# American Institute of Mathematical Sciences

March  2022, 18(2): 825-841. doi: 10.3934/jimo.2020180

## Energy management method for an unpowered landing

 School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

* Corresponding author: Yingjing Shi (shiyingjing@uestc.edu.cn)

Received  June 2020 Revised  September 2020 Published  March 2022 Early access  December 2020

Fund Project: This work is supported in part by the National Natural Science Foundation of China under grant (No. 61973055), the Fundamental Research Funds for the Central Universities (No. ZYGX2019J062) and a grant from the applied basic research programs of Sichuan province (No. 2019YJ0206).

The unpowered landing of unmanned aerial vehicle (UAV) is a critical stage, which affects the safety of flight. To solve the problem of the unpowered landing of UAV, an energy management scheme is proposed. After the cruise is over, the aircraft shuts down the engine and begins to land. When the aircraft is in the high altitude, the dynamic pressure is too large, and it is difficult to open the speed brake. When the aircraft is in the low altitude, it is close to the runway. The method of S-turn may make the aircraft veer off the runway and may be unable to land. So two different schemes of high altitude and low altitude are designed to control energy. In the high altitude, when the energy is too high, it takes the S-turn scheme to consume excess energy. At the same time, the availability and reasonability of the S-turn scheme is demonstrated. In the low altitude, the open angle of speed brake is controlled to adjust the energy consumption. Finally, the simulation results are given to illustrate the availability of energy management.

Citation: Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial and Management Optimization, 2022, 18 (2) : 825-841. doi: 10.3934/jimo.2020180
##### References:
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Campbell, A vision based forced landing site selection system for an autonomous UAV, in 2005 International Conference on Intelligent Sensors, Sensor Networks and Information Processing, (2005), Melbourne, Australia, 397–402. [6] R. Frezza and C. Altafini, Autonomous landing by computer vision: An application of path following in SE, in Proceedings of the 39th IEEE Conference on Decision and Control, (2000), Sydney, NSW, Australia, 2527–2532. doi: 10.1109/CDC.2000.914183. [7] R. Ghanem and B. Zireg, Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide, Numerical Algebra, Control and Optimization, 10 (2020), 275-300. [8] D. Gu and D. W. Zhang, Parametric control to second-order linear time-varying systems based on dynamic compensator and multi-objective optimization, Applied Mathematics and Computation, 365 (2020), 124681, 25pp. doi: 10.1016/j.amc.2019.124681. [9] D. Gu and D. W. Zhang, A parametric method to design dynamic compensator for high-order quasi-linear systems, Nonlinear Dynamics, 100 (2020), 1379-1400.  doi: 10.1007/s11071-020-05555-0. [10] X. Guo, S. Denman and C. Fookes, et al., Automatic UAV forced landing site detection using machine learning, in 2014 International Conference on Digital Image Computing: Techniques and Applications (DICTA), (2014), Wollongong, NSW, Australia, 1–7. doi: 10.1109/DICTA.2014.7008097. [11] K. R. Horneman and C. A. Kluever, Terminal area energy management trajectory planning for an unpowered reusable launch vehicle, in Collection of Technical Papers - AIAA Atmospheric Flight Mechanics Conference, (2004), Providence, RI, United States, 1103–1120. doi: 10.2514/6.2004-5183. [12] D. Jung and P. Tsiotras, On-line path generation for unmanned aerial vehicles using B-spline path templates, Journal of Guidance, 36 (2013), 1642-1653.  doi: 10.2514/1.60780. [13] R. Li and Y. J. Shi, The fuel optimal control problem of a hypersonic aircraft with periodic cruising mode, J Mathematical and Computer Modelling, 55 (2012), 2141-2150.  doi: 10.1016/j.mcm.2011.12.052. [14] B. Li, X. Guo, X. Zeng, S. Dian and M. Guo, An optimal PID tuning method for a single link manipulator based on the control parametrization technique, Discrete and Continuous Dynamical Systems Series S, 13 (2020), 1813-1823.  doi: 10.3934/dcdss.2020107. [15] R. Li, Y. J. Shi and K. L. Teo, Coordination arrival control for multi-agent systems, International Journal of Robust and Nonlinear Control, 26 (2016), 1456-1474.  doi: 10.1002/rnc.3359. [16] L. Mejias and D. L. Fitzgerald, A multi-layered approach for site detection in UAS emergency landing scenarios using geometry-based image segmentation, in 2013 International Conference on Unmanned Aircraft Systems (ICUAS), (2013), Atlanta, GA, USA, 366–372. doi: 10.1109/ICUAS.2013.6564710. [17] T. E. Moore, Space shuttle entry terminal area energy management, Houston, TX, United States [Online]. Available: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920010688.pdf [18] H. Niu, Z. J. Feng, Q. J. Xiao and Y. J. Zhang, A PID control method based on optimal control strategy, Numerical Algebra, Control and Optimization, (2020). [19] M. Ohno, Y. Yamaguchi and T. Hata, Robust flight control law design for an automatic landing flight experiment, Control Engineering Practice, 7 (1999), 1143-1151. [20] D. Orfanus, E. D. Freitas and F. Eliassen, Self-organization as a supporting paradigm for military UAV relay networks, IEEE Communications Letters, 20 (2016), 804-807.  doi: 10.1109/LCOMM.2016.2524405. [21] R. K. Rangel, C. A. D. Oliveira and K. H. Kienitz, Development of a multipurpose tactical surveillance system using UAV's, in 2014 IEEE Aerospace Conference, (2014), Big Sky, MT, USA, 1–9. doi: 10.1109/AERO.2014.6836300. [22] S. D. Ridder and E. Mooij, Terminal area trajectory planning using the energy-tube concept for reusable launch vehicles, Acta Astronautica, 68 (2011), 915-930.  doi: 10.1016/j.actaastro.2010.08.032. [23] S. D. Ridder and E. Mooij, Optimal longitudinal trajectories for reusable space vehicles in the terminal area, Journal of Spacecraft and Rockets, 48 (2011), 642-653.  doi: 10.2514/1.51083. [24] S. D. Ridder and E. Mooij, Terminal area trajectory planning using the energy-tube concept for reusable launch, Acta Astronautica, 68 (2011), 915-930.  doi: 10.1016/j.actaastro.2010.08.032. [25] M. Senpheng and M. Ruchanurucks, Automatic landing assistant system based on stripe lines on runway using computer vision, in 2015 International Conference on Science and Technology (TICST), (2015), Pathum Thani, Thailand, 35–39. doi: 10.1109/TICST.2015.7369336. [26] I. Shapira and J. Ben-Asher, Range maximization for emergency landing after engine cut-off, Journal of Guidance, 42 (2005), 1296-1306. [27] Y. F. Shen, Z. Rahman and D. Krusienski, A vision-based automatic safe landing-site detection system, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 294-311. [28] Y. J. Shi, R. Li and K. L. Teo, Design of a band-stop filter for a space shuttle vehicle, IEEE Transactions on Circuits and Systems-II: Express Briefs, 62 (2015), 1174-1178.  doi: 10.1109/TCSII.2015.2468922. [29] Y. J. Shi, R. Li and H. L. Xu, Control augmentation design of UAVs based on deviation modification of aerodynamic focus, Journal of Industrial and Management Optimization, 11 (2015), 231-240.  doi: 10.3934/jimo.2015.11.231. [30] D. G. Ward, J. F. Monaco and J. D. Schierman, Reconfigurable control for VTOL UAV shipboard landing, in Guidance, Navigation, and Control Conference and Exhibit, Portland, OR, United States, (1999), 499–509. doi: 10.2514/6.1999-4045. [31] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [32] J. F. Zhang and C. J. Jia, Automatic landing controller design and simulation of flying-wing unmanned aerial vehicle, in Proceedings of 2013 2nd International Conference on Measurement, Information and Control, 2 (2013), Harbin, China, 893–896. doi: 10.1109/MIC.2013.6758104. [33] M. Zhou, J. Zhou and J. G. Guo, Terminal area energy management trajectory planning for an unpowered reusable launch vehicle with gliding limitations, Applied Mechanics and Materials, 446 (2014), 611-615.  doi: 10.4028/www.scientific.net/AMM.446-447.611.

show all references

##### References:
 [1] V. Artale, C. L. R. Milazzo and C. Orlando, Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller, Journal of Industrial and Management Optimization, 13 (2017), 2067-2091.  doi: 10.3934/jimo.2017032. [2] A. R. Babaei and M. Mortazavi, Three-dimensional curvature-constrained trajectory planning based on in-flight waypoints, Journal of Aircraft, 47 (2010), 1391-1398.  doi: 10.2514/1.47711. [3] J. Benasher and D. K. Maya, Pseudo-spectral-method based optimal glide in the event of engine cut-off, in Navigation, and Control Conference, (2011), Tel-Aviv and Haifa, Israel, 1201–1218. doi: 10.2514/6.2011-6596. [4] P. Eng, L. Mejias and X. Liu, Automating human thought processes for a uav forced landing, Journal of Intelligent and Robotic Systems, 57 (2010), 329-349.  doi: 10.1007/978-90-481-8764-5_17. [5] D. L. Fitzgerald, R. A. Walker and D. A. Campbell, A vision based forced landing site selection system for an autonomous UAV, in 2005 International Conference on Intelligent Sensors, Sensor Networks and Information Processing, (2005), Melbourne, Australia, 397–402. [6] R. Frezza and C. Altafini, Autonomous landing by computer vision: An application of path following in SE, in Proceedings of the 39th IEEE Conference on Decision and Control, (2000), Sydney, NSW, Australia, 2527–2532. doi: 10.1109/CDC.2000.914183. [7] R. Ghanem and B. Zireg, Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide, Numerical Algebra, Control and Optimization, 10 (2020), 275-300. [8] D. Gu and D. W. Zhang, Parametric control to second-order linear time-varying systems based on dynamic compensator and multi-objective optimization, Applied Mathematics and Computation, 365 (2020), 124681, 25pp. doi: 10.1016/j.amc.2019.124681. [9] D. Gu and D. W. Zhang, A parametric method to design dynamic compensator for high-order quasi-linear systems, Nonlinear Dynamics, 100 (2020), 1379-1400.  doi: 10.1007/s11071-020-05555-0. [10] X. Guo, S. Denman and C. Fookes, et al., Automatic UAV forced landing site detection using machine learning, in 2014 International Conference on Digital Image Computing: Techniques and Applications (DICTA), (2014), Wollongong, NSW, Australia, 1–7. doi: 10.1109/DICTA.2014.7008097. [11] K. R. Horneman and C. A. Kluever, Terminal area energy management trajectory planning for an unpowered reusable launch vehicle, in Collection of Technical Papers - AIAA Atmospheric Flight Mechanics Conference, (2004), Providence, RI, United States, 1103–1120. doi: 10.2514/6.2004-5183. [12] D. Jung and P. Tsiotras, On-line path generation for unmanned aerial vehicles using B-spline path templates, Journal of Guidance, 36 (2013), 1642-1653.  doi: 10.2514/1.60780. [13] R. Li and Y. J. Shi, The fuel optimal control problem of a hypersonic aircraft with periodic cruising mode, J Mathematical and Computer Modelling, 55 (2012), 2141-2150.  doi: 10.1016/j.mcm.2011.12.052. [14] B. Li, X. Guo, X. Zeng, S. Dian and M. Guo, An optimal PID tuning method for a single link manipulator based on the control parametrization technique, Discrete and Continuous Dynamical Systems Series S, 13 (2020), 1813-1823.  doi: 10.3934/dcdss.2020107. [15] R. Li, Y. J. Shi and K. L. Teo, Coordination arrival control for multi-agent systems, International Journal of Robust and Nonlinear Control, 26 (2016), 1456-1474.  doi: 10.1002/rnc.3359. [16] L. Mejias and D. L. Fitzgerald, A multi-layered approach for site detection in UAS emergency landing scenarios using geometry-based image segmentation, in 2013 International Conference on Unmanned Aircraft Systems (ICUAS), (2013), Atlanta, GA, USA, 366–372. doi: 10.1109/ICUAS.2013.6564710. [17] T. E. Moore, Space shuttle entry terminal area energy management, Houston, TX, United States [Online]. Available: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920010688.pdf [18] H. Niu, Z. J. Feng, Q. J. Xiao and Y. J. Zhang, A PID control method based on optimal control strategy, Numerical Algebra, Control and Optimization, (2020). [19] M. Ohno, Y. Yamaguchi and T. Hata, Robust flight control law design for an automatic landing flight experiment, Control Engineering Practice, 7 (1999), 1143-1151. [20] D. Orfanus, E. D. Freitas and F. Eliassen, Self-organization as a supporting paradigm for military UAV relay networks, IEEE Communications Letters, 20 (2016), 804-807.  doi: 10.1109/LCOMM.2016.2524405. [21] R. K. Rangel, C. A. D. Oliveira and K. H. Kienitz, Development of a multipurpose tactical surveillance system using UAV's, in 2014 IEEE Aerospace Conference, (2014), Big Sky, MT, USA, 1–9. doi: 10.1109/AERO.2014.6836300. [22] S. D. Ridder and E. Mooij, Terminal area trajectory planning using the energy-tube concept for reusable launch vehicles, Acta Astronautica, 68 (2011), 915-930.  doi: 10.1016/j.actaastro.2010.08.032. [23] S. D. Ridder and E. Mooij, Optimal longitudinal trajectories for reusable space vehicles in the terminal area, Journal of Spacecraft and Rockets, 48 (2011), 642-653.  doi: 10.2514/1.51083. [24] S. D. Ridder and E. Mooij, Terminal area trajectory planning using the energy-tube concept for reusable launch, Acta Astronautica, 68 (2011), 915-930.  doi: 10.1016/j.actaastro.2010.08.032. [25] M. Senpheng and M. Ruchanurucks, Automatic landing assistant system based on stripe lines on runway using computer vision, in 2015 International Conference on Science and Technology (TICST), (2015), Pathum Thani, Thailand, 35–39. doi: 10.1109/TICST.2015.7369336. [26] I. Shapira and J. Ben-Asher, Range maximization for emergency landing after engine cut-off, Journal of Guidance, 42 (2005), 1296-1306. [27] Y. F. Shen, Z. Rahman and D. Krusienski, A vision-based automatic safe landing-site detection system, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 294-311. [28] Y. J. Shi, R. Li and K. L. Teo, Design of a band-stop filter for a space shuttle vehicle, IEEE Transactions on Circuits and Systems-II: Express Briefs, 62 (2015), 1174-1178.  doi: 10.1109/TCSII.2015.2468922. [29] Y. J. Shi, R. Li and H. L. Xu, Control augmentation design of UAVs based on deviation modification of aerodynamic focus, Journal of Industrial and Management Optimization, 11 (2015), 231-240.  doi: 10.3934/jimo.2015.11.231. [30] D. G. Ward, J. F. Monaco and J. D. Schierman, Reconfigurable control for VTOL UAV shipboard landing, in Guidance, Navigation, and Control Conference and Exhibit, Portland, OR, United States, (1999), 499–509. doi: 10.2514/6.1999-4045. [31] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [32] J. F. Zhang and C. J. Jia, Automatic landing controller design and simulation of flying-wing unmanned aerial vehicle, in Proceedings of 2013 2nd International Conference on Measurement, Information and Control, 2 (2013), Harbin, China, 893–896. doi: 10.1109/MIC.2013.6758104. [33] M. Zhou, J. Zhou and J. G. Guo, Terminal area energy management trajectory planning for an unpowered reusable launch vehicle with gliding limitations, Applied Mechanics and Materials, 446 (2014), 611-615.  doi: 10.4028/www.scientific.net/AMM.446-447.611.
The full ground track of S-turn scheme
Flow chart of terminal energy management at various phase
The prediction of range in acquisition phase
The prediction of range in HAC-Turn phase
Track angle of aircraft
The judgment nominal line of aircraft in pre-approach phase
The force direction of the aircraft
The high energy change of the S-turning scheme
The range of the S-turning scheme at high energy
The low energy change of the S-turning scheme
The range of the S-turning scheme at low energy
The angle of speed brake in the nominal case
The angle of speed brake at high energy
The angle of speed brake at low energy
The resistance of closed speed brake and opened speed brake
 h(m) $\rho(kg/m^3)$ $\alpha (^\circ)$ $D_c$(N) $D_o$(N) 4500 0.7768 4.86 5541 7565 4000 0.8191 4.74 5643 7782 3500 0.8632 4.46 5769 8028 3000 0.9091 4.36 5913 8296 2500 0.9569 4.12 6071 8558 2000 1.0065 3.76 11760 14290 1500 1.0580 3.60 12290 14950 1000 1.1116 3.44 12840 15630
 h(m) $\rho(kg/m^3)$ $\alpha (^\circ)$ $D_c$(N) $D_o$(N) 4500 0.7768 4.86 5541 7565 4000 0.8191 4.74 5643 7782 3500 0.8632 4.46 5769 8028 3000 0.9091 4.36 5913 8296 2500 0.9569 4.12 6071 8558 2000 1.0065 3.76 11760 14290 1500 1.0580 3.60 12290 14950 1000 1.1116 3.44 12840 15630
The control effect of the speed brake
 Speed brake Glide angle (high/low) Touchdown velocity Closed $-9^{\circ}/-15^{\circ}/-19^{\circ}$ 293.4km/h Fully opened $-29^{\circ}/-15^{\circ}/-19^{\circ}$ 290.2km/h
 Speed brake Glide angle (high/low) Touchdown velocity Closed $-9^{\circ}/-15^{\circ}/-19^{\circ}$ 293.4km/h Fully opened $-29^{\circ}/-15^{\circ}/-19^{\circ}$ 290.2km/h
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