J(6e + Tv(o,6) = TA(t) + Tg(e) + Td (w(t),e)          (6-1)


For a manipulator having N links, e(t), 6(t) and '6(t) are the vectors of length N defining angular positions, velocities, and accelerations of the actuator joints. The matrix J(e) . RNx   is the inertia matrix which depends on the manipulator's configuration (i.e., the joint angles e(t)).  It can be shown that J(o) is positive definite for all e (see [24]) and is thus always invertible.        The inertia torque vector

TV(6,6) : RN  corresponds to dynamic torques caused by the velocities of the manipulator's links. Denoting the j-th component of the intertia torque vector as Tv(e,6).   we have the following


                           TV(0,6) = 6'PJ(e)b                       (6-2)


where PJ(e) . RNxN is a purely configuration dependent matrix referred to in [28] as the intertia power modeling matrix. The term TA(t) 6 RN is the control torque vector which is typically supplied to the actuator joints by electric motors.    The torques resulting from gravitational loading are designated by     Tg (e)   e RN which is a configuration

dependent vector. Finally, Td(w(t),O) E RN is a torque vector resulting from external uncontrollable forces.   It is possible to write Td(w(t), e) in the following form
                          Td(w(t),e) = D(e)w(t)                     (6-3)


where D(e) : RNxd depends only on the manipulator's configuration and w(t) e Rd denotes the disturbance force vector.