Geometric Analysis of the Formation Problem for Autonomous Robots

• Published in: IEEE transactions on automatic control Vol. 55; no. 10; pp. 2379 - 2384
• Main Authors: Dörfler, Florian, Francis, Bruce
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Autonomous; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control theory. Systems; Convergence; Distributed control; Dynamic tests; Dynamics; Exact sciences and technology; Formation control; global stability analysis; hyperbolic invariant manifolds; Invariants; Lyapunov method; Mathematics; Mobile robots; Robot control; Robot kinematics; Robot sensing systems; Robotics; Robots; Sensor arrays; Sensor phenomena and characterization; Stability; Stability analysis; Surveillance; Autonomous system; Invariant; Center manifold; Differential geometry; Global analysis; Control program; Formation control; Robot dynamics; Invariant set; Local effect; Geometrical model; Robotics; Closed feedback; Global stabilization; global stability analysis; Distributed control; Instability; hyperbolic invariant manifolds; Formation; Lyapunov function; Linearization
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2010.2053735

In the formation control problem for autonomous robots, a distributed control law steers the robots to the desired target formation. A local stability result of the target formation can be derived by methods of linearization and center manifold theory or via a Lyapunov-based approach. Besides the target formation, the closed-loop dynamics of the robots feature various other undesired invariant sets such as nonrigid formations. This note addresses a global stability analysis of the closed-loop formation control dynamics. We pursue a differential geometric approach and derive purely algebraic conditions for local stability of invariant embedded submanifolds. These theoretical results are then applied to the well-known example of a cyclic triangular formation and result in instability of all invariant sets other than the target formation.