卫星跟踪卫星任务的引力谱分析和状态估计方法
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摘要
本文从谱分析和非线性系统状态估计的角度对卫卫跟踪技术进行探索，所做的主要工作和创新有两部分：引力向量的谱分析理论和新型状态估计方法的提出。具体内容如下：
在地球引力场量的表示中引入了直角坐标，建立了一套完整的引力位、引力向量和引力张量的公式体系，从而便于直接利用地心地固系中GPS对低轨卫星的跟踪数据。基于引力场量的直角坐标表示，推导出了这些引力场量的直接计算公式和具有更稳定数值计算性能的递推公式。此外还给出了引力场量在不同直角坐标系之间的转换关系。
将引力位、引力向量纳入傅立叶谱分析的框架。借助于傅立叶分析理论，得到了引力位和引力向量在经纬线方向和沿卫星轨道方向的傅立叶级数表达式，并且针对低低卫星跟踪卫星的实际情况，推导出了进行低低跟踪的两颗卫星之间的引力向量之差的傅立叶级数表达式。根据这些表达式得到了集总系数与引力位系数之间的关系。
根据卫卫跟踪观测的时间序列获得所需的引力向量是一个复杂的非线性系统状态估计问题，为此建立了卫卫跟踪观测系统的状态空间模型，对所研究系统的可观测性从解析法的角度进行了分析，从而为应用状态估计方法奠定了基础。
为了满足轨道模拟计算和状态参数估计的需要，提出了改进的扩展卡尔曼滤波方法，包括Sigma点卡尔曼滤波(SPKF)方法、基于高斯-赫尔默特积分规则的滤波方法，推导了基于Stirling内插公式的中心差分卡尔曼滤波(CDKF)方法。这几种方法利用了统计意义上的线性化逼近模型方程，而EKF仅仅是通过泰勒级数展开式的一阶线性化逼近复杂的非线性模型。通过研究还发现了CDKF与SPKF方法在本质上的相似性，并且将它们从形式上统一起来，提出了利用矩阵分解技术的实现算法：平方根SPKF和平方根CDKF算法。实验结果表明，本文提出的改进型扩展卡尔曼滤波方法易于实现，因为它们不需要计算非线性系统模型方程的雅可比矩阵；它们还具有良好的稳定性，比扩展卡尔曼滤波高出一阶的精度，而计算量并没有显著的增长。
模拟计算了GRACE两颗卫星35转的轨道弧段，利用本文提出的非线性系统状态估计方法计算出了轨道引力向量的时间序列，再根据引力谱分析理论对两颗卫星29转弧段的引力向量之差进行了分析，得到了关于引力信号的频谱，并从中选取了对整个信号功率贡献最大的频率。借助于傅立叶谱的解析表达式，确定了集总系数即傅立叶系数与引力位球谐系数之间的联系，所得到的结果可作为实际数据处理的参考。
In this dissertation there are mainly two problems discussed. One is the spectrum analysis theory of gravity vector of satellites in high-low or low-low mode, the other is the derivation of new state-estimation methods based on the Extended Kalman Filter. The main work and research are listed as follows.The aim of spectrum analysis is to determine the very parts of the Earth's gravitational field that have the strongest effect on the orbit of an artificial satellite. To attain this goal, this dissertation presents a spectrum analysis method which is based on an analytical Fourier series analysis of the commonly used spherical harmonic series expansion which is used for the representation of the Earth's gravitational field. By means of the spectrum expansion the connection between the gravity potential coefficients and the frequencies in the field's representation is determined.Within the Cartesian coordinates framework, recursive formulae for the computation of not only gravitational vector but also the gravitational tensor are derived which have more stable numerical performance than direct formulae. And the transformation of both gravitational vector and tensor is also discussed in detail.The computation of gravitational acceleration vectors of a satellite from its position and velocity observables, as well as its orbit determination, can be reduced to a state estimation problem for nonlinear systems. Firstly, a state-space model for satellite gravity observation system is set up and its observability is analyzed. Then, as far as state estimation of highly non-linear systems like the satellite gravity observation system is concerned, this dissertation is also dedicated to improve the existing Extended Kalman Filter. There are mainly two new state estimation methods put forward: Sigma Point Kalman Filter and Central Difference Kalman Filter. Although these two new methods are similar to the Extended Kalman Filter, they overcome its inherent defects through statistical linearization approximation, and thus gain higher accuracy, more stable numerical stability, while the computation cost can be kept low by means of efficient linear algebraic techniques. And the methods are essentially same and both can be called the Sigma Point Kalman Filter. Results of experiments show that the new methods can be used in place of the Extended Kalman Filter in most applications.Base on the above-mentioned spectrum analysis and state estimation methods, the dedicated gravitational field mission, GRACE, serves as an example with special conditions highlighted that arise for low earth orbit missions. The determination of the gravitational acceleration vectors acting on the GRACE satellites along the orbit from the observables position and velocity is performed by means of the Sigma Point Kalman Filter for filtering and parameter estimation for non-linear systems. The results can serve as reference for practical data processing.
在地球引力场量的表示中引入了直角坐标，建立了一套完整的引力位、引力向量和引力张量的公式体系，从而便于直接利用地心地固系中GPS对低轨卫星的跟踪数据。基于引力场量的直角坐标表示，推导出了这些引力场量的直接计算公式和具有更稳定数值计算性能的递推公式。此外还给出了引力场量在不同直角坐标系之间的转换关系。
将引力位、引力向量纳入傅立叶谱分析的框架。借助于傅立叶分析理论，得到了引力位和引力向量在经纬线方向和沿卫星轨道方向的傅立叶级数表达式，并且针对低低卫星跟踪卫星的实际情况，推导出了进行低低跟踪的两颗卫星之间的引力向量之差的傅立叶级数表达式。根据这些表达式得到了集总系数与引力位系数之间的关系。
根据卫卫跟踪观测的时间序列获得所需的引力向量是一个复杂的非线性系统状态估计问题，为此建立了卫卫跟踪观测系统的状态空间模型，对所研究系统的可观测性从解析法的角度进行了分析，从而为应用状态估计方法奠定了基础。
为了满足轨道模拟计算和状态参数估计的需要，提出了改进的扩展卡尔曼滤波方法，包括Sigma点卡尔曼滤波(SPKF)方法、基于高斯-赫尔默特积分规则的滤波方法，推导了基于Stirling内插公式的中心差分卡尔曼滤波(CDKF)方法。这几种方法利用了统计意义上的线性化逼近模型方程，而EKF仅仅是通过泰勒级数展开式的一阶线性化逼近复杂的非线性模型。通过研究还发现了CDKF与SPKF方法在本质上的相似性，并且将它们从形式上统一起来，提出了利用矩阵分解技术的实现算法：平方根SPKF和平方根CDKF算法。实验结果表明，本文提出的改进型扩展卡尔曼滤波方法易于实现，因为它们不需要计算非线性系统模型方程的雅可比矩阵；它们还具有良好的稳定性，比扩展卡尔曼滤波高出一阶的精度，而计算量并没有显著的增长。
模拟计算了GRACE两颗卫星35转的轨道弧段，利用本文提出的非线性系统状态估计方法计算出了轨道引力向量的时间序列，再根据引力谱分析理论对两颗卫星29转弧段的引力向量之差进行了分析，得到了关于引力信号的频谱，并从中选取了对整个信号功率贡献最大的频率。借助于傅立叶谱的解析表达式，确定了集总系数即傅立叶系数与引力位球谐系数之间的联系，所得到的结果可作为实际数据处理的参考。
In this dissertation there are mainly two problems discussed. One is the spectrum analysis theory of gravity vector of satellites in high-low or low-low mode, the other is the derivation of new state-estimation methods based on the Extended Kalman Filter. The main work and research are listed as follows.The aim of spectrum analysis is to determine the very parts of the Earth's gravitational field that have the strongest effect on the orbit of an artificial satellite. To attain this goal, this dissertation presents a spectrum analysis method which is based on an analytical Fourier series analysis of the commonly used spherical harmonic series expansion which is used for the representation of the Earth's gravitational field. By means of the spectrum expansion the connection between the gravity potential coefficients and the frequencies in the field's representation is determined.Within the Cartesian coordinates framework, recursive formulae for the computation of not only gravitational vector but also the gravitational tensor are derived which have more stable numerical performance than direct formulae. And the transformation of both gravitational vector and tensor is also discussed in detail.The computation of gravitational acceleration vectors of a satellite from its position and velocity observables, as well as its orbit determination, can be reduced to a state estimation problem for nonlinear systems. Firstly, a state-space model for satellite gravity observation system is set up and its observability is analyzed. Then, as far as state estimation of highly non-linear systems like the satellite gravity observation system is concerned, this dissertation is also dedicated to improve the existing Extended Kalman Filter. There are mainly two new state estimation methods put forward: Sigma Point Kalman Filter and Central Difference Kalman Filter. Although these two new methods are similar to the Extended Kalman Filter, they overcome its inherent defects through statistical linearization approximation, and thus gain higher accuracy, more stable numerical stability, while the computation cost can be kept low by means of efficient linear algebraic techniques. And the methods are essentially same and both can be called the Sigma Point Kalman Filter. Results of experiments show that the new methods can be used in place of the Extended Kalman Filter in most applications.Base on the above-mentioned spectrum analysis and state estimation methods, the dedicated gravitational field mission, GRACE, serves as an example with special conditions highlighted that arise for low earth orbit missions. The determination of the gravitational acceleration vectors acting on the GRACE satellites along the orbit from the observables position and velocity is performed by means of the Sigma Point Kalman Filter for filtering and parameter estimation for non-linear systems. The results can serve as reference for practical data processing.
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