Iterative RNDOP-Optimal Anchor Placement for Beyond Convex Hull ToA-based Localization: Performance Bounds and Heuristic Algorithms

TL;DR

The work addresses robust localization of far-away targets outside the anchors’ convex hull in ToA-based systems by introducing Range-Normalized DOP (RNDOP) to decouple distance from geometry. It derives closed-form RNDOP expressions and asymptotic bounds, then formulates a minimax optimization to place anchors for worst-case performance, which is non-convex under practical constraints. To make the problem tractable, the authors propose iterative single-anchor addition schemes and three practical heuristics (RNDOP-driven, trace-based, and eigenvector-based), along with 2D extensions, bounds, and an uncertainty analysis for anchor positions. They provide computational complexity results and extensive simulations demonstrating how the trace-based heuristic often yields the best localization accuracy while maintaining reasonable computation times, as well as universal and problem-specific bounds that guide deployment. The framework offers actionable design insights for beyond-convex-hull localization in UAV-based and vehicle-centric scenarios, including robustness to anchor-position errors and guidance on algorithm choice based on performance-time tradeoffs.

Abstract

Localizing targets outside the anchors' convex hull is an understudied but prevalent scenario in vehicle-centric, UAV-based, and self-localization applications. Considering such scenarios, this paper studies the optimal anchor placement problem for Time-of-Arrival (ToA)-based localization schemes such that the worst-case Dilution of Precision (DOP) is minimized. Building on prior results on DOP scaling laws for beyond convex hull ToA-based localization, we propose a novel metric termed the Range-Normalized DOP (RNDOP). We show that the worst-case DOP-optimal anchor placement problem simplifies to a min-max RNDOP-optimal anchor placement problem. Unfortunately, this formulation results in a non-convex and intractable problem under realistic constraints. To overcome this, we propose iterative anchor addition schemes, which result in a tractable albeit non-convex problem. By exploiting the structure arising from the resultant rank-1 update, we devise three heuristic schemes with varying performance-complexity tradeoffs. In addition, we also derive the upper and lower bounds for scenarios where we are placing anchors to optimize the worst-case (a) 3D positioning error and (b) 2D positioning error. We build on these results to design a cohesive iterative algorithmic framework for robust anchor placement, characterize the impact of anchor position uncertainty, and then discuss the computational complexity of the proposed schemes. Using numerical results, we validate the accuracy of our theoretical results. We also present comprehensive Monte-Carlo simulation results to compare the positioning error and execution time performance of each iterative scheme, discuss the tradeoffs, and provide valuable system design insights for beyond convex hull localization scenarios.

Figures (6)

• Figure 1: 3D ToA-based localization of far-away targets. The anchors are mounted on UAVs, and the $i^{\rm th}$ anchor is at $\bm{r_i}=(x_i,y_i,z_i), i=1,2,\cdots,N$ such that all anchors are confined in a relatively small volume enclosed by their convex hull, such that $\sum_{i=1}^N \bm{r_i}= \bm{0}$. The target is far-away from the origin such that $r_t \approx \| \bm{r_t} - \bm{r_i} \|_2$.
• Figure 2: Variation of ${\mathsf{R}^{(+)}_\mathsf{xyz}}$ and its bounds as a function of the number of total anchors ($M=N + N_{\rm a}$) under the (a) Minimax RNDOP-optimal scheme (Lemma \ref{['lemma:iterative_3D_RNDOP_based']}), (b) Minimax Trace-based scheme (Lemma \ref{['lemma:iterative_trace_based_3d']}), and (c) Minimax Eigenvector-based scheme (Lemma \ref{['lemma:min_eigvalue_anch_plcmnt_3D']}), for beyond convex hull 3D localization.
• Figure 3: Variation of ${\mathsf{R}^{(+)}_\mathsf{xy}}$ and its bounds as a function of the number of anchors ($M=N + N_{\rm a}$) under the (a) Minimax RNDOP-optimal scheme (Lemma \ref{['lemma:iterative_rndop_2d']}), (b) Minimax Trace-based scheme (Lemma \ref{['lemma:iterative_trace_based_2d']}), and (c) Minimax Eigenvector-based (Lemma \ref{['lemma:iterative_eigvec_based_2d']}) scheme, for 2D (XY) beyond convex hull localization.
• Figure 4: (a) CDF of 3D positioning error for the $\mathcal{R}_{i_{10}}$ (good, denoted by dashed lines) and $\mathcal{R}_{i_{90}}$ (bad, denoted by solid lines) anchor configurations obtained by each iterative method (3D optimal schemes), for targets distributed uniformly at random in a sphere of radius 200m. (b) CDF of 2D positioning error for the $\mathcal{R}_{i_{10}}$ (good, denoted by dashed lines) and $\mathcal{R}_{i_{90}}$ (bad, denoted by solid lines) anchor configurations obtained by each iterative method (2D optimal schemes), for targets distributed uniformly on the XY-plane at random in a circle of radius 200m.
• Figure 5: Distribution of the fractional change in minimax RNDOP $(\Delta \mathsf{R}^{(+)}/\mathsf{R}^{(+)}(\mathcal{R}_a) )$ due anchor position uncertainty of $\bm{\Delta r_i} = [\Delta x_i, \Delta y_i, \Delta z_i]^T$, for (a) 3D (blue) , and (b) 2D (red) beyond convex hull localization scenarios. The anchor position uncertainty is distributed as $\Delta x_i \sim {\rm U}[-1 \text{m},1\text{m}], \Delta y_i \sim {\rm U}[-1 \text{m},1\text{m}]$, and $\Delta z_i \sim {\rm U}[-1 \text{m},1\text{m}]$ for $i=1,2,\cdots,N$.

Theorems & Definitions (50)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• proof
• Theorem 1
• proof
• Definition 1
• Lemma 1