Page 106 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Control for Manipulation Systems 99

Figure 6.2 Approximation of y 1 (solid: y 1 ; dotted: DPPR).

Figure 6.3 Approximation of T (v) (solid: T (v); dotted: DPPR).
f
f
Moreover, the DPPR is used to approximate a well-known and more
realistic friction model (e.g., LuGre-like model), which is given by

T f (v) = f c sgn(v) + (f s − f c )e −(v/v c ) 2 sgn(v) + f vv, (6.7)

where the friction coefficients are set as f s = 2.4N, f c = 0.8N, v c = 1.05 /s,
◦
f v = 0.8 N. The result is shown in Fig. 6.3, which provides very good ap-
proximation response. In particular, the Stribeck effect can be captured by
using this DPPR friction model.
In the above simulations, we collect the friction torque T f offline and
then use it for the friction model identification. However, in the online
implementation, it may not be able to directly measure this torque and thus