for e3. The unused equation is a constraint on the endeffector pose parameter tz.

 When a3 = 0 or d2 = 0, the conditions for dH=O obtained have all been already discussed, hence there are only ten minimal pose- independent arm geometry conditions for which the reduced system is singular. In all ten cases, we found at most two distinct inverse kinematic solution sets.

 To summarize the above cases, we find that a four-DOF robot manipulator will in general have a unique inverse kinematic solution set. At most two solution sets can be found when the arm has one of the following special structures:

 1. Three consecutive joint axes that are parallel.

 2. Three consecutive joint axes intersect.

 3. Two consecutive pairs of parallel or intersecting joint axes.

 4. Three consecutive joint axes such that two intersect and two are parallel.

 5. Three consecutive joint axes such that the firsttwo intersect and the last two intersect.

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