A Celestial Analytic Positioning Method by Stellar Horizon Atmospheric Refraction

This article, in allusion to the limitation of conventional stellar horizon atmospheric refraction based on orbital dynamics model and nonlinear Kalman filter in practical applications, proposes a new celestial analytic positioning method by stellar horizon atmospheric refraction for high-altitude flight vehicles, such as spacecraft, airplanes and ballistic missiles. First, by setting up the geometric connexion among the flight vehicle, the Earth and the altitude of starlight refraction, an expression for the relationship of starlight refraction angle and atmospheric density is deduced. Second, there are produced a novel measurement model of starlight refraction in a continuous range of altitudes (CRA) from 20 km to 50 km on the basis of the standard atmospheric data in stratosphere, and an empirical formula of stellar horizon atmospheric refraction in the same altitudes against the tangent altitude. Third, there is introduced a celestial analytic positioning algorithm, which uses the least square differential correction instead of nonlinear Kalman filter. The information about positions of a flight vehicle can be obtained directly by solving a set of nonlinear measurement equations. The stellar positioning algorithm adopts the characteristics of stellar horizon atmospheric refraction thereby removing needs for orbit dynamics models and priori knowledge of flight vehicles. The simulation results evidence the validity of the proposed stellar positioning algorithm.     

空间平台机动变轨自主导航研究

针对空间平台在高轨道机动变轨过程中自主导航的需求,采用了基于Kalman滤波器的捷联惯导与星敏感器的组合导航方案。结合Kalman滤波中协方差更新的误差分配分析方法,分析了影响空间平台状态估计误差的主要因素。采用适用于高轨道的球谐重力模型,运用STK工具包设计了变轨机动轨迹,将该轨迹应用于组合导航方案的仿真验证。仿真结果表明,量测噪声是影响空间平台姿态精度的主要因素,加速度计零偏对变轨过程速度精度有决定性影响,改善两者的精度可以实现空间平台机动变轨的高精度自主导航。     

Spacecraft orbit determination using GPS navigation solutions

The orbit determination using the GPS navigation solutions for the KOMPSAT-1 spacecraft has been studied. The Cowell method of special perturbation theories was employed to develop a precision orbit propagation, and the perturbations due to geopotential, the gravity of the Sun and the Moon, solid Earth tides, ocean tides, the Earth's dynamic polar motion, solar radiation pressure, and atmospheric drag were modeled. Specifically, the satellite box-wing macro model was applied to minimize the drag errors at low altitude. The estimation scheme consisted of an extended Kalman filter and Bayesian least square method. To investigate the applicability of the method to the KOMPSAT-1 spacecraft, the orbit determination was accomplished using the GPS navigation solutions for the TOPEX/POSEIDON and TAOS satellites. The orbit determination results were compared with NASA POE generated by global laser tracking. The position and velocity accuracy was estimated about 16∼7 m and 0.0157∼0.0074 m·s⁻¹ RMS, respectively, for the two satellites in the presence of SA. These results verify that an orbit determination scheme using GPS navigation solutions can provide the static orbit information and reduce conspicuously the position and velocity errors of navigation solutions. It can be suggested that the sequential and batch orbit determination using the GPS navigation solutions be the most appropriate method in the KOMPSAT-1 type mission.     

单通道测量条件下DORIS实时定轨方法研究

重点研究了单通道测量条件下的DORIS实时定轨方法。推导了实时定轨的扩展卡尔曼滤波（EKF）公式,仿真生成了47个测站的观测数据文件,分析了单通道测量条件下EKF滤波器的收敛性及其精度。仿真实验表明,仅有单个测站的观测数据时,滤波难以收敛;但在不同测站的观测数据切换时,滤波收敛效果显著。对于轨道高度为800km的卫星,初轨各轴位置误差小于500m、速度误差小于5m/s时,滤波平均收敛时间约55min;若各项系统误差能准确建模,实时轨道的位置精度将优于40cm。     

Determination des orbites des satellites artificiels

Summary The problem of the determination of the orbit of a celestial body is an old astronomical problem, of which solutions can be found in many classical books on elementary celestial mechanics. However, the introduction of new (radioelectric) means of observations for the artificial satellites have brought up new solutions of this old problem.The author reminds the definitions of the six elliptic elements of an orbit. The problem of their determination is usually divided into two separate steps: a preliminary orbit determination and the improvement of the preliminary elements. Two principal types of preliminary orbit determination exist: the Gaussian type, purely geometrical in which the positions of the body at two different times are determined, and the Laplacian type, of a more dynamical character, in which both position and velocity vectors are found for a given time. The improvement of the preliminary elements is usually obtained by a numerical solution of equations of variations of the elements, minimizing the sum of the squares of the angular distances between the computed and the observed points. The elements on which these variations are applied can be quite various. The coefficients of the equations are usually obtained numerically, although their analytical expressions can also be derived.Most of the modification proposed to the classical methods of orbit determination are more technical improvements in connection with the computation on electronic machines. The most interesting modifications are inspired by the fact that time measurements are less precise than the position (Batrakov, Iszak), the effects of this difference in errors are discussed in this paper. In some cases, using Laplace's method, no preliminary orbit is computed, and all the observations can be used at once (Barlier, Kovalevsky).The determination of orbits from radar measurements is discussed. The proposed methods are quite different in principle from the classical ones. They are actually improvements of a circular orbit whose determination is made by a method of undetermined coefficients (Baker). Some of the elements determined by these methods are quite unstable, and it is always wiser to use other informations together with radar measurements.The determination of orbits from Doppler data alone has been worked out by many investigators. It is the inverse problem of the main problem of all navigational systems. The method proposed by Patton is summarized. The principles of others are quite analogous. The results are improved when more than one receiver are used. The different types of corrections: refraction, perturbations, etc., should be introduced in later stages of the determination, and the whole frequency curve is to be used if a good determination of all elements is desired.Finally, the author quotes a method (Baker) using both Doppler and radar data, and the solution of a new problem: the determination of the orbit of a satellite of another planet from Doppler data alone (Deutsch).     

惯性+星光折射定位组合导航技术研究

由于现有惯性器件精度水平有限,纯惯性导航误差较大,因此需要采用组合导航的方式来提高导航精度。目前,全自主组合导航方式中传统的惯性+星光定姿组合导航方法只能实现定姿,不能实现定位,无法修正加速度表测量误差引起的惯性导航误差,故在精度上可提升空间有限。为此,提出了一种惯性+星光折射定位组合导航方法,重点从星光折射定位原理、大气折射模型、非线性滤波和选星策略几个方面进行论证及分析。通过理论分析与数学仿真相结合的手段,验证了星光折射定位原理的正确性及工程可行性,可将自主导航精度提升至100m,从而为进一步提高自主导航精度提供了一种技术途径。     

Simultaneous spacecraft attitude and orbit estimation using magnetic field vector measurements

Based on magnetometer measurements only, three-axis attitude, rate, and orbit estimation are successfully achieved. A single Augmented Dynamics Extended Kalman Filter (ADEKF) is configured by combining the spacecraft nonlinear attitude dynamics and quaternion kinematics with orbital mechanics. The filter design is adopted for three-axis stabilized spacecraft in low Earth orbits where the aerodynamic drag is the dominant source of disturbances in addition to the spacecraft magnetic residuals. To reduce the computational burden, another Interlaced Extended Kalman Filter (IEKF) is developed to uncouple the attitude/rate from the orbit dynamics. Both filters are implemented using the magnetometer measurements and their corresponding time derivatives. As a part of EgyptSat-1 flight scenario, detumbling and standby modes are used for performance testing of the ADEKF. The concept of local observability is applied to the basic filter and the stability is investigated by incorporating extensive Monte Carlo simulations with uniformly distributed initial conditions. The filter shows the capability of estimating the attitude better than 5 deg and rate of order 0.03 deg/s in each axis. In orbit estimation, the filter is capable of estimating the position with accuracy less than 8 km and velocity upto 5 m/s in each axis.     

月球重力场模型误差对CE-5T1探测器定轨的影响

利用嫦娥五号再入返回飞行试验拓展任务期间获取的探测器(CE-5T1)实测数据,采用内符合方法比较了3种重力场模型的实测数据定轨结果,发现采用GRAIL(Gravity Recovery And Interior Laboratory,重力恢复与内部实验室)重力场模型进行定轨的结果最优。相比于之前的嫦娥系列探测器定轨常用的LP(Lunar Prospector,月球勘探者)重力场模型,采用GRAIL重力场模型定轨后测距数据的残差降低了1个量级。进一步采用不同重力场模型进行轨道外推,定量分析重力场模型对不同类型轨道的影响,结果表明,对于倾角为90°的环月极轨道,不同重力场模型的轨道外推结果差异较小;而对于倾角为20°和40°的环月轨道,不同重力场模型的外推星历的偏差均方根可达到2km,大于当前环月探测器的定轨精度。为此,建议在后续探月任务中使用GRAIL重力场模型进行轨道确定。     

基于磁强计测量的微小卫星轨道姿态 耦合确定方法

针对基于磁强计测量的微小卫星, 提出了基于串并联混合策略的轨道姿态 耦合确定方法。滤波初期,考虑到估计误差较大,采用先地磁场大小测量的轨道确定、 后地磁场矢量测量的姿态确定的串联策略;稳定后,采用基于矢量测量的轨道、姿态同 步确定的并联策略。为降低轨道、姿态确定的相互影响,设计了基于信息的鲁棒Kalman 滤波,通过自动调节增益矩阵处理两个滤波系统间的影响。仿真表明,该方法在提高鲁 棒性的同时,还能适当提高状态估计的精度。     

Quick position determination using 1 or 2 LEO satellites

We describe an approach for a medium accuracy position determination of a user terminal (UT) on the Earth surface, using one or two low Earth orbit (LEO) satellites. The positioning approach is intended to meet the requirements of a worldwide personal communications system using LEO satellites. The basic two requirements are: (1) immediate positioning, and (2) horizontal position accuracy of the order of 10 km. Those requirements stem from the need of the system to know the user's approximate location before it connects his call. The approach makes use of the two-way communication with the UT, which can receive, transmit, and make its own measurements. Delay and Doppler measurements are used in order to enable instantaneous positioning with one satellite, and in order to achieve unambiguous positioning with two satellites. A simplified Globalstar satellite constellation and the expected Globalstar delay and frequency measurement accuracy are used to demonstrate the concept and to evaluate its performances     

绕月探测器的自主光学导航研究

提出了一种利用高斯-马尔科夫过程和Unscented卡尔曼滤波的绕月探测器自主光学导航算法。针对很难事先确定精确地绕月探测器轨道动力学模型问题,提出利用高斯-马尔科夫过程来近似轨道动力学中的无模型加速度,进而提高了轨道动力学模型的精度;考虑到基于扩展卡尔曼滤波的轨道确定存在的问题,提出利用基于Unscented卡尔曼滤波来估计探测器的位置、速度及无模型加速度,提高了轨道估计精度和保证了算法的稳定性。最后,通过数学仿真验证了自主光学导航算法的有效性。     

地球卫星星光折射导航量测量及其性能对比

星光折射天文导航是一种重要的地球卫星自主导航方式,量测量是影响其导航精度的重要因素。在地球卫星星光折射导航中,折射视高度、星光折射角、折射星像素坐标（折射星矢量）是3种常用的量测量,结合星光折射导航的基本原理重点介绍了这3种量测量的获取方法和量测模型,通过仿真和可观性分析比较了相同条件下3种量测量的导航性能。仿真结果表明,由于折射星像素坐标可以同时反映星光折射的大小和方向可观性高,而星光折射角和折射视高度仅能反映星光折射的大小,无法反映其方向可观性低,因此折射星像素坐标的导航性能优于星光折射角和折射视高度。此外,本文也对星敏感器精度、卫星轨道高度、星敏感器安装夹角3种因素对3种方法导航性能的影响进行了分析。     

Geostationary orbit determination for time synchronization using analytical dynamic models

A real time analytical orbit determination method has been developed for precision national time synchronization. The one-way time transfer technique via a geostationary TV satellite standard time and frequency signal (STFS) dissemination system was considered. The differential method was also applied for mitigating errors in geostationary satellite STFS dissemination system. Analytical dynamic orbit determination with extended Kalman filter (EKF) was implemented to improve differential mode STFS (DSTFS) service accuracy by acquiring better accuracy of a geostationary satellite position. The perturbation force models applied for satellite dynamics include the geopotential perturbation up to fifth degree and order harmonics, luni-solar perturbations, and solar radiation pressure. All of the perturbation effects were analyzed by secular, short, and long period variations for equinoctial orbit elements such as semimajor axis, eccentricity vector, inclination vector, and mean right ascension of the geostationary satellite. The reference stations for orbit determination were composed of four calibrated stations. Simulations were performed to evaluate the performance of real time analytical orbit determination in Korea. The simulation results demonstrated that it is possible to determine real time position of geostationary satellite with the accuracy of 300 m rms. This performance implies that the time accuracy is better than 25 ns all over the Korean peninsula. The real time analytical orbit determination method developed in this research can provide a reliable, extremely high accurate time synchronization service through setting up domestic-only benchmarks.     

Phase Residual Estimations for PCVs of Spaceborne GPS Receiver Antenna and Their Impacts on Precise Orbit Determination of GRACE Satellites

In-flight phase center systematic errors of global positioning system(GPS) receiver antenna are the main restriction for improving the precision of precise orbit determination using dual-frequency GPS.Residual approach is one of the valid methods for in-flight calibration of GPS receiver antenna phase center variations(PCVs) from ground calibration.In this paper,followed by the correction model of spaceborne GPS receiver antenna phase center,ionosphere-free PCVs can be directly estimated by ionosphere-free carrier phase post-fit residuals of reduced dynamic orbit determination.By the data processing of gravity recovery and climate experiment(GRACE) satellites,the following conclusions are drawn.Firstly,the distributions of ionosphere-free carrier phase post-fit residuals from different periods have the similar systematic characteristics.Secondly,simulations show that the influence of phase residual estimations for ionosphere-free PCVs on orbit determination can reach the centimeter level.Finally,it is shown by in-flight data processing that phase residual estimations of current period could not only be used for the calibration for GPS receiver antenna phase center of foretime and current period,but also be used for the forecast of ionosphere-free PCVs in future period,and the accuracy of orbit determination can be well improved.     

An epoch state filter for use with analytical orbit models of low earth satellites

Kalman filters provide a well established means for satellite orbit determination. In combination with space based sensors like GPS, DORIS or PRARE, accurate estimates of the spacecraft position and velocity can be obtained in real-time on-board the space vehicle. Traditionally, numerical methods of varying complexity are applied for propagating the state vector between measurements and updates of the state vector are referred to the epoch of the latest sensor output. In the present study a different approach is followed, which offers increased on-board autonomy and is particularly promising for small satellites with moderate accuracy requirements. An analytical orbit model is used to describe the spacecraft trajectory and mean elements at epoch are estimated instead of the instantaneous, osculating state vector. This adds the capability of performing on-board orbit prediction over time scales of up to one week, which is required, for instance, for the autonomous forecast of eclipse times or station contacts. Making use of the SGP4 orbit model that is compatible with NORAD twoline elements, an epoch state Kalman filter has been implemented and tested with GPS flight data of GPS/MET (MicroLab-1) and MOMSNAV (MIR). It is demonstrated that the proposed method provides an accuracy compatible with that of the analytical model and is robust enough to handle large data gaps in case of limited on-board resources for GPS operations. By adjusting the ballistic coefficient along with the mean elements, a considerable improvement of mid-term orbit predictions is achieved over methods that are restricted to the estimation of the state vector alone.     

单点弹道精度与轨道半长轴远地点高度计算精度的映射关系

在仅使用单点位置、速度信息计算轨道的奈件下,针对轨道半长轴、远地点高度的精度问题,在轨道面内,应用活力公式和二体运动学理论推导得出了轨道计算精度与弹道测量精度间映射关系的解析表达式,并采用数值分析方法给出了不同的位置、速度误差与半长轴、远地点高度最大误差之间的数值关系.仿真结果表明,对于位置误差和速度误差大小分别为100 m和1 m/s的算例,半长轴最大误差和远地点高度最大误差分别约为2 km和4 km.基于此方法,可以将弹道误差传递至轨道参数误差,进一步分析故障误判和漏判概率;也可根据轨道参数精度要求反算弹道测量精度要求,以作为地面测量系统建设的技术依据.     

基于卫星和星敏感器的冗余激光惯组在轨标定

采用高精度卫星导航速度、位置信息以及星敏感器提供的姿态信息设计十表冗余捷联惯组的标定模型,包含陀螺和加速度计的零次项和标度因数,对卫星和星敏感器辅助的冗余激光陀螺捷联惯组进行实时在轨标定.利用标准Kalman滤波和Sage-Husa自适应滤波作为估计算法,对十表冗余捷联惯组参数进行在线估计.数值仿真结果表明:参数标定精度均在7％以内,是一种实时的在轨标定方法,满足误差补偿要求.冗余惯组在轨标定方法为航天器高精度定姿和定轨提供了一种理论参考.     

Cubesat solar sail 3-axis stabilization using panel translation and magnetic torquing

A Cubesat mission with a deployable solar sail of 5 meter by 5 meter in a sun-synchronous low earth orbit is analyzed to demonstrate solar sailing using active attitude stabilization of the sail panel. The sail panel is kept parallel to the orbital plane to minimize aerodynamic drag and optimize the orbit inclination change caused by the solar pressure force normal to the sail surface. A practical control system is proposed, using a combination of small 2-dimensional translation of the sail panel and 3-axis magnetic torquing which is proved to have sufficient control authority over the gravity gradient and aerodynamic disturbance torques. Miniaturized CMOS cameras are used as sun and nadir vector attitude sensors and a robust Kalman filter is used to accurately estimate the inertially referenced body rates from only the sun vector measurements. It is shown through realistic simulation tests that the proposed control system, although inactive during eclipse, will be able to stabilize the sail panel to within ±2° in all attitude angles during the sunlit part of the orbit, when solar sailing is possible.     

基于我国深空站三向测量模式的数据精度分析

为了验证我国深空站三向测量模式的正确性,以同步星跟踪试验中的测量数据为基础,建立了站间同步修正算法和三向测量观测模型,通过与同步卫星的精密星历反算测量值比较,得到了测量数据的标定参数,结果表明,我国深空站测控能够实现dm级的测量精度,明显优于“嫦娥二号”测量的水平;同时利用测量数据进行定轨策略分析,最终实现了10 m量级的同步卫星定轨精度.分析结果为“嫦娥三号”探测器实施有效测控提供了依据.     

Gravity Field Recovery from GRACE: Unique Aspects of the High Precision Inter-Satellite Data and Analysis Methods

The very high accuracy of the Doppler and range measurements between the two low-flying and co-orbiting spacecraft of the GRACE mission, which will be at the μm/sec and ≈10 μm levels respectively, requires that special procedures be applied in the processing of these data. Parts of the existing orbit determination and gravity field parameters retrieval methods and software must be modified in order to fully benefit from the capabilities of this mission. This is being done in the following areas: (i) numerical integration of the equations of motion (summed form, accuracy of the predictor-corrector loop, Encke's formulation): (ii) special inter-satellite dynamical parameterization for very short arcs; (iii) accurate solution of large least-squares problems (normal equations vs. orthogonal decomposition of observation equations); (iv) handling the observation equations with high accuracy. Theoretical concepts and first tests of some of the newly implemented algorithms are presented. This revised version was published online in August 2006 with corrections to the Cover Date.