Here we have used (6-2), (6-3) and the fact that e* = e* = 0. Since e* is a constant, both Tg(e*) and D(e*) are also constant. This means that the frequencies of TA(t) are identical to the frequencies of w(t) and hence only the dynamics of the disturbance signal need to be included in the internal model system. In this case, assumption (A.2) is satisfied exactly. Note that the internal model system should also include dynamics (i.e., integrators) to accomodate for the constant gravitational torque Tg(e*) if this torque is not supplied separately.
Feedback Gain Calculation
In this section we discuss the feedback gain calculation for the manipulator system when the linearized model is used. In conjunction
with the chosen control scheme, this linearized model is evaluated over the nominal trajectory. The linearized closed-loop system takes the following form
() *()] -BH¢ A n(t) - n*(t)
(6-10)
Since only the first N components of (i.e., e) are taken as the output, H¢ is defined as
H [INxN, 0] (6-11)
where INxN is the NxN identity matrix. Evaluation of the nominal
Jacobian matrices using a linearization of (6-6) leads to