Beyond Collision Cones: Dynamic Obstacle Avoidance for Nonholonomic Robots via Dynamic Parabolic Control Barrier Functions

TL;DR

This work introduces the Dynamic Parabolic CBF (DPCBF) to tackle safety-critical dynamic obstacle avoidance for nonholonomic robots. By replacing fixed collision cones with a state-dependent parabolic safety boundary in a Line-of-Sight frame, the method adaptively modulates curvature and vertex via $\lambda(x)$ and $\mu(x)$, improving QP feasibility under input constraints. The authors prove DPCBF validity for the kinematic bicycle model and demonstrate via extensive simulations that DPCBF achieves higher navigation success and lower control intervention than baseline CBF methods, even with up to 100 dynamic obstacles. The approach offers a principled, less-conservative mechanism for safe, efficient navigation in dense, dynamic environments with practical potential for real-world deployment.

Abstract

Control Barrier Functions (CBFs) are a powerful tool for ensuring the safety of autonomous systems, yet applying them to nonholonomic robots in cluttered, dynamic environments remains an open challenge. State-of-the-art methods often rely on collision-cone or velocity-obstacle constraints which, by only considering the angle of the relative velocity, are inherently conservative and can render the CBF-based quadratic program infeasible, particularly in dense scenarios. To address this issue, we propose a Dynamic Parabolic Control Barrier Function (DPCBF) that defines the safe set using a parabolic boundary. The parabola's vertex and curvature dynamically adapt based on both the distance to an obstacle and the magnitude of the relative velocity, creating a less restrictive safety constraint. We prove that the proposed DPCBF is valid for a kinematic bicycle model subject to input constraints. Extensive comparative simulations demonstrate that our DPCBF-based controller significantly enhances navigation success rates and QP feasibility compared to baseline methods. Our approach successfully navigates through dense environments with up to 100 dynamic obstacles, scenarios where collision cone-based methods fail due to infeasibility.

Figures (13)

• Figure 1: Illustrative comparison of two CBF mechanisms in dynamic obstacle avoidance scenarios. (a) Since the Collision Cone CBF (C3BF) evaluates only the heading of the relative velocity, it may classify the robot as unsafe regardless of its actual distance from the obstacle. (b) Our Dynamic Parabolic CBF (DPCBF) establishes a more flexible safety condition by evaluating both relative position and the magnitude of relative velocity, which avoids unnecessary restrictions when clearance is large. As shown in (b), the parabola's vertex shifts away from the robot's origin by $\mu({\boldsymbol x})$. This relaxes the safety constraint, allowing for less restrictive movements that approach the boundary of the unsafe set while remaining provably safe.
• Figure 2: Schematic of the kinematic bicycle model. The robot's state defined by its Center of Mass (CoM) position $(x,y)$, heading angle $\theta$, and forward velocity $v$. The distances from the CoM to the front and rear axles are $\ell_{f}$ and $\ell_{r}$, respectively. The front wheel steering angle is $\delta$, and $\beta$ is the resulting vehicle slip angle.
• Figure 3: A closer look on collision cone-based CBF. (a) Single obstacle: if $h_{\textup{cc}}({\boldsymbol x}(t_0)) \geq 0$, the CBF constraint keeps $h_{\textup{cc}}({\boldsymbol x}(t)) \ge 0$ for all $t \ge t_0$. (b) Five obstacles: if $h_{j, \textup{cc}}({\boldsymbol x}(t)) \geq 0$ for all $j$-th obstacle and the CBF-QP is feasible at $t$, safety is at least maintained at a given time $t$. (c) Even with $h_{j,\textup{cc}}({\boldsymbol x}(t_0)) \ge 0$ for all $j$-th obstacle, cone intersections can leave no admissible relative velocity direction, leading the CBF-QP infeasible. (d) Given the initial configuration where the robot is surrounded by the union of the collision cones, there is no feasible solution to the CBF-QP even though a large collision-free area exists nearby.
• Figure 4: Visualization of the global world frame $(x,y)$ and the rotated Line-of-Sight (LoS) frame $(\tilde{x},\tilde{y})$ used in our formulation. By rotating the coordinates by an angle $\alpha$\ref{['eq:dpcbf_alpha_def']}, the $\tilde{x}$-axis of the LoS frame is aligned with the vector from the robot to the obstacle, ${\boldsymbol p}_{\textup{rel}}$. This transformation simplifies the definition of the parabolic safety boundary \ref{['eq:parabolic_region']}, allowing its position and curvature to adapt online based on the relative velocity components in this new frame.
• Figure 5: Three examples describe how \ref{['eq:parabolic_region']} shapes the safety boundary. (a) When $k_{\lambda}=0$ ($k_{\mu}$ being active), as the obstacle gets closer, the parabola's vertex moves toward the robot, shrinking the safe region. (b) Where $k_{\mu}=0$ ($k_{\lambda}$ being active), as the obstacle approaches or the relative velocity magnitude increases, the curvature of the parabola decrease, leading a larger unsafe set. (c) In DPCBF, where $k_{\lambda}\neq0$ and $k_{\mu}\neq0$, both the vertex and the curvature of the parabola adapts dynamically.

Theorems & Definitions (19)

• Theorem 1
• proof
• Remark 1
• Proposition 1: Infimum sub-additivity
• Proposition 2: Triangle Inequality
• Corollary 1: Supremum form of Proposition \ref{['prop:triangle']}
• Proposition 3: Reverse Triangle Inequality
• Corollary 2: Infimum form of Proposition \ref{['prop:reverse']}
• Proposition 4
• proof