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A Taylor Series Approach to Correct Localization Errors in Robotic Field Mapping using Gaussian Processes

Muzaffar Qureshi, Tochukwu Elijah Ogri, Kyle Volle, Rushikesh Kamalapurkar

TL;DR

This work tackles robust Gaussian Process mapping under deterministic localization errors in robotic field mapping. It introduces a two-stage gradient-based correction framework that performs second-order Taylor expansions of the GP mean and covariance around planned locations, with offline precomputation of Jacobians and Hessians to enable real-time updates without full retraining. The key contributions are (i) analytic expressions for mean and covariance corrections under input perturbations, (ii) a decomposition that separates dependence on measurements from location details to allow offline storage, and (iii) a complexity analysis and simulations showing substantial speedups and accuracy gains in 1D and 2D scenarios. The method is practically impactful for GPS-denied or drift-prone robotics, enabling accurate GP-based maps as better localization information becomes available, and is extensible to other kernels and larger-scale problems with memory-efficient representations.

Abstract

Gaussian Processes (GPs) are powerful non-parametric Bayesian models for regression of scalar fields, formulated under the assumption that measurement locations are perfectly known and the corresponding field measurements have Gaussian noise. However, many real-world scalar field mapping applications rely on sensor-equipped mobile robots to collect field measurements, where imperfect localization introduces state uncertainty. Such discrepancies between the estimated and true measurement locations degrade GP mean and covariance estimates. To address this challenge, we propose a method for updating the GP models when improved estimates become available. Leveraging the differentiability of the kernel function, a second-order correction algorithm is developed using the precomputed Jacobians and Hessians of the GP mean and covariance functions for real-time refinement based on measurement location discrepancy data. Simulation results demonstrate improved prediction accuracy and computational efficiency compared to full model retraining.

A Taylor Series Approach to Correct Localization Errors in Robotic Field Mapping using Gaussian Processes

TL;DR

This work tackles robust Gaussian Process mapping under deterministic localization errors in robotic field mapping. It introduces a two-stage gradient-based correction framework that performs second-order Taylor expansions of the GP mean and covariance around planned locations, with offline precomputation of Jacobians and Hessians to enable real-time updates without full retraining. The key contributions are (i) analytic expressions for mean and covariance corrections under input perturbations, (ii) a decomposition that separates dependence on measurements from location details to allow offline storage, and (iii) a complexity analysis and simulations showing substantial speedups and accuracy gains in 1D and 2D scenarios. The method is practically impactful for GPS-denied or drift-prone robotics, enabling accurate GP-based maps as better localization information becomes available, and is extensible to other kernels and larger-scale problems with memory-efficient representations.

Abstract

Gaussian Processes (GPs) are powerful non-parametric Bayesian models for regression of scalar fields, formulated under the assumption that measurement locations are perfectly known and the corresponding field measurements have Gaussian noise. However, many real-world scalar field mapping applications rely on sensor-equipped mobile robots to collect field measurements, where imperfect localization introduces state uncertainty. Such discrepancies between the estimated and true measurement locations degrade GP mean and covariance estimates. To address this challenge, we propose a method for updating the GP models when improved estimates become available. Leveraging the differentiability of the kernel function, a second-order correction algorithm is developed using the precomputed Jacobians and Hessians of the GP mean and covariance functions for real-time refinement based on measurement location discrepancy data. Simulation results demonstrate improved prediction accuracy and computational efficiency compared to full model retraining.
Paper Structure (20 sections, 2 theorems, 28 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 2 theorems, 28 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Gaussian process Regression
  4. Taylor Series Approximation of Mean and Covariance functions
  5. Gradient of the Test-Train Kernel Matrix
  6. Gradient of the Training Kernel Matrix
  7. Gradient of the GP Mean and Covariance
  8. Gradient of the GP Mean Function
  9. Gradient of the GP Covariance Function
  10. Offline Computation of GP Gradients for Efficient Online Updates
  11. Decomposition and Precomputation of Gradients
  12. Mean Function Update
  13. Covariance Function Update
  14. Computational Complexity Analysis
  15. Convergence Analysis
  16. ...and 5 more sections

Key Result

Theorem 1

Given the GP mean function $\mathbf{m}: \mathcal{D} \to \mathbb{R}^M$, if the kernel function $k$ is analytic on the domain $\mathcal{X} \subset \mathbb{R}^n$, then the mean function $\mathbf{m}$ is also analytic on $\mathcal{D}$. The true mean value $\mathcal{M} \coloneqq \mathbf{m}(\boldsymbol{x}) where $\nabla^N \mathbf{m}(\hat{\boldsymbol{x}})$ is the $N$-th order gradient tensor of $\mathbf{m

Figures (7)

  • Figure 1: Baseline GP model prediction trained on the original, non-corrupted 1D measurement locations. The true function is shown in blue.
  • Figure 2: GP model prediction trained on corrupted 1D measurement locations, demonstrating the model shift relative to the true function and original training points.
  • Figure 3: Corrected GP prediction for the 1D simulation. The model, updated using the gradient-based correction, shows the mean function (red line) and $2\sigma$ confidence interval realigned with the true function (blue line).
  • Figure 4: Absolute error of the mean for the 1D simulation, comparing the corrupted GP against the corrected GP across the test domain.
  • Figure 5: Visualization of the true scalar field for the 2D simulation.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 2