give a desired end-effector location. In fact, 6 is not unique [29]. Once 0 is known, obtaining the joint velocities and joint accelerations given the end-effector velocities and accelerations is rather easy.
Suppose that the desired end-effector trajectory has been specified
and converted into the corresponding trajectory of joint positions, velocities, and accelerations. The symbols O*(t), 6*(t), and U*(t) will be used to denote this trajectory where we are assuming that e*(t) is twice differentiable. The torque TA(t) which must be supplied to the actuators to produce the desired trajectory can be calculated from the relationship
T A(t) = J(e )e* + TV(0*,0*) - Tg(0*) - Td(w(t), ) (6-7)
Here it is assumed that the disturbance w(t) is known. If this is not
the case, the expected (nominal) disturbance ;i*(t) would generally be substituted into (6-7).
Feedback Control System for Tracking and Disturbance Rejection
The control strategy discussed in Chapter Five shall be adopted for the control of the robotic manipulator. Thus, it will be necessary to know in advance the nominal trajectory as well as the input which gives this trajectory. The nominal trajectory needed for the control algorithm consists of only the joint angles and the joint velocities. If
the nominal joint angles and joint velocities are denoted T*(t) and o*(t) respectively, it is appropriate to define the nominal state trajectory as (t)6
(t) : (t)J (6-8)