103
Controllability is used in condition (1) since we wish to have complete
freedom in selecting the eigenvalue locations.
It is not difficult to show that for the robotic manipulator,
conditions (1) and (2) always hold true. To verify condition (1) we
evaluate the first two blocks of the standard controllability matrix
[20] for the pair [F,(t), G*(t)]. This gives
9 <p
[G*(t), F*(t)G*(t)]
9 9 9
0 J"1(0*)
J-1(0*) F2(t)J'1(0*)_
(6-14)
Since J(0 ) is nonsingular, the controllability matrix has full rank
(i.e., rank 2N for the N-link manipulator) and hence the pair
[F (t), G,(t)] is controllable.
9 9
Now consider condition (2). Using (6-11) through (6-13) it is
possible to write
*
X1!2Nx2N V1'
*<*>"
_
Xi!NxN
-F^t)
-INxN
Vnxn 
0
_1 *
J )
- -H*
0 .
_*NxN
0
0
(6-15)
The right-hand side of (6-15) is a 3Nx3N matrix which is readily seen to
have full rank for all Aj. The fact that this matrix is full rank for
any A^ means that the stability condition can be achieved regardless of
the eigenvalues of the internal model system. In otherwords, the
frequencies of the reference and disturbance signals will not be a
factor in deciding whether or not the stability condition can be met.
By showing that both conditions (1) and (2) hold, we have proven
that it is possible to stabilize the linearized manipulator system
(assuming the slowly time-varying approach will apply).