空间目标轨道预报误差与碰撞概率问题研究
|
摘要
随着人类航天活动的日益频繁，在轨空间目标数量快速增长，已对空间环境和航天事业的可持续发展产生了重大影响，必须加强对空间目标碰撞预警问题的研究。本文以空间目标碰撞预警过程为研究对象，针对碰撞预警中涉及的轨道预报误差和碰撞概率问题进行研究。主要成果如下：
基于相对运动理论研究了空间目标初始误差传播特性和位置速度误差的负相关特性。考虑到轨道预报误差与目标矢径相比是小量，将目标的真实状态和预报状态分别对应于一“实”一“虚”两个近距离空间目标，将轨道预报误差看作它们之间的相对运动。基于代数法模型的C-W方程和T-H方程分别进行了圆轨道和椭圆轨道的初始误差传播分析。基于几何法模型在近圆轨道假设下对位置速度误差的负相关特性进行了分析，讨论了该特性在初始误差协方差选取中的应用。利用历史轨道数据对负相关特性进行了验证。
提出了基于历史轨道数据的轨道预报误差周期特性分析方法和泊松级数拟合方法。选取轨道数据历元时刻前后半个周期内的状态预报值为参考值，在全轨道周期内求差以反映周期特性。利用最小二乘方法拟合得到了泊松系数矩阵，分别讨论了多项式项、三角函数项、混合项的作用。作为误差拟合函数，泊松级数可以描述误差随预报时间的长期变化和随在轨位置的周期变化，而且泊松系数矩阵可以在进行碰撞预警分析之前就得到。
在接近几何分析的基础上推导了圆轨道和一般轨道情形下碰撞概率的显式表达式。在圆轨道情形下推导了接近距离的RSW分量表示和接近几何关系（过轨道面交线高度差和时间差、轨道夹角等）表示的碰撞概率显式表达式。在一般轨道情形下推导了接近几何关系（过速度公垂线高度差、时间差、速度夹角、速度大小比）表示的和接近距离的NTW分量表示的碰撞概率显式表达式。分析了圆轨道情形下显式表达式适用的偏心率范围，对于大多数LEO目标，圆轨道情形下显式表达式的精度对于碰撞风险评估和预警决策而言是足够的。
基于碰撞概率的显式表达式分析了碰撞概率灵敏度和最大碰撞概率的计算方法，给出了计算最大碰撞概率的完整步骤。根据碰撞概率的显式表达式，分析了碰撞概率对接近距离的RSW分量、轨道误差标准差、接近角度和目标大小的两类灵敏度。在误差椭球形状固定和不定两种情况下推导得到了以接近几何关系或接近距离的分量表示的最大碰撞概率的解析表达式。讨论了当接近距离的一个分量为零时的特殊情况，给出了计算最大碰撞概率的完整步骤。
基于碰撞概率的显式表达式对碰撞预警的漏警率和虚警率进行了分析，介绍并实现了考虑多因素的碰撞风险综合评估方法。碰撞预警实质上是一个判别分析问题。根据碰撞概率的显式表达式定义了碰撞预警的安全区域和危险区域，给出了漏警率和虚警率的定义、计算公式和基本规律。介绍并实现了综合考虑风险参数和轨道品质参数的碰撞风险综合评估方法。
介绍了空间目标碰撞预警软件系统，利用编目目标的两行轨道根数（TLE）数据给出了系统应用实例。介绍了空间目标碰撞预警软件系统的模块组成和并行计算环境，本文提出的轨道预报误差分析方法和碰撞概率分析方法在软件系统中得到了应用。利用编目目标的TLE数据给出碰撞预警软件系统的应用实例。
论文以我国空间目标碰撞预警工程需求为牵引，研究了碰撞预警涉及的轨道预报误差和碰撞概率问题，解决了制约工程实践的关键问题，发展了碰撞预警分析方法，可为空间目标碰撞预警工程系统的建立和完善提供技术支持。
The amount of on-orbit space objects has been growing continuously because ofthe increasing and frequent aerospace activities, which has been strongly affecting thespace environment and the sustainable development of aerospace industry. Thishighlights the practical necessity of conjunction assessment and collision avoidance.Taking the process of conjunction assessment as research object, this dissertation mainlystudies the orbital prediction error and collision probability problems involved inconjunction assessment. The main achievements are summarized as follows:
The propagation characteristics of initial orbital error and negativecorrelation characteristics of position and velocity error are studied based on therelative motion theory. Considering that the orbital prediction error is much smallerthan the radius of object, the true state and predicted state of space object can be seen astwo objects: a real one and a virtual one, and the prediction error can be seen as therelative motion between them. The propagation characteristics of initial error of circularand eccentric orbit are respectively analyzed based on C-W Equations and T-HEquations, which belong to the algebraic model. The negative correlation characteristicsof position and velocity error in the case of near-circular orbit are studied based on thegeometrical model of relative motion. The application of negative correlationcharacteristics in the determination of initial error covariance is discussed. Thecorrelation characteristics are validated by historical orbital data.
A methodology for periodicity characterization and Poisson series fitting fororbital prediction error based on historical orbital data is presented. The predictedstates in an orbital period centered at its epoch are taken as reference states. Thecomparison is conducted in a whole period so that the residuals can reflect theperiodicity. The Poisson coefficient matrices of each error components are fitted usingleast squares method. Effects of polynomial, trigonometric, and mixed terms of Poissonseries are discussed. As error-fitting function, the Poisson series can describe variationof error with respect to propagation duration and on-orbit position of objects. ThePoisson coefficient matrices can be obtained before close approach analysis.
The explicit expressions of collision probability (Pc) in the cases of circularorbit and general orbit are derived based on analysis of conjunction geometry. Inthe case of circular orbit, Pcis expressed as explicit functions of the RSW componentsof relative position or the conjunction geometries (crossing altitude difference and timedifference of the line of intersection of two orbital planes, the angle between orbitalplanes, etc.). In the case of general orbit, Pcis expressed as explicit functions of theconjunction geometries (crossing altitude difference and time difference of the commonperpendicular line to two velocities, the angle between two velocities, etc.) or the NTW components of relative position. The explicit expression relates Pcwith components ofrelative position. The eccentricity’s bound of explicit expression in the case of circularis determined. The precision of explicit expression is sufficient for conjunction riskassessment and decision-making for most LEO objects.
The sensitivity of Pcand maximum collision probability (Pcmax) is analyzedbased on the explicit expression of Pc, the integrated procedure for estimating Pcmaxis provided. The sensitivities of Pcto RSW components of relative position, orbitalerror standard deviations, conjunction angle and object’s size are analyzed based on theexplicit expression of Pc. Analytical expressions of Pcmaxin terms of conjunctiongeometries or components of relative position are deduced in both cases of fixed andarbitrary error ellipsoid shape. Special cases when one of the coordinates in conjunctionplane tends to zero are discussed. The integrated procedure for estimating Pcmaxandcorresponding error standard deviations is provided.
The probabilities of missing alarm and false alarm are studied based on theexplicit expressions of Pc, the comprehensive assessment of collision riskconsidering multi-factors is researched. The conjunction assessment is substantially adiscriminant analysis problem. The safety-region and danger-region of conjunctionassessment are defined by using explicit expression of Pc. The definition, formulae, andbasic property of probabilities of missing alarm and false alarm are provided. Thecomprehensive assessment of collision risk considering both risk assessment and qualityassessment parameters is introduced and implemented.
The conjunction assessment software system is introduced and demonstratedby Two Line Element (TLE) data of space catalogued objects. The constituentmodules and parallel computing environment of conjunction assessment softwaresystem are introduced, in which the methods of orbital error analysis and collisionprobability analysis achieved in this dissertation have been applied. The softwaresystem is demonstrated by TLE data of space catalogued objects.
Motivated by practical requirement of conjunction assessment, this dissertationfocuses on the involved orbit error and collision probability problem. This dissertationhas developed the conjunction assessment methodology. The achievements will providetechnical support for the establishment and improvement of conjunction assessmentengineering system.
基于相对运动理论研究了空间目标初始误差传播特性和位置速度误差的负相关特性。考虑到轨道预报误差与目标矢径相比是小量，将目标的真实状态和预报状态分别对应于一“实”一“虚”两个近距离空间目标，将轨道预报误差看作它们之间的相对运动。基于代数法模型的C-W方程和T-H方程分别进行了圆轨道和椭圆轨道的初始误差传播分析。基于几何法模型在近圆轨道假设下对位置速度误差的负相关特性进行了分析，讨论了该特性在初始误差协方差选取中的应用。利用历史轨道数据对负相关特性进行了验证。
提出了基于历史轨道数据的轨道预报误差周期特性分析方法和泊松级数拟合方法。选取轨道数据历元时刻前后半个周期内的状态预报值为参考值，在全轨道周期内求差以反映周期特性。利用最小二乘方法拟合得到了泊松系数矩阵，分别讨论了多项式项、三角函数项、混合项的作用。作为误差拟合函数，泊松级数可以描述误差随预报时间的长期变化和随在轨位置的周期变化，而且泊松系数矩阵可以在进行碰撞预警分析之前就得到。
在接近几何分析的基础上推导了圆轨道和一般轨道情形下碰撞概率的显式表达式。在圆轨道情形下推导了接近距离的RSW分量表示和接近几何关系（过轨道面交线高度差和时间差、轨道夹角等）表示的碰撞概率显式表达式。在一般轨道情形下推导了接近几何关系（过速度公垂线高度差、时间差、速度夹角、速度大小比）表示的和接近距离的NTW分量表示的碰撞概率显式表达式。分析了圆轨道情形下显式表达式适用的偏心率范围，对于大多数LEO目标，圆轨道情形下显式表达式的精度对于碰撞风险评估和预警决策而言是足够的。
基于碰撞概率的显式表达式分析了碰撞概率灵敏度和最大碰撞概率的计算方法，给出了计算最大碰撞概率的完整步骤。根据碰撞概率的显式表达式，分析了碰撞概率对接近距离的RSW分量、轨道误差标准差、接近角度和目标大小的两类灵敏度。在误差椭球形状固定和不定两种情况下推导得到了以接近几何关系或接近距离的分量表示的最大碰撞概率的解析表达式。讨论了当接近距离的一个分量为零时的特殊情况，给出了计算最大碰撞概率的完整步骤。
基于碰撞概率的显式表达式对碰撞预警的漏警率和虚警率进行了分析，介绍并实现了考虑多因素的碰撞风险综合评估方法。碰撞预警实质上是一个判别分析问题。根据碰撞概率的显式表达式定义了碰撞预警的安全区域和危险区域，给出了漏警率和虚警率的定义、计算公式和基本规律。介绍并实现了综合考虑风险参数和轨道品质参数的碰撞风险综合评估方法。
介绍了空间目标碰撞预警软件系统，利用编目目标的两行轨道根数（TLE）数据给出了系统应用实例。介绍了空间目标碰撞预警软件系统的模块组成和并行计算环境，本文提出的轨道预报误差分析方法和碰撞概率分析方法在软件系统中得到了应用。利用编目目标的TLE数据给出碰撞预警软件系统的应用实例。
论文以我国空间目标碰撞预警工程需求为牵引，研究了碰撞预警涉及的轨道预报误差和碰撞概率问题，解决了制约工程实践的关键问题，发展了碰撞预警分析方法，可为空间目标碰撞预警工程系统的建立和完善提供技术支持。
The amount of on-orbit space objects has been growing continuously because ofthe increasing and frequent aerospace activities, which has been strongly affecting thespace environment and the sustainable development of aerospace industry. Thishighlights the practical necessity of conjunction assessment and collision avoidance.Taking the process of conjunction assessment as research object, this dissertation mainlystudies the orbital prediction error and collision probability problems involved inconjunction assessment. The main achievements are summarized as follows:
The propagation characteristics of initial orbital error and negativecorrelation characteristics of position and velocity error are studied based on therelative motion theory. Considering that the orbital prediction error is much smallerthan the radius of object, the true state and predicted state of space object can be seen astwo objects: a real one and a virtual one, and the prediction error can be seen as therelative motion between them. The propagation characteristics of initial error of circularand eccentric orbit are respectively analyzed based on C-W Equations and T-HEquations, which belong to the algebraic model. The negative correlation characteristicsof position and velocity error in the case of near-circular orbit are studied based on thegeometrical model of relative motion. The application of negative correlationcharacteristics in the determination of initial error covariance is discussed. Thecorrelation characteristics are validated by historical orbital data.
A methodology for periodicity characterization and Poisson series fitting fororbital prediction error based on historical orbital data is presented. The predictedstates in an orbital period centered at its epoch are taken as reference states. Thecomparison is conducted in a whole period so that the residuals can reflect theperiodicity. The Poisson coefficient matrices of each error components are fitted usingleast squares method. Effects of polynomial, trigonometric, and mixed terms of Poissonseries are discussed. As error-fitting function, the Poisson series can describe variationof error with respect to propagation duration and on-orbit position of objects. ThePoisson coefficient matrices can be obtained before close approach analysis.
The explicit expressions of collision probability (Pc) in the cases of circularorbit and general orbit are derived based on analysis of conjunction geometry. Inthe case of circular orbit, Pcis expressed as explicit functions of the RSW componentsof relative position or the conjunction geometries (crossing altitude difference and timedifference of the line of intersection of two orbital planes, the angle between orbitalplanes, etc.). In the case of general orbit, Pcis expressed as explicit functions of theconjunction geometries (crossing altitude difference and time difference of the commonperpendicular line to two velocities, the angle between two velocities, etc.) or the NTW components of relative position. The explicit expression relates Pcwith components ofrelative position. The eccentricity’s bound of explicit expression in the case of circularis determined. The precision of explicit expression is sufficient for conjunction riskassessment and decision-making for most LEO objects.
The sensitivity of Pcand maximum collision probability (Pcmax) is analyzedbased on the explicit expression of Pc, the integrated procedure for estimating Pcmaxis provided. The sensitivities of Pcto RSW components of relative position, orbitalerror standard deviations, conjunction angle and object’s size are analyzed based on theexplicit expression of Pc. Analytical expressions of Pcmaxin terms of conjunctiongeometries or components of relative position are deduced in both cases of fixed andarbitrary error ellipsoid shape. Special cases when one of the coordinates in conjunctionplane tends to zero are discussed. The integrated procedure for estimating Pcmaxandcorresponding error standard deviations is provided.
The probabilities of missing alarm and false alarm are studied based on theexplicit expressions of Pc, the comprehensive assessment of collision riskconsidering multi-factors is researched. The conjunction assessment is substantially adiscriminant analysis problem. The safety-region and danger-region of conjunctionassessment are defined by using explicit expression of Pc. The definition, formulae, andbasic property of probabilities of missing alarm and false alarm are provided. Thecomprehensive assessment of collision risk considering both risk assessment and qualityassessment parameters is introduced and implemented.
The conjunction assessment software system is introduced and demonstratedby Two Line Element (TLE) data of space catalogued objects. The constituentmodules and parallel computing environment of conjunction assessment softwaresystem are introduced, in which the methods of orbital error analysis and collisionprobability analysis achieved in this dissertation have been applied. The softwaresystem is demonstrated by TLE data of space catalogued objects.
Motivated by practical requirement of conjunction assessment, this dissertationfocuses on the involved orbit error and collision probability problem. This dissertationhas developed the conjunction assessment methodology. The achievements will providetechnical support for the establishment and improvement of conjunction assessmentengineering system.
引文
[1] Liou J C. Satellite box score [J]. Orbital Debris Quarterly News,2013,17(1):7.
[2] Crowther R. Space junk-protecting space for future generations [J]. Science,2002,(296):1241~1242.
[3] Liou J C, Johnson N L. Risk in space from orbiting debris [J]. Science,2006,(311):340~341.
[4]刘林.航天器轨道理论[M].北京:国防工业出版社,2000.6.
[5] Hoots F R, Roehrich R L. Space track report No.3: models for propagation ofNORAD element sets [R]. Peterson：Aerospace Defense Command，United StatesAir Force,1980:1~79.
[6] Hoots F R, Schumacher P W, Glover R A. History of analytical orbit modeling inthe U.S. space surveillance system [J]. Journal of Guidance, Control and Dynamic,2004,27(2):174~185.
[7] Vallado D A, Crawford P, Hujsak R. Revisiting spacetrack No.3[C]. AIAA2006-6753. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Keystone, Colorado,2006.8.
[8]李济生.航天器轨道确定[M].北京:国防工业出版社,2003.
[9] Vallado D A. An analysis of state vector propagation using differing flightdynamics programs [C]. AAS05-199.
[10]Fonte D J. Comparison of orbit propagation in the research and developmentGoddard trajectory determination system. AAS95-431.
[11]吴连大.人造卫星与空间碎片的轨道和探测[M].北京:中国科学技术出版社,2011.9.
[12]郑勤余,吴连大.卫星与空间碎片碰撞预警的快速算法[J].天文学报,2004,45(4):422~427.
[13]刘静,王荣兰,张宏博.空间碎片碰撞预警研究[J].空间科学学报,2004,24(6):462~469.
[14]Hoots F R, Crawford L L, Roehrich R L. An analytic method to determine futureclose approaches between satellites [J]. Celestial Mechanics,1984,33(2):143~158.
[15]秋宏兴,祝转民,吴连大.航天器碰撞分析方法研究[J].宇航学报,2005,26(3):257~261.
[16]Dybczynski P A, Jopek T J, Serafin R A. On the minimum distance between twoKeplerian orbits with a common focus [J]. Celestial Mechanics,1986(38):345~356.
[17]Kholshevnikov K V, Vassiliev N N. On the distance function between twoKeplerian elliptic orbits [J]. Celestial Mechanics and Dynamical Astronomy,1999(75):75~83.
[18]Gronchi G F. An algebraic method to compute the critical points of the distancefunction between two Keplerian orbits [J]. Celestial Mechanics and DynamicalAstronomy,2005(93):295~329.
[19]Gronchi G F, Tommei G. On the uncertainty of the minimal distance between twoconfocal Keplerian orbits [J]. Discrete and Continuous Dynamical Systems-SeriesB,2007,7(4):755~778.
[20]Murison M A, Munteanu A. On the distance function between two confocalKeplerian orbits [J].2006.
[21]Armellin R, Di Lizia P, Berz M, et al. Computing the critical points of the distancefunction between two Keplerian orbits via rigorous global optimization [J].Celestial Mechanics and Dynamical Astronomy,2010(107):377~395.
[22]Chen L, Zhou B Z, Han L. An analytic method of collision detection for activespacecrafts [C]. The55th International Astronautical Congress, Vancouver,Canada,2004.
[23]Alfano S, Negron D. Determining satellite close approach [J]. Journal of theAstronautical Sciences,1993,41(2):217~225.
[24]Alfano S. Determining satellite close approach, Part II [J]. Journal of theAstronautical Sciences,1994,42(2):143~152.
[25]Vallado D A. Fundamentals of astrodynamics and applications [M]. Second Edition.El Segundo: Microcosm Press,2004.
[26]李鉴,肖业伦.一种改进的空间目标接近分析快速算法[J].航空学报,2007,28(增刊): s42~s48.
[27]Alarcon-Rodriguez J R, Martinez-Fadrique F M, Klinkrad H. Collision riskassessment with a smart sieve method [C]. Proceedings of joint ESA/NASASpace-Flight Safety Conference, Noordwijk, Netherlands,2002.6.
[28]Klinkrad H, Alarcon J R, Sanchez N. Collision avoidance for operational ESAsatellite [C]. Proceedings of the Fourth European Conference on Space Debris,Darmstadt, Germany,2005.4.
[29]Klinkrad H. Space debris–models and risk analysis [M]. New York:Springer-Praxis,2006.
[30]汪颢,黄海.在轨物体接近算法研究[J].宇航学报,2008,29(6):1747~1751.
[31]Faulds A L, Spencer D B. Satellite close-approach filtering using geneticalgorithms [J]. Journal of Spacecraft and Rockets,2003,40(2):248~252.
[32]Gelb A, Warren R S. Direct statistical analysis of nonlinear system–CADET [J].AIAA Guidance and Control Conference,1972:72~875.
[33]Taylor J H, Price C F. Direct statistical analysis of missile guidance systems viaCADET [R]. Massachusetts: Analytic Sciences Corporation,1974.8.
[34]Taylor J H. Handbook for the direct statistical analysis of missile guidance systemvia CADET [R]. Massachusetts: Analytic Sciences Corporation,1975.5.
[35]Liang L B, Luo Y Z, Zhang J, Li H Y, Tang G J. Rendezvous-Phasing errorspropagation using quasi-linearization method[C]. AIAA2010-7594. AIAAGuidance, Navigation, and Control Conference, Toronto, Ontario Canada,2010.9.
[36]梁立波,罗亚中,杏建军,唐国金.基于协方差分析描述函数法的非线性交会精度分析[J].系统工程与电子技术,2010,32(9):1977~1981.
[37]Vallado D A. A preliminary analysis of state vector prediction accuracy. AAS07-358. AAS/AIAA Space Flight Mechanics Meeting, Sedona, Arizona,28January-01February2007.
[38]Vallado D A. An analysis of state vector propagation using differing flightdynamics programs [C]. AAS05-199. AAS/AIAA Space Flight MechanicsConference, Copper Mountain, Colorado,2005.1.
[39]Chan J C, Navarro D. Comparisons of NORAD Two-line elements withINTELSAT orbital elements [C]. Proceedings of the Third European Conferenceon Space Debris, Darmstadt, Germany,2001.3.
[40]Kelso T S. Validation of SGP4and IS-GPS-200D against GPS precisionephemerides [C]. AAS07-127. AAS/AIAA Space Flight Mechanics Meeting,Sedona, Arizona,28January-01February2007.
[41]Boyce W H. Examination of NORAD TLE accuracy using the Iridium constellation[J]. Spaceflight mechanics2004. Advances in the Astronautical Science,2004,(119):2133~2142.
[42]Snow D, Kaya D. Element set prediction accuracy assessment [J]. Astrodynamics1999. Advances in the Astronautical Science,1999,(103):1937~1958.
[43]Muldoon A R, Elkaim G H. Improved orbit estimation ssing GPS measurements forconjunction analysis [C].
[44]Krag H, Klinkrad H, Alarcon-Rodriguez J R. Assessment of orbit uncertainties forcollision risk predictions at ESA [C]. Proceedings of Second IAASS Conference,Chicago, USA,2007.5.
[45]Flohrer T, Krag H, Klinkrad H. Assessment and categorization of TLE orbit errorsfor the US SSN catalogue [C]. Proceedings of the Advanced Maui Optical andSpace Surveillance Technologies Conference, Maui, US,2008.9.
[46]Wang R, Liu J, Zhang Q M. Propagation errors analysis of TLE data [J]. Advancesin Space Research,2009(43):1065~1069.
[47]Peterson G E, Gist R G, Oltrogge D L. Covariance generation for space objectsusing public data [J]. Advances in the Astronautical Sciences,2001(108I):201~214.
[48]Deguine B, Foliard J, Alby F, et al. Covariance modeling in satellite collision riskactivities [C]. AIAA2002-4631. AIAA/AAS Astrodynamics Specialist Conferenceand Exhibit, Monterey, California,2002.8.
[49]Osweiler V P. Covariance estimation and autocorrelation of NORAD Two-LineElement Sets [D]. Ohio: Air Force Institute of Technology, US Air University,2006.3.
[50]Hirose C, Kudo N, Matsuda I, et al. Evaluation of the TLE prediction errors forconjunction assessment [C]. IAC-10-A6.2.8. The61st International AstronauticalCongress, Prague, Czech,2010.9.
[51]Duncan M, Long A. Realistic covariance prediction for the Earth scienceconstellation [C]. AIAA2006-6293. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Keystone, Colorado,2006.8.
[52]Legendre P, Deguine B, Garmier R, et al. Two Line Element accuracy assessmentbased on a mixture of Gaussian laws [C]. AIAA2006-6518. AIAA/AASAstrodynamics Specialist Conference and Exhibit, Keystone, Colorado,2006.8.
[53]Legendre P, Garmier R, Prat G, et al. Improvement of the Two Line Elementaccuracy assessment based on a mixture of Gaussian laws [C]. AAS07-390.AAS/AIAA Astrodynamics Specialist Conference, Mackinac Island, Michigan,2007.8.
[54]Legendre P, Garmier R, Revelin B, et al. Improvement of the TLE accuracy modelbased on a Gaussian mixture depending on the propagation duration [C]. AIAA2008-6772. AIAA Guidance, Navigation, and Control Conference and Exhibit,2008.
[55]Levit C, Marshall W. Improved orbit predictions using Two-Line Elements [J].Advances in Space Research,2011,(47):1107~1115.
[56]Muldoon A R, Elkaim G H, Rickard I F, et al. Improved orbital debris trajectoryestimation based on sequential TLE processing [C]. IAC-09.A6.2.9. The60thInternational Astronautical Congress, Daejeon, R. Korea,2009.10.
[57]Matney M J, Anz-Meador P, Foster J L. Covariance correlations in collisionavoidance probability calculations [J]. Advances in Space Research,2004,(34):1109~1114.
[58]Coppola V T, Woodburn J, Hujsak R. Effects of cross correlated covariance onspacecraft collision probability [C]. AAS04-181. The AAS/AIAA Space FlightMechanics Meeting,2004.
[59]Foster J L, Estes H S. A parametric analysis of orbital debris collision probabilityand maneuver rate for space vehicles [R]. NASA/JSC-25898. Houston: NASAJohnson Space Flight Center,1992.8.
[60]Patera R P. General method for calculating satellite collision probability [J]. Journalof Guidance, Control, and Dynamics,2001,24(4):716~722.
[61]Patera R P. Quick method to determine long-term orbital collision risk [C]. AIAA2002-1809. SatMax2002-Satellite Performance Workshop, Arlington, VA,2002.4.
[62]Patera R P. Conventional form of the collision probability integral for arbitraryspace vehicle shape [C]. AIAA2004-5218. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Providence, Rhode Island,2004.8.
[63]Patera R P. Calculating collision probability for arbitrary space-vehicle shapes vianumerical quadrature [J]. Journal of Guidance, Control, and Dynamics,2005,28(6):1326~1328.
[64]Alfano S. Accommodating rectangular objects in probability calculations [C].AIAA2004-5217. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Providence, Rhode Island,2004.8.
[65]Alfano S. A numerical implementation of spherical object collision probability [J].The Journal of the Astronautical Sciences,2005,53(1):103~109.
[66]Alfano S. Satellite collision probability enhancements [J]. Journal of Guidance,Control and Dynamics,2006,29(3):588~512.
[67]Alfano S. Beta conjunction analysis tool [C]. AAS07393. AAS/AIAA SpaceFlight Mechanics Meeting, Sedona, Arizona,28January-01February2007.
[68]Chan F K. Spacecraft collision probability [M]. El Segundo: The Aerospace Press,2008.
[69]Chan F K. Collision probability analyses for Earth-orbiting satellites [J]. Advancesin the Astronautical Sciences,1997(96):1033~1048.
[70]Chan F K. Determination of minimum spacecraft separation at conjunction [C].
[71]Chan F K. Improved analytical expressions for computing spacecraft collisionprobabilities [C]. AAS03-184. AAS/AIAA Space Flight Mechanics Meeting,Ponce, Puerto,2003.2.
[72]Akella M R, Alfriend K T. The probability of collision between space objects [J].Journal of Guidance, Control and Dynamics,2000,23(5):769~772.
[73]Alfano S. Review of conjunction probability methods for short-term encounters [C].AAS07-148. AAS/AIAA Space Flight Mechanics Meeting, Sedona, Arizona,28January-01February2007.
[74]CCSDS Secretariat. Draft recommendation for space data system standards:Conjunction Data Message. Draft recommended standard CCSDS508.0-R-1,2012.
[75]王华.交会对接的控制与轨迹安全[D].长沙:国防科学技术大学研究生院博士学位论文,2007.10.
[76]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.
[77]王华,李海阳,唐国金.视场约束的交会对接V-bar撤离控制研究[J].宇航学报,2007,28(1):28~32.
[78]王华,唐国金,李海阳.基于碰撞概率的交会对接近距离导引段的轨迹安全[J].宇航学报,2007,28(3):648~652.
[79]王华,李海阳,唐国金.交会对接近距离导引段的轨迹安全[J].宇航学报,2007,28(6):1554~1558.
[80]王华,李海阳,唐国金.基于碰撞概率的交会对接最优碰撞规避机动[J].宇航学报,2008,29(1):220~223.
[81]程陶,刘静,王荣兰,于有成.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.
[82]程陶.编目空间碎片的碰撞概率方法研究及应用[D].北京:中国科学院研究生院硕士学位论文,2006.6.
[83]冯昊.空间碎片碰撞概率及其阈值分析和研究[D].北京:中国科学院研究生院硕士学位论文,2008.6.
[84]杨旭.空间碎片碰撞概率及其敏感度分析研究[D].北京:中国科学院研究生院硕士学位论文,2010.6.
[85]张明选.航天器碰撞概率的计算方法研究[D].哈尔滨:哈尔滨工业大学硕士学位论文,2010.6.
[86]吴波.空间目标交会期间碰撞概率研究[D].郑州:信息工程大学硕士学位论文,2011.6.
[87]Chan F K. Short-term vs. long-term spacecraft encounters [C]. AIAA2004-5460.AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, RhodeIsland,2004.8.
[88]Chan F K. Spacecraft collision probability for long-term encounters [C]. AAS03-549. AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Montana,3-7Aug2003.
[89]Patera R P. Satellite collision probability for non-linear relative motion [J]. Journalof Guidance, Control, and Dynamics,2003,26(5):728~733.
[90]Patera R P. Collision probability for larger bodies having nonlinear relative motion[J]. Journal of Guidance, Control, and Dynamics,2006,29(6):1468~1471.
[91]Slater G L, Byram S M, Williams T W. Collision avoidance for satellites information flight [J]. Journal of Guidance, Control, and Dynamics,2006,29(5):1139~1146.
[92]Alfano S. Addressing nonlinear relative motion for spacecraft collision probability
[C]. AIAA2006-6760. AIAA/AAS Astrodynamics Specialist Conference andExhibit, Keystone, Colorado,2006.8.
[93]McKinley D P. Development of a nonlinear probability of collision tool for theEarth observing system [C]. AIAA/AAS Astrodynamics Specialist Conference andExhibit, Keystone, Colorado,2006.8.
[94]许晓丽,熊永清.非线性相对运动下空间碎片碰撞概率计算的研究[J].天文学报,2011,52(1):73~85.
[95]张戈.卫星编队飞行碰撞概率预测与避撞措施研究[D].大连:大连理工大学硕士学位论文,2008.6.
[96]Luo Y Z, Liang L B, Wang H, Tang G J. Quantitative performance for spacecraftrendezvous trajectory safety [J]. Journal of Guidance, Control, and Dynamics,2011,34(4):1264~1269.
[97]Liang L B, Luo Y Z, Tang G J, Li H Y. Safety-Optimal impulsive rendezvous withtrajectory uncertainties [C].61st International Astronautical Congress, Prague,Czech Republic,2010.10.
[98]梁立波,罗亚中,王华,唐国金.空间交会轨迹安全特性分析[J].国防科技大学学报,2010,32(5):17~22.
[99]梁立波.近距离导引段交会轨迹安全性的定量评价和设计优化方法[D].长沙:国防科学技术大学研究生院博士学位论文,2011.4.
[100]梁立波,罗亚中,王华,唐国金.空间交会轨迹安全性定量评价指标研究[J].宇航学报,2010,31(10):2239~2245.
[101] Bird D. Conjunction summary message guide [C].
[102] Alfriend K T, Akella M R, Frisbee J, et al. Probability of collision erroranalysis [J]. Space Debris,1999,(1):21~35.
[103] Jenkin A B. Effect of orbit data quality on the feasibility of collision riskmanagement [J]. Journal of Spacecraft and Rockets,2004,41(4):677~683.
[104] Newman L K, Duncan M. Establishment and implementation of a closeapproach evaluation and avoidance process for Earth observing system missions
[C]. AIAA2006-6291. AIAA/AAS Astrodynamics Specialist Conference andExhibit, Keystone, Colorado,21-24Aug2006.
[105] Frigm R C, Levi J A, Mantziaras D C. Assessment, planning, and executionconsiderations for conjunction risk assessment and mitigation operations [C].AIAA2010-1926. SpaceOps2010Conference: Delivering on the Dream,Huntsville, Alabama,25-30Apr2010.
[106] Alfano S. Relating Position Uncertainty to Maximum Conjunction Probability
[C]. AAS03-548. AAS/AIAA Astrodynamics Specialists Conference, Big Sky,Montana,2003.8.
[107] Alfano S. Determining Probability Upper Bounds for NEO Close Approaches
[C]. AIAA2004-1478.2004Planetary Defense Conference: Protecting Earth fromAsteroids, Orange County, California,2004.2
[108] Gottlieb R G, Sponaugle S J, Gaylor D E. Orbit Determination AccuracyRequirements for Collision Avoidance [C]. AAS01-181. AAS/AIAA Space FlightMechanics Meeting, Santa Barbara, California,2001.2.
[109] Peterson G E. NEO Orbit Uncertainties and their Effect on Risk Assessment
[C]. AIAA2004-1420.2004Planetary Defense Conference: Protecting Earth fromAsteroids, Orange County, California,2004.2
[110] Frigm R C. A single conjunction risk assessment metric: the F-value [C]. AAS09-373,9Aug2009.
[111] Chan F K. Spacecraft maneuvers to mitigate potential collision threats [C].AIAA2002-4629. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Monterey, California,2002.8.
[112] Leleux D, Spencer R, Zimmerman P, et al. Probability-based space shuttlecollision avoidance[C]. Space OPS2002Conference, Houston,2002.10.
[113] Peterson G E. Maneuver selection for probability reduction of near-circularorbit conjunction [C]. AIAA/AAS2002-4630. AIAA/AAS AstrodynamicsSpecialist Conference and Exhibit, Monterey, California,2002.8.
[114] Patera R P, Peterson G E. Space vehicle maneuver method to lower collisionrisk to an acceptable level [J]. Journal of Guidance, Control, and Dynamics,2003,26(2):233~237.
[115] Patera R P. Space vehicle conflict avoidance analysis [J]. Journal of Guidance,Control, and Dynamics,2007,30(2):492~498.
[116] Slater G L, Byram S M, Williams T W. Collision avoidance for satellites information flight [C]. AIAA2004-5216. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Providence, Rhode Island,2004.8.
[117] Kelly B D, Picciotto S D. Probability based optimal collision avoidancemaneuvers [C]. AIAA2005-6775. Space2005, Long Beach, California,2005.8.
[118] Alfano S. Collision avoidance maneuver planning tool [C]. AAS05-308.AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, California,2005.8.
[119] Sanchez-Ortiz N, Bello-Mora M, Klinkrad H. Collision avoidance maneuversduring spacecraft mission lifetime: risk reduction and required delta-V [J].Advances in Space Research,2006,(38):2107~2116.
[120] Kim H D, Kim H J. Optimal collision avoidance maneuver to maintain a LEOstation keeping [C]. IAC-10-A6.2.9. The61st International Astronautical Congress,Prague, Czech,2010.9.
[121] Chen L, Han L, Ma Z H. Approach to collision avoidance optimal maneuverswith perturbation analysis [C]. The57th International Astronautical Congress,Valencia, Spain,2006.10.
[122] Gavin R T. NASA’s orbital debris conjunction assessment and collisionavoidance strategy [R]. JSC-CN-19799. Houston: NASA Johnson Space CenterFlight Dynamics Division,2010.2.
[123] Newman L K. The NASA robotic conjunction assessment process: overviewand operational experiences [C]. IAC-08-A.6.2.6. The59th InternationalAstronautical Congress, IAC2008, Glasgow, Scotland,2008.9.
[124] Kelso T S, Alfano S. Satellite orbital conjunction reports assessing threateningencounters in space (SOCRATES)[C]. AAS05-124.15th AAS/AIAA SpaceFlight Mechanics Conference, Copper Mountain, Colorado,2005.1.
[125] Klinkrad H, Alarcon J R, Sanchez N. Collision avoidance for operational ESAsatellite [C]. Proceedings of the Fourth European Conference on Space Debris,Darmstadt, Germany,2005.4.
[126] Alarcon-Rodriguez J R, Martinez-Fadrique F M, Klinkrad H. Development of acollision risk assessment tool [J]. Advances in Space Research,2004,(34):1120~1124.
[127] Flohrer T, Krag H, Klinkrad, H. ESA’s process for the identification andassessment of high-risk conjunction events [J]. Advances in Space Research,2009,(44):355~363.
[128] Laporte F, Sasot E. Operational management of collision risks for LEOsatellites at CNES [C]. AIAA2008-3409. SpaceOps2008Conference, Heidelberg,Germany,2008.5.
[129] Aida S, Kirschner M, Wermuth M, et al. Collision avoidance operations forLEO satellites controlled by GSOC [C]. AIAA2010-2298. SpaceOps2010Conference, Huntsville, Alabama,2010.4.
[130] Matsuda I, Hirose C, Kudo N. The JAXA conjunction assessment process [C].AIAA2010-2039. SpaceOps2010Conference, Huntsville, Alabama,2010.4.
[131] Choi S J, Jung O C, Chung D W, etc. Operational results of automatedconjunction analysis system using in-house generated KOMPSAT-2ephemeris [C].AIAA2010-2297. SpaceOps2010Conference, Huntsville, Alabama,2010.4.
[132]杨乐平,朱彦伟,黄涣.航天器相对运动轨迹规划与控制[M].北京:国防工业出版社,2010.8.
[133]张玉锟.卫星编队飞行的动力学与控制技术研究[D].长沙:国防科学技术大学研究生院博士学位论文,2002.
[134]孟云鹤.近地轨道航天器编队飞行控制与应用研究[D].长沙:国防科学技术大学研究生院博士学位论文,2006.
[135]安雪滢.椭圆轨道航天器编队飞行动力学及应用研究[D].长沙:国防科学技术大学研究生院博士学位论文,2006.
[136]李俊峰,高云峰,宝音贺西.卫星编队飞行动力学与控制研究[J].力学与实践,2002,24(2):1~6.
[137]高云峰,宝音贺西,李俊峰.卫星编队飞行的动力学特性与相对轨道构形仿真[J].清华大学学报(自然科学版),2002,42(4):458~461.
[138] Sparks A. Satellites formationkeeping control in the presence of gravityperturbations [C]. Proceedings of the American Control Conference, Chicago,Illinois,2000.6.
[139] Alfriend K T, Schaub H, Gim D. Gravitational perturbations nonlinearity andcircular orbit assumption effects on formation flying control strategies [C]. AAS00-012. AAS Guidance and Control Conference, Breckenridge, CO,2000.2.
[140]何焱,谈细春.空间交会对接中Hill方程的计算误差特性研究[J].中国空间科学技术,1998,18(6):7~14.
[141]张振民,许滨,张珩.航天器相对运动与Hill解的适用性分析[J].中国空间科学技术,2002,22(2):1~8.
[142]张振民,林来兴.小卫星编队飞行动力学及其应用[J].航天控制,2002,(3):44~50.
[143]林来兴.卫星编队飞行动力学仿真及其应用[J].中国空间科学技术,2005,25(2):26~33.
[144] Lawden D F. Optimal trajectories for space navigation[M]. Butterworths,London,1963,79~86.
[145] Carter T E. New form for the optimal rendezvous equations near a Keplerianorbit[J]. Journal of Guidance,1990,13(1):183~186.
[146] Carter T E. Linearized impulsive rendezvous problem [J]. Journal ofOptimization Theory and Applications,1995,86(3):553~584.
[147] Inalhan G, Tillerson M, How J P. Relative dynamics and control of spacecraftformations in eccentric orbits [J]. Journal of Guidance, Control and Dynamics,2002,25(1):48~59.
[148] Yamanaka K, Ankersen F. New state transition matrix for relative motion on anarbitrary elliptical orbit[J]. Journal of Guidance, Control, and Dynamics,2002,25(1):60~66.
[149]岳晓奎,苑云霞.椭圆轨道相对动力学状态转移矩阵[J].中国空间科学技术,2011,(1):42~47.
[150]杏建军.编队卫星周期性相对运动轨道设计与构形保持研究[D].长沙:国防科学技术大学研究生院博士学位论文,2007.10.
[151] Xing J J, Tang G J, Xi X N, Li H Y. Satellite formation design and optimalstationkeeping considering nonlinearity and eccentricity [J]. Journal of Guidance,Control, and Dynamics,2007,30(5):1523~1528.
[152]杏建军,李海阳,唐国金,郗晓宁.非线性条件下编队卫星周期性相对运动条件[J].宇航学报,2006,27(3):359~362.
[153]杏建军,唐国金,郗晓宁,张翼,李海阳.椭圆参考轨道编队卫星非线性周期性相对运动条件[J].清华大学学报(自然科学版),2006,46(8):1462~1465.
[154] Xing J J, Tang G J, Li H Y. New method of the analytic periodic solution forspacecraft formation in elliptical orbits [C].57th International AstronauticalCongress, Spain,2006.
[155]杏建军,李海阳,唐国金,郗晓宁.利用数值优化技术设计周期性绕飞的编队轨道[J].国防科技大学学报,2006,28(1):13~16.
[156]杏建军,郗晓宁,王威等.双星编队星座相对状态自主确定[J].宇航学报,2003,24(3):254~258.
[157] Schaub H, Alfriend K T. Hybrid Cartesian and orbit element feedback law forformation flying spacecraft [J]. Journal of Guidance, Control, and Dynamics,2002,25(2):387~393.
[158] Schaub H. Relative orbit geometry through classical orbit element differences[J]. Journal of Guidance, Control, and Dynamics,2004,27(5):839~848.
[159]陈磊,韩蕾,白显宗,周伯昭.空间目标轨道力学与误差分析[M].北京:国防工业出版社,2010.11.
[160] Ghrist R W, Plakalovic D. Impact of non-Gaussian error volumes onconjunction assessment risk analysis [C]. AIAA2012-4965. AIAA/AASAstrodynamics Specialist Conference, Minneapolis, Minnesota,2012.8.
[161]白显宗.空间目标碰撞预警中的碰撞概率问题研究[D].长沙:国防科学技术大学研究生院硕士学位论文,2008.12.
[162]柳仲贵.卫星轨道误差的相关性[J].飞行器测控学报,2011,30(5):45~49.
[163]马振华.现代应用数学手册:概率统计与随机过程卷[M].北京:清华大学出版社,2000.7.
[164] Hodge V J, Austin J. A survey of outlier detection methodologies [J]. ArtificialIntelligence Review,2004,(22):85~126.
[165] Jefferys W H. A Fortran-based list processor for Poisson series [J]. CelestialMechanics,1970,(2):474~480.
[166] Cherniack J R. A more general system for Poisson series manipulation [J].Celestial Mechanics,1973,(7):107~121.
[167] Ivanova T. A new echeloned Poisson series processor (EPSP)[J]. CelestialMechanics and Dynamical Astronomy,2001,(80):167~176.
[168] Broucke R, Garthwaite K. A programming system for analytical seriesexpansions on a computer [J]. Celestial Mechanics,1969,(1):271~284.
[169] Henrard J. A survey of Poisson series processors [J]. Celestial Mechanics,1989,(45):245~253.
[170] San-Juan J F, Abada A. Algebraic and symbolic manipulation of Poisson series[J]. Journal of Symbolic Computation,2001,(32):565~572.
[171] Martínez M C, Navarro J F, Ferrándiz J M. An improved algorithm to computecircular functions of Poisson series [J]. Celestial Mechanics and DynamicalAstronomy,2007,(99):59~68.
[172] San-Juan J F, Abada A. Communications between the Poisson series processorPSPC and general scientific software [J]. Mathematics and Computers inSimulation,2001,(57):307~315.
[173] Fateman R J. On The multiplication of Poisson series [J]. Celestial Mechanics,1974,(10):243~247.
[174] Fekken A, Hilhorst D, Kerckhoffs E J H, Water P R. Parallel computer algebraon Poisson series [J]. Simulation Practice and Theory,1996,(4):219~243.
[175] Richardson D L. PARSEC: An interactive Poisson series processor for personalcomputing systems [J]. Celestial Mechanics,1989,(45):267~274.
[176] Liou J C. Satellite collision leaves significant debris clouds [J]. Orbital DebrisQuarterly News,2009,13(4):1~2.
[177]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.
[178]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.
[179] Liou J C. Accidental collisions of cataloged satellites identified [J]. OrbitalDebris Quarterly News,2005,9(2):1~2.
[180] Liou J C. ISS crew seeks safe haven during debris flyby [J]. Orbital DebrisQuarterly News,2009,13(4):3.
[181]白显宗,陈磊.空间目标碰撞概率的显式表达式及其影响因素分析[J].空间科学学报,2009,29(4):422~431.
[182] Bai X Z, Chen L, Tang G J. Explicit expression of collision probability forspace objects in arbitrary-shape orbits [C]. The61st International AstronauticalCongress, Prague, Czech, Sep27-Oct1,2010.
[183]白显宗,陈磊,唐国金.基于显式表达式的空间目标最大碰撞概率分析[J].宇航学报,2010,31(3):880-887.
[184]邱月明.基于安全性的交会对接近程段运动控制策略研究[D].长沙:国防科学技术大学研究生院硕士学位论文,2011.11.
[185]谭跃进,陈英武,易进先.系统工程原理[M].长沙:国防科技大学出版社,1999.
[186] Abernathy B, Harvey S, Surka D M, et al. The CAOS-D architecture forconjunction analysis [C]. AIAA2011-1434. Infotech@Aerospace2011, St. Louis,Missouri,2011.3.
[187] Escobar D, Paskowitz M, águeda A, et al. Efficient all vs. all collision riskanalyses [C].
[188] Escobar D, águeda A, Martín L, et al. Predicting collision risk for EuropeanSSA system with closeap [C]. The First European Space Surveillance Conference,Madrid, Spain,2011.6.
[189] Healy L M. Close conjunction detection on parallel computer [J]. Journal ofGuidance, Control, and Dynamics,1995,18(4):824~829.
[190] Mercurio M, Singla P, Patra A. A hierarchical tree code based approach forefficient conjunction analysis [C]. AIAA2012-5017. AIAA/AAS AstrodynamicsSpecialist Conference, Minneapolis, Minnesota,2012.8.
[191] Mockel M, Wiedemann C, Flegel S, et al. Using parallel computing for thedisplay and simulation of the space debris environment [J]. Advances in SpaceResearch,2011,(48):173~183.
[192]潘伟全.卫星碰撞分析预报系统高效并行化技术研究[D].长沙:国防科学技术大学研究生院硕士学位论文,2007.4.
[193] Grama A, Gupta A, Karypis G, et al. Introduction to Parallel Computing [M].2ed. Addison Wesley,2003.1.
[2] Crowther R. Space junk-protecting space for future generations [J]. Science,2002,(296):1241~1242.
[3] Liou J C, Johnson N L. Risk in space from orbiting debris [J]. Science,2006,(311):340~341.
[4]刘林.航天器轨道理论[M].北京:国防工业出版社,2000.6.
[5] Hoots F R, Roehrich R L. Space track report No.3: models for propagation ofNORAD element sets [R]. Peterson：Aerospace Defense Command，United StatesAir Force,1980:1~79.
[6] Hoots F R, Schumacher P W, Glover R A. History of analytical orbit modeling inthe U.S. space surveillance system [J]. Journal of Guidance, Control and Dynamic,2004,27(2):174~185.
[7] Vallado D A, Crawford P, Hujsak R. Revisiting spacetrack No.3[C]. AIAA2006-6753. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Keystone, Colorado,2006.8.
[8]李济生.航天器轨道确定[M].北京:国防工业出版社,2003.
[9] Vallado D A. An analysis of state vector propagation using differing flightdynamics programs [C]. AAS05-199.
[10]Fonte D J. Comparison of orbit propagation in the research and developmentGoddard trajectory determination system. AAS95-431.
[11]吴连大.人造卫星与空间碎片的轨道和探测[M].北京:中国科学技术出版社,2011.9.
[12]郑勤余,吴连大.卫星与空间碎片碰撞预警的快速算法[J].天文学报,2004,45(4):422~427.
[13]刘静,王荣兰,张宏博.空间碎片碰撞预警研究[J].空间科学学报,2004,24(6):462~469.
[14]Hoots F R, Crawford L L, Roehrich R L. An analytic method to determine futureclose approaches between satellites [J]. Celestial Mechanics,1984,33(2):143~158.
[15]秋宏兴,祝转民,吴连大.航天器碰撞分析方法研究[J].宇航学报,2005,26(3):257~261.
[16]Dybczynski P A, Jopek T J, Serafin R A. On the minimum distance between twoKeplerian orbits with a common focus [J]. Celestial Mechanics,1986(38):345~356.
[17]Kholshevnikov K V, Vassiliev N N. On the distance function between twoKeplerian elliptic orbits [J]. Celestial Mechanics and Dynamical Astronomy,1999(75):75~83.
[18]Gronchi G F. An algebraic method to compute the critical points of the distancefunction between two Keplerian orbits [J]. Celestial Mechanics and DynamicalAstronomy,2005(93):295~329.
[19]Gronchi G F, Tommei G. On the uncertainty of the minimal distance between twoconfocal Keplerian orbits [J]. Discrete and Continuous Dynamical Systems-SeriesB,2007,7(4):755~778.
[20]Murison M A, Munteanu A. On the distance function between two confocalKeplerian orbits [J].2006.
[21]Armellin R, Di Lizia P, Berz M, et al. Computing the critical points of the distancefunction between two Keplerian orbits via rigorous global optimization [J].Celestial Mechanics and Dynamical Astronomy,2010(107):377~395.
[22]Chen L, Zhou B Z, Han L. An analytic method of collision detection for activespacecrafts [C]. The55th International Astronautical Congress, Vancouver,Canada,2004.
[23]Alfano S, Negron D. Determining satellite close approach [J]. Journal of theAstronautical Sciences,1993,41(2):217~225.
[24]Alfano S. Determining satellite close approach, Part II [J]. Journal of theAstronautical Sciences,1994,42(2):143~152.
[25]Vallado D A. Fundamentals of astrodynamics and applications [M]. Second Edition.El Segundo: Microcosm Press,2004.
[26]李鉴,肖业伦.一种改进的空间目标接近分析快速算法[J].航空学报,2007,28(增刊): s42~s48.
[27]Alarcon-Rodriguez J R, Martinez-Fadrique F M, Klinkrad H. Collision riskassessment with a smart sieve method [C]. Proceedings of joint ESA/NASASpace-Flight Safety Conference, Noordwijk, Netherlands,2002.6.
[28]Klinkrad H, Alarcon J R, Sanchez N. Collision avoidance for operational ESAsatellite [C]. Proceedings of the Fourth European Conference on Space Debris,Darmstadt, Germany,2005.4.
[29]Klinkrad H. Space debris–models and risk analysis [M]. New York:Springer-Praxis,2006.
[30]汪颢,黄海.在轨物体接近算法研究[J].宇航学报,2008,29(6):1747~1751.
[31]Faulds A L, Spencer D B. Satellite close-approach filtering using geneticalgorithms [J]. Journal of Spacecraft and Rockets,2003,40(2):248~252.
[32]Gelb A, Warren R S. Direct statistical analysis of nonlinear system–CADET [J].AIAA Guidance and Control Conference,1972:72~875.
[33]Taylor J H, Price C F. Direct statistical analysis of missile guidance systems viaCADET [R]. Massachusetts: Analytic Sciences Corporation,1974.8.
[34]Taylor J H. Handbook for the direct statistical analysis of missile guidance systemvia CADET [R]. Massachusetts: Analytic Sciences Corporation,1975.5.
[35]Liang L B, Luo Y Z, Zhang J, Li H Y, Tang G J. Rendezvous-Phasing errorspropagation using quasi-linearization method[C]. AIAA2010-7594. AIAAGuidance, Navigation, and Control Conference, Toronto, Ontario Canada,2010.9.
[36]梁立波,罗亚中,杏建军,唐国金.基于协方差分析描述函数法的非线性交会精度分析[J].系统工程与电子技术,2010,32(9):1977~1981.
[37]Vallado D A. A preliminary analysis of state vector prediction accuracy. AAS07-358. AAS/AIAA Space Flight Mechanics Meeting, Sedona, Arizona,28January-01February2007.
[38]Vallado D A. An analysis of state vector propagation using differing flightdynamics programs [C]. AAS05-199. AAS/AIAA Space Flight MechanicsConference, Copper Mountain, Colorado,2005.1.
[39]Chan J C, Navarro D. Comparisons of NORAD Two-line elements withINTELSAT orbital elements [C]. Proceedings of the Third European Conferenceon Space Debris, Darmstadt, Germany,2001.3.
[40]Kelso T S. Validation of SGP4and IS-GPS-200D against GPS precisionephemerides [C]. AAS07-127. AAS/AIAA Space Flight Mechanics Meeting,Sedona, Arizona,28January-01February2007.
[41]Boyce W H. Examination of NORAD TLE accuracy using the Iridium constellation[J]. Spaceflight mechanics2004. Advances in the Astronautical Science,2004,(119):2133~2142.
[42]Snow D, Kaya D. Element set prediction accuracy assessment [J]. Astrodynamics1999. Advances in the Astronautical Science,1999,(103):1937~1958.
[43]Muldoon A R, Elkaim G H. Improved orbit estimation ssing GPS measurements forconjunction analysis [C].
[44]Krag H, Klinkrad H, Alarcon-Rodriguez J R. Assessment of orbit uncertainties forcollision risk predictions at ESA [C]. Proceedings of Second IAASS Conference,Chicago, USA,2007.5.
[45]Flohrer T, Krag H, Klinkrad H. Assessment and categorization of TLE orbit errorsfor the US SSN catalogue [C]. Proceedings of the Advanced Maui Optical andSpace Surveillance Technologies Conference, Maui, US,2008.9.
[46]Wang R, Liu J, Zhang Q M. Propagation errors analysis of TLE data [J]. Advancesin Space Research,2009(43):1065~1069.
[47]Peterson G E, Gist R G, Oltrogge D L. Covariance generation for space objectsusing public data [J]. Advances in the Astronautical Sciences,2001(108I):201~214.
[48]Deguine B, Foliard J, Alby F, et al. Covariance modeling in satellite collision riskactivities [C]. AIAA2002-4631. AIAA/AAS Astrodynamics Specialist Conferenceand Exhibit, Monterey, California,2002.8.
[49]Osweiler V P. Covariance estimation and autocorrelation of NORAD Two-LineElement Sets [D]. Ohio: Air Force Institute of Technology, US Air University,2006.3.
[50]Hirose C, Kudo N, Matsuda I, et al. Evaluation of the TLE prediction errors forconjunction assessment [C]. IAC-10-A6.2.8. The61st International AstronauticalCongress, Prague, Czech,2010.9.
[51]Duncan M, Long A. Realistic covariance prediction for the Earth scienceconstellation [C]. AIAA2006-6293. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Keystone, Colorado,2006.8.
[52]Legendre P, Deguine B, Garmier R, et al. Two Line Element accuracy assessmentbased on a mixture of Gaussian laws [C]. AIAA2006-6518. AIAA/AASAstrodynamics Specialist Conference and Exhibit, Keystone, Colorado,2006.8.
[53]Legendre P, Garmier R, Prat G, et al. Improvement of the Two Line Elementaccuracy assessment based on a mixture of Gaussian laws [C]. AAS07-390.AAS/AIAA Astrodynamics Specialist Conference, Mackinac Island, Michigan,2007.8.
[54]Legendre P, Garmier R, Revelin B, et al. Improvement of the TLE accuracy modelbased on a Gaussian mixture depending on the propagation duration [C]. AIAA2008-6772. AIAA Guidance, Navigation, and Control Conference and Exhibit,2008.
[55]Levit C, Marshall W. Improved orbit predictions using Two-Line Elements [J].Advances in Space Research,2011,(47):1107~1115.
[56]Muldoon A R, Elkaim G H, Rickard I F, et al. Improved orbital debris trajectoryestimation based on sequential TLE processing [C]. IAC-09.A6.2.9. The60thInternational Astronautical Congress, Daejeon, R. Korea,2009.10.
[57]Matney M J, Anz-Meador P, Foster J L. Covariance correlations in collisionavoidance probability calculations [J]. Advances in Space Research,2004,(34):1109~1114.
[58]Coppola V T, Woodburn J, Hujsak R. Effects of cross correlated covariance onspacecraft collision probability [C]. AAS04-181. The AAS/AIAA Space FlightMechanics Meeting,2004.
[59]Foster J L, Estes H S. A parametric analysis of orbital debris collision probabilityand maneuver rate for space vehicles [R]. NASA/JSC-25898. Houston: NASAJohnson Space Flight Center,1992.8.
[60]Patera R P. General method for calculating satellite collision probability [J]. Journalof Guidance, Control, and Dynamics,2001,24(4):716~722.
[61]Patera R P. Quick method to determine long-term orbital collision risk [C]. AIAA2002-1809. SatMax2002-Satellite Performance Workshop, Arlington, VA,2002.4.
[62]Patera R P. Conventional form of the collision probability integral for arbitraryspace vehicle shape [C]. AIAA2004-5218. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Providence, Rhode Island,2004.8.
[63]Patera R P. Calculating collision probability for arbitrary space-vehicle shapes vianumerical quadrature [J]. Journal of Guidance, Control, and Dynamics,2005,28(6):1326~1328.
[64]Alfano S. Accommodating rectangular objects in probability calculations [C].AIAA2004-5217. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Providence, Rhode Island,2004.8.
[65]Alfano S. A numerical implementation of spherical object collision probability [J].The Journal of the Astronautical Sciences,2005,53(1):103~109.
[66]Alfano S. Satellite collision probability enhancements [J]. Journal of Guidance,Control and Dynamics,2006,29(3):588~512.
[67]Alfano S. Beta conjunction analysis tool [C]. AAS07393. AAS/AIAA SpaceFlight Mechanics Meeting, Sedona, Arizona,28January-01February2007.
[68]Chan F K. Spacecraft collision probability [M]. El Segundo: The Aerospace Press,2008.
[69]Chan F K. Collision probability analyses for Earth-orbiting satellites [J]. Advancesin the Astronautical Sciences,1997(96):1033~1048.
[70]Chan F K. Determination of minimum spacecraft separation at conjunction [C].
[71]Chan F K. Improved analytical expressions for computing spacecraft collisionprobabilities [C]. AAS03-184. AAS/AIAA Space Flight Mechanics Meeting,Ponce, Puerto,2003.2.
[72]Akella M R, Alfriend K T. The probability of collision between space objects [J].Journal of Guidance, Control and Dynamics,2000,23(5):769~772.
[73]Alfano S. Review of conjunction probability methods for short-term encounters [C].AAS07-148. AAS/AIAA Space Flight Mechanics Meeting, Sedona, Arizona,28January-01February2007.
[74]CCSDS Secretariat. Draft recommendation for space data system standards:Conjunction Data Message. Draft recommended standard CCSDS508.0-R-1,2012.
[75]王华.交会对接的控制与轨迹安全[D].长沙:国防科学技术大学研究生院博士学位论文,2007.10.
[76]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.
[77]王华,李海阳,唐国金.视场约束的交会对接V-bar撤离控制研究[J].宇航学报,2007,28(1):28~32.
[78]王华,唐国金,李海阳.基于碰撞概率的交会对接近距离导引段的轨迹安全[J].宇航学报,2007,28(3):648~652.
[79]王华,李海阳,唐国金.交会对接近距离导引段的轨迹安全[J].宇航学报,2007,28(6):1554~1558.
[80]王华,李海阳,唐国金.基于碰撞概率的交会对接最优碰撞规避机动[J].宇航学报,2008,29(1):220~223.
[81]程陶,刘静,王荣兰,于有成.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.
[82]程陶.编目空间碎片的碰撞概率方法研究及应用[D].北京:中国科学院研究生院硕士学位论文,2006.6.
[83]冯昊.空间碎片碰撞概率及其阈值分析和研究[D].北京:中国科学院研究生院硕士学位论文,2008.6.
[84]杨旭.空间碎片碰撞概率及其敏感度分析研究[D].北京:中国科学院研究生院硕士学位论文,2010.6.
[85]张明选.航天器碰撞概率的计算方法研究[D].哈尔滨:哈尔滨工业大学硕士学位论文,2010.6.
[86]吴波.空间目标交会期间碰撞概率研究[D].郑州:信息工程大学硕士学位论文,2011.6.
[87]Chan F K. Short-term vs. long-term spacecraft encounters [C]. AIAA2004-5460.AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, RhodeIsland,2004.8.
[88]Chan F K. Spacecraft collision probability for long-term encounters [C]. AAS03-549. AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Montana,3-7Aug2003.
[89]Patera R P. Satellite collision probability for non-linear relative motion [J]. Journalof Guidance, Control, and Dynamics,2003,26(5):728~733.
[90]Patera R P. Collision probability for larger bodies having nonlinear relative motion[J]. Journal of Guidance, Control, and Dynamics,2006,29(6):1468~1471.
[91]Slater G L, Byram S M, Williams T W. Collision avoidance for satellites information flight [J]. Journal of Guidance, Control, and Dynamics,2006,29(5):1139~1146.
[92]Alfano S. Addressing nonlinear relative motion for spacecraft collision probability
[C]. AIAA2006-6760. AIAA/AAS Astrodynamics Specialist Conference andExhibit, Keystone, Colorado,2006.8.
[93]McKinley D P. Development of a nonlinear probability of collision tool for theEarth observing system [C]. AIAA/AAS Astrodynamics Specialist Conference andExhibit, Keystone, Colorado,2006.8.
[94]许晓丽,熊永清.非线性相对运动下空间碎片碰撞概率计算的研究[J].天文学报,2011,52(1):73~85.
[95]张戈.卫星编队飞行碰撞概率预测与避撞措施研究[D].大连:大连理工大学硕士学位论文,2008.6.
[96]Luo Y Z, Liang L B, Wang H, Tang G J. Quantitative performance for spacecraftrendezvous trajectory safety [J]. Journal of Guidance, Control, and Dynamics,2011,34(4):1264~1269.
[97]Liang L B, Luo Y Z, Tang G J, Li H Y. Safety-Optimal impulsive rendezvous withtrajectory uncertainties [C].61st International Astronautical Congress, Prague,Czech Republic,2010.10.
[98]梁立波,罗亚中,王华,唐国金.空间交会轨迹安全特性分析[J].国防科技大学学报,2010,32(5):17~22.
[99]梁立波.近距离导引段交会轨迹安全性的定量评价和设计优化方法[D].长沙:国防科学技术大学研究生院博士学位论文,2011.4.
[100]梁立波,罗亚中,王华,唐国金.空间交会轨迹安全性定量评价指标研究[J].宇航学报,2010,31(10):2239~2245.
[101] Bird D. Conjunction summary message guide [C].
[102] Alfriend K T, Akella M R, Frisbee J, et al. Probability of collision erroranalysis [J]. Space Debris,1999,(1):21~35.
[103] Jenkin A B. Effect of orbit data quality on the feasibility of collision riskmanagement [J]. Journal of Spacecraft and Rockets,2004,41(4):677~683.
[104] Newman L K, Duncan M. Establishment and implementation of a closeapproach evaluation and avoidance process for Earth observing system missions
[C]. AIAA2006-6291. AIAA/AAS Astrodynamics Specialist Conference andExhibit, Keystone, Colorado,21-24Aug2006.
[105] Frigm R C, Levi J A, Mantziaras D C. Assessment, planning, and executionconsiderations for conjunction risk assessment and mitigation operations [C].AIAA2010-1926. SpaceOps2010Conference: Delivering on the Dream,Huntsville, Alabama,25-30Apr2010.
[106] Alfano S. Relating Position Uncertainty to Maximum Conjunction Probability
[C]. AAS03-548. AAS/AIAA Astrodynamics Specialists Conference, Big Sky,Montana,2003.8.
[107] Alfano S. Determining Probability Upper Bounds for NEO Close Approaches
[C]. AIAA2004-1478.2004Planetary Defense Conference: Protecting Earth fromAsteroids, Orange County, California,2004.2
[108] Gottlieb R G, Sponaugle S J, Gaylor D E. Orbit Determination AccuracyRequirements for Collision Avoidance [C]. AAS01-181. AAS/AIAA Space FlightMechanics Meeting, Santa Barbara, California,2001.2.
[109] Peterson G E. NEO Orbit Uncertainties and their Effect on Risk Assessment
[C]. AIAA2004-1420.2004Planetary Defense Conference: Protecting Earth fromAsteroids, Orange County, California,2004.2
[110] Frigm R C. A single conjunction risk assessment metric: the F-value [C]. AAS09-373,9Aug2009.
[111] Chan F K. Spacecraft maneuvers to mitigate potential collision threats [C].AIAA2002-4629. AIAA/AAS Astrodynamics Specialist Conference and Exhibit,Monterey, California,2002.8.
[112] Leleux D, Spencer R, Zimmerman P, et al. Probability-based space shuttlecollision avoidance[C]. Space OPS2002Conference, Houston,2002.10.
[113] Peterson G E. Maneuver selection for probability reduction of near-circularorbit conjunction [C]. AIAA/AAS2002-4630. AIAA/AAS AstrodynamicsSpecialist Conference and Exhibit, Monterey, California,2002.8.
[114] Patera R P, Peterson G E. Space vehicle maneuver method to lower collisionrisk to an acceptable level [J]. Journal of Guidance, Control, and Dynamics,2003,26(2):233~237.
[115] Patera R P. Space vehicle conflict avoidance analysis [J]. Journal of Guidance,Control, and Dynamics,2007,30(2):492~498.
[116] Slater G L, Byram S M, Williams T W. Collision avoidance for satellites information flight [C]. AIAA2004-5216. AIAA/AAS Astrodynamics SpecialistConference and Exhibit, Providence, Rhode Island,2004.8.
[117] Kelly B D, Picciotto S D. Probability based optimal collision avoidancemaneuvers [C]. AIAA2005-6775. Space2005, Long Beach, California,2005.8.
[118] Alfano S. Collision avoidance maneuver planning tool [C]. AAS05-308.AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, California,2005.8.
[119] Sanchez-Ortiz N, Bello-Mora M, Klinkrad H. Collision avoidance maneuversduring spacecraft mission lifetime: risk reduction and required delta-V [J].Advances in Space Research,2006,(38):2107~2116.
[120] Kim H D, Kim H J. Optimal collision avoidance maneuver to maintain a LEOstation keeping [C]. IAC-10-A6.2.9. The61st International Astronautical Congress,Prague, Czech,2010.9.
[121] Chen L, Han L, Ma Z H. Approach to collision avoidance optimal maneuverswith perturbation analysis [C]. The57th International Astronautical Congress,Valencia, Spain,2006.10.
[122] Gavin R T. NASA’s orbital debris conjunction assessment and collisionavoidance strategy [R]. JSC-CN-19799. Houston: NASA Johnson Space CenterFlight Dynamics Division,2010.2.
[123] Newman L K. The NASA robotic conjunction assessment process: overviewand operational experiences [C]. IAC-08-A.6.2.6. The59th InternationalAstronautical Congress, IAC2008, Glasgow, Scotland,2008.9.
[124] Kelso T S, Alfano S. Satellite orbital conjunction reports assessing threateningencounters in space (SOCRATES)[C]. AAS05-124.15th AAS/AIAA SpaceFlight Mechanics Conference, Copper Mountain, Colorado,2005.1.
[125] Klinkrad H, Alarcon J R, Sanchez N. Collision avoidance for operational ESAsatellite [C]. Proceedings of the Fourth European Conference on Space Debris,Darmstadt, Germany,2005.4.
[126] Alarcon-Rodriguez J R, Martinez-Fadrique F M, Klinkrad H. Development of acollision risk assessment tool [J]. Advances in Space Research,2004,(34):1120~1124.
[127] Flohrer T, Krag H, Klinkrad, H. ESA’s process for the identification andassessment of high-risk conjunction events [J]. Advances in Space Research,2009,(44):355~363.
[128] Laporte F, Sasot E. Operational management of collision risks for LEOsatellites at CNES [C]. AIAA2008-3409. SpaceOps2008Conference, Heidelberg,Germany,2008.5.
[129] Aida S, Kirschner M, Wermuth M, et al. Collision avoidance operations forLEO satellites controlled by GSOC [C]. AIAA2010-2298. SpaceOps2010Conference, Huntsville, Alabama,2010.4.
[130] Matsuda I, Hirose C, Kudo N. The JAXA conjunction assessment process [C].AIAA2010-2039. SpaceOps2010Conference, Huntsville, Alabama,2010.4.
[131] Choi S J, Jung O C, Chung D W, etc. Operational results of automatedconjunction analysis system using in-house generated KOMPSAT-2ephemeris [C].AIAA2010-2297. SpaceOps2010Conference, Huntsville, Alabama,2010.4.
[132]杨乐平,朱彦伟,黄涣.航天器相对运动轨迹规划与控制[M].北京:国防工业出版社,2010.8.
[133]张玉锟.卫星编队飞行的动力学与控制技术研究[D].长沙:国防科学技术大学研究生院博士学位论文,2002.
[134]孟云鹤.近地轨道航天器编队飞行控制与应用研究[D].长沙:国防科学技术大学研究生院博士学位论文,2006.
[135]安雪滢.椭圆轨道航天器编队飞行动力学及应用研究[D].长沙:国防科学技术大学研究生院博士学位论文,2006.
[136]李俊峰,高云峰,宝音贺西.卫星编队飞行动力学与控制研究[J].力学与实践,2002,24(2):1~6.
[137]高云峰,宝音贺西,李俊峰.卫星编队飞行的动力学特性与相对轨道构形仿真[J].清华大学学报(自然科学版),2002,42(4):458~461.
[138] Sparks A. Satellites formationkeeping control in the presence of gravityperturbations [C]. Proceedings of the American Control Conference, Chicago,Illinois,2000.6.
[139] Alfriend K T, Schaub H, Gim D. Gravitational perturbations nonlinearity andcircular orbit assumption effects on formation flying control strategies [C]. AAS00-012. AAS Guidance and Control Conference, Breckenridge, CO,2000.2.
[140]何焱,谈细春.空间交会对接中Hill方程的计算误差特性研究[J].中国空间科学技术,1998,18(6):7~14.
[141]张振民,许滨,张珩.航天器相对运动与Hill解的适用性分析[J].中国空间科学技术,2002,22(2):1~8.
[142]张振民,林来兴.小卫星编队飞行动力学及其应用[J].航天控制,2002,(3):44~50.
[143]林来兴.卫星编队飞行动力学仿真及其应用[J].中国空间科学技术,2005,25(2):26~33.
[144] Lawden D F. Optimal trajectories for space navigation[M]. Butterworths,London,1963,79~86.
[145] Carter T E. New form for the optimal rendezvous equations near a Keplerianorbit[J]. Journal of Guidance,1990,13(1):183~186.
[146] Carter T E. Linearized impulsive rendezvous problem [J]. Journal ofOptimization Theory and Applications,1995,86(3):553~584.
[147] Inalhan G, Tillerson M, How J P. Relative dynamics and control of spacecraftformations in eccentric orbits [J]. Journal of Guidance, Control and Dynamics,2002,25(1):48~59.
[148] Yamanaka K, Ankersen F. New state transition matrix for relative motion on anarbitrary elliptical orbit[J]. Journal of Guidance, Control, and Dynamics,2002,25(1):60~66.
[149]岳晓奎,苑云霞.椭圆轨道相对动力学状态转移矩阵[J].中国空间科学技术,2011,(1):42~47.
[150]杏建军.编队卫星周期性相对运动轨道设计与构形保持研究[D].长沙:国防科学技术大学研究生院博士学位论文,2007.10.
[151] Xing J J, Tang G J, Xi X N, Li H Y. Satellite formation design and optimalstationkeeping considering nonlinearity and eccentricity [J]. Journal of Guidance,Control, and Dynamics,2007,30(5):1523~1528.
[152]杏建军,李海阳,唐国金,郗晓宁.非线性条件下编队卫星周期性相对运动条件[J].宇航学报,2006,27(3):359~362.
[153]杏建军,唐国金,郗晓宁,张翼,李海阳.椭圆参考轨道编队卫星非线性周期性相对运动条件[J].清华大学学报(自然科学版),2006,46(8):1462~1465.
[154] Xing J J, Tang G J, Li H Y. New method of the analytic periodic solution forspacecraft formation in elliptical orbits [C].57th International AstronauticalCongress, Spain,2006.
[155]杏建军,李海阳,唐国金,郗晓宁.利用数值优化技术设计周期性绕飞的编队轨道[J].国防科技大学学报,2006,28(1):13~16.
[156]杏建军,郗晓宁,王威等.双星编队星座相对状态自主确定[J].宇航学报,2003,24(3):254~258.
[157] Schaub H, Alfriend K T. Hybrid Cartesian and orbit element feedback law forformation flying spacecraft [J]. Journal of Guidance, Control, and Dynamics,2002,25(2):387~393.
[158] Schaub H. Relative orbit geometry through classical orbit element differences[J]. Journal of Guidance, Control, and Dynamics,2004,27(5):839~848.
[159]陈磊,韩蕾,白显宗,周伯昭.空间目标轨道力学与误差分析[M].北京:国防工业出版社,2010.11.
[160] Ghrist R W, Plakalovic D. Impact of non-Gaussian error volumes onconjunction assessment risk analysis [C]. AIAA2012-4965. AIAA/AASAstrodynamics Specialist Conference, Minneapolis, Minnesota,2012.8.
[161]白显宗.空间目标碰撞预警中的碰撞概率问题研究[D].长沙:国防科学技术大学研究生院硕士学位论文,2008.12.
[162]柳仲贵.卫星轨道误差的相关性[J].飞行器测控学报,2011,30(5):45~49.
[163]马振华.现代应用数学手册:概率统计与随机过程卷[M].北京:清华大学出版社,2000.7.
[164] Hodge V J, Austin J. A survey of outlier detection methodologies [J]. ArtificialIntelligence Review,2004,(22):85~126.
[165] Jefferys W H. A Fortran-based list processor for Poisson series [J]. CelestialMechanics,1970,(2):474~480.
[166] Cherniack J R. A more general system for Poisson series manipulation [J].Celestial Mechanics,1973,(7):107~121.
[167] Ivanova T. A new echeloned Poisson series processor (EPSP)[J]. CelestialMechanics and Dynamical Astronomy,2001,(80):167~176.
[168] Broucke R, Garthwaite K. A programming system for analytical seriesexpansions on a computer [J]. Celestial Mechanics,1969,(1):271~284.
[169] Henrard J. A survey of Poisson series processors [J]. Celestial Mechanics,1989,(45):245~253.
[170] San-Juan J F, Abada A. Algebraic and symbolic manipulation of Poisson series[J]. Journal of Symbolic Computation,2001,(32):565~572.
[171] Martínez M C, Navarro J F, Ferrándiz J M. An improved algorithm to computecircular functions of Poisson series [J]. Celestial Mechanics and DynamicalAstronomy,2007,(99):59~68.
[172] San-Juan J F, Abada A. Communications between the Poisson series processorPSPC and general scientific software [J]. Mathematics and Computers inSimulation,2001,(57):307~315.
[173] Fateman R J. On The multiplication of Poisson series [J]. Celestial Mechanics,1974,(10):243~247.
[174] Fekken A, Hilhorst D, Kerckhoffs E J H, Water P R. Parallel computer algebraon Poisson series [J]. Simulation Practice and Theory,1996,(4):219~243.
[175] Richardson D L. PARSEC: An interactive Poisson series processor for personalcomputing systems [J]. Celestial Mechanics,1989,(45):267~274.
[176] Liou J C. Satellite collision leaves significant debris clouds [J]. Orbital DebrisQuarterly News,2009,13(4):1~2.
[177]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.
[178]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.
[179] Liou J C. Accidental collisions of cataloged satellites identified [J]. OrbitalDebris Quarterly News,2005,9(2):1~2.
[180] Liou J C. ISS crew seeks safe haven during debris flyby [J]. Orbital DebrisQuarterly News,2009,13(4):3.
[181]白显宗,陈磊.空间目标碰撞概率的显式表达式及其影响因素分析[J].空间科学学报,2009,29(4):422~431.
[182] Bai X Z, Chen L, Tang G J. Explicit expression of collision probability forspace objects in arbitrary-shape orbits [C]. The61st International AstronauticalCongress, Prague, Czech, Sep27-Oct1,2010.
[183]白显宗,陈磊,唐国金.基于显式表达式的空间目标最大碰撞概率分析[J].宇航学报,2010,31(3):880-887.
[184]邱月明.基于安全性的交会对接近程段运动控制策略研究[D].长沙:国防科学技术大学研究生院硕士学位论文,2011.11.
[185]谭跃进,陈英武,易进先.系统工程原理[M].长沙:国防科技大学出版社,1999.
[186] Abernathy B, Harvey S, Surka D M, et al. The CAOS-D architecture forconjunction analysis [C]. AIAA2011-1434. Infotech@Aerospace2011, St. Louis,Missouri,2011.3.
[187] Escobar D, Paskowitz M, águeda A, et al. Efficient all vs. all collision riskanalyses [C].
[188] Escobar D, águeda A, Martín L, et al. Predicting collision risk for EuropeanSSA system with closeap [C]. The First European Space Surveillance Conference,Madrid, Spain,2011.6.
[189] Healy L M. Close conjunction detection on parallel computer [J]. Journal ofGuidance, Control, and Dynamics,1995,18(4):824~829.
[190] Mercurio M, Singla P, Patra A. A hierarchical tree code based approach forefficient conjunction analysis [C]. AIAA2012-5017. AIAA/AAS AstrodynamicsSpecialist Conference, Minneapolis, Minnesota,2012.8.
[191] Mockel M, Wiedemann C, Flegel S, et al. Using parallel computing for thedisplay and simulation of the space debris environment [J]. Advances in SpaceResearch,2011,(48):173~183.
[192]潘伟全.卫星碰撞分析预报系统高效并行化技术研究[D].长沙:国防科学技术大学研究生院硕士学位论文,2007.4.
[193] Grama A, Gupta A, Karypis G, et al. Introduction to Parallel Computing [M].2ed. Addison Wesley,2003.1.