Itzhack Y. Bar-Itzhack

Attitude Determination from Vector Observations: Quaternion Estimation

Two recursive estimation algorithms, which use pairs of measured vectors to yield minimum variance estimates of the quaternion of rotation, are presented. The nonlinear relations between the direction cosine matrix and the quaternion are linearized, and a variant of the extended Kalman filter is used to estimate the difference between the quaternion and its estimate. With each measurement this estimate is updated and added to the whole quaternion estimate. This operation constitutes a full state reset in the estimation process. Filter tuning is needed to obtain a converging filter. The second algorithm presented uses the normality property of the quaternion of rotation to obtain, in a straightforward design, a filter which converges, with a smaller error, to a normal quaternion. This algorithm changes the state but not the covariance computation of the original algorithm and implies only a partial reset. Results of Monte-Carlo simulation runs are presented which demonstrate the superiority of the normalized quaternion.

REQUEST: A Recursive QUEST Algorithm for Sequential Attitude Determination

In order to find the attitude of a spacecraft with respect to a reference coordinate system, vector measurements are taken. The vectors are pairs of measurements of the same generalized vector, taken in the spacecraft body coordinates, as well as in the reference coordinate system. We are interested in finding the best estimate of the transformation between these coordinate system.s The algorithm called QUEST yields that estimate where attitude is expressed by a quarternion. Quest is an efficient algorithm which provides a least squares fit of the quaternion of rotation to the vector measurements. Quest however, is a single time point (single frame) batch algorithm, thus measurements that were taken at previous time points are discarded. The algorithm presented in this work provides a recursive routine which considers all past measurements. The algorithm is based on on the fact that the, so called, K matrix, one of whose eigenvectors is the sought quaternion, is linerly related to the measured pairs, and on the ability to propagate K. The extraction of the appropriate eigenvector is done according to the classical QUEST algorithm. This stage, however, can be eliminated, and the computation simplified, if a standard eigenvalue-eigenvector solver algorithm is used. The development of the recursive algorithm is presented and illustrated via a numerical example.

Unified approach to inertial navigation system error modeling

Several inertial navigation system error models have been developed and used in the literature. Most of the models are ad hoc models which were needed to solve certain particular problems and were developed for that purpose only. Consequently, the relationship, correspondence, and equivalence between the various models is not evident. This paper presents a new methodology for developing inertial navigation systems error models which also puts all of the known models in the same framework and shows the equivalence between them. The new methodology is based on several choices the developer has to make which uniquely define the error model. This new approach enables the development of all existing models in a unified way, hence the equivalence and correspondence between them is obvious. Moreover, any new model which is of interest can be developed using the methodology presented in this work. In fact, any new model which will ever be developed for the class of systems considered here will fit into the framework described in this paper.

Optimal-REQUEST Algorithm for Attitude Determination

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.

Evaluation of Attitude and Orbit Estimation Using Actual Earth Magnetic Field Data

Asingle,augmentedextendedKalmane lter (EKF),whichsimultaneouslyandautonomouslyestimatesspacecraft attitude and orbit, hasbeen developed and successfully tested with real magnetometer and gyro data only. Because theEarthmagnetice eld isa functionoftimeand position,andbecausetimeisknownquiteprecisely,thedifferences between the computed and measured magnetic e eld components, as measured by the magnetometers throughout the entire spacecraft orbit, are a function of both orbit and attitude errors. Thus, conceivably these differences could be used to estimate both orbit and attitude; an observability study validated this assumption. The results of testing the EKF with actual magnetometer and gyro data, from four satellites supported by the NASA Goddard Space Flight Center Guidance, Navigation, and Control Center, are presented and evaluated. They cone rm the assumption that a single EKF can estimate both attitude and orbit when using gyros and magnetometers only.

SATELLITE ANGULAR RATE ESTIMATION FROM VECTOR MEASUREMENTS

Analgorithm ispresented forestimating theangularratevectorofa satellitethat isbased on the timederivatives of vector measurements expressed in a reference and in body coordinates. The computed derivatives are fed into a special Kalman e lter, which yields an estimateof the spacecraft angularvelocity. Thise lter, an extended interlaced Kalman e lter (EIKF), is an extension of the interlaced Kalman e lter (IKF) presented in the literature. Like the IKF, the EIKF is a suboptimal Kalman e lter that, although being linear, estimates the stateof a nonlineardynamic system. It consists of two or three parallel Kalman e lters whose individual estimates are fed to one another and are considered as known inputs by the other parallel e lter (s). The nonlinear dynamics stem from the nonlinear differential equation that describes the rotation of a three-dimensional body. Tests using simulated as well as real Rossi X-Ray Timing Explore satellite data indicate that the algorithm works satisfactorily.

Optimized TRIAD Algorithm for Attitude Determination

TRIAD is a well known simple algorithm that generates the attitude matrix between two coordinate systems when the components of two abstract vectors are given in the two systems. TRIAD however, is sensitive to the order in which the algorithm handles the vectors, such that the resulting attitude matrix is influenced more by the vector processed first. In this work we present a new algorithm, which we call Optimized TRIAD, that blends in a specified manner the two matrices generated by TRIAD when processing one vector first, and then when processing the other vector first. On the average, Optimized TRIAD yields a matrix which is better than either one of the two matrices in that is ti the closest to the correct matrix. This result is demonstrated through simulation.

The Psi-Angle Error Equation in Strapdown Inertial Navigation Systems

A detailed development is presented of the psi-angle vector differential equation as applied to the error analysis of strapdown inertial navigation systems. The coordinate systems involved and the psi misalignment vector are clearly defined. It is proven that apart from a sign change the psi-angle differential equation in the error analysis of strapdown inertial navigation systems is identical to the one used in conventional gimbaled inertial navigation systems.

Classification of Algorithms for Angular Velocity Estimation

Scanning the known algorithms for deriving the angular velocity vector from either vector measurements or attitude measurements, it is realized that the algorithms are divided into two categories, each category employing a different approach. One category requires differentiation of the measurements, and one does not. A superficial inspection of the two categories may lead to the conclusion that the two are not related to one another; however, a deeper examination reveals that the two are closely related. In this paper the two categories are formulated in general terms, which enables the presentation of the connection between them. The connection is demonstrated through examples.

Angular-Rate Estimation Using Delayed Quaternion Measurements

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared one that uses differentiated quaternion measurements to yield coarse rate measurements, which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear part of the rotas rotational dynamics equation of a body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This non unique decomposition, enables the treatment of the nonlinear spacecraft (SC) dynamics model as a linear one and, thus, the application of a PseudoLinear 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 gain matrix and thus eliminates the need to compute recursively the filter covariance matrix. The replacement of the rotational dynamics by a simple Markov model is also examined. In this paper special consideration is given to the problem of delayed quaternion 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 are presented.