Decentralized trajectory optimization using virtual motion camouflage and particle swarm optimization


This paper investigates a decentralized trajectory optimization method to solve a nonlinear constrained trajectory optimization problem. Especially, we consider a problem constrained on the terminal time and angle in a multi-robot application. The proposed algorithm is based on virtual motion camouflage (VMC) and particle swarm optimization (PSO). VMC changes a typical full space optimal problem to a subspace optimal problem, so it can reduce the dimension of the original problem by using path control parameters (PCPs). If PCPs are optimized, then the optimal path can be obtained. In this work, PSO is used to optimize these PCPs. In multi-robot path planning, each robot generates its own optimal path by using VMC and PSO, and sends its path information to the other robots. Then, the other robots use this path information when planning their own paths. Simulation and experimental results show that the optimal paths considering the terminal time and angle constraints are effectively generated by decentralized VMC and PSO.

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  1. Anisi, D., Robinson, J., Gren, P. (2006) On-line trajectory planning for aerial vehicles: a safe approach with guaranteed task completion. In AIAA Guidance, Navigation and Control Conference and Exhibit.

  2. Bernard, M., Kondak, K., Maza, I., & Ollero, A. (2011). Autonomous transportation and deployment with aerial robots for search and rescue missions. Journal of Field Robotics, 28(6), 914–931.

  3. Betts, J. T. (1998). Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, Dynamics, 21(2), 193–207.

  4. Bhattacharya, S., Kumar, V., & Likachev, M. (2010). Distributed optimization with pairwise constraints and its application to multi-robot path planning. In Proceedings of Robotics: Science and Systems.

  5. Bollino, K., & Lewis, L. R. (2008). Collision-free multi-uav optimal path planning and cooperative control for tactical applications. In AIAA Guidance, Navigation and Control Conference and Exhibit.

  6. Borrelli, F., Keviczky, T., & Balas, G. (2004). Collision-free uav formation flight using decentralized optimization and invariant sets. In CDC. 43rd IEEE Conference on Decision and Control, 2004, (vol. 1, pp. 1099–1104).

  7. Clerc, M., & Kennedy, J. (2002). The particle swarm explosion, stability, and convergence in a multidimensional complex space. IEEE Trans on Evolutionary Computation, 6(1), 58–73.

  8. Darby, C. L., Hager, W. W., & Rao, A. V. (2011). Direct trajectory optimization using a variable low-order adaptive pseudospectral method. Journal of Spacecraft and Rockets, 48(3), 433–445.

  9. Desaraju, V. R., & How, J. P. (2012). Decentralized path planning for multi-agent teams with complex constraints. Autonomous Robots, 32(4), 385–403.

  10. Fahroo, F., & Ross, I. M. (2001). Costate estimation by a legendre pseudospectral method. Journal of Guidance, Control, and Dynamics, 24(2), 270–277.

  11. Harl, N., & Balakrishnan, S. (2012). Impact time and angle guidance with sliding mode control. IEEE Transactions on Control Systems Technology, 20(6), 1436–1449.

  12. Inalhan, G., Stipanovic, D., & Tomlin, C. (2002). Decentralized optimization, with application to multiple aircraft coordination. In Proceedings of the 41st IEEE Conference on Decision and Control, 2002, (vol. 1, pp. 1147–1155).

  13. Jeon, I. S., Lee, J. I., & Tahk, M. J. (2006). Impact-time-control guidance law for anti-ship missiles. IEEE Transactions on Control Systems Technology, 14(2), 260–266.

  14. Jorris, T. R., & Cobb, R. G. (2009). Three-dimensional trajectory optimization satisfying waypoint and no-fly zone constraints. Journal of Guidance, Control, and Dynamics, 32(2), 551–572.

  15. Kennedy, J., & Eberhart, R. C. (1995). Particle swarm optimization. In Proceedings the 1995 IEEE International Conference on Neural Networks, (pp. 1942–1948).

  16. Keviczky, T., Borrelli, F., & Balas, G. J. (2004). A study on decentralized receding horizon control for decoupled systems. In IEEE Proceedings of the 2004 American Control Conference, (vol. 6, pp. 4921–4926).

  17. Kim, M., & Grider, K. V. (1973). Terminal guidance for impact attitude angle constrained flight trajectories. IEEE Transactions on Aerospace and Electronic Systems AES, 9(6), 852–859.

  18. Koh, B. I., George, A. D., Haftka, R. T., & Fregly, B. J. (2006). Parallel asynchronous particle swarm optimization. International Journal for Numerical Methods in Engineering, 67(4), 578–595.

  19. Kuwata, Y., & How, J. (2011). Cooperative distributed robust trajectory optimization using receding horizon milp. IEEE Transactions on Control Systems Technologies, 19(2), 423–431.

  20. Kuwata, Y., Teo, J., Fiore, G., Karaman, S., Frazzoli, E., & How, J. P. (2009). Real-time motion planning with applications to autonomous urban driving. IEEE Transactions on Control Systems Technology, 17(5), 1105–1118.

  21. Kwak, D., Choi, B., & Kim, H. (2013). Trajectory optimization using virtual motion camouflage and particle swarm optimization. In J. Lee, M. Lee, H. Liu, & J. H. Ryu (Eds.), Intelligent Robotics and Applications, Lecture Notes in Computer Science, (Vol. 8102, pp. 594–604). Berlin, Heidelberg: Springer.

  22. Lee, J. I., Jeon, I. S., & Tahk, M. J. (2007). Guidance law to control impact time and angle. IEEE Transactions on Aerospace Electronic Systems, 43(1), 301–310.

  23. Luca, A., Oriolo, G., & Vendittelli, M. (2001). Control of wheeled mobile robots: An experimental overview. In Ramsete Lecture Notes in Control and Information Sciences, (vol. 270, pp. 181–226). Berlin, Heidelberg: Springer.

  24. Mondada, F., Bonani, M., Raemy, X., Pugh, J., Cianci, C., Klaptocz, A., Magnenat, S., Zufferey, J., Floreano, D., & Martinoli, A. (2009). The e-puck, a robot designed for education in engineering. In Proceedings of the 9th Conference on Autonomous Robot Systems and Competitions, (pp. 59–65).

  25. Olson, E., Strom, J., Morton, R., Richardson, A., Ranganathan, P., Goeddel, R., et al. (2012). Progress toward multi-robot reconnaissance and the magic 2010 competition. Journal of Field Robotics, 29(5), 762–792.

  26. Parker, L. E. (2002). Distributed algorithms for multi-robot observation of multiple moving targets. Autonomous Robots, 12(3), 231–255.

  27. Richards, A., & How, J. (2004). A decentralized algorithm for robust constrained model predictive control. In Proceedings of the American Control Conference, 2004. (vol. 5, pp. 4261–4266). IEEE, New York.

  28. Ryoo, C. K., Cho, H., & Tahk, M. J. (2006). Time-to-go weighted optimal guidance with impact angle constraints. IEEE Transactions on Control Systems Technology, 14(3), 483–492.

  29. Schouwenaars, T., How, J., & Feron, E. (2004). Decentralized cooperative trajectory planning of multiple aircraft with hard safety guarantees. In AIAA Guidance, Navigation and Control Conference and Exhibit.

  30. Schutte, J. F., Reinbolt, J. A., Fregly, B. J., Haftka, R. T., & George, A. D. (2004). Parallel global optimization with the particle swarm algorithm. International Journal of Numerical Methods in Engineering, 61(13), 2296–2315.

  31. Song, T. L., & Shin, S. (1999). Time-optimal impact angle control for vertical plane engagements. IEEE Transactions on Aerospace Electronic Systems, 35(2), 738–742.

  32. Srinivasan, M. V., & Davey, M. (1995). Strategies for active camouflage of motion. Proceedings of the Royal Society of London. Series B: Biological Sciences, 259(1354), 19–25.

  33. Stryk, O., & Bulirsch, R. (1992). Direct and indirect methods for trajectory optimization. Annals of Operations Research, 37(1), 357–373.

  34. Xu, Y. (2007). Virtual motion camouflage and suboptimal trajectory design. In AIAA Guidance, Navigation and Control Conference and Exhibit.

  35. Xu, Y. (2008). Subspace optimal control and motion camouflage. In AIAA Guidance, Navigation and Control Conference and Exhibit.

  36. Xu, Y., & Basset, G. (2009). Pre and post optimality checking of the virtual motion camouflage based nonlinear constrained subspace optimal control. In AIAA Guidance, Navigation, and Control Conference.

  37. Xu, Y., & Basset, G. (2012). Sequential virtual motion camouflage method for nonlinear constrained optimal trajectory control. Automatica, 48(7), 1273–1285.

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This research was financially supported by a grant to Unmanned Technology Research Center funded by Defense Acquisition Program Administration, and by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT & Future Planning (MSIP) (no. 2009-0083495).

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Correspondence to H. Jin Kim.

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Kwak, D.J., Choi, B., Cho, D. et al. Decentralized trajectory optimization using virtual motion camouflage and particle swarm optimization. Auton Robot 38, 161–177 (2015) doi:10.1007/s10514-014-9399-7

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  • Trajectory optimization
  • Virtual motion camouflage
  • Particle swarm optimization
  • Terminal time and angle control
  • Multi-robot systems