A Multiplicative Residual Approach to Attitude Kalman Filtering with Unit-vector Measurements

Using direction vectors of unit length as measurements for attitude estimation in an extended Kalman filter inevitably results in a singular measurement covariance matrix. Singularity of the measurement covariance means no noise is present in one component of the measurement. Unit vector measurements have no noise in the radial component. Singular measurement covariances can be dealt with by the classic Kalman filter formulation as long as the estimated measurement covariance is non singular in the same direction. Unit vector measurements violate this condition since both the true measurement and the estimated measurement have perfectly known lengths. Minimum variance estimation for the unit vector attitude Kalman filter is studied in this work. An optimal multiplicative residual approach is presented. The proposed approach is compared with the classic additive residual attitude Kalman filter. INTRODUCTION The Kalman filter [1, 2] is a widely used algorithm in spacecraft navigation. While the Kalman filter is usually employed to estimate vector quantities such as position or velocity, modifications to the classic algorithm exist to estimate attitude. One popular spacecraft attitude representation is the quaternion-of-rotation. [3, 4] Two common approaches to enforce the unit-norm constraint of the quaternion-of-rotation in the Kalman filter are the multiplicative extended Kalman filter (MEKF) [5] and the additive extended Kalman filter (AEKF) [6]. Projection techniques and constrained Kalman filtering to enforce the quaternion normalization also exist [7, 8]. Direction measurements from attitude sensors are often provided as bearing angles. A unit vector can be created from the angles. While the Kalman filter can easily process the angular measurements, processing unit vectors is a widely adopted technique [6, 9]. The QUEST measurement model [9] is a unit vector measurement model. More recently Cheng et al. [10] introduced a new measurement model. Both these models are additive; Mortari and Majji [11] introduced a multiplicative measurement model. The covariance matrix of the additive measurement models is obtained through linearization assuming the measurement errors are small. This assumption is equivalent to making the measurement error perpendicular to the measure∗Senior Member of the Technical Staff, GN&C Manned Space Systems, The Charles Stark Draper Laboratory, 17629 El Camino Real, Suite 470, Houston, Texas, 77058. E-mail: [email protected]