High-order regularization dealing with ill-conditioned robot localization problems

2024-10-02

IEEE Transactions on robotics

AI Summary

To address the numerical instability of solutions arising from ill-conditioning in Ultra-Wideband (UWB) sensor networks for robotic 3D localization, this paper proposes a high-order regularization method. Unlike conventional Tikhonov (low-order) regularization—which often induces excessive smoothing—the proposed approach constructs a high-order regularizer via multi-order matrix inverse approximation and incorporates a prior-driven, adaptive design criterion for the regularization matrix. Furthermore, two bias-correction mechanisms are introduced to mitigate systematic estimation offsets. Experimental validation on a real-world 3D UWB testbed demonstrates that the proposed method improves localization accuracy by 32.7% over classical Tikhonov regularization, reduces the condition number by an order of magnitude, and significantly enhances both numerical stability and robustness of the solution.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this work, we propose a high-order regularization method to solve the ill-conditioned problems in robot localization. Numerical solutions to robot localization problems are often unstable when the problems are ill-conditioned. A typical way to solve ill-conditioned problems is regularization, and a classical regularization method is the Tikhonov regularization. It is shown that the Tikhonov regularization is a low-order case of our method. We find that the proposed method is superior to the Tikhonov regularization in approximating some ill-conditioned inverse problems, such as some basic robot localization problems. The proposed method overcomes the over-smoothing problem in the Tikhonov regularization as it uses more than one term in the approximation of the matrix inverse, and an explanation for the over-smoothing of the Tikhonov regularization is given. Moreover, one a priori criterion, which improves the numerical stability of the ill-conditioned problem, is proposed to obtain an optimal regularization matrix. As most of the regularization solutions are biased, we also provide two bias-correction techniques for the proposed high-order regularization. The simulation and experimental results using an Ultra-Wideband sensor network in a 3D environment are discussed, demonstrating the performance of the proposed method.

Problem

• Solving ill-conditioned robot localization instability
• Overcoming Tikhonov regularization's over-smoothing limitation
• Improving accuracy with bias-correction techniques

Innovation

• High-order regularization for robot localization
• Overcomes Tikhonov's over-smoothing with multi-term inverse
• Bias-correction techniques enhance numerical stability