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

Abstract

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.

Angle-Constrained Formation Control for Circular Mobile Robots

“…By some algebraic computations, we obtain 1 2 c ij f ij if and only if cos θ ij 5 6 . When cos θ ij < 5 6 , we have the region for which k ij e ij is positive is a subset of e feas ij , whereas when cos θ ij ≥ 5 6 , we have k ij e ij is positive over the entire domain e feas ij . The properties of (8) will be used later for deriving the exponential convergence of the error dynamics.…”

“…The robots use available relative position or distance (or relative bearing) measurements in the design and execution of the distributed control laws. Recently, new rigidity theories, such as angle rigidity [5], ratio-of-distance-rigidity [6], and bearingratio-of-distance-rigidity theory [7] have also been developed for characterizing a (at least locally) unique target formation shape using a set of angle, ratio-of-distance, and bearing-ratioof-distance constraints, respectively. These theories focus on providing more flexibility to the target deployment by allowing scaling or rotational motions.…”

Securing and Maneuvering Heterogeneous Mobile Robot Formations: Distributed Control Design and Stability Analysis

“…Prior to designing control laws for the mobile robots, we need a way to describe the desired geometrical shape. In the existing literature, this desired shape can be given by a set of fixed positions [38], relative positions [42], distances [28,48], bearings 2 [54], or angles [16]. Each specification leads to several sensing requirements for the team of mobile robots.…”

“…Roughly speaking, this means that during a motion, the shape should not be able to change its form while satisfying the given set of distance constraints. Rigidity theory has also been developed for shapes described by a set of bearing [55] or angle [16] constraints. Fig.…”

“…When the formation shape is specified by inter-robot relative positions [41], then the consensus-based formation control approaches, which are linear, can be used directly. For realizing a formation shape using only distance [23,28], bearing [53], or angle constraints [16], the notion of rigidity [6,16,55] for the underlying interconnection topology is required. The condition of rigidity describes the motions for the whole formation which preserve the formation shape.…”