Motion and Inertia Estimation for Non-Cooperative Space Objects During Long-Term Occlusion Based on UKF-GP
Abstract
:1. Introduction
- We utilize multi-output GP models to predict projection measurements from a stereo-camera system so that the UKF-GP model can be updated during occlusion. GP models are trained using the projection measurements provided by stereo-camera systems. A product kernel, consisting of two periodic kernels, is designed to capture periodic trends in the non-periodic and noisy training data.
- The initial guesses for the periodicity hyper-parameters are intelligently obtained from the FFT analysis of the training data, enhancing the hyper-parameter training procedure.
- The projection predictions from the GP models and their derivatives obtained from the finite difference method (FDM) are used as pseudo-measurements in the UKF-GP fusion algorithm during long-term occlusion.
- Monte Carlo simulations across multiple tumbling frequencies demonstrate that GP models accurately predict projections for thousands of seconds under occlusion. Furthermore, the UKF-GP algorithm outperforms the conventional UFK in estimating motion and inertia parameters.
2. Methodology
2.1. Problem Scenario
2.2. Unscented Kalman Filter (UKF)
2.2.1. Dynamics Model
2.2.2. Observation Model
2.3. The Gaussian Process (GP)
Algorithm 1 Periodicity hyper-parameters initial guess determination algorithm for . |
|
2.4. UKF-GP Algorithm
2.5. Simulation Workflow
Algorithm 2 Simulation workflow. |
|
3. Results and Discussion
- Discrete-time intervals: s;
- Sensor measurement sampling rate: Hz;
- Principal axes inertia parameters of the target: , , ;
- Tumbling frequencies: , , , and Hz (the corresponding polhode periods = 441.045 s, 147.015 s, 88.209 s, and 63.006 s, respectively);
- Initial angular velocities in : rad/s;
- Initial attitude quaternion: ;
- Standard deviation of the projection measurement: rad.
3.1. Prediction Performance of the GP
- Number of features: ;
- Position of the feature in : m;
- Training data time-span: , 100, 200, 500, 1000, 1500, 2000, 2500, and 3000 s;
- Duration of prediction: s;
- Number of runs: .
3.2. Prediction Performance of UKF-GP
- Total simulation duration: s.
- Duration of the sensor data availability: s.
- Duration of the occlusion: s.
- Number of features: .
- User-defined constant parameters: , , (from [26]).
- Position of the features in :
- –
- Feature 1: m;
- –
- Feature 2: m;
- –
- Feature 3: m;
- –
- Feature 4: m;
- –
- Feature 5: m.
- Initial guesses of the state variables:
- –
- rad/s;
- –
- ;
- –
- ;
- –
- m.
- Standard deviation of the measurement noise of the optical flow:
- –
- For Hz, rad/s;
- –
- For Hz, rad/s;
- –
- For Hz, rad/s;
- –
- For Hz, rad/s.
- Standard deviation of the measurement noise of the disparity: rad.
- Variances for the initial state error covariance matrix:
- –
- Variance of : ;
- –
- Variance of : ;
- –
- Variance of and : ;
- –
- Variance of : .
- Variances in the process noise covariance matrix:
- –
- Variance in :
- ∗
- For Hz, ;
- ∗
- For Hz, ;
- ∗
- For Hz, ;
- ∗
- For Hz, .
- –
- Variance in :
- ∗
- For Hz, ;
- ∗
- For Hz, ;
- ∗
- For Hz, ;
- ∗
- For Hz, .
- –
- Variance in and : .
- –
- Variance in : .
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
UKF | Unscented Kalman filter |
GP | Gaussian process |
FFT | Fast Fourier transform |
LCS | Laser camera system |
EKF | Extended Kalman filter |
IEKF | Iterated extended Kalman filter |
MC | Monte Carlo |
CoMBiNa | Coarse model-based relative navigation |
SLAM | Simultaneous localization and mapping |
ML | Machine learning |
SPGP | Sparse pseudo-input Gaussian process |
RBF | Radial basis function |
DVQ-MANN | Dual-vector quaternion-based mixed artificial neural network |
DVQ-EKF | Dual-vector quaternion-based extended Kalman filter |
CM | Center of mass |
COP | Center of projection |
SURF | Sped-up robust features |
SIFT | Scale invariant feature transform |
L-BFGS | Limited-memory Broyden–Fletcher–Goldfarb–Shanno |
RMSE | Root mean square error |
Appendix A. Determination of Expressions of and
Appendix B. Determination of the Polhode Period
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Input | Output | ||
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Time, t | Projection Components, [rad] | ||
[sec] | Component | Component | Component |
⋮ | ⋮ | ⋮ | ⋮ |
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Kabir, R.H.; Bai, X. Motion and Inertia Estimation for Non-Cooperative Space Objects During Long-Term Occlusion Based on UKF-GP. Sensors 2025, 25, 647. https://doi.org/10.3390/s25030647
Kabir RH, Bai X. Motion and Inertia Estimation for Non-Cooperative Space Objects During Long-Term Occlusion Based on UKF-GP. Sensors. 2025; 25(3):647. https://doi.org/10.3390/s25030647
Chicago/Turabian StyleKabir, Rabiul Hasan, and Xiaoli Bai. 2025. "Motion and Inertia Estimation for Non-Cooperative Space Objects During Long-Term Occlusion Based on UKF-GP" Sensors 25, no. 3: 647. https://doi.org/10.3390/s25030647
APA StyleKabir, R. H., & Bai, X. (2025). Motion and Inertia Estimation for Non-Cooperative Space Objects During Long-Term Occlusion Based on UKF-GP. Sensors, 25(3), 647. https://doi.org/10.3390/s25030647