PATENT 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.

PATENT DESCRIPTION
TECHNICAL FIELD 
     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 
     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. 
     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. 
     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. 
     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 
     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. 
     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. 
     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. 
     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. 
     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 
       The present invention will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements, and wherein: 
         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; 
         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 ; 
         FIG. 3  depicts a block diagram of a complementary filter that may be used to implement the motor controller of  FIG. 1 ; 
         FIG. 4  depicts a 10 second calibration of rotor rotational rate for the complementary filter of  FIG. 3 ; 
         FIG. 5  a square wave characteristic of derived rotational rate that may occur when individual magnet rotor poles exhibit differing magnetic flux intensities; 
         FIG. 6  an exemplary sensor alignment that may be used to implement magnetic flux intensity variation compensation; 
         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 
         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 
     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. 
     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. 
     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 . 
     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: 
     
       
         
           
             
               θ 
               . 
             
             = 
             
               
                 
                   2 
                   
                     3 
                     ⁢ 
                     
                       3 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     X 
                     - 
                     Y 
                   
                   ) 
                 
                 ⁢ 
                 
                   Z 
                   . 
                 
               
               + 
               
                 
                   ( 
                   
                     Y 
                     - 
                     Z 
                   
                   ) 
                 
                 ⁢ 
                 
                   X 
                   . 
                 
               
               + 
               
                 
                   ( 
                   
                     Z 
                     - 
                     X 
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     Y 
                     . 
                   
                   . 
                 
               
             
           
         
       
     
     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 . 
     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). 
     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. 
     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)}). 
     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 
               (       e   .   g   .     ,     2     3   ⁢     3           )     ,         
to thereby generate a rate signal  136  that is proportional to the rotational rate of the rotor  112 .
 
     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. 
     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. 
     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 . 
     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. 
     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 . 
     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. 
     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. 
     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 . 
     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 
               1     3   ⁢     3         ,         
rather than
 
               2     3   ⁢     3         .         
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 .
 
     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. 
     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 
     It is initially assumed that sinusoidal waveforms of the following form are supplied:
 
 X =sin(θ+30),
 
 Y =sin(θ+150)
 
 Z =sin(θ+270).
 
     When the first derivative of each of these sinusoids is taken, the result is:
 
 {dot over (X)}= {dot over (θ)} cos(θ+30),
 
 {dot over (Y)}= {dot over (θ)} cos(θ+150),
 
 Ż= {dot over (θ)} cos(θ+270).
 
     Using known trigonometric identities, Equation 1 below may be derived as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                         ⁢ 
                         
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                               ⁡ 
                               
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 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 
                   
                   ) 
                 
               
             
           
         
       
     
     Now, if rate is computed again, but with the addition of 3 rd  harmonic signal distortion, then:
 
 X=A  sin(θ+30)+ B  sin(3(θ+30))
 
 Y=A  sin(θ+150)+ B  sin(3(θ+150))
 
 Z=A  sin(θ+270)+ B  sin(3(θ+270)),
 
and:
 
 {dot over (X)}=A {dot over (θ)} cos(θ+30)+ B 3{dot over (θ)} cos(3(θ+30))
 
 {dot over (Y)}=A {dot over (θ)} cos(θ+150)+ B 3{dot over (θ)} cos(3(θ+150))
 
 Ż=A {dot over (θ)} cos(θ+270)+ B 3{dot over (θ)} cos(3(θ+270)).
 
     As a result: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       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 
                   
                   . 
                 
               
             
           
         
       
     
     Hence, if a nominal signal amplitude of A=1 is assumed, then: 
                   (     X   -   Y     )     ⁢     Z   .       +       (     Y   -   Z     )     ⁢     X   .       +       (     Z   -   X     )     ⁢     Y   .         =       θ   .     ⁢         3   ⁢     3       2     .             
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.