Patent Publication Number: US-7221316-B2

Title: Control segment-based lever-arm correction via curve fitting for high accuracy navigation

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   Application Ser. No. 11/247,641, entitled “SPACE-BASED LEVER ARM CORRECTION IN NAVIGATIONAL SYSTEMS EMPLOYING SPOT BEAMS,” by Jonathan A. Tekawy and Kevin M. O&#39;Brien, and filed on same date herewith; and 
   Application Ser. No. 11/247,493, entitled “USER SEGMENT-BASED LEVER ARM CORRECTION VIA PRESCRIBED MANEUVER FOR HIGH-ACCURACY NAVIGATION,” by Jonathan A. Tekawy and Kevin M. O&#39;Brien and filed on same date herewith. 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates generally to space-based navigational systems, and in particular, to a control segment-based lever arm correction system applicable to navigational systems employing spot beams. 
   2. Description of the Related Art 
   The Global Positioning System (GPS) is a satellite system that transmits navigation signals that are received by ground-based GPS receivers and used to determine the position of the GPS receiver to a high degree of accuracy. GPS currently provides standard service to commercial receivers, and a higher accuracy service to military receivers authorized to receive such signals. 
   In current systems, the GPS navigation signal is transmitted via a wide beam satellite antenna disposed on each GPS satellite. The wide beam antenna permits any GPS receiver having a line-of-sight to the GPS satellite to receive the navigational signal, and when the navigation signal from a sufficient number of GPS satellites has been acquired, the GPS receiver can determine its position via a precision clock and well-known triangulation techniques. 
   Because GPS signals are also used in military applications, countermeasures can be expected to be applied in an attempt to reduce the effectiveness of the GPS system. One such countermeasure is jamming. To increase the effectiveness of the GPS signals in a jamming environment, a steerable high gain antenna may be used to transmit high intensity GPS signals via spot beams to areas where needed. 
   One difficulty with this approach is that the high gain spot beam antenna is typically physically displaced from the wide beam antenna, and consequently, the phase center of each antenna is also displaced as well. This displacement is known as the “lever arm” between the antennas, and left uncorrected, can negatively affect the ability of the GPS receivers to determine their position. Without any correction, the lever arm between the wide beam antenna and a 7 meter diameter spot beam antenna can contribute up to 4.4 meters of user range error (URE). Depending on the GPS satellite constellation, this uncompensated URE can produce up to nine meters (RMS) of vertical (altitude) navigation error, which is a factor of 10 higher than the performance of the current GPS constellation of 0.9 meters (RMS). In civil aviation applications, such errors are sufficient to result in loss of life, and in military applications, they can result in increased collateral damage, and increased sortie and weapon consumption to perform the same mission. 
   Further exacerbating this problem is the fact that in order to maintain proper Sun and Earth pointing, the GPS satellites are required to perform attitude maneuvers. Such maneuvers can be very large, particularly about yaw axis. 
   To achieve such high-accuracy navigation demanded for many missions, the GPS system must provide users with real-time corrections of the spot beam antenna phase center location relative to the Earth coverage antenna phase center, even while the spot beam antenna is moving to track specific terrestrial locations. 
   What is needed is an apparatus and method for computing a correction for lever arm related errors, and for incorporating this correction in navigation computations. The present invention satisfies these needs. 
   SUMMARY OF THE INVENTION 
   To address the requirements described above, the present invention discloses a method and apparatus for estimating a lever arm correction between a wide beam antenna and a spot beam antenna. The method comprises the steps of generating a predicted satellite attitude over the time period T from a predicted position of the at least one satellite over the time period T and a profile of a prescribed satellite attitude maneuver over the time period T; generating predicted gimbal angles of the spot beam antenna over the time period T from the predicted satellite attitude over the time period T and the terrestrial location of the of the spot beam on Earth when illuminating a navigational receiver over the time period T; predicting the lever arm correction between the wide beam antenna and the spot beam antenna over the time period T from the predicted gimbal angles of the spot beam antenna over the time period T and the predicted satellite attitude over the time period T; and transmitting the predicted lever arm correction from the command station to the navigational receiver. The apparatus comprises a lever arm correction module, for predicting a lever arm correction between the first satellite antenna and the second satellite antenna, and a transmitter for transmitting the predicted lever arm correction to a navigational receiver via the navigational satellite. In one embodiment, the lever arm correction module comprises a satellite position predictor module, for generating a predicted navigational satellite bus position over the time period T, a satellite attitude predictor module, for generating a predicted navigational satellite bus attitude over the time period T from the predicted navigational satellite bus position over the time period T and a profile of a prescribed navigational satellite bus attitude maneuver over the time period T, and a lever arm prediction module, for predicting the lever arm correction between the first antenna and the second antenna over the time period T from a terrestrial location of the spot beam illuminating the navigational receiver over the time period T, the predicted navigational satellite bus attitude over the time period T, and a geometry between the navigational satellite bus and the second satellite antenna. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Referring now to the drawings in which like reference numbers represent corresponding parts throughout: 
       FIG. 1A  is an illustration of a three-axis stabilized satellite; 
       FIG. 1B  is an illustration of an embodiment of a gimbal assembly; 
       FIG. 2  is a diagram depicting a functional architecture of a representative satellite attitude control system; 
       FIG. 3  is a diagram depicting geometrical relationships between the satellite, the Earth and the Sun; 
       FIGS. 4A–4E  are diagrams depicting the yaw noon maneuver at orbital midnight for a zero Sun angle; 
       FIG. 5  is a diagram depicting the yaw attitude angle as a function of the orbit angle and the Sun angle; 
       FIG. 6  is a diagram illustrating an exemplary embodiment of a technique for estimating the lever arm correction between the wide coverage antenna and the spot beam antenna; 
       FIG. 7  is a diagram illustrating architectural elements of a space-based navigational system; 
       FIGS. 8A and 8B  are diagrams illustrating exemplary process steps that can be used to implement one embodiment of the invention, and how these process steps can be allocated to the architectural elements; 
       FIG. 9  is a plot of the yaw attitude angle of the satellite  100  as a function of the orbit angle and the Sun angle for an exemplary prescribed satellite  100  maneuver profile; 
       FIG. 10  is a diagram showing typical lever arm motion over time; and 
       FIG. 11  is a diagram illustrating one technique that can be used to arrive at the prescribed maneuver profile. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   In the following description, reference is made to the accompanying drawings which form a part hereof, and which is shown, by way of illustration, several embodiments of the present invention. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
     FIG. 1A  illustrates a three-axis stabilized satellite or spacecraft  100 . The satellite  100  has a main body  102  (which may be referred to as the “satellite bus”), one or more solar panels  104 , one or more navigation beam antennas  106 E and  106 S, and a telemetry and command antenna  108  which is used to communicate with a control ground station. The satellite  100  may also include one or more sensors  110  to measure the attitude of the satellite  100 . These sensors may include sun sensors, earth sensors, and star sensors. Since the solar panels are often referred to by the designations “North” and “South”, the solar panels in  FIG. 1A  are referred to by the numerals  104 N and  104 S for the “North” and “South” solar panels, respectively. 
   The three axes of the spacecraft  100  are shown in  FIG. 1A . The pitch axis lies along the line of the solar panels&#39;  140 N and  140 S mutual rotation axes. The roll and yaw axes are perpendicular to the pitch axis and lie in the directions and planes shown. 
   In the illustrated embodiment, the satellite  100  includes a first navigation beam antenna  106 E and a second navigation beam antenna  106 S. The first navigation beam antenna  106 E is a wide-beam antenna which transmits a navigation signal with a beamwidth covering the widest range of the Earth&#39;s surface possible from that satellite&#39;s altitude at any time, and is directed toward the Earth along the yaw axis. Since this antenna  106 E offers the widest coverage of the Earth&#39;s surface, it is typically not steerable. The navigation system uses a constellation of such satellites  100  to provide coverage anywhere on the Earth&#39;s surface by at least 4 different satellites at all times, thus permitting the navigational signal transmitted by the satellites to be used to determine the location and clock bias of the receiver using triangulation techniques. 
   The second navigation beam antenna  106 S is a steerable spot beam antenna that provides a second navigation signal in a much narrower navigation beam. This allows transmission of a higher-strength beam to selected points on the ground without requiring excessive transmitter power, thus reducing the effectiveness of countermeasures such as jamming. Since the required service area includes substantially the entire surface of the Earth and the beamwidth of the spot beam antenna  106 S is not wide enough to cover the entire surface area, the boresight of the spot beam antenna  106 S can be steered about to direct the spot beam where desired. Such steering can be accomplished mechanically, by use of a gimbal structure driven by gimbal motors, or electronically, using phased arrays, for example. 
     FIG. 1A  also shows the phase center  112 E of the wide beam antenna  106 E and the phase center  112 S of the spot beam antenna  106 S. Since the spot beam antenna  106 S is offset from the wide beam antenna  106 E, the phase centers  112 S,  112 E of the antennas are separated by an antenna lever arm  114 , which is represented as a vector originating at the phase center  112 E of the wide beam antenna  106 E and extending to the phase center  112 S of the spot beam antenna  106 S. Due to motion of the satellite bus  102  and the spot beam antenna  106 S and other factors, the antenna lever arm  114  does not remain fixed, but can vary substantially over time. This variance is enough to add a significant uncertainty in the ability of a navigation receiver (such as a GPS receiver) to determine its location when a navigation signal is received via the spot beam. 
   The spot beam antenna  106 S may be steered electronically (by appropriate phasing of elements in a scanning array) or mechanically (by use of a non-scanning antenna and a gimbal assembly), or a combination of both.  FIG. 1B  illustrates an embodiment using a gimbal assembly  152  having an inner gimbal  154 A and an outer gimbal  154 B. The inner gimbal  154 A is associated with inner gimbal coordinate frame  156 B while the outer gimbal  154 B is associated with an outer gimbal coordinate frame  156 B. Driven by gimbal motors or other devices (not illustrated), the inner and outer gimbals  154 A,  154 B angularly direct the antenna  106 S boresight (which is represented in antenna signal boresight coordinate frame  156 D) where desired to transmit the navigation signal spot beam using the antenna  106 S. Both inner and outer gimbals  154  typically include a potentiometer or other means to measure the gimbal angle. 
   To determine the angle at which the inner and outer gimbals  154  should be positioned to direct the spot beam antenna  106 S to the desired scan locations (to perform a specific mission profile), the spacecraft  100  determines its attitude via the attitude sensors  110 , which may be mounted on the satellite bus or body  102 . Using the measured satellite attitude and the angular and translational displacement between the satellite body  102  and the spot beam antenna  106 S (as expressed by coordinate systems  156 A– 156 D), the satellite  100  can determine the appropriate gimbal angles, and commands the gimbal motors to move the gimbals  154  to the appropriate positions. 
     FIG. 2  is a diagram depicting the functional architecture of a representative attitude control system  200 . Control of the spacecraft is provided by a computer or spacecraft control processor (SCP)  202 . The SCP  202  performs a number of functions which may include post ejection sequencing, transfer orbit processing, acquisition control, station-keeping control, normal mode control, mechanisms control, fault protection, and spacecraft systems support, among others. 
   The SCP  202  may implement one or more processing modules such as antenna control module  276 , which is used to control the satellite spot beam antenna drive  274  to slew the spot beam antenna  106 S to the appropriate orientation and to transmit a navigation signal. Alternatively, the antenna control module  276  can be implemented in a different processor or in dedicated circuitry. 
   Input to the spacecraft control processor  202  may come from any combination of a number of spacecraft components and subsystems, such as a transfer orbit sun sensor  204 , an acquisition sun sensor  206 , an inertial reference unit  208 , a transfer orbit Earth sensor  210 , an operational orbit Earth sensor  212 , a normal mode wide angle sun sensor  214 , a magnetometer  216 , and one or more star sensors  218 . 
   The SCP  202  generates control signal commands  220  which are directed to a command decoder unit  222 . The command decoder unit operates the load shedding and battery charging systems  224 . The command decoder unit also sends signals to the magnetic torque control unit (MTCU)  226  and the torque coil  228 . 
   Wheel torque commands  262  are generated by the SCP  202  and are communicated to the wheel drive electronics  240  which command the speed of the reaction wheels in reaction wheel assembly(s)  244 . Typically, the spacecraft  100  includes four reaction wheels, at least one in each orthogonal direction, and one for redundancy purposes. The speed of the reaction wheels is also measured and fed back to the SCP  202  by feedback control signal  264 . The SCP  202  also communicates commands and data  254  with command stations (further described in connection with  FIG. 7 ) via a satellite transmitter/receiver (or transceiver)  258 . 
   The foregoing describes an exemplary space stabilized satellite attitude control system. The present invention can be implemented with other attitude control system designs as well. 
   The SCP  202  may include or have access to memory  270 , such as a random access memory (RAM). Generally, the SCP  202  operates under control of an operating system  272  stored in the memory  270 , and interfaces with the other system components to accept inputs and generate outputs, including commands. Applications running in the SCP  202  access and manipulate data stored in the memory  270 . The spacecraft  100  may also comprise an external communication device such as a satellite link for communicating with other computers at, for example, a ground station. If necessary, operation instructions for new applications can be uploaded from ground stations. 
   In one embodiment, instructions implementing the operating system  272 , application programs, and other modules are tangibly embodied in a computer-readable medium, e.g., data storage device, which could include a RAM, EEPROM, or other memory device. Further, the operating system  272  and the computer program are comprised of instructions which, when read and executed by the SCP  202 , cause the spacecraft processor  202  to perform the steps necessary to implement and/or use the present invention. Computer program and/or operating instructions may also be tangibly embodied in memory  270  and/or data communications devices (e.g. other devices in the spacecraft  100  or on the ground), thereby making a computer program product or article of manufacture according to the invention. As such, the terms “program storage device,” “article of manufacture” and “computer program product” as used herein are intended to encompass a computer program accessible from any computer readable device or media. 
   As described above, it is desirable to correct for navigation errors caused by the lever arm  114  between the wide beam antenna and the steerable spot beam antenna. One approach taken to correct for such errors is to compute the antenna lever arm  114  in the satellite coordinate frame, and transmit this information to the GPS receivers in the navigational message. The GPS receiver then uses this information to compute corrections in the GPS (e.g. Earth Centered Earth Fixed) coordinate frame. However, this technique is complicated by the maneuvers that the satellite  100  must undergo to direct the solar panels  104  at the Sun and the wide beam antenna  106 E at the Earth. Depending on the Sun angle, this requires a rather substantial yaw maneuver twice each orbit (at orbit noon and midnight). 
     FIG. 3  is a diagram depicting the geometrical relationships between the satellite  100 , the Earth  308  and the Sun  306 . Each satellite  100  of the GPS system is in an orbit  302  around the Earth  308 , thus defining an orbital plane  304 . The angle θ between the orbital plane  304  and the Sun  306  is the Sun angle. The satellite  100  orbits the Earth  308 , always directing the antenna  106 E at the Earth  308 . The satellite  100  attitude and solar panels  104  are adjusted to keep the solar panels  104  directed at the Sun  306  and the antenna  106 E directed at the Earth  308  at the same time. Typically, this requires the satellite  100  to perform a large rotation about the yaw axis at the orbital midnight (0 degree orbit angle) and the orbital noon (180 degree orbit angle). The smaller the Sun angle θ, the larger the required rotation about the yaw axis. 
     FIGS. 4A–4E  are diagrams depicting the yaw maneuver at orbital midnight for a zero Sun angle (wherein the Sun  306  is in the orbital plane  304 ).  FIG. 4A  illustrates the satellite  100  attitude and solar panel  104  orientation as the orbital midnight is approached. Note that the solar panels  104  are canted at an angle to direct them to be perpendicular to the Sun&#39;s rays. Shortly before the Sun  306  begins to be eclipsed by the Earth  308 , the satellite  100  begins its maneuver and yaws counter clockwise as shown in  FIG. 4B  through  FIG. 4D . When the Sun  306  fully emerges from the eclipse, the satellite  100  has yawed 180 degrees from its initial position. 
     FIG. 5  is a diagram plotting the yaw angle as a function of the orbit angle. At higher Sun  306  angles θ, the yaw angle movement is gradual, as shown in trace  502 . However, at low Sun  306  angles θ (e.g. Sun angles less than about 5 degrees), the yaw angle abruptly transitions 180 degrees at orbital angles of 0 and 180 degrees, with resulting high yaw angular rates, as shown in trace  504 . Such low Sun angle conditions occur over a significant portion of the year. 
     FIG. 6  is a diagram illustrating an exemplary embodiment of a technique for estimating the lever arm  114  correction between the wide coverage antenna  106 E and the spot beam antenna  106 S. In block  602 , a prediction of the satellite  100  position over a time period T is generated. Block  604  illustrates the generation of a prediction of the satellite  100  attitude profile over the time period T from a prescribed navigational satellite attitude maneuver. Finally, block  606  illustrates predicting the lever arm correction between the wide coverage antenna and the spot beam antenna over the time period T using the profile of the predicted satellite attitude maneuver over the time period T and a position of the terrestrial location of the spot beam on the Earth (e.g. when illuminating one or more of the navigational receivers). 
     FIG. 7  is a diagram illustrating architectural elements of a space-based navigational system  700 . The navigational system  700  comprises a space segment  704  having a plurality of navigational satellites  100  (hereinafter alternatively referred to as spacecraft) in orbit around the Earth. Each of the satellites  100  includes a first transmitter  710 , which transmits a first navigational signal using the wide beam antenna  106 E and a second transmitter  712 , which, using the spot beam antenna  106 S, transmits a spot beam  719  having a second navigational signal to one or more of the navigational receivers  716  that together form the user segment  706 . The spot beam  719  covers an area  715  on the Earth&#39;s surface, and is centered on the target spot beam position  717 . Both the first navigational signal and the second navigational signal include information that can be used to determine the location of the navigational receiver  716 . In a jamming environment, a navigation receiver may lose lock on the first navigation signal and will not be able to produce any positional solution. The second navigational signal, however, is typically of greater power by virtue of being transmitted via a spot beam, and, will be used for positional determination using the navigational receiver  716  in the jamming environment. The satellites  100  in the space segment  704  are controlled via commands from the ground segment  702 , which comprises one or more ground-based command stations  707 , each including a command center  708  and a communicatively coupled antenna  714  for transmitting and receiving information and commands to and from the satellites  100  of the space segment  704 . 
   The operations illustrated in  FIG. 6  can be performed on-board the satellite  100 , the ground segment  702  controlling the satellite  100 , by the navigational receivers  716  making up the user segment  706 , or any combination thereof. 
   However, while a fixed antenna lever arm  114  can be computed in the satellite coordinate frame  156  and transmitted to the GPS receiver in the navigational message, this technique has serious disadvantages. First, this approach also cannot be applied to GPS systems that use steerable spot beam antennas, because the GPS receivers do not know the satellite  100  attitude, very accurately, and thus have insufficient information to compute an accurate estimate of the lever arm  114  for all extended times. Second, even if the GPS receivers were provided with the satellite  100  attitude, this technique requires that the GPS receivers perform the majority of the correction processing. Since the GPS receivers are often required to be very small, lightweight, and low power consumption devices, such processing can be a significant problem. Also, because of satellite attitude control system  200  limitations, the high yaw rate maneuvers that occur at low Sun  306  angles (such as shown in  FIG. 5 ) add substantial uncertainty to the antenna lever arm  114  corrections transmitted to the GPS receivers. 
   One solution to these problems is to compute discrete lever arm  114  corrections in the GPS coordinate frame in the satellite  100  or in the ground segment  702  and periodically transmit the corrections to the GPS receivers. Since the correction data is in the GPS coordinate frame, it can be applied to the satellite orbital motion data already provided by the GPS system, and thus requires only minimal GPS receiver processing. However, such discrete corrections do not provide solutions in between the discrete points, and adding additional discrete points would consume a prohibitively large portion of the transmission bandwidth in the navigation message. 
   Finally, a problem with either of the foregoing correction approaches is that the corrections must be extrapolated for some period into the future. The first approach (that of transmitting the fixed antenna lever arm position in the satellite  100  coordinate frame in the navigational message and relying upon the GPS receivers to use this information along with a satellite yaw steering maneuver to enable computation of the corrections in a GPS coordinate frame), relies on an idealized (and often inaccurate) prediction of the satellite  100  attitude for all relevant times. The second approach makes predictions at least 15 minutes into the future, and during those 15 minutes, without additional information or constraints, this solution can yield large errors. 
     FIGS. 8A and 8B  are diagrams illustrating exemplary process steps that can be used to implement one embodiment of the invention, and how these process steps can be allocated to the architectural elements (e.g. the control segment  702 , the space segment  704 , and the user segment  706 ). 
   Referring first to  FIG. 8A , the predicted position of the satellite  100  over a time period T is computed, as shown in block  802 . The predicted position of the satellite  100  can be determined using a variety of methods well known in the art, and can include parameters to model satellite maneuvers (whether for mission specific or station keeping purposes). 
   Next, as shown in block  806 , a prediction of the satellite&#39;s attitude over the same time period T is computed. This is accomplished using the computed satellite position (from the computation shown in block  802 ), and a prescribed satellite maneuver  808 . The prescribed satellite attitude maneuver  808  is performed to direct the yaw axis of the satellite bus  102  at the Earth  308  while directing the satellite solar panels  104  at the Sun  306 , as illustrated in  FIGS. 4A–4E , while requiring satellite angular rates and/or accelerations only within the satellite&#39;s capabilities. Typically, the maneuver is required about only the yaw axis. 
     FIG. 9  is a plot of the yaw attitude angle of the satellite  100  as a function of the orbit angle  310  and the Sun  306  angle for an exemplary profile of a prescribed satellite  100  maneuver. The prescribed maneuver can be defined such that, when accounting for the capabilities and limitations of the satellite&#39;s attitude control system  200 , the satellite  100  can execute the prescribed attitude maneuver within a specific angular tolerance, the selection of such tolerance is a function of the desired lever arm estimation accuracy. In one embodiment, the value of this tolerance, expressed as the difference between the predicted satellite attitude and the actual satellite attitude is less than 0.2 degrees. 
   As was true in  FIG. 5 , trace  502  illustrates yaw angle profile for a high Sun  306  angle θ. At high Sun  306  angles θ, the required yaw rates and accelerations are relatively small, so that ideal and prescribed yaw profiles are the same. However, at low Sun  306  angles θ (e.g. less than about 5 degrees), the required yaw rates and accelerations are much higher for the ideal profile, as shown in trace  504 . These yaw rates are high enough so that the satellite&#39;s attitude control system  200  cannot follow the ideal profile, and thus, the deviation from the expected satellite  100  attitude angle and the actual satellite  100  attitude angle is significant enough to negatively impact the estimation of the lever arm  114 . Also, the maneuver is initiated based upon sensor (e.g. Sun, Earth, and/or star sensors) measurements, hence the time the maneuver is initiated is another source of unpredictability. To prevent this, the satellite is commanded to perform a prescribed satellite maneuver  808 . 
   This prescribed satellite maneuver  808  can be defined such that, when accounting for the capabilities and limitation of the satellite&#39;s attitude control system  200 , the satellite  100  can execute the prescribed attitude maneuver to within specified tolerances. The maneuver profile can be arrived at by (1) limiting the satellite attitude angular rate and/or acceleration in all axes (e.g. pitch, roll, and yaw) or a single axis (e.g. yaw only) to a maximum value, (2) limiting the error between the predicted satellite attitude and the actual satellite attitude (when attitude control system  200  limitations are considered), (3) and/or placing other suitable restrictions upon the commanded satellite motion. For example, the prescribed satellite maneuver  808  can be defined so as to limit the error between the predicted satellite attitude and the actual satellite attitude to less than 0.2 degrees and/or limit the yaw attitude rate to a particular value. 
   An exemplary profile of a prescribed satellite maneuver is illustrated in  FIG. 9  as trace  902 . Note that the difference between the profile of the ideal motion (trace  504 ) and the profile of the prescribed maneuver (trace  902 ) is small enough so that the solar panels  104  remain adequately directed at the Sun, yet by commanding the satellite attitude control system  200  to perform the prescribed maneuver instead of the ideal maneuver, the resulting actual satellite  100  attitude becomes predictable enough to allow the lever arm to be accurately predicted. 
   Returning to  FIG. 8A , spot beam gimbal angles are computed, as shown in block  816 . This is accomplished using the predicted satellite  100  attitude over the time period T (computed in block  806 ), the gimbal geometry  810 , and the target position of the spot beam (e.g. where the spot beam is aimed to illuminate one or more terrestrial targets such as navigational receivers) over the time period T, as computed in block  814 . The position of the target spot beam over time period T may be computed from a mission profile  812 , which is the mission that one or more of the satellites  100  in the navigation system  700  is to perform (for example, which areas on the Earth&#39;s surface the spot beam antenna will be directed at). 
   The gimbal geometry  810  (which includes, for example, a description of the dimensions and orientation of the gimbal structures and joints used to rotate the gimbals to direct the spot beam antenna  106 S or equations describing a relationship between pointing directions and gimbal angles) can be loaded into a memory of the satellites  100  of the space segment  704 , or can be uplinked to the space segment  704  from the ground segment, if desired. The geometry can include, for example, positional vectors which together extend from the wide coverage antenna  106 E phase center  112 E to the spot beam antenna  106 S and its phase center  112 S. This may include, for example, a vector between the phase center  112 E of the wide coverage antenna  106 E to the satellite bus  102 , from the satellite bus  102  to the rotational axis of the spot beam antenna  106 S inner gimbal motor  156 B, and from this point to the rotational axis of the outer spot beam antenna  106 S gimbal motor  156 C, and from the second spot beam antenna  106 S gimbal motor to the spot beam antenna  106  phase center. This geometry may also be expressed in terms of coordinate transformations. For example, from the wide-coverage antenna  106 E phase center  112 E to the satellite bus coordinate frame  156 A, through a coordinate transformation from the satellite bus coordinate frame  156 A to the inner gimbal coordinate frame  156 B, through an additional coordinate transformation from the inner gimbal coordinate frame  156 B to the outer gimbal coordinate frame  156 C and finally through a coordinate transformation from the outer gimbal coordinate frame  156 C to the spot beam antenna coordinate frame  156 D. 
   Turning to  FIG. 8B , the spot beam gimbal angles are used to predict the lever arm  114  vector. In one embodiment, the lever arm vector is predicted in the satellite bus coordinate system  156 A. Via appropriate coordinate transformations, this predicted lever arm vector  114  can be transformed from any of the foregoing coordinate systems (including the satellite bus  156 A and the spot beam antenna  156 D) to a desired coordinate frame. In one embodiment, the predicted lever arm vector  114  is transformed from the satellite bus coordinate frame  156 A into an Earth Centered Inertial or Earth Centered Earth Fixed (ECEF) coordinate frame so as to permit the navigational receiver  716  to apply the lever arm correction to its derived position estimates with a minimum of computations. This embodiment is shown in  FIG. 8B . 
   In block  828 , the lever arm  114  vector
 
 LA   SV ( x ( t ), y ( t ), z ( t ))
 
is transformed into Earth-Centered-Earth-Fixed (ECEF) coordinates so that this information may be used by the navigational receivers  716  without further conversion. The conversion can be performed with an estimate of the satellite&#39;s attitude computed in block  806 , or can use the actual measured satellite attitude, transmitted from the satellite  100  to the ground segment  702 .
 
     FIG. 10  is a diagram illustrating the behavior of the lever arm expressed in component parts LA ECEF (x(t)), LA ECEF (y(t)), LA ECEF (z(t)). Because the lever arm  114  vector is time dependent, the transmission of the lever arm to the navigational receivers  716  would consume a significant amount of available transmission bandwidth. In a preferred embodiment of the invention, this difficulty is overcome by using curve-fitting techniques (including, for example linear and non-linear regression) to identify closed form equations that describe the behavior of the lever arm over time during the time period T. In particular, the time-varying lever arm vector is expressed into its component parts (e.g. x(t), y(t), and z(t)), and each time-varying component part is curve-fit to an equation definable by combination of functions, each defined by one or more functional coefficients. This is shown in block  824 . 
   For example, each time-varying component can be fit to an n th  order polynomial 
   
     
       
         
           
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   The appropriate curve-fitting technique (e.g. the characteristics and order of the equation) can be selected according to the characteristics of the component part of the time varying lever arm  114  vector. For example, after z(t) for the time period T has been computed, z(t) can be evaluated to determine whether it is representable more accurately with a second, third, or fourth order polynomial. This can be accomplished, for example, by computing the mean squared error for each representation, and selecting the characteristics of the curve fit to minimize mean squared error. It may also be desirable to use different curve-fitting techniques and characteristics for the different component parts of the lever arm  114  vector. 
   Next, the lever arm vector coefficients are sent to the satellite  100 , as shown in block  832 . The satellite receives the vector, as shown in block  833 , and transmits the lever arm vector coefficients to the navigational receivers  716 , as shown in block  834 . The navigation receivers receive the coefficients, convert those coefficients to a time-varying lever arm correction prediction in ECEF coordinates, and apply the lever arm corrections at the appropriate time, as shown in blocks  835 – 838 . 
   The operations illustrated blocks  802 ,  806 ,  814 ,  816 ,  820 ,  822 ,  824 , and  832  of  FIGS. 8A and 8B  as being performed by the ground segment  702  can be implemented by software modules executed by a processor resident in the command station  707 , software modules executed by special or general purpose auxiliary processors in the command station  707  according to instructions stored in a memory communicatively coupled to the processor, or by dedicated electronic circuitry in the command station. For example, in one embodiment, the operations shown in blocks  802 ,  806 ,  814 ,  816 ,  820 ,  822 , and  824  are performed by a lever arm correction module  801 , which can be implemented in the command station processor  718 , an auxiliary processor, or dedicated electronic circuitry. Further, the lever arm correction module  801  can include a satellite position predictor module  804 , a satellite attitude predictor module  806 , and a lever arm prediction module  826 , which may further comprise a coordinate transformer  828  and curve fitting module  830  to perform the associated tasks shown in  FIGS. 8A and 8B . These modules may also be implemented in the command station processor  718 , an auxiliary processor, or dedicated electronic circuitry. 
   Generation of Prescribed Maneuver Profile 
     FIG. 11  is a diagram illustrating one technique that can be used to arrive at the prescribed maneuver profile. In block  1102 , the performance of the satellite attitude control system (ACS) is determined, using specifications and/or test data of the applicable components, including, for example, the reaction or momentum wheels, ACS thrusters, and attitude sensors. In block  1104 , this data is used to determine the maximum angular rate and/or acceleration that can be achieved by the spacecraft  100  under the most demanding maneuvers it will be required to perform, typically the yaw maneuver at orbital noon and midnight for low Sun angles, as depicted in  FIGS. 4A–4E . Since the yaw channel is typically most critical, a yaw steering profile can be examined. This can be accomplished using digital, analog, or hybrid control simulations of the satellite ACS. In block  1106 , an ideal attitude profile is generated. This profile does not consider the limitations of the ACS, and yields zero pointing errors about all axes and zero pointing errors of the solar panels at all times. In block  1108 , a candidate prescribed attitude profile is selected. The candidate profile is selected so as to attempt to match or closely approximate the ideal profile generated in block  1106 . Preferably, the candidate attitude profile is computed using closed form equations with parametric variability. The candidate profile will generally deviate from the ideal profile by an error, and the goal is to minimize this error. 
   Using this information, the candidate prescribed attitude profile is tested to determine if the spacecraft  100  can perform the maneuver described in the profile with sufficient (and typically nearly perfect) accuracy. Again, this can be performed with digital, analog, or hybrid simulations. If the test indicates that the satellite  100  can follow the prescribed maneuver profile, even better performance can be achieved by selecting a more demanding prescribed maneuver profile (using perhaps another candidate profile that follows the ideal profile more closely) that yields smaller errors when compared to the ideal profile, processing can return to block  1106  for the generation and eventual evaluation of that profile. If the test indicates that the satellite  100  cannot follow the prescribed maneuver profile, then the candidate prescribed maneuver profile is too demanding. The profile is adjusted in block  1114  and re-evaluated in steps  1108 – 1110 . The foregoing process is iteratively repeated until a prescribed maneuver profile is found that achieves the least error between itself and the ideal maneuver profile, while still allowing the spacecraft to follow it essentially error free. 
   An example candidate prescribed yaw attitude profile is given below
 
Ψ p   =a  tan 2└−sin(β p ),−cos(β p )sin(α)┘
 
where Ψ p  is the prescribed yaw attitude (angle), α is the spacecraft orbit angle (i.e. its position), a tan 2 is the two-argument arctangent function and
 
             β   p     =       β   a     +         (            α        -     π   2         π   2       )     n     ⁡     [         sign   ⁡     (     β   a     )       ⁢     β   min       -     β   a       ]               
where β a  is the actual current sun beta angle (i.e. the angle between the orbit plane and the sun), β min  is a minimum sun beta angle below which the satellite cannot perfectly follow the ideal profile, and the orbit angle α is normalized (“unwrapped”) so that its value always falls between −π and +π. The exponent n is the parameter of the above equations, and it would be adjusted on each pass of the blocks  1008 – 1012  to achieve the best performance. Note that for this example there are some constraints on n, such as it must be even and positive. While other forms of the candidate profile may yield better performance, the above technique and candidate profile yields a value of n=10.
 
   CONCLUSION 
   This concludes the description of the preferred embodiments of the present invention. The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. The above specification, examples and data provide a complete description of the manufacture and use of the composition of the invention. Since many embodiments of the invention can be made without departing from the spirit and scope of the invention, the invention resides in the claims hereinafter appended.