structures for which dH is always zero (independent of the end-effector pose) have DH--parameters 01, 02, 03, a,, a2, a3, and d2 for which each of the six terms in Eq. (5.28) is individually zero. These terms, in turn, are zero when particular structure parameters are equal to zero. For

example, al=a02=0 will make the determinant zero. In order to enumerate the minimum number of distinct combinations of zero parameters that make d11 = 0, we examine all possible cases when a particular parameter is zero. We get seven

simpler expressions of dH, listed in Table 5-1., by

separately assuming each relevant parameter to be equal to zero.

 Table 5-1 provides a simple mean for finding all (poseindependent) 4-DOF robot geometries for which matrix H will be singular. In the next section, we show that the inverse

kinematics problem for such robots can still be solved by use of the reduced system of equations (5.13)-(5.16). Special 4-DOF Arm Structures

 A trivial condition occurs when two consecutive joint axes coincide somewhere along the arm. Such a degenerate

condition is detected by Iai = ai= 0 for some joint i. In

this case, the manipulator loses one degree of freedom and becomes a redundant 3-DOF arm. If a solution set exists for such an arm, there will be an infinite number of solution sets. A careful analysis of Table 5-1 shows that there are only ten minimal, non-trivial, conditions on the arm

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