不规则小行星引力场内的飞行动力学

李俊峰; 曾祥远

doi:10.6052/1000-0992-16-042

不规则小行星引力场内的飞行动力学

doi: 10.6052/1000-0992-16-042

李俊峰^1,,
曾祥远^2, ,

1.
清华大学航天航空学院, 北京 100084
2.
北京理工大学自动化学院, 北京 100081

详细信息

作者简介:
李俊峰, 1964年出生, 1987年获北京大学力学系学士学位, 1993年获莫斯科大学数学力学系博士学位, 1995年在清华大学工程力学系博士后出站, 现任清华大学航天航空学院教授.近30年从事航天动力学、运动稳定性、天体力学、卫星姿态控制、充液系统晃动等方面的科研工作, 主讲理论力学、运动稳定性、航天器动力学、飞行器姿态控制系统、高等动力学、现代航天技术概论等课程. E-mail: lijunf@tsinghua.edu.cn

通讯作者:
曾祥远, E-mail: zeng@bit.edu.cn

中图分类号: P185.7
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出版历程
- 收稿日期: 2016-11-15
- 网络出版日期: 2017-01-20
- 刊出日期: 2017-02-24

Flight dynamics in the gravitational fields of irregular asteroids

LI Junfeng^1
,,
ZENG Xiangyuan^{2
, ,}

1.
School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
2.
School of Automation, Beijing Institute of Technology, Beijing 100081, China

More Information

Author Bio:
LI Junfeng, E-mail: lijunf@tsinghua.edu.cn

Corresponding author: ZENG Xiangyuan, E-mail: zeng@bit.edu.cn

摘要

摘要: 小行星探测是当前深空探测的主要方向之一, 具有重要的科学意义.绝大多数小行星引力场极不规则, 探测器在小行星附近运动形态复杂多样.由于同时受到中心引力、快速自旋的不规则形状摄动力、以及光压摄动等作用, 探测器容易与小行星发生碰撞或逃逸.概述小行星研究现状和不规则引力场建模方法.重点介绍不规则引力场内动力学特性, 包括引力平衡点、局部流形、自然周期轨道和悬停探测轨道等, 尝试提出新的研究方向.
- 小行星探测 /
- 不规则引力场 /
- 局部流形 /
- 周期轨道 /
- 悬停轨道
Abstract: Asteroid exploration is one of the main topics of deep space exploration, and has great scientific significance. Due to the irregular gravitational field of most asteroids, dynamical behaviors of spacecraft close to them are very complicated. Affected by the central gravity, the perturbing forces from the fast spin of the irregular shapes of asteroids, the solar radiation pressure force, etc., the spacecraft easily collide with the asteroids or escape from them. In this article, an overview of the status of asteroid research and the irregular gravitational field modeling methods is given. Particularly, this article focuses on the dynamic characteristics of irregular gravitational fields, including equilibrium points, local manifolds, natural periodic orbits and body-fixed hovering flights, and tries to propose new research directions.
- asteroid exploration /
- irregular gravitational field /
- local manifold /
- periodic orbits /
- body-fixed hovering

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图 1 人类航天器探测过的小天体等比缩放排列图

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图 2 小行星Itokawa不同引力场建模方法.(a) 级数展开法参考球, (b) 级数展开法参考椭球, (c) 质点群法示意图, (d) Itokawa多面体模型

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图 3 旋转偶极子近似细长小行星及其改进模型.(a) 偶极子近似Itokawa引力场, (b) 考虑二阶项的改进偶极子模型

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图 4 小行星Geographos引力场内周期轨道示例.(a) 共线平动点局部周期轨道, (b) 非共线平动点局部周期轨道, (c) 草帽轨道, (d) 类8字形轨道

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图 5 小行星引力场内太阳帆航天器本体悬停轨道.(a) 球形小行星附近悬停轨道, (b) 细长小行星本体悬停轨道

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参考文献(88)

[1]	崔平远, 乔栋. 2013.小天体附近轨道动力学与控制研究现状与展望.力学进展, 43: 526-539 Gui P Y, Qiao D. 2013. Research progress and prospect of orbital dynamics and control near small bodies.Advances in Mechanics, 43: 526-539
[2]	龚胜平. 2008.太阳帆航天器动力学与控制研究[博士论文].北京:清华大学 Gong S P. 2008. Study on dynamics and control of sailcraft. [PhD Thesis]. Beijing: Tsinghua University.
[3]	高扬. 2011.电火箭星际航行:技术进展、轨道设计与综合优化.力学学报, 43: 991-1019 http://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201106005.htm Gao Y. 2011. Interplanetary travel with electric propulsion: Technological progress, trajectory design, and comprehensive optimization. Chinese Journal of Theoretical and Applied Mechanics, 43: 991-1019. http://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201106005.htm
[4]	李俊峰, 宝音贺西. 2007.深空探测中的动力学与控制.力学与实践, 29: 1-9 http://cpfd.cnki.com.cn/Article/CPFDTOTAL-AGLU200812002011.htm Li J F, Baoyin H X. 2007. Dynamics and control in deep space exploration. Mechanics in Engineering, 29: 1-9. http://cpfd.cnki.com.cn/Article/CPFDTOTAL-AGLU200812002011.htm
[5]	李俊峰, 曾祥远, 张韵. 2016.小行星的奇特动力学.力学与实践, 38: 603-611 http://www.cnki.com.cn/Article/CJFDTOTAL-LXYS201606002.htm Li J F, Zeng X Y, Zhang Y. 2016. Unique dynamics of asteroids. Mechanics in Engineering, 38: 603-611. http://www.cnki.com.cn/Article/CJFDTOTAL-LXYS201606002.htm
[6]	李九天, 罗亚中, 唐国金. 2015.小行星探测多脉冲交会轨道多目标优化.国防科技大学学报, 36: 3910-3918 http://www.cnki.com.cn/Article/CJFDTOTAL-GFKJ201103003.htm Li J T, Luo Y Z, Tang G J. 2015. Multi-objective optimization of multi-impulse rendezvous trajectory for exploring asteroids. Journal of National University of Defense Technology, 36: 3910-3918. http://www.cnki.com.cn/Article/CJFDTOTAL-GFKJ201103003.htm
[7]	刘林, 侯锡云. 2012.深空探测器轨道力学.北京:电子工业出版社 Liu L, Hou X Y. 2012. Orbital Mechanics of Deep Space Explorer. Beijing: Publishing House of Electronics Industry.
[8]	徐明. 2009.平动点轨道的动力学与控制研究综述.宇航学报, 30: 1299-1313 http://www.cnki.com.cn/Article/CJFDTOTAL-YHXB200904000.htm Xu M. 2009. Overview of orbital dynamics and control for libration point orbits. Journal of Astronautics, 30: 1299-1313. http://www.cnki.com.cn/Article/CJFDTOTAL-YHXB200904000.htm
[9]	于洋. 2014.小天体引力场中的轨道动力学研究. [博士论文].北京:清华大学 Yu Y. 2014. Research on orbital dynamics in the gravitational field of small bodies. [PhD Thesis]. Beijing: Tsinghua University.
[10]	郑永春, 欧阳自远. 2014.太阳系探测的发展趋势与科学问题分析.深空探测学报, 1: 83-92 http://www.cnki.com.cn/Article/CJFDTOTAL-SKTC201402001.htm Zheng Y C, Ouyang Z Y. 2014. Development trend analysis of solar system exploration and the scientific vision for future missions. Journal of Deep Space Exploration, 1: 83-92. http://www.cnki.com.cn/Article/CJFDTOTAL-SKTC201402001.htm
[11]	曾祥远. 2013.深空探测太阳帆航天器新型轨道设计. [博士论文].北京:清华大学 Zeng X Y. 2013. Solar sail spacecraft novel trajectory design in deep space exploration. [PhD Thesis]. Beijing: Tsinghua University.
[12]	Belbruno E A, Miller J. 1993. Sun-perturbed Earth-to-Moon transfers with ballistic capture. Journal of Guidance, Control, and Dynamics, 16: 770-775. doi: 10.2514/3.21079
[13]	Barden B T, and Howell K C. 1998. Fundamental motions near collinear libration points and their transitions. The Journal of the Astronautical Sciences, 46: 361-378. http://cat.inist.fr/?aModele=afficheN&cpsidt=10293927
[14]	Bartczak P, Breiter S. 2003. Double material segment as the model of irregular bodies. Celestial Mechanics and Dynamical Astronomy, 86: 131-141. doi: 10.1023/A:1024115015470
[15]	Broschart S B. 2006. Close proximity spacecraft maneuvers near irregularly shaped small-bodies: hovering, translation, and descent. [PhD Thesis]. USA: The University of Michigan.
[16]	Chermnykh S V. 1987. On the stability of libration points in a certain gravitational field. Vest. Leningrad Univ., 2: 73-77.
[17]	Chanut T G G, Aljbaae S, and Carruba V. 2015. Mascon gravitation model using a shaped polyhedral source. Monthly Notices of the Royal Astronomical Society, 450: 3742-3749. doi: 10.1093/mnras/stv845
[18]	Chapman C R, Veverka J, Thomas P C, Klaasen K, Belton M J S, Harch A, Mcewen A, Johnson T V, Helfenstein P, Davies M E, Merline W J, Denk T. 1995. Discovery and physical properties of Dactyl, a satellite of asteroid 243 Ida. Nature, 374: 783-785. doi: 10.1038/374783a0
[19]	Colagrossi A, Fabio F, Lavagna M, Howell K. 2015. Dynamical evolution about asteroids with high fidelity gravity field and perturbations modeling//AAS 15-637, Vail, CO, USA. https://www.researchgate.net/publication/281967488_DYNAMICAL_EVOLUTION_ABOUT_ASTEROIDS_WITH_HIGH_FIDELITY_GRAVITY_FIELD_AND_PERTURBATIONS_MODELING
[20]	Deaconu G, Louembet C, and Theron A. 2015. Designing continuously constrained spacecraft relative trajectories for proximity operations. Journal of Guidance, Control, and Dynamics, 38: 1208-1217. doi: 10.2514/1.G000283
[21]	Deprit A, Henrard J. 1967. Natural families of periodic orbits. The Astronomical Journal, 72: 158-172. doi: 10.1086/110212
[22]	Duboshin G N. 1959. On one particular case of the problem of the translational-rotational motion of two bodies. Soviet Astron., 3: 154-165. http://adsabs.harvard.edu/abs/1959AZh....36..153D
[23]	Elipe A, Lara M. 2003. A simple model for the chaotic motion around (433) Eros. Journal of the Astronautical Sciences, 51: 391-404. https://www.researchgate.net/publication/28130105_A_Simple_Model_for_the_Chaotic_Motion_Around_433_Eros
[24]	Farrés A, Jorbá A. 2012. Orbital dynamics of a solar sail near L1 and L2 in the elliptic hill problem. IAC-12.C1.6.4, 63rd International Astronautical Congeress, Naples, Italy.
[25]	Fornasier S, Clark B E, Dotto E. 2011. Spectroscopic survey of X-type asteroids. Icarus, 214: 131-146. doi: 10.1016/j.icarus.2011.04.022
[26]	Furfaro R. 2015. Hovering in asteroid dynamical environments using higher-order sliding control. Journal of Guidance, Control, and Dynamics, 38: 263-279. doi: 10.2514/1.G000631
[27]	Geissler P, Petit J M, Durda D D, Moore J. 1996. Erosion and ejecta reaccretion of 243 Ida and its Moon. Icarus, 120: 140-157. doi: 10.1006/icar.1996.0042
[28]	Goźiewski K, Maciejewski A J. 1998. Nonlinear stability of the Lagrangian libration points in the Chermnykh problem. Celestial Mechanics and Dynamical Astronomy, 70: 41-58. doi: 10.1023/A:1008250207046
[29]	Goźiewski K, Maciejewski A J. 1999. Unrestricted planar problem of a symmetric body and a point mass: Triangular libration points and their stability. Celestial Mechanics and Dynamical Astronomy, 75: 251-285. doi: 10.1023/A:1008337017789
[30]	Gomez G, Koon W S, Lo M W, Ross S D. 2000. Connections between periodic orbits and resonance transitions in celestial mechanics. Chaos: An interdisciplinary Journal of Nonlinear Science, 10: 427-469. doi: 10.1063/1.166509
[31]	Guelman M. 2015. Closed-loop control of close orbits around asteroids. Journal of Guidance, Control, and Dynamics, 38: 854-860. doi: 10.2514/1.G000158
[32]	Harris A W. 1996. The rotation rates of very small asteroids: Evidence for "rubble pile" structure//Lunar and Planetary Science XXVII, 27: 493-494.
[33]	Hénon M. 1997. Generating families in the restricted three-body problem. Springer-Verlag Berlin Heidelberg, Germany.
[34]	Herrera-Sucarrat E, Palmer P L, Roberts R M. 2014. Asteroid observation and landing trajectories using invariant manifolds. Journal of Guidance, Control, and Dynamics, 37: 907-920. doi: 10.2514/1.59594
[35]	Hirabayashi M, Morimoto M Y, Yano H, Kawaguchi J, Bellerose J. 2010. Linear stability of collinear equilibrium points around an asteroid as a two-connected-mass: Application to fast rotating asteroid 2000EB14. Icarus, 206, 780-782. doi: 10.1016/j.icarus.2009.12.023
[36]	Hobson E W. 1955. The theory of spherical and ellipsoidal harmonics. Vermont: Chelsea Publishing Company.
[37]	Hu W D. 2002. Orbital motion in uniformly rotating second degree and order gravity fields. [PhD Thesis]. USA: University of Michigan.
[38]	Jiang Y, Baoyin H, Li J F, Li H. 2014. Orbits and manifolds near the equilibrium points around a rotating asteroid. Astrophysics and Space Science, 349: 83-106. doi: 10.1007/s10509-013-1618-8
[39]	Jiang Y, Yu Y, Baoyin H X. 2015. Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies. Nonlinear Dynamics, 81: 119-140. doi: 10.1007/s11071-015-1977-5
[40]	Kirpichnikov S N, Kokoriev A A. 1988. On the stability of stationary collinear Lagrangian motions in the system of two attracting bodies: an axisymmetrical, peer-like and spherically symmetric. Vest. Leningrad Univ., 3: 72-84.
[41]	Kokoriev A A, and Kirpichnikov S N. 1988. On the stability of stationary triangular Lagrangian motions in the system of two attracting bodies: an axisymmetrical, peer-like and spherically symmetric. Vest. Leningrad Univ., 1: 75-84.
[42]	Koon W S, Lo M W, Marsden J E, Ross S D. 2011. Dynamical systems, the three-body problem and space mission design. World Scientific.
[43]	Kushvah B S. 2008. Linear stability of equilibrium points in the generalized photogravitational Chermnykh's problem. Astrophysics and Space Science, 318: 41-50. doi: 10.1007/s10509-008-9898-0
[44]	Lara M, Pelaez J. 2002. On the numerical continuation of periodic orbits: An intrinsic, 3-dimensional, differential, predictor-corrector algorithm. Astronomy & Astrophysics, 389: 692-701. https://www.researchgate.net/publication/266215956_On_the_numerical_continuation_of_periodic_orbits_An_intrinsic_3-dimensional_differential_predictor-corrector_algorithm
[45]	Lee D, Sanyal A K, Butcher E A, Scheeres D J. 2015. Finite-time control for spacecraft body-fixed hovering over an asteroid. IEEE Transactions on Aerospace and Electronic Systems, 51: 506-520. doi: 10.1109/TAES.2014.140197
[46]	Lee D, Vukovich G. 2015. Adaptive sliding mode control for spacecraft body-fixed hovering in the proximity of an asteroid. Aerospace Science and Technology, 46: 471-483. doi: 10.1016/j.ast.2015.09.001
[47]	Li X Y, Qiao D, Cui P Y. 2013. The equilibria and periodic orbits around a dumbbell-shaped body. Astrophysics and Space Science, 348: 417-426. doi: 10.1007/s10509-013-1592-1
[48]	Liu X D, Baoyin H X, Ma X R. 2011. Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube. Astrophysics and Space Science, 333: 409-418. doi: 10.1007/s10509-011-0669-y
[49]	Llanos P J, Miller J K, Hintz G R. 2014. Orbital evolution and environmental analysis around asteroid 2008 EV5//24th AAS/AIAA Space Flight Mechanics Meeting, New Mexico, USA.
[50]	Lo M W, Ross S D. 1998. Low energy interplanetary transfers using invariant manifolds of L1, L2 and halo orbits. Advances in the Astronautical Sciences, 99: 551-557. https://www.researchgate.net/publication/266038546_Low_Energy_Interplanetary_Transfers_Using_the_Invariant_Manifolds_of_L1_L2_and_Halo_Orbits
[51]	Lowry S C, Weissman P R, Duddy S R, RozitisB, FitzsimmonsA, GreenSF, HicksMD, SnodgrassC, WoltersSD, ChesleySR, vanOersP. 2014. The internal structure of asteroid (25143) Itokawa as revealed by detection of YORP spin-up. Astronomy & Astrophysics, 562: A48: 1-9.
[52]	MacMillan W D. 1930. The theory of the potential. New York: McGraw-Hill.
[53]	McInnes C R. 1999. Solar sailing: Technology, dynamics and mission applications. 1st edition, Springer Praxis, England, UK.
[54]	Morrow E, Scheeres D J, Lubin D. 2001. Solar sail orbit operations at asteroids. Journal of Spacecraft and Rockets, 38: 279-286. doi: 10.2514/2.3682
[55]	Morrow E, Scheeres D J, Lubin D. 2002. Solar sail orbit operations at asteroids: Exploring the coupled effect of an imperfectly reflecting sail and a nonspherical asteroid//AIAA 2002-4991, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, CA, USA, August 5-8.
[56]	Najid N E, Elourabi E H, Zegoumou M. 2011. Potential generated by a massive inhomogeneous straight segment. Research in Astronomy and Astrophysics, 11: 345-352. doi: 10.1088/1674-4527/11/3/008
[57]	Nazari M, Wauson R, Critz T, Butcher E A, Scheeres D J. 2014. Observer-based body-fixed hovering control over a tumbling asteroid. Acta Astronautica, 102: 124-139. doi: 10.1016/j.actaastro.2014.05.016
[58]	Prieto-Llanos T, Gmez-Tierno M A. 1994. Stationkeeping at Libration Points of Natural Elongated Bodies. Journal of Guidance, Control, and Dynamics, 17: 787-794. doi: 10.2514/3.21268
[59]	Riaguas A, Elipe A, Lara M. 1999. Periodic orbits around a massive straight segment. Celestial Mechanics and Dynamical Astronomy, 73: 169-178. doi: 10.1023/A:1008399030624
[60]	Szebehely V. 1967. Theory of orbits: The restricted problem of three bodies. New York: Academic Press
[61]	Sawai S, Scheeres D J, Broschart S B. 2002. Control of hovering spacecraft using altimetry. Journal of Guidance, Control, and Dynamics, 25: 786-795. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/77306/AIAA-2000-4421-987.pdf;sequence=1
[62]	Scheeres D J, Ostro S J, Hudson R S, Werner R A. 1996. Orbits close to asteroid 4769 Castalia. Icarus, 121: 67-87. doi: 10.1006/icar.1996.0072
[63]	Scheeres D J, Ostro S J, Hudson R S, Dejong E M, Suzuki S. 1998. Dynamics of orbits close to asteroid 4179 Toutatis. Icarus, 132: 53-79. doi: 10.1006/icar.1997.5870
[64]	Scheeres D J. 1999. Stability of hovering orbits around small bodies. 9th Spaceflight Mechanics Meeting, Advances in the Astronautical Sciences, 102: 855-873.
[65]	Scheeres D. 2002. Orbital motion in strongly perturbed environments: applications to asteroid, comet and planetary satellite orbiters. London (UK): Springer-Praxis.
[66]	Scheeres D J, Broschart S, Ostro S J, Benner L A. 2004. The dynamical environment about asteroid 25143 Itokawa: Target of the Hayabusa mission//AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA 2004-4864, Providence, Rhode Island.
[67]	Scheeres D J, Gaskell R, Abe S, Barnouin-Jha O, Hashimoto T, Kawaguchi J, Kubota T, Saito J, Yoshikawa M, Hirata N, Mukai T, Ishiguro M, Kominato T, Shirkawa K, Uo M. 2006. The actual dynamical environment about Itokawa//AIAA 2006-6661, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, CO, August 21-24.
[68]	Scheeres D J. 2012. Orbital mechanics about small bodies. Acta Astronautica, 72: 1-14. doi: 10.1016/j.actaastro.2011.10.021
[69]	Scheeres D J. 2014. Close proximity dynamics and control about asteroids//2014 American Control Conference, Portland, Oregon, US, June 4-6: 1584-1598.
[70]	Tsirogiannis G A, Perdios E A, Markellos V V. 2009. Improved grid search method: an efficient tool for global computation of periodic orbits. Celestial Mechanics and Dynamical Astronomy, 103: 49-78. doi: 10.1007/s10569-008-9165-2
[71]	Wang X Y, Jiang Y, Gong S P. 2014. Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies. Astrophysics and Space Science, 353: 105-121. doi: 10.1007/s10509-014-2022-8
[72]	Wang X Y, Li J F, Gong S P. 2016. Bifurcation of equilibrium points in the potential field of asteroid 101955 Bennu. Monthly Notices of the Royal Astronomical Society, 455: 3724-3734. doi: 10.1093/mnras/stv2426
[73]	Wang Y, Xu S J. 2013. Gravity gradient torque of spacecraft orbiting asteroids. Aircraft Engineering and Aerospace Technology, 85: 72-81. doi: 10.1108/00022661311294049
[74]	Werner R A. 1994. The gravitational potential of a homogeneous polyhedron or don't cut corners. Celestial Mechanics and Dynamical Astronomy, 59: 253-278. doi: 10.1007/BF00692875
[75]	Werner R A, Scheeres D J. 1996. Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celestial Mechanics and Dynamical Astronomy, 65: 313-344. doi: 10.1007%2FBF00053511
[76]	Williams T, Abate M. 2009. Capabilities of furlable solar sails for asteroid proximity operations. Journal of Spacecraft and Rockets, 46: 967-975. doi: 10.2514/1.30355
[77]	Yang H W, Zeng X Y, Baoyin H X. 2015. Feasible region and stability analysis for hovering around elongated asteroids with low thrust. Research in Astronomy and Astrophysics, 15: 1571-1586. doi: 10.1088/1674-4527/15/9/013
[78]	Yoshimitsu T, Kubota T, Nakatani I. 2006. MINERVA rover which became a small artificial solar satellite//20th Annual AIAA/USU Conference on Small Satellites, SSC06-IV-4.
[79]	Yu Y, Baoyin H X. 2012a. Orbital dynamics in the vicinity of asteroid 216 Kleopatra. The Astronomical Journal, 143: 62-70. doi: 10.1088/0004-6256/143/3/62
[80]	Yu Y, Baoyin H X. 2012b. Generating families of 3D periodic orbits about asteroids. Monthly Notices of the Royal Astronomical Society, 427: 872-881. doi: 10.1111/(ISSN)1365-2966
[81]	Yu Y, Baoyin H X, Jiang Y. 2015. Constructing the natural families of periodic orbits near irregular bodies. Monthly Notices of the Royal Astronomical Society, 453: 3269-3277. https://www.researchgate.net/profile/Yu_Jiang22/publication/280621457_Construct_the_natural_families_of_periodic_orbits_near_irregular_bodies/links/55efdceb08aef559dc44eb9c.pdf?inViewer=true&disableCoverPage=true&origin=publication_detail
[82]	Yu Y, Baoyin H X. 2015. Modeling of migrating grains on asteroid's surface. Astrophysics and Space Science, 355: 43-56. https://www.researchgate.net/publication/269290538_Modeling_of_migrating_grains_on_asteroid's_surface
[83]	Zeng X Y, Jiang F H, Li J F, Baoyin H X. 2015a. Study on the connection between the rotating mass dipole and natural elongated bodies. Astrophysics and Space Science, 356: 29-42. doi: 10.1007/s10509-014-2187-1
[84]	Zeng X Y, Jiang F H, Li J F. 2015b. Asteroid body-fixed hovering using nonideal solar sails. Research in Astronomy and Astrophysics, 15: 597-607. doi: 10.1088/1674-4527/15/4/011
[85]	Zeng X Y, Baoyin H X, Li J F. 2016a. Updated rotating mass dipole with oblateness of one primary (Ⅰ) : equilibria in the equator and their stability. Astrophysics and Space Science, 361, 14. doi: 10.1007/s10509-015-2598-7
[86]	Zeng X Y, Baoyin H X, Li J F. 2016b. Updated rotating mass dipole with oblateness of one primary (Ⅱ) : out-of-plane equilibria and their stability. Astrophysics and Space Science, 361, 15. doi: 10.1007/s10509-015-2599-6
[87]	Zeng X Y, Gong S P, Li J F, et al. 2016c. Solar sail body-fixed hovering over elongated asteroids. Journal of Guidance, Control, and Dynamics, 39: 1223-1231. doi: 10.2514/1.G001061
[88]	Zeng X Y, Liu X D. 2017. Searching for time-optimal periodic orbits near irregularly shaped asteroids by using an indirect method. IEEE Transactions on Aerospace and Electronic Systems. (To be published) http://ieeexplore.ieee.org/servlet/opac?punumber=7