MANIPULATORS

Transcription

1 215 IEEE/RSJ International onference on Intelligent Robots and Systems (IROS) ongress enter Hamburg Sept 28 - Oct 2, 215. Hamburg, Germany FRITION MODELING WITH TEMPERATURE EFFETS FOR INDUSTRIAL ROBOT MANIPULATORS Luca Simoni 1, Manuel Beschi 2, Giovanni Legnani 1, Antonio Visioli 1 1 Department of Mechanical and Industrial Engineering, University of Brescia, Brescia, Italy 2 Institute of Industrial and Automation Technologies (ITIA-NR), National Research ouncil, Milan, Italy Abstract In this paper we present a new friction model for industrial robot manipulators that takes into account temperature effects. In particular, after having shown that friction might change very significantly during robot operations, two solutions based on a polynomial description of the joint friction are proposed and compared. In both cases the models proposed do not need a measurement of the joint temperature, but just of the environmental temperature, so as to be easily applied in industry. Experimental results demonstrate the effectiveness of the applied methodology. I. INTRODUTION It is well known that friction is one of the most relevant problem in the design of effective control strategies for industrial robot manipulators and for mechanical systems in general. Actually, friction is a complex phenomenon that arises when there is a relative motion between two surfaces in contact and depends on many factors such as the geometry and materials of the two surfaces, their relative velocity, the presence of lubricants, and so on. As such, it introduces nonlinearities that, if not properly compensated, might cause tracking errors, limit cycles and stick-slip motions. For this reason, many friction models have been devised in the literature (see, for example, [1]) and the vast majority of them are dependent on the speed of the joint, if a manipulator is considered. They can be divided into two categories: static models and dynamic models [2], [3]. In the first category, the friction force is a static function of the velocity. The best-known models in this context are the oulomb friction model, in which the friction force is assumed to be independent of the velocity, and the viscous friction in which force is proportional to velocity. An effective alternative is to use a polynomial expression to better take into account the complexity of the phenomenon [4]. Regarding dynamic models, relevant examples are the Dahl model and its extension [5], [6], the generalized Maxwell-slip model [7], and the more recent LuGre model [8], which has been extended and applied in different forms [9], [1]. An excellent review of friction compensation techniques in robotics has been presented in [11]. In general, friction models are employed to compute a feedforward action that compensates for the friction effect (see, for example, [12], [13]). In order to take into account the approximate knowledge of friction model parameters or their time-variant characteristic, many adaptive schemes have been proposed (see, for example, [14], [15], [16], [17]). It is in any case recognized that friction is still a relevant issue for the control of industrial robots and it depends on other factors than speed. Recently, research has especially focused on investigating and modeling the role of temperature in friction [18], [19]. Indeed, temperature plays a key role especially for those robot operations that require the robot to often stop for a relatively long time, so that the joint temperature cools down, changing the friction characteristics (mainly because the properties of the lubricant change). In this paper we propose a simple friction model that explicitly takes into account the temperature of the joint. In particular, the temperature is estimated by means of a basic first principle thermodynamic model which is then used to update a third-order polynomial expression. Two methods based on this approach are devised and compared. Experimental results are obtained by means of an industrial omau robot manipulator. It is worth stressing that, differently from the methods proposed in [18], [19], here the measurement of the joint temperature, that is of the lubricant temperature, is not necessary, so that the technique is easier to implement in practice. The paper is organized as follows. The robot manipulator employed for the experiments is described in Section 2. Section 3 is devoted to show the influence of temperature on friction effect. The new temperature based friction models are presented in Section 4. Identification and validation results are given in Sections 5 and 6 respectively. oncluding remarks are in section 7. II. THE MANIPULATOR USED FOR THE EXPERIMENTS The robot employed in this work is a omau SMART NS AR (see Figure 1), and it has been designed to perform arc welding tasks. It is a serial manipulator built with an anthropomorphic kinematic structure, with six degrees of freedom and a maximum payload of 16 kilograms, placed on the ground for a standard working configuration. Joints are actuated by means of six A brushless servomotors with speed reducers. The maximum stall torque for each motor is shown in Table I, while the type and reduction rate of the speed reducers are shown in Table II. The limits of position and speed of each joint are displayed in Table III /15/$ IEEE 3524

2 Fig. 1. The robot used in the experiments. Brakes are mounted directly inside the servomotors. Oil is used as lubricant for the first four axes while grease for the last two joints. Axis Stall Torque [Nm] Voltage [V] Power [kw] TABLE I HARATERISTIS OF THE MOTORS. Axis Reduction rate Type Lubrication Gears oil ycloidal oil ycloidal oil Gears oil Harmonic Drive grease 6 5. Harmonic Drive grease TABLE II REDUTION RATES OF THE SPEED REDUERS FOR EAH JOINT. Joint Position limits [deg] Speed limits [deg/s] 1 ± / / ± ± ± TABLE III POSITION AND SPEED JOINTS LIMITS. It is worth stressing that the control architecture used for this work is the standard industrial one, called 4G, so that the proposed friction models can be implemented with a typical industrial setup. The controller exploits only the data related to the motors position (which is measured by using encoders) and their current, with a sampling frequency of 1 khz. Velocity and acceleration of the joints are determined by using an approach based on a Savitzky-Golay filter that uses a nth-order polynomial interpolation, while joint torque is determined by multiplying the motor current by the torque constant of the motor. III. POLYNOMIAL FRITION MODEL As already mentioned, friction is a complex phenomenon that depends on many factors, including temperature as we will show in the next sections, but, most of all, on velocity. For this reason many models have been proposed in the literature. Among them, we have selected the polynomial friction model, see [3] [4], because it is the model that best fits the obtained experimental friction torque data. Indeed, a third order polynomial function can represent both Stribeck and oulomb effects. Further, it is also able to represent a viscous effect that is not necessarily linear. The friction torque function can be therefore represented as [ τ f = c + c 1 ω + c 2 ω 2 + c 3 ω 3] sgn(ω) (1) where ω is the joint velocity and it has to be noted that a symmetric function is considered, that is, the coefficients c,...,c 3 are the same for positive and negative velocities. The polynomial coefficients can be determined easily by moving only one joint of the robot at a time, so that a one degree-of-freedom system can be considered for each joint. In this case the dynamic equation of the joint can be written as τ + τ f + τ w = J ω (2) where J is the inertia seen by the motor, τ is the motor torque, τ f is the friction torque and τ w is the torque related to the gravitational force. From (2) it is immediate to write τ f = J ω τ τ w (3) Now, a standard least-squares based method can be applied to estimate friction torque coefficients. In fact, equation (2) can be rewritten by expressing all the terms as: τ = J ω τ f τ w = J ω c sgn(ω) c 1 ω c 2 ω 2 sgn(ω) c 3 ω 3 P x cos(θ) + P y sin(θ) where the gravity compensation torque has been expressed as τ w = P x cos(θ) P y sin(θ) (5) where θ is the joint (measured) position and (4) P x = mgl cos(γ) P y = mgl sin(γ), (6) where m is the mass of the moving part of the robot, g is the gravity acceleration, l is the distance between the joint axis of rotation and the centre of mass of the moving part of the manipulator and γ is the angle of the centre of the mass position when θ =. Thus, equation (4) can be written in matrix form where in each row there are all the samples collected at a given sampling time: where τ = M X (7) τ = [ τ 1 τ 2... τ n ] T (8) 3525

3 is the vector of the measured motor torque at n sampling instants, M = ω 1 sgn(ω 1 ) ω 1 ω1 2 sgn(ω 1) ω1 3 cos(θ 1 ) +sin(θ 1 ) ω 2 sgn(ω 2 ) ω 2 ω2 2 sgn(ω 2) ω2 3 cos(θ 2 ) +sin(θ 2 ) ω n sgn(ω n ) ω n ω 2 n sgn(ω n ) ω 3 n cos(θ n ) +sin(θ n ) is the matrix containing the other measured data and (9) X = [ J c c 1 c 2 c 3 P x P y ] T (1) is the vector of the coefficients to be determined. Then, coefficients can be estimated as X = M + τ (11) when M + denotes the pseudo-inverse of the matrix M. It is obvious that, in order to provide a sensible result, the trajectory used in the experiments have to span all the possible motor velocities for many times in order to effectively cope with measurement noise and excite all the friction dynamics. In this paper, as a standard controller is employed, the performed trajectories are point-to-point motions with S-curve velocity profiles, as imposed by the controller, repeated many times. However, the friction torque strongly depends also on the joint temperature, as it will be explained in the next section. IV. MODELING TEMPERATURE EFFET An important issue that can be noted during the experiments is that friction torque changes during robot operation, in particular, considering a series of point to point motions performed after the robot has not been used for a long time. If the friction function is determined after different time intervals, it appears that the friction term decreases, until a steady-state value is obtained for a given velocity. An illustrative example related to the second joint of the robot is shown in Figure 2, where the friction curves have been determined every six minutes and it appears that, after the robot warm-up, friction torque decreases of about 33% at high velocities. This behaviour can be associated with the temperature of the joint, which increases accordingly to the joint employment. It is therefore necessary to take into account temperature effect in the friction model in order to compensate for the warm-up of the robot during its operations, especially if operations start after a time interval during which the robot is at rest. For this purpose, two new models are proposed hereafter. Both models are derived starting from the thermal balance of the joint, that is, W acc = W in W out (12) where W acc is the accumulating thermal power in the joint, that is, W acc = dt (13) dt where T is the internal temperature of the joint and is the thermal capacity of the joint; W out is the dissipated thermal power W out = K (T T env ) (14) Fig. 2. Modification of the friction torque curve during robot warm-up (intervals of six minutes). where K is the coefficient of thermal exchange between joint and air and T env is the environmental temperature (which is assumed to be constant during a robot operation); W in is the thermal power injected to the joint, which is assumed to be the product between the friction torque and the speed of motion, that is, W in = τ f ω (15) A. Four parameters model The first developed model, named four parameters model, is based on the assumption that friction coefficients c...c 3 change linearly with temperature T as c i = c i [α (T T ) + β] (16) where α and β are the coefficients representing the assumed linear relation between friction torque and temperature, c i is the value of c i for T = T and β = 1. Thus, this model assumes a linear relation between joint internal temperature and friction torque expressed as dt = [ τ f,rms ω K(T T env ) ] 1 (17) dt τ f,rms = τ f,rms [α(t T ) + β] (18) where τ f,rms is the RMS value of the friction torque in a given robot operation cycle and τ f,rms is the RMS value of the friction torque obtained when a robot cycle is performed for the first time when the manipulator is used after it has been stopped for a long period (T = T = Tenv ). Finally, ω is the velocity that, by applying the RMS torque, would give the mean friction power and can be called equivalent thermal velocity. Namely, it is the ratio between the mean friction power and the RMS friction torque, that is, ω = mean( τ f ω ) τ f,rms. (19) The friction torque at a given temperature is then expressed as ( τ f (T ) = c + c 1 ω + c 2 ω 2 + c 3 ω 3) sgn(ω)[α(t T ) + β] (2) 3526

4 Thus, once the four model parameters α, β, K, (in addition to c, c 1, c 2, c 3 ) are estimated by performing identification experiments (see Section V) at a given environmental temperature T env = T, the model can be employed to obtain the friction torque for new robot operations, starting from an environmental temperature T env. It is worth stressing that, in case of perfect modeling, it is β = 1; however, a slightly different value is expected because other possible friction causes have not been considered and because of the unavoidable presence of noise in the identification phase. In practice, at the beginning of a new robot operation, after the robot has been at rest the friction torque is given by (2) with T = T env. Then, at the end of the first cycle, the value of τ f,rms is computed and the value of τ f,rms is computed by means of (18). By also computing the value of ω by means of (19), the change of the temperature in the joint can be calculated by means of (17). Then, if the cycle does not change during robot operations, the value of the joint temperature can be obtained at the end of a cycle by simply applying (18) and (17) iteratively (note that τ f,rms is the RMS value obtained at the first iteration). Friction torque can then be estimated by applying (2) to the updated value of the joint temperature. The differential equation (17) can be solved by considering T () = T as initial condition. It results: T (t) = βτ f,rms ω + KT env ατ f,rms T env ω K ατ f,rms ω where +he ατ f,rms ω K t (21) h = KT ατ f,rms T ω βτ f,rms ω KT env + ατ f,rms T ω. K ατ f,rms ω (22) It appears that the time constant of the first order system is t c = ατ f,rms ω K (23) and its knowledge can be employed to determine the duration of the time interval after which the joint temperature has to be updated. A sensible value for this time interval is t c /1. B. Six parameters model The second model developed is a modification of the previous one in which each term of the type ω k have independent coefficients that relate friction to temperature. Assuming that, it is possible to express the relation between joint internal temperature and friction as dt = [ τ f,rms ω K(T T env ) ] 1 (24) dt τ f,rms = ( c + c (T T ) ) + ( c 1 + c 1(T T ) ) ω + ( c2 + c 2(T T ) ) ω 2 + ( c 3 + c 3(T T ) ) ω 3 (25) where c, c 1, c 2 and c 3 are the coefficients representing the assumed linear relation between friction torque coefficients and temperature, and c, c 1, c 2 and c 3 are the polynomial friction coefficients at T = T. The friction torque at a given temperature is: τ f (T ) = [ ( c + c (T T ) ) + ( c 1 + c 1 (T T ) ) ω + ( c2 + c 2 (T T ) ) ω 2 + ( c 3 + c 3 (T T ) ) ω 3 ] sgn(ω). (26) The approach is therefore similar to the four parameters model, but in this case the parameters to be estimated in the identification experiment are c, c 1, c 2, c 3, K, and (in addition to c, c 1, c 2, c 3 ). The solution of the differential equation with T () = T yields where and T (t) = A ω + KT env B ωt K B ω + h e B ω K t (27) h = KT BT ω A ω KT env + BT ω, (28) K B ω [ A = τ f,rms = c + c 1 ω + c 2 ω 2 + c 3 ω 3] (29) [ B = c + c 1 ω + c 2 ω 2 + c 3 ω 3] (T T ). (3) In this case the time constant is equal to t c = B ω K (31) It is worth stressing that τ f,rms is actually contained in the right-hand side of (29) and that τ f,rms = A + B(T T ). V. IDENTIFIATION EXPERIMENTS This section shows the results of the second joint of the robot as it is the most relevant to gravitational force; the other joints have in any case a similar behaviour. Identification procedure has been performed, for both models, by applying four different duty cycles (Fig. 3), where for each operation the robot starts after a long period (one night) of inactivity (T = T = Tenv ). In particular, cycle times of four minutes have been considered and each one has been divided in subintervals of one minute. The motion during each subinterval is the same, but the number of subintervals in which the robot is moved in each cycle is different. In this way, the thermal power injected into the joint is different. In each subinterval of the cycle time, point to point motions from -45 deg to 7 deg and viceversa are performed with six different maximum velocities, namely, 5%, 24%, 43%, 62%, 81%, and 1% of the maximum velocity allowed by the motor. The temperature estimation is updated at the end of each cycle, that is, every four minutes. The parameters of the models are estimated by considering all the data collected during experiments, and, using the Matlab function fmincon, by minimizing the sum of the square errors, that is the difference between the friction force calculated by means of the method described in Section III, and the one estimated separately at the end of each cycle using only the data collected during the last cycle. The obtained values of the parameters are α =.896, β =.91796, K = , = for the four parameters model and c = 1, c 1 =.578, c 2 = , 3527

5 Fig. 3. The different cycles robot operations for the identification of te models (W in = 25%, 5%, 75%, 1%). c 3 =.17453, K =.1 and = for the six parameters model. The obtained results are shown in Figure 4 and in Figure 5 for the four and six parameters model respectively, where the duty cycle of 1% is considered and the measured and estimated friction torques are compared at different time intervals (in particular, every 25 cycles, that is after 4, 14, 24, and 34 minutes). Similar results are obtained for the other duty cycles. It appears that the six parameters model provides better results. The temperature evolution determined for the four different duty cycles are plotted in Figure 6 where it can be seen that, as expected, the temperature increases with the duty cycle. The determined values of the time constants are t c = min for robot cycles where the robot is moving for one minute and it is stopped for three minutes (that is, W in = 25%), t c = min for W in = 5%, t c = min for W in = 75%, and t c = min for W in = 1%, when the robot is always in motion. It is therefore evident that updating the value of the temperature every 4 minutes is sufficient to take into account the dynamics of the system. From the identification results it appears in any case that both models are capable of positively describing temperature effect on the friction torque. Indeed, friction torque decreases from the beginning of the operations after a period of rest, until a given time when the temperature function becomes almost constant and therefore the friction does not change significantly anymore. It has not been possible, in the specific set-up, to check if the actual joint temperature coincided to that previously estimated; however, it is not important to have an exact temperature estimation, but it is sufficient to have a parameter that roughly represents it and fits the experimental results. This parameter proved to be the estimated temperature evaluated in the proposed model. VI. VALIDATION EXPERIMENTS The obtained models have then been validated by considering again the second joint of the robot and by applying to it point-to-point motions different from those applied in section V. These have been repeated for many cycles of three 1 1 friction torque [Nm] 1 1 Fig. 4. Four parameters model, duty cycle W in = 1%. Torque versus velocity plot, estimated with least square (blue line) and identified with the proposed model (red line) friction torque. Top left: results after 4 minutes. Top right: after 14 minutes. Bottom left: after 24 minutes. Bottom right: after 34 minutes Fig. 5. Six parameters model, duty cycle W in = 1%. Torque versus velocity plot, estimated with least square (blue line) and identified with the proposed model (red line) friction torque. Top left: results after 4 minutes. Top right: after 14 minutes. Bottom left: after 24 minutes. Bottom right: after 34 minutes. Temperature [ ] Win = 25% Win = 5% 25 Win = 75% Win = 1% time [min] Fig. 6. Estimated joint temperature for different robot operations, that is, for different input thermal powers injected into the joint. 3528

6 minutes, never stopping the robot, with velocities equal to 1%, 25%, 4%, 55%, 7%, 85% and 1% of the maximum velocity in each cycle. The results predicted by calculating the friction torque with the method in Section III have then been compared with those obtained by applying the four and the six parameters models estimated in the previous section. Results are shown in Figures 7 and 8 for the four and six parameters models, where the plots are related to the comparison at different time intervals every 25 cycles, that is, after 3, 78, 153, and 228 minutes. These results confirm again the effectiveness of the proposed models and, in particular, of the six parameters model which provides better results Fig. 7. Validation of the four parameters model. Estimated with least square (blue line) and identified with the proposed model (red line) friction torque. Top left: after 3 minutes. Top right: after 78 minutes. Bottom left: after 153 minutes. Bottom right: after 228 minutes Fig. 8. Validation of the six parameters model. Estimated with least square (blue line) and identified with the proposed model (red line) friction torque. Top left: after 3 minutes. Top right: after 78 minutes. Bottom left: after 153 minutes. Bottom right: after 228 minutes. VII. ONLUSIONS In this paper we have proposed a friction model for industrial robot manipulators that takes into account joint temperature variation due to thermal power injected into the joint by friction itself. Two models have been proposed and their effectiveness has been shown. The main feature of the models is that they do not require any internal measurement of the joint. In fact, once an identification campaign has been performed on a given manipulator, only environmental temperature measurement is needed to obtain the friction torque for a given motion. Future work will consist in exploiting the devised model in order to improve the motion control of the robot and to optimize estimation of internal temperature of the joints when operating in environmental conditions different from those used in initial calibration. AKNOWLEDGMENTS The work was supported in part by HAF - Hybrid Aluminium Forging (grant , Joint Funded Project ARIPLO Foundation and Lombardy Region). REFERENES [1] S. Andersson, A. Söderberg, and S. Björklund, Friction models for sliding dry, boundary and mixed lubricated contacts, Tribology International, vol. 4, no. 4, pp , 7. [2] H. Olsson, K. J. Åström,.. de Wit, M. Gäfvert, and P. Lischinsky, Friction models and friction compensation, European Journal of ontrol, vol. 4, pp , [3] V. van Geffen, A study of friction models and friction compensation, Technische Universiteit Eindhoven, Department Mechanical Engineering, Dynamics and ontrol Technology Group, Eindhoven (NL), Tech. Rep. DT 9.118, 9. [4] A. Visioli and G. 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