Generalised Rigidity and Path-Rigidity for Agent Formations

The classical concept of rigidity characterises conditions under which distance constraints between agents in R enforce a rigid structure on the whole collection of agents. The present paper has two goals. Firstly, we propose a generalised theory for rigidity to allow heterogeneous agent states on non-Euclidean spaces and general non-linear relative state constraints. To do this, we characterise rigidity as an invariance property with respect to a topological group action that is introduced as a natural structure in the problem formulation. Secondly, we use this new framework to formulate a new concept of path-rigidity, which captures the property that allows a rigid formation to be steered continuously between any two configurations that are congruent. This is an important property for path planning and control of rigid formations. The main result of the second part of the paper provides a simple and easily checked condition to determine if a globally rigid formation is also path-rigid.