Abstract:
A method for calibration of a digital sun sensor is disclosed. The method comprises following steps. First, an integrated mathematic model for imaging of a sun sensor is established according to the external and internal parameters of the calibration system of the sun sensor. Next, the two axis of the rotator are rotated by different angles. Then, calibration points&#39; 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 the calibration points&#39; data to the integrated mathematic model. The disclosure also relates to an application device of the calibration method. The device comprises: a sun simulator to provide the incident sunlight, a two-axis rotator to acquire different the calibration points&#39; data, and 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 sun sensors. By integrated external and internal parameters modeling, the disclosure improves 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 claims priority from Chinese Patent Application Serial No. 200710118498.1 filed Jul. 6, 2007, the disclosure of which, including the specification, drawings and claims, is incorporated herein by reference in its entirety. 
       TECHNICAL FIELD 
       [0002]    The disclosure relates to the measurement techniques for a sun sensor, and more particularly to a kind of method and device for calibration of a digital sun sensor. 
       BACKGROUND 
       [0003]    A sun sensor is a kind of attitude sensor for measuring the angle between the sun light and a certain axis or plane of a moving vehicle. Sun sensors are widely used in many areas such as, for example, 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 a CMOS (Complementary Metal Oxide Semiconductor) or a CCD (Charge Coupled Device), and an information processing circuit. The principle of a sun sensor is as follows: the 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. Finally, the attitude of spacecraft is obtained. 
         [0004]    Before the sun sensor is put into use, the internal parameters of it must be precisely calibrated to guarantee high measurement precision. The internal parameters include the focal length F, which equals to distance between the optical mask and the image sensor, the origin coordinate where the pin hole is projected to the image sensor (also called as main point) and the distortion coefficients, etc. The calibration of such internal parameters is referred to as sun sensor calibration. Currently, there are two kinds of calibration methods. The first utilizes the real sunlight and performs data acquisition and calibration when the sun is at or near its zenith. The second uses sun simulators to provide sun light, perform data acquisition, and calibration with the help of rotator. For the latter, only focal length F and an origin coordinate are used in the calibration model, and the calibration precision is higher than the former. Moreover, the calibration process is more convenient. However, there are some disadvantages with this method: 
         [0005]    The sunlight vector from the sun simulator is not strictly vertical to the plane formed by the two rotation axes of the rotator coordinate frame. Moreover, there is installation error between the sun sensor and the rotator, such that the sun 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 a sun sensor. Therefore, the precision of estimation of internal parameters is adversely influenced. 
         [0006]    Generally, the optical mask of a sun sensor is shaped by etching a pinhole on a glass base. Because of the refraction of glass base and the limitation of the processing technique, there is nonlinear distortion in the pinhole imaging model of a sun sensor. Accordingly, errors are introduced into the calibration method which only includes internal parameters of focal length F and origin coordinate. 
       SUMMARY 
       [0007]    To address the problems mentioned above, the disclosure aims at providing a high precision calibration method and device for a digital sun sensor. 
         [0008]    To reach the aims above, the technical scheme of the invention is as follows. 
         [0009]    A calibration method for a digital sun sensor is disclosed, which includes the following steps. 
         [0010]    A. First, an integrated mathematic model for imaging of a sun sensor is established according to external and internal parameters of a calibration system of a sun sensor. 
         [0011]    B. 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. 
         [0012]    C. A two-step calibration program is then implemented to calculate the calibration parameters after substituting calibration points&#39; data to the integrated mathematic model. 
         [0013]    In one embodiment of the disclosed method, Step A further comprises: 
         [0014]    A1. Establishing a rotator coordinate frame and a sun sensor coordinate frame, and establishing an external parameters&#39; modeling equation according to a rotation matrix from the rotator coordinate frame to the sun sensor coordinate frame and a pitch and yaw angle of simulated sun light in the rotator coordinate frame. 
         [0015]    A2. Establishing an internal parameters&#39; modeling equation, wherein the internal parameters include: the origin coordinate where a pin hole on an optical mask of a sun sensor is projected to an image sensor, a focal length which equals to a distance between the optical mask and the image sensor, and a radial and tangential distortion coefficient of the optical mask. 
         [0016]    A3. Establishing an integrated external and internal parameters imaging modeling equation of the sun sensor according to the external parameters modeling equation and the internal parameters modeling equation of calibration system. 
         [0017]    In one embodiment, Step C of the disclosed method further includes: 
         [0018]    C1. Assuming that the radial and tangential distortion coefficients of internal parameters are zeros, the origin coordinate where the pin hole is projected to image sensor is determined by a nonlinear least square iteration. 
         [0019]    C2. Based on the results from step C1, the rest of the parameters are calculated by a nonlinear least square iteration. 
         [0020]    An embodiment of a calibration device for digital sun sensor comprises: a sun simulator to provide incident sun light, a two-axis rotator with internal and external frames, a bracket on which the sun sensor is installed, an optical platform to uphold the sun simulator and two-axis rotator, and a processing computer connected to the sun sensor to perform calibration data acquisition and processing. The sun simulator and two-axis rotator are installed on the each side of the optical platform respectively. 
         [0021]    The processing computer, which comprises a data acquisition module and a data processing module, calculates calibration parameters by a data processing program. 
         [0022]    The data acquisition module acquires the calibration points&#39; data, which includes the rotating angle of internal frame of the two-axis rotator, the rotating angle of external frame of the two-axis rotator and the centroid coordinate of imaging spot at this position. 
         [0023]    The data processing module calculates the final calibration parameters based on the calibration points&#39; data acquired above. 
         [0024]    The embodiments disclosed have following advantages: 
         [0025]    An integrated external and internal parameters modeling is adopted in the disclosure, which avoids the introduction of the error of external parameters into the estimation process of internal parameters. Therefore, the calibration precision of the internal parameters is improved. 
         [0026]    The calibration precision is improved by considering the distortion coefficients as a part of the internal parameters. 
         [0027]    No complicated installation and adjustment is needed, so that the calibration process is simplified noticeably. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0028]    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 embodiment given below, serve to explain the principles of the present disclosure. 
           [0029]      FIG. 1  is a flow chart illustrating the method of the present embodiment; 
           [0030]      FIG. 2  is a schematic diagram illustrating the distribution of the calibration points on the image sensor; 
           [0031]      FIG. 3  is a schematic diagram showing a structure of the calibration device of the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0032]    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. 
         [0033]    The basic principle of the disclosure is establishing an integrated external and internal parameters imaging modeling of sun sensor, which takes into account the errors such as the installation error of sun simulator, the installation error of sun sensor on the two-axis rotator, the installation error of optical mask and the distortion of optical mask, etc; a two-step calibration method is implemented to solve the parameters and high precision of calibration is achieved. 
         [0034]    The disclosure uses an integrated external and internal parameters modeling method to establish the mathematic imaging model of a sun sensor. The detailed steps are as follows. 
         [0035]    Step  1 : The integrated imaging model of sun sensor is established according to the external and internal parameters of the calibration system of a sun sensor, 
         [0036]    Step  101 : Coordinate frames are established. 
         [0037]    Before the description of the external parameters modeling, the coordinate frames involved in the disclosure are explained as follows. 
         [0038]    The sun sensor coordinate frame (marked as Sun) is defined. That is, its X-axis and Y-axis are the row and column of the image sensor respectively, and the Z-axis is vertical to the X-Y plane. 
         [0039]    The rotator coordinate frame (Marked as Rot) is defined such 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. 
         [0040]    The sun sensor coordinate frame and rotator coordinate frame defined in the disclosure are both right-hand coordinate (either left-hand coordinate). 
         [0041]    Step  102 : External parameters modeling 
         [0042]    The external parameters that have effect on the calibration precision of the internal parameters of the sun sensor include: 
         [0043]    (1) Sunlight vector e from a sun simulator is not strictly vertical to the 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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                               e 
                                
                               
                                   
                               
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                                
                               
                                   
                               
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                     = 
                     
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                   ( 
                   1 
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         [0044]    Here, e 1 , e 2 , e 3  are three direction components of vector e in the coordinate frame Rot, and α, β are pitch and yaw angles in the coordinate frame Rot respectively. 
         [0045]    (2) There is installation error between the sun sensor and the rotator which results in the difference of sun sensor coordinate frame Sun and rotator coordinate frame Rot. Assuming that 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) 
         [0046]    Here, Rot(X′,α 1 ), Rot(Y′,β 1 ) and Rot(Z′,φ 1 ) are rotation matrixes which denote rotation an angle of α 1  about axis X′, rotation an angle of β 1  about axis Y′ and rotation an angle of φ 1  about axis Z′ respectively- 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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         [0047]    From above formulas, it can be seen that there are five external parameters, in total, in the calibration system of sun sensor; namely α, β, α 1 , β 1 , φ 1 . 
         [0048]    Step  103 : Internal parameters modeling 
         [0049]    There are errors in the installation of an optical mask of a sun sensor: 
         [0050]    (3) The distance between the optical mask and the imaging plane of image sensor is not the ideal value F but the real value of F′. 
         [0051]    (4) The point where the pin hole on the optical mask of sun sensor is projected to an image sensor is not the origin of the sun sensor coordinate, and assuming the coordinate of the real projected origin is (x 0 , y 0 ) 
         [0052]    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 the distortion in the x and y direction respectively, the radial distortion coefficients and tangential distortion coefficients are expressed as: 
         [0000]    
       
         
           
             
               
                 
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                                       2 
                                     
                                   
                                 
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                   ( 
                   4 
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                   { 
                   
                     
                       
                         
                           x 
                           = 
                           
                             
                               x 
                               C 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           y 
                           = 
                           
                             
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                               C 
                             
                             - 
                             
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                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             r 
                             2 
                           
                           = 
                           
                             
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                               2 
                             
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                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0053]    Here, x c  and y c  are the centroid coordinates of a measured spot; x 0  and y 0  are the coordinates of the 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 . 
         [0054]    Step  104 : Establishing the integrated external and internal parameters imaging model of a sun sensor 
         [0055]    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 
                         = 
                         
                           
                             Rot 
                              
                             
                               ( 
                               
                                 
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                                    
                                   
                                       
                                   
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                                   1 
                                 
                               
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                         = 
                         
                           
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                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0056]    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]    
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         = 
                         
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                         = 
                         
                           Rsr 
                           * 
                           Rrot 
                           * 
                           e 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
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                           * 
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                                     3 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             C 
                           
                           = 
                           
                             
                               
                                 F 
                                 ′ 
                               
                               * 
                               
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                             
                             + 
                             
                               x 
                               0 
                             
                             + 
                             dx 
                           
                         
                       
                     
                     
                       
                         
                           
                             y 
                             C 
                           
                           = 
                           
                             
                               
                                 F 
                                 ′ 
                               
                               * 
                               
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                             
                             + 
                             
                               y 
                               0 
                             
                             + 
                             dy 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0057]    In the above formula, V is the expression of 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. 
         [0058]    The integrated external and internal parameters imaging model of a sun sensor is obtained by substituting equations (1)-(7) into equation (8). The calibration of 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. 
         [0059]    Step  2 : Acquisition of calibration points&#39; data 
         [0060]    The two axes of the rotator are rotated by different angles to make sure the imaging spots spread over the whole plane of image sensor with the sunlight within the field of view of ±55° (as shown in  FIG. 2 ). The interface circuit of the sun sensor transfers the centroid coordinates (x 0 , y 0 ) of the maging 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, 
         [0061]    Step  3 : Data processing 
         [0062]    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. 
         [0063]    Step  301 : Determination of the internal parameters x 0  and y 0    
         [0064]    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 
                   ) 
                 
               
             
           
         
       
     
         [0065]    Here, n is a parameter vector which consists of the 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 the measured values while {circumflex over (x)} c  and ŷ c  are the estimated values, and Δn is the estimated deviation of the parameter vector, and Δx and Δy are 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 
                   ) 
                 
               
             
           
         
       
     
         [0066]    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 
                   ) 
                 
               
             
           
         
       
     
         [0067]    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 following: 
         [0000]      Δ n   (k+1)   =Δn   (k) −( M   k   T   M   k ) −1   M   k   T   P   (k)    (12) 
         [0068]    In the above equation, P consists of 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 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0069]    Here, k is 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. 
         [0070]    Step  302 : Determination of internal parameters F′, q 1 , q 2 , q 3 , p 1 , p 2 , p 3  and external parameters 
         [0071]    Substituting (x 0 , y 0 ) calculated from the previous step into the model equation (8), and using vector j to denote the 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 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0072]    Correspondingly the sensitive matrixes C and D change to: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           A 
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∂ 
                                       
                                         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 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           B 
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       ∂ 
                                       
                                         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 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0073]    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)=Δ   j   (k) −( N   k   T   N   k   T ) −1   N   k   T   P   (k)    (13) 
         [0074]    In the above equation, N consists of sensitive matrixes C and D, and their expressions are: 
         [0000]    
       
         
           
             N 
             = 
             
               [ 
               
                 
                   
                     
                       C 
                       1 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       C 
                       m 
                     
                   
                 
                 
                   
                     
                       D 
                       1 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       D 
                       m 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0075]    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. 
         [0076]    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 
         [0077]    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, and the precise attitude angle of the sunlight in 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. 
         [0078]    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 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. 
         [0079]    The processing computer  5  includes a data acquisition module and data processing module. The data acquisition module acquires the calibration points&#39; data which includes the rotating angle θ 1  of external frame, the rotating angle θ 2  of the internal frame and the centroid coordinate (x c , y c ) of the imaging spot 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 frame of rotator by different angles. The processing computer  5  records these calibration points&#39; data and calculates the corresponding calibration parameters. 
         [0080]    The rotator used in the invention 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 0.1solar constant. The diameter of effective radiation area is 200 mm, and the collimation angle of light beam is 32′. 
         [0081]    Totally 84 groups of recorded calibration point&#39;s data are listed 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 
               
               
                   
               
             
          
         
       
     
         [0082]    The 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 
               
               
                   
               
             
          
         
       
     
         [0083]    The total statistical square root error of x c  and y c  are 5.09 pixel and 4.27 pixel respectively. Substituting the parameters calibrated by the method of the present disclosure into the attitude computing formula of sun sensor, an attitude precision of 0.02 (1σ) is obtained. Because 14 total external and internal parameters are used in the disclosure, 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 are 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. 
         [0084]    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.