APPENDIX
 SOLVING FOR 82



 .In the inverse kinematic- solution for four-degree-offreedom robot manipulators described in Chapter 4, after solving for EI and 83, we said that the angle e2 can be obtained by solving one the two linear systems of equations in S2 and C2 given by Eqs. (5.23) and (5.24) or Eqs. (5.25) and (5.26). In this Appendix, we show that e2 can always be computed from one of those two systems when the 4-DOF arm is not in a degenerate configuration.

 The absolute value of the determinant of the system of Eqs. (5.23)and (5.24), D1 is given by


 D1 = (62d3 T2a3S3)2 + (a2 + a3C3)2 (Al)


and, in absolute value, the determinant of the system of Eqs. (5.24) and (5.25) yields


 D2= (62T3 + 7203C3) + (o3S3)2. (A2)


 A value of 82 is obtained as long as D1 and D2 are not simultaneously equal to zero. However, both determinants equal zero requires that the four conditions


 a2d3 r2a3S3 = 0 (A3)

 a2 + a3C3 = 0 (A4)


133

..