A Graph Laplacian Approach to Coordinate-Free Formation Stabilization for Directed Networks

This paper concentrates on coordinate-free formation control for directed networks, for which the dynamic motion of each agent is assumed to be governed only by a local control. We develop a graph Laplacian approach to solve the global and exponential formation stabilization problem using merely relative position measurements between neighbors. First, to capture the sensing and control architectures that are needed to maintain the shape of a formation, a necessary and sufficient topological condition is proposed. Second, a Laplacian-based control law is developed for the stabilization problem of a group of mobile agents to a desired formation shape under both fixed and switching topologies due to temporal node failures. Simulation results are provided to demonstrate that our Laplacian-based formation control strategy is inherently fault-tolerant and robust to node failures.

[1]  Manfredi Maggiore,et al.  Necessary and sufficient graphical conditions for formation control of unicycles , 2005, IEEE Transactions on Automatic Control.

[2]  Brian D. O. Anderson,et al.  Formation control using range-only measurements , 2011, Autom..

[3]  Christian Commault,et al.  Generic properties and control of linear structured systems: a survey , 2003, Autom..

[4]  George J. Pappas,et al.  Distributed formation control with permutation symmetries , 2007, 2007 46th IEEE Conference on Decision and Control.

[5]  C. Loan.  The Sensitivity of the Matrix Exponential , 1977 .

[6]  Vijay Kumar,et al.  Decentralized formation control with variable shapes for aerial robots , 2012, 2012 IEEE International Conference on Robotics and Automation.

[7]  Jorge Cortés,et al.  Global and robust formation-shape stabilization of relative sensing networks , 2009, Autom..

[8]  Lili Wang,et al.  Affine formation of multi-agent systems over directed graphs , 2014, 53rd IEEE Conference on Decision and Control.

[9]  J. Hendrickx,et al.  Rigid graph control architectures for autonomous formations , 2008, IEEE Control Systems.

[10]  Naomi Ehrich Leonard,et al.  Cooperative Control for Ocean Sampling: The Glider Coordinated Control System , 2008, IEEE Transactions on Control Systems Technology.

[11]  Steven J. Gortler,et al.  Characterizing generic global rigidity , 2007, Ad Hoc Networks.

[12]  Minyue Fu,et al.  A Barycentric Coordinate Based Distributed Localization Algorithm for Sensor Networks , 2014, IEEE Transactions on Signal Processing.

[13]  Vincent D. Blondel,et al.  Three and higher dimensional autonomous formations: Rigidity, persistence and structural persistence , 2007, Autom..

[14]  Zhiyun Lin,et al.  Complex Laplacian and pattern formation in multi-agent systems , 2012, 2012 24th Chinese Control and Decision Conference (CCDC).

[15]  Minyue Fu,et al.  Distributed Self Localization for Relative Position Sensing Networks in 2D Space , 2015, IEEE Transactions on Signal Processing.

[16]  Hyo-Sung Ahn,et al.  Distance‐based undirected formations of single‐integrator and double‐integrator modeled agents in n‐dimensional space , 2014 .

[17]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[18]  O. Venjakob,et al.  Localizations and completions of skew power series rings , 2010 .

[19]  Brian D. O. Anderson,et al.  Combining distance-based formation shape control with formation translation , 2012 .

[20]  Minyue Fu,et al.  A linear control approach to distributed multi-agent formations in d-dimensional space , 2013, 52nd IEEE Conference on Decision and Control.

[21]  Lorenzo Sabattini,et al.  Arbitrarily shaped formations of mobile robots: artificial potential fields and coordinate transformation , 2011, Auton. Robots.

[22]  Minyue Fu,et al.  Localizability and Distributed Localization of Sensor Networks using Relative Position Measurements , 2013 .

[23]  Dusan M. Stipanovic,et al.  Formation Control and Collision Avoidance for Multi-agent Non-holonomic Systems: Theory and Experiments , 2008, Int. J. Robotics Res..

[24]  Lili Wang,et al.  Formation control of directed multi-agent networks based on complex Laplacian , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[25]  Yu-Ping Tian,et al.  Global stabilization of rigid formations in the plane , 2013, Autom..

[26]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[27]  Karl H. Johansson,et al.  Further results on the stability of distance-based multi-robot formations , 2009, 2009 American Control Conference.

[28]  Pierre F. J. Lermusiaux,et al.  Science of Autonomy: Time-Optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles , 2016 .

[29]  Maurizio Porfiri,et al.  Tracking and formation control of multiple autonomous agents: A two-level consensus approach , 2007, Autom..

[30]  Brian D. O. Anderson,et al.  Control of Minimally Persistent Formations in the Plane , 2009, SIAM J. Control. Optim..

[31]  Pierre F. J. Lermusiaux,et al.  Path planning in time dependent flow fields using level set methods , 2012, 2012 IEEE International Conference on Robotics and Automation.

[32]  Randal W. Beard,et al.  Multiple UAV Coalitions for a Search and Prosecute Mission , 2011, J. Intell. Robotic Syst..

[33]  W. Ren.  Consensus strategies for cooperative control of vehicle formations , 2007 .

[34]  Florian Dörfler,et al.  Geometric Analysis of the Formation Problem for Autonomous Robots , 2010, IEEE Transactions on Automatic Control.

[35]  George J. Pappas,et al.  Flocking in Fixed and Switching Networks , 2007, IEEE Transactions on Automatic Control.

[36]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[37]  Wei Ding,et al.  Collective motions and formations under pursuit strategies on directed acyclic graphs , 2010, Autom..

[38]  Soura Dasgupta,et al.  Coordinated fault tolerant optimal control of autonomous agents: geometry and communication architecture , 2011, Commun. Inf. Syst..

[39]  Lili Wang,et al.  Distributed Formation Control of Multi-Agent Systems Using Complex Laplacian , 2014, IEEE Transactions on Automatic Control.

[40]  Murat Arcak,et al.  Pattern Formation by Lateral Inhibition in Large-Scale Networks of Cells , 2013, IEEE Transactions on Automatic Control.

[41]  Mireille E. Broucke,et al.  Stabilisation of infinitesimally rigid formations of multi-robot networks , 2009, Int. J. Control.

[42]  Lili Wang,et al.  Realizability of similar formation and local control of directed multi-agent networks in discrete-time , 2013, 52nd IEEE Conference on Decision and Control.

[43]  Hyo-Sung Ahn,et al.  Formation control of mobile agents based on inter-agent distance dynamics , 2011, Autom..

[44]  Alessandro Giua,et al.  Leader-follower formation via complex Laplacian , 2013, Autom..

[45]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion With Limited Communication , 2008, IEEE Transactions on Automatic Control.

[46]  Murat Arcak,et al.  Scaling the size of a formation using relative position feedback , 2012, Autom..

[47]  Shmuel Friedland,et al.  On inverse multiplicative eigenvalue problems for matrices , 1975 .