115
This search program was run for each of the four possible combinations of values for the sign ambiguities u45 and u4 (1,1), (1,-i), (-1,1), and (-1,-i)).
The 16 solution sets found are listed in Table 9-5. This result is of importance because it provides the first tangible proof -that a six-DOF manipulator can actually achieve a given pose with 16 different configurations.
Some manipulators can achieve a particular end-effector pose in an infinite number of configurations when two or more joint axes coincide. Such degenerate conditions, where the manipulator loses degrees of freedom, force the manipulator jacobian to become singular. To verify that the manipulator under discussion is non-degenerate at each of the sixteen configurations found, we compute the determinant of the manipulator Jacobian.
The symbolic Jacobian of a 6-DOF manipulator has its simplest expression in frame 3 (mid-frame) (Renaud, 1980a).
For the robot described here, the mid-frame Jacobian 3J, computed using tables given in Doty (1987), is
0.2 C23 S3 0 0 1.5 S4 0
-0.3-C2 0 0 0 -1.5 C4 0
0.2 S23 -C3 0 0 0 -1.5 C5 (9.58)
$23 0 0 0 0 S45
0 1 1 0 0 -C45
-C23 0 0 1 1 0
..