切比雪夫多项式拟合GPS轨道坐标的改进算法  

An improved algorithm to fit GPS orbital coordinates with Chebyshev polynomials

作　　者：谢孟辛 张捍卫 XIE Mengxin;ZHANG Hanwei(School of Surveying and Land Information Engineering,Henan Polytechnic University,Jiaozuo,Henan 454003,China;Institute of Resources and Environment,Henan Polytechnic University,Jiaozuo,Henan454003,China)

机构地区：[1]河南理工大学测绘与国土信息工程学院,河南焦作454003 [2]河南理工大学资源与环境学院,河南焦作454003

出　　处：《测绘科学》2021年第6期53-58,共6页Science of Surveying and Mapping

基　　金：国家自然科学基金项目(42074002,41931075)。

摘　　要：针对利用切比雪夫多项式拟合卫星轨道,常规算法在高阶次拟合时,由于数值计算不稳定,拟合误差增大趋于发散的问题,该文使用2种改进算法来求解轨道坐标拟合。考虑在高阶次拟合时法方程为病态方程且系数矩阵接近奇异矩阵,改进算法以矩阵分解为基础,在求解法方程时避免对奇异矩阵求逆和病态方程求解,从而获得较为精确的计算结果。基于IGS精密星历数据的拟合实验中,分别使用了常规算法、LU分解算法和QR分解算法,结果显示在6和12 h弧段的轨道拟合中,无论是在算法的稳定性还是在轨道拟合精度方面,QR分解算法都有一定的优越性。GPS orbital coordinates are usually fitted by Chebyshev polynomials. However, due to the instability of numerical calculation, the fitting error of the conventional algorithm increases and tends to diverge when it comes to the higher order fitting. In order to solve this problem, two improved algorithms were used in this paper. Considering that the normal equation was ill-conditioned and the coefficient matrix approached the singular matrix in the case of higher order fitting, the improved algorithm was based on the matrix decomposition, avoiding the inverse of singular matrix and ill-conditioned equation solving, so as to obtain more accurate calculation results. In the fitting experiment based on International GNSS Service(IGS) precise ephemeris data, the conventional algorithm, LU decomposition algorithm and QR decomposition algorithm were used respectively.The results showed that in the orbit fitting of the 6 and 12 harcs,the QR decomposition algorithm had certain advantages in terms of both the stability of the algorithm and the accuracy of the orbit fitting.

关 键 词：切比雪夫多项式拟合 奇异矩阵 病态方程 高斯消去法 LU分解 格拉姆-施密特正交化 QR分解 

分 类 号：P228[天文地球—大地测量学与测量工程]