﻿ 火星引力捕获动力学与动态误差分析
«上一篇
 文章快速检索 高级检索

 力学学报  2015, Vol. 47 Issue (1): 15-23  DOI: 10.6052/0459-1879-14-327 0

### 引用本文 [复制中英文]

[复制中文]
Fang Baodong, Wu Meiping, Zhang Wei. MARS GRAVITY CAPTURE DYNAMIC MODEL AND ERROR ANALYSIS[J]. Chinese Journal of Ship Research, 2015, 47(1): 15-23. DOI: 10.6052/0459-1879-14-327.
[复制英文]

### 文章历史

2014-10-23收稿
2014-11-10录用
2014-12-12网络版发表

1. 国防科技大学, 长沙 410073;
2. 上海卫星工程研究所, 上海 200240;
3. 上海市深空探测技术重点实验室, 上海 200240

1 火星引力捕获动力学 1.1 工程定义

1.2 引力球、作用球与希尔球

 图 1 限制性三体引力关系示意图 Fig. 1 Restricted three-body problem diagram

1.2.1 引力球计算公式推导

 ${\pmb F}_s^{(v)} = - \dfrac{Gm_v m_s }{{ R}_v^3}{\pmb R}_v$ (1)
 ${\pmb F}_p^{(v)} = - \dfrac{Gm_v m_p }{r^3}{\pmb r}$ (2)

 $\dfrac{Gm_v m_s }{R^2_v}=\dfrac{Gm_vm_p}{r^2}$ (3)

 $r_1 \approx \left( {\dfrac{m_p }{m_s }} \right)^{\tfrac 12}R$ (4)

1.2.2 作用球计算公式推导

 $m_v \ddot{\pmb R}_v = {\pmb F}_s^{(v)} + {\pmb F}_p^{(v)} = - \dfrac{Gm_v m_s }{R_v ^3}{\pmb R}_v - \dfrac{Gm_v m_p }{r^3}{\pmb r}$ (5)

 $\ddot{\pmb R}_v = - \dfrac{Gm_s }{R_v ^3}{\pmb R}_v - \dfrac{Gm_p }{r^3}{\pmb r}$ (6)

 $\dfrac{ | {\pmb P}_p | }{ | {\pmb A}_s | } = \dfrac{m_p }{m_s }\left( {\dfrac{R_v }{r}} \right)^2 \approx \dfrac{m_p }{m_s }\left( {\dfrac{R}{r}} \right)^2$ (7)

 $\ddot{\pmb r} = \ddot{\pmb R}_v - \ddot{\pmb R} = - \dfrac{Gm_p }{r^3}{\pmb r} - \left( {\dfrac{Gm_s }{R_v ^3}{\pmb R}_v -\dfrac{Gm_s }{R^3}{\pmb R}} \right)$ (8)

 $| {\pmb p}_s | = \left| { - Gm_s \left( {\dfrac{1}{R_v ^2} - \dfrac{1}{R^3}} \right)\dfrac{{\pmb R}}{R} } \right| = Gm_s \left( {\dfrac{1}{R_v^2} - \dfrac{1}{R^2}} \right)$ (9)
 $\frac{{|{p_s}|}}{{|{a_p}|}} = \frac{{{m_s}}}{{{m_p}}}\left( {\frac{1}{{R_v^2}} - \frac{1}{{{R^2}}}} \right){r^2}{\rm{ }}$ (10)

 $\dfrac{ | {\pmb p}_s | }{ | {\pmb a}_p | } = \dfrac{m_s }{m_p }\dfrac{(2R - r)r^3}{(R - r)^2R^2}$ (11)

 $\rho = 2^{ - \tfrac 15}\left( {\dfrac{m_p }{m_s }} \right)^{\tfrac 25}R$ (12)
1.2.3 希尔球计算公式推导

 $\rho _{\rm Hill} = \left( {\dfrac{m_p }{3m_s }} \right)^{\tfrac{1}{3}}\left[{1 - \sum\limits_{n = 1}^\infty {\left( {\dfrac{1}{3}} \right)^n\left( {\dfrac{m_p }{3m_s }} \right)^{\tfrac{n}{3}}} } \right]R$ (13)

 $\rho _{\rm Hill} \approx \left( {\dfrac{m_p }{3m_s }} \right)^{\tfrac{1}{3}}R$ (14)
1.2.4 引力球、作用球与希尔球比较

1.3 理论建模

 图 2 火星制动捕获过程示意图 Fig. 2 Mars orbit insertion process

 $\ddot{\pmb r} = - \dfrac{\mu }{r^3}{\pmb r} + {\pmb a }_N + {\pmb a }_{NS} + {\pmb a }_R + {\pmb a }_A + {\pmb a }_M$ (15)

1.4 工程化处理

 $\ddot{\pmb r} = - \dfrac{\mu }{r^3}{\pmb r} + {\pmb a }_N + {\pmb a }_M$ (16)
 ${\pmb a}_M = {\pmb F} / m$ (17)

 $\dot {m} = \dfrac{F}{I_{sp} g_0 }$ (18)

2 仿真分析 2.1 捕获误差源

 图 3 制动捕获误差源分解框图 Fig. 3 Mars orbit insertion error constitution diagram

2.2 仿真条件捕获误差源

 图 4 标称捕获轨道 Fig. 4 Nominal capture orbit
 图 5 到达近火点前12d内的器火距离变化 Fig. 5 Mars-probe distance since 12 days before orbit capture

2.3 仿真结果与分析 2.3.1 导航定位初始误差拉偏

 图 6 导航定位初始误差影响 Fig. 6 Navigation position initial error effects

2.3.2 导航定速初始误差拉偏

 图 7 导航定速初始误差影响 Fig. 7 Navigation velocity initial error effects

2.3.3 发动机推力误差拉偏

 图 8 制动发动机推力大小误差影响 Fig. 8 Capture engine thrust magnitude error effects

 图 9 制动发动机推力方向误差影响 Fig. 9 Capture engine thrust direction error effects

2.3.4 制动点火时间误差拉偏

 图 10 制动点火时间误差影响 Fig. 10 Capture engine ignition time error effects

2.3.5 综合影响分析

3 结 论

 [1] 刘林, 汤靖师. 火星轨道器运动的轨道变化特征. 宇航学报, 2008, 29(2): 461-466 (Liu Lin, Tang Jingshi. Orbit variation characteristics of the Mars' orbiters. Journal of Astronautics, 2008, 29(2): 461-466 (in Chinese)) [2] 刘林, 赵玉晖, 张巍等. 环火卫星运动的坐标系附加摄动及相应坐标系的选择. 天文学报, 2010, 51(4): 412-421 (Liu Lin, Zhao Yuhui, Zhang Wei, et al. The coordinate additional perturbations to Mars orbiters and the choice of corresponding coordinate system. Acta Astronomica Sinica, 2010, 51(4): 412-421 (in Chinese)) [3] 陈杨, 赵国强, 宝音贺西等. 精确动力学模型下的火星探测轨道设计. 中国空间科学技术, 2012, 1: 8-15 (Chen Yang, Zhao Guoqiang, Baoyin Hexi, et al. Orbit design for Mars exploration by the accurate dynamic model. Chinese Space Science and Technology, 2012, 1: 8-15 (in Chinese)) [4] 李俊峰, 宝音贺西. 深空探测中的动力学与控制. 力学与实践, 2007, 29(4): 1-8 (Li Junfeng, Baoyin Hexi. Dynamics and control in deep space exploration. Mechanics in Engineering, 2007, 29(4): 1-8 (in Chinese)) [5] Fischer J, Denis M. Mars orbit insertion —— a new challenge for Europe success with ESA's Mars Express. European Space Operations Centre. IEEEAC, 2004, 1370 [6] Liever P, Habchi S, Burnell S, et al. Computational fluid dynamics prediction of the Beagle 2 aerodynamic database. Journal of Spacecraft and Rocket, 2003, 40(5): 632-638 [7] Jai B, Wenker D, Hammer B, et al. An overview of Mars reconnaissance orbiter mission and operations challenges. AIAA SPACE Conference & Exposition, Long Beach, California. 2007, AIAA 2007-6090 [8] Liechty D. Aeroheating analysis for the Mars reconnaissance orbiter with comparison to flight data. Journal of Spacecraft and Rocket, 2007, 44(6): 1226-1231 [9] Lyons D, Beerer J, Esposito P, et al. Mars global surveyor: aerobraking mission overview. Journal of Spacecraft and Rocket, 1999, 36(3): 307-313 [10] Jet Propulsion Laboratory. Orbit insertion phase. http://www.msss. com/mars/global_surveyor/mgs_msn_plan/section5/section5.html, 2010 [11] 尚海滨, 崔平远, 栾恩杰. 地球-火星的燃料最省小推力转移轨道的设计与优化. 宇航学报, 2006, 27(6): 1168-1173 (Shang Haibin, Cui Pingyuan, Luan Enjie. Design and optimization of Earth-Mars optimal fuel low thrust trajectory. Journal of Astronautics, 2006, 27(6): 1168-1173 (in Chinese)) [12] 杨嘉摨. 航天器轨道动力学与控制(上). 北京: 中国宇航出版社, 2004 (Yang Jiaxi. Spacecraft Orbital Dynamics and Control (Vol.1). Beijing: China Aerospace Press, 2004 (in Chinese)) [13] Chebotarev G. On the dynamical limits of the solar system. Soviet Astronomy, 1965, 8(5): 787-796 [14] 刘林, 侯锡云. 深空探测器轨道力学. 北京: 电子工业出版社, 2012 (Liu Lin, Hou Xiyun. Deep Space Probe Orbital Mechanics. Beijing: Electronic Industry Press, 2012 (in Chinese)) [15] 郝岩. 深空测控网. 北京: 国防工业出版社, 2004 (Hao Yan. Deep Space Telemetry Network. Beijing: National Defense Industry Press, 2004 (in Chinese)) [16] 徐福祥. 卫星工程概论. 北京: 中国宇航出版社, 2004 (Xu Fuxiang. Introduction of Satellite Engineering. Beijing: China Aerospace Press, 2004 (in Chinese)"
MARS GRAVITY CAPTURE DYNAMIC MODEL AND ERROR ANALYSIS
Fang Baodong, Wu Meiping, Zhang Wei
1. National University of Defense Technology, Changsha 410073;
2. Shanghai Institute of Satellite Engineering, Shanghai 200240, China;
3. Shanghai Key Laboratory of Deep Space Exploration Technology, Shanghai 200240, China
Fund: The project was supported by the the National Basic Research Program of China (2014CB744200) and Shanghai Key Laboratory of Deep SpaceExploration Technology (13dz2260100).
Abstract: Mars orbit capture is a one and only opportunity for Mars probes and the key factor to determine whether the mission is successful. Starting with the constrained three-body problem, equations for calculating the Mars gravity sphere, influence sphere and Hill sphere are derived. Their property and applicability are discussed. Based on the definition and physical significance of influence sphere, an engineering definition of capture phase is proposed. The orbital dynamic model was built inside the influence sphere and the error sources that may affect the accuracy of capture orbit are presented. Finally, the influences on perigee and apogee of the capture orbit caused by the position and velocity navigation error, engine thrust error and timing errors are analyzed through Monte Carlo simulations. The limit exceed possibility caused by different error sources are also discussed and the dominating sources is pointed out. The result can be used as a reference for the orbit capture implementation of future Chinese Mars orbiters.
Key words: Mars orbit capture    gravity sphere    influence sphere    Hill sphere    error analysis