Clock bias prediction algorithm for navigation satellites based on a supervised learning long short-term memory neural network

In a satellite navigation system, high-precision prediction of satellite clock bias directly determines the accuracy of navigation, positioning, and time synchronization and is the key to realizing autonomous navigation. To further improve satellite clock bias prediction accuracy, we establish a satellite clock bias prediction model by using long short-term memory (LSTM) that can accurately express the nonlinear characteristics of the navigation satellite clock bias. Outliers in the original clock bias should be preprocessed before using the clock bias for prediction. By analyzing the working principle of the traditional median absolute deviations method, the ambiguity of the mathematical model of that method was improved. Experimental results show that the improved method is better than the traditional method at detecting gross errors. The single difference sequence of the preprocessed satellite clock bias was taken as the research object. First, a quadratic polynomial model was fit to the trend term of the single difference sequence. Second, based on the LSTM neural network model and the basic principles of supervised learning, a supervised learning LSTM network model (SL-LSTM) was proposed that models cyclic and random terms. Finally, the prediction function of the satellite clock bias was realized by extrapolating the model by adding a trend term. We adopt the GPS precision satellite clock bias of International GNSS Service data forecast experiments and apply wavelet neural network (WNN), autoregressive integrated moving average (ARIMA), and quadratic polynomial (QP) models to compare their prediction effects. The average prediction RMSE for 3 h, 6 h, 12 h, 1 d, and 3 d based on the SL-LSTM improved by approximately −21.80, −1.85, 8.57, 2.27, and 40.79%, respectively, compared with the results of the WNN. The average prediction RMSE based on the SL-LSTM improved by approximately 38.23, 65.48, 80.22, 85.18, and 94.51% compared with the ARIMA results. The average prediction RMSE based on the SL-LSTM improved by approximately 82.37, 75.88, 67.24, 45.71, and 58.22% compared with the QP results. Compared with the WNN, the SL-LSTM method has no obvious advantages in the prediction accuracy and stability in short-term prediction but achieves a better long-term prediction accuracy and stability. With an increased prediction duration, the SL-LSTM method is clearly better than the other three methods in terms of the prediction accuracy and stability. The results indicated that the quality of satellite clock bias prediction by the SL-LSTM method is better than that of the above three methods and is more suitable for the middle- and long-term prediction of satellite clock bias.

Improving precise point positioning performance based on Prophet model

Precision point positioning (PPP) is widely used in maritime navigation and other scenarios because it does not require a reference station. In PPP, the satellite clock bias (SCB) cannot be eliminated by differential, thus leading to an increase in positioning error. The prediction accuracy of SCB has become one of the key factors restricting positioning accuracy. Although International GNSS Service (IGS) provides the ultra-rapid ephemeris prediction part (IGU-P), its quality and real-time performance can not meet the practical application. In order to improve the accuracy of PPP, this paper proposes to use the Prophet model to predict SCB. Specifically, SCB sequence is read from the observation part in the ultra-rapid ephemeris (IGU-O) released by IGS. Next, the SCB sequence between adjacent epochs are subtracted to obtain the corresponding SCB single difference sequence. Then using the Prophet model to predict SCB single difference sequence. Finally, the prediction result is substituted into the PPP positioning observation equation to obtain the positioning result. This paper uses the final ephemeris (IGF) published by IGS as a benchmark and compares the experimental results with IGU-P. For the selected four satellites, compared with the results of the IGU-P, the accuracy of SCB prediction of the model in this paper is improved by about 50.3%, 61.7%, 60.4%, and 48.8%. In terms of PPP positioning results, we use Real-time kinematic (RTK) measurements as a benchmark in this paper. Positioning accuracy has increased by 26%, 35%, and 19% in the N, E, and U directions, respectively. The results show that the Prophet model can improve the performance of PPP.

A New BDS-2 Satellite Clock Bias Prediction Algorithm with an Improved Exponential Smoothing Method

High-accuracy and dependable prediction of the bias of space-borne atomic clocks is extremely crucial for the normal operation of the satellites in the case of interrupted communication. Currently, the clock bias prediction for the Chinese BeiDou Navigation Satellite System (BDS) remains still a huge challenge. To develop a high-precision approach for forecasting satellite clock bias (SCB) in allusion to analyze the shortcomings of the exponential smoothing (ES) model, a modified ES model is proposed hereof, especially for BDS-2 satellites. Firstly, the basic ES models and their prediction mechanism are introduced. As the smoothing coefficient is difficult to determine, this leads to increasing fitting errors and poor forecast results. This issue is addressed by introducing a dynamic “thick near thin far (TNTF)” principle based on the sliding windows (SW) to optimize the best smoothing coefficient. Furthermore, to enhance the short-term forecasted accuracy of the ES model, the gray model (GM) is adopted to learn the fitting residuals of the ES model and combine the forecasted results of the ES model with the predicted results of the GM model from error learning (ES + GM). Compared with the single ES models, the experimental results show that the short-term forecast based on the ES + GM models is improved remarkably, especially for the combination of the three ES model and GM model (ES3 + GM). To further improve the medium-term prediction accuracy of the ES model, the new algorithms in ES with GM error learning based on the SW (ES + GM + SW) are presented. Through examples analysis, compared with the single ES2 (ES3) model, results indicate that (1) the average forecast precision of the new algorithms ES2 + GM + SW (ES3 + GM + SW) can be dramatically enhanced by 49.10% (56.40%) from 5.56 ns (6.77 ns) to 2.83 ns (2.95 ns); (2) the average forecast stability of the new algorithms ES2 + GM + SW (ES3 + GM + SW) is also observably boosted by 53.40% (49.60%) from 8.99 ns (16.13 ns) to 4.19 ns (8.13 ns). These new coupling forecast models proposed in this contribution are more effective in clock bias prediction both forecast accuracy and forecast stability.

Fusion-based Satellite Clock Bias Prediction Considering Characteristics and Fitted Residue

As Satellite Clock Bias (SCB) prediction may be affected by various factors such as periodic items, sampling length, and stochastic items, a fusion-based prediction method is proposed by considering characteristics of SCB and fitted residue. On this basis, an instance algorithm is presented by fusing four typical prediction models. First, we use Empirical Mode Decomposition (EMD) to pre-process and decompose the SCB series into multiple components with various characteristics. Then, we analyse the fitting performance of each model for different components and prediction length, namely short-, mid- and long-term prediction, and select models with the best performance. Next, we analyse fitted residue of the reconstructed SCB, and select the model with the best fitting results. Finally, we fuse the multiple selected models for SCB prediction. The method is tested using Global Positioning System (GPS) precise clock products provided by the International Global Navigation Satellite System Service (IGS). Experimental results show that, compared with single prediction models and existing combination models, the proposed fusion-based prediction method improves accuracy and stability. In particular, the proposed method is more stable and has better performance for mid- and long-term prediction.