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.