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.
Computer generation of variants of cutting of long items into pieces of desired sizes
