卫星轨道确定模型误差的深度加权核估计方法

卫星动力学模型误差是客观存在的事实,动力学模型误差传递到轨道确定算法中构成部分形式未知的模型误差,并且与测量系统自身的系统误差和随机误差耦合在一起形成定轨模型误差,严重影响轨道确定精度.详细推导了存在动力学模型误差的轨道改进方程,对模型中能准确描述的部分建立了参数化模型,对不能准确描述的误差部分,建立了非参数模型.构建了部分线性轨道改进模型,利用二阶段估计法和核函数估计法对模型误差进行拟合估计,并在轨道改进中予以补偿.根据数据深度理论,建立了非参数模型误差的深度加权核估计方法,提高了模型误差估计的抗差性.最后结合天基空间目标监视系统进行了轨道确定仿真实验.实验结果表明,模型误差是影响轨道确定精度的重要因素,核函数估计法可以有效估计定轨中的模型误差,窗宽是提高模型估计精度的重要变量,通过深度加权处理可以明显提高核函数估计的抗差性,提高轨道确定精度.     

关于数值求解天体运动方程的几个问题

本文讨论三个问题：1.在采用各种非辛（Symplectic）的数值积分器积分天体运动方程时，截断误差将引起人为的能量耗散，这一问题是不能用简单地在相应的力模型中加进一个人为的阻力因子而得以解决的，被歪曲的能量（或数值轨道）必须在积分过程的每一步用能量关系来进行校正，此即能量控制方法．2.当摄动加速度涉及到坐标轴的旋转时，如何在各种积分器中采用能量控制方法．3.对于大偏心率轨道，用数值方法求解相应运动方程时，积分步长必须随运动天体与中心天体之间的距离变化而改变，显然，这对所有积分器都是不方便的，特别是多步积分器．本义给出了一种步长均匀化的处理，可以使上述大偏心率轨道积分问题按定步长计算．     

基于自适应稳健最小p范数估计的轨道确定方法研究

当测轨数据误差不服从正态分布时,传统的最小二乘(LSE)轨道确定方法将不是最优的.为了获得高精度的定轨结果,一种可行的策略是采用基于最小p范数(Lp)的轨道确定方法.通过分析Lp估计的相关性质,得出普通Lp估计不具有良好的抗差性的结论.为抑制模型误差和异常值的影响,提出了基于数据深度加权的稳健最小p范数估计方法,并证明了相关性质,得出了其崩溃点可以达到1/2的结论.最后,通过残差分析和矩估计法自适应估计相关参数,使得估计达到最大效率.以天基空间目标监视系统为背景进行了仿真试验.结果表明,当观测数据存在系统误差或异常值时,或者当目标动力学模型存在误差或者天基观测平台存在系统误差时,即使观测数据服从正态分布,LSE也不是最优的,在这种意义下自适应稳健Lp估计轨道确定方法比传统轨道确定方法更加稳健,定轨精度也更高.     

小倾角LEO多星天基平台光学跟踪GEO精密定轨

针对地基卫星测控系统(Tracking Telemetry and Command, TT&C)系统对地球静止轨道(Geostation-\lk ary Earth Orbit, GEO)卫星在空间和时间覆盖上的局限性, 提出小倾角低地球轨道(Low Earth Orbit, LEO)多星组网天基平台对GEO卫星进行跟踪定轨的方法. 根据空间环境和光学可视条件对仿真数据进行筛选以模拟真实的观测场景, 利用光学测角数据, 使用数值方法对GEO卫星的轨道进行确定. 结果与参考轨道进行重叠对比, 在平台轨道精度5 m、测量精度5rq\rq、 定轨弧长12 h的情况下, 两颗LEO卫星对GEO卫星进行跟踪定轨的精度可达到千米量级, 4颗LEO卫星对GEO目标进行跟踪定轨的精度可达到百米量级. 随着LEO组网卫星数量的增加, 定轨精度得到了较大的提高.     

天文动力学方程数值积分中的一种有效变步法

利用积分曲线的曲率控制步长的技巧，使天文动力学方程数值解法的精度和速度有较大提高，这种方法适用于天体精密定轨以及一些精度要求高的常微分方程初值问题的数值积分。     

稳定化方法与后稳定化方法的比较

对BaumgaLrte的稳定化和Chin的后稳定化进行了详尽讨论与数值比较．用经典数值方法并结合这两种稳定化方式都能提高数值精度和改善数值稳定性．在最佳稳定参数下稳定化精度一般不等价于后稳定化．两者精度优劣并无常定．考虑到Baumgarte的稳定化使得数值积分的右函数更复杂和增加计算耗费,尤其是存在稳定参数最佳选取的麻烦,故推荐后稳定化投入实算．但值得注意的是用后稳定化与没有经过稳定化处理的经典积分器来比不宜扩大积分步长．     

辛方法的校正公式

1996年Wisdom等提出了对辛方法进行校正的概念和实践，现在继续对辛校正进行详尽讨论和数值比较，尤其对哈密顿函数可分解为一个主要部分和多个次要部分的一般情形，用Lie级数推导任意阶的各种辛算法的一次和二次辛校正公式并对一些算法给出具体的辛校正公式。又以日、木、土三体问题为模型进行数值实验，结果表明一次辛校正能提高精度，改善数值稳定性。计算效率也比较高，因而值得推荐使用，辛方法通常用大步长数值积分，这时二次辛校正并没有显著提高结果的精度，却大大增加了计算时间，不应予以推荐。     

基于历史TLE的空间目标轨道预报误差演化规律研究

北美防空司令部(North American Aerospace Defense Command, NORAD)发布的双行根数(Two Line Element, TLE)是广大航天工作者最常用的轨道根数,与其对应的轨道模型是SGP4/SDP4 (Simplified General Perturbation Version 4/Simplified Deep-space Perturbation Version 4)解析模型.由于TLE中并没有包含相应的轨道精度信息,编目轨道的应用范围受到很大的限制.基于Space-Track网站发布的历史TLE数据和配套的SGP4/SDP4动力学模型,采用定轨标预报的方法统计并生成了大量目标轨道的预报误差,通过对预报轨道的时间区间划分给出了每个目标的预报误差随预报时间变化的拟合系数,并进一步对不同类型轨道预报误差的演化规律和特征进行了分类讨论,给出了4种轨道类型目标的轨道预报误差随时间演化的平均解析模型,为拓展双行根数的应用提供有价值的参考.     

关于登月习行轨道数值积分的讨论

讨论了登月飞行轨道的数值积分问题，根据两个标准－－积分精度和计算CPU时间，我们选择了合适的积分器，同时还讨论了登月飞行轨道的数值积分的误差传播问题，指出了登月飞行轨道与人造地球卫星轨道的差别。     

多星混合资料的定轨

在空间目标光学观测资料定轨中常常会遇到多个目标的观测被标记为同一个目标的情况,由于包含了多个目标的数据,定轨过程无法收敛或者完全错误.从极大似然估计角度,采用EM(Expectation Maximum)方法提出一种将轨道改进和识别过程相互融合的处理方法,并在具体实现过程中给出一种稳健估计方法.数值模拟表明方法简便、有效、可行.采用站间差历元差的手段,对当前电离层变化值进行求解与预测,可以将电离层延时误差的变化控制在一定范围,满足频率传递的要求.     

The Long-term Error Estimation Method for the Numerical Integrations of Celestial Orbits

Numerical methods have become a very important type of tool for celestial mechanics, especially in the study of planetary ephemerides. The errors generated during the computation are hard to know beforehand when applying a certain numerical integrator to solve a certain orbit. In that case, it is not easy to design a certain integrator for a certain celestial case when the requirement of accuracy were extremely high or the time-span of the integration were extremely large. Especially when a fixed-step method is applied, the caution and effort it takes would always be tremendous in finding a suitable time-step, because it is about whether the accuracy and time-cost of the final result are acceptable. Thus, finding the best balance between efficiency and accuracy with the least time cost appeared to be a major obstruction in the face of both numerical integrator designers and their users. To solve this problem, we investigate the variation pattern of truncation error and the pattern of rounding error distributions with time-step and time-span of the integration. According to those patterns, we promote an error estimation method that could predict the distribution of rounding errors and the total truncation errors with any time-step at any time-spot with little experimental cost, and test it with the Adams-Cowell method in the calculation of circular periodic orbits. This error estimation method is expected to be applied to the comparison of the performance of different numerical integrators, and also it can be of great help for finding the best solution to certain cases of complex celestial orbits calculations.     

On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC) ⁿ Hermite scheme for planetary N -body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N -body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N -body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC) ⁿ Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)² Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N  = 100,  m  = 10²⁵ g planetesimal system with the rms eccentricity 〈 e ²〉^1/2 ≲0.03.     

Integration error over very long time spans

The main limit to the time span of a numerical integration of the planetary orbits is no longer set by the availability of computer resources, but rather by the accumulation of the integration error. By the latter we mean the difference between the computed orbit and the dynamical behaviour of the real physical system, whatever the causes. The analysis of these causes requires an interdisciplinary effort: there are physical model and parameters errors, algorithm and discretisation errors, rounding off errors and reliability problems in the computer hardware and system software, as well as instabilities in the dynamical system. We list all the sources of integration error we are aware of and discuss their relevance in determining the present limit to the time span of a meaningful integration of the orbit of the planets. At present this limit is of the order of 10⁸ years for the outer planets. We discuss in more detail the truncation error of multistep algorithms (when applied to eccentric orbits), the coefficient error, the method of Encke and the associated coordinate change error, the procedures used to test the numerical integration software and their limitations. Many problems remain open, including the one of a realistic statistical model of the rounding off error; at present, the latter can only be described by a semiempirical model based upon the simpleN ² formula (N=number of steps, =machine accuracy), with an unknown numerical coefficient which is determined only a posteriori.     

辛积分器中沿迹误差的一种补偿方法

辛积分器严格描述了一摄动Ｈａｍｉｌｔｏｎ系统的流，因而导致天体轨道的沿迹误差随时间呈线性增长趋势。本文利用这一特点，提出了一种对其沿迹误差进行估算的数值方法，从而达到了对数值结果进行沿迹误差补偿的目的，数值结果证实了此方法在较大积分步长和较长积分时间的数值计算中是有效的。     

Algorithmic regularization of the few-body problem

Using a modified leapfrog method as a basic mapping, we produce a new numerical integrator for the stellar dynamical few-body problem. We do not use coordinate transformation and the differential equations are not regularized, but the leapfrog algorithm gives regular results even for collision orbits. For this reason, application of extrapolation methods gives high precision. We compare the new integrator with several others and find it promising. Especially interesting is its efficiency for some potentials that differ from the Newtonian one at small distances.     

Behaviour of the SMF method for the numerical integration of satellite orbits

The SMF algorithms were recently developed by the authors as a multistep generalization of the ScheifeleG-functions one-step method. Like the last, the proposed codes integrate harmonic oscillations without truncation error and the perturbing parameter appears as a factor of that error when integrating perturbed oscillations. Therefore they seemed to be convenient for the accurate integration of orbital problems after the application of linearizing transformations, such as KS or BF. In this paper we present several numerical experiments concerning the propagation of Earth satellite orbits, that illustrate the performance of the the SMF method. In general, it provides greater accuracy than the usual standard algorithms for similar computational cost.     

Time regularization of an Adams-Moulton-Cowell algorithm

This paper deals with the Adams-Moulton-Cowell multistep integrator, as described by Oestwinter and Cohen (1972). In order to evaluate the accuracy of the method, we started to test it in the case of the unperturbed two-body motion; numerical instability may arise by integrating first order systems. The accuracy is improved by applying a Sundmann transformation of the independent variable. The algorithm is then modified such that the equations of pure keplerian motion are integrated with respect to the new independent variable without truncation error; numerical experiments show the considerable improvement of accuracy and the reduction of computing time for Keplerian motion.If terms of the disturbing function of the Earth are added to the central potential, the time-transformation is less effective. With a modification of this time-transformation as given by Moynot in 1971, it is possible to reduce the propagation of the truncation error in the J2 problem.     

Beware λ-truncation! Sample truncation and bias in luminosity calibration using trigonometric parallaxes

The common practice in luminosity calibration of sample truncation according to relative parallax error λ can lead to bias with indirect methods such as reduced parallaxes as well as with direct methods. This bias is not cancelled by the Lutz–Kelker corrections and in fact can be either negative or positive. Making the selection stricter can actually lead to a larger absolute amount of bias and lower accuracy in certain cases.
The degree to which this bias is present depends upon whether the sample is more nearly specified by the relative parallax error or by the limiting apparent magnitude when both limits formally apply; when the latter limit dominates it is absent. The difference between the means for the two extreme cases is what is customarily termed the Malmquist bias. However, it is not truly bias, but rather what we call here an offset .
For a sample to be effectively magnitude-limited, there is a lower bound imposed on the mean absolute magnitude which depends on the limiting magnitude. If a wide-ranging luminosity relation such as the Wilson–Bappu relation is to be calibrated, some portion of the relation may be magnitude-limited and the rest not. In that case there will be offsets between the different parts of the relation, including the transition region between the two extremes, as well as bias outside the magnitude-limited part.
Another, less common, practice is truncation according to weight, specifically with the reduced parallax method. Such truncation can also bias the calibration with one variant of the method. Indeed, the weighting scheme used with that variant introduces bias even without truncation.
For calibration it is probably best to use a general maximum likelihood method such as the grid method with a magnitude-limited sample and no limit on relative parallax error. The Malmquist shift could then be applied to obtain an estimate of the volume-limited mean.     

A Time-Transformed Leapfrog Scheme

We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included.     

可用于带耗散力卫星轨道长间隔积分的数值积分方法

本文分析了对称方法不适合带耗散力的卫星轨道长间隔积分的缺陷和本质原因,并针对这个问题,采用文所介绍的方法,构造并推荐了一组积分公式。