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178 Computed-Torque Control

Figure 4.2.4: (Cont.)(c) position.

Minimum-Time Trajectories

There is an important special class of LFPB trajectories. Suppose the
acceleration is limited by a maximum value of a M and it is desired for the
robot arm to get from one position to another in minimum time. For simplicity
assume that the initial and final velocities are equal to zero. The general case
is covered in [Lewis 1986a] (see the Problems).
A minimum-time trajectory is shown in Figure 4.2.5. To drive joint variable
i from a rest position of q 0 =q i (t 0 ) to a desired final rest position of q f =q i (t f ) in a
minimum time t f , the maximum acceleration a M should be applied until the
switching time t s , after which time the maximum deceleration -a M should be
applied until t f . Note that both t s and t f depend on q 0 and q f . We may write

Then the velocity equations yield

Copyright © 2004 by Marcel Dekker, Inc.