A Unified Dissertation on Bearing Rigidity Theory

Accounting for the current state-of-the-art, this work aims at summarizing the main notions about the bearing rigidity theory, namely the branch of knowledge investigating the structural properties for multi-element systems necessary to preserve the inter-units bearings when exposed to deformations. Our original contribution consists in the definition of a unified framework for the statement of the principal definitions and results on the bearing rigidity theory that are then particularized by evaluating the most studied metric spaces.

[1]  John Baillieul,et al.  The combinatorial graph theory of structured formations , 2007, 2007 46th IEEE Conference on Decision and Control.

[2]  T. Eren,et al.  Using Angle of Arrival (Bearing) Information in Network Localization , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[3]  Antonio Franchi,et al.  Bearing rigidity theory in SE(3) , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[4]  Robert Mahony,et al.  The Role of Symmetry in Rigidity Analysis: A Tool for Network Localization and Formation Control , 2018, IEEE Transactions on Automatic Control.

[5]  Robert Mahony,et al.  Generalised Rigidity and Path-Rigidity for Agent Formations , 2016 .

[6]  Shiyu Zhao,et al.  Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization , 2014, IEEE Transactions on Automatic Control.

[7]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[8]  Riccardo Spica,et al.  Active decentralized scale estimation for bearing-based localization , 2016, 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[9]  Hyo-Sung Ahn,et al.  Formation Control and Network Localization via Orientation Alignment , 2014, IEEE Transactions on Automatic Control.

[10]  Roberto Tron,et al.  Theory and methods for bearing rigidity recovery , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[11]  P. Giordano,et al.  Rigidity theory in SE(2) for unscaled relative position estimation using only bearing measurements , 2013, 2014 European Control Conference (ECC).

[12]  Marcio de Queiroz,et al.  Adaptive Rigidity-Based Formation Control for Multirobotic Vehicles With Dynamics , 2015, IEEE Transactions on Control Systems Technology.

[13]  Abdo Y. Alfakih,et al.  On the universal rigidity of generic bar frameworks , 2010, Contributions Discret. Math..

[14]  Antonio Franchi,et al.  Modeling and Control of UAV Bearing Formations with Bilateral High-level Steering , 2012, Int. J. Robotics Res..

[15]  Brian D. O. Anderson,et al.  Stabilization of rigid formations with direction-only constraints , 2011, IEEE Conference on Decision and Control and European Control Conference.

[16]  B. Anderson,et al.  Directed graphs for the analysis of rigidity and persistence in autonomous agent systems , 2007 .

[17]  Robert Connelly,et al.  Generic Global Rigidity , 2005, Discret. Comput. Geom..

[18]  Walter Whiteley,et al.  The Union of Matroids and the Rigidity of Frameworks , 1988, SIAM J. Discret. Math..

[19]  Antonio Franchi,et al.  Decentralized rigidity maintenance control with range measurements for multi-robot systems , 2013, Int. J. Robotics Res..

[20]  Ligang Liu,et al.  On affine rigidity , 2010, J. Comput. Geom..

[21]  B. Roth.  Rigid and Flexible Frameworks , 1981 .

[22]  Shiyu Zhao,et al.  Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions , 2015, Autom..

[23]  Gaurav S. Sukhatme,et al.  Evaluating Network Rigidity in Realistic Systems: Decentralization, Asynchronicity, and Parallelization , 2014, IEEE Transactions on Robotics.

[24]  Antonio Franchi,et al.  Bearing-only formation control using an SE(2) rigidity theory , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[25]  I. Sokolov,et al.  Rigidity , 2020, Geometric Regular Polytopes.

[26]  B. Roth,et al.  The rigidity of graphs , 1978 .

[27]  Shiyu Zhao,et al.  Bearing-based distributed control and estimation of multi-agent systems , 2015, 2015 European Control Conference (ECC).

[28]  Giulia Michieletto,et al.  Formation Control for Fully Actuated Systems: a Quaternion-based Bearing Rigidity Approach , 2019, 2019 18th European Control Conference (ECC).

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

[30]  Tolga Eren.  Using Angle of Arrival (Bearing) Information for Localization in Robot Networks , 2007 .

[31]  Zhiyong Sun,et al.  Laman graphs are generically bearing rigid in arbitrary dimensions , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[32]  Ying Zhang,et al.  Rigidity guided localisation for mobile robotic sensor networks , 2010, Int. J. Ad Hoc Ubiquitous Comput..

[33]  Steven J. Gortler,et al.  Characterizing the Universal Rigidity of Generic Frameworks , 2014, Discret. Comput. Geom..

[34]  Monique Laurent,et al.  Cuts, matrix completions and graph rigidity , 1997, Math. Program..

[35]  Frank Dellaert,et al.  Rigid components identification and rigidity control in bearing-only localization using the graph cycle basis , 2015, 2015 American Control Conference (ACC).

[36]  Anthony Nixon,et al.  Rigidity of Frameworks Supported on Surfaces , 2010, SIAM J. Discret. Math..

[37]  Shiyu Zhao,et al.  Bearing Rigidity Theory and Its Applications for Control and Estimation of Network Systems: Life Beyond Distance Rigidity , 2018, IEEE Control Systems.

[38]  B. Anderson,et al.  Sensor and network topologies of formations with direction, bearing, and angle information between agents , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[39]  B. Anderson,et al.  A FRAMEWORK FOR MAINTAINING FORMATIONS BASED ON RIGIDITY , 2002 .

[40]  Gordon M. Crippen,et al.  Distance Geometry and Molecular Conformation , 1988 .

[41]  Zhiyong Sun,et al.  Quantization effects and convergence properties of rigid formation control systems with quantized distance measurements , 2018, International Journal of Robust and Nonlinear Control.

[42]  A. Alfakih.  Graph rigidity via Euclidean distance matrices , 2000 .

[43]  Antonio Franchi,et al.  Rigidity Maintenance Control for Multi-Robot Systems , 2012, Robotics: Science and Systems.

[44]  Saba Ramazani,et al.  Rigidity-Based Multiagent Layered Formation Control , 2017, IEEE Transactions on Cybernetics.

[45]  Frank Allgöwer,et al.  Growing optimally rigid formations , 2012, 2012 American Control Conference (ACC).

[46]  Antonio Franchi,et al.  Decentralized control of parallel rigid formations with direction constraints and bearing measurements , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[47]  Jianghai Hu,et al.  Stiffness matrix and quantitative measure of formation rigidity , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[48]  Jianghai Hu,et al.  Optimizing formation rigidity under connectivity constraints , 2010, 49th IEEE Conference on Decision and Control (CDC).

[49]  B. Jackson.  Notes on the Rigidity of Graphs , 2007 .

[50]  Tolga Eren,et al.  Formation shape control based on bearing rigidity , 2012, Int. J. Control.

[51]  Antonio Franchi,et al.  A rigidity-based decentralized bearing formation controller for groups of quadrotor UAVs , 2016, 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).