Abstract:
A satellite ( 30 ) programmed for sun-nadir steering and having a solar wing ( 36 ) mounted to the satellite body ( 35 ) and being selectively moveable about two mutually orthogonal axes (A, B). The solar wing ( 36 ) is mounted to the satellite body ( 35 ) by a pair of gimbals ( 43, 45 ) thus allowing two degrees of freedom and thereby permitting the solar wing ( 36 ) to be rotated about the two mutually orthogonal axes (A, B). A first motor ( 42 ) in operative engagement with the gimbal ( 43 ) selectively rotates the solar wing about the axis (A), while a second motor ( 46 ) in operative engagement with the gimbal ( 45 ) selectively rotates the solar wing about the second axis (B). A control circuit ( 40 ) is in operative engagement with the first and second motors and selectively causes the first motor to rotate the solar wing about the first axis to a predetermined position, and selectively causes the second motor to rotate the solar wing about the second axis to a predetermined position.

Description:
STATEMENT REGARDING RELATED APPLICATIONS 
     This application claims domestic priority from an earlier filed provisional application, Ser. No. 60/095,387, filed Aug. 5, 1998. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to an improved system and method for reducing solar array sun-pointing error on satellites programmed for sun-nadir steering. 
     BACKGROUND OF THE INVENTION 
     A line between a satellite and the earth center of mass is typically called the nadir, while a line between the satellite and the sun is called a sunline. In a satellite programmed for sun-nadir steering these two lines are used as references to control the position of the satellite. In sun-nadir steering the spacecraft yaw axis is oriented toward the earth, generally coincident with the nadir. Any axis in the spacecraft can be pointed in any direction by rotating the spacecraft through two angles about any two fixed spacecraft axes. In conventional sun-nadir steering, the yaw axis is maintained earth pointed and the two rotation axes are chosen as yaw and pitch. 
     Conventional sun-nadir steering may be explained very simply as follows: 1) the spacecraft is yawed until the sun comes into the roll-yaw plane; 2) the solar array is pointed at the sun by rotating the solar array about a pitch gimbal until the solar array is normal to the sun. A more detailed description of sun-nadir steering may be found in  Effects of Solar Radiation Pressure on Satellite Attitude Control  by R. J. McElvain, published in Progress in Astronautics and Rocketry, Volume 8, Guidance and Control, published by Academic Press, 1962. The McElvain reference gives the body and wing steering equations, cited therein as equations 19 and 20. 
     The term sun-nadir steering may be used broadly, and may encompass steering laws that follow conventional sun-nadir steering as described above with substantive yawing and array pitch rotation over a significant period of time. Some examples include sun-target steering, rate limited sun-nadir steering, declination-limited sun-nadir steering, and the method disclosed in U.S. Pat. No. 5,794,891 issued to Polle et al. In sun-target steering, the yaw axis is pointed at a target other than nadir, such as a ground-fixed point. In rate-limited sun-nadir steering, the spacecraft yaw rate is limited, and the yaw rate is allowed to lag or lead the conventional sun-nadir profile to accommodate this yaw rate-limited configuration. In declination-limited steering, when the sunline is inconveniently close to the orbital plane, the spacecraft body is held orbit normal, and the solar array is pointed by a pitch gimbal. 
     On many spacecraft it is desirable to employ concentrator solar arrays, which provide more power per solar cell, thereby giving more power on a per unit cost basis. These concentrator arrays use mirrors or lenses that focus the sun&#39;s rays on small, high-temperature photovoltaic cells. However, these concentrator arrays typically must be pointed at the sun with a very high degree of accuracy in order to generate enough power to meet the bus requirements. The required pointing accuracy typically renders concentrator arrays unsuitable for use on satellites programmed for sun-nadir steering due to the sun tracking pointing error inherent when effectuating the noon turn (simply put, the satellite must “flip” at solar noon and solar midnight). The closer the sun lies to the orbit plane, the faster this “flip” must be done to point the arrays exactly at the sun. 
     Ideally, a spacecraft programmed for sun-nadir steering would effectuate the noon turn instantaneously when the sun is in the orbital plane. However, in practice the noon turn is both rate and acceleration limited, and thus there will always be sun tracking pointing error during portions of the turn in this case. Nevertheless, it would be highly desirable to have a spacecraft programmed for sun-nadir steering that is equipped with concentrator arrays, and which minimizes sun-tracking pointing error during portions of the noon-turn. 
     The solar arrays for satellites programmed for sun-nadir steering are conventionally sized to account for the fact that the solar arrays will experience power loss due to sun tracking pointing error during portions of the noon turn. This power loss is a especially problematic on high power satellites or on satellites in a low earth orbit having high orbital rates, which sometimes have orbital periods of as low as ninety minutes. The power loss when the solar arrays are not perpendicular to the sunlight is not significant for non-concentrated solar arrays because the power loss due to the non-perpendicularity goes roughly with the cosine of the angle away from perpendicularity. Therefore, an error angle of 25 degrees still allows for cosine(25 degrees)=0.906 of the power—over 90%. 
     For concentrator panels, however, the reduction in power with error angle is typically linear. Consequently, conventional sun-nadir satellites with concentrator arrays would require relatively large solar arrays and additional batteries, all of which increases weight, in order to account for the resulting loss in sunlight exposure due to sun tracking pointing error. Unfortunately, the extra weight increases the rotational moment of inertia of the spacecraft, which in turn necessitates the use of larger reaction wheels to perform the noon turn. The larger reaction wheels in turn increase the spacecraft weight even more. As the weight increases, the achievable slew rate is reduced, which negatively impacts the sun tracking pointing error during the noon turn. 
     It is known to point a solar array accurately at the sun by means of a two-axis gimbal between the solar array and the spacecraft body. Such a system is described, for example, in Fisher et al., “Magnetic Momentum Bias Control With Two-Gimballed Appendages”, Paper No. AAS 95-005, at page 72, which can be found in Volume 88 of  Advances in the Astronautical Sciences,  published for the American Astronautical Society. As discussed therein, the body is held fixed with respect to nadir and the orbit, the inner gimbal rotates at orbit rate, and the outer gimbal tracks out the angle between the orbit and the sunline. This approach has many disadvantages. First, the outer gimbal deflections required can be very large (the Fisher article shows 90 degrees) and can stay that way for many orbits. Such large and persistent gimbal travel sweeps the array through a spacecraft body field of view, potentially intruding into the fields of view of sensors, the payload field of view, thermal radiators, and even into thruster plumes. It also creates large variations in the spacecraft inertia matrix and can create severe gravity gradient torques in low orbits. Furthermore, the benefits of sun-nadir steering in limiting the momentum buildup from solar torques, and of limiting the directions that sunlight can intrude on radiators, payloads, etc., are lost. Thus, it would be desirable to avoid, minimize, or even eliminate one or more of the above-cited problems. 
     SUMMARY OF THE INVENTION 
     According to one aspect of the invention, a satellite is programmed for sun-nadir steering and includes a solar wing mounted to the satellite body. The solar wing is mounted to the satellite body by a gimbal having two degrees of freedom to thereby permit the solar wing to be rotated about two mutually orthogonal axes. A first motor in operative engagement with the gimbal selectively rotates the solar wing about a first axis, while a second motor in operative engagement with the gimbal selectively rotates the solar wing about a second axis. A control circuit in operative engagement with the first and second motors selectively causes the first motor to rotate the solar wing about the first axis to a predetermined position, and selectively causes the second motor to rotate the solar wing about the second axis to a predetermined position. 
     In further accordance with a preferred embodiment of the invention, the first and second motors are stepper motors. The control circuit causes the first and second motors to move the solar wing about the first and second axes to a position wherein the solar array is substantially perpendicular to sunlight. Preferably, the control circuit is programmed to maintain the solar array within 1° of perpendicular to the sunlight, and the solar array comprises a concentrator array. The control circuit may select the position of the solar wing by monitoring the amount of power produced by the solar array, by means such as a wing-mounted sun sensor, or may select the position of the solar wing based on ephemeris data stored in memory. The control circuit preferably comprises a microprocessor, and includes a closed loop circuit or an open loop circuit for controlling the movement of the solar wing about the first and second axes. A closed loop circuit comprises the control circuit, the stepper motors, the solar wing, and a power sensing circuit. 
     The satellite may include a second solar wing having attached thereto a second solar array, with the first and second solar wings being located on opposite sides of the satellite body. The second solar wing is also mounted to the satellite body by a gimbal having two degrees of freedom, to thereby permit the second solar wing to be rotated about two mutually orthogonal axes. The second gimbal is in operative engagement with another pair of motors for selectively rotating the second solar wing about the first and second axes, respectively. The control circuit is adapted to pivot the solar wings in substantially equal directions about the first axis, and is further adapted to pivot the solar wings in substantially equal directions about the second axis. 
     In accordance with another aspect of the invention, a satellite for use in a non-geostationary orbit includes a satellite body, an onboard attitude control system programmed for sun-nadir steering, and a solar wing mounted to the satellite body by a yoke rotatable about a first axis. The yoke includes a gimbal rotatable about a second axis perpendicular to the first axis. The solar wing includes at least one solar array. A first motor is in operative engagement with the yoke for selectively rotating the solar wing about a first axis, and a second motor is in operative engagement with the gimbal for selectively rotating the solar wing about a second axis. A control circuit is in operative communication with the first and second motors for selectively causing the first motor to rotate the solar wing about the first axis and for selectively causing the second motor to rotate the solar wing about the second axis. 
     In accordance with yet another aspect of the invention, a method for decreasing the power requirements of a satellite in a low earth orbit and having a solar wing comprises the steps of programming an onboard attitude control system for sun-nadir steering, rotating the solar wing about a first axis and pivoting the solar wing about a second axis to maintain the solar array substantially normal to the sunline. 
     In accordance with a still further aspect of the invention, a method for decreasing the power requirements of a satellite in a low earth orbit, the satellite including a power generating solar wing, comprises the steps of programming an onboard attitude control system for sun-nadir steering, rotating the solar wing about a first axis, and pivoting the solar wing about a second axis to substantially maximize the solar wing output. 
    
    
     These and other objects, features and advantages of the present invention will become readily apparent to those skilled in the art upon a reading of the following description with reference being had to the accompanying drawings. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a perspective view of a satellite constructed in accordance with the teachings of the present invention; 
     FIG. 2 is an enlarged schematic illustration of the satellite illustrated in FIG. 1; 
     FIG. 3 is a schematic illustration representing the orientation of a satellite orbiting the Earth with the Sun shown positioned in the orbital plane; 
     FIG. 4 is a schematic illustration representing the orientation of a satellite orbiting the Earth with the Sun shown positioned above the orbital plane; 
     FIG. 5 is a graphical illustration of representative yaw steering and pitch requirements for a satellite in various orbits; 
     FIG. 6 is a schematic illustration of an open loop control system for pointing a solar wing of the satellite shown in FIGS. 1 and 2; 
     FIG. 7 is a schematic illustration of a closed loop control system for pointing the solar wing of the satellite shown in FIGS. 1 and 2; 
     FIG. 8 is a schematic illustration of Sun-Nadir steering geometry of a satellite in an equatorial orbit in the summer (i.e., the sun being above the plane of the page); 
     FIG. 9 is a schematic illustration of Sun-Nadir steering geometry of a satellite in an equatorial orbit in the winter (i.e., the sun being below the plane of the page); 
     FIG. 10 is a schematic illustration of the position of the sun relative to the axes of a satellite in an inclined orbit having an orbital period of approximately 115 minutes and illustrating the sun tracking pointing error during the first turn; 
     FIG. 11 is a schematic illustration of the position of the sun relative to the axes of a satellite in an inclined orbit having an orbital period of approximately 115 minutes and illustrating the sun tracking pointing error during the second turn; 
     FIG. 12 is a graphical representation of the yaw angle as a function of time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 13 is a graphical representation of the body angular rates for roll, pitch and yaw over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 14 is a graphical representation of the solar panel angular rate about the inner gimbal over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 15 is a graphical representation of the solar panel angle about the inner gimbal over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 16 is a graphical representation of the solar panel angular rate about the outer gimbal over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 17 is a graphical representation of the solar panel angle about the outer gimbal over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 18 is a graphical representation of the sun elevation angle relative to the satellite for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 0°; 
     FIG. 19 is a graphical representation of the yaw angle as a function of time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 1°; 
     FIG. 20 is a graphical representation of the body angular rates for roll, pitch and yaw over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 1°; 
     FIG. 21 is a graphical representation of the solar panel angular rate about the first gimbal over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 1°; and 
     FIG. 22 is a graphical representation of the solar panel angle over time for a satellite performing sun-nadir, rate limited steering in a given orbit with a Beta angle of 1°. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     A satellite  30  assembled in accordance with the teachings of the present invention is illustrated in FIG.  1 . The satellite has an onboard attitude control system (ACS)  32  (FIGS. 6 and 7) programmed for sun-nadir steering in a manner well known to those of skill in the art, and further in a manner that requires turning of the satellite  30  at local noon and local midnight in order to prevent certain portions of the satellite, (i.e., the “cold” side of the satellite), from being exposed to solar irradiation. In so doing, the negative roll axis is maintained away from the sun. The concept as well as the effectuation of the “noon-turn” is well known to those of skill in the art and will not be discussed further herein. 
     As shown in FIG. 2, for purposes of explanation, the satellite  30  will have three reference axes, X, Y, and Z. The yaw or Z axis is coincident with the satellite boresight, the pitch or Y axis is generally parallel to the yokes which support the solar wings (i.e., the yokes  38  and  38   a  described in greater detail below), and the roll or X axis is generally mutually perpendicular to the Y and Z axes. The nadir is indicated by the reference numeral  31  as shown in FIGS. 2,  3  and  4 . The sunline is indicated by the reference numeral  33  and is also shown in FIGS. 3 and 4. In sun-nadir steering, the Z axis is generally coincident with the nadir  31 , and the Y axis is generally perpendicular to the sunline. 
     In sun-nadir steering, it is desirable at all times to maintain the Sun in the X-Z plane. This is accomplished by maneuvering the satellite  30  about any two fixed axes as is well known. It will be noted that the orbit rate and the angular rate of the nadir will be the same (i.e., the rate of the nadir  31  rotating about the earth will depend on the rate of the satellite  30  orbiting the earth). 
     It will be noted that the derivation of the array gimbal angles, the body yaw angles, mechanism travel requirements, etc., may be accomplished using known and well accepted methods which are within the knowledge of one skilled in the art. Accordingly, the derivation of such angles will not be repeated herein. A discussion of sun-nadir steering may be found in U.S. Pat. No. 5,794,891 issued to Polle et al. 
     As shown in that FIG. 1, the satellite  30  includes a bus or body  35  which typically holds the control electronics, the navigational and telemetry electronics, and the antennas for the satellite. The satellite  30  also includes a pair of solar panels or wings  36 . Each of the solar wings  36  includes a plurality of solar cells and is independently mounted on a yoke  38 . As shown in FIG. 1, the yokes  38  typically extend from opposite sides of the body  35 . The mounting yokes  38  and the satellite body  35  may be of a conventional design and will not be further discussed herein Each of the solar wings  36  is preferably a concentrator or Fresnel Lens array  39 , in which a plurality of solar cells are located on a generally concave collection surface. Such an array  39  collects solar illumination over an area greater than the area of the solar cells, and then concentrates that gathered illumination onto the solar cells for power generation. As is known to those skilled in the art, a concentrator or Fresnel Lens array requires much greater pointing accuracy than do conventional solar arrays. As shown in FIG. 2, each of the solar wings  36  includes a normal vector  37 . As would be known to those skilled in the art, the solar cells will generate the most power when the normal vector  37  is aligned with the sunline  33 . 
     A control circuit  40  (FIG. 6) is operatively connected to the ACS  32  as well as to a pair of gimbals  43 ,  45  as shown in FIG.  2 . Although the satellite  30  will preferably include a pair of solar wings  36 ,  36   a , each of which is mounted to a yoke  38 ,  38   a , respectively, only the structure and operation of a single solar wing  36  will be discussed in detail herein. It will be understood, however, that a second solar wing having the same or similar components may be attached to the satellite, typically to the opposite side of the body  35 . The control circuit  40  is preferably a microprocessor of the type commercially available, as would be known to those skilled in the art. 
     The gimbals  43 ,  45  are provided in order to adjust the position of the solar wing  36  relative to the body  35 . The gimbal  43  is moveable about an axis “A”, while the gimbal  45  is moveable about an axis “B”, as shown in FIG.  2 . It will be understood that the axes A and B are mutually perpendicular, and that the axis A is generally parallel and coincident with a longitudinal axis of the yoke  38 . For the purpose of rotating the solar wing  36  about the axis A, i.e., about an axis parallel to a longitudinal axis of the yoke  38 , the satellite  30  includes a rotational motor  42 . Preferably, the rotational motor  42  is controlled to pivot the solar wing  36  about the A axis to a position determined by the control circuit  40 . 
     The gimbal  45  is provided for purposes of rotating the solar wing  36  about the B axis, i.e., about an axis perpendicular to the A axis and the longitudinal axis of the yoke  38 . The satellite  30  includes a rotational motor  46  for rotating the gimbal  45  about the axis B. Preferably, the rotational motor  46  is controlled to pivot the wing  36  about the B axis to a position determined by the control circuit  40 . As an alternative to using two gimbals  43 ,  45  on each yoke  38 , a single gimbal having two degrees of freedom may be employed (i.e., a gimbal moveable about both axes A and B. 
     As an example, and referring to FIG. 3, if the Sun were located in the orbital plane  49 , the sun would lie in the X-Z plane of the satellite  30 . Accordingly, the solar wing  36  could be pointed toward the sun by rotating the gimbal  43  about the axis A using the rotational motor  42  (which rotation would effectively rotate the wing  36  purely in pitch—about the Y axis). However, in the event the Sun were located above the orbital plane  49 , such as is shown in FIG. 4, then pointing of the wings  36  would be accomplished by also rotating the gimbal  45  about the B axis using the rotational motor  46 . The same would hold true for circumstances in which the sun lies below the orbital plane, although accurate pointing would be effectuated by rotating the gimbals  43 ,  45  in the opposite directions. 
     As shown in FIG. 2, the angular position of the wing  36  about the axis A due to rotation of the gimbal  43  may be referred to as the solar panel angle φ 2 , while the yaw angle of the satellite  30  is referred to as the angle φ 3 . Finally, the sun elevation angle φ 4  is the angular position of the wing  36  when rotated about the axis B using the gimbal  45 . For a given orbit, the pertinent angles are calculated in accordance with well accepted spacecraft attitude control principles. As shown in FIG. 4, the angle between the sunline and the orbital plane is referred to as the declination angle C. Also shown in FIG. 4 is a line between the earth and the sun, which line forms an angle Beta (β), referred to as the sun angle. It will be noted that, due to the distance of the earth from the sun, the declination angle C and β may be interchangeable, as they differ at most on the order of 4 hundredths of a degree (0.04 degrees). 
     As explained in detail below, the ACS  32  of the satellite  30  is programmed for sun-nadir steering, and the satellite  30  is adapted to position the solar wing(s)  36  of the satellite  30  in order to maximize the amount of power produced by the solar cells located thereon. The ACS  32  is preferably mounted within or on the body  35  of the satellite  30 . 
     Preferably, the motors  42  and  46  are implemented as conventional stepper motors such as are commercially available from such vendors as Ducommon-AEI of Carson, Calif. or Tecstar Electro-Systems Division of Durham, N.C. However, persons of ordinary skill in the art will appreciate that other motors can be implemented as well. Similarly, while persons of ordinary skill in the art will appreciate from the above disclosure that only one pivoting motor and one rotational motor are needed to maintain the surface of a solar panel substantially perpendicular to the solar illumination arriving from the sun, in the preferred embodiment two rotational motors and two pivoting motors are used for each yoke  38  for purposes of redundancy. Preferably, these paired motors are redundantly arranged such that either one of the motors  42  or either one of the motors  46  can pivot the solar wing  36  about the required axis if the other motor in the pair fails. 
     Persons of ordinary skill in the art will further appreciate that, although the above description has assumed that the satellite  30  includes two solar wings  36  and two yokes  38 , the teachings of the invention can be applied to satellites having any number of wings  36  and yokes  38  including, but not limited to one wing  36  and one yoke  38 . In instances where two wings  36  and two yokes  38  are employed, the wing/yoke pairs are preferably located on opposite sides of the satellite body  35 , and the pivoting motors  46  are preferably controlled to pivot the solar wings  36  in substantially equal but opposite directions so that the wings  36 ,  36   a  remain substantially parallel, while the rotational motors are preferably controlled to rotate the solar wings  36  in substantially equal directions. 
     Referring to FIG. 6, an open loop control circuit  57  is shown. The open loop control circuit  57  includes the control circuitry  40 , which may be operatively connected to interface electronics  48 . The control circuit  40  is also operatively connected to the ACS  32 . The interface electronics  48  preferably include the circuitry necessary to permit communication between the control circuit  40  and the motors  42 ,  46 , which are connected to the solar wing  36 . For example, the interface electronics  48  preferably include a digital to analog converter and/or voltage conversion circuitry to convert the output of the control circuit  40  to a level and format usable by the motors  42 ,  46 . The interface electronics  48  thus permit the control circuit  40  to control the position of the solar wing  36  using the motors  42 ,  46 . 
     Referring to FIG. 7, a closed loop control circuit  55  is shown. The closed loop control circuit  55  includes the control circuit  40 , which may be operatively connected to a power sensing circuit  51  and telemetry conditioning electronics  53 , as well as to the interface electronics  48  and the motors  42 ,  46 . The motors  42 ,  46  are in turn operatively connected to the solar wing  36 , and the solar array  39  on the solar wing  36  is operatively connected to the power sensing circuit  51 . The closed loop control circuit  55  is operatively connected to the ACS  32 , and establishes the desired position of the solar wing  36 . The power sensing circuit  51  is preferably adapted to sense the amount of power being produced by the array  39 , and will generate a signal which is communicated to the telemetry conditioning electronics  53 , processes the signal to generate a signal indicative of the optimum position for the solar wing  36 , which signal is then communicated to the control circuit, which then effectuates the desired position change for the solar wing  36  via the interface electronics  48  and the motors  42 ,  46 . Thus, the closed loop circuit  55 , based on sensed increases or decreases in the amount of power being produced, makes appropriate changes to the position of the solar wing  36 . 
     As an alternative, a sun sensor  59  (illustrated schematically in phantom in FIG. 6) may be mounted the solar wing  36  to sense the solar wing attitude, in place of the power sensing circuit  51  and solar array  39  arrangement shown. 
     Quantitative Example 
     The gimbal angle profiles for a satellite in a 1400 km orbit having an orbital period of approximately 115 minutes were calculated. FIG. 12 is a graphical representation of the spacecraft yaw angle over time for a Beta angle of 0° (zero degrees), while FIG. 13 is a graphical representation of the body angular rates in roll pitch and yaw for the same orbit. FIGS. 14 and 15 are graphical representations of the angular rate and the angle, respectively, of the solar panel about the A axis (i.e., rotation of the gimbal  43 ) for the same orbit. FIG. 16 and 17 are graphical representations of the angular rate and the angle, respectively, of the solar panel about the B axis (i.e., rotation of the gimbal  45 ) for the same orbit. FIG. 18 is a graphical representation of the sun elevation angle on the -X face of the satellite body for the same orbit. 
     Referring now to FIGS. 19-22, the gimbal angle profiles are shown for a similar orbit, but the Beta angle has been increased to 1°. FIG. 19 is a graphical representation of the spacecraft yaw angle over time for the Beta angle of 1° (one degree), while FIG. 20 is a graphical representation of the body angular rates in roll pitch and yaw for the same orbit. FIGS. 21 and 22 represent the angular rate and the angle, respectively, of the solar panel about the A axis (i.e., rotation of the gimbal  43 ) for the same orbit. 
     In both of the above illustrations, it can be seen that the angular rates are substantially reduced over prior art systems, while also providing for more accurate pointing of the solar arrays toward the sun. 
     In operation, the satellite body  35  is rotated in yaw about the Z axis (i.e., about nadir  31 ), in order to keep the angle between the sunline  33  and the axis A close to perpendicular. The solar wing  36  is then rotated about axis A to orient the normal vector  37  as close as possible to the sunline  33 . Preferably, the rotation of the gimbal  43  about the axis A (φ 2 ) will be at an angular rate not substantially greater than the orbit angular rate, so as to maintain the normal vector  37  close to the sunline  33 . The solar wing  36  is then rotated relative to the B axis by rotating the gimbal  45 . Preferably, the angular rotation of the gimbal  45  about the axis B (φ 4 ) may be less than, and in some cases substantially less than, 20° (twenty degrees), and will bring the normal vector  37  even closer to the sunline  33 . The Fresnel Lens array on the solar wing  36  will then collect solar power from the solar illumination over an area substantially greater than the area of the solar cells, and will focus the solar power onto the solar cells in order to produce electrical power. The position of the solar wing  36 , and thus the angular rotation of the solar wing  36  about the axes A and B, will be controlled by the control circuit  40  using well known principles. 
     FIGS. 8 and 9 illustrate the interplay between the spacecraft body yaw attitude and the solar array relative pitch angle for various spacecraft positions in the orbit. 
     Although certain instantiations of the teachings of the invention have been described herein, the scope of coverage of this patent is not limited thereto. On the contrary, this patent covers all instantiations of the teachings of the invention fairly falling within the scope of the appended claims either literally or under the doctrine of equivalents.