Static friction model structures

Static models of friction capture friction as a function of the velocity relative to both surfaces. Under the static friction the following models will be reviewed: Classical, Karnopp, model proposed by Lorinc, exponential, and neural network based models.

Classical model variations

This friction model is made up different component which individually capture different aspects of the friction phenomena. The component make-up of the classical model are: coulomb, viscous, static, and Stribeck friction.

Coulomb friction

This model simply states that the friction force is directly proportional to the normal loading between the surfaces and acts in such a way as to oppose the motion of these bodies. It is not dependent on the relative velocity. Mathematically the friction force is

Ff = µkNsgn(v) (2.3)

and with Fc= µkN then

F_f = Fcsgn(v) (2.4)

where Ff is the friction force, µk is the coefficient of kinetic friction, N is normal loading

and v is relative velocity, Fcis the Coulomb friction force. sgn is a signum function.

The coulomb force takes on either of two values of equal magnitude, but of opposite direction depending on the direction of the motion as shown in Fig 2.7a. This model finds application in many engineering systems because of its simplicity, and yields approximate values of the friction force mostly at increasing velocities. Because of this it has been used in friction modelling and compensation [40], [41], [42]. However it does not capture the dynamics of the friction phenomena especially in the zero velocity regions.

Viscous friction

The viscous friction force is proportional to the velocity of relative motion, hence as velocity increases or decreases this friction force increases or decreases in proportion. This is friction as a result of contact lubrication. However, the viscous friction coefficient is quite small such that friction due to viscosity is very small for low velocities. The friction force as a result of viscosity is often modelled as linear function of the velocity as

with fvbeing the coefficient of viscous friction and other terms as previously defined.

The viscous friction is usually incorporated into the coulomb model yielding the Coulomb+viscous model as in eqn. 2.6 to account for the fact that friction actually is dependent on the relative velocity of motion. This is represented as in [45], mathematically .

F_f = µNsgn(v) + fvv (2.6)

This implies that during sliding the friction force is the sum of the coulomb and viscous forces, figure 2.7b.

Static friction

The friction force in the pre-sliding and the gross-sliding regimes are different, it is shown that for most materials the sliding friction is lower than the friction at no sliding. The static friction (or stiction) is the amount of force required to initiate motion. The stiction value is dependent on the applied external force and not on the velocity. It increases with this force till a threshold is reached beyond which there is sliding. This force threshold is called the breakaway or Stiction force. The underlying concept is that the friction force is independent of the contact area and the relative velocity of motion and it opposes motion. So the stiction force is modelled thus

Ff =    Fe if v = 0, |Fe| < Fs F_ssgn(Fe) if v = 0, |Fe| ≥ Fs (2.7)

where Fs is the static friction, v relative velocity, Fe the applied external force. The static

friction is related to the normal force through

F_s= µsN

where µs is the coefficient of static friction.

In order to reflect this in the coulomb base model the friction at no sliding is depicted as static friction while that during sliding is called the Coulomb (kinetic) friction as in figure 2.7c. The modified classical structure is then of the form

F_f =          F_csgn(v) + fvv if v ̸= 0 F_e if v = 0, |Fe| < Fs Fssgn(Fe) if v = 0, |Fe| ≥ Fs (2.8)

Stribeck friction

From experiments it was observed that the transition of friction from the static to the kinetic regime is not sudden but gradual with a continuous drop in the friction value from the static value towards the coulomb friction value, as relative velocity is increased from zero. This effect is called the Stribeck effect, and the friction-velocity relation is called the Stribeck curve [17], see figure 2.7d.

The Stribeck curve is usually modelled by exponential functions of the form

F_f = Fcsgn(v) + (Fs− Fc)e−(

v

vs)δ _(2.9)

where vs is the transition velocity called the Stribeck velocity and the δ a curve shaping

parameter.

Hence the combination of eqn. 2.8 with the exponential Stribeck friction eqn. 2.9 gives rise to a more complex but better representation of the friction features of (2.10). Mathematically it is represented as [46]. F_f =          F(v) if v ̸= 0 F_e if v = 0, |Fe| < Fs F_ssgn(Fe) if v = 0, |Fe| ≥ Fs (2.10)

with F(v) being the friction force as a function of the velocity.

F(v) = Fcsgn(v) + (Fs− Fc)e−(

v

vs)δ_{+ fv}v (2.11)

The eqn. 2.11, which is a Gaussian variant of the exponential models [5], is often useful for the steady state identification of friction model parameters as will be shown in subsequent chapters.

The classical model of friction takes on a continuum of values bounded by the stiction at the upper and lower bounds, and is discontinuous at zero velocity. This discontinuity posses simulation and implementation challenges.

In the paper [47], a modification of the stiction force was proposed to reflect its dependence on the rate of change of the applied force (section 2.2). This modification closely captures observed features of the friction phenomena before gross sliding begins as the velocity varies as suggested in the paper. In the paper the stiction is replaced by a function

Φs( ˙τ ) = Fc+ (Fs− Fc)e−|

˙ τ

Fig. 2.7 Variations of the classical friction model: (a) Coulomb friction, (b) Coulomb+Viscous friction, (c) Stiction+Coulomb+Viscous friction, and (d) Stribeck+Viscous friction

where Φs( ˙τ ) is the static friction force as a function of the applied torque rate of change, ˙τ the

rate of change of applied torque, ts is the torque rate coefficient, δ a curve fitting parameter.

This exponential function is bounded between the stiction Fs and the Coulomb friction Fc

forces.

Karnopp model

The variations of the classical friction model discussed above are not capable of handling friction dynamics resulting in the pre-sliding regime. The Karnopp model [reference], was developed to address some of the shortcomings of the classical models namely the friction force at zero velocity. To achieve this, Karnopp introduced a low velocity range Dvsuch that

if |v| ≤ |Dv| then the system is assumed to be in stick mode. The friction force within this

band is equivalent to the static friction Fs or the applied external force Fe. As the applied

external force exceeds the stiction, the body experiences an increasing velocity from the rest position beyond Dv. Beyond Dvthe model switches to the slip mode. This model is easy to

2.13 by F_f =          F_csgn(v) + fvv if |v| ≥ |Dv| min(Fe, Fs) if Fe≥ 0 max(Fe, −Fs) if Fe≤ 0, |v| < |Dv| (2.13)

Some of the limitations of the Karnopp friction model are;

1. It is application specific; the model is bespoke for every configuration so one cannot use the same model for different applications, [46].

2. System coupling; the model forms part of the system configuration.

3. Zero velocity deviation; there is no cohesion between the model output and the output of observed real systems.

4. It’s complexity; the model complexity increases with system complexity geometrically and this possess strong limitation to its applicability, [33].

Lorinc model

In his paper [5], Lorinc presents a static model for friction non-linearity. The model is essentially a linearised version of the exponential model (Tustin variation) of the friction force. The formulation assumes a static mapping of the friction force as a function of velocity and is subject to implementation problems at zero velocities. For Positive velocities the friction force (Ff) representation is

F_f = ΦT_fξf(ω) (2.14)

where the parameter vector is Φf, and ξf(ω) the regressor vector, The Lorinc friction model

has been applied in control systems problems as in [7], [39]. Model realisation uses two different linear equations of the Tustin model to reproduce the low and high velocity regimes and a switching function capable of detecting the Stribeck switch velocity. Other dynamic features of friction like frictional lag, pre-sliding hysteresis are difficult to reproduce using this model. Being an approximate linear model though simple, it does not reflect true friction at low velocities.

Neural network based models

Soft approaches based on the neural networks (NN) are increasingly being explored in the modelling of system friction, [10]. This is driven by the ease by which the networks can

easily approximate any continuous function to a high degree of accuracy so long as the network is large enough. This approximation property is used in generating non-linear mapping of the inputs and outputs data obtained from experiments, on the basis of the Stone-Weierstrass theorem [48], with the use of activation functions. Various options for these function activators are; Radial Basis Function (RBF), the hyperbolic tangent and the sigmoid. It should be noted however that though neural networks are easy to train and use, and simple structured with high approximation properties, it is not without its own issues some of which are: lack of network dynamism since it is a static approximator, it yields opaque models often without mathematical interpretation, difficulties in model analysis, isolation of cause and effect, and determination of dynamic characteristics of the system from the model. Friction modelling and compensation using this approach has been proposed in many articles [49], [50]. In [25] application to dc model parameter identification using compound evolution techniques was investigated.