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

Abstract

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.

Long-term autonomous time-keeping of navigation constellations based on sparse sampling LSTM algorithm

“…• Gross error removal and interpolation: Serious gross errors will affect the accuracy of clock error forecasting. The Median Absolute Deviation (MAD) can indicate gross errors in the data sequence (Huang et al, 2021a(Huang et al, , 2021b. After all gross errors are removed, the cubic spline interpolation method can be used to smooth fill in missing values.…”

“…Since space-based atomic clocks are disturbed by factors such as mechanical vibration, electromagnetic radiation, and temperature changes, their clock error prediction models also include nonlinear random terms. Many studies have shown that neural network models have better forecasting effects on nonlinear data (Huang et al, 2021a(Huang et al, , 2021b. Among them, the Long Short-Term Memory (LSTM) neural network model exhibits a better forecasting effect on time series data (Greff et al, 2016;Wang et al, 2018).…”

A Systematic Review of Machine Learning Techniques for GNSS Use Cases

“…7). The ML models have been compared with several conventional non-ML models: regression model [14,80,101,159,160], brute force approach [143], traditional statistical approaches [60,94,[161][162][163][164], classical KF [129], Bayes-optimal rule [118], least square (LS)-based approach [40], Saastamoinen model [110], autoregressive model and a traditional LEO propagation model (EKF-STAN) [146], conventional wind speed retrieval method [43], Maximum-Likelihood Power-Distortion (PD-ML) [165], BERNESE 5.2 [114], CYGNSS [44], Hydrostaticseasonal-time (HST) model [49], Statistical Theta method [51][52][53]166], MAPGEO2004 geoid model [73], GNSS-IR soil moisture [58], Autoregressive (AR) and Autoregressive Moving Average (ARMA) [167], ERA-Interima global atmospheric reanalysis (now ERA5 reanalysis) [107], Empirical linear algorithms (LRM and LLM) [59], International Reference Ionosphere (IRI) 2016 model [168], NeQuick and IRI-2001 global TEC model [169][170][171], EKF-based integration scheme [172], CODE GIMs (Global Ionospheric Maps) [173], autoregressive integrated moving average (ARIMA), and quadratic polynomial (QP) models [174], least square regression algorithms (LSR) and bi-ha...…”

Improved SSA-Based GRU Neural Network for BDS-3 Satellite Clock Bias Forecasting

“…Their experimental results suggested that the accuracy and stability of the model were significantly better than those of traditional models such as the quadratic polynomial model and gray model; in particular, the advantage of this model was obvious when forecasting a cesium atomic clock. A further study [ 27 ] proposed a clock bias forecasting model that combined supervised learning and a long short-term memory neural network. The experimental results showed that the model had a significant advantage in controlling the accumulation of the forecasting error over time and that it was suitable for medium- and long-term clock bias forecasting.…”