空间目标交会期间碰撞概率研究
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摘要
碰撞概率计算是实现轨道预警,确保航天器飞行安全的一种重要方法。现有短时碰撞概率算法求解二维概率密度函数时计算复杂,在交会平面进行积分区域转化时引入误差较大,空间目标等效模型只采用了简单的球体模型。针对上述问题,本文以现有碰撞模型为基础,给出了计算二维概率密度函数的简化方法和高精度的碰撞概率快速算法,从概率分布和空间目标形状等效两个方面给出改进模型及相应的碰撞概率算法,并对长时交会的碰撞情况进行了分析。
本文主要工作如下:
1.研究了短时交会模型。通过轨道预测数据分析验证了交会区域空间目标的运动为线性运动。给出了一种能同时计算相对位置误差二维概率密度函数均值和方差的方法,该方法利用正交投影和特征分解,在将三维问题转化到交会平面的过程中,无需联立方程组就能求得二维概率密度函数参数。
2.给出了一种基于圆域的碰撞概率快速计算方法。该方法采用近似积分法则,将二维问题转化为多项式求解,避免了积分区域转化对计算精度的影响,实验表明其计算精度与二维积分直接计算相当,近似值与精确值之比达到了0.99。
3.结合碰撞预警需求,从概率分布和空间目标形状等效两个方面改进了原有模型。位置误差服从均匀分布时的碰撞概率算法在计算效率上得到较大提高,可以用作先期筛选。空间目标椭球体等效模型的碰撞概率方法,对空间目标的描述更符合实际,并为进一步建立符合空间目标实际形状的高精度碰撞模型提供了参考。
4.给出了长时交会情况下的碰撞概率算法,该算法利用相对运动方程和蒙特卡罗积分,以及碰撞概率模型中目标函数与重要性函数的一致性,得到了方差最小的碰撞概率,并与短时交会碰撞概率算法进行了分析比较。
The calculation of collision probability is an important method to trajectory early warning and the safety of spacecraft. Currently, there are some problems with the algorithm of calculating collision probability, the computing efficiency is not ideal to acquire planar probability density function; the incoming error cannot be ignored when changing the integral area on the encounter plane; and the description to the equivalent model only gives one simple spherical model. Based on the short-term encounter model, a method that can compute the value of mean and variance simultaneity is proposed and a precise algorithm of collision probability at encounter plane is also put forward; the improvement of the old model is presented in detail from the attribution of density and equivalent model of space objects with corresponding algorithm. At last, the collision in the case of long-term encounter is analyzed particularly.
The primary work of the paper is as follows:
Firstly, the short-term encounter model is studied and the linear movement of space objects is validated in the encounter area with orbit forecast data. Experiments show that it is feasible to approximate curve with beeline in short arc. A method is put forward to obtain mean and variance of the planar probability density function simultaneously. Based on orthogonal projection and eigenvalue decomposition, it can be obtained without simultaneous equations in the process of changing the three-dimension problem to encounter plane.
Secondly, one fast algorithm of collision probability is proposed based on round area. It adopts approximate integral theorem, turns the two-dimension problem to multinomial, and avoids the infection of the change of integral area to the compute precision. Experiments show that the precision of the method is close to the result of direct integral and the ratio between approximation and exact value arrives at 0.99.
Thirdly, the original model is improved on probability distribution and equivalent model of space objects. A collision model is developed with corresponding algorithm when the position error submits to uniformity. Experiments show that the method can get the collision probability more efficiency and can be used as early filtration. The ellipsoid equivalent model of space objects and corresponding algorithm is developed. The experiment shows that the description of space object is more accurate to its characteristic and the model can provide an important reference to establish the actual shape model of space objects.
Finally, algorithm for computing the collision probability of long-term encounter is proposed with Clohessey-Whiltshire equation and Monte Carlo integral, and the minimal variance is acquired because of the coherence of objective function and important function. At last the method of long-term encounter is compared with the algorithm of short-term encounter.
本文主要工作如下:
1.研究了短时交会模型。通过轨道预测数据分析验证了交会区域空间目标的运动为线性运动。给出了一种能同时计算相对位置误差二维概率密度函数均值和方差的方法,该方法利用正交投影和特征分解,在将三维问题转化到交会平面的过程中,无需联立方程组就能求得二维概率密度函数参数。
2.给出了一种基于圆域的碰撞概率快速计算方法。该方法采用近似积分法则,将二维问题转化为多项式求解,避免了积分区域转化对计算精度的影响,实验表明其计算精度与二维积分直接计算相当,近似值与精确值之比达到了0.99。
3.结合碰撞预警需求,从概率分布和空间目标形状等效两个方面改进了原有模型。位置误差服从均匀分布时的碰撞概率算法在计算效率上得到较大提高,可以用作先期筛选。空间目标椭球体等效模型的碰撞概率方法,对空间目标的描述更符合实际,并为进一步建立符合空间目标实际形状的高精度碰撞模型提供了参考。
4.给出了长时交会情况下的碰撞概率算法,该算法利用相对运动方程和蒙特卡罗积分,以及碰撞概率模型中目标函数与重要性函数的一致性,得到了方差最小的碰撞概率,并与短时交会碰撞概率算法进行了分析比较。
The calculation of collision probability is an important method to trajectory early warning and the safety of spacecraft. Currently, there are some problems with the algorithm of calculating collision probability, the computing efficiency is not ideal to acquire planar probability density function; the incoming error cannot be ignored when changing the integral area on the encounter plane; and the description to the equivalent model only gives one simple spherical model. Based on the short-term encounter model, a method that can compute the value of mean and variance simultaneity is proposed and a precise algorithm of collision probability at encounter plane is also put forward; the improvement of the old model is presented in detail from the attribution of density and equivalent model of space objects with corresponding algorithm. At last, the collision in the case of long-term encounter is analyzed particularly.
The primary work of the paper is as follows:
Firstly, the short-term encounter model is studied and the linear movement of space objects is validated in the encounter area with orbit forecast data. Experiments show that it is feasible to approximate curve with beeline in short arc. A method is put forward to obtain mean and variance of the planar probability density function simultaneously. Based on orthogonal projection and eigenvalue decomposition, it can be obtained without simultaneous equations in the process of changing the three-dimension problem to encounter plane.
Secondly, one fast algorithm of collision probability is proposed based on round area. It adopts approximate integral theorem, turns the two-dimension problem to multinomial, and avoids the infection of the change of integral area to the compute precision. Experiments show that the precision of the method is close to the result of direct integral and the ratio between approximation and exact value arrives at 0.99.
Thirdly, the original model is improved on probability distribution and equivalent model of space objects. A collision model is developed with corresponding algorithm when the position error submits to uniformity. Experiments show that the method can get the collision probability more efficiency and can be used as early filtration. The ellipsoid equivalent model of space objects and corresponding algorithm is developed. The experiment shows that the description of space object is more accurate to its characteristic and the model can provide an important reference to establish the actual shape model of space objects.
Finally, algorithm for computing the collision probability of long-term encounter is proposed with Clohessey-Whiltshire equation and Monte Carlo integral, and the minimal variance is acquired because of the coherence of objective function and important function. At last the method of long-term encounter is compared with the algorithm of short-term encounter.
引文
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[23]王华,李海阳,唐国金.飞行器碰撞概率计算的一般方法[J].国防科技大学学报, 2006, 28(4): 27-31.
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[26]白显宗,陈磊.空间目标碰撞概率的显式表达式及影响因素分析[J].空间科学学报, 2009, 29(4): 422-431.
[27]白显宗,陈磊.基于空间压缩和无穷级数的空间碎片碰撞概率快速算法[J].应用数学学报, 2009, 32(2): 336-353.
[28]王华,唐国金.非线性相对运动的飞行器碰撞概率研究[J].宇航学报, 2006, 27(Sup.): 160-165.
[29]李春来,欧阳自远,都亨.空间碎片与空间环境[J].第四纪研究, 2002, 22(6): 540-550
[30]罗刚桥.空间碎片减缓措施及其研究对策[J].航天器工程, 2000, 9(2): 45-56.
[31] McDonnell J A M, Ratcliff P R, Green S F, etc. Microparticle populations at LEO altitudes: recent spacecraft measurements [J]. Icarus, 1997, 127(1): 55-64.
[32]郭荣.近地轨道航天器的空间碎片碰撞预警与轨道规避策略研究[D].长沙:国防科学技术大学研究生院, 2005.
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[38]刘静,王荣兰等. CZ-4火箭碎片与美国碎片的碰撞事件分析[C].第三届全国空间碎片专题研讨会论文集,北京, 2005.
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[42]王建峰,刘静.规避时段选择与机动变轨研究[C].第三届全国空间碎片专题研讨会,北京, 2005.
[43]白显宗,陈磊.空间目标碰撞概率计算方法研究[J].宇航学报, 2008, 29(4): 1435-1442.
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[2] Bernhard R P, Christiansen E L, Kerr J H. Space shuttle meteoroid and orbital debris impact damage [J]. International Journal of Impact Engineering, 2001, 26: 33-38.
[3]李寿平.美俄卫星相撞的若干法律问题[J].中国航天, 2009, (5): 36-38.
[4]张伟,黄文波,管公顺等.航天器微流星体及空间碎片环境与风险分析[J].中国空间科学技术, 2003, 23(6):58-63.
[5] Berend N. A deterministic approach for collision risk assessment and determination of an optimal avoidance maneuver [C]. The Second European Conference on Space Debris, Darmstadt Germany, March 1997, 619-624.
[6] Concha Marco a. Implementation of a collision probability prediction technique for constellation maneuver planning [R]. Goddard Space Flight Center, February 2007.
[7] John D Vedder, Jill L Tabor. New method for estimating low-earth-orbit collision probabilities [J]. Journal of Spacecraft and Rockets, 1991, 28(2), 210-215.
[8] Lunde Alfred, Foster James Lee Jr. A 160-day simulation of space station debris avoidance operations with the United States space command (USSPACECOM) [R]. NASA/TM-2006- 213720, May 2006.
[9] Foster J L, Estes H S. A parametric analysis of orbital debris collision probability and maneuver rate for space vehicles [R]. NASA JSC-25898, August 1992.
[10] Khutorovsky Z N, Boikov V and Kamensky S Y. Direct method for the analysis of collision probability of artificial space objects in LEO: techniques, methods and applications [C]. Proceedings of the First European Conference on Space Debris, ESA SD-01, 1993, 491-508.
[11] Akella M R, Alfriend K T. Probability of collision between space objects [J]. Journal of Guidance Control and Dynamics, 2000, 23(5): 769-772.
[12] Alfriend K T, Akella M R, Frisbee J, etc. Probability of collision error analysis [C].AIAA Guidance Navigation and Control Conference, Boston Massachusetts, 1999, 21-35.
[13]龚建村,韩蕾,马志昊,陈磊.载人飞船碰撞检测解析方法研究[J].飞行器测控学报, 2007, 26(2):11-14.
[14]白显宗,陈磊.空间目标碰撞概率的置信度问题研究[C].第24界飞行器测控学术年会,南昌, 2008 11.
[15] Chan K F. Collision probability analyses for earth orbiting satellites [J]. Advances in the Astronautical Sciences, 1997, 96: 1033-1048.
[16] Chan K F. Analytical expression for computing spacecraft collision probabilities [C]. AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara California, February 2001, 11-15.
[17] Berend N. Estimation of the probability of collision between two catalogued orbiting objects[J]. Advances in Space Research, 1999, 23(1): 243-247.
[18] Patera R P. General method for calculating satellite collision probability [J]. Journal of Guidance Control and Dynamics, 2001, 24(4): 716-722.
[19] Patera R P. Method for calculating collision probability between a satellite and a space tether [J]. Journal of Guidance Control and Dynamics, 2002, 25 (5): 940-945.
[20] Patera R P. Satellite collision probability for nonlinear relative motion [J]. Journal of Guidance Control and Dynamics, 2003, 26(5): 728-733.
[21] Patera R P, Glenn E Peterson. Space vehicle maneuver method to lower collision risk to acceptable level [J]. Journal of Guidance Control and Dynamics, 2003, 26(2):233-237.
[22] Salvatore Alfano. A numerical implementation of spherical object collision probability [J]. The Journal of Astronautical Sciences, 2005, 53(1): 103-109.
[23]王华,李海阳,唐国金.飞行器碰撞概率计算的一般方法[J].国防科技大学学报, 2006, 28(4): 27-31.
[24]程陶,刘静,王荣兰等.空间碎片预警中的碰撞概率方法研究[J].空间科学学报, 2006, 26(6): 452-458.
[25]程陶.编目空间碎片的碰撞概率方法研究及应用[D].北京:中国科学院空间科学与应用研究中心硕士学位论文, 2006.
[26]白显宗,陈磊.空间目标碰撞概率的显式表达式及影响因素分析[J].空间科学学报, 2009, 29(4): 422-431.
[27]白显宗,陈磊.基于空间压缩和无穷级数的空间碎片碰撞概率快速算法[J].应用数学学报, 2009, 32(2): 336-353.
[28]王华,唐国金.非线性相对运动的飞行器碰撞概率研究[J].宇航学报, 2006, 27(Sup.): 160-165.
[29]李春来,欧阳自远,都亨.空间碎片与空间环境[J].第四纪研究, 2002, 22(6): 540-550
[30]罗刚桥.空间碎片减缓措施及其研究对策[J].航天器工程, 2000, 9(2): 45-56.
[31] McDonnell J A M, Ratcliff P R, Green S F, etc. Microparticle populations at LEO altitudes: recent spacecraft measurements [J]. Icarus, 1997, 127(1): 55-64.
[32]郭荣.近地轨道航天器的空间碎片碰撞预警与轨道规避策略研究[D].长沙:国防科学技术大学研究生院, 2005.
[33] Flury W, Klinkrad H, Janin G, etc. Measurements and modeling of the space debris environment [C]. IAF, 45th International Astronautical Congress, Jerusalem Israel, 1994.
[34]倪迎红译.天基雷达轨道碎片检测[J].电子工程信息, 2009, (1): 17-20.
[35]常伯浚等.飞行力学手册[M].北京:国防工业出版社, 1974.
[36] Johnson N. First natural collision of cataloged earth satellites [J/OL].The Orbital Debris Quarterly News, 1996, 1(2): 1-2. http://orbitaldebris.jsc.nasa.gov/newsletter/pdfs/ODQNV 1i2.pdf.
[37] Accidental collisions of cataloged satellites identified [J/OL].The Orbital Debris Quarterly News, 2005, 9(2): 1-2. http://orbitaldebris.jsc.nasa.gov/newsletter/pdfs/ODQNV9i2.pdf.
[38]刘静,王荣兰等. CZ-4火箭碎片与美国碎片的碰撞事件分析[C].第三届全国空间碎片专题研讨会论文集,北京, 2005.
[39] Johnson N. Growth of official satellite catalog [C]. Orbital Debris Program Office, NASA Johnson Space Center, Houston TX, March 28 2002.
[40] Foster J L Jr, Wortham M B, Frisbee J H Jr. ISS debris avoidance maneuver threshold analysis [R]. Johnson Space Center, May 2007.
[41] Liou J C. NASA’s long-term debris environment and active debris removal modeling activities [R]. NASA Orbital Debris Program Office Johnson Space Center, 8-10 December 2009.
[42]王建峰,刘静.规避时段选择与机动变轨研究[C].第三届全国空间碎片专题研讨会,北京, 2005.
[43]白显宗,陈磊.空间目标碰撞概率计算方法研究[J].宇航学报, 2008, 29(4): 1435-1442.
[44] Kelso T S. Iridium 33/Cosmos 2251 collision [R]. Celestrak, July 2009.
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