Fundamental Limits of Rigid Body Localization

TL;DR

This paper introduces a flexible framework for computing the fundamental limits of rigid body localization accuracy using the Cramér-Rao Lower Bound, explicitly accounting for measurement types and error distributions.

Contribution

It develops a generic, information-centric method to derive CRLBs for RBL, capturing translation and rotation uncertainties, and provides closed-form bounds including orthonormality constraints.

Findings

The proposed CRLB framework accurately bounds estimation errors.

Numerical results validate the bounds against state-of-the-art estimators.

The approach enables straightforward updates when measurements change.

Abstract

We consider a novel and general approach to easily compute the Cram\'er-Rao Lower Bounds (CRLBs) of rigid body localization (RBL) problem using arbitrary types of information. To that end, we adopt an information-centric construction of the Fisher information matrix (FIM), which allows capturing the contribution of each measurement towards the FIM explicitly, both in terms of input measurement types, as well as of their error distributions. Taking advantage of this approach, we derive a generic framework for evaluating the CRLB, which is applicable to arbitrary rigid body localization scenarios, and which, unlike the formulation of FIM commonly used in point-target localization, is better suited to RBL problems as it explicitly allows capturing the precision in both the translation vector and the rotation matrix (or alternative the rotation angles) of the rigid body, with respect to a reference. Examples of CRLBs obtained via the proposed approach are given in closed form, including the bound incorporating an orthonormality constraint onto the rotation matrix, which enables a straightforward adjustment of the derived bound when new measurements are added or removed. Numerical results illustrate that the derived expression correctly lower-bounds the errors of estimated localization parameters obtained via various related state-of-the-art (SotA) estimators, revealing their accuracies and suggesting that SotA RBL algorithms can still be improved.