Page 165 -

3.4 State-Variable Representations and Feedback Linearization 147

(3.4.28)

Given the function h(q), it is straightforward to compute the Jacobian J(q)
associated with h(q). In the special case where represents the Cartesian velocity,
J(q) is the arm Jacobian discussed in Appendix A. Then, if all joints are
revolute, the units of J are those of length.
According to (3.4.2),

(3.4.29)
so that (3.4.26) yields

(3.4.30)

Define the control input function
(3.4.31)

and the disturbance function
(3.4.32)

2p
Now we may define a state x(t) R by
(3.4.33)

and write the robot dynamics as

(3.4.34)

This is a linear state-space system of the form
(3.4.35)

driven both by the control input u(t) and the disturbance v(t). Due to the
special form of A and B, this system is said to be in Brunovsky canonical
form (Chapter 2). The reader should determine the controllability matrix to
verify that it is always controllable from u(t).
Equation (3.4.31) is said to be a linearizing transformation for the robot
dynamical equation. We may invert this transformation to obtain

(3.4.36)

Copyright © 2004 by Marcel Dekker, Inc.