Safe Leader-Follower Density Control via Higher-Order Mean-Field Control Barrier Functions

Abstract

Safety filters for large-scale multi-agent systems can be formulated at the population level through mean-field control barrier functions (MF-CBFs). Most existing formulations are first-order, requiring the control input to appear explicitly in the first time derivative of the safety functional; higher-order extensions have only very recently been proposed for a single, directly actuated population. We consider instead the case in which the population to be kept safe is not directly actuated, but is transported by a field generated by a second, controlled population. For this setting, we develop a higher-order MF-CBF framework for density dynamics with safety constraints of relative degree larger than one. Our main result is that indirect actuation of this kind does not destroy the control-affine structure of the safety condition: the control-dependent part of the second time derivative of the safety functional is obtained in closed form and is linear in the control velocity field, with a kernel given by the actuated density times the spatial gradient of a first variation. Relative-degree-two safety constraints can therefore still be enforced through a convex quadratic program. We then specialize the framework to a leader-follower density control problem in the presence of dangerous regions, where safety is ensured by keeping the overlap of each population with those regions below a prescribed threshold. We show that the follower safety functional has relative degree two with respect to the leader control field, while the leader safety functional has relative degree one. We further characterize exactly when the two simultaneous constraints are infeasible, introduce an adaptive-gain mechanism to mitigate potential conflicts, and bound the tracking errors induced by the safety correction. Numerical simulations in cluttered environments illustrate the resulting safety-performance trade-off.