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The equations for various spin particles with oscillator-like interactions are discussed in this talk. Topics discussed include: (1) comment on 'The Klein-Gordon Oscillator'; (2) the Dirac oscillator in quaternion form; (3) the Dirac-Dowker oscillator; (4) the Weinberg oscillator; and (5) note on the two-body Dirac oscillator.
This paper presents an extended Kalman filter for estimating attitude sensor timing errors. Spacecraft attitude is determined by finding the mean rotation from a set of reference vectors in inertial space to the corresponding observed vectors in the body frame. Any timing errors in the observations can lead to attitude errors if either the spacecraft is rotating or the reference vectors themselves vary with time. The state vector here consists of the attitude quaternion, timetag biases, and, optionally, gyro drift rate biases. The filter models the timetags as random walk processes: their expectation values propagate as constants and white noise contributes to their covariance. Thus, this filter is applicable to cases where the true timing errors are constant or slowly varying. The observability of the state vector is studied first through an examination of the algebraic observability condition and then through several examples with simulated star tracker timing errors. The examples use both simulated and actual flight data from the Extreme Ultraviolet Explorer (EUVE). The flight data come from times when EUVE had a constant rotation rate, while the simulated data feature large angle attitude maneuvers. The tests include cases with timetag errors on one or two sensors, both constant and time-varying, and with and without gyro bias errors. Due to EUVE's sensor geometry, the observability of the state vector is severely limited when the spacecraft rotation rate is constant. In the absence of attitude maneuvers, the state elements are highly correlated, and the state estimate is unreliable. The estimates are particularly sensitive to filter mistuning in this case. The EUVE geometry, though, is a degenerate case having coplanar sensors and rotation vector. Observability is much improved and the filter performs well when the rate is either varying or noncoplanar with the sensors, as during a slew. Even with bad geometry and constant rates, if gyro biases are independently known, the timetag error for a single sensor can be accurately estimated as long as its boresight is not too close to the spacecraft rotation axis.
LaRCsim is a set of ANSI C routines that implement a full set of equations of motion for a rigid-body aircraft in atmospheric and low-earth orbital flight, suitable for pilot-in-the-loop simulations on a workstation-class computer. All six rigid-body degrees of freedom are modeled. The modules provided include calculations of the typical aircraft rigid-body simulation variables, earth geodesy, gravity and atmospheric models, and support several data recording options. Features/limitations of the current version include English units of measure, a 1962 atmosphere model in cubic spline function lookup form, ranging from sea level to 75,000 feet, rotating oblate spheroidal earth model, with aircraft C.G. coordinates in both geocentric and geodetic axes. Angular integrations are done using quaternion state variables Vehicle X-Z symmetry is assumed.
A nonlinear filtering theory from a deterministic point of view is presented and an application to attitude determination is considered. The approach that is taken in this paper is motivated largely by the H(sun infinity) control and estimation theory for linear systems which has been evolved within the last decade. Rather than formulating the estimation problem as a game played by two adversaries, as has been done in the linear case, we employ in this work some notions from the theory of dissipative systems as a vehicle for arriving at certain Hamilton-Jacobi inequality, which in turn, provides a solution to the filtering problem, whenever it is satisfied. Application of this method to a linear estimation problem and to the problem of estimating a spacecraft attitude quaternion and gyro drift bias vector are presented. In limiting cases, these give the Kalman filter and the extended Kalman filter, respectively. The main advantages of this approach over its probabilistic counterpart are that this approach does not require a prior knowledge of any statistics, and that in general it is more amenable to a quantitative assessment regarding approximations such as linearization, and that in certain cases this approach yields an exact solution to the nonlinear filtering.
This paper introduces two new seven-parameter representations for spacecraft attitude dynamics modeling. The seven parameters are the three components of the total system angular momentum in the spacecraft body frame; the three components of the angular momentum in the inertial reference frame; and an angle variable. These obey a single constraint as do parameterizations that include a quaternion; in this case the constraint is the equality of the sum of the squares of the angular momentum components in the two frames. The two representations are nonsingular if the system angular momentum is non-zero and obeys certain orientation constraints. The new parameterizations of the attitude matrix, the equations of motion, and the relation of the solution of these equations to Euler angles for torque-free motion are developed and analyzed. The superiority of the new parameterizations for numerical integration is shown in a specific example.
Science requirements for the GLAS mission demand that the laser altimeter be pointed to within 50 m of the location of the previous repeat ground track. The satellite will be flown in a repeat orbit of 182 days. Operationally, the required pointing information will be determined on the ground using the nominal ground track, to which pointing is desired, and the current propagated orbit of the satellite as inputs to the roll computation algorithm developed by CCAR. The roll profile will be used to generate a set of fit coefficients which can be uploaded on a daily basis and used by the on-board attitude control system. In addition, an algorithm has been developed for computation of the associated command quaternions which will be necessary when pointing at targets of opportunity. It may be desirable in the future to perform the roll calculation in an autonomous real-time mode on-board the spacecraft. GPS can provide near real-time tracking of the satellite, and the nominal ground track can be stored in the on-board computer. It will be necessary to choose the spacing of this nominal ground track to meet storage requirements in the on-board environment. Several methods for generating the roll profile from a sparse reference ground track are presented.
This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quatemion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quatemion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quatemion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.
This document is presentation in viewgraph form, which outlines the methods of determining spacecraft attitude. The presentation reviews several parameterizations relating to spacecraft attitude, such as Euler's Theorem, Rodriques parameters, and Euler-Rodriques parameters or Quaternion. Onboard attitude determination is the norm, using either single frame or filtering methods. The presentation reviews several mathematical representations of attitude. The mechanisms for determining attitude on board the Hubble Space Telescope, the Tropical Rainfall and Measuring Mission and the Solar Anomalous and Magnetospheric Particle Explorer are reviewed. Wahba's problem, Procrustes Problem, and some solutions are also summarized.
The general problem of gyro calibration can be stated as the estimation of the scale factors, misalignments, and drift-rate biases of the gyro using the on-orbit sensor measurements. These gyro parameters have been traditionally treated as temperature-independent in the operational flight dynamics ground systems at NASA Goddard Space Flight Center (GSFC), a scenario which has been successfully applied in the gyro calibration of a large number of missions. A significant departure from this is the Microwave Anisotropy Probe (MAP) mission where, due to the high thermal variations expected during the mission phase, it is necessary to model the scale factors as functions of temperature. This paper addresses the issue of gyro calibration for the MAP gyro model using a manufacturer-supplied model of the variation of scale factors with temperature. The problem is formulated as a least squares problem and solved using the Levenberg-Marquardt algorithm in the MATLAB(R) library function NLSQ. The algorithm was tested on simulated data with Gaussian noise for the quaternions as well as the gyro rates and was found to consistently converge close to the true values. Significant improvement in accuracy was noticed due to the estimation of the temperature-dependent scale factors as against constant scale factors.
This paper presents the results of a performance study of the autonomous star trackers (ASTs) on the IMAGE and the EO-1 spacecraft. IMAGE is a spinning spacecraft without gyros or redundant precision attitude sensors, so the statistical properties of the AST are estimated simply by comparing the output observed quaternions with a rigid rotator model with constant angular momentum. The initial conditions are determined by a least-squares fit to minimize the AST residuals. An additional fit is used to remove the remaining systematic error and to obtain the inherent sensor noise. Gyro rate data are available for the EO-1 mission, so the AST noise statistics are obtained from the residuals after solving for an epoch attitude and gyro bias also using a least-squares method.
Accurate spacecraft attitudes may only be obtained if the primary attitude sensors are well calibrated. Launch shock, relaxation of gravitational stresses and similar effects often produce large enough alignment shifts so that on-orbit alignment calibration is necessary if attitude accuracy requirements are to be met. A variety of attitude sensor alignment algorithms have been developed to meet the need for on-orbit calibration. Two new algorithms are presented here: ALICAL and ALIQUEST. Each of these has advantages in particular circumstances. ALICAL is an attitude independent algorithm that uses near simultaneous measurements from two or more sensors to produce accurate sensor alignments. For each set of simultaneous observations the attitude is overdetermined. The information content of the extra degrees of freedom can be combined over numerous sets to provide the sensor alignments. ALIQUEST is an attitude dependent algorithm that combines sensor and attitude data into a loss function that has the same mathematical form as the Wahba problem. Alignments can then be determined using any of the algorithms (such as the QUEST quaternion estimator) that have been developed to solve the Wahba problem for attitude. Results from the use of these methods on active missions are presented.
The Mars Odyssey spacecraft made use of multipass aerobraking to gradually reduce its orbit period from a highly elliptical insertion orbit to its final science orbit. Aerobraking operations provided an opportunity to apply advanced thermal analysis techniques to predict the temperature of the spacecraft's solar array for each drag pass. Odyssey telemetry data was used to correlate the thermal model. The thermal analysis was tightly coupled to the flight mechanics, aerodynamics, and atmospheric modeling efforts being performed during operations. Specifically, the thermal analysis predictions required a calculation of the spacecraft's velocity relative to the atmosphere, a prediction of the atmospheric density, and a prediction of the heat transfer coefficients due to aerodynamic heating. Temperature correlations were performed by comparing predicted temperatures of the thermocouples to the actual thermocouple readings from the spacecraft. Time histories of the spacecraft relative velocity, atmospheric density, and heat transfer coefficients, calculated using flight accelerometer and quaternion data, were used to calculate the aerodynamic heating. During aerobraking operations, the correlations were used to continually update the thermal model, thus increasing confidence in the predictions. This paper describes the thermal analysis that was performed and presents the correlations to the flight data.
The Geosynchronous Imaging Fourier Transform Spectrometer (GIFTS) Instrument is the next generation spectrometer for remote sensing weather satellites. The GIFTS instrument will be used to perform scans of the Earth s atmosphere by assembling a series of field-of- views (FOV) into a larger pattern. Realization of this process is achieved by step scanning the instrument FOV in a contiguous fashion across any desired portion of the visible Earth. A 2.3 arc second pointing stability, with respect to the scanning instrument, must be maintained for the duration of the FOV scan. A star tracker producing attitude data at 100 Hz rate will be used by the autonomous pointing algorithm to precisely track target FOV s on the surface of the Earth. The main objective is to validate the pointing algorithm in the presence of spacecraft disturbances and determine acceptable disturbance limits from expected noise sources. Proof of concept validation of the pointing system algorithm is carried out with a full system simulation developed using Matlab Simulink. Models for the following components function within the full system simulation: inertial reference unit (IRU), attitude control system (ACS), reaction wheels, star tracker, and mirror controller. With the spacecraft orbital position and attitude maintained to within specified limits the pointing algorithm receives quaternion, ephemeris, and initialization data that are used to construct the required mirror pointing commands at a 100 Hz rate. This comprehensive simulation will also aid in obtaining a thorough understanding of spacecraft disturbances and other sources of pointing system errors. Parameter sensitivity studies and disturbance analysis will be used to obtain limits of operability for the GIFTS instrument. The culmination of this simulation development and analysis will be used to validate the specified performance requirements outlined for this instrument.
The Inertial Stellar Compass (ISC) is a miniature, low power, stellar inertial attitude determination system with an accuracy of better than 0.1 degree (1 sigma) in three axes. The ISC consumes only 3.5 Watts of power and is contained in a 2.5 kg package. With its embedded on-board processor, the ISC provides attitude quaternion information and has Lost-in-Space (LIS) initialization capability. The attitude accuracy and LIS capability are provided by combining a wide field of view Active Pixel Sensor (APS) star camera and Micro- ElectroMechanical System (MEMS) inertial sensor information in an integrated sensor system. The performance and small form factor make the ISC a useful sensor for a wide range of missions. In particular, the ISC represents an enabling, fully integrated, micro-satellite attitude determination system. Other applications include using the ISC as a single sensor solution for attitude determination on medium performance spacecraft and as a bolt on independent safe-hold sensor or coarse acquisition sensor for many other spacecraft. NASA's New Millennium Program (NMP) has selected the ISC technology for a Space Technology 6 (ST6) flight validation experiment scheduled for 2004. NMP missions, such a s ST6, are intended to validate advanced technologies that have not flown in space in order to reduce the risk associated with their infusion into future NASA missions. This paper describes the design, operation, and performance of the ISC and outlines the technology validation plan. A number of mission applications for the ISC technology are highlighted, both for the baseline ST6 ISC configuration and more ambitious applications where ISC hardware and software modifications would be required. These applications demonstrate the wide range of Space and Earth Science missions that would benefit from infusion of the ISC technology.
Previous high fidelity onboard attitude algorithms estimated only the spacecraft attitude and gyro bias. The desire to promote spacecraft and ground autonomy and improvements in onboard computing power has spurred development of more sophisticated calibration algorithms. Namely, there is a desire to provide for sensor calibration through calibration parameter estimation onboard the spacecraft as well as autonomous estimation on the ground. Gyro calibration is a particularly challenging area of research. There are a variety of gyro devices available for any prospective mission ranging from inexpensive low fidelity gyros with potentially unstable scale factors to much more expensive extremely stable high fidelity units. Much research has been devoted to designing dedicated estimators such as particular Extended Kalman Filter (EKF) algorithms or Square Root Information Filters. This paper builds upon previous attitude, rate, and specialized gyro parameter estimation work performed with Pseudo Linear Kalman Filter (PSELIKA). The PSELIKA advantage is the use of the standard linear Kalman Filter algorithm. A PSELIKA algorithm for an orthogonal gyro set which includes estimates of attitude, rate, gyro misalignments, gyro scale factors, and gyro bias is developed and tested using simulated and flight data. The measurements PSELIKA uses include gyro and quaternion tracker data.
This paper presents the overall mathematical model and results from pseudo linear recursive estimators of attitude and rate for a spinning spacecraft. The measurements considered are vector measurements obtained by sun-sensors, fixed head star trackers, horizon sensors, and three axis magnetometers. Two filters are proposed for estimating the attitude as well as the angular rate vector. One filter, called the q-Filter, yields the attitude estimate as a quaternion estimate, and the other filter, called the D-Filter, yields the estimated direction cosine matrix. Because the spacecraft is gyro-less, Euler's equation of angular motion of rigid bodies is used to enable the estimation of the angular velocity. A simpler Markov model is suggested as a replacement for Euler's equation in the case where the vector measurements are obtained at high rates relative to the spacecraft angular rate.
The absence of a globally nonsingular three-parameter representation of rotations forces attitude Kalman filters to estimate either a singular or a redundant attitude representation. We compare two filtering strategies using simplified kinematics and measurement models. Our favored strategy estimates a three-parameter representation of attitude deviations from a reference attitude specified by a higher- dimensional nonsingular parameterization. The deviations from the reference are assumed to be small enough to avoid any singularity or discontinuity of the three-dimensional parameterization. We point out some disadvantages of the other strategy, which directly estimates the four-parameter quaternion representation.
During Aqua launch support, attitude analysts noticed several anomalies in Onboard Computer (OBC) rates and in rates computed by the ground Attitude Determination System (ADS). These included: 1) periodic jumps in the OBC pitch rate every 2 minutes; 2) spikes in ADS pitch rate every 4 minutes; 3) close agreement between pitch rates computed by ADS and those derived from telemetered OBC quaternions (in contrast to the step-wise pattern observed for telemetered OBC rates); 4) spikes of +/- 10 milliseconds in telemetered IRU integration time every 4 minutes (despite the fact that telemetered time tags of any two sequential IRU measurements were always 1 second apart from each other). An analysis presented in the paper explains this anomalous behavior by a small average offset of about 0.5 +/- 0.05 microsec in the time interval between two sequential accumulated angle measurements. It is shown that errors in the estimated pitch angle due to neglecting the aforementioned variations in the integration time interval by the OBC is within +/- 2 arcseconds. Ground attitude solutions are found to be accurate enough to see the effect of the variations on the accuracy of the estimated pitch angle.