Rigid formation shape control in general dimensions: an invariance principle and open problems

The control systems resulting from seeking to control the shape of a rigid formation are frequently described by gradient flows which are designed to minimize some predefined and relevant potential functions linked to the desired formation shape. This paper discusses and establishes a rank-preserving property of such formation shape stabilization systems. We further show some properties of the degenerate critical formations that live in a lower dimensional space, and prove that they are unstable. The implication of these results is that if all the agents start with generic initial positions, then their trajectories will be strictly bounded away from the set of degenerate formations at any finite or infinite time. By establishing this invariance principle, some previous results on equilibrium analysis for rigid formation systems can be greatly simplified. The results also have applications in other fields such as the multidimensional scaling study. The results are obtained from a joint analysis of rank-preserving flow theory, graph rigidity theory and invariant manifold theory.