Robust Egoistic Rigid Body Localization

TL;DR

This paper tackles egoistic rigid body localization (RBL), where a primary object estimates both the translation and orientation of a target using only inter‑body distance measurements and without any infrastructure or prior knowledge of the target’s shape. It introduces two complementary translation estimators and a rotation estimator that operate without knowing the target conformation, accommodating arbitrary shapes and unequal landmark counts, and it addresses incomplete observations through matrix completion and constrained optimization. The first translation method uses Nyström‑based matrix completion to reconstruct the target’s distance matrix and applies MDS followed by Procrustes alignment to recover the translation; the second robust method refines a corrected translation estimator with a constraint to improve resilience to missing data. A separate egoistic rotation estimator computes the relative rotation from distance data alone, with a permutation‑aware eigenstructure handling to avoid misordering, and complexity analyses show cubic scaling in the number of landmarks. Overall, the proposed egoistic RBL framework offers robust, infrastructure‑free pose estimation suitable for V2X/IoV scenarios, with matrix completion providing practical improvements under incomplete data and future work aimed at reducing complexity and enabling dynamic tracking.

Abstract

We consider a robust and self-reliant (or "egoistic") variation of the rigid body localization (RBL) problem, in which a primary rigid body seeks to estimate the pose (i.e., location and orientation) of another rigid body (or "target"), relative to its own, without the assistance of external infrastructure, without prior knowledge of the shape of the target, and taking into account the possibility that the available observations are incomplete. Three complementary contributions are then offered for such a scenario. The first is a method to estimate the translation vector between the center point of both rigid bodies, which unlike existing techniques does not require that both objects have the same shape or even the same number of landmark points. This technique is shown to significantly outperform the state-of-the-art (SotA) under complete information, but to be sensitive to data erasures, even when enhanced by matrix completion methods. The second contribution, designed to offer improved performance in the presence of incomplete information, offers a robust alternative to the latter, at the expense of a slight relative loss under complete information. Finally, the third contribution is a scheme for the estimation of the rotation matrix describing the relative orientation of the target rigid body with respect to the primary. Comparisons of the proposed schemes and SotA techniques demonstrate the advantage of the contributed methods in terms of root mean square error (RMSE) performance under fully complete information and incomplete conditions.

Figures (10)

• Figure 1: Illustration of a rigid body transformation at two distinct locations $\boldsymbol{S}^{(0)}$ and $\boldsymbol{S}^{(1)}$. Without loss of generality, we set the initial position to be identical to the matrix $\boldsymbol{C}$, which defines the shape and orientation of the rigid body. The second location $\boldsymbol{S}^{(1)}$ of the body is then determined according to equation \ref{['eq:basic_model_one_body']}, and is obtained by the transformation of $\boldsymbol{S}^{(0)}$ via a rotation matrix $\bm{Q}$ and a translation vector $\boldsymbol{t}$.
• Figure 2: Illustration of a two-body egoistic RBL scenario. Each rigid body has a different shape and orientation, defined by distinct conformation matrices $\boldsymbol{C}_1$ and $\boldsymbol{C}_2$, respectively. The translation vector $\boldsymbol{t}$ between the bodies, highlighted in yellow, is defined by the difference between the geometric centers of the two rigid bodies.
• Figure 3: RMSE of the translation estimate of the GA proposed methods and the SotA, over the range error $\sigma$.
• Figure 4: RMSE of the translation estimate of the proposed MDS-based method for different levels of available information, over the range error $\sigma$.
• Figure 5: RMSE of the translation estimate of the proposed method aided by matrix completion for different levels of available information, over the range error $\sigma$.