Abstract:
When correcting for velocity aberration in satellite imagery, a closed-form error covariance propagation model can produce more easily calculable error terms than a corresponding Monte Carlo analysis. The closed-form error covariance propagation model is symbolic, rather than numeric. The symbolic error covariance propagation model relates input parameters to one another pairwise and in closed form. For a particular image, the symbolic error covariance propagation model receives an input measurement value and an input error value for each input parameter. The symbolic error covariance propagation model operates on the input values to produce a set of output correction values, which correct for velocity aberration. The output correction values can be used to convert apparent coordinate values to corrected coordinate values. The symbolic error covariance matrix operates on the input error values to produce a set of output error values, which identify a reliability of the corrected coordinate values.

Description:
TECHNICAL FIELD 
       [0001]    Examples pertain generally to correcting registration errors in satellite imagery (errors in mapping from a point in an image to its corresponding point on the earth&#39;s surface), and more particularly to calculating error terms when correcting for velocity aberration in satellite images. 
       BACKGROUND 
       [0002]    There are several near earth orbiting commercial satellites that can provide images of structures or targets on the ground. For applications that rely on these images, it is important to accurately register the images to respective coordinates on the ground. 
         [0003]    For satellite-generated images of earth-based targets, one known source of misregistration is referred to as velocity aberration. Velocity aberration can arise in an optical system with a sufficiently large velocity relative to the point being imaged. A typical velocity of a near earth orbiting commercial satellite can be on the order of 7.5 kilometers per second, with respect to a location on the earth directly beneath the satellite. This velocity is large enough to produce a registration error of several detector pixels at the satellite-based camera. 
         [0004]    The correction for velocity aberration is generally well-known. However, it is generally challenging to calculate error terms associated with the correction. These error terms estimate the confidence level, or reliability, of the velocity aberration correction. 
         [0005]    Historically, error calculation for velocity aberration correction has been treated statistically with a Monte Carlo analysis. In general, these Monte Carlo analyses can be time-consuming and computationally expensive. In order to produce statistically significant results, a Monte Carlo analysis can require that a large number of simulated cases be executed and analyzed, which can be difficult or impossible due to the limitations of computational resources and processing time requirements. As a result, error estimation for velocity aberration correction can be lacking. 
       SUMMARY 
       [0006]    When correcting for velocity aberration in satellite imagery, a closed-form covariance matrix propagation can produce more reliable and more easily calculable error terms than a corresponding matrix generated by a Monte Carlo analysis. Performing calculations with the closed form covariance matrix can be significantly faster than with a corresponding Monte Carlo analysis, can provide greater immunity to data outliers, and can provide immediate checks for statistical consistency. 
         [0007]    The closed-form covariance matrix propagation is symbolic, rather than numeric. The symbolic covariance matrix propagation relates the known covariance matrix of the input parameters to the resulting covariance matrix of the aberration correction terms in closed form. For a particular image, one employs the required input parameters to compute the velocity aberration correction terms at a selected point in the image. The covariance matrix of the velocity aberration correction terms is then computed from the required input parameters, calculated velocity aberration terms, and the input parameter covariance matrix. The numerical values for the input parameter covariance matrix are received and the symbolic formulas are used to calculate the numerical values for the covariance matrix of the velocity aberration correction terms. 
         [0008]    The input and output error values are numerical values that indicate a reliability, or confidence level, of a corresponding numerical coordinate value or correction value. For instance, for a set of coordinates (x, y, z), the corresponding error values can be (σ x , σ y , σ z ). The value of σ x  represents the standard deviation of x, while σ x   2  denotes the variance of x. In general, as the error value σ x  decreases, the confidence in the reported value of x increases. 
         [0009]    This summary is intended to provide an overview of subject matter of the present patent application. It is not intended to provide an exclusive or exhaustive explanation of the invention. The Detailed Description is included to provide further information about the present patent application. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    In the drawings, which are not necessarily drawn to scale, like numerals may describe similar components in different views. Like numerals having different letter suffixes may represent different instances of similar components. The drawings illustrate generally, by way of example, but not by way of limitation, various embodiments discussed in the present document. 
           [0011]      FIG. 1  is a schematic drawing of an example of a system for receiving and processing imagery, such as satellite imagery, in accordance with some embodiments. 
           [0012]      FIG. 2  is a flow chart of an example of a method for calculating numeric output error values for velocity aberration correction of an image, in accordance with some embodiments. 
       
    
    
     DETAILED DESCRIPTION 
       [0013]      FIG. 1  is a schematic drawing of an example of a system  100  for receiving and processing imagery, such as satellite imagery. A satellite  102 , such as an IKONOS or another near earth orbiting commercial satellite, captures an image  104  of a target  106  on earth  108 . The system  100  downloads the captured image  104 , along with associated metadata  110  corresponding to conditions under which the image  104  was taken. 
         [0014]    The metadata  110  includes a set of apparent coordinate values  112  corresponding to a selected point in the image. The apparent coordinate values  112  suffer from velocity aberration. Velocity aberration is generally well-understood in the fields of astronomy and space-based imaging. Velocity aberration produces a pointing error, so that the image  104  formed at the sensor on the satellite is translated away from its expected location. If velocity aberration is left uncorrected, the image  104  and corresponding location on the target  106  can be misregistered with respect to each other. Velocity aberration does not degrade the image  104 . The system  100  provides a correction for the velocity aberration, and additionally provides a measure of reliability of the correction. 
         [0015]    The metadata  110  also includes plurality of input values  114  for corresponding input parameters  116 . The input parameters  116  are geometric quantities, such as distances and angular rates that define the satellite sensor&#39;s position and velocity. The input parameters  116  are defined symbolically and not numerically. The input values  114  are numeric, with numerical values that correspond to the input parameters  116 . 
         [0016]    Each input value  114  includes an input measured value  118 , an input mean error  120 , and an input error standard deviation value  122 . Each input measured value  118  represents a measurement of a corresponding input parameter  116  via onboard satellite instruments; this can be a considered a best estimate of the value of the input parameter. Each input mean error  120  represents an inherent bias of the corresponding input measured value  118 ; if there is no bias, then the input mean error  120  is zero. Each input error standard deviation value  122  represents a reliability of the best estimate. A relatively low input error standard deviation value  122  implies a relatively high confidence in the corresponding input measured value  118 , and a relatively high input error standard deviation value  122  implies a relatively low confidence in the corresponding input measured value  118 . In some examples, at least one mean input error standard deviation value  122  remains invariant for multiple images taken with a particular telescope in the satellite  102 . In some examples, all the input error standard deviation values  122  remain invariant for multiple images taken with a particular telescope in the satellite  102 . 
         [0017]    A symbolic error covariance propagation model  124  receives the input values  114  in the metadata  110 . The symbolic error covariance propagation model  124  includes a symbolic covariance matrix that relates the input parameters  116  to one another pairwise and in closed form. The symbolic covariance matrix can be image-independent, and can be used for other optical systems having the same configuration of input parameters  116 . The symbolic covariance matrix is symbolic, not numeric. In some examples, the input parameters  116  and the symbolic covariance matrix remain invariant for multiple images taken with a particular telescope in the satellite  102 . The Appendix to this document includes a mathematical derivation of an example of a suitable symbolic covariance matrix. 
         [0018]    Velocity aberration correction generates a set of output correction values  126  from the required input values  114 . The output correction values  126  relate the apparent coordinate values  112  to a set of corrected coordinate values  128 , and can therefore correct for velocity aberration in the image  104 . The corrected coordinate values  128  can be stored, along with the image  104 , and can be presented to a user as a best estimate of coordinates within the image  104 . 
         [0019]    The symbolic error covariance propagation model  124  generates a set of output error values  130  from the input values  114 . The output error values  130  identify a reliability of the output correction values  126 . The output error values  130  can also be stored, along with the image  104 , and can be presented to a user as a measure of reliability of the coordinate correction. 
         [0020]    As an example, a user of the system  100  can download an image  104  with metadata  110 . The system  100  can extract the suitable input values  114  from the metadata  110 . The system  100  can use extracted input values  114  to calculate output correction values  126  to correct for velocity aberration in the image  104 . The system  100  can apply the output correction values  124  to a set of apparent coordinate values  112  to generate a set of corrected coordinate values  128 . The system  100  can use the input values  114  to calculate output error values  130  that estimate a confidence level or reliability of the corrected coordinate values  128 . The system can present to the user the image  104 , the corrected coordinate values  128 , and the output error values  130 . 
         [0021]    The system  100  can be a computer system that includes hardware, firmware and software. Examples may also be implemented as instructions stored on a computer-readable storage device, which may be read and executed by at least one processor to perform the operations described herein. A computer-readable storage device may include any non-transitory mechanism for storing information in a form readable by a machine (e.g., a computer). For example, a computer-readable storage device may include read-only memory (ROM), random-access memory (RAM), magnetic disk storage media, optical storage media, flash-memory devices, and other storage devices and media. In some examples, computer systems can include one or more processors, optionally connected to a network, and may be configured with instructions stored on a computer-readable storage device. 
         [0022]    The input parameters  116  can include a separation between a center of mass of a telescope and a vertex of a primary mirror of the telescope, and include x, y, and z components of: a geocentric radius vector to a center of mass of the telescope; a velocity vector to the center of mass of the telescope; a geocentric radius vector to a target ground point; an angular rate vector of a body reference frame of the telescope; and a unit vector along a z-axis of the body reference frame of the telescope. These input parameters are but one example; other suitable input parameters  116  can also be used. 
         [0023]      FIG. 2  is a flow chart of an example of a method  200  for calculating numeric output error values for an image. The image has a corresponding set of output correction values that correct for velocity aberration. The method  200  can be executed by system  100  of  FIG. 1 , or by another suitable system. 
         [0024]    At  202 , method  200  receives metadata corresponding to conditions under which an image was taken. The metadata includes a plurality of input values for corresponding input parameters. Each input value includes an input measured value, an input mean error, and an input error standard deviation value. At  204 , method  200  provides the plurality of input values to a symbolic error covariance propagation model. The symbolic error covariance propagation model includes a symbolic covariance matrix that relates the plurality of input parameters to one another pairwise and in closed form. At  206 , method  200  generates a set of output error values from the symbolic error covariance propagation model and the plurality of input values. The set of output error values identifies a reliability of the set of output correction values. 
         [0025]    The remainder of the Detailed Description is an Appendix that includes a mathematical derivation of an example of a symbolic error covariance propagation model that is suitable for use in the system  100  of  FIG. 1 . The derived symbolic error covariance propagation model uses the input parameters  116 , as noted above. 
       APPENDIX 
       [0026]    Beginning with the well-known Lorentz transformations from special relativity, one can relativistically formulate a true line of sight correction process, then use a first-order differential approximation to construct a covariance propagation model of the aberration correction process. The first-order differential approximation acknowledges that relative velocities between reference frames are significantly less than the speed of light, which is the case for near earth orbiting commercial satellites. 
         [0027]    There are sixteen input parameters employed in the calculation of the corrected line-of-sight. Each input parameter has an input measured value (X), an input mean error value (μ X ), and an input error term (σ X ). 
         [0028]    The sixteen input error terms are defined symbolically as follows. Quantities Δx cm , Δy cm , Δz cm  are three random error components of a geocentric radius vector            CM  to a telescope center of mass. Quantities Δv xcm , Δv ycm , Δv zcm  are three random error components of a velocity vector            CM  to the telescope center of mass. Quantities Δx p , Δy p , Δz p  are three random error components of a geocentric radius vector            P  to a target ground point. Quantity Δd Z  is a random error component of a distance from a center of mass to the vertex of the telescope&#39;s primary mirror. Quantities Δω xb , Δω yb , Δω zb  are three random error components of an angular rate vector            B  of the telescope&#39;s body reference frame. Quantities Δz xb , Δz yb , Δz zb  are three random error components of a unit vector {circumflex over (Z)} B  along a Z axis of the telescope&#39;s body frame. In some examples, one or more of the sixteen input error terms can be omitted. 
         [0029]    Quantity Δ{circumflex over (q)}′ 3×1  is an error vector for corrected (true) line-of-sight components: 
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         [0031]    Quantity Δ           16×1  is an input error vector, formed from the sixteen input error terms: 
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         [0032]    A first-order differential approximation can linearly relate the quantities Δ{circumflex over (q)}′ 3×1 , H 3×16 , and Δ           16×1  to one other: 
         [0000]      Δ {circumflex over (q)}′   3×1   =H   3×16 Δ           16×1  
 
         [0033]    Applying an expectation operator leads to: 
         [0000]                Δ {circumflex over (q)}′   3×1             =H   3×16           Δ           16×1           ,  (1)
 
         [0034]    Quantity P Δ{circumflex over (q)}′  is a covariance P Δ{circumflex over (q)}′  of resulting errors in the calculated line-of-sight components, and can be calculated by: 
         [0000]        P   Δ{circumflex over (q)}′ ≡         (Δ {circumflex over (q)}′−             Δ{circumflex over (q)} ′         )(Δ {circumflex over (q)}′−             Δ{circumflex over (q)} ′         ) T           
 
         [0000]        P   Δ{circumflex over (q)}′   =H           (Δ         −         Δ                   )(Δ         −         Δ                   ) T             H   T  
 
         [0000]        P   Δ{circumflex over (q)}′   =H             H   T ,  (2)
 
         [0035]    where           is a 16-by-16 covariance matrix of input parameter errors. Equations (1) and (2) form a covariance propagation model. The covariance propagation model relates the means and covariance of the random measurement errors in the parameters input into the aberration correction process to the means and covariance of the components of the corrected true line of sight. 
         [0036]    The preceding is but one example of a symbolic error covariance propagation model; other suitable symbolic error covariance propagation models can also be used. 
         [0037]    The indicated partial derivatives that constitute the matrix H 3×16  are now evaluated in detail. The first step is to give detailed expressions for q′ x , q′ y , q′ z : 
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                             k 
                             ^ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Let ξ represent any of the sixteen scalar input parameters. Direct differentiation of equations 3, 4, and 5 yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         q 
                         x 
                         ′ 
                       
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 ∂ 
                                                 
                                                   q 
                                                   ^ 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                             · 
                                             
                                               u 
                                               ^ 
                                             
                                           
                                           ) 
                                         
                                         + 
                                         
                                           ( 
                                           
                                             
                                               q 
                                               ^ 
                                             
                                             · 
                                             
                                               
                                                 ∂ 
                                                 
                                                   u 
                                                   ^ 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       
                                         1 
                                         - 
                                         
                                           β 
                                           2 
                                         
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         - 
                                         
                                           u 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         1 
                                         - 
                                         
                                           β 
                                           2 
                                         
                                       
                                     
                                   
                                    
                                   β 
                                    
                                   
                                     
                                       ∂ 
                                       β 
                                     
                                     
                                       ∂ 
                                       ξ 
                                     
                                   
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           
                             u 
                             ^ 
                           
                           · 
                           
                             i 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   1 
                                   - 
                                   
                                     β 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 { 
                                 
                                   1 
                                   - 
                                   
                                     β 
                                      
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                               2 
                             
                           
                           ] 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ∂ 
                                         β 
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   β 
                                    
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               ∂ 
                                               
                                                 q 
                                                 ^ 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                           · 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         ( 
                                         
                                           
                                             q 
                                             ^ 
                                           
                                           · 
                                           
                                             
                                               ∂ 
                                               
                                                 v 
                                                 ^ 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             u 
                             ^ 
                           
                           · 
                           
                             i 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   v 
                                   ^ 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 1 
                                 - 
                                 
                                   β 
                                   2 
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 
                                   u 
                                   ^ 
                                 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                             · 
                             
                               i 
                               ^ 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         
                                           q 
                                           ^ 
                                         
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       q 
                                       ^ 
                                     
                                     · 
                                     
                                       
                                         ∂ 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             - 
                             
                               
                                 ∂ 
                                 β 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           
                             v 
                             ^ 
                           
                           · 
                           
                             i 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                               - 
                               β 
                             
                             ) 
                           
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   β 
                                    
                                   
                                     ( 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ∂ 
                                   β 
                                 
                                 
                                   ∂ 
                                   ξ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               β 
                                
                               
                                 { 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ∂ 
                                           
                                             q 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     ( 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         
                                           ∂ 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             ^ 
                           
                           · 
                           
                             i 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   v 
                                   ^ 
                                 
                               
                               ) 
                             
                             - 
                             β 
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 
                                   v 
                                   ^ 
                                 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                             · 
                             
                               i 
                               ^ 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         q 
                         y 
                         ′ 
                       
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 ∂ 
                                                 
                                                   q 
                                                   ^ 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                             · 
                                             
                                               u 
                                               ^ 
                                             
                                           
                                           ) 
                                         
                                         + 
                                         
                                           ( 
                                           
                                             
                                               q 
                                               ^ 
                                             
                                             · 
                                             
                                               
                                                 ∂ 
                                                 
                                                   u 
                                                   ^ 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       
                                         1 
                                         - 
                                         
                                           β 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                           
                                         
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           u 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         1 
                                         - 
                                         
                                           β 
                                           2 
                                         
                                       
                                     
                                   
                                    
                                   β 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∂ 
                                       β 
                                     
                                     
                                       ∂ 
                                       ξ 
                                     
                                   
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           
                             u 
                             ^ 
                           
                           · 
                           
                             j 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     u 
                                     ^ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   1 
                                   - 
                                   
                                     β 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 { 
                                 
                                   1 
                                   - 
                                   
                                     β 
                                      
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                               2 
                             
                           
                           ] 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ∂ 
                                         β 
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   β 
                                    
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               ∂ 
                                               
                                                 q 
                                                 ^ 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                           · 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         ( 
                                         
                                           
                                             q 
                                             ^ 
                                           
                                           · 
                                           
                                             
                                               ∂ 
                                               
                                                 v 
                                                 ^ 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             u 
                             ^ 
                           
                           · 
                           
                             j 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   u 
                                   ^ 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 1 
                                 - 
                                 
                                   β 
                                   2 
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 
                                   u 
                                   ^ 
                                 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                             · 
                             
                               j 
                               ^ 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         
                                           q 
                                           ^ 
                                         
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     q 
                                     · 
                                     
                                       
                                         ∂ 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             - 
                             
                               
                                 ∂ 
                                 β 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           
                             v 
                             ^ 
                           
                           · 
                           
                             j 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                               - 
                               β 
                             
                             ) 
                           
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   β 
                                    
                                   
                                     ( 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ∂ 
                                   β 
                                 
                                 
                                   ∂ 
                                   ξ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               β 
                                
                               
                                 { 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ∂ 
                                           
                                             q 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     ( 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         
                                           ∂ 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             ^ 
                           
                           · 
                           
                             j 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   v 
                                   ^ 
                                 
                               
                               ) 
                             
                             - 
                             β 
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 
                                   v 
                                   ^ 
                                 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                             · 
                             
                               j 
                               ^ 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         q 
                         z 
                         ′ 
                       
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 ∂ 
                                                 
                                                   q 
                                                   ^ 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                             · 
                                             
                                               u 
                                               ^ 
                                             
                                           
                                           ) 
                                         
                                         + 
                                         
                                           ( 
                                           
                                             
                                               q 
                                               ^ 
                                             
                                             · 
                                             
                                               
                                                 ∂ 
                                                 
                                                   u 
                                                   ^ 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ] 
                                     
                                      
                                     
                                       
                                         1 
                                         - 
                                         
                                           β 
                                           2 
                                         
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           u 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         1 
                                         - 
                                         
                                           β 
                                           2 
                                         
                                       
                                     
                                   
                                    
                                   β 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∂ 
                                       β 
                                     
                                     
                                       ∂ 
                                       ξ 
                                     
                                   
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           
                             u 
                             ^ 
                           
                           · 
                           
                             k 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     u 
                                     ^ 
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   1 
                                   - 
                                   
                                     β 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 { 
                                 
                                   1 
                                   - 
                                   
                                     β 
                                      
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                               2 
                             
                           
                           ] 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ∂ 
                                         β 
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   β 
                                    
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               ∂ 
                                               
                                                 q 
                                                 ^ 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                           · 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         ( 
                                         
                                           
                                             q 
                                             ^ 
                                           
                                           · 
                                           
                                             
                                               ∂ 
                                               
                                                 v 
                                                 ^ 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             u 
                             ^ 
                           
                           · 
                           
                             k 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   u 
                                   ^ 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 1 
                                 - 
                                 
                                   β 
                                   2 
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 
                                   u 
                                   ^ 
                                 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                             · 
                             
                               k 
                               ^ 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ∂ 
                                         
                                           q 
                                           ^ 
                                         
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       q 
                                       ^ 
                                     
                                     · 
                                     
                                       
                                         ∂ 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       
                                         ∂ 
                                         ξ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                             - 
                             
                               
                                 ∂ 
                                 β 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           
                             v 
                             ^ 
                           
                           · 
                           
                             k 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                               - 
                               β 
                             
                             ) 
                           
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                     
                                 
                                  
                                 
                                   β 
                                    
                                   
                                     ( 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ∂ 
                                   β 
                                 
                                 
                                   ∂ 
                                   ξ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               β 
                                
                               
                                 { 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ∂ 
                                           
                                             q 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     ( 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         
                                           ∂ 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             ^ 
                           
                           · 
                           
                             k 
                             ^ 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   v 
                                   ^ 
                                 
                               
                               ) 
                             
                             - 
                             β 
                           
                           
                             1 
                             - 
                             
                               β 
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   · 
                                   
                                     v 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 
                                   v 
                                   ^ 
                                 
                               
                               
                                 ∂ 
                                 ξ 
                               
                             
                             · 
                             
                               k 
                               ^ 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The next indicated partial derivatives to be evaluated are: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 β 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 ∂ 
                 
                   u 
                   ^ 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 ∂ 
                 
                   v 
                   ^ 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 
                   ∂ 
                   
                     q 
                     ^ 
                   
                 
                 
                   ∂ 
                   ξ 
                 
               
               . 
             
           
         
       
     
         [0000]    Direct differentiation yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       β 
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     
                                       
                                         R 
                                         
                                           ⇀ 
                                           . 
                                         
                                       
                                       CM 
                                     
                                   
                                   
                                     ∂ 
                                     ξ 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ∂ 
                                       
                                         d 
                                         Z 
                                       
                                     
                                     
                                       ∂ 
                                       ξ 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           ⇀ 
                                         
                                         B 
                                       
                                       × 
                                       
                                         
                                           Z 
                                           ^ 
                                         
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   d 
                                   Z 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ∂ 
                                           
                                             
                                               ω 
                                               ⇀ 
                                             
                                             B 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                       × 
                                       
                                         
                                           Z 
                                           ⇀ 
                                         
                                         B 
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           ω 
                                           ⇀ 
                                         
                                         B 
                                       
                                       × 
                                       
                                         
                                           ∂ 
                                           
                                             
                                               Z 
                                               ⇀ 
                                             
                                             B 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         } 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               R 
                               
                                 ⇀ 
                                 . 
                               
                             
                             CM 
                           
                           + 
                           
                             
                               d 
                               Z 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ω 
                                     ⇀ 
                                   
                                   B 
                                 
                                 × 
                                 
                                   
                                     Z 
                                     ^ 
                                   
                                   B 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       2 
                        
                       
                         
                           
                             [ 
                             
                               
                                 - 
                                 
                                   
                                     R 
                                     
                                       ⇀ 
                                       . 
                                     
                                   
                                   CM 
                                 
                               
                               - 
                               
                                 
                                   d 
                                   Z 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ω 
                                         ⇀ 
                                       
                                       B 
                                     
                                     × 
                                     
                                       
                                         Z 
                                         ^ 
                                       
                                       B 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             [ 
                             
                               
                                 - 
                                 
                                   
                                     R 
                                     
                                       ⇀ 
                                       . 
                                     
                                   
                                   CM 
                                 
                               
                               - 
                               
                                 
                                   d 
                                   Z 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ω 
                                         ⇀ 
                                       
                                       B 
                                     
                                     × 
                                     
                                       
                                         Z 
                                         ^ 
                                       
                                       B 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         u 
                         ^ 
                       
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ∂ 
                             
                               q 
                               ^ 
                             
                           
                           
                             ∂ 
                             ξ 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     ∂ 
                                     
                                       q 
                                       ^ 
                                     
                                   
                                   
                                     ∂ 
                                     ξ 
                                   
                                 
                                 · 
                                 
                                   v 
                                   ^ 
                                 
                               
                               + 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   
                                     ∂ 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   
                                     ∂ 
                                     ξ 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                             v 
                             ^ 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 q 
                                 ^ 
                               
                               · 
                               
                                 v 
                                 ^ 
                               
                             
                             ) 
                           
                            
                           
                             
                               ∂ 
                               
                                 v 
                                 ^ 
                               
                             
                             
                               ∂ 
                               ξ 
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 q 
                                 ^ 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       q 
                                       ^ 
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   v 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 q 
                                 ^ 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       q 
                                       ^ 
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   v 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             q 
                             ^ 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   q 
                                   ^ 
                                 
                                 · 
                                 
                                   v 
                                   ^ 
                                 
                               
                               ) 
                             
                              
                             
                               v 
                               ^ 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           
                             { 
                             
                               
                                 
                                   ∂ 
                                   
                                     q 
                                     ^ 
                                   
                                 
                                 
                                   ∂ 
                                   ξ 
                                 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ∂ 
                                           
                                             q 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                       · 
                                       
                                         v 
                                         ^ 
                                       
                                     
                                     + 
                                     
                                       
                                         q 
                                         ^ 
                                       
                                       · 
                                       
                                         
                                           ∂ 
                                           
                                             v 
                                             ^ 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   v 
                                   ^ 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       q 
                                       ^ 
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     ∂ 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   
                                     ∂ 
                                     ξ 
                                   
                                 
                               
                             
                             } 
                           
                           · 
                           
                             ( 
                             
                               
                                 q 
                                 ^ 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       q 
                                       ^ 
                                     
                                     · 
                                     
                                       v 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   v 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                         
                         
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       v 
                                       ^ 
                                     
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     q 
                                     ^ 
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           q 
                                           ^ 
                                         
                                         · 
                                         
                                           v 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       v 
                                       ^ 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           
                             3 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         v 
                         ^ 
                       
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             
                               ∂ 
                               
                                 
                                   R 
                                   
                                     ⇀ 
                                     . 
                                   
                                 
                                 CM 
                               
                             
                             
                               ∂ 
                               ξ 
                             
                           
                         
                         - 
                         
                           
                             
                               ∂ 
                               
                                 d 
                                 Z 
                               
                             
                             
                               ∂ 
                               ξ 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ω 
                                   ⇀ 
                                 
                                 B 
                               
                               × 
                               
                                 
                                   Z 
                                   ^ 
                                 
                                 B 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             d 
                             Z 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     ∂ 
                                     
                                       
                                         ω 
                                         ⇀ 
                                       
                                       B 
                                     
                                   
                                   
                                     ∂ 
                                     ξ 
                                   
                                 
                                 × 
                                 
                                   
                                     Z 
                                     ^ 
                                   
                                   B 
                                 
                               
                               + 
                               
                                 
                                   
                                     ω 
                                     ⇀ 
                                   
                                   B 
                                 
                                 × 
                                 
                                   
                                     ∂ 
                                     
                                       
                                         Z 
                                         ^ 
                                       
                                       B 
                                     
                                   
                                   
                                     ∂ 
                                     ξ 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   
                                     R 
                                     
                                       ⇀ 
                                       . 
                                     
                                   
                                   CM 
                                 
                                 - 
                                 
                                   
                                     d 
                                     Z 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           ⇀ 
                                         
                                         B 
                                       
                                       × 
                                       
                                         
                                           Z 
                                           ^ 
                                         
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   
                                     R 
                                     
                                       ⇀ 
                                       . 
                                     
                                   
                                   CM 
                                 
                               
                               - 
                               
                                 
                                   d 
                                   Z 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ω 
                                         ⇀ 
                                       
                                       B 
                                     
                                     × 
                                     
                                       
                                         Z 
                                         ^ 
                                       
                                       B 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     R 
                                     
                                       ⇀ 
                                       . 
                                     
                                   
                                   CM 
                                 
                                 + 
                                 
                                   
                                     d 
                                     Z 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           ⇀ 
                                         
                                         B 
                                       
                                       × 
                                       
                                         
                                           Z 
                                           ^ 
                                         
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               { 
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             
                                               ∂ 
                                               
                                                 
                                                   R 
                                                   
                                                     ⇀ 
                                                     . 
                                                   
                                                 
                                                 CM 
                                               
                                             
                                             
                                               ∂ 
                                               ξ 
                                             
                                           
                                           + 
                                           
                                             
                                               
                                                 ∂ 
                                                 
                                                   d 
                                                   Z 
                                                 
                                               
                                               
                                                 ∂ 
                                                 ξ 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     ω 
                                                     ⇀ 
                                                   
                                                   B 
                                                 
                                                 × 
                                                 
                                                   
                                                     Z 
                                                     ^ 
                                                   
                                                   B 
                                                 
                                               
                                               ) 
                                             
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             d 
                                             Z 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   
                                                     ∂ 
                                                     
                                                       
                                                         ω 
                                                         ⇀ 
                                                       
                                                       B 
                                                     
                                                   
                                                   
                                                     ∂ 
                                                     ξ 
                                                   
                                                 
                                                 × 
                                                 
                                                   
                                                     Z 
                                                     ^ 
                                                   
                                                   B 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     ω 
                                                     ⇀ 
                                                   
                                                   B 
                                                 
                                                 × 
                                                 
                                                   
                                                     ∂ 
                                                     
                                                       
                                                         Z 
                                                         ^ 
                                                       
                                                       B 
                                                     
                                                   
                                                   
                                                     ∂ 
                                                     ξ 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       
                                         R 
                                         
                                           ⇀ 
                                           . 
                                         
                                       
                                       CM 
                                     
                                     + 
                                     
                                       
                                         d 
                                         Z 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               ω 
                                               ⇀ 
                                             
                                             B 
                                           
                                           × 
                                           
                                             
                                               Z 
                                               ^ 
                                             
                                             B 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               } 
                             
                           
                         
                       
                       
                         
                           { 
                           
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     
                                       R 
                                       
                                         ⇀ 
                                         . 
                                       
                                     
                                     CM 
                                   
                                 
                                 - 
                                 
                                   
                                     d 
                                     Z 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           ⇀ 
                                         
                                         B 
                                       
                                       × 
                                       
                                         
                                           Z 
                                           ^ 
                                         
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     
                                       R 
                                       
                                         ⇀ 
                                         . 
                                       
                                     
                                     CM 
                                   
                                 
                                 - 
                                 
                                   
                                     d 
                                     Z 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           ω 
                                           ⇀ 
                                         
                                         B 
                                       
                                       × 
                                       
                                         
                                           Z 
                                           ^ 
                                         
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           } 
                         
                         
                           3 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         q 
                         ^ 
                       
                     
                     
                       ∂ 
                       ξ 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ∂ 
                             
                               
                                 R 
                                 ⇀ 
                               
                               P 
                             
                           
                           
                             ∂ 
                             ξ 
                           
                         
                         - 
                         
                           
                             
                               ∂ 
                               
                                 d 
                                 Z 
                               
                             
                             
                               ∂ 
                               ξ 
                             
                           
                            
                           
                             
                               Z 
                               ^ 
                             
                             B 
                           
                         
                         - 
                         
                           
                             d 
                             Z 
                           
                            
                           
                             
                               ∂ 
                               
                                 
                                   Z 
                                   ^ 
                                 
                                 B 
                               
                             
                             
                               ∂ 
                               ξ 
                             
                           
                         
                         - 
                         
                           
                             ∂ 
                             
                               
                                 R 
                                 ⇀ 
                               
                               CM 
                             
                           
                           
                             ∂ 
                             ξ 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 
                                   R 
                                   ⇀ 
                                 
                                 P 
                               
                               - 
                               
                                 
                                   d 
                                   Z 
                                 
                                  
                                 
                                   
                                     Z 
                                     ^ 
                                   
                                   B 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   ⇀ 
                                 
                                 CM 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 
                                   R 
                                   ⇀ 
                                 
                                 P 
                               
                               - 
                               
                                 
                                   d 
                                   Z 
                                 
                                  
                                 
                                   
                                     Z 
                                     ^ 
                                   
                                   B 
                                 
                               
                               - 
                               
                                 
                                   R 
                                   ⇀ 
                                 
                                 CM 
                               
                             
                             ) 
                           
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               R 
                               ⇀ 
                             
                             P 
                           
                           - 
                           
                             
                               d 
                               Z 
                             
                              
                             
                               
                                 Z 
                                 ^ 
                               
                               B 
                             
                           
                           - 
                           
                             
                               R 
                               ⇀ 
                             
                             CM 
                           
                         
                         ) 
                       
                        
                       
                         
                           { 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ∂ 
                                           
                                             
                                               R 
                                               ⇀ 
                                             
                                             P 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             ∂ 
                                             
                                               d 
                                               Z 
                                             
                                           
                                           
                                             ∂ 
                                             ξ 
                                           
                                         
                                          
                                         
                                           
                                             Z 
                                             ^ 
                                           
                                           B 
                                         
                                       
                                       - 
                                       
                                         
                                           d 
                                           Z 
                                         
                                          
                                         
                                           
                                             ∂ 
                                             
                                               
                                                 Z 
                                                 ^ 
                                               
                                               B 
                                             
                                           
                                           
                                             ∂ 
                                             ξ 
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           ∂ 
                                           
                                             
                                               R 
                                               ⇀ 
                                             
                                             CM 
                                           
                                         
                                         
                                           ∂ 
                                           ξ 
                                         
                                       
                                     
                                     ) 
                                   
                                   · 
                                 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         R 
                                         ⇀ 
                                       
                                       P 
                                     
                                     - 
                                     
                                       
                                         d 
                                         Z 
                                       
                                        
                                       
                                         
                                           Z 
                                           ^ 
                                         
                                         B 
                                       
                                     
                                     - 
                                     
                                       
                                         R 
                                         ⇀ 
                                       
                                       CM 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           } 
                         
                         
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       R 
                                       ⇀ 
                                     
                                     P 
                                   
                                   - 
                                   
                                     
                                       d 
                                       Z 
                                     
                                      
                                     
                                       
                                         Z 
                                         ^ 
                                       
                                       B 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       ⇀ 
                                     
                                     CM 
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     
                                       R 
                                       ⇀ 
                                     
                                     P 
                                   
                                   - 
                                   
                                     
                                       d 
                                       Z 
                                     
                                      
                                     
                                       
                                         Z 
                                         ^ 
                                       
                                       B 
                                     
                                   
                                   - 
                                   
                                     
                                       R 
                                       ⇀ 
                                     
                                     CM 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                           
                             3 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The remaining partial derivatives to be evaluated are: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   
                     R 
                     ⇀ 
                   
                   CM 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 ∂ 
                 
                   
                     R 
                     
                       ⇀ 
                       . 
                     
                   
                   CM 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 ∂ 
                 
                   
                     R 
                     ⇀ 
                   
                   P 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 ∂ 
                 
                   d 
                   Z 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 ∂ 
                 
                   
                     ω 
                     ⇀ 
                   
                   B 
                 
               
               
                 ∂ 
                 ξ 
               
             
             , 
             
               
                 
                   ∂ 
                   
                     
                       Z 
                       ^ 
                     
                     B 
                   
                 
                 
                   ∂ 
                   ξ 
                 
               
               . 
             
           
         
       
     
         [0000]    Given the possible values that the dummy variable ξ may assume, only sixteen of the partial derivatives above are non-zero. These partial derivatives are: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           R 
                           ⇀ 
                         
                         CM 
                       
                     
                     
                       ∂ 
                       
                         x 
                         
                           c 
                            
                           
                               
                           
                            
                           m 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           
                             R 
                             
                               ⇀ 
                               . 
                             
                           
                           CM 
                         
                       
                       
                         ∂ 
                         
                           v 
                           xcm 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             
                               R 
                               ⇀ 
                             
                             P 
                           
                         
                         
                           ∂ 
                           
                             x 
                             p 
                           
                         
                       
                       = 
                       
                         
                           
                             ∂ 
                             
                               
                                 ω 
                                 ⇀ 
                               
                               B 
                             
                           
                           
                             ∂ 
                             
                               ω 
                               xb 
                             
                           
                         
                         = 
                         
                           
                             
                               ∂ 
                               
                                 
                                   Z 
                                   ^ 
                                 
                                 B 
                               
                             
                             
                               ∂ 
                               
                                 Z 
                                 xb 
                               
                             
                           
                           = 
                           
                             i 
                             ^ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           R 
                           ⇀ 
                         
                         CM 
                       
                     
                     
                       ∂ 
                       
                         y 
                         
                           c 
                            
                           
                               
                           
                            
                           m 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           
                             R 
                             
                               ⇀ 
                               . 
                             
                           
                           CM 
                         
                       
                       
                         ∂ 
                         
                           v 
                           ycm 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             
                               R 
                               ⇀ 
                             
                             P 
                           
                         
                         
                           ∂ 
                           
                             y 
                             p 
                           
                         
                       
                       = 
                       
                         
                           
                             ∂ 
                             
                               
                                 ω 
                                 ⇀ 
                               
                               B 
                             
                           
                           
                             ∂ 
                             
                               ω 
                               yb 
                             
                           
                         
                         = 
                         
                           
                             
                               ∂ 
                               
                                 
                                   Z 
                                   ^ 
                                 
                                 B 
                               
                             
                             
                               ∂ 
                               
                                 Z 
                                 yb 
                               
                             
                           
                           = 
                           
                             j 
                             ^ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           R 
                           ⇀ 
                         
                         CM 
                       
                     
                     
                       ∂ 
                       
                         z 
                         
                           c 
                            
                           
                               
                           
                            
                           m 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           
                             R 
                             
                               ⇀ 
                               . 
                             
                           
                           CM 
                         
                       
                       
                         ∂ 
                         
                           v 
                           zcm 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             
                               R 
                               ⇀ 
                             
                             P 
                           
                         
                         
                           ∂ 
                           
                             z 
                             p 
                           
                         
                       
                       = 
                       
                         
                           
                             ∂ 
                             
                               
                                 ω 
                                 ⇀ 
                               
                               B 
                             
                           
                           
                             ∂ 
                             
                               ω 
                               zb 
                             
                           
                         
                         = 
                         
                           
                             
                               ∂ 
                               
                                 
                                   Z 
                                   ^ 
                                 
                                 B 
                               
                             
                             
                               ∂ 
                               
                                 Z 
                                 
                                   zb 
                                    
                                   
                                       
                                   
                                 
                               
                             
                           
                           = 
                           
                             k 
                             ^ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         d 
                         Z 
                       
                     
                     
                       ∂ 
                       
                         d 
                         Z 
                       
                     
                   
                   = 
                   1 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This constitutes all the partial derivatives needed to evaluate the matrix H 3×16 .