Patent Publication Number: US-10763579-B2

Title: Mobile terminal antenna alignment using arbitrary orientation attitude

Description:
RELATED APPLICATIONS 
     This present application for patent is a continuation of U.S. patent application Ser. No. 14/595,025 by Sleight, et al., entitled “Mobile Terminal Antenna Alignment Using Arbitrary Orientation Attitude” filed Jan. 12, 2015, and claims the benefit of U.S. Provisional Application No. 61/927,322 entitled “Mobile Terminal Antenna Alignment Using Arbitrary Orientation Attitude”, filed Jan. 14, 2014, which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The disclosed method and apparatus relates to aligning an antenna and more specifically to aligning an antenna mounted on a mobile platform to the platform. 
     BACKGROUND 
     Satellite communication systems provide a means by which data, including audio, video and various other sorts of data, can be communicated from a transmitter at one location to a receiver at another location. Satellite communication systems are currently being used on mobile platforms, such as civilian airlines and privately owned aircraft to provide entertainment and internet access to the passengers. Military platforms, such as aircraft and ships, currently use satellite communication systems to receive and transmit various types of information, including strategic and tactical information. 
     Satellite communication systems require an antenna to receive signals from, and transmit signals to, a satellite. The antenna typically must be pointed accurately at the satellite. A satellite antenna positioner is typically used to point the antenna at the satellite. It is common for these antenna positioners to have two axes of motion (e.g., elevation and azimuth). In the case of a system mounted on an aircraft, the elevation and azimuth that will point the antenna to the satellite can be calculated if the following information is known: (1) the location and attitude of the aircraft; and (2) the location of the satellite, assuming the relative alignment of the antenna to the body of the aircraft is known. In most commercial airliners and military aircraft, the attitude of an aircraft is determined by a position and attitude measuring device (PAMD), such as an inertial reference unit (IRU). 
     Such systems typically provide the attitude of the aircraft in terms of three orthogonal axes: roll, pitch, and yaw. Errors in alignment of the antenna with respect to the PAMD will cause pointing errors (i.e., the antenna will not be pointed accurately at the desired satellite when using information from the PAMD to calculate the parameters, such as azimuth and elevation, for pointing the antenna). These alignment errors can be defined as roll, pitch and yaw errors. The antenna can be “peaked” to correct for these errors for a particular orientation. Peaking involves finding the antenna direction that results in the greatest signal strength received from the satellite through the antenna. These corrections are determined within the antenna positioner. Accordingly, such corrections will be determined in the two axes of elevation and azimuth used by the antenna positioner. 
     While the corrections can be converted from azimuth and elevation to a three dimensional Cartesian coordinate system, a problem exists in that such corrections will only be accurate for that particular orientation of the mobile platform. Applying these corrections to the azimuth and elevation calculated for other orientations will not accurately point the antenna. In fact, applying such corrections may result in even greater pointing errors in some orientations. 
     It can be seen that accurately aligning the antenna to the PAMD of an aircraft is important when using the PAMD output to position a satellite antenna. However, performing the alignment poses challenges. Therefore, there is currently a need for a simple and accurate means by which to align a satellite antenna to a mobile platform, such as an aircraft frame or PAMD within an aircraft. 
     SUMMARY 
     Various embodiments of a method and apparatus for accurately aligning a satellite antenna mounted on a mobile platform are disclosed. In one embodiment of the disclosed method and apparatus, the platform is an aircraft. However, the disclosed concepts can be applied to other mobile platforms as well, such as ships, trucks, trains, automobiles, and the like. In accordance with one embodiment of the disclosed method and apparatus, the platform is placed in a first orientation, which may be arbitrarily selected for convenience. Measurements are made in the first orientation. In the case in which the platform is an aircraft, the platform can be placed in the first orientation during a pre-flight alignment procedure, or while the aircraft is undergoing ground movement (e.g., taxiing), or during flight. 
     The measurements are made by receiving the location of the platform and the location of a satellite of interest. The location of the platform and the satellite are used to determine a first vector {right arrow over (d)} from the platform to the satellite. The first vector {right arrow over (d)} is represented in coordinates defined with respect to a topocentric reference frame. An output from a Position and Attitude Measuring Device (PAMD), such as an inertial reference unit (IRU), provides the attitude of the platform. It should be noted that there may be an offset between the platform reference frame and the PAMD reference frame. However, for the purpose of this discussion, the platform reference frame is assumed to be aligned with the PAMD reference frame. Any such offset will be irrelevant, so long as the relationship between the PAMD and the antenna reference frames remains fixed. A second vector {right arrow over (d)}′ i  is determined by performing an alias transformation on the first vector {right arrow over (d)} based on the attitude output from the PAMD to transform the first vector {right arrow over (d)} from the topocentric reference frame to the second vector {right arrow over (d)}′ i  having coordinates defined with respect to the platform reference frame (i.e., PAMD reference frame). 
     In addition, an antenna control unit (ACU) peaks the antenna. The orientation of the antenna when peaked is determined based on the output from an antenna positioning motor or sensors used to assist in positioning the antenna (i.e., directing the antenna to a satellite). For example, in one embodiment in which the antenna is positioned using an antenna positioning motor having motion in azimuth and elevation, the azimuth and elevation that result in the antenna receiving the strongest signal are used as the orientation of the antenna. A third vector {right arrow over (d)}″ i  pointing from the antenna to the satellite represented in coordinates defined with respect to the antenna reference frame is determined based on the orientation of the antenna when peaked (i.e., the azimuth and elevation in the embodiment in which the antenna motor operates in these two axes). 
     The measurements are repeated for several orientations. Once measurements for an adequate number of orientations have been collected, a matrix comprising the collection of second vectors {right arrow over (d)}′ i  and a matrix comprising the collection of third vectors {right arrow over (d)}″ i  are used to determine a first rotation matrix. The first rotation matrix can then be used to determine roll, pitch and yaw offsets between the PAMD reference frame and the antenna reference frame. 
     A second rotation matrix is derived from the roll, pitch and yaw offsets. The second rotation matrix is used to perform an alias transformation on a vector in the PAMD reference frame to a vector in the antenna reference frame. Accordingly, a vector calculated to point from the platform to the satellite can be transformed to a vector pointing from the antenna to the satellite in the antenna reference frame. The vector in the antenna reference frame can be used to generate coordinates (i.e., azimuth and elevation) to be used in the antenna position motor to accurately point the antenna to the satellite. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The disclosed method and apparatus, in accordance with one or more various embodiments, is described with reference to the following figures. The drawings are provided for purposes of illustration only and merely depict examples of some embodiments of the disclosed method and apparatus. These drawings are provided to facilitate the reader&#39;s understanding of the disclosed method and apparatus. They should not be considered to limit the breadth, scope, or applicability of the claimed invention. It should be noted that for clarity and ease of illustration these drawings are not necessarily made to scale. 
         FIG. 1 a    is an illustration of the relevant components of a satellite communication system in accordance with one embodiment of the presently disclosed method and apparatus. 
         FIG. 1 b    is an illustration of an alternative embodiment of the presently disclosed method and apparatus in which a Position and Attitude Measurement Device (PAMD) is included within an Antenna Alignment Module (AAM). 
         FIG. 2  is an illustration of a three-dimensional Cartesian coordinate frame set in a topocentric reference frame. 
         FIG. 3  is an illustration of the aircraft and the associated PAMD reference frame associated with the PAMD on board the aircraft. 
         FIG. 4  is an illustration of a vector in a first reference frame comprising an X 1 , Y 1 , and Z axis. 
         FIG. 5  is a simplified flow chart of the procedure used in accordance with one embodiment of the disclosed method and apparatus for determining the roll, pitch and yaw rotational offsets between an antenna and a positioning and attitude measurement device (PAMD) mounted in an aircraft. 
         FIG. 6  is a simplified flow chart of a procedure for using the calculated roll, pitch and yaw offsets to direct an antenna at a satellite. 
     
    
    
     The figures are not intended to be exhaustive or to limit the claimed invention to the precise form disclosed. It should be understood that the disclosed method and apparatus can be practiced with modification and alteration, and that the invention should be limited only by the claims and the equivalents thereof. 
     DETAILED DESCRIPTION 
       FIG. 1 a    is an illustration of the relevant components of a satellite communication system  100  in accordance with one embodiment of the presently disclosed method and apparatus. In the illustrated embodiment, an antenna  102  is mounted on a mobile platform. For the sake of illustration, the platform shown in  FIG. 1 a    is an aircraft  103 . However, it should be noted that the platform could be any mobile platform, such as a truck, automobile, ship, train or other such mobile platform. 
       FIG. 1 a    is intended to identify the relevant components of a system and not to accurately represent the relative location or size of the equipment within an aircraft. Furthermore, only those components that are relevant to the presently disclosed method and apparatus are depicted in  FIG. 1 a    for the sake of simplicity. Accordingly, the scale and relative location of the equipment within an actual aircraft may vary significantly from what is depicted in  FIG. 1 a   . Furthermore, some components that are necessary for a satellite communication system, but which are not necessary for the disclosed method and apparatus for aligning an antenna, are not shown in  FIG. 1   a.    
     An antenna positioning module, such as an antenna positioning motor  104  is coupled to the antenna  102  to move the antenna  102 . Alternatively, the antenna positioning module is an electronically steering module that directs the antenna beam. In accordance with one embodiment of the disclosed method and apparatus, the motor  104  moves the antenna in azimuth and elevation. In an alternative embodiment, the positioning motor  104  may move the antenna in three axes or in different axes, such as yaw and pitch. An antenna alignment module AAM  108  comprising an antenna control unit (ACU)  115  provides control signals to the motor  104  through a first output port  107 . In one embodiment, a radome  105  covers the antenna  102  and motor  104 . Alternatively, the motor  104  may be below the antenna  102  and inside the fuselage of the aircraft. In another alternative embodiment, the antenna is connected remotely by linkage that allows the motor  104  to control the movement of the antenna  102 . It will be understood by those skilled in the art that any manner by which the antenna can be positioned, including electronically steering the antenna, would be within the scope of the disclosed method and apparatus. The motor  104  or electronic steering module may provide information regarding the position of the antenna  102  back to the AAM  108  through an input port  111 . 
     In accordance with one embodiment of the disclosed method and apparatus, signals received by the antenna  102  are coupled to a low noise block (LNB)  110 . In one such embodiment, the LNB  110  amplifies the signals. In one such embodiment, the LNB also performs front end processing, such as filtering and/or frequency down-conversion. The output of the LNB  110  is coupled to a modem  112 . In one such embodiment, the modem  112  measures the received power and provides an output signal  117  through an input port  109  to the AAM  108  indicating the received power. Alternatively, the received power is measured within the LNB  110  or by another component within the receive chain. Any device and manner can be used to measure the received power and would be within the scope of the disclosed method and apparatus. For the purposes of this disclosure, received power is measured to provide feedback to assist in pointing the antenna, as is discussed in greater detail below. 
     In accordance with one embodiment of the presently disclosed method and apparatus, an attitude determining device is present. In one such embodiment, the attitude determining device is included within a position and attitude measuring device (PAMD)  114  is present (illustrated as being on board the aircraft  103 ). In accordance with one embodiment of the disclosed method and apparatus, the PAMD  114  is an inertial reference unit (IRU). Alternatively, the PAMD  114  may be an inertial measurement unit (IMU) or any other device capable of providing information regarding position and attitude. It should be further noted that in one embodiment of the disclosed method and apparatus, the PAMD  114  comprises two independent devices or systems, the attitude determining device that determines attitude and a position determining device that determines position. For example, a set of gyroscopes can provide information regarding attitude. An independent global positioning system (GPS) can provide information regarding position. In any case, the PAMD  114  provides the attitude and position of the aircraft  103  to the AAM  108 . For the purposes of this discussion, it is assumed that the PAMD  114  is aligned with the platform (i.e., the aircraft  103 ). Any offset between the platform and the PAMD  114  will be irrelevant, since all measurements are made with respect to the PAMD  114 , as long as the relationship between the PAMD and the antenna remain unchanged. In one embodiment, in addition to providing information that assists with pointing and alignment of the antenna  102 , the PAMD  114  provides real-time information that helps the pilot navigate and operate the aircraft  103 . Alternatively, two independent systems are provided. The first such system provides information used for alignment of the antenna  102  and the second for navigation. In either case, in one embodiment, the PAMD  114  used for alignment of the antenna  102  is assumed to be aligned with a topocentric frame of reference. Alternatively, the PAMD  114  is aligned with a reference frame that has a relationship with the topocentric reference frame that is either known or that can be determined. In yet another alternative embodiment, the PAMD  114  is aligned with a reference frame that has a relationship with a reference frame in which a satellite  106  can be located. In accordance with one embodiment of the disclosed method and apparatus, the attitude of the platform, the PAMD  114  and the antenna  102  remain essentially unchanged as the platform changes attitude. It will be understood that some change will occur due to flexing of the platform and the structural components of the antenna mount, etc. In cases in which the offset between the PAMD reference frame and the antenna reference frame change over time due to structural changes due to loading or aging, such differences can be accounted for by re-aligning the antenna to the PAMD using the process disclosed herein. 
     The AAM  108  receives information from the PAMD  114  through an input port  109 . In the embodiment shown in  FIG. 1 a   , the information is provided through the modem  112 . However, in an alternative embodiment, the PAMD  114  is directly connected to the AAM  108 . In one such embodiment, the information is provided over a standard ARINC  429  bus. Routing the information provided by the PAMD  114  through the modem allows the connection that is otherwise required between the modem and the AAM  108  to be advantageously exploited.  FIG. 1 b    illustrates an embodiment in which the LNB  110 , modem  112 , PAMD  114  and ACU  115 , is all located within the AAM  108 . Alternatively, some, but not all, of these components are located within the AAM  108 . It should be noted that the functions of each of these components can be performed by others of the components as well. For example, in one embodiment of the disclosed method and apparatus, a processor within the modem  112  determines the values of some of the vectors associated with the alignment procedure. Additionally, the functions associated with the PAMD  114  can be performed by a PAMD within the AAM  108 . An additional PAMD can also be provided within the platform to assist with navigation of the platform. In one embodiment, the additional PAMD also provides information that is used by the AAM  108 . 
       FIG. 2  is an illustration of a three dimensional Cartesian coordinate frame  200  set in a topocentric reference frame. In this example, the X axis  202  is aligned with the compass heading North, the Y axis  204  is aligned with the compass heading East and the Z axis  206  is aligned with an earth radian that emanates from the origin of the reference frame and extends through the center of the earth. This alignment is commonly known as North, East, Down (NED). Each axis is orthogonal and forms a 90 degree angle with each of the other axes. In accordance with one embodiment of the disclosed method and apparatus, the origin of the topocentric reference frame used by the PAMD  114  is the latitude and longitude of the aircraft  103 . Altitude is assumed to be zero (i.e., the origin of the topocentric reference frame is at earth surface). 
       FIG. 3  is an illustration of the aircraft  103  and an associated PAMD reference frame  300  associated with the PAMD  114  on board the aircraft  103 . In this example, the X axis of the PAMD reference frame  300  is along the longitudinal axis  302  of the aircraft  103 . The Y axis is along the lateral axis  304  of the aircraft  103  The Z axis is along the vertical axis  306  of the aircraft  103 . Unlike the topocentric reference frame which remains fixed in attitude with respect to earth, the PAMD reference frame  300  moves along with the aircraft  103 . The attitude of the aircraft  103  is defined by the set of rotations in roll, pitch and yaw between the PAMD reference frame  300  and the topocentric reference frame  200 . Roll is the rotation of the aircraft  301  about the X axis. Pitch is the rotation of the aircraft  301  about the Y axis. Yaw is the rotation of the aircraft  301  about the Z axis. 
     In one embodiment of the disclosed method and apparatus, information indicating the attitude of the aircraft  103  is output from the PAMD  114  in the form of three angular displacements. A first angular displacement represents the rotation in roll, the second represents the rotation in pitch and the third represents the rotation in yaw. 
     In order to receive the satellite signals through the antenna  102  with the maximum possible signal strength, the antenna  102  must be positioned to point at a transmitting satellite  106  (similarly for transmission from the antenna  102  to the satellite  106 ). When attempting to point an antenna  102  at a satellite  106 , a vector can be calculated from the antenna  102  to the satellite  106 , assuming known values for (1) the location of the satellite  106 , (2) the location of the antenna  102  and (3) the attitude of the antenna with respect to the satellite  106 . All of these factors can be measured or computed. In particular, the locations of satellites are well known and available in coordinates that are typically represented in a topocentric reference frame. In accordance with one embodiment of the disclosed method and apparatus, the location of the satellite is provided to the AAM  108  from the modem  112  through the input port  109 . Alternatively, the PAMD  114  is within the AAM  108 . In some embodiments of the presently disclosed method and apparatus, the origin of the reference frame used to define the location of the satellite  106  will be displaced from the origin of the topocentric reference frame having an origin at the latitude and longitude of the aircraft  103 . 
     The location of the antenna  102  can be assumed to be the location that is output by the PAMD  114  (i.e., any error due to the fact that the antenna  102  may not be exactly collocated with the PAMD  114  are assumed to be negligible and are thus ignored). With this information, a unit vector {right arrow over (d)} can be calculated which points from the antenna  102  to the satellite  106 . The vector {right arrow over (d)} is composed of three components, dx, dy, dz, with respect to the topocentric reference frame having its origin at the latitude and longitude of the aircraft  103 . If the aircraft  103  (and so the PAMD  114 ) is aligned with the topocentric reference frame (i.e., the aircraft  103  is pointing north with no pitch or roll with respect to the topocentric reference frame), then the azimuth and elevation of the antenna  102  can be easily calculated directly from the vector {right arrow over (d)}. 
     However, in the more general case, the aircraft  103  has an attitude that is not aligned with the topocentric reference frame. That is, the aircraft  103  has a heading other than North and may have a pitch and roll offset as well. In this case, the vector {right arrow over (d)} must be transformed using an alias transformation. An alias transformation is defined as a transformation of the coordinates of a vector from a first coordinate system to a second coordinate system. The vector remains in the same place and only the coordinate system changes (i.e., the frame of reference used to represent the vector). Accordingly, a vector having coordinates defined with respect to a first reference frame can be represented as a vector having coordinates defined with respect to a second reference frame. 
       FIG. 4  is an illustration of a vector  401  in a first reference frame  200  comprising an X 1  axis  202 , a Y 1  axis  204 , and a Z axis  206 .  FIG. 4  further shows a rotation of the first reference frame  200 . In this example, rotating the first reference frame  200  forms a second reference frame comprising an X 2  axis  402 , a Y 2  axis  404 , and the same Z axis  206 . In the case shown in  FIG. 4 , the first reference frame is rotated about only one axis (i.e., the Z axis  206 ) in order to simplify the example. Therefore, the Z axis  206  is common to both reference frames. In the example of  FIG. 4 , the vector  401  lies in the X, Y plane of both the first and second reference frames (i.e., the Z component of the vector  401  is zero in both frames of reference). In the first reference frame  200 , the vector  401  has a projection to the X axis of approximately −0.707 (assuming the vector  401  to be a unit vector forming an angle of 45 degrees between the X and Y axis). The projection of the vector  401  on the Y axis is approximately 0.707. If the first reference frame  200  is rotated −45 degrees about the Z axis  206  (using the right-hand rule of thumb convention), then the vector  401  has a projection on the X axis of −1.0 and a projection on the Y axis of 0.0 in the second reference frame. 
     While in this example the transformation is easy to see, it is typically necessary to determine the transformation more generally. The alias transformation of a vector from a first reference frame to a second reference frame can be calculated as:
 
 {right arrow over (d)}′   i   =M   i   {right arrow over (d)}   Eq. 1
 
     where {right arrow over (d)}′ i  is the vector  401  in the second reference frame, {right arrow over (d)} is the vector  401  in the first reference frame and M i  is the rotation matrix shown in Eq. 2 below. Thus, the rotation matrix M i  of Eq. 2 is used to perform the alias transformation of the vector  401  from the first to the second reference frame. The index i is used to distinguish a first orientation from subsequent orientations, each orientation being referenced to the first reference frame. 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           R 
                           i 
                         
                         , 
                         
                           P 
                           i 
                         
                         , 
                         
                           Y 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     P 
                                     i 
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     Y 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     P 
                                     i 
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     Y 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     P 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         P 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         P 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     R 
                                     i 
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     P 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         P 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         P 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         R 
                                         i 
                                       
                                       ) 
                                     
                                     * 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Y 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     R 
                                     i 
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     P 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     In the case of the example shown in  FIG. 4  in which R i  is 0°, P i  is 0° and Y i  is −45°, the rotation matrix times the vector {right arrow over (d)} is equal to: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       i 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       d 
                       → 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     45 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     45 
                                   
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 - 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       - 
                                       45 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     45 
                                   
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             
                               - 
                               .707 
                             
                           
                         
                         
                           
                             .707 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   .707 
                                 
                                 
                                   
                                     - 
                                     .707 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   .707 
                                 
                                 
                                   .707 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   - 
                                   .707 
                                 
                               
                             
                             
                               
                                 .707 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       - 
                                       .5 
                                     
                                     + 
                                     
                                       - 
                                       .5 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       .5 
                                     
                                     + 
                                     .5 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                           = 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                             = 
                             
                               
                                 d 
                                 → 
                               
                               i 
                               ′ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     If both the location of the satellite  106  and the antenna  102  were known and the antenna  102  were well aligned to the PAMD coordinate frame, the vector from the antenna  102  to the satellite  106  could be easily calculated by applying Eq. 2 and using the roll, pitch and yaw output from the PAMD  114 . However, when the antenna  102  is mounted on an aircraft  103 , as is the case in one embodiment of the disclosed method and apparatus, the attitude of the aircraft  103  (and so, typically the PAMD  114 ) will typically be offset from the antenna frame of reference. That is, the antenna  102  typically will not be perfectly aligned with the PAMD  114 . 
     Calculating the azimuth and elevation of the antenna required to point the antenna  102  at the satellite  106  requires the antenna  102  to be aligned with the PAMD  114 . In accordance with one procedure for aligning the antenna  102  to the PAMD  114 , the aircraft  103  must be taken onto as level a surface as possible. The aircraft is oriented so that the output of the PAMD  114  in yaw (heading) is 0° (i.e., North). The antenna  102  is then peaked to determine the azimuth and elevation that yields the strongest signal from the satellite  106 . Additional measurements are made by physically repositioning the aircraft  103  to headings of 90°, 180° and 270° based on heading readings from the PAMD  114 . If the aircraft is resting on perfectly flat terrain, then the azimuth and elevation measurements will directly translate to the roll, pitch and yaw offsets between the antenna reference frame and the topocentric reference frame. However, this method requires that the aircraft  103  be perfectly level and that it be oriented very precisely to 0°, 90°, 180° and 270°. 
     Alternatively, an alignment procedure according to the disclosed method and apparatus can be used in which the aircraft  103  is initially in any orientation. In accordance with this procedure, a first vector {right arrow over (d)} from the antenna  102  to the satellite  106  is calculated in the topocentric reference frame. Since the location of the satellite  106  and the location of the aircraft  103  are both known in the topocentric reference frame, this is easily accomplished. For the purpose of determining the first vector {right arrow over (d)}, the difference between the location of the aircraft  103  and the location of the antenna  102  is considered negligible. Any difference in the location of the origin of the topocentric reference frame used to define the location of the satellite  106  and the origin of the reference frame used to define the location of the aircraft  103  (i.e., the location output of the PAMD  114 ) is easily managed by a simple translation of the coordinates from one reference frame to the other. 
     Next, the first vector {right arrow over (d)} is transformed by an alias transformation to the PAMD reference frame to determine a second vector {right arrow over (d)}′ i . This is done using the alias transformation noted above in Eq. 1. The first vector {right arrow over (d)} is multiplied with the rotation matrix M i  (R i , P i , Y i ), of Eq. 2, where R i , is the amount of roll as indicated by the PAMD  114 , P i , is the amount of pitch as indicated by the PAMD  114 , and Y i  is the amount of yaw as indicated by the PAMD  114 . 
     If the antenna  102  is aligned with the PAMD  114 , the second vector {right arrow over (d)}′ i  in the PAMD reference frame could be directly converted to azimuth and elevation. However, assuming there is a rotational offset between the reference frame of the PAMD  114  and the reference frame of the antenna  102 , directly converting the vector in the PAMD reference frame to an azimuth and elevation will result in an error in the calculation of the azimuth and elevation of the antenna  102 . The result is that the antenna  102  will not be pointed directly at the satellite  106 . The error can be measured by peaking the antenna  102  and reading the resulting azimuth and elevation directly from the antenna positioning motor  104  or a sensor on the antenna  102 . However, correcting the error in this manner is only valid for that particular orientation. 
     In order to provide a more general solution that will be valid in all orientations, the following method and apparatus is disclosed for providing a best fit rotation matrix between the PAMD reference frame and the antenna reference frame. 
     In accordance with one embodiment of the disclosed method and apparatus, the antenna  102  is peaked to determine the azimuth and elevation setting of the positioning motor  104  that results in the maximum signal strength being received in a signal from the satellite  106  with the aircraft in a first orientation. Signal strength can be determined based on the amplitude, signal to noise ratio (SNR), amount of received power, or other such metric. In accordance with one embodiment, the azimuth and elevation are determined by the control signals provided to the antenna positioning motor  104 . Alternatively, the azimuth and elevation are read directly from the motor  104 . In an alternative embodiment, the azimuth and elevation are read from an antenna position sensor (not shown) coupled to the antenna  102  or to the antenna positioning motor  104 . 
     In accordance with one embodiment of the disclosed method and apparatus, a step track technique is used to “peak” the antenna. In one such step track peaking scheme, the antenna  102  is positioned roughly toward the satellite  106 . This is done using a rough estimate of the pointing elevation and azimuth to be applied to the motor  104 . In one embodiment of the disclosed method and apparatus, the offset between the PAMD  114  reference frame and the antenna  102  will not be so great that the satellite signal is not detectable. Therefore, in accordance with one embodiment, the azimuth and elevation calculated under the assumption that there is no offset between the PAMD reference frame and the antenna reference frame is a sufficiently accurate estimate at which to begin the peaking procedure. 
     A measurement is made of the power received through the antenna. The position of the antenna  102  is then changed in elevation by one “step”. The AAM  108  directs the antenna  102  to implement the peaking technique based on the received power measurements provided from the modem  112 . In an alternative embodiment, the received power is measured by a device other than the modem  112 . It will be understood by those skilled in the art that a device placed essentially anywhere along the receive chain can be used to measure the received power. 
     For example, if the amount of received power drops after changing the elevation of the antenna  102 , the antenna  102  is moved in the opposite direction. In one embodiment, the antenna  102  moves by two steps. If the amount of received power increases, the antenna is moved another step further in that direction. Another power measurement is made. Each time the amount of receive power increases, the antenna is moved another “step” in the same direction. Upon measuring a drop in the power, the antenna direction is reversed and moved one step back. Once the peak power measurement for elevation has been detected, the antenna begins a similar search for the peak in azimuth. If the initial azimuth position was not the peak, then the search in elevation is repeated. If the antenna was not at the peak elevation, then the search for the peak in azimuth is again repeated. This process will continue until both the elevation and the azimuth are at the peak received power. 
     It will be clear to those skilled in the art that this is a simplistic step track peaking algorithm. Many modifications to this procedure can be implemented to improve the likelihood that the antenna is at the best pointing elevation and azimuth. Furthermore, other peaking techniques can be employed, such as, but not limited to, one technique known commonly as conical scan (conscan). 
     In addition to determining the azimuth and elevation of the antenna  102  at peak for the first orientation of the aircraft  103 , an attitude reading from the PAMD  114  is taken. In accordance with one embodiment of the disclosed method and apparatus, the aircraft  103  is positioned in various additional orientations. In one embodiment of the disclosed method and apparatus, the additional orientations are achieved by rotating the aircraft on the ground. Alternatively, the additional orientations could be achieved by a relative change in orientation with respect to the satellite, such as using a different satellite with the aircraft remaining in a fixed orientation with respect to the earth. In one case in which the aircraft is moved, the heading of the aircraft  103  is changed for each additional orientation. This can be done by taxiing the aircraft or towing the aircraft to move the aircraft to the new orientation. In yet another embodiment, the aircraft  103  can be in flight during the procedure. Accordingly, as the aircraft  103  maneuvers over the course of the flight, the orientation will change, allowing additional measurements to be made. In one such embodiment, the rate of change of the attitude output from the PAMD  114  is determined and used to estimate the attitude of the aircraft  103  at particular times when the antenna  102  is peaked. For example, a determination of the attitude of the platform is made at a first time prior to the aircraft being in the first orientation (i=1) (i.e., the first orientation at which the antenna  102  is peaked). In addition, a determination as to the rate of change of the attitude of the aircraft  103  is made at the first time. A determination is then made as to the attitude of the aircraft  103  at a second time when the aircraft is in the first orientation. The determination is made from the attitude and rate of change of the aircraft  103  determined at the first time. Accordingly, from the attitude and the rate of change in the attitude at a first time, an extrapolation can be made to determine the attitude at a second time that occurs either before or after the first time. For each particular orientation, the antenna  102  is peaked to determine the azimuth and elevation that results in the highest received signal level. The attitude output of the PAMD  114  is associated with the azimuth and elevation for that particular orientation. The azimuth and elevation at each orientation are converted to a third vector {right arrow over (d)}″ i  in a Cartesian coordinate system in the antenna reference frame using the following relationship, where α is azimuth and ε is elevation:
 
 d″   ix =cos ε i  cos α i  
 
 d″   iy =cos ε i  sin α i  
 
 d″   iz =sin ε i   Eq. 4
 
     Accordingly, for each attitude there is a first vector {right arrow over (d)} determined by the location of the platform  103  and the location of the satellite  106  and represented in coordinates defined with respect to the first reference frame (i.e., the topocentric reference frame). In addition, there is a second vector {right arrow over (d)}′ i  represented by coordinates defined with respect to the second reference frame (i.e., PAMD reference frame) and a third vector {right arrow over (d)}″ i  represented by coordinates defined with respect to the third reference frame (i.e., antenna reference frame). The collection of second vectors {right arrow over (d)}′ i  forms a first matrix  D ′ and the collection of third vectors {right arrow over (d)}″ i  forms a second matrix  D ″. If each collection of second and third vectors has no measurement noise or other source of error or inconsistency, then the first matrix is related to the second by the following equation, where T is a rotation matrix:
 
   D ″=T D ′   Eq. 5
 
     Once a sufficient number of measurements for {right arrow over (d)}′ i  and {right arrow over (d)}″ i  have been gathered, the rotation matrix T can be solved. By solving for T, the general transformation from the PAMD reference frame to the antenna reference frame can be calculated (i.e., the offset in each of the three axes, roll, pitch and yaw can be determined and used to calculate an alias transformation). Thus, the output of the PAMD  114  can be used to calculate the azimuth and elevation needed to point the antenna  102  to the satellite  106 . 
     Solving for T matches the form of Wahba&#39;s Problem. There are several ways known to solve Wahba&#39;s Problem. One way is to use Singular Value Decomposition to determine the pseudoinverse of the collection of vectors  D ′ in the PAMD reference frame. By multiplying each side of equation Eq. 5 by the pseudoinverse  D ′ +  of  D ′, the following equations result:
 
   D ″=T D ′ 
 
   D ″ D ′   +   =T D ′ D ′   + 
 
 T= D ″ D ′   −   Eq. 6
 
     The pseudoinverse can be calculated by using the elements of the singular value decomposition (SVD) of  D ′.
 
   D ′=USV*  
 
   D ′   +   =VS   +   U*   Eq. 7
 
     The pseudoinverse of S may be computed by taking the transpose of the matrix formed with diagonal elements equal to the reciprocal of the diagonal elements of S. For a collection of measurements that are noisy or that have other errors, use of the pseudoinverse will produce a least-squares estimate of the rotation.
 
 {circumflex over (T)}= D ″ D ′   + , where  {circumflex over (T)}  is the least squares estimate.  Eq. 8
 
     The elements of {circumflex over (T)} may be used to derive the roll, pitch and yaw offsets to the vector {right arrow over (d)}′ i  output from the PAMD  114  using the relationships of Eq. 9 and Eq. 10. {circumflex over (T)} is interpreted as the product of Roll, Pitch, and Yaw rotations. The composite rotation matrix is given as: 
     
       
         
           
             
               
                 
                   
                     T 
                     ⁡ 
                     
                       ( 
                       
                         
                           R 
                           0 
                         
                         , 
                         
                           P 
                           0 
                         
                         , 
                         
                           Y 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               r 
                               11 
                             
                           
                           
                             
                               r 
                               12 
                             
                           
                           
                             
                               r 
                               13 
                             
                           
                         
                         
                           
                             
                               r 
                               21 
                             
                           
                           
                             
                               r 
                               22 
                             
                           
                           
                             
                               r 
                               23 
                             
                           
                         
                         
                           
                             
                               r 
                               31 
                             
                           
                           
                             
                               r 
                               32 
                             
                           
                           
                             
                               r 
                               33 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Y 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Y 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       R 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       R 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     From Eq. 9, one can see that the solutions to the Roll, Pitch and Yaw rotations are:
 
 Y   0 =tan −1 ( r   12   /r   11 )
 
 P   0 =tan −1 (− r   13 /√{square root over ( r   23   2   +r   33   2 )})
 
 R   0 =tan −1 ( r   23   /r   33 )  Eq. 10
 
     A vector that is initially in the PADM reference frame (i.e., a vector derived in the topocentric reference frame and translated to the PADM reference frame) can be further transformed by an alias transform to the antenna reference frame using knowledge of the roll, pitch and yaw rotations provided in Eq. 10. 
       FIG. 5  is a simplified flow chart of the procedure used in accordance with one embodiment of the disclosed method and apparatus for determining the roll, pitch and yaw rotational offsets between an antenna  102  and a PAMD  114  mounted in an aircraft. 
     In STEP  501 , an initial measurement is made with the aircraft  103  in a first orientation. Taking the initial measurement includes having a processor within the AAM  108  determine a first vector {right arrow over (d)}. The first vector {right arrow over (d)} is represented using coordinates defined with respect to a first reference frame (i.e., a topocentric reference frame). The first vector {right arrow over (d)} is determined from the location of the aircraft  103  and the location of the satellite  106 . In accordance with one embodiment of the disclosed method and apparatus, both the location of the aircraft  103  and the location of the satellite  106  are represented by coordinates defined with respect to a first reference frame (e.g., a topocentric reference frame). In an alternative embodiment, the determination of the first vector can be done by a processor that is not on board the aircraft. Information regarding the location of the aircraft  103  and the location of the satellite  106  are provided to such a processor. 
     The process of taking the initial measurement also includes the AAM  108  using information from the PAMD  114  indicating the attitude of the aircraft with the aircraft  103  in the first orientation. The information from the PAMD  114  is represented with coordinates defined with respect to a first reference frame. The processor within the PAMD  114  determines a second vector {right arrow over (d)}′ i  by performing an alias transformation on the first vector {right arrow over (d)}. The alias transformation transforms the representation of the first vector {right arrow over (d)} from coordinates defined with respect to the first reference frame to coordinates defined with respect to a second reference frame (i.e., the PAMD reference frame). The transformation is performed based on the relative rotation of the second reference frame with respect to the first reference frame. The relative rotation is determined by the attitude of the aircraft  103  in the first orientation (see Eq. 1 above). The attitude of the aircraft is provided by the PAMD  114 . In one embodiment, the transformation is performed within the AAM  108 . In an alternative embodiment, the transformation is performed by a processor that is not on board the aircraft  103 . Information necessary to perform the transformation is provided to such a process to enable the transformation to be performed. 
     It should be noted that the first vector {right arrow over (d)} does not take into account the orientation of the aircraft  103 , but is determined based only on the location of the aircraft  103  and the location of the satellite  106 . Therefore, in the case in which the aircraft  103  remains at the same location for each orientation, there is no index i associated with the first vector {right arrow over (d)}. However, if the location of the aircraft or the satellite changes from one orientation to another, the change in location can be taken into account. In that case, the first vector {right arrow over (d)} would be represented as {right arrow over (d)} i  to indicate the value of the first vector at each orientation i. 
     In addition, during the initial measurement, the processor records the azimuth and elevation of the antenna  102  when the antenna  102  is directed at the satellite  106 . In accordance with one embodiment of the disclosed method and apparatus, the antenna  102  is directed at the satellite  106  by peaking the antenna  102  to receive the strongest signal possible from the satellite  106 . In one embodiment of the disclosed method and apparatus, information regarding the attitude of the antenna  102  is provided to the AAM  108 . For example, in one embodiment of the disclosed method and apparatus, the azimuth and elevation of the positioning motor  104  that results in the antenna  102  receiving the strongest signal from the satellite  106  is provided to the AAM  108 . In another embodiment in which the antenna is electronically steered, the attitude of the antenna  102  is the direction of the electronic bore sight or information from which the direction of the antenna bore sight can be derived. Based on the information received by the AAM  108 , the processor within the AAM  108  determines a third vector {right arrow over (d)}″ i  that points from the antenna  102  to the satellite  106 . The third vector {right arrow over (d)}″ i  is represented in Cartesian coordinates defined with respect to a third reference frame (i.e., the antenna reference frame). In an alternative embodiment, the attitude of the antenna is provided to a processor that is not on-board the aircraft  103 . In accordance with such an embodiment, the third vector {right arrow over (d)}″ i  is determined by such a processor. 
     An additional measurement is taken with the aircraft  103  in a second orientation (STEP  503 ). In similar fashion to the initial measurement, the additional measurement is taken by peaking the antenna  102 , determining the antenna azimuth and elevation and recording the output of the PAMD  114  at the second orientation and determining first, second and third vectors {right arrow over (d 2 )}, {right arrow over (d 2 )}′, {right arrow over (d 2 )}″. Note that for the case in which the aircraft  103  remains essentially in the same location, but only changes attitude from one orientation to another, the value {right arrow over (d)}={right arrow over (d 1 )}={right arrow over (d 2 )}={right arrow over (d i )}, for all i orientations. 
     A determination is made as to whether enough measurements have been taken (STEP  505 ). If more measurements are desired, then the process repeats STEP  503  with the aircraft  103  in different orientations. It should be noted that in accordance with one embodiment of the disclosed method and apparatus, the particular orientations at which measurements are taken are essentially arbitrary. In addition, the particular number of measurements to be made will depend upon the desired accuracy. In accordance with one embodiment, eight measurements are made at various orientations distributed approximately evenly about the yaw axis  306  of the aircraft  103  (see  FIG. 3 ). Alternatively, various factors are used to influence the selection of orientations at which to take each of the measurements. 
     One such factor is the relative deviation from the other orientations at which measurements have been (or are to be) taken. In some embodiments, measurements are taken at orientations that are spaced relatively evenly over the 360° of rotation possible in each axis (roll, pitch and yaw). In other embodiments, measurements are taken at relatively arbitrary orientations during operation of the aircraft  103 , including while taxiing, or in flight, or both. The measurements may be taken over a span of time. It should be noted that a reasonably accurate determination of the offsets in roll, pitch and yaw between the reference frame of the antenna  102  and the aircraft  103  (or PAMD  114 ) can be made based on orientations resulting from rotating the aircraft  103  about only one axis, such as yaw. The offsets can be determined initially prior to operation of the satellite communication system, early in the operation of that system, or at periodic intervals during operation. In one embodiment in which offsets are updated periodically, the updates can be used to learn and correct minor changes in alignment over time, including changes in the frame of the aircraft, differing conditions (e.g., when the aircraft is on landing gear and when in the air, when the aircraft has differing loads, etc.). 
     As noted above, an alignment procedure may be performed in which the aircraft  103  is placed in 8 different orientations, each orientation having a heading spaced evenly around the 360° of the compass. The roll and pitch of the aircraft  103  need not be tightly controlled. Accordingly, in one such embodiment, the aircraft  103  is turned to each compass heading at which a measurement is to be taken. In one embodiment of the disclosed method, the particular orientations selected are not critical, allowing for a relatively fast and simply procedure to be implemented for determining the offsets in roll, pitch and yaw between the reference frame of the antenna  102  and the aircraft  103  (or PAMD  114 ). 
     Once a sufficient number of measurements (i.e., vectors {right arrow over (d)}′, {right arrow over (d)}″) have been collected, a composite rotation matrix {circumflex over (T)} is calculated based on the relationships shown above in Eq. 5 through Eq. 8 (SPEP  507 ). The roll, pitch and yaw offsets of the antenna reference frame with respect to the PAMD reference frame are then calculated based on the values presented in the composite rotation matrix {circumflex over (T)} (SPEP  509 ). In one embodiment of the disclosed method and apparatus, the calculation of the composite rotation matrix {circumflex over (T)} is made by a processor that is not on-board the aircraft  103 . The resulting roll, pitch and yaw offsets are then transmitted back to the aircraft  103  to be used to direct an antenna  102  or they are used to perform a correction to the antenna positioning information and then transmitted to the aircraft  103 . 
       FIG. 6  is a simplified flow chart of a procedure for using the calculated roll, pitch and yaw offsets determined from the first, second and third vectors of  FIG. 5  to direct an antenna at a satellite. Initially, the output of the PAMD  114  is received (STEP  601 ). The output of the PAMD  114  includes the location and attitude of the PAMD  114  in coordinates defined with respect to the PAMD reference frame. From the location of the PAMD  114  and the location of the satellite  106 , a fourth vector {right arrow over (d)} from the PAMD  114  to the satellite  106  can be calculated in coordinates defined with respect to the topocentric reference frame (SPEP  603 ). An alias transformation is then performed on the fourth vector {right arrow over (d)} to transform the coordinates of the vector {right arrow over (d)} to the PAMD reference frame. The alias transformation is performed by applying Eq. 2 to the attitude information provided from the PAMD  114  to generate a first rotation matrix M i  (STEP  605 ). 
     The vector {right arrow over (d)} represented by coordinates defined with respect to the topocentric reference frame is then multiplied by the first rotation matrix M i . The result is a fifth vector {right arrow over (d)}′ i  that points from the PAMD  114  to the satellite  106 . The fifth vector {right arrow over (d)}′ i  is represented using coordinates defined with respect to the PAMD reference frame (STEP  607 ). 
     A second rotation matrix {circumflex over (T)} is generated (STEP  609 ) by applying the roll, pitch and yaw offsets determined in STEP  509  of  FIG. 5  to Eq. 9. The fifth vector {right arrow over (d)}′ i  is then multiplied by the second rotation matrix {circumflex over (T)} to transform coordinates of the fifth vector {circumflex over (d)}′ i  to the antenna reference frame (STEP  611 ). The result is a sixth vector {circumflex over (d)}″ i  that points from the antenna  102  to the satellite  106  represented in Cartesian coordinates defined with respect to the antenna reference frame. The sixth vector {right arrow over (d)}″ i  is then converted to coordinates represented with respect to azimuth and elevation. The azimuth and elevation of the vector {right arrow over (d)}″ i  are then provided to the positioning motor  104  to point the antenna  102  to the satellite  106  (SPEP  613 ). 
     It should be noted that in addition to the offsets in roll, pitch and yaw, a constant error in the elevation positioner may exist which produces an error in the elevation and azimuth determined by the procedure of  FIG. 5  and  FIG. 6 . In accordance with one embodiment of the disclosed method and apparatus, it is desirable to account for this error as well. One source of such a constant elevation error is a misalignment of a motor stop in the positioning motor  104 . Such a constant elevation error introduces a translation error. The translation error comes from the fact that the elevation offset will corrupt the collection of measurements used to derive the vector {right arrow over (d)}″ i . As noted above, the vector {right arrow over (d)}″ i  points from the antenna  102  to the satellite  106  in Cartesian coordinates in the antenna reference frame. 
     One way to estimate the error in the elevation measurements made when the antenna is peaked to the satellite is to multiply the vector {right arrow over (d)}″ i  by the transpose of an orthogonalized version of {circumflex over (T)} and further by the transpose of the rotation matrix M i . Accordingly, the error in elevation measurements, {tilde over (w)} i  is:
 
 {tilde over (w)}   i   =M   i   T   {tilde over (T)}   T   {right arrow over (d)}″   i   Eq. 11
 
     Orthogonalizing the matrix {circumflex over (T)} effectively strips out the elevation offset information from the matrix {circumflex over (T)}. One way to orthogonalize the matrix is to compute the roll, pitch and yaw offsets. The roll, pitch and yaw offsets are then used to construct a rotation matrix as follows: 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       ~ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           R 
                           0 
                         
                         , 
                         
                           P 
                           0 
                         
                         , 
                         
                           Y 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               r 
                               11 
                             
                           
                           
                             
                               r 
                               12 
                             
                           
                           
                             
                               r 
                               13 
                             
                           
                         
                         
                           
                             
                               r 
                               21 
                             
                           
                           
                             
                               r 
                               22 
                             
                           
                           
                             
                               r 
                               23 
                             
                           
                         
                         
                           
                             
                               r 
                               31 
                             
                           
                           
                             
                               r 
                               32 
                             
                           
                           
                             
                               r 
                               33 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Y 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Y 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       R 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       R 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     Another way to orthogonalized the matrix {tilde over (T)} is to apply the Singular Value Decomposition to {tilde over (T)}. Omitting the S component will result in an orthogonal matrix {tilde over (T)}.
 
 {circumflex over (T)}=USV*  
 
 {tilde over (T)}=UV*   Eq. 13
 
     Other statistical shape analysis procedures may be used, such as Procrustes analysis. For example, the Matlab® statistics toolbox function PROCRUSTES may be used to estimate {tilde over (T)}. In accordance with this approach:
 
[ dd,Z,tr ]=procrustes( X,Y ,‘Scaling’,false,‘Reflection’,false)  Eq. 14
 
     where X is the transpose of the matrix with columns that are the measured direction vectors in the antenna reference frame; and Y is the transpose of the matrix with columns that are the direction vectors in the PAMD reference frame. The best fit rotation matrix will be returned as ‘tr.T’. Note that the Procrustes function determines the matrix that fits the form:
 
   D ″   T   = D ′   T   {circumflex over (T)}   T   Eq. 15
 
     In one embodiment in which Procrustes or other statistical shape analysis methods are used, the roll, pitch and yaw offsets may be computed directly from the orthogonal matrix as well. The elements of T may be used to derive the roll, pitch and yaw offsets of the antenna positioner relative to the PAMD  114 . 
     
       
         
           
             
               
                 
                   
                     T 
                     ⁡ 
                     
                       ( 
                       
                         
                           R 
                           0 
                         
                         , 
                         
                           P 
                           0 
                         
                         , 
                         
                           Y 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               r 
                               11 
                             
                           
                           
                             
                               r 
                               12 
                             
                           
                           
                             
                               r 
                               13 
                             
                           
                         
                         
                           
                             
                               r 
                               21 
                             
                           
                           
                             
                               r 
                               22 
                             
                           
                           
                             
                               r 
                               23 
                             
                           
                         
                         
                           
                             
                               r 
                               31 
                             
                           
                           
                             
                               r 
                               32 
                             
                           
                           
                             
                               r 
                               33 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Y 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Y 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       R 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           P 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       sin 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           R 
                                           0 
                                         
                                         ) 
                                       
                                       * 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             Y 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       R 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 * 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       0 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   16 
                 
               
             
           
         
       
     
     The solutions are then
 
 Y   0 =tan −1 ( r   12   /r   11 )
 
 P   0 =tan −1 (− r   13 /√{square root over ( r   23   2   +r   33   2 )})
 
 R   0 =tan −1 ( r   23   /r   33 )  Eq. 17
 
     Alternatively, since the vector {right arrow over (d)}″ i  has the elevation error, multiplying it as noted in Eq. 11 will result in a vector {tilde over (w)} i . The z component of the vector {tilde over (w)} i  can be used to calculate the elevation angle {tilde over (ε)} i  for each measurement i as follows:
 
{tilde over (ε)} i =sin −1 ( w   iz )  Eq. 18
 
     The elevation offset can be determined by taking the average of the difference between the elevations {tilde over (ε)} i  for each measurement i and the ideal topocentric elevation angle {tilde over (ε)} o  (i.e., the elevation from the PAMD  114  to the satellite absent any offset or constant elevation error). 
     
       
         
           
             
               
                 
                   Δɛ 
                   = 
                   
                     
                       
                         1 
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             ɛ 
                             ~ 
                           
                           i 
                         
                       
                     
                     - 
                     
                       ɛ 
                       0 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   19 
                 
               
             
           
         
       
     
     Once the constant elevation error Δε is determined, the roll, pitch and yaw offsets can be recalculated using elevation angles that have been corrected for the constant elevation error Δε. 
     Although the disclosed method and apparatus is described above in terms of various examples of embodiments and implementations, it should be understood that the particular features, aspects and functionality described in one or more of the individual embodiments are not limited in their applicability to the particular embodiment with which they are described. For example, it is possible to use quaternions to express the relationships of the different reference frames to one another and thus determine the offset between the antenna reference frame and the PAMD reference frame by taking measurements at various orientations as described above. Furthermore, some or all aspects of the disclosed method and apparatus may be implemented in hardware or software, or a combination of both (e.g., programmable logic arrays). Various general purpose computing machines may be used with programs written in accordance with the teachings herein. Alternatively, a special purpose computer or special-purpose hardware (such as integrated circuits) may be used to perform particular functions. Thus, the disclosed method and apparatus may be implemented in one or more computer programs executing on one or more programmed or programmable computer systems. 
     Each such computer program may be stored on or downloaded to (for example, by being encoded in a propagated signal and delivered over a communication medium such as a network) a tangible, non-transitory storage media or device (e.g., solid state memory or media, or magnetic or optical media) readable by a general or special purpose programmable computer, for configuring and operating the computer when the storage media or device is read by the computer system to perform the procedures described herein. The inventive system may also be considered to be implemented as a computer-readable storage medium, configured with a computer program, where the storage medium so configured causes a computer system to operate in a specific and predefined manner to perform the functions described herein. Thus, the breadth and scope of the claimed invention should not be limited by any of the examples provided in describing the above disclosed embodiments. 
     Terms and phrases used in this document, and variations thereof, unless otherwise expressly stated, should be construed as open ended as opposed to limiting. As examples of the foregoing: the term “including” should be read as meaning “including, without limitation” or the like; the term “example” is used to provide examples of instances of the item in discussion, not an exhaustive or limiting list thereof; the terms “a” or “an” should be read as meaning “at least one,” “one or more” or the like; and adjectives such as “conventional,” “traditional,” “normal,” “standard,” “known” and terms of similar meaning should not be construed as limiting the item described to a given time period or to an item available as of a given time, but instead should be read to encompass conventional, traditional, normal, or standard technologies that may be available or known now or at any time in the future. Likewise, where this document refers to technologies that would be apparent or known to one of ordinary skill in the art, such technologies encompass those apparent or known to the skilled artisan now or at any time in the future. 
     A group of items linked with the conjunction “and” should not be read as requiring that each and every one of those items be present in the grouping, but rather should be read as “and/or” unless expressly stated otherwise. Similarly, a group of items linked with the conjunction “or” should not be read as requiring mutual exclusivity among that group, but rather should also be read as “and/or” unless expressly stated otherwise. Furthermore, although items, elements or components of the disclosed method and apparatus may be described or claimed in the singular, the plural is contemplated to be within the scope thereof unless limitation to the singular is explicitly stated. 
     Additionally, the various embodiments set forth herein are described with the aid of block diagrams, flow charts and other illustrations. As will become apparent to one of ordinary skill in the art after reading this document, the illustrated embodiments and their various alternatives can be implemented without confinement to the illustrated examples. For example, block diagrams and their accompanying description should not be construed as mandating a particular architecture or configuration. Further, some of the steps described above may be order independent, and thus can be performed in an order different from that described. Various activities described with respect to the embodiments identified above can be executed in repetitive, serial, or parallel fashion.