- Email: [email protected]
An Integrated Friction Model with Improved Presliding Behaviour
Copyright © IFAC Robot Control, Nantes, France, 1997
AN INTEGRATED FRICTION MODEL WITH IMPROVED PRESLIDING BEHAVIOUR C. Ganseman J. Swevers 1 T. Prajogo F. AI-Bender
Mechanical Engineering Department Katholieke Universiteit Leuven Celestijnenlaan 300 B, B3001 Heverlee, Belgium E-mail: [email protected]
Abstract: This paper presents a new dynamical friction model which allows accurate modelling both in the sliding and the presliding regions. Transition between these two regions is accomplished without a switching function. The model incorporates a hysteresis function with local memory. This last aspect proves essential for modelling presliding friction that is encountered in real physical situations. The model as a whole can also handle the Stribeck effect and stick-slip behaviour as has been demonstrated by validation on a KUKA IR 361 robot . In this sense, this model can be considered as more complete in comparison with others found in literature. Keywords: Friction, Friction modelling, Identification, Friction compensation
1. INTRODUCTION
arms), these micro-movements may correspond with movements of the order of millimeters in other parts of a mechanism, e.g. in robots (Armstrong-Helouvry et al. , 1994) . 2 the sliding region in which the two contacting surfaces or objects are moving over longer distances.
Friction in mechanical systems is a nonlinear phenomenon which gives rise to a lot of control problems such as static errors when integral control is not present, limit cycles when integral control is too high or stick-slip when the proportional control is too slow. The majority of controllers consider usually only the linear viscous friction and regard the nonlinear friction as a disturbance. Extending this linear viscous friction model by including nonlinear Coulomb friction may yield unsatisfactory results if very accurate control at velocity reversal is required.
The classical description of friction in the sliding region is a static relation between velocity and friction force. The simplest model consists of Coulomb and viscous friction. The inclusion of the Stribeck effect with one or more break points gives a better approximation at low velocities (Armstrong-Helouvry, 1991).
In order to model friction, two different working regions can be distinguished :
Classical friction models do not consider the presliding region: the system does not move as long as the applied force is smaller than the maximum static force . Experiments have however shown that there is a deflection or relative movement in the presliding region, so that the relationship between the deflection and the applied force resembles a nonlinear spring and exhibits a hysteretic behaviour (Futami et al., 1990; Prajogo et al. , 1995) . Armstrong (Armstrong-Helouvry,
1 the presliding region in which the two contacting surfaces or objects are moving only over very short distances (2 - 5IJ.m in steel junctions (Armstrong-Helouvry et al., 1994)) . Due to motion amplification (gears, lever 1 Senior Research Assistant with the N .F .W .O . (Belgian National Fund for Scientific Research)
153
1991) simplifies the presliding friction behaviour: the deflection in presliding is modelled by a linear spring and does not include hysteretic behaviour.
where x represents the position (displacement) of the mass. The friction force F is modelled by a set of equations which depend on a state variable z . This z-variable can be visualised as the average deflection of two sets of contacting bristles which model the local topology of the contacting surfaces (Canudas de Wit et al., 1995).
The displacement-force and velocity-force relationships considered in the classical friction models are static. However, experiments show that friction is a dynamic process. There is e.g. a time lag between a change in displacement or velocity and a change in friction force (Hess and Soom, 1990) . Dabl (Dabl, 1986) proposes a dynamic friction model that captures the nonlinear spring-like behaviour in the presliding region. Canudas de Wit et al. (Canudas de Wit et al., 1995) extend this model. Their model resembles the Dabl model in the presliding region, but in addition it allows arbitrary constant-velocity characteristics, e.g. the Stribeck effect, for the sliding region. Friction in sliding and presliding region are described by the same set of differential equations without switching functions such that the transition between presliding and sliding is smooth. The differential equations depend on a state variable which represents the average deflection of elastic bristles which visualize the topology of the contacting rigid bodies. The resulting model shows dynamical characteristics such as frictionallag, varying break-away force and stickslip motion. However, hysteretic behaviour with local memory between displacement and applied force in presliding as measured by (Futami et al., 1990) cannot be accounted for by this model. This may result in an overestimation of the energy dissipation.
1 The friction force equation yields the friction force based on the current hysteresis transition curve, the derivative of the state variable z and the current velocity:
F
Fd is a point-symmetrical strictly increasing function of z, e.g. a piecewise-linear spring characteristic (Futami et al., 1990). Section 3.2 discusses further details about the implementation and properties of the hysteresis model. 2 The nonlinear state equation is based on the current hysteresis transition curve and the current velocity.
2. FRICTION MODEL EQUATIONS This section describes the dynamic friction equations for a 1-DOF system. The extension to multiple DOF systems is straightforward and does not have any influence on the equations of the friction model.
S(v) models the constant velocity behaviour in sliding. Parameter n is similar to the exponent of the general Dabl model (Dabl, 1986) . It allows to modify the influence of s~d/~k
Consider a mass m subject to a driving force f and a friction force F. Its dynamical equation is:
rPx
(2)
Fh (z) is the hysteresis friction force, i.e. the part of the friction force exhibiting hysteretic behaviour. It is a function of the current value of Z and the friction force history in presliding. Fh(z) is described in detail below. (11 is a micro-viscous damping coefficient, (12 is the viscous damping coefficient, v is the velocity of the mass m. Friction exhibits hysteretic behaviour in presliding between velocity reversals. This part of the friction is modelled by a hysteresis function Fh (z) consisting of transition curves (curves between two reversal points or extrema). Each velocity reversal initiates a new transition curve, adds a new extremum to the hysteresis memory, and resets the state variable z to zero. The transition curve which is active at a certain time will be called the current transition curve, and is represented by Fd(Z). The value of Fh(Z) at the beginning of a transition curve is represented by Fb (see figure 1):
This paper describes a new dynamic friction model which extends the Canudas de Wit model with more accurate modelling in the presliding region: a hysteresis model with arbitrary displacement-force transition curves in the presliding region. Section 2 describes the model equations. Section 3 describes the behaviour of the model in the sliding and the presliding region, and shows that this behaviour corresponds to known friction characteristics.
m=f-F dt 2
dz
= Fh(Z) + (11 dt + (12 V
on the difference between ~~ and v, such that the model behaviour corresponds better to friction measurements in the transition from
(1)
154
The function S(v) determines the constant velocity characteristics in the sliding region near zero velocity while 0"2 V becomes significant at high velocity. Choosing:
Fr..Fb+Fd 0.2
- - _ ._ - - - - _ ._ - - - - - - - - .- - . - - - - .
-
Z
S(v) = Fe
Fd
j~." -
Fb
,
- - _ ._ -
10 position (microns)
-5
15
20
- Fe)
* e-(u";) "
(6)
with Fe the Coulomb friction , Fs the static friction , Vs the Stribeck velocity and ~ an exponential parameter, yields the classical Stribeck effect (Bo, 1982). Extra flexibility can be built into (5) by assigning different values of Fs , Fe, Vs, ~, 0"2 for positive (Ft, Fe+, v;, ~+, an and for negative (Fs-, Fe-, v;, ~- , a;) velocities.
-
-1
+ (Fs
25
Fig. 1. Fb and Fd on the present hysteresis transition curve presliding to sliding. For high values of n , ~~ will be different from v only when Fd(Z) is close to S(v) - Fb ·
From equation 4 it follows that dz / dt = Z reaches an extreme value when Fh(Z) = Fd(Z) + H = S(v) . Since Fh(z) is a strictly increasing function of z , it follows that Fh(Z) also reaches an extreme value at this point. The extreme values of Fh(Z) depend on the velocity v, and approach Fs (Fs- or Ft) as v approaches zero. Remark
o and
3. FRlCTION MODEL BEHAVIOUR This section describes the behaviour of the model for two steady-state solutions (one in sliding (constant velocity) and one in presliding (zero velocity)) and for stick-slip conditions. The analysis of the steady-state solutions allows to gain insight into the different parts of the model equations. It is shown that the behaviour of the model corresponds closely to the measured friction characteristics. Obviously constant velocity and stick-slip characteristics are similar to those of the Canudas de Wit model (Canudas de Wit et al., 1995). The presliding behaviour however shows a significant improvement over the Canudas de Wit model due to the inclusion of a hysteresis model with local memory.
Figure 2 shows results of a constant velocity friction torque measurement on the first joint of an industrial KUKA IR 361 robot. The stars in the figure indicate the measured friction torques at different positive and negative velocities. The full line represents model (5) fitted to the measurements with different parameter values for positive and negative velocities. The figure shows that there is a good correspondence between the measured characteristic and the model, indicating that the proposed parameterization of S(v) (equation (6)) is appropriate.
20
3.1 Constant velocity behaviour in the sliding region
For constant velocity different from zero and in steady state (~~ = 0) , the state equation (4) reduces to: ~L S --~--~ -()7 S ---70--~07 S --~--~"
dz dt
~i.-.ll'l l
.
Fd(Z) Fd(Z) In) = 0 = v(1 - stgn( S(v) - Fb ) * IS(v) - Fb
Fig. 2. Measured friction forceand modelled friction for constant velocities
~
1=
Fd( Z) S(v) - Fb 3.2 Zero velocity behaviour in the presliding region
~
S(v)
= Fd(Z) + Fb
This section shows the improvement in the presliding region of the present friction model over that of Canudas de Wit (Canudas de Wit et al., 1995). The hysteresis model allows to reproduce the experimentally observed friction characteristics in the presliding region. The Canudas
The friction force equation (2) reduces to :
F = Fb
+ Fd(Z) + a2 V
F=S(v)+a2 v
(5) 155
from B to C and then ramped up again from C to D . The displacement-force relation that is obtained does not correspond with experimentally observed friction characteristics ((FUtami et al., 1990) , figure 4) . Figure 4 shows presliding friction torque measurements on the first joint of an industrial KUKA IR 361. During the experiment, the applied force was ramped up and down just like in the simulation. The friction curve in figure 4 is clearly different from the curve in figure 3: the measured curve closes when the applied torque is ramped up from C to D and the friction force returns to the original hysteresis curve. This is typical of hysteresis with local memory and is an essential characteristic of the proposed model that puts it closer to physical reality and distinguishes it from hitherto existing models, e.g. the Canudas de Wit model. The energy dissipation, which is equal to the area below this displacementfriction curve, is much larger with the Canudas de Wit model since the displacement-friction curve is not closed: the resulting displacement with the Canudas de Wit model is much larger.
de Wit model is too dissipative with respect to experimentally observed characteristics. In the presliding region , friction is a function of the relative displacement x. This relationship is often related to the tangential stiffness of the asperities. For small forces and corresponding displacements, the deformation of the asperities is elastic. For larger forces and displacements, the asperities and the surface layer deform plastically. This surface layer may be an oxide film on the metal surface or a layer due to the reaction with additives in the lubricant (ArmstrongHelouvry et al. , 1994). The resulting behaviour is hysteretic (FUtami et al., 1990; Rogers and Boothroyd, 1975) , and can be modelled by a hysteresis model with transition curves which are non linear relations between relative displacement and friction force : e.g. piecewise linear functions (FUtami et al., 1990) . Neither the Dahl model nor the Canudas de Wit model, which resembles the Dahl model in the presliding region , include a hysteresis model with local memory. Both models only show a hysteresis like behaviour when the extremal points lie symmetric with respect to the origin.
10
D
E
e-
The slope of the displacement-force relationship is governed by the velocity and the current value of the force in the Dahl model or by the velocity and the current value of the state variable Z in the Canudas de Wit model. Figure 3 shows the simulation results obtained from the Canudas de Wit model (simulation model and parameter values are taken from (Canudas de Wit et al., 1995)) . In the simulation, the driving force f is
A -20 0
10
5 time (s)
10
§
§. c:
g
A
' (ij
&'-50 0
1.5
10 D
~ 1
5
'0 50 !!!
E
e-
Cl>
0
Cl>
0
.E 0 .5
~
!! -10 .
£ 1 2 time (s)
~40
-20~----------~
3
-50
0 50 position (micro-rad)
Cl)
c:
e0
D
Fig. 4. Whisker-shape measured on KUKA 361 IR robot
§.20 c:
g
C
'(ij
0
c.
0 1.5
~
For zero velocity, which corresponds to steadystate in presliding, equations (2) and (4) reduce to:
0 1 2 time (s)
3
1
F = Fb
+ Fd(Z)
= Fh( z )
dz =0
dt 20 40 poSition (microns)
The hysteresis model relates the state variable Z and the hystersis force F h . The implementation of the hysteresis model requires two memory stacks: one for the minima of Fh in ascending order (stack m) , and one for the maxima of Fh (stack
Fig. 3. Simulation of Canudas de Wit model ramped up from A to B, then ramped down 156
M). The stacks grow at a velocity reversal and shrink when an internal hysteresis loop is closed. The stacks are reset when the system goes from presliding to sliding. The value of Fb equals the most recent element of stack M if the transition curve is descending and of stack m if the transition curve is ascending. The value of the state variable Z is reset to zero at each velocity reversal and recalculated at the closing of an internal loop.
Fi 1 is the inverse function of the strictly increasing function Fd , i.e. z = Fi 1 (Fd (z)). This mechanism of the closing of an internal loop corresponds to the experimentally measured behaviour as shown in figure 4.
The following three mechanisms govern the hysteresis model: 0 .1
1 Velocity reversal: Velocity reversal results in a new extreme value for Fh which has to be added to one of the stacks: a maximum value for Fh is added to stack M, a minimum value for Fh is added to stack m. After velocity reversal a new transition curve is started by setting Fb equal to Fh at the velocity reversal, i.e. the most recent element of the updated stack, and by resetting Fd(Z) and Z to zero. Figure 5 illustrates a reversal from a negative to a positive velocity, and show Fd and Fb before and after velocity reversal.
10
20
~("""'I
30
D.'
-20
-10
0
~(rNauw.)
Fig. 6. Fd and Fb before and after closing of internal loop
3 Resetting of the hysteresis model: The hysteresis behaviour disapears upon going from presliding to sliding. The hysteresis model is reset for strictly positive (strictly negative) velocities v when the hysteresis friction force Fdz) reaches a maximum (minimum) in presliding, i.e. when ~~ becomes zero. This mechanism does not apply to extreme values of Fh(Z) reached at velocity reversals (v = 0) since then the system does not leave the presliding region. Resetting of the hysteresis model corresponds to wiping out the hysteretic history, i.e. wiping out all the hysteresis transition curves which lie within the largest possible hysteresis loop. The largest possible hysteresis loop connects Fs- to Fs+ (see remark 1 in section 3.1) . This is accomplished by resetting stack m to Fs- and stack M to Ft. This corresponds to setting Fb equal to Fs- (F;) for positive (negative) velocities. At the same time, z is set to Fi 1 (S(v) - Fs-) (Fd- 1 (S(v) - F;)), so that the value of Fh(Z) remains unchanged. As a result, this resetting mechanism does not change dz/dt (= 0) and the total friction force F (equations (2) and (4)) .
10 20 30 p(IIIIiIIion (rrDoN.l
Fig. 5. Fd and Fb before and after velocity reversal 2 Closing of an internal loop: At the closing of an internal loop, the extreme values associated with this internal loop are removed from the stack. This is called the wiping out effect of hysteretic behaviour (Mayergoyz, 1991): if a hysteresis loop is closed, this loop is removed from the hysteresis memory, and the future hysteresis behaves as if this closed loop never occured. Figure 6 illustrates the closing of an internal loop by an ascending transition curve. SM (i) and Sm (i) represent the values of stacks M and m respectively. When the internal loop is closed, SM(i) and Sm (i) are removed from these stack, such that the SM(i - 1) and Sm(i - 1) appear at the top of the stacks. The value of Sm (i - 1) becomes the Fb value for the ascending curve. The value of Z is recalculated such that the new ascending curve continues the transition curve that started from Sm (i - 1) and ended at SM(i) (i.e. the beginning of the internal loop), i.e.:
Figure 7 shows measured (solid line) and simulated (dashed line) presliding friction hysteresis loops for the first joint of a KUKA robot. The hysterisis model is based on a simple piece-wise linear transition curve consisting of three linear parts. The figure shows good correspondence between measurments and model. 157
Future work will include application of the model in friction identification and compensation for robots and machine tools which is the underlined objective of this work. The physical reality of the hysteresis behaviour and its model equivalent with resetting of states and updating of stacks might difficult the formal statement of friction compensation schemes.
20
"
-(-
IiI080
;,00
8120
8140
8160
Acknowledgement
8110
Fig. 7. Presliding displacement-torque curve measured on a KUKA IR 361 robot (full line) and predicted by the described hysteresis friction model (dashed line).
This text presents research results of the Belgian programme on Interuniversity Poles of attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibiliby is assumed by its authors.
3.3 Stick-slip behaviour
The model allows to simulate stick-slip behaviour. Stick-slip is caused by the fact that friction at rest is larger than that during motion and is determined mainly by the low-velocity behaviour (function S(v) in equation (4)). It is observed in the first joint of the mentioned KUKA robot when this joint is controlled with a low gain proportional position feedback controller. Figure 8 shows the simulated and the measured stick-slip behaviour for a desired velocity of 0.1 radj s and a proportional feedback gain of 1000Nmjrad. The maximum relative error between a simulated slip distance and a measured slip distance is 6%.
5. REFERENCES Armstrong-Helouvry, B. (1991). Control of Machines with friction. Kluwer Acacemic Publishers. Armstrong-Helouvry, B., P. Dupont and C. Canudas de Wit (1994). A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30(7), 1083-1138. Bo, L. (1982). The friction-speed relation and its influence on the critical velocity of stick-slip motion. Wear 82(3), 277- 289. Canudas de Wit, C., H. Olsson, K. Astrom and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Transactions on Automatic Control 40(3), 419-425. Dahl, P.R. (1986) . A solid friction model. Technical Report TOR-158(3107-18) . The Aerospace Corporation. El Segundo, CA . Futami, S., A. Furutani and S. Yoshida (1990). Nanometer positioning and its microdynamics. Nanotechnology 1(1), 31-37. Hess, D. and A. So om (1990). Friction at a lubricated line contact operating at oscilaating sliding velocities. ASME Journal of Tribology 112(1), 147-152. Mayergoyz, I.D. (1991). Mathematical models of hysteresis. Springer-Verlag. New-York. Prajogo, T ., F . AI-Bender and H. Van Brussel (1995). Identification of pre-rolling friction dynamics of rolling element bearings: modelling and application to precise positioning systems. In: Proc. of the 8th International precision engineering seminar. Compiegne. pp. 229-232. Rogers, P.F. and G. Boothroyd (1975). Damping at metallic interfaces subjected to oscillating tangential loads. ASME Journal of Engineering for Industry pp. 1087-1093.
0 .8 0 .7
0 .3 0 .2 0 .1
°0~L-~----~--~'----~--~'0~~'2 1Ine(1)
Fig. 8. Stick-slip behaviour for model (full line) and KUKA 361 (dashed line)
4. CONCLUSION This paper presents a dynamical friction model which is valid both in the sliding and the presliding regions. Presliding friction is modelled by means of an hysteresis model with local memory. The model can account accurately for experimentally obtained friction characteristics: Stribeck friction in sliding, hysteretic behaviour in presliding, and stick-slip behaviour. 158
AN INTEGRATED FRICTION MODEL WITH IMPROVED PRESLIDING BEHAVIOUR C. Ganseman J. Swevers 1 T. Prajogo F. AI-Bender
Mechanical Engineering Department Katholieke Universiteit Leuven Celestijnenlaan 300 B, B3001 Heverlee, Belgium E-mail: [email protected]
Abstract: This paper presents a new dynamical friction model which allows accurate modelling both in the sliding and the presliding regions. Transition between these two regions is accomplished without a switching function. The model incorporates a hysteresis function with local memory. This last aspect proves essential for modelling presliding friction that is encountered in real physical situations. The model as a whole can also handle the Stribeck effect and stick-slip behaviour as has been demonstrated by validation on a KUKA IR 361 robot . In this sense, this model can be considered as more complete in comparison with others found in literature. Keywords: Friction, Friction modelling, Identification, Friction compensation
1. INTRODUCTION
arms), these micro-movements may correspond with movements of the order of millimeters in other parts of a mechanism, e.g. in robots (Armstrong-Helouvry et al. , 1994) . 2 the sliding region in which the two contacting surfaces or objects are moving over longer distances.
Friction in mechanical systems is a nonlinear phenomenon which gives rise to a lot of control problems such as static errors when integral control is not present, limit cycles when integral control is too high or stick-slip when the proportional control is too slow. The majority of controllers consider usually only the linear viscous friction and regard the nonlinear friction as a disturbance. Extending this linear viscous friction model by including nonlinear Coulomb friction may yield unsatisfactory results if very accurate control at velocity reversal is required.
The classical description of friction in the sliding region is a static relation between velocity and friction force. The simplest model consists of Coulomb and viscous friction. The inclusion of the Stribeck effect with one or more break points gives a better approximation at low velocities (Armstrong-Helouvry, 1991).
In order to model friction, two different working regions can be distinguished :
Classical friction models do not consider the presliding region: the system does not move as long as the applied force is smaller than the maximum static force . Experiments have however shown that there is a deflection or relative movement in the presliding region, so that the relationship between the deflection and the applied force resembles a nonlinear spring and exhibits a hysteretic behaviour (Futami et al., 1990; Prajogo et al. , 1995) . Armstrong (Armstrong-Helouvry,
1 the presliding region in which the two contacting surfaces or objects are moving only over very short distances (2 - 5IJ.m in steel junctions (Armstrong-Helouvry et al., 1994)) . Due to motion amplification (gears, lever 1 Senior Research Assistant with the N .F .W .O . (Belgian National Fund for Scientific Research)
153
1991) simplifies the presliding friction behaviour: the deflection in presliding is modelled by a linear spring and does not include hysteretic behaviour.
where x represents the position (displacement) of the mass. The friction force F is modelled by a set of equations which depend on a state variable z . This z-variable can be visualised as the average deflection of two sets of contacting bristles which model the local topology of the contacting surfaces (Canudas de Wit et al., 1995).
The displacement-force and velocity-force relationships considered in the classical friction models are static. However, experiments show that friction is a dynamic process. There is e.g. a time lag between a change in displacement or velocity and a change in friction force (Hess and Soom, 1990) . Dabl (Dabl, 1986) proposes a dynamic friction model that captures the nonlinear spring-like behaviour in the presliding region. Canudas de Wit et al. (Canudas de Wit et al., 1995) extend this model. Their model resembles the Dabl model in the presliding region, but in addition it allows arbitrary constant-velocity characteristics, e.g. the Stribeck effect, for the sliding region. Friction in sliding and presliding region are described by the same set of differential equations without switching functions such that the transition between presliding and sliding is smooth. The differential equations depend on a state variable which represents the average deflection of elastic bristles which visualize the topology of the contacting rigid bodies. The resulting model shows dynamical characteristics such as frictionallag, varying break-away force and stickslip motion. However, hysteretic behaviour with local memory between displacement and applied force in presliding as measured by (Futami et al., 1990) cannot be accounted for by this model. This may result in an overestimation of the energy dissipation.
1 The friction force equation yields the friction force based on the current hysteresis transition curve, the derivative of the state variable z and the current velocity:
F
Fd is a point-symmetrical strictly increasing function of z, e.g. a piecewise-linear spring characteristic (Futami et al., 1990). Section 3.2 discusses further details about the implementation and properties of the hysteresis model. 2 The nonlinear state equation is based on the current hysteresis transition curve and the current velocity.
2. FRICTION MODEL EQUATIONS This section describes the dynamic friction equations for a 1-DOF system. The extension to multiple DOF systems is straightforward and does not have any influence on the equations of the friction model.
S(v) models the constant velocity behaviour in sliding. Parameter n is similar to the exponent of the general Dabl model (Dabl, 1986) . It allows to modify the influence of s~d/~k
Consider a mass m subject to a driving force f and a friction force F. Its dynamical equation is:
rPx
(2)
Fh (z) is the hysteresis friction force, i.e. the part of the friction force exhibiting hysteretic behaviour. It is a function of the current value of Z and the friction force history in presliding. Fh(z) is described in detail below. (11 is a micro-viscous damping coefficient, (12 is the viscous damping coefficient, v is the velocity of the mass m. Friction exhibits hysteretic behaviour in presliding between velocity reversals. This part of the friction is modelled by a hysteresis function Fh (z) consisting of transition curves (curves between two reversal points or extrema). Each velocity reversal initiates a new transition curve, adds a new extremum to the hysteresis memory, and resets the state variable z to zero. The transition curve which is active at a certain time will be called the current transition curve, and is represented by Fd(Z). The value of Fh(Z) at the beginning of a transition curve is represented by Fb (see figure 1):
This paper describes a new dynamic friction model which extends the Canudas de Wit model with more accurate modelling in the presliding region: a hysteresis model with arbitrary displacement-force transition curves in the presliding region. Section 2 describes the model equations. Section 3 describes the behaviour of the model in the sliding and the presliding region, and shows that this behaviour corresponds to known friction characteristics.
m=f-F dt 2
dz
= Fh(Z) + (11 dt + (12 V
on the difference between ~~ and v, such that the model behaviour corresponds better to friction measurements in the transition from
(1)
154
The function S(v) determines the constant velocity characteristics in the sliding region near zero velocity while 0"2 V becomes significant at high velocity. Choosing:
Fr..Fb+Fd 0.2
- - _ ._ - - - - _ ._ - - - - - - - - .- - . - - - - .
-
Z
S(v) = Fe
Fd
j~." -
Fb
,
- - _ ._ -
10 position (microns)
-5
15
20
- Fe)
* e-(u";) "
(6)
with Fe the Coulomb friction , Fs the static friction , Vs the Stribeck velocity and ~ an exponential parameter, yields the classical Stribeck effect (Bo, 1982). Extra flexibility can be built into (5) by assigning different values of Fs , Fe, Vs, ~, 0"2 for positive (Ft, Fe+, v;, ~+, an and for negative (Fs-, Fe-, v;, ~- , a;) velocities.
-
-1
+ (Fs
25
Fig. 1. Fb and Fd on the present hysteresis transition curve presliding to sliding. For high values of n , ~~ will be different from v only when Fd(Z) is close to S(v) - Fb ·
From equation 4 it follows that dz / dt = Z reaches an extreme value when Fh(Z) = Fd(Z) + H = S(v) . Since Fh(z) is a strictly increasing function of z , it follows that Fh(Z) also reaches an extreme value at this point. The extreme values of Fh(Z) depend on the velocity v, and approach Fs (Fs- or Ft) as v approaches zero. Remark
o and
3. FRlCTION MODEL BEHAVIOUR This section describes the behaviour of the model for two steady-state solutions (one in sliding (constant velocity) and one in presliding (zero velocity)) and for stick-slip conditions. The analysis of the steady-state solutions allows to gain insight into the different parts of the model equations. It is shown that the behaviour of the model corresponds closely to the measured friction characteristics. Obviously constant velocity and stick-slip characteristics are similar to those of the Canudas de Wit model (Canudas de Wit et al., 1995). The presliding behaviour however shows a significant improvement over the Canudas de Wit model due to the inclusion of a hysteresis model with local memory.
Figure 2 shows results of a constant velocity friction torque measurement on the first joint of an industrial KUKA IR 361 robot. The stars in the figure indicate the measured friction torques at different positive and negative velocities. The full line represents model (5) fitted to the measurements with different parameter values for positive and negative velocities. The figure shows that there is a good correspondence between the measured characteristic and the model, indicating that the proposed parameterization of S(v) (equation (6)) is appropriate.
20
3.1 Constant velocity behaviour in the sliding region
For constant velocity different from zero and in steady state (~~ = 0) , the state equation (4) reduces to: ~L S --~--~ -()7 S ---70--~07 S --~--~"
dz dt
~i.-.ll'l l
.
Fd(Z) Fd(Z) In) = 0 = v(1 - stgn( S(v) - Fb ) * IS(v) - Fb
Fig. 2. Measured friction forceand modelled friction for constant velocities
~
1=
Fd( Z) S(v) - Fb 3.2 Zero velocity behaviour in the presliding region
~
S(v)
= Fd(Z) + Fb
This section shows the improvement in the presliding region of the present friction model over that of Canudas de Wit (Canudas de Wit et al., 1995). The hysteresis model allows to reproduce the experimentally observed friction characteristics in the presliding region. The Canudas
The friction force equation (2) reduces to :
F = Fb
+ Fd(Z) + a2 V
F=S(v)+a2 v
(5) 155
from B to C and then ramped up again from C to D . The displacement-force relation that is obtained does not correspond with experimentally observed friction characteristics ((FUtami et al., 1990) , figure 4) . Figure 4 shows presliding friction torque measurements on the first joint of an industrial KUKA IR 361. During the experiment, the applied force was ramped up and down just like in the simulation. The friction curve in figure 4 is clearly different from the curve in figure 3: the measured curve closes when the applied torque is ramped up from C to D and the friction force returns to the original hysteresis curve. This is typical of hysteresis with local memory and is an essential characteristic of the proposed model that puts it closer to physical reality and distinguishes it from hitherto existing models, e.g. the Canudas de Wit model. The energy dissipation, which is equal to the area below this displacementfriction curve, is much larger with the Canudas de Wit model since the displacement-friction curve is not closed: the resulting displacement with the Canudas de Wit model is much larger.
de Wit model is too dissipative with respect to experimentally observed characteristics. In the presliding region , friction is a function of the relative displacement x. This relationship is often related to the tangential stiffness of the asperities. For small forces and corresponding displacements, the deformation of the asperities is elastic. For larger forces and displacements, the asperities and the surface layer deform plastically. This surface layer may be an oxide film on the metal surface or a layer due to the reaction with additives in the lubricant (ArmstrongHelouvry et al. , 1994). The resulting behaviour is hysteretic (FUtami et al., 1990; Rogers and Boothroyd, 1975) , and can be modelled by a hysteresis model with transition curves which are non linear relations between relative displacement and friction force : e.g. piecewise linear functions (FUtami et al., 1990) . Neither the Dahl model nor the Canudas de Wit model, which resembles the Dahl model in the presliding region , include a hysteresis model with local memory. Both models only show a hysteresis like behaviour when the extremal points lie symmetric with respect to the origin.
10
D
E
e-
The slope of the displacement-force relationship is governed by the velocity and the current value of the force in the Dahl model or by the velocity and the current value of the state variable Z in the Canudas de Wit model. Figure 3 shows the simulation results obtained from the Canudas de Wit model (simulation model and parameter values are taken from (Canudas de Wit et al., 1995)) . In the simulation, the driving force f is
A -20 0
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Fig. 4. Whisker-shape measured on KUKA 361 IR robot
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For zero velocity, which corresponds to steadystate in presliding, equations (2) and (4) reduce to:
0 1 2 time (s)
3
1
F = Fb
+ Fd(Z)
= Fh( z )
dz =0
dt 20 40 poSition (microns)
The hysteresis model relates the state variable Z and the hystersis force F h . The implementation of the hysteresis model requires two memory stacks: one for the minima of Fh in ascending order (stack m) , and one for the maxima of Fh (stack
Fig. 3. Simulation of Canudas de Wit model ramped up from A to B, then ramped down 156
M). The stacks grow at a velocity reversal and shrink when an internal hysteresis loop is closed. The stacks are reset when the system goes from presliding to sliding. The value of Fb equals the most recent element of stack M if the transition curve is descending and of stack m if the transition curve is ascending. The value of the state variable Z is reset to zero at each velocity reversal and recalculated at the closing of an internal loop.
Fi 1 is the inverse function of the strictly increasing function Fd , i.e. z = Fi 1 (Fd (z)). This mechanism of the closing of an internal loop corresponds to the experimentally measured behaviour as shown in figure 4.
The following three mechanisms govern the hysteresis model: 0 .1
1 Velocity reversal: Velocity reversal results in a new extreme value for Fh which has to be added to one of the stacks: a maximum value for Fh is added to stack M, a minimum value for Fh is added to stack m. After velocity reversal a new transition curve is started by setting Fb equal to Fh at the velocity reversal, i.e. the most recent element of the updated stack, and by resetting Fd(Z) and Z to zero. Figure 5 illustrates a reversal from a negative to a positive velocity, and show Fd and Fb before and after velocity reversal.
10
20
~("""'I
30
D.'
-20
-10
0
~(rNauw.)
Fig. 6. Fd and Fb before and after closing of internal loop
3 Resetting of the hysteresis model: The hysteresis behaviour disapears upon going from presliding to sliding. The hysteresis model is reset for strictly positive (strictly negative) velocities v when the hysteresis friction force Fdz) reaches a maximum (minimum) in presliding, i.e. when ~~ becomes zero. This mechanism does not apply to extreme values of Fh(Z) reached at velocity reversals (v = 0) since then the system does not leave the presliding region. Resetting of the hysteresis model corresponds to wiping out the hysteretic history, i.e. wiping out all the hysteresis transition curves which lie within the largest possible hysteresis loop. The largest possible hysteresis loop connects Fs- to Fs+ (see remark 1 in section 3.1) . This is accomplished by resetting stack m to Fs- and stack M to Ft. This corresponds to setting Fb equal to Fs- (F;) for positive (negative) velocities. At the same time, z is set to Fi 1 (S(v) - Fs-) (Fd- 1 (S(v) - F;)), so that the value of Fh(Z) remains unchanged. As a result, this resetting mechanism does not change dz/dt (= 0) and the total friction force F (equations (2) and (4)) .
10 20 30 p(IIIIiIIion (rrDoN.l
Fig. 5. Fd and Fb before and after velocity reversal 2 Closing of an internal loop: At the closing of an internal loop, the extreme values associated with this internal loop are removed from the stack. This is called the wiping out effect of hysteretic behaviour (Mayergoyz, 1991): if a hysteresis loop is closed, this loop is removed from the hysteresis memory, and the future hysteresis behaves as if this closed loop never occured. Figure 6 illustrates the closing of an internal loop by an ascending transition curve. SM (i) and Sm (i) represent the values of stacks M and m respectively. When the internal loop is closed, SM(i) and Sm (i) are removed from these stack, such that the SM(i - 1) and Sm(i - 1) appear at the top of the stacks. The value of Sm (i - 1) becomes the Fb value for the ascending curve. The value of Z is recalculated such that the new ascending curve continues the transition curve that started from Sm (i - 1) and ended at SM(i) (i.e. the beginning of the internal loop), i.e.:
Figure 7 shows measured (solid line) and simulated (dashed line) presliding friction hysteresis loops for the first joint of a KUKA robot. The hysterisis model is based on a simple piece-wise linear transition curve consisting of three linear parts. The figure shows good correspondence between measurments and model. 157
Future work will include application of the model in friction identification and compensation for robots and machine tools which is the underlined objective of this work. The physical reality of the hysteresis behaviour and its model equivalent with resetting of states and updating of stacks might difficult the formal statement of friction compensation schemes.
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Acknowledgement
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Fig. 7. Presliding displacement-torque curve measured on a KUKA IR 361 robot (full line) and predicted by the described hysteresis friction model (dashed line).
This text presents research results of the Belgian programme on Interuniversity Poles of attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibiliby is assumed by its authors.
3.3 Stick-slip behaviour
The model allows to simulate stick-slip behaviour. Stick-slip is caused by the fact that friction at rest is larger than that during motion and is determined mainly by the low-velocity behaviour (function S(v) in equation (4)). It is observed in the first joint of the mentioned KUKA robot when this joint is controlled with a low gain proportional position feedback controller. Figure 8 shows the simulated and the measured stick-slip behaviour for a desired velocity of 0.1 radj s and a proportional feedback gain of 1000Nmjrad. The maximum relative error between a simulated slip distance and a measured slip distance is 6%.
5. REFERENCES Armstrong-Helouvry, B. (1991). Control of Machines with friction. Kluwer Acacemic Publishers. Armstrong-Helouvry, B., P. Dupont and C. Canudas de Wit (1994). A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30(7), 1083-1138. Bo, L. (1982). The friction-speed relation and its influence on the critical velocity of stick-slip motion. Wear 82(3), 277- 289. Canudas de Wit, C., H. Olsson, K. Astrom and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Transactions on Automatic Control 40(3), 419-425. Dahl, P.R. (1986) . A solid friction model. Technical Report TOR-158(3107-18) . The Aerospace Corporation. El Segundo, CA . Futami, S., A. Furutani and S. Yoshida (1990). Nanometer positioning and its microdynamics. Nanotechnology 1(1), 31-37. Hess, D. and A. So om (1990). Friction at a lubricated line contact operating at oscilaating sliding velocities. ASME Journal of Tribology 112(1), 147-152. Mayergoyz, I.D. (1991). Mathematical models of hysteresis. Springer-Verlag. New-York. Prajogo, T ., F . AI-Bender and H. Van Brussel (1995). Identification of pre-rolling friction dynamics of rolling element bearings: modelling and application to precise positioning systems. In: Proc. of the 8th International precision engineering seminar. Compiegne. pp. 229-232. Rogers, P.F. and G. Boothroyd (1975). Damping at metallic interfaces subjected to oscillating tangential loads. ASME Journal of Engineering for Industry pp. 1087-1093.
0 .8 0 .7
0 .3 0 .2 0 .1
°0~L-~----~--~'----~--~'0~~'2 1Ine(1)
Fig. 8. Stick-slip behaviour for model (full line) and KUKA 361 (dashed line)
4. CONCLUSION This paper presents a dynamical friction model which is valid both in the sliding and the presliding regions. Presliding friction is modelled by means of an hysteresis model with local memory. The model can account accurately for experimentally obtained friction characteristics: Stribeck friction in sliding, hysteretic behaviour in presliding, and stick-slip behaviour. 158