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14-20 Section 14
© 1999 by CRC Press LLC
The Jacobian can be used to perform inverse kinematics at the velocity level as follows. If we define
[J
–1
] to be the inverse of the Jacobian (assuming J is square and nonsingular), then
(14.3.10)
and the above expression can be solved iteratively for (and hence q by numerical integration) given
a desired end effector trajectory and the current state q of the manipulator. This method for determining
joint trajectories given desired end effector trajectories is known as Resolved Rate Control and has
become increasingly popular. The technique is particularly useful when the positional inverse kinematics
is difficult or intractable for a given manipulator.
Notice, however, that the above expression requires that J is both nonsingular and square. Violation
of the nonsingularity assumption means that the robot is in a singular configuration, and if J has more
columns than rows, then the robot is kinematically redundant. These two issues will be discussed in the
following subsections.
Example 14.3.4
By direct differentiation of the forward kinematics derived earlier for our example,
(14.3.11)
Notice that each column of the Jacobian represents the (instantaneous) effect of the corresponding
joint on the end effector motions. Thus, considering the third column of the Jacobian, we confirm that
the third joint (with variable d
3
) cannot cause any change in the orientation (φ) of the end effector.
Singularities
A significant issue in kinematic analysis surrounds so-called singular configurations. These are defined
to be configurations q
s
at which J(q
s
) has less than full rank (Spong and Vidyasagar, 1989). Physically,