An engineer's guide to Particle Filtering on the Stiefel manifold

Abstract

In many engineering applications the state of a dynamical system is modelled by a Stochastic Differential Equation (SDE) evolving in a “curved” (non-Euclidean) space such as the Stiefel manifold - the set of n × p real matrices with orthonormal columns, (n ≥ p). Due to the advances in computing power, the problem of state estimation can be efficiently addressed by the Particle Filter (PF). However, PF algorithms have to be completely reworked to handle the geometry, and the very few papers that properly deal with either the geometry or the stochastics of the problem are in the mathematics literature and are not accessible to an engineering audience. With this in mind and motivated by deterministic schemes on the Stiefel manifold, we give a direct accessible derivation of a novel PF algorithm for state estimation on the Stiefel manifold such that the resulting estimators always remain on the manifold. Our method can be applied to ANY dynamical system (SDE) on the Stiefel manifold. We do not rely on differential geometry or advanced stochastic calculus. Simulation examples are provided.