Hybrid rigidity theory with signed constraints and its application to formation shape control in 2-D space

In this paper, we develop a hybrid rigidity theory that involves distances (or unsigned angles) and signed angle constraints for a framework in 2-D space. The new rigidity theory determines a rigid formation shape up to a translation and a rotation by a set of distance and signed angle constraints, or up to a translation, a rotation and, additionally, a scaling factor by a set of unsigned angle and signed angle constraints. In particular, the hybrid rigidity theory provides insights on choosing heterogeneous constraints to address issues associated with flip (or reflection), flex and ordering ambiguities for a target formation. We then apply the rigidity theory to formation shape control with the minimal number of heterogeneous constraints in 2-D space. It is shown that a developed gradient-based control system guarantees local exponential convergence to a desired formation, where each single-integrator modeled agent only requires relative position measurements with respect to its neighbors.