97
give a desired end-effector location. In fact, e 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 6*(t), 0*(t), and e*(t) will
be used to denote this trajectory where we are assuming that 0*(t) is
twice differentiable. The torque T^(t) which must be supplied to the
actuators to produce the desired trajectory can be calculated from the
relationship
k 99k \t k *k n k A k
TA(t) = J(9 )0 + TV(0 ,0 ) T9(0 ) T(w(t),0 ) (6-7)
Here it is assumed that the disturbance w(t) is known. If this is not
the case, the expected (nominal) disturbance w*(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 algo
rithm consists of only the joint angles and the joint velocities. If
m~k
the nominal joint angles and joint velocities are denoted 0 (t) and
Xk
0 (t) respectively, it is appropriate to define the nominal state
trajectory as
$*(t) =
0*(t)
0*(t)
(6-8)