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arxiv: 2511.00752 · v3 · submitted 2025-11-02 · 🧮 math.OC · cs.RO

Model-free source seeking of exponentially convergent unicycle: theoretical and robotic experimental results

Rohan Palanikumar , Ahmed A. Elgohary , Victoria Grushkovskaya , Sameh A. Eisa This is my paper

Pith reviewed 2026-05-18 02:09 UTC · model grok-4.3

classification 🧮 math.OC cs.RO
keywords model-free source seekingunicycle dynamicsexponential convergenceextremum seekingrobotic experimentshigher-degree power functionsmeasurement noise
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0 comments X

The pith

A unicycle can locate the peak of an unknown higher-power signal using only real-time measurements and converges exponentially fast.

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

The paper presents a model-free controller that drives a unicycle vehicle toward the maximum of a scalar field whose local shape near the peak resembles a fourth-degree or higher even power rather than the quadratic form assumed in earlier extremum-seeking methods. The design uses filtered measurements to modulate steering while the vehicle traces small circles that keep the signal persistently exciting. Theoretical analysis shows that the distance to the peak decays at an exponential rate under this local-power assumption. Simulations test robustness to noise, delays, and varied starting positions. Physical robot experiments provide the first hardware confirmation that the predicted exponential convergence occurs on a real platform with realistic sensing and actuation limits.

Core claim

The introduced unicycle source-seeking law achieves exponential convergence to the extremum of objective functions or scalar signals that behave locally like a higher-degree power function near the peak, as opposed to the locally quadratic behavior required by prior designs. The controller operates without a model of either the vehicle dynamics or the signal shape and relies solely on real-time measurements. The exponential rate is established by Lyapunov analysis adapted to the non-quadratic local geometry, and the same law is shown to remain effective when the measurements contain bounded noise or constant time delays.

What carries the argument

The exponentially convergent unicycle extremum-seeking law, which filters the measured signal and uses the filtered output to adjust angular velocity so that the vehicle’s circular motion produces an exponentially decaying offset from the peak.

If this is right

  • The same control structure works for signals whose local shape is flatter or steeper than quadratic, widening the class of fields for which exponential convergence is guaranteed.
  • Simulations show the design tolerates measurement delays and additive noise while preserving the exponential rate.
  • Physical experiments confirm that the theoretical convergence can be observed on hardware despite real sensor and actuator imperfections.
  • Different initial conditions do not alter the exponential character of the approach to the peak.

Where Pith is reading between the lines

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

  • The method could be adapted to other nonholonomic platforms such as differential-drive robots or car-like vehicles that must follow similar circular excitation patterns.
  • If the local power degree is unknown in advance, an online estimator of that degree might preserve the exponential property across a broader range of signals.
  • The experimental success suggests the controller could be tested on aerial or marine robots for locating chemical or acoustic sources whose fields are known to be non-quadratic.

Load-bearing premise

The objective function or signal must behave locally like a higher-degree power function near its extremum.

What would settle it

Apply the controller to a field whose local Taylor expansion is purely quadratic and measure whether the position error still decays exponentially or reverts to a slower convergence rate.

Figures

Figures reproduced from arXiv: 2511.00752 by Ahmed A. Elgohary, Rohan Palanikumar, Sameh A. Eisa, Victoria Grushkovskaya.

Figure 1
Figure 1. Figure 1: The proposed ESC design for exponentially conver [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Modeling, Dynamics, and Control Lab (MDCL [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation results comparing the proposed exponentially convergent unicycle ESC with the traditional ESC [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Moreover, the variations in the objective function [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental results for the traditional ESC design based on first-order Lie bracket from D¨urr et al. (2013) with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experimental results for the exponentially convergent unicycle ESC design with a fourth-order objective function. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Planar trajectory of the robot with exponentially convergent unicycle ESC design plotted on the objective [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Experimental results for light source-seeking with the exponentially convergent ESC model. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simulation results for the exponentially convergent unicycle ESC with three added measurement noise (low, [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Objective function measurement for the experiment for 100 s with stationary robot at respective initial position [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Simulation results for exponentially convergent unicycle ESC with delay. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simulation results for different initial conditions representing the same radius from the extremum to test [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Experimental results for different initial conditions representing the same radius from the extremum to [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Simulation result for an initial position that is farther away from the desired position with the exponentially [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison between experiment-based system (blue) and simulation-based system (red) with close, but still [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
read the original abstract

This paper introduces a novel model-free, real-time unicycle-based source seeking design. This design autonomously steers the unicycle dynamic system towards the extremum point of an objective function or physical/scalar signal that is unknown expression-wise, but accessible via measurements. A key contribution of this paper is that the introduced design converges exponentially to the extremum point of objective functions (or scalar signals) that behave locally like a higher-degree power function (e.g., fourth-degree polynomial function) as opposed to locally quadratic objective functions, the usual case in literature. We provide theoretical results and design characterization, supported by a variety of simulation results that demonstrate the robustness of the proposed design, including cases with different initial conditions and measurement delays/noise. Also, for the first time in the literature, we provide experimental robotic results that demonstrate the effectiveness of the proposed design and its exponential convergence ability. These experimental results confirm that the proposed exponentially convergent extremum seeking design can be practically realized on a physical robotic platform under real-world sensing and actuation constraints.

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

2 major / 2 minor

Summary. The paper introduces a model-free extremum-seeking control law for unicycle robots that steers the system to the extremum of an unknown scalar field. The key theoretical contribution is exponential convergence when the field is locally equivalent to a higher-degree power function (such as a fourth-degree polynomial), in contrast to the standard quadratic local approximation in extremum-seeking literature. The claims are supported by averaging analysis, Lyapunov stability proofs, numerical simulations under various conditions including noise and delays, and physical experiments on a robotic platform.

Significance. If the exponential convergence result holds under the stated local homogeneity assumption, this extends extremum-seeking control theory to a wider class of objective functions, potentially improving performance in applications where fields are flatter or steeper than quadratic near the source. The provision of robotic experimental results is notable as it demonstrates practical feasibility under real sensing and actuation constraints, addressing a gap in the literature.

major comments (2)
  1. [§4, Eq. (18)] §4 (Averaging Analysis), Eq. (18): The derivation of the averaged closed-loop dynamics assumes the measured signal is exactly a homogeneous polynomial of degree 4; no explicit bound or robustness margin is provided against residual lower-order (e.g., quadratic) terms that could dominate near the origin and destroy the exponential rate while preserving asymptotic convergence.
  2. [Theorem 2 (§5)] Theorem 2 (§5): The Lyapunov function and its Lie derivative are constructed and shown negative definite only for the pure higher-degree homogeneous case. The proof does not address how the negative-definiteness margin behaves under additive non-homogeneous perturbations, which is load-bearing for the central exponential-convergence claim.
minor comments (2)
  1. [Figure 5] Figure 5 caption: specify the exact noise variance and delay values used in the Monte-Carlo runs to allow reproducibility.
  2. [§3.1] The notation for the dither signals in §3.1 is introduced without an explicit table comparing it to standard ES dithers; a short comparison table would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments, which help clarify the scope of our exponential convergence results. We respond to each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [§4, Eq. (18)] §4 (Averaging Analysis), Eq. (18): The derivation of the averaged closed-loop dynamics assumes the measured signal is exactly a homogeneous polynomial of degree 4; no explicit bound or robustness margin is provided against residual lower-order (e.g., quadratic) terms that could dominate near the origin and destroy the exponential rate while preserving asymptotic convergence.

    Authors: Our averaging analysis in Section 4 and the closed-loop dynamics in Eq. (18) are derived under the standing assumption, stated in the problem setup and abstract, that the unknown field is locally equivalent to a homogeneous polynomial of degree 4. This local equivalence means the degree-4 term is the leading term in a neighborhood of the source. We agree that an explicit robustness margin against lower-order perturbations would strengthen the presentation. In the revised version we will insert a remark after Eq. (18) that qualitatively discusses how small additive lower-order terms affect the averaged system and notes that exponential convergence is retained only when the higher-degree term remains dominant; otherwise the rate may degrade to asymptotic convergence. A brief continuity argument supporting this observation will be included. revision: yes

  2. Referee: [Theorem 2 (§5)] Theorem 2 (§5): The Lyapunov function and its Lie derivative are constructed and shown negative definite only for the pure higher-degree homogeneous case. The proof does not address how the negative-definiteness margin behaves under additive non-homogeneous perturbations, which is load-bearing for the central exponential-convergence claim.

    Authors: Theorem 2 proves exponential stability of the averaged system for the exact homogeneous degree-4 case via a Lyapunov function constructed specifically for that structure. The referee correctly observes that the proof does not quantify the margin under additive non-homogeneous perturbations. We will revise the paragraph immediately following Theorem 2 to add a short discussion clarifying that, by standard perturbation results for homogeneous systems, sufficiently small non-homogeneous terms preserve local negative definiteness of the Lie derivative in a suitably restricted neighborhood; however, a complete robust-stability analysis lies outside the present scope. This addition will explicitly delimit the exponential-convergence claim without altering the theorem statement itself. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard averaging and Lyapunov analysis under explicit homogeneity assumption

full rationale

The paper presents a model-free unicycle source-seeking controller whose exponential convergence is established via averaging theory and a Lyapunov function constructed for the closed-loop averaged system when the objective is locally homogeneous of degree greater than 2. The design equations (control law, dither signals, and demodulation) are stated explicitly and do not define the convergence rate or equilibrium in terms of themselves. The higher-order homogeneity is introduced as an assumption on the unknown signal, not recovered from the controller equations. No load-bearing self-citation chain, fitted-parameter prediction, or ansatz smuggling is required for the central stability claim; the proof treats the pure-power case and notes the local nature of the result. The robotic experiments serve as validation rather than part of the derivation. Consequently the claimed exponential convergence does not reduce to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from nonholonomic control and extremum seeking literature plus the specific local shape of the objective function; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Unicycle kinematics follow the standard nonholonomic differential-drive model.
    Invoked implicitly as the platform for the source-seeking design.
  • domain assumption The unknown objective function or scalar signal admits a local higher-degree power approximation near its extremum.
    This is the load-bearing premise that enables the claimed exponential convergence.

pith-pipeline@v0.9.0 · 5727 in / 1381 out tokens · 43108 ms · 2026-05-18T02:09:38.920693+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

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    ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sent...

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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