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Synchronous Observer Design for Landmark-Inertial SLAM with Almost-Global Convergence

Arkadeep Saha, Pieter van Goor, Antonio Franchi, Ravi Banavar

TL;DR

The paper addresses landmark-inertial SLAM (LI-SLAM) by formulating a continuous-time nonlinear geometric observer on Lie groups. It uses a quotient-manifold base-space representation to factor out frame invariances, enabling a synchronous observer on SE_{n+2}(3) with an auxiliary SIM_{n+2}(3) state that can be kept constant to yield a minimal estimator. The main results prove almost-global asymptotic stability for the base-space error and local exponential stability for the attitude error, with simulations validating convergence and robustness to poor initializations. The approach offers a provably convergent and computationally efficient alternative to EKF and optimization-based SLAM for landmark-inertial scenarios.

Abstract

Landmark Inertial Simultaneous Localisation and Mapping (LI-SLAM) is the problem of estimating the locations of landmarks in the environment and the robot's pose relative to those landmarks using landmark position measurements and measurements from Inertial Measurement Unit (IMU). This paper proposes a nonlinear observer for LI-SLAM posed in continuous time and analyses the observer in a base space that encodes all the observable states of LI-SLAM. The local exponential stability and almost-global asymptotic stability of the error dynamics in base space is established in the proof section and validated using simulations.

Synchronous Observer Design for Landmark-Inertial SLAM with Almost-Global Convergence

TL;DR

The paper addresses landmark-inertial SLAM (LI-SLAM) by formulating a continuous-time nonlinear geometric observer on Lie groups. It uses a quotient-manifold base-space representation to factor out frame invariances, enabling a synchronous observer on SE_{n+2}(3) with an auxiliary SIM_{n+2}(3) state that can be kept constant to yield a minimal estimator. The main results prove almost-global asymptotic stability for the base-space error and local exponential stability for the attitude error, with simulations validating convergence and robustness to poor initializations. The approach offers a provably convergent and computationally efficient alternative to EKF and optimization-based SLAM for landmark-inertial scenarios.

Abstract

Landmark Inertial Simultaneous Localisation and Mapping (LI-SLAM) is the problem of estimating the locations of landmarks in the environment and the robot's pose relative to those landmarks using landmark position measurements and measurements from Inertial Measurement Unit (IMU). This paper proposes a nonlinear observer for LI-SLAM posed in continuous time and analyses the observer in a base space that encodes all the observable states of LI-SLAM. The local exponential stability and almost-global asymptotic stability of the error dynamics in base space is established in the proof section and validated using simulations.

Paper Structure

This paper contains 11 sections, 1 theorem, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Description
  4. LI-SLAM Dynamics and Measurements
  5. LI-SLAM invariance
  6. Lie Group Interpretation
  7. Observer Design
  8. Synchronous Observer Architecture
  9. Observer Design
  10. Simulations
  11. Conclusion

Key Result

Theorem 1

Let $X \in \mathbf{SE}_{n+2}(3)$ denote the LI-SLAM state with dynamics eq:system_dynamics and measurements meas, and let $\hat{X}\in \mathbf{SE}_{n+2}(3)$ denote the observer state with dynamics eq:observer_arch. The estimated measurements are written as $\hat{y}_i = h_i(\hat{X}) = \hat{R}^\top(\ha Define the correction terms $\Delta = (\Omega_\Delta, W_\Delta)\in \mathfrak{se}_{n+2}(3)$ and $W_\

Figures (3)

  • Figure 1: The trajectories of the true and estimated robot and landmark positions over time. All the initial and final positions are marked with $\circ$ and $*$ respectively. The trajectories are aligned as described in Section \ref{['sec:simulations']}
  • Figure 2: The reduced attitude error, body frame velocity error and the errors in relative landmark positions in body frame converge to zero. The value of the Lyapunov function decreases steadily over time.
  • Figure 3: The attitude errors in roll and pitch converge to zero while the attitude error in yaw converges to a constant. The errors in robot and landmark positions also converge to a constant.

Theorems & Definitions (2)

  • Theorem 1
  • proof