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