Page 25 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

h 1 (v) is introduced to represent the reversal behavior while h 2 (v) depicts
the Stribeck effect, which are defined as

2f s , ˙ x ≥ 0
h 1 (˙x) = (1.15)
0, ˙ x < 0

and

h 2 (˙x) = e −(˙x/˙x c ) 2 (1.16)

where ˙x c is the critical velocity at which the friction torque is minimum.
The above friction model (1.14) is developed based on the fact that
any arbitrary continuous piecewise linear function can be effectively rep-
resented by using a discontinuous piecewise parametric representation
(DPPR) as shown in [14]. However, different to standard DPPR, the static
friction has a jumping behavior at zero velocity where the direction of mo-
tion changes. Moreover, in the low velocity regime, the Stribeck effect and
the Coulomb friction force mainly contribute to the friction, which makes
the friction highly non-linear and non-smoothing in low-speed especially
near zero crossings. In order to address the Stribeck effect and the jumping
behavior at zero velocity, two additional terms are introduced in DPPR:
1) A jump term h 1 (˙x) related to the maximum static force, which is used to
represent the reversal behavior of friction force when the motion direction
is changed; 2) An exponential component h 2 (˙x) to denote the Stribeck
effect. As for an example, Fig. 1.2 provides theprofileofDPPR friction
model.

1.3 CONCLUSION
This chapter describes different friction dynamics and several friction mod-
els, which will be used in the control designs to be presented in this book.
Classical models are developed to show the dominant friction components,
among which LuGre model is able to represent most dynamic behaviors
of friction. Moreover, a continuously differentiable friction model is re-
cently proposed to facilitate continuous control designs. The introduced
discontinuous piecewise parametric representation (DPPR) friction model
is particularly suitable for model parameter identification.