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

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.