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Trajectory Based Observer Design: A Framework for Lightweight Sensor Fusion

Federico Oliva, Tom Shaked, Daniele Carnevale, Amir Degani

TL;DR

Trajectory Based Optimization Design (TBOD) presents an optimization-driven framework to design general nonlinear observers for multi-sensor systems by offline-tuning observer parameters over a set of nominal trajectories. The method integrates a plant-replica term with an injection term, and solves a trajectory-based optimization to minimize output-mismatch across trajectories, enabling lightweight, modular sensor fusion. The rover case study demonstrates stability guarantees and competitive position accuracy with a substantial orientation improvement over an Extended Kalman Filter, while maintaining real-time feasibility compared to particle filtering. Overall, TBOD offers a practical design pathway for robust, trajectory-aware observers applicable to a broad range of nonlinear, multi-sensor estimation problems.

Abstract

Efficient observer design and accurate sensor fusion are key in state estimation. This work proposes an optimization-based methodology, termed Trajectory Based Optimization Design (TBOD), allowing the user to easily design observers for general nonlinear systems and multi-sensor setups. Starting from parametrized observer dynamics, the proposed method considers a finite set of pre-recorded measurement trajectories from the nominal plant and exploits them to tune the observer parameters through numerical optimization. This research hinges on the classic observer's theory and Moving Horizon Estimators methodology. Optimization is exploited to ease the observer's design, providing the user with a lightweight, general-purpose sensor fusion methodology. TBOD's main characteristics are the capability to handle general sensors efficiently and in a modular way and, most importantly, its straightforward tuning procedure. The TBOD's performance is tested on a terrestrial rover localization problem, combining IMU and ranging sensors provided by Ultra Wide Band antennas, and validated through a motion-capture system. Comparison with an Extended Kalman Filter is also provided, matching its position estimation accuracy and significantly improving in the orientation.

Trajectory Based Observer Design: A Framework for Lightweight Sensor Fusion

TL;DR

Trajectory Based Optimization Design (TBOD) presents an optimization-driven framework to design general nonlinear observers for multi-sensor systems by offline-tuning observer parameters over a set of nominal trajectories. The method integrates a plant-replica term with an injection term, and solves a trajectory-based optimization to minimize output-mismatch across trajectories, enabling lightweight, modular sensor fusion. The rover case study demonstrates stability guarantees and competitive position accuracy with a substantial orientation improvement over an Extended Kalman Filter, while maintaining real-time feasibility compared to particle filtering. Overall, TBOD offers a practical design pathway for robust, trajectory-aware observers applicable to a broad range of nonlinear, multi-sensor estimation problems.

Abstract

Efficient observer design and accurate sensor fusion are key in state estimation. This work proposes an optimization-based methodology, termed Trajectory Based Optimization Design (TBOD), allowing the user to easily design observers for general nonlinear systems and multi-sensor setups. Starting from parametrized observer dynamics, the proposed method considers a finite set of pre-recorded measurement trajectories from the nominal plant and exploits them to tune the observer parameters through numerical optimization. This research hinges on the classic observer's theory and Moving Horizon Estimators methodology. Optimization is exploited to ease the observer's design, providing the user with a lightweight, general-purpose sensor fusion methodology. TBOD's main characteristics are the capability to handle general sensors efficiently and in a modular way and, most importantly, its straightforward tuning procedure. The TBOD's performance is tested on a terrestrial rover localization problem, combining IMU and ranging sensors provided by Ultra Wide Band antennas, and validated through a motion-capture system. Comparison with an Extended Kalman Filter is also provided, matching its position estimation accuracy and significantly improving in the orientation.

Paper Structure

This paper contains 20 sections, 2 theorems, 36 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Novelty
  3. Performance evaluation
  4. Structure
  5. The Observation Problem
  6. The TBOD method
  7. The observer's structure
  8. The optimization problem
  9. Comments
  10. Case study
  11. Observer design
  12. The plant as a hybrid system
  13. The hybrid observer
  14. The position trilateration
  15. Computing the orientation
  16. ...and 5 more sections

Key Result

Theorem 1

Consider the hybrid plant $\mathcal{P}_h$ in expset:design:eqn:plant_dyn_hybrid and observer in expset:design:eqn:obs_dyn_hybrid, where assumption expset:design:ass:noisetrilateration holds. Consider the noise-free scenario, i.e., $(\bm{\nu}_a, \bm{\nu}_\omega, \bm{\nu}_d) = 0$. Assume that there ex Then, the origin of the estimation error system $\mathcal{E}_h$ in expset:eqn:errDyn is Uniformly G

Figures (3)

  • Figure 1: Graphical representation of the rovers' sensor setup.
  • Figure 2: Representation of one of the trajectories used for the tuning (red-dashed) and one of the trajectories used to test the observer performance (solid-gray). The blue-filled circles are the anchors.
  • Figure 3: $(X, Y,\varphi)$ estimation on a test trajectory. The ground truth is in solid blue, the EKF is in dotted red, and the TBOD is in dashed green.

Theorems & Definitions (3)

  • Theorem 1: Full observer - stability
  • proof
  • Corollary 1: Noise scenario