A Simple Attitude Unscented Kalman Filter: Theory and Evaluation in a Magnetometer-Only Spacecraft Scenario

Abstract

A quaternion-based attitude unscented Kalman filter is formulated with quaternion errors parameterized by small angle approximations and is applied to a filter with a state vector consisting of the attitude quaternion and the gyro bias vector. The filter is evaluated using extensive Monte Carlo data in a simulated lost-in-space scenario of a low-Earth orbiting spacecraft processing only three-axis magnetometer and gyro measurements. The filter is found to be robust, accurate, and rapidly convergent in this scenario for small true gyro biases and small initial uncertainties in their values, often converging in only one half of an orbit period to an attitude accuracy of 0.1 degrees. The filter convergence is found to depend significantly on the value of the true gyro biases as well as the initial gyro bias covariances. Monte Carlo results also indicate that this unscented Kalman filter is significantly less robust than an extended Kalman filter with the same attitude approach, but performs slightly better than another unscented Kalman filter with a generalized Rodrigues parameter approach to quaternion errors.

Unscented and square‐root unscented Kalman filters for quaternionic systems

“…In the UKFs of group 2), the unit quaternion norms are preserved in some steps but not in others; they are preserved in problematic item 8 in the work of Kraft; item 2 in the work of Vartiainen et al; in items 1, 3, and 6 in the work of Larsen and Vinther; and items 1, 3, 6, and 8 in other works …”

“…Within the literature, more than one solution has been given to the problem of creating UFs for quaternionic systems (see, eg, other works). Some works use the same algorithms of the UKFs for Euclidean systems to estimate the state of quaternionic systems (these works are not studied in this work), that is, they do not take unit quaternions norm constraint into account (see, eg, the works of Enayati et al and Vaccarella et al); others do take this constraint into account and can be divided into three groups: a first treats unit quaternion norms but does not preserve them in any problematic item (first row of Table ); a second preserves unit quaternion norms only in some (but not all) problematic items (second row of Table ); and a third preserves unit quaternion norms in all problematic items (third row of Table ). …”

“…In this work, we consider quaternionic systems , that is, stochastic dynamic systems containing state variables taking values on S 3 (see for the specific model studied in this paper). Quaternionic systems can model many kinds of applications, such as in aerospace systems, robotics, computer vision, and biomedical systems …”

“…We restrict ourselves to additive UKFs because the majority of the UKFs for rotating systems based on unit quaternions are additive filters (cf) and because augmented UKFs can be obtained by simple manipulations of the additive UKFs (cf the works of Julier and Uhlmann, and Menegaz).…”

Approximate Chernoff fusion of Gaussian mixtures for ballistic target tracking in the re-entry phase

“…Technically speaking, the problem is a typical recursive nonlinear state estimation problem, where the nonlinearity stems from both the target dynamics [5] and measurement functions [5]. Several nonlinear estimation methods have been used for ballistic target tracking, including extended Kalman filter [6], unscented Kalman filter [7], particle filter [8] and ensemble Kalman filter [2]. One disadvantage of these methods is that they assume the ballistic coefficient, which is one of key parameters for target dynamics, is known with sufficient accuracy.…”