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

TL;DR

This paper introduces a high-order regularization method for robot localization that outperforms Tikhonov regularization by reducing over-smoothing and enhancing numerical stability, validated through simulations and experiments.

Contribution

The paper proposes a novel high-order regularization technique that improves stability and accuracy in ill-conditioned robot localization problems, surpassing traditional Tikhonov regularization.

Findings

The proposed method reduces over-smoothing in ill-conditioned inverse problems.

It provides an a priori criterion for optimal regularization matrix selection.

Experimental results demonstrate improved localization accuracy in 3D environments.

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.