American Institute of Mathematical Sciences

2013, 3(1): 31-48. doi: 10.3934/naco.2013.3.31

Safe and reliable coverage control

 1 Coordinated Science Laboratory, Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, United States 2 Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, California, United States 3 Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, California, United States

Received  December 2011 Revised  November 2012 Published  January 2013

In this paper we consider a problem of designing control laws for multiple mobile agents trying to accomplish three objectives. One of the objectives is to sense a given compact domain while satisfying the other objective which is to avoid collisions between the agents themselves as well as with the obstacles. To keep the communication links between the agents reliable, the agents need to stay relatively close during the sensing operation which is the third and final objective. The design of control laws is based on carefully constructed objective functions and on an assumption that the agents' dynamic models are nonlinear yet affine in control laws. As an illustration of some performance characteristics of the proposed control laws, a numerical example is provided.
Citation: Dušan M. Stipanović, Christopher Valicka, Claire J. Tomlin, Thomas R. Bewley. Safe and reliable coverage control. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 31-48. doi: 10.3934/naco.2013.3.31
References:
 [1] A. Bacciotti and L. Rosier, "Liapunov Functions and Stability in Control Theory," 2ndedition, Springer-Verlag, Berlin, Germany, 2005. [2] J. V. Breakwell and P. Hagedorn, Point capture of two evaders in succession, Journal of Optimization Theory and Applications, 27 (1979), 89-97. doi: 10.1007/BF00933327. [3] C. R. Burns, R. F. Wang and D. M. Stipanović, A study of human and receding horizon controller perfomance of a remote navigation task with obstacles and feedback delays, Journal of Behavioral Robotics, 2 (2011), 44-63. [4] F. L. Chernousko, Controlled search of a moving object, Prikladnia Matematika i Mekhanika (in Russian), 44 (1981), 3-12. [5] N. Chopra, D. M. Stipanovićand M. W. Spong, On synchronization and collision avoidance for mechanical systems, Proceedings of the 2008 American Control Conference, (2008), 3713-3718. doi: 10.1109/ACC.2008.4587071. [6] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," Mc-Graw Hill, New York, NY, 1955. [7] M. Corless, G. Leitmann, and J. M. Skowronski, Adaptive control for avoidance or evasion in an uncertain environment, Computers & Mathematics with Applications, 13 (1987), 1-11. doi: 10.1016/0898-1221(87)90090-3. [8] M. Corless and G. Leitmann, Adaptive controllers for avoidance or evasion in an uncertain environment: some examples, Computers & Mathematics with Applications, 18 (1989), 161-170. doi: 10.1016/0898-1221(89)90133-8. [9] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [10] V. V. Filippov, "Basic Topological Structures of Ordinary Differential Equations," Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. [11] H. Flanders, Differentiation under the integral sign, The American Mathematical Monthly, 80 (1973), 615-627. doi: 10.2307/2319163. [12] R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques," Birkhäuser, Boston, MA, 1996. [13] J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, NY, 1993. [14] P. F. Hokayem, D. M. Stipanović and M. W. Spong, Semiautonomous control of multiple networked Langrangian systems, International Journal of Robust and Nonlinear Control, 19 (2009), 2040-2055. doi: 10.1002/rnc.1391. [15] I. I. Hussein and D. M. Stipanović, Effective coverage control for mobile sensor networks with guaranteed collision avoidance, IEEE Transactions on Control Systems Technology, 15 (2007), 642-657. doi: 10.1109/TCST.2007.899155. [16] I. I. Hussein and D. M. Stipanović, Effective coverage control using dynamic sensor networks with flocking and guaranteed collision avoidance, Proceedings of the 2007 American Control Conference, (2007), 3420-3425. doi: 10.1109/ACC.2007.4282310. [17] R. Isaacs, "Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization," John Wiley and Sons, Inc., New York, NY, 1965. [18] A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover Publications, New York, NY, 1975. [19] V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications," Mathematics in Science and Engineering, vol. 55, Academic Press, New York, NY, 1969. [20] V. Lakshmikantham, S. Leela and A. A. Martinyuk, "Stability Analysis of Nonlinear Systems," Marcel Dekker, New York, NY, 1989. [21] I. B. Lazarević, "Multidimensional Mathematical Analysis," Orion-Art, Belgrade, Serbia, 2005. [22] G. Leitmann and J. Skowronski, Avoidance control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298. [23] G. Leitmann, Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, 32 (1980), 569-576. doi: 10.1007/BF00934040. [24] G. Leitmann, Guaranteed avoidance feedback control, IEEE Transactions on Automatic Control, 25 (1980), 850-851. doi: 10.1109/TAC.1980.1102408. [25] G. Leitmann and J. Skowronski, A note on avoidance control, Optimal Control Applications & Methods, 4 (1983), 335-342. doi: 10.1002/oca.4660040406. [26] S. Mastellone, D. M. Stipanović, C. R. Graunke, K. A. Intlekofer and M. W. Spong, Formation control and collision avoidance for multi-agent nonholonomic systems: theory and experiments, International Journal of Robotics Research, 13 (2008), 107-126. doi: 10.1177/0278364907084441. [27] A. A. Melikyan, The problem of time-optimal control with the search for a target point, Prikladnia Matematika i Mekhanika (In Russian), 54 (1990), 1-7. [28] K. M. Miettinen, "Nonlinear Multiobjective Optimization," Kluwer Academic Publishers, Norwell, MA, 1998. doi: 10.1007/978-1-4615-5563-6. [29] I. Mitchell, A. M. Bayen and C. J. Tomlin, A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games, IEEE Transactions on Automatic Control, 50 (2005), 947-957. doi: 10.1109/TAC.2005.851439. [30] L. A. Petrosjan, "Differential Games of Pursuit," Series on Optimization, vol. 2, World Scientific, Singapore, 1993. doi: 10.1142/1670. [31] E. J. Rodríguez-Seda, J. J. Troy, C. A. Erignac, P. Murray, D. M. Stipanović and M. W. Spong, Bilateral teleoperation of multiple mobile agents: Formation control and collision avoidance, IEEE Transactions on Control Systems Technology, 18 (2010), 984-992. doi: 10.1109/TCST.2009.2030176. [32] E. J. Rodríguez-Seda, D. M. Stipanović and M. W. Spong, Collision avoidance control with sensing uncertainties, Proceedings of the 2011 American Control Conference, 2011. [33] M. Saska, J. S. Mejía, D. M. Stipanović and K. Schilling, Control and navigation of formations of car-like robots on a receding horizon, Proceedings of the 2009 IEEE Multi-conference on Systems and Control. St Petersburg, Russia, 2009. [34] R. Siegwart and I. R. Nourbakhsh, "Introduction to Autonomous Mobile Robots," The MIT Press, Cambridge, Massachusetts, 2004. [35] D. M. Stipanović, Sriram and C. J. Tomlin, Strategies for agents in multi-player pursuit-evasion games, Proceedings of the Eleventh International Symposium on Dynamic Games and Applications, Tucson, Arizona, 2004. [36] D. M. Stipanović, P. F. Hokayem, M. W. Spong and D. D. Šiljak, Cooperative avoidance control for multi-agent systems, Journal of Dynamic Systems, Measurement, and Control, 129 (2007), 699-707. doi: 10.1115/1.2764510. [37] D. M. Stipanović, A. Melikyan and N. Hovakimyan, Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations, Annals of Dynamic Games, 10 (2009), 133-145. [38] D. M. Stipanović, A survey and some new results in avoidance control, in "Proceedings of the 15th International Workshop on Dynamics and Control," Tossa de Mar, Spain, (2009), 166-173. [39] D. M. Stipanović, A. Melikyan and N. Hovakimyan, Guaranteed strategies for nonlinear multi-player pursuit-evasion games, International Game Theory Review, 12 (2010), 1-17. doi: 10.1142/S0219198910002489. [40] D. M. Stipanović, C. J. Tomlin and G. Leitmann, Monotone approximations of minimum and maximum functions and multi-objective problems, Applied Mathematics & Optimization, 66 (2012), 455-473. [41] D. M. Stipanović, C. J. Tomlin and G. Leitmann, A note on monotone approximations of minimum and maximum functions and multi-objective problems, Numerical Algebra, Control and Optimization, 1 (2011), 487-493. [42] D. M. Stipanović, C. J. Tomlin and C. Valicka, Collision free coverage control with multiple agents, Proceedings of the RoMoCo'11 Conference, Bukowy Dworek, Poland, 2011. [43] E. M. Vaisbord and V. I. Zhukovskiy, "Introduction to Multi-Player Differential Games and Their Applications," Gordon and Breach, New York, NY, 1988. [44] C. G. Valicka, S. R. Bieniawski, J. Vian, and D. M. Stipanović, Cooperative avoidance control for UAVs, Proceedings of the Tenth International Conference on Control, Automation, Robotics and Vision (ICARCV 2008), Hanoi, Vietnam (2008), 1462-1468. [45] T. L. Vincent and W. J. Grantham, "Nonlinear and Optimal Control Systems," John Wiley & Sons, Inc., New York, NY, 1997. [46] M. M. Zavlanos and G. J. Pappas, Potential fields for maintaining connectivity of mobile networks, IEEE Transactions on Robotics, 23 (2007), 812-816. doi: 10.1109/TRO.2007.900642. [47] V. A. Zorich, "Mathematical Analysis II," Springer-Verlag, Berlin, Germany, 2004.

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References:
 [1] A. Bacciotti and L. Rosier, "Liapunov Functions and Stability in Control Theory," 2ndedition, Springer-Verlag, Berlin, Germany, 2005. [2] J. V. Breakwell and P. Hagedorn, Point capture of two evaders in succession, Journal of Optimization Theory and Applications, 27 (1979), 89-97. doi: 10.1007/BF00933327. [3] C. R. Burns, R. F. Wang and D. M. Stipanović, A study of human and receding horizon controller perfomance of a remote navigation task with obstacles and feedback delays, Journal of Behavioral Robotics, 2 (2011), 44-63. [4] F. L. Chernousko, Controlled search of a moving object, Prikladnia Matematika i Mekhanika (in Russian), 44 (1981), 3-12. [5] N. Chopra, D. M. Stipanovićand M. W. Spong, On synchronization and collision avoidance for mechanical systems, Proceedings of the 2008 American Control Conference, (2008), 3713-3718. doi: 10.1109/ACC.2008.4587071. [6] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," Mc-Graw Hill, New York, NY, 1955. [7] M. Corless, G. Leitmann, and J. M. Skowronski, Adaptive control for avoidance or evasion in an uncertain environment, Computers & Mathematics with Applications, 13 (1987), 1-11. doi: 10.1016/0898-1221(87)90090-3. [8] M. Corless and G. Leitmann, Adaptive controllers for avoidance or evasion in an uncertain environment: some examples, Computers & Mathematics with Applications, 18 (1989), 161-170. doi: 10.1016/0898-1221(89)90133-8. [9] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [10] V. V. Filippov, "Basic Topological Structures of Ordinary Differential Equations," Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. [11] H. Flanders, Differentiation under the integral sign, The American Mathematical Monthly, 80 (1973), 615-627. doi: 10.2307/2319163. [12] R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques," Birkhäuser, Boston, MA, 1996. [13] J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, NY, 1993. [14] P. F. Hokayem, D. M. Stipanović and M. W. Spong, Semiautonomous control of multiple networked Langrangian systems, International Journal of Robust and Nonlinear Control, 19 (2009), 2040-2055. doi: 10.1002/rnc.1391. [15] I. I. Hussein and D. M. Stipanović, Effective coverage control for mobile sensor networks with guaranteed collision avoidance, IEEE Transactions on Control Systems Technology, 15 (2007), 642-657. doi: 10.1109/TCST.2007.899155. [16] I. I. Hussein and D. M. Stipanović, Effective coverage control using dynamic sensor networks with flocking and guaranteed collision avoidance, Proceedings of the 2007 American Control Conference, (2007), 3420-3425. doi: 10.1109/ACC.2007.4282310. [17] R. Isaacs, "Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization," John Wiley and Sons, Inc., New York, NY, 1965. [18] A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover Publications, New York, NY, 1975. [19] V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications," Mathematics in Science and Engineering, vol. 55, Academic Press, New York, NY, 1969. [20] V. Lakshmikantham, S. Leela and A. A. Martinyuk, "Stability Analysis of Nonlinear Systems," Marcel Dekker, New York, NY, 1989. [21] I. B. Lazarević, "Multidimensional Mathematical Analysis," Orion-Art, Belgrade, Serbia, 2005. [22] G. Leitmann and J. Skowronski, Avoidance control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298. [23] G. Leitmann, Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, 32 (1980), 569-576. doi: 10.1007/BF00934040. [24] G. Leitmann, Guaranteed avoidance feedback control, IEEE Transactions on Automatic Control, 25 (1980), 850-851. doi: 10.1109/TAC.1980.1102408. [25] G. Leitmann and J. Skowronski, A note on avoidance control, Optimal Control Applications & Methods, 4 (1983), 335-342. doi: 10.1002/oca.4660040406. [26] S. Mastellone, D. M. Stipanović, C. R. Graunke, K. A. Intlekofer and M. W. Spong, Formation control and collision avoidance for multi-agent nonholonomic systems: theory and experiments, International Journal of Robotics Research, 13 (2008), 107-126. doi: 10.1177/0278364907084441. [27] A. A. Melikyan, The problem of time-optimal control with the search for a target point, Prikladnia Matematika i Mekhanika (In Russian), 54 (1990), 1-7. [28] K. M. Miettinen, "Nonlinear Multiobjective Optimization," Kluwer Academic Publishers, Norwell, MA, 1998. doi: 10.1007/978-1-4615-5563-6. [29] I. Mitchell, A. M. Bayen and C. J. Tomlin, A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games, IEEE Transactions on Automatic Control, 50 (2005), 947-957. doi: 10.1109/TAC.2005.851439. [30] L. A. Petrosjan, "Differential Games of Pursuit," Series on Optimization, vol. 2, World Scientific, Singapore, 1993. doi: 10.1142/1670. [31] E. J. Rodríguez-Seda, J. J. Troy, C. A. Erignac, P. Murray, D. M. Stipanović and M. W. Spong, Bilateral teleoperation of multiple mobile agents: Formation control and collision avoidance, IEEE Transactions on Control Systems Technology, 18 (2010), 984-992. doi: 10.1109/TCST.2009.2030176. [32] E. J. Rodríguez-Seda, D. M. Stipanović and M. W. Spong, Collision avoidance control with sensing uncertainties, Proceedings of the 2011 American Control Conference, 2011. [33] M. Saska, J. S. Mejía, D. M. Stipanović and K. Schilling, Control and navigation of formations of car-like robots on a receding horizon, Proceedings of the 2009 IEEE Multi-conference on Systems and Control. St Petersburg, Russia, 2009. [34] R. Siegwart and I. R. Nourbakhsh, "Introduction to Autonomous Mobile Robots," The MIT Press, Cambridge, Massachusetts, 2004. [35] D. M. Stipanović, Sriram and C. J. Tomlin, Strategies for agents in multi-player pursuit-evasion games, Proceedings of the Eleventh International Symposium on Dynamic Games and Applications, Tucson, Arizona, 2004. [36] D. M. Stipanović, P. F. Hokayem, M. W. Spong and D. D. Šiljak, Cooperative avoidance control for multi-agent systems, Journal of Dynamic Systems, Measurement, and Control, 129 (2007), 699-707. doi: 10.1115/1.2764510. [37] D. M. Stipanović, A. Melikyan and N. Hovakimyan, Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations, Annals of Dynamic Games, 10 (2009), 133-145. [38] D. M. Stipanović, A survey and some new results in avoidance control, in "Proceedings of the 15th International Workshop on Dynamics and Control," Tossa de Mar, Spain, (2009), 166-173. [39] D. M. Stipanović, A. Melikyan and N. Hovakimyan, Guaranteed strategies for nonlinear multi-player pursuit-evasion games, International Game Theory Review, 12 (2010), 1-17. doi: 10.1142/S0219198910002489. [40] D. M. Stipanović, C. J. Tomlin and G. Leitmann, Monotone approximations of minimum and maximum functions and multi-objective problems, Applied Mathematics & Optimization, 66 (2012), 455-473. [41] D. M. Stipanović, C. J. Tomlin and G. Leitmann, A note on monotone approximations of minimum and maximum functions and multi-objective problems, Numerical Algebra, Control and Optimization, 1 (2011), 487-493. [42] D. M. Stipanović, C. J. Tomlin and C. Valicka, Collision free coverage control with multiple agents, Proceedings of the RoMoCo'11 Conference, Bukowy Dworek, Poland, 2011. [43] E. M. Vaisbord and V. I. Zhukovskiy, "Introduction to Multi-Player Differential Games and Their Applications," Gordon and Breach, New York, NY, 1988. [44] C. G. Valicka, S. R. Bieniawski, J. Vian, and D. M. Stipanović, Cooperative avoidance control for UAVs, Proceedings of the Tenth International Conference on Control, Automation, Robotics and Vision (ICARCV 2008), Hanoi, Vietnam (2008), 1462-1468. [45] T. L. Vincent and W. J. Grantham, "Nonlinear and Optimal Control Systems," John Wiley & Sons, Inc., New York, NY, 1997. [46] M. M. Zavlanos and G. J. Pappas, Potential fields for maintaining connectivity of mobile networks, IEEE Transactions on Robotics, 23 (2007), 812-816. doi: 10.1109/TRO.2007.900642. [47] V. A. Zorich, "Mathematical Analysis II," Springer-Verlag, Berlin, Germany, 2004.
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