# Initial Results of Modeling and Improvement of BDS-2/GPS Broadcast Ephemeris Satellite Orbit Based on BP and PSO-BP Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Orbit Error of Broadcast Ephemeris

#### 2.2. Impact of AODE of BDS

#### 2.3. Model of BP Neural Network

#### 2.4. PSO–BP Neural Network

_{1}and $c$

_{2}are the learning factors, generally 2; $rand$

_{1}and $rand$

_{2}are random numbers between (0,1); ${v}_{i}$ and ${x}_{i\text{}}$ represent the velocity and position of the ith dimension of the particle; $pbes{t}_{i}$ and $gbes{t}_{i}$ represent the value of the ith dimension at the best position of a particle and the value of the ith dimension at the best position of the entire population. These two formulas are updated for a certain dimension of particles. For each dimension of the particles, the formula must be used to update.

_{2}= $c$

_{2}= 2, and the inertia weight is 0.95.

#### 2.5. Experiment Process

## 3. Results

#### 3.1. BDS Satellite Orbit Error Model Output

#### 3.1.1. GEO Orbit

#### 3.1.2. IGSO Orbit

#### 3.1.3. MEO Orbit

## 4. Discussion

## 5. Conclusions and Suggestions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Variation in the orbit errors of BDS satellites in different orbits: (

**a**) BDS GEO satellite orbit errors of C01\C03\C04; (

**b**) BDS IGSO satellite orbit errors of C06\C07\C08; (

**c**) BDS MEO satellite orbit errors of C11\C12\C14.

**Figure 2.**Variation of the orbit errors of the GPS satellites: (

**a**) G03/G13/G16 orbit error; (

**b**) G20/G24/G30 orbit error.

**Figure 3.**Variation in the AODE for the BDS satellite: (

**a**) BDS MEO satellite AODE; (

**b**) BDS IGSO satellite AODE.

**Figure 5.**Three-layer BP neural network model error propagation. Layers 1, 2, and 3 are the input, hidden, and output layers, respectively.

**Figure 7.**BDS C03 satellite orbit modeling outcome: (

**a**) BDS C11 satellite orbit modeling outcome on the 57th day of 2020; (

**b**) BDS C11 satellite orbit modeling outcome on the 57th–59th days of 2020.

**Figure 8.**BDS C08 satellite orbit modeling outcome: (

**a**) BDS C08 satellite orbit modeling outcome on the 57th day of 2020; (

**b**) BDS C08 satellite orbit modeling outcome on the 57th–59th days of 2020.

**Figure 9.**BDS C11 satellite orbit modeling outcome: (

**a**) BDS C11 satellite orbit modeling outcome on the 57th day of 2020; (

**b**) BDS C11 satellite orbit modeling outcome on the 57th–59th days of 2020.

**Figure 10.**GPS G02/ G16 satellite orbit modeling outcome: (

**a**) the satellite orbit modeling outcome of G02 on the 57th day of 2020; (

**b**) the satellite orbit modeling outcome of G16 on the 57th day of 2020; (

**c**) GPS G02 satellite orbit modeling outcome on the 57th–59th days of 2020; (

**d**) GPS G16 satellite orbit modeling outcome on the 57th–59th days of 2020.

Days | Direction | Along-Track | Cross-Track | Radial-Track | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Model | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | |

1 d | Real | −9.06 | 0.47 | 9.07 | −0.43 | 0.78 | 0.89 | 0.26 | 0.61 | 0.66 |

BP | 0.18 | 1.92 | 1.91 | 0.10 | 0.28 | 0.29 | 0.01 | 0.40 | 0.40 | |

PSO–BP | −0.73 | 0.46 | 0.87 | −0.05 | 0.14 | 0.15 | −0.02 | 0.10 | 0.10 | |

3 d | Real | −8.48 | 1.42 | 8.60 | −0.39 | 0.77 | 0.86 | 0.32 | 0.62 | 0.70 |

BP | −1.43 | 1.45 | 2.03 | 0.52 | 0.29 | 0.60 | −0.09 | 0.19 | 0.21 | |

PSO–BP | −0.38 | 1.41 | 1.46 | 0.55 | 0.29 | 0.62 | −0.01 | 0.22 | 0.22 |

Days | Direction | Along-Track | Cross-Track | Radial-Track | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Model | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | |

1 d | Real | −2.33 | 0.74 | 2.44 | −0.03 | 2.01 | 2.00 | 1.27 | 0.19 | 1.29 |

BP | −0.28 | 0.15 | 0.31 | 0.08 | 0.19 | 0.21 | 0.04 | 0.08 | 0.09 | |

PSO–BP | −0.27 | 0.15 | 0.30 | 0.10 | 0.18 | 0.21 | 0.03 | 0.07 | 0.08 | |

3 d | Real | −2.26 | 0.80 | 2.39 | 0.03 | 1.93 | 1.93 | 1.32 | 0.22 | 1.33 |

BP | −1.27 | 0.63 | 1.41 | 0.52 | 0.76 | 0.92 | 0.35 | 0.37 | 0.51 | |

PSO–BP | −0.47 | 0.24 | 0.53 | 0.06 | 0.33 | 0.33 | −0.08 | 0.17 | 0.19 |

Days | Direction | Along-Track | Cross-Track | Radial-Track | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Model | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | |

1 d | Real | 0.40 | 1.30 | 1.35 | −0.33 | 0.32 | 0.46 | 0.74 | 0.28 | 0.79 |

BP | −0.38 | 1.14 | 1.20 | 0.01 | 0.35 | 0.35 | −0.21 | 0.17 | 0.27 | |

PSO–BP | 0.11 | 1.03 | 1.03 | 0.05 | 0.35 | 0.35 | −0.21 | 0.09 | 0.23 | |

3 d | Real | 0.22 | 1.40 | 1.41 | −0.39 | 0.42 | 0.57 | 0.97 | 0.47 | 1.08 |

BP | 0.56 | 1.40 | 1.51 | −0.08 | 0.36 | 0.37 | 0.01 | 0.35 | 0.35 | |

PSO–BP | 0.52 | 1.33 | 1.43 | −0.04 | 0.35 | 0.35 | 0.12 | 0.29 | 0.32 |

**Table 4.**BDS satellite broadcast ephemeris and model-refined ephemeris orbit errors RMS(3D) and improvement rate.

PRN | RMS(3D)/m (1 Day) | Improvement Rate (1 Day) | RMS(3D)/m (3 Days) | Improvement Rate (3 Days) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Real | BP | PSO–BP | BP | PSO–BP | Real | BP | PSO–BP | BP | PSO–BP | |

C01 | 2.57 | 1.37 | 1.57 | 47% | 39% | 2.88 | 1.79 | 1.67 | 38% | 42% |

C03 | 9.14 | 1.98 | 0.77 | 78% | 92% | 8.67 | 2.13 | 1.60 | 75% | 82% |

C04 | 9.77 | 3.70 | 2.54 | 62% | 74% | 9.26 | 2.93 | 1.79 | 68% | 81% |

C06 | 1.85 | 0.41 | 0.40 | 78% | 78% | 1.91 | 0.67 | 0.74 | 65% | 61% |

C07 | 3.42 | 0.34 | 0.36 | 90% | 89% | 3.09 | 1.33 | 1.18 | 57% | 62% |

C08 | 3.41 | 0.38 | 0.38 | 89% | 89% | 3.35 | 1.76 | 0.65 | 47% | 81% |

C09 | 2.47 | 0.33 | 0.34 | 87% | 86% | 2.43 | 0.70 | 0.57 | 71% | 77% |

C10 | 3.48 | 0.78 | 0.77 | 78% | 78% | 3.21 | 1.01 | 0.99 | 68% | 69% |

C11 | 1.63 | 1.28 | 1.12 | 21% | 32% | 1.87 | 1.59 | 1.51 | 15% | 19% |

C12 | 3.38 | 7.14 | 3.24 | −111% | 4% | 3.37 | 2.96 | 2.92 | 12% | 14% |

C14 | 1.77 | 4.61 | 0.93 | −161% | 48% | 3.74 | 3.37 | 3.39 | 10% | 9% |

C16 | 1.87 | 0.80 | 0.67 | 57% | 64% | 1.83 | 0.89 | 0.87 | 51% | 53% |

Days | PRN | Direction | Along-Track | Cross-Track | Radial-Track | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Model | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | Mean/m | STD/m | RMS/m | ||

1 d | G02 | Real | −0.93 | 0.88 | 1.28 | −0.06 | 0.48 | 0.48 | −0.05 | 0.16 | 0.16 |

BP | 0.17 | 1.02 | 1.03 | 0.01 | 0.31 | 0.31 | 0.01 | 0.10 | 0.10 | ||

PSO–BP | 0.20 | 1.00 | 1.01 | −0.01 | 0.32 | 0.31 | 0.01 | 0.08 | 0.08 | ||

G16 | Real | −1.42 | 0.76 | 1.61 | 0.01 | 0.43 | 0.42 | 1.70 | 0.09 | 1.70 | |

BP | −1.42 | 0.82 | 1.64 | −0.06 | 0.20 | 0.21 | 0.09 | 0.06 | 0.10 | ||

PSO–BP | −1.40 | 0.78 | 1.60 | −0.03 | 0.20 | 0.20 | 0.10 | 0.06 | 0.11 | ||

3 d | G02 | Real | −1.28 | 1.04 | 1.65 | −0.08 | 0.43 | 0.43 | −0.03 | 0.15 | 0.16 |

BP | −0.12 | 0.85 | 0.86 | −0.02 | 0.24 | 0.24 | 0.01 | 0.11 | 0.11 | ||

PSO–BP | 0.00 | 0.84 | 0.84 | 0.02 | 0.27 | 0.27 | 0.00 | 0.10 | 0.10 | ||

G16 | Real | −1.36 | 0.98 | 1.67 | 0.03 | 0.31 | 0.32 | 1.71 | 0.11 | 1.72 | |

BP | −1.10 | 0.99 | 1.48 | −0.03 | 0.22 | 0.22 | 0.09 | 0.08 | 0.11 | ||

PSO–BP | −1.01 | 0.98 | 1.41 | −0.02 | 0.22 | 0.22 | 0.11 | 0.07 | 0.13 |

**Table 6.**GPS satellite broadcast ephemeris orbit error and model-refined ephemeris orbit errors of RMS(3D) and the improvement rate.

PRN | RMS(3D)/m (1 Day) | Improvement Rate (1 Day) | RMS(3D)/m (3 Days) | Improvement Rate (3 Days) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Real | BP | PSO–BP | BP | PSO–BP | Real | BP | PSO–BP | BP | PSO–BP | |

G01 | 1.66 | 1.18 | 1.09 | 29% | 34% | 1.58 | 1.50 | 1.29 | 5% | 18% |

G02 | 1.37 | 1.08 | 1.06 | 21% | 23% | 1.71 | 0.90 | 0.89 | 47% | 48% |

G03 | 1.65 | 1.25 | 1.17 | 24% | 29% | 1.93 | 1.02 | 1.02 | 47% | 47% |

G04 | 1.68 | 0.82 | 0.81 | 51% | 52% | 1.75 | 0.85 | 0.87 | 51% | 50% |

G05 | 0.94 | 0.81 | 0.77 | 14% | 18% | 1.06 | 0.93 | 0.89 | 12% | 17% |

G06 | 1.68 | 0.95 | 0.94 | 43% | 44% | 1.69 | 0.99 | 0.97 | 41% | 43% |

G07 | 1.56 | 0.96 | 0.86 | 38% | 45% | 1.33 | 1.24 | 1.23 | 7% | 8% |

G08 | 1.66 | 1.14 | 1.15 | 31% | 30% | 1.79 | 1.07 | 1.08 | 40% | 39% |

G09 | 1.56 | 0.88 | 0.89 | 43% | 43% | 1.62 | 0.84 | 0.83 | 48% | 49% |

G10 | 1.63 | 1.19 | 1.18 | 26% | 28% | 1.85 | 1.11 | 1.14 | 40% | 38% |

G11 | 1.64 | 1.04 | 1.05 | 36% | 36% | 1.79 | 1.03 | 1.07 | 42% | 40% |

G12 | 0.85 | 0.70 | 0.63 | 18% | 25% | 0.97 | 0.81 | 0.83 | 16% | 14% |

G13 | 1.86 | 0.92 | 0.81 | 51% | 56% | 2.16 | 1.13 | 1.17 | 48% | 46% |

G14 | 1.85 | 0.93 | 0.99 | 50% | 47% | 1.83 | 0.92 | 0.95 | 50% | 48% |

G16 | 2.39 | 1.66 | 1.62 | 30% | 32% | 2.42 | 1.50 | 1.43 | 38% | 41% |

G18 | 1.77 | 1.05 | 0.95 | 41% | 47% | 1.71 | 1.06 | 0.94 | 38% | 45% |

G19 | 3.30 | 1.49 | 1.89 | 55% | 43% | 2.99 | 2.26 | 2.32 | 24% | 23% |

G20 | 2.13 | 0.63 | 0.64 | 71% | 70% | 1.94 | 1.02 | 1.07 | 47% | 45% |

G21 | 1.98 | 1.05 | 1.05 | 47% | 47% | 2.00 | 0.98 | 0.96 | 51% | 52% |

G22 | 0.85 | 0.85 | 0.86 | 0% | −1% | 1.33 | 1.20 | 1.19 | 9% | 10% |

G23 | 1.34 | 0.83 | 0.83 | 38% | 38% | 1.15 | 0.94 | 1.09 | 18% | 5% |

G24 | 1.80 | 1.28 | 1.30 | 29% | 28% | 1.76 | 1.56 | 1.52 | 11% | 14% |

G25 | 1.68 | 0.98 | 0.94 | 42% | 44% | 1.72 | 1.21 | 1.19 | 30% | 31% |

G26 | 1.41 | 0.92 | 0.90 | 35% | 37% | 1.48 | 1.28 | 1.23 | 13% | 16% |

G27 | 1.51 | 0.89 | 0.76 | 41% | 50% | 1.54 | 0.88 | 0.85 | 43% | 45% |

G28 | 1.78 | 0.78 | 0.83 | 56% | 53% | 1.80 | 0.88 | 0.98 | 51% | 46% |

G29 | 0.80 | 1.07 | 0.98 | −34% | −22% | 1.00 | 1.24 | 1.23 | −24% | −23% |

G30 | 1.98 | 1.28 | 1.19 | 35% | 40% | 1.72 | 1.27 | 1.29 | 26% | 25% |

G31 | 0.98 | 0.99 | 1.01 | −2% | −3% | 0.91 | 0.86 | 0.87 | 5% | 4% |

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## Share and Cite

**MDPI and ACS Style**

Chen, H.; Niu, F.; Su, X.; Geng, T.; Liu, Z.; Li, Q.
Initial Results of Modeling and Improvement of BDS-2/GPS Broadcast Ephemeris Satellite Orbit Based on BP and PSO-BP Neural Networks. *Remote Sens.* **2021**, *13*, 4801.
https://doi.org/10.3390/rs13234801

**AMA Style**

Chen H, Niu F, Su X, Geng T, Liu Z, Li Q.
Initial Results of Modeling and Improvement of BDS-2/GPS Broadcast Ephemeris Satellite Orbit Based on BP and PSO-BP Neural Networks. *Remote Sensing*. 2021; 13(23):4801.
https://doi.org/10.3390/rs13234801

**Chicago/Turabian Style**

Chen, Hanlin, Fei Niu, Xing Su, Tao Geng, Zhimin Liu, and Qiang Li.
2021. "Initial Results of Modeling and Improvement of BDS-2/GPS Broadcast Ephemeris Satellite Orbit Based on BP and PSO-BP Neural Networks" *Remote Sensing* 13, no. 23: 4801.
https://doi.org/10.3390/rs13234801