Dynamical Update Maps for Particle Flow with Differential Algebra

2025-05-02

AI Summary

To address the excessive computational burden of particle flow filters in real-time nonlinear attitude estimation for resource-constrained platforms (e.g., CubeSats), this paper proposes a novel particle flow filtering framework based on differential algebra (DA). For the first time, DA is uniformly applied to both prediction and update stages, replacing conventional numerical integration with analytic polynomial mappings to jointly propagate particles under system dynamics and measurement updates, thereby enabling efficient evolution of the posterior density. The method integrates homotopy-based transformation and stochastic differential equation modeling, employing bias vectors to precisely characterize uncertainty propagation. Numerical experiments demonstrate that the proposed filter achieves accuracy comparable to state-of-the-art methods while reducing computation time by an order of magnitude—significantly enhancing real-time performance and practical feasibility. This work establishes a new paradigm for high-precision autonomous attitude determination on small spacecraft.

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Abstract

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.

Problem

• Improving computational efficiency of particle flow filters
• Solving prediction and update equations using Differential Algebra
• Enabling real-time PDF updates for small systems like CubeSats

Innovation

• Uses Differential Algebra for particle flow
• Solves differential equations with polynomial maps
• Reduces computational time for small systems