Abstract:
A method and apparatus for controlling a plurality of solar panels of a spacecraft is described. The method comprises the steps of providing a first step command to a first solar panel, and providing a second step command to a second solar panel at a time of a transient zero-crossing of a dynamic response of the spacecraft body to the first step command, wherein the second solar panel is disposed on an opposite side of the spacecraft from the first solar panel. The apparatus comprises a processor, a first solar panel driver, communicatively coupled to the processor, for providing a first step command to a first solar panel, and a second solar panel driver, communicatively coupled to the processor, for providing a second step command to a second solar panel at a time of a transient zero-crossing of a dynamic response of the spacecraft body to the first step command.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     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. 10/386,796, entitled “METHOD AND APPARATUS FOR STEPPING SPACECRAFT MECHANISMS AT LOW DISTURBANCE RATES,” by Ketao Liu, filed on Mar. 12, 2003, attorney&#39;s docket number PD-02-0237; and     application Ser. No. 10/348,663, entitled “METHOD AND APPARATUS FOR MINIMIZING SOLAR ARRAY SUN TRACKING DISTURBANCE VIA NORTH AND SOUTH SOLAR ARRAY WING CANCELLATION,” by Ketao Liu, filed Mar. 12, 2003, attorney&#39;s docket number PD-02-0470.       
 
     
    
     BACKGROUND OF THE INVENTION  
       [0004]     1. Field of the Invention  
         [0005]     The present invention relates generally to systems and methods for controlling spacecraft or satellites, and in particular to a system and method for stepping spacecraft mechanisms to minimize disturbances generated by the stepping mechanism.  
         [0006]     2. Description of the Related Art  
         [0007]     Three-axis stabilized spacecraft or satellites often include mechanisms for manipulating appendages. These mechanisms include mechanisms that rotate the antenna reflectors to specific targets, gimbals that scan a payload image frame across a specific area of the Earth&#39;s surface, and solar array drivers that step solar arrays to track the Sun. Typically, such mechanisms use simple, reliable stepper motors coupled to the spacecraft component (payloads or solar arrays) via gear-driven transmissions. Stepper motors are desirable because they are relatively simple to control, reliable, lightweight and well adapted to continuous use. The stepper motors and transmissions are used to rotate the solar array along its longitudinal axis to track the sun while the spacecraft orbits about the Earth. The rate that the solar array must be rotated is a function of the satellite orbital period. At geosynchronous orbit, this rate is about 0.004 degrees per second.  
         [0008]     The use of a stepper motor in spacecraft with highly flexible structural components such as large deployable payload booms, antenna reflectors, and solar arrays may potentially excite some structural modes of these components and generate significant oscillation disturbances in the spacecraft itself. This disturbance can degrade the spacecraft pointing, cause excessive activity of the spacecraft control actuators, and make autonomous spacecraft momentum dumping difficult. The induced oscillation is particularly critical in spacecraft where absolute platform stability is desirable. Vibrations can cause deterioration of any inertia-sensitive operations of a spacecraft.  
         [0009]     This disturbance problem can be alleviated by a number of techniques. One technique is to employ high bandwidth control loops to mitigate the impact of this disturbance to the spacecraft pointing. This technique, however, has significant limitations. For many spacecraft, the structural modes that are excited by the stepping mechanisms are outside of the spacecraft control bandwidth. Consequently, these high-bandwidth control loops have only very limited effects on the disturbance. Further extension of the bandwidth of the control loops to include these structural modes will very often result in control loop stability problems. Furthermore, high-bandwidth control also unnecessarily increases actuator operation, which can increase wear and result in excess energy consumption.  
         [0010]     Another technique for mining the solar array drive stepping disturbance is disclosed in U.S. Pat. No. 4,843,294, entitled “Solar Array Stepping to Minimize Array Excitation,” issued Jun. 27, 1989 to Bhat et al, which is hereby incorporated by reference herein. In this reference, mechanical oscillations of a mechanism containing a stepper motor, such as a solar array powered spacecraft, are reduced and minimized by the execution of step movements in pairs of steps. The period between steps is equal to one-half of the period of torsional oscillation of the mechanism. While this method can reduce structural disturbances, it is not very effective when the mechanism has significant backlash and stiction. This is because the backlash and stiction can significantly interrupt the two-step pattern of this method.  
         [0011]     Another technique is described in co-pending and commonly-assigned patent application Ser. No. 10/386,796, entitled “METHOD AND APPARATUS FOR STEPPING SPACECRAFT MECHANISMS AT LOW DISTURBANCE RATES,” by Ketao Liu, filed on Mar. 12, 2003, attorney&#39;s docket number PD-02-0237, in which transients due to the interaction between appendage stepping and resonances are reduced by deadbeating at a half resonance cycle between the North and South wings. This technique, however, is subject to frequency sensitivities and uncertainties, and cannot be implemented in all existing spacecraft.  
         [0012]     Still another technique is described in co-pending and commonly-assigned patent application Ser. No. 10/348,663, entitled “METHOD AND APPARATUS FOR MINIMIZING SOLAR ARRAY SUN TRACKING DISTURBANCE VIA NORTH AND SOUTH SOLAR ARRAY WING CANCELLATION,” by Ketao Liu, filed Mar. 12, 2003, attorney&#39;s docket number PD-02-0470. The technique, however, is a more cumbersome implementation.  
         [0013]     There is therefore a need for a system and method for minimizing disturbances in stepper-motor driven mechanisms that are more robust to mechanism backlash and stiction. The present invention satisfies that need.  
       SUMMARY OF THE INVENTION  
       [0014]     The present invention is a method and apparatus for controlling a plurality of solar panels of a spacecraft. The method comprises the steps of providing a first step command to a first solar panel, and providing a second step command to a second solar panel at a time of a transient zero-crossing of a dynamic response of the spacecraft to the first step command, wherein the second solar panel is disposed on an opposite side of the spacecraft from the first solar panel. The apparatus comprises a processor, a first solar panel driver, communicatively coupled to the processor, for providing a first step command to a first solar panel, and a second solar panel driver, communicatively coupled to the processor, for providing a second step command to a second solar panel at a time of a transient zero-crossing of a dynamic response of the first solar panel to the first step command. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0015]     Referring now to the drawings in which like reference numbers represent corresponding parts throughout:  
         [0016]      FIG. 1  is a diagram depicting a typical spacecraft;  
         [0017]      FIG. 2  is a block diagram depicting a satellite control system;  
         [0018]      FIG. 3  is a block diagram depicting a stepper motor mechanism usable to manipulate a solar panel to track the Sun;  
         [0019]      FIG. 4  is a flow chart depicting exemplary process steps used to reduce step-induced transients;  
         [0020]      FIG. 5  is a diagram showing the transient response of a solar panel to a step command;  
         [0021]      FIG. 6A  is a diagram showing a solar wing position estimation and control system;  
         [0022]      FIG. 6B  is a diagram showing a view of the spacecraft from above;  
         [0023]      FIGS. 7 and 8  are diagrams illustrating how transient cancellation bias may be implemented by appropriate quantization of the solar panel commands. 
     
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS  
       [0024]     In the following description of the preferred embodiment, reference is made to the accompanying drawings, which form a part hereof, and in which is shown byway of illustration a specific embodiment in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention.  
         [0025]      FIG. 1  depicts a three-axis stabilized satellite or spacecraft  100 . The spacecraft  100  is preferably situated in a stationary orbit about the Earth. The spacecraft  100  has a main body  102 , a pair of solar panels or wings  104 N and  104 S (hereinafter referred to collectively as solar panel(s) or wing(s)  104 ), a pair of high gain narrow beam antennas and their reflectors  106 , and a scanning payload  108  which can be used to scan a specific area of Earth surface. The spacecraft  100  may also include one or more sensors  110  to measure the attitude of the spacecraft  100 . These sensors  110  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  104  in  FIG. 1  are referred to by the numerals  104 N and  104 S for the “North” and “South” solar panels, respectively.  
         [0026]     The three axes of the spacecraft  100  are shown in  FIG. 1 . 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. The antenna  108  points to the Earth along the yaw axis Z.  
         [0027]     One or more of the solar panels  104  can be rotated about the pitch axis and can be tilted towards the plane formed by the roll and yaw axes. This is depicted in  FIG. 1  as tilted position  104 N′.  
         [0028]      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  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 post ejection sequencing could include initializing to ascent mode and thruster active nutation control (TAN %. 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 station keeping control could include auto mode sequencing, gyro calibration, station keeping 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&#39; 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.  
         [0029]     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 .  
         [0030]     The SCP  202  generates control signal commands  220  which are directed to a command decoder unit  222 . The command decoder unit  222  operates the load shedding and battery charging systems  224 . The command decoder unit  222  also sends signals to the magnetic torque control unit (MTCU)  226  and the torque coil  228 .  
         [0031]     The SCP  202  also sends control commands  230  to the thruster valve driver unit  232  which in turn controls the liquid apogee motor (LAM) thrusters  234  and the attitude control thrusters  236 .  
         [0032]     Generally, the spacecraft  100  may use thrusters, momentum/reaction wheels, or a combination thereof to perform spacecraft attitude control.  
         [0033]     Wheel torque commands  262  are generated by the SCP  202  and are communicated to the wheel drive speed electronics  238 . These effect changes in the wheel speeds for wheels in reaction wheel assembly  242 . The speed of the wheels is also measured and fed back to the SCP  202  by feedback control signal  264 .  
         [0034]     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 .  
         [0035]     The wheel drive electronics  238  receive signals from the SCP  202  and control the rotational speed of the reaction wheels.  
         [0036]     Other spacecraft could employ momentum wheels, external torquers, chemical or electric thrusters, magnetic torquers, solar pressure, etc. This includes momentum bias spacecraft that attempt to maintain the spacecraft body fixed and steer payload elements with payload gimbals.  
         [0037]     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.  
         [0038]     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 control 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 present 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.  
         [0039]      FIG. 3  is a block diagram of a appendage driver  300  that can be used to implement the North solar wing driver (SWD)  246  or the South SWD  248 , the East RPM  250  or the West RPM  252 , or the payload scanning gimbals (not shown). An appendage  324  such as the antenna reflector  106 E or  106 W, solar panel  104 N or  104 S, or scan payload  108  (hereinafter alternately referred to as antenna reflector(s)  106  and solar panel(s)  104 ) is driven by the transmission  324 , which in turn is coupled to a stepper motor  322 . The transmission  324  converts the rotational output of the stepper motor  322  into appropriate appendage motion. The stepper motor  322  is provided with a suitable power source such as the spacecraft power subsystem to supply driving force to the gear assembly  324 . Power from the spacecraft power subsystem may be conditioned by power conditioner  320 . A stepper motor driver  318  provides the input signal by appropriate signal lines  326  to the stepper motor  322 . The stepper motor driver  318  is controlled by the SCP  202  from which necessary step timing and step rate can be derived for the function applied to the stepper motor  322 . The driver  318  may include the North SWD  246 , the South SWD  248 , the East RPM  250 , the West RPM  252  or the payload scanning gimbals.  
         [0040]     During flight, both the North solar wing  104 N and the south solar wing  104 S are rotated about their longitudinal axes to direct the planar surface of the solar collectors in the direction of the Sun. Typically, the north solar wing  104 N and the south solar wing  104  are stepped simultaneously. This is not problematic if the solar wings  104  are diametrically opposed from one another relative to the spacecraft  100  center of mass and are not tilted, and the solar wing  104  pitch inertia is small.  
         [0041]     In some applications, however, in order to derive maximum energy from the Sun, the solar wings  104  must also be tilted away from the pitch axis and toward the plane defined by the roll and yaw axes by as much as 23 degrees plus orbit inclination. When the solar wings  104  are tilted, there is a relatively large inertia in the pitch axis, and a relatively large inertia coupling from the pitch axis to the roll and yaw axes. In these circumstances, solar wing drive stepping in the pitch axis causes a relatively large momentum exchange between the solar wing  104  and the spacecraft main body  102 . This induces larger spacecraft pointing transients and larger wheel torque dithering in the roll, pitch, and yaw axes. Large pointing transients degrade payload performance, and larger wheel torques reduce the reliability of the momentum wheels.  
         [0042]     These problems can be alleviated by spreading the stepping of the solar wings such that the induced transients are temporally spread and do not sum together. Further, an optimal time lag between the solar wing stepping commands can be derived such that transients due to the stepping of one of the solar wings  104  cancels the transients induced by the stepping of the other solar wings. The optimal time lag depends on the time constant of the spacecraft  100  attitude control system using the reaction wheels and the time constant of the appendage driver  300  and the appendage  324 . This is equivalent to the time the spacecraft attitude control system brings the transient in the reverse direction (e.g., the transient zero crossing).  
         [0043]     The relative stepping time lag between a first solar wing (e.g. the North solar wing  104 N) and a second solar wing ( 104 S) can be implemented by applying a bias angle to the angular command θ cmd  to the solar wing driver  246 ,  248 , or by applying a similar bias angle to the measured angular position of the solar wing  104 . This bias angle, θ bias , can be computed as θ bias =Δt·ω str , wherein θ bias  is the transient cancellation bias angle, Δt is the time lag of the transient zero crossing and ω str  is the nominal Sun tracking rate.  
         [0044]      FIG. 4  is a flow chart depicting exemplary process steps used to reduce step-induced transients of the spacecraft body  102 . A first step command is provided to a first solar panel  104  such as solar panel  104 N, as shown in block  402 .  
         [0045]      FIG. 5  is a diagram showing the transient response of the first solar panel  104 N to the step command. The first step command  502  is provided at time t=t 0 , and the dynamic response of the solar panel  104 N is indicated as trace  504 . The dynamic response includes a steady state value (nominally equal to the step command N (1)   swd ) and a transient response.  
         [0046]     Because the solar panel  104 N is coupled to the spacecraft body  102  and may be tilted, the dynamic response of the solar panel  104 N induces spacecraft  100  motion as well. This motion, the spacecraft body  102  transient response,  506  includes one or more zero crossings such as zero crossings  508 A and  508 B.  
         [0047]     Returning to  FIG. 4 , a second step command  510  is provided to a second solar panel such as solar panel  104 S. The step command is provided at a time t=t 0 +Δt that the spacecraft body  102  transient response  506  to the first step command N (1)   swd  crosses zero at location  508 A. This is shown in block  404 , and results in a dynamic response of the second solar panel  104 S as shown in trace  512 .  
         [0048]      FIG. 6A  is a block diagram illustrating a solar wing  104  position estimation and control system  600  that can be used to implement the foregoing technique. In  FIG. 6A : 
        θ (i)w     —     b   est    614  is an estimated solar wing to spacecraft body  102  angle at each step (i);     θ (i)b     —     s   est    616  is an estimated Sun to spacecraft body  102  angle at each step (i);     θ (i)   cmd    602  is an existing commanded bias angle, for example, for environmental torque balancing, at each step (i) (this is not a final command sent to the solar wing drive);     C w    608  is an angular error to angular rate update gain;     ω str    610  is a solar wing Sun tracking rate; and     t SOL  is a time period between instantiations (i) of the estimation and control system  600 ; and     z is the inherent variable of the Z-transform operator.          
         [0056]      FIG. 6B  is a view of the spacecraft  100  from above, illustrating solar wing to spacecraft body and the Sun to spacecraft body angles (θ w     —     b , and θ b     —     s , respectively) in the roll/yaw plane.  
         [0057]     Returning to  FIG. 6A , an (existing) commanded wing bias angle θ cmd    602  is quantized by quantizer  603 . As described further below, in one embodiment, the commanded wing bias angle θ cmd    602  is quantized to a least significant bit equal to the step size, and to the value of the nearest step. The quantized commanded wing bias angle is compared to an estimated position of the solar wing relative to the Sun, or the wing to Sun angle θ (i)w     —     s   est    618  to determine the wing position error θ (i)w     —     s   err . The wing to Sun angle θ (i)w     —     s   est    618  is the difference between an estimated Sun to spacecraft  100  body angle θ (i)b     —     s   est    616  and the desired wing to body angle θ (i)w     —     b   est    614 .  
         [0058]     The estimated Sun to spacecraft  100  body angle θ (i)b     —     s   est    616  is typically determined by a separate subsystem or algorithm in the SCP  202 , and can be computed by measuring the position of the Sun (with a Sun sensor  206 , for example), by computing the Sun position from ephemeris data, or a combination of both techniques.  
         [0059]     The wing position error θ (i)   err  is applied to a position gain C w    608 . This value is added to the solar wing Sun tracking rate ω str    610 , which is a nominal (error free) solar wing  104  angular rate required to keep the solar wing  104  directed at the Sun. The value of ω str    610  is typically computed from the orbital rate of the spacecraft  100 . The desired angular rate, ω (i)   cmd , is converted to the desired solar wing angular position θ (i)w     —     b   est    614  by integrating through the step interval t SOL , as shown in block  612 . The desired solar wing angular position θ (i)w     —     b   est    614  is quantized by second quantizer  615 .  
         [0060]     A current solar wing angular position relative to the spacecraft body  102 , S (i)   SWD    620  is subtracted from the quantized desired solar wing angular position θ (i)w     —     b   est    614  to arrive at the step command provided to the solar wing drives  246 ,  248 . The current solar wing angular position can be measured by appropriate sensors in the transmission  324 , on a shaft coupling the solar wing  104  to the transmission  324 , or may be determined by bookeeping the number of step commands provided to the drive  246 ,  248 .  
         [0061]     Since the angular position of the solar wing is commanded in term of a plurality of steps (i), the equations used to determine the number of steps to be taken for both the North solar wing  104 N and the South solar wing  104 S are as follows:  
               θ       (   i   )     ⁢   w_s     est     =       θ       (   i   )     ⁢   w_b     est     -     θ       (   i   )     ⁢   b_s     est               Equation   ⁢           ⁢     (   1   )                   θ     (   i   )     err     =       θ     (   i   )     cmd     -     θ       (   i   )     ⁢   w_s     est               Equation   ⁢           ⁢     (   2   )                   ω     (   i   )     cmd     =         C   w     ⁢     θ     (   i   )     err       +     ω   str               Equation   ⁢           ⁢     (   3   )                   θ     (   i   )     est     =       θ     (     i   -   1     )     est     +       ω     (   i   )     cmd     ⁢     t   SOL                 Equation   ⁢           ⁢     (   4   )                   N     (   i   )     swd     =       N     (     i   -   1     )     swd     +       ω     (   i   )     cmd     ⁢       t   SOL       Δ   ⁢           ⁢     θ   swd           -     S     (   i   )     swd               Equation   ⁢           ⁢     (   5   )                                           
 
 where N (i)   swd  is the number of steps the solar wing  104  must take to achieve the desired solar wing angle (e.g. the steps sent to the solar wing drive); Δθ swd  is the angular displacement of the solar wing per step (i), in radians. 
 
         [0062]     Both solar wings  104 N and  104 S use the same Sun position (whether obtained by Sun sensor measurement or ephemeris prediction) to determine the spacecraft body to sun angle θ (i)b     —     s   est    616 . This ordinarily results in SWD  246 ,  248  commands that step each solar wing  104 N and  104 S at the same time. This technique is referred to as synched stepping. As described above, synched stepping, however, can result in undesirable momentum exchanges between the solar wings  104  and the spacecraft body  102 .  
         [0063]     To alleviate this problem, the step commands applied to the second solar wing (e.g. solar wing  104 S) are time-delayed by a value Δt so that the spacecraft body  102  transients that result from stepping the first solar wing (e.g. solar wing  104 N) are effectively canceled by spacecraft body  102  transients arising from appropriately timed stepping of the second solar wing  104 S. This time delay can be implemented by adding a transient cancellation bias angle θ bias    604  to the commanded wing bias angle θ cmd  for second solar wing  104 , as shown in  FIG. 6A .  
         [0064]      FIGS. 7 and 8  are diagrams illustrating how the requisite transient cancellation bias angle θ bias    604  can be implemented by appropriately quantizing the solar panel bias commands. A first wing bias angle command θ cmd    602  is computed (for example, for environmental torque balance), as shown in block  702 . The wing bias angle command θ cmd    602  for the first solar panel  104 N is then quantized to the nearest step such that the least significant bit (LSB) equals the SWD  246 , 248  step size Δθ swd . This is shown in block  704 . The computed wing bias angle command θ cmd    602  is quantized to the nearest steps  803  with the LSB equal to the step size Δθ swd . This quantized first solar panel bias command is provided to the solar wing position estimation and control system  600 , as shown in block  706 . The solar wing position estimation and control system  600  computes a stepping command θ (i)w     —     b   est  for the first solar wing  104 N. This plot is shown in  FIG. 8  as plot  806 .  
         [0065]     A second wing bias angle command θ cmd    602  for the second solar wing  104 S is computed, as shown in block  508  of  FIG. 7 . The second wing bias angle command θ cmd    602  is quantized by the quantizer  603 , as shown in block  710 . The quantized second solar panel command is then biased by θ bias  to create a biased second wing bias angle command, as shown in block  712 . The second solar panel bias θ bias  can be determined from terrestrially based processors simulating the dynamic response of the spacecraft body to the first step command, by terrestrially based testing of the dynamic response of the spacecraft body to the first step command, or can be determined by estimations derived by the SCP or by space-based testing.  
         [0066]     The biased second wing bias angle command is provided to the solar wing position estimation and control system  600  as shown in block  714 . The final solar wing stepping command N (i)   SWD  is computed by the solar wing position estimation and control system  600 . The resulting stepping command θ (i)w     —     b   est  for the second solar wing  104 S is shown as plot  806  shown in  FIG. 8 . Note that the angular bias θ bias  implements a time delay Δt between the first solar wing  104 N command and the second solar wing  104 S command. This time delay Δt is chosen to allow the spacecraft body  102  transients induced by stepping the second solar wing  104 S to cancel those induced by stepping the first solar wing  104 N. Although the foregoing has been described with respect to an embodiment in which the transient cancellation bias angle is applied to the second solar panel command in order to implement the time delay θt, it is noted that the present invention may also be implemented by adding a suitable bias anywhere in the estimation and control system  600  illustrated in  FIG. 6A . For example, similar results can be obtained if the transient cancellation bias angle θ bias  is added to the solar wing  104 S position (e.g. S (i)   SWD ) instead of θ (i)   cmd . Suitable biases may also be added elsewhere in the estimation and control loop with similar results (e.g. θ (i)b     —     s   est , for example).  
       Conclusion  
       [0067]     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 form the spirit and scope of the invention, the invention resides in the claims hereinafter appended.