Abstract:
Methods and apparatus are provided for deriving precision position and rate information for motors using relatively low precision analog sensors, and for implementing compensation techniques that overcome inherent sensor errors and rotor magnet flux tolerances.

Description:
TECHNICAL FIELD 
       [0001]    The present invention generally relates to motor control and, more particularly, to a system and method for implementing fine resolution motor control using analog Hall-effect sensors. 
       BACKGROUND 
       [0002]    Electric motors, such as brushless motors, may be used for many different applications. In many instances, analog Hall-effect sensors are used for sensing rotor position in brushless motors where, for example, trapezoidal commutation is employed. This can be a cost effective solution when precision motor control is not needed. However, for precision motor (e.g., fine resolution) control applications, which employ sinusoidal or field oriented control commutation, relatively higher resolution position and rate sensing may be needed. Historically, this higher resolution sensing has been accomplished with relatively more expensive and complex resolver type AC sensors and resolver to digital conversion circuitry. 
         [0003]    Recently, other techniques have been implemented that use relatively low-cost analog Hall-effect sensors. In accordance with these techniques, analog Hall-effect sensors generate three sinusoidal signals representative of the position of the motor rotor. Using a rigorous mathematical approach, the rotor position and rate may then be derived. One example of a mathematical approach, which is disclosed in U.S. Pat. No. 6,744,230, combines the three sinusoidal signals into two orthogonal vectors using known trigonometric identities. Standard arctangent algorithms and other trigonometric identities may then be used to derive the rotational position and rate from the orthogonal vectors. 
         [0004]    While the above described mathematical technique works quite well, it can exhibit certain drawbacks. For example, the derived position and rate can be subject to a number of errors. These errors may be associated with individual Hall-effect sensors or specific magnet poles. In particular, the Hall-effect sensor errors may arise from positioning errors (phase and amplitude) and/or from electrical offset of the sensors. The magnet pole errors may appear as overall magnetic flux intensity variations or distortions in the sinusoidal waveform. 
         [0005]    Hence, there is a need for a relatively low-cost motor control scheme that may implement fine resolution motor control and that overcomes inherent sensor errors and rotor magnetic flux intensity variation errors. The present invention addresses at least this need. 
       BRIEF SUMMARY 
       [0006]    In one embodiment, and by way of example only, a method of generating a motor control signal for a motor having a rotor that rotates at a rotational rate includes generating first, second, and third sinusoidal signals, with each sinusoidal signal being 120-degrees out of phase with the other sinusoidal signals. the first, second, and third sinusoidal signals are differentiated to obtain first, second, and third cosine signals, respectively. The first sinusoidal signal is subtracted from the third sinusoidal signal to obtain a first sinusoidal difference signal, the second sinusoidal signal is subtracted from the first sinusoidal signal to obtain a second sinusoidal difference signal, and the third sinusoidal signal is subtracted from the second sinusoidal signal to obtain a third sinusoidal difference signal. The first sinusoidal difference signal and the second cosine signal are multiplied to obtain a first product signal, the second sinusoidal difference signal and the third cosine signal are multiplied to obtain a second product signal, and the third sinusoidal difference signal and the first cosine signal are multiplied to obtain a third product signal. The first, second, and third product signals are summed to obtain a summation signal, and the summation signal is multiplied by a constant to obtain a rate signal that is proportional to the rotational rate of the rotor. 
         [0007]    In another exemplary embodiment, a motor controller includes first, second, and third sensor means, first, second, and third differentiation means, first, second, and third, subtraction means, first, second, third, and fourth multiplication means, and summing means. The first, second, and third sensor means generate first, second, and third sinusoidal signals, respectively, with each sinusoidal signal being 120-degrees out of phase with the other sinusoidal signals. The first, second, and third differentiation means differentiate the first, second, and third sinusoidal signals, respectively, to generate first, second, and third cosine signals, respectively. The first subtraction means subtracts the first sinusoidal signal from the third sinusoidal signal to generate a first sinusoidal difference signal, the second subtraction means subtracts the second sinusoidal signal from the first sinusoidal signal to generate a second sinusoidal difference signal, and the third subtraction means subtracts the third sinusoidal signal from the second sinusoidal signal to generate a third sinusoidal difference signal. The first multiplication means multiplies the first sinusoidal difference signal and the second cosine signal to generate a first product signal, the second multiplication means multiplies the second sinusoidal difference signal and the third cosine signal to generate a second product signal, and the third multiplication means multiplies the third sinusoidal difference signal and the first cosine signal to generate a third product signal. The summing means sums the first, second, and third product signals to generate a summation signal, and the fourth multiplication means multiplies the summation signal by a constant to generate a rate signal that is proportional to the rotational rate of the rotor. 
         [0008]    In still another exemplary embodiment, a motor control system includes a motor, first, second, and third sensors, and a motor controller. The motor includes a stator and a rotationally mounted rotor. The rotor is configured, upon the stator being controllably energized, to rotate to a rotational position at a rotational rate. The first, second, and third sensors are evenly spaced about the rotor to sense the rotational position thereof and supply first, second, and third sinusoidal sensor signals, respectively. Each sinusoidal sensor signal is 120-degrees out of phase with the other sinusoidal sensor signals. The motor controller is coupled to receive each of the sinusoidal sensor signals and is configured to controllably energize the stator from a power source. The motor controller includes first, second, and third differentiation means, first, second, and third, subtraction means, first, second, third, and fourth multiplication means, and summing means. The first, second, and third differentiation means differentiate the first, second, and third sinusoidal signals, respectively, to generate first, second, and third cosine signals, respectively. The first subtraction means subtracts the first sinusoidal signal from the third sinusoidal signal to generate a first sinusoidal difference signal, the second subtraction means subtracts the second sinusoidal signal from the first sinusoidal signal to generate a second sinusoidal difference signal, and the third subtraction means subtracts the third sinusoidal signal from the second sinusoidal signal to generate a third sinusoidal difference signal. The first multiplication means multiplies the first sinusoidal difference signal and the second cosine signal to generate a first product signal, the second multiplication means multiplies the second sinusoidal difference signal and the third cosine signal to generate a second product signal, and the third multiplication means multiplies the third sinusoidal difference signal and the first cosine signal to generate a third product signal. The summing means sums the first, second, and third product signals to generate a summation signal, and the fourth multiplication means multiplies the summation signal by a constant to generate a rate signal that is proportional to the rotational rate of the rotor. 
         [0009]    In yet another exemplary embodiment, a self-calibrating complementary filter includes subtraction means, gain adjustment means, first and second integration means, multiplication means, and summing means. The subtraction means receives a filtered position signal and a position signal and subtracts the filtered position signal from the position signal to supply a position error signal. The gain adjustment means applies a predetermined gain to the position error signal to supply an adjusted position error signal. The first integration means integrates the position error signal and supplies a rate adjustment signal. The multiplication means for receives a rate signal and the rate adjustment signal and multiplies the rate signal and the rate adjustment signal to generate an adjusted rate signal. The summing means sums the adjusted rate signal and the adjusted position error signal to generate the filtered rate signal. The second integration means integrates the filtered rate signal to obtain the filtered position signal. 
         [0010]    Other desirable features and characteristics of the present invention will become apparent from the subsequent detailed description and the appended claims, taken in conjunction with the accompanying drawings and the preceding background. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]    The present invention will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements, and wherein: 
           [0012]      FIG. 1  is a block diagram of a motor control system including a portion of a motor controller that may be used to derive motor rotational rate according to an embodiment of the present invention; 
           [0013]      FIG. 2  depicts three ideal sinusoidal waveforms that are generated by three position sensors that may be used to implement the motor control system of  FIG. 1 ; 
           [0014]      FIG. 3  depicts a block diagram of a complementary filter that may be used to implement the motor controller of  FIG. 1 ; 
           [0015]      FIG. 4  depicts a 10 second calibration of rotor rotational rate for the complementary filter of  FIG. 3 ; 
           [0016]      FIG. 5  a square wave characteristic of derived rotational rate that may occur when individual magnet rotor poles exhibit differing magnetic flux intensities; 
           [0017]      FIG. 6  an exemplary sensor alignment that may be used to implement magnetic flux intensity variation compensation; 
           [0018]      FIG. 7  depicts a portion of a motor controller that may be used to derive motor rotational rate and implement magnetic flux intensity variation compensation; and 
           [0019]      FIG. 8  depicts a derived rotor rotational rate, using the motor controller of  FIG. 7 , which is automatically compensated for individual magnet intensity variations. 
       
    
    
     DETAILED DESCRIPTION 
       [0020]    The following detailed description is merely exemplary in nature and is not intended to limit the invention or the application and uses of the invention. Furthermore, there is no intention to be bound by any theory presented in the preceding background or the following detailed description. 
         [0021]    Referring to  FIG. 1 , a functional block diagram of a motor control system  100  is depicted and includes a motor  102 , a plurality of sensors  104 , and a motor controller  106 . In the depicted embodiment the motor  102  is implemented as a three-phase, two-pole brushless machine that includes a stator  108  and a rotor  112 . As is generally known, the stator  108 , when controllably energized, generates a rotating magnetic field that interacts with the rotor  112 . The rotor  112  in turn rotates, generating a torque. 
         [0022]    The plurality of sensors  104  are evenly spaced about the rotor  112  and are configured to sense the rotational position thereof. In the depicted embodiment three analog Hall-effect sensors  104 - 1 ,  104 - 2 ,  104 - 3  spaced 120-degrees apart from each other are used. Analog Hall-effect sensors, as is generally known, generate a voltage signal having an amplitude that is proportional to sensed magnetic field strength. Thus, for the configuration depicted in  FIG. 1  the sensors  104  generate three sinusoidal signals. As depicted in  FIG. 2  these sinusoidal signals  202 , referred to herein as a first sinusoidal signal  202 - 1 , a second sinusoidal signal  202 - 2 , and a third sinusoidal  202 - 3 , are 120-degrees out of phase with each other. These sinusoidal signals are supplied to the motor controller  106 . 
         [0023]    The motor controller  106  receives the first, second, and third sinusoidal signals  202  and, implementing position and rate control, controllably energizes the motor stator  108  from a non-illustrated power source. As noted above, several mathematical approaches can be used to derive both position and rate from the three sinusoidal signals  202 . In the depicted embodiment the motor controller  106  implements a derived rate algorithm that directly compensates for the previously-mentioned 3 rd  harmonic magnet intensity distortion. Specifically, the rate algorithm is the sum of three derived rates in accordance with the following equation: 
         [0000]    
       
         
           
             
               θ 
               . 
             
             = 
             
               
                 
                   2 
                   
                     3 
                      
                     
                       3 
                     
                   
                 
                  
                 
                   ( 
                   
                     X 
                     - 
                     Y 
                   
                   ) 
                 
                  
                 
                   Z 
                   . 
                 
               
               + 
               
                 
                   ( 
                   
                     Y 
                     - 
                     Z 
                   
                   ) 
                 
                  
                 
                   X 
                   . 
                 
               
               + 
               
                 
                   ( 
                   
                     Z 
                     - 
                     X 
                   
                   ) 
                 
                  
                 
                   
                     Y 
                     . 
                   
                   . 
                 
               
             
           
         
       
     
         [0024]    The derivation of the above equation, for nominal sine wave magnets and with 3 rd  harmonic distortion, is provided in the attached Appendix I. As is shown therein, the resulting equations are identical both with and without 3 rd  harmonic distortion, which shows that 3 rd  harmonic distortion is completely eliminated with this algorithm. Moreover, this algorithm provides uniform rate output for phase or amplitude errors that may be present due to Hall-effect sensor position errors. Returning once again to  FIG. 1 , it is seen that the motor controller  106 , to implement the above algorithm, includes first, second, and third subtraction functions  114 - 1 ,  114 - 2 ,  114 - 3 , first, second, and third differentiation functions  116 - 1 ,  116 - 2 ,  116 - 3 , first, second, and third multiplication functions  118 - 1 ,  118 -,  118 - 3 , an addition function  122 , and a fourth multiplication function  124 . 
         [0025]    The first subtraction function  114 - 1  is coupled to receive the first sinusoidal signal  202 - 1  and the third sinusoidal signal  202 - 3  and generates a first sinusoidal difference signal  126 - 1  representative of the mathematical difference between these two sinusoidal signals (e.g., (Z−X)). The second subtraction function  114 - 2  is coupled to receive the second sinusoidal signal  202 - 2  and the first sinusoidal signal  202 - 1  and to generates a second sinusoidal difference signal  126 - 2  representative of the mathematical difference between these two sinusoidal signals (e.g., (X−Y)). The third subtraction function  114 - 3  is coupled to receive the third sinusoidal signal  202 - 3  and the second sinusoidal signal  202 - 2  and generates a third sinusoidal difference signal  126 - 3  representative of the mathematical difference between these two sinusoidal signals (e.g., (Y−Z). 
         [0026]    The first differentiation function  116 - 1  is coupled to receive the first sinusoidal signal  202 - 1  and generates a first cosine signal  128 - 1  (e.g., {dot over (X)}) representative of the derivative of the first sinusoidal signal. The second differentiation function  116 - 2  is coupled to receive the second sinusoidal signal  202 - 2  and generates a second cosine signal  128 - 2  (e.g., {dot over (Y)}) representative of the derivative of the second sinusoidal signal. The third differentiation function  116 - 3  is coupled to receive the third sinusoidal signal  202 - 3  and generates a third cosine signal  128 - 3  (e.g., Ż) representative of the derivative of the third sinusoidal signal. 
         [0027]    The first multiplication function  118 - 1  is coupled to receive the first sinusoidal difference signal  126 - 1  and the second cosine signal  128 - 2  and generates a first product signal  132 - 1  representative of the mathematical product of these two signals (e.g., (Z−X){dot over (Y)}). The second multiplication  118 - 2  is coupled to receive the second sinusoidal difference signal  126 - 2  and the third cosine signal  128 - 3  and generates a second product signal  132 - 2  representative of the mathematical product of these two signals (e.g., (X−Y)Ż). The third multiplication function  118 - 3  is coupled to receive the third sinusoidal difference signal  126 - 3  and the first cosine signal  128 - 1  and generates a third product signal  132 - 3  representative of the mathematical product of these two signals (e.g., (Y−Z){dot over (X)}). 
         [0028]    The addition function  122  is coupled to receive the first, second, and third product signals from the first, second, and third multiplication functions  118 - 1 ,  118 - 2 , and  118 - 3 , respectively. The first addition function  122  is configured, upon receipt of these signals, to generate a summation signal  134  that is representative of the mathematical sum of the three signals (e.g., (X−Y)Ż+(Y−Z){dot over (X)}+(Z−X){dot over (Y)}). The summation signal  134  is supplied to the fourth multiplication function  124 . The fourth multiplication function  124  is coupled to receive the summation signal  134  from the addition function  122 . The fourth multiplication function  124  is configured, upon receipt of the summation signal  134 , to multiply the summation signal  134  by a constant 
         [0000]    
       
         
           
             
               ( 
               
                 
                   e 
                   . 
                   g 
                   . 
                 
                 , 
                 
                   2 
                   
                     3 
                      
                     
                       3 
                     
                   
                 
               
               ) 
             
             , 
           
         
       
     
         [0000]    to thereby generate a rate signal  136  that is proportional to the rotational rate of the rotor  112 . 
         [0029]    In addition to deriving the rotational rate of the rotor  112 , the motor controller  106  may also implement one or more algorithms to derive the rotational position of the rotor  112 . For example, standard arctangent algorithms can be applied to determine 360-degree rotor position from sine and cosine signals derived from sinusoidal signals supplied from the plurality of sensors  104 . One exemplary algorithm is disclosed in U.S. Pat. No. 6,744,230, which is assigned to the assignee of the instant application, and the entirety of which is incorporated herein by reference. In accordance with that algorithm, the three sinusoidal signals from the sensors  104  may be combined into two orthogonal vectors representative of the sine and the cosine of the instantaneous rotational position of the rotor  112 . The instantaneous rotational position may be computed from the arctangent of these two signals. 
         [0030]    No matter the specific manner in which the rotational position of the rotor  112  is derived, it is noted that residual errors in derived rotational rate are quite different from errors induced in computed rotational position, and most notably in an arctangent computed rotational position. In such instances, a complementary filter may be used to compute estimated rotor position and rotor rate. Such a complementary filter benefits from both the low frequency accuracy of arctangent computed rotor position and the high frequency stability of derived rotor rate. A first order complementary filter, such as the one disclosed in the above-mentioned U.S. Pat. No. 6,744,230, provides relatively smooth estimated rotor position and rotor rate. However, the dynamic response of this filter exhibits errors due to derived rate gain error. Because precision motor control relies on accurate feedback signals under dynamic conditions, this complementary filter may not be suitable. Instead, a complementary filter, such as the one depicted in  FIG. 3 , is preferably used. 
         [0031]    As  FIG. 3  depicts, the complementary filter  300  is coupled to receive a derived rotor rotational position signal and a derived rotor rotational rate signal and is configured to supply a filtered rotor rotational position signal and a filtered rotor rotational rate signal. The complementary filter  300  is a self-calibrating filter, and includes a subtraction function  302 , a gain adjustment function  304 , a first integration function  306 , a multiplication function  308 , a summation function  312 , and a second integration function  314 . The first subtraction function  302  is coupled to receive the derived rotational position and the filtered rotor rotational position and supplies a position error representative of the mathematical difference between the derived rotor rotational position and the filtered rotor rotational position. The position error is supplied to the gain adjustment function  304  and the first integration function  306 . 
         [0032]    The gain adjustment function  304  is configured, upon receipt of the position error, to apply a predetermined gain (1/τ) to the position error, and thereby generate an adjusted position error. The adjusted position error is supplied to the summation function  312 . As is generally known, the parameter (τ) is representative of the smoothing function of the filter  300 , and its value may be selected to meet desired filter characteristics. 
         [0033]    The first integration function  306 , as was noted above, is coupled to receive the position error. The first integration function  306 , upon receipt of the position error, integrates the position error and supplies a rate adjustment to the multiplication function  308 . The multiplication function  308 , which also receives the derived rotor rotational rate, multiplies the derived rotor rotational rate and the rate adjustment to generate an adjusted rate. The adjusted rate is supplied to the summation function  312 . 
         [0034]    The summation function  312  receives the adjusted position error from the gain adjustment function  304  and the adjusted rate from the multiplication function  308 . The summation function  312  sums the adjusted rate and the adjusted position error to generate the filtered rotor rotational rate. The filtered rotor rotational rate is also supplied to the second integration function  314 . The second integration function  314  integrates the filtered rotor rotational rate to generate the filtered rotor rotational position. 
         [0035]    The complementary filter  300  depicted in  FIG. 3  and described above provides, via the first integration function  306  and the multiplication function  308 , a calibration mechanism. That is, the position error is integrated to adjust the derived rate gain. Once this calibration is performed, the integrator value can be frozen or the calibration can be ongoing. In either case, once the calibration establishes the gain correction on the derived rotor rotational rate the filtered rotor rotational rate and position are accurate under static and dynamic conditions. This capability is illustrated in  FIG. 4 , which graphically depicts a 10 second calibration by the complementary filter  300 . As may be seen, the filtered rotor rotational rate is initially around 110 rad/sec, but settles to the correct value of 100 rad/sec. 
         [0036]    The derived rate algorithm implemented by the motor controller  106 , and which was described above, directly compensates for the 3 rd  harmonic magnet intensity distortion, which was also previously described. Nonetheless, it is noted that individual magnet poles may exhibit differing magnetic flux intensities. Such variations between the individual magnet poles can result in the derived rotor rotational rate exhibiting a square wave characteristic, most notably with a two-pole rotor  112 . More specifically, as the rotor  112  rotates through a full rotation, the alignment of the sensors  104  alternates, with two of the three sensors being aligned with one pole or the other. This alternating pattern results in the square wave rotor rotational rate signal, such as the one depicted in  FIG. 5 . 
         [0037]    One method of compensating for the above-described characteristic, which is depicted in  FIG. 6 , is to position a second set of sensors (e.g.,  104 - 1 ′,  104 - 2 ′,  104 - 3 ′) 180-degrees apart from the first set of sensors (e.g.,  104 - 1 ,  104 - 2 ,  104 - 3 ). The motor controller  106  is additionally configured to implement the derived rate algorithm for the second set of sensors. As depicted in  FIG. 7 , it may thus be appreciated that the motor controller  106  preferably further includes fourth, fifth, and sixth subtraction functions  114 - 4 ,  114 - 5 ,  114 - 6 , fourth, fifth, and sixth differentiation functions  116 - 4 ,  116 - 5 ,  116 - 6 , fifth, sixth, and seventh multiplication functions  118 - 5 ,  118 - 6 ,  118 - 7 , a second addition function  122 - 2 , and an eighth multiplication function  124 - 2 . Also, the constant that the fourth and eighth multiplication functions  124 - 1 ,  124 - 2  each implement is 
         [0000]    
       
         
           
             
               1 
               
                 3 
                  
                 
                   3 
                 
               
             
             , 
           
         
       
     
         [0000]    rather than 
         [0000]    
       
         
           
             
               2 
               
                 3 
                  
                 
                   3 
                 
               
             
             . 
           
         
       
     
         [0000]    Turning to  FIG. 8 , it is seen that the rotor rotational rate derived from the second set of sensors  104 - 1 ′,  104 - 2 ′,  104 - 3 ′ produces a square wave rate signal  802  that is 180-degrees out of phase with the first derived rotor rotational rate signal  804 . Thus, by summing these two derived rates together, via a third addition function  122 - 3  (see  FIG. 7 ), individual magnet intensity variations are automatically compensated for in producing a relatively constant rate signal  806 . 
         [0038]    The system described herein provides a relatively low-cost, fine resolution motor control scheme that overcomes inherent sensor errors and rotor magnetic flux intensity variation errors. It will be appreciated that some or all of the described motor controller  106  may be implemented in hardware, software, firmware, or various combinations thereof. It will additionally be appreciated that some or all of the individual functions described herein may be implemented using analog components, digital components, or various combinations thereof. 
         [0039]    While at least one exemplary embodiment has been presented in the foregoing detailed description of the invention, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing an exemplary embodiment of the invention. It being understood that various changes may be made in the function and arrangement of elements described in an exemplary embodiment without departing from the scope of the invention as set forth in the appended claims. 
       APPENDIX I 
       [0040]    It is initially assumed that sinusoidal waveforms of the following form are supplied: 
         [0000]        X =sin(θ+30) 
         [0000]        Y =sin(θ+150) 
         [0000]        Z =sin(θ+270) 
         [0041]    When the first derivative of each of these sinusoids is taken, the result is: 
         [0000]        {dot over (X)}= {dot over (θ)} cos(θ+30) 
         [0000]        {dot over (Y)}= {dot over (θ)} cos(θ+150) 
         [0000]        Ż= {dot over (θ)} cos(θ+270) 
         [0042]    Using known trigonometric identities, Equation 1 below may be derived as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             X 
                             - 
                             Y 
                           
                           ) 
                         
                          
                         
                           Z 
                           . 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             Y 
                             - 
                             Z 
                           
                           ) 
                         
                          
                         
                           X 
                           . 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             Z 
                             - 
                             X 
                           
                           ) 
                         
                          
                         
                           Y 
                           . 
                         
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   30 
                                 
                                 ) 
                               
                             
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   150 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                          
                         
                           θ 
                           . 
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               270 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                               
                           
                            
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   150 
                                 
                                 ) 
                               
                             
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   270 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                          
                         
                           θ 
                           . 
                         
                          
                         
                           cos 
                           ( 
                           
                               
                           
                            
                           
                             θ 
                             + 
                             30 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           [ 
                           
                               
                           
                            
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   270 
                                 
                                 ) 
                               
                             
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   30 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                          
                         
                           θ 
                           . 
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               150 
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           ( 
                           
                             X 
                             - 
                             Y 
                           
                           ) 
                         
                          
                         
                           Z 
                           . 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             Y 
                             - 
                             Z 
                           
                           ) 
                         
                          
                         
                           X 
                           . 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             Z 
                             - 
                             X 
                           
                           ) 
                         
                          
                         
                           Y 
                           . 
                         
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   30 
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   30 
                                 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   150 
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   150 
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                          
                         
                           
                             θ 
                             . 
                           
                            
                           
                             [ 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 270 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 270 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   150 
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   150 
                                 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   270 
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   270 
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                          
                         
                           
                             θ 
                             . 
                           
                            
                           
                             [ 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 30 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 30 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   270 
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   270 
                                 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   30 
                                 
                                 + 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   30 
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                          
                         
                           
                             θ 
                             . 
                           
                            
                           
                             [ 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 150 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                                  
                                 
                                     
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 150 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ( 
                         
                           X 
                           - 
                           Y 
                         
                         ) 
                       
                        
                       
                         Z 
                         . 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           Y 
                           - 
                           Z 
                         
                         ) 
                       
                        
                       
                         X 
                         . 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           Z 
                           - 
                           X 
                         
                         ) 
                       
                        
                       
                         Y 
                         . 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   
                                     
                                       3 
                                     
                                     2 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   
                                     - 
                                     
                                       
                                         3 
                                       
                                       2 
                                     
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                          
                         
                           
                             θ 
                             . 
                           
                            
                           
                             [ 
                             
                               sin 
                                
                               
                                   
                               
                                
                               θ 
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               [ 
                               
                                 
                                   
                                     - 
                                     
                                       
                                         3 
                                       
                                       2 
                                     
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               ] 
                             
                           
                           ] 
                         
                          
                         
                           
                             θ 
                             . 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     3 
                                   
                                   2 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               - 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               [ 
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                   
                               
                                
                               
                                 
                                   
                                     
                                       3 
                                     
                                     2 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                          
                         
                           
                             θ 
                             . 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     
                                       3 
                                     
                                     2 
                                   
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               - 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                             ] 
                           
                         
                          
                         
                           
 
                         
                          
                         
                           
                             
                               ( 
                               
                                 X 
                                 - 
                                 Y 
                               
                               ) 
                             
                              
                             
                               Z 
                               . 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 Y 
                                 - 
                                 Z 
                               
                               ) 
                             
                              
                             
                               X 
                               . 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 Z 
                                 - 
                                 X 
                               
                               ) 
                             
                              
                             
                               Y 
                               . 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             [ 
                             
                               
                                 3 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               θ 
                             
                             ] 
                           
                            
                           
                             
                               θ 
                               . 
                             
                              
                             
                               [ 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               ] 
                             
                           
                         
                         + 
                         
                           
                             [ 
                             
                                 
                             
                              
                             
                               
                                 
                                   - 
                                   
                                     
                                       3 
                                     
                                     2 
                                   
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               + 
                               
                                 
                                   3 
                                   2 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                             ] 
                           
                            
                           
                             
                               θ 
                               . 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       3 
                                     
                                     2 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 - 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                           
                         
                         + 
                         
                           
                             [ 
                             
                                 
                             
                              
                             
                               
                                 
                                   - 
                                   
                                     
                                       3 
                                     
                                     2 
                                   
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               - 
                               
                                 
                                   3 
                                   2 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                             
                             ] 
                           
                            
                           
                             
                               θ 
                               . 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     - 
                                     
                                       
                                         3 
                                       
                                       2 
                                     
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 - 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                           
                            
                           
                             
 
                           
                            
                           
                             ( 
                             
                               X 
                               - 
                               Y 
                             
                             ) 
                           
                            
                           
                             Z 
                             . 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               Y 
                               - 
                               Z 
                             
                             ) 
                           
                            
                           
                             X 
                             . 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               Z 
                               - 
                               X 
                             
                             ) 
                           
                            
                           
                             Y 
                             . 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               3 
                             
                              
                             
                               sin 
                               2 
                             
                              
                             θ 
                              
                             
                               θ 
                               . 
                             
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     - 
                                     
                                       3 
                                     
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     
                                       3 
                                        
                                       
                                         3 
                                       
                                     
                                     4 
                                   
                                    
                                   
                                     cos 
                                     2 
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     
                                       3 
                                     
                                     4 
                                   
                                    
                                   
                                     sin 
                                     2 
                                   
                                    
                                   θ 
                                 
                                 - 
                                 
                                   
                                     3 
                                     4 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                              
                             
                               θ 
                               . 
                             
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     3 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     
                                       3 
                                        
                                       
                                         3 
                                       
                                     
                                     4 
                                   
                                    
                                   
                                     cos 
                                     2 
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     
                                       3 
                                     
                                     4 
                                   
                                    
                                   
                                     sin 
                                     2 
                                   
                                    
                                   θ 
                                 
                                 + 
                                 
                                   
                                     3 
                                     4 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                    
                                   
                                       
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                               
                               ] 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                                
                               
                                 
 
                               
                               ( 
                               
                                 X 
                                 - 
                                 Y 
                               
                               ) 
                             
                              
                             
                               Z 
                               . 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 Y 
                                 - 
                                 Z 
                               
                               ) 
                             
                              
                             
                               X 
                               . 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 Z 
                                 - 
                                 X 
                               
                               ) 
                             
                              
                             
                               Y 
                               . 
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 3 
                               
                                
                               
                                 sin 
                                 2 
                               
                                
                               θ 
                                
                               
                                 θ 
                                 . 
                               
                             
                             + 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         3 
                                          
                                         
                                           3 
                                         
                                       
                                       4 
                                     
                                      
                                     
                                       cos 
                                       2 
                                     
                                      
                                     θ 
                                   
                                   + 
                                   
                                     
                                       
                                         3 
                                       
                                       4 
                                     
                                      
                                     
                                       sin 
                                       2 
                                     
                                      
                                     θ 
                                   
                                 
                                 ] 
                               
                                
                               
                                 θ 
                                 . 
                               
                             
                             + 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         3 
                                          
                                         
                                           3 
                                         
                                       
                                       4 
                                     
                                      
                                     
                                       cos 
                                       2 
                                     
                                      
                                     θ 
                                   
                                   + 
                                   
                                     
                                       
                                         3 
                                       
                                       4 
                                     
                                      
                                     
                                       sin 
                                       2 
                                     
                                      
                                     θ 
                                   
                                 
                                 ] 
                               
                                
                               
                                 
                                   θ 
                                   . 
                                 
                                  
                                 
                                   
 
                                 
                                 ( 
                                 
                                   X 
                                   - 
                                   Y 
                                 
                                 ) 
                               
                                
                               
                                 Z 
                                 . 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   Y 
                                   - 
                                   Z 
                                 
                                 ) 
                               
                                
                               
                                 X 
                                 . 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   X 
                                 
                                 ) 
                               
                                
                               
                                 Y 
                                 . 
                               
                             
                           
                           = 
                           
                             
                               
                                 
                                   3 
                                 
                                  
                                 
                                   sin 
                                   2 
                                 
                                  
                                 θ 
                                  
                                 
                                   θ 
                                   . 
                                 
                               
                               + 
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           3 
                                            
                                           
                                             3 
                                           
                                         
                                         2 
                                       
                                        
                                       
                                         cos 
                                         2 
                                       
                                        
                                       θ 
                                     
                                     + 
                                     
                                       
                                         
                                           3 
                                         
                                         2 
                                       
                                        
                                       
                                         sin 
                                         2 
                                       
                                        
                                       θ 
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                   
                                     θ 
                                     . 
                                   
                                    
                                   
                                     
 
                                   
                                   ( 
                                   
                                     X 
                                     - 
                                     Y 
                                   
                                   ) 
                                 
                                  
                                 
                                   Z 
                                   . 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     Y 
                                     - 
                                     Z 
                                   
                                   ) 
                                 
                                  
                                 
                                   X 
                                   . 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     Z 
                                     - 
                                     X 
                                   
                                   ) 
                                 
                                  
                                 
                                   Y 
                                   . 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             3 
                                              
                                             
                                               3 
                                             
                                           
                                           2 
                                         
                                          
                                         
                                           sin 
                                           2 
                                         
                                          
                                         θ 
                                       
                                       + 
                                       
                                         
                                           
                                             3 
                                              
                                             
                                               3 
                                             
                                           
                                           2 
                                         
                                          
                                         
                                           cos 
                                           2 
                                         
                                          
                                         θ 
                                       
                                     
                                     ] 
                                   
                                    
                                   
                                     
                                       θ 
                                       . 
                                     
                                      
                                     
                                       
 
                                     
                                     ( 
                                     
                                       X 
                                       - 
                                       Y 
                                     
                                     ) 
                                   
                                    
                                   
                                     Z 
                                     . 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       Y 
                                       - 
                                       Z 
                                     
                                     ) 
                                   
                                    
                                   
                                     X 
                                     . 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       Z 
                                       - 
                                       X 
                                     
                                     ) 
                                   
                                    
                                   
                                     Y 
                                     . 
                                   
                                 
                               
                               = 
                               
                                 
                                   
                                     3 
                                      
                                     
                                       3 
                                     
                                   
                                   2 
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
         [0043]    Now, if rate is computed again, but with the addition of 3 rd  harmonic signal distortion, then: 
         [0000]        X=A  sin(θ+30)+ B  sin(3(θ+30)) 
         [0000]        Y=A  sin(θ+150)+ B  sin(3(θ+150)) 
         [0000]        Z=A  sin(θ+270)+ B  sin(3(θ+270)), 
         [0000]      and: 
         [0000]        {dot over (X)}=A {dot over (θ)} cos(θ+30)+ B 3{dot over (θ)} cos(3(θ+30)) 
         [0000]        {dot over (Y)}=A {dot over (θ)} cos(θ+150)+ B 3{dot over (θ)} cos(3(θ+150)) 
         [0000]        Ż=A {dot over (θ)} cos(θ+270)+ B 3{dot over (θ)} cos(3(θ+270)). 
         [0044]    As a result: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         A 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               30 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               3 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   30 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                           ( 
                           
                             θ 
                             + 
                             150 
                           
                           ) 
                         
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                           ( 
                           
                             3 
                              
                             
                               ( 
                               
                                 θ 
                                 + 
                                 150 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             θ 
                             + 
                             270 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             3 
                              
                             
                               ( 
                               
                                 θ 
                                 + 
                                 270 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         A 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               150 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               3 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   150 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               270 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               3 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   270 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             θ 
                             + 
                             30 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             3 
                              
                             
                               ( 
                               
                                 θ 
                                 + 
                                 30 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         A 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               270 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               3 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   270 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               θ 
                               + 
                               30 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               3 
                                
                               
                                 ( 
                                 
                                   θ 
                                   + 
                                   30 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                         θ 
                         . 
                       
                        
                       
                           
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             θ 
                             + 
                             150 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             3 
                              
                             
                               ( 
                               
                                 θ 
                                 + 
                                 150 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         30 
                       
                       + 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         30 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θcos90 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θsin 
                          
                         
                             
                         
                          
                         90 
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         150 
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         150 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         3 
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         450 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         450 
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       270 
                     
                     - 
                     
                       A 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       270 
                     
                     + 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                         
                           θ 
                            
                           
                               
                           
                         
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       810 
                     
                     - 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       810 
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         150 
                       
                       + 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         150 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θcos450 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θsin 
                          
                         
                             
                         
                          
                         450 
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         270 
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         270 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         3 
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         810 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         810 
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       30 
                     
                     - 
                     
                       A 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       30 
                     
                     + 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                         
                           θ 
                            
                           
                               
                           
                         
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       90 
                     
                     - 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       90 
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         270 
                       
                       + 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         270 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θcos810 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θsin 
                          
                         
                             
                         
                          
                         810 
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         30 
                       
                       - 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         30 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         3 
                          
                         θ 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         90 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         90 
                       
                     
                     ] 
                   
                   [ 
                   
                     
                       A 
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       150 
                     
                     - 
                     
                       A 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       150 
                     
                     + 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                         θ 
                         . 
                       
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       450 
                     
                     - 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       450 
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         A 
                          
                         
                           
                             3 
                           
                           2 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         A 
                          
                         
                           1 
                           2 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         θ 
                       
                       + 
                       
                         A 
                          
                         
                           
                             3 
                           
                           2 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                       - 
                       
                         A 
                          
                         
                           1 
                           2 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         θ 
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                         θ 
                         . 
                       
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         
                           - 
                           A 
                         
                          
                         
                           
                             3 
                           
                           2 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                        
                       
                           
                       
                       + 
                       
                         A 
                          
                         
                           1 
                           2 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         A 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                      
                     
                         
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       A 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       A 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         
                           - 
                           A 
                         
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                       
                       - 
                       
                         A 
                          
                         
                           
                             3 
                           
                           2 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                        
                       
                           
                       
                       - 
                       
                         A 
                          
                         
                           1 
                           2 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       - 
                       
                         B 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         3 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         - 
                         A 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       A 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       B 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       
                           
                       
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       
                           
                       
                        
                       
                         B 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                           
                       
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       θcos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                           
                       
                        
                       
                         B 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   ] 
                 
                 + 
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         - 
                         
                           A 
                           2 
                         
                       
                        
                       
                         3 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θcos3θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         4 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       3 
                        
                       θ3 
                        
                       
                           
                       
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       θ 
                     
                   
                    
                   
                       
                   
                   ] 
                 
                 + 
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                           
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         3 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       
                         θA 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         4 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ3 
                        
                       
                           
                       
                        
                       AB 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θcos3θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θsin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       
                         θ 
                         . 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       θcos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   θ 
                   . 
                 
                  
                 
                   [ 
                   
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         3 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         4 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcosθ 
                     
                     - 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         3 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         1 
                         4 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θsin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θsin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       AB 
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         
                           2 
                            
                           
                               
                           
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       AB 
                        
                       
                         1 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       
                         θ3B 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θcos3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θcos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     - 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       3 
                        
                       
                         B 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       3 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   
                     θ 
                       
                   
                   . 
                 
                  
                 
                   [ 
                   
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                      
                     
                         
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                         
                         4 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   θ 
                   . 
                 
                  
                 
                   [ 
                   
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                            
                           
                             3 
                           
                         
                         2 
                       
                        
                       
                         sin 
                         2 
                       
                        
                       θ 
                     
                     + 
                     
                       
                         A 
                         2 
                       
                        
                       
                         
                           3 
                            
                           
                             3 
                           
                         
                         2 
                       
                        
                       
                         cos 
                         2 
                       
                        
                       θ 
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     and 
                      
                     
                       
 
                     
                     ( 
                     
                       X 
                       - 
                       Y 
                     
                     ) 
                   
                    
                   
                     Z 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Y 
                       - 
                       Z 
                     
                     ) 
                   
                    
                   
                     X 
                     . 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       Z 
                       - 
                       X 
                     
                     ) 
                   
                    
                   
                     Y 
                     . 
                   
                 
               
               = 
               
                 
                   θ 
                   . 
                 
                  
                 
                   A 
                   2 
                 
                  
                 
                   
                     
                       3 
                        
                       
                         3 
                       
                     
                     2 
                   
                   . 
                 
               
             
           
         
       
     
         [0045]    Hence, if a nominal signal amplitude of A=1 is assumed, then: 
         [0000]    
       
         
           
             
               
                 
                   ( 
                   
                     X 
                     - 
                     Y 
                   
                   ) 
                 
                  
                 
                   Z 
                   . 
                 
               
               + 
               
                 
                   ( 
                   
                     Y 
                     - 
                     Z 
                   
                   ) 
                 
                  
                 
                   X 
                   . 
                 
               
               + 
               
                 
                   ( 
                   
                     Z 
                     - 
                     X 
                   
                   ) 
                 
                  
                 
                   Y 
                   . 
                 
               
             
             = 
             
               
                 θ 
                 . 
               
                
               
                 
                   
                     3 
                      
                     
                       3 
                     
                   
                   2 
                 
                 . 
               
             
           
         
       
     
         [0000]    This equation is identical to Equation 1, which proves that 3 rd  harmonic signal distortion does not affect the output when this algorithm is implemented.