Variety and connectivity in kinematic chains

The enumeration of kinematic chains, also known as number synthesis, has been used for at least the past four decades as a means of finding better mechanisms for some predefined purpose. In practice, however, enumeration can be difficult to implement since the number of kinematic chains generated is often too large to manually consider the individual merits of each chain. For this reason, the concepts of variety and connectivity can be used to classify kinematic chains according to the constraints required as described in the literature. In this regard, this paper presents three new results. First, a redefinition of the concepts of variety and connectivity in an algorithmic form is introduced. Second, a set of important relations between connectivity and variety, introduced by Tischler et al. in [C.R. Tischler, A.E. Samuel, K.H. Hunt, Kinematic chains for robot hands: Part 2 kinematic constraints, classification, connectivity, and actuation, Mechanism and Machine Theory 30 (8) (1995) 1217–1239] as conjectures lacking formal proofs, are stated as theorems in this paper and formally proved. Finally, a new algorithm is proposed for the computation of the variety of a kinematic chain; this algorithm includes other parameters such as connectivity and redundancy.