Friction Problems in Servomechanisms: Modeling and ... - CiteSeerX

Friction Problems in Servomechanisms: Modeling and Compensation Techniques

Jan Tommy Gravdahl,

Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim

Outline of this presentation

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

Friction

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

Magni ed section of a photo of a highly polished steel surface, Halliday and Resnick (1988):

Schematic drawing of two surfaces in contact, Gafvert (1996):

Surfaces built up by asperities. True contact occurs between asperities, asperity junction Asperity widht: typically 10 m, slope: 5 ; 10 (steel).

Friction - denition of terms

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Static friction (stiction): The force (torque) needed to initiate motion from rest Dynamic (Coulumb) friction: A friction component independent of velocity 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 like a linear spring for small displacements Stribeck eect: Decreasing friction with increasing velocity at low velocities. Caused by uid lubricants.

Friction models for constant velocity: Static models

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

Other static models

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Karnopp (1985): Stribeck friction with a dead zone around zero velocity to make simuations0less time consuming. 1 ( F ; F ) Hess and Soom (1990): F (v) = @Fc + 1+(Sv=vsc)2 + Fv vA sgn(v) Armstrong-Helvoury(1990): 0 1 2 ; ( v=v ) F (v) = @FC + (Fs ; FC )e s + Fv vA sgn(v) The (v=vs)2-term model the Stribeck eect Canudas de Wit et.al (1991): F (v) = (Fc + 1jvj1=2 + 2v)sgn(v) Modeling for adaption, linear in parameters. All these models are discontinious for v = 0. An approximation with a nite slope through the origin would not reect the physical phenomena, Karnopp (1985).

Various static models a)

F

b)

F

v

d)

c)

v

v

F

F

e)

v

F

f)

v

F

v

a) Coulumb d) Stribeck eect b) Coulumb+viscous e) Karnopp c) Coulumb+viscous+stiction f) Hess and Soom Armstrong, etc

The generalized Stribeck curve F

I

II

III

IV

v

I II III IV

No sliding, elastic deformation Boundary lubrication Partial uid lubriacation Full uid lubrication

Models with time delay

These models include the phenomenon known as frictional memory (or lag) by using Ff (v t) = Fvel(v(t ; l)) lag

Friction Friction

velocity time



velocity !

Hess and Soom (1990): F (v t) = F + ;; + F v sgn(v) The Armstrong (1994) seven parameter model: c

8 > < 0 > : C

(FS Fc ) 1+(v (t l )=vs )2

v

 x if v = 0 (pre sliding !displacement) F (x v t) = F + Fs( td) sgn(v) + Fv v if v 6= 0 v t; =v 1+( (

1

l)

s )2

Includes stiction, Stribeck eect, Dahl eect and lag, but requires switching and many parameters

Dynamic friction models

Static models do not capture observed friction phenomena like  the hysteresis observed experimentally by Hess and Soom (1990). Low velocity ) time delay not accurate enough  position dependence like the Dahl eect. Asperity junctions behave like linear springs before break-away.  variations in the break-way force. Friction

Break-away force

Dispalcement

Force rate

) Friction models involving dynamics are neccessary to describe the friction phenomena accurately

The Dahl model

Inspired by the stress-strain characteristic from solid mechanics, Dahl (1968) proposed the model:

dF =  1 ; F sgn(v)   dx FC where x is displacement. Friction depends only on position. In the time domain ( = 1): F = z  j v j z_ = v ; F z C 0 BB @

1 CC A

A generalization of Coulomb friction: dF = 0 ) F = F sgn(v) c dx The Dahl model models pre-sliding displacement and frictional lag, but not stiction or the Stribeck eect.

The bristle model (Norw.: bust)

Proposed by Hessig and Friedland (1991). Models the microscopic contact points of the asperity junctions. N bonded bristles Sliding body (x_i-b_i) Stationary surface

Uses an algorithm to calculate NX F = i=1 0(xi ; bi) Where N is the number of bristles, 0 is the stiness and (xi ;bi) is the deection. As jxi ; bij = s, the bound snaps, and a new is formed. Inecient due to complexity

The reset integrator model

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Proposed by Hessig and Friedland (1991) to make the bristle model computationally feasible Instead of snapping a bristle, the bond is kept constant at the point of rupture Strain in bond: dz = 8>< 0 if (v > 0 and z  z0) or (v < 0 and z  ;z0) dt >: v otherwise dz Friction force: F = (1 + a(z))  ( v ) z +  0 1 dt 8 > < a if jz j < z0 Stiction achieved by a(z) = >: 0 otherwise Much easier to simulate than the bristle model, but care must be taken in handling the discontinuities

The models of Bliman and Sorine

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Bliman and Sorine (1993,1995) stress rate independence 4 Rt F depends on sgn(v) and s = 0 jv( )jd F a function of path and not velocity Model given by F = C Txs dxs = Ax + Bv s s ds 1. order model can be reduced to Dahl model and further to Coulomb 2. order: A 2 IR22 B C xs 2 IR2. Correctly models stiction. Emulates Stribeck eect by using two Dahl models in parallel. Olson et.al (1998): not true Stribeck eect

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 F = 0z + 1 dt + 2v

dz = v ; jvj z dt g(v) ;( vv )2 g(v) = FC + (FS ; FC )e 0

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Only one rst order di. equation

Other friction modeling techniques

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Neural networks Dominguez et.al (1997) model the dynamic friction of a servomotor using neural networks. The resulting model includes Stribeck eect and frictional lag, but failed to model other known friction phenomena. Du and Nair (1997,1998) more promising. Spectral analysis Popovic and Goldenberg (1998) use spectral analysis to model the position- and velocity dependent friction of a servo motor. Friction force is represented by a Fourier series: 0 1

NX BB 2

Ff (q q_) = A0(q_) + 2 j =1 Aj (q_) sin B@ C ; Bj (q_)CCCA j Veri ed experimentally. Accuracy can be improved by increasing N , the number of DFT components

Comparative studies of friction models

Haessig and Friedland (1991) compare the bristle model, the reset integrator model, the Dahl model, the static Karnopp model and the classical Stribeck model. Results:  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 Gafvert (1997) compares the Bliman and Sorine models to the LuGre model. Conclusion: LuGre includes more friction phenomena than Bliman and Sorine. Conclusion: The LuGre model is probably the most accurate dynamic friction model avaliable

Friction compensation

Tasks in servomechanisms that require friction compensation: Precision positioning, Velocity reversal and Velocity tracking Approaches to solve the friction problem: ❏ 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. ❏ Model based friction compensation  Estimating the friction force F by F^ using a friction model and compensating for friction by adding F^ to the control  The estimate F^ can be xed (identi ed oine) or adaptive

Non model based compensation techniques

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Dither  Introduction of a high freq. oscillation keeps the system in motion, avoiding sticktion. (in use on e.g. gun mounts)  Analysis with describing functions (Balchen 1967) or averaging (Mossaheb 1983)  Normal dither (external vibrator) modi es friction, tangential dither (control input) modi es the inuence of friction Impulsive control  Achieve high precision positioning by applying a series of small impacts, when in stick. Yang and Tomizuka (1988): Adaptive pulse width control. 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

Model based compensation techniques

Overview Friction Model

Application

Problem Method

Static

Servo Motors

N DOF Manipulator

Regulation Tracking PID

Dynamic

Reg

Other

Track

adaptive estimation

Servo Motors

Reg

N DOF Manipulator

Track

Passivity

Reg

Nonlinear

Track Robust

System models

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

Position regulation for 1DOF mass system

Southward et.al (1991): ❏ System has similar eq. of motion as a servomotor ❏ Uses static Stribeck and Karnopp friction models ❏ Control law: PD + nonlinear (discontinuous) friction compensation. F K_p x+F_c(x)

Fs x Fs

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Global asymptotic stability proved by LaSalle's theorem using Dini deriviatives Position regulation con rmed by experimental results

Adaptive position tracking for servo

Friedland and Park (1991,1992): ❏ Position tracking of servo with static Coulomb friction ❏ Adaption of unknown Coulomb friction: F^c = z ; kjvj  !  ; 1 z_ = k jvj u ; F (v F^c) sgn(v) ) F^c ! Fc asymptotically. ❏ Control law: u = PD + F (v F^c) ❏ Position tracking con rmed in simulations, also when including viscous friction. ❏ Experimentally con rmed by Mentzelopoulou and Friedland (1994) for Coulomb friction, and by Amin et.al (1997) for viscous friction ❏ Extended to two DOF manipulator by Yazdizadeh and Khoasani (1996)

Friction compensation with static friction models Adaptive velocity tracking for a tracking telescope

Gilbart and Winston (1974): ❏ The rst result on adaptive friction compensation ❏ System: a motor driving an optical telescope ❏ Friction modeled as classical Coulomb friction ❏ MRAC: u = K1(t) _p + K2(t)( m + 12 _m) + K3(t)sgn( _p) {z } | Friction est: ❏ GAS proven by Lyapunov ❏ 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.

High precision position control

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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 small distance  from the reference. Then fuzzy+PD Trackin contr. y m

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Plant Fuzzy

y

PD

Tracking controller: Adaptive sliding mode + friction comp. Paramters in friction model found using Evolution Strategies. Fuzzy rules tuned by Experimental Evolutionary Progr. Experiments con rm position error less than 1 m. Position error in an area dominated by the Dahl eect, emphasizing the need for accurate friction models Extended by Kim et.al. (1996b) to tracking

Adaptive friction compensation in manipulators

Canudas de Wit et.al (1991): ❏ Low velocity tracking of the last link in 2DOF manipulator ❏ Uses the static friction model F (v) = (Fc + 1jvj1=2 + 2v)sgn(v) Linear in unknown parameters ❏ Controller structure u = (m^ + I^)v + mgr ^ cos(q) + F^f v = feed forward + PID F^f = T (t) estimated estimation algorithm: Exponentially weighted least squares. ❏ < 1 used to avoid friction overcompensation, which is shown to cause oscillations

Position regulation of N DOF robot manipulator

Cai and Song (1993): ❏ 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 = ;Kv q_ ; Kpq + G(q) ; ;non ;noni = msi tanh iqi ❏ ❏ ❏

Similar in spirit as Southward et.al (1991), but continuous. Convergence to a set by LaSalle's theorem. Size of the set depends on i. Con rmed by simulations

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

Friction compensation with dynamic friction models Position tracking of airborn servo

Walrath (1984): ❏ Studied stabilization of airborn pointing and tracking telescope ❏ A servomotor produces a corrective torque to compensate for gimbal bearing friction ❏ Observed that friction responds continuously to velocity reversal. Static model not sucient ) Dahl's model ❏ Probably the rst reference to employ a dynamic friction model in control design ❏ Controller : u = Proportional + F^ ❏ The estimate F^ was calculated using the Dahl model. Adaption on model parameter. ❏ Experimentally veri ed. Reported of a factor ve improvement in RMS pointing error

Position control of servomotor

Khorrami et.al (1997): ❏ Considers a servomotor driving a load with friction ❏ Dynamic friction modeled with LuGre model ❏ Uses a robust adaptive variable structure controller: ^ TPx ; T tanh((a + bt)BTPx) u = ;BTPx ; B with update law ^_ = ;kBTPxk2 ; > 0 ❏ The unmeasurable friction states are treated as bounded disturbances ❏ Globally asymptotically stable ❏ Also give similar results using backstepping when considering compliant transmission with friction at both motor and load side. ❏ Result extended in Sankaranarayanan and Khorrami (1997) to the low velocity tracking problem

Position tracking for servomotor

Canudas de Wit and Lischinsky (1997): ❏ Study position tracking for a servomotor ❏ Use dynamic LuGre model ❏ The parameters of the LuGre model estimated numerically from experiments ❏ Fixed friction compensation: u = Js2 xr ; JH (s)e + F^ where H (s) depends on controller choise (PD, ltered PID) ❏ Adapting normal force variations: d z ^ 2 u = Js xr ; JH (s)e ; 0z^ + 1 dt + 2v dz^ = v ; ^0jvj z^ ; ke  d ^ = ; 0jvj z^(z ; z^) dt g(v) dt g(v) m ❏ ❏

Adaption also for temperature changes Convergence e ! 0 by Lyapunov. Con rmed experimentally.

Position tracking for manipulators

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

Position tracking of manipulator

Panteley et.al (1997,1998): ❏ Study position tracking in manipulators using LuGre ❏ Main point: Treat the friction compensation problem as a disturbance rejection problem ❏ The part of the friction force dependent on z is regarded a disturbance, and the linear (viscous) component is compensated by the adaptive controller ❏ Designs an adaptive controller using passivity arguments. ❏ An adaptive Slotine and Li controller strictly passi es the system and an outer loop (tanh) rejects the disturbance: u = ;Kds + ^ = Y1(q q_ qr  q_r ) Y2(q_) Y3(q_ s)] ^_ = ; Ts ; = ;T > 0 ❏

Results con rmed experimentally

Comparative studies of compensation schemes

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

Comparasions, contn.

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

Concluding remarks

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

References on modelling and control of friction Updated May 28, 1998

The literature was collected during the PhD-study of:

Jan Tommy Gravdahl Dept. of Engineering Cybernetics Norwegian University of Science and Technology N-7034 Trondheim, Norway

Telephone +47 73594393 Fax: +47 73594399 E-mail: [email protected]

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