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Dynamical Update Maps for Particle Flow with Differential Algebra

Simone Servadio

TL;DR

This work addresses the computational burden of particle flow filters in nonlinear state estimation by introducing a Differential Algebra (DA) based polynomial representation of the flow. By expressing both the prediction and update steps as state-transition and update polynomials, the method forms a single composite map (DARUFF) that maps prior deviations directly to the updated state using polynomial evaluation, dramatically reducing computation. The approach yields comparable accuracy to traditional ODE-based flows in both a range measurement example and a CubeSat attitude determination task, while delivering substantial speed-ups that enable on-board, low-power filtering for small spacecraft. The results suggest that DA-based polynomial maps are a practical and scalable alternative for high-dimensional, real-time estimation problems, with clear avenues for incorporating diffusion and direct moment computation in future work.

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.

Dynamical Update Maps for Particle Flow with Differential Algebra

TL;DR

This work addresses the computational burden of particle flow filters in nonlinear state estimation by introducing a Differential Algebra (DA) based polynomial representation of the flow. By expressing both the prediction and update steps as state-transition and update polynomials, the method forms a single composite map (DARUFF) that maps prior deviations directly to the updated state using polynomial evaluation, dramatically reducing computation. The approach yields comparable accuracy to traditional ODE-based flows in both a range measurement example and a CubeSat attitude determination task, while delivering substantial speed-ups that enable on-board, low-power filtering for small spacecraft. The results suggest that DA-based polynomial maps are a practical and scalable alternative for high-dimensional, real-time estimation problems, with clear avenues for incorporating diffusion and direct moment computation in future work.

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.
Paper Structure (10 sections, 41 equations, 9 figures, 1 algorithm)

This paper contains 10 sections, 41 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Particle Flow Update
  3. Particle Flowing with Recursive Update
  4. The Differential Algebra Approach
  5. The DA Recursive Particle Flow
  6. The DA Ensemble Propagation
  7. Map Combination
  8. Range Measurement Example
  9. Attitude Determination
  10. Conclusion

Figures (9)

  • Figure 1: ODE vs. DA Flow solution
  • Figure 2: DA Flow Analysis with respect to the expansion order
  • Figure 3: Computational Time Comparison
  • Figure 4: Monte Carlo analysis for the quaternions error.
  • Figure 5: Monte Carlo analysis for the angular velocity error.
  • ...and 4 more figures