Bearing rigidity and formation stabilization for multiple rigid bodies in SE(3)

In this work, we first distinguish different notions related to bearing rigidity in graph theory and then further investigate the formation stabilization problem for multiple rigid bodies. Different from many previous works on formation control using bearing rigidity, we do not require the use of a shared global coordinate system, which is enabled by extending bearing rigidity theory to multi-agent frameworks embedded in the three dimensional special Euclidean group SE(3) and expressing the needed bearing information in each agent's local coordinate system. Here, each agent is modeled by a rigid body with 3 DOFs in translation and 3 DOFs in rotation. One key step in our approach is to define the bearing rigidity matrix in SE(3) and construct the necessary and sufficient conditions for infinitesimal bearing rigidity. In the end, a gradient-based bearing formation control algorithm is proposed to stabilize formations of multiple rigid bodies in SE(3)

Angle rigidity and its usage to stabilize multi-agent formations in 2D

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 multi-point 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 condit

Maneuvering formations of mobile agents using designed mismatched angles

This paper investigates how to maneuver a planar formation of mobile agents using designed mismatched angles. The desired formation shape is specified by a set of interior angle constraints. To realize the maneuver of translation, rotation and scaling of the formation as a whole, we intentionally force the agents to maintain mismatched desired angles by introducing a pair of mismatch parameters for each angle constraint. To allow different information requirements in the design and implementation stages, we consider both measurement-dependent and 10 measurement-independent mismatches. Starting from a triangular formation, we consider generically angle rigid formations that can be constructed from the triangular formation by adding new agents in sequence, each having two angle constraints associated with some existing three agents. The control law for each newly added agent arises naturally from the angle constraints and makes full use of the angle mismatch parameters. We show that the control can effectively stabilize the formations while simultaneously realizing maneuvering. Simulations are conducted to validate the theoretical results

Global stabilization for triangular formations under mixed distance and bearing constraints

This paper addresses the triangular formation control problem for a system of three agents under mixed distance and bearing constraints. The main challenge is to find a fully distributed control law for each agent to guarantee the global convergence towards a desired triangular formation. To solve this problem, we invoke the property that a triangle can be uniquely determined by the lengths of its two sides together with the magnitude of the corresponding included angle. Based on this feature, we design a class of control strategies, under which each agent is only responsible for a single control variable, i.e., a distance or an angle, such that the control laws can be implemented in local coordinate frames. The global convergence is shown by analyzing the dynamics of the closed-loop system in its cascade form. Then we discuss some extensions on more general formation shapes and give the quadrilateral formation as an example. Simulation results are provided to validate the effectiveness of the proposed control strategies

Stabilizing and maneuvering angle rigid multi-agent formations with double-integrator agent dynamics

This paper studies formation stabilization and maneuvering of mobile agents governed by double-integrator dynamics. The desired formation is described by a set of triple-agent angles. A carefully chosen such set of angle constraints guarantees that the desired formation is angle rigid. To achieve the desired angle rigid formation, a stabilization control law is proposed using only local velocity and direction measurements. We show that the closed-loop dynamics of the formation, when each agent is modeled by a double-integrator, are closely related to the corresponding one in single-integrator agent dynamics. Sufficient conditions are constructed to guarantee the closed-loop stability for identical and distinct velocity damping gains, respectively. To guide an angle rigid formation to move with the desired translational velocity, orientation and scale, formation maneuvering laws are then proposed. Simulation examples are also provided to validate the results

Stabilizing a mobile agent under two angle constraints

Recently, angle-based formation control laws have been proposed assuming that each agent only aims at maintaining one desired angle with respect to its neighbors. To extend the angle-based formation control algorithms to more general cases, we propose for each agent to maintain simultaneously multiple prescribed angles. As a first step, we investigate how an agent can be enabled to maintain two desired angles taking three static and collinear markers as references. We first show the condition for the existence of such positions for any given two angle constraints. Then, we propose a formation control law that uses local bearing measurements with respect to the markers, under which the local convergence of the moving agent to the desired position is guaranteed while satisfying the two constraints of the desired angles. A simulation example is provided to demonstrate the correctness of the theoretical results