Encircling General 2-D Boundaries by Mobile Robots with Collision Avoidance: A Vector Field Guided Approach

TL;DR

This work addresses encircling general 2-D boundaries when boundary equations are unknown. It introduces a Fourier-based method to fit sampled boundary points into a parametric curve $\tilde{\bm{c}}(\rho)$, enabling a VF-guided encirclement that maintains constant speed and tangent alignment, with a polar radius error $e(x,y)$ to measure proximity. To ensure safety and actuation limits, a CBF-based safety layer is embedded into a QP that minimally alters the VF reference $\bm{u}_r$, yielding feasible controls under obstacle avoidance and input constraints. The approach is validated through numerical simulations and hardware experiments on an e-puck2 platform, showing successful encirclement of irregular and non-star-shaped boundaries while avoiding obstacles and respecting velocity limits. The method provides a practical, computation-friendly solution for tasks like environmental monitoring and chemical spill containment where boundary shapes are complex and not analytically given.

Abstract

The ability to automatically encircle boundaries with mobile robots is crucial for tasks such as border tracking and object enclosing. Previous research has primarily focused on regular boundaries, often assuming that their geometric equations are known in advance, which is not often the case in practice. In this paper, we investigate a more general case and propose an algorithm that addresses geometric irregularities of boundaries without requiring prior knowledge of their analytical expressions. To achieve this, we develop a Fourier-based curve fitting method for boundary approximation using sampled points, enabling parametric characterization of general 2-D boundaries. This approach allows star-shaped boundaries to be fitted into polar-angle-based parametric curves, while boundaries of other shapes are handled through decomposition. Then, we design a vector field (VF) to achieve the encirclement of the parameterized boundary, wherein a polar radius error is introduced to measure the robot's ``distance'' to the boundary. The controller is finally synthesized using a control barrier function and quadratic programming to mediate some potentially conflicting specifications: boundary encirclement, obstacle avoidance, and limited actuation. In this manner, the VF-guided reference control not only guides the boundary encircling action, but can also be minimally modified to satisfy obstacle avoidance and input saturation constraints. Simulations and experiments are presented to verify the performance of our new method, which can be applied to mobile robots to perform practical tasks such as cleaning chemical spills and environment monitoring.

Figures (22)

• Figure 1: General boundaries in robot encircling tasks. (a) Encircling the boundary of chemical spills to clean hazardous chemicals. (b) Encircling the boundary of wildfire for fire-fighting. (c) Encircling an isoline of sensory data (e.g., sea temperature, air contaminant concentration) for environment monitoring.
• Figure 2: Illustration of nonholonomic mobile robot and velocity constraint.
• Figure 3: Framework of the proposed algorithm.
• Figure 4: Boundary representation.
• Figure 5: Left: Tangent vector and tracking error. Right: Switching mechanism for non star-shaped boundary, and two VFs work in different regions.