A novel two-stage quaternion estimator from vector observations that is a synthesis between Wahba's approach and the Kalman filtering approach is presented. The first stage features an optimal denoising procedure of the elements of a time-varying noisy K-matrix. The second stage produces a quaternion estimate from the filtered K-matrix via any eigenvalue-eigenvector solver. This work's contribution consists in performing the denoising via Kalman filtering. For that purpose, a matrix Kalman filter (MKF) is developed, which has the advantage of preserving the natural formulation of the matrix plant equations. As a result, two aspects of a previous algorithm, called Optimal-REQUEST (OPREQ), are improved: the K-matrix update estimation stage uses a matrix gain rather than a scalar gain, and that gain is optimized with respect to the classical minimum-variance cost. This work assumes that the sensed lines of sight (LOS) are time invariant as seen in the chosen reference frame. This assumption fits in various operational mission architectures. An exact Kalman filter is developed that accounts for the state-multiplicative noise in the process equation. A reduced estimator is also developed assuming simple expressions for the filter covariance matrices. A constrained estimator, which enforces the symmetry and null-trace of the estimated matrix, is designed using the pseudomeasurement (PM) technique. Extensive Monte-Carlo simulations illustrate the performance of the novel filters with a spinning and nutating spacecraft (SC) as a case study. Extensive Monte-Carlo simulations show that the proposed estimator outperforms OPREQ. As illustrated by additional Monte-Carlo simulations, the constrained MKF exhibits a better transient and a better steady-state accuracy than the unconstrained filter for large initial disturbances in the symmetry and null-trace properties.

This work presents several algorithms that use vector observations in order to estimate the direction cosine matrix (DCM) as well as three constant biases and three time-varying drifts in body-mounted gyro output errors. All the algorithms use the matrix Kalman filter (MKF) paradigm, which preserves the natural formulation of the DCM state-space model equations. Focusing on the DCM estimation problem, the assumption of white noise in the gyro and in the vector observations errors yields reduced and efficient filter covariance computations. The orthogonality constraint on the DCM is handled via the technique of pseudomeasurement, which is naturally embedded in the MKF. Two additional known "brute-force" procedures are implemented for the sake of comparison. Extensive Monte-Carlo simulations illustrate the performances of the different estimators. When estimating only the DCM, it is shown that all the proposed orthogonalization procedures accelerate the estimation convergence. Nevertheless, the pseudomeasurement technique shows a smoother and shorter transient than the brute-force procedures, which on the other hand yield more accurate steady-states. The reduced covariance computations yield a more accurate steady-state than the full covariance computations but show a slower transient. When estimating the DCM as well as the gyro biases and drifts, enforcing orthogonalization seems to penalize the DCM estimation as long as the biases are not correctly identified. For the sake of computation savings during long duration missions, a mixed estimator, switching between long periods of DCM-only estimation and short periods of DCM-biases estimation, appears to be a promising strategy.

[Abstract] This paper presents a simple Kalman filter for continuously estimating the full attitude of a spinning spacecraft. This algorithm is comprised of two low-order decoupled Kalman filters; one estimates the spin axis orientation, and the other estimates the spin rate and the spin (phase) angle. The filters are ambiguity free and do not rely on the spacecraft dynamics. They were successfully tested using data obtained from one of the Space Technology (ST)-5 satellites.

This paper presents a single frame algorithm for the spin-axis orientation-determination of spinning spacecraft that encounters no ambiguity problems, as well as a simple Kalman filter for continuously estimating the full attitude of a spinning spacecraft. The later algorithm is comprised of two low order decoupled Kalman filters; one estimates the spin axis orientation, and the other estimates the spin rate and the spin (phase) angle. The filters are ambiguity free and do not rely on the spacecraft dynamics. They were successfully tested using data obtained from one of the ST5 satellites.

In this paper we research the extraction of the angular rate vector from attitude information without differentiation, in particular from quaternion measurements. We show that instead of using a Kalman filter of some kind, it is possible to obtain good rate estimates, suitable for spacecraft attitude control loop damping, using simple feedback loops, thereby eliminating the need for recurrent covariance computation performed when a Kalman filter is used. This considerably simplifies the computations required for rate estimation in gyro-less spacecraft. Some interesting qualities of the Kalman filter gain are explored, proven and utilized. We examine two kinds of feedback loops, one with varying gain that is proportional to the well known Q matrix, which is computed using the measured quaternion, and the other type of feedback loop is one with constant coefficients. The latter type includes two kinds; namely, a proportional feedback loop, and a proportional-integral feedback loop. The various schemes are examined through simulations and their performance is compared. It is shown that all schemes are adequate for extracting the angular velocity at an accuracy suitable for control loop damping.

In this paper we research the extraction of the angular rate vector from attitude information without differentiation, in particular from quaternion measurements. We show that instead of using a Kalman filter of some kind, it is possible to obtain good rate estimates, suitable for spacecraft attitude control loop damping, using simple feedback loops, thereby eliminating the need for recurrent covariance computation performed when a Kalman filter is used. This simplification considerably decreases the computations required for rate estimation in gyro-less spacecraft. Some interesting qualities of the Kalman filter gain are explored, proven and utilized. We examine two kinds of feedback loops, one with varying gain that is proportional to the well known Q matrix, which is computed using the measured quaternion, and the other type of feedback loop is one with constant coefficients. The latter type includes two kinds; namely, a proportional feedback loop, and a proportional- integral feedback loop. The various schemes are examined through simulations and their performance is compared. It is shown that all schemes are adequate for extracting the angular velocity at accuracy suitable for control loop damping.

A general discrete-time Kalman filter (KF) for state matrix estimation using matrix measurements is presented. The new algorithm evaluates the state matrix estimate and the estimation error covariance matrix in terms of the original system matrices. The proposed algorithm naturally fits systems which are most conveniently described by matrix process and measurement equations. Its formulation uses a compact notation for aiding both intuition and mathematical manipulation. It is a straightforward extension of the classical KF, and includes as special cases other matrix filters that were developed in the past. Beyond the analytical value of the matrix filter, it is shown through various examples arising in engineering problems that this filter can be computationally more efficient than its vectorized version.

This paper presents a novel Kalman filter (KF) for estimating the attitude-quaternion as well as gyro random drifts from vector measurements. Employing a special manipulation on the measurement equation results in a linear pseudo-measurement equation whose error is state-dependent. Because the quaternion kinematics equation is linear, the combination of the two yields a linear KF that eliminates the usual linearization procedure and is less sensitive to initial estimation errors. General accurate expressions for the covariance matrices of the system state-dependent noises are developed. In addition, an analysis shows how to compute these covariance matrices efficiently. An adaptive version of the filter is also developed to handle modeling errors of the dynamic system noise statistics. Monte-Carlo simulations are carried out that demonstrate the efficiency of both versions of the filter. In the particular case of high initial estimation errors, a typical extended Kalman filter (EKF) fails to converge whereas the proposed filter succeeds.

This paper presents a new approach to recursive estimation of the Euler-vector and of the quaternion of rotation from vector observations. This new approach is based on geometric considerations. We examine the geometry of the family of Euler-vectors and quaternions that are determined by a single vector and its transformation to another coordinate system. Then, using this single vector pair, an Euler-vector of maximum rotation and an Euler-vector of minimum rotation are defined. Similarly a corresponding quaternion of maximum rotation and a corresponding quaternion of minimum rotation are defined too. Based on these minimum and maximum elements, a measurement Eulervector and a measurement quaternion are defined. These vector and quaternion measurements enable the use of linear measurement models in Kalman filters. Consequently, a linear or a pseudo-linear filter can be used for estimating the Eulervector, and most importantly, a linear and a pseudo-linear Kalman filters can be used for estimating the attitude quaternion, all from vector measurements, thereby alleviating the convergence problem that may be encountered with the use of nonlinear filters. Simulation results are presented which validate the design of the filters.

This paper presents a comparison between two approaches to sensor calibration. According to one approach, called explicit, an estimator compares the sensor readings to reference readings, and uses the difference between the two to estimate the calibration parameters. According to the other approach, called implicit, the sensor error is integrated to form a different entity, which is then compared with a reference quantity of this entity, and the calibration parameters are inferred from the difference. In particular this paper presents the comparison between these approaches when applied to in-flight spacecraft gyro calibration. Reference spacecraft rate is needed for gyro calibration when using the explicit approach; however, such reference rates are not readily available for in-flight calibration. Therefore the calibration parameter-estimator is expanded to include the estimation of that reference rate, which is based on attitude measurements in the form of attitude-quaternion. A comparison between the two approaches is made using simulated data. It is concluded that the performances of the two approaches are basically comparable. Sensitivity tests indicate that the explicit filter results are essentially insensitive to variations in given spacecraft dynamics model parameters.

identical to the Schuler-period found in terrestrial inertial navigation systems (INS), and in the Schuler Pendulum model. On the other hand, the existence of the Schuler-period in INS is well known to engineers and technicians performing INS work, but the fact that this period is identical to the period of LEO satellites is barely known in the INS community, if at all, let alone its existence in the Hill-Clohessy-Wiltshire equations. In this paper we examine the four phenomena; namely, Schuler pendulum, INS Schuler oscillations, LEO satellite orbital period, and the period of the relative motion between two adjacent LEO satellites. In particular we show why the LEO satellite’s orbital period is identical to the well-known INS Schuler period. We also show that if the INS error equations are generalized to a non-terrestrial case, a generalized form of the Schuler oscillation exists, which takes the form of the Hill-Clohessy-Wiltshire equations for satellite relative motion.

REQUEST is a recursive algorithm for least-squares estimation of the attitude quaternion of a rigid body using vector measurements. It uses a constant, empirically chosen gain and is, therefore, suboptimal when filtering propagation noises. The algorithm presented here is an optimized REQUEST procedure, which optimally filters measurement as well as propagation noises. The special case of zero-mean white noises is considered. The solution approach is based on state-space modeling of the K-matrix system and uses Kalman-filtering techniques to estimate the optimal K matrix. Then, the attitude quaternion is extracted from the estimated K matrix. A simulation study is used to demonstrate the performance of the algorithm.

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 performance of the two filters is examined using simulated data.

The Magnetometer Navigation (MAGNAV) algorithm is currently running as a flight experiment as part of the Wide Field Infrared Explorer (WIRE) Post-Science Engineering Testbed. Initialization of MAGNAV occurred on September 4, 2003. MAGNAV is designed to autonomously estimate the spacecraft orbit, attitude, and rate using magnetometer and sun sensor data. Since the Earth's magnetic field is a function of time and position, and since time is known quite precisely, the differences between the computed magnetic field and measured magnetic field components, as measured by the magnetometer throughout the entire spacecraft orbit, are a function of the spacecraft trajectory and attitude errors. Therefore, these errors are used to estimate both trajectory and attitude. In addition, the time rate of change of the magnetic field vector is used to estimate the spacecraft rotation rate. The estimation of the attitude and trajectory is augmented with the rate estimation into an Extended Kalman filter blended with a pseudo-linear Kalman filter. Sun sensor data is also used to improve the accuracy and observability of the attitude and rate estimates. This test serves to validate MAGNAV as a single low cost navigation system which utilizes reliable, flight qualified sensors. MAGNAV is intended as a backup algorithm, an initialization algorithm, or possibly a prime navigation algorithm for a mission with coarse requirements. Results from the first six months of operation are presented.

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 an implicit algorithm for spacecraft onboard instrument calibration, particularly to onboard gyro calibration. This work is an extension of previous work that was done where an explicit gyro calibration algorithm was applied to the AQUA spacecraft gyros. The algorithm presented in this paper was tested using simulated data and real data that were downloaded from the Microwave Anisotropy Probe (MAP) spacecraft. The calibration tests gave very good results. A comparison between the use of the implicit calibration algorithm used here with the explicit algorithm used for AQUA spacecraft indicates that both provide an excellent estimation of the gyro calibration parameters with similar accuracies.

The paper presents a general discrete-time Kalman filter for state matrix estimation using matrix measurements. The new algorithm evaluates the state matrix estimate and the estimation error covariance matrix in terms of the original system matrices. The proposed algorithm naturally fits systems which are most conveniently described by matrix process and measurement equations. Its formulation uses a compact notation for aiding both intuition and mathematical manipulation. It is a straightforward extension of the classical Kalman filter, and includes as special cases other matrix filters that were developed in the past.