Model-Free Optimization and Control of Rigid Body Dynamics: An Extremum Seeking for Vibrational Stabilization Approach

Ahmed A. Elgohary; Rohan Palanikumar; Sameh A. Eisa; Simone Martini

arxiv: 2510.22402 · v5 · submitted 2025-10-25 · 🧮 math.OC

Model-Free Optimization and Control of Rigid Body Dynamics: An Extremum Seeking for Vibrational Stabilization Approach

Rohan Palanikumar , Ahmed A. Elgohary , Simone Martini , Sameh A. Eisa This is my paper

Pith reviewed 2026-05-18 03:59 UTC · model grok-4.3

classification 🧮 math.OC

keywords model-free optimizationextremum seeking controlvibrational stabilizationrigid body dynamicssatellite attitude controlquadcopter dynamicsunicycle motion

0 comments

The pith

A single vibrational signal can stabilize rigid body systems like satellites and quadcopters at the optimum of a measurable but unknown objective function.

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

The paper introduces a model-free method that uses extremum seeking control combined with vibrational stabilization to optimize and control rigid body dynamics in real time. It establishes that one perturbation signal suffices to drive the system to the best measured performance state without knowing the system equations or the exact form of the objective. A sympathetic reader would care because many mechanical systems, from spacecraft to drones, have uncertain or hard-to-model dynamics, and this approach bypasses the need for detailed models while still achieving both optimization and stabilization. The demonstrations on satellite attitude, quadcopter attitude, and unicycle motion show the method works even when measurements include delays or noise.

Core claim

The ESC-VS method stabilizes a rigid body dynamic system about the optimal state of an objective function that can be unknown expression-wise but assessable through measurements; the ESC-VS is operable by using only one perturbation/vibrational signal, and this holds for a class of second-order mechanical systems as shown through simulations of satellite attitude dynamics, quadcopter attitude dynamics, and acceleration-controlled unicycle dynamics, including cases with measurement delays and noise.

What carries the argument

The ESC-VS approach, which combines extremum seeking with vibrational stabilization to apply a single perturbation signal that simultaneously optimizes the measured objective and stabilizes the second-order rigid body dynamics.

If this is right

Real-time model-free optimization becomes possible for satellite attitude control using only measurements of a performance index.
Quadcopter attitude can be driven to an optimal state without explicit knowledge of its rotational dynamics.
Acceleration-controlled unicycle motion can be optimized and stabilized with the same single-signal method.
The approach continues to function when sensor data contains delays or additive noise.
A unified methodology emerges for both optimization and control of rigid body systems that belong to the second-order class.

Where Pith is reading between the lines

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

The single-signal design may lower implementation cost and actuator effort compared with multi-frequency extremum seeking schemes on similar hardware.
If the second-order assumption holds, the method could transfer directly to other underactuated mechanical systems such as certain robotic joints or vehicle suspensions.
Hardware experiments on physical satellites or drones would provide a direct test of robustness beyond the reported simulations.

Load-bearing premise

The objective function must remain accurately measurable even while the vibrational perturbation is actively applied to the system.

What would settle it

In simulation or hardware test of one of the three example systems, the states fail to converge to the measured optimum or the closed-loop system loses stability when the single vibrational signal is applied.

read the original abstract

In this paper, we introduce a model-free, real-time, dynamic optimization and control method for a class of rigid body dynamics. Our method is based on a recent extremum seeking control for vibrational stabilization (ESC-VS) approach that is applicable to a class of second-order mechanical systems. The new ESC-VS method is able to stabilize a rigid body dynamic system about the optimal state of an objective function that can be unknown expression-wise, but assessable through measurements; the ESC-VS is operable by using only one perturbation/vibrational signal. We demonstrate the effectiveness and the applicability of our ESC-VS approach via three rigid-body systems: (1) satellite attitude dynamics, (2) quadcopter attitude dynamics, and (3) acceleration-controlled unicycle dynamics. The results, including simulations with and without measurement delays/noise, illustrate the ability of our ESC-VS to operate successfully as a new methodology of optimization and control for rigid body dynamics.

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.

Desk Editor's Note private letter to a colleague

The paper adapts single-signal ESC-VS to rigid-body systems and shows workable simulations, but the excitation story for 3-DOF attitude looks thin.

read the letter

The core move here is taking an existing extremum-seeking-plus-vibrational-stabilization scheme that was already framed for second-order mechanical systems and specializing it to rigid-body attitude and motion problems. They run it on satellite attitude, quadcopter attitude, and an acceleration-controlled unicycle, all with a single perturbation signal, and they include runs with measurement noise and delay. That is the actual new piece: the concrete mapping to these three rigid-body examples and the claim that one scalar dither is enough. The simulations appear to track the unknown optimum reasonably well in the reported cases, which is useful evidence for people who want model-free real-time tuning on vehicles where full dynamics are hard to maintain.

Referee Report

2 major / 1 minor

Summary. The paper introduces a model-free extremum seeking control for vibrational stabilization (ESC-VS) method applicable to a class of second-order rigid body dynamics. It claims that the approach stabilizes the system at the optimum of an unknown but measurable objective function using only a single perturbation/vibrational signal, and demonstrates this via simulations on satellite attitude dynamics, quadcopter attitude dynamics, and acceleration-controlled unicycle dynamics, including cases with noise and delays.

Significance. If the single-signal claim and stability properties hold with rigorous support, the method could offer a simplified model-free real-time optimization framework for mechanical systems, reducing the need for multiple dithers in applications such as aerospace attitude control.

major comments (2)

[Abstract] Abstract: The claim that a single vibrational signal suffices to drive the averaged dynamics to the unknown optimum for 3-DOF rigid-body attitude systems (satellite and quadcopter) is load-bearing but unsupported. Standard multi-variable ESC requires independent or orthogonal dithers to ensure Hessian identifiability in the averaged system; a single scalar torque perturbation may leave rotational modes unexcited or produce a singular averaged Hessian, especially for non-quadratic objectives on SO(3). No averaging analysis or persistence-of-excitation verification is referenced.
[Abstract] Abstract (simulation claims): The effectiveness assertions rest on simulations with and without noise/delays, yet no quantitative performance metrics, stability proofs, or derivation details for the rigid-body ESC-VS extension are provided. This leaves the central claim without analytical grounding.

minor comments (1)

[Abstract] Abstract: The description of the 'class of second-order mechanical systems' could be made more precise to clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below, with clarifications and indications of where we will revise the paper to strengthen the presentation.

read point-by-point responses

Referee: [Abstract] The claim that a single vibrational signal suffices to drive the averaged dynamics to the unknown optimum for 3-DOF rigid-body attitude systems (satellite and quadcopter) is load-bearing but unsupported. Standard multi-variable ESC requires independent or orthogonal dithers to ensure Hessian identifiability in the averaged system; a single scalar torque perturbation may leave rotational modes unexcited or produce a singular averaged Hessian, especially for non-quadratic objectives on SO(3). No averaging analysis or persistence-of-excitation verification is referenced.

Authors: We appreciate this detailed observation. The ESC-VS method builds directly on a recent framework for second-order mechanical systems in which a single high-frequency vibrational signal is shown to be sufficient for averaged convergence to the optimum through the vibrational stabilization mechanism. For rigid-body attitude dynamics, the single torque perturbation interacts with the nonlinear rotational equations to provide the necessary excitation across the degrees of freedom. The averaging analysis and persistence-of-excitation conditions follow from the base ESC-VS results, adapted to the rigid-body case. We agree, however, that the abstract does not explicitly reference this analysis. In the revision we will add a concise subsection outlining the application of averaging theory to the single-signal rigid-body setting and the relevant persistence-of-excitation verification on SO(3). revision: yes
Referee: [Abstract] The effectiveness assertions rest on simulations with and without noise/delays, yet no quantitative performance metrics, stability proofs, or derivation details for the rigid-body ESC-VS extension are provided. This leaves the central claim without analytical grounding.

Authors: We thank the referee for noting this. The full manuscript presents the simulation results (including cases with noise and delays) and grounds stability in the averaging theory of the underlying ESC-VS approach, with the rigid-body extension derived from the second-order mechanical-system framework. The abstract is necessarily brief. To improve clarity and analytical grounding we will (i) incorporate quantitative performance metrics (e.g., convergence times, steady-state errors, and RMS values) into the simulation figures and tables and (ii) add a short outline of the derivation steps for the rigid-body ESC-VS extension in the methods section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: ESC-VS presented as independent construction for rigid-body systems

full rationale

The paper introduces ESC-VS as a model-free method for stabilizing rigid-body dynamics (satellite attitude, quadcopter, unicycle) about an unknown optimum using a single vibrational signal. The derivation applies the approach to second-order mechanical systems with measurement-based objective functions and validates via simulations including noise/delays. No load-bearing step reduces a claimed prediction or result to a fitted input, self-definition, or self-citation chain by construction. The central claim rests on the novel applicability and single-signal operation, which is externally falsifiable through the reported simulations and does not rely on prior author results as an unverified uniqueness theorem. This is a standard honest non-finding for a control-theory construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the unproven applicability of the ESC-VS construction to the class of second-order rigid-body systems and on the assumption that objective-function measurements remain usable under vibrational excitation.

axioms (2)

domain assumption The rigid-body dynamics belong to a class of second-order mechanical systems for which the ESC-VS method is applicable.
Stated in the abstract as the basis for extending the method to satellite, quadcopter, and unicycle examples.
domain assumption The objective function can be evaluated through real-time measurements even while the vibrational perturbation is applied.
Required for the model-free operation described.

pith-pipeline@v0.9.0 · 5715 in / 1312 out tokens · 24052 ms · 2026-05-18T03:59:30.403265+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The new ESC-VS method is able to stabilize a rigid body dynamic system about the optimal state of an objective function ... using only one perturbation/vibrational signal.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.