  Safety filters based on control barrier functions (CBFs) have become a
popular method to guarantee safety for uncertified control policies, e.g., as
resulting from reinforcement learning. Here, safety is defined as staying in a
pre-defined set, the safe set, that adheres to the system's state constraints,
e.g., as given by lane boundaries for a self-driving vehicle. In this paper, we
examine one commonly overlooked problem that arises in practical
implementations of continuous-time CBF-based safety filters. In particular, we
look at the issues caused by discrete-time implementations of the
continuous-time CBF-based safety filter, especially for cases where the
magnitude of the Lie derivative of the CBF with respect to the control input is
zero or close to zero. When overlooked, this filter can result in undesirable
chattering effects or constraint violations. In this work, we propose three
mitigation strategies that allow us to use a continuous-time safety filter in a
discrete-time implementation with a local relative degree. Using these
strategies in augmented CBF-based safety filters, we achieve safety for all
states in the safe set by either using an additional penalty term in the safety
filtering objective or modifying the CBF such that those undesired states are
not encountered during closed-loop operation. We demonstrate the presented
issue and validate our three proposed mitigation strategies in simulation and
on a real-world quadrotor.
