This paper investigates cooperative trajectory planning of multiple unmanned combat aerial vehicles (multi-UCAV) in performing autonomous cooperative air-to-ground target attack missions. By integrating an approximate allowable attack region model, several constraint models, and a multicriterion objective function, the problem is formulated as a cooperative trajectory optimal control problem (CTOCP). Then, a virtual motion camouflage (VMC) for cooperative trajectory planning of multi-UCAV, combining with the differential flatness theory, Gauss pseudospectral method (GPM), and nonlinear programming, is designed to solve the CTOCP. In particular, the notion of the virtual time is introduced to the VMC problem formulation to handle the temporal cooperative constraints. The simulation experiments validate that the CTOCP can be effectively solved by the cooperative trajectory planning algorithm based on VMC which integrates the spatial and temporal constraints on the trajectory level, and the comparative experiments illustrate that VMC based algorithm is more efficient than GPM based direct collocation method in tackling the CTOCP. 1. Introduction Nowadays, it is an active research area to perform autonomous cooperative air-to-ground target attack (CA/GTA) missions using multiple unmanned combat aerial vehicles (multi-UCAV) [1]. However, compared with single UCAV planning and coordinated formation control problems [2], new technical challenges in the CA/GTA missions are emerging. The cooperative trajectory planning is one of the key challenging technologies, due to its high dimensionality, severe equality and inequality constraints involved, and the requirement of spatial-temporal cooperation of multi-UCAV. Recently, various algorithms have been developed to solve this cooperative trajectory planning problem [3, 4], including artificial neural network methods [5], sample-based planning methods [6], maneuver automation (MA) [7], and optimal control methods. There is no doubt that the optimal control theory is the most natural framework for this type of problem with dynamic constraints [8]. However, the rapid solution to optimal control problems (OCPs) for complicated nonlinear systems, such as UCAVs, is a challenging task [9]. Analytical solutions are seldom available or even possible. As a result, one usually resorts to numerical techniques [3]. The techniques can be classified into two general types, namely, indirect and direct methods. Indirect methods [10] solve the OCPs by formulating the first-order optimality conditions, applying Pontryagin’s Maximum
References
[1]
R. M. Murray, “Recent research in cooperative control of multivehicle systems,” Journal of Dynamic Systems, Measurement and Control, vol. 129, no. 5, pp. 571–583, 2007.
[2]
M. Fu and J. Jiao, “A hybrid approach for coordinated formation control of multiple surface vessels,” Mathematical Problems in Engineering, vol. 2013, Article ID 794284, 8 pages, 2013.
[3]
J. T. Betts, “Survey of numerical methods for trajectory optimization,” Journal of Guidance, Control, and Dynamics, vol. 21, no. 2, pp. 193–207, 1998.
[4]
C. Goerzen, Z. Kong, and B. Mettler, “A survey of motion planning algorithms from the perspective of autonomous UAV guidance,” Journal of Intelligent and Robotic Systems, vol. 57, no. 1–4, pp. 65–100, 2010.
[5]
J. F. Horn, E. M. Schmidt, B. R. Geiger, and M. P. DeAngelo, “Neural network-based trajectory optimization for unmanned aerial vehicles,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 2, pp. 548–562, 2012.
[6]
S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal motion planning,” The International Journal of Robotics Research, vol. 30, no. 7, pp. 846–894, 2011.
[7]
P. Narayan, P. Meyer, and D. Campbell, “Embedding human expert cognition into autonomous UAS trajectory planning,” IEEE Transaction on Cybernetics, vol. 43, no. 2, pp. 530–543, 2013.
[8]
Q. Gong, L. R. Lewis, and I. M. Ross, “Pseudospectral motion planning for autonomous vehicles,” Journal of Guidance, Control, and Dynamics, vol. 32, no. 3, pp. 1039–1045, 2009.
[9]
H. Liu, S. Chen, L. Shen, et al., “An integrated multi-criterion hp-Adaptive pseudospectral method for direct optimal control problems solving,” Mathematical Problems in Engineering, vol. 2012, Article ID 760890, 22 pages, 2012.
[10]
K. Yang, S. Sukkarieh, and Y. Kang, “Adaptive nonlinear model predictive path tracking control for a fixed-wing unmanned aerial vehicle,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Chicago, Ill, USA, August 2009.
[11]
O. A. Yakimenko, Y. Xu, and G. Basset, “Computing short-time aircraft maneuvers using direct methods,” in Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii, USA, August 2008.
[12]
D. Benson, A Gauss pseudospectral transcription for optimal control [Dissertation for the doctoral degree], Massachusettes Institute of Technology, Cambridge, Mass, USA, 2005.
[13]
M. Soler, A. Olivares, and E. Staffetti, “Hybrid optimal control approach to commercial aircraft trajectory planning,” Journal of Guidance, Control, and Dynamics, vol. 33, no. 3, pp. 985–991, 2010.
[14]
J. Li, H. An, H. Zhu, et al., “Geometric pseudospectral method on SE(3) for rigid-body dynamics with application to aircraft,” Mathematical Problems in Engineering, vol. 2013, Article ID 595086, 16 pages, 2013.
[15]
A. Verma and J. L. Junkins, “Inverse dynamics approach for real-time determination of feasible aircraft reference trajectories,” in Proceedings of the 24th Atmospheric Flight Mechanics Conference, 1999.
[16]
R. G. Drury, “Performance of NLP algorithms with inverse dynamics for near-real time trajectory generation,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Portland, Ore, USA, 2011.
[17]
F.-L. Lian, “Cooperative path planning of dynamical multi-agent systems using differential flatness approach,” International Journal of Control, Automation and Systems, vol. 6, no. 3, pp. 401–412, 2008.
[18]
A. Chamseddine, Y. Zhang, C. A. Rabbath, et al., “Trajectory planning and peplanning strategies applied to a quadrotor unmanned aerial vehicle,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 5, pp. 1667–1671, 2012.
[19]
X. Gu, Y. Zhang, S. Chen, et al., “Cooperative trajectory planning for multi-UCAV performing air-to-ground target attack missions,” Advanced Materials Research, vol. 718-720, pp. 1329–1334, 2013.
[20]
X. Gu, Y. Zhang, J. Chen, et al., “Real-time cooperative trajectory planning using differential flatness approach and B-splines,” Applied Mechanics and Materials, vol. 333-335, pp. 1338–1343, 2013.
[21]
Y. Xu, “Analytical solutions to spacecraft formation-flying guidance using virtual motion camouflage,” Journal of Guidance, Control, and Dynamics, vol. 33, no. 5, pp. 1376–1386, 2010.
[22]
Y. Xu and G. Basset, “Sequential virtual motion camouflage method for nonlinear constrained optimal trajectory control,” Automatica, vol. 48, no. 7, pp. 1273–1285, 2012.
[23]
Y. Xu, “Virtual motion camouflage and suboptimal trajectory design,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, pp. 555–571, August 2007.
[24]
Y. Xu and G. Basset, “Pre and post optimality checking of the virtual motion camouflage based nonlinear constrained subspace optimal control,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Chicago, Ill, USA, August 2009.
[25]
K. P. Bollino and L. R. Lewis, “Collision-free multi-UAV optimal path planning and cooperative control for tactical applications,” in Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii, USA, August 2008.
[26]
Y. Kuwata and J. P. How Sr., “Cooperative distributed robust trajectory optimization using receding horizon MILP,” IEEE Transactions on Control Systems Technology, vol. 19, no. 2, pp. 423–431, 2011.
[27]
Y. Xu and G. Basset, “Virtual motion camouflage based phantom track generation through cooperative electronic combat air vehicles,” Automatica, vol. 46, no. 9, pp. 1454–1461, 2010.
[28]
Y. Xu, M. Xin, and J. Wang, “Unmanned aerial vehicle formation flight via a hierarchical cooperative control approach,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Portland, Ore, USA, 2011.
[29]
T. W. McLain and R. W. Beard, “Coordination variables, coordination functions, and cooperative-timing missions,” Journal of Guidance, Control, and Dynamics, vol. 28, no. 1, pp. 150–161, 2005.
[30]
I. Kamlner, O. Yakimenko, V. Dobrokhodov et al., “Coordinated path following for time-critical missions of multiple UAVs via ？1 adaptive output feedback controllers,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, pp. 915–948, August 2007.
[31]
B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, NY, USA, 2003.
[32]
Y. Zhang, J. Chen, and L. Shen, “Hybrid hierarchical trajectory planning for a fixed-wing UCAV performing air-to-surface multi-target attack,” Journal of Systems Engineering and Electronics, vol. 23, no. 4, pp. 256–264, 2012.
[33]
X. Gu, Y. Zhang, N. Wang, et al., “Robust multi-objective optimization based on NSGA-II for UCAV weapon delivery,” in Proceedings of the 4th International Conference on Intelligent Human-Machine Systems and Cybernetics, 2012.
[34]
L. P. R. Lewis, Rapid motion planning and autonomous obstacle avoidance for unmanned vehicles [M.S. thesis], Naval Postgraduate School, Monterey, Calif, USA, 2006.
[35]
E. Besada-Portas, L. de la Torre, J. M. de la Cruz, and B. de Andrés-Toro, “Evolutionary trajectory planner for multiple UAVs in realistic scenarios,” IEEE Transactions on Robotics, vol. 26, no. 4, pp. 619–634, 2010.
[36]
M. A. Hurni, P. Sekhavat, M. Karpenko, and I. M. Ross, “A pseudospectral optimal motion planner for autonomous unmanned vehicles,” in Proceedings of the American Control Conference (ACC '10), pp. 1591–1598, Baltimore, Md, USA, July 2010.
[37]
Y. Wang, T. Wei, and X. Qu, “Study of multi-objective fuzzy optimization for path planning,” Chinese Journal of Aeronautics, vol. 25, no. 1, pp. 51–56, 2012.
[38]
M. V. Srinivasan and M. Davey, “Strategies for active camouflage of motion,” Proceedings of the Royal Society B, vol. 259, no. 1354, pp. 19–25, 1995.
[39]
M. Fliess, J. Levine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and examples,” International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995.
[40]
K. Holmstr？m, A. O. G？ran, and M. M. Edvall, USer'S Guide For TOMLAB /SNOPT, Department of Mathematics and Physics, M？lardalen University, V？ster？s, Sweden, 2008.
[41]
S. Chen, H. Liu, J. Chen, et al., “Penetration trajectory planning based on radar tracking features for UAV,” Aircraft Engineering and Aerospace Technology, vol. 85, no. 1, pp. 62–71, 2013.
[42]
I. M. Ross and Q. Gong, “Guess-free trajectory optimization,” in Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, USA, August 2008.
