On-Orbit Self-Calibration Method for Star Sensors Based on Models and Splines

Table 1. Simulation parameters for star sensors
View table
Table 1. Simulation parameters for star sensors
Parameter Value
Focal length /mm 43.3
Field of view /［（°）×（°）］ 20×20
Resolution /（pixel×pixel） 1024×1024
Pixel size /mm 0.015
Maximum magnitude /mV 6.0

Table 2. Distortion simulation parameters

View table

Table 2. Distortion simulation parameters

Distortion type	Distortion formula	Distortion parameter
Thin prism distortion	$\begin{array}{l} Δ x = s_{1} ({\bar{x}}^{2} + {\bar{y}}^{2}), \\ Δ y = s_{2} ({\bar{x}}^{2} + {\bar{y}}^{2}) \end{array}$	$s_{1} = 2.3 \times 10^{- 7}, s_{2} = 4.5 \times 10^{- 7}$
Radial distortion	$\begin{array}{l} Δ x = k_{1} \bar{x} r^{2} + k_{2} \bar{x} r^{4}, \\ Δ y = k_{1} \bar{y} r^{2} + k_{2} \bar{y} r^{4} \end{array}$	$k_{1} = 1.5 \times 10^{- 8}, k_{2} = 3.5 \times 10^{- 15}$
Decentering distortion	$\begin{array}{l} Δ x = p_{1} (r^{2} + 2 {\bar{x}}^{2}) + 2 p_{2} \bar{x} \bar{y}, \\ Δ y = p_{2} (r^{2} + 2 {\bar{y}}^{2}) + 2 p_{1} \bar{x} \bar{y} \end{array}$	$p_{1} = 1.5 \times 10^{- 13}, p_{2} = 4.0 \times 10^{- 13}$
Local distortion	$\begin{array}{l} Δ x = m {\bar{x}}^{'} e x p \{- [{(r^{'})}^{2} / σ^{2}]\}, \\ Δ y = m {\bar{y}}^{'} e x p \{- [{(r^{'})}^{2} / σ^{2}]\}, \\ {\bar{x}}^{'} = s g n (x - x_{0}^{'}), \\ {\bar{y}}^{'} = s g n (y - y_{0}^{'}), \\ r^{'} = \sqrt[]{{(x - x_{0}^{'})}^{2} + {(y - y_{0}^{'})}^{2}} \end{array}$	$\begin{array}{l} m = 2.4 p i x e l, \\ x_{0}^{'} = 250 p i x e l, \\ y_{0}^{'} = 250 p i x e l, \\ σ = 75 p i x e l \end{array}$
Polynomial distortion	$\begin{array}{l} Δ x = a_{1} \bar{x} + a_{2} \bar{y} + a_{3} {\bar{x}}^{2} + a_{4} \bar{x} \bar{y} + a_{5} {\bar{y}}^{2}, \\ Δ y = b_{1} \bar{x} + b_{2} \bar{y} + b_{3} {\bar{x}}^{2} + b_{4} \bar{x} \bar{y} + b_{5} {\bar{y}}^{2} \end{array}$	$\begin{matrix} \begin{array}{l} a_{1} = 3.0 \times 10^{- 5}, \\ a_{2} = - 5.0 \times 10^{- 5}, \\ a_{3} = 9.5 \times 10^{- 6}, \\ a_{4} = - 5.8 \times 10^{- 7}, \\ a_{5} = - 9.6 \times 10^{- 6}, \end{array} & \begin{array}{l} b_{1} = 3.0 \times 10^{- 5}, \\ b_{2} = 5.0 \times 10^{- 5}, \\ b_{3} = - 9.5 \times 10^{- 6}, \\ b_{4} = 3.8 \times 10^{- 7}, \\ b_{5} = 5.8 \times 10^{- 6} \end{array} \end{matrix}$

Table 3. Spacing and size of control points in each layer of B-spline grid
View table
Table 3. Spacing and size of control points in each layer of B-spline grid
Numberof layers Control point spacing Number ofcontrol points
x /pixel y /pixel
1 512 512 6×6
2 256 256 8×8
3 128 128 12×12
4 64 64 20×20
5 32 32 36×36

Table 4. Comparison of calibration accuracy and time in distortion simulation
View table
Table 4. Comparison of calibration accuracy and time in distortion simulation
Method Corrected starposition error /pixel
Training
time /s
Testtime /s
Neural network method 0.0774 528.1738 0.01896
GA-optimized
BP algorithm
0.0977 2202.3726 0.01815
Proposed
algorithm
0.0511 0.5778 0.00280

Table 5. Comparison of attitude output results of simulated star sensor
View table
Table 5. Comparison of attitude output results of simulated star sensor
Method DEC RA Roll
Before calibration 57.1308 17.2407 145.1491
Neural network method 5.8894 0.6573 11.3444
GA-optimized
BP algorithm
50.5346 5.6079 85.7682
Proposed algorithm 2.0068 0.6573 7.9778

Table 6. Comparison of calibration accuracy and time of in-orbit star sensor

View table

Table 6. Comparison of calibration accuracy and time of in-orbit star sensor

Method

Corrected star position error /pixel

Trainingtime /s

Testtime /s

method

Neural network

0.27924

510.361000

0.041100

GA-optimized

BP algorithm

0.34405

1234.852000

0.008600

Proposedalgorithm

0.23706

0.457767

0.004524

Table 7. Comparison of attitude output results after in-orbit star sensor calibration
View table
Table 7. Comparison of attitude output results after in-orbit star sensor calibration
Method DEC /（″） RA /（″） Roll /（″）
Neural network method 4.3704 3.11400 27.1800
GA-optimized
BP algorithm
4.9140 4.30920 31.5324
Proposed algorithm 2.7756 3.33684 20.3364

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Haodong Yan, Shuai Zhi, Xurui Chen, Zhaoxiong Li, Guopeng Ding, Yangyang Zhang, Yonghe Zhang, Zhencai Zhu. On-Orbit Self-Calibration Method for Star Sensors Based on Models and Splines[J]. Acta Optica Sinica, 2024, 44(12): 1212004

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