Tracking Control of an Ultrasonic Linear Motor Actuated Stage Using a Sliding-mode Controller with Friction Compensation

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1 Vol. 3, No., pp. 3-39() Tracking Control of an Ultrasonic Linear Motor Actuated Stage Using a Sliding-mode Controller with Friction Compensation Chih-Jer Lin,*, Ming-Jia Li and Kai-Ren Liu Graduate Institute of Automation Technology, National Taipei University of Technology, Taipei, Taiwan, ROC * Corresponding Author / cjlin@ntut.edu.tw KEYWORDS : Ultrasonic motor(usm), Particle swarm optimization(pso), Charge search system(css), Feed forward controller, High-gain observer, Disturbance observer, Sliding mode control, Back-propagation neural network controller The purpose of this paper is to investigate the nonlinear friction and compensation for a piezoelectric ceramic ultrasonic motor (USM). Although the architecture of the USM is different from the general electric-mechanical motor, the mathematic model for the USM motor can use the same friction model to formulate the friction phenomenon. To establish the feedforward controller, the system identification for the USM is needed to study to design the model-based controller. To obtain the optimal system parameters of the USM, PSO and CSS algorithms are studied to identify the system parameters for the nonlinear friction model. For the controller design, a non-model based controller, using back-propagation neural network controller to perform tracking tasks, and the model-based controller, which consists of the feed-forward controller based on the system identification and the sliding-mode control, are discussed in this paper. Finally, the two real-time tracking tasks are used to validate the proposed method. Manuscript received: April 8, / Accepted: May,. Introduction. Modeling of the USM system The movement principle of an USM relies on the piezoelectric effect. Curie brothers found the crystal piezoelectric effect in 88. Since then, the piezoelectric material field has been developed. An USM has also been widely used in our daily applications, e.g. the camera lenses, micro optical stage or the medical engineering. Also, the USM is currently a micro-moving component being aggressively researched and developed in the industries. In general, the sound frequency range of ~ khz can be sensed by human ears. An ultrasonic wave is a vibration with vibration frequency of more than khz. Therefore, the driving force of an USM is obtained by way of ultrasonic vibration and the stage s slider is driven by using the friction []. The USM has become a product in more demands in the last decade and many different types of USM have been developed. However, its motion mode is similar with each other without considering the mechanism design and the movements are based on the longitudinal mode or the lateral mode. After these two modes are combined, an elliptical motion appears between the stator and the rotor. Although the moving distance is not large per step, the speed of the moving stage depends on the vibration frequency. However, the heat due to the friction effect may cause a problem for the precise positioning of the stage. In this study, a USM stage is developed as shown in Fig. and the friction compensation is studied in this paper. Fig. shows the experimental positioning stage which consists of a single-axis stage and a piezoelectric ceramic ultrasonic motor (USM). As the ceramic structure is twisted from deformation of the electrode through the electric field, the deformation makes the rotor push the stage and actuate the stage to move laterally. For the driving control architecture, a motor driver for the USM is set up in velocity mode. In addition, the moving stage is equipped with an optical encoder and a linear scale to feedback the displacement of the stage. The optical scale has the resolution of.µm in order to meet the requirement of micrometer or submicron. For the controller design, a real-time control system (dspace DS) is used to realize the proposed control. Fig. Schematics of the experimental stage SoildWorks 3

2 Vol. 3, No., pp. 3-39() Consider the simplified mathematical model for the USM and Fig. shows the equivalent free-body diagram, where m s is the stage mass; x s is the differential of the stage displacement, i.e. the stage moving speed; f f is the friction occurs between the slides of the sliding stages; f t is the tangential force caused by the stator. As the voltage is applied to the actuator, the stator of the USM is deformed. If the tangential force f t which pushes the sliding stage is less than the friction f f, the sliding stage will not move. Otherwise, if the tangential force f t is larger than the friction, the stator will make the stage move as the friction f f is overcome. To compensate the friction effect, we must measure the friction model for designing the feed-forward controller Fig. Stick slip response of the stage s velocity Fig. The actuation principle of the USM 3. Parameter identification for the friction model with algorithms 3. Measurement of system friction To measure the nonlinear friction phenomenon, a dynamic friction measurement experiment is used to investigate the stick-slip effect which usually appears at the zero-speed crossover point []. The physical phenomenon of dynamic friction is described by the relationship between the input force (voltage) and the stage s velocity. In this study, a sinusoid voltage with amplitude of (V) at the frequency of (HZ) is shown in Fig. 3. The measurement result of the stage s velocity is shown in Fig.. Figs. 3 and are used to identify the system parameters in the next section. 3. System parameter identification using the CSS and PSO To identify the system parameters, two algorithms are studied to improve the modeling precision. One is the charge search system (CSS) [3] and the other one is the particle swarm optimization (PSO) []. In this study, we integrated the CSS with the PSO algorithm and combined the advantages of each algorithm. First, the CSS is used in the first iterations and then the PSO takes over the work after the th iteration. Fig. shows the comparison of the simulation result using the obtained system parameter with the actual velocity response. The results in Fig. show that the identified model is very similar to the actual model. - - sim exp Fig. Comparison of the simulation with the actual velocity response 3 voltage(v) Fig. 3 Sinusoid input with the amplitude of (V) at Hz. Controllers design. Controller architecture In this paper, we implemented the controllers including the controller, the high-gain observer, the feed-forward controller and the disturbance observer []. The controller was used to improve the system stability in the position loop and the transient response. The high-gain observer was used to solve the discontinuous problem of the velocity which is obtained from the displacement signal. The feed-forward controller was used to compensate for the nonlinear due 36

3 Vol. 3, No., pp. 3-39() to the friction. The disturbance observer was responsible for handling the modeling uncertainties and external disturbance... High-gain observer For the tracking control problem of the mechanical system, a positioning sensor is usually used for the feedback control. However, the velocity signal is obtained using the difference relation between the position signal and the sampling time. As the noise of the sensor system appears, there is a discontinuous problem for the velocity computation. Therefore, a high-gain observer is used to estimate the velocity of the mechanical system to deal the above problem. For the mechanical system in this paper, the input force is denoted as Tt () and the motion equation can be described as follows. My t Tt T yt () * () () f ( ()) x x x M ( T) () n n where M R n is the system inertia matrix; T R is the system * n control torque; Tf ( y ( t)) R is the nonlinear friction term; x y[ y y ] T n is the position of the mechanical system; x y [ y y ] T n is the velocity. If the velocity x is unknown, the estimated velocity can be obtained using the following Equations (3) and () []: performance of the feed-forward compensation depends on the accuracy of the friction model. If the estimated friction term ˆF is not the same as the actual friction F (shown in Fig. 6), it will lead to some problem. Therefore, the disturbance observer is introduced to deal with this problem. Fig. 6 Schematics of feed-forward compensation..3 Disturbance observer As the above mentioned, the disturbance observer is used to deal with the mismatched term of the system modelling. The block diagram of the disturbance observer is shown in Fig. 7. x ˆ xˆ ( ˆ ) Hp xx xˆ H ( ˆ ) v xx (3) () n n, where xr xr is the estimated values for x ; x, is a small positive scalar; H p H v are the positive definite matrice, and H H p I v is a Hurwitz matrix. Define the error of the observor is described as follows. x x xˆ, x x xˆ, z x, z x () Fig. 7 Block diagram of the disturbance observer For convenience of explanation, let us first assume that Q(s) =, and we can derive the following: P n ˆ ˆ d ( ) u d (8) P P The observer error equations can be determined from (3), () and () and we have: z z Hpz (6) z Hvz M ( T) (7).. Feed-forward controller In the tracking experiments, the nonlinear friction makes the moving stage have stick-slip phenomena at the zero-speed crossover point as shown in Fig.. Therefore, the tracking error cannot be dismissed completely by the feedback controller. Therefore, a feedforward compensation is necessary for this case study. As discussed in the above, the LuGre friction model is identified in Section 3. Therefore, the feed-forward compensation can be designed using the LuGre model with the identified parameters. However, the where P is the actual model; P n is the nominal model; d is the external disturbance; ˆd is the observed disturbance; is the measurement noise. From the equation, we can find that as Pn P, the item ˆd obtained from the observer is the sum of disturbance and noise, which can be used to effectively estimate the influence of the actual disturbance and the noise. Conversely, if the item P n is not approximate to the system transfer function, there exists a difference between the estimated disturbance and noise and actual ones.. Sliding mode control The sliding mode control is divided into two stages. As any initial state does not contract with the sliding surface, this case is called the reaching mode. After the state locates at the sliding surface, it is called the sliding mode [6]. Ideally, the time for the reaching mode may influence the robustness of the control system. After the state 37

4 Vol. 3, No., pp. 3-39() falls in the sliding surface, the robustness of the system is guaranteed, because the state does not leave the sliding surface and it will be attracted to the origin; this condition is called the sliding-mode condition. To make the state arrive at the sliding surface as soon as possible, a switching control should be applied to make the state fall into the sliding surface and the equivalent control is used to make the state move to the origin along with the sliding surface..3 Integral type sliding mode control Define a new sliding-mode surface is defined as follows [7]: s e ce c edt (9) On the other hand, we apply the simulated controller in the experiment and Fig. 9 shows the comparison of the actual control voltage with the simulated control output. From Fig. 9, the experimental results validate that the simulation is correct. voltage(v) 6 3 sim exp - Consider the estimated friction as f according to the LuGre friction model, the modeling uncertainty is defined as follows. f f f d where f d is the actual friction. () As the same as the design for the traditional sliding mode controller, the equivalent control can be obtained as Eq. (). Therefore, the total control is described as Eq. (). ueq f m( xd ce ce) () uf m( x ce c e) wsgn( s) () d Result and discussion. Simulation and validation Before the feedback controllers are implemented to the USM system, the feed-forward compensation is discussed firstly. Fig. 8 shows the comparison of the open-loop response with the experimental results using the proposed feed-forward compensation. Although the compensated result is different from the reference, the stick slip phenomena is disappear Fig. 9 Comparison of the simulation with the actual input voltage. Experimental results After the discussion of the feed-forward compensation, the six proposed controllers are implemented to the USM system. Table shows the tracking root-mean-square error (RMSE) for these six controllers. From the experimental results in Table, the proposed ISMC+FF+HGO+DOB has the best performance of the tracking task for the USM system. Fig. shows the compassion between the six different controllers and Fig. shows the differences between the tradition sliding-mode controller and the integral sliding-mode controller. From the results of Fig., the proposed ISMC+FF+HGO+DOB controller has the smallest tracking error among these six different controllers. Fig. shows that the tracking error of the integral sliding-mode controller is smaller than the one of the traditional sliding-mode controller Table Comparison of each controller (Unit: µm) Controller Frequency Travel +FF +FF+ HGO mm FF Open loop Hz mm mm mm Controller +FF+ SMC+FF+ ISMC+FF+HGO+DOB Frequency Travel HGO+DOB HGO+DOB mm Fig. 8 Feed-forward compensation for the USM stage Hz mm mm.8...mm

5 Vol. 3, No., pp. 3-39() Displacement(mm) 6 - Ref +FF +FF+HGO +FF+HGO+DOB VSS+FF+HGO+DOB In this paper, to compensate the nonlinear friction of the USM, the LuGue model is used to describe the stick-slip friction phenomena. Then, the optimization algorithm is used to identify the system friction parameter; the CSS+PSO algorithm shows that the identified model is very similar to the actual model from Fig.. For the control architecture, a controller is used to guarantee the system stability and ensure that the system has good transient state error. Then, a feedforward controller is used to compensate for the system friction and the high-gain observer is used to solve the discontinuity of the velocity estimation. The disturbance observer is used to overcome the external disturbance to achieve the precision tracking. The experimental shows that the proposed ISMC+FF+HGO+DOB controller has the best performance. REFERENCES Displacement(mm) error(mm) Ref +FF +FF+HGO Fig. The tracking response of each controller 6 x -3 +FF+HGO+DOB VSS+FF+HGO+DOB SMC ISMC Fig. Tracking error comparison between the SMC and ISMC [] M. Umeda, T. Nakazawa, K. Ohnishi, K. Nakamura, M. Kurosawa, and S. Ueha, Positioning characteristic of ultrasonic rotary actuator with two mode operation, Proc. IEEE Ultrasonics Symposium, pp., December (99) DOI:.9/ULTSYM.99.7 [] D. Karnopp, Computer simulation of slip-stick friction in mechanical dynamic systems ASME Journal of Dynamic Systems, Measurement and Control, 7, -3 (98) DOI:./.3698 [3] A. Kaveh, S. Talatahari, A novel heuristic optimization method: charged system search Acta Mechanica, 3, () DOI:.7/s [] J. Kennedy, R. Eberhart, Particle swarm optimization, Proceedings of the IEEE, International Conference on Neural Networks Perth (ICNN 9), pp.9 98, (99) DOI:.9/ICNN [] J. A. Heredia, W. Yu, A high-gain observer-based PD control for robot manipulator, Proceedings of the American Control Conference (ACC ), IEEE Press, pp.8-, June () DOI:.9/ACC [6] J. J. E. Slotine, W. Li, Applied nonlinear control (Prentice-Hall, New Jersey, New Jersey 99) [7] H. C. Liaw, B. Shirinzadeh, Constrained motion tracking control of piezo-actuated flexure-based four-bar mechanisms for Micro/Nano manipulation IEEE Transactions on Automation Science and Engeering, 7,699-7 () DOI:.9/TASE.9.36 [8] B. Armstrong-Hélouvry, P. Dupont, and C. C. De Wit, A survey of models, analysis tools and compensation methods for the control of machines with friction Automatica, 3, (99) DOI:.6/-98(9) Conclusion 39