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The star tracker is a highaccuracy attitude measurement device widely used in spacecraft. Its performance depends largely on the precision of the optical system parameters. Therefore, the analysis of the optical system parameter errors and a precise calibration model are crucial to the accuracy of the star tracker. Research in this field is relatively lacking a systematic and universal analysis up to now. This paper proposes in detail an approach for the synthetic error analysis of the star tracker, without the complicated theoretical derivation. This approach can determine the error propagation relationship of the star tracker, and can build intuitively and systematically an error model. The analysis results can be used as a foundation and a guide for the optical design, calibration, and compensation of the star tracker. A calibration experiment is designed and conducted. Excellent calibration results are achieved based on the calibration model. To summarize, the error analysis approach and the calibration method are proved to be adequate and precise, and could provide an important guarantee for the design, manufacture, and measurement of highaccuracy star trackers.
With the development of Earthobserving satellites and deepspace exploration satellites, requirements for attitude measurement accuracy are increasing. Thus, error analysis of the accuracy and calibration of the star tracker have become particularly important.
At present, research and analysis of the effect factors on the star tracker accuracy are being conducted. References [
Factors such as misalignment, aberration, instrument aging and temperature effects [
The groundbased calibration of star trackers generally includes real night sky observation and laboratory calibration. Real night sky observation can take advantage of the characteristics of the star tracker utilizing the star angular distance. This method is relatively easy to apply, whereas the model parameters interact with one another. Obtaining the global maximum is difficult, and this method is greatly influenced by the environment. Laboratory calibration could use a star simulator as the source. However, it is not easy to manufacture a highaccuracy star simulator.
Camera calibration techniques [
Last but not the least, the optical imaging principle and focus matters of the star tracker and the camera are not the same due to their functions. The camera uses a finite distance imaging mode, while the star tracker adopts an infinite distance imaging mode. General camera calibration methods are not suitable for the star tracker. Taking reference [
Therefore, the calibration method provided in literature [
To summarize, the literature on the analysis and evaluation of the error sources of star trackers has not been adequate until now. This paper proposes a systematic method for weight analysis of the error source. Optical parameters that play key roles in the accuracy of the star tracker (such as the principal point deviation, focal length error, imaging plane inclination error and distortion) [
The star tracker is a highaccuracy attitude measurement device, which considers the stars as the measuring object. It obtains the direction vector from the celestial inertial coordinate system by detecting the different locations of the stars on the celestial sphere. After many years of astronomical observations, star positions on the celestial sphere are predictable. Stars in the celestial sphere coordinate system can be expressed in the right ascension and declination (
Navigation stars are selected from the star catalog to meet the imaging requirement, and their data are stored in the memory of the star tracker.
When a star tracker with attitude matrix
The position of the principal point of the star tracker on the image plane is (
The relationship between
When the number of navigation stars is more than two, the attitude matrix can be solved by the QUEST algorithm [
The existence of errors and noise in the system are inevitable. According to the pinhole model shown in
Star vector measurement error concerns the accuracy of vector
Extraction error of the star point position
The process in which the star tracker detects the navigation stars includes background radiation, optical systems, photoelectric detectors and signal processing. Each segment affects the extraction quality of the target signal.
Stars are far from the Earth, thus, starlight rays are considered as parallel light rays and can converge to a point on the focal plane. However, most star trackers adopt the defocus form [
Star tracker optical parameter errors
The star tracker system cannot achieve the ideal image model because of the principal point deviation, focal length error, inclination of the image plane, distortion in practical use. Therefore, it is necessary to establish a calibration model for above parameter errors, and analyze parameter error and model error.
Star catalog errors concern the accuracy of vector
Star tracker internal algorithm error concerns the accuracy of the final attitude matrix
In the following analysis, we use the angle measurement error (
Angle
The
When the actual optical axis is along the positive direction of the ideal optical axis, Δ
We define
We adopt two methods to discuss the error effects. First, we use the Monte Carlo (MC) stochastic modeling method. In this method, it is assumed that the errors after calibration, such as noise, inclination angle, focal length and principal point errors are random errors. These errors are considered to coincide with the normal distribution. In addition, the number of stars that can be captured in the sky is more than 6,000. Therefore, it is reasonable to consider the incident angle in the FOV as uniformly distributed. Based on the two statistical assumptions, we can combine the geometry and MC random models, and develop the complete error effect analysis. Second, we use the maximum error method to prove the simulation result of the MC method. This method can easily identify the error distribution of the different incident angles. Analysis of the position of the maximum error can also provide information for further study. The object analyzed in this paper is a star tracker of 7″ accuracy and the FOV is 17°. The focal length of the system is approximately 49.74 mm. The star tracker adopts the APS CMOS image sensor with 1,024 × 1,024 pixels, and the size of each pixel is 0.015 mm.
The MC method [
Then, combination error analysis using the MC stochastic simulation is conducted. The
We conduct a simulation using Maximum Error Method to compare with the method in Section 3.3.1 based on the error propagation model in Section 3.2 and the system parameters of the star tracker, as well as the range of the star point extraction error, the principal point error, focal length error, inclination of the image plane and distortion. The incident angle
Under the simulation conditions, the effect of the error of the principal point on the star tracker accuracy, along with the incident angle is shown in
Under the simulation conditions, the effect of the error of focal length on the star tracker accuracy, along with the incident angle is shown in
Under the simulation conditions, the effect of the error of inclination image plane on the star tracker accuracy, along with the incident angle is shown in
Under the simulation conditions, the effect of the error due to distortion on the star tracker accuracy, along with the incident angle is shown in
The simulation results show that the error effects obtained by the maximum error method agree with the results obtained by the MC simulation method. We can also find that inclination of the image plane and the distortion are two key factors that need to be calibrated. The calibration method will be elaborated in the next section.
Optical systematic error analysis method proposed in this paper can perform analysis on the sensitivity of factors (such as the error of star position extraction error, position error of principal point, error of focal length, inclination of the image plane and the distortion) that may influence the accuracy of the star tracker.
The above analysis of the system error can be applied in the following areas: (1) reference for the calibration target,
For application (1) above, because of the limitations in the calibration method, the effect factors cannot be separated from one another. Thus, determining whether the five indicators are all satisfied is difficult. Therefore, the proposed method is used primarily in the determination and demonstration of the design indicators. For application (3), some of the restrictions in certain error range can easily meet, such as the principal point position error, whereas satisfying the others are more difficult. These error factors need to be calibrated elaborately, such as the inclination angle of the image plane and the distortion. Therefore, emphasis on the calibration method is related to the system parameters. The calibration method and the processes are designed according to the characteristics of the system, so that the
The calibration object of this paper is a star tracker with 7″ accuracy. Based on the result of the above analysis, optical parameters of the system are calibrated. The laboratory calibration of the star tracker can be performed using a threeaxis turntable and a collimator or an autocollimator theodolite. In essence, their operating principle is the same. However, because the collimator does not have a selfcollimation function, which could introduce trouble to the calibration of the principal point, we adopt the autocollimator theodolite.
The autocollimator theodolite we employ in the experiment is the Leica 6100A.
According to the analysis in Section 3, the calibration objective should be focused on the inclination of the image plane and the distortion. The basic block diagram of the calibration process is shown in
For better representation of the location and the relationship, we create the coordinate systems as in
Angle
First, we adjust the theodolite to ensure that its outgoing light travels only along the longitude direction. The latitude value does not change during this process. Imaging conducted at every 0.5° can yield a series of measured values that determines the external parameters.
Then, according to the 17° FOV of the star tracker, we adjust the theodolite so that its light travels along the two orthogonal directions as shown in
The image obtained by the star tracker is shown in
The series of point coordinate values obtained in Section 4.2.2 can be used to solve angle
Linear fitting can also be adopted to solve the value of
The positions of the series of specific points are obtained according to the above discussion and preparation. The characteristics of these points include the following: the test points are distributed in two orthogonal directions and the test points in the same direction are almost centrosymmetrical. Taking advantage of relationship between the symmetric points, we build a calibration model, shown in
The longitude and latitude values (
Firstly, we adjust the autocollimation theodolite, and ensure that the crossline in the theodolite eyepiece is coincident with the specular reflection image of the star tracker glass shield, as shown in
Secondly, we can obtain a series of focal lengths utilizing the incident light in different directions and their image point (
The simulation results shown in
From the multiple groups of symmetrical points, we can obtain optimized distortion coefficients
When the principal point, focal length and distortion coefficients are determined, the inclination angle of the image plane can be obtained using the geometric relationship, and the average value can be calculated using the multipleset of symmetrical points.
So far, the principal point, the focal length, the radial distortion coefficients and the inclination angle of image plane in one direction are obtained. The principal point, focal length, and radial distortion coefficients are suitable for the entire plane. Inclination angle in the other measurement direction can be obtained in the same manner.
The inclination angle in the two measurement directions is not sufficient. The ultimate goal of calibration experiment is to obtain an ideal image position of incident rays from any direction in the FOV. Based on the parameters obtained above, there are many methods to solve this problem. We adopt a coordinate transformation method, and establish a coordinate transformation framework for incident rays. Therefore, we make the following analysis.
We can obtain the position of point
Considering the angle ∠
Thus, the position of point
The transformation matrix from coordinate system
Thus, the whole parameter estimation model of the optical system of the star tracker is completed.
For any point on the image plane of the incident ray, we can determine its position in
The equation for straight line
In the calibration process described in Section 4.2.4.2, the principal point position is obtained and considered as its true position. In reality however, due to lens installation, accuracy in manufacture as well as the limitations in the eyepiece alignment when using the theodolite, a deviation error of the optical axis is inevitable. That is to say, there may be an error
As shown in
The
The calibration experiment is conducted in the laboratory. From the above analysis, we can know the longitude angle
As described in Section 4.2.4.1, we can firstly obtain external parameter
Solving the slope of the line can determine the value of
As described in Section 4.2.4.2—(1), we can obtain the position of the principal point as (515.1859, 514.2069) pixels.
As described in Section 4.2.4.2—(2), we can obtain a series of focal lengths utilizing the incident light in different directions and their image point (
Theoretically, the values of the calculated focal lengths are equal. However, distortion can cause departure of the calculated focal lengths in different incident angles. Inclination of the image plane may cause deviation even when points are in centrosymmetry incident angles. Therefore, the estimations of
As described in Section 4.2.4.2—(3), we can obtain the distortion values Δ
We use linear least squares fitting to obtain the distortion curve. The fitting result is shown as follows, and
Calibration result is summarized in
After calibration, we can obtain the calibrated image point position of the incident light in different directions. The estimations of
We can see from
A synthetic error analysis approach for the star tracker has been proposed in detail in this paper. This approach can provide the error propagation relationship of the star tracker. Based on the analysis results, a calibration experiment is designed and conducted. Excellent calibration results are achieved. The calibration experiment can not only guarantee the accuracy to meet the design requirement, but can even improve the accuracy of the star tracker to a higher level. To summarize, the error analysis approach and the calibration method are proved to be adequate and precise, and are very important for the design, manufacture, and measurement of highaccuracy star trackers.
This work is partially supported by a grant from the National High Technology Research and Development Program of China (863 Program) (No. 2012AA121503). The laboratory calibration is performed in the State Key Laboratory of Precision Measurement Technology and Instruments at Tsinghua University. Both of them are gratefully acknowledged.
Star tracker ideal imaging model.
Sketch of complete error propagation.
Synthetic analysis result using MC stochastic simulation.
Influence of the star point extraction error on the star tracker accuracy. (
Influence of the principal point error on the star tracker accuracy. (
Influence of the focal length error on the star tracker accuracy. (
Influence of the inclination angle error on the star tracker accuracy. (
Influence of the distortion on the star sensor accuracy. (
(
Calibration flow diagram.
Coordinate systems in the ideal pinhole imaging model.
Coordinate systems in the case of errors.
(
Image of the emergent crossline of the theodolite. (
Sketch map of the solution of the gravity center of the crossline.
Calibration schematic diagram.
Principal point measurement principle.
Error between the ideal and real distances of the symmetric points. The
Relationship of the different distortion expressions.
(
Measurement points and linear fitting curve (
Estimations of
Distortion curve. (
Reestimations of
Calibration residual error.
Singleerror factor analysis using MC stochastic simulation.
Error of star point extraction:  μ=0, σ=0.1/3 pixels (Gaussian distribution)  μ = 0.0073, σ = 2.0281( 
Error of principal point:  μ = 0, σ = 4.5/3 pixels (Gaussian distribution)  μ = −0.0139, σ = 2.0400( 
Error of focal length:  μ = 0, σ = 0.6/3 pixels (Gaussian distribution)  μ = −0.0079, σ = 1.8182( 
Error of inclination angle:  μ = 0, σ = 0.075°/3 (Gaussian distribution)  μ = 0.0097, σ = 1.9703( 
Distortion:  μ = 0, σ = 0.1/3 pixels (Gaussian distribution)  μ = −0.0185, σ = 2.0325( 
Distortion values.

 

Δ 
Δ  
1  29.0039  −0.021951  28.9778  −0.041757 
2  58.0162  −0.050355  57.9644  −0.042748 
3  87.0237  −0.065911  86.9740  −0.061112 
4  116.0016  −0.076456  115.9721  −0.061893 
5  145.0044  −0.079762  144.9961  −0.067778 
6  174.0553  −0.082395  174.0393  −0.065835 
7  203.1132  −0.079229  203.0897  −0.048521 
8  232.1929  −0.063527  232.1630  −0.019044 
9  261.2868  −0.037760  261.2486  0.018263 
10  290.4139  0.007676  290.4194  0.039629 
11  319.5688  0.084626  319.6433  0.061659 
12  348.7565  0.164076  348.8699  0.095181 
13  378.0469  0.194652  378.0526  0.226332 
14  407.3608  0.258506  407.3291  0.330992 
15  436.6681  0.383765  436.7842  0.321828 
16  466.1053  0.461704  466.1020  0.503454 
17  495.6124  0.545419  495.6529  0.536663 
Calibration result.
0.2268  

Principal point (pixels)  (515.1859, 514.2069) 
Focal length 
3319.15 
−7.038e − 04  
−2.344e − 06  
2.4705e − 08  
−2.7031e − 11  
3.1804e − 15  
Inclination angle of the image plane(°) ( 
0.1469 (σ = 0.0413) 
Inclination angle of the image plane(°) ( 
0.0524 (σ = 0.0221) 