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May  2022, 18(3): 1935-1948. doi: 10.3934/jimo.2021050

Multi-aircraft cooperative path planning for maneuvering target detection

AVIC LEIHUA Electronic Technology Institute, Wuxi 214063, China

* Corresponding author: Yongkun Wang

Received  July 2020 Revised  January 2021 Published  May 2022 Early access  March 2021

Multi-aircraft cooperative path planning is a key problem in modern and future air combat scenario. In this paper, this problem is studied in aspect of airborne radar detection to maintain a continuous tracking of a manoeuvring air target. Firstly, the objective function is established in combination with multiple constraints considered, including Doppler blind zone constraint, radar viewing aspect constraint, baseline constraint, and so on. Then, the above optimal control problem is transformed into a nonlinear programming problem with a series of algebraic constraints by hp-adaptive Gauss pseudospectral method (HPAGPM). And it is solved by GPOPS software package based on MATLAB. Simulation results show that the optimized cooperative paths can be got to achieve continuous tracking of maneuvering air target by HPAGPM.

Citation: Yongkun Wang, Fengshou He, Xiaobo Deng. Multi-aircraft cooperative path planning for maneuvering target detection. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1935-1948. doi: 10.3934/jimo.2021050
References:
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D. Benson, Gauss Pseudospectral Transcription for Optimal Control, Massachusetts Institute of Technology, 2005.

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J. T. Betts and W. P. Huffman, Application of sparse nonlinear programming to trajectory optimization, Journal of Guidance, Control, and Dynamics, 15 (1992), 198-206.  doi: 10.2514/3.20819.

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K. Bousson, Single gridpoint dynamic programming for trajectory optimization, in 2005 AIAA Atmospheric Flight Mechanics Conference and Exhibit, San Francisco, USA, (2005), 1–8. doi: 10.2514/6.2005-5902.

[4]

X. T. Chen and J. Z. Wang, Sliding-mode guidance for simultaneous control of impact time and angle, Journal of Guidance, Control, and Dynamics, 42 (2019), 394-401.  doi: 10.2514/1.G003893.

[5]

X. M. ChengH. F. Li and R. Zhang, Efficient ascent trajectory optimization using convex models based on the Newton-Kantorovich/Pseudospectral approach, Aerospace Science and Technology, 66 (2017), 140-151.  doi: 10.1016/j.ast.2017.02.023.

[6]

Y. Cherfaoui and M. Moulai, Biobjective optimization over the efficient set of multiobjective integer programming problem, Journal of Industrial and Management Optimization, 17 (2021), 117-131.  doi: 10.3934/jimo.2019102.

[7]

J. M. C. Clark, P. A. Kountouriotis and R. B. Vinter, A methodology for incorporating the Doppler blind zone in target tracking algorithms, in 2008 11th International Conference on Information Fusion, Cologne, Germany, (2008), 1–8.

[8]

H. B. DanX. X. Wei and Z. M. Dong, Multiple UCAVs cooperative air combat simulation platform based on PSO, ACO, and game theory, IEEE Transactions on Aerospace and Electronic System Magazine, 28 (2013), 12-19.  doi: 10.1109/MAES.2013.6678487.

[9]

C. L. Darby, W. W. Hager and A. V. Rao, An improved adaptive hp algorithm using pseudospectral methods for optimal control, in 2010 AIAA Guidance, Navigation, and Control Conference, Reston, USA, 2012. doi: 10.2514/6.2010-8272.

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C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.

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M. Gandhi and E. Theodorou, A comparison between trajectory optimization methods: Differential dynamic programming and pseudospectral optimal control, in 2016 AIAA Guidance, Navigation, and Control Conference, San Diego, California, USA, (2016), 1–16.

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C. GoerzenZ. Kong and B. Mettler, A survey of motion planning algorithms from the perspective of autonomous UAV guidance, Journal of Intelligent and Robotic Systems, 57 (2010), 65-100.  doi: 10.1007/978-90-481-8764-5_5.

[13]

Y. F. Guo, D. Z. Feng and X. Wang, The earth-mars transfer trajectory optimization of solar sail based on hp-adaptive pseudospectral method, Discrete Dynamics in Nature and Society, 2018 (2018), Art. ID 6916848, 14 pp. doi: 10.1155/2018/6916848.

[14]

R. P. HuangS. J. QuX. G. Yang and Z. M. Liu, Multi-stage distributionally robust optimization with risk aversion, Journal of Industrial and Management Optimization, 17 (2021), 233-259.  doi: 10.3934/jimo.2019109.

[15]

G. Q. HuangY. P. Lu and Y. Nan, A survey of numerical algorithms for trajectory optimization of flight vehicles, Science China Technological Sciences, 55 (2012), 2538-2560.  doi: 10.1007/s11431-012-4946-y.

[16]

T. H. KimC. H. LeeI. S. Jeon and M. J. Tahk, Augmented polynomial guidance with impact time and angle constraints, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 2806-2817. 

[17]

S. KangR. Tekin and F. Holzapfel, Generalized impact time and angle control via look-angle shaping, Journal of Guidance, Control, and Dynamics, 42 (2019), 695-702.  doi: 10.2514/1.G003765.

[18]

A. KhatamiS. Mirghasemi and A. Khosravi, A new PSO-based approach to fire flame detection using K-Medoids clustering, Expert Systems with Applications, 68 (2017), 69-80.  doi: 10.1016/j.eswa.2016.09.021.

[19]

M. MertensW. Koch and T. Kirubarajan, Exploiting Doppler blind zone information for ground moving target tracking with bistatic airborne radar, IEEE Transactions on Aerospace and Electronic Systems, 50 (2014), 130-148.  doi: 10.1109/TAES.2013.120718.

[20]

F. W. Moore, Radar cross-section reduction via route planning and intelligent control, IEEE Transactions on Control Systems Technology, 10 (2016), 696-700.  doi: 10.1109/TCST.2002.801879.

[21]

L. H. Nam, L. Huang, X. J. Li and J. F. Xu, An approach for coverage path planning for UAVs, in 2016 IEEE 14th International Workshop on Advanced Motion Control, Auckland, New Zealand, (2016), 411–416. doi: 10.1109/AMC.2016.7496385.

[22]

N. Ozalo and O. K. Sahingoz, Optimal UAV path planning in a 3D threat environment by using parallel evolutionary algorithms, in 2013 International Conference on Unmanned Aircraft Systems, Grand Hyatt Atlanta, Atlanta, (2013), 308–317.

[23]

N. OzakiS. CampagnolaR. Funase and C. H. Yam, Stochastic differential dynamic programming with unscented transform for low-thrust trajectory design, Journal of Guidance, Control, and Dynamics, 41 (2018), 377-381.  doi: 10.2514/1.G002367.

[24]

M. Patterson and A. Rao, Gpops-Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Transactions on Mathematical Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904.

[25]

Y. H. Qu, Y. T. Zhang and Y. M. Zhang, Optimal flight path planning for UAVs in a 3-D threat environment, in 2014 International Conference on Unmanned Aircraft systems, Orlando, FL, USA, (2014), 149–155. doi: 10.1109/ICUAS.2014.6842250.

[26]

A. V. Rao, D. A. Benson and C. Darby, Algorithm 902: GPOPS A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method, ACM Transactions on Mathematical Software, 37 (2010), Article 22. doi: 10.1145/1731022.1731032.

[27]

J. R. RiehlG. E. Collins and J. P. Hespanha, Cooperative search by UAV teams: A model predictive approach using dynamic graphs, IEEE Transactions on Aerospace and Electronic systems, 47 (2011), 2637-2656.  doi: 10.1109/TAES.2011.6034656.

[28]

V. RobergeM. Tarbouchi and G. Labonte, Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning, IEEE Transactions on Industrial Information, 9 (2013), 132-141.  doi: 10.1109/TII.2012.2198665.

[29]

B. M. SathyarajL. C. JainA. Finn and S. Drake, Multiple UAVs path planning algorithms: A comparative study, Fuzzy Optimization and Decision Making, 7 (2008), 257-267.  doi: 10.1007/s10700-008-9035-0.

[30]

P. Y. Volkan, A new vibrational genetic algorithm enhanced with a Voronoi diagram for path planning of autonomous UAV, Aerospace Science and Technology, 16 (2012), 47-55. 

[31]

B. Z. Xu, Y. J. Wang and L. Liu, Multi-stage boost aircraft trajectory optimization strategy based on hp adaptive Gauss pseudo spectral method, in 10th International Conference on Modelling, Identification and Control, Guiyang, China, 2018, 1–7. doi: 10.1109/ICMIC.2018.8529869.

[32]

P. YaoZ. X. Xie and P. Ren, Optimal UAV route planning for coverage search of stationary target in river, IEEE Transactions on Control Systems Technology, 27 (2019), 822-829.  doi: 10.1109/TCST.2017.2781655.

[33]

M. Zhang, Z. Zhu, Z. Zhao and X. Li, Trajectory optimization for missile-borne SAR imaging phase via Gauss Pseudospectral Method, in 2011 IEEE CIE International Conference on Radar, Chengdu, China, (2011), 867–870. doi: 10.1109/CIE-Radar.2011.6159678.

show all references

References:
[1]

D. Benson, Gauss Pseudospectral Transcription for Optimal Control, Massachusetts Institute of Technology, 2005.

[2]

J. T. Betts and W. P. Huffman, Application of sparse nonlinear programming to trajectory optimization, Journal of Guidance, Control, and Dynamics, 15 (1992), 198-206.  doi: 10.2514/3.20819.

[3]

K. Bousson, Single gridpoint dynamic programming for trajectory optimization, in 2005 AIAA Atmospheric Flight Mechanics Conference and Exhibit, San Francisco, USA, (2005), 1–8. doi: 10.2514/6.2005-5902.

[4]

X. T. Chen and J. Z. Wang, Sliding-mode guidance for simultaneous control of impact time and angle, Journal of Guidance, Control, and Dynamics, 42 (2019), 394-401.  doi: 10.2514/1.G003893.

[5]

X. M. ChengH. F. Li and R. Zhang, Efficient ascent trajectory optimization using convex models based on the Newton-Kantorovich/Pseudospectral approach, Aerospace Science and Technology, 66 (2017), 140-151.  doi: 10.1016/j.ast.2017.02.023.

[6]

Y. Cherfaoui and M. Moulai, Biobjective optimization over the efficient set of multiobjective integer programming problem, Journal of Industrial and Management Optimization, 17 (2021), 117-131.  doi: 10.3934/jimo.2019102.

[7]

J. M. C. Clark, P. A. Kountouriotis and R. B. Vinter, A methodology for incorporating the Doppler blind zone in target tracking algorithms, in 2008 11th International Conference on Information Fusion, Cologne, Germany, (2008), 1–8.

[8]

H. B. DanX. X. Wei and Z. M. Dong, Multiple UCAVs cooperative air combat simulation platform based on PSO, ACO, and game theory, IEEE Transactions on Aerospace and Electronic System Magazine, 28 (2013), 12-19.  doi: 10.1109/MAES.2013.6678487.

[9]

C. L. Darby, W. W. Hager and A. V. Rao, An improved adaptive hp algorithm using pseudospectral methods for optimal control, in 2010 AIAA Guidance, Navigation, and Control Conference, Reston, USA, 2012. doi: 10.2514/6.2010-8272.

[10]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.

[11]

M. Gandhi and E. Theodorou, A comparison between trajectory optimization methods: Differential dynamic programming and pseudospectral optimal control, in 2016 AIAA Guidance, Navigation, and Control Conference, San Diego, California, USA, (2016), 1–16.

[12]

C. GoerzenZ. Kong and B. Mettler, A survey of motion planning algorithms from the perspective of autonomous UAV guidance, Journal of Intelligent and Robotic Systems, 57 (2010), 65-100.  doi: 10.1007/978-90-481-8764-5_5.

[13]

Y. F. Guo, D. Z. Feng and X. Wang, The earth-mars transfer trajectory optimization of solar sail based on hp-adaptive pseudospectral method, Discrete Dynamics in Nature and Society, 2018 (2018), Art. ID 6916848, 14 pp. doi: 10.1155/2018/6916848.

[14]

R. P. HuangS. J. QuX. G. Yang and Z. M. Liu, Multi-stage distributionally robust optimization with risk aversion, Journal of Industrial and Management Optimization, 17 (2021), 233-259.  doi: 10.3934/jimo.2019109.

[15]

G. Q. HuangY. P. Lu and Y. Nan, A survey of numerical algorithms for trajectory optimization of flight vehicles, Science China Technological Sciences, 55 (2012), 2538-2560.  doi: 10.1007/s11431-012-4946-y.

[16]

T. H. KimC. H. LeeI. S. Jeon and M. J. Tahk, Augmented polynomial guidance with impact time and angle constraints, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 2806-2817. 

[17]

S. KangR. Tekin and F. Holzapfel, Generalized impact time and angle control via look-angle shaping, Journal of Guidance, Control, and Dynamics, 42 (2019), 695-702.  doi: 10.2514/1.G003765.

[18]

A. KhatamiS. Mirghasemi and A. Khosravi, A new PSO-based approach to fire flame detection using K-Medoids clustering, Expert Systems with Applications, 68 (2017), 69-80.  doi: 10.1016/j.eswa.2016.09.021.

[19]

M. MertensW. Koch and T. Kirubarajan, Exploiting Doppler blind zone information for ground moving target tracking with bistatic airborne radar, IEEE Transactions on Aerospace and Electronic Systems, 50 (2014), 130-148.  doi: 10.1109/TAES.2013.120718.

[20]

F. W. Moore, Radar cross-section reduction via route planning and intelligent control, IEEE Transactions on Control Systems Technology, 10 (2016), 696-700.  doi: 10.1109/TCST.2002.801879.

[21]

L. H. Nam, L. Huang, X. J. Li and J. F. Xu, An approach for coverage path planning for UAVs, in 2016 IEEE 14th International Workshop on Advanced Motion Control, Auckland, New Zealand, (2016), 411–416. doi: 10.1109/AMC.2016.7496385.

[22]

N. Ozalo and O. K. Sahingoz, Optimal UAV path planning in a 3D threat environment by using parallel evolutionary algorithms, in 2013 International Conference on Unmanned Aircraft Systems, Grand Hyatt Atlanta, Atlanta, (2013), 308–317.

[23]

N. OzakiS. CampagnolaR. Funase and C. H. Yam, Stochastic differential dynamic programming with unscented transform for low-thrust trajectory design, Journal of Guidance, Control, and Dynamics, 41 (2018), 377-381.  doi: 10.2514/1.G002367.

[24]

M. Patterson and A. Rao, Gpops-Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Transactions on Mathematical Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904.

[25]

Y. H. Qu, Y. T. Zhang and Y. M. Zhang, Optimal flight path planning for UAVs in a 3-D threat environment, in 2014 International Conference on Unmanned Aircraft systems, Orlando, FL, USA, (2014), 149–155. doi: 10.1109/ICUAS.2014.6842250.

[26]

A. V. Rao, D. A. Benson and C. Darby, Algorithm 902: GPOPS A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method, ACM Transactions on Mathematical Software, 37 (2010), Article 22. doi: 10.1145/1731022.1731032.

[27]

J. R. RiehlG. E. Collins and J. P. Hespanha, Cooperative search by UAV teams: A model predictive approach using dynamic graphs, IEEE Transactions on Aerospace and Electronic systems, 47 (2011), 2637-2656.  doi: 10.1109/TAES.2011.6034656.

[28]

V. RobergeM. Tarbouchi and G. Labonte, Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning, IEEE Transactions on Industrial Information, 9 (2013), 132-141.  doi: 10.1109/TII.2012.2198665.

[29]

B. M. SathyarajL. C. JainA. Finn and S. Drake, Multiple UAVs path planning algorithms: A comparative study, Fuzzy Optimization and Decision Making, 7 (2008), 257-267.  doi: 10.1007/s10700-008-9035-0.

[30]

P. Y. Volkan, A new vibrational genetic algorithm enhanced with a Voronoi diagram for path planning of autonomous UAV, Aerospace Science and Technology, 16 (2012), 47-55. 

[31]

B. Z. Xu, Y. J. Wang and L. Liu, Multi-stage boost aircraft trajectory optimization strategy based on hp adaptive Gauss pseudo spectral method, in 10th International Conference on Modelling, Identification and Control, Guiyang, China, 2018, 1–7. doi: 10.1109/ICMIC.2018.8529869.

[32]

P. YaoZ. X. Xie and P. Ren, Optimal UAV route planning for coverage search of stationary target in river, IEEE Transactions on Control Systems Technology, 27 (2019), 822-829.  doi: 10.1109/TCST.2017.2781655.

[33]

M. Zhang, Z. Zhu, Z. Zhao and X. Li, Trajectory optimization for missile-borne SAR imaging phase via Gauss Pseudospectral Method, in 2011 IEEE CIE International Conference on Radar, Chengdu, China, (2011), 867–870. doi: 10.1109/CIE-Radar.2011.6159678.

Figure 1.  An illustration for multi-aircraft air combat
Figure 2.  Target radial velocity results without path planning in two-to-one scenario
Figure 3.  Flight trajcetories in two-to-one scenario
Figure 4.  Target radial velocity in two-to-one scenario
Figure 5.  The airborne radar blind zone in two-to-one scenario with HPAGPM
Figure 6.  The airborne radar blind zone in two-to-one scenario with GPM
Figure 7.  Azimuth angle of the target relative to the aircraft in two-to-one scenario
Figure 8.  The normal accelerations in two-to-one scenario
Figure 9.  Flight trajcetories in four-to-one scenario
Figure 10.  Target radial velocity in four-to-one scenario
Figure 11.  Azimuth angle of the target relative to the aircraft in four-to-one scenario
Figure 12.  The normal accelerations in four-to-one scenario
Table 1.  Compraison results for theree methods
Method Aboved 2v1 scenario 50 scenarios
run time blind zone time solution probability
PSO 195s 58s 80%
GPM 176s 51s 92%
HPAGPM 109s 40s 96%
Method Aboved 2v1 scenario 50 scenarios
run time blind zone time solution probability
PSO 195s 58s 80%
GPM 176s 51s 92%
HPAGPM 109s 40s 96%
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