Constraint graphs for combinatorial mobility determination

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The mobility is a fundamental property of mechanisms and any design methodology needs to estimate at least the generic topological mobility, i.e. the mobility that almost all mechanisms with a given kinematic topology exhibit. The determination of the topological mobility requires combinatorial algorithms operating upon a graph representation of the kinematic constraints. A faithful graph representation is thus crucial. The lack of such a graph representation was a key obstacle for applying combinatorial methods.As solution to problem, in this paper a general kinematic constraint graph, called elementary graph (EG), is introduced allowing for a unique representation of constraints including multiple-joints. From this a reduced graph is derived, called mixed graph (MG), which contains the minimal number of elements while ensuring uniqueness. It shown that from the MG can be derived the well-known body-bar (BB) and bar-joint (BJ) graphs as special cases.The novel MG provides a unique basis for combinatorial algorithms for mobility determination. The most efficient algorithm is the Pebble Game, widely used in biology and chemistry, which exists for BB and BJ. Using the MG requires an adapted variant of the Pebble Game. This will be reported in a forthcoming paper.As prerequisite, the terminology used to describe the mobility of mechanisms is reviewed and a consistent general concept for topological mobility is introduced. Also the shortcomings of structure mobility criteria are briefly discussed.