Second-Order Cone Programming (SOCP) Techniques for Coordinating Large-Scale Robot Teams in Polygonal Environments

  • Jason C. Derenick
  • John R. Spletzer
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 369)

Abstract

In this paper, we present an online optimization approach for coordinating large-scale robot teams in both convex and non-convex polygonal environments. In the former, we investigate the problem of moving a team of m robots from an initial shape to an objective shape while minimizing the total distance the team must travel within the specified workspace. Employing SOCP techniques, we establish a theoretical complexity of O(k1.5m1.5) for this problem with O(km) performance in practice – where k denotes the number of linear inequalities used to model the workspace. Regarding the latter, we present a multi-phase hybrid optimization approach. In Phase I, an optimal path is generated over an appropriate tessellation of the workspace. In Phase II, model predictive control techniques are used to identify optimal formation trajectories over said path while guaranteeing against collisions with obstacles and workspace boundaries. Once again employing SOCP, we establish complementary complexity measures of O(l3.5m1.5) and O(l1.5m3.5) for this problem with O(l3m) and O(lm3) performance in practice – where l denotes the length of the optimization horizon.

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References

  1. 1.
    Belta, C., Isler, V., Pappas, G.: Discrete abstractions for robot motion planning and control in polygonal environments. IEEE Transactions on Robotics 17(6), 864–875 (2005)CrossRefGoogle Scholar
  2. 2.
    Belta, C., Kumar, V.: Abstraction and control for groups of robots. IEEE Trans. on Robotics and Automation (2004)Google Scholar
  3. 3.
    Belta, C., Kumar, V.: Optimal motion generation for groups of robots: a geometric approach. ASME Journal of Mechanical Design 126 (2004)Google Scholar
  4. 4.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Unviersity Press, Cambridge (2004)MATHGoogle Scholar
  5. 5.
    Choset, H., Lynch, K., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L., Thrun, S.: Principles of Robot Motion Planning. MIT Press, Cambridge (2005)Google Scholar
  6. 6.
    Conner, D.C., Rizzi, A.A., Choset, H.: Composition of local potential functions for global robot control and navigation. In: IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Las Vegas, Nevada, USA (October 2003)Google Scholar
  7. 7.
    Cortés, J., Martínez, S., Karatas, T., Bullo, F.: Coverage control for mobile sensing networks. IEEE Trans. on Robotics and Automation 20(2), 243–255 (2004)CrossRefGoogle Scholar
  8. 8.
    Das, A.K., Fierro, R., Kumar, V., Ostrowski, J.P., Spletzer, J., Taylor, C.J.: A vision-based formation control framework. IEEE Trans. on Robotics and Automation 18(5), 813–825 (2002)CrossRefGoogle Scholar
  9. 9.
    Derenick, J., Mansley, C., Spletzer, J.: Efficient motion planning strategies for large-scale sensor networks. In: Proceedings of the Seventh International Workshop on the Algorithmic Foundations of Robotics (WAFR 2006), New York, NY, USA (July 2006)Google Scholar
  10. 10.
    Dijkstra, E.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–272 (1959)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. John Wiley, Chichester (1998)MATHGoogle Scholar
  12. 12.
    Feddema, J.T., Robinett, R.D., Byrne, R.H.: An optimization approach to distributed controls of multiple robot vehicles. In: Workshop on Control and Cooperation of Intelligent Miniature Robots, IEEE/RSJ IROS, October 31, 2003, Las Vegas, Nevada (2003)Google Scholar
  13. 13.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Springer, Heidelberg (2000)MATHGoogle Scholar
  14. 14.
    Jadbabaie, A., Yu, J., Hauser, J.: Unconstrained receding-horizon control of nonlinear systems. IEEE Trans. on Automatic Control 46(5), 776–783 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kloetzer, M., Belta, C.: A framework for automatic deployment of robots in 2d and 3d environments. In: IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Beijing, China, October 2006, pp. 953–958 (2006)Google Scholar
  16. 16.
    Lindemann, S., LaValle, S.: Smooth feedback for car-like vehicles in polygonal environments. In: Proc. IEEE Int. Conf. Robot. Automat, Roma, Italy (April 2007)Google Scholar
  17. 17.
    Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. In: Linear Algebra and Applications, Special Issue on Linear Algebra in Control, Signals and Image Processing (1998)Google Scholar
  18. 18.
    Mayne, D., Rawings, J., Rao, C., Scokaert, P.: Constrained model predictive control: Stability and optimality. Automatics 36(6), 789–814 (2000)MATHCrossRefGoogle Scholar
  19. 19.
    MOSEK ApS: The MOSEK Optimization Tools Version 3.2 (Revision 8) User’s Manual and Reference., http://www.mosek.com
  20. 20.
    Press, W., et al.: Numerical Recipes in C. Cambridge University Press, Cambridge (1993)Google Scholar
  21. 21.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (2003)MATHGoogle Scholar
  22. 22.
    Zhang, F., Goldgeier, M., Krishnaprasad, P.S.: Control of small formations using shape coordinates. In: Proc. IEEE Int. Conf. Robot. Automat., vol. 2, Taipei, Sep. 2003, IEEE Computer Society Press, Los Alamitos (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jason C. Derenick
    • 1
  • John R. Spletzer
    • 1
  1. 1.Lehigh University, Bethlehem PA 18015USA

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