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We describe a new space gravitational wave observatory design called LAGRANGE that maintains all important LISA science at about half the cost and with reduced technical risk. It consists of three drag-free spacecraft in the most stable geocentric formation, the Earth-Moon L3, L4, and L5 Lagrange points. Fixed antennas allow continuous contact with the Earth, solving the problem of communications bandwidth and latency. A 70 mm diameter AuPt sphere with a 35 mm gap to its enclosure serves as a single inertial reference per spacecraft, which is operated in "true" drag-free mode (no test mass forcing). This is the core of the Modular Gravitational Reference Sensor whose other advantages are: a simple caging design based on the DISCOS 1972 drag-free mission, an all optical read-out with pm fine and nm coarse sensors, and the extensive technology heritage from the Honeywell gyroscopes, and the DISCOS and Gravity Probe B drag-free sensors. An Interferometric Measurement System, designed with reflective optics and a highly stabilized frequency standard, performs the inter-test mass ranging and requires a single optical bench with one laser per spacecraft. Two 20 cm diameter telescopes per spacecraft, each with in-field pointing, incorporate novel technology developed for advanced optical systems by Lockheed Martin, who also designed the spacecraft based on a multi-flight proven bus structure. Additional technological advancements include the drag-free propulsion, thermal control, charge management systems, and materials. LAGRANGE sub-systems are designed to be scalable and modular, making them interchangeable with those of LISA or other gravitational science missions. We plan to space qualify critical technologies on small and nano satellite flights, with the first launch (UV-LED Sat) in 2013., Comment: 24 pages, to be submitted to Classical and Quantum Gravity

Gravity Probe B, launched 20 April 2004, is a space experiment testing two fundamental predictions of Einstein's theory of General Relativity (GR), the geodetic and frame-dragging effects, by means of cryogenic gyroscopes in Earth orbit. Data collection started 28 August 2004 and ended 14 August 2005. Analysis of the data from all four gyroscopes results in a geodetic drift rate of -6,601.8+/- 18.3 mas/yr and a frame-dragging drift rate of -37.2 +/- 7.2 mas/yr, to be compared with the GR predictions of -6,606.1 mas/yr and -39.2 mas/yr, respectively (`mas' is milliarc-second; 1mas = 4.848 x 10-9 rad).

Helium gas flows continuously from a low-temperature dewar supplying propellant for satellite thrusters. Thrust is obtained by movement of a spool located between two opposing nozzles, thus differentially restricting the helium flow. The design procedure and test results of a high bandwidth, low power, linear electromagnetic actuator for the pool movement are presented. Two mathematical models are used to describe the helium flow through the thruster under different flow conditions. The experimental data measured in the vacuum chamber and the computational results from the mathematical models are compared. The models agree with the experiment to 3 percent except near a Knudsen number of 1.

Methods of limiting perturbations in microgravity experiments are proposed. An acceleration level below 10 to the -4th m/s-squared is necessary to maintain an undisturbed microgravity environment. Machinery vibrations, crew motion, and the firing of vernier thrusters produce acceleration levels greate than 10 to the -4th m/s-squared. The use of a weak spring system or simple electromagnets to isolate an experimental table from these factors is described. The manners in which crew motion and vernier firing are countered by the springs are examined. The steady acceleration caused by atmospheric drag, gravity gradient force, and steady rotation can be maintained below 10 to the -th m/s-squared; however, the springs can protect the table from these accelerations if required.

Dynamic simulation of the Gravsat's attitude and translation control system is used to provide an upper bound for the fuel supply and test the feasibility of the preliminary design. A preliminary design is made for the disturbance compensation system (DISCOS) sensor, the thruster control laws, reaction wheel control laws, and the onboard state estimators. The sensor analysis and noise measurements show no problems in scaling the Triad navigation satellite sensor design up to meet the Gravsat requirements, except for proof mass center-of-mass offset. A promising technique is proposed to measure and eliminate this error. The covariance analysis confirms that a sophisticated post-flight data fit will be necessary to reconstruct a scientifically useful proof mass state. The DISCOS sensor will have to be continuously calibrated from the inflight data to achieve this reconstruction.

Experimental verification of the theory describing arbitrary motions of an airfoil is reported. The experimental apparatus is described. A mechanism was designed to provide two separate degrees of freedom without friction or backlash to mask the small but important aerodynamic effects of interest.

According to certain theoretical predictions, the Newtonian laws of motion must be corrected for the effect of a gravitational field. Schiff (1960) proposed an experiment which would demonstrate the effect predicted by Einstein's Theory of General Relativity on a gyroscope. The experiment has been under development at Stanford University since 1961. The requirements involved make it necessary that the test be performed in a satellite to take advantage of weightlessness in space. In a discussion of engineering developments related to the experiment, attention is given to the development of proportional helium thrusters, the simulation of the attitude control system, aspects of inner loop control, the mechanization of the two-loop attitude control system, the effects of helium slosh on spacecraft pointing, and the data instrumentation system.

The theory for aerodynamic loading associated with transient motion of an airfoil in a wind tunnel has been extended and verified experimentally. A generalized Theodorsen Function which includes wall effects is described, and finite-state approximations are developed for large wing-semichord to wall-spacing ratios. Finally, experimental results are presented which verify the theoretical predictions for transient airfoil motions. These results were obtained using a small, low-subsonic wind tunnel with a unique airfoil suspension design that provides uncoupled sensing and actuation for two degrees of freedom.

The parameters and relations associated with optimal systems are examined, taking into account a quadratic performance index and a root square locus plot, including the characteristic roots of the optimal system and its adjoint system as a function of the cost function weights. The calculation of the locus is described and the employment of the considered relations in studies of a drag-free satellite is discussed. Attention is given to weights regarding the initial states, questions of rotating integral control, approaches for experimental verification, and the performance of various methods for the reduction of fuel consumption due to center of spin offsets.

The progress accomplished in the Stanford Gyro Relativity program during the period November 1974 to October 1975 was described. Gyro developments were continued in the main laboratory dewar, concentrating on the operation of a three axis gyro readout and on improvements to the methods of canceling trapped fields in the rotor; these efforts culminated in the first successful observation of the London moment in the spinning gyro rotor in March 1975. Following a review meeting at that time, a new goal was formulated for the next 12 to 18 months, namely to operate a gyroscope in the new ultra-low field facility with readout resolution approaching 1 arc-second. The following other tasks were also completed: (1) sputtering work, (2) magnetometry, (3) construction and installation of the North Star simulator, (4) analysis of torques on the gyro, especially in inclined orbits, (5) equivalence principle accelerometer, and (6) analysis of a twin-satellite test of relativity.

The translational and attitude control of a drag-free, rotating satellite are usually designed to be independent of each other. However, coupling may exist which can produce instability. The different types of coupling and the stability conditions are studied, using root locus type techniques. The resulting root loci have unusual shapes due to the nonlinearity of the characteristic polynomial in terms of the coupling parameters. It is possible, however, to use standard root loci techniques by appropriate factoring and subsequent superposition of loci. The effect of the main parameters upon the overall stability of the system is studied. Both, attitude and translation instabilities eventually arise - the former being more critical for typical satellite designs. An analytical expression for the stability boundary is developed.

The mass center location of a spinning drag-free satellite can be estimated because there is control required to accelerate the mass center along the axis of spin as long as there is some nutation in the spinning motion. Linear and nonlinear models are compared and observability discussed. Online estimation fails when nutation is damped so an offline mechanization is proposed. A new sensor has been designed to permit greater relative motion than was possible on the drag-free satellite flown in 1972 (JH-1). Experimental laboratory results using a spinning vehicle with the new sensor mounted 30 cm from a spherical air bearing support are presented which confirm earlier simulation results.

Performance tests of gyroscope operations and gyroscope readout equipment are discussed. The gyroscope was tested for 400 hours at liquid helium temperatures with spin speeds up to 30 Hz. Readout by observing trapped magnetic flux in the spinning rotor with a sensitive magnetometer was accomplished. Application of the gyroscope to space probes and shuttle vehicles.

The original motivation for studying control laws for pulse plasma systems was based on the improved life characteristics possible with pulse plasma jets. These pulse plasma units are relatively massive compared with cold gas thrustors. As a result, therefore, significant mass savings can be achieved by minimizing the the number of thrustors. The control laws, therefore, were developed for thrust available from two thrustors only. In a spinning satellite, these thrustors are sufficient to completely control the vehicle as long as the spin rate is sufficiently high for a given level of external disturbance. The thrustors are canted so that a component of each is along the plus and minus spin axis. The other component of each thrustor acts in the radial direction. It is sufficient to analyze the behavior in the plane of spin assuming a single thrustor.

A spacecraft going within 4 solar radii of the Sun experiences an acceleration up to .00005 g from solar radiation pressure, and significant (though smaller) accelerations from the solar wind. To obtain satisfactory information about relativistic effects and the Sun's quadrupole mass coefficient, these non-gravitational accelerations are either measured or compensated out by making the spacecraft drag free. A proof mass inside the spacecraft structure is shielded from the external forces, so that it follows a nearly ideal gravitational orbit, and a control system activates gas jets (or other translational forcers) to make the vehicle follow the mass. The problems are mechanizing the control laws and minimizing extraneous effects such as the self gravitational pull of the spacecraft. The extraneous forces can be averaged in one plane by having a spinning vehicle.

The primary goal of the current gravity gradiometer research at Stanford has been to establish the feasibility of using a gravity gradiometer with 1 E accuracy, as the primary sensor in various applications. The two applications considered here in detail are geodesy missions and inertial navigation systems. Preliminary sections on gravity models and gravity gradiometer bias estimation are also included.

A satellite configuration having two gyroscopes with axes parallel to the boresight of a telescope and two at right angles to the telescope and approximately parallel and perpendicular to the earth's axis is proposed for measuring geodetic precessions due to the earth's motion about the sun, higher order geodetic terms calculated from the earth's quadrapole mass moment (0.010 arc-sec/year in a 400 nautical mile polar orbit), and deflection by the sun of the starlight signal for the reference telescope. Data from the experiment also contain large periodic signals due to the annual and orbital aberrations of starlight which are useful in providing a built in reference signal of known amplitude for scaling the relativity signals, and should yield a singularly precise measurement of the parallax of the reference star. The development of the gyroscope and its readout system are discussed, as well as signal integration, drag-free control, and attitude control.

This paper presents several different methods that can be used to determine the aircraft orientation (attitude) without direct measurement. The methods combine state estimation techniques with measurements from solid state pressure sensors, accelerometers, and magnetometers to determine the aircraft state. The paper demonstrates how the estimation algorithms were validated and compared using flight test data, and it presents results of performance sensitivity analyses of sensor error, modeling inaccuracies, and wind disturbance effects on the attitude estimation errors. If implemented, the methods presented could make it possible to eliminate directional and vertical gyros and to change from many isolated sensors to an integrated, small, reliable sensing package for determining the aircraft state.

The criteria of the significant value of the panel instruments used in general aviation were examined and kinematic equations were added for comparison. An instrument survey was performed to establish the present state of the art in linear and angular accelerometers, pressure transducers, and magnetometers. A very preliminary evaluation was done of the computers available for data evaluation and estimator mechanization. The mathematical model of a light twin aircraft employed in the evaluation was documented, the results of the sensor survey and the results of the design studies were presented.

Candidate autopilot control laws that control the engine failure transient sink rates by demonstrating the engineering application of modern state variable control theory were defined. The results of approximate modal analysis were compared to those derived from full state analyses provided from computer design solutions. The aircraft was described, and a state variable model of its longitudinal dynamic motion due to engine and control variations was defined. The classical fast and slow modes were assumed to be sufficiently different to define reduced order approximations of the aircraft motion amendable to hand analysis control definition methods. The original state equations of motion were also applied to a large scale state variable control design program, in particular OPTSYS. The resulting control laws were compared with respect to their relative responses, ease of application, and meeting the desired performance objectives.

The Triad satellite is maintained within 1 mm of its proof mass (a small metal ball) in a purely gravitational orbit, using a drag-free control system, named Discos, as the disturbance compensation system. The Discos proof mass is surrounded by an outer shell which holds fuel tanks and cold gas jets. The shell shields the proof mass from such nongravitational forces as radiation pressure, atmospheric drag, and micrometeorite impact. Whenever these forces displace the outer shell relative to the proof mass, an error signal is generated by a capacitive bridge sensor, and the satellite is propelled by gas jets to remain centered on the proof mass when the error reaches 1 mm. Local atmospheric densities near 800 km were measured, using ball position data, and the observed accelerations were corrected for the effects of solar radiation pressure. The measured densities reveal a greater dependence on latitude than that indicated by balloon satellites, and also a dependence on longitude. Only a small dependence on Kp, however, was observed in the region where the measurements were made.

A disturbance compensation system for satellites based on the drag-free concept was mechanized and flown, using a spherical proof mass and a cam-guided caging mechanism. The caging mechanism controls the location of the proof mass for testing and constrains it during launch. Design requirements, design details, and hardware are described.

Some of the technology developed in building a drag-free satellite simulator for laboratory use is described. The design decisions made in order to achieve a 10 to the -11th power g perturbation level for the navigation satellite is discussed. The control system development that will make possible drag-free operation of spinning satellites to reduce perturbation levels by averaging is described.

Drag-free satellite design and propulsion requirements, noting orbit perturbation mechanisms

The use of estimators or observers is discussed as applied to satellite attitude control and the control of drag-free satellites. The practical problems of implementation are discussed, and the relative advantages of full and reduced state estimators are compared, particularly in terms of their effectiveness and bandwidth as filters. Three applications are used to illustrate the principles. They are: (1) a reaction wheel control system, (2) a spinning attitude control system, and (3) a drag-free satellite translational control system. Fixed estimator gains are shown to be adequate for these (and many other) applications. Our experience in the hardware realization of estimators has led to categorize the error sources in terms of those that improve with increased estimator gains and those that get worse with increased estimator gains.

Drag-free satellite design and propulsion requirements, noting orbit perturbation mechanisms

Attitude-translation motion coupling effect on stability of gravity-stabilized drag-free satellites

Drag measuring requirements and comparison of accelerometer or drag-free satellite control systems usage for atmospheric density determination

The design of a drag-free satellite and its application to measuring tidal interaction of the earth and tesseral harmonics are discussed. Principle areas of discussion are: (1) the feasibility of making geophysical measurements which are not possible with conventional satellites, and (2) design of attitude and translation control systems for spinning vehicle and possible coupling of attitude and translation control for gravity stabilized vehicles.

Automatic feedback table leveling system, noting tilt to simulate aerodynamic drag

Martian radius determined by measuring gravity field at known surface points

Automatic feedback table leveling system, noting tilt to simulate aerodynamic drag

Work was conducted to develop an extremely low drift rate gyroscope and a very precise star tracker. A proposed relativity satellite will measure very accurately the theoretically predicted 'relativistic' precession of the gyroscope relative to an inertial reference frame provided by the star tracker. Aspects of precision spinning attitude control are discussed together with questions of gyro operation, and the hopping mode for lunar transportation. For the attitude control system of the lunar hopper, a number of control laws were investigated. The studies indicated that some suboptimal controls should be adequate for the system.

Methods are presented for representing unsteady aerodynamic loadings, valid for arbitrary motion, for the two-dimensional typical section with a trailing-edge control surface in incompressible flow and a three-dimensional lifting surface with leading- or trailing-edge control surfaces in subsonic compressible flow. Loads for the two-dimensional incompressible case are obtained via analytic continuation of the Theodorsen function into the complex plane, with guidance from a time-domain approximation of the Wagner indicial function. For the three-dimensional case, it is proposed that oscillatory generalized aerodynamic forces be approximated by Pade fractions, thereby permitting an obvious continuation into the complex plane. A theoretical justification for this procedure is briefly outlined, and some practical aspects of its implementation are discussed.

A satellite experiment to test the predicted relativistic precession of a gyroscope moving through a gravitational field has been under development at Stanford University for a number of years. The instruments will be located in a liquid helium bath to insure dimensional stability and to permit using superconducting types of circuitry for readout of the gyro orientation. Heat leaks in the system cause the helium to boil and the resultant helium gas is used for attitude control. The principal subject of the paper is the design and experimental evaluation of an electromagnetically actuated differential thrustor which was built and tested at Stanford University. The results are unique because most propulsion systems operate on-off in order to utilize propellant efficiently. In this case, the gas must flow continually to provide cooling, and the requirements are primarily for low power, small volume, and high reliability.

A satellite being designed by the Stanford University to measure (with extremely high precision) the effect of General Relativity is described. Specifically, the satellite will measure two relativistic precessions predicted by the theory: the geodetic effect (6.9 arcsec/yr), due solely to motion about the earth, and the motional effect (0.05 arcsec/yr), due to rotation of the earth. The gyro design requirements, including the requirement for precise attitude control and a dynamic model for attitude control synthesis, are discussed. Closed loop simulation of the satellite's natural dynamics on an analog computer is described.

Analysis of the gravity gradiometer developed by Forward and Bell (1970) suggest that an accuracy, in the range 0.1 to 0.5 EU can be expected in a lunar orbiter application. This accuracy will allow gradient anomalies associated with mascons to be mapped with 1% accuracy and should reveal a great deal of new information about the lunar gravity field. The proposed experiment calls for putting such a gradiometer into a closely circular polar orbit at an average height of about 30 km above the lunar surface. This orbit allows the entire lunar surface to be covered in fourteen days, the gradiometer to be checked twice per revolution and results in successive passes above the lunar surface being spaced at about the resolution limit of about 30 km set both by the satellite altitude and instrumental integration time.