The reduced system of equations (5.13)-(5.16) can still be efficiently used to find all the solution sets of Eq. (5.1) when matrix H is singular.
Case 1: al=a2=0. First three joint axes are parallel (Entry 1 in Table 5-2).The reduced system of equations becomes
alty S1 + altx C1 + a2a3 S3 = rI (5.29)
0 = r2 (5.30)
0 = r3 (5.31)
alPy S1 + alpx C1 + a2a3 C3 = r4. (5.32)
Equations (5.30) and (5.31) are constraints on pose parameters tz and pz respectively. Only end-effector poses that satisfy pz = dl + d2 + d3 and tz = 73 (Eqs. (5.18) and (5.19)) are solvable with this arm geometry.
Equations (5.29) and (5.32) still allow a solution in the style of Pieper (1968) by first eliminating S3 and C3 from the equations. This can be done by solving for S3 and C3 and substituting in the Pythagorean identity (4.17) to get
{[rl (alty S1 + altx Cl)]/a2a3)2 +
{[rl (alPy S1 + alPx Cl1)]/a2a3) 2 1. (5.33) With the trigonometric identities
Sl = 2 tl/(l + t12) and C1 = (1 t12)/(l + tl2)
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