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4.5 Digital Robot Control 227

A continuous PID controller that only uses joint position measurements q(t)
is given by

(4.5.8)

where k is the proportional gain, T 1 is the integration time constant or “reset”
time, T D is the derivative time constant. Rather than use pure differentiation,
a “filtered derivative” is used which has a pole far left in the s-plane at s=-N/
T D . The value for N is often in the range 3 to 10; it is usually fixed by the
manufacturer of the controller [Åström and Wittenmark 1984]. A special case
of the PID controller, of course, is the PD controller, which is therefore also
covered by this discussion.
A common approximate discretization technique for converting continuous-
time controllers K (s) to digital controllers K(z) is the bilinear transform (BLT),
c
where
(4.5.9)

(4.5.10)

This corresponds to approximating integration by the trapezoidal rule. Under
this mapping, stable continuous systems with poles at s are mapped into
stable discrete systems with poles at

(4.5.11)

The finite zeros also map according to this transformation. However, the
zeros at infinity in the s-plane map into zeros at z=-1.
Using the BLT to discretize (4.5.8) yields

(4.5.12)

with the discrete integral and derivative time constants

(4.5.13)

(4.5.14)

Copyright © 2004 by Marcel Dekker, Inc.