fact, obvious since complete controllability of the linearized system has already been asserted.
One method of obtaining M1, M2 and M3 is simply to select
stabilizing feedback gains for some arbitrary time, say to. This yields
MI = Fl(to) - J()Kl(to)
M2 = F2(to) - Jl*)K12(to) (6-19)
- 1 _*
M3 = -J (e )K2(to)
Then, the feedback proposed to make the system behave as though it were time-invariant is the following
Kl'l(t) = a(e*(t)) [Tl(t) - M1]
K1,2(t) = J(O*(t)) [F2(t) - M21 (6-20)
K2(t) = -J(O*(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.