In order to write the dynamic equation modeling the manipulator in standard state-space form, we make the definition
(t) :: ( (6-4)
From equation (6-1) it then follows that
$(t) = f (4(t), TAM w(t)) (6-5)
where
f ( (t), TA(t), w(t)) := 1 E T t) + Tg( ) + Td (w(t)O) - TV( , ) 1]
(6-6)
Equation (6-5) is a nonlinear dynamic equation which can be used either in performing simulations or in controller design.
Acuator Driving Torques
Often one wishes to compute the torques required to drive the manipulator over some trajectory. This trajectory is usually specified
in terms of the position, velocity, and acceleration of the manipulator's end-effector (i.e., hand). It is usually necessary to convert the end-effector trajectory into the corresponding trajectory for the joint angles, joint velocities, and joint accelerations. This task
shall not be discussed in detail here since it will not be of major importance in demonstrating the control algorithm. We mention, however,
that it can be difficult to obtain the joint angle vector 0 which will