Dimensional perturbation of rigidity and mobility

Mechanisms, defined as assemblies of dimensioned rigid bodies linked by ideal joints, can be partitioned in three mobility states: the rigid state (where bodies can have only one position relative to each other), the mobile state (where bodies can move relatively to each other) and the impossible state (where bodies dimensions and specified joints cannot lead to a feasible assembly). It is also clear that although bodies dimensions can vary in a continuous way, assemblies may experience quite abrupt changes across those states. This paper proposes a new approach to this problem with the goal of being able to predict the mobility class of an assembly of arbitrary complexity, and how it can be affected by a perturbation of the dimensions of its bodies. It does so by proposing a simple and general state transition framework including the three above defined states and seven transitions describing how a dimensional perturbation can affect them. Using this framework, the mobility of a mechanism is easier to capture and predict, using only dimensional (u) and positional (p) parameters involved in an appropriate equation (F(u,p)=0). This is achieved by focusing on how F() behaves when u and p get perturbed, and the impact of this reaction on the mobility state of the assembly. As a result of this more mathematic approach to the problem, previously used notions of iso-constraint, over-constraint and paradoxical assembly, traditionally used to describe such assemblies, can be rigorously defined and thus clarified.