BubbleCBF: Constructive Safety for Second-Order Systems and Application to Autonomous Robot Navigation in Unknown Environments

Satisfying safety constraints for systems with second-order dynamics is challenging because a position constraint may require infeasible deceleration. Higher-order control barrier functions (CBFs) provide an approach for expressing safety conditions for such systems but feasibility depends on the system dynamics and available control authority. We develop a constructive approach for designing feasible safety constraints for second-order control-affine systems. Starting from a signed distance function (SDF), we construct collision-free spheres, i.e., bubbles, in position space, and use their quadratic geometry and overlap to construct a velocity-dependent safety constraint. The associated safe set is represented as a union of intersections and enforced through a quadratic-program safety filter. Under mild assumptions on the dynamics, bubble overlap, and design parameters, we prove that the quadratic-program-based controller is well defined, continuous in the system state, and renders the set forward invariant. We also derive sufficient conditions for feasibility under bounded inputs. We demonstrate that our BubbleCBF, applied to a differential-drive robot, achieves safe autonomous navigation in unknown environments.

• SDF-to-HOCBF construction through BubbleCBF. For robot navigation, we convert SDF queries into quadratic balls whose analytic derivatives enable a feasible higher-order CBF design.
• Feasibility of the union-of-balls CBF-QP. We construct a QP over the union of velocity-dependent bubble sets and prove feasibility and forward invariance under bounded control inputs.
• Unknown-environment validation. We apply BubbleCBF to differential-drive navigation with online SDF estimation and demonstrate safe, deadlock-free operation in simulation and on hardware.

LiDAR → online SDF → bubble cover → BubbleCBF safety filter.