Motion Planning and Control of a Swarm of Boids in a 3-Dimensional Space
In this paper, we propose a solution to the motion
planning and control problem for a swarm of three-dimensional
boids. The swarm exhibit collective emergent behaviors within the
vicinity of the workspace. The capability of biological systems
to autonomously maneuver, track and pursue evasive targets in a
cluttered environment is vastly superior to any engineered system. It
is considered an emergent behavior arising from simple rules that are
followed by individuals and may not involve any central coordination.
A generalized, yet scalable algorithm for attraction to the centroid
and inter-individual swarm avoidance is proposed. We present a set
of new continuous time-invariant velocity control laws, formulated via
the Lyapunov-based control scheme for target attraction and collision
avoidance. The controllers provide a collision-free trajectory. The
control laws proposed in this paper also ensures practical stability
of the system. The effectiveness of the control laws is demonstrated
via computer simulations.
[1] A. Okubo. Diffusion and Ecological problems: Mathematical Models.
Springer - Verla, New York, 1980.
[2] C. Blum and D. Merkle. Swarm Intelligence: Introduction and
Applications. Springer - Verlag Berlin Heidelberg, Germany, 2008.
[3] E. Bonebeau. Swarm Intelligence: From Natural to Artificial Sytems.
Oxford University Press, New York, 1999.
[4] A. Martinoli, K. Easton, and W. Agassounon. Modeling swarm
robotic systems: A case study in collaborative distributed manipulation.
International Journal of Robotics Research, 23(4):415–436, 2004.
Special Issue on Experimental Robotics, P. Dario and B. Siciliano,
editors. Invited paper.
[5] L. Edelstein-Keshet. Mathematical models of swarming and
social aggregation. In Procs. 2001 International Symposium on
Nonlinear Theory and Its Applications, pages 1–7, Miyagi, Japan,
October-November 2001.
[6] E. Forgoston and I. B. Schwartz. Delay-induced instabilities in
self-propelling swarms. Phys. Rev. E, 77(3):035203, Mar 2008.
[7] M. Dorigo, L.M. Gambardella, M. Birattari, A. Martinoli, R. Poli,
and T. Stützle. Ant Colony Optimization and Swarm Intelligence: 5th
International Workshop, ANTS 2006, Brussels, Belgium, September 4-7,
2006, Proceedings, volume 4150. Springer, 2006.
[8] Q.K. Pan, M. Fatih Tasgetiren, and Y.C. Liang. A discrete particle swarm
optimization algorithm for the no-wait flowshop scheduling problem.
Computers & Operations Research, 35(9):2807–2839, 2008.
[9] J. Raj, B. Sharma, J. Vanualailai, and S. Singh. Swarm navigation in a
complex environment. In International Conference on Mathematical,
Computational and Statistical Sciences, and Engineering, WASET,
Phuket, Thailand, Issue 72, pages 1157 – 1163, December 2012.
[10] G.J. Gelderblom, G. Cremers, M. de Wilt, W. Kortekaas, A. Thielmann,
K. Cuhls, A. Sachinopoulou, and I. Korhonen. The opinions expressed
in this study are those of the authors and do not necessarily reflect the
views of the european commission. 2008.
[11] C. W. Reynolds. Flocks, herds, and schools: A distributed behavioral
model, in computer graphics. In Proceedings of the 14th annual
conference on Computer graphics and interactive techniques, pages
25–34, New York, USA, 1987.
[12] C. W. Reynolds. Steering behaviors for autonomous characters. In
Proceedings of Game Developers Conference, pages 763–782, Miller
Freeman Game Group, San Francisco, California, USA, 1999.
[13] B. Sharma, J. Vanualailai, and A. Prasad. Formation control of a
swarm of mobile manipulators. Rocky Mountain Journal of Mathematics,
41(3):909–940, 2011.
[14] H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable flocking of mobile
agents, part I: fixed topology. volume 2, pages 2010–2015, 2003.
[15] A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros. Mutual
interactions, potentials, and individual distance in a social aggregation.
Journal of Mathematical Biology, 47:353–389, 2003.
[16] V. Gazi and K.M. Passino. Stability analysis of social foraging swarms.
In IEEE Transactions on Systems, Man and Cybernetics - Part B, volume
34(1), pages 539–557, 2004.
[17] V. Lakshmikantham, S. Leela, and A. A. Martynyuk. Practical Stability
of Nonlinear Systems. World Scientific, Singapore, 1990.
[18] E. W. Justh and P. S. Krishnaprasad. Equilibria and steering laws for
planar formations. Systems & Control Letters, 52(1):25 – 38, 2004.
[19] David S. Morgan and Ira B. Schwartz. Dynamic coordinated control
laws in multiple agent models. Physics Letters A, 340(1-4):121 – 131,
2005.
[20] V. Lakshmikantham, V. M. Matrosov, and S. Sivasundaram. Vector
Lyapunov Functions and Stability Analysis of Nonlinear Systems. Kluwer
Academic, Dordrecht / Boston / London, 1991.
[1] A. Okubo. Diffusion and Ecological problems: Mathematical Models.
Springer - Verla, New York, 1980.
[2] C. Blum and D. Merkle. Swarm Intelligence: Introduction and
Applications. Springer - Verlag Berlin Heidelberg, Germany, 2008.
[3] E. Bonebeau. Swarm Intelligence: From Natural to Artificial Sytems.
Oxford University Press, New York, 1999.
[4] A. Martinoli, K. Easton, and W. Agassounon. Modeling swarm
robotic systems: A case study in collaborative distributed manipulation.
International Journal of Robotics Research, 23(4):415–436, 2004.
Special Issue on Experimental Robotics, P. Dario and B. Siciliano,
editors. Invited paper.
[5] L. Edelstein-Keshet. Mathematical models of swarming and
social aggregation. In Procs. 2001 International Symposium on
Nonlinear Theory and Its Applications, pages 1–7, Miyagi, Japan,
October-November 2001.
[6] E. Forgoston and I. B. Schwartz. Delay-induced instabilities in
self-propelling swarms. Phys. Rev. E, 77(3):035203, Mar 2008.
[7] M. Dorigo, L.M. Gambardella, M. Birattari, A. Martinoli, R. Poli,
and T. Stützle. Ant Colony Optimization and Swarm Intelligence: 5th
International Workshop, ANTS 2006, Brussels, Belgium, September 4-7,
2006, Proceedings, volume 4150. Springer, 2006.
[8] Q.K. Pan, M. Fatih Tasgetiren, and Y.C. Liang. A discrete particle swarm
optimization algorithm for the no-wait flowshop scheduling problem.
Computers & Operations Research, 35(9):2807–2839, 2008.
[9] J. Raj, B. Sharma, J. Vanualailai, and S. Singh. Swarm navigation in a
complex environment. In International Conference on Mathematical,
Computational and Statistical Sciences, and Engineering, WASET,
Phuket, Thailand, Issue 72, pages 1157 – 1163, December 2012.
[10] G.J. Gelderblom, G. Cremers, M. de Wilt, W. Kortekaas, A. Thielmann,
K. Cuhls, A. Sachinopoulou, and I. Korhonen. The opinions expressed
in this study are those of the authors and do not necessarily reflect the
views of the european commission. 2008.
[11] C. W. Reynolds. Flocks, herds, and schools: A distributed behavioral
model, in computer graphics. In Proceedings of the 14th annual
conference on Computer graphics and interactive techniques, pages
25–34, New York, USA, 1987.
[12] C. W. Reynolds. Steering behaviors for autonomous characters. In
Proceedings of Game Developers Conference, pages 763–782, Miller
Freeman Game Group, San Francisco, California, USA, 1999.
[13] B. Sharma, J. Vanualailai, and A. Prasad. Formation control of a
swarm of mobile manipulators. Rocky Mountain Journal of Mathematics,
41(3):909–940, 2011.
[14] H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable flocking of mobile
agents, part I: fixed topology. volume 2, pages 2010–2015, 2003.
[15] A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros. Mutual
interactions, potentials, and individual distance in a social aggregation.
Journal of Mathematical Biology, 47:353–389, 2003.
[16] V. Gazi and K.M. Passino. Stability analysis of social foraging swarms.
In IEEE Transactions on Systems, Man and Cybernetics - Part B, volume
34(1), pages 539–557, 2004.
[17] V. Lakshmikantham, S. Leela, and A. A. Martynyuk. Practical Stability
of Nonlinear Systems. World Scientific, Singapore, 1990.
[18] E. W. Justh and P. S. Krishnaprasad. Equilibria and steering laws for
planar formations. Systems & Control Letters, 52(1):25 – 38, 2004.
[19] David S. Morgan and Ira B. Schwartz. Dynamic coordinated control
laws in multiple agent models. Physics Letters A, 340(1-4):121 – 131,
2005.
[20] V. Lakshmikantham, V. M. Matrosov, and S. Sivasundaram. Vector
Lyapunov Functions and Stability Analysis of Nonlinear Systems. Kluwer
Academic, Dordrecht / Boston / London, 1991.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:66352", author = "Bibhya Sharma and Jito Vanualailai and Jai Raj", title = "Motion Planning and Control of a Swarm of Boids in a 3-Dimensional Space", abstract = "In this paper, we propose a solution to the motion
planning and control problem for a swarm of three-dimensional
boids. The swarm exhibit collective emergent behaviors within the
vicinity of the workspace. The capability of biological systems
to autonomously maneuver, track and pursue evasive targets in a
cluttered environment is vastly superior to any engineered system. It
is considered an emergent behavior arising from simple rules that are
followed by individuals and may not involve any central coordination.
A generalized, yet scalable algorithm for attraction to the centroid
and inter-individual swarm avoidance is proposed. We present a set
of new continuous time-invariant velocity control laws, formulated via
the Lyapunov-based control scheme for target attraction and collision
avoidance. The controllers provide a collision-free trajectory. The
control laws proposed in this paper also ensures practical stability
of the system. The effectiveness of the control laws is demonstrated
via computer simulations.
", keywords = "Swarm, Practical stability, Motion planning.", volume = "8", number = "2", pages = "266-6", }
