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3.4 State-Variable Representations and Feedback Linearization 143

(3.4.3)

In this section we intend to show some equivalent formulations of the arm
dynamical equation.
The nonlinear state-variable representation discussed in Chapter 2,

(3.4.4)

has many properties which are useful from a controls point of view. The
function u(t) is the control input and x(t) is the state vector, which describes
how the energy is stored in a system. We show here how to place (3.4.1) into
such a form. In Chapter 4 we show how to use computers to simulate the
behavior of a robot arm using this nonlinear state-variable form. Throughout
the book we shall use the state-space formulation repeatedly for controls design,
either in the nonlinear form or in the linear form

(3.4.5)

In this section we also present a general approach to feedback linearization
for the nonlinear robot equation, which involves redefining variables in a
methodical way to yield a linear state equation in terms of a dynamical
variable we are interested in. This variable could be, for instance, the joint
variable q(t), a Cartesian position, or the position in a camera frame of
reference.

Hamiltonian Formulation

The arm equation was derived using Lagrangian mechanics. Here, let us use
Hamiltonian mechanics [Marion 1965] to derive a state-variable formulation
of the manipulator dynamics [Arimoto and Miyazaki 1984], [Gu and Loh
1985]. Let us neglect the friction terms and the
disturbance τ d for simplicity; they may easily be added at the end of our
development.
In Section 3.2 we expressed the arm Lagrangian as

(3.4.6)

n
with q(t) R the joint variable, K the kinetic energy, P the potential energy,
and M(q) the arm inertia matrix. Define the generalized momentum by

(3.4.7)

Copyright © 2004 by Marcel Dekker, Inc.