[Paper Review] The Recursive Gauss-Newton Filter

Roaldje Nadjiasngar, Michael Inggs|arXiv (Cornell University)|Oct 24, 2011

Neural Networks and Applications8 references3 citations

TL;DR

This paper introduces a compact, memory-efficient recursive Gauss-Newton filter (RGNF) derived from the Gauss-Newton method, combining local linearization and weighted least squares for nonlinear state estimation. By integrating the Levenberg-Maquardt damping strategy, the RGNF achieves robust performance across all four combinations of linear/nonlinear process and observation models, significantly reducing computational memory while maintaining accuracy in tracking maneuvering targets.

ABSTRACT

This paper presents a compact, recursive, non-linear, filter, derived from the Gauss-Newton (GNF), which is an algorithm that is based on weighted least squares and the Newton method of local linearisation. The recursive form (RGNF), which is then adapted to the Levenberg-Maquardt method is applicable to linear / nonlinear of process state models, coupled with the linear / nonlinear observation schemes. Simulation studies have demonstrated the robustness of the RGNF, and a large reduction in the amount of computational memory required, identified in the past as a major limitation on the use of the GNF.

Motivation & Objective

• To develop a recursive, memory-efficient formulation of the Gauss-Newton filter (GNF) to overcome the high computational memory demands of the original non-recursive GNF.
• To extend the applicability of the GNF to all four combinations of linear and nonlinear process dynamics and observation models.
• To enhance robustness and convergence stability by integrating the Levenberg-Maquardt damping strategy into the recursive filter framework.
• To enable real-time tracking of maneuvering targets using a single filter model with adaptive damping and forgetting factors.

Proposed method

• Derives a recursive form of the Gauss-Newton filter using local linearization around a nominal trajectory, avoiding explicit computation of the state transition matrix.
• Introduces a recursive update for the Hessian matrix approximation using weighted least squares, enabling efficient state estimation without storing past data.
• Applies the Levenberg-Maquardt damping factor to balance between Gauss-Newton and steepest descent steps, improving convergence and robustness.
• Incorporates a forgetting factor into the recursive equations to enhance tracking performance for maneuvering targets.
• Uses a compact, unified algorithm applicable to all four cases: linear/linear, linear/nonlinear, nonlinear/linear, and nonlinear/nonlinear process and observation models.
• Employs numerical integration to compute the sensitivity matrix A(Ẋ(t)) and transition matrix Φ for the perturbation dynamics.

Experimental results

Research questions

• RQ1 Can a recursive, memory-efficient form of the Gauss-Newton filter be derived that avoids storing past data or transition matrices?
• RQ2 How does the integration of the Levenberg-Maquardt damping strategy improve the robustness and convergence of the recursive Gauss-Newton filter?
• RQ3 To what extent can the recursive Gauss-Newton filter track maneuvering targets across all combinations of linear and nonlinear process and observation models?
• RQ4 What is the impact of the damping factor and forgetting factor on estimation accuracy and convergence speed in nonlinear tracking scenarios?

Key findings

• The recursive Gauss-Newton filter (RGNF) achieves zero memory overhead by eliminating the need to store past state transition matrices or historical data.
• The RGNF demonstrates robust performance across all four model combinations: linear/linear, linear/nonlinear, nonlinear/linear, and nonlinear/nonlinear dynamics and observations.
• The velocity RMSE varies with target maneuvers, while the position RMSE remains stable, indicating effective tracking of target states despite dynamic changes.
• The damping factor increases sharply at maneuver transitions, indicating adaptive response to changes in target dynamics.
• The average number of iterations per cycle increases during maneuvers, reflecting higher computational demand during state transitions.
• The filter achieves convergence with a small number of iterations (average k ≈ 10–15) and maintains low root mean squared error (RMSE) across 250 Monte Carlo runs.