Page 22 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Friction Dynamics and Modeling 13

T f = f c sgn(˙x) (1.4)

where f c = μ|f n |. μ and f n are the friction coefficient and the normal force.

Viscous friction
Viscous friction is proportional to the motion velocity ˙x, which is described
by

T f = f v ˙x (1.5)

where f v is the viscous coefficient.

Stribeck friction
The Stribeck friction shows the velocity dependence property. This phe-
nomenon can be described by

⎧
⎪ f z , ˙ x = 0
⎨
T f = f e , ˙ x = 0and |f e | < f s (1.6)
f ssgn(f e ), otherwise
⎪
⎩
where f z is an arbitrary function and can be chosen as the following form

f z = f c + (f s − f c )e −(˙x/˙x s ) 2 (1.7)

where ˙x s is the Stribeck velocity.

1.2.2 Classical Friction Models

Based on the previous formulation of friction dynamics, one can also
obtain a variety of classical models combining different frictions aforemen-
tioned.

Exponential model
In [8], an exponential friction model incorporating both the Coulomb and
viscous frictions is given as

T f = f c sgn(˙x) + (f s − f c )e −(˙x/˙x s ) 2 + f v ˙x (1.8)

where ˙x s is an empirical parameter, f c is the Coulomb friction level, f s is the
level of the stiction force and f υ is the viscous coefficient.