SCIENTIA SINICA Physica, Mechanica & Astronomica, Volume 49 , Issue 8 : 084505(2019) https://doi.org/10.1360/SSPMA2018-00351

## Convergence properties analysis of the indirect optimization techniques for solar sail trajectory optimization

• AcceptedDec 11, 2018
• PublishedJun 3, 2019
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### Abstract

The indirect method is commonly used for solving the solar sail spacecraft trajectory optimization problem. However, the convergence difficulty usually tends to happen due to the sensitivity of the initial costates. In order to improve the numerical convergence properties, many indirect optimization techniques, such as the normalization of the costate vector, have been proposed in the research of deep space trajectory optimization problem. However, there is little literature focused on the analysis of the convergence properties for different indirect optimization techniques. This paper takes the asteroid rendezvous mission with solar sail spacecraft as background, and analyzes the relationship between the indirect optimization techniques which includes the normalization of the costate vector, the homotopic approach and different coordinate systems (the Cartesian and the Spherical coordinate system) for solar sail dynamic models and the convergence properties. First, dynamic models based on different indirect optimization techniques are developed. The shooting method is used to solve for the optimal costates. Then, the convergence properties of the three techniques are compared and analyzed according to the times of initial guesses and the accuracy of the guessed costates. The results of the numerical simulations show that the homotopic approach can achieve the highest accuracy of initial guesses and the best convergence properties.

### References

[1] Li J F, Baoyin H X, Jiang F H. Dynamics and Control of Interplanetary Flight (in Chinese). Beijing: Tsinghua University Press, 2014 [李俊峰, 宝音贺西, 蒋方华. 深空探测动力学与控制. 北京: 清华大学出版社, 2014]. Google Scholar

[2] Dachwald B, Wie B. Solar sail trajectory optimization for intercepting, impacting, and deflecting near-Earth asteroids. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit. San Francisco: AIAA, 2005. Google Scholar

[3] Dachwald B. Optimal solar sail trajectories for missions to the outer solar system. J Guid Control Dyn, 2005, 28: 1187-1193 CrossRef ADS Google Scholar

[4] Ball A J, Ulamec S, Dachwald B, et al. A small mission for in situ exploration of a primitive binary near-Earth asteroid. Adv Space Res, 2009, 43: 317-324 CrossRef ADS Google Scholar

[5] Brophy J R, Friedman L, Culick F. Asteroid retrieval feasibility. In: 2012 IEEE Aerospace Conference. Big Sky, MT: IEEE, 2012. 1–16. Google Scholar

[6] Zeng X, Gong S, Li J. Fast solar sail rendezvous mission to near Earth asteroids. Acta Astronaut, 2014, 105: 40-56 CrossRef ADS Google Scholar

[7] Peloni A, Ceriotti M, Dachwald B. Solar-sail trajectory design for a multiple near-Earth-asteroid rendezvous mission. J Guid Control Dyn, 2016, 39: 2712-2724 CrossRef ADS Google Scholar

[8] Peloni A, Dachwald B, Ceriotti M. Multiple near-Earth asteroid rendezvous mission: Solar-sailing options. Adv Space Res, 2018, 62: 2084-2098 CrossRef ADS Google Scholar

[9] Heiligers J, Scheeres D J. Solar-sail orbital motion about asteroids and binary asteroid systems. J Guid Control Dyn, 2018, 41: 1947-1962 CrossRef ADS Google Scholar

[10] Mengali G, Quarta A A. Rapid solar sail rendezvous missions to asteroid 99942 Apophis. J Spacecraft Rockets, 2006, 46: 134-140 CrossRef ADS Google Scholar

[11] Betts J T. Survey of numerical methods for trajectory optimization. J Guid Control Dyn, 1998, 21: 193-207 CrossRef ADS Google Scholar

[12] Conway B A. A survey of methods available for the numerical optimization of continuous dynamic systems. J Optim Theor Appl, 2012, 152: 271-306 CrossRef Google Scholar

[13] Gong S P, Gao Y F, Li J F. Solar sail time-optimal interplanetary transfer trajectory design. Res Astron Astrophys, 2011, 11: 981-996 CrossRef ADS Google Scholar

[14] Zeng X, Alfriend K T, Li J, et al. Optimal solar sail trajectory analysis for interstellar missions. J Astronaut Sci, 2012, 59: 502-516 CrossRef ADS Google Scholar

[15] Ma Y Y, Pan B F. Solar sail time-optimal trajectory optimization using Kustaanheimo-Stiefel transformation. In: The Fourth International Symposium on Solar Sailing. Kyoto, 2017. Google Scholar

[16] Bertrand R, Epenoy R. New smoothing techniques for solving bang-bang optimal control problems—Numerical results and statistical interpretation. Optim Control Appl Meth, 2002, 23: 171-197 CrossRef Google Scholar

[17] Guo T, Jiang F, Li J. Homotopic approach and pseudospectral method applied jointly to low thrust trajectory optimization. Acta Astronaut, 2012, 71: 38-50 CrossRef ADS Google Scholar

[18] Jiang F, Baoyin H, Li J. Practical techniques for low-thrust trajectory optimization with homotopic approach. J Guid Control Dyn, 2012, 35: 245-258 CrossRef ADS Google Scholar

[19] Yang H, Li J, Baoyin H. Low-cost transfer between asteroids with distant orbits using multiple gravity assists. Adv Space Res, 2015, 56: 837-847 CrossRef ADS Google Scholar

[20] Gong S, Li J, Jiang F. Interplanetary trajectory design for a hybrid propulsion system. Aerospace Sci Tech, 2015, 45: 104-113 CrossRef Google Scholar

[21] Sullo N, Peloni A, Ceriotti M. Low-thrust to solar-sail trajectories: A homotopic approach. J Guid Control Dyn, 2017, 40: 2796-2806 CrossRef ADS Google Scholar

[22] Junkins J L, Taheri E. Exploration of alternative state vector choices for low-thrust trajectory optimization. J Guid Control Dyn, 2019, 42: 47-64 CrossRef Google Scholar

[23] Zeng X Y. Solar Sail Spacecraft Novel Trajectory Design in Deep Space Exploration (in Chinese). Dissertation for Doctoral Degree. Beijing: Tsinghua University, 2013 [曾祥远. 深空探测太阳帆航天器新型轨道设计. 博士学位论文. 北京: 清华大学, 2013]. Google Scholar

[24] McInnes C R. Solar Sailing: Technology, Dynamics and Mission Applications. London: Springer-Verlag, 1999. 34–170. Google Scholar

[25] Moré J J, Garbow B S, Hillstrom K E. User guide for MINPACK-1. Technical Report. Argonne: Argonne National Laboratory, 1980. Google Scholar

[26] Chi Z, Yang H, Chen S, et al. Homotopy method for optimization of variable-specific-impulse low-thrust trajectories. Astrophys Space Sci, 2017, 362: 216 CrossRef ADS Google Scholar

• Figure 1

(Color online) The transfer trajectory from Earth to Apophis (ac=0.6 mm/s2).

• Figure 2

(Color online) The cone angle and the clock angle along the trajectory (ac=0.6 mm/s2).

• Figure 3

(Color online) The convergence results with and without normalization.

• Figure 4

(Color online) The value of guessed and optimal costate vector without and with the normalization.

• Figure 5

(Color online) The comparison of the convergence properties in the Cartesian and Spherical coordinate system (ac=0.6 mm/s2).

• Figure 6

(Color online) The relative error of the guessed costate vector in the Cartesian and the Spherical coordinate system (ac=0.6 mm/s2).

• Figure 7

(Color online) The comparison of the convergence properties in the Cartesian and Spherical coordinate system (ac=0.12 mm/s2).

• Figure 8

(Color online) The relative error of the guessed costate vector in the Cartesian and the Spherical coordinate system (ac=0.12 mm/s2).

• Figure 9

(Color online) The transfer trajectory from Earth to Apophis (ac=0.12 mm/s2).

• Figure 10

(Color online) The transfer time as a function of the sail characteristic acceleration.

• Figure 11

(Color online) The transfer time and the equivalent characteristic acceleration as functions of the homotopic parameter.

• Table 1   Prescribed orbital elements of Earth and Apophis at the departure time
 参数 地球 Apophis 半长轴 (AU) 0.9999880 0.9222942 偏心率 0.0167168 0.1911153 轨道倾角 (°) 0.00088544 3.33190072 升交点赤经 (°) 175.406477 204.431287 近地点幅角 (°) 287.61578 126.42502 平近点角 (°) 64.50968 152.73163
• Table 2   The relative error of the guessed costate vector in the Spherical and Cartesian coordinate system
 坐标系 λ 协态变量猜值 协态变量最优解 相对误差 (%) 球系 λ1 5.648761990 5.687399726 6.8×10−1 λ2 0.188500373 0.189491922 5.2×10−1 λ3 1.281364474 1.273299290 6.3×10−1 λ4 9.272879111 9.336672806 6.8×10−1 λ5 2.627394826 2.635509524 3.1×10−1 λ6 −3.568097892 −3.592479453 6.8×10−1 直系 λ1 −0.882922640 −0.882922641 2.4×10−8 λ2 −0.048028677 −0.048028677 1.6×10−7 λ3 0.167845952 0.167845953 2.7×10−7 λ4 −0.690479170 −0.690479171 2.6×10−8 λ5 −0.144033770 −0.144033770 1.7×10−7 λ6 0.312089311 0.312089312 2.0×10−7

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