Abstract:
For an antenna on a satellite in an inclined orbit about the Earth, cross-track motion resulting from the rotation of the Earth can be reduced in antenna coordinates by yawing the antenna (preferably by yawing the entire satellite, for example, by using a reaction wheel system) by an appropriate angle, which varies throughout the orbit.

Description:
BACKGROUND OF THE INVENTION 
     (a) Field of the Invention 
     The present invention relates generally to a method for steering a satellite antenna beam, and more particularly, to a method for simplifying the steering of an antenna beam on a satellite in an inclined earth orbit in order to compensate for cross-track motion of earth-based terminals that is caused by rotation of the Earth. 
     (b) Description of Related Art 
     Antenna systems for communication satellites that are in non-geostationary orbits may require continuous adjustment of beam steering directions relative to the satellite to maintain coverage of users located within an earth-fixed cell during the pass of the satellite over the cell. The direction from the satellite to the users in satellite coordinates is affected by the rotation of the Earth as well as by the orbital motion of the satellite. The surface speed of the Earth due to rotation is proportional to the cosine of the latitude of the satellite, which varies throughout the orbit for all but equatorial (zero-inclination) orbits. This variation of relative velocity as a function of latitude normally requires beam steering in the cross-track direction (i.e., orthogonal to the velocity vector of the satellite) as well as in the along-track direction (i.e., along the velocity vector of the satellite), which in turn results in excessively complicated and cumbersome beam steering systems. 
     If the satellite antenna system is an electronically steered, high gain, low side lobe multibeam array, antenna steering may involve the control of the phase and amplitude of many elements. The number of active control elements required is substantially increased when beam steering is required in the cross-track direction as well as the along-track direction. This is normally the case, since for an antenna array aligned with the satellite geometric axes, cross-track motion results from the rotation of the Earth. 
     SUMMARY OF THE INVENTION 
     By using a fairly simple yaw steering method for the satellite, cross-track beam steering can be avoided, thereby greatly simplifying the antenna beam control steering system 
     In accordance with the present invention, cross-track motion of ground targets resulting from the rotation of the Earth can be dramatically reduced in antenna coordinates by yawing the antenna (preferably by yawing the entire satellite, for example, by using a reaction wheel system) by an appropriate angle, which varies throughout the orbit. The yaw steering method in accordance with the present invention, which is easy to implement, results in a considerable simplification of the antenna beam steering system. 
     The yaw steering method of the present invention uses a yaw angle φ, which is a function of the time from the ascending node of the orbit of the satellite, the period of the orbit and the inclination of the orbit. Assuming a circular orbit the desired yaw angle φ, in accordance with the present invention, is given by the expression: 
     
       
         tan(φ))=[sin(i)cos(2π/P)]/[(D/P)-cos(i)] 
       
     
     where: 
     φis the desired yaw angle; 
     i is the inclination of the orbit; 
     t is the time in the orbit of the satellite from the ascending node of the orbit; 
     P is the period of the orbit; and 
     D is the period of the rotation of the Earth. 
     For polar orbits, the inclination, i, is 90° and the expression for the desired yaw angle, φ, reduces to: 
     tan(φ)=(P/D) cos(2π/P). 
     In accordance with one aspect of the present invention, a method is provided for steering a satellite antenna mounted to a satellite. The satellite has a pitch axis, a roll axis, and a yaw axis and travels in an orbit around a rotating object. The orbit has an inclination and an ascending node. The method comprises the steps of: determining the inclination of the orbit; determining the time in the orbit from the ascending node; determining the period of the orbit; determining the period of the rotation of the object; and steering the antenna about the yaw axis by an angle, φ, wherein φ is a function of the inclination of the orbit, the time in the orbit from the ascending node, the period of the orbit, and the period of the rotation of the object. 
     In accordance with a further aspect of the present invention, the steering step includes a step of calculating the angle φ using the formula: φ=arctan [[sin(i)cos(2πt/P)]/[(D/P)-cos(i)]], where i is the inclination of the orbit, t is the time in the orbit from the ascending node, P is the period of the orbit, and D is the period of the rotation of the object. 
     In accordance with yet another aspect of the present invention, a method is provided for steering a satellite antenna mounted to a satellite. The satellite has a pitch axis, a roll axis, and a yaw axis, and travels in an orbit around the Earth, the orbit having an inclination and an ascending node. The method comprises the steps of: determining the inclination of the orbit; determining the time in the orbit from the ascending node; determining the period of the orbit; and steering the antenna about the yaw axis by an angle, φ, wherein φ is a function of the inclination of the orbit, the time in the orbit from the ascending node, the period of the orbit, and the period of the rotation of the Earth. 
     The use of the present invention for an antenna mounted to a satellite in a low-earth orbit inclined at about 85 degrees at an altitude of about 1400 km (about 870 miles) is predicted to reduce the cross-track path of a ground target relative to the antenna from about 4.5 degrees to about 0.04 degrees. 
     The invention itself, together with further objects and attendant advantages, will be best understood by reference to the following detailed description, taken in conjunction with the accompanying drawing. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     FIG. 1 is a schematic view of an exemplary satellite capable of being used for carrying out the method of the present invention, and the ground track of the satellite on a quadrant of the Earth; 
     FIG. 2 is a more detailed schematic diagram of the satellite shown in FIG. 1, further illustrating an apparatus for carrying out the method of the present invention; 
     FIG. 3 is a schematic vectorial representation of the velocity of an earth-based terminal in a satellite-based frame of reference; and 
     FIG. 4 is a flow diagram illustrating the method of the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The invention will now be described in connection with a current application of the inventive method of yaw steering. 
     FIG. 1 schematically shows a satellite  10  in an inclined, circular low-earth orbit over a quadrant  12  of the northern hemisphere of the Earth  13 . A curve in FIG. 1 designated  14 , represents the path of a sub-satellite point  15  (i.e., a point on the surface of the Earth 13 directly below the satellite  10 ), as the sub-satellite point  15  travels across the quadrant  12  of the Earth  13 . The sub-satellite point  15  travels at a velocity v in a direction indicated by an arrow  17 . An antenna  16  is mounted to the satellite  10 . A satellite-fixed coordinate system is depicted in FIG. 1 as follows: a yaw axis  18 , a roll axis  20 , and a pitch axis  22 . 
     Referring now to FIG. 2, the satellite  10  in which the present invention may be implemented is shown in further detail. In addition to the antenna  16 , the satellite  10  further includes an earth sensor  24 , a reaction wheel system  25 , a sun sensor  26 , a spacecraft control processor  28 . The earth sensor  24  and the sun sensor  26  sense the attitude of the satellite  10  with respect to the Earth  13  and the Sun  30 . The antenna  16  projects an electromagnetic beam  32  onto a coverage area  34  on the Earth  13 . 
     The earth sensor  24  and the sun sensor  26  are only two examples of attitude sensors, and other types of attitude sensors may be used in attitude determination systems. Alternatively, attitude sensors may use beacons, constellations, or other heavenly bodies as reference objects. Output signals from the attitude sensors  24 ,  26  are fed to the spacecraft control processor  28 , which is responsible for attitude determination and adjustment. 
     Command signals from the spacecraft control processor  28  are sent to the reaction wheel system  25  to adjust the attitude of the satellite  10 . The reaction wheel system  25  is only one example of a device that can be used to adjust the attitude of the satellite  10 . Alternatively, other devices, such as, for example, chemical or electrical thrusters could be used to adjust the attitude of the satellite  10  in response to commands from the spacecraft control processor  28 . 
     FIG. 3 shows a vectorial derivation of the velocity of an earth-based terminal in a satellite-based frame of reference. The expression for the desired yaw angle, φ, is derived as follows. When the satellite  10  is traveling in a northeasterly direction, as illustrated by the arrow  17  in FIG. 1, the orbital motion of the satellite  10  causes a user on the Earth  13  to appear to be moving southwesterly in a frame of reference fixed to the satellite  10 . The motion of the Earth  13  due to its rotation adds a relatively small eastward component, giving a clockwise rotation of the resultant motion vector as shown in FIG.  3 . As set forth in further detail below, the application of standard trigonometric identities to this construction shown in FIG. 3, results in the expression for the desired yaw angle, φ. 
     Even though the desired yaw angle φ is relatively small for low altitude satellites, it necessitates a component of beam steering in a direction perpendicular to the motion of the satellite. If many narrow beams are used to enhance the communication capacity of the satellites, many thousands of electronically controlled adjustments may be required to steer these narrow beams. 
     Reducing the beam steering complexity by implementing a one-dimensional system of yaw steering according to the present invention can reduce the number of active control elements to a small fraction of those required for a two-dimensional steering case. The implementation of the yaw steering method of the present invention adds no more complexity to the satellite than that of a single active antenna control element, versus thousands of such control elements eliminated by the yaw steering method. Because the frequency of the yaw steering method is so low, one cycle per orbit, the mechanical power associated with the yaw steering method is negligible. 
     With reference to FIGS. 1 and 3, the desired yaw angle, φ is derived as follows:        φ   =         tan     -   1            s     w   -   e         -       tan     -   1            s   w                   tan                 φ     =         s                 e         s   2     +     w   2     -     w                 e         =       s                 e         V   2     -     w                 e                   w   =     V                 sin                 α             s   =     V                 cos                 α             e   =     E                 cos                 b               cos                 i     =     cos                 b                 sin                 α               cot                 α     =     cos                 c                 tan                 i               tan                 φ     =       cos                 c                 sin                 i         V   E     -     cos                 i                                
     where: 
     i is the orbital inclination angle (FIG.  1 ); 
     c is the orbital arc from the ascending node to the sub-satellite point  15  (FIG.  1 ); 
     b is the latitude of the satellite  10  (FIG.  1 ); 
     a is the angle between the velocity vector of the sub-satellite point  15  and the meridian (FIGS.  1  and  3 ); 
     V is the velocity of the sub-satellite point  15  in the satellite frame of reference (FIGS.  1  and  3 ); 
     s is the south component of V (FIG.  3 ); 
     w is the west component of V (IG.  3 ); and 
     E is the surface speed of the Earth  13  at the equator (equal to about 1,524 feet per second or about 465 meters per second). 
     V′, shown in FIG. 3, is the velocity of an earth-based terminal in the satellite frame of reference after performing the yaw correction by an angle of φ. 
     FIG. 4 is a flow diagram illustrating how the method of the present invention can be carried out using the spacecraft control module  28  on the satellite  10 . As will be readily understood by those skilled in the art, the method can be carried out using either software or hardware programmed appropriately. First, at block  36 , the spacecraft control module  28  determines the position of the satellite  10  and the yaw angle of the satellite  10 . Next, at block  38 , the spacecraft control module  28  calculates the desired yaw angle, φ. At block  40 , the spacecraft control module  28  then compares the yaw angle of the satellite  10  to the desired yaw angle, φ. If no yaw angle adjustment is needed, the spacecraft control module  28  returns to block  36  and repeats the determination of the position of the satellite  10  and the yaw angle of the satellite  10 . If yaw angle adjustment is needed, the spacecraft control module  28  sends an appropriate set of commands to the reaction wheel system  25  in order to adjust the yaw angle of the satellite  10  to the desired yaw angle, φ, as indicated at block  42 . The spacecraft control module  28  then returns to block  36  and repeats the determination of the position of the satellite  10  and the yaw angle of the satellite  10 . 
     If desired, the antenna  16  could be steered by appropriate actuators mounted to the satellite  10 , instead of adjusting the yaw angle of the entire satellite  10 . However, and as noted above, because the frequency of the yaw steering method is so low, one cycle per orbit, the mechanical power associated with the yaw steering method is negligible. Accordingly, to minimize the complexity of the satellite  10 , it may be preferable to adjust the yaw angle of the entire satellite  10  using the reaction wheel system  25  instead of using an additional mechanism to steer the antenna  16 . 
     The present invention has been described with reference to specific examples, which are intended to be illustrative only, and not to be limiting of the invention, as it will be apparent to those of ordinary skill in the art that changes, additions and/or deletions may be made to the disclosed embodiments without departing from the spirit and scope of the invention.