System and Method for the Improvement of Attitude Control System Testbeds for Small Satellites

Abstract

A rotational negative-inertia converter (RNIC) has a housing enclosing a flywheel configured to rotate around an axis of symmetry; a motor with a stator attached to the housing and a rotor attached to the flywheel to rotate it around the axis of symmetry; a housing angular accelerometer attached to said housing; a flywheel angular accelerometer; and a controller configured to receive measured accelerometer values from the accelerometers. The controller is configured to drive the motor to maintain the angular acceleration of the flywheel at a value proportional to the housing angular acceleration, with a predetermined proportionality constant.

A method for calibrating an ADCS testbed comprising a DUT holder with three RNICs includes: using measured angular velocities of the DUT holder and RNIC flywheels, and ZGT data, to compute moments of inertia of the DUT holder with and without a satellite with ADCS, allowing compensation for those moments by the RNICs.

Claims

1: A rotational negative-inertia converter (RNIC) comprising: a housing; a flywheel having an axis of symmetry, the flywheel being operatively configured to rotate within the housing around the axis of symmetry; a first motor having a stator and a rotor, said stator attached to said housing and said rotor attached to said flywheel, operatively configured to rotate said flywheel in any direction around said axis of symmetry; a housing angular accelerometer attached to said housing, providing a housing angular acceleration value; a flywheel angular accelerometer providing a flywheel angular acceleration value; and a controller electrically coupled to said first motor, said housing angular accelerometer, and said flywheel angular accelerometer, said controller being operatively configured to receive said housing angular accelerometer value and said flywheel angular acceleration value; wherein said controller is operatively configured to drive said first motor to maintain said flywheel angular acceleration value proportional to said housing angular acceleration value with a predetermined proportionality constant.

2: The RNIC of claim 1, wherein said flywheel, said first motor, said housing angular accelerometer, and said flywheel angular accelerometer are concealed within said housing.

3: The RNIC of claim 1 further comprising a wireless interface operatively configured to allow said controller to communicate with an external controller.

4: The RNIC of claim 1. Wherein said axis of symmetry passes through a center of mass characterizing the RNIC.

5: The RNIC of claim 1, wherein the predetermined proportionality constant is less than or equal to −1.

6: The RNIC of claim 1, further comprising a second motor, said second motor being operatively configured to controllably move said flywheel along said axis of symmetry.

7: The RNIC of claim 6 further comprising a wireless interface operatively configured to allow said controller to communicate with an external controller.

8: A method for calibrating an ADCS testbed comprising a DUT holder with three rotational negative-inertial converters, the method comprising: gravitational torque zeroing of the DUT holder by adjusting positions of three weight centering modules included within the DUT holder along three mutually orthogonal axes; collecting ZGT data comprising mass and distance values characterizing each adjusted weight centering module for a corresponding one of the orthogonal axes; setting a flywheel within each of the rotational negative-inertia converters (RNICs) into motion, spinning about a corresponding one of the mutually orthogonal axes; measuring angular velocities of the DUT holder and of the spinning flywheels; computing, using the measured angular velocities and collected ZGT data, moments of inertia of passive components of the DUT holder; loading a satellite into the DUT holder of the testbed; gravitational torque zeroing of the loaded DUT holder by re-adjusting positions of the weight centering modules; collecting new ZGT data comprising mass and distance values characterizing each re-adjusted weight centering module; computing, using the new ZGT data and the computed moments of inertia of the passive components of the DUT holder, total moments of inertia of the DUT holder of the testbed without the satellite loaded therein; and computing coefficients k.sub.x, k.sub.y, and k.sub.z as a first output, such that values of the coefficients may be set into controllers in the DUT holder and used to cancel out DUT holder moments of inertia that would otherwise affect the satellite ADCS in subsequent characterization of the satellite by the testbed.

9: The method of claim 8, further comprising: after collecting new ZGT data comprising mass and distance values characterizing the re-adjusted weight centering modules, calculating a center of mass of the satellite as a second output; and after computing, using the new ZGT data and the computed moments of inertia of the passive components of the DUT holder, total moments of inertia of the DUT holder of the testbed without the satellite loaded therein: setting a flywheel within each of the rotational negative-inertia converters (RNICs) into motion; measuring angular velocities of spinning flywheels and the DUT holder; and calculating moments of inertia of the satellite as a third output.

Description

DESCRIPTIONS OF THE FIGURES

[0042] The accompanying drawings are not necessarily drawn to scale.

[0043] FIG. 1 (PRIOR ART) illustrates an air-bearing testbed used to characterize ADCS system for small satellites in which rotation in all directions is possible.

[0044] FIG. 2 (PRIOR ART) illustrates the adjustment of center of mass of the loaded testbed system.

[0045] FIG. 3 illustrates an RNIC according to one embodiment.

[0046] FIG. 4 illustrates a testbed for characterization of nanosatellite ADCS system employing three RNICs, according to one embodiment.

[0047] FIG. 5 is a pictorial view used to explain the procedure for calibration of an unloaded testbed according to one embodiment.

[0048] FIG. 6 is a pictorial view used to explain the procedure for calibration of one of the embodiments representing a loaded testbed according to one embodiment.

[0049] FIG. 7 illustrates a method for calibration of nanosatellite testbed employing three RNICs, according to one embodiment.

[0050] FIG. 8 illustrates the method of FIG. 7, with additional steps for the extraction of mechanical properties of nanosatellite, according to one embodiment.

DETAILED DESCRIPTION OF THE INVENTION

[0051] Various embodiments of the present invention will now be described more fully with reference to the accompanying figures, in which some, but not all embodiments of the inventions are shown. These inventions may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein.

Embodiment 1: RNIC

[0052] FIG. 3 illustrates a non-limiting example of an embodiment of a rotation negative-inertia converter 300 (“RNIC”) having a moment-compensating axis 311 and comprising a housing 301, a motor 308 attached to housing 301 and operatively configured to actuate (rotate) a motor shaft 313, the motor shaft 313 has an axis of rotation that coincides with the moment compensating axis 311, a flywheel 306 and a encoder disk 307, both attached to the motor shaft 313 and operatively configured to rotate the shaft 313 around the moment compensating axis 311 when powered by the motor 308. The RNIC 300 further comprises a controller 304, a housing gyroscope 303, and a wireless transceiver 305 all disposed on a printed circuit board 309 and all being mutually electrically connected through electrical connections within the printer circuit board 309 which in turn is connected to the power source 310 via electrical connection 314. The motor 308 is electrically connected to the printer circuit board via connection 315. A quadrature optical encoder 302 attached to housing 301 is operatively configured to sense the rotation of encoder disk 307 relative to housing 301 and is electrically coupled to controller 304 via electrical connection 312. Housing gyroscope 303 is operatively configured to sense rotation of housing 301 relative to an inertial frame of reference. An inertial frame of reference is a frame of reference not undergoing acceleration. In other words, an inertial frame of reference for a physical object with zero net force acting on it moves with a constant velocity and the velocity may be zero. The RNIC 300 has a dimensional reference point 316, a point of reference for installing the RNIC and the point through which the axis of rotation passes. The reference point 316 is needed to be able to determine the distance between the RNIC 500 and other elements in the spherical DUT holder.

[0053] The elements of the RNIC 300 that are operatively configured to rotate, meaning the flywheel 306, encoder disk 307, motor shaft 313 and the rotor (rotating part) (not shown in FIG. 3) of the motor 308, are jointly referred to as the rotatable part of the RNIC 300 as they measurably contribute to the mechanics of motion of the RNIC 300. The rest of the RNIC is referred to as the fixed part of the RNIC, the fixed part also includes elements of the RNIC 300 that may be moveable, but their movement contributes negligibly to the mechanics of the RNIC. An example of an element whose mechanical movement is assumed to make negligible contribution to the mechanics of the RNIC is a gyroscope.

[0054] The angular velocity ω.sub.h and acceleration ah of the housing relative to an inertial frame of reference is measured using housing gyroscope 303 rigidly attached to the housing 301 via the printed circuit board 309. The encoder-based rotational sensor (“EBR sensor”), comprising the encoder disk 307, optical encoder 302 and controller 304 is used for measuring the angular velocity ω.sub.fw and acceleration α.sub.fw, of the flywheel 306 relative to the housing 301. Data from both sensors are delivered to the controller 304 and the controller 304 is operatively configured to use the angular velocity ω.sub.h of housing rotation relative to an inertial frame and the angular velocity ω.sub.fw of flywheel rotation relative to housing 301 as feedback to maintain the ratio k of angular velocities ω.sub.fw=k.Math.ω.sub.h and angular acceleration α.sub.fw=k.Math.α.sub.h to a predetermined value, where the coefficient k is a predetermined parameter that is input to the controller. The control is achieved using a feedback control algorithm (such as PID controller) digitally implemented within the controller 304. Note that because the coefficient k represents the ratio between the flywheel and housing angular accelerations, it must also represent the ratio between the angular velocities of the flywheel and the housing. It is clear that by adjusting the ratio k, one can compensate for the spherical DUT holder moment of inertia. The ultimate goal of this control is to free the satellite ADCS from having to exert torque in order to rotate the spherical DUT holder on the test bed. In a sense, the testbed itself exerts the required torque using the RNIC.

[0055] Therefore, judging by the torque an observer sees as having to be applied, the effective moment of inertia of the RNIC I.sub.eff is the sum of the moment of inertia I.sub.h of the housing 310 and the apparent moment of inertia of the rotating flywheel 306, which is the sum of its intrinsic moment of inertia I.sub.fw and the additional moment of inertia kI.sub.fw imposed due to the feedback imposed by the controller. Hence, I.sub.eff=I.sub.h+(k+1) I.sub.fw. The flywheel 306, rotor of the motor 308, motor shaft 313 and encoder disc 302 contribute to I.sub.fw, while all other parts of the device contribute to I.sub.h. Moments of inertia I.sub.fw and I.sub.h are constant and do not change during the operation of the device, but the proportionality coefficient k is adjustable and is the controlling parameter. The constant k can be positive, zero or negative, and it is arbitrarily chosen by a user to achieve the desired effective moment of inertia of the system. If k<−1, the effective moment of inertia is less than the moment of inertia of the housing 301. The effective moment of inertia may even become negative for lower values of k. The chosen value of k is implemented as a part of the control algorithm.

[0056] It is clear that, while selecting the proportionality coefficient k gives a certain freedom in mechanical control of the spherical holder and the device under test, the objective is to ensure that the moment of inertia I.sub.u, of the unloaded spherical DUT holder is made negligible or zero relative to the moment of inertia of the satellite. This means that the moment of inertia in each axis must be adjusted so that I.sub.u=I.sub.h+(k+1)I.sub.fw equals zero. The only controllable factor in this equation is the gain coefficient k so an appropriate, predetermined value of k must be set into the RNIC controller. In this way, the RNIC can completely cancel the moment of inertia of the unloaded spherical DUT holder. Three RNICs allow this cancellation to be done in each of the three orthogonal axes.

[0057] An RNIC constructed according to the present invention is characterized by a moment compensating axis, which defines the direction and the line on the surface and body of the RNIC. The operation of RNIC is furthermore characterized by its gain coefficient k, which determines the amount of moment of inertia that RNIC will compensate. The coefficient is predetermined and is generally not changed during the operation, but in some embodiments may be altered without departing from the spirit of the invention.

[0058] In some embodiments the power source 310 is an inductively coupled transformer, an element of a wireless power system.

[0059] In some embodiments, the weight-centering module function is integrated into the RNIC by employing the flywheel as both the weight and a flywheel. This is realized by allowing the flywheel to move linearly along the axis of symmetry to be used in weight-centering, while simultaneously being allowed to rotate to be used in the inertial zeroing method.

Embodiment 2: ADCS Testbed

[0060] The purpose of an ADCS testbed is to emulate space conditions (primarily a frictionless and zero gravity environment) for optimization and verification of a satellite.sup.1 ADCS system on the surface of Earth, prior to launching the system into orbit. In one embodiment, the ADCS testbed comprises at least one RNIC to compensate and eliminate the moment of inertia of the spherical DUT holder, when loaded with the satellite under test. .sup.1The term satellite used in the remainder of this disclosure should be understood as a term of convenience, that may mean “nanosatellite” in some applications.

[0061] FIG. 4 illustrates a cross section view of one embodiment of a testbed 400 for testing the ADCS system of a nanosatellite according to the present invention. The testbed comprises a spherical air bearing base 401, which is powered by pressured air inserted through pneumatic inlet 402 (the compressed air is provided by a compressor which is not shown). The spherical DUT holder 409 allows complete freedom of rotation while inserted into the air bearing 401. The spherical DUT holder 409 comprises an x-axis weight-centering module 404, a y-axis weight-centering module 403 and, not visible in the drawing since it is positioned behind the z-axis RNIC 415, a z-axis weight-centering module. The weight-centering modules are positioned along and define three orthogonal axes of the spherical DUT holder 409: the x-axis 410 and the y-axis 411 are visible and perpendicular to each other and lie in the plane of the paper, while z-axis is not shown but is perpendicular to the paper and points towards the reader, and passes through the intersection between the x-axis 410 and the y-axis 411. The spherical DUT holder 409 further comprises a controller 407, a power source 408 and a DUT space 406, which is prepared to fit a specific satellite format by design. Finally, the spherical DUT holder 409 is equipped with three RNICs, one for each of the three orthogonal axes: x-axis RNIC 414 positioned so that its axis of symmetry is parallel to and coincides with the x-axis 410. The y-axis RNIC 413 is positioned so that its axis of symmetry is parallel to and coincides with the y-axis 411, and finally, the z-axis RNIC 415 is positioned so that its axis of symmetry is parallel to and coincides with the z-axis (not shown).

[0062] In this application, the satellite under test, of an exemplary cubic shape, is placed prior to characterization into the DUT space 406 in the center of the spherical holder 409. In FIG. 4, a silhouette of the inserted satellite is shown by the dashed square 416. When the satellite (device under test) is not present in the DUT space 406, the ADCS testbed is referred to as “unloaded”, while when the satellite is inserted into the DUT space 406 (as shown by 416), the testbed is referred to as “loaded”.

[0063] When the satellite 416 is inserted into the spherical holder 409, an effort is made to ensure that the center of the mass (not shown) of the satellite 416 coincides with the center of the spherical holder 409. It is well known that designing a satellite with its center of mass at its geometrical center is uncommon and that some additional means are necessary to accomplish this. The centering of the center of mass requires moving the center of mass in three orthogonal directions and hence this is generally accomplished using three weight-centering modules provided for this purpose within the spherical holder 409. In FIG. 4, x-axis weight-centering module 404 and y-axis weight-centering module 403 are visible, while the z-axis weight-centering module is not visible because it is hidden behind the z-axis RNIC 415. Each of the weight-centering modules comprises a weight that can be moved along its respective axis using a linear actuator. All three weight-centering modules are electrically coupled to a controller 407.

[0064] Once the weight-centering has been completed, namely, gravitational torque has been substantially removed, the preferred ADCS testbed compensates for the moments of inertia added to the satellite 416 by the existence of the spherical DUT holder 409 which comprises three weight-centering modules and three RNICs. The RNICs are controlled with their respective coefficients k, which for three orthogonal axes are referred to as coefficients k.sub.x, k.sub.y, and k.sub.z. With proper choice of values for these three coefficients inserted to the RNICs, as described above, the moments of inertia along three orthogonal axes I.sub.x, I.sub.y, and I.sub.z can all be substantially eliminated. In this way, the moments of inertia of the loaded spherical DUT holder 409 (with satellite 416) are substantially equal to the axis-respective moments of inertia of the satellite 416. Stated differently, the ADCS on-board the satellite is now subjected to moments of inertia that are substantially equal to those of the satellite alone, without the influence of the spherical DUT holder inertia.

[0065] The temporal transient in the operation of the RNICs within the testbed depend on the bandwidth of the control loops and the speed by which the ADCS on the satellite operates. In the present disclosure it is assumed that the RNICs respond to the slowly changing orientation of the satellite. If the rate of change is so fast that the RNIC controller lags behind, it may produce disturbance in the response. This effect can be substantially minimized by increasing the sampling rate or the rated power of RNIC motors. Because of that, the presented testbed can be easily scaled for larger satellites. Minimizing the moment of inertia of the testbed is no longer an issue since it can be reduced using RNICs.

Embodiment 3: Method for Calibration

[0066] In addition to the apparatuses disclosed in embodiments 1 and 2, we disclose a method for calibrating the ADCS testbed for both weight-centering and inertial zeroing. Calibrating the testbed for inertial zeroing amounts to determining the correct values of the coefficients k.sub.x, k.sub.y, and k.sub.z, that result in eliminating the moments of inertia added to the satellite by placing the satellite within the spherical DUT holder.

[0067] FIG. 5 illustrates a simplified pictorial view of an unloaded spherical DUT holder 500 having a smooth spherical surface 501 that enables the holder 500 to slide on an air bearing such as 401 shown in FIG. 4. The interior of the spherical DUT holder 500 is simplified to show only the elements that are actuated or relevant to the calibration method explained next.

[0068] To assist with the description, an orthogonal coordinate system is defined, consisting essentially of three orthogonal axes shown in FIG. 5 with x-axis 502, y-axis 503, and z-axis 504. The origin of the orthogonal coordinate system is defined as the intersection of the three orthogonal axes. The center 550 of the spherical shape 501 of the spherical DUT holder 500 coincides with the origin of the coordinate system.

[0069] One embodiment of a method 700 to calibrate the testbed is disclosed with the help of FIG. 5, FIG. 6, and the flow diagram shown in FIG. 7. The method comprises two groups of steps. The first group of steps, up to and including step 704, is performed without the spherical DUT holder being loaded with the satellite (device under test) and the second group of tests, from step 705 through 709 in FIG. 7, is performed while a satellite is present within the spherical DUT holder.

[0070] It should be noted that during those calibration method steps in which the flywheels in the RNICs are switched on, they are operated “open-loop” i.e., with no feedback applied via any controllers.

[0071] The purpose of the calibration method disclosed below is to (i) determine testbed input parameters that substantially eliminate gravitational torque when a device under test is loaded and (ii) substantially eliminate the contribution of the moment of inertia of spherical DUT holder when a satellite is loaded by providing the coefficients k.sub.x, k.sub.y, and k.sub.z to the RNICs. The goal as discussed above is to prevent the satellite ADCS from being burdened by the moment of inertia added by the spherical DUT holder during the characterization of the performance of satellite's ADCS. Furthermore, the same setup can be used to determine the moment of inertia and the position of the center of mass of the device under test.

[0072] Step 701 in FIG. 7 achieves gravitational torque zeroing, as follows. In the unloaded spherical DUT holder 500, the center of mass 506 will generally be displaced from the center of the sphere 505 as seen in FIG. 5. The reason for this is that adjusting the location of all components within the unloaded spherical DUT holder 500 is never perfect. Once the unloaded spherical DUT holder 500 is placed onto the air bearing base (not shown in FIG. 5, but corresponding to 401 shown in FIG. 4) the product of the weight of unloaded spherical DUT holder 500 and the distance between the center of mass 506 and the center of the sphere 505 will produce a torque that will want to turn the spherical DUT Holder 500 towards a position of minimal potential energy, but without friction will keep on oscillating around the position of lowest potential energy. This added torque is referred to as gravitational torque and is undesirable because it adds additional torque that the satellite ADCS will have to correct for, which does not represent the reality the ADCS will face when the satellite is in orbit. A correction is needed to null this torque and is referred to as weight centering.

[0073] The fine correction of the displacement (distance between the center of mass 506 and center of sphere 505) is performed by using three weights (x-axis weight 512, y-axis weight 513, and z-axis weight 514) operatively configured to move parallel to their respective axes as is indicated by the straight double-headed arrows shown next to the weights. The mechanisms that move each of the weights are omitted from the drawing in FIG. 5 for simplicity, and because there are multiple ways to accomplish this task known in the art. The method for adjusting the center of mass comprises of shifting each of the weights along its respective axis until the center of mass 506 coincides with the center of sphere 505. This condition is referred to as “unloaded holder with zero gravitational torque”. FIG. 5. illustrates the beginning of this method in which there is a displacement between center of mass 506 and the center of sphere 505. FIG. 5 does not show the end of the procedure.

[0074] At Step 702, the distances a.sub.x, a.sub.y, a.sub.z, between three weights and the center of the sphere (shown in FIGS. 5 as 522, 523, and 524) are determined. The masses of the weights 512, 513, and 514, noted as M.sub.x, M.sub.y, M.sub.z, respectively, are presumably known by design of the testbed. This collection of distance and mass data defines a zero gravitational torque (“ZGT”) condition for unloaded testbed.

[0075] Before disclosing the method for inertial zeroing, it is important to describe the inertial changes that occur during the gravitational torque zeroing. This will be explained with the help of FIG. 5 for an example involving an x-axis weight. Consider the state in which all the weights (512, 513, and 514) are positioned at some arbitrary distances from the center of the sphere 505. These distances are referred to as a.sub.x, a.sub.y, and a.sub.z, respectively, and shown in FIG. 5, labelled as 522, 523, and 524, respectively. We refer to this state as the initial condition and this condition is characterized by three spherical holder 500 moments of inertia: I.sub.x, I.sub.y, and I.sub.z. Here I.sub.x is the moment of inertia of the unloaded spherical DUT holder 500 for rotation around x-axis 502, and the rest follow cyclically.

[0076] When the x-axis weight 512 is moved along the x-axis 502 by an amount equal to d.sub.x for the purpose of displacing the center of mass 506 of the spherical DUT holder 500 (during the gravitational torque zeroing), the moments of inertia I.sub.y and I.sub.z change, but ideally, the moment of inertia I.sub.x does not change. This is true when the x-axis of the coordinate system defining the spherical DUT holder passes through the center of mass of x-axis weight 512 as stated at the beginning of this method.

[0077] In other embodiments of weight-centering modules described in the literature such as “Automatic Mass Balancing of a Spacecraft Three-Axis Simulator: Analysis and Experimentation” written by Simone Chesi and coworkers and published in Journal of Guidance, Control, and Dynamics in 2014, the axes along which the balancing weight is actuated does not pass through the center of rotation (i.e. center of sphere 505). In such a scenario, moving any of the weights affects all three moment of inertia I.sub.x, I.sub.y, and I.sub.z, and must be taken into account.

[0078] The inertial characteristics of q-axis weight (where q can be x, y, or z) are given by moments of inertia along three directions I.sub.qx, I.sub.qy, and I.sub.qz. The moments of inertia are given for a weight-centered axis for each of the weights and axes. The components of the unloaded spherical DUT holder 500 can be divided into components that move to alter the holder's center of mass or alter its inertial properties, which we refer to as the active components, and the components or elements of components that do not move or rotate, which we shall refer to as passive/fixed elements.

[0079] To characterize the moment of inertia I.sub.x of unloaded spherical DUT holder 500 around x-axis 502, one considers the moment of inertia of passive components around x-axis 502 I.sub.xo and the contributions of active components at specified position after gravitational torque zeroing. x-axis weight 512, y-axis weight 513, and z-axis weight 514 contribute to I.sub.x with their inherent moments of inertia around x-axis 502 I.sub.xx, I.sub.yx, and I.sub.zx, respectively. Additionally, y-axis weight 513, and z-axis weight 514 contribute to I.sub.x with additional inertia M.sub.ya.sub.y.sup.2 and M.sub.za.sub.z.sup.2, respectively. This is a result of rotating y-axis weight 513, and z-axis weight 514 around x-axis 502, with their own centers of mass at the distance from x-axis 502 a.sub.y 523 and a.sub.z 524, respectively. Since the x-axis 502 passes through the center of mass of x-axis weight 512, the distance between the two is equal to zero (a.sub.x=0). Thus, the term M.sub.xa.sub.x.sup.2 is also equal to zero and does not contribute to I.sub.x. Even though M.sub.xa.sub.x.sup.2 does not contribute to I.sub.x in this embodiment, it is included in the equations below for the sake of generality. The same approach is applied to calculate contributions to the moments of inertia I.sub.y and I.sub.z.

I.sub.x=I.sub.xo+I.sub.xx+I.sub.yx+I.sub.zx+M.sub.xa.sub.x.sup.2 +M.sub.ya.sub.y.sup.2 +M.sub.za.sub.z.sup.2

I.sub.y=I.sub.yo+I.sub.xy+I.sub.yy+I.sub.zy+M.sub.xa.sub.x.sup.2 +M.sub.ya.sub.y.sup.2 +M.sub.za.sub.z.sup.2

I.sub.z=I.sub.zo+I.sub.xz+I.sub.yz+I.sub.zz+M.sub.xa.sub.x.sup.2 +M.sub.ya.sub.y.sup.2 +M.sub.za.sub.z.sup.2   (A)

[0080] These expressions make use of the Parallel Axis Theorem, also referred to as the Huygens-Steiner Theorem or just Steiner Theorem, which tells how to compute the moment of inertia of a rigid body for rotation around an axis parallel to the axis through the center of mass, but at a passing through the rigid body at a distance equal to a.sub.x, a.sub.y or a.sub.z in the above expressions. The description of this theorem can be found in textbooks on classical mechanics, such as, Classical Mechanics by Herbert Goldstein published by Addison-Wesley in 1980.

[0081] In one embodiment of the method to zero the inertial forces imposed on the satellite (DUT) by the holder, one must know the following physical parameters of the unloaded spherical DUT holder 500: mass of the each of the weights, the distance from the center of mass of each of the weights to the respective axis, and the moments of inertia of each of the weights around a weight-centered axis for each of the weights which passes through the center mass of the weight and coincides with the respective axis of the spherical DUT holder 500. The details of this are not described in connection with FIG. 5. The weights shown in FIG. 5, namely, x-axis weight 512, y-axis weight 512, and z-axis weight 514, are all illustratively depicted as circular disks and the coordinate axes 502, 503 and 504 of the spherical DUT holder 500 pass through the centers of these disks which is consistent with the above definition.

[0082] At this step the unknown variables in equations (A) are I.sub.x, I.sub.y, I.sub.z, I.sub.xo, I.sub.yo, I.sub.zo. In the next step of the method the moments of inertia I.sub.x, I.sub.y, I.sub.z are computed.

[0083] At Step 703, after ensuring that the flywheels within the RNICs and the unloaded spherical DUT holder are in the resting state, i.e., their angular velocities are equal to zero, the flywheels within the RNICs are turned on, either sequentially or simultaneously. When the x-axis RNIC gradually increases the angular velocity of its flywheel to a constant predetermined angular velocity ω.sub.xo expressed in an inertial frame, the unloaded spherical DUT holder also accelerates, due to the conservation of angular momentum, and remains rotating at an angular velocity ω.sub.x1 assuming that there is no friction, or the friction is negligible. This angular velocity ω.sub.x1 is measured by the housing gyroscope in the RNICs, while the angular velocity ω.sub.xo is calculated from the flywheel rotation sensor within the x-axis RNIC. Recall that the flywheel rotation sensor measures the angular velocity of the flywheel with respect to the reference frame fixed to the housing. Thus, to express the angular velocity of the flywheel in the inertial frame, one simply sums up the angular velocity measured by the gyroscope ω.sub.x1 and the angular velocity measured by the flywheel rotation sensor. This sum is the angular velocity ω.sub.xo.

[0084] At Step 704, from the known moment of inertia of the flywheel I.sub.xfw the moment of inertia I.sub.x of the unloaded spherical

[0085] DUT holder is computed using the conservation of angular momentum principle: I.sub.xfwω.sub.xo =I.sub.xω.sub.x1. Notice that the ratio between the angular velocities is constant since it depends only on the ratio of I.sub.xfw to I.sub.x which does not change in time. The same procedure is repeated to calculate I.sub.y and I.sub.z. At this point all three moments of inertia of the unloaded spherical DUT holder I.sub.x, I.sub.y, and I.sub.z are known and the only remaining unknowns I.sub.xo, I.sub.yo, and I.sub.zo (the moments of inertia of the passive/fixed components of the spherical DUT holder) can be calculated from equations (A), shown above. Note that the passive/fixed section of the unloaded spherical DUT holder is identical to the passive/fixed section of the loaded spherical DUT holder because no part of the inserted satellite adds to the passive/fixed section of the spherical DUT holder.

[0086] At Step 705, a device under test (a satellite) is loaded into the spherical DUT holder, using any of various well-known methods for doing so. At this point the spherical holder is referred to as a loaded spherical DUT holder.

[0087] FIG. 6 illustrates a simplified view of a loaded spherical DUT holder 600, having a smooth spherical surface 601 that enables the holder 600 to slide on an air bearing like bearing 401 shown in FIG. 4. The interior of the spherical DUT holder 600 is simplified to show only the elements that are actuated or relevant to the calibration method explained next.

[0088] To assist with the description, an orthogonal coordinate system is defined, consisting essentially of three orthogonal axes shown in FIG. 6 with x-axis 602, y-axis 603, and z-axis 604. The origin of the orthogonal coordinate system is defined as the intersection of the three orthogonal axes. The center of the sphere is not shown in FIG. 6. because it is obscured by satellite 630. The loaded spherical DUT holder 600 further comprises moveable weights—x-axis weight 612, y-axis weight 613, z-axis weight 614—and the axis-associated RNICs: x-axis RNIC 632, y-axis RNIC 633, and z-axis RNIC 634.

[0089] At Step 706, the gravitational torque zeroing process previously performed on the unloaded DUT holder is repeated, but this time with satellite 630 installed into the DUT space.

[0090] At Step 707, the ZGT data for the loaded testbed, i.e., the distances of the weights from the sphere center b.sub.x, b.sub.y, and b.sub.z. are acquired. These distances are not indicated in FIG. 6., but they correspond to the distances 522, 523 and 524 shown in FIG. 5. Masses M.sub.x, M.sub.y, M.sub.z of the weights 612, 613, 614 remain unchanged.

[0091] At Step 708, the moment of inertia of the spherical DUT holder without the device under test (satellite) is computed, by summing up the previously computed moments of inertia of the passive components I.sub.xo, I.sub.yo, and I.sub.zo, and of the active components from the acquired ZGT data in step 707, using the Parallel Axis Theorem.

I.sub.x′=I.sub.xo+I.sub.xx+I.sub.yx+I.sub.zx+M.sub.xb.sub.x.sup.2 +M.sub.yb.sub.y.sup.2 +M.sub.zb.sub.z.sup.2

I.sub.y′=I.sub.yo+I.sub.xy+I.sub.yy+I.sub.zy+M.sub.xb.sub.x.sup.2 +M.sub.yb.sub.y.sup.2 +M.sub.zb.sub.z.sup.2

I.sub.z′=I.sub.zo+I.sub.xz+I.sub.yz+I.sub.zz+M.sub.xb.sub.x.sup.2 +M.sub.yb.sub.y.sup.2 +M.sub.zb.sub.z.sup.2   (B)

[0092] At Step 709, the coefficients k.sub.x, k.sub.y, and k.sub.z required to ensure that the testbed inertia is eliminated or substantially reduced are computed. Recall that the effective inertia of the RNIC in the x-axis can be expressed as I.sub.xeff=I.sub.x′+(k.sub.x+1) I.sub.xfw. Requiring that I.sub.xeff=I.sub.yeff=I.sub.zeff=0 leads to the equations (C)

0=I.sub.x′+(k.sub.x+1)I.sub.xfw

0=I.sub.y′+(k.sub.y+1)I.sub.yfw

0=I.sub.z′+(k.sub.z+1)I.sub.zfw   (C)

[0093] Since all moments of inertia in equations {C} are previously calculated or known, the equations are solved for the coefficients k.sub.x, k.sub.y, and k.sub.z. Having established these coefficients, their values may be set as predetermined parameters, programming them into the RNIC controllers (not shown in the figure), so that the satellite may henceforward be characterized in a testbed that has no moment of inertia added by the spherical DUT holder and other testbed components.

[0094] One of the important advantages of the presented method for calibrating of the testbed is that it does not presume any knowledge of the mechanical properties of the satellite under test, specifically, the moments of inertia and the position of its own center of mass. However, the same setup could be used to extract the same mechanical properties of the satellite, as will now be described.

Embodiment 4: Method for Extraction of Mechanical Properties of the Satellite

[0095] In addition to the method for calibration just described, we disclose a method 800 in FIG. 8 for the extraction of mechanical properties of the satellite, namely the position of its center of mass and its own moments of inertia. It is assumed that the mass of the unloaded spherical DUT holder M.sub.uh and mass of the satellite M.sub.s are known.

[0096] Steps 801 through 808 in FIG. 8 are identical to steps 701 through 708 described above in detail with respect to FIG. 7, so the content will not be repeated here. In brief, in steps 801-804, for the unloaded testbed, gravitational torque centering is followed by computation of moments of inertia of passive components of the DUT holder. Next, in steps 805-808 with the satellite in place, gravitational torque centering is re-established, and total moments of inertia of the DUT holder are computed. As in the case of method 700, it should be noted that during those calibration method steps in which the flywheels in the RNICs are switched on, they are operated “open-loop” i.e. with no feedback applied via any controllers.

[0097] At Step 809, after ensuring that the flywheels within the RNICs and the loaded spherical DUT holder are in the resting state, i.e., their angular velocities are equal to zero. Then the flywheels within the RNICs are turned on, either sequentially or simultaneously. When the x-axis RNIC gradually increases the angular velocity of its flywheel to a constant predetermined angular velocity ω.sub.xo expressed in inertial frame, due to the conservation of angular momentum, the loaded spherical DUT holder also accelerates and remains rotating at an angular velocity ω.sub.x2 assuming that there is no friction, or the friction is negligible. This angular velocity is measured by the housing gyroscope in the RNICs, while the frequency ω.sub.xo is calculated from the flywheel rotation sensor within the x-axis RNIC. Recall that the flywheel rotation sensor measures the angular velocity of the flywheel with respect to the reference frame fixed to the housing. Thus, to express the angular velocity of the flywheel in the inertial frame, one simply sums up the angular velocity measured by the gyroscope ω.sub.x2 and the angular velocity measured by the flywheel rotation sensor. This sum represents the angular velocity ω.sub.xo . From the known moment of inertia of the flywheel I.sub.xfw, compute the moment of inertia I.sub.x″of the loaded spherical DUT holder from conservation of angular momentum principle: I.sub.xfwω.sub.xo=I.sub.x″ω.sub.x2 . Notice that the ratio between the angular velocities is constant since it depends only on the ratio of I.sub.xfw to I.sub.x″ which does not change in time. The same procedure is repeated to calculate I.sub.y″ and I.sub.z″.

[0098] Step 810. Subtract the moments of inertia of the spherical DUT holder without the device under test testbed I.sub.x′, I.sub.y″, and I.sub.z′ from the computed moments of inertia of the loaded spherical DUT holder I.sub.x″, I.sub.y″, and I.sub.z′ to obtain the moments of inertia of the satellite I.sub.xs, I.sub.ysand I.sub.zs, around x-axis, y-axis, and z-axis, respectively.

I.sub.xs=I.sub.x″−I.sub.x′

I.sub.ys=I.sub.y″−I.sub.y′

I.sub.ys=I.sub.y″−I.sub.y′  (E)

[0099] At Step 811, to determine the position of the center of mass of the device under test (satellite), the distances of the weights from the sphere center of the unloaded spherical DUT holder a.sub.x, a.sub.y, and a.sub.z, are compared with those of the loaded spherical DUT holder b.sub.x, b.sub.y, and b.sub.zz. The position of the center of mass of the satellite with respect to the center of the sphere is given by c.sub.x, c.sub.y, and c.sub.z, in the direction of x-axis, y-axis, and z-axis, respectively, where:

c.sub.x=M.sub.x(a.sub.x/M.sub.uh−b.sub.x)/M.sub.s

c.sub.y=M.sub.y(a.sub.y/M.sub.uh−b.sub.y)/M.sub.s

c.sub.z=M.sub.z(a.sub.z/M.sub.uh−b.sub.z)/M.sub.s   (D)

The equations (D) are derived using the definition of the center of mass for a system of particles.

[0100] A primary benefit of embodiments of the present invention discussed in this disclosure is the ability to test the performance of small satellites more accurately and realistically than hitherto possible. Indeed, small satellites are prone to minute disturbances, and the example testing systems described herein are designed to measure these minute disturbances. Moreover, they render the issues associated with miniature mass balancing systems and attitude feedback devices practically irrelevant.

[0101] Aspects of the invention discussed herein may be applied to compensate for moments of inertia not only of testbeds used to assess performance of ACDS systems of satellites, but of systems or devices with no connection to ACDS or satellite control at all. Moreover, they may be applied to systems with just a single or two degrees of rotational freedom. Such “1-D” and “2-D” systems are restricted to rotate only around one axis (e.g., any structure mounted on a shaft) or around two axes lying in a plane, respectively. If such a “1-D” system has a center of mass that does not lie on the axis of rotation, it vibrates while rotating. Similarly, the center of mass of the “2-D” systems must lie at the line that passes through the intersection of the axes of rotation and is orthogonal to the plane containing the two axes of rotation. A different kind of method than those described above may be used to translate the center of mass to the desired location. However, if there is a need to compensate for excessive moment of inertia (either due to the center-of-mass-centering system or for any other reason), the same kind of RNIC described for the “3-D” cases considered above can be used.

[0102] Many modifications and other embodiments of the inventions set forth herein will come to mind to one skilled in the art to which these inventions pertain, having the benefit of the teachings presented in the foregoing descriptions and the associated drawings. Therefore, it is to be understood that the inventions are not to be limited to the specific embodiments disclosed and that modifications and other embodiments are intended to be included within the scope of the appended inventive concepts. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.