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 & 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, (2005).   Google Scholar

[2]

J. V. Breakwell and P. Hagedorn, Point capture of two evaders in succession,, Journal of Optimization Theory and Applications, 27 (1979), 89.  doi: 10.1007/BF00933327.  Google Scholar

[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.   Google Scholar

[4]

F. L. Chernousko, Controlled search of a moving object,, Prikladnia Matematika i Mekhanika (in Russian), 44 (1981), 3.   Google Scholar

[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.  doi: 10.1109/ACC.2008.4587071.  Google Scholar

[6]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", Mc-Graw Hill, (1955).   Google Scholar

[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.  doi: 10.1016/0898-1221(87)90090-3.  Google Scholar

[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.  doi: 10.1016/0898-1221(89)90133-8.  Google Scholar

[9]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Kluwer Academic Publishers, (1988).   Google Scholar

[10]

V. V. Filippov, "Basic Topological Structures of Ordinary Differential Equations,", Kluwer Academic Publishers, (1998).   Google Scholar

[11]

H. Flanders, Differentiation under the integral sign,, The American Mathematical Monthly, 80 (1973), 615.  doi: 10.2307/2319163.  Google Scholar

[12]

R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques,", Birkhäuser, (1996).   Google Scholar

[13]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar

[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.  doi: 10.1002/rnc.1391.  Google Scholar

[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.  doi: 10.1109/TCST.2007.899155.  Google Scholar

[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.  doi: 10.1109/ACC.2007.4282310.  Google Scholar

[17]

R. Isaacs, "Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization,", John Wiley and Sons, (1965).   Google Scholar

[18]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Dover Publications, (1975).   Google Scholar

[19]

V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications,", Mathematics in Science and Engineering, (1969).   Google Scholar

[20]

V. Lakshmikantham, S. Leela and A. A. Martinyuk, "Stability Analysis of Nonlinear Systems,", Marcel Dekker, (1989).   Google Scholar

[21]

I. B. Lazarević, "Multidimensional Mathematical Analysis,", Orion-Art, (2005).   Google Scholar

[22]

G. Leitmann and J. Skowronski, Avoidance control,, Journal of Optimization Theory and Applications, 23 (1977), 581.  doi: 10.1007/BF00933298.  Google Scholar

[23]

G. Leitmann, Guaranteed avoidance strategies,, Journal of Optimization Theory and Applications, 32 (1980), 569.  doi: 10.1007/BF00934040.  Google Scholar

[24]

G. Leitmann, Guaranteed avoidance feedback control,, IEEE Transactions on Automatic Control, 25 (1980), 850.  doi: 10.1109/TAC.1980.1102408.  Google Scholar

[25]

G. Leitmann and J. Skowronski, A note on avoidance control,, Optimal Control Applications & Methods, 4 (1983), 335.  doi: 10.1002/oca.4660040406.  Google Scholar

[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.  doi: 10.1177/0278364907084441.  Google Scholar

[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.   Google Scholar

[28]

K. M. Miettinen, "Nonlinear Multiobjective Optimization,", Kluwer Academic Publishers, (1998).  doi: 10.1007/978-1-4615-5563-6.  Google Scholar

[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.  doi: 10.1109/TAC.2005.851439.  Google Scholar

[30]

L. A. Petrosjan, "Differential Games of Pursuit,", Series on Optimization, (1993).  doi: 10.1142/1670.  Google Scholar

[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.  doi: 10.1109/TCST.2009.2030176.  Google Scholar

[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).   Google Scholar

[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, (2009).   Google Scholar

[34]

R. Siegwart and I. R. Nourbakhsh, "Introduction to Autonomous Mobile Robots,", The MIT Press, (2004).   Google Scholar

[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, (2004).   Google Scholar

[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, 129 (2007), 699.  doi: 10.1115/1.2764510.  Google Scholar

[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.   Google Scholar

[38]

D. M. Stipanović, A survey and some new results in avoidance control,, in, (2009), 166.   Google Scholar

[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.  doi: 10.1142/S0219198910002489.  Google Scholar

[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.   Google Scholar

[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, 1 (2011), 487.   Google Scholar

[42]

D. M. Stipanović, C. J. Tomlin and C. Valicka, Collision free coverage control with multiple agents,, Proceedings of the RoMoCo'11 Conference, (2011).   Google Scholar

[43]

E. M. Vaisbord and V. I. Zhukovskiy, "Introduction to Multi-Player Differential Games and Their Applications,", Gordon and Breach, (1988).   Google Scholar

[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, (2008), 1462.   Google Scholar

[45]

T. L. Vincent and W. J. Grantham, "Nonlinear and Optimal Control Systems,", John Wiley & Sons, (1997).   Google Scholar

[46]

M. M. Zavlanos and G. J. Pappas, Potential fields for maintaining connectivity of mobile networks,, IEEE Transactions on Robotics, 23 (2007), 812.  doi: 10.1109/TRO.2007.900642.  Google Scholar

[47]

V. A. Zorich, "Mathematical Analysis II,", Springer-Verlag, (2004).   Google Scholar

show all references

References:
[1]

A. Bacciotti and L. Rosier, "Liapunov Functions and Stability in Control Theory,", 2ndedition, (2005).   Google Scholar

[2]

J. V. Breakwell and P. Hagedorn, Point capture of two evaders in succession,, Journal of Optimization Theory and Applications, 27 (1979), 89.  doi: 10.1007/BF00933327.  Google Scholar

[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.   Google Scholar

[4]

F. L. Chernousko, Controlled search of a moving object,, Prikladnia Matematika i Mekhanika (in Russian), 44 (1981), 3.   Google Scholar

[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.  doi: 10.1109/ACC.2008.4587071.  Google Scholar

[6]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", Mc-Graw Hill, (1955).   Google Scholar

[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.  doi: 10.1016/0898-1221(87)90090-3.  Google Scholar

[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.  doi: 10.1016/0898-1221(89)90133-8.  Google Scholar

[9]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Kluwer Academic Publishers, (1988).   Google Scholar

[10]

V. V. Filippov, "Basic Topological Structures of Ordinary Differential Equations,", Kluwer Academic Publishers, (1998).   Google Scholar

[11]

H. Flanders, Differentiation under the integral sign,, The American Mathematical Monthly, 80 (1973), 615.  doi: 10.2307/2319163.  Google Scholar

[12]

R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques,", Birkhäuser, (1996).   Google Scholar

[13]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar

[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.  doi: 10.1002/rnc.1391.  Google Scholar

[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.  doi: 10.1109/TCST.2007.899155.  Google Scholar

[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.  doi: 10.1109/ACC.2007.4282310.  Google Scholar

[17]

R. Isaacs, "Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization,", John Wiley and Sons, (1965).   Google Scholar

[18]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Dover Publications, (1975).   Google Scholar

[19]

V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications,", Mathematics in Science and Engineering, (1969).   Google Scholar

[20]

V. Lakshmikantham, S. Leela and A. A. Martinyuk, "Stability Analysis of Nonlinear Systems,", Marcel Dekker, (1989).   Google Scholar

[21]

I. B. Lazarević, "Multidimensional Mathematical Analysis,", Orion-Art, (2005).   Google Scholar

[22]

G. Leitmann and J. Skowronski, Avoidance control,, Journal of Optimization Theory and Applications, 23 (1977), 581.  doi: 10.1007/BF00933298.  Google Scholar

[23]

G. Leitmann, Guaranteed avoidance strategies,, Journal of Optimization Theory and Applications, 32 (1980), 569.  doi: 10.1007/BF00934040.  Google Scholar

[24]

G. Leitmann, Guaranteed avoidance feedback control,, IEEE Transactions on Automatic Control, 25 (1980), 850.  doi: 10.1109/TAC.1980.1102408.  Google Scholar

[25]

G. Leitmann and J. Skowronski, A note on avoidance control,, Optimal Control Applications & Methods, 4 (1983), 335.  doi: 10.1002/oca.4660040406.  Google Scholar

[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.  doi: 10.1177/0278364907084441.  Google Scholar

[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.   Google Scholar

[28]

K. M. Miettinen, "Nonlinear Multiobjective Optimization,", Kluwer Academic Publishers, (1998).  doi: 10.1007/978-1-4615-5563-6.  Google Scholar

[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.  doi: 10.1109/TAC.2005.851439.  Google Scholar

[30]

L. A. Petrosjan, "Differential Games of Pursuit,", Series on Optimization, (1993).  doi: 10.1142/1670.  Google Scholar

[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.  doi: 10.1109/TCST.2009.2030176.  Google Scholar

[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).   Google Scholar

[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, (2009).   Google Scholar

[34]

R. Siegwart and I. R. Nourbakhsh, "Introduction to Autonomous Mobile Robots,", The MIT Press, (2004).   Google Scholar

[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, (2004).   Google Scholar

[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, 129 (2007), 699.  doi: 10.1115/1.2764510.  Google Scholar

[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.   Google Scholar

[38]

D. M. Stipanović, A survey and some new results in avoidance control,, in, (2009), 166.   Google Scholar

[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.  doi: 10.1142/S0219198910002489.  Google Scholar

[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.   Google Scholar

[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, 1 (2011), 487.   Google Scholar

[42]

D. M. Stipanović, C. J. Tomlin and C. Valicka, Collision free coverage control with multiple agents,, Proceedings of the RoMoCo'11 Conference, (2011).   Google Scholar

[43]

E. M. Vaisbord and V. I. Zhukovskiy, "Introduction to Multi-Player Differential Games and Their Applications,", Gordon and Breach, (1988).   Google Scholar

[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, (2008), 1462.   Google Scholar

[45]

T. L. Vincent and W. J. Grantham, "Nonlinear and Optimal Control Systems,", John Wiley & Sons, (1997).   Google Scholar

[46]

M. M. Zavlanos and G. J. Pappas, Potential fields for maintaining connectivity of mobile networks,, IEEE Transactions on Robotics, 23 (2007), 812.  doi: 10.1109/TRO.2007.900642.  Google Scholar

[47]

V. A. Zorich, "Mathematical Analysis II,", Springer-Verlag, (2004).   Google Scholar

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