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

Abstract: This paper introduces a novel model-free, real-time unicycle-based source seeking design. This design steers autonomously the unicycle dynamic system towards the extremum point of an objective function or physical/scaler 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 scaler signals) that behave locally like a higher-degree power functions (e.g., fourth degree polynomial function) as opposed to locally quadratic objective functions, the usual case in literature. We provide theoretical and simulation results to support out theoretical results. 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.

• The paper presents a novel model-free ESC architecture that uses high-order Lie bracket averaging to achieve exponential convergence for quartic objective functions.
• The paper demonstrates through simulations and robotic experiments that the proposed controller converges significantly faster than traditional methods, reducing convergence time from 100 seconds to approximately 20 seconds in simulations and from 1200 seconds to 350 seconds in experiments.
• The paper validates the method for light source seeking, showing robustness to sensor noise and flat regions in the objective landscape while maintaining real-time performance.

Exponentially Convergent Model-Free Source Seeking for Unicycle Systems: Theory and Robotic Validation

Introduction and Motivation

This work presents a novel model-free extremum seeking control (ESC) architecture for unicycle-type robotic systems, targeting exponential convergence to the extremum of objective functions that locally exhibit higher-order polynomial behavior, specifically quartic forms. The design leverages recent advances in high-order Lie bracket averaging, enabling the system to autonomously seek unknown signal sources (e.g., light intensity peaks) using only real-time measurements, without explicit knowledge of the objective function's analytical form or system model. The approach is validated both theoretically and through differential drive robotic experiments, marking a significant step in the practical deployment of ESC for non-quadratic objective landscapes.

Theoretical Framework: High-Order Lie Bracket ESC

The proposed ESC law is constructed for control-affine systems with two inputs, utilizing specialized dither signals to excite third-order Lie brackets, which are necessary for exponential convergence when the objective function is locally quartic. The control inputs are defined as:

$$u_1^\varepsilon(t) = \varepsilon^{1/N -1}v_1(t/\varepsilon), \quad u_2^\varepsilon(t) = \varepsilon^{1/N -1}v_2(t/\varepsilon)$$

where $N$ is the order of the Lie bracket (here, $N=3$ for quartic objectives), and $v_1$, $v_2$ are periodic dithers chosen to isolate the desired bracket order. The vector fields $g_1$, $g_2$ are selected such that the $N$-th order Lie bracket of the system dynamics yields a negative multiple of the $(N-1)$-th derivative of the objective function, ensuring descent toward the extremum.

For the unicycle kinematics:

$$\dot{x} = v \cos(\Omega t), \quad \dot{y} = v \sin(\Omega t)$$

the control law for the linear velocity $v$ is synthesized as:

$$v = 2(2\pi / \varepsilon)^{3/4} \left[3cJ(x,y) \sin(6\pi t/\varepsilon) + a\cos(2\pi t/\varepsilon)\right]$$

where $J(x,y)$ is the measured objective (e.g., light intensity), $c$ and $a$ are design parameters, and $\varepsilon$ is the dither period. An optional high-pass filter (HPF) can be incorporated to attenuate low-frequency disturbances and improve transient response.

Figure 1: The proposed ESC design for exponentially convergent unicycle, used in both simulation and differential drive robotic experiments.

Stability Analysis

The stability of the proposed ESC system is rigorously established via reduction to its Lie bracket system (LBS), which, for the quartic objective, is third-order. The analysis employs a time-varying coordinate transformation and a Lyapunov function with cross terms, yielding sufficient conditions for exponential stability of the equilibrium at the extremum point. The derived bounds on the system parameters ($c_1$, $c_2$, $\Omega$) guarantee that the state converges exponentially to a neighborhood of the extremum, with the neighborhood size tunable via the dither parameter $\varepsilon$.

The practical exponential stability is formalized as:

$$\|\mathbf{X} - \mathbf{X}_d\| \leq q_1 e^{-q_2 t} + d(\varepsilon)$$

where $d(\varepsilon) \to 0$ as $\varepsilon \to 0$, ensuring arbitrarily tight convergence for sufficiently small dithers.

Simulation Results

Simulations compare the proposed third-order Lie bracket ESC with the traditional first-order design for a quartic objective function $J(x, y) = (x-1)^4 + (y+2)^4$. Identical initial conditions and parameters are used for both controllers. The results demonstrate that the traditional ESC fails to converge within 100 seconds, while the proposed design achieves convergence in approximately 20 seconds, confirming the theoretical predictions and the necessity of higher-order Lie bracket excitation for non-quadratic objectives.

Experimental Validation

Experimental Setup

Experiments are conducted using TurtleBot3 robots equipped with motion capture systems for position feedback and analog light sensors for source-seeking tasks. The ESC controller operates in real time, receiving only sensor measurements as feedback.

Figure 2: Experimental setup in the Modeling, Dynamics, and Control Lab, showing the robot, light source, motion capture system, and sensor integration.

Results for Known Objective Function

In experiments with a known quartic objective, the traditional first-order ESC converges to a point with non-negligible error, failing to reach the true minimum even after 1200 seconds. In contrast, the exponentially convergent ESC reaches the minimum in approximately 350 seconds and maintains proximity to the extremum, even in the presence of flat regions in the objective landscape and sensor resolution limitations.

Model-Free Light Source Seeking

The controller is further validated in a model-free light source-seeking task, where the robot uses only the light sensor output as feedback. The system successfully navigates to the location of maximum light intensity, with the measured signal minimized and the robot oscillating around the source due to the dithering action. The HPF effectively mitigates sensor noise and environmental fluctuations.

Implementation Considerations

• Parameter Selection: The convergence rate and neighborhood size are sensitive to the dither period $\varepsilon$ and the design parameters $c$, $a$, and $\Omega$. These must be tuned to balance convergence speed, robustness to noise, and actuator limitations.
• Computational Requirements: The controller is lightweight, requiring only real-time measurement acquisition and basic trigonometric computations for dither generation.
• Sensor Integration: The approach is agnostic to the type of sensor used for feedback, enabling application to diverse source-seeking tasks (chemical, thermal, etc.).
• Robustness: The design demonstrates resilience to measurement noise and flat objective regions, outperforming traditional ESC in both simulation and hardware.

Implications and Future Directions

The results establish that high-order Lie bracket ESC enables exponential convergence for non-quadratic objective functions in model-free robotic source-seeking. This expands the applicability of ESC to environments where the signal landscape is not well-approximated by quadratic forms, such as biological, chemical, or environmental fields. The demonstrated robustness and real-time performance suggest suitability for deployment in autonomous exploration, environmental monitoring, and adaptive search tasks.

Future work will generalize the approach to arbitrary polynomial objectives, investigate bounded update rate designs to reduce oscillations, and explore multi-agent extensions for cooperative source-seeking.

Conclusion

This study introduces and validates a model-free, exponentially convergent ESC architecture for unicycle systems, capable of seeking sources in higher-order objective landscapes using only real-time measurements. Theoretical analysis, simulation, and robotic experiments confirm the superiority of high-order Lie bracket ESC over traditional methods, with strong implications for autonomous robotics and adaptive control in complex environments.