Page 163 -

3.4 State-Variable Representations and Feedback Linearization 145

with the control input defined by

(3.4.18)

This is a nonlinear state equation of the form (3.4.4). It is important to note
that this dynamical equation is linear in the control input u, which excites
each component of the generalized momentum p(t).
This Hamiltonian state-space formulation was used to derive a PID control,
law using the Lyapunov approach in [Arimoto and Miyazaki 1984] and to
derive a trajectory-following control in [Gu and Loh 1985].

Position/Velocity Formulations
Alternative state-space formulations of the arm dynamics may be obtained by
defining the position/velocity state x R as
2n
(3.4.19)

For simplicity, neglect the disturbance τ d and friction F v +F d ( )and note that
according to (3.4.2), we may write

(3.4.20)

Now, we may directly write the position/velocity state-space representation

(3.4.21)

which is in the form of (3.4.4) with u(t)=τ(t)
An alternative linear state equation of the form (3.4.5) may be written as

(3.4.22)

with control input defined by
(3.4.23)

Both of these position/velocity state-space formulations will prove useful in
later chapters.
Feedback Linearization

Let us now develop a general approach to the determination of linear state-
space representations of the arm dynamics (3.4.1)–(3.4.2). The technique
involves a linearization transformation that removes the manipulator

Copyright © 2004 by Marcel Dekker, Inc.