GPS衛星軌道預測方法之研究

Abstract

長時間未使用的GPS接收機在開機時，因缺乏星曆資料而需時甚久方能完成首次定位。為減少開機時間，本論文利用GPS衛星軌道的估測及數值方法，使得接收機能迅速取得衛星軌道資訊，得以維持在熱開機的狀態。 由於GPS定位採用之座標系統並非慣性座標系，故須建立兩者之間的轉換，方可套用力學模型。力學計算除了地球重力效應外還需修正許多擾動項，才能進行軌道估測。 完整的力學模型配合4階Runge-Kutta數值方法解算微分方程即可進行軌道的預測，這些擾動項修正可以將預估三天之均方根誤差由數萬公尺的誤差修正到數百公尺不等的誤差。 但我們發現，這些誤差的型態相當特殊，類似一個振幅逐漸增大且週期穩定的正弦函數，且對於每顆衛星都存在類似的情形，於是本論文對於這樣的誤差進行修正，並將其稱之為顯誤差函數。顯誤差函數的成因據推測是由離心率極小的橢圓軌道之旋轉所引起，經過模擬類似真實衛星軌跡進行旋轉，亦發生類似的狀況，藉此可建立顯誤差函數的數學型態。我們利用過去8小時的星曆資料來輔助預測，決定顯誤差函數的各項參數，再透過差分的方法來確定顯誤差函數的相位。顯誤差函數對於衛星軌道的修正具有顯著的效果，可將原來未修正的均方根誤差再下修40%至80%，使得軌道的預測更加精確。

GPS receivers may not be able to provide positioning information timely because of the lack of ephemeris data if the receivers are used after being shut down for a long time. The orbital prediction and numerical method may be used to get more orbital information so that the receivers’ TTFF(Time-To-First-Fix) can be reduced i.e. the receivers can be maintained in the condition of hot start. Since the coordinate system used in GPS positioning is not inertial, transformation between ECI frame and ECEF frame should be established. In Orbital prediction, not only the Earth’s central force but also other perturbations should be taken into account. Considering these perturbations, the RMS error can be reduced from ten of kilometers to hundreds of meters in three days by using 4-th order Runge-Kutta numerical method. The error pattern between the actual orbit and the estimated orbit of each satellite can be approximated by a sine function with increasing amplitude and constant periods, which is named apparent error function since here. It is observed that the cause of the apparent error may be due to the rotation of elliptical orbit with extremely small eccentricity. The parameters of the apparent error function are computed based on the ephemeris data in past 8 hours. With the assistance of apparent error function, we can improve further performance of error estimation by 40% to 80%. As a result, the prediction of the satellite orbit becomes more accurate.