105
fact, obvious since complete controllability of the linearized system
has already been asserted.
One method of obtaining Mj, M2 and M3 is simply to select
stabilizing feedback gains for some arbitrary time, say tQ. This yields
MX = F1(t0) -
M2 F2(t0) (6-19)
M3 -J_1(e*)K2(t0)
Then, the feedback proposed to make the system behave as though it were
time-invariant is the following
Kl,l(t) = J(e*(t)) [Fx(t) Mx]
K1>2(t) = J(0*(t)) [F2(t) M2] (6-20)
K2(t) = -J(0*(t)) M3
Substituting the above gains into (6-17) will immediately verify that
they are indeed correct.
We note that the feedback gains derived here can be evaluated at
any instant of time using the nominal position and velocity of the
manipulator. One method of computing the gains in real-time, however,
might be based on actual rather than nominal trajectories. This is
especially true if the nominal trajectory is not known in advance.