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246 Computed-Torque Control

all the nm feedback loops simultaneously by solving the matrix design
equations.

Linear Quadratic Computed-Torque Design
Now we apply these results to the control of the robot manipulator dynamics

(4.6.9)
According to Section 4.4, the computed-torque control law

(4.6.10)

yields the error system

(4.6.11)

which we may write as
(4.6.12)
with the state defined as

(4.6.13)

Now, select the outer-loop PD feedback

(4.6.14)
To find a stabilizing gain K, select the design parameter Q in the PI as

so that the position and velocity errors are independently weighted. Then,
due to the simple form of the A and B matrices, which represent n de-coupled
Newton’s law (i.e., double integrator) systems, the solution of the Riccati
equation is easily found (see the Problems). Using this solution in (4.6.5)
yields the formula for the optimal stabilizing gains

(4.6.15)

This LQ approach reveals the relation between the PD gains and some design
parameters Q and R that determine the total energy in the closed-loop system.

Copyright © 2004 by Marcel Dekker, Inc.