Addressing Relative Degree Issues in Control Barrier Function Synthesis with Physics-Informed Neural Networks

TL;DR

This work tackles safety filtering under control-affine dynamics when the relative degree of a single CBF can vary across the state space, causing inactivity and potential safety violations. It reframes CBF synthesis as solving boundary value problems, designing CBF gradients via multiple functions and boundary conditions to preserve the safe-set geometry; solutions are obtained with physics-informed neural networks (PINNs). The proposed multi-CBF framework ensures nonzero Lie derivatives for all input channels, enabling feasible safety filters without conservative safe-set approximations, and demonstrates effectiveness in both simulation (including nonconvex and convex safe sets) and real quadrotor experiments. The approach provides robust, geometry-preserving safety guarantees for learning-based controllers in robotics, with practical impact on reducing chattering and ensuring forward invariance of safety sets.

Abstract

In robotics, control barrier function (CBF)-based safety filters are commonly used to enforce state constraints. A critical challenge arises when the relative degree of the CBF varies across the state space. This variability can create regions within the safe set where the control input becomes unconstrained. When implemented as a safety filter, this may result in chattering near the safety boundary and ultimately compromise system safety. To address this issue, we propose a novel approach for CBF synthesis by formulating it as solving a set of boundary value problems. The solutions to the boundary value problems are determined using physics-informed neural networks (PINNs). Our approach ensures that the synthesized CBFs maintain a constant relative degree across the set of admissible states, thereby preventing unconstrained control scenarios. We illustrate the approach in simulation and further verify it through real-world quadrotor experiments, demonstrating its effectiveness in preserving desired system safety properties.

Figures (5)

• Figure 1: An illustration of our proposed approach where we leverage multiple control barrier functions (CBFs) to mitigate the varying relative degree issue in certifying compact safe sets brunke2024practical. In this work, we introduce an alternative perspective by formulating CBF synthesis as boundary value problems, which are solved using physics-informed neural networks (PINNs). This approach allows us to mitigate the relative degree issue without conservative safe set approximations. (a) As an example, a non-convex, compact safe set is parameterized by multiple CBFs, each covering a segment of the safe set boundary. (b) The level sets of two representative CBFs are shown. The flexibility of PINNs allows us to closely approximate the original safe set boundary for both convex segments and non-convex segments (highlighted in red between two crosses in the top panel).
• Figure 2: A block diagram of a typical safety filter control architecture. Given an uncertified policy $\pi(x)$, a safety filter $\pi_\text{s}(x)$ is designed to safeguard the system by making minimal adjustments to the control inputs when they are deemed unsafe.
• Figure 3: The use of multiple CBFs has been proposed to mitigate issues arising from varying relative degrees. Here, we illustrate the boundaries of approximated safe sets for the approach from brunke2024practical and our proposed method. In these illustrations, the true safe set $\mathcal{C}$ and the approximated safe set $\mathcal{C}_Q$ are shown in blue and green, respectively. For each case, the boundary of an individual CBF, $\partial \mathcal{C}_q$, is shown as a dark gray solid line, and the corresponding level sets are illustrated in light gray. The arrows on the boundary $\partial \mathcal{C}_q$ indicate the positive gradient directions. (a) In brunke2024practical, we proposed using a polytopic set to under-approximate $\mathbb{C}$. In this case, the safety boundaries are hyperplanes. This approach is conservative and is restricted to convex sets. (b) In this work, we instead use multiple CBFs, where the individual CBFs can have nonlinear boundaries to better preserve the geometry of the original safe set. This approach can be generalized to certain non-convex safe sets where the condition in \ref{['eqn:surface_normal_condition']} can be satisfied for all pairs of points on $\partial \mathcal{C}_q$ for all $q$.
• Figure 4: Simulation results of our proposed PINN-based CBF synthesis approach for rendering a non-convex set safe. We can accurately approximate the desired safe set (gray shaded area) using ten PINNs (zero-level sets in green). We compare the single-CBF and our proposed multi-CBF approach starting from the same initial condition (blue circle). The system's state in closed-loop with the single-CBF safety filter (black crosses) violates the desired safety constraint. In contrast, the state of the system in closed-loop with the CBF safety filter using our PINN-based CBFs (red crosses) stays inside the safe set for all future time, although the system converges to a state where $L_g h(x) = 0$ for the single CBF (black solid line).
• Figure 5: Experimental results of our proposed PINN-based CBF synthesis approach for rendering a convex set safe. Similar to the simulation example, we approximate the desired safe set (gray shaded area) using eight PINNs (zero-level sets in green). The single-CBF baseline uses a quadratic CBF, which results in a set where $L_g h(x) = 0$ inside of $\mathbb{C}$ (black solid line). The quadrotor is initialized at similar states (blue circle), and an uncertified policy (blue dashed line) is used to drive the quadrotor from the interior of the safe set to the unsafe region. With the standard single-CBF safety filter, as the quadrotor approaches the safety boundary (black crosses in the first panel), we observe large input oscillations near the boundary of the safe set due to filter inactivity (second panel). In contrast, with our proposed multi-CBF method, as the quadrotor approaches the safety boundary (red crosses in the third panel), the input oscillations are mitigated through proper CBF gradient design (fourth panel).

Theorems & Definitions (9)

• Definition 1: Forward Invariant Set
• Definition 2: Extended Class-$\mathcal{K}$ Function
• Definition 3: Relative Degree ames2019a
• Definition 4: CBF ames2019a
• Corollary 1: Forward Invariance of Safe Set ames2019a
• Example 1: Standard CBF Parametrization
• proof
• Example 2: CBF Synthesis as Boundary Value Problems
• Definition 5: Multi-CBFs