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146 Robot Dynamics

nonlinearities. It is a simplified version of the feedback linearization technique
in [Hunt et al. 1983, Gilbert and Ha 1984]. See also [Kreutz 1989].
n
The robot dynamics are given by (3.4.2) with q R Let us define a general
sort of output by

(3.4.24)

n
with h(q) a general predetermined function of the joint variable q R and s(t)
a general predetermined time function. The control problem, then, will be to
select the joint torque and force inputs τ(t) in order to make the output y(t) go
to zero.
The selection of h(q) and s(t) is based on the control objectives we have in
mind. For instance, if h(q)=-q and s(t)=q d(t), the desired joint space trajectory
we would like the arm to follow, then y(t)=q d(t)-q(t)=e(t) the joint space tracking
error. Forcing y(t) to zero in this case would cause the joint variables q(t) to
track their desired values q d(t), resulting in arm trajectory following.
As another example, could represent the Cartesian space
3
tracking error, with the position error and e 0 R the orientation
error. Controlling y(t) to zero would then result in trajectory following directly
in Cartesian space, which is, after all, where the desired motion is usually
specified.
Finally, -h(q) could represent the nonlinear transformation to a camera
frame of reference and s(t) the desired trajectory in that frame. Then y(t) is the
camera frame tracking error. Forcing y(t) to zero would then result in tracking
motion in camera space.

Feedback Linearizing Transformation. To determine a linear state-variable
model for robot controller design, let us simply differentiate the output y(t)
twice to obtain

(3.4.25)

(3.4.26)

where we have defined the Jacobian

(3.4.27)

p
If y R , the Jacobian is a p×n matrix of the form

Copyright © 2004 by Marcel Dekker, Inc.