Page 21 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

12 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

proposed, which is suitable for high-performance continuous control de-
sign. Moreover, a new discontinuous piecewise parametric representation
(DPPR) friction model was also reported in our previous work [7], which
can be used for fraction identification.
In this chapter, the generic friction dynamics and several friction models
are presented, which will be used in the different friction compensation
control designs to be presented in this book.

1.2 FRICTION DYNAMICS AND MODELS

In high precision systems, an accurate friction model is essential to cap-
ture various friction characteristics. In the following, some typical friction
models will be described.

1.2.1 Friction Dynamics
Static friction (Stiction)
The friction for zero velocity is a function of the external torque f e.When
the magnitude of f e is smaller than the maximum stiction force f s,the static
friction opposes the motion x (˙x is the motion velocity). The stiction model
is given by

f e , |f e | < f s
T f = (1.1)
f s δ(˙x)sgn(f e ), |f e |≥ f s
where

1, ˙ x = 0 +1, f e > 0
δ(˙x) = , sgn(f e ) = (1.2)
0, ˙ x = 0 −1, f e < 0
In practical implementations, stiction can be seen as a force of constraint
in presliding, and the applied force T f can be described as the following
form
T f = f t xδ(˙x) (1.3)

where f t denotes the tangential stiffness, x is the displacement and the phe-
nomenon that stiction occurs only at zero velocity is described by δ(˙x).

Coulomb friction (Dry friction)
Coulomb friction is proportional to the normal force of contact. It opposes
the relative motion and is described by