117
The determinant of the manipulator Jacobian, d., is
-independent of the frame of expression and can be easily obtained from matrix 3j;
= 1.5 (C3 S45 IS4 (0.3 + C2)-- 0;-2C23 C43
S3 $23 C4 (1.5 C5 + 0.2 S45)). (9.59)
The values of dj, listed in Table 9-5, prove that all sixteen solutions found correspond to non-degenerate configurations of the 0M25 robot arm.
Figure 9.4 shows photographs of a computer graphics simulation of the 0M25 manipulator in the sixteen configurations listed in Table 9-5. Figure 9.3 is a hand drawing of this manipulator in configuration 1 of Table 9-5 (i.e. corresponding to the first solution set) with all link frames clearly indicated and, to help differentiate between solutions with common values of 81 and 82, we have attempted to indicate the direction of axis vectors zl, z3, z4, and z5 on the photographs. The position and orientation of the end-effector and the base frame (as shown on Fig. 9.3) are the same for all sixteen representations of Figure 9.4.
It is also interesting to note that this large number of solutions can be realized by an orthogonal manipulator with a fairly simple geometry. Finally, this example shows that the techniques developed in this dissertation can be used to implement simple and efficient inverse kinematic analysis tools such as the search algorithm just discussed
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