Joint-Feasibility Margin Feedback for Bounded-Input Second-Order Control Barrier Functions

Optimal control for safety-critical systems is often dependent on the conservativeness of constraints. Control Barrier Functions (CBFs) serve as a medium to represent such constraints, but constructing a minimally conservative CBF is a computationally intractable problem. Therefore, approaches that can guarantee safety while reducing conservatism will help improve the optimality of the system under consideration. Here, we present a Model Predictive Control (MPC) formulation using CBF as a terminal constraint, which is proven to improve feasibility and reachable sets with increasing prediction horizon. The constructive nature of the proofs allows for warm-starting the nonlinear optimization problem, thereby reducing the computational time substantially. Simulations are set up for a simple nonholonomic system to numerically validate the results, and it is observed that the number of infeasible points decreased by a factor of 1.7 to 2.7. The increase in reachable state space was demonstrated by the ability of the system to track trajectories that are entirely inside the unsafe region of the control barrier function.

Control Barrier Functions (CBFs) can provide provable safety guarantees for dynamic systems. However, finding a valid CBF for a system of interest is often non-trivial, especially for systems with low computational resources, higher-order dynamics, and moving close to obstacles of complex shape. A common solution to this problem is to use a purely distance-based CBF. In this paper, we study Hyperplane CBFs (H-CBFs), where a hyperplane separates the agent from the obstacle. First, we note that the common distance-based CBF is a special case of an H-CBF where the hyperplane is a supporting hyperplane of the obstacle that is orthogonal to a line between the agent and the closest point of the obstacle. We then show that a less conservative CBF can be found by optimising over the orientation of the supporting hyperplane, in order to find the Least Restrictive Hyperplane CBF. This enables us to maintain the safety guarantees while allowing controls that are closer to the desired ones, especially when moving fast and passing close to obstacles. We illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles, and show that the proposed approach reduces the average gap between desired and safe controls by an order of magnitude.

Multi-robot Control Barrier Function (CBF) safety filters can become infeasible, but a failed quadratic program (QP) does not indicate why the conflict occurred or how to resolve it. To address this, we develop an exact feasibility certificate for multi-agent CBF filters with heterogeneous control-affine dynamics and convex input sets. The certificate quantifies a feasibility reserve by separating the demand imposed by safety constraints from the available actuator supply. This decomposition shows when CBF gain tuning or increased actuation can and cannot resolve infeasibility, and identifies the agents and interactions responsible for the conflict. We further propose an algorithm to optimally allocate shared safety constraints by maximizing the worst local feasibility margin, yielding a linear program for polyhedral input sets. In $320$ paired closed-loop simulations, the proposed allocation reduces infeasible control steps from roughly $50\%$ to $6.2\%$, and reduces safety-violating runs from $118/160$ to $24/160$. In addition, across $52$ infeasibility events, the certificate identifies an interaction whose relaxation restores feasibility in $94\%$ of cases.