Combined stationkeeping and momentum management

[0002] The present invention relates generally to spacecraft attitude control and momentum management, and specifically to a system and method of simultaneously performing north-south stationkeeping maneuvers and 3-axis momentum management.

[0003] It is important, in a geosynchronous communication spacecraft, to control the spacecraft's attitude in order to orient communication hardware relative to a planetary location. It is also important for momentum stored in a momentum wheel stabilization system of the spacecraft to be periodically unloaded or desaturated. Mechanisms to accomplish attitude correction and momentum desaturation ensure the geosynchronous spacecraft is kept on-station.

[0004] Several mechanisms are used by orbiting spacecraft for stationkeeping, attitude control and momentum desaturation. Typically, bi-propellant chemical thrusters are used to take a spacecraft from a transfer orbit to a synchronous orbit. The bi-propellant thrusters could also be used for stationkeeping, but such thrusters produce relatively powerful disturbance torques due to thruster misalignment, center of mass offsets, and plume impingements. As a result, the bi-propellant thrusters are not preferred for the fine adjustments which are typically made during stationkeeping maneuvers, attitude control, and momentum desaturation.

[0005] Alternatively, a spacecraft may use magnetic torquers for attitude control and momentum desaturation. In operation, magnetic torquers alter a spacecraft's attitude by applying the force of an electric coil's dipole moment with respect to the earth's magnetic field. However, magnetic torquers are not preferred for stationkeeping for they produce torque without lateral force. There are other disadvantages associated with the use of magnetic torquers in orbiting spacecraft. By example, magnetic torquers are heavy, provide little control power, and cannot be used to control the pitch attitude of the spacecraft due to the alignment of the dipole moment section with the magnetic field of the earth.

[0006] Conventional orbiting spacecraft may employ additional techniques for stationkeeping, attitude control, and momentum desaturation. The additional techniques include, for example, the use of ion propulsion thrusters. In fact, ion propulsion thrusters are particularly desirable for north-south stationkeeping on spacecraft with long mission lifetimes. However, a disadvantage of ion propulsion thrusters is their weight. Also, as with bi-propellant thrusters, the ion propulsion thrusters may produce disturbance torques on the spacecraft resulting in undesirable attitude motion.

[0007] It can be appreciated that in addition to the undesirable attitude motion caused by the conventional stationkeeping, attitude control and momentum desaturation techniques discussed above, undesirable attitude motion may also be caused by solar pressure imparted upon the orbiting spacecraft.

[0008] Momentum wheel stabilization systems are commonly used to counteract sources of disturbance torque. Typically, such stabilization systems include one or more momentum wheels and control loops to sense changes in the spacecraft's attitude. Sensors on the spacecraft may detect deviations on any of three orthogonal spacecraft axes; the x-axis (roll), the y-axis (pitch), or the z-axis (yaw). The control loops determine the required speed of the wheels to absorb pitch and yaw momentum based on the sensed attitude. Commonly, yaw and pitch momentum are absorbed directly by the momentum wheels, while roll momentum is absorbed as a change in yaw body angle which results from cross-coupling dynamics of momentum biased spacecraft. However, the roll momentum can be stored in the wheels if the wheel system design permits.

[0009] As was stated above, momentum stored in the momentum wheels must be periodically relieved, or desaturated. Desaturation is required in order to keep the momentum wheels within a predetermined speed range. Desaturation is typically accomplished by applying an external torque to the spacecraft through chemical propulsion thrusting or magnetic torquing. The external torque is applied to counteract, and thus reduce the stored momentum. However, chemical propulsion thrusting is not preferred for maintaining highly accurate spacecraft attitude during stationkeeping maneuvers because of the inherit disturbance torques introduced. Magnetic torques cannot be produced parallel to the magnetic field vector.

[0010] U.S. Pat. No. 5,349,532 Tilley et al. disclose a system and method to maintain a desired spacecraft attitude and unload accumulated momentum while performing north-south stationkeeping maneuvers. Tilley et al. disclose a device for sensing the spacecraft's position, attitude, and stored wheel momentum, and a device for determining the torques required to produce the desired attitude for the spacecraft and desaturate the momentum wheels. Tilley et al. teach in one embodiment the use of ion propulsion thrusters which are gimballed and throttled to produce the desired torques. Ideally, a spacecraft's stationkeeping and momentum management system would operate so as to maximize fuel efficiency during the mission lifetime, without sacrificing reliability. The inventors have determined that this goal would be realized with a system for simultaneously maintaining the spacecraft's stationkeeping maneuvers and momentum management, which employs a thruster firing cycle wherein thrusters are independently fired at predetermined locations along the spacecraft's orbit.

[0011] It is a first object and advantage of this invention to provide a system and method that simultaneously performs a spacecraft's north-south stationkeeping maneuvers and 3-axis momentum management.

[0012] It is another object and advantage of this invention to provide a system and method that simultaneously performs a spacecraft's north-south stationkeeping maneuvers and 3-axis momentum management, without the use of magnetic torquers, where one of a pair of thrusters is fired at predetermined locations along the spacecraft's orbital path.

[0013] It is another object and advantage of this invention to provide a system and method that simultaneously performs a spacecraft's north-south stationkeeping maneuvers and 3-axis momentum management by independently firing one of a pair of thrusters, thus reducing the spacecraft's power consumption.

[0014] Further objects and advantages of this invention will become more apparent from a consideration of the drawings and ensuing description.

[0015] The foregoing and other problems are overcome and the objects and advantages are realized by methods and methods and apparatus in accordance with embodiments of this invention, wherein simultaneous north-south stationkeeping and 3-axis momentum management is provided for a spacecraft, such as a geosynchronous orbiting spacecraft.

[0016] In the present invention, a method of simultaneous north-south stationkeeping and 3-axis momentum management for a geosynchronous orbiting spacecraft having a first thruster, a second thruster, and at least three momentum wheels mounted on-board the spacecraft is disclosed. In a first-pass of a thruster firing cycle, the method includes steps of: firing the first thruster at a first point along an orbital path of the spacecraft; and firing the second thruster at a second point along the orbital path of the spacecraft, where a thrust vector at the second point is non-parallel with a thrust vector at the first point. In a second-pass of the thruster firing cycle, the method fires the first thruster at a third point along the orbital path of the spacecraft, where a thrust vector at the third point is parallel with and on an opposite side of the orbital path from the thrust vector at the second point. The second-pass further includes firing the second thruster at a fourth point along the orbital path of the spacecraft, where a thrust vector at the fourth point is parallel with and on an opposite side of the orbital path from the thrust vector at the first point. When the first and the second thrusters are fired over the four points described previously on the orbital path, the radial components of a change in velocity produced by firing the first and the second thrusters are cancelled, and angular momentum stored in the at least three momentum wheels is desaturated.

[0017] In one embodiment, the first-pass and the second-pass of the thruster firing cycle are performed over two spacecraft orbits. Additionally, the angular momentum stored in the at least three momentum wheels is desaturated by firing, at a given time, a single thruster to decrease the speed of the at least three momentum wheels along a respective orthogonal spacecraft axis.

[0018] In another embodiment, the method includes initial steps of: defining a point A along an orbital path of the spacecraft where the orbital path crosses a line of nodes coplanar with a z-axis of the spacecraft; and defining a point B along the orbital path of the spacecraft where the orbital path crosses the line of nodes and wherein the point B is oriented 180° from the point A. In the another embodiment a first-pass of a thruster firing cycle includes a step of firing the first thruster at a point A′ along the orbital path of the spacecraft, where the point A′ is a first predetermined angle from the point A. The first-pass also includes a step of firing the second thruster at a point B′ along the orbital path of the spacecraft, where the point B′ is a second predetermined angle from the point B and a second predetermined time of motion from the point A′, and where the point B′ is non-collinear with the point A′ and the earth's center, i.e. a point O.

[0019] In a second-pass of the thruster firing cycle of the another embodiment, the method further includes a step of firing the first thruster at a point A″ along the orbital path of the spacecraft, where the point A″ is a third predetermined angle from the point A and a third predetermined time of motion from the point B′, and where the point A″ is collinear with and on an opposite side of the orbital path from the point B′ and the earth's center at the point O. The second-pass also includes a step of firing the second thruster at a point B″ along the orbital path of the spacecraft, where the point B″ is a fourth predetermined angle from the point B and a fourth predetermined time of motion from the point A″, and where the point B″ is collinear with and on an opposite side of the orbital path from the point A′ and the earth's center at the point O.

[0020] Preferably, values of the first, the second, the third and the fourth predetermined angles and the second, the third and the fourth predetermined times of motion within the thruster firing cycle are programmable. In one embodiment, the programmable values are transmitted to the spacecraft by ground command from a ground station. For example, the first predetermined angle is about 15°, and about 1 hour of orbital motion at geosynchronous altitude, from the point A; the second predetermined angle is about 15° from the point B and the second predetermined time of motion is about 10 hours from the point A′; the third predetermined angle is about 15° from the point A in a direction opposite the point A′ and the third predetermined time of motion is about 12 hours from the point B′; and the fourth predetermined angle is about 15° from the point B in a direction opposite the point B′, and the fourth predetermined time of motion is about 14 hours from the point A″.

[0021] Additionally, the present invention discloses a system for simultaneous north-south stationkeeping and 3-axis momentum management for a geosynchronous orbiting spacecraft that operates in accordance with the foregoing methods. The spacecraft has a center of mass, a north face, a south face and an anti-earth side facing opposite earth during the geosynchronous orbit. The system includes a first thruster mounted on the anti-earth side of the spacecraft and adjacent to the north face of the spacecraft. The first thruster is aligned to produce a first thrust vector nearly through the center of mass. The system also includes a second thruster mounted on the anti-earth side of the spacecraft and adjacent to the south face of the spacecraft. The second thruster is aligned to produce a second thrust vector nearly through the center of mass.

[0022] Also included in the system are at least three momentum wheels. The momentum wheels are mounted on-board the spacecraft for maintaining an orbital attitude of the spacecraft with respect to a corresponding one of three spacecraft axes, and in a non-planar manner such that angular momentum of the corresponding one of the three spacecraft axes is independently stored in at least one of the at least three momentum wheels. The angular momentum stored in each of the momentum wheels is desaturated by firing, at a given time, a single one of the first or the second thrusters to release the stored angular momentum.

[0023] The system further includes a controller that determines an attitude adjustment in the north-south direction of the geosynchronous orbit of the spacecraft. The controller produces an attitude adjustment control signal. The controller also determines if any of the at least three momentum wheels are saturated. In response to this determination, the controller produces a desaturation control signal. Preferably, the first thruster and the second thruster are responsive to the attitude adjustment control signal and the desaturation control signal for maintaining a predetermined orbital position along the geosynchronous orbital path of the spacecraft.

[0024] The above set forth and other features of the invention are made more apparent in the ensuing Detailed Description of the Invention when read in conjunction with the attached Drawings, wherein:

[0025] FIG. 1 is a block diagram of a spacecraft configuration according to the present invention;

[0026] FIG. 2 is a block diagram which illustrates a gimballed mounting of a thruster to the spacecraft of FIG. 1;

[0027] FIG. 3 is a plan view of a spacecraft's geosynchronous orbit about the earth in accordance with the present invention;

[0028] FIG. 4 is a side view of the spacecraft's geosynchronous orbit about the earth in accordance with the present invention;

[0029] FIGS. 5A-5D are plan views of the spacecraft's geosynchronous orbit about the earth in which are illustrated two successive passes of a thruster firing cycle in accordance with an aspect of this invention; and

[0030] FIG. 6 is a logic flow diagram of the two successive passes of the thruster firing cycle performed in accordance with the present invention;

[0031] FIG. 7 is a logic flow diagram depicting a method of thruster firing in order to dump a desired amount of stored momentum.

[0032] Identically labelled elements appearing in different ones of the above described figures refer to the same elements but may not be referenced in the description for all figures.

[0033] FIG. 1 illustrates an exemplary embodiment of a spacecraft 1 suitable for being placed in a geosynchronous orbit about the earth. The spacecraft 1 includes a north solar array 2, a south solar array 3, a first thruster 4, a second thruster 5, and at least three momentum wheels, for example, a momentum wheel 6. The at least three momentum wheels are mounted on-board the spacecraft 1 for maintaining the spacecraft's attitude with respect to a particular axis. In accordance with an aspect of this invention the at least three momentum wheels are not coplanar; thus momentum of the three spacecraft axes, the x-axis (roll), the y-axis (pitch), or the z-axis (yaw), can be stored independently in the at least three momentum wheels. Preferably, the total number of momentum wheels mounted on-board the spacecraft 1 are determined by reliability and mission requirements.

[0034] In conventional systems, the spacecraft 1 operates in one of two modes. In a first mode, the spacecraft 1 performs its intended mission, i.e. the purpose in which it was deployed, for example, to operate as a communication satellite. In a second mode, the spacecraft 1 performs any necessary attitude control or stationkeeping maneuvers required to maintain the spacecraft's geosynchronous orbital position. The teaching of this invention is intended to address the operation of the spacecraft 1 during the second mode.

[0035] It can be appreciated that the determination to enter the second mode and to perform attitude control or stationkeeping maneuvers can be initiated by logic devices and sensors located on-board the spacecraft 1, or alternatively, by command from a ground station 10. In one embodiment of the present invention the ground station 10 transmits ground commands 12 to the spacecraft 1 that instruct the spacecraft 1 to enter the second mode. Additionally, the ground commands 12 may include certain parameters, as will be discussed below, that are employed during the attitude control or the stationkeeping maneuvers.

[0036] In order to maintain a desired geosynchronous orbital attitude, periodic spacecraft attitude correction is required. In particular, the momentum wheels are used to compensate for any disturbance torque along the three orthogonal axes of the spacecraft 1. Over time, the momentum wheels are eventually forced into saturation, and thus can not operate to maintain the orbital attitude. Before saturation occurs, the momentum wheels are desaturated. The desaturation is performed, in accordance with the present invention, by firing, at a given time in conjunction with stationkeeping activity, one of a pair of thrusters. The firing of the thruster is such that it subtracts from the angular momentum of the momentum wheels, and the thruster firings decrease the speed of the wheels along a respective orthogonal spacecraft axis to achieve the desaturation of the momentum wheels. Additionally, the thrusters may be fired independently from the desaturation process to compensate for deviations along an axis.

[0037] As illustrated in FIG. 1, a first thruster 4 and a second thruster 5 are mounted on an anti-earth side of the spacecraft 1, i.e. on the side opposite the earth during orbit. The first thruster 4 and the second thruster 5 are aligned to each deliver a thrust vector slightly off from the spacecraft's center of mass. That is, the thrusters 4 and 5 each deliver a thrust vector at an angle of Φ₁and Φ₂respectively, between the spacecraft y and z axes. Also, the first thruster 4 is mounted adjacent to the spacecraft's north face 7, while the second thruster 5 is mounted adjacent to the spacecraft's south face 8. When either the first thruster 4 or the second thruster 5 is fired a resulting change of velocity, ΔV1 or ΔV2 respectively, is realized. The change of velocity is represented by a component in two of the three orthogonal spacecraft axes, i.e. the y-axis (pitch) and the z-axis (yaw). These components are discussed in detail below.

[0038] It can be appreciated that if the thrusters 4 and 5 were aligned such that the thrust vectors ΔV1 and ΔV2 were directed through the center of mass (i.e., Φ₁=Φ₂=45° when a=b) no torque on the spacecraft 1 is produced.

[0039] Therefore, by misaligning the thrusters 4 and 5 such that the thrust vectors are directed slightly off from the center of mass, torque is created in a direction perpendicular to the applied thrust to desaturate wheel momentum stored in the perpendicular direction. The magnitude of the momentum dumped, i.e. desaturated, can be determined by measuring a time period during which the misaligned thrusters were fired since momentum is represented by a product of torque multiplied by the time period.

[0040] In FIG. 2, a detailed block diagram of the thruster 4 is shown. As depicted in FIG. 2, the thruster 4 is gimballed to enable desaturation of the roll momentum (i.e., momentum in the spacecraft x axis) and of the combined pitch/yaw momentum (i.e., momentum in the combined spacecraft y and z axes). A first gimbal 14 rotates about the spacecraft's x axis to generate a roll torque T_X. The rotation of the first gimbal 14 is illustrated in FIG. 2 as δ₁. A second gimbal 16 rotates about an axis that is defined about a 45° angle Φ₃between the spacecraft's y and z axes. The second gimbal 16 generates a pitch/yaw torque T_y/z. The rotation of the second gimbal 16 is illustrated in FIG. 2 as δ₂.

[0041] Assuming that the thruster 4 is fired through the spacecraft's center of mass (i.e. Φ₁=45°), the radius Δr can be represented by the hypothenuse of a right triangle formed from Ar and sides a and b (FIG. 1). Assuming, for example, that the sides a and b are each 1.7 m in length, the radius Ar is calculated as:

[0042] Δr={square root over (2*(1.7+L )²+L )}

[0043] Δr=1.7*2  (1)

[0044] The thrusters 4 and 5 are fired and velocity vectors are directed, in accordance with the present invention, slightly off from the center of mass to include an angular offset ΔΦ, for example, 1° per north-south station keeping maneuver, or, in radians, 1°/57.3. Assuming a thruster firing force of, for example, 0.08 N, the magnitude of the roll torque T_Xis defined as:

[0045] T_x=F_SPT*Δr*ΔΦ

[0046] T_x=0.08 N*(1.7 m*{square root over (2+L )})*{fraction (1°/57.3)}  (2)

[0047] So, T_x=3.4×10⁻³Nm per 1° of δ₁.

[0048] Similarly, the magnitude of the pitch/yaw torque T_y/zis defined as:

[0049] T_y/z=F_SPT*Δr*ΔΦ

[0050] T_y/z=0.08 N*(1.7 m*{square root over (2+L )})*{fraction (1°/57.3)}  (3)

[0051] So, T_y/z=3.4×10⁻³Nm per 1° of δ₂.

[0052] It should be appreciated that the thruster 5 is also gimballed to enable desaturation of the roll momentum and of the combined pitch/yaw momentum.

[0053] In one aspect of the present invention, the thruster 4 or 5 is fired over a predetermined time frame of, for example, about 24.5 minutes, according to a series of method steps, in order to dump a specified amount of stored momentum, for example, about 2.5 Nms of stored momentum. Referring also to FIG. 7, the series of method steps are as follows.

[0054] Step A places the gimbals (e.g., gimbals 14 and 16 when thruster 4 is to be fired) within a predetermined angle of a (ground) calculated zero-torque alignment (hereinafter referred to as a preset gimbal angle).

[0055] Step B ignites the thrusters and allows transients to settle and be absorbed by the momentum wheel system.

[0056] Step C calculates and steps the gimbals to near a zero-torque position.

[0057] Step D calculates the momentum dumping required and steps the gimbals by a predetermined angle to create a momentum desaturation torque for a variable time period (e.g., time period length * torque=momentum dumped).

[0058] Step E removes the preset gimbal angle to return to the zero-torque position.

[0059] Step F repeats Steps D and E as needed, until the stationkeeping activity is complete and the thrusters are off. Preferably there is not more than one such series executed per hour.

[0060] In one embodiment, the preset gimbal angle is preferably about 0.5° and the predetermined angle for stepping the gimbals is about 0.01°. These values can be substituted into Equations 2 and 3 to yield the roll torque T_xand the pitch/yaw torque T_y/xgenerated for each north-south stationkeeping maneuver with a 0.5° gimbal offset.

[0061] T_x=F_SPT*Δr*ΔΦ

[0062] T_x=0.08 N*(1.7 m*{square root over (2+L )})*{fraction (0.5°/57.3)}  (4)

[0063] So, T_x=0.0017 Nm per 0.5° of δ₁and the pitch/yaw torque T_y/zis calculated as:

[0064] T_y/z=F_SPT*Δr*ΔΦ

[0065] T_y/z=0.08 N*(1.7 m*{square root over (2+L )})*{fraction (0.5°/57.3)}  (5)

[0066] The resulting pitch/yaw torque is, T_y/z=0.0017 Nm per 0.5° of δ₂. Since the momentum dumped is determined by the formula: torque * time period, each north-south station keeping maneuver performed with a time period of at least 1470 seconds or about 24.5 minutes can desaturate 2.5 Nms in the roll and pitch/yaw axes of the wheels.

[0067] In addition to the influences of torque and the duration of firing time, the amount of momentum desaturated is also influenced by the firing cycle of the thrusters 4 and 5. As shown in FIG. 3, the first thruster 4 and the second thruster 5 are separately fired at predetermined intervals along the spacecraft's geosynchronous orbital path about the earth. The firing of the first thruster 4 is represented by a point A along the spacecraft's path of orbit. Similarly, a predetermined firing of the second thruster 5 is represented by a point B. The points A and B represent points where the spacecraft's geosynchronous orbital path crosses a line of nodes which is coplanar with the z-axis of the spacecraft. Further, the point B is oriented 180° from the point A. Thus, the point B is on an opposite side of the orbit from the point A. Assuming that the geosynchronous orbit around the earth is of about a 24 hour duration, then the distance between the points A and B would represent a 12 hour difference along the spacecraft's orbital path. By firing the thrusters 4 and 5 at points which represent the 12 hour difference along the spacecraft's orbital path, radial effects in the change of velocity are effectively cancelled. The cancelling effect is apparent upon an examination of the components of the change in velocity.

[0068] Referring to FIG. 4, the components of the change in velocity due to the firing of the first thruster 4 and the second thruster 5 at points A and B are illustrated. As shown, both thrust vectors ΔV1 and ΔV2 yield thrust vector components along the y-axis and the z-axis, ΔV1_y, ΔV1_z, ΔV2_y, and ΔV2_z, respectively. Of note, the y-axis components, ΔV1_yand ΔV_2y, are reversed. That is, the y-axis component of thrust vector ΔV1 produces a southward change in velocity, while the y-axis component of thrust vector ΔV2 produces a northward change in velocity. The reverse, or opposite, components of velocity are a direct result of the alignment of the thrusters 4 and 5 with respect to the spacecraft's center of mass. Since the first thruster 4 is mounted adjacent the north face 7 of the spacecraft 1, the ΔV1 thrust vector, when fired in the direction of the center of mass, produces a south component. Similarly, the second thruster 5 is mounted adjacent the south face 8 of the spacecraft 1 and produces a north component when fired in the direction of the center of mass.

[0069] During firing, the thrusters 4 and 5 are preferably gimballed (as discussed above) to null disturbance torques and relieve two axes of wheel momentum accumulation. Thus, the orientation of the thrust vectors and the firing at opposite intervals along the spacecraft's geosynchronous orbit about the earth (at points A and B) enables 2-axes of inertial momentum to be relieved, i.e. desaturated, from the momentum wheel stabilization system. However, and in accordance with the present invention, 3-axis momentum management is realized by modifying the timing of the thruster firings.

[0070] Referring again to FIG. 3, points A′, B′, A″, and B″ are now emphasized. As discussed above, the thrusters 4 and 5 are fired at collinear points (A and B with the earth's center, i.e. point O) along the spacecraft's orbital path. As such, radial effects of the thrust vectors are cancelled and momentum is dumped in two of the three spacecraft axes. In accordance with the present invention, however, the thrusters 4 and 5 are fired within a thruster firing cycle in which, at various points, firing occurs at offset, non-collinear points along the spacecraft's orbital path. That is, the firing of the thrusters 4 and 5 is offset so that 3-axis momentum control is realized and the radial effects of the thrust vectors are cancelled over, for example, two orbits.

[0071] As is illustrated in FIG. 3, the north-south stationkeeping on a first orbit is offset to points A′ and B′. The points A′, B′ and O (the earth's center) are offset from, i.e., are not collinear with, the collinear firing points A, B and O by a first angle θ₁and a second angle 62, respectively. Firing the first thruster 4 at the point A′ and then firing the second thruster 5 at the point B′ ensures that the thrust vectors (ΔV1 and ΔV2) are not parallel. Since the thrust vectors are not parallel, 2-axis momentum control by the second thruster 5 at the point B′ desaturates momentum that was unaffected by the firing of the first thruster 4 at point A′. The offset firing of the thrusters during this first orbit, however, does not result in the cancellation of the radial components of the change in velocity due to the firing of the first thruster 4 and the second thruster 5. To cancel the radial components of the change in velocity a second firing of each of the thrusters is required. The second firings of the thrusters 4 and 5 are performed at points A″ and B″, respectively. The points A″, B″ and O are offset from the collinear firing at the points A, B and O by a third angle θ₃and a fourth angle θ₄, respectively. It is noted that the points A″ and B″ are offset an equal and opposite way from the first pair of thruster firings at points A′ and B′. In other words, angle θ₁is equal to angle θ₄and angle θ₂is equal to angle θ₃. By repeating the above thruster firing cycle all 3-axes of wheel momentum are desaturated.

[0072] In accordance with a thruster firing cycle 20 of the present invention, north-south stationkeeping and 3-axis momentum management are simultaneously provided. The thruster firing cycle 20, illustrated in FIGS. 5A-5D, is accomplished by varying the pattern of the firing of the first thruster 4 and the second thruster 5 according to the steps outlined in the flow chart of FIG. 6.

[0073] At Block 22 of FIG. 6 the thruster firing cycle 20 begins. In a first step and during a first orbit, the first thruster 4 is fired (Block 24) at the point A′ while maintaining attitude with wheel control and gimballing the first thruster to desaturate momentum wheels as needed. Preferably, the point A′ is a first predetermined angle from the point A (θ₁), or a first predetermined time of motion from the point A (FIG. 5A). In one embodiment, the first predetermined angle (θ₁) is about 15° and the first predetermined time of motion is about 1 hour of motion from the point A.

[0074] At Block 26, the second thruster 5 is fired. The firing of the second thruster 5 occurs at the point B′ which, preferably, is a second predetermined angle from the point B (θ₂) and a second predetermined time of motion from the point A′. The firing of the second thruster occurs while maintaining attitude with wheel control and gimballing the second thruster to desaturate momentum wheels as needed. In one embodiment, the second predetermined angle (θ₂) is about 15° from the point B and the second predetermined time of motion is about 10 hours of motion from the point A′. It is particularly noted that, in accordance with the present invention, the points A′, B′ and O are non-collinear (FIG. 5B). Since the points A′, B′ and O are non-collinear, the radial components of the change of velocity produced upon the firing of the first and the second thrusters 4 and 5 are not cancelled. Additionally, since the points A′, B′ and O are non-collinear, the components of the velocity vectors produced by the firing of the first thruster 4 at the point A′ and the firing of the second thruster 5 at the point B′ desaturate momentum stored along differing thrust vector directions in inertial space.

[0075] At Block 28, a second pair of thruster firings are begun in order to cancel the effects of the radial components of the change of velocity produced by the first pair of firings of the first and the second thrusters 4 and 5. At Block 28, the first thruster 4 is again fired. The second firing of the first thruster 4 occurs at the point A″ (FIG. 5C). The point A″ is a third predetermined angle from the point A (θ₃), or a third predetermined time of motion from the point B′ such that the points B′, A″ and O are collinear. As a result of the second firing of the first thruster 4 at the point A″, the radial components of the change of velocity produced by the first firing of the second thruster 5 at the point B′ are cancelled. In one embodiment, the third predetermined angle (θ₃) is about 15° from the point A and the third predetermined time of motion is about 12 hours of motion from the point B′.

[0076] Similarly, at Block 30, the second thruster 5 is again fired. The second thruster 5 is now fired at the point B″ (FIG. 5D). The point B″ is a fourth predetermined angle from the point B (θ₄), and a fourth predetermined time of motion from the point A′ such that the points B″, A′ and O are collinear. As a result of the second firing of the second thruster 5 at the point B″, the radial components of the change of velocity produced upon the first firing of the first thruster 4 at point A′ are cancelled. Preferably, the fourth predetermined angle (θ₄) is about 15° from the point B and the fourth predetermined time of motion is about 14 hours of motion from the point A″.

[0077] As is demonstrated above, the thruster firing cycle 20 achieves both 3-axis momentum control and balanced north-south stationkeeping while producing no radial velocity change effect with two pairs of thruster firings performed over, for example, two orbits of the spacecraft.

[0078] In accordance with the present invention, differing values for the predetermined angles (θ₁, θ₂, θ₃and θ₄) and the predetermined times of motion from points A′ to B′, B′ to A″ and A″ to B″ can be calculated on-board and/or received by ground commands 12 from the ground station 10. As such, the predetermined angles and times of motions can be modified prior to the start of a thrust firing cycle to account for changing orbital mechanics and/or fuel optimization considerations.

[0079] Although described in the context of preferred embodiments, it should be realized that a number of modifications to these teachings may occur to one skilled in the art. By example, and as discussed above, the teachings of this invention are not intended to be limited to any specific predetermined angles and/or predetermined times of motion between the alternating firings of the thrusters in the thruster firing cycle disclosed in the present invention.

[0080] Thus, while the invention has been particularly shown and described with respect to preferred embodiments thereof, it will be understood by those skilled in the art that changes in form and details may be made therein without departing from the scope and spirit of the invention.