M.Sc Thesis

M.Sc StudentCohen Yizhar
SubjectGeometry Approach for Recursive Attitude Estimation from
Vector Observation
DepartmentDepartment of Aerospace Engineering
Supervisors PROF. Yaakov Oshman
PROFESSOR EMERITUS Y. Bar-Itzhack (Deceased)


New methods are presented for attitude determination from a sequence of single direction measurements.  First presented by G. Wahba in 1965, the attitude determination problem is as follows: given a set of n unit vectors, measured in body axes system (bi), and the same vectors represented in some reference system (ri), find the transformation from the reference system to the body system based on these measurements.

The estimation of the quaternion of rotation using Kalman filtering is not simple, as it leads to a nonlinear measurement update model, and requires special handling of the quaternion unit norm.  Known algorithms for attitude determination using the Kalman filter, e.g., the Additive Extended Kalman Filter (AEKF) or the Multiplicative Extended Kalman Filter (MEKF), use a linearized measurement model.

The new geometric method proposes a “direct measurement” approach for the measurement of the quaternion of rotation.  The “measured quaternion” is computed using a single direction measurement and the a priory estimated quaternion.  First proposed by R. G. Reynolds in 1998, the quaternion measurement process is as follows: from a single direction measurement, that is, the vector pair b and r, two quaternions that span the “rotation plane” in the four dimensional quaternion space are computed.  Then, the a priory estimated quaternion is projected onto this space, thus forming the measured quaternion.  The novelty of the present research is in the geometric interpretation of the projection, and the incorporation of the “measured quaternion” into the Kalman filter.

The “direct measurement” of the quaternion leads to a very simple, and linear, measurement model, allowing the use of linear, simpler dynamic models in the filter.  Several estimators were examined using the new measurement model: 1) quaternion estimation from direct measurement of the angular rates, 2) quaternion estimation from rate measurements with constant bias, and 3) three different types of estimators for cases with no direct rate measurements for which the rates also need to be estimated.

The idea of using geometric projection was tested also for the Euler vector. Not surprisingly, the filter based on this representation was found to be inferior to the quaternion-based filters.  The superiority of the new estimators over the AEKF and the MEKF, in terms of both robustness to initial errors and computational efficiency, was demonstrated using extensive Monte Carlo simulations.