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The Earth Observing System (EOS) Aqua satellite was successfully launched on May 4,2002. Aqua is the second in the series of EOS satellites. EOS is part of NASA s Earth Science Enterprise Program, whose goals are to advance the scientific understanding of the Earth system. Aqua is a three-axis stabilized, Earth-pointing spacecraft in a nearly circular, sun-synchronous orbit at an altitude of 705 km. The Goddard Space Flight Center (GSFC) Flight Dynamics attitude team supported all phases of the launch and early mission. This paper presents the main results and lessons learned during this period, including: real-time attitude mode transition support, sensor calibration, onboard computer attitude validation, response to spacecraft emergencies, postlaunch attitude analyses, and anomaly resolution. In particular, Flight Dynamics support proved to be invaluable for successful Earth acquisition, fine-point mode transition, and recognition and correction of several anomalies, including support for the resolution of problems observed with the MODIS instrument.

The fine Sun sensor used on many spacecraft consists of two independent single-axis sensor heads, nominally mounted perpendicularly. These detect the Sun angle over a field of view typically of +32 deg. (There is a trade-off between accuracy and size of the field of view that allows for much latitude in any numbers quoted.) The nonlinear "transfer" function that maps the telemetered counts into observed angles consists of 9 adjustable parameters for each axis (1 8 total). An augmented transfer function has previously been reported that achieves a significant accuracy improvement across the entire field of view. That function expands the parameter set to 12 coefficients per axis (24 total) and includes cross terms combining counts from both axes. To make the best use of the Sun sensor for attitude determination, it must be calibrated after launch. However, the large number of parameters and the nonlinearity of the problem make this a challenging task. The purpose of this paper is to examine ways to improve convergence of the parameter search algorithm. In particular, experience has shown that the problem should be broken down into several steps, solving for a selected subset of the parameters at each step. This approach has now been incorporated as an option in the calibration utility.

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.

Spacecraft attitude estimation by means of an extended Kalman filter requires a reasonably true model of the inherent noise of each sensor. For some sensors, the largest uncorrected noise comes not from the sensor itself, but from errors in the model for the predicted observations. This is certainly the case for Earth horizon sensors. The Earth horizon as seen from low Earth orbit is nearly a circle whose radius depends primarily on altitude. A straightforward latitude-dependent correction is added to this to account for the oblateness of the Earth. There also are both seasonal and stochastic variations in the horizon height. The seasonal variations can be predicted to some limited degree based on models derived from historical data. The stochastic component characteristically shows variations that are correlated both in time and space but which are unpredictable over long time spans. This work investigates whether Earth horizon sensor performance can be improved by solving for its systematic error as an augmentation of an attitude Kalman filter. It is found that using only Earth and Sun sensors, the augmented state is not fully observable. Even when magnetometer data is included, only the pitch axis component of the error can be improved; the roll component is unobservable.

The Submillimeter Wave Astronomy Satellite (SWAS) was successfully launched on December 6, 1998 at 00:58 UTC. The two year mission is the fourth in the series of Small Explorer (SMEX) missions. SWAS is dedicated to the study of star formation and interstellar chemistry. SWAS was injected into a 635 km by 650 km orbit with an inclination of nearly 70 deg by an Orbital Sciences Corporation Pegasus XL launch vehicle. The Flight Dynamics attitude and navigation teams supported all phases of the early mission. This support included orbit determination, attitude determination, real-time monitoring, and sensor calibration. This paper reports the main results and lessons learned concerning navigation, support software, star tracker performance, magnetometer and gyroscope calibrations, and anomaly resolution. This includes information on spacecraft tip-off rates, first-day navigation problems, target acquisition anomalies, star tracker anomalies, and significant sensor improvements due to calibration efforts.

The Submillimeter Wave Astronomy Satellite (SWAS) was successfully launched on December 6, 1998 at 00:58 UTC. The two year mission is the fourth in the series of Small Explorer (SMEX) missions. SWAS is dedicated to the study of star formation and interstellar chemistry. SWAS was injected into a 635 km by 650 km orbit with an inclination of nearly 70 deg by an Orbital Sciences Corporation Pegasus XL launch vehicle. The Flight Dynamics attitude and navigation teams supported all phases of the early mission. This support included orbit determination, attitude determination, real-time monitoring, and sensor calibration. This paper reports the main results and lessons learned concerning navigation, support software, star tracker performance, magnetometer and gyroscope calibrations, and anomaly resolution. This includes information on spacecraft tip-off rates, first-day navigation problems, target acquisition anomalies, star tracker anomalies, and significant sensor improvements due to calibration efforts.

Magnetometers and gyroscopes have been shown to provide very accurate attitudes for a variety of spacecraft. These results have been obtained, however, using a batch-least-squares algorithm and long periods of data. For use in onboard applications, attitudes are best determined using sequential estimators such as the Kalman filter. When a filter is used to determine attitudes using magnetometer and gyroscope data for input, the resulting accuracy is limited by both the sensor accuracies and errors inherent in the Earth magnetic field model. The Kalman filter accounts for the random component by modeling the magnetometer and gyroscope errors as white noise processes. However, even when these tuning parameters are physically realistic, the rate biases (included in the state vector) have been found to show systematic oscillations. These are attributed to the field model errors. If the gyroscope noise is sufficiently small, the tuned filter 'memory' will be long compared to the orbital period. In this case, the variations in the rate bias induced by field model errors are substantially reduced. Mistuning the filter to have a short memory time leads to strongly oscillating rate biases and increased attitude errors. To reduce the effect of the magnetic field model errors, these errors are estimated within the filter and used to correct the reference model. An exponentially-correlated noise model is used to represent the filter estimate of the systematic error. Results from several test cases using in-flight data from the Compton Gamma Ray Observatory are presented. These tests emphasize magnetometer errors, but the method is generally applicable to any sensor subject to a combination of random and systematic noise.

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.

Attitude estimator use observations from different times to reduce the effects of noise. If the vehicle is rotating, the attitude at one time needs to be propagated to that at another time. If the vehicle measures its angular velocity, attitude propagating entails integrating a rotational kinematics equation only. If a measured angular velocity is not available, torques can be computed and an additional rotational dynamics equation integrated to give the angular velocity. Initial conditions for either of these integrations come from the estimation process. Sometimes additional quantities, such as gyro and torque parameters, are also solved for. Although the partial derivatives of attitude with respect to initial attitude and gyro parameters are well known, the corresponding partial derivatives with respect to initial angular velocity and torque parameters are less familiar. They can be derived and computed numerically in a way that is analogous to that used for the initial attitude and gyro parameters. Previous papers have demonstrated the feasibility of using dynamics models for attitude estimation but have not provided details of how each angular velocity and torque parameters can be estimated. This tutorial paper provides some of that detail, notably how to compute the state transition matrix when closed form expressions are not available. It also attempts to put dynamics estimation in perspective by showing the progression from constant to gyro-propagated to dynamics-propagated attitude motion models. Readers not already familiar with attitude estimation will find this paper an introduction to the subject, and attitude specialists may appreciate the collection of heretofore scattered results brought together in a single place.

Given a valid system model and adequate observability, a Kalman filter will converge toward the true system state with error statistics given by the estimated error covariance matrix. The errors generally do not continue to decrease. Rather, a balance is reached between the gain of information from new measurements and the loss of information during propagation. The errors can be further reduced, however, by a second pass through the data with an optimal smoother. This algorithm obtains the optimally weighted average of forward and backward propagating Kalman filters. It roughly halves the error covariance by including future as well as past measurements in each estimate. This paper investigates whether such benefits actually accrue in the application of an optimal smoother to spacecraft attitude determination. Tests are performed both with actual spacecraft data from the Extreme Ultraviolet Explorer (EUVE) and with simulated data for which the true state vector and noise statistics are exactly known.

We present results for two real-time filters prototyped for the Compton Gamma Ray Observatory (GRO), the Extreme Ultraviolet Explorer (EUVE), the Cosmic Background Explorer (COBE), and the next generation of Geostationary Operational Environmental Satellites (GOES). Both real and simulated data were used to solve for attitude and gyro biases. These filters promise advantages over single-frame and batch methods for missions like GOES, where startup and transfer-orbit operations require quick knowledge of attitude and gyro biases.

The loss function for the Wahba attitude estimation problem employs unit vector observations with scalar weights. It is usually associated with the QUEST observation model, where the actual sensor noise is assumed the same for all components of the observed vector, regardless of the position in the field of view. The QUEST model has the great advantage of being sensor-independent except for the scalar parameter that characterizes the sensor errors. Although efficient algorithms for solving the Wahba problem exist, extending these algorithms to estimate gyro biases or sensor alignments has had mixed success. However, it is straightforward to estimate bias and alignment parameters with a Kalman filter. This paper investigates the use of an extended Kalman filter for the attitude and gyro bias that incorporates the QUEST observation model, to be referred to as the Unit Vector Filter (UVF). The UVF results are compared with those from a more conventional filter, the Real-Time Sequential Filter (RTSF), for which the residual is the two-dimensional projection of the unit vector onto the plane perpendicular to the sensor boresight. The RTSF is similar in design to that used by Multimission Modular Spacecraft for onboard attitude determination. An apparent obstacle to the use of unit vectors as measurements is their singular noise covariance matrix. Shuster has shown that this problem should not affect filter performance. The UVF and RTSF are tested using actual flight data from the Extreme Ultraviolet Explorer (EUVE). It is found that these filters generate nearly identical attitude and gyro bias estimates, thus validating the use of unit vectors and the QUEST noise model.

The goal of the Flight Dynamics Facility (FDF) attitude support is twofold: to determine spacecraft attitude and to explain deviations from nominal attitude behavior. Attitude determination often requires resolving contradictions in the sensor observations. This may be accomplished by applying calibration corrections or by revising the observation models. After accounting for all known sources of error, solution accuracy should be limited only by observation and propagation noise. The second half of the goal is to explain why the attitude may not be as originally intended. Reasons for such deviations include sensor or actuator misalignments and control system performance. In these cases, the ability to explain the behavior should, in principle, be limited only by knowledge of the sensor and actuator data and external torques. Documented here are some results obtained to date in support of the Cosmic Background Explorer (COBE). Advantages and shortcomings of the integrated attitude determination/sensor calibration software are discussed. Some preliminary attitude solutions using data from the Diffuse Infrared Background Experiment (DIRBE) instrument are presented and compared to solutions using Sun and Earth sensors. A dynamical model is constructed to illustrate the relative importance of the various sensor imprefections. This model also shows the connection between the high- and low-frequency attitude oscillations.