YNAMIC HYSTERESIS MODEL DERIVATED FROM LuGre

MODEL

Sinuhe Benitez

CITEDI-IPN

2498 Roll Dr. 757 Otay Mesa San Diego, California, 92154 USA

Leonardo Acho and Ricardo Guerra

CITEDI-IPN

2498 Roll Dr. 757 Otay Mesa San Diego, California, 92154 USA

Keywords:

Hysteresis, Dynamic Model, Friction.

Abstract:

This paper presents a dynamic hysteresis model; which is a modiﬁcation of the well known LuGre model. This

model has been based on a modiﬁcation to the LuGre model dynamic, which could be seen as a forward and

backward displacement in the steady state solution of the dynamic LuGre model. The LuGre friction model

is based on the average deﬂection of the bristles; implicitly, it is based on the relationship between stress and

strain of the bristles under deformation. From the friction model point of view, this dynamic hysteresis model

can capture the deformation behaviour between stress and strain beyond the elasticity region for the material

(the bristles), a region where the relationship between stress and strain is no longer linear. So, our model can

capture the friction phenomena of the original LuGre model and presents a new behaviour in the pre-sliding

regime. Simulation results are presented to support our contribution.

1 INTRODUCTION

While working in mechanical applications, it is im-

portant to have an accurate model of the system in

question. One of the more important considerations

that must be taken into account in any mechanical sys-

tem is the friction phenomenon. Many attempts have

been made to reproduce the effects caused by friction,

which is a non-linear effect. The importance of mod-

eling friction lies in the simple fact that all mechani-

cal systems present this phenomenon, moreover, be-

ing that it has non-linear behaviour, linear controllers

have very poor performance when trying to overcome

friction.

The ﬁrst attempts made to model friction were sta-

tic models, such as the Coulomb model and the Vis-

cous friction model. The performance of these mod-

els was surpassed by the Dahl friction model (Dahl,

1968) and, more recently the LuGre friction model

published by Carlos Canudas et al. (C. Canudas de

Wit, 1995), which is at present one of the more widely

used models for friction compensation. It has the

ability to reproduce behaviours such as stick-slip mo-

tion, limit cycles, pre-sliding displacement and vary-

ing break away force. The basis for this model is an

abstraction that supposes the existence of microscopic

bristles on the surfaces in contact where friction is

present, these bristles are elastic and hence present

deformation which is captured in the model by the

introduction of an internal state which represents the

average deﬂection of said bristles. The interactions

between the bristles are responsible for the effects col-

lectively referred to as the friction phenomenon.

Research has been carried out to extend the LuGre

model but it relies on the usage of memory stacks

to reproduce the hysteresis phenomenon (J. Swevers,

2000). The main disadvantage of this approach is that

the hysteresis behaviour is modeled by a static func-

tion.

All the friction models presented recreate the elastic

behaviours of contact surfaces and the breakaway be-

haviours, but there is no model that can reproduce the

effects that occur between the elastic region and the

breakaway region, such as the segment of the stress-

strain curve (Young’s modulus and Hooke’s law) that

corresponds to non-elastic deformation region.

This paper presents a new dynamic hysteresis model

derived from the LuGre model. The inclusion of

the hysteresis friction model within the LuGre fric-

tion model yields results that approximate physical

closer to the reality than the previous models. The

new model shows a ﬂat segment at the corners of the

pre-sliding displacement behaviour, produced by the

non-elastic deformation shown in Young’s modulus.

179

Benitez S., Acho L. and Guerra R. (2005).

DYNAMIC HYSTERESIS MODEL DERIVATED FROM LuGre MODEL.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 179-183

DOI: 10.5220/0001156201790183

 SciTePress

The paper is structured as follows: Section II reviews

the LuGre friction model, Section III presents the

Hysteresis model, Section IV shows the numeric sim-

ulation results, ﬁnally, the conclusions are given in

Section V.

2 THE LUGRE MODEL

The LuGre model consists of an internal state, which

is used to produce friction force, (C. Canudas de Wit,

1995), this model is as follow.

˙z = v −

|v|

g(v)

z (1)

f = σ

z + σ

˙z + σ

v (2)

g(v) = (F

+ (F

− F

−(v/v

)

))/σ

(3)

where z is the state, v is the relative velocity between

surfaces, g(v) a function that models the constant ve-

locity behaviour, σ

an equivalent stiffness for the

position-force relationship at velocity reversal, σ

the

micro-viscous friction coefﬁcient, and σ

the viscous

friction coefﬁcient. F

and F

are the Coulomb fric-

tion level and the level of stiction, respectively.

This model can capture most of the known friction

behaviours like pre-sliding displacement, friction lag,

varying breakaway force and stick-slip motion, and

limit cycles, (C. Canudas de Wit, 1995).

Table 1: Parameters values used in all simulations.

Parameters Values Unit

s 10

[N/m]

√

[N–S/m]

0.4 [N–S/m]

1 [N]

1.5 [N]

0.001 [m/s]

2.1 Pre-sliding effect

If a force is applied to a pair of surfaces in contact

there will be a displacement. A simulation was per-

formed in (C. Canudas de Wit, 1995), with an applied

force slowly ramped up to 1.425 N which is 95% of

. The force was then kept constant for a while and

later ramped down to -1.425 N, where it was kept con-

stant and then ramped up to 1.425 N again. The re-

sults of the simulation are shown in (Fig. 1), this be-

haviour agrees qualitatively with the experiment re-

sults in (J. Courtney-Pratt, 1957).

Figure 1: Pre-sliding displacement.

2.2 Friction Lag

Hysteresis behaviour in the relationship between fric-

tion and velocity was shown by Hess and Soom

(D.P. Hess, 1990) while they were studying the dy-

namic behaviour of friction when velocity is varied

during unidirectional motion. While friction force is

lower for decreasing velocities than for increasing ve-

locities, the hysteresis loop becomes wider at higher

rates of velocity change. The LuGre model can cap-

ture this effect as is shown in (Fig. 2).

Figure 2: Friction Lag

2.3 Breakaway Force

The breakaway force can be investigated through ex-

periments with stick-slip motion. In (C. Canudas de

Wit, 1995) simulations were performed using the dy-

namic model, where a force was applied to a unit mass

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180

and ramped up at different rates, and the friction force

when the mass started to slide was determined. The

Breakaway force was therefore determined at the time

where a sharp increase in the velocity could be ob-

served, (Fig. 3) shows the force at breakaway as a

function of the rate of increase of the applied force.

Figure 3: Breakaway Force

2.4 Stick-Slip Motion

Stick-slip motion is a typical behaviour for systems

with friction. It is caused by the fact that friction is

larger at rest than during motion. In (C. Canudas de

Wit, 1995) the author used in his experiments a unit

mass attached to a spring with stiffness k = 2N/m,

and a constant velocity dy/dt = 0.1m/s, experimen-

tal results are shown in (Fig. 4).

Figure 4: Stick-Slip motion

2.5 Limit Cycle Cause by Friction

Limit cycles are another phenomenon produce by

friction, this behaviour is shown in (C. Canudas de

Wit, 1995) applying a friction force to a unit mass

= u − F (4)

with F as a friction force given by (Eq. 2), and u as

control force given by a PID controller

u = −k

v − K

(x − xd) −K

(x − xd) (5)

simulation results are shown in (Fig. 5)

Figure 5: Limit cycle

In (C. Canudas de Wit, 1995) it has been proven that

the LuGre model can capture almost all the effects

caused by friction, but still some phenomena are miss-

ing for instance, what happens when a bristle is out

of the elastic region but not yet in the breakaway re-

gion, i.e. when it is in the non-elastic deformation

region, such kind of phenomena are not captured by

LuGre model, here the motivation is to improve this

dynamic friction model in order to be able to capture

more phenomena produced by friction.

3 HYSTERESIS MODEL

Some papers have been published improving the

pre-sliding behaviour adding to the friction equation

(Eq. 1) a static hysteresis friction force, see (J. Sw-

evers, 2000) (C. Ganseman, 1997), which is a static

function, these improvements can reproduce all the

effects shown by the LuGre model.

But neither the LuGre model nor its improvement

DYNAMIC HYSTERESIS MODEL DERIVATED FROM LuGre MODEL

181

(J. Swevers, 2000) can reproduce the pre-sliding

displacement in the non-elastic deformation region

shown on the stress-strain curve of Young’s modulus

(F. W. Sears, 1957) (Fig. 6).

Figure 6: Stress-Strain Curve.

The dynamic hysteresis friction model proposed is

derived from the forward and backward displacement

of the steady state solution of the LuGre friction

model. The steady-state solution is (C. Canudas de

Wit, 1995).

|v|

g(v) = g(v)sgn(v) (6)

If we add a displacement a, we get

v ± a

|v ± a|

g(v) = g(v)sgn(v ± a) (7)

After this displacement, represented by the parameter

a, it is possible to obtain the behaviour shown in (Fig.

˙z =

(

(v + a) −

|v + a|

g (v)

z ˙v > 0

(v − a) −

|v −a|

g (v)

z ˙v < 0

(8)

with a as the displacement factor that can be a con-

stant or could be a function of the state variable z,

where a, in some way has a relation with the non-

elastic deformation width (non-elastic deformation

region Fig. 6), this region is located between the elas-

ticity and breakaway regions of the stress-strain curve

(Fig. 6), with this forward and backward displace-

ment it is possible to see the deformation segment of

the stress-strain curve, and reproduce all the friction

effects reproduced by the LuGre model.

This improvement comes as it was mentioned, from

the steady state solution of (Eq. 1)

Figure 7: Hysteresis Behavior.

4 NUMERICAL SIMULATION

In order to show the validity of this model, simula-

tions were run on Matlab version 6.0 and the results

are shown in the following ﬁgures. The same para-

meters used in (C. Canudas de Wit, 1995) were used

in these simulations, where it was found that pro-

posed model can capture the same phenomena that

LuGre, such as Frictional lag, varying Breakaway

force, Stick-slip motion, and limit cycles, but a new

phenomenon was found in the pre-sliding regime,

which is assumed to be the non-elastic deformation

of the bristles while a force is applied before entering

the rupture zone where the displacement starts.

Figure (8) shows the pre-sliding displacement for

the model considering that a is a constant value of

0.000002

Figure (9) shows the friction force in the pre-sliding

regime behaviour plotted against displacement, for

the model considering that a is a minimum function

between a constant value of 2 × 10

−6

and the norm

of the internal state z; i.e., a = min{2 × 10

−6

, kzk}.

It is possible to see a high correlation between (Fig.

9) and (J. Swevers, 2000, Fig. 10), which is an exper-

imental result of displacement-torque curve from the

ﬁrst axis of the KUKA IR 361 robot, the non-elastic

deformation region can be seen in both ﬁgures.

Figure (10) shows the friction force in the pre-sliding

regime behaviour plotted against displacement, for

the model considering that a is a maximum function

between a constant value of 2×10

−6

and the norm of

the internal state z;i.e., a = max{2 × 10

−6

, kzk}.

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182

Figure 8: Pre-Sliding displacement.

5 CONCLUSION

A dynamic hysteresis friction model was presented

which is valid in both the sliding and pre-sliding

regimes. The system is able to accurately capture fric-

tion characteristics, experimentally obtained such as:

Stribeck effect during sliding, hysteresis behaviour in

pre-sliding, frictional lag, varying breakaway force,

and stick-slip behaviour, as well as limit cycles. It

is concluded that the developed dynamic hysteresis

friction model can capture any friction behaviour de-

scribed in the literature or experimentally observed,

but the main improvement with respect to the LuGre

model is that the new model can even capture non-

elastic deformation, a phenomenon not captured by

any dynamic friction model to our knowledge. It is

for these reasons that we consider that this model will

be of great use, particularly in mechanical applica-

tions such as system modeling and control by friction

compensation; all though, future work is needed to

achieve implementation in the afore mentioned areas.

REFERENCES

C. Canudas de Wit, H. Olsson, K. A. P. L. (1995). A new

model for control of system with friction. IEEE Trans.

on Automatic Control, (3):419–425.

C. Ganseman, J. Swevers, T. P. F. A. B. (1997). An in-

tegrated friction model with improved presliding be-

havior. In Proc. Symp. Robot Control, pages 221–226,

Nates, France.

Dahl, P. (1968). A solid friction model. Technical Report

TOR-0158(3107-18)-1,1968, Aerospace Corp.

Figure 9: Pre-Sliding displacement.

Figure 10: Pre-sliding displacement.

D.P. Hess, A. S. (1990). Friction at a lubricated line contact

operating at oscillating sliding velocities. J. Tribology,

pages 147–152.

F. W. Sears, M. W. Zemansky, H. D. Y. R. A. F. (1957).

Fisica Universitaria. Addison Wesley Longman,

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J. Courtney-Pratt, E. E. (1957). The effect of a tangential

force on the contact of metallic bodies. In Proc. Royal

Society, pages 529–550, Nates, France.

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grated friction model structure with improved preslid-

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Trans. on Automatic Control, (4):675–686.

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