Composition of Control Barrier Functions With Differing Relative Degrees for Safety Under Input Constraints

TL;DR

This work addresses guaranteed safety for control systems subject to input (actuator) constraints by composing multiple control barrier functions (CBFs) that may have different relative degrees into a single composite CBF via a soft-min operator. It introduces control dynamics so input constraints become CBFs in the closed-loop state and couples them with safety CBFs using the composite soft-minimum construction. A real-time quadratic program then minimizes a performance cost while enforcing the composite CBF constraint, guaranteeing forward invariance of the safe set and adherence to input limits; the approach is demonstrated on a nonholonomic ground robot with position, velocity, and input bounds. The results provide a scalable, online safety mechanism that handles multiple, possibly heterogeneous CBFs and is applicable to systems with actuator saturation and other input constraints. The contribution advances safe, constraint-compliant operation in robotics and related domains where real-time safety under input limits is critical.

Abstract

This paper presents a new approach for guaranteed safety subject to input constraints (e.g., actuator limits) using a composition of multiple control barrier functions (CBFs). First, we present a method for constructing a single CBF from multiple CBFs, which can have different relative degrees. This construction relies on a soft minimum function and yields a CBF whose $0$-superlevel set is a subset of the union of the $0$-superlevel sets of all the CBFs used in the construction. Next, we extend the approach to systems with input constraints. Specifically, we introduce control dynamics that allow us to express the input constraints as CBFs in the closed-loop state (i.e., the state of the system and the controller). The CBFs constructed from input constraints do not have the same relative degree as the safety constraints. Thus, the composite soft-minimum CBF construction is used to combine the input-constraint CBFs with the safety-constraint CBFs. Finally, we present a feasible real-time-optimization control that guarantees that the state remains in the $0$-superlevel set of the composite soft-minimum CBF. We demonstrate these approaches on a nonholonomic ground robot example.

Figures (6)

• Figure 1: ${\mathcal{S}}_{\rm s}$, and closed-loop trajectories for 4 goal locations.
• Figure 2: $q_{\rm x}$, $q_{\rm y}$, $v$, $\theta$, $u$ and $u_{\rm d}$ for $q_{{\rm d}}=[\,3 \quad 4.5\,]^{\rm T}$.
• Figure 3: $h$, $\min b_{j,i}$, and $\min h_j$ for $q_{{\rm d}}=[\,3 \quad 4.5\,]^{\rm T}$.
• Figure 4: ${\mathcal{S}}_{\rm s}$ and closed-loop trajectories for 4 goal locations.
• Figure 5: $q_{\rm x}$, $q_{\rm y}$, $v$, $\theta$, $\hat{u}$, $\hat{u}_{\rm d}$, $e$, $u$ and $u_{\rm d}$ for $q_{{\rm d}}=[\,3 \quad 4.5\,]^{\rm T}$.