Roundabout Constrained Convex Generators: A Unified Framework for Multiply-Connected Reachable Sets

TL;DR

The paper addresses the challenge of representing multiply-connected feasible regions with a single hole in control/verification tasks. It introduces Roundabout Constrained Convex Generators ($RCG$), defined as the set difference between an outer $CG$ and an inner $CG$, and provides a unified exact formulation that supports solver-friendly operations. Key contributions include closure results under linear transformations, Minkowski sums with a $CCG$, and intersections with a $CCG$, plus specialized cases such as Roundabout Zonotopes and Ellipsotopes and an open-source implementation. This framework enables compact, topologically-aware reachability analyses for safety-critical systems, with potential extensions to hybrid, multi-hole scenarios in future work.

Abstract

This paper introduces Roundabout Constrained Convex Generators (RCGs), a set representation framework for modeling multiply connected regions in control and verification applications. The RCG representation extends the constrained convex generators framework by incorporating an inner exclusion zone, creating sets with topological holes that naturally arise in collision avoidance and safety-critical control problems. We present two equivalent formulations: a set difference representation that provides geometric intuition and a unified parametric representation that facilitates computational implementation. The paper establishes closure properties under fundamental operations, including linear transformations, Minkowski sums, and intersections with convex generator sets. We derive special cases, including roundabout zonotopes and roundabout ellipsotopes, which offer computational advantages for specific norm selections. The framework maintains compatibility with existing optimization solvers while enabling the representation of non-convex feasible regions that were previously challenging to model efficiently.

Figures (5)

• Figure 1: A traffic roundabout: admissible motion (gray) circulates around an inaccessible island (green).
• Figure 2: A $\mathcal{RE}$ with elliptical outer boundary ($p = 2$) and elliptical inner boundary ($p = 2$). The blue region represents the feasible set $\mathcal{RE}$.
• Figure 3: RCG with elliptical outer boundary ($p = 2$) and polygonal inner boundary ($p = \infty$). The orange region represents the feasible set $\mathcal{R}$.
• Figure 4: Nine common RCG configurations obtained by varying outer and inner $p$-norms. Columns correspond to outer boundary norms ($p = 1, 2, \infty$) and rows to inner boundary norms ($p = 1, 2, \infty$).
• Figure 5: Visualization of the roundabout zonotope intersection. Left: Parameter space $(\beta_1, \beta_2)$ showing the feasible region $\mathcal{B}_{\text{intersection}}$ (blue) as the intersection of the outer constraint $\mathcal{B}_1: \|\bm{\beta}\|_\infty \leq 1$ (black dashed box) with the constraint from $\mathcal{Y}$, while excluding the inner region $\mathcal{B}_2: \|\bm{\beta}\|_\infty \leq r$ (red dashed box). Right: The resulting geometric sets in $\mathbb{R}^2$ showing $\mathcal{Z}_{\text{out}}$ (green), $\mathcal{Z}_{\text{in}}$ (orange dashed), $\mathcal{Y}$ (magenta), and the final intersection $(\mathcal{Z}_{\text{out}} \cap \mathcal{Y}) \setminus (\mathcal{Z}_{\text{in}} \cap \mathcal{Y})$ (blue filled region).

Theorems & Definitions (25)

• Definition II.1: Constrained Convex Generator silverccg2022
• Proposition II.2: Fundamental Operations silvestre2023exact
• Proposition II.3: Halfspace Intersection
• proof
• Definition III.1: Roundabout Constrained Convex Generators
• Lemma III.2: Derivation of Unified RCG Representation
• proof
• Proposition IV.1: Linear Transformation of RCG
• proof
• Proposition IV.2: Minkowski Sum of RCG and CCG