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2.11 Linear Controller Design 93

THEOREM 2.10–3: Let V(x) be a Lyapunov function of a continuous-time
system that satisfies the following properties:

then x(t) is uniformlt bounded

THEOREM 2.10–4: Let V(x) be a Lyapunov function of a continuous-time
system that satisfies the following properties:

where η is a positive constant, γ 1 and γ 2 are continuous, strictly increasing
functions, and γ 3 is a continuous, nondecreasing function. Then, if

x(t) is uniformly ultimately bounded. In addition, if x(0)=0, x(t) is uniformly
bounded.

2.11 Linear Controller Design

The purpose of control design is to make the robot respond in a predictable
and desirable fashion to a set of input signals. In this section we review how
a linear controller may be designed in order that the behavior of the robot-
controller combination is acceptable. It seems obvious that the firs requirement
on the robot-controller is its stability. Therefore, the first function of the
controller is to stabilize the robot when it is moving in space. In fact, we can
now pose the problem we wish to solve:

Copyright © 2004 by Marcel Dekker, Inc.