Formations on directed cycles with bearing‐only measurements

Abstract

Summary This paper studies a bearing‐only–based formation control problem for a group of single‐integrator agents with directed cycle sensing topology. In a 2‐dimensional space, a necessary and sufficient condition for the set of desired bearing vectors to be feasible is derived. Then, we propose a bearing‐only control law for every agent and prove that the formation asymptotically converges to a formation specified by a set of feasible desired bearing vectors. Analysis of the equilibrium formations in the plane for a 3‐agent system and subsequent extension to an n‐agent system is provided. We further extend the analysis on directed triangular formation into a 3‐dimensional space. Finally, simulations validate the theoretical results.

Bearing-Based Formation Control of A Group of Agents with Leader-First Follower Structure

“…Therefore, the equilibrium p 2 = p * 2a is globally asymptotically stable and almost globally exponentially stable. (27) and agent i (3 ≤ i ≤ n) employs the control law (24), the formation globally asymptotically reaches the desired formation satisfying all bearing vectors in B.…”

“…Specifically, we focus on the leader-first follower (LFF) graphs that can be generated from a bearing-based Henneberg construction. It is worth remarking that the analysis in the undirected case cannot be used in the directed case due to the asymmetry in the sensing graph [24]. The lack of symmetry raises difficulties in analysis, for example, the formation's centroid and scale are not invariant as in the undirected case.…”

“…For instance, in [25], by assuming the existence of three stationary beacons in the plane, it was proved that any n-agent system with an acyclic directed sensing graph is locally asymptotically stable. The local stability of planar formations with directed cycle graphs was studied in [24], [26]. The authors in [27] introduced the bearing Laplacian from a set of desired bearing vectors and defined bearing persistence based on the null space of the bearing Laplacian.…”

Physical Interpretation of Rigidity for Bearing Formations: Application to Mobility and Singularity Analyses

“…In this context, a powerful theoretical framework for addressing decentralized formation control/localization from onboard sensing is that of formation rigidity [8]. The tools from rigidity theory (and, in particular, the notion of infinitesimal rigidity) have been widely exploited in the community, by first focusing on the cases of distance rigidity [9][10][11][12][13][14] (i.e., assuming that the robots are equipped with a distance sensor), and, more recently, by considering the case of bearing rigidity [6,[15][16][17][18][19][20][21][22][23] (which is, instead, representative of onboard cameras).…”

A scalable algorithm for the decomposition of minimally rigid graph

“…, and the edges in E c are assigned to n À 2 distinct vertices belong to V s in a scalable way via communication. We choose the initialized edge as (1,15; the initialized edge is not unique), and the output of the algorithm is as follows (3,9), (1, 10), (2, 11), (8,12), (4, 13), (7,14), (1,16), (11,17), (9,18), (3,19), (8,20)g 3, 4, 5, 6, 7, 15, 8, 9, 10, 11, 12, f 13, 14, 16, 17, 18, 19, 20g E c = f(3, 2), (4, 3), (5,4), (6,4), (7, 6), (7,15), (8,5), (9,8), (10,3), (11,3), (12,11), (13,11), (14,13), (16,13), (17,12), (18,11), (19,12), (20,…”