The Status of Passive-Gravity-Gradient Stabilization

THE STATUS OF PASSIVE-GRAVITY-GRADIENT STABILIZATION

by Bruce E. Tinling and Vernon K. Merrick Research Scientists NASA, Ames Research Center Moffett Field, California ABSTRACT In the last few years significant advances have been made in the theory and implementation of passive attitude control systems for earth pointing satellites. In each of the systems that have evolved, the gravity-gradient provides a preferred orientation such that the axis of minimum moment of inertia points toward the center of the earth. Many techniques have been developed for damping attitude motions about this preferred orientation. This is usually accomplished by absorbing the energy of the relative motion between the stabilized body and one or more auxiliary bodies. The various damping techniques differ basically in the ambient field that determines the position of the auxiliary bodies. Systems have been devised in which the auxiliary bodies are oriented by solar, magnetic, aerodynamic, or graNity fields. Hardware for some of these systems is in an advanced state of development. The choice of a system for passive attitude control will depend upon the orbital altitude and inclination, the anticipated eccentricity, and acceptable steady-state pointing accuracy, whether or not control about the earth-pointing axis is required, the acceptable time to damp transient motions, and the probability of successful deployment of the system. The manner in which the choice of the ambient field utilized to effect the damping enters into the fulfillment of these requirements is discussed to point out the advantages and disadvantages of each system. To date, passive graVity-gradient stabilized satellites have been flown only at low orbital altitudes. This flight experience is discussed and the performance compared with theoretical predictions. For many applications, it is desirable to place earth-pointing satellites at much higher altitudes, particularly in stationary, or synchronous, orbits. An experimental flight program which could prove the practicability of passive stabilization at this altitude is described.

earth has been known since the publication of Lagrange's famous paper on the librations of the moon. 1 This phenomenon can be utilized to orient an artificial earth satellite if a means is provided for damping librations. Many damping systems have been devised. These systems can be divided into three classes; those which require both sensors and power, those which require power, but no sensors, and those which require neither power nor sensors. The first two classes might be considered either semiactive or semipassive. Typical examples are the active damping system which accelerates inertia wheels in response to sensor signals,2 and the control-moment gyro which requires no sensors. 3 This paper is concerned solely with the completely passive third class, which requires neither sensors nor power. The first recorded concept of a passive-gravitygradient control system is believed to be contained in a United States patent filed by Roberson and Breakwell in 1956. 4 The principal activity in the 8 years since this patent declaration has been in theoretical studies leading to practical mechanizations of gravity-gradient control systems. Only within the last 2 years has hardware been developed to the stage of providing solutions to the principal practical problems. As a result we are now on the threshold of developing earthpointing stabilization systems which promise to be simpler, lighter, cheaper, and more reliable than any preViously devised. The recent advances in this technology have stimulated considerable commercial, scientific, and military interest, and plans are being executed to extend the flight experience with these systems to cover the complete range of practical orbits. The principal difference in the various mechanizations of passive-gravity-gradient control schemes lies in the choice of the ambient field that is used as a reference for the damping system. Systems have been devised that are referenced to either the solar, magnetic, aerodynamic, or graVity fields. The choice of a system for a given application depends upon the altitude, inclination, and anticipated eccentricity of the orbit, the acceptable steady-state pointing accuracy, whether or not control about the earthpointing axis is required, the acceptable time to damp transient motions, and the probability of

INTROIUCTION The fact that the graVity-gradient can cause an earth satellite to point one axis toward the

Superior numbers refer to similarly-numbered references at the end of this paper. 234

successful deployment of the system. The manner in which the choice of ambient field for effecting the damping enters into the fulfillment of mission requirements is discussed to point out the advantages and disadvantages of each system.

to maintain excellent pointing accuracy. However, because the gravity-gradient diminishes as the cube of the orbit radius, balancing of disturbance torques created by solar pressure forces becomes more important with increasing altitude. One method of balancing these forces is to make the satellite symmetrical. Unfortunately, even when deployed symmetrically, gravity-gradient rods will have asymmetries introduced by thermal distortion. The source of this asymmetry is illustrated in ~ig. 1. Differential expansion between the illuminated and dark sides of the rod cause it to assume apprOXimately a circular shape. Solar radiation pressure forces will then be greater on the end of the rod which is more nearly normal to the incident rays and a torque about the center of mass will exist. The predominant component of this disturbance varies as even harmonics of the angle W with the second harmonic being predominant. The magnitude of the disturbance increases as the cube of the rod length, whereas, since most of the mass is at the rod tips, the restoring torque increases approximately as the square of the rod length. Thus, there is a limit beyond which additional rod length will impair, rather than improve, pointing accuracy.

The more useful applications of earth oriented satellites are for high altitude orbits, particularly the synchronous orbit. To date, passive systems have been flown only at orbital altitudes of the order of 800 km. This flight experience is discussed and the performance of the passive control systems is compared with theoretical predictions. Means for extending the flight experience to include the synchronous orbit are discussed.

RESTORING TORQUES Before proceeding to a discussion of the influence of damping technique on the performance of a gravity-gradient control system, it is well to review the state-of-the-art of the technique used to create the necessary restoring torques. It is well known that the magnitude of the gravity-gradient restoring torque varies directly with the difference between the principal moment of inertia about the earth-pointing axis and axes transverse to it and inversely as the cube of the radius of the orbit. If the transver~e principal moments of inertia differ, a gyroscop1C torque will exist which will orient the satellite about the earth-pointing axis. Furthermore, by choosing the proper ratios of the principal moments of inertia, it is possible to select, within certain bounds, the natural oscillation frequencies of the satellite. 5 ,6,7

The situation is alleviated by silver plating the surface of the rods. This serves two functions. The more important function is to reflect most of the incident radiation. The thin coating of silver also improves the thermal conductivity, thereby reducing the thermal gradient. Some research conducted at General Electric Co., sponsored by NASA, has been directed toward obtaining the best possible reflective and conductive properties by forming the rods from a silver alloy. Short rods were formed having suitable structural properties, but manufacturing techniques have not been perfected to provide silver alloy rods of sufficient length and straightness for flight hardware.

The satellite structure, as stored in a booster prior to launch, cannot provide the differences in moment of inertia required to limit pointing errors to acceptable levels. Only one technique that has been proposed for creating the required moment of inertia differences has met with any success. This technique was originally proposed by Kamm in a paper published in 1962 which described a practical passive control system. a The technique consists in unrolling a prestressed tape, usually manufactured from beryllium-copper, which assumes the shape of an overlapped circular tube. Extending these tubes or rods from the satellite, with or without masses at their tips, affords a method of increasing the moment of inertia by factors as great as 1000 with little expenditure of satellite mass. The technology of manufacturing these elements is considerably older than gravitygradient technology. They have been manufactured as storable retractable antennas for about 20 years by de Havilland Aircraft Ltd. of Canada.

A possible alternate solution to the thermal bending problem is to provide an equal and opposite torque by attaching conical reflectors to each rod tip. The size of reflector required is modest, being only slightly larger than the usual tip mass. However, this technique requires that the radius of curvature be known, and that the rod lie in the plane defined by the sun and the position of the undeflected rod. The radius of curvature is calculable, but recent experimental measurements have indicated that the plane of bending deviates as much as 20 0 from the plane of symmetry. The reason for this is that the tube is asymmetric in cross section (see sketch in Fig. 1) and that the coldest and hottest points are not necessarily diametrically opposed. At present, no solution to the thermal bending problem is in sight. For rod lengths to 30 meters or less tip masses can prOVide the required moments of inertia. Limiting the length of the rods is not a serious restriction for orbits up to several earth radii, but the additional tip

For orbital altitudes of the order of an earth radius, a single rod of the order of 10 to 20 meters in length with a tip mass of several kilograms will provide sufficient gravity torque 235

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mass increases the control system weight considerably for the synchronous mission.

bismuth shell, in conjunction with the horseshoe magnets in the inner sphere, prOVides the centering force which prevents contact between the two spheres. The centering force is caused by the repulsive action of a magnetic field upon a diamagnetic material such as bismuth. Therefore, no friction between the two spheres can eXist, so any coupling between them is purely viscous.

One other source of error is attributable to the gravity-gradient rods, namely, errors introduced by initial structural curvature. The accepted figure for this error is roughly 1/2 0 . This error can possibly be reduced for multirod configurations if the curvature is determined on the ground and the rods are paired so that the principal axes of inertia lie in the desired directions.

The viscous fluid between the inner and outer spheres dissipates energy when relative motion takes place between the inner and outer spheres. It also serves to cushion the inner sphere against shock and acceleration during boost. Further mechanical details of this unit as well as planned improvements to secure greater lifetime and reliability can be found in reference 7.

DAMPING Techniques dependent on the geomagnetic field Simple and effective techniques have been devised for damPing librational motions by utilizing the geomagnetic field. The ultimate in simplicity is achieved in a system devised by the Applied Physics Laboratories of Johns Hopkins University.s The damping device in this instance consists of permeable rods rigidly attached to the satellite structure. As the librational motion causes changes in the direction of the magnetic field imposed on these rods, hysteresis losses occur within the material, thereby dissipating the librational energy. The mass of magnetic material required is small at low altitudes. For instance, the mass of material for a satellite designed for an altitude of 650 km is approximately 200 g. As the altitude is increased, the strength of the geomagnetic field varies inversely as the cube of the orbit radius, and the moment of inertia of the satellite is likely to be required to increase by the same amount to prOVide sufficient restoring torque. These two factors will cause the required mass of magnetic hysteresis material to increase rapidly with orbit radius.

the its rod tip

The damPer unit can be placed anywhere within satellite. However, the obvious choice for location is at the tip of a graVity-gradient where it serves the additional function of a mass to increase the moment of inertia.

The number of gravity-gradient rods required will depend upon whether two- or three-axis stabilization is required and upon the orbital altitude. For a low altitude satellite, only a single rod will be required to point an axis toward the earth. This is the particular system built by General Electric for a Naval Research Laboratory Satellite (see Fig. 3). As the altitude is increased, the ratio of solar pressure disturbing torque to gravity-gradient restoring torque will increase in proportion to the cube of the orbit radius and the satellite must be made symmetrical by extending rods in opposite directions from the stabilized package. Additional rods must be deployed to provide dissimilar transverse principal moments of inertia if the vehicle is to be oriented about the earth-pointing axis. An additional pair erected along the desired direction of velocity vector will suffice to provide the three dissimilar moments of inertia. However, except for extremely large satellites, the moment of inertia of the graVity-gradient rods contributes nearly 100 percent of the total moment of inertia. Therefore, when only four rods are deployed, the mass distribution will be nearly planar. For this mass distribution, the satellite will have natural frequencies of orbital and twice orbital frequency. Since solar pressure disturbances and disturbance caused by orbital eccentricity also occur at these frequencies, it will usually be necessary to erect a third and shorter pair of rods orthogonal to the others so that the disturbance and natural frequencies differ. A three-axis magnetically damped system will therefore probably require six rods.

Another successful damping scheme has been devised in which an auxiliary body, containing a bar magnet, follows the geomagnetic field. The auxiliary body is coupled to the satellite through a spherical viscous damPer. This system has been studied by several industrial and governmental organizations within the United States and a successful mechanization has been developed by the General Electric Co. and flown successfully at an altitude of 800 km. The unit weighed approximately 4.5 kg. Units for higher altitudes will be somewhat heavier because the dipole strength must be increased for the auxiliary body to follow the geomagnetic field. A sketch of the damper unit is shown in Fig. 2. It consists of two concentric spheres separated by a viscous fluid. The internal sphere contains a bar magnet and secondary horseshoe magnets mounted circumferentially within the sphere. The outer sphere consists of two integral concentric shells; an inner shell of bismuth and an outer shell of aluminum alloy. The

The magnetic damper can also be used to advantage for satellites at altitudes of less than 600 km where aerodynamic forces become significant, particularly when a payload must be positioned for retrofiring into the atmosphere. 7 Stabilization in this case is prOVided by

237

Fig. 3. - Artist's drawing of Naval Research Laboratories Satellite.

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239

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240

fraction of a satellite lifetime which may be 5 years or more. However, it is estimated that attitude disturbances as large as 150 caused by micrometeorites will occur about once per year, with smaller disturbances more frequent and larger disturbances less frequent.~o If this estimation proves to be correct, a considerable fraction of the lifetime of a synchronous, magnetically damped satellite will be spent recovering from micrometeorite hits.

extending two rods about 50 feet in length with the magnetic dampers at their tips. As shown in Fig. 4, the rods are swept back slightly so that the aerodynamic forces provide restoring torques that augment the gravitational torque about the pitch axis and the gyroscopic torque about the local vertical. The angle between the rods must be greater than 90 0 , otherwise the gravitational and aerodynamic torques will be in opposition and the stable position will be a function of altitude.

The magnetic field at synchronous altitudes will not be stationary because of the distortion of the magnetic field by the solar activity. Satellite probes have indicated that the solar wind deforms the magnetosphere, compressing it on the sunlit side of the earth and elongating it on the dark side. To a synchronous satellite, this deformation would appear as a diurnal variation in the field direction. A recent model of the distorted field (Mead~~) indicates this distortion to be small at synchronous altitude. Large nonperiodic variations caused by magnetic storms represent an additional disturbance. Measurements of field strength and direction from Explorer 6 have indicated that the direction of the field can change by as much as 60 0 and the field strength can be halved during a storm. L2 The occurrence of magnetic storms and the steady-state diurnal variation will cause damper induced errors and, therefore, will influence the selection of damping levels.

All systems that depend upon the geomagnetic field for damping will have a pointing error induced by the damper. The underlying reason is that the magnetic field is nongeocentric so that relative motion exists between the satellite and the direction of the magnetic field vector even for zero attitude error. A part of this motion will be transmitted to the satellite, depending upon the damping level. Results given by Moyer et al. for a system with an oriented auxiliary body are shown in Fig. 5. 7 These results are for polar orbits where the relative motion of the magnetic field vector is the greatest. For a given induced error the decay rate is constant when expressed in orbits and is very nearly independent of altitude. At low altitudes, or high orbital rate, a small damper induced error will not imply an excessive number of days to damp. For example, for a 650 km orbit, an induced error of only 0.50 implies a time constant (time to damp to lie times the initial amplitude) of the order of 3 days. On the other hand, the same induced error for a 24 hr. orbit implies a time constant on the order of 45 days. To reduce the time to only 10 days implies an induced pointing error on the order of 20 •

Techniques dependent upon the gravitational field The damping as well as the restoring torque can be made to depend upon the gravitational field. Passive control schemes based upon this concept have an advantage in that there are no damperinduced pointing errors. Two distinct types of damping systems have been devised. Each relies on attitude errors to induce relative motion between the satellite and an auxiliary body.

The damper induced error is smallest for equatorial orbits and would vanish for equatorial orbits if the geomagnetic and geocentric poles were coincident. It is fortunate that they are not coincident since, in this event, damping of librations within the orbital plane would also vanish. The synchronous equatorial orbit is an interesting limiting case. Here, the satellite is stationary relative to the nominal geomagnetic field so that the induced error does not exist. The magnetic field vector lies between 100 and 200 from the normal to the plane of the orbit, depending upon geomagnetic longitude. There is a component of the field, therefore, that permits damping of the pitching motion within the orbital plane. Damping of the pitching motion within a reasonable period, 20 orbital periods per time constant, for example, implies that the motion in roll about the velocity vector will be critically damped.

One of the damping techniques dissipates the energy of the pumping motion of an auxiliary mass attached to the satellite. Theoretical studies of systems of this type have been reported by Newton~3 and Paul.~4 This damping system is effective in damping motions within the orbital plane, but damping of the cross-plane motion vanishes for small attitude motions. Some other technique must therefore be provided to damp the roll motions for a complete system. In a system of this type built by the Applied Physics Laboratories, the cross-plane motion was damped by magnetic hysteresis rods. An artist's conception of this satellite is shown in Fig. 6. 9 The auxiliary body is attached to the end of a gravitygradient boom by a weak spring. Damping is afforded by mechanical hysteresis within a cadmium coating on the spring. The system is lightly damped, requiring about 50 orbits to reduce the amplitude to lie times the initial value. It is

The long time constants characteristic of magnetically damped satellites at synchronous altitude are not particularly severe if only the initial transient motion is considered. A 20-day time constant for small angles usually implies stabilization within 50 days which is not a large

241

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Fig. 10. - Solar-orien ted damping system.

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postulated by Newton that damping times of the order of 20 orbital periods are possible with this system.

axes of rotation at right angles - in effect, a universal joint. A sketch of this arrangement is shown in Fig. 8(a). Another version of this system is possible, if the satellite and damper mass distribution are interchanged (see Fig. 8(b)). The principal difficulty with this inverted arrangement is that the body to be stabilized is the body of least inertia, and, in the presence of a disturbance, will have a greater deviation from the vertical than does the vertical rod.

It is possible to improve the damping afforded by pumping motion of a spring-mass system and to provide damping about all axes if the coupling between the torsional and longitudinal modes of the spring is exploited. This coupling exists if the spring wire is noncircular in cross section. The torsional motion is damped by dissipating the energy of the relative motion between the tip mass and an additional body within it. The additional body is suspended in a viscous fluid. A system having these features, known as the Rice/Wilberforce damper, has been developed by Goodyear Aerospace COrp.15 The damping time constant is about 5 orbital periods.

In the Vertistat configuration, the equilibrium positions of the principal axes of all three bodies lie either within the orbital plane or normal to it. For this situation, the pitch motion within the orbital plane and the roll-yaw motion about the other two axes are independent insofar as the small angle motion is concerned. This simplifies the mathematics of analysis and optimization. However, if two connected bodies are oriented so that at least two of the principal axes are skewed to the orbital plane, it will be found that any attitude motion will cause relative motion of the damper rod. It is possible, therefore, to prOVide damping about all three axes with a single auxiliary body that has a single degree of freedom. 17 A sketch of a system of this type is shown in Fig. 9(a). The principal axes of the satellite body are either in the plane of the crossed booms or normal to them. The earthpointing axis bisects the acute angle formed by the booms. The cross products of inertia of the satellite body and the damper rod, relative to a reference frame iIDbedded in and normal to the orbital plane, are equal and opposite. This is dictated by nature because the principal axes of inertia of the entire satellite must be either normal to or within the orbital plane when the satellite is in equilibrium. For low altitudes, where the gravity-gradient torques are large compared to solar torques, it is possible to simplify the mechanization. This is done, in effect, by cutting the symmetrical configuration in half. The damper rod can be supported from one end as well as from the center when it lies in the horizontal plane since it is in equilibrium regardless of where it is supported. If the satellite body, exclusive of graVity-gradient rods, is reasonably slender, configurations such as shown in Fig. 9(b) are possible. Thus, for this case, three-axis stabilization is possible with the erection of only two gravity-gradient rods, one of which has a single degree of rotational freedom. If the satellite body is a slender cylinder, as shown in Fig. 9(b), its axis will be skewed to both the horizontal and orbital planes.

Many systems have been devised in which attitude errors cause relative rotational motion between the satellite and an auxiliary body. This idea originated with Roberson and Breakwell who applied for a United States Patent in 1956. 4 The first practical mechanization was by Kamm. 8 A sketch of an arrangement of this type is shown in Fig. 7. Kamm suggested the use of' the storable tubular rods and recognized the optimum orientation of the auxiliary bodies. The auxiliary body, which in this case is a simple rod, is located in the horizontal plane. In this location, the damper rod is unstable in the gravitational field and must be stabilized with a spring. The reason for placing the damper rod in the horizontal plane can be seen intuitively from the motion of two rods, both alined with the vertical and connected by a viscous damper. If one of the rods is disturbed, the relative motion between them will be damped until they are both oscillating with the same phase and amplitude. This mode of motion will be undamped. The greatest possible damping, therefore, might be expected when the damper rod lies in the horizontal plane. This observation has been rigorously proved by Zajac for simple pitching motion within the orbital plane. 16 In the Vertistat configuration, two damper rods are required. Each is connected to the stabilized body through a frictionless spring-damper mechanism. The rod with the greater moment of inertia lies in the orbital plane and will damp only motion within that plane. The other damper rod is normal to the orbital plane and, because of gyroscopic coupling, is effective in damping motions about the other two axes. As long as these two rods have different inertias, a preferred orientation about the earth-pointing axis will exist.

The damping of the "Vertistat" and all of its derivatives is much better than any of the other systems which have been devised. It is possible for most of the configurations to achieve damping constants (time to reduce amplitude to lie) of less than one orbital period for all modes of motion. However, a system designed for the maximum possible damping will usually suffer from reduced restoring torques, greater sensitivity to

A system similar to "Vertistat" has been proposed by Bell Laboratories and others. The Bell System,10 uses a single auxiliary body consisting of a pair of crossed rods. This body is connected to the satellite through two single-degreeof-freedom spring-damper mechanisms with their

243

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off-design conditions, degradation in the response to the principal oscillatory disturbances torques which occur at once and twice orbital frequency, and degradation of the response to orbital eccentricity. For this reason, a balance between damping time and steady-state pointing accuracy is usually sought in the design. A realistic damping time constant for a practical system, therefore, appears to lie between one and two orbital periods.

restraint. Two dampers are coupled together at the end of each support rod to prOVide 3600 of freedom about any axis. The disadvantage of this system is its extreme complexity, requiring six rods, four damper units, and two inflatable structures. A possible future development of this design principle is to place a solar power system within the auxiliary bodies. This would mean further complication since a way must be found to deliver the power through the coupling without incurring either spring or friction torques.

The key to a successful mechanization of a "Vertistat" type system is a spring-damper device. There are two devices in an advanced state of development, one at Bell Laboratories lO and one at General Electric Co. lS The Bell Laboratories' device relies on a torsional flexure to produce the required spring torques. Damping torque is produced by rotational hysteresis from relative displacement between a magnet, attached to the rotor, and magnetic material attached to the case. The damping torque for this device is amplitude dependent since it depends on magnetic hysteresis. The General Electric device relies on diamagnetic forces to prOVide a frictionless suspension of the rotor. Damping is achieved by the generation of eddy currents in an aluminum disk as it rotates between poles of permanent magnets. The torsional restraint is obtained from ferromagnetic inclusions in the aluminum disk. These inclusions interact with the magnets to restore the disk to a given rotational position. The inclusions are shaped to give a linear torsional restraint over ±45° of rotation. This device differs dynamically from that developed by Bell Laboratories in that the damping is viscous, depending upon angular velocity rather than amplitude.

Inflatable structures have also been suggested for use at low altitudes where their orientation will be dictated by aerodynamic forces. The stabilization method appears to hold no advantage over the magnetic technique described earlier.

STABILIZATION SYSTEM WEIGHT The weight of a passive attitude control system is determined principally by the moment of inertia required to achieve the desired pointing accuracy. For this reason, the weight of the stabilized body influences control system weight only in that larger bodies are likely to have larger disturbance torques. About 5 kg were attributable to the control system of the passively oriented satellites flown to date. 7 ,8 This figure can be taken as a current lower limit since it is representative of lowaltitude satellites with no stabilization about the earth-pointing axis. The heaviest control systems will be required for three-axis stabilization at synchronous altitude. Estimates of the weight of these systems by Moyer and Foulke are summarized in Fig. 11.lS For satellites of the weight presently contemplated, the control system weight will be between 12 and 20 kg, depending upon the type of damping. A gravity-oriented damper is shown to be superior to the other types in this respect. However, if stabilization about the earth-pointing axis is not required, there will be little difference between the weights of the magnetic and gravity-oriented damping systems.

Technique dependent upon solar field General Electric Co. has proposed using solar pressure to orient the damping bodies. lS In this technique, the satellite body is coupled through an eddy current disk to inflatable auxiliary bodies similar in structure to Echo II. The auxiliary bodies are oriented by solar pressure and prOVide an inertial reference for the damper (see Fig. 10). Since the auxiliary bodies are stationary in inertial space, a predictable steady-state bias from the zero gravity-torque position is introduced which is dependent upon orbital rate (or altitude) and the damping coefficient. Oscillations are induced when the solar reference is lost through eclipse. This effect is less severe as the altitude is increased. In this system, therefore, the damping time associated with a given damper-induced error decreases as the altitude increases. The time constant for errors of the order of 2 0 is about five days at synchronous altitude.

FLIGHT PROGRAMS Two earth-pointing, passively stabilized satellites have been flown by the United States. The excellent results obtained from these flights, which were at altitudes of the order of 600 km, have added increased stimulus to proceed with experiments at altitudes which will lead to successful passive attitude control for synchronous orbits. The first gravity-gradient-stabilized satellite was conceived and built by the Applied Physics Laboratories of Johns Hopkins University and was

The dampers are similar in concept to the eddy current damper described for the pure gravity system except that there is no torsional 245

launched in mid 1963. 9 As noted earlier, this satellite was damped through pumping librations of an attached spring mass and by magnetic hysteresis rods. The transient motion following deployment of the stabilization system is shown in Fig. 12. The motion damped in about 16 days or about 2 0 orbits. The final pointing accuracy was from 5 to 100. Part of this error was caused by thermal bending of gravity-gradient rods. The rods were formed from beryllium-copper and had no reflective coating. When the satellite passed from the earth's shadow into the sunlight, the rapid heating of the rod induced attitude oscillations which had a Eeriod of several minutes and an amplitude of 5. Berylliumcopper has little internal damping and it is reported that these oscillations did not dissipate during an orbital period.

The design of gravity-gradient-stabilized Applied Technology Satellites launched into a synchronous orbit will be influenced by the extrapolated results of the earlier flight at 10,000 km. The synchronous satellite would determine whether east-west station keeping is compatible with gravity-gradient stabilization. A successful flight of the synchronous satellite will demonstrate that passive graVity-gradient stabilization is feasible for all practical orbits.

6

REFERENCES

(1) (2)

The second graVity-gradient satellite was launched by the Naval Research Laboratories and used the General Electric magnetically oriented damping system (see Fig. 4).7 The attitude instrumentation for this flight determined when the deviation from the local vertical was within one of several ranges. The smallest of these determined when the error was less than 6 0 • The librations were damped to within this smallest range after 40 orbits. A precise correlation between measured and predicted damping was not possible because of the limited instrumentation. However, the altitude measurements usually fell within the predicted attitude envelope.

(6)

(7)

Several future flight experiments are being considered for synchronous and near synchronous altitudes. For a system which uses the General Electric magnetically oriented damper at an altitude of 20,000 to 28,000 km, the deployed satellite Imlst be sY!llllletrical. The goal of this experiment would be to demonstrate an earthpointing system at near synchronous altitude. Specific design goals are to damp to within ±300 within 20 orbits and to achieve a final pointing accuracy of 40 or better for a circular orbit with additional errors due to eccentricity held to less than 1 0 per percent eccentricity.

(8)

(10)

(11)

By far the most comprehensive graVity-gradient stabilization experiments are embodied in the Applications Technology Satellite program of the NASA. This program includes satellites stabilized by the inertially coupled system that relies solely upon the graVity-gradient. A deployed satellite similar to that shown in Fig. 9(a) in a 10,000 km orbit could verify the performance predicted by theory. The design may include the ability to vary disturbances by introducing known additional torques through a magnet or possibly a small inertia wheel. Other experiments include varying the length of the rods and the angle between them, viewing the bending of the gravity-gradient rods with a TV camera, and measuring the effectiveness of both the eddy current and magnetic hysteresis dampers.

(12)

(14) (15)

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