利用曲線擬合方法改善由粗估星曆計算之GPS衛星位置精確度

Abstract

一般由粗估星曆(Almanac)計算出全球衛星定位系統(Global Positioning System，簡稱GPS)之衛星軌道，在一週的時間內，其位置誤差約在1~3km，如何改善其誤差，使得由粗估星曆所計算出來之衛星位置更加準確，是我們的主要目標。 首先，我們針對第1天，該天之粗估星曆，計算出之後7天的衛星位置，再由精確星曆亦求出此7天的衛星位置，將此兩份衛星位置相減，可得到衛星位置誤差，而分析其趨勢行為，可以發現在第1天之誤差大約在2至3公里左右，然後緩慢收斂，在第4天時其誤差達到最小極限，約在1公里左右，接著其誤差行為則成持續發散的現象。 根據以上所描述衛星位置誤差的行為，我們觀察到，在第1到4天時其衛星位置誤差雖然相對較小，但就規律性並不佳，而觀察在第5至8天的時候，其誤差雖然較第1到4天大，但相對來說其趨勢較為規律。所以我們利用此特性，設計出兩種預估衛星位置之演算法來改進粗估星曆所計算出的衛星位置精準度。衛星位置預估演算法I為取出第1天之位置誤差曲線，利用曲線擬合(curve fitting)的方式，建立一誤差模型，再利用此誤差模型外插推算第2天之誤差；再由第1天之粗估星曆計算出第2天的GPS衛星位置，將兩者結合，進而估計出第2天之衛星位置。衛星位置預估演算法I則是取出第5天的衛星位置誤差曲線，同樣的作曲線擬合建立誤差模型，然後再由此誤差模型去外插推算第6天的誤差；再由原先的粗估星曆計算出第6天的衛星位置，兩者結合，進而得到第6天估計的衛星位置。 從計算結果可以得到，利用衛星位置預估演算法I的粗估星曆加上誤差預估模型，所估計出來的衛星位置誤差約可以保持在0.8公里左右，衛星位置預估演算法II的粗估星曆加上誤差預估模型，其求出來的衛星位置誤差約可以保持在0.3公里左右，比單純的使用該天的粗估星曆所計算出來的衛星位置誤差約在2~3公里有所改進，亦比使用前4天的粗估星曆所計算出來的衛星位置誤差約在1公里左右有所改進。

In general , the GPS satellite positions computed from Almanac Data are not accurate enough. Their error are about 1~3 km within 1 week. How to improve the position errors is the main object of this thesis.. First we get the Almanac of the 1st day. Then we compute the satellite positions of the next days base on the Almanac(inaccurate satellite position). Also , we compute these ones based on the Ephemeris(accurate satellite position). Let the satellite position errors , are obtained by taking the difference between them. Some interesting trends are found . First , the errors of the first day are about 2~3 km. They will converge slowly to the minimal values about 1km at day4. Then , the errors will diverge. Based on the characters of the position errors , we also find that although the position errors during day1~4 are relative smaller , they are not regular. During day5~8 , the position errors are bigger but the regularity is better . We use the property of the trends , two algorithms to improve the satellite position errors computed by Almanac are developed . In Algorithm I a curve fitting method is used to establish the model parameters of the position errors of day1 This model is extrapolated to day2 for computing the satellite position errors . The estimated satellite positions of day2 are obtainded by adding the day2 positions by Almanac and the errors. In Algorithm II , a curve fitting method is used to establish the best model parameters of the position error of day5. This model is extrapolated to day 66 for computing the satellite position errors. The estimated satellite positions of day6 ard obtained by adding the day6 positions by Almanac and the errors. In the simulation result we can find out whether we use Algorithm I or Algorithm II, Almanac+Error Model Estimation , the position error can be less than 500m , which is better than just by Almanac , also is better than the Almanac which is 4 days ago, the position error is about 1km, to improve the position error of satellite position.