In this paper the algorithm is completed for generation of envelope-surfaces for plane-workspaces of generally proportioned manipulators. Then the ruled surface Ψ is used for adapting the algorithm to 3-R manipulators for which the outermost two axes intersect (a2 = 0). The discriminant D is further developed, and it is used to classify 3-R manipulators, having a2 = 0, into seven Types. Manipulators, which are of Type 7, (i) can provide any orientation to a tool plane σ or (ii), with a fourth appropriately placed R joint and tool plane Σ, can also provide any attitude to the end effector. Design conditions are developed and presented which ensure that a manipulator will possess these properties of dexterity. The conditions are based on coupled motions at all three, or four, axes.
Effective divisor classes on a ruled surface
