Angle Rigidity and Its Usage to Stabilize Multiagent Formations in 2-D

Motivated by the challenging formation stabilization problem for mobile robotic teams wherein no distance or relative position measurements are available but each robot can only measure some of relative angles with respect to its neighbors in its local coordinate frame, we develop the notion of “angle rigidity” for a multipoint framework, named “angularity”, consisting of a set of nodes embedded in a Euclidean space, and a set of angle constraints among them. Different from bearings or angles defined in a global frame, the angles we use do not rely on the knowledge of a global frame, and are signed according to the counter-clockwise direction. Here, angle rigidity refers to the property specifying that under proper angle constraints, the angularity can only translate, rotate, or scale as a whole when one or more of its nodes are perturbed locally. We first demonstrate that this angle rigidity property, in sharp comparison to bearing rigidity or other reported rigidity related to angles of frameworks in the literature, is not a global property since an angle rigid angularity may allow flex ambiguity. We then construct necessary and sufficient conditions for infinitesimal angle rigidity by checking the rank of an angularity's rigidity matrix. We develop a combinatorial necessary condition for infinitesimal minimal angle rigidity. Using the developed theories, a formation stabilization algorithm is designed for a robotic team to achieve an angle rigid formation, in which only angle measurements are needed. Simulation examples demonstrate the advantages of the proposed angle-only formation control approach.