Nonlinear state estimation on unit spheres using manifold particle filtering

In many applications in engineering, one is interested in tracking a dynamic system whose state evolves on a manifold. Solutions to such problems frequently must resort to nonlinear filtering techniques as many manifolds can be described as equality restrictions on higher-dimensional embedding spaces. We propose in this paper a new particle filtering (PF) method to track the states of dynamic systems that evolve according to a random walk on the unit sphere. We derive an approximation to the intractable optimal importance function and develop a Markov Chain Monte Carlo (MCMC) method to sample from it. The system state variable is then estimated via a Monte Carlo approximation of its intrinsic mean on the sphere, obtained from the Karcher mean of the particle set. As we verify via computer simulations, the proposed method shows improved performance compared to previous Constrained Extended Kalman filters and Bootstrap PF solutions.