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Unified Constrained Geometric Configuration Optimization for Source Localization Systems: A Riemannian Manifold-Based Approach

Xin Cheng, Feng Shu, Gangle Sun, Xinrui Li, Yuqi Chen, Guangjie Han

arXiv:2609.12670v1eess.SP

TL;DR

Sensor geometry affects localization accuracy, yet constrained-geometry optimization has not unified five commonly used localization methods. The paper formulates a shared constrained PEB objective and proposes RM-CGCOA; simulations report lower PEB than existing strategies and near-optimal-search performance at much faster running time.

  • Problem

    Sensor geometry influences localization performance, but prior constrained-geometry methods covered individual systems rather than unifying TOA, TDOA, RSS, AOA, and RSSD under general position constraints.

    Source: §I · pp. 1–2

  • Method

    The paper derives a unified A-optimal PEB formulation and proposes RM-CGCOA, optimizing orientation on a unit-circle product manifold with an adaptive pullback for angle constraints.

    Source: §I · p. 2

  • Results

    For TOA, TDOA, RSS, and AOA, RM-CGCOA beats random and uniform strategies and matches near-optimal search PEB faster; at N = 18, it reduces uniform-strategy PEB by 33.8% (TDOA).

    Source: §I · p. 3, B. Results of TOA, TDOA, AOA and RSS… · p. 9

  • Takeaways & Limitations

    Simulations report PEB reduction and low computational cost for RM-CGCOA across the five localization systems.

    Source: §VI · p. 10

Abstract

from arXiv · show

Time of arrival (TOA), time difference of arrival (TDOA), received signal strength (RSS), received signal strength difference (RSSD) and angle of arrival (AOA) are commonly used techniques for source localization. The positioning accuracy of these systems depends heavily on the geometric configuration of sensors, which is typically constrained by practical conditions. This paper presents a unified framework for optimizing sensor geometry across all five localization systems, explicitly incorporating both distance and angle constraints on sensor positions. First, the Cram\'er-Rao lower bounds (CRLBs) of these systems are transformed to obtain a unified expression. Based on this expression, a unified constrained geometric configuration optimization problem is formulated. The problem is then simplified into a compact form by replacing the sensor-target angles with an orientation matrix. Subsequently, a Riemannian manifold-based constrained geometric configuration optimization algorithm (RM-CGCOA) is proposed to optimize the sensor-target distances and the orientation matrix. This algorithm casts the orientation matrix onto a product manifold of unit circles. An adaptive pullback is further proposed to strictly enforce angle-related inequality constraints, ensuring that all iterates remain feasible when updating the orientation matrix via the Riemannian gradient. Within RM-CGCOA, analytical optimal distances are derived for TOA, TDOA, RSS and AOA, whereas for RSSD, the distances are updated using projected gradient descent (PGD) together with the orientation matrix. Experimental results demonstrate that the proposed RM-CGCOA consistently achieves a significantly lower position error bound (PEB) compared with the existing strategies and yields a similar PEB to the near-optimal search algorithm, but with a much faster running time.

I. INTRODUCTION

The introduction motivates constrained sensor placement and presents a unified PEB-based optimization framework for five localization systems. It outlines the system model and the TOA measurement model used to derive a CRLB.

I. INTRODUCTION

  • Earlier constrained-geometry methods addressed individual localization systems, whereas this work formulates a unified A-optimality framework covering TOA, TDOA, RSS, AOA, and RSSD.The introduction identifies distance and angle constraints as part of the unified formulation.Source: §I · p. 2
  • RM-CGCOA updates the orientation matrix on a product manifold of unit circles and uses an adaptive pullback to enforce angle-related inequality constraints; RSSD distances use projected gradient descent.Analytical optimal distances are derived for TOA, TDOA, RSS, and AOA.Source: §I · pp. 2–3
  • Across TOA, TDOA, RSS, and AOA, RM-CGCOA achieves lower PEB than random and uniform strategies and matches near-optimal search while running much faster; for RSSD, it achieves the lowest PEB among considered methods.The simulations also find a threshold beyond which a larger spread angle is unnecessary and indicate that RSSD requires distance optimization.Source: §I · p. 3

II. SYSTEM MODEL

  • The system model constrains each sensor's source distance and limits sensor positions to an angular sector, with CRLB used to benchmark unbiased-estimator accuracy.The sensor-target geometry is represented by distances and relative angles.Source: §II · p. 3

A. CRLB of TOA-Based Localization System

  • For TOA, synchronized sensors measure signal propagation time with Gaussian measurement noise, which is converted to distance measurements to form the measurement vector and derive the FIM.The orientation matrix comprises unit orientation vectors from the target to the sensors.Source: A. CRLB of TOA-Based Localization System · pp. 3–4

B. CRLB of TDOA-Based Localization System

The TDOA model handles an unknown target transmission time by differencing sensor measurements against a reference sensor. Its transformed measurement noise remains Gaussian and determines the covariance used in the FIM.

B. CRLB of TDOA-Based Localization System

  • TDOA assumes synchronized sensors that are not synchronized with the target, and models sensor measurement errors as Gaussian.The unknown target transmission time is denoted t0.Source: B. CRLB of TDOA-Based Localization… · p. 4
  • Because the target transmission time is unknown, TDOA differences the distance measurements, using the first sensor as the reference.Each difference is the range to sensor n minus the range to sensor 1, plus the corresponding noise difference.Source: B. CRLB of TDOA-Based Localization… · p. 4
  • The differenced noise is a linear transformation of Gaussian errors, so it remains Gaussian with covariance Rtdoa for the FIM calculation.The transformation is represented by K.Source: B. CRLB of TDOA-Based Localization… · p. 4

C. CRLB of RSS-Based Localization System

The RSS and RSSD models use signal-strength measurements to derive their respective CRLBs. RSSD differences remove dependence on unknown source power, while RSS assumes that power is known.

C. CRLB of RSS-Based Localization System

  • RSS models received signal strength with log-normal shadow fading and Gaussian measurement noise, using known source power and path-loss exponent to form the measurement vector and FIM.The source power and path-loss exponent are assumed known through calibration or prior knowledge.Source: C. CRLB of RSS-Based Localization System · p. 4

D. CRLB of RSSD-Based Localization System

  • When source power is unknown, RSSD differences between sensors eliminate that power from the RSS measurements.The model takes the first sensor as the reference.Source: D. CRLB of RSSD-Based Localization… · p. 4
  • RSSD difference noise is a linear transformation of Gaussian measurement errors, yielding a Gaussian noise vector with covariance Rrssd for the FIM and CRLB.The noise vector is expressed as ηrssd = Kνrssd.Source: D. CRLB of RSSD-Based Localization… · p. 5

E. CRLB of AOA-Based Localization System

The AOA section describes its angle measurement model, while the constrained-configuration objective uses PEB and the unified characterization puts the systems’ CRLBs into a common form. This representation supports a more concise optimization formulation.

E. CRLB of AOA-Based Localization System

  • For AOA, each sensor measures an incoming signal angle from phase differences across its antenna elements.The model’s angle is computed from the target and sensor coordinates using a four-quadrant arctangent.Source: E. CRLB of AOA-Based Localization System · p. 5

F. Constrained Geometric Configuration Optimization Problem

A. Unified Characterization

  • A linear transformation expresses the different systems’ CRLBs in a unified form, separating orientation, distance, noise, and system-parameter components.The resulting form is used to express the optimization problem uniformly and simplify its formulation.Source: A. Unified Characterization · p. 5

B. Problem Transformation

The transformation simplifies a difficult geometry optimization by replacing sensor-target angles with orientation vectors and expressing angle limits as inequalities. It also yields optimal-distance conditions for several localization systems.

Problem transformation

  • The unified objective remains difficult to solve because its relationship to the optimization variables is too complex, motivating further simplification.Source: B. Problem Transformation · p. 6
  • Replacing sensor-target angles with orientation vectors simplifies the formulation, and the angle constraint becomes the component-wise inequality h_n ⪰ h_0.The equivalent formulation retains the distance bounds d_min ≤ d_n ≤ d_max.Source: B. Problem Transformation · p. 6

Distance optimization

  • The objective increases with sensor-target distance in RSS and AOA, so setting every distance to d_min is optimal for those systems.For TOA and TDOA, the CRLB is distance-independent, so d_n = d_min also satisfies the global optimum condition.Source: B. Problem Transformation · p. 6
  • In RSSD, the minimum distance is not optimal because matrix K couples distances across sensors.Source: B. Problem Transformation · p. 6

IV. PROPOSED CONSTRAINED GEOMETRIC CONFIGURATION OPTIMIZATION ALGORITHM

RM-CGCOA optimizes orientation vectors on a product manifold of unit circles while enforcing angle-related inequalities. It holds analytically optimal distances fixed for four systems and updates RSSD distances by projected gradient descent.

Algorithm design

  • RM-CGCOA uses a product manifold of unit circles to maintain orientation-vector norm constraints while optimizing the constrained geometry problems.Riemannian gradients, vector transport, and retraction operations are used for manifold updates.Source: §IV · pp. 6–7

Feasibility enforcement

  • An adaptive pullback after retraction strictly enforces the orientation inequality constraints at each update.The pullback is needed because a step-size update may otherwise violate those constraints.Source: §IV · pp. 7–8

Distance updates

  • For RSSD, RM-CGCOA updates distances by projected gradient descent onto [d_min, d_max], synchronized with orientation updates; other systems keep optimal distances fixed.The fixed-distance systems are RSS, TOA, TDOA, and AOA.Source: §IV · p. 7

A. Experimental Setup

The experimental setup specifies default localization parameters and compares RM-CGCOA with random, uniform, and search strategies. It also reports a running-time comparison with the search algorithm.

A. Experimental Setup

  • Default source-localization settings use target position p = [0, 0]^T, six sensors (N = 6), minimum distance d_min = 200 m, maximum distance d_max = 400 m, and maximum spread angle β_max = 2π.Source: A. Experimental Setup · p. 8
  • Table I compares RM-CGCOA and the search algorithm by running time in seconds.Source: Table I · p. 8
  • The comparison set comprises random, uniform, and search strategies across all localization scenarios.The random strategy’s performance is averaged over 200 independent runs.Source: A. Experimental Setup · p. 8

B. Results of TOA, TDOA, AOA and RSS Systems

For TOA, TDOA, AOA, and RSS, RM-CGCOA achieves lower PEB than random and uniform strategies and performs nearly as well as near-optimal search. The results also show that larger sensor counts and spread angles have diminishing benefits, while optimized layouts vary by system.

B. Results of TOA, TDOA, AOA and RSS Systems

C. Results of RSSD System

In RSSD, RM-CGCOA achieves lower PEB than the compared methods, with performance depending on jointly optimized sensor distances and orientations. The conclusion places this result within a unified framework for five localization systems, using analytical distances for four and projected gradient descent for RSSD.

C. Results of RSSD System

  • RM-CGCOA yields lower PEB than the other three algorithms as spread angle varies; search catches up at larger angles when both methods use minimum sensor distances.The comparison is shown in Fig. 4.Source: C. Results of RSSD System · p. 10, Figure 4 · p. 10
  • At N = 18, RM-CGCOA reduces PEB versus uniform, random, and search by 46.1%, 35.7%, and 6.0%, respectively.The sensor-count comparison is shown in Fig. 5.Source: C. Results of RSSD System · p. 10, Figure 5 · p. 10
  • Optimized RSSD distances do not always reach the minimum: some sensors are placed at the maximum allowable distance when βmax is small.Increasing the maximum distance lowers RM-CGCOA PEB, with pronounced diminishing returns; Fig. 7 highlights the need to optimize distances.Source: C. Results of RSSD System · p. 10

VI. CONCLUSION

  • The conclusion formulates one constrained geometry problem for TOA, TDOA, AOA, RSS, and RSSD, then uses RM-CGCOA to optimize orientation and distance.It derives analytical optimal distances for TOA, TDOA, RSS, and AOA; RSSD distances are jointly updated with orientation using PGD.Source: §VI · p. 10

APPENDIX A DERIVATION OF DISTANCE GRADIENT IN RSSD SYSTEM

Appendix A derives the RSSD PEB gradient with respect to sensor distance; Appendix B derives the Euclidean gradient given in Eq. (45).

APPENDIX A DERIVATION OF DISTANCE GRADIENT IN RSSD SYSTEM

  • Figure 6 depicts RM-CGCOA-optimized sensor positions in the RSSD system, with the target marked by a black hexagram and sensors by colored diamonds.Figure 7 plots PEB against maximum distance in the RSSD system.Source: Figure 6 · p. 11, Figure 7 · p. 11
  • The RSSD derivation rewrites the Fisher information matrix and then obtains the PEB derivative with respect to sensor distance.It introduces Y = K^T(KRνK^T)^−1K in the distance-derivative calculation.Source: Appendix A · pp. 10–12

APPENDIX B DERIVATION OF EQ. (45)

  • The Appendix B derivation uses C = F^−1 and the differential of the inverse matrix to obtain the Euclidean gradient of PEB at h_n.It identifies A(n, :) as the n-th row vector of A.Source: Appendix B · p. 12
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