Zero-Order Control Barrier Functions for Sampled-Data Systems with State and Input Dependent Safety Constraints

TL;DR

The paper addresses safety-critical control for continuous-time affine systems with state–input dependent constraints under discrete control updates by introducing Zero-Order Control Barrier Functions (ZOCBFs). ZOCBF uses a one-step prediction condition, $h(\phi(T; x_0,u), u) - h(x_0,u_0) \ge -\gamma(h(x_0,u_0)) + \delta$, to guarantee safety over the inter-sampling interval without requiring differentiation, and it remains robust to sampling through the margin $\delta$. Three numerical implementations are proposed—dynamics linearization, numerical integration, and parallel simulation—to enforce the ZOCBF condition online as a safety filter via $u \in U_{zocbf}(x_0,u_0)$. The methods are demonstrated on collision-avoidance and rollover-prevention problems, showing that ZOCBF can handle high-relative-degree constraints and state–input dependent safety functions while producing smoother, safer behavior. This framework provides a flexible, practically implementable approach to safety filtering in sampled-data robotic systems, with potential extensions to more complex dynamics and higher dimensions.

Abstract

We propose a novel zero-order control barrier function (ZOCBF) for sampled-data systems to ensure system safety. Our formulation generalizes conventional control barrier functions and straightforwardly handles safety constraints with high-relative degrees or those that explicitly depend on both system states and inputs. The proposed ZOCBF condition does not require any differentiation operation. Instead, it involves computing the difference of the ZOCBF values at two consecutive sampling instants. We propose three numerical approaches to enforce the ZOCBF condition, tailored to different problem settings and available computational resources. We demonstrate the effectiveness of our approach through a collision avoidance example and a rollover prevention example on uneven terrains.

Figures (3)

• Figure 1: Numerical simulation of the double integrator system \ref{['eq:integrator']} with ${T \!=\! 0.1~s}$, ${\gamma_c \!=\! 1}$, ${u_{nom} \!=\! 0}$, ${\delta \!=\! 0.01}$, and the initial state ${x_0 \!=\! [0~2]^\top}$. The positions and velocities of the double integrator (Top). The evolution of the ZOCBFs $h_1$ and $h_2$ along the system’s trajectories and the control input over time (Bottom). In each plot, the dashed lines represent the results with discrete-time higher-order CBFs (HOCBFs) for comparison. All controllers effectively ensure safety.
• Figure 2: Simulation of a differential drive vehicle model \ref{['eq:uni_nom']} navigating on uneven terrain, comparing the safety-filtered trajectory using the proposed ZOCBF filter (blue) with the unsafe nominal controller (green) while following a reference trajectory (black). The initial position is denoted as $x_0$, while the final position is denoted as $x_f$. The red areas along the blue path indicate unsafe routes on which the vehicle rollovers when under the nominal (unfiltered) controller. The vehicle is able to drive on rough terrain safely with the ZOCBF, whereas it is unsafe under the nominal controller.
• Figure 3: Control inputs with the proposed ZOCBF-based safety filter and the nominal controller over time (Top). Evolution of the rollover CBF \ref{['eq:yzmp']} values for both controllers (Bottom).

Theorems & Definitions (11)

• Definition 1: Zero-Order Control Barrier Functions
• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2: ZOCBFs, discrete-time CBFs, and conventional CBFs
• Remark 3: ZOCBF and vanilla CBF constraint
• Proposition 1
• proof