relationship of the manipulator's links [33] we can write the following equation
r(t) = h(o(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 6(t) and )(t).
When (6-21) is linearized about a nominal trajectory, the following result is obtained
H1(t) Hl(t)e(t) + H2(t)f(t) (6-22)
where
H Hh(',f) ah(e,f)
H1(t) ae le,f=nominals 2(t) af le,f=nominals
(6-23)
Here F(t), '(t), and ?(t) denote the deviations of r(t), e(t) and f(t) from their nominal values. We note that H1(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 **(t) equal W(t). Provided that H1(t) is nonsingular this gives