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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

TL;DR

This work addresses model-free optimization and control for rigid body dynamics using Extremum Seeking for Vibrational Stabilization (ESC-VS). It formulates rigid body motion as a second-order system, applies a single high-frequency perturbation with an adaptive estimate, and leverages VOC averaging to steer the system toward the unknown objective minimum $J(\mathbf{q})$ while guaranteeing stability. The authors prove asymptotic stability for the averaged ESC-VS and demonstrate practical stability for the actual rigid body dynamics, validating the approach on satellite attitude, quadcopter attitude, and acceleration-controlled unicycle scenarios, including cases with measurement delay and noise. This delivers a real-time, model-free control paradigm for complex rigid-body platforms where full dynamic models are uncertain or unavailable, with potential impact on aerospace and robotics applications.

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.

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

TL;DR

This work addresses model-free optimization and control for rigid body dynamics using Extremum Seeking for Vibrational Stabilization (ESC-VS). It formulates rigid body motion as a second-order system, applies a single high-frequency perturbation with an adaptive estimate, and leverages VOC averaging to steer the system toward the unknown objective minimum while guaranteeing stability. The authors prove asymptotic stability for the averaged ESC-VS and demonstrate practical stability for the actual rigid body dynamics, validating the approach on satellite attitude, quadcopter attitude, and acceleration-controlled unicycle scenarios, including cases with measurement delay and noise. This delivers a real-time, model-free control paradigm for complex rigid-body platforms where full dynamic models are uncertain or unavailable, with potential impact on aerospace and robotics applications.

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.
Paper Structure (11 sections, 5 theorems, 24 equations, 8 figures, 3 tables)

This paper contains 11 sections, 5 theorems, 24 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background on Extremum Seeking for Vibrational Stabilization (ESC-VS)
  3. Main Results: ESC-VS Rigid Body System
  4. Application 1: Satellite Attitude Control
  5. Application 2: Quadcopter Attitude Control
  6. Application 3: Acceleration-controlled Unicycle
  7. Numerical Simulations and Discussion
  8. ESC-VS Response to Delay and Noise
  9. ESC-VS Response to Measurement Delay
  10. ESC-VS Response to Measurement Noise
  11. Conclusion

Key Result

Corollary 2.3

elgohary2025letters If $\bar{\mathbf{x}}(t) \in \mathscr{C}_0, \forall t \in [0,\ \omega t_{f}]$ with $t_f > 0$ and $\bar{\mathbf{x}}(0)=\mathbf{x}(0)$, we have $|\mathbf{x}(t)-\bar{\mathbf{x}}(t)|=O(1/\omega)$ for $t \in [0,\ \omega t_{f}]$.

Figures (8)

  • Figure 1: The proposed ESC-VS rigid body structure, which we demonstrate its effectiveness by three major applications.
  • Figure 2: Simulation Results for the satellite. (a) Quaternion, $\bm{Q}$, (b) Satellite angular velocity, $\bm{\Omega}$, (c) Reaction wheel angular velocity, $\bm{\Omega}_{RW}$, (d) Control estimate, $\hat{u}$, (e) Objective function measurements, $J$.
  • Figure 3: Simulation Results for the quadcopter. (a) Euler angles, $\phi, \ \theta, \ \psi$, (b) Quadcopter angular velocity, $\bm{\Omega}$, (c) Control estimate, $\hat{u}$, (d) Objective function measurements, $J$.
  • Figure 4: Simulation Results for the unicycle model. (a) X position, $x$, (b) Y position, $y$, (c) Path of unicycle, (d) Unicycle velocity, $v$, (e) Unicycle angular velocity, $\Omega$, (f) Control estimate, $\hat{u}$, (g) Objective function, $J$.
  • Figure 5: Delay analysis. (a) Satellite objective function, (b) Quadcopter objective function, (c) Unicycle objective function.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Remark 2.1
  • Remark 2.2
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 3 more