Formation shape control based on bearing rigidity. (English) Zbl 1417.93112

Summary: Distance measurements are not the only geometric quantities that can be used for multi-agent formation shape control. Bearing measurements can be used in conjunction with distances. This article employs bearing rigidity for mobile formations, which was developed for robot and sensor network localisation, so that bearings can be used for shape control in mobile formations. The first part of this article examines graph theoretical models for formation network analysis and control law design that are needed to maintain the shape of a formation in two-dimensional space, while the formation moves as a cohesive whole. Bearing-based shape control for a formation of mobile agents involves the design of distributed control laws that ensure the formation moves, so that bearing constraints maintain some desired values. The second part of this article focuses on the design of a distributed control scheme for nonholonomic agents to solve the bearing-based formation shape control problem. In particular, a control law using feedback linearisation is proposed based on shape variables. We simulate the shape control behaviour on differential drive agents for an exemplary bearing rigid formation using the results obtained in the first and second parts of this article.


93B27 Geometric methods
93A14 Decentralized systems
68T42 Agent technology and artificial intelligence
93B18 Linearizations
93B52 Feedback control
Full Text: DOI


[1] Alur, R, Das, A, Esposito, J, Fierro, R, Hur, Y, Grudic, G, Kumar, V, Lee, I, Ostrowski, JP, Pappas, G, Southall, J, Spletzer, J and Taylor, CJ. 2001. “A Framework and Architecture for Multirobot Coordination”. In Experimental Robotics VII, Edited by: Rus, D and Singh, S. 303-312. New York: Springer-Verlag.
[2] Anderson, BDO. Belhumeur, P.N., Eren, T., Goldenberg, D.K., Morse, A.S., Whiteley, W., and Yang, Y.R. (2009), ‘Graphical Properties of Easily Localizable Sensor Networks’, Wireless Networks, 15, pp. 177-191
[3] Anderson, BDO, Yu, C, Fidan, B and Hendrickx, JM. 2008. Rigid Graph Control Architectures for Autonomous Formations. Control Systems Magazine, IEEE, 28: 48-63. · Zbl 1395.93383
[4] Aspnes, J, Eren, T, Goldenberg, D, Whiteley, W, Yang, YR, Morse, AS, Anderson, BDO and Belhumeur, PN. 2006. A Theory of Network Localization. IEEE Transactions on Mobile Computing, 5: 1663-1678.
[5] Baker, S and Nayar, SK. 1999. A Theory of Single-Viewpoint Catadioptric Image Formation. International Journal on Computer Vision, 35: 175-196.
[6] Consolini, L, Morbidi, F, Prattichizzo, D and Tosques, M. 2009. Stabilization of a Hierarchical Formation of Unicycle Robots with Velocity and Curvature Constraints. IEEE Transactions on Robotics, 25: 1176-1184.
[7] Das, A, Fierro, R, Kumar, V and Ostrowski, J. 2002. A Vision-based Formation Control Framework. IEEE Transactions on Robotics and Automation, 18: 813-825.
[8] Desai, J. 2002. A Graph Theoretic Approach for Modeling Mobile Robot Team Formations. Journal of Robotic Systems (now called: Journal of Field Robotics), 19: 511-525. · Zbl 1020.68089
[9] Desai, J, Kumar, V and Ostrowski, J. 1998. Controlling Formations of Multiple Mobile Robots. Proceedings of 1998 IEEE International Conference on Robotics and Automation. 1998. pp.2864-2869.
[10] Desai, J, Ostrowski, J and Kumar, V. 2001. Modeling and Control of Formations of Nonholonomic Mobile Robots. IEEE Transactions on Robotics and Automation, 17: 905-908.
[11] Desai, J, Wang, CC, Zefran, M and Kumar, V. 1996. Motion Planning for Multiple Mobile Manipulators. Proceedings of 1996 IEEE International Conference on Robotics and Automation. 1996. pp.2073-2078. Minneapolis, , Minnesota
[12] Dieudonne, Y, Labbani-Igbida, O and Petit, F. 2010. Deterministic Robot-network Localization is Hard. IEEE Transactions on Robotics, 26: 331-339.
[13] Dubins, LE. 1957. On Curves of Minimal Length with a Constraint on Average Curvature and with Presribed Initial and Terminal Positions and Tangents. American Journal of Mathematics, 79: 497-516. · Zbl 0098.35401
[14] Eren, T. 2007. Using Angle of Arrival (Bearing) Information for Localization in Robot Networks. Turkish Journal of Electrical Engineering and Computer Sciences, 15: 169-186.
[15] Eren, T, Anderson, BDO, Morse, AS, Whiteley, W and Belhumeur, PN. 2004. Operations on Rigid Formations of Autonomous Agents. Communications in Information and Systems, 3: 223-258.
[16] Eren, T. Belhumeur, P.N., Anderson, B.D.O., and Morse, A.S. (2002), ‘A Framework for Maintaining Formations Based on Rigidity’, in Proceedings of the 15Th IFAC World Congress, July, Barcelona, Spain, pp. 2752-2757
[17] Eren, T, Goldenberg, DK, Whiteley, W, Morse, AS, Anderson, BDO and Belhumeur, PN. 2004. Rigidity, Computation, and Randomization in Network Localization. Proceedings of the International Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM). March2004. pp.2673-2684. Hong Kong
[18] Eren, T, Whiteley, W and Belhumeur, PN. 2006. Using Angle of Arrival (Bearing) Information in Network Localization. Proceedings of the 45th IEEE Conference on Decision and Control. 2006. pp.4676-4681. December, San Diego, , California
[19] Eren, T, Whiteley, W, Morse, AS, Belhumeur, PN and Anderson, BDO. 2003. Sensor and Network Topologies of Formations with Direction, Bearing and Angle Information Between Agents. Proceedings of the 42nd IEEE Conference on Decision and Control. 2003. pp.3064-3069. December, Maui, , Hawaii
[20] Fidan, B and Anderson, BDO. 2007. Switching Control for Robust Autonomous Robot and Vehicle Platoon Formation Maintenance. Proceedings of the 15th Mediterranean Conference on Control and Automation. 2007. pp.1-6. Athens, , Greece
[21] Fidan, B, Yu, C and Anderson, BDO. 2007. Acquiring and Maintaining Persistence of Autonomous Multi-vehicle Formations. IET Control Theory and Applications, 1: 452-460.
[22] Fierro, R, Das, A, Spletzer, J, Alur, R, Esposito, J, Hur, Y, Grudic, G, Kumar, V, Lee, I, Ostrowski, JP, Pappas, G, Southall, J and Taylor, CJ. 2002b. Framework and Architecture for Multi-robot Coordination. The International Journal of Robotics Research, 21: 977-995.
[23] Fierro, R, Song, P, Das, A and Kumar, V. 2002a. “Cooperative control of robot formations”. In Cooperative Control and Optimization, Applied Optimization, Edited by: Murphey, R and Pardalos, P. Vol. 66, 73-93. Dordrecht, The Netherlands: Kluwer Academic Press.
[24] Gazi, V. 2005. Formation Control of a Multi-agent System using Nonlinear Servomechanism. International Journal of Control, 78: 554-565. · Zbl 1134.93333
[25] Gazi, V and Passino, KM. 2011. Swarm Stability and Optimization, Springer Publishing Company, Incorporated. · Zbl 1303.68008
[26] Hendrickx, JM, Anderson, BDO, Delvenne, JC and Blondel, VD. 2007. Directed Graphs for the Analysis of Rigidity and Persistence in Autonomous Agents Systems. International Journal of Robust and Nonlinear Control, 17: 960-981. · Zbl 1266.68182
[27] Jackson, B and Jordán, T. 2005. Connected Rigidity Matroids and Unique Realisations of Graphs. Journal of Combinatorial Theory, Series B, 94(1): 1-29. · Zbl 1076.05021
[28] Kendall, DG. 1984. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces. Bulletin of the London Mathematical Society, 16: 81-121. · Zbl 0579.62100
[29] Khalil, H. 2002. Nonlinear Systems, 3rd, Upper Saddle River, , NJ: Prentice Hall.
[30] Krick, L, Broucke, ME and Francis, BA. 2009. Stabilisation of Infinitesimally Rigid Formations of Multi-robot Networks. International Journal of Control, 82: 423-439. · Zbl 1168.93306
[31] Latombe, JC. 1991. Robot Motion Planning, Boston: Kluwer Academic Publishers.
[32] Mariottini, GL, Morbidi, F, Prattichizzo, D, Pappas, GJ and Daniilidis, K. 2007. Leader Follower Formations: Uncalibrated Vision-based Localization and Control. Proceedings of IEEE International Conference on Robotics and Automation. 2007. pp.2403-2408. Rome, , Italy
[33] Mariottini, GL, Pappas, GJ, Prattichizzo, D and Daniilidis, K. 2005. Vision-based Localization of Leader-follower Formations. Proceedings of IEEE Conference on Decision and Control. 2005. pp.635-640. Seville, , Spain
[34] Morbidi, F, Mariottini, GL and Prattichizzo, D. 2010. Observer Design via Immersion and Invariance for Vision-based Leader-follower Formation Control. Automatica, 46: 148-154. · Zbl 1214.93011
[35] Olfati-Saber, R and Murray, RM. 2002. Graph Rigidity and Distributed Formation Stabilization of Multi-vehicle Systems. Proceedings of the 41st IEEE Conference on Decision and Control. 2002. pp.2965-2971. December, Las Vegas, , NV, USA
[36] Reeds, JA and Shepp, LA. 1990. Optimal Paths for a Car That Goes Both Forwards and Backwards. Pacific Journal of Mathematics, 145: 367-393.
[37] Servatius, B and Whiteley, W. 1999. Constraining Plane Configurations in Computer Aided Design: Combinatorics of Directions and Lengths. SIAM Journal of Discrete Mathematics, 12: 136-153. · Zbl 0916.68182
[38] Slotine, JJE and Li, W. 1991. Applied Nonlinear Control, New Jersey: Prentice Hall.
[39] Whiteley, W. 1997. “Rigidity and Scene Analysis”. In Handbook of Discrete and Computational Geometry, Edited by: Goodman, J and O’Rourke, J. 893-916. Boca Raton, FL, , USA: CRC Press.
[40] Yu, C, Anderson, BDO, Dasgupta, S and Fidan, B. 2009. Control of Minimally Persistent Formations in the Plane. SIAM Journal on Control and Optimization, 48: 206-233. · Zbl 1182.93013
[41] Yu, C, Hendrickx, JM, Fidan, B, Anderson, BDO and Blondel, VD. 2007. Three and Higher Dimensional Autonomous Formations: Rigidity, Persistence and Structural Persistence. Automatica, 43: 387-402. · Zbl 1137.93309
[42] Yun, X and Yamamoto, Y. 1993. Internal Dynamics of a Wheeled Mobile Robot. Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems. 1993. pp.1288-1294. Yokohama, , Japan
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