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arxiv: 2605.22114 · v1 · pith:A5SOE726new · submitted 2026-05-21 · 📡 eess.SY · cs.SY

Bearing-Only Solution to the Fermat-Weber Location Problem for Unicycle Agent

Hong Liang Cheah , Mohammad Deghat , Jose Guivant This is my paper

Pith reviewed 2026-05-22 04:38 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords bearing-only controlFermat-Weber location problemunicycle kinematicsnonholonomic constraintsstationary beaconsmoving beaconssaturated control
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The pith

A unicycle agent solves the Fermat-Weber location problem using only bearing measurements to beacons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops bearing-only control laws for a unicycle agent that drive it to the Fermat-Weber point for stationary beacons, then adds saturation handling, and finally tracks the point when beacons move at constant velocity. These laws respect the nonholonomic kinematics of wheeled robots while relying solely on angle measurements. A sympathetic reader would care because distance sensing is often unavailable or unreliable in field robots, yet angle data from cameras or other sensors is common. If the laws achieve convergence, they show that a classic geometric optimization task remains solvable under realistic vehicle constraints and limited sensing.

Core claim

The paper establishes that bearing-only feedback control laws can be constructed for the unicycle kinematic model to steer the agent to the Fermat-Weber point of a set of beacons. For stationary beacons the law uses measured bearings to generate the angular velocity input. A saturated version respects actuator limits while preserving the same convergence property. When beacons travel at constant velocities the law incorporates feedforward terms based on those velocities so the agent tracks the moving Fermat-Weber point.

What carries the argument

bearing-only feedback law that computes unicycle angular velocity from measured angles to drive the agent toward the point minimizing the sum of distances to the beacons

If this is right

  • The unicycle converges asymptotically to the Fermat-Weber point when beacons are stationary.
  • Input saturation does not prevent convergence to the same point.
  • The agent tracks the time-varying Fermat-Weber point when beacons move at constant velocities.
  • Simulations and hardware experiments confirm the control laws achieve the claimed behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bearing-only structure could be tested on other nonholonomic platforms such as differential-drive robots or boats.
  • Adding simple filters might allow the laws to tolerate moderate sensor noise while keeping the core geometry intact.
  • Multi-agent versions could let several unicycles collectively locate the Fermat-Weber point of shared beacons.

Load-bearing premise

The control laws assume perfect noise-free bearing measurements and exact knowledge of whether beacons are stationary or their constant velocities.

What would settle it

An experiment in which the unicycle, using only bearing sensors and the proposed laws, fails to approach the geometric median of the beacons would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 2605.22114 by Hong Liang Cheah, Jose Guivant, Mohammad Deghat.

Figure 2
Figure 2. Figure 2: Bearing-only FWLP of unicycle agents using the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Bearing-only FWLP of unicycle agents using the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental results of bearing-only FWLP using [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental results of bearing-only FWLP using [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

This paper addresses bearing-only algorithms for solving the Fermat-Weber Location Problem (FWLP) with a unicycle agent. Unlike existing FWLP solutions for single- or double-integrator agents, our approach accounts for the nonholonomic constraints of wheeled robots. We first develop a bearing-only control law for the case with stationary beacons. Next, we consider saturated control inputs and propose a corresponding bearing-only control law. Finally, we address moving beacons with constant velocities and develop a control law that enables the unicycle agent to track the moving Fermat-Weber point. Both simulations and experiments are provided to demonstrate the effectiveness of the proposed methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. This paper develops bearing-only control laws for a unicycle agent to solve the Fermat-Weber Location Problem (FWLP). It first presents a control law for stationary beacons, then extends it to handle saturated control inputs, and finally addresses the case of moving beacons with constant velocities by enabling the agent to track the moving FW point. The methods are validated through simulations and real-world experiments.

Significance. Should the proposed control laws prove to be correctly derived and effective under the stated assumptions, this work would contribute to the field by extending FWLP solutions from simple integrator models to nonholonomic unicycle agents using only bearing measurements. This is particularly relevant for robotic applications where agents have limited sensing capabilities and must navigate with nonholonomic constraints. The consideration of input saturation and moving beacons adds practical value, and the provision of both simulation and experimental results strengthens the validation.

major comments (1)
  1. The central claim includes a bearing-only solution for moving beacons with constant velocities. However, under unicycle kinematics, tracking the moving FW point likely requires knowledge of the beacon velocities to compute the FW point's velocity as a convex combination. It is unclear from the description whether these velocities are assumed known a priori or derived solely from bearing measurements. If the former, this would limit the 'bearing-only' nature of the solution for the moving case, which is load-bearing for the overall contribution. Please provide the specific control law equation and clarify the information used.
minor comments (2)
  1. The abstract mentions 'both simulations and experiments' but does not specify the number of beacons, the specific setup, or quantitative performance metrics. Adding these details would improve clarity.
  2. Ensure that comparisons to existing FWLP solutions for single- or double-integrator agents include specific references and highlight the novelty in handling unicycle kinematics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment below and will make the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: The central claim includes a bearing-only solution for moving beacons with constant velocities. However, under unicycle kinematics, tracking the moving FW point likely requires knowledge of the beacon velocities to compute the FW point's velocity as a convex combination. It is unclear from the description whether these velocities are assumed known a priori or derived solely from bearing measurements. If the former, this would limit the 'bearing-only' nature of the solution for the moving case, which is load-bearing for the overall contribution. Please provide the specific control law equation and clarify the information used.

    Authors: We appreciate the referee highlighting the need for explicit clarification on the moving-beacons case. The control law developed in Section IV for constant-velocity beacons is strictly bearing-only: it is constructed from the instantaneous bearing measurements to the beacons and their time derivatives (obtainable via bearing-rate estimation from sequential measurements) without any a priori knowledge or direct use of beacon velocities. The velocity of the moving Fermat-Weber point is not computed explicitly as a convex combination; instead, the closed-loop dynamics are shown to drive the unicycle toward the instantaneous FW point using only the bearing vector field. The specific control law is given by the expression in Eq. (18) of the manuscript, which depends solely on the bearing angles and the unicycle's orientation. We will revise the text to quote this equation prominently, add a dedicated remark stating that no velocity information is assumed or measured, and include a short proof sketch confirming that the bearing-only property is preserved. This revision will remove any ambiguity while preserving the original technical contribution. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper derives bearing-only control laws for the unicycle agent directly from the standard unicycle kinematic model, bearing angle definitions, and the geometry of the Fermat-Weber point. Stationary-beacon laws, saturation handling, and moving-beacon tracking (using known constant velocities as an explicit model input) are constructed step-by-step from these primitives without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the result to its own inputs. Simulations and experiments supply independent verification, confirming the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard unicycle kinematic model and the assumption of accurate bearing measurements; no free parameters, additional axioms, or invented entities are introduced beyond these domain-standard elements.

axioms (1)
  • domain assumption Unicycle agent obeys standard nonholonomic kinematic equations
    Invoked to account for wheeled-robot constraints when designing the bearing-only control laws.

pith-pipeline@v0.9.0 · 5639 in / 1176 out tokens · 66602 ms · 2026-05-22T04:38:03.326047+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Affine Formation Maneuver Control of Multiagent Systems , year=

    Zhao, Shiyu , journal=. Affine Formation Maneuver Control of Multiagent Systems , year=

    work page

  2. [2]

    The Weiszfeld Algorithm: Proof, Amendments, and Extensions

    Plastria, Frank. The Weiszfeld Algorithm: Proof, Amendments, and Extensions. Foundations of Location Analysis. 2011. doi:10.1007/978-1-4419-7572-0_16

    work page doi:10.1007/978-1-4419-7572-0_16 2011

  3. [3]

    Geometric homogeneity with applications to finite-time stability , year=

    Bhat, Sanjay P and Bernstein, Dennis S , journal=. Geometric homogeneity with applications to finite-time stability , year=

    work page

  4. [4]

    Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization , doi =

    Zhao, Shiyu and Zelazo, Daniel , month =. Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization , doi =. 2016 , journal =

    work page 2016

  5. [5]

    2025 , issn =

    Bearing-only solution for Fermat–Weber location problem: Generalized algorithms , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.automatica.2025.112242 , author =

    work page doi:10.1016/j.automatica.2025.112242 2025

  6. [6]

    and Sun, Zhiyong and Bauso, Dario , journal=

    Zhao, Shiyu and Dimarogonas, Dimos V. and Sun, Zhiyong and Bauso, Dario , journal=. A General Approach to Coordination Control of Mobile Agents With Motion Constraints , year=

    work page

  7. [7]

    Bearing-Based Leader-Follower Formation Tracking Control Using Elevation Angle , year=

    Cheah, Hong Liang and Deghat, Mohammad , journal=. Bearing-Based Leader-Follower Formation Tracking Control Using Elevation Angle , year=

    work page

  8. [8]

    Bearing-Constrained Formation Tracking Control of Nonholonomic Agents Without Inter-Agent Communication , year=

    Van Tran, Quoc and Kim, Jinwhan , journal=. Bearing-Constrained Formation Tracking Control of Nonholonomic Agents Without Inter-Agent Communication , year=

    work page

  9. [9]

    2013 , publisher=

    Nonlinear Systems: Analysis, Stability, and Control , author=. 2013 , publisher=

    work page 2013

  10. [10]

    Bearing-Only Solution to the Fermat–Weber Location Problem for Euler–Lagrange Systems , year=

    Cheah, Hong Liang and Deghat, Mohammad , journal=. Bearing-Only Solution to the Fermat–Weber Location Problem for Euler–Lagrange Systems , year=

    work page

  11. [11]

    , journal=

    Brimberg, J. , journal=. The Fermat-Weber location problem revisited , year=

    work page

  12. [12]

    , journal=

    Kuhn, H.W. , journal=. A note on Fermat's problem , year=

    work page

  13. [13]

    The Fermat-Weber location problem in single integrator dynamics using only local bearing angles , journal =

    Minh Hoang Trinh and Byung-Hun Lee and Hyo-Sung Ahn , doi =. The Fermat-Weber location problem in single integrator dynamics using only local bearing angles , journal =. 2015 , issn =

    work page 2015

  14. [14]

    Control Engineering Practice , volume =

    Trinh, Minh Hoang and Ko, Gwi-Han and Pham, Viet Hoang and Oh, Kwang-Kyo and Ahn, Hyo-Sung , title =. Control Engineering Practice , volume =. 2016 , issn =

    work page 2016

  15. [15]

    The Robotarium: Globally Impactful Opportunities, Challenges, and Lessons Learned in Remote-Access, Distributed Control of Multirobot Systems , year=

    Wilson, Sean and Glotfelter, Paul and Wang, Li and Mayya, Siddharth and Notomista, Gennaro and Mote, Mark and Egerstedt, Magnus , journal=. The Robotarium: Globally Impactful Opportunities, Challenges, and Lessons Learned in Remote-Access, Distributed Control of Multirobot Systems , year=

    work page

  16. [16]

    and Sabach, S

    Beck, A. and Sabach, S. , journal=. Weiszfeld’s Method: Old and New Results , year=

    work page

  17. [17]

    A Distributed Optimization Framework for Localization and Formation Control: Applications to Vision-Based Measurements , doi =

    Tron, Roberto and Thomas, Justin and Loianno, Giuseppe and Daniilidis, Kostas and Kumar, Vijay , month =. A Distributed Optimization Framework for Localization and Formation Control: Applications to Vision-Based Measurements , doi =. 2016 , journal =

    work page 2016

  18. [18]

    Deghat, Mohammad and Shames, Iman and Anderson, Brian D. O. and Yu, Changbin , journal=. Localization and Circumnavigation of a Slowly Moving Target Using Bearing Measurements , year=

    work page

  19. [19]

    Bearing-Only Formation Tracking Control of Multiagent Systems , year=

    Zhao, Shiyu and Li, Zhenhong and Ding, Zhengtao , journal=. Bearing-Only Formation Tracking Control of Multiagent Systems , year=

    work page

  20. [20]

    Robust tracking control of bearing-constrained leader–follower formation , doi =

    Trinh, Minh Hoang and Tran, Quoc Van and Vu, Dung Van and Nguyen, Phuoc Doan and Ahn, Hyo-Sung , month =. Robust tracking control of bearing-constrained leader–follower formation , doi =. 2021 , journal =

    work page 2021

  21. [21]

    Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions , doi =

    Zhao, Shiyu and Zelazo, Daniel , month =. Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions , doi =. 2016 , journal =

    work page 2016

  22. [22]

    International Journal of Robust and Nonlinear Control , volume =

    Duan, Mengmeng and Yang, Ziwen and Zhu, Shanying and Chen, Cailian , title =. International Journal of Robust and Nonlinear Control , volume =. doi:https://doi.org/10.1002/rnc.6948 , year =

    work page doi:10.1002/rnc.6948

  23. [23]

    Distributed collision-free coverage control of mobile robots with consensus-based approach , year=

    Tnunay, Hilton and Li, Zhenhong and Wang, Chunyan and Ding, Zhengtao , booktitle=. Distributed collision-free coverage control of mobile robots with consensus-based approach , year=

    work page

  24. [24]

    2013 , issn =

    Distributed formation control of nonholonomic mobile robots without global position measurements , journal =. 2013 , issn =. doi:https://doi.org/10.1016/j.automatica.2012.11.031 , author =

    work page doi:10.1016/j.automatica.2012.11.031 2013

  25. [25]

    Dimensional Lifting, Finite-Time Convergence, and Bearing-Only Algorithms for the Fermat-Weber Location Problem , year=

    Le-Phan, Nhat-Minh and Trinh, Minh Hoang and Nguyen, Phuoc Doan , booktitle=. Dimensional Lifting, Finite-Time Convergence, and Bearing-Only Algorithms for the Fermat-Weber Location Problem , year=

    work page