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(k 1.5 m 1.5) 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(l 3.5 m 1.5) and O(l 1.5 m 3.5) for this problem with O(l 3 m) and O(lm 3) performance in practice – where l denotes the length of the optimization horizon.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Belta, C., Kumar, V.: Abstraction and control for groups of robots. IEEE Trans. on Robotics and Automation (2004)
Belta, C., Kumar, V.: Optimal motion generation for groups of robots: a geometric approach. ASME Journal of Mechanical Design 126 (2004)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Unviersity Press, Cambridge (2004)
Choset, H., Lynch, K., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L., Thrun, S.: Principles of Robot Motion Planning. MIT Press, Cambridge (2005)
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)
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)
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)
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)
Dijkstra, E.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–272 (1959)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. John Wiley, Chichester (1998)
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)
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Springer, Heidelberg (2000)
Jadbabaie, A., Yu, J., Hauser, J.: Unconstrained receding-horizon control of nonlinear systems. IEEE Trans. on Automatic Control 46(5), 776–783 (2001)
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)
Lindemann, S., LaValle, S.: Smooth feedback for car-like vehicles in polygonal environments. In: Proc. IEEE Int. Conf. Robot. Automat, Roma, Italy (April 2007)
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)
Mayne, D., Rawings, J., Rao, C., Scokaert, P.: Constrained model predictive control: Stability and optimality. Automatics 36(6), 789–814 (2000)
MOSEK ApS: The MOSEK Optimization Tools Version 3.2 (Revision 8) User’s Manual and Reference., http://www.mosek.com
Press, W., et al.: Numerical Recipes in C. Cambridge University Press, Cambridge (1993)
Saad, Y.: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (2003)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Derenick, J.C., Spletzer, J.R. (2007). Second-Order Cone Programming (SOCP) Techniques for Coordinating Large-Scale Robot Teams in Polygonal Environments. In: Pardalos, P.M., Murphey, R., Grundel, D., Hirsch, M.J. (eds) Advances in Cooperative Control and Optimization. Lecture Notes in Control and Information Sciences, vol 369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74356-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-74356-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74354-5
Online ISBN: 978-3-540-74356-9
eBook Packages: EngineeringEngineering (R0)