manipulator. Although di can be assumed zero without loss of generality for a 4-DOF manipulator, this system of equations will be used for 4-DOF sections of larger manipulators (next chapters) for which the parameter corresponding to dI will, in general, not be zero. Hence, dI is assumed not equal to zero at this point.

 A unique solution to the reduced system is given by



 S1 r1

 C1 = H-1 r2 (5.21)
 H1 2 (5.21)
 S3 r3

 C3 r4


where


 alty altx a2G3 -a2a3d2

 altx -alty 0 a3
 H = y, (5.22)
 0alPx -lPy -a2a3 0
 alpy alPx a2a3d2 a2a3



when matrix H is nonsingular. Unique values of 81 and 83 are thus obtained from the values of Sl, Cl, S3, and C3. The case where H is not invertible is discussed in the next sections because of its interesting implications.

 With 81 and 83 known, Eq. (5.7) provides away to solve for e2. Indeed, when expanded, the first 2 components yield


 (o2d3-'2a3S3) S2 + (a2+a3C3) C2 = ClPx + SlPy al (5.23)

..