Frequency domain identification of dynamic friction model parameters ...

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 2, MARCH 2002

191

Frequency Domain Identification of Dynamic Friction Model Parameters Ron H. A. Hensen, Marinus (René) J. G. van de Molengraft, and Maarten Steinbuch, Member, IEEE

Abstract—This paper presents a frequency domain identification of dynamic model parameters for frictional presliding behavior. The identification procedure for the dynamic model parameters, i.e., 1) the stiffness and 2) the damping of the presliding phenomenon, is reduced from performing several dedicated experiments to one experiment where the system is excited with random noise and the frequency response function (FRF) of the phenomenon is measured. Time domain validation experiments on a servomechanism show accurate estimates of the dynamic model parameters for the linearized presliding behavior. Index Terms—Frequency domain analysis, friction, identification, linear approximation.

I. INTRODUCTION

O

VER THE past decade the use of dynamic friction models has grown immensely [2], [9], [10]. The LuGre model [2], that is closely related to the work of [11], [10], is a commonly used model for friction 1) compensation [2], [14], [15]; 2) simulation [5], [10]; and 3) observer design [6], [16]. The strength of the dynamic LuGre friction model is the ability to describe a large number of practically observed friction phenomena; for references of these phenomena see [1]. One of the interesting observed frictional properties is the presliding displacement [3], i.e., spring-like behavior for near zero relative velocity (stiction). Here, we are interested in the identification of this phenomenon, which is also described by the dynamic LuGre model. The identification of the LuGre model is described in [12], [5]. The idea is to estimate the model parameters in different friction regimes, i.e., 1) in the sliding phase and 2) in the stick phase, by performing appropriate experiments in each regime. The estimation of the sliding parameters in the Stribeck friction curve can be done by various techniques, e.g., a least-squares method [12] or extended Kalman filtering [7]. However, the identification of the presliding phenomenon is far from trivial and performing suitable experiments is time-consuming as discussed in [12]. Furthermore, the dynamic parameter corresponding to the damping of the elastic bristles is often given a value instead of being estimated such that a well-damped behavior is obtained for zero velocity crossing [13]. Another drawback of the proposed identification procedures [12], [13] is the need for measurement or reconstruction of the relative Manuscript received April 19, 2000; revised December 22, 2000. Manuscript received in final form April 10, 2001. Recommended by Associate Editor K. Kozlowski. The authors are with the Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology (EUT), Eindhoven, 5600 MB, The Netherlands. Publisher Item Identifier S 1063-6536(02)00084-2.

Fig. 1. The friction interface between two surfaces is thought of as a contact between bristles, where the bristles on one surface are shown as being rigid.

velocity. Here, the second-order description of the linearized LuGre model in the stiction regime will be used to perform a frequency domain identification of the dynamic parameters. The advantage of this technique is that the necessary measurements are solely the sampled system position and input. Thus time-consuming experiments are replaced by a single experiment where the system is excited with random noise and the frequency response function (FRF) of the system is measured. Moreover, both the stiffness and damping of the presliding behavior can be estimated from the measured FRF. To perform this technique a high-resolution encoder is used to observe the presliding behavior. Furthermore, a comment on the linearization of the LuGre friction model in the stick phase will be given and the notion of generalized differentials [8] will be addressed to obtain the linearization. The outline of this paper is as follows. In Section II, we will give a short description of the dynamic LuGre model, the linearization and the presliding phenomenon. The experimental setup used for the presliding measurements will be described in Section III. The frequency domain identification and time domain validation will be discussed in Section IV. In Section V the paper will be concluded and further research topics will be addressed. II. THE LUGRE MODEL AND PRESLIDING BEHAVIOR In the LuGre model, the friction force during stiction is modeled as the average force applied by a set of elastic springs under a tangential microscopic displacement. An interpretation of these elastic springs can be given under the assumption that the two moving surfaces are in contact by a large number of bristles with a certain stiffness [10], which can be represented as depicted in Fig. 1. To incorporate this phenomenon in a continuous friction model extra dynamics describing the average bristle displacement is needed. Hence, extra model parameters, i.e., bristle stiffness and damping, are introduced to model these dynamics. The LuGre friction model implements it as

1063-6536/02$17.00 © 2002 IEEE

(1) (2)

192


TABLE I PARAMETERS VALUES USED IN SIMULATION

The vector field for this simple mass system subjected to the LuGre friction model reads (a)

where , and as in the LuGre friction description given above, the applied force and the Stribeck is defined as curve

(b) Fig. 2. Presliding behavior of dynamic LuGre friction model. (a) Friction force F against presliding displacement x . (b) Time response of the presliding behavior.

where is the tangential friction force; the average bristle deflection; the relative velocity between the two surfaces; the Stribeck curve for steady-state velocities; the bristle stiffness; the bristle damping. the viscous damping-coefficient and the dynamic model parameters; In [2], the presliding behavior of the model is investigated and it is concluded that the phenomenon qualitatively describes the experimentally observed results in [3]. The model lacks the plastic deformation property that is hard to capture in one model describing both the sliding phase and stick phase. A simulated presliding displacement of the LuGre model is shown in Fig. 2(a) and (b) for model parameters as given in Table I. A unit mass subjected to friction is considered where an external force —slowly ramped up to 95% of the static friction —is applied with an initial state of the system equal . Then the force is decreased slowly to to zero the negative counterpart of the maximal applied force, i.e., [N], and this cycle is repeated.

with the static friction, the Coulomb friction and the Stribeck velocity. An interesting point in the stress-strain curve of Fig. 2(a) is . In this initial state, where Fig. 2(b) shows the initial state and the time responses of the states, the time derivatives of are visibly equal. Furthermore, the state after one cycle of is again zero and illustrates the lack of the ability to model plastic deformations. Due to the nonsmoothness of the LuGre model, i.e., the presence of an absolute-value operator on the relative velocity in (2), the derivation of the linearization for the initial state needs special attention. Obviously, the right-hand side of (2) is not ab. However, it possesses a left and solutely differentiable at right derivative defined as

The subdifferentials of Clarke [8], also called generalized differentials, can be used in this case and states that the generalized derivative of vectorfield at state is declared as any value included between its left and right derivative. The closed convex hull of the derivative extremes is called the generalized differential of at co

Now, the generalized differential of the vector field with respect to can be regarded as the generalized Jacobian in the sense of Clarke

HENSEN et al.: FREQUENCY DOMAIN IDENTIFICATION OF DYNAMIC FRICTION MODEL PARAMETERS

193

where and are, respectively, the left and right Jacobian matrices. The generalized Jacobian matrix in the zero state becomes

which is independent of , i.e., no convex combination of left and right Jacobian matrices. This is due to the fact that for this special situation the left and right Jacobian are equal. However, it should be emphasized that this is not always the case for nonsmooth differential equations and then the notion of subdifferentials is essential. Returning to the linearized system, the oband in Fig. 2(b) served initially equal time derivatives of . Now the substiis mathematically shown since is possible and the linearization of the unit tution of mass subjected to LuGre friction for zero state reads

Fig. 3. Experimental setup. TABLE II SPECIFICATION OF THE APPARATUS

(3) (4) represents the stiffness and where the linearized system.

the damping of

III. EXPERIMENTAL SETUP A rotating arm system consisting of 1) an induction motor; 2) a low backlash planetary transmission; and 3) a rotating arm will be considered here. Due to the bearings and seals in the motor and transmission, the inertia of the total system, i.e., the combined inertia of the separate elements, is subjected to friction. The angular displacement of the system is measured with a high-resolution encoder that produces two sinusoidal signals as output. These two 90 in phase-shifted signals, i.e., an analog sine and an analog cosine, are interpolated and digitized into two 90 in phase-shifted square-wave pulse trains, i.e., two TTL signals. The interpolation increments degree is set to 40, which results in a resolution of per revolution of the motor shaft. Due to a gear ratio of the transmission of 8.192 the resolution of the angular arm displacement increments per revolution. The measurements becomes induction motor is supplied by a pulse width modulation source inverter which translates the input signal, i.e., the desired torque expressed in a voltage, intothreephase signals with a fundamental frequency. This source inverter actually controls the torque produced by the motor to the desired torque. The input signal of the source inverter and the TTL encoder signals are, respectively, sent and read by a dSPACE system [4]. During the experiments the sample frequency of the dSPACE system is set to 10 [kHz]. To perform on-line frequency domain measurements, the input signal and the angular displacement are processed by a SigLab system [18]. A schematic representation of the setup is given in Fig. 3 and the specifications of the separate experimental elements are given in Table II. IV. FREQUENCY DOMAIN IDENTIFICATION AND TIME DOMAIN VALIDATION The system under consideration can be modeled as (5)

where is the effective inertia of the motor-transmission-rotating arm combination; the angular displacement; the input voltage; the motor constant, which is known and equal to 16 [Nm/V]; the friction torque that is modeled by the dynamic LuGre model (1) and (2). Linearizing (5) as presented in Section II for the stiction regime and zero state the linearized system in frequency domain reads (6) , the To estimate this frequency response function (FRF) system is excited with a PRBS signal of a bandwidth up to 500 [Hz] and a root mean square (rms) level below the static fricis obtained by averaging 50 tion . The measured FRF time series of 8192 samples at a sample frequency of 10 [kHz] with a Hanning window and 50% overlap. Since the linearization (3) and (4) is only locally valid, the nonlinear behavior is investigated on by varying the RMS level of the noise within the static friction of the system. In Fig. 2(a) the dynamic parameter is depicted as the stiffness of the friction torque for small variations of the presliding displacement under small variations of the applied torque . Hence, the noise level used during the experiment should be very small to obtain valid measurements for the linearized model. A noise level up to the static friction will give an “averaged” stiffness

194


(a) Fig. 5.

(b) Fig. 4. Frequency response functions of presliding behavior. (a) Stiffness nonlinearity. (b) Measured and estimated FRF.

STIFFNESS

TABLE III DIFFERENT NOISE LEVELS

FOR

lower than the real stiffness. In Fig. 4(a), this nonlinear behavior is shown for increasing noise levels. The magnitude of the various FRFs for frequencies approaching zero represent the gain of the presliding behavior. In Table III, the rough estimates of the stiffness for the different noise levels are decreases, i.e., ingiven. As expected, the stiffness creases, as the level of excitation increases. from the measured To estimate a second-order FRF with the smallest noise level, an iterative procedure of convex combination steps similar to the SK-iteration of Sanathanan and Koerner [17] is used. Inspection of the measured FRF shows high-order behavior for frequencies above 150 [Hz]. Since it is not our goal to identify this behavior, the focus will be on the second —order dynamics up to 150 [Hz]. and estimated are depicted in The measured

Break-away experiment.

Fig. 4(b). The estimated dynamic LuGre parameters become [Nm/rad] and 2) [Nms/rad] 1) [kgm /rad]. Here, the and the estimated inertia parameters have to be considered as the lumped compliance and impedance of the total system rather than the stiffness and damping of the friction alone. Hence, in the sequel of the paper the parameters will be addressed as the dynamic model parameters for the frictional presliding behavior. The obtained dynamic model parameters are validated by two time domain experiments, i.e., 1) a break-away experiment and 2) sinusoidal excitation of the system in the stick regime. First a ramp input is used to perform a break-away experiment, where . The measured and the voltage applied is given by simulated responses are depicted in Fig. 5, where in the upper part the macroscopic displacement is given and in the lower part the microscopic or presliding displacement. From the lower part can be concluded that the estimated dynamic model parameters and the LuGre model are valid for the linearization described in Section II, since the slope of the measured response and the LuGre model response are equal for small input torques. In comparison to the LuGre model shows the system response larger presliding displacements for larger input torques. The reason is yet unclear but might be explained by: 1) plastic deformation or creep of the system is not incorporated in the LuGre model and 2) the LuGre model structure is too limited to describe the presliding behavior accurately, e.g., the stiffness might be a nonlinear function in or . On the other hand the macroscopic differences are very small. To investigate the obtained dynamic model parameters and the LuGre model for the presliding behavior more extensively, the system is excited with two sinusoidal input torques. In Fig. 6(a), the applied torque has a frequency of 20 [Hz] and an amplitude of 0.01 [V] that is equal to the noise level used for the identification procedure. Again the estimated dynamic model parameters and the LuGre model are able to predict the presliding behavior very accurately. In Fig. 6(b), the results of a second sinusoidal experi[Hz] and an ampliment is shown with a frequency of tude of 0.035 [V], i.e., 90% of the static Friction . For these larger input torques the system response gives again larger microscopic displacement than the LuGre model. The reason for

HENSEN et al.: FREQUENCY DOMAIN IDENTIFICATION OF DYNAMIC FRICTION MODEL PARAMETERS

195

Time domain validation experiments show accurate estimation of the dynamic model parameters for the linearized presliding behavior, i.e., locally valid around zero presliding displacement and zero applied force. However, the applicability of the dynamic LuGre friction model for the description of the entire presliding phenomenon, i.e., the total stick regime for an applied force lower than the static friction , is limited. Extensive evaluation of the presliding behavior will be a topic of future research as well as the modification of the LuGre model to obtain a more accurate dynamic friction model for the description of the presliding phenomenon. REFERENCES

(a)

(b) Fig. 6. Sinusoidal validation. (a) Sinusoidal input torque u = 0:01 sin(40t). (b) Sinusoidal input voltage u = 0:035 sin(t).

this difference in the presliding behavior can be sought in the arguments given above. V. CONCLUSION This paper demonstrates a frequency domain identification technique for the dynamic model parameters in the frictional presliding behavior. To perform this frequency domain identification the dynamic LuGre friction model is linearized to obtain a linear second-order description that is locally valid in the stick regime. The identification procedure is reduced from performing several dedicated experiments to one experiment where the system is excited with random noise and the FRF of the system is measured. The measured FRF is used to estimate both and 2) the dynamic model parameters, i.e., 1) the stiffness . Excitation of the presliding behavior outside damping the linearization shows a nonlinear behavior that describes a decreasing stiffness for an increasing applied torque. This phenomenon is also incorporated in the dynamic LuGre friction model.

[1] B. Armstrong-Hélouvry, P. Dupont, and C. Canudas de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, pp. 1083–1138, 1994. [2] C. Canudas de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, pp. 419–425, 1995. [3] J. Courtney-Pratt and E. Eisner, “The effect of a tangential force on the contact of metallic bodies,” in Proc. Roy. Soc., vol. A238, 1957, pp. 529–550. [4] E. dSPACE digital signal processing and control engineering GmbH, “DS1102 user’s guide,”, Paderborn, Germany. [5] M. Gäfvert, “Comparisons of two dynamic friction models,” in Proc. 6th IEEE Conf. Contr. Applicat., Hartford, CT, 1997. [6] H. Olsson and K. J. Åström, “Observer-based friction compensation,” in Proc. 35th Conf. Decision Contr., vol. 4, Kobe, Dec. 1996, pp. 4345–4350. [7] R. H. A. Hensen and G. Z. Angelis, et al., “Grey-box modeling of friction: An experimental case-study,” Europ. J. Contr., vol. 6, no. 3, pp. 258–267, 2000. [8] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, “Nonsmooth analysis and control theory,” in Graduate Texts in Mathematics. New York: Springer-Verlag, 1998. [9] P.-A. Bliman and M. Sorine, “Easy-to-use realistic dry friction models for automatic control,” in Proc. 3rd Europ. Contr. Conf., Rome, Italy, 1995, pp. 3788–3794. [10] D. A. Haessig, Jr. and B. Friedland, “On the modeling and simulation of friction,” J. Dyn. Syst., Measurement, Contr., vol. 113, no. 3, pp. 354–362, Sept. 1991. [11] P. Dahl, “A solid friction model,” Aerospace Corp., El Segundo, CA, Tech. Rep. TOR-0158(3107-18)-1, 1968. [12] C. Canudas de Wit and P. Lischinsky, “Adaptive friction compensation with partially known dynamic friction model,” Int. J. Adaptive Contr. Signal Processing, vol. 11, pp. 65–80, 1997. [13] M. Gäfvert, “Comparison of two friction models,” Master’s thesis, LUTFD2/TFRT-5561-SE, Lund Inst. Technol., Lund, Sweden, 1996. [14] C. Canudas de Wit, “Control of friction-driven systems,” in Proc. Europ. Contr. Conf., Sept. 1999. [15] M. Gäfvert, J. Svensson, and K. J. Åström, “Friction and friction compensation in the Furuta pendulum,” in Proc. Europ. Contr. Conf., Sept. 1999. [16] R. Nitsche and L. Gaul, “Vibration control using semiactive friction damping,” in Proc. Europ. Contr. Conf., Sept. 1999. [17] R. de Callafon, “Feedback oriented identification for enhanced and robust control,” Ph.D. dissertation, Delft Univ. Press, Oct. 1998. [18] SigLab Model 20-42, DSP Technology Inc., “User guide,”, Fremont, CA.

Ron H. A. Hensen received the M.Sc. degree in mechanical engineering from Eindhoven University of Technology (EUT), Eindhoven, The Netherlands, in 1997. Currently, he is pursuing the Ph.D. degree in the Control Systems Technology Group at the EUT. The project includes the modeling of friction in mechanical systems and the prediction of friction induced limit cycles.

196


Marinus (René) J. G. van de Molengraft was born in Eindhoven, The Netherlands, in 1963. He received the M.Sc. degree (cum laude) in mechanical engineering in 1986 at the Eindhoven University of Technology (EUT). He received the Ph.D. degree in 1990 with a thesis on the identification of nonlinear mechanical systems. Since 1991 he has been an Assistant Professor in the Control Systems Technology Group at the Mechanical Engineering department of the EUT. His main interests at the moment are control of high-performance motion systems and gray-box modeling of nonlinear systems.

Maarten Steinbuch (S’83–M’89) received the M.Sc. degree (cum laude) in mechanical engineering from Delft University of Technology, Delft, The Netherlands, in 1984. In 1989, he received the Ph.D. degree from Delft University of Technology on the subject of modeling and control of wind energy conversion systems. From 1984 to 1987, he was a Research Assistant at Delft University of Technology and KEMA (Power Industry Research Institute), Arnhem, The Netherlands. From 1987 to 1998, he was with Philips Research Labs., Eindhoven as a Member of the Scientific Staff, working on modeling and control of mechatronic applications. From 1998 to 1999, he was a Manager of the Dynamics and Control Group (18 M.Sc. and Ph.D. engineers) at Philips Center for Manufacturing Technology. Since 1999, he has been Full Professor of the Control Systems Technology Group at the Mechanical Engineering Department of Eindhoven University of Technology. He has more than 70 refereed journal and conference publications, and holds two patents. His research interests include modeling and control of motion systems. Dr. Steinbuch was an Associate Editor of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY from 1993 to 1997 and of IFAC Control Engineering Practice from 1994 to 1996. He is currently Associate Editor of IEEE CONTROL SYSTEMS MAGAZINE, of Journal A (Belgium) and Editor-at-Large of the European Journal of Control.