Improving Angle-Only Orbit Determination Accuracy for Earth–Moon Libration Orbits Using a Neural-Network-Based Approach
Abstract
:1. Introduction
2. Preliminaries
2.1. Least Square Method
Algorithm 1: Pseudocode of least squares method [55] |
2.2. Networks For Addressing Sequential Inputs
2.2.1. RNN
2.2.2. LSTM
2.2.3. BiLSTM
3. Methodology
3.1. Dynamics and Measurement Model
3.2. Sample Construction
- True measurement error . Note that the true measurement error is equivalent to the measurement noise defined in Equation (1).
- Predicted measurement error .
- Relative measurement error .
- The relative measurement error . The relative measurement error is assumed to contain information related to the true measurement error .
- The variables related to the measurements, including the true measurements and the position of the ground-based station (in the ECI coordinate) . These variables may reflect the sensitivity of the estimated errors to measurement errors.
- The estimated state obtained using the LSM. The estimated state can reflect the physical information of the orbit, for example, the orientation and the distance of the spacecraft.
- The measurement interval (). When , in this work. The measurement interval is an important variable, as the longer the measurement interval is, the weaker the relationship between two successive measurements.
3.3. Modified BiLSTM Structure
3.4. Training Process and Results
3.5. Overall Methodology
Algorithm 2: Pseudocode of the proposed method |
|
4. Numerical Simulation
4.1. Scenario Descriptions
4.2. Simulation Results
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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LSRM Cell | Input Vectors | Output Vectors |
---|---|---|
k-th forward LSTM cell | current input state previous hidden state previous cell state | current forward hidden state current forward cell state |
k-th reverse LSTM cell | current input state future hidden state future cell state | current reverse hidden state current reverse cell state |
Parameters | Value |
---|---|
Station | Eglin, FL |
Latitude (deg) | 30.57 |
Longitude (deg) | −86.21 |
Altitude (m) | 34.7 |
Feasible elevation (deg) | 0–90 |
Orbit | T | J | |||
---|---|---|---|---|---|
1 | 1.1809 | 0 | −0.1558 | 3.4155 | 3.1641 |
2 | 1.1807 | −0.0139 | −0.1569 | 3.4139 | 3.1632 |
3 | 1.1802 | −0.0259 | −0.1596 | 3.4100 | 3.1611 |
4 | 1.1794 | −0.0379 | −0.1636 | 3.4037 | 3.1579 |
5 | 1.1782 | −0.0499 | −0.1687 | 3.3949 | 3.1534 |
6 | 1.1766 | −0.0619 | −0.1747 | 3.3835 | 3.1479 |
7 | 1.1746 | −0.0739 | −0.1811 | 3.3694 | 3.1414 |
8 | 1.1721 | −0.0859 | −0.1878 | 3.3521 | 3.1340 |
9 | 1.1690 | −0.0979 | −0.1945 | 3.3314 | 3.1257 |
10 | 1.1653 | −0.1099 | −0.2010 | 3.3066 | 3.1166 |
11 | 1.1611 | −0.1219 | −0.2072 | 3.2768 | 3.1068 |
12 | 1.1561 | −0.1339 | −0.2129 | 3.2406 | 3.0963 |
13 | 1.1503 | −0.1459 | −0.2179 | 3.1960 | 3.0853 |
14 | 1.1435 | −0.1579 | −0.2219 | 3.1395 | 3.0737 |
15 | 1.1354 | −0.1699 | −0.2247 | 3.0647 | 3.0617 |
16 | 1.1234 | −0.1837 | −0.2252 | 2.9380 | 3.0472 |
17 | 1.1113 | −0.1934 | −0.2220 | 2.7897 | 3.0367 |
18 | 1.0992 | −0.1993 | −0.2152 | 2.6245 | 3.0302 |
19 | 1.0872 | −0.2020 | −0.2053 | 2.4486 | 3.0273 |
20 | 1.0759 | −0.2022 | −0.1932 | 2.2778 | 3.0276 |
21 | 1.0633 | −0.2003 | −0.1769 | 2.0876 | 3.0308 |
22 | 1.0512 | −0.1968 | −0.1586 | 1.9106 | 3.0363 |
23 | 1.0392 | −0.1919 | −0.1379 | 1.7420 | 3.0439 |
24 | 1.0274 | −0.1856 | −0.1147 | 1.5827 | 3.0533 |
25 | 1.0118 | −0.1739 | −0.0798 | 1.3741 | 3.0700 |
Orbit | T | J | ||
---|---|---|---|---|
1 | 1.1762 | −0.1228 | 3.3979 | 3.1718 |
2 | 1.1809 | −0.1558 | 3.4155 | 3.1641 |
3 | 1.1843 | −0.1815 | 3.4339 | 3.1565 |
4 | 1.1924 | −0.2498 | 3.5122 | 3.1302 |
5 | 1.2005 | −0.3235 | 3.6831 | 3.0918 |
6 | 1.2086 | −0.3801 | 3.9409 | 3.0565 |
7 | 1.2167 | −0.4162 | 4.2071 | 3.0329 |
8 | 1.2200 | −0.4275 | 4.3112 | 3.0257 |
9 | 1.2248 | −0.4418 | 4.4573 | 3.0166 |
10 | 1.2329 | −0.4620 | 4.6912 | 3.0045 |
11 | 1.2410 | −0.4792 | 4.9113 | 2.9948 |
12 | 1.2491 | −0.4947 | 5.1195 | 2.9869 |
13 | 1.2572 | −0.5090 | 5.3172 | 2.9799 |
14 | 1.2653 | −0.5226 | 5.5053 | 2.9739 |
15 | 1.2734 | −0.5356 | 5.6848 | 2.9685 |
16 | 1.2815 | −0.5482 | 5.8560 | 2.9634 |
17 | 1.2896 | −0.5606 | 6.0196 | 2.9589 |
18 | 1.2977 | −0.5728 | 6.1758 | 2.9545 |
19 | 1.3058 | −0.5848 | 6.3251 | 2.9505 |
20 | 1.3139 | −0.5967 | 6.4678 | 2.9466 |
21 | 1.3220 | −0.6086 | 6.6042 | 2.9427 |
Parameters | Symbol | Lower Bounds | Upper Bounds |
---|---|---|---|
Time of flight (h) | 2 | 6 | |
Number of measurements (-) | 121 | 361 | |
Initial position estimated error (km) | |||
Initial velocity estimated error (m/s) |
Number of Neurons of Input Layer | Number of Hidden Layers | Number of Neurons of Hidden Layer | Number of Neurons of Output Layer | Mapping Relationship | |
---|---|---|---|---|---|
Initialization lager | 14 | L | M | M | |
14 | L | M | M | ||
Pre-BiLSTM layer | 14 | L | M | M | |
Post-BiLSTM layer | M | L | M | M | |
Sigmoid layer | M | 1 | M | 2 |
L | M | Activation Function | Minimal Validation MCE | |
---|---|---|---|---|
RA | DEC | |||
1 | 16 | Leaky ReLU | 2.5320 | 1.1904 |
ReLU | 1.1746 | 2.5989 | ||
Tanh | 2.7256 | 1.1799 | ||
32 | Leaky ReLU | 2.292 | 2.5028 | |
ReLU | 2.4641 | 3.3524 | ||
Tanh | 2.4527 | 2.6765 | ||
64 | Leaky ReLU | 2.3771 | 2.7930 | |
ReLU | 2.3828 | 2.8491 | ||
Tanh | 2.8891 | 1.1901 | ||
128 | Leaky ReLU | 2.6264 | 2.3285 | |
ReLU | 2.4589 | 2.3786 | ||
Tanh | 2.4291 | 2.3836 | ||
2 | 16 | Leaky ReLU | 1.2170 | 2.6885 |
ReLU | 2.4529 | 1.2117 | ||
Tanh | 2.5764 | 1.8663 | ||
32 | Leaky ReLU | 1.1737 | 1.1863 | |
ReLU | 2.4806 | 1.1755 | ||
Tanh | 2.3103 | 1.8573 | ||
64 | Leaky ReLU | 2.4558 | 2.4421 | |
ReLU | 2.5516 | 1.8231 | ||
Tanh | 2.6790 | 1.8164 | ||
128 | Leaky ReLU | 2.7573 | 3.3301 | |
ReLU | 2.6885 | 2.3767 | ||
Tanh | 2.7142 | 1.1875 |
Actual Class | Total | Predicted Class | True Positive Rate (%) | False Negative Rate (%) | |
---|---|---|---|---|---|
Accurate | Inaccurate | ||||
Accurate | 864,022 | 860,716 | 3306 | 99.6173 | 0.3826 |
Inaccurate | 40,996 | 5171 | 35,825 | 87.3865 | 12.6134 |
Actual Class | Total | Predicted Class | True Positive Rate (%) | False Negative Rate (%) | |
---|---|---|---|---|---|
Accurate | Inaccurate | ||||
Accurate | 863,856 | 860,438 | 3418 | 99.6043 | 0.3956 |
Inaccurate | 41,162 | 5596 | 35,566 | 86.4049 | 13.5950 |
Accurate Class | Inaccurate Class | ||
---|---|---|---|
BiLSTM | RA | 99.6173 | 87.3865 |
DEC | 99.6043 | 86.4049 | |
Forward LSTM | RA | 99.4742 | 83.5989 |
DEC | 95.4365 | 34.5017 | |
Backward LSTM | RA | 99.5834 | 84.4987 |
DEC | 95.5991 | 64.3418 |
Parameter | Value | |
---|---|---|
Initial epoch | 2,458,860.75 Julian Date | |
Time of flight | 6 h | |
Number of measurements | 361 | |
Initial state (ECI frame) | −238,078.6112 km | |
251,708.0350 km | ||
132,135.5595 km | ||
−1.5244 km/s | ||
−0.8960 km/s | ||
−0.8935 km/s |
Method | Index | Measurement Duration (Hour) | ||||
---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | ||
BiLSTM-LSM | 108.0226 | 52.2082 | 26.4953 | 18.6093 | 13.3092 | |
30.3310 | 14.1457 | 7.9966 | 5.1123 | 4.1821 | ||
LSM | 122.7295 | 57.0202 | 28.7276 | 20.4697 | 14.4617 | |
34.0102 | 15.4605 | 8.8635 | 5.6236 | 4.6241 | ||
LSAR | 149.8743 | 69.5296 | 35.8228 | 25.4228 | 18.7784 | |
39.0967 | 18.6796 | 11.4537 | 7.0075 | 6.0896 |
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Share and Cite
Zhang, Z.; Shi, Y.; Zheng, Z. Improving Angle-Only Orbit Determination Accuracy for Earth–Moon Libration Orbits Using a Neural-Network-Based Approach. Remote Sens. 2024, 16, 3287. https://doi.org/10.3390/rs16173287
Zhang Z, Shi Y, Zheng Z. Improving Angle-Only Orbit Determination Accuracy for Earth–Moon Libration Orbits Using a Neural-Network-Based Approach. Remote Sensing. 2024; 16(17):3287. https://doi.org/10.3390/rs16173287
Chicago/Turabian StyleZhang, Zhe, Yishuai Shi, and Zuoxiu Zheng. 2024. "Improving Angle-Only Orbit Determination Accuracy for Earth–Moon Libration Orbits Using a Neural-Network-Based Approach" Remote Sensing 16, no. 17: 3287. https://doi.org/10.3390/rs16173287