On the Stabilization of Rigid Formations on Regular Curves

This work deals with the problem of stabilizing a multi-agent rigid formation on a general class of planar curves. Namely, we seek to stabilize an equilateral polygonal formation on closed planar differentiable curves after a path sweep. The task of finding an inscribed regular polygon centered at the point of interest is solved via a randomized multi-start Newton-Like algorithm for which one is able to ascertain the existence of a minimizer. Then we design a continuous feedback law that guarantees convergence to, and sufficient sweeping of the curve, followed by convergence to the desired formation vertices while ensuring inter-agent avoidance. The proposed approach is validated through numerical simulations for different classes of curves and different rigid formations. Code: https://github.com/mebbaid/paper-elobaid-ifacwc-2026

💡 Research Summary

The paper addresses the problem of driving a team of autonomous agents to form a regular polygonal “rigid” formation that lies on a prescribed closed planar curve. The authors consider a broad class of curves that are at least continuously differentiable (C¹) and may contain isolated points where the tangent vanishes (corners or cusps) as well as self‑intersections. Standard path‑following techniques such as transverse feedback linearization (TFL) fail on such non‑simple curves because the parameter space becomes singular. To overcome this, each agent’s state is augmented with a virtual lifted coordinate, effectively embedding the original curve into a higher‑dimensional manifold that is simple (no self‑intersections). On this lifted manifold a conventional TFL controller can be designed to stabilize the transverse error and guarantee convergence of every agent to the curve.

The second major contribution is a systematic method for locating an inscribed regular polygon on the curve. The authors formulate the geometric constraints (equal side lengths and equal interior angles) as residual functions rL,i and rA,i of the parameters θ = (θ0,…,θn‑1), where each θi is a curve parameter that maps to a vertex γ(θi). A weighted sum of squared residuals defines a cost functional J(θ). They prove that J is continuous on the compact hyper‑cube