Outline of this presentation Introduction Friction models Static models 1. Models with time delay 2. Dynamic models 3. Friction compensation Non-model

Transcription

1 of Engineering Cybernetics Department University of Science and Technology Norwegian Problems in Servomechanisms: Friction and Compensation Techniques Modeling Jan Tommy Gravdahl, Trondheim

2 Outline of this presentation Introduction Friction models Static models 1. Models with time delay 2. Dynamic models 3. Friction compensation Non-model based compensation 1. Compensation based on static friction models 2. Compensation based on dynamic friction models 3. Comparision of compensation schemes 4. Concluding remarks

3 Very complex phenomenon, composed of several physical in combination. Modeling most often empirical. phonomena Friction is the tangential reaction force between two surfaces Friction in contact Friction depends on contact geometry and topology, proper- of the surface materials, displacement, relative velocity ties lubrication and Friction in servomechanisms can cause limit oscillations, as stick-slip, and regulation/tracking errors. known Friction causes wear in the system and reduces lifetime. Friction is dissipative, that is, it can only extract energy from the system

4 section of a photo of a highly polished steel surface, Magnied and Resnick (1988): Halliday Schematic drawing of two surfaces in contact, Gafvert (1996): built up by asperities. Surfaces contact occurs between asperities, asperity junction True Asperity widht: typically 10m, slope: 5 ; 10 (steel).

5 Friction - denition of terms Static friction (stiction): The force (torque) needed to initiate motion from rest Dynamic (Coulumb) friction: A friction component indepen- of velocity dent Viscous friction: Velocity dependent friction between solid and lubricant Break-away: The transition from rest (stiction) to motion (dynamic friction) Break-Away force: The amount of force needed to overcome static friction Dahl-eect: Elastic deformation of asperity junctions behaves a linear spring for small displacements like Stribeck eect: Decreasing friction with increasing velocity at low velocities. Caused by uid lubricants.

6 Static friction=stiction by Morin (1833). Might be greater than Coulumb Intruduced Friction models for constant velocity: Static models results for constant velocity: Classic Coulomb friction force proportional to normal load: Friction = F c sgn(v) F c = F N. Known by L. da Vinci (1519), F by Amontons (1699) and developed by Coulomb rediscovered Not neccesarily symmetric. (1785). friction Viscous friction dependent friction, Ex: Friction force proportional Velocity velocity: to = F v v, Reynolds (1886). Caused by viscosity of lubricants. F Negative viscous friction Introduced by Stribeck (1902). The Stribeck eect.

7 @F C + (F s ; F C )e ;(v=v s) 2 + F v v 1 sgn(v) A Karnopp (1985): Stribeck friction with a dead zone around Other static models Hess and Soom (1990): F (v) = Armstrong-Helvoury(1990): 0 c + (F S;F c ) ) 2 +F v va sgn(v) s 1 zero velocity to make simuations less time consuming. 0 F (v) = The (v=v s ) 2 -term model the Stribeck eect Canudas de Wit et.al (1991): F (v) = (F c + 1 jvj 1=2 + 2 v)sgn(v) for adaption, linear in parameters. Modeling All these models are discontinious for v = 0. An approxima- with a nite slope through the origin would not reect tion physical phenomena, Karnopp (1985). the

8 Coulumb d) Stribeck eect a) Coulumb+viscous e) Karnopp b) Coulumb+viscous+stiction f) Hess and Soom c) etc Armstrong, Various static models F F F a) b) c) v v v d) F e) F f) F v v v

9 F I The generalized Stribeck curve II III IV v No sliding, elastic deformation I Boundary lubrication II Partial uid lubriacation III Full uid lubrication IV

10 and Soom (1990): F (v t) = F c + (F S;F c ) 1+(v(t; Hess )=v s ) 2 l 8 ><! v v sgn(v) F + Models with time delay models include the phenomenon known as frictional memory These (or lag) by using F f (v t) = F vel (v(t ; l )) lag Friction Friction velocity time velocity The Armstrong (1994) seven parameter model: F (x v t) = 0 x if v = 0 (pre sliding displacement) C + F s ( t d ) 1 F >: 1+(v(t; l )=v s ) 2! sgn(v) + F v v ifv 6= 0 stiction, Stribeck eect, Dahl eect and lag, but requires Includes switching and many parameters

11 Dynamic friction models models do not capture observed friction phenomena like Static the hysteresis observed experimentally by Hess and Soom Low velocity ) time delay not accurate enough (1990). position dependence like the Dahl eect. Asperity junctions like linear springs before break-away. behave variations in the break-way force. Friction Break-away force Dispalcement Force rate Friction models involving dynamics are neccessary to describe ) friction phenomena accurately the

12 df = dx df = 0 ) F = F csgn(v) dx The Dahl model by the stress-strain characteristic from solid mechanics, Inspired (1968) proposed the model: Dahl 1 0 B C B C ; F F C sgn(v) x is displacement. Friction depends only on position. In where time domain ( = 1): the = z F = v ; jvj _z z F C of Coulomb friction: generalization A Dahl model models pre-sliding displacement and frictional The but not stiction or the Stribeck eect. lag,

13 F = N X N is the number of bristles, 0 is the stiness and (x i ;b i ) Where the deection. is The bristle model (Norw.: bust) Proposed by Hessig and Friedland (1991). Sliding body N bonded bristles Models the microscopic contact points of the asperity junctions. (x_i-b_i) Stationary surface Uses an algorithm to calculate 0 (x i ; b i ) i=1 jx i ; b i j = s, the bound snaps, and a new is formed. As due to complexity Inecient

14 dz = dt 8 >< >: Proposed by Hessig and Friedland (1991) to make the bristle The reset integrator model computationally feasible model Instead of snapping a bristle, the bond is kept constant at point of rupture the Strain in bond: if (v > 0 and z z 0 ) or (v < 0 and z ;z 0 ) 0 otherwise v Friction F = (1 + a(z)) (v)z + dz 1 dt force: 0 Stiction achieved by a(z) = 8 >< Much easier to simulate than the bristle model, but care must >: if jzj < z 0 a otherwise 0 be taken in handling the discontinuities

15 = C T x s F s dx The models of Bliman and Sorine Bliman and Sorine (1993,1995) stress rate independence F depends on sgn(v) and s = R t 4 jv( )jd 0 F a function of path and not velocity Model given by = Ax s + Bv s ds 1. order model can be reduced to Dahl model and further to Coulomb 2. order: A 2 IR 22 B C x s 2 IR 2. models stiction. Emulates Stribeck eect by using Correctly Dahl models in parallel. Olson et.al (1998): not true two Stribeck eect

16 dz 0 z + 1 F + 2v = dt v ; jvj = z g(v) The LuGre (Lund-Grenoble) dynamic friction model Introduced by Canudas de Wit et.al. (1995) An extension of the Dahl model Based on bristle deection in an average sense Models both Stribeck eect, stiction, frictional lag and varying break-away force dz dt = F C + (F S ; F C )e ;( v v 0 )2 g(v) Only one rst order di. equation

17 Spectral analysis and Goldenberg (1998) use spectral analysis to Popovic Other friction modeling techniques Neural networks et.al (1997) model the dynamic friction of a ser- Dominguez using neural networks. The resulting model includes vomotor eect and frictional lag, but failed to model other Stribeck friction phenomena. Du and Nair (1997,1998) more known promising. the position- and velocity dependent friction of a servo model Friction force is represented by a Fourier series: motor. 0 1 C B C N X B A j ( _q) sin B C F f (q _q) = A 0 ( _q) + 2 j=1 C j ; B j ( _q) A Veried experimentally. Accuracy can be improved by increasing N, the number of DFT components

18 Comparative studies of friction models and Friedland (1991) compare the bristle model, the reset Haessig integrator model, the Dahl model, the static Karnopp model the classical Stribeck model. Results: and Dahl: No stiction Karnopp fast, bristle and classical slow (in simulations) The classical model wrongly predicts limit cycles Implementation: Karnopp hard, Dahl and reset integrator easy (1997) compares the Bliman and Sorine models to the Gafvert model. Conclusion: LuGre includes more friction phenomena LuGre than Bliman and Sorine. Conclusion: The LuGre model is probably the most accurate dynamic friction model avaliable

19 Model based friction compensation Estimating the friction force F by ^F using a friction model Friction compensation in servomechanisms that require friction compensation: Tasks positioning, Velocity reversal and Velocity tracking Precision to solve the friction problem: Approaches Friction avoidance, design for control Lubricant selection, uid (oil, grease) or dry (teon, diamond) (Ball) bearings, active control, magnetic, piezoelectric Redesign of physical system, inertia reduction Non-model based friction compensation PD/PID Dither. compensating for friction by adding ^F to the control and The estimate ^F can be xed (identied oine) or adaptive

20 Dither Introduction of a high freq. oscillation keeps the system in Non model based compensation techniques avoiding sticktion. (in use on e.g. gun mounts) motion, Analysis with describing functions (Balchen 1967) or av- (Mossaheb 1983) eraging Normal dither (external vibrator) modies friction, tangen- dither (control input) modies the inuence of friction tial Impulsive control Achieve high precision positioning by applying a series of impacts, when in stick. Yang and Tomizuka (1988): small pulse width control. Adaptive PD/PID. The regulator problem is stable under PD control. Tracking may lead to stick-slip limit cycles. Integral action reduces steady-state errors ) hunting

21 Model based compensation techniques

22 Overview Friction Model Static Dynamic Application Servo Motors N DOF Manipulator Other Servo Motors N DOF Manipulator Problem Regulation Tracking Reg Track Reg Track Reg Track Method PID adaptive estimation Passivity Nonlinear Robust

23 J _! = u ; F f n : control torques and F f 2 IR n : friction torques. Only IR in joints included. Can also have friction when in friction System models study of compensation techniques will be restricted to the The two applications following 1. Servomotors driving a load with friction: J is the moment of inertia,! is the angular velocity, where is the input torque and F f is the friction torque. u 2. Robotic manipulators in N DOF with friction in the joints: D(q)q + C(q _q) _q + g(q) = u ; F f q 2 IR n : joint angles, D(q): inertia matrix, C(q _q) _q: where of Coriolis and centrifugal terms, g(q) : gravity, u 2 vector contact with environment (force control).

24 Position regulation for 1DOF mass system et.al (1991): Southward System has similar eq. of motion as a servomotor Uses static Stribeck and Karnopp friction models Control law: PD + nonlinear (discontinuous) friction com- pensation. F K_p x+f_c(x) F s x F s Global asymptotic stability proved by LaSalle's theorem using Dini deriviatives Position regulation conrmed by experimental results

25 ^F c = z ; kjvj = ;1! u ; F (v ^F c ) sgn(v) kjvj _z Adaptive position tracking for servo and Park (1991,1992): Friedland Position tracking of servo with static Coulomb friction Adaption of unknown Coulomb friction: ^F c! F c asymptotically. ) Control law: u = PD + F (v c ) ^F tracking conrmed in simulations, also when includ- Position ing viscous friction. Experimentally conrmed by Mentzelopoulou and Friedland for Coulomb friction, and by Amin et.al (1997) for (1994) friction viscous Extended to two DOF manipulator by Yazdizadeh and Khoasani (1996)

26 Friction compensation with static friction models Adaptive velocity tracking for a tracking telescope and Winston (1974): Gilbart The rst result on adaptive friction compensation System: a motor driving an optical telescope Friction modeled as classical Coulomb friction MRAC: u = K (t) _ p + K 2 (t)( m + 12 _ m ) + K 3 (t)sgn( _ p ) 1 GAS proven by Lyapunov {z } est: Friction Controller was implemented on a 24in optical telescope used for tracking satellites The application requires velocity of motor to pass through zero Adaption eliminated dead zone encountered in zero crossings RMS tracking error reduced by a factor of six due to friction compensation.

27 Tracking controller: Adaptive sliding mode + friction comp. in friction model found using Evolution Strategies. Paramters Kim et.al. (1996a) study a servo motor driving a xy table The friction model used is the Armstrong 7-parameter model A \tracking controller" brings the system within a High precision position control Trackin contr. y m Fuzzy PD Plant y small distance from the reference. Then fuzzy+pd Fuzzy rules tuned by Experimental Evolutionary Progr. Experiments conrm position error less than 1m. Position error in an area dominated by the Dahl eect, em- the need for accurate friction models phasizing Extended by Kim et.al. (1996b) to tracking

28 ( ^m + ^I)v + ^mgr cos(q) + ^F f = feed forward + PID v f = T (t) estimated ^F Adaptive friction compensation in manipulators de Wit et.al (1991): Canudas Low velocity tracking of the last link in 2DOF manipulator Uses the static friction model F (v) = (F c + 1 jvj 1=2 + 2 v)sgn(v) in unknown parameters Linear Controller structure u = algorithm: Exponentially weighted least squares. estimation < 1 used to avoid friction overcompensation, which is shown to cause oscillations

29 ; non i = msi tanh i q i Position regulation of N DOF robot manipulator and Song (1993): Cai Position regulation of manipulator with Coloumb friction and stiction Uses the Karnopp friction model, with zero dead band in stability analysis Control law, PD+adaptive gravity compensation and robust friction compensation: u = ;K v _q ; K p q + G(q) ; ; non Similar in spirit as Southward et.al (1991), but continuous. Convergence to a set by LaSalle's theorem. Size of the set on i. depends Conrmed by simulations

30 Song et.al (1997): Tracking of N DOF manipulator

31 Friction compensation with dynamic friction models Position tracking of airborn servo (1984): Walrath Studied stabilization of airborn pointing and tracking tele- scope A servomotor produces a corrective torque to compensate for gimbal bearing friction Observed that friction responds continuously to velocity re- Static model not sucient ) Dahl's model versal. Probably the rst reference to employ a dynamic friction in control design model Controller : u = Proportional + ^F estimate ^F was calculated using the Dahl model. Adap- The on model parameter. tion Experimentally veried. Reported of a factor ve improve- ment in RMS pointing error

32 Position control of servomotor et.al (1997): Khorrami Considers a servomotor driving a load with friction Dynamic friction modeled with LuGre model Uses a robust adaptive variable structure controller: u = ;B T Px; ^B T Px; T tanh((a + bt)b T Px) with update law _^ = ;kb T Pxk 2 ; > 0 The unmeasurable friction states are treated as bounded disturbances Globally asymptotically stable Also give similar results using backstepping when considering transmission with friction at both motor and load compliant side. Result extended in Sankaranarayanan and Khorrami (1997) to the low velocity tracking problem

33 2 d^z r ; JH(s)e ; 0^z + 1 Js + 2v x dt v ; ^ 0jvj = g(v) = ; 0jvj g(v) ^z(z m ; ^z) dt Position tracking for servomotor de Wit and Lischinsky (1997): Canudas Study position tracking for a servomotor Use dynamic LuGre model The parameters of the LuGre model estimated numerically from experiments Fixed friction compensation: u = Js 2 x r ; JH(s)e + ^F where depends on controller choise (PD, ltered PID) H(s) Adapting normal force variations: u = d^z ^z ; ke d^ dt Adaption also for temperature changes Convergence e! 0 by Lyapunov. Conrmed experimentally.

34 Control law: u = w + ( _q)^z + k r, where w is represents c to be cancelled out, ( _q)^z is the estimated friction dynamics Position tracking for manipulators et.al (1997): Vedagarbha Consider the problem of position tracking in a manipulator dynamic friction in the joints with Use the dynamic LuGre model in each joint Employ nonlinear observers to estimate the unmeasurable states, and presents convergence results both for friction and non-adaptive controllers. adaptive and r is the ltered tracking error torque Adaption is studied for unknown (linear) parameters in LuGre model and for variations in the normal force.

35 _^ = ; T s ; = ; T > 0 Position tracking of manipulator et.al (1997,1998): Panteley Study position tracking in manipulators using LuGre Main point: Treat the friction compensation problem as a rejection problem disturbance The part of the friction force dependent on z is regarded a and the linear (viscous) component is compensated disturbance, by the adaptive controller Designs an adaptive controller using passivity arguments. An adaptive Slotine and Li controller strictly passies the system and an outer loop (tanh) rejects the disturbance: u = ;K d s + ^ = [Y 1 (q _q q r _q r ) Y 2 ( _q) Y 3 ( _q s)] Results conrmed experimentally

36 Comparative studies of compensation schemes and Krisnaprasad (1992): position tracking of servo Leonard Compares 5 dierent controllers: PID 1. Dither 2. MRAC based on Gilbart and Winston (1974), asymmetric 3. AS Coulumb+Stribeck, Computed torque + int.+friction comp., four dierent 4. models of friction static Based on Walrath (1984), Computed torque + int.+ dynamic 5. Dahl friction comp. Compared experimentally Problem: track a sinusoidal reference trajectory Conclusions: Model based controllers better than PID and dither 1. The Dahl based controller outperformed the other 2.

37 Du and Nair (1997) their NN controller with the adaptive schemes of Compare Canudas de Wit and Lischinsky (1997) their adaptive LuGre based controller with a PID. compare Comparasions, contn. de vit et.al. (1991) and Friedland and Park (1991). Canudas perform better, but computationally intensive NN contest. No Panteley et.al (1998) approach compared to Amin et.al. (1997), based on Passivity scheme of Friedland and Park (1991). Performance adaptive for sinusoidal trajectories, passivity based better for similar trajectories. Amin requires knowledge of J complicated

38 Friction is a complicated phenomenon. Many approaches to Concluding remarks friction. Classic: F f = F f (v). model The trend is toward using dynamic friction models, where the of the art is the LuGre model. state Dynamics are required to explain observed friction phenom- ena. Two main approaches to friction compensation: I) Non model and II) Model based based Model based : u = u + ^F f, where ^F f can be calculated nom by the use of adaption and estimation. Recent publications e.g. also employ disturbance rejection schemes.