Five-Degree-of-Freedom Pose Estimation from an Imaged Ellipsoid of Revolution

In this Note, I present an analytical solution to 5 degrees of freedom pose estimation from imaged ellipsoids of revolution. A theory is developed which shows that the range, the latitude of observation, and three attitude angles can be retrieved from the image of an ellipsoid of revolution (the longitude being unobservable due to the axial symmetry of the target). Starting from an image of the ellipsoid, the limb is extracted and fitted to an ellipse. Then, by making use of some analytical results from perspective geometry, the range and the latitude of observation are computed as the solution of a system of equations involving the eigenvalues of the fitted ellipse. Finally, the attitude matrix is computed by solving a modified orthogonal Procrustes problem. The algorithm is verified through numerical simulations with synthetically generated images of a spheroid and both synthetic and real images of Ceres taken by Dawn spacecraft. Results for the spheroid indicate the very good performance of the method in retrieving the 5 pose DOF. When applied to a target departing from an ideal spheroid as Ceres, performance degrades slightly for the range and attitude angles; the latitude of observation is instead estimated with poor accuracy. The method developed can be exploited for coarse OPNAV, when the relative attitude between the spacecraft and the target is poorly constrained or, conversely, for relative attitude determination when the position between the spacecraft and the target is not well known.