Designing Barrier Functions for Graceful Safety Control

Abstract

This paper examines the problem of achieving "grace" when controlling dynamical systems for safety, which is defined in terms of providing multi-layered safety assurances. Namely, two safety layers are created: a primary layer that represents a desirable degree of safety, and a secondary failsafe layer. Graceful control then involves ensuring that even if the primary layer is breached, the failsafe layer remains forward invariant. The paper pursues this goal by constructing a safety constraint that combines the concepts of zeroing and reciprocal control barrier functions with regard to the primary and secondary safe sets, respectively. This constraint is analogous to a stiffening spring, making it possible to construct energy-based analytical proofs of the resulting graceful safety guarantees. The proposed approach is developed for systems with a relative degree of either 1 or 2, the latter case being particularly useful for mechanical systems. We demonstrate the applicability of the method using a wall collision avoidance example. This demonstration highlights the benefits of the proposed approach compared to traditional benchmarks from the literature.

Figures (14)

• Figure 1: Mechanical model used for the illustrative examples in this paper.
• Figure 2: Position & velocity for baseline zeroing CBF
• Figure 3: Value of the barrier function & velocity vs. position plot for baseline zeroing CBF
• Figure 4: State & input trajectories for the controller based on the exponential CBF
• Figure 5: Barrier & phase plot for the controller based on the exponential CBF

Theorems & Definitions (13)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Definition 3
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5