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

Mani Amani; Mohammed Alyaseen; Zhirui Dai; Ki Myung Brian Lee; Nikolay Atanasov; Jorge Cortes

cs.ROOct 14, 2026

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

Authors: Mani Amani, Mohammed Alyaseen, Zhirui Dai, Ki Myung Brian Lee, Nikolay Atanasov, Jorge Cortes

Organizations: Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, CA, USA. · Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA.

Abstract

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. Videos and results are available at: https://existentialrobotics.org/bubblecbf/.

Figures & tables

Fig. 1 : BubbleCBF guides a second-order system from start to goal through a safe corridor consisting of overlapping bubbles. Each bubble is a ball with radius underestimating the distance to the nearest obstacle. The union of bubbles is lifted to a velocity-dependent safe set to account for the relative-degree-two dynamics. A quadratic program with a control barrier function constraint defined from the lifted safe set is used to modify a nominal controller to restrict the system’s outward motion near bubble boundaries and ensure sufficient control authority for safety. The nominal system trajectory (orange) and the BubbleCBF -filtered trajectory (green) are shown.

Fig. 2 : Application of BubbleCBF to differential-drive robot navigation. Using measurements from range sensors on the robot (1st panel), we estimate an SDF model of the obstacles (2nd panel) in real-time. We construct safe bubbles from the SDF (3rd panel) and define a nominal controller and constraints for the BubbleCBF optimization (4th panel).

Fig. 3 : All planners share the same start and goal positions. Cyan denotes the bubble cover, red the path bubbles, and black the planned trajectory. The nominal controller collides at speeds above $1$ m/s (panels (b) and (c)), while SDF-CBF deadlocks at $5$ m/s (panel (c)).

Fig. 4 : Navigation experiment in an a priori unknown environment. Left: the robot and cluttered test environment. Right: the generated bubble cover, with the executed robot trajectory in green and the reference Bézier trajectory in white.

$v$	Method	SR (%)	CR (%)	DR (%)	TR (%)
5.0	Nominal	38	62	0	0
	BubbleCBF	100	0	0	0
	SDF-CBF	81	0	19	0
2.5	Nominal	84	16	0	0
	BubbleCBF	96	0	0	4
	SDF-CBF	87	0	13	0

TABLE I : Simulation performance at increasing reference speeds. SR, CR, TR, and DR are reported over 100 randomly sampled initial positions for each method and speed. navigation at all tested speeds.

Metric	BubbleCBF	Nominal	SDF CBF
CR (%) $\downarrow$	0.0	83.3	0.0
SR (%) $\uparrow$	100.0	16.7	50.0
DR (%) $\downarrow$	0.0	0.0	50.0
CA (m 2 ) $\uparrow$	8.42 $\pm$ 0.89	7.86 $\pm$ 1.75	8.75 $\pm$ 0.80
OD (m) $\uparrow$	0.720 $\pm$ 0.030	0.714 $\pm$ 0.052	0.687 $\pm$ 0.028

TABLE II : Results for real-world experiments. CR, SR, DR, CA, and OD from obstacles. Start goals were identical, each planner had an exploration budget of 2 minutes. Success is defined when the robot does not experience a collision or deadlock in the time budget.

Fig. 5 : Autonomous navigation in a large unknown environment. Top: the robot at three times during the experiment. Bottom: the online-constructed bubble cover (red), reference Bézier trajectory (white), and executed trajectory (green). White arrows indicate the robot positions shown in the top panels.

CR (%) $\downarrow$	SR (%) $\uparrow$	DR (%) $\downarrow$	CA (m 2 ) $\uparrow$	OD (m) $\uparrow$
0.0	100.0	0.0	$126.99\pm 43.13$	$1.247\pm 0.31$

TABLE III : Performance of BubbleCBF in the large-scale unknown environment. CR, SR, DR, CA, and OD from obstacles.

Control Barrier Functions

Control barrier functions encode safe states as constraints that keep a controlled system away from hazards.

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Is Your Safe Controller Actually Safe? A Critical Review of CBF Tautologies and Hidden Assumptions

Taekyung Kim

Department of Robotics, University of Michigan, Ann Arbor, MI, 48109, USA

Control Barrier Functions Safety-Critical Control Robot Safety Safe Robot Navigation

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This tutorial provides a critical review of the practical application of Control Barrier Functions (CBFs) in robotic safety. While the theoretical foundations of CBFs are well-established, I identify a recurring gap between the mathematical assumption of a safe controller's existence and its constructive realization in systems with input constraints. I highlight the distinction between candidate and valid CBFs by analyzing the interplay of system dynamics, actuation limits, and class-K functions. I further show that some purported demonstrations of safe robot policies or controllers are limited to passively safe systems, such as single integrators or kinematic manipulators, where safety is already inherited from the underlying physics and even naive geometric hard constraints suffice to prevent collisions. By revisiting simple low-dimensional examples, I show when CBF formulations provide valid safety guarantees and when they fail due to common misuses. I then provide practical guidelines for constructing realizable safety arguments for systems without such passive safety. A crowd-navigation simulation study further illustrates that CBF-derived reward shaping in reinforcement learning can improve empirical behavior without establishing formal safety. The goal of this tutorial is to bridge the gap between theoretical guarantees and actual implementation, supported by an open-source interactive web demonstration that visualizes these concepts intuitively.

Abstract

Control Barrier Functions

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