Least Restrictive Hyperplane Control Barrier Functions

TL;DR

The paper tackles safety guarantees for dynamic systems by improving how hyperplane-based CBFs are selected. It introduces a least restrictive H-CBF by jointly optimizing the control input $\mathbf{u}$ and a hyperplane orientation parameter $\theta$ through a smooth parameterization, solving a P-CBF-QP. The approach is demonstrated on a 2D double integrator with static and moving, arbitrarily shaped obstacles, including discs and ellipses. Results show that the method can produce less conservative, faster trajectories while preserving safety, suggesting practical benefits for real autonomous systems.

Abstract

Control Barrier Functions (CBFs) can provide provable safety guarantees for dynamic systems. However, finding a valid CBF for a system of interest is often non-trivial, especially if the shape of the unsafe region is complex and the CBFs are of higher order. A common solution to this problem is to make a conservative approximation of the unsafe region in the form of a line/hyperplane, and use the corresponding conservative Hyperplane-CBF when deciding on safe control actions. In this letter, we note that conservative constraints are only a problem if they prevent us from doing what we want. Thus, instead of first choosing a CBF and then choosing a safe control with respect to the CBF, we optimize over a combination of CBFs and safe controls to get as close as possible to our desired control, while still having the safety guarantee provided by the CBF. We call the corresponding CBF the least restrictive Hyperplane-CBF. Finally, we also provide a way of creating a smooth parameterization of the CBF-family for the optimization, and illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles.

Figures (8)

• Figure 1: A robot at position $(0,0)$ with velocity $(1,2)$ in front of a polygonal obstacle. Five of the infinitely many supporting hyperplanes, parameterized by some $\theta$, are shown (A-E). If the goal is at G$_1$, an H-CBF based on D is the least restrictive, whereas if the goal is at G$_2$, an H-CBF based on A or B might be the least restrictive. Thus, we propose to optimize over $\theta$ at the same time as the control $\mathbf{u}$, to find the least restrictive H-CBF.
• Figure 2: The geometry of the simple disc world. The green circle represents the agent ($A$), and the red circle represents the obstacle ($O$). Many of the vectors used to define the CBF hyperplane are shown as arrows.
• Figure 3: The geometrics of a general 2D obstacle. The light blue area is the obstacle $\mathcal{O}$, and combined with the red areas, we get $\text{conv}(\mathcal{O})$.
• Figure 4: The setup of example \ref{['exp: A']}. The agent moves on a straight line, safely passing an obstacle. The corresponding value of the CBF constraint \ref{['eq_CBF_constr']} for the two different cases is shown in Figure \ref{['fig: Exp A constraint']}.
• Figure 5: The CBF constraint \ref{['eq_CBF_constr']} with $\alpha(x) = x$ for the agent in Example \ref{['exp: A']}. Note that this safe trajectory would not be allowed by the orthogonal CBF of borrmann_control_2015 (dashed), as it is negative for $t\in [3.5,4.5]$, right before passing the obstacle. However, it would be allowed by the least restrictive H-CBF (dash-dotted). Note that the two CBFs are identical when the agent is moving away from the obstacle, after passing the "safe line" at $t = 5$ s, since then $b_{ij}=0$ for both, see Equation \ref{['eq_bij']}.

Theorems & Definitions (15)

• Definition 1
• Theorem 1: Supporting Hyperplane in Every Direction
• proof
• Theorem 2: Separating Hyperplane
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1