Abstract:
A method for calibration of a digital celestial sensor is disclosed. The method comprises the following steps: firstly, an integrated mathematic model for imaging of a celestial sensor is established according to external and internal parameters of the calibration system of the celestial sensor. Secondly, by rotating two axes of a rotator by different angles, calibration points data are acquired and sent to a processing computer through an interface circuit. Finally, a two-step calibration program is implemented to calculate the calibration parameters by substituting calibration points&#39; data to the integrated mathematic model. The disclosure also relates to an application device of the calibration method, wherein the device comprises: a celestial simulator to provide simulated sunlight or starlight, a two-axis rotator to acquire different the calibration points&#39; data, a processing computer to record the calibration points&#39; data and calculate the calibration parameters. The calibration method and device apply to many kinds of digital celestial sensors. By integrating external and internal parameters modeling, the disclosure improves the calibration precision. Meanwhile, the whole calibration process is simplified because precise installation and adjustment is not required.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a Continuation-in-part application of U.S. Ser. No. 12/019,539 filed Jan. 24, 2008, which claims priority from Chinese Patent Application Serial No. 200710118498.1 filed Jul. 6, 2007, the disclosures of which, including the respective specifications, drawings and claims, are incorporated herein by reference in their entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    The disclosure relates to measurement techniques for a celestial sensor, especially to a kind of method and device for calibration of a digital celestial sensor. 
       BACKGROUND 
       [0003]    Sun sensors and star sensors are two kinds of important celestial sensors and are widely used in many spacecrafts for attitude measurement. Sun sensors are a kind of attitude sensor for measuring the angle between the sun light and a certain axis or plane of a moving vehicle, and is widely used in many areas such as solar energy utilization and attitude control of spacecraft. New digital sun sensors mainly include: an optical mask with single pinhole or pinhole array, an image sensor such as CMOS (Complementary Metal Oxide Semiconductor) or CCD (Charge Coupled Device), and an information processing circuit. 
         [0004]    The principle of a sun sensor is as follows: sun light is projected onto the image sensor though the pinhole on the optical mask and a spot is formed. The position of the spot changes with the incident angle of sun light. Then, spot image processing and attitude computing are executed by an information processing circuit, and finally, the attitude of the spacecraft is obtained. 
         [0005]    A star sensor is a kind of high precision attitude measurement device for spacecraft that operates by observing stars. The principle of the star senor is as follows: a sky image is captured by an image sensor such as CCD or CMOS, and an image processing procedure is carried out to extract the centroid coordinates and brightness of the stars in the sky image. Next, a star identification program uses this information to find corresponding matches between measured stars (in sky image) and guide stars (in a star catalog). Finally a 3-axis attitude is obtained. 
         [0006]    Before the celestial sensor is put into use, its internal parameters must be precisely calibrated to guarantee high measurement precision. The internal parameters include the focal length F of the optical system, an origin coordinate where the optical axis crosses the image sensor (also called as main point) and distortion coefficients, etc. The calibration of such internal parameters is referred to as celestial sensor calibration. Currently, there are two kinds of calibration methods. The first method is to utilize real sunlight or starlight and perform data acquisition and calibration. The second method uses a celestial simulator to provide simulated sunlight or starlight, and performs data acquisition and calibration with the help of a rotator. For the latter, only the focal length F and the main point coordinate are used in the calibration model, and the calibration precision is higher than the former. Further, the calibration process is more convenient. However, there are some disadvantages with this method. 
         [0007]    For example, the sunlight or starlight vector from the celestial simulator is not strictly vertical to the plane formed by the two rotation axis of the rotator coordinate frame. Moreover, there is installation error between the celestial sensor and the rotator, such that the celestial sensor coordinate frame can not be identical to the rotator coordinate frame. Because of those external factors, such as installation error and adjustment error, there is error in the calibration method which uses only internal parameters in imaging modeling of celestial sensor. Therefore, the precision of estimation of internal parameters is influenced. 
         [0008]    Generally, there is nonlinear distortion in the pinhole imaging model of the celestial sensor. So, errors are introduced into the calibration method which only includes internal parameters of the focal length F and the main point coordinate. 
       SUMMARY 
       [0009]    To solve the problem mentioned above, the disclosure aims at providing a high precision calibration method for a digital celestial sensor. Another purpose of the disclosure is to provide a calibration device to implement the method. 
         [0010]    To reach the aims above, an embodiment of the technical scheme of the disclosure is as follows. 
         [0011]    A calibration method for a digital celestial sensor includes the following steps. 
         [0012]    An integrated mathematic model for imaging of a celestial sensor is established according to external and internal parameters of a calibration system of the celestial sensor. 
         [0013]    By rotating two axes of a rotator by different angles, calibration points&#39; data is acquired and sent to a processing computer through an interface circuit. 
         [0014]    A two-step calibration program is implemented to calculate calibration parameters after substituting the calibration points&#39; data to the integrated mathematic model. 
         [0015]    Step A Includes: 
         [0016]    A1. Establishing a rotator coordinate frame and a celestial sensor coordinate frame, and establishing an external parameters modeling equation according to a rotation matrix from the rotator coordinate frame to the celestial sensor coordinate frame and the pitch and yaw angle of simulated sunlight or starlight in the rotator coordinate frame. 
         [0017]    A2. Establishing an internal parameters modeling equation, wherein the internal parameters include: an origin coordinate wherein an optical axis crosses an image sensor (also referred to as a main point), a focal length of an optical system, and a radial and tangential distortion coefficient of the optical system. 
         [0018]    A3. Establishing an integrated external and internal parameters imaging modeling equation of the celestial sensor according to the external parameters modeling equation and the internal parameters modeling equation of calibration system. 
         [0019]    Step C Includes: 
         [0020]    C1. Assuming that the radial and tangential distortion coefficients of the internal parameters are zeros, the main point coordinate is determined by a nonlinear least square iteration. 
         [0021]    C2. Based on the results from step Cl, the rest of the parameters are calculated by a nonlinear least square iteration. 
         [0022]    A calibration device for digital celestial sensor comprises: a celestial simulator to provide simulated sunlight or starlight, a two-axis rotator with internal and external frames, a bracket on which the celestial sensor is installed, an optical platform to uphold the celestial simulator and the two-axis rotator, and a processing computer connecting with the celestial sensor to perform calibration data acquisition and processing. The celestial simulator and two-axis rotator are installed on each side of the optical platform respectively. 
         [0023]    The processing computer, which comprises a data acquisition module and a data processing module, calculates calibration parameters by a data processing program. 
         [0024]    The data acquisition module acquires the calibration points&#39; data, which includes the rotating angle of the internal frame of the two-axis rotator, the rotating angle of the external frame of the two-axis rotator, and the centroid coordinate of imaging spot at this position. 
         [0025]    The data processing module calculates the final calibration parameters based on the calibration points&#39; data acquired above. 
         [0026]    The calibration device and method of the disclosure has following advantages: 
         [0027]    An integrated external and internal parameters modeling is adopted in disclosure, which avoids the introduction of the error of external parameters into the estimation process of internal parameters. Therefore, the calibration precision of internal parameters is improved. 
         [0028]    The calibration precision of the celestial sensor is improved by considering the distortion coefficients as a part of internal parameters. 
         [0029]    No complicated installation and adjustment is needed, so that the calibration process is simplified noticeably. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0030]    The accompanying drawings, which are incorporated in and constitute a part of specification, illustrate an exemplary embodiment of the present disclosure and, together with the general description given above and the detailed description of the preferred embodiment given below, serve to explain the principles of the present disclosure. 
           [0031]      FIG. 1  is a flow chart for the calibration method of the present disclosure; 
           [0032]      FIG. 2  is a schematic diagram showing the distribution of calibration points on an image sensor in the calibration of a sun sensor; 
           [0033]      FIG. 3  is a schematic diagram showing a structure of a calibration device of the present disclosure; and 
           [0034]      FIG. 4  is a schematic diagram showing the distribution of the calibration points on the image sensor in the calibration of the star sensor. 
       
    
    
     DETAILED DESCRIPTION 
       [0035]    While the claims are not limited to the illustrated embodiments, an appreciation of various aspects of the present disclosure is best gained through a discussion of various examples thereof Referring now to the drawings, illustrative embodiments will be described in detail. Although the drawings represent the embodiments, the drawings are not necessarily to scale and certain features may be exaggerated to better illustrate and explain an innovative aspect of an embodiment. Further, the embodiments described herein are not intended to be exhaustive or otherwise limiting or restricting to the precise form and configuration shown in the drawings and disclosed in the following detailed description. 
         [0036]    The basic principle of the disclosure is as follows. First, an integrated external and internal parameters imaging modeling of a celestial sensor is established, which takes into account the errors such as the installation error of a celestial simulator, the installation error of the celestial sensor on a two-axis rotator, the installation error and the distortion of an optical system, etc. Next, a two-step calibration method is implemented to solve the parameters and high precision of calibration is achieved. 
         [0037]    The disclosure utilizes a sun sensor, for example, to describe the calibration method and device of a celestial sensor. However, it is understood that the present disclosure is not limited to a sun sensor. 
         [0038]    In one embodiment, the disclosure uses an integrated external and internal parameters modeling method to establish a mathematic imaging model of a sun sensor. The detailed steps are as follows. 
         [0039]    Step  1 : The Integrated Imaging Model of Sun Sensor is Established According to External and Internal Parameters of the Calibration System of the Sun Sensor. 
         [0040]    Step  101 : Coordinate Frames are then Established. 
         [0041]    Before describing the external parameters modeling, the coordinate frames involved in the disclosure are explained as follows. 
         [0042]    A sun sensor coordinate frame (marked as Sun) is defined that its X-axis and Y-axis are the row and column of the image sensor respectively. The Z-axis is vertical to the X-Y plane. 
         [0043]    A rotator coordinate frame (Marked as Rot) is defined that its X′-axis and Y′-axis are the horizontal rotation axis and vertical rotation axis of the rotator on which the sun sensor is installed, and the Z′ axis of Rot is vertical to the X′-Y′ plane. 
         [0044]    The sun sensor coordinate frame and rotator coordinate frame defined in the disclosure are both right-hand coordinates (or left-hand coordinates). 
         [0045]    Step  102 : External Parameters Modeling 
         [0046]    External parameters that have effect on the calibration precision of the internal parameters of the sun sensor include: 
         [0047]    A sunlight vector e from a sun simulator is not strictly vertical to a plane formed by the two rotation axis of the rotator coordinate frame; assuming that the expression of vector e in the rotator coordinate frame is: 
         [0000]    
       
         
           
             
               
                 
                   e 
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         [0048]    Here, e 1 , e 2 , e 3  are three direction components of vector e in the coordinate frame Rot, and α, β are the pitch and yaw angles in the coordinate frame Rot, respectively. 
         [0049]    (2) There is installation error between the sun sensor and the rotator, which results in a difference of sun sensor coordinate frame Sun and rotator coordinate frame Rot. Assuming the rotation matrix Rsr denotes the rotation from rotator coordinate frame Rot to sun sensor coordinate frame Sun, is expressed as follows: 
         [0000]        Rsr=Rot ( Z′, φ 1)* Rot ( Y′, β 1)* Rot ( X′, α 1)   (2) 
         [0050]    Here, Rot(X′, α 1 ), Rot(Y′, β 1 ) and Rot(Z′, Φ 1 ) are rotation matrices, which denote rotation angle of α 1  about axis X′, rotation angle of β 1  about axis Y′ and rotation angle of φ 1  about axis Z′ correspondingly. The rotator coordinate frame is transformed to the sun sensor frame coordinate by these rotations. The expressions of these rotations are: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0051]    From above formulas, it can be seen that there are five total external parameters in the calibration system of the sun sensor, namely α, β, α 1 , β 1 , φ 1 . 
         [0052]    Step  103 : Internal Parameters Modeling 
         [0053]    There are errors in the installation of the optical mask of the sun sensor: 
         [0054]    The distance between the optical mask and the imaging plane of the image sensor is not the ideal value F but the real value of F′. 
         [0055]    The point where the pin hole on the optical mask of the sun sensor is projected to image sensor is not the origin of the sun sensor coordinate, and assuming that the coordinate of the real projected origin is (x 0 , y 0 ). 
         [0056]    Moreover, there is distortion in the pinhole imaging because of the glass base of the optical mask of the sun sensor. Assuming that dx and dy represent distortion in x and y direction respectively The radial distortion coefficients and tangential distortion coefficients are expressed as: 
         [0000]    
       
         
           
             
               
                 
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                   { 
                   
                     
                       
                         
                           x 
                           = 
                           
                             
                               x 
                               C 
                             
                             - 
                             
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                               0 
                             
                           
                         
                       
                     
                     
                       
                         
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                           = 
                           
                             
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                   5 
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         [0057]    Here, x c  and y c  are the centroid coordinates of a measured spot; x 0  and y 0  are the coordinates of an origin corresponding to the pinhole; q 1 , q 2 , q 3  are radial distortion coefficients; p 1 , p 2 , p 3  are tangential distortion coefficients. So, there are a total of nine internal parameters, namely x 0 , y 0 , F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3 . 
         [0058]    Step  104 : Establishing the Integrated External and Internal Parameters Imaging Model of the Sun Sensor 
         [0059]    The rotator is rotated to acquire different calibration points&#39; data. Assuming that the real rotation angle about the Y′ axis of rotator is θ 1  and the rotation angle about the X′ axis of rotator is θ 2 , the corresponding rotation matrix Rrot can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Rrot 
                         = 
                         
                           
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                   ( 
                   6 
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         [0060]    According to the external and internal parameters of the calibration system and the real rotation angles of the rotator in the calibration process, the integrated imaging model of sun sensor can be established as following: 
         [0000]    
       
         
           
             
               
                 
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                           [ 
                           
                             
                               
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             C 
                           
                           = 
                           
                             
                               
                                 F 
                                 ′ 
                               
                               * 
                               
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                             
                             + 
                             
                               x 
                               0 
                             
                             + 
                             dx 
                           
                         
                       
                     
                     
                       
                         
                           
                             y 
                             C 
                           
                           = 
                           
                             
                               
                                 F 
                                 ′ 
                               
                               * 
                               
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                             
                             + 
                             
                               y 
                               0 
                             
                             + 
                             dy 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0061]    In the above formula, V is the expression of the sunlight vector e in the current sun sensor coordinate frame when the internal and external frames of the rotator are rotated by θ 1  and θ 2  respectively. 
         [0062]    The integrated external and internal parameters imaging model of the sun sensor is obtained by substituting equations (1)˜(7) into equation (8). The calibration of the sun sensor in the disclosure is to determine the internal parameters (x 0 , y 0 , F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3 ) and external parameters (α, β, α 1 , β 1 , φ 1 ) in the modeling equation according to the calibration points&#39; data. 
         [0063]    Step  2 : Acquisition of Calibration Points&#39; Data 
         [0064]    The two axes of the rotator are rotated by different angles to make sure the imaging spots spread over the whole plane of the image sensor with the sunlight within the field of view of ±55° (as shown in  FIG. 2 ). An interface circuit of the sun sensor transfers the centroid coordinates (x c , y c ) of the imaging spot to the processing computer at each rotation position of the rotator. The processing computer records the rotation angle θ 1  of the external frame and the rotation angle θ 2  of the internal frame simultaneously. When the rotator has rotated for m different positions, m groups of calibration points&#39; data are acquired. 
         [0065]    Step  3 : Data Processing 
         [0066]    It can be seen from the model equation that there are a total of 14 calibration parameters in the calibration system. The precision of these parameters are relatively low and the iteration can&#39;t easily converge if all 14 parameters are determined by a one-time least square method. Therefore, a two-step method is adopted to calculate the 14 parameters. 
         [0067]    Step  301 : Determination of the Internal Parameters x 0  and y 0    
         [0068]    Firstly, assume that the distortion coefficients q 1 , q 2 , q 3 , p 1 , p 2 , p 3  are all equal to zero, so the model equation (8) can be simplified as: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             C 
                           
                           = 
                           
                             
                               
                                 
                                   F 
                                   ′ 
                                 
                                 * 
                                 
                                   
                                     f 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   
                                     f 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                               
                               + 
                               
                                 x 
                                 0 
                               
                             
                             = 
                             
                               
                                 f 
                                 x 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             y 
                             C 
                           
                           = 
                           
                             
                               
                                 
                                   F 
                                   ′ 
                                 
                                 * 
                                 
                                   
                                     f 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   
                                     f 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                               
                               + 
                               
                                 y 
                                 0 
                               
                             
                             = 
                             
                               
                                 f 
                                 y 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0069]    Here, n is a parameter vector which consists of model parameters [x 0 , y 0 , F′, α, β, α 1 , β 1 , φ 1 ]. Since f x  and f y  are both nonlinear functions, a nonlinear least square iteration method is adopted to estimate the parameter vector n. Assuming that x c  and y c  are a measured value while {circumflex over (x)} c  and ŷ c  are estimated values, and Δn is the estimated deviation of the parameter vector, and Δx and Δy are an estimated deviation of x c  and y c  respectively, it gets 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             x 
                           
                           = 
                           
                             
                               
                                 x 
                                 C 
                               
                               - 
                               
                                 
                                   x 
                                   ^ 
                                 
                                 C 
                               
                             
                             ≈ 
                             
                               A 
                                
                               
                                   
                               
                                
                               Δ 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Δ 
                              
                             
                                 
                             
                              
                             y 
                           
                           = 
                           
                             
                               
                                 y 
                                 C 
                               
                               - 
                               
                                 
                                   y 
                                   ^ 
                                 
                                 C 
                               
                             
                             ≈ 
                             
                               B 
                                
                               
                                   
                               
                                
                               Δ 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0070]    Here, A and B are sensitive matrixes, and their expressions are: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           A 
                           = 
                           
                             [ 
                             
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     x 
                                     0 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     y 
                                     0 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     F 
                                     ′ 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   ∂ 
                                   α 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   ∂ 
                                   β 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     α 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     β 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     x 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     ϕ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           B 
                           = 
                           
                             [ 
                             
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     x 
                                     0 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     y 
                                     0 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     F 
                                     ′ 
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   ∂ 
                                   α 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   ∂ 
                                   β 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     α 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     β 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     f 
                                     y 
                                   
                                 
                                 
                                   
                                     ∂ 
                                     ϕ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0071]    Assuming that the number of calibration points&#39; data is m, combining the estimated deviation Δx and Δy and the sensitive matrixes, the iteration equation of parameter vector is established as follows. 
         [0000]      Δ n   (k+1)   =Δn   (k) −( M   k   T   M   k ) −1   M   k   T   P   (k)    (12) 
         [0072]    In the above equation, P consists of an estimated deviation Δx and Δy, and M consists of two sensitive matrixes A and B. Their expressions are: 
         [0000]    
       
         
           
             P 
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             x 
                             1 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             x 
                             m 
                           
                         
                       
                     
                     
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             y 
                             1 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             y 
                             m 
                           
                         
                       
                     
                   
                   ] 
                 
                 · 
                 M 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         A 
                         1 
                       
                     
                   
                   
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         A 
                         m 
                       
                     
                   
                   
                     
                       
                         B 
                         1 
                       
                     
                   
                   
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         B 
                         m 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0073]    Here, k is the iteration times and can be set between 5 and 10. Among the calculated model parameters, when iteration ends, only (x 0 , y 0 ) is chosen as the final calibration result to be used in next step to determinate the other parameters. 
         [0074]    Step  302 : Determination of Internal Parameters F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3  and External Parameters 
         [0075]    Substituting (x 0 , y 0 ) calculated from the previous step into the model equation (8), and using vector j to denote model parameters [F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3 , α, β, α 1 , β 1 , φ 1 ], it Gets: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         x 
                       
                       = 
                       
                         
                           
                             x 
                             C 
                           
                           - 
                           
                             
                               x 
                               ^ 
                             
                             C 
                           
                         
                         ≈ 
                         
                           C 
                            
                           
                               
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         y 
                       
                       = 
                       
                         
                           
                             y 
                             C 
                           
                           - 
                           
                             
                               y 
                               ^ 
                             
                             C 
                           
                         
                         ≈ 
                         
                           D 
                            
                           
                               
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                 
               
                 
             
           
         
       
     
         [0076]    Correspondingly, the sensitive matrixes C and D change to: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     C 
                     = 
                     
                       [ 
                       
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             ∂ 
                             
                               F 
                               ′ 
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               q 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               q 
                             
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               q 
                             
                              
                             
                                 
                             
                              
                             3 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               p 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               p 
                             
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               p 
                             
                              
                             
                                 
                             
                              
                             3 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             ∂ 
                             α 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             ∂ 
                             β 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               α 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               β 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               x 
                             
                           
                           
                             
                               ∂ 
                               ϕ 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     D 
                     = 
                     
                       [ 
                       
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             ∂ 
                             
                               F 
                               ′ 
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               q 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               q 
                             
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               q 
                             
                              
                             
                                 
                             
                              
                             3 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               p 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               p 
                             
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               p 
                             
                              
                             
                                 
                             
                              
                             3 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             ∂ 
                             α 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             ∂ 
                             β 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               α 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               β 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             ∂ 
                             
                               f 
                               y 
                             
                           
                           
                             
                               ∂ 
                               ϕ 
                             
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0077]    A same nonlinear least square iteration method is adopted to estimate the parameter vector j, and a similar iteration equation of parameter vector is established 
         [0000]      Δ j   (k+1)   =Δj   (k) −( N   k   T   N   k ) −1   N   k   T   P   (k)    (13) 
         [0078]    In the above equation, N comprises sensitive matrixes C and D, and their expressions are: 
         [0000]    
       
         
           
             N 
             = 
             
               [ 
               
                 
                   
                     
                       C 
                       1 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       C 
                       m 
                     
                   
                 
                 
                   
                     
                       D 
                       1 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       D 
                       m 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0079]    Here, k is iteration times and can be set between 5 and 10. When the iteration ends, the model parameters F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3 , α, β, α 1 , β 1  and φ 1  are determined and chosen as the final calibration result. 
         [0080]    Combining (x 0 , y 0 ) determined in the first step and F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3 , α, β, α 1 , β 1 , φ 1  determined in the second step, all the calibration parameters in the calibration system are determined. 
         [0081]    Finally, all calibrated internal parameters x 0 , y 0 , F′, q 1 , q 2 , q 3 , p 1 , p 2  and p 3  are substituted into corresponding attitude conversion formulas of the sun sensor, then precise attitude angle of the sunlight in the sun sensor coordinate frame will be calculated. Thereby, the attitude information of the satellites or spacecraft on which the sun sensor is installed is determined. 
         [0082]    As shown in  FIG. 3 , the calibration device in the disclosure comprises a sun simulator  1  to provide sunlight, a two-axis rotator  2  with external and internal frames, a bracket  3  to install the sun sensor, an optical platform  4  to uphold the sun simulator  1  and two-axis rotator  2 , and a processing computer  5  to perform data acquisition and computing. The sun simulator  1  and the two-axis rotator  2  are installed on the each side of the optical platform respectively, and the sun simulator is used to provide needed sunlight. 
         [0083]    The processing computer  5  includes a data acquisition module and a data processing module. The data acquisition module acquires the calibration points&#39; data, which includes the rotating angle θ 1  of the external frame, the rotating angle θ 2  of the internal frame and the centroid coordinate (x c , y c ) of imaging spots at this position. A two-step method and nonlinear least square method are used by the data processing module to determine the final calibration parameters. During the calibration process using the calibration device, the sun sensor  6  is installed on the bracket  3 . Different calibration points&#39; data is acquired by rotating the external and internal frames of the rotator by different angles. The processing computer  5  records these calibration points&#39; data and calculates the corresponding calibration parameters. 
         [0084]    In one specific embodiment, the rotator used in the disclosure has the precision of ±0.4″ for the external frame and ±0.3″ for the internal frame. The radiation intensity of the sun simulator is a 0.1 solar constant. The diameter of the effective radiation area is 200 mm, and the collimation angle of light beam is 32′. 
         [0085]    A total of 84 groups of recorded calibration point&#39;s data are listed below in Table 1. 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
             
               
                   
                 m 
               
             
          
           
               
                   
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
               
                 θ1(°) 
                 8 
                 4 
                 −4 
                 −8 
                 −4 
                 4 
                 16 
               
               
                 θ2(°) 
                 0 
                 7 
                 7 
                 0 
                 −7 
                 −7 
                 0 
               
               
                 x c (pixel) 
                 490.5938 
                 507.8291 
                 543.1906 
                 561.3750 
                 544.1250 
                 508.5344 
                 454.1313 
               
               
                 y c (pixel) 
                 518.7813 
                 549.8000 
                 550.2844 
                 519.7188 
                 488.4250 
                 488.0000 
                 518.3125 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
               
               
                   
               
               
                 θ1(°) 
                 8 
                 −8 
                 −16 
                 −8 
                 8 
                 25 
                 23 
               
               
                 θ2(°) 
                 14 
                 14 
                 0 
                 −14 
                 −14 
                 0 
                 9 
               
               
                 x c (pixel) 
                 488.9906 
                 561.3781 
                 598.2000 
                 563.4437 
                 490.2813 
                 410.2156 
                 419.0313 
               
               
                 y c (pixel) 
                 580.9938 
                 582.0313 
                 520.2000 
                 456.7437 
                 455.8750 
                 517.9063 
                 557.2813 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
               
               
                   
               
               
                 θ1(°) 
                 19 
                 12 
                 4 
                 −4 
                 −12 
                 −19 
                 −23 
               
               
                 θ2(°) 
                 17 
                 22 
                 25 
                 25 
                 22 
                 17 
                 9 
               
               
                 x c (pixel) 
                 436.1250 
                 468.1187 
                 505.5250 
                 543.6625 
                 581.6875 
                 614.8469 
                 633.0531 
               
               
                 y c (pixel) 
                 593.8063 
                 618.4063 
                 634.1969 
                 634.8125 
                 620.2500 
                 596.7156 
                 560.3156 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
               
               
                   
               
               
                 θ1(°) 
                 −25 
                 −23 
                 −19 
                 −12 
                 −4 
                 4 
                 12 
               
               
                 θ2(°) 
                 0 
                 −9 
                 −17 
                 −22 
                 −25 
                 −25 
                 −22 
               
               
                 x c (pixel) 
                 643.1250 
                 634.8219 
                 617.9156 
                 585.3406 
                 547.1594 
                 508.2531 
                 469.8563 
               
               
                 y c (pixel) 
                 520.9688 
                 481.0000 
                 443.5812 
                 418.2250 
                 402.0125 
                 401.6594 
                 417.0469 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
                 35 
               
               
                   
               
               
                 θ1(°) 
                 19 
                 23 
                 35 
                 33 
                 26 
                 17 
                 6 
               
               
                 θ2(°) 
                 −17 
                 −9 
                 0 
                 13 
                 24 
                 31 
                 35 
               
               
                 x c (pixel) 
                 437.0000 
                 419.3781 
                 354.5000 
                 362.7813 
                 394.7813 
                 437.6250 
                 492.9063 
               
               
                 y c (pixel) 
                 441.6469 
                 478.4469 
                 517.4063 
                 574.0000 
                 626.4375 
                 664.5938 
                 689.2500 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
               
               
                   
               
               
                 θ1(°) 
                 −6 
                 −17 
                 −26 
                 −33 
                 −35 
                 −33 
                 −26 
               
               
                 θ2(°) 
                 35 
                 31 
                 24 
                 13 
                 0 
                 −13 
                 −24 
               
               
                 x c (pixel) 
                 555.3750 
                 611.7813 
                 656.3750 
                 690.6563 
                 700.6437 
                 693.6906 
                 661.6844 
               
               
                 y c (pixel) 
                 690.3750 
                 667.5625 
                 630.6000 
                 578.7875 
                 521.9219 
                 464.2500 
                 409.8344 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
               
               
                   
               
               
                 θ1(°) 
                 −17 
                 −6 
                 6 
                 17 
                 26 
                 33 
                 45 
               
               
                 θ2(°) 
                 −31 
                 −35 
                 −35 
                 −31 
                 −24 
                 −13 
                 0 
               
               
                 x c (pixel) 
                 617.9688 
                 560.8750 
                 496.5656 
                 439.3875 
                 395.3531 
                 362.8438 
                 285.8062 
               
               
                 y c (pixel) 
                 369.9438 
                 344.2781 
                 343.6938 
                 368.3125 
                 407.1563 
                 460.4375 
                 516.8438 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
               
               
                   
               
               
                 θ1(°) 
                 42 
                 33 
                 21 
                 7 
                 −7 
                 −21 
                 −33 
               
               
                 θ2(°) 
                 19 
                 33 
                 41 
                 45 
                 45 
                 41 
                 33 
               
               
                 x c (pixel) 
                 297.7500 
                 340.0313 
                 402.8937 
                 482.1781 
                 564.8781 
                 645.9562 
                 711.8312 
               
               
                 y c (pixel) 
                 599.5625 
                 672.4063 
                 724.9406 
                 756.8063 
                 758.4031 
                 729.5563 
                 678.7188 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
               
               
                   
               
               
                 θ1(°) 
                 −42 
                 −45 
                 −42 
                 −33 
                 −21 
                 −7 
                 7 
               
               
                 θ2(°) 
                 19 
                 0 
                 −19 
                 −33 
                 −41 
                 −45 
                 −45 
               
               
                 x c (pixel) 
                 757.7719 
                 772.5313 
                 763.1094 
                 720.9344 
                 656.031 
                 573.1188 
                 486.9376 
               
               
                 y c (pixel) 
                 606.5938 
                 523.2250 
                 438.1875 
                 361.4688 
                 305.3438 
                 271.6875 
                 271.1250 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 64 
                 65 
                 66 
                 67 
                 68 
                 69 
                 70 
               
               
                   
               
               
                 θ1(°) 
                 21 
                 33 
                 42 
                 55 
                 50 
                 39 
                 24 
               
               
                 θ2(°) 
                 −41 
                 −33 
                 −19 
                 0 
                 26 
                 43 
                 51 
               
               
                 x c (pixel) 
                 404.1875 
                 339.4063 
                 296.8750 
                 192.3687 
                 216.7094 
                 268.3406 
                 360.0000 
               
               
                 y c (pixel) 
                 303.5313 
                 358.1250 
                 433.1563 
                 516.1063 
                 630.3469 
                 732.7344 
                 802.9063 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
               
               
                   
               
               
                 θ1(°) 
                 8 
                 −8 
                 −24 
                 −39 
                 −50 
                 −55 
                 −50 
               
               
                 θ2(°) 
                 55 
                 55 
                 51 
                 43 
                 26 
                 0 
                 −26 
               
               
                 x c (pixel) 
                 466.2719 
                 579.3750 
                 688.5125 
                 785.5000 
                 842.8563 
                 872.4375 
                 851.6000 
               
               
                 y c (pixel) 
                 847.9688 
                 850.5031 
                 809.8906 
                 742.6594 
                 640.5406 
                 525.0313 
                 406.3125 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
               
               
                 θ1(°) 
                 −39 
                 −24 
                 −8 
                 8 
                 24 
                 39 
                 50 
               
               
                 θ2(°) 
                 −43 
                 −51 
                 −55 
                 −55 
                 −51 
                 −43 
                 −26 
               
               
                 x c (pixel) 
                 801.0313 
                 704.8688 
                 591.8750 
                 471.9031 
                 359.5313 
                 264.8906 
                 214.3094 
               
               
                 y c (pixel) 
                 296.0969 
                 219.2594 
                 170.4094 
                 169.8906 
                 217.6375 
                 292.5125 
                 400.0000 
               
               
                   
               
             
          
         
       
     
         [0086]    A calibration result is obtained by processing the calibration points&#39; data listed in Table 1 using the calibration method described above. The calibration result is listed in Table 2. 
         [0000]    
       
         
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 x 0 (pixel) 
                 y 0 (pixel) 
                 F′(pixel) 
                 q1 
                 q2 
                 q3 
                 p1 
               
               
                   
               
             
             
               
                 523 
                 525 
                 251 
                 −7.27e−7 
                 3.28e−12 
                 −7.826e−18 
                 4.85e−6 
               
               
                 p2 
                 p3 
                 α(°) 
                 β(°) 
                 α1(°) 
                 β1(°) 
                 φ1(°) 
               
               
                 1.98e−7 
                 −1.6e−6 
                 44.832 
                 93.798 
                 1.686 
                 0.354 
                 0.786 
               
               
                   
               
             
          
         
       
     
         [0087]    The total statistical square root error of x c  and y c  are 5.09 pixels and 4.27 pixels respectively. Substituting the parameters calibrated by the method of the present disclosure into the attitude computing formula of the sun sensor, an attitude precision of 0.02 ( 1 σ) is obtained. Because totally 14 external and internal parameters are used in the invention, theoretically at least 14 groups of calibration data are needed to solve the calibration parameters. Generally, in order to obtain more precise parameters, 50-100 groups of calibration data are acquired. Meanwhile, the calibration points spread over the field of view of sun sensor as widely as possible. Of course, the more the calibration points are used, the more precise the calibration result are, but at the cost of computing. 
         [0088]    Replacing the sun sensor and sun simulator with a star sensor and star simulator respectively, the star sensor can be calibrated in the same way with high precision independent of complicated installment and adjustment. 
         [0089]    An example of calibration of the star sensor using the calibration method in the disclosure is provided as follows. 
         [0090]    The star sensor to be calibrated has a field of view of 10.9′×10.9′ and a 1024×1024 image sensor with a pixel element size of 0.015×0.015 mm. The star simulator can provide starlight with a magnitude from 4 Mv to 8 Mv. 
         [0091]    Similar to the data acquisition process of the calibration of the sun sensor, rotating the two axes of the rotator by different angles to make sure the imaging (star) spots spread over the whole plane of the image sensor within the field of view of ±4 (as shown in  FIG. 4 ), a total of 81 groups of the calibration point&#39;s data are acquired (as shown in Table. 3). 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
             
               
               
               
               
               
               
             
           
               
                   
                 TABLE 3 
               
               
                   
                   
               
             
             
               
                   
                 m 
               
             
          
           
               
                   
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
               
                 θ1(°) 
                 −4 
                 −3 
                 −2 
                 −1 
                 0 
                 1 
                 2 
               
               
                 θ2(°) 
                 4 
                 4 
                 4 
                 4 
                 4 
                 4 
                 4 
               
               
                 x c (pixel) 
                 846.023 
                 846.0038 
                 846.0034 
                 845.991 
                 846.0664 
                 846.1506 
                 846.2582 
               
               
                 y c (pixel) 
                 87.5376 
                 181.2595 
                 274.7782 
                 368.2523 
                 461.5407 
                 554.8719 
                 648.2764 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
               
               
                   
               
               
                 θ1(°) 
                 3 
                 4 
                 4 
                 3 
                 2 
                 1 
                 0 
               
               
                 θ2(°) 
                 4 
                 4 
                 3 
                 3 
                 3 
                 3 
                 3 
               
               
                 x c (pixel) 
                 846.4803 
                 846.6616 
                 753.1895 
                 753.0588 
                 752.9397 
                 752.8297 
                 752.6954 
               
               
                 y c (pixel) 
                 741.6939 
                 835.3656 
                 835.1762 
                 741.5762 
                 648.1623 
                 554.9021 
                 461.6425 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
               
               
                   
               
               
                 θ1(°) 
                 −1 
                 −2 
                 −3 
                 −4 
                 −4 
                 −3 
                 −2 
               
               
                 θ2(°) 
                 3 
                 3 
                 3 
                 3 
                 2 
                 2 
                 2 
               
               
                 x c (pixel) 
                 752.6478 
                 752.6125 
                 752.6111 
                 752.6793 
                 659.4337 
                 659.3499 
                 659.3284 
               
               
                 y c (pixel) 
                 368.3785 
                 275.0673 
                 181.5721 
                 87.8912 
                 88.1736 
                 181.7621 
                 275.2395 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
               
               
                   
               
               
                 θ1(°) 
                 −1 
                 0 
                 1 
                 2 
                 3 
                 4 
                 4 
               
               
                 θ2(°) 
                 2 
                 2 
                 2 
                 2 
                 2 
                 2 
                 1 
               
               
                 x c (pixel) 
                 659.3432 
                 659.4309 
                 659.5709 
                 659.72 
                 659.8921 
                 660.0049 
                 566.8324 
               
               
                 y c (pixel) 
                 368.5127 
                 461.7081 
                 554.9435 
                 648.1823 
                 741.5134 
                 835.044 
                 835.0712 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
                 35 
               
               
                   
               
               
                 θ1(°) 
                 3 
                 2 
                 1 
                 0 
                 −1 
                 −2 
                 −3 
               
               
                 θ2(°) 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
                 1 
               
               
                 x c (pixel) 
                 566.6722 
                 566.5035 
                 566.2811 
                 566.1572 
                 566.133 
                 566.1131 
                 566.0962 
               
               
                 y c (pixel) 
                 741.5379 
                 648.2287 
                 554.9968 
                 461.7978 
                 368.5974 
                 275.3243 
                 181.8933 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
               
               
                   
               
               
                 θ1(°) 
                 −4 
                 −4 
                 −3 
                 −2 
                 −1 
                 0 
                 1 
               
               
                 θ2(°) 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 x c (pixel) 
                 566.1277 
                 473.0167 
                 472.988 
                 473.036 
                 473.0403 
                 473.0709 
                 473.137 
               
               
                 y c (pixel) 
                 88.2568 
                 88.2656 
                 181.9414 
                 275.3722 
                 368.6592 
                 461.866 
                 555.1176 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
               
               
                   
               
               
                 θ1(°) 
                 2 
                 3 
                 4 
                 4 
                 3 
                 2 
                 1 
               
               
                 θ2(°) 
                 0 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 x c (pixel) 
                 473.3198 
                 473.4968 
                 473.6655 
                 380.4244 
                 380.245 
                 380.0918 
                 379.9698 
               
               
                 y c (pixel) 
                 648.3413 
                 741.6483 
                 835.2176 
                 835.4632 
                 741.859 
                 648.5022 
                 555.2623 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
               
               
                   
               
               
                 θ1(°) 
                 0 
                 −1 
                 −2 
                 −3 
                 −4 
                 −4 
                 −3 
               
               
                 θ2(°) 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −2 
                 −2 
               
               
                 x c (pixel) 
                 379.8922 
                 379.8083 
                 379.7638 
                 379.778 
                 379.8063 
                 286.3837 
                 286.3974 
               
               
                 y c (pixel) 
                 461.9989 
                 368.7132 
                 275.4092 
                 181.9218 
                 88.2488 
                 88.0188 
                 181.7662 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
               
               
                   
               
               
                 θ1(°) 
                 −2 
                 −1 
                 0 
                 1 
                 2 
                 3 
                 4 
               
               
                 θ2(°) 
                 −2 
                 −2 
                 −2 
                 −2 
                 −2 
                 −2 
                 −2 
               
               
                 x c (pixel) 
                 286.41 
                 286.4695 
                 286.5712 
                 286.612 
                 286.7459 
                 286.907 
                 287.0918 
               
               
                 y c (pixel) 
                 275.3348 
                 368.7194 
                 462.0791 
                 555.3959 
                 648.7273 
                 742.1938 
                 835.8131 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 64 
                 65 
                 66 
                 67 
                 68 
                 69 
                 70 
               
               
                   
               
               
                 θ1(°) 
                 4 
                 3 
                 2 
                 1 
                 0 
                 −1 
                 −2 
               
               
                 θ2(°) 
                 −3 
                 −3 
                 −3 
                 −3 
                 −3 
                 −3 
                 −3 
               
               
                 x c (pixel) 
                 193.6289 
                 193.4583 
                 193.3174 
                 193.1805 
                 193.0724 
                 192.9838 
                 192.9098 
               
               
                 y c (pixel) 
                 836.2621 
                 742.5339 
                 648.9849 
                 555.5375 
                 462.1253 
                 368.668 
                 275.1848 
               
               
                   
               
             
          
           
               
                   
                 m 
               
             
          
           
               
                   
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
               
               
                   
               
               
                 θ1(°) 
                 −3 
                 −4 
                 −4 
                 −3 
                 −2 
                 −1 
                 0 
               
               
                 θ2(°) 
                 −3 
                 −3 
                 −4 
                 −4 
                 −4 
                 −4 
                 −4 
               
               
                 x c (pixel) 
                 192.8675 
                 192.9053 
                 99.1793 
                 99.1509 
                 99.1857 
                 99.2415 
                 99.3311 
               
               
                 y c (pixel) 
                 181.5179 
                 87.6925 
                 87.3046 
                 181.2268 
                 275.0071 
                 368.6396 
                 462.224 
               
               
                   
               
             
          
           
               
                   
                 m 
                   
               
             
          
           
               
                   
                   
                 78 
                 79 
                 80 
                 81 
               
               
                   
                   
               
               
                   
                 θ1(°) 
                 1 
                 2 
                 3 
                 4 
               
               
                   
                 θ2(°) 
                 −4 
                 −4 
                 −4 
                 −4 
               
               
                   
                 x c (pixel) 
                 99.4387 
                 99.5852 
                 99.7679 
                 99.9614 
               
               
                   
                 y c (pixel) 
                 555.7563 
                 649.3598 
                 743.0164 
                 836.8736 
               
               
                   
                   
               
             
          
         
       
     
         [0092]    The calibration result is listed in Table 4. 
         [0000]    
       
         
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 x 0 (pixel) 
                 y 0 (pixel) 
                 F′(mm) 
                 q1 
                 q2 
                 q3 
                 p1 
               
               
                   
               
             
             
               
                 495 
                 506 
                 80.0891 
                 3.2184e−6 
                 8.0914e−9 
                 −3.2403e−10 
                 −1.7636e−5 
               
               
                 p2 
                 p3 
                 α(°) 
                 β(°) 
                 α1(°) 
                 β1(°) 
                 φ1(°) 
               
               
                 −7.5307e−5 
                 −6.4e−3 
                 74.7025 
                 89.0048 
                 −1.1935 
                 −0.2108 
                 0.0536 
               
               
                   
               
             
          
         
       
     
         [0093]    The root of mean square (RMS) error of x c  and y c  are 0.0702 pixel and 0.0707 pixel respectively. Substituting the calibrated parameters into the attitude computing formula of the star sensor, an attitude precision of 3.8″ (1σ) is obtained. 
         [0094]    The foregoing description of various embodiments of the disclosure has been present for purpose of illustration and description. It is not intent to be exhaustive or to limit the disclosure to the precise embodiments disclosed. Numerous modifications or variations are possible in light of the above teachings. The embodiments discussed where chosen and described to provide the best illustration of the principles of the disclosure and its practical application to thereby enable one of ordinary skill in the art to utilize the disclosure in various embodiments and with various modifications as are suited to the particular use contemplated. All such modifications and variations are within the scope of the disclosure as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled.