and
(a2+a3C3) S2 (a2d3-r2a3S3) C2 =
-1lSlPx + T1C1Py + al(pz-dl). (5.24) When the determinant of this linear system of equations in S2 and C2 is not 0, a unique value of e2 can be computed. Otherwise, we can obtain e2 uniquely from another linear system of equations in S2 and C2,
(r2a3C3+a2r3) S2 + a3S3 C2 = Cltx + Slty (5.25)
and
C3S3 S2 (T263C3+a2T3) C2
= 1Sltx + iClty + aitz, (5.26)
derived from Eq. (5.5). Note that 92 can also be computed using a system of two equations formed by Eq. (5.23) or Eq. (5.24) and one of Eqs. (5.25) and (5.26). The Appendix
shows that the determinants of the two systems of equations above are simultaneously zero only when joint axis 2 aligns with another joint axis which puts the arm in a degenerate configuration.
To complete the 4-DOF solution set, we use Eq. (5.2) which can be rewritten as
R4 = RI1 R21 RI-1 R.
The first column vector of R4, obtained by multiplying both sides by the first canonical unit vector x = [1,0,0] T
..