Optimization-free Smooth Control Barrier Function for Polygonal Collision Avoidance

TL;DR

This work tackles polygonal collision avoidance by introducing an optimization-free, smooth CBF framework. It constructs a nonconservative lower bound h_a(x) to the SDF using a Boolean structure, then smooths it with a log-sum-exp approximation to obtain a differentiable hat h_a that serves as a CBF; a distributed safety filter is derived as a closed-form QP to guarantee safety for two planar agents. The authors prove that h_a shares the same boundary zero-set as the true SDF, establish conditions under which hat h_a is a valid CBF, and demonstrate the method on two nonlinear 2D systems (underactuated nonholonomic vehicles and an underactuated crane), showing substantial computational efficiency compared with optimization-embedded alternatives. The results indicate strong potential for fast, scalable polygonal PCA in 2D with possible extensions to 3D, higher-order dynamics, and constrained inputs, albeit current work is limited to planar scenarios.

Abstract

Polygonal collision avoidance (PCA) is short for the problem of collision avoidance between two polygons (i.e., polytopes in planar) that own their dynamic equations. This problem suffers the inherent difficulty in dealing with non-smooth boundaries and recently optimization-defined metrics, such as signed distance field (SDF) and its variants, have been proposed as control barrier functions (CBFs) to tackle PCA problems. In contrast, we propose an optimization-free smooth CBF method in this paper, which is computationally efficient and proved to be nonconservative. It is achieved by three main steps: a lower bound of SDF is expressed as a nested Boolean logic composition first, then its smooth approximation is established by applying the latest log-sum-exp method, after which a specified CBF-based safety filter is proposed to address this class of problems. To illustrate its wide applications, the optimization-free smooth CBF method is extended to solve distributed collision avoidance of two underactuated nonholonomic vehicles and drive an underactuated container crane to avoid a moving obstacle respectively, for which numerical simulations are also performed.

Figures (9)

• Figure 1: The illustration of SDF between polygons (taking triangles for example), where $\mathcal{P}(x) = \mathcal{P}^j(x^j) - \mathcal{P}^i(x^i)$ denoting the Minkowski difference of $\mathcal{P}^i(x^i)$ and $\mathcal{P}^j(x^j)$.
• Figure 2: The illustration of the relation between polygons $\mathcal{P}^j(x^j), -\mathcal{P}^i(x^i), \mathcal{P}(x)$, where $\mathcal{N}^{j}(x^j), \mathcal{N}_{-}^{i}(x^i),\mathcal{N}(x)$ are their outer normal vectors. Viewed at the figure, it holds that $\mathcal{N}(x)= \mathcal{N}^{j}(x^j)\cup \mathcal{N}_{-}^{i}(x^i)$.
• Figure 3: The organization of the main theoretical results.
• Figure 4: Illustration of the two simulation examples. (a) Coordinated collision avoidance of two polygonal nonholonomic vehicles. (b) Collision avoidance of an underactuated container crane with a polygonal obstacle.
• Figure 5: Illustration of the moving process of vehicles $i,j$ using the nominal control law (without the safety filter). By observing that $\mathcal{P}^{i}$ (blue region) intersects $\mathcal{P}^{j}$ (green region), as well as the origin (red dot) is inside their Minkowski difference $\mathcal{P}^{ij}=\mathcal{P}^{j}- \mathcal{P}^{i}$ (red region), one can know that the collision happens around $1.9~\mathrm{s}$.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Lemma 1
• Example 1: Polygonal rigid bodies
• Example 2: Polygonal formation
• Remark 1
• Proposition 1
• Lemma 2
• proof
• Remark 2