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# Modeling and Simulation of Nonlinear Friction in XY AC Servo Table

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### Modeling and Simulation of Nonlinear Friction in XY AC Servo Table

1. 1. Modeling and Simulation of Nonlinear Friction in XY AC Servo Table Zhang Tao, Lu Changhou, Xi Zhuyan School of Mechanical Engineering Shandong University Jinan, Shandong 250061, China victor21c@126.com Abstract-The non-linear friction is an important factor affecting the machine-finishing precision. The microscopic structure of friction contact surface is analysed, which is supposed as the viscoelasticity material. The modelling method of static friction is discussed. A partition friction model has been established separately in the pre-sliding regime and dynamic friction region. The friction in XY AC servo table is measured based on the new designed strain sensor, which is used to estimate the related parameter of the model. The compensation effect of two models is compared using the simulation software. The result indicates the tracking errors using the partition friction model proposed in this article is smaller than that of the static Stribeck model, which shows that the new model can reflect the friction characteristic of the static friction stage. Index Terms-Friction Modeling, Compensation, tracking, viscoelasticity I. INTRODUCTION Accurate tracking and precise positioning is needed in precise processing. However, the friction existed in the mechanical system would limit the performance of the controlled system. Friction shows Stribeck effect, hysteresis, spring-like behaviour, and varying breakaway force [1]. Friction also shows different behaviours during velocity reversal, which is called stick-slip. In system consideration, XY table, which is a ball-screw- driven mechanism actuated by AC servo drives, has widely been used in many applications due to its low cost and generality. For ideal linear systems, accurate tracking may be achieved using linear control techniques such as PID control etc. Unfortunately, the nonlinear characteristic introduced by friction has a significant influence on the performance. So these nonlinear disturbances in XY AC servo table have to be compensated in order to achieve effective tracking and accurate positioning. In control applications, compensation for friction is the paradigm to achieve high-precision performance [2]. Lots of works have been done to investigate friction dynamics and different methods have been proposed [3,4,5,6,7,8]. With the rapid development of computer technology, friction compensation based on model is more feasible to improve control precision. In the past years, large numbers of friction models have been researched, and some of them have been used successful in practical applications. Good friction models can explain nonlinear friction phenomena, which have been studied by many researchers. These friction models, which include Coulomb model, Karnopp model [9], LuGre model [10], Leuven model [11,12], etc., consist of different components such as Coulomb, viscous, static, Stribeck friction, friction lag and stick-slip motion, each of which considers a certain aspect of friction. All the models presented in these researches illustrate the importance of choosing an appropriate friction model to predict the system performance. The purpose of this paper is to propose a new model that includes all the various aspects in static friction regime and dynamic friction regime. Part I introduces the friction problem in general and the importance of appropriate model. Part II analyses the character of friction and presents a new friction model. The experiment equipment is introduced in part III. Part IV gives the parameter estimate of the friction model. The simulation of friction compensation is presented in part V. A conclusion is presented in Part VI. II. MODELING OF FRICTION Conventionally, friction can be divided into two regimes: static friction and dynamic friction. If we consider two objects in frictional contact, then there will always be a displacement resulting from an applied force, unless the contact is infinitely stiff. Now, below a certain force threshold, if the force is held constant, the displacement will likewise remain constant. When the force is decreased to zero, not all displacement will be recovered, i.e., that there will, in general, be a residual displacement. This is the static friction or pre-sliding regime, in which, although there is relative motion, there are still points of unbroken contact and points of micro-slip on the two surfaces of the objects resulting in hysteresis of the force in the displacement that marks the frictional behaviour in that regime. Above that force threshold, the system will be critically stable; displacement will not remain constant for a constant applied force: the object will suddenly accelerate; all connections are broken, and we have true or gross sliding. The term “friction force” is usually taken to mean ‘‘the resistance to the motion during true sliding; it usually has its maximum value at the commencement of motion and usually decreases with increasing relative velocity [13]. It has been observed that the force is predominantly a function of the displacement in the pre-sliding regime showing quasi rate-independent hysteresis with nonlocal memory [14] and predominantly a function of the velocity in the true sliding regime, showing as the Stribeck curve [15]. Many other investigators [16,17,18] have also found similar results for metal-to-metal contacts. 1-4244-0466-5/06/\$20.00 ©2006 IEEE 618 Proceedings of the 2006 IEEE International Conference on Mechatronics and Automation June 25 - 28, 2006, Luoyang, China
2. 2. The borderline distinguishing the two regimes is the “pre- sliding distance” and the “breakaway force” or the “static friction force” threshold. The friction is modelled separately in pre-sliding regime and sliding regime because of the different characters in two regimes. A. Modeling of static fiction Generally, there is a certain extent roughness at the contact surface. Borderline lubrication is the main lubrication mode. Fig. 1 shows the borderline lubrication model. During the borderline lubrication, the oil film is very thin; the thickness of film can even decrease to one or two molecule- deep. At the same time, the distance between friction surfaces reduces and the mutual function strengthens between the rough peaks. The friction characteristics are fully decided on the physical chemistry function of the surface film and the contact mechanics of the rough peaks [19]. base borderline lubrication film lubricant oil Fig. 1. Borderline lubrication model Taking into account the borderline lubrication mechanism and properties of static friction, we can analyse the friction contact surface as the viscoelasticity body. So the Kelvin chain model of viscoelasticity body is adopted to describe the static friction model, which is shown in Fig. 2. This model consists of two elements: a nonlinear spring module (k) and a viscous camper (c) [20]. These elements are massless and, hence, do not really exist. They are phenomenological elements. Each element exhibits a special mechanical property and is described by a simple mathematical expression. The combined result then matches the complete pre-sliding behaviour. For convenience, this model is described as a system in linear motion and the following analysis that is based on this model gives the relations between force, mass, linear displacement and velocity [21]. Considering the modules shown in Fig.2, in which F is applied force and x is the elongation of the spring module, at time t, the governing equation is described as follows )()( txktxcF iiii (1) If F(t) is input, we have FtJtx ii )()( (2) where iiii kctktJ /)/exp(1)( principle of superposition and using heredity integral, the final pre-sliding displacement can be represented as follows (3) From the characteristic of linear viscoelasticity body, we know that the contribution of each stage exerted force to the final deformation is independent. According to the Boltzmann K1 C1 F X K2 C2 Kn Cn … nxxx1 2 Fig. 2. Kelvin chain model d d dFt tJtJFt iii 0 )( )()()0()( (4) Equation (4) can be expressed as the convolut rm: x ion fo )(*)()()0()( tdFtJtJFtx iii (5) The total deformation is n i 1 (6) From above equ tween applied force and the pre-sliding displacement can be obtained at time t, in whi ynamic friction regime, the friction force is a namic friction behaviour, the i txtx )()( ations, the relation be ch there are only two kinds of parameters: stiffness coefficient (ki) and viscous coefficient (ci). So if the applied force has been known, it is convenient to identify the parameter from the experiment data with curve fitting technology. B. Modeling of dynamic fiction In the d function of velocity. To describe dy LuGre friction model proposed by Canudas et al. [10] was adopted. At the constant velocity, the general friction force can be written as vvvFFFvs scsc 2/exp)()( (6) Wit he Coulomb friction, the static frich cF t sF tion, s the Stri elbeck v ocity, 1 an shape fact and 2or, viscous fr ion II EXPERIMENTAL SET-UP ict coefficient. I. A. X-Y Table Fig. 3(a) shows the XY AC servo table, in which the riments were performed with a computer con friction expe trolled system. The XY table is connected to ac servo motors through ball-screws with a 5mm pitch. The servo motor is MSMA082AIG permanent synchronous motor of Panasonic, equipped with a MSMA083AIA driver. It generates 10,000 pulses per revolution. The input is given as a velocity command by the computer through a motion controller of Googol Technology Ltd. The computer exchanges information with the controller through a host (a) (b) Fig. 3. Th perimental set-upe ex 619
3. 3. computer com sending motion surement earch on friction compensation and TION munication interface, including commands to the controller and acquiring the present state and control parameters. Current measurement was carried out with a K25 current-type Hall current senor. Its full range is 25A, while one phase rated current of ac motor is 4.3A, so with allowance considered, the lead was wound four loops around the Hall device. The measured current values are read in the computer by a 12-bit A/D converter. When the computer sends out the slope velocity command, servo motor will speed up from rest to the given velocity with the given acceleration. The linear grating resolution was employed to measure displacement of the table dynamically. Actual acceleration measurement was performed with an accelerator of Brül&Kjær, by adjustment; the reading 1V is equal to 8.92857m/s2 . B. Torque Mea At present, most res control for servo table is based on input and output signal. Generally, input signal is the output torque or servo current of motor. For DC motor, it is easy to measure and control current. So the servo current is always treated as the input signal in most research. However, in AC servo system, the angle between excitation magnetic field and armature magnetomotive force is not fixed. The method of vector control and magnetic field direction detection is usually adopted. So it is more difficult to control output torque of motor through adjusting armature current. In this situation, it is a good idea to adopt the output torque of AC servo system as the input signal. To deal with this problem, measurement of motor output torque is the first work to do. To measure the output torque of motor, a newly developed torque sensor based on strain gauge is designed, show as Fig. 3(b). Motor and basement of XY table are connected with a sleeve. Four semiconductor strain gauges with high sensitivities are glued at the sleeve, which compose the Huygens' construction electric bridge. When the table is driven by motor, a torque signal will be produced in the sleeve whose size is equal and direction is opposite to the motor output torque signal. This strain signal may transmit directly through the dynamic strain gauge to computer. The strain voltage signal is obtained through processing. Compared with classical sensor using carbon brush and collecting ring, this torque sensor can obtain torque signal more precisely. The measured torque data is processed to obtain the friction with related dynamical equation. IV. PARAMETERS IDENTIFICA (a) (b) Fig. 4. The measured and estimated curve It is also no rts contribute the city V. SIMULATION OF FRICTION COMPENSATION The oposed mod THE ESTIMATED VALUES OF PARAMETERS ted that, in this system, many pa friction force. They include the friction of ball bearings, the friction between brushes and commutator bars and the friction in the encoders. These frictions can be divided into two categories, rolling friction and sliding friction. However, the former is of an order much smaller than that of the latter and can be neglected. In this research we did not try to distinguish the individual contribution of each friction source but treated them as a combined result as what happens in most machines. Hsieh C. et al.[21] refers that there is no difference between such kind of friction and those frictions in other cases as reported in the literature as long as sliding friction is concerned. The parameter set for sliding regime depends on the choice of )(vS . The parameters are determined based on constant velo tests over the full velocity range of the table. Fig. 4(a) shows the mean measured friction for different constant positive and negative velocities, in which the friction force corresponding to each measurement point is obtained by averaging data samples, measured during five experiments. Table 1 shows the identified parameters for positive and negative velocities. The parameters are identified using a nonlinear least squares identification algorithm in the Matlab optimization toolbox. The full line in Fig. 4(a) shows the estimated Stribeck curve. To estimate the parameters of pre-sliding regime, a sine displacement signal with small amplitude and low frequency is applied to the system. The amplitude of this reference signal is chosen in such a way that the system operates in the pre- sliding regime. The low frequency of signal can guarantee the movement inertia is minute. The displacement of worktable is recorded. At the same time, the current of servo motor is measured. To model the hysteresis curve a static friction model containing six units of nonlinear spring module and viscous camper is used. So the numbers of elements in Kelvin chain are chosen to be 6. Stiffness coefficient (ki) and viscous coefficient (ci) of each element is estimated using Matlab identification tools. The estimated curve which represents the relation of pre-sliding displacement and friction force is shown in Fig. 4(b). The figure shows the hysteresis effect clearly. effect of friction compensation using the pr el is simulated in this section. TABLE I Parameter Positive Negative Fs (N) 240.6770 -290.2720 Fc (N) 152.9965 -200.8617 Vs (m/s) 0.0448 -0.2868 1.9954 2.0561 2 (N·s/m) 803.9993 831.7143 620
4. 4. This friction el discu ve is a n model. The transitio ecides wh part o n model should be use t is neces cert lue of the transition tim research nce [ s that the transition tim rsely p nal to are-root of the acceleration at the start of h has been verified in their experiments is rela used in our simulation. mod n time d ssed abo ich partitio f frictio d. So i sary to as ain the va e. The of refere 22] show e is inve roportio the squ the worktabl . So th e, whic tion is The method of PID-controller and feed forward control is applied to compensate the errors caused by friction. The friction behaviour is compensated using a feed forward control. The real position is fed back and the tracking error is used a PID-controller. The form of PID-controller is dtxxKxxKxxKu didddp )()()( The total control scheme and simulation programme are shown in Fig.5. The reference signal of displacement used in this simulation is )5.0sin(25)sin(50 ttxd (mm). In Fig. 6, the curve of ‘line 1’ shows the tracking errors using friction compensation based on the proposed model in this paper. The curve of ‘line 2’ shows the tracking errors obtained only with the Stribeck friction model. Obviously, the tracking effect of form he velocity reversal point. The reason is the disc e latter model. proposed in this paper has a more outstanding per l the tracking errors. Bec er is better than that of the latter. The largest position errors appear in t ontinuity in the friction model at velocity reversal, especially in th VI. CONCLUSIONS This paper analyses the friction characteristics in two regimes: pre-sliding and dynamic. A partition friction model is built. Pre-sliding friction is modelled by means of hysteresis model with nonlocal memory based on the viscoelasticity theory. Two friction models are used to compensation tracking error in which the effects are compared. The simulation result shows that the new friction model formance to contro ause it reflects the static friction characteristic in pre- sliding stage. In this way, the developed model is a effective friction model for description of experimentally observed friction behaviour. Being flexible, the obtained model structure should include other friction effects such as position dependence of friction behaviour. This will be the subject of future work that will also include practical application of the model in friction compensation of machine tools. Fig. 5. The control scheme of friction compensation Fig. 6. Tracking errors using two models ACKNOWLEDGMENT The part of this research was supported by the Key Project of Chinese Ministry of Education (No.104111). 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