Dynamical Update Maps for Particle Flow with Differential Algebra

Particle Flow Filters estimate the ``a posteriori" probability density function (PDF) by moving an ensemble of particles according to the likelihood. Particles are propagated under the system dynamics until a measurement becomes available when each particle undergoes an additional stochastic differential equation in a pseudo-time that updates the distribution following a homotopy transformation. This flow of particles can be represented as a recursive update step of the filter. In this work, we leverage the Differential Algebra (DA) representation of the solution flow of dynamics to improve the computational burden of particle flow filters. Thanks to this approximation, both the prediction and the update differential equations are solved in the DA framework, creating two sets of polynomial maps: the first propagates particles forward in time while the second updates particles, achieving the flow. The final result is a new particle flow filter that rapidly propagates and updates PDFs using mathematics based on deviation vectors. Numerical applications show the benefits of the proposed technique, especially in reducing computational time, so that small systems such as CubeSats can run the filter for attitude determination.