Research Papers

Control of Directed Formations Using Interconnected Systems Stability

[+] Author and Article Information
Pengpeng Zhang, Milad Khaledyan, Tairan Liu

Department of Mechanical and
Industrial Engineering,
Louisiana State University,
Baton Rouge, LA 70803

Marcio de Queiroz

Department of Mechanical and
Industrial Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: mdeque1@lsu.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received July 24, 2017; final manuscript received October 4, 2018; published online December 6, 2018. Assoc. Editor: Suman Chakravorty.

J. Dyn. Sys., Meas., Control 141(4), 041003 (Dec 06, 2018) (11 pages) Paper No: DS-17-1381; doi: 10.1115/1.4041703 History: Received July 24, 2017; Revised October 04, 2018

This paper deals with the problem of rigid formation control using directed graphs in both two-dimensional (2D) and three-dimensional (3D) spaces. Directed graphs reduce the number of communication, sensing, and/or control channels of the multi-agent system. We show that the directed version of the gradient descent control law asymptotically stabilizes the interagent distance error dynamics of minimally persistent formation graphs. The control analysis begins with a (possibly cyclic) primitive formation that is grown consecutively by Henneberg-type insertions, resulting at each step in two interconnected nonlinear systems, which are recursively analyzed using the stability of interconnected systems. Simulation and experimental results are presented for the directed formation controller in comparison to the standard undirected controller.

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Cheng, L. , Wang, Y. , Ren, W. , Hou, Z.-G. , and Tan, M. , 2016, “ Containment Control of Multiagent Systems With Dynamic Leaders Based on PIn-Type Approach,” IEEE Trans. Cyber, 46(12), pp. 3004–3017. [CrossRef]
Gazi, V. , and Passino, K. M. , 2011, Swarm Stability and Optimization, Springer, Berlin.
Gutiérrez, H. , Morales, A. , and Nijmeijer, H. , 2017, “ Synchronization Control for a Swarm of Unicycle Robots: Analysis of Different Controller Topologies,” Asian J. Control, 19(6), pp. 1–12.
Ren, W. , and Beard, R. W. , 2008, Distributed Consensus in Multi-Vehicle Cooperative Control, Springer, London.
Ren, W. , and Cao, Y. , 2011, Distributed Coordination of Multi-Agent Networks: Emergent Problems, Models, and Issues, Springer-Verlag, London.
Su, S. , Lin, Z. , and Garcia, A. , 2016, “ Distributed Synchronization Control of Multiagent Systems With Unknown Nonlinearities,” IEEE Trans. Cyber., 46(1), pp. 325–338. [CrossRef]
Summers, T. H. , Yu, C. , Dasgupta, S. , and Anderson, B. D. O. , 2011, “ Control of Minimally Persistent Leader-Remote-Follower and Coleader Formations in the Plane,” IEEE Trans. Autom. Control, 56(12), pp. 2778–2792. [CrossRef]
Eren, T. , Belhumeur, P. N. , Anderson, B. D. O. , and Morse, A. S. , 2002, “ A Framework for Maintaining Formations Based on Rigidity,” IFAC Congress, Barcelona, Spain, July 21–26, pp. 2752–2757.
Anderson, B. D. O. , Yu, C. , Fidan, B. , and Hendrickx, J. M. , 2008, “ Rigid Graph Control Architectures for Autonomous Formations,” IEEE Control Syst. Mag., 28(6), pp. 48–63. [CrossRef]
Cai, X. , and de Queiroz, M. , 2014, “ Rigidity-Based Stabilization of Multi-Agent Formations,” ASME J. Dyn. Syst. Meas. Control, 136(1), p. 014502.
Cai, X. , and de Queiroz, M. , 2015, “ Adaptive Rigidity-Based Formation Control for Multi-Robotic Vehicles With Dynamics,” IEEE Trans. Control Syst. Tech., 23(1), pp. 389–396. [CrossRef]
Cao, M. , Morse, A. S. , Yu, C. , Anderson, B. D. O. , and Dasgupta, S. , 2011, “ Maintaining a Directed, Triangular Formation of Mobile Autonomous Agents,” Commun. Inf. Syst., 11(1), pp. 1–16.
Dimarogonas, D. V. , and Johansson, K. H. , 2009, “ Further Results on the Stability of Distance-Based Multi-Robot Formations,” American Control Conference, St. Louis, MO, June 10–12, pp. 2972–2977.
Dimarogonas, D. V. , and Johansson, K. H. , 2010, “ Stability Analysis for Multi-Agent Systems Using the Incidence Matrix: Quantized Communication and Formation Control,” Automatica, 46(4), pp. 695–700. [CrossRef]
Dörfler, F. , and Francis, B. , 2010, “ Geometric Analysis of the Formation Problem for Autonomous Robots,” IEEE Trans. Autom. Control, 55(10), pp. 2379–2384. [CrossRef]
Krick, L. , Broucke, M. E. , and Francis, B. A. , 2009, “ Stabilization of Infinitesimally Rigid Formations of Multi-Robot Networks,” Intl. J. Control, 82(3), pp. 423–439. [CrossRef]
Oh, K.-K. , and Ahn, H.-S. , 2011, “ Formation Control of Mobile Agents Based on Inter-Agent Distance Dynamics,” Automatica, 47(10), pp. 2306–2312. [CrossRef]
Oh, K.-K. , and Ahn, H.-S. , 2014, “ Distance-Based Undirected Formations of Single-Integrator and Double-Integrator Modeled Agents in n-Dimensional Space,” Int. J. Rob. Nonlinear Control, 24(12), pp. 1809–1820. [CrossRef]
Sun, Z. , Helmke, U. , and Anderson, B. D. O. , 2015, “ Rigid Formation Shape Control in General Dimensions: An Invariance Principle and Open Problems,” IEEE 54th Conference Decision and Control (CDC), Osaka, Japan, Dec. 15–18, pp. 6095–6100.
Yu, C. , Hendrickx, J. M. , Fidan, B. , Anderson, B. D. O. , and Blondel, V. D. , 2007, “ Three and Higher Dimensional Autonomous Formations: Rigidity, Persistence and Structural Persistence,” Automatica, 43(3), pp. 387–402. [CrossRef]
Zhang, P. , de Queiroz, M. , and Cai, X. , 2015, “ Three-Dimensional Dynamic Formation Control of Multi-Agent Systems Using Rigid Graphs,” ASME J. Dyn. Syst. Meas. Control, 137(11), p. 111006.
Baillieul, J. , and Suri, A. , 2003, “ Information Patterns and Hedging Brockett's Theorem in Controlling Vehicle Formations,” IEEE Conference Decision and Control, Maui, HI, Dec. 9–12, pp. 556–563.
Mou, S. , Belabbas, M.-A. , Morse, A. S. , Sun, Z. , and Anderson, B. D. O. , 2016, “ Undirected Rigid Formations Are Problematic,” IEEE Trans. Autom. Control, 61(10), pp. 2821–2836. [CrossRef]
Tanner, H. G. , Pappas, G. J. , and Kumar, V. , 2002, “ Input-to-State Stability on Formation Graphs,” 41st IEEE Conference Decision and Control, Las Vegas, NV, Dec. 10–13, pp. 2439–2444.
Eren, T. , Whiteley, W. , Anderson, B. D. O. , Morse, A. S. , and Bellhumeur, P. N. , 2005, “ Information Structures to Secure Control of Rigid Formations With Leader-Follower Structure,” American Control Conference, Portland, OR, June 8–10, pp. 2966–2971.
Hendrickx, J. M. , Anderson, B. D. O. , Delvenne, J. , and Blondel, V. D. , 2007, “ Directed Graphs for the Analysis of Rigidity and Persistence in Autonomous Agent Systems,” Intl. J. Robust Nonlinear Control, 17(10–11), pp. 960–981. [CrossRef]
Anderson, B. D. O. , Yu, C. , Dasgupta, S. , and Morse, A. S. , 2007, “ Control of a Three-Coleader Formation in the Plane,” Syst. Control Lett., 56(9–10), pp. 573–578. [CrossRef]
Yu, C. , Anderson, B. D. O. , Dasgupta, S. , and Fidan, B. , 2009, “ Control of Minimally Persistent Formations in the Plane,” SIAM J. Control Optim., 48(1), pp. 206–233. [CrossRef]
Park, M.-C. , Sun, Z. , Oh, K.-K. , Anderson, B. D. O. , and Ahn, H.-S. , 2014, “ Finite-Time Convergence Control for Acyclic Persistent Formations,” IEEE International Symposium Intelligent Control (ISIC), Juan Les Pins, France, Oct. 8–10, pp. 1608–1613.
Oh, K.-K. , and Ahn, H.-S. , 2011, “ Distance-Based Control of Cycle-Free Persistent Formations,” IEEE Multi-Conference Systems and Control, Denver, CO, Sept. 28–30, pp. 816–821.
Graver, J. , Servatius, B. , and Servatius, H. , 1993, Combinatorial Rigidity, American Mathematical Society, Providence, RI.
Asimow, L. , and Roth, B. , 1979, “ The Rigidity of Graphs II,” J. Math. Anal. Appl., 68(1), pp. 171–190. [CrossRef]
Izmestiev, I. , 2009, Infinitesimal Rigidity of Frameworks and Surfaces (Lectures on Infinitesimal Rigidity), Kyushu University, Fukuoka, Japan.
Jackson, B. , 2007, “ Notes on the Rigidity of Graphs,” Notes of the Levico Conference, Levico Terme, Italy, Vol. 4.
Tay, T. , and Whiteley, W. , 1985, “ Generating Isostatic Frameworks,” Struc. Topol., 11, pp. 21–69.
Aspnes, J. , Egen, J. , Goldenberg, D. K. , Morse, A. S. , Whiteley, W. , Yang, Y. R. , Anderson, B. D. O. , and Belhumeur, P. N. , 2006, “ A Theory of Network Localization,” IEEE Trans. Mob. Comput., 5(12), pp. 1663–1678. [CrossRef]
Hendrickx, J. M. , Fidan, B. , Yu, C. , Anderson, B. D. O. , and Blondel, V. D. , 2006, “ Elementary Operations for the Reorganization of Minimally Persistent Formations,” International Symposium Mathematical Theory Networks and Systems, Kyoto, Japan, July 24–28, pp. 859–873.
Horn, R. A. , and Johnson, C. R. , 1985, Matrix Analysis, Cambridge University Press, Cambridge, UK.
Cai, X. , and de Queiroz, M. , 2015, “ Formation Maneuvering and Target Interception for Multi-Agent Systems Via Rigid Graphs,” Asian J. Control, 17(4), pp. 1174–1186. [CrossRef]
Khalil, H. K. , 2015, Nonlinear Control, Pearson Education, Harlow, UK.
Marquez, J. H. , 2003, Nonlinear Control Systems Analysis and Design, Wiley, Hoboken, NJ.
Pickem, D. , Wang, L. , Glotfelter, P. , Mote, M. , Ames, A. , Feron, E. , and Egerstedt, M. , 2016, “ The Robotarium: A Remotely Accessible Swarm Robotics Research Testbed,” e-print .
Pickem, D. , Lee, M. , and Egerstedt, M. , 2015, “ The GRITSBot in Its Natural habitat - A Multi-Robot Testbed,” IEEE International Conference Robotic Automation (ICRA), Seattle, WA, May 26–30, pp. 4062–4067.


Grahic Jump Location
Fig. 1

Directed framework example

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Fig. 8

Experiment: robot trajectories qi(t), i=1,…5

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Fig. 9

Experiment: interagent distance errors, eij(t), (i,j)∈Ed∗

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Fig. 2

Basis step for the 3D formation case

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Fig. 3

Simulation: desired framework F* in ℝ3

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Fig. 4

Simulation: Agent trajectories qi(t), i=1,…8

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Fig. 5

Simulation: interagent distance errors eij(t), (i,j)∈Ed∗

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Fig. 6

Simulation: control inputs ui(t), i=1,…,8 along each direction

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Fig. 7

Experiment: desired framework F∗ in ℝ2

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Fig. 10

Experiment: control inputs, ui(t), i=1,…,5 along each direction

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Fig. 11

Tetrahedron for the nth step of the 3D formation case



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