Fundamental Limits of Rigid Body Localization

TL;DR

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.

Abstract

We consider a novel and general approach to easily compute the Cramér-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.

Figures (6)

• Figure 1: Illustration of a rigid body at an initial location aligned with a reference frame $\bm{C}$, and at another location $\bm{\Theta}$, as obtained after the linear transformation described in equation \ref{['eq:basic_model_one_body']}, comprising a3D rotation via the matrix$\bm{Q}$ and a translation by the vector $\bm{t}$. Notice that by force of the latter, the rotation matrix $\bm{Q}$ reduces to an identity $\bm{I}=[\bm{e}_1, \bm{e}_2, \bm{e}_3]$ if the body is at its canonic location such that ${\color{black}\bm{\Theta}}=\bm{C}$.
• Figure 3: RMSE of the translation vector estimate of the SotA and the CRLB variations, over the range error $\sigma$.
• Figure 4: RMSE of the rotation matrix estimate of the SotA and the CRLB variations, over the range error $\sigma$.
• Figure 5: Illustration of the angle-of-arrival measurements dissimilarity function calculation. Notice that the orthonormal vectors $\mathbf{a}$ and $\mathbf{b}$ are centered at $\bm{\theta}_a$, and act as a base for the angle of arrival measurement. Specifically, the plane onto which $\mathbf{d}_{na}$ is projected (giving $\mathbf{d}^\dagger_{na}$) is defined by (normal to) $\mathbf{a}$ and contains the reference vector $\mathbf{b}$ against which the angle of arrival is measured.
• Figure 6: RMSE of the translation vector estimate of the SotA and the CRLB variations, over various noise levels.

Theorems & Definitions (1)

• Lemma 1: Sum-product Formulation of FIM