Patent Publication Number: US-6990396-B2

Title: Method and apparatus for reaction wheel dynamic compensation in long-duration deployment of a large reflector

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application is related to the following co-pending and commonly assigned patent application(s), all of which applications are incorporated by reference herein: 
   Application Ser. No. 10/318,715 entitled “METHOD AND APPARATUS FOR SOLAR TACKING MOMENTUM MAINTENANCE IN LONG-DURATION DEPLOYMENT OF A LARGE REFLECTOR,” filed on same date herewith, by Hanching G. Wang, Richard A. Fowell, and Che-Hang C. Ih. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to systems and methods for controlling spacecraft, and in particular to a system and method for controlling the spacecraft while deploying a large appendage such as a reflector. 
   2. Description of the Related Art 
   Satellite systems typically include appendages that are stowed during launch and deployed sometime thereafter. Typically, these appendages include solar wings (having the solar cells and the supporting structures), passive and/or active sensors, and antennas used for satellite-to-ground or satellite-to-satellite communications. 
   A typical goal during satellite appendage deployment is to maintain a spacecraft dynamic state close to that of rotating about an axis fixed in both the body and in inertial space. A rotation axis fixed in both the spacecraft and inertial space implies that the satellite angular rate vector direction is fixed in the satellite. This is an idealized state, and deployments are designed to keep the spacecraft state within acceptable bounds of this state. 
   One purpose of this goal is to keep the sun close to the desirable direction for the body, for solar power or thermal control reasons. Another is to keep the axis of symmetry of the toroidal telemetry and command antenna close to a known direction in space. The rotation itself helps largely to average out momentum buildup due to environmental torques, and ensures that any blockages of telemetry and command antenna line of sight by spacecraft structures are cleared by rotation. Such appendages must also be deployed in such a way that they are not damaged, and such that the long term stability of the satellite is not adversely affected. 
   Fortunately, since the inertia of each of the appendages is typically much smaller than that of the satellite itself, the attitude control system of the host satellite is typically adequate to assure the stability of the satellite during and/or after deployment. 
   A number of solar array deployment techniques have been developed. A first technique uses only the design variables of the satellite spin speed and the initial orientation of the spin vector in space, and the spacecraft North and South solar arrays are deployed in turn without any other attitude control. A second technique times the first deployment with respect to the sun rotational phase, and the second deployment with respect to the first, in order to meet the design constraints. A third technique initiates nutation with a thruster pulse or a wheel maneuver prior to the release of the first solar array, timed so that the constraints of the deployment are met over the deployment sequence. A fourth technique uses an internal wheel momentum to alter the dynamic motions during deployment to favorable effect. This includes using an internal wheel momentum bias along the spin axis (“superspin”) to make the effective inertia ratio σ eff &gt;1, thus rendering the spacecraft passively stable during the deployment. A fifth technique uses active wheel control during deployment to help damp out unwanted spacecraft body rates. Other deployment techniques have used similar methods, even for deployment of asymmetrical antennas measuring 49 feet in maximum dimension, and over long duration deployments lasting several hours. 
   In all of these techniques, the effects of momentum buildup due to environmental torques (including solar torques, thermal emission torques, atmospheric drag torques, and radio frequency (RF) emission torques) are small enough so that the effect on the satellite attitude is acceptably small, and are therefore neglected. 
   However, in some circumstances, the effect of environmental torques and the momentum of the deployed appendage(s) cannot be adequately accommodated by the attitude control system of the satellite. This can be the case, for example, when the satellite appendage is a large antenna. 
   Satellites have long been used to communicate information with terrestrially based ground stations. Communications systems have also been devised to permit worldwide communications to mobile receivers, but such designs have met with limited success. One of the reasons for this limited success is that it is difficult to design a lightweight, mobile receiver that can communicate with the satellite constellation, even a constellation in mid or low Earth orbits. 
   One solution to this problem is to equip one or more of the satellites with one or more high-sensitivity transmit/receive antennas. The problem with this solution is that to provide high sensitivity over wide beamwidths, the antennas must be quite large. This raises difficulties in three respects. First, the larger antennas have larger inertia than the smaller prior art antennas discussed above, thus imposing more stringent requirements on the design of the satellite control system, including the sensors (e.g. star and sun sensors, gyros and accelerometers), the attitude control thrusters, and the algorithms implemented by the control system. Second, since the larger antenna must be designed to fit within the storage bay of the launch vehicle deploying the satellite, the structures relating to its deployment are typically more complex than those of smaller appendages. For example, a foldable large antenna will generally have more joints and more structural elements than a smaller prior art antenna, and each of these structural elements must be folded out during deployment. Expressing the dynamics of the deployment of such antennas can be a daunting task, making it difficult for the satellite designer to assure that such deployment does not compromise satellite stability. 
   It is also desirable to perform the deployment over an extended period of time to avoid mesh tangling, and to even the thermal condition of truss joints. However, doing so subjects the satellite and the deployed appendage to a variety of disturbance torques (including solar torques) that further complicate the deployment process. It is possible, of course, to simply design the attitude control system to permit the rapid deployment of the appendage (e.g. by thruster torques), but such a solution would require a satellite control system that would be far more robust than is required at any time after the appendage is deployed, increasing the weight and cost of the satellite. Also, using thrusters during deployment raises the possibility that the appendage will be damaged during deployment. 
   Solar torquing techniques using satellite appendages have been applied to operational non-spinning satellites with relatively constant mass properties. For example, such techniques are disclosed in “Survey of Solar Sailing Configurations for Satellite Attitude Control,” AAS 91-486, by George A. Kyroudis, which is hereby incorporated by reference herein. U.S. Pat. No. 5,816,540, issued to Murphy, U.S. Pat. No. 4,949,922, issued to Rosen, and U.S. Pat. No. 4,325,124, (also incorporated by reference herein) also disclose oscillating one or more of the solar arrays to apply solar torque (a technique sometimes known as “solar tacking”). While useful, such techniques are thought to be only applicable in operational, non-spinning satellites with relatively constant mass properties (e.g. not during deployment of a large appendage). Further, such techniques are limited in application, because the manipulation of the solar arrays can have a deleterious effect on the ability of the solar arrays to generate power for satellite operation. 
   Transverse momentum has been used to produce cross-product gyroscopic torques in other contexts, namely to oppose external boost motor torques. This technique is disclosed in U.S. Pat. No. 6,032,903, which is hereby incorporated by reference. However, this technique relies on a combination of gyroscopic torque and thruster torque, and, as described above, it is undesirable to use thruster torque during appendage deployment. Further, this technique is typically appropriate for different and much larger disturbance torques. 
   Magnetic torquing momentum management systems have also been used. Such techniques use a small gimbal offset in a gimbaled momentum wheel, such that, as the satellite body rotates 360 degrees in pitch over 24 hours, the peak transverse angular momentum over the day is largely minimized. However, this technique is not generally applicable to the problem of large appendage deployment, where the dynamic state is expected to be a rotation perpendicular to the pitch axis, at a rotational rate that is typically an order of magnitude higher. 
   What is needed is a system and method for controlling a satellite while deploying an appendage having a large moment of inertia. What is also needed is that the method be sufficiently robust to permit the design of a deployment sequence without requiring a detailed model of the inertia of the appendage as a function of time during the deployment sequence. The present invention satisfies that need. 
   SUMMARY OF THE INVENTION 
   To address the requirements described above, the present invention discloses a method, apparatus for controlling an attitude of a spacecraft while deploying an appendage from the spacecraft. The present invention initializes the transverse momentum perpendicular to the environmental torques experienced during appendage deployment to oppose them by cross-product gyroscopic torque. The method comprises the steps of determining a direction and amount of momentum to add to the spacecraft via the plurality of momentum wheels, storing momentum in at least one of the momentum wheels to provide momentum in the determined direction and in the determined amount, and deploying the appendage while controlling the attitude of the spacecraft at least in part by orienting the solar wings to apply differential solar torque to the spacecraft. 
   The present invention applies two elements (direct solar control and gyroscopic torque due to internal momentum perpendicular to the spacecraft rotational rate) to counter the effects of environmental torques on deployment. Unlike the prior art, the present invention recognizes and takes advantage of the fact the payload is inactive during satellite and appendage deployment. This reduces the power consumption and heat dissipation of the satellite, permitting greater flexibility in the manipulation of the solar arrays (because there is less need to keep the surfaces of the solar arrays directed at the sun and because the thermal radiators remain in the stowed position, permitting greater solar array movement without interference). 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Referring now to the drawings in which like reference numbers represent corresponding parts throughout: 
       FIGS. 1A and 1B  are diagrams depicting a three axis stabilized satellite; 
       FIG. 2  is a block diagram depicting a satellite control system; 
       FIG. 3  is a block diagram of one embodiment of the spacecraft appendage control system; 
       FIGS. 4A-4C  are a diagrams illustrating the deployment of a spacecraft appendage; 
       FIG. 5  is a flow chart presenting exemplary process steps used to practice one embodiment of the invention; 
       FIG. 6  is a flow chart presenting exemplary process steps used to compute the magnitude and direction of the momentum to add to the spacecraft before deploying the appendage; 
       FIG. 7  is a flow chart further illustrating the computation of the momentum to add to the spacecraft before deploying the appendage; 
       FIG. 8  is a diagram presenting illustrative method steps that can be used to control the attitude of the spacecraft and/or add or subtract momentum from spacecraft; and 
       FIGS. 9A and 9B  are diagrams showing techniques for applying differential solar torque to the spacecraft. 
   

   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. 
     FIGS. 1A and 1B  illustrate a three-axis stabilized satellite or spacecraft  100 .  FIG. 1A  illustrates an appendage  114  in a deployed position, while  FIG. 1B  illustrates the appendage  114  in a stowed position. The satellite  100  has a main body  102 , a pair of solar wings or panels  104 N and  104 S. The satellite  100  may also have a number of instruments and sensors such as one or more high gain narrow beam antennas  106 , a telemetry and command omni-directional antenna which is aimed at a control ground station, a boom-mounted magnetometer, or an optical imager that is aimed at various ground targets. 
   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. The satellite  100  also includes one or more attitude control thrusters  112 , for applying rotation torques to the satellite  100 . The satellite also includes an antenna reflector  114 A and a boom  114 B with an associated deployment mechanism and driver components such as a combination of motors, springs, and damping mechanisms. Collectively, these elements are alternatively referred to hereinafter as the appendage  114 . 
   The three axes of the spacecraft  100  are shown in FIG.  1 B. The pitch axis Y lies along the plane of the solar panels  104 N and  104 S. The roll axis X and yaw axis Z are perpendicular to the pitch axis Y and lie in the directions and planes shown. 
     FIG. 2  is a diagram depicting the functional architecture of a representative attitude control system. Control of the spacecraft is provided by a computer or spacecraft control processor (SCP)  202 . The SCP performs a number of functions which may include post ejection sequencing, transfer orbit processing, acquisition control, stationkeeping control, normal mode control, mechanisms control, fault protection, and spacecraft systems support, among others. The post ejection sequencing could include initializing to ascent mode and thruster active nutation control (TANC). The transfer orbit processing could include attitude data processing, thruster pulse firing, perigee assist maneuvers, and liquid apogee motor (LAM) thruster firing. The acquisition control could include idle mode sequencing, sun search/acquisition, and Earth search/acquisition. The stationkeeping control could include auto mode sequencing, gyro calibration, stationkeeping attitude control and transition to normal mode. The normal mode control could include attitude estimation, attitude and solar array steering, momentum bias control, magnetic torquing, and thruster momentum dumping (H-dumping). The mechanisms mode control could include solar panel control and reflector positioning control. The spacecraft control systems support could include tracking and command processing, battery charge management and pressure transducer processing. 
   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 . Ground commands are also input into the spacecraft control processor. These commands determine the control functions of the processor and the scan patterns of some instruments and sensors. 
   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 torquer control unit (MTCU)  226  and the torque coil  228 . 
   The SCP  202  also sends control commands  230  to the thruster valve driver unit  232  which in turn controls the liquid apogee motor (LAM) thruster  234  and the attitude control thrusters  236 . 
   Generally, the spacecraft  100  may use thrusters, momentum/reaction wheels, or a combination thereof to perform spacecraft  100  attitude control. 
   Wheel torque commands  262  are generated by the SCP  202  and are communicated to the wheel drive electronics  238  and  240 . These effect changes in the wheel speeds for wheels in momentum/reaction wheel assemblies  242  and  244 , respectively. The speed of the wheels is also measured and fed back to the SCP  202  by feedback control signal  264 . 
   When momentum wheel assemblies are used, the spacecraft control processor also sends jackscrew drive signals  266  to the momentum wheel assemblies  242  and  244 . These signals control the operation of the jackscrews individually and thus the amount of tilt of the momentum wheels. The position of the jackscrews is then fed back through command signal  268  to the spacecraft control processor. The signals  268  are also sent to the telemetry encoder unit  258  and in turn to the ground station  260 . The spacecraft typically includes 4 reaction wheels, disposed to permit application of torques in any direction, and permitting for a backup torque wheel, however, different numbers of momentum wheels and momentum wheels of other design may be used. For the sake of simplification, these wheels will be referred to as momentum wheels, reaction wheel(s), or torque wheel(s) hereinafter. 
   For some satellites, the spacecraft control processor  202  also commands the scan motions of various sensors and instruments. The scan timings and patterns  276  generated by the SCP  202  are communicated to the scan motor drivers  278 . 
   The SCP  202  also provides commands to the solar wing drives  246 ,  248 , which manipulate solar wings  104 N and  104 S respectively. The solar wings  104 N and  104 S can be manipulated about the X axis and about the Y axis shown in FIG.  1 . The SCP  202  can also step reflector positioning mechanisms (RPMs)  250  and  252  to adjust the antenna orientation. Modules  250  and  252  provide the mechanism positions to the TM encoder unit  258 . 
   The SCP  202  also sends command signals  254  to the telemetry encoder unit  258  which in turn sends feedback signals  256  to the SCP  202 . This feedback loop, as with the other feedback loops to the SCP  202  described earlier, assist in the overall control of the spacecraft. The SCP  202  communicates with the telemetry encoder unit  258 , which receives the signals from various spacecraft components and subsystems indicating current operating conditions, and then relays them to the ground station  260 . 
   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 , causes 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. 
     FIG. 3  is a block diagram of one embodiment of a spacecraft  100  appendage control system  300 . The spacecraft appendage control system  300  comprises a deployment module (DM)  302 . The DM  302  can be implemented by separate electronic circuits, by the SCP  202  (as shown in FIG.  2 ), or by a separate processor dedicated to the functions described below. 
   The attitude control sensors  310 , (which may include, by way of example, data from gyros and accelerometers in the inertial reference unit  208 , operational earth orbit sensors  210 , normal mode wide angle sun sensor(s)  214 , magnetometers  216 , star sensors  218 , and or telemetered data), provides information to the attitude control module  312  to control the attitude of the spacecraft  100  under normal (non-deployment) conditions. The attitude control sensors  310  also provide such measurements to the deployment module (DM)  302 . The DM  302  provides commands to the deployment mechanism  304 , the solar wing drives  246 ,  248 , and the attitude control actuators  306  (which include, by way of example, the ACS thruster(s)  236  and associated thruster drive units  232 , and the momentum or reaction wheel assembly  242 ,  244  and associated modules). Such commands result in spacecraft  100  motion, which is represented by spacecraft dynamics  308 . 
     FIGS. 4A-4C  are a diagram presenting an illustration of the spacecraft  100  during deployment of the appendage  114 . In one embodiment the illustrated steps are performed in a sun-hold phase of satellite deployment after the sun has been acquired and tracked by either of the sun sensors  206  or  214 . 
     FIG. 4A  is a diagram of the spacecraft  100  during boom  114 B deployment. When the spacecraft enters the sun-hold phase, the sun sensor boresight (on the X-Z plane) points toward the Sun  402 . In the illustrated embodiment, the sun sensor  206  boresight is on the X-Z plane, 49 degrees away from the +X axis, and 41 degrees away from the −Z axis. 
     FIG. 4B  is an illustration of the spacecraft  100  after the boom  114 B is deployed (in one embodiment, the total deployment angle is 210 degrees). After the boom is deployed, the attitude control system supplies commands to the momentum wheels to recover the pre-deployment attitude of the spacecraft  100 . The spacecraft  100  is then reoriented to the desired attitude for reflector deployment, as shown in FIG.  4 C. This attitude is chosen to optimize the reflector  114 A temperature and avoid sun shadowing on the reflector elements while maintaining the Sun  402  in the sun sensor field of view. In one embodiment, the Sun is 6 degrees away from the −Z axis and toward the +Y axis. 
   The reflector bundle is then released allowing it to bloom to full size. After blooming, the reflector is deployed by appropriate commands from the ground or the spacecraft itself, using stepping motors or other devices. 
   Spacecraft Control During Appendage Deployment 
   As described above, it is desirable to hold the spacecraft  100  in a specific orientation during the boom  114 B and/or reflector  114 A deployment. However, environmental torques as well as the boom  114 B and reflector  114 A deployment itself can cause significant attitude perturbations. Such perturbations can be compensated for by (1) use of gyroscopic torques using the momentum wheels  244  and  242 , (2) orienting the solar wings  104  to apply differential torque to the spacecraft, or (3) by use of both techniques. These techniques are described below. 
   A dynamic equation for the deployment of the appendage  114  is 
                 T   og     +     T   sp     +     Δ   ⁢           ⁢     T   st         =         ⅆ     ⅆ   t       ⁢     (       I   ⁢           ⁢   ω     +   H     )       +     ω   ×     (       I   ⁢           ⁢   ω     +   H     )                 Equation   ⁢           ⁢     (   1   )               
 
wherein T og  and T sp  are environmental (outgassing and nominal solar pressure, respectively) torques, ΔT st  is the controlled solar tacking torque (discussed further below), I is the total inertia of the spacecraft (typically expressed in a matrix as in Equation (2) below), ω is the spacecraft body spin rate (typically expressed as a vector), and H represents the momentum in the reaction wheels  244  and  242  (also typically expressed as a vector).
 
   As discussed above, typically, the spin axis of the spacecraft  100  is along the sun-line (the line from the spacecraft  100  to the Sun  402 ). Typically, however, the spin axis of the spacecraft is not a major axis, and the spacecraft  100  is therefore not passively stable. 
   A desired amount and direction of H is computed to correct the environmental torques T og  and T sp , using Equation (1). As described above, certain spacecraft  100  to Sun  402  geometries are desirable (in order to meet thermal and power requirements). Based on a selected spacecraft  100  to Sun  402  geometry, a predicted outgassing T og  and solar torque T sp  are generated. These predicted values are used to compute H. The direction of H is perpendicular to the sun line and to the sum of the environmental torques T og  and T sp . 
   Since the spacecraft  100  is not typically symmetrical about the sun line (represented by off-diagonal terms in the matrix of Equation (2) below), the spacecraft  100  will wobble when rotated about the sun line, In one embodiment, the momentum H is also selected to balance this spacecraft  100  wobble, so that when the spacecraft  100  is rotated, its dynamics are like those of a symmetrical object being rotated along a major axis. 
   The wheel momentum can be resolved into two components: super-spin momentum, H s , and transverse momentum, H t . The wheel momentum along the spin axis is the super-spin momentum, H s . H s  is selected such that the effective inertia ratio σ eff &gt;1, thus rendering the spacecraft  100  passively stable during the deployment. 
   Assuming the spacecraft  100  is spinning along the axis- 3  with spacecraft inertia: 
             [           I   11         0       0           0         I   22         0           0       0         I   33           ]           Equation   ⁢           ⁢     (   2   )               
 
and spin rate ω s  (ω s  is the spin rate of the spacecraft along the spin axis  3 ), with the added super-spin wheel momentum H s , the equivalent dynamic inertia along the spin axis, I s , can be computed by Equation (3) below. 
               I   s     =       I   33     +       H   s       ω   s                 Equation   ⁢           ⁢     (   3   )               
 
   The effective inertia ratio can be computed by: 
               σ   eff     =     1   +         (         I   s       I   22       -   1     )     ⁢     (         I   s       I   11       -   1     )                   Equation   ⁢           ⁢     (   4   )               
 
   Note that in a case where the spacecraft is symmetric and spinning about an axis of symmetry,
 
 I   11   =I   22   =I   transverse   =I   t   Equation (5)
 
and
 
 I   33   =I   spinline   Equation (6)
 
   Substituting Equation (5) into Equation (4) yields Equation (7) below. 
               σ   eff     =       1   +         (         I   s       I   t       -   1     )     ⁢     (         I   s       I   t       -   1     )           =       I   s       I   t                 Equation   ⁢           ⁢     (   7   )               
 
   Further substitution of Equation (3) into Equation (7) results in Equation (8), which represents a simplified representation of the effective inertia ratio for the case of a symmetric spacecraft spinning about an axis of symmetry. 
               σ   eff     =           I   spinline     +       H   s       ω   s           I   t       =         I   spinline     +                   M   _     spin                 ω   s           I   t                 Equation   ⁢           ⁢     (   8   )               
 
   The wheel momentum placed in the transverse plane, H t  is used to cancel the dynamic imbalance torque and the predicted mean out-gassing torque and mean solar torque, i.e.,
 
 T   og   +T   sp   −ω×Iω=ω×H   t   Equation (9)
 
   A transverse momentum, H t , is computed so as to counter external torque based on equation (9) above, and a superspin momentum, H s , is selected that maximizes the effective inertia ratios and does not saturate the plurality of momentum wheels  242 , and  244  (although two momentum wheels are shown in  FIG. 2 , the spacecraft  100  typically includes at least 3 momentum wheels and, for purposes of redundancy, an additional momentum wheel). 
   An optimal design for momentum wheel dynamic compensation for deployment of the appendage requires appropriate selection of both H s  and H t . In this passive deployment, the coning angle due to residual transverse torque (the transverse torque after the reduction by wheel transverse momentum) is proportional to the residual transverse torque, and inversely proportional to the total spacecraft momentum. This favors higher projected H s  in the spin axis and higher spin rate ω s . The coning angle is also inversely proportional to the nutation frequency which favors higher σ eff  (i.e., lower spin rate for fixed H s ) and favors higher ω s . Therefore, there exists an optimal selection of the spin rate, ω s , given the allocated super-spin H s  from the momentum wheels  242  and  244 . For the special case of a spacecraft  100  which is symmetric with respect to the spin axis, it can be shown, to the first order approximation, that the resultant optimal spin rate, ω s     —     opt , and optimal effective inertia ratio, σ eff     —     opt , are 
               ω   s_opt     =         H   s       2   ⁢     I   spinline         ⁢       (         2   ⁢     I   spinline         I   t       -   1     )       (     1   -       I   spinline       I   t         )                 Equation   ⁢           ⁢     (   10   )                   σ   eff_opt     =     1     2   -       I   t       I   spinline                   Equation   ⁢           ⁢     (   11   )               
 
   Note that the above optimal values only exist when I spinline &lt;I t &lt;2I spinline . For an asymmetric spacecraft, the wobble angle increases as spin rate increases (while σ eff  is reduced). The optimal spin rate ω s     —     opt  and the optimal effective inertia ratio, σ eff     —     opt  can be computed by a numerical optimization algorithm. 
     FIG. 5  is a flow chart presenting illustrative process steps that can be used to reduce the effect of appendage deployment perturbations, as described above. A prediction of the environmental torque that the spacecraft  100  will be subjected to during deployment of the appendage  114  is computed, as shown in block  502 . This environmental torque, and other perturbations caused by the deployment of the appendage  114  will be compensated for by orienting the solar wings  104 , by storing sufficient momentum in the momentum wheel(s)  242 ,  244 , or a combination of both techniques. In one embodiment, the momentum wheel(s)  242 ,  244  are used to stabilize the spacecraft  100  during the appendage  114  deployment, and the solar wings  104  are oriented to prevent saturation of the momentum wheels  242 ,  244  while this is accomplished. 
   A direction and amount of momentum to be added to the spacecraft {right arrow over (M)} is then determined, as shown in block  504 . 
     FIG. 6  is a flow chart presenting exemplary method steps that can be used to determine the direction and amount of momentum to be added to the spacecraft {right arrow over (M)}. In blocks  602  and  604 , pre-deployment spacecraft momentum {right arrow over (M)} b , and a desired post-appendage deployment momentum {right arrow over (M)} a  is determined. Momentum changes during deployment, {right arrow over (M)} l  are determined, as shown in block  606 . Finally, desired momentum to be added to the spacecraft  100  is determined from {right arrow over (M)} a , {right arrow over (M)} b , and {right arrow over (M)} l  according to Equations (12)-(15) below:
   {right arrow over (M)}={right arrow over (M)}   a   −{right arrow over (M)}   b   −{right arrow over (M)}   l   Equation (12)   {right arrow over (M)}={right arrow over (M)}   spin   +{right arrow over (M)}   transverse   Equation (13)   {right arrow over (M)}   spin   ={right arrow over (M)}·{right arrow over (u)}=H   s   Equation (14)   {right arrow over (M)}   transverse   ={right arrow over (M)}·{right arrow over (v)}=H   t   Equation (15) 
   wherein {right arrow over (u)} is the unit vector of the spin axis (i.e. {right arrow over (M)} spin  is the projection of the total momentum into the spin axis, or the momentum component in the direction of the spin axis), and {right arrow over (v)} is the unit vector of the transverse axis. From these values, the amount of and direction of momentum to add to the spacecraft  100  is determined, as shown in block  608 . 
   The pre-deployment spacecraft momentum {right arrow over (M)} b  can be computed from known or measured spacecraft parameters (e.g., rotational inertia) and data obtained from the attitude control sensors  310  and other navigational and attitude sensors such as those shown in FIG.  2 . The desired post-deployment spacecraft momentum {right arrow over (M)} a  can be determined based on desired dynamic condition during and post the deployment, and can be computed using spacecraft spin rate, wheel superspin, wheel dynamic balancing, and other spacecraft paramters. Similarly, the momentum change during deployment {right arrow over (M)} l  can be estimated from spacecraft geometry and mass properties (including environmental torque sources such as ambient magnetic field, atmospheric drag, gravity gradient, and radiation pressure), navigational data, the deployment procedures and predicted spacecraft  100  responses to commands to perform such procedures. 
     FIG. 7  is a flow chart further illustrating the computation of the momentum {right arrow over (M)} to add to the spacecraft  100  before deploying the appendage  114 . As described in Equation (13) above, the total momentum {right arrow over (M)} to be added to the spacecraft includes a transverse momentum, H t  (which is computed so as to counter external torque based on equation (9) above), and a superspin momentum, H s  (which is selected to render the spacecraft passively stable during deployment). These values are computed as show in blocks  702  and  704 . Further, an value for the spin rate of the spacecraft ω s  that optimally minimizes the resulting coning angle of the spacecraft can be established using the relationships described in Equations (10) and (11). 
   In one embodiment, the process of determining the amount and direction of momentum to add to the spacecraft is performed by the deployment module  302  of SCP  202 , however, this process can be performed in other modules of the SCP  202 , in another spacecraft processor, or in dedicated circuitry. Typically, the momentum is added to the spacecraft  100  by adding momentum to the momentum wheels  242 ,  244 . This can be accomplished by spinning up the momentum wheels  242 ,  244 , while operating ACS thrusters  236  on the spacecraft  100 . 
   Returning to  FIG. 5 , momentum is now stored, as shown in block  506 . This is accomplished by providing one or more commands from the DM  302  one or more of the attitude control actuators  306 , which include the momentum wheels  242  and  244 . In one embodiment, the momentum is stored in the momentum wheels  242  and  244 . The amount of momentum stored in each particular wheel is determined by the magnitude and direction of {right arrow over (M)} the orientation and inertia of each respective momentum wheel. 
   To stabilize the spacecraft  100  and to permit the storage of momentum in the momentum wheels  242 , selected attitude control thrusters may be activated. Because the large torque from the attitude control thrusters may negatively affect the deployment of the appendage, the momentum wheel spin up is typically performed before any portion of the appendage is deployed. However, this need not be the case. In the exemplary illustration presented in  FIGS. 4A-4C , for example, it is possible to activate the attitude control thrusters when the spacecraft has boom deployed as shown in FIG.  4 A and/or  FIG. 4B , as the torque may not substantially effect the appendage before the reflector  114  is deployed. 
   Once the desired amount of momentum is stored in the momentum wheels, the appendage  114  is deployed, as shown in block  510 . While the appendage  114  is being deployed, the attitude of the spacecraft  100  is controlled by the reaction wheels, further more, the attitude and/or momentum of the spacecraft  100  is controlled by orienting the solar wings to apply differential solar torque to the spacecraft  100 , as shown in block  508 . This process continues until the deployment of the appendage is completed, as shown in block  512 . 
     FIG. 8  is a diagram presenting illustrative method steps that can be used to control the attitude of the spacecraft and/or add or subtract momentum from spacecraft  100 . 
   As shown in block  802 , at least one of the solar wings  104  is oriented to apply differential torque to control the spin momentum H s  along the spin axis of the spacecraft  100 . This torque is known as windmill torque. 
   As shown in block  804 , at least one of the solar wings  104  is also oriented to apply differential torque to control the transverse momentum H t . Generally, these operations are performed concurrently, but this need not be the case. 
     FIGS. 9A and 9B  are diagrams showing techniques for applying differential solar torque to the spacecraft  100 .  FIG. 9A  depicts “solar tacking.” Solar tacking is implemented by rotating one or more of the solar wings  104  about an axis perpendicular to the longitudinal axis of the solar wing  104  such as the X (roll) axis of the spacecraft (in the X-Z plane). As shown in  FIG. 9A , when solar wing  104 S is moved from its nominal position at  104 S′ to position  104 S, there is reduced solar pressure on this side of the spacecraft  100 , while the other side of the spacecraft receives the nominal amount of solar pressure. As a result, there is a differential “overturning” torque, and the spacecraft will rotate (roll) in the direction of the arrow shown in FIG.  9 A. These solar wing  104  manipulations are useful in countering the environmental torques described above. 
     FIG. 9B  depicts an implementation of “windmilling,” in which the spacecraft rotates about the Z (yaw) axis. Here, rotation of the solar wings  104  about the Y axis of the spacecraft  100  (and the longitudinal axis of the wings) causes differential solar torques on each of the solar wings, causing the spacecraft to yaw. If each of the solar wings,  104 N or  104 S, are rotated in opposite directions from its nominal position  104 N′ or  104 S′, substantially equal and opposite solar pressure on each solar wings is theoretically imparted to the spacecraft  100 , causing the spacecraft to increase or decrease its momentum along the sun line (extending from the spacecraft to the Sun. These solar wing  104  manipulations can be used to control the spin momentum H s  along the spin axis of the spacecraft. 
   The amount of differential solar torque that can be applied to the spacecraft  100  by the solar wings  104  is a function of parameters (including solar wing  104  and spacecraft  100  shape and reflectivity as well as the relative orientation of the Sun) which can change over time and be difficult to predict. In one embodiment of the present invention, the amount of differential solar torque resulting from solar wing  104  manipulations is estimated. This can be accomplished by estimating spacecraft angular accelerations using spacecraft angular rate measurements. These angular acceleration estimates are used to more accurately determine the relationship between solar wing  104  manipulations and the resulting differential solar torque. Further solar wing  104  orientation commands can then be suitably adjusted so that the desired differential solar torque is provided. 
   Conclusion 
   This concludes the description of the preferred embodiments of the present invention. In summary, the present invention describes a method, apparatus, and memory for controlling the attitude of a spacecraft while deploying an appendage. 
   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. For example, while the foregoing discloses performing many operations using the SCP  202 , such operations can be performed by or with the help of auxiliary processors or special purpose hardware modules. Further, such operations can be performed by processors disposed on another satellite or on the ground, with the required data uploaded to the satellite  100  when required. 
   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.