107
relationship of the manipulator's links [33] we can write the following
equation
r(t) = h(e(t), f(t)) (6-21)
where r(t) is the vector locating the end-effector of the manipulator
and h(e,f) is a function of the manipulator's joint angle vector e(t)
and of the force vector f(t), both of which are assumed measurable.
This equation is only valid in the quasi-static case since otherwise,
the end-effector location would also depend on e(t) and e(t).
When (6-21) is linearized about a nominal trajectory, the following
result is obtained
r(t) HjUJeU) + H2(t)f(t) (6-22)
where
Hx(t) =
9h(e,f)
ae
e,f=nominals
H2(t) =
ah(e,f)
af
e,f=nominaIs
(6-23)
Here r(t), Q{t), and f(t) denote the deviations of r(t), e(t) and f(t)
from their nominal values. We note that H^(t) is actually the standard
Jacobian matrix [28] often used to relate the end-effector velocity to
the joint-angle velocities.
If the end-effector is to maintain a specified trajectory, r(t)
must be zero. Consequently, e (t) can be obtained by setting r(t) equal
to zero, solving for e(t), and letting e*(t) equal e(t). Provided that
Hj(t) is nonsingular this gives