Page 101 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

fective and the training phases of these algorithms before achieving con-
vergence may be sluggish. Hence, there is still a need to develop advanced
possibly continuous or piecewise continuous friction models, which are
more suitable for control designs.
This chapter presents an alternative modeling and feedforward compen-
sation control method of joint friction for a multi-link manipulator system.
A new discontinuous piecewise parametric representation (DPPR) is fur-
ther tailored to capture the generic friction dynamics, which could facilitate
the modeling and control implementation. In particular, the friction torque
can be estimated online using only the experimental data, while no prior
knowledge of the unknown frictions is required. Unlike other data based
modeling [6], [9], this modeling method can approximate the friction dy-
namics in either positive or negative velocity regions rather than dealing
with them separately. The estimated friction is then incorporated into the
control as a feedforward compensation. The closed-loop stability is proved
by using the Lyapunov theory. Numerical simulations are also given to
show the validity of this new modeling and control scheme.

6.2 SYSTEM DYNAMICS AND PROBLEM STATEMENT

6.2.1 Manipulation System Dynamics
Consider the dynamics of a manipulator system given in the Lagrange
form [10]
M(x)¨x + C(x, ˙x) + T f = τ (6.1)

n
with x, ˙x, ¨x ∈ R are the joint position, velocity, and acceleration vectors,
respectively. M(x) ∈ R n×n is the unknown inertia matrix and C(x, ˙x)¨x ∈ R n
is the unknown vector resulting from Coriolis, centripetal accelerations
n
n
and gravity. τ ∈ R and T f ∈ R are the control input and friction torque,
respectively.
Two widely used properties of the manipulation system (6.1)aregiven
as [11,12]:
Property 1: M(x) is a positive definite symmetric matrix bounded by
m 1I ≤ M(x) ≤ m 2I,where m 1, m 2 are positive constants;
Property 2:The matrix ˙ M(x) − 2C(x, ˙x) is skew symmetric.

6.2.2 Problem Formulation
The objective of this chapter is to develop a composite control for manipu-
lation system (6.1), which makes the system output x track a given position