Pareto Control Barrier Function for Inner Safe Set Maximization Under Input Constraints

TL;DR

This work addresses maximizing an inner safe set for continuous-time, control-affine systems under input constraints by introducing the Pareto Control Barrier Function (PCBF). PCBF integrates Pareto multi-task learning with neural CBFs to balance safety (feasibility) and inner safe-set expansion (volume) using Gaussian interior sampling and a region-based descent strategy with analytically solvable subproblems. The method achieves a larger, safer inner safe set than prior neural CBFs (e.g., LCCBF), closely matches Hamilton-Jacobi reachability results on a low-dimensional inverted pendulum, and scales to high-dimensional systems like a 12‑D quadrotor, where it substantially enlarges the safe region while ensuring obstacle avoidance. The approach offers a scalable, less conservative alternative to HJ-based methods and traditional ICCBF/ZCBF approaches, with future work aimed at handling model uncertainty and disturbances.

Abstract

This article introduces the Pareto Control Barrier Function (PCBF) algorithm to maximize the inner safe set of dynamical systems under input constraints. Traditional Control Barrier Functions (CBFs) ensure safety by maintaining system trajectories within a safe set but often fail to account for realistic input constraints. To address this problem, we leverage the Pareto multi-task learning framework to balance competing objectives of safety and safe set volume. The PCBF algorithm is applicable to high-dimensional systems and is computationally efficient. We validate its effectiveness through comparison with Hamilton-Jacobi reachability for an inverted pendulum and through simulations on a 12-dimensional quadrotor system. Results show that the PCBF consistently outperforms existing methods, yielding larger safe sets and ensuring safety under input constraints.

Figures (5)

• Figure 1: Pareto front of the Inner Safe Set Maximization Problem. The blue line represents the Pareto front, which delineates the boundary between the feasible and infeasible regions in the objective space. No solution $\theta$ can produce objective values $[\mathcal{L}_{\text{feas}}(\theta) \quad \mathcal{L}_{\text{vol}}(\theta)]^T$ that lie below this curve. The vertical axis represents solutions with zero feasibility loss. All feasible CBFs lie on this axis. The optimal CBF, with parameter $\theta^*$, lies at the intersection of the Pareto front and the vertical axis.
• Figure 2: The blue solid curve represents the Pareto front, while the black dashed lines indicate the upper and lower bounds. The green point denotes a feasible solution (i.e., a solution $\theta$ with $\mathbf{L}(\theta)$ lies between the two bounds), the red point represents an infeasible solution, and the blue point corresponds to $\theta^*$, the solution of problem \ref{['eq:problem-ism-loss']}. If the update causes $\theta$ to fall outside the feasible region, the PCBF algorithm will gradually guide $\theta$ back into the feasible region bounded by the upper and lower bounds.
• Figure 3: The inner safe sets of the inverted pendulum problem obtained by the LCCBF (blue solid line) and PCBF (orange solid line) methods, alongside the infinite-time viability kernel (black dashed line) computed using HJ reachability. The region enclosed by the black dashed line represents the true safe states.
• Figure 4: The left, middle, and right subfigures present 2D slices of the learned inner safe sets obtained by the PCBF and LCCBF methods corresponding to the $x-y$, $x-v_x$, and $\psi-\omega_z$ planes. All other states are set to zero in each slice. In the left and middle subfigures, the obstacle's projection in the corresponding 2D plane is indicated by black dashed lines, representing regions unsafe for the quadrotor.
• Figure 5: Trajectories of the quadrotor with and without the PCBF safety filter in the presence of a rectangle obstacle.

Theorems & Definitions (9)

• Definition 1: Safe Set
• Definition 2: Forward Invariant
• Definition 3: Inner Safe Set agrawal2021safe
• Definition 4: Control Barrier Functions (CBFs)
• Definition 5: Pareto Dominance lin2019pareto
• Definition 6: Pareto Optimality lin2019pareto
• Definition 7: Pareto Front lin2019pareto
• Lemma 1: Fliege and Svaiter fliege2000steepest
• Lemma 2