[Paper Review] Model-Free Source Seeking by a Novel Single-Integrator with Attenuating Oscillations and Better Convergence Rate: Robotic Experiments

Shivam Bajpai, Ahmed A. Elgohary+1 more|arXiv (Cornell University)|Nov 7, 2023

Extremum Seeking Control Systems4 citations

TL;DR

This paper proposes a novel single-integrator control-affine extremum seeking control (ESC) design with a Geometric-Based Kalman Filter (GEKF) to enable model-free, real-time source seeking in mobile robots. By attenuating oscillations through adaptive gain modulation, the method achieves asymptotic convergence to the source, demonstrated experimentally using a TurtleBot3 in a light source seeking task with significantly improved convergence and reduced steady-state oscillations compared to conventional ESC approaches.

ABSTRACT

In this paper we validate, including experimentally, the effectiveness of a recent theoretical developments made by our group on control-affine Extremum Seeking Control (ESC) systems. In particular, our validation is concerned with the problem of source seeking by a mobile robot to the unknown source of a scalar signal (e.g., light). Our recent theoretical results made it possible to estimate the gradient of the unknown objective function (i.e., the scalar signal) incorporated in the ESC and use such information to apply an adaptation law which attenuates the oscillations of the ESC system while converging to the extremum (i.e., source). Based on our previous results, we propose here an amended design of the simple single-integrator control-affine structure known in ESC literature and show that it can functions effectively to achieve a model-free, real-time source seeking of light with attenuated oscillations using only local measurements of the light intensity. Results imply that the proposed design has significant potential as it also demonstrated much better convergence rate. We hope this paper encourages expansion of the proposed design in other fields, problems and experiments.

Motivation & Objective

• To address the persistent oscillation problem in control-affine extremum seeking control (ESC) systems, especially in single-integrator designs, which traditionally exhibit limit-cycle behavior around the source.
• To experimentally validate recent theoretical advances in GEKF-based ESC that enable attenuating oscillations and asymptotic convergence, overcoming limitations of prior control-affine ESC implementations.
• To demonstrate the feasibility and superiority of the proposed design in real-world robotic source seeking using a TurtleBot3, particularly in dynamic environments with time-varying sources.
• To encourage broader adoption of control-affine ESC systems by showing that simple, low-parameter designs can outperform classic ESC structures when enhanced with GEKF.
• To provide empirical evidence that adaptive gain modulation based on distance to the source enhances gradient estimation and convergence speed.

Proposed method

• The proposed method uses a modified single-integrator control law: $ \dot{x} = u = f(x)\sqrt{\omega}u_1 + \sqrt{\omega}u_2 $, where $ u_1 $ and $ u_2 $ are sinusoidal perturbations, but with adaptive amplitudes $ a_x $ and $ a_y $ that vary based on the robot’s distance from the source.
• A Geometric-Based Kalman Filter (GEKF) is integrated into the control loop to estimate the gradient of the objective function (e.g., light intensity) from noisy sensor measurements, improving estimation accuracy and enabling convergence.
• The adaptation law dynamically adjusts the perturbation amplitudes: larger when far from the source (to enhance gradient estimation) and smaller when near (to reduce oscillations and enable convergence).
• The system uses a motion capture system to track the TurtleBot3’s trajectory in real time, with sensor data from a light intensity sensor used to estimate the objective function.
• Theoretical stability is supported by verifying that the gradient components $ |J_x| $ and $ |J_y| $ decay as $ 1/t^P $ with $ P > 1 $, satisfying conditions from Theorem 1 in prior work [22].
• The experimental setup includes a light source whose position can be changed dynamically, allowing testing of the system’s ability to track moving sources through adaptive gain modulation.

Experimental results

Research questions

• RQ1 Can a simple control-affine single-integrator ESC system with GEKF achieve attenuating oscillations and asymptotic convergence in real robotic experiments?
• RQ2 Does the proposed adaptive gain modulation strategy improve convergence speed and reduce steady-state oscillations compared to standard ESC implementations?
• RQ3 Can the GEKF-enhanced design effectively track a time-varying source by dynamically adjusting perturbation amplitudes based on proximity?
• RQ4 Is the theoretical stability condition—gradient components decaying as $ 1/t^P $ with $ P > 1 $—verified empirically in real-world robotic experiments?
• RQ5 Can a low-parameter, simple control law outperform more complex classic ESC structures in source seeking when enhanced with GEKF?

Key findings

• The proposed GEKF-enhanced single-integrator ESC design successfully reduced steady-state oscillations to near-zero levels as the robot approached the light source, demonstrating attenuating oscillations.
• The system achieved significantly faster convergence to the source compared to traditional ESC methods, as evidenced by shorter time-to-peak intensity and reduced trajectory dispersion.
• The adaptation law effectively increased perturbation amplitudes when the robot was far from the source and reduced them when close, enabling robust gradient estimation and convergence.
• Experimental results confirmed that the gradient components $ |J_x| $ and $ |J_y| $ decayed as $ 1/t^P $ with $ P > 1 $, satisfying the theoretical stability condition from Theorem 1.
• The robot successfully tracked a moving light source by dynamically adjusting its perturbation amplitude, demonstrating the system’s capability for time-varying extremum tracking.
• The method outperformed standard single-integrator ESC in both convergence speed and oscillation attenuation, validating the effectiveness of the GEKF integration.