Dynamic friction modelling without drift and its application in the simulation of a valve controlled hydraulic cylinder system.

Transcription

1 Dynamic riction modelling without drit and its application in the simulation o a valve controlled hydraulic cylinder system. Junhong Yang 1, Andrew Plummer, Yong Xue 1 1 School o Mechatronics Engineering and Automation, National University o Deense Technology, Changsha, China Centre or Power Transmission and Motion Control, Department o Mechanical Engineering, University o Bath, Bath BA 5DR, UK The ull correspondence details or the corresponding author, Junhong Yang. Tel: ; addresses: yangjunhong@nudt.edu.cn Junhong Yang received the B.S. degree rom Xi an Jiaotong University in Xi an and the M.S. and PH.D. degrees rom National University o Deence Technology in Changsha. He is currently a lecture in School o Mechatronics Engineering & Automation at the National University o Deence Technology. Dr. Yang s researches include robotics, high eicient hydraulic diving system or mobile robot, and portable luid power source. Andrew Plummer received his PH.D degree rom the University o Bath in He currently is the proessor and director o the centre or Power Transmission and Motion Control at the University o Bath. He has a variety o research interests in the ield o motion and orce control, including inverse-model based control o electrohydraulic servo systems, control o parallel kinematic mechanisms, hybrid hydraulic / piezoelectric actuation, and active vehicle control. Yong Xue received the B.S. and M.S. degrees rom National University o Deence Technology in Changsha. He is currently a Ph.D. student in Mechatronics Engineering & Automation at the National University o Deence Technology. Mr.Xue s research interests include design o robotic mechanism and high eicient hydraulic diving system or mobile robot.

2 Abstract: The rictional modelling literature is reviewed, and it is demonstrated that unrealistic drit results when the shape coeicient is 1.0 or the LuGre and the Ferretti riction models. Drit will not occur but other dynamic riction characteristics can t be represented when the shape coeicient is 0. Based on the above, the LuGre riction model and the Ferretti riction model are improved. The velocity-riction characteristic, the stick-slip and the cycling caused by riction, and the drit are compared in simulation. The results show that the improved riction model well relects realistic riction dynamic characteristics and avoids drit. Finally, the improved riction model is used in a nonlinear mathematic model o a valve controlled hydraulic cylinder system. The cylinder s motion at low velocity is simulated and the related eperimental results are presented. The results show that the improved riction model gives realistic low velocity motion o the cylinder. Keywords: riction model; stick-slip; limit cycles; drit; valve controlled cylinder 1. Introduction Friction is inevitable in mechanical systems. The nonlinear behaviour caused by riction has an adverse inluence on the ultra-low velocity and high precision position control o servo mechanisms and hydraulic systems. Friction severely aects the stability o the control system designed to obtain very small steady-state error and could lead to limit cycles and stick-slip, and aect the requency response bandwidth o the closed loop system [1, ]. It is important to establish an accurate riction model or both understanding the riction phenomenon and compensating or riction. Until now, a lot o research on riction modelling has been undertaken [3-]. Many eperiments on riction show that there eist two riction regimes [8]: the pre-sliding regime and the sliding regime. In the pre-sliding regime the riction orce appears to be a unction o relative micro-

3 displacement (elastic deormation and plastic deormation), and its characteristics are similar to a nonlinear spring. As the displacement becomes larger, the spring suddenly ruptures leading to the relative motion between two contact suraces, and the sliding regime begins. In the sliding regime the riction orce appears to be a unction o relative velocity. Up to now, many riction models have been proposed, and they can be classiied into two categories: static riction models and dynamic riction models. Among static riction models the representatives are the Coulomb plus viscous riction model [9] and the eponential riction model [10], these models can t predict dynamic riction. Though the seven parameters model [11-1] can relect the riction s static and dynamic characteristics, in essence it is only the crude combination o static and dynamic models and has no eplicit physical content, and moreover it contains redundant parameters. The dynamic riction models include: the Dahl model [13-15], the LuGre model [16], the Elastic-Plastic model [17,18], the Ferretti model [19], the Leuven model [0], and the GMS model [].. The development o dynamic riction modelling.1 Dahl riction model The Dahl model, which was developed in the late 1950s, is a dynamic model with one state, and is widely used to simulate aerospace systems [13, 14]. The Dahl dynamic model essentially describes the riction s pre-sliding regime, and in this regime the riction orce is the unction o relative micro-displacement between two riction suraces. The mathematic epression o the Dahl model is as ollows [13]. d d 0(1 sgn( v)) (1) c Where is the riction orce, is the relative micro-displacement between two riction

4 suraces, σ0 is the stiness coeicient, c is the Coulomb riction orce, v is the relative velocity o two riction suraces, α is the coeicient determining the shape o the curve between riction orce and relative micro-displacement and is always larger than zero. Let v d / dt, z / 0, and the riction orce can be epressed as equation () when 1. dz dt z 0 v v (a) c z 0 (b). LuGre riction model The Dahl dynamic model does not take into account the riction s sliding regime. In the sliding regime the lubricant ilm plays a dominant pole, and it is appropriate to describe the riction orce as a unction o relative velocity between two riction suraces. In the case that the velocity is very low, the lubricant ilm isn t ormed ully, and the riction orce would decreased when the relative velocity increases, as per the Stribeck phenomenon. The Stribeck phenomenon can be described by the ollowing equation: sgn( v) g( v) (3a) s g v) ( )ep[ ( v / v ) ] (3b) ( c s c s Where s is the maimum static riction orce, c is the Coulomb riction orce, v is the relative velocity o two riction suraces, vs is the velocity at the turning point o Stribeck curve, and s is the corrective coeicient o curve. The literature [16] integrates the Dahl model () and the Stribeck equation (3) to derive the ollowing LuGre riction model: dz 0z v v (4a) dt g( v)

5 g( v) ( )ep[ ( v / v ) ] (4b) c s c s dz 0z 1 v (4c) dt Where is the total riction orce, c is the Coulomb riction orce, s is the maimum static riction orce, v is the relative velocity o two riction suraces, vs is the velocity at the turning point o Stribeck curve, 0 is the equivalent stiness coeicient between the riction orce and the relative displacement o two riction suraces when the relative velocity s direction changes, 1 is the micro-viscous riction coeicient, and is the viscous riction coeicient..3 Elastic-plastic riction model The LuGre model produces steady-state drit when a tiny eternal vibratory stimulation is applied [17], whereas in practice there isn t relative motion between two the riction suraces because the vibratory stimulation is smaller than the maimum breakout riction. In the same case, though steady-state drit doesn t arise in the Karnopp riction model, micro-displacement which should arise does not. Based on the above problem, [17] and [18] propose the elastic-plastic riction model as a development o the LuGre riction model. dz dt 0z v ( z, v) v (5a) g( v) g( v) ( )ep[ ( v / v ) ] (5b) c s c s dz 0z 1 v (5c) dt 0, i z zbaor sgn( v) sgn( z) ( zv, ) 1, i z zss( v) zba zss 1 z ( ) 1 else sin( ) zss zba (5d)

6 Where, c, s, v, v s, 0, 1, are the same as the parameters in the LuGre riction model shown in section.. 0 z z ( v) (6) ba ss z ( v) g( v) / / (7) ss 0 ma s 0 where z ba represents the range o riction state variable z when riction is characterized by linear damping and a linear spring, and z ss represents the range o riction state variable z when the riction contact surace is in the pre-sliding regime in which there are only elastic deormation and plastic deormation..4 The Ferretti Model Ferretti proposes an integral riction model [19], and the simulation results show that the model is consistent with the LuGre model in terms o relecting stick-slip, limit cycles, and so on, while being computationally more eicient and avoiding nonphysical drit through letting α=0 in the Dahl model. The integral riction model is given in the ollowing. (8a) z p v z d d p p 0(1 sgn( v)) (8b) s s v sgn v s c ep v / v s 1 v (8c) p 1(1 sgn( v) ) v (8d) s Where p is the term which relates riction to micro-displacement in the pre-sliding regime, v is the term which relates riction to velocity in the sliding regime, and z is the micro-viscous riction term when riction transitions rom the pre-sliding regime to

7 the sliding regime, and the other parameters are the same as the parameters in the LuGre model shown in section...5 Leuven riction model In [0], the pre-sliding regime riction orce is modelled as a hysteresis unction o relative micro-displacement, with nonlocal memory. Considering this actor, it corrects the LuGre model and derives the Leuven riction model. dz dt 1 h( z) h ( z) v(1 sgn( ) ) (9a) g( v) g( v) s g v) ( )ep[ ( v / v ) ] (9b) ( c s c s dz h ( z) 1 v (9c) dt Where () h z is the hysteresis unction with non-local memory, and it is the point symmetrical and strictly increasing unction with the input state variable z. () h z can be ound by eperimental identiication[0] or by theoretical modelling[1]. δ 1 is similar to the shape coeicient α in the Dahl model (1). The other parameters are the same as the parameters in the LuGre model shown in section...6 GMS (Generalized Mawell-Slip) riction model The literature [8, ] takes into account advantages and disadvantages o the elastic-plastic riction model and the Leuven riction model and proposes the GMS riction model. It can not only relect the non-local memory hysteresis loops and the Stribeck phenomenon in the pre-sliding regime, but also avoid the same steady drit as the elastic-plastic model. N i (10a) i1 v

8 k v i g() v di dt sgn( v) C( ) else i i i F i i gv () c s c s (10b) g( v) ( )ep[ ( v / v ) s ] (10c) Where N is the number o riction model units, k i is the contact stiness, i is the split coeicient o the riction model, C is the coeicient concerning the velocity v, i is the riction orce unit, is the viscous riction coeicient, and is the total riction orce. The other parameters are the same as the parameters in the LuGre model shown in section.. 3. Improvement o the LuGre riction model and the Ferretti riction model Though the elastic-plastic model, the Leuven model and the GMS model can accurately relect the riction s dynamic characteristics, they are comple. And so it is very diicult to utilize them in closed loop control system analysis and design. Most o the literature about riction or control purpose has adopted the LuGre riction model [3-5]. The literature [15] indicates by simulation that the Ferretti riction model has the same characteristics as the LuGre riction model, and also it is computationally more eicient. The Ferretti riction model can avoid unrealistic drit when α=0 [15], while the LuGre riction model and the Ferretti riction model can t accurately relect the riction s dynamic characteristics. The dierence in the riction orce as a unction o velocity is shown in igure 1 and igure when α=0 and α=1, and it is ound that the riction model can ully relect the riction s dynamic characteristics when α=1, but can t relect the riction s dynamic characteristics when α=0. Through the above summary o the literature, the ollowing conclusion can be reached: (1) the drit o the riction model mainly occurs in the pre-sliding regime in

9 which there is no obvious relative motion in reality, and urthermore, the drit mainly occurs in the elastic deormation stage o the pre-sliding regime; () the basis o the LuGre riction model and Ferretti riction model is the Dahl riction model, and the parameter α o the Dahl riction model is the coeicient determining the shape o curve between riction orce and relative micro-displacement in the pre-sliding regime; (3) when α=0, drit can be avoided, and when α=1, the riction s dynamic characteristics can be well relected. Thereore the ollowing improvement o the LuGre and the Ferretti riction models is derived. The riction process is divided into the two stages. The irst stage is the elastic deormation, in which the riction orce is not larger than the coulomb riction orce c, and by letting α=0 in this stage drit is avoided. The second stage is the plastic deormation and sliding riction, and by letting α=1 in this stage ensures the riction s dynamic characteristics are modelled correctly. 3.1 The improved LuGre riction model dz z dt g( v) 0 v(1 sgn( v)) (11a) c s s c s (11b) g( v) ( )ep[ ( v / v ) ] dz 0z 1 v (11c) dt 0 z z 1 z z es es (11d) Where zes c / 0, c is the coulomb riction orce, 0 is the stiness coeicient. The term z es represents the range o riction state variable z when the riction contact surace is in the pre-sliding regime in which there is only elastic deormation.

10 3. The improved Ferretti riction model (1a) p v z d d p p 0(1 sgn( v)) (1b) s s v sgn v s c ep v / v s 1 v (1c) z p 1(1 sgn( v) ) v (1d) s 0 1 p p c c (1e) Where p is the term which relates riction to micro-displacement in the pre-sliding regime, c is the coulomb riction orce, and p c means that the riction is in the elastic deormation stage and p c means that the riction is in the plastic deormation and sliding regime. 4. Analysis o simulation eperiment Simulation results [15] already indicate the consistency o the LuGre model and the Ferretti model, and when α=0 both can avoid drit, while through simulation it is ound that the model can t relect stick-slip and limit cycles caused by riction nonlinearity when α=0. For brevity, in the ollowing only results or the Ferretti model and the improved Ferretti model are given, and the simulation parameters which are the same as [15] are shown in table 1. Table 1. The parameters o riction simulation σ0 σ1 σ c s vs δs N/m 10 Ns/m 0.4 Ns/m 1N 1.5N m/s

11 4.1 Simulation o the relation between the riction orce and velocity Let v=0.001t, set the integrator and let its absolute tolerance equal to , eternal reset is none, initial condition source is internal, initial condition is zero, and not limit output and not ignore limit and reset when linearizing. When adopting variable simulating step, it is needed to add a two-order ilter whose cut o requency is 10KHz beore the derivate o z goes into the equation (11a) when simulating the improved LuGre model, the solver is ode45 (Dormand-Prince), and Relative tolerance is 1e-3. I the ied simulating step is adopted or the two improved riction models, the simulating step is equal to and the solver is ode3 (Bogacki-Shampine). Separately solving the improved Ferretti model, the Ferretti model (α=1) and the Ferretti model (α=0) gives the curves between the riction orce and the relative velocity shown in Figure 1 and Figure. Figure 1. Friction characteristics with varying velocity (dashed line: Ferretti model with α=1,solid line: improved Ferretti model)

12 Figure. Friction characteristics with varying velocity (Ferretti model with α=0) The simulation result shows that the riction characteristics relected by the improved Ferretti model and the Ferretti model with α=1 are very similar. 4. Simulation o Stick-slip and Limit Cycles The simulation o stick-slip adopts the model shown in Figure 3, where the spring stiness k=n m -1, the moving speed o the spring s end is v =0.1 m s -1, m is unit mass, and other simulation parameters are shown in table 1. The stick-slip simulation results are shown in Figure 4 and Figure 5. The dynamic model or simulation is as ollows: t k ( v ) dt 0 m (13) m k v Figure 3. Simulation model or stick-slip and limited cycles Figure 4. Stick-slip simulation eperiment(dashed line: Ferretti model with α=1,solid line: improved Ferretti model )

13 Figure 5. Stick-slip simulation eperiment (Ferretti model with α=0) Figure 6 and Figure 7 show the simulation o a Proportional-Integral-Derivative closed loop position control o the unit mass m (taking out the spring) with riction. The igures show this control system s step response, where the input step reerence signal =1m, the output signal is the displacement o unit mass, and the parameters o the PID controller are k p =3 N m -1, k i =4 N m -1 s-1, k d =6 N s m -1. The dynamic model or simulation is as ollows: kp( ) ki ( ) dt kd m (14) Figure 6. Hunting simulation eperiment (dashed line: Ferretti model with α=1,solid line: improved Ferretti model )

14 Figure 7. Hunting simulation eperiment (Ferretti model with α=0) The simulation results show that both the improved Ferretti model and the Ferretti model with α=1 can relect stick-slip and limit cycles caused by riction, while the Ferretti model with α=0 can t relect these phenomena. 4.3 Simulation o non-physical drit With a horizontal vibratory stimulation on the unit mass m in Figure 3, and when the maimum o the vibratory stimulation s amplitude doesn t eceed the maimum o breakout riction (the maimum static riction orce), in reality the unit mass wouldn t ehibit micro-motion, but when adopting the LuGre riction model or the Ferretti riction model with α=1 in the control system, the simulation result shows that there is a steady drit o the unit mass. In the simulation, the input signal is the imposed eternal orce on the mass, and the output signal is the displacement o unit mass. The imposed eternal orce is u( t) a bsin( wt), where a=0.5 N, b=0.5 N, w=6л/5 rad s -1. So the maimum eternal orce is 0.75N, and it is samller than the maimum static riction orce s showed in Table 1. Other parameters are as shown in table 1.

15 Figure 8. Simulation eperiment with eternal vibration orce (dashed line: Ferretti model with α=0,solid line: improved Ferretti model ) Figure 9. Simulation eperiment with eternal vibration orce (Ferretti model with α=1) The displacements o the unit mass or dierent riction models are shown in Figure 8 and Figure 9. It can been seen that the Ferretti model with α=1 produces steady drit as showed in Figure 9, and this drit would not occur in reality because the stimulating orce is less than the maimum static riction orce. Figure 8 shows that both the Ferretti model with α=0 and the improved Ferretti model don t produce the steady drit as showed in Figure 9 From the above results, it can been seen that the improved Ferretti model can relect the stick-slip and limit cycle phenomena caused by riction, and at the same time avoid drit. The authors have ound that the improved LuGre model has the same characteristics as the improved Ferretti model, but results are not included here.

16 5. Application o the improved LuGre riction model in the simulation o a valve controlled hydraulic cylinder system Friction nonlinearity in valve controlled hydraulic cylinders degrades the position tracking precision, and can result in limited cycles, stick-slip, and reduces the requency response bandwidth o the closed loop system [6, 7]. Building a model which can relect the real dynamic riction characteristic is very useul or simulation analysis and riction compensation control. Figure 10. Valve controlled cylinder system A valve controlled cylinder system is shown in Figure 10. When the riction orce is not considered, the plant s nonlinear state space model is given by equation (15) [8], where the state variable 1 is the displacement o the cylinder y, is the velocity o the cylinder, 3 is the pressure o the cylinder rodless chamber, 4 is the pressure o the cylinder rod chamber, and other parameters is shown in table. e 3 4 i 4 1 e i ) ( ) ( ) ( ) ( V C V C V A u g V C V C V A u g M F M A M B M A L (15)

17 Where g ( ) sgn((1 sgn( u)) p / sgn( u) ) Cw ((1 sgn( u)) p / sgn( u) ) d 3 1 s 3 s 3 V1 g ( ) sgn((1 sgn( u)) p / sgn( u) ) Cw ((1 sgn( u)) p / sgn( u) ) d 5 s 4 s 4 V1 When considering riction, the second equation o state space model (15) should be changed into the ollowing equation. A1 B A F L 3 4 sgn( ) (16) M M M M M Where is the riction orce and can be ound through the improved Ferretti riction model. Firstly, the Coulomb riction orce c and the maimum static riction orce s o the improved Ferretti riction model need to be determined eperimentally. Lay the cylinder horizontally, and give the servo valve a small opening signal. The piston starts to move at a very small constant velocity. The inertial orce caused by the piston rod can be ignored due to the small mass o the piston rod and its very small acceleration. Measure the pressure o the rodless chamber and the rod chamber, and the riction orce o the cylinder can be derived by the orce balance ormula p1 A1 p1 A. The curves in Figure 11 are the riction orce and the velocity when the piston rod is etending, and the curves in Figure 1 are the riction orce and the velocity when the piston rod is retracting. The dynamic riction characteristic is obvious in igure 11 and not in Figure 1. It can be ound in igure 11 that c 100N, s 145N and 4 1 v s 310 m s.

18 Figure 11. Friction orce and velocity when etending Figure 1. Friction orce and velocity when contracting Table. Simulating parameters o valve controlled cylinders Parameter name Value Parameter name Value Valve opening area gradient w3/m Bulk modulus K/ MPa 750 Valve opening area gradient w5/m Viscous damping coeicient B/(N s m -1 ) 800 Flow coeicient Cd Mass o the piston rod M /Kg 5 Area o rodless chamber A1/ m Density / (kg m -3 ) Area o rod chamber Initial volume o rodless chamber Initial volume o rod chamber A/ m Coeicient o internal leakage V10/m Coeicient o eternal leakage V0/m Pressure o power source Ci/(m s -1 Pa -1 ) Ce/(m s -1 Pa -1 ) p s / MPa 7 Secondly, the equivalent stiness coeicient σ 0, the micro-viscous riction coeicient σ 1 and the viscous riction coeicient σ need to be determined. Because the viscous riction orce has been considered in the second equation o the state space model (15), let σ =0. The values o σ 0, σ 1 and δs are rom the reerence [13]: σ 0 =105N m , 1 10 N s m and δ s =. For the closed loop controller, let the commanded displacement o the piston rod be r=0.05sin(6.8t) m, and the proportional position controller is given by u= 0.04(r-y). The output displacement o the piston rod is shown

19 in Figure 13 and Figure 14. The corresponding eperiment is done on the actual valve controlled cylinder system, and the displacement o the cylinder is showed in Figure 15. Figure 13. Displacement o the cylinder with riction model Figure 14. Zoom in on the zone A o Fig.13 Figure 15. The hunting displacement o the piston rod near zero velocity Figure 13 and Figure 14 show that the state space model including the improved riction model can relect the limit cycles when the direction o the piston rod movement changes and its velocity is very small. Finally, the drit is simulated on the valve controlled cylinder system with the Ferretti riction and with the improved Ferretti riction. The simulation model is shown in Figure 16, and a very small hunting stimulation is eerted on the piston rod through a simple closed loop proportional orce control. Let the command orce be Fe=40+0sin(6л/5) N, and the actual orce eerted on the piston rod is very close to Fe, its maimum value o 60N is less than the Coulomb riction orce (100N) and the

20 maimum static riction orce (145N). Because o the sign unction and absolute value unction in the state space equation (15), there are some high-requency components in the eedback signal, so a two-order ilter with 0 rad/s cut o requency is added as shown in Figure16. The proportional coeicient is Kp= Figure 16. Simulation o the valve controlled hydraulic cylinder system with eternal vibration orce Figure 17. Displacement o the piston rod with the LuGre riction model Figure 17 shows the displacement o the piston rod when the riction part o equation (16) adopts the Ferretti riction model. This shows that the displacement drits up over time. In act, the displacement shouldn t drit up because the orce eerted on the piston rod is less than the Coulomb riction orce and the maimum static riction orce, and so the mathematic model is not correct under this situation.

21 Figure 18. Displacement o the piston rod with the improved LuGre riction model When adopting the improved riction model, the displacement o the piston rod is shown in Figure18, and no drit occurs. 6. Conclusions The development process o a dynamic riction model has been systemically summarized, resulting in a riction model which is simple and well relects the dynamic behaviour o riction in reality. It was ound that the LuGre and Ferretti models can well relect the riction dynamic characteristics when the coeicient α determining the shape o curve between riction orce and relative micro-displacement equals to 1.0, but drit occurs; when α=0, there is no drit in the model, but the riction dynamic characteristics are not well relected. Thereore in this work the value o α is varied, so that α=0 to avoid drit when riction is in the pre-sliding regime (elastic deormation), and α=1 to relect sliding riction when riction is in the plastic deormation stage and the sliding regime. The corresponding improved LuGre and Ferretti riction models were developed, and the simulation results show that the improved riction models can well relect the riction dynamic characteristics and don t produce drit. Finally, the improved LuGre riction model was applied to the modelling o a nonlinear valve controlled hydraulic cylinder system, and the related simulation

22 eperiments were done. The results show that the model o the hydraulic system including the improved Ferretti riction model ehibits realistic low velocity riction dynamic characteristics and at the same time doesn t produce drit. Acknowledgements This work is supported by the National Natural Science Foundation o China (No ). Reerences [1] Friedland B and Park Y J, On adaptive riction compensation, IEEE Transaction Automatic Control. 37(199)37, pp [] Pierre E.Dupont and Eric P.Dunlap, Friction modelling and PD compensation at very low velocities, Journal o Dynamic System, Measurement, and Control. 117(1995), pp [3] Susanne V.Krichel and Oliver Sawodny, Non-linear riction modelling and simulation o long pneumatic transmission lines, Mathematical and Computer Modelling o Dynamic Systems. 6(013), pp.1- [4] Daniel Helmick and William Messner, Describing unction analysis o Dahl model riction, 009 American Control conerence. 6(009), pp [5] H. Olsson, K. J. Astrom, C.Canudas de Wit, M. Gavert, and P. Lischinsky, Friction models and riction compensation, European Journal o Control. 4(1998), pp [6] A. Bonsignore, G.Ferretti, and G.Magnani, Analytical ormulation o the classical riction model or motion analysis and simulation, Mathematical and Computer Modelling o Dynamical systems. 5(1)(1999), pp [7] C.L. Chen, K. C.Lin, and C. Hsieh, Presliding riction mode: modelling and eperimental study with a ball-screw-driven set-up, Mathematical and Computer Modelling o Dynamical Systems. 11(4)( 005), pp [8] Lampaert V, Al-Bender F, and Swevers J, A generalized mawell-slip riction model appropriate or control purposes, Physics and Control International Conerence. (003), pp [9] Wei-wei Shang, Shuang Cong, and Shi-long Jiang. Dynamic model based nonlinear tracking control o a planar parallel manipulator, Nonlinear Dynamic. 60(4)(010), pp [10] Lorinc Marton, Szabolcs Fodor, and Nariman Sepehri, A practical method or riction identiication in hydraulic actuators, Mechatronics. 1(011), pp [11] Armstrong B, Dupont P, and Canudas de Wit C, A survey o models, analysis tools and compensation methods or the control o machines with riction, Automatica. 30(7)(1994), pp

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