Attracting Manifolds for Attitude Estimation in Flatland and Otherlands∗

Non-convex and non-affine parameterizations of uncertainty are intrinsic within every attitude estimation problem given the fact that minimal and/or nonsingular representations of the attitude matrix are invariably nonlinear functions of the unknown attitude variables. Of course, this fact remains true for rotation matrices both in the 2-D plane (flatland) and in higher dimensional spaces (otherlands). Therefore, estimation problems involving minimal nonsingular representations of unknown attitude matrices bring significant challenges to the adaptive estimation community. This paper develops a novel algorithm for attitude estimation. The proposed algorithm relies upon the design of an adaptive update law for the attitude estimate while preserving its inherent orthogonal structure. The underlying approach borrows from the classical Poisson differential equation in rigid-body rotational kinematics and endows certain manifold attractivity features within the adaptive estimation algorithm. Consequently, we are not only able to efficiently handle the non-affine and non-convex nature of the parameter uncertainty, but are also ensured of estimation algorithm stability and robustness under bounded measurement noise. In addition to a rigorous discussion on the overall methodology, the paper provides example simulations that help demonstrate the effectiveness of the attracting manifolds design.