Malcolm D. Shuster

Kalman Filtering for Spacecraft Attitude Estimation

HIS report reviews the methods of Kalman filtering in attitude estimation and their development over the last two decades. This review is not intended to be complete but is limited to algorithms suitable for spacecraft equipped with three-axis gyros as well as attitude sensors. These are the systems to which we feel that Kalman filtering is most ap- plicable. The Kalman filter uses a dynamical model for the time development of the system and a model of the sensor measurements to obtain the most accurate estimate possible of the system state using a linear estimator based on present and past measurements. It is, thus, ideally suited to both ground-based and on-board attitude determination. However, the applicability of the Kalman filtering technique rests on the availability of an accurate dynamical model. The dynamic equations for the spacecraft attitude pose many difficulties in the filter modeling. In particular, the external torques and the distribution of momentum internally due to the use of rotating or rastering instruments lead to significant uncertainties in the modeling. For autonomous spacecraft the use of inertial reference units as a model replacement permits the circumvention of these problems. In this representation the angular velocity of the spacecraft is obtained from the gyro data. The kinematic equations are used to obtain the attitude state and this is augmented by means of additional state-vector components for the gyro biases. Thus, gyro data are not treated as observations and the gyro noise appears as state noise rather than as observation noise. It is theoretically possible that a spacecraft is three-axis stabilized with such rigidity that the time development of the system can be described accurately without gyro information, or that it is one-axis stabilized so that only a single gyro is needed to provide information on the time history of the system. The modification of the algorithms presented here in order to apply to those cases is slight. However, this is of little practical importance because a control system capable of such

Three-axis attitude determination from vector observations

Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations. The first of these, the TRIAD algorithm, provides a deterministic (i.e., nonoptimal) solution for the attitude based on two vector observations. The second, the QUEST algorithm, is an optimal algorithm which determines the attitude that achieves the best weighted overlap of an arbitrary number of reference and observation vectors. Analytical expressions are given for the covariance matrices for the two algorithms using a fairly realistic model for the measurement errors. The mathematical relationship of the two algorithms and their relative merits are discussed and numerical examples are given. The advantage of computing the covariance matrix in the body frame rather than in the inertial frame (e.g., in terms of Euler angles) is emphasized. These results are valuable when a single-frame attitude must be computed frequently. They will also be useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.

The quest for better attitudes

This article is adapted from my Brouwer lecture of February 2001 [1]. It has been shortened by about one quarter and updated. Given the occasion (in 2001) I had thought it inappropriate to give a talk in my usual style, which is rich in equations and mathematical derivation. Instead, I attempted to give a talk rich in perspective and personal anecdotes (and maybe a few equations). Since the original lecture, my appreciation of the subject matter has grown and some addenda and corrigenda have become necessary.

The Generalized Wahba Problem

The Generalized Wahba Problem, which can accept as input both measured directions and measured attitudes, is defined and examined in terms of both the attitude profile matrix B and the Davenport matrix K. The possibility of extending the generalization to scalar measurements is also examined. We obtain a number of new results relating these two matrices to the attitude estimate and to the attitude-error covariance matrix. We compare the generalized Wahba problem also with a less restrictive approach to attitude estimation.

The TRIAD Algorithm as Maximum Likelihood Estimation

The TRIAD algorithm is shown to be derivable as a maximum-likelihood estimator. In particular, using the QUEST measurement model, the TRIAD attitude error covariance matrix can be derived as the inverse of the Fisher information matrix. The treatment here gives a microscopic analysis of the algorithm and its connection to the QUEST algorithm. It also sheds valuable light on the origin of discrete degeneracies in deterministic attitude estimation.

Spin-Axis Attitude Estimation

Spin-axis attitude estimation is examined in a manner analogous to the study of three-axis attitude estimation. Measurement modeling issues are given careful consideration, as are those of representation, frame, and constraint. Three approaches to spin-axis attitude estimation are presented and compared numerically. A thorough covariance analysis of all algorithms is performed.

The nature of the quaternion

Some of the confusions concerning quaternions as they are employed in spacecraft attitude work are discussed. The order of quaternion multiplication is discussed in terms of its historical development and its consequences for the quaternion imaginaries. The different formulations for the quaternions are also contrasted. It is shown that the three Hamilton imaginaries cannot be interpreted as the basis of the vector space of physical vectors but only as constant numerical column vectors, the autorepresentation of a physical basis.

The TASTE Test

The TASTE test, which has been an important component of spacecraft attitude mission support for more than a quarter-century, is documented here. The TASTE test permitted data validation and editing for direction sensors to be automated for the first time, greatly decreasing data processing time, and was an important reason for the rapid adoption of QUEST. The statistical properties of the TASTE test are derived. The value of the TASTE test for implementation in modern CCD star trackers is presented.

The Optimization of TRIAD

The optimized TRIAD algorithm of Bar-Itzhack and Harman, proposed in 1997 on the basis of insupportable premises, is shown to provide, nonetheless, the optimal solution of the Wahba problem to within terms on the order of the measurement variances. A careful rigorous derivation of the algorithm is presented as well as a comparison with the arguments of Bar-Itzhack and Harman. The algorithm is compared to a result of Markley’s.

A partitioned recursive algorithm for the estimation of dynamical and initial-condition parameters from cross-sectional data

Many practical applications require the simultaneous estimation of unknown dynamical parameters and unknown initial means and covariances from an ensemble of tests. A recursive algorithm which asymptotically obtains the maximum likelihood estimate of both sets of unknown parameters is presented. The computational requirements of the algorithm are greatly reduced by partitioning the parameter vector into initial and dynamical parameters and making use of a sufficient statistic as an intermediate variable for the estimation of initial condition parameters. This partitioning leads to a two-tier filter for calculating some of the required parameter sensitivities. The results are illustrated by an application to a simplified robotic system.