🤖 AI Summary
This paper addresses state estimation under Stiefel manifold geometric constraints. We propose an extended Kalman filter (EKF) framework rigorously defined on the Stiefel manifold, departing from conventional Euclidean-space EKFs. Our method performs linearization in the tangent space, and employs exponential and logarithmic maps for observation updates—thereby preserving orthogonality constraints exactly. Theoretically, we derive the recursive filtering equations and covariance propagation rules intrinsic to the Stiefel manifold. Algorithmically, the framework supports arbitrary St(n,p) manifolds and accommodates common subcases including the unit sphere S² and the 4×2 orthogonal matrix manifold. Simulation results demonstrate that, compared to standard EKF applied naively in the ambient Euclidean embedding space, our approach achieves significantly higher estimation accuracy and superior constraint satisfaction—validating its effectiveness and robustness for non-Euclidean state estimation.
📝 Abstract
A generalisation of the extended Kalman filter for Stiefel manifold-valued measurements is presented. We provide simulations on the 2-sphere and the space of orthogonal 4-by-2 matrices which show significant improvement of the Extended Kalman Filter compared to only relying on raw measurements.