Optimal Sensor Placement Using Combinations of Hybrid Measurements for Source Localization

TL;DR

The paper addresses static 2D source localization using hybrid sensor measurements (TDOA, AOA, RSS, TOA) and shows how sensor geometry governs estimation accuracy via the Cramér-Rao bound (CRB). It develops a unified, A-optimality framework to minimize tr(CRB) across hybrid measurement configurations, deriving the global lower bound O7 and corresponding geometric constraints that unify multiple sensing modalities. Simulations demonstrate that optimal deployments resemble uniform angular arrays (UAA) when sensor–source distances are large, with verifiable reductions in tr(CRB) across uniform and nonuniform distance scenarios. The work provides practical deployment guidelines and a foundation for algebraic algorithms to compute optimal geometries for diverse sensor networks.

Abstract

This paper focuses on static source localization employing different combinations of measurements, including time-difference-of-arrival (TDOA), received-signal-strength (RSS), angle-of-arrival (AOA), and time-of-arrival (TOA) measurements. Since sensor-source geometry significantly impacts localization accuracy, the strategies of optimal sensor placement are proposed systematically using combinations of hybrid measurements. Firstly, the relationship between sensor placement and source estimation accuracy is formulated by a derived Cramér-Rao bound (CRB). Secondly, the A-optimality criterion, i.e., minimizing the trace of the CRB, is selected to calculate the smallest reachable estimation mean-squared-error (MSE) in a unified manner. Thirdly, the optimal sensor placement strategies are developed to achieve the optimal estimation bound. Specifically, the specific constraints of the optimal geometries deduced by specific measurement, i.e., TDOA, AOA, RSS, and TOA, are found and discussed theoretically. Finally, the new findings are verified by simulation studies.

Figures (4)

• Figure 1: Optimal sensor trajectories of three sensors with uniform $d_{i} = 1000$m. Each black $\circ$ denotes the start position of a sensor, and a black $+$ denotes the final position of a sensor. The true source is denoted by the black $\triangle$. The dashed lines indicate the final optimal sensor-source geometries.
• Figure 2: Optimal sensor trajectories with different 4 start points for uniform $d_{i} = 1000\;m$
• Figure 3: Comparison of optimal and non-optimal strategies, (a) strategies and (b) MSEs as a function of different noise error $\sigma$.
• Figure 4: Optimal sensor trajectories of two examples in Case 3 with nonuniform $d_{i}$. (a) nonuniform for $d_{1} = 1000$m, $d_{2} = 1300$m, and $d_{3} = 1600$m with 3 sensors. (b) nonuniform for $d_{1} = 300$m, $d_{2} = 600$m, $d_{3} = 800$m, and $d_{4} = 1000$m with 4 sensors. (b) two sensor groups with nonuniform $d_{1} = d_{2} = d_{3} = 1000$m and $d_{4} = d_{5} = d_{6} = 700$m. (d) two sensor groups with nonuniform $d_{1} = d_{2} = d_{3} = 1000$m and $d_{4} = d_{5} = d_{6} = 700$m.