Dynamic Collision Avoidance Using Velocity Obstacle-Based Control Barrier Functions

TL;DR

The paper tackles safe navigation for acceleration-controlled unicycle robots in dynamic environments by formulating a CLF-VOCBF-MIQP that unifies navigation and safety under velocity-space constraints. It replaces high-order CBFs with velocity obstacle-based CBFs (VOCBFs), enabling relative-degree-1 safety constraints and leveraging VO variants for reactive avoidance in distributed multi-robot settings. Computational efficiency is achieved by splitting the MIQP into CLF-VOCBF-QPs and by deploying a decision network to select a single subproblem, reducing real-time burden while maintaining safety guarantees. Numerical simulations show improved dynamic obstacle avoidance over HOCBFs and robust performance with static/dynamic obstacles, plus a successful extension to distributed multi-robot scenarios. The work advances practical safety-critical control by integrating VO geometry with CLF-based navigation in an efficient optimization framework suitable for real-time robotic deployment.

Abstract

Designing safety-critical controllers for acceleration-controlled unicycle robots is challenging, as control inputs may not appear in the constraints of control Lyapunov functions(CLFs) and control barrier functions (CBFs), leading to invalid controllers. Existing methods often rely on state-feedback-based CLFs and high-order CBFs (HOCBFs), which are computationally expensive to construct and fail to maintain effectiveness in dynamic environments with fast-moving, nearby obstacles. To address these challenges, we propose constructing velocity obstacle-based CBFs (VOCBFs) in the velocity space to enhance dynamic collision avoidance capabilities, instead of relying on distance-based CBFs that require the introduction of HOCBFs. Additionally, by extending VOCBFs using variants of VO, we enable reactive collision avoidance between robots. We formulate a safety-critical controller for acceleration-controlled unicycle robots as a mixed-integer quadratic programming (MIQP), integrating state-feedback-based CLFs for navigation and VOCBFs for collision avoidance. To enhance the efficiency of solving the MIQP, we split the MIQP into multiple sub-optimization problems and employ a decision network to reduce computational costs. Numerical simulations demonstrate that our approach effectively guides the robot to its target while avoiding collisions. Compared to HOCBFs, VOCBFs exhibit significantly improved dynamic obstacle avoidance performance, especially when obstacles are fast-moving and close to the robot. Furthermore, we extend our method to distributed multi-robot systems.

Figures (10)

• Figure 1: Collision avoidance between the robot and obstacles is achieved using a VO-based approach, where the VO is depicted as the shaded area. The obstacle is enlarged by inflating it with the robot's radius, which is then used to construct the VO. Here, $(x_\text{p}, y_\text{p})$ denote the coordinates of the rear axle axis, $(x_\text{c}, y_\text{c})$ represents the robot's center, and $l$ is the distance between them.
• Figure 2: Velocity obstacle $\text{VO}_{\text{R}_i|\text{O}_j}(\bm{v}_{\text{O}_j})$ of robot $\text{R}_i$ induced by the obstacle $\text{O}_j$, and $\text{CC}_{\text{R}_i|\text{O}_j}$ represents the collision cone between them. If the relative velocity $\bm{v}_{\text{R}_i} - \bm{v}_{\text{O}_j} \in \text{CC}_{\text{R}_i|\text{O}_j}$ or the robot's velocity $\bm{v}_{\text{R}_i} \in \text{VO}_{\text{R}_i|\text{O}_j}(\bm{v}_{\text{O}_j})$, then a collision will occur between $\text{R}_i$ and $\text{O}_j$ at some future time with the assumption that both maintain their current velocities.
• Figure 3: The safe range of the robot's velocity is denoted by $\text{VO}_{\text{R}_i|\text{O}_j}^{\text{C}}(\bm{v}_{\text{O}_j})$, and it can be represented by the disjunction of two linear constraints, where each linear constraint requires the robot's velocity is within a half-space. Each half-space corresponds to a direction in which the robot can navigate around the obstacle, and satisfying both leads to backward avoidance.
• Figure 4: Structure of the decision network. The decision network takes as input the states of $M$ obstacles and the robot's target state, both represented in the robot's local frame. It outputs the probabilities for the robot to navigate around each obstacle in three possible directions (left, right, or backward). These probabilities are used to determine the active constraints in the formulation of the CLF-VOCBF-QP-DecNet optimization problem.
• Figure 5: Simulation results of guiding the robot to its destination while avoiding collisions with static and dynamic obstacles. The robot's start and target positions are represented by the light blue circle and purple star, respectively, while the black squares indicate the start positions of the dynamic obstacles. All obstacles are represented by black dashed circles. In (b), the robot’ positions over time, along with those of the dynamic obstacles, are illustrated using color gradients.

Theorems & Definitions (11)

• Definition 1: Class $\mathcal{K}$ and $\mathcal{K}_{\infty}$ functions ames2019control
• Definition 2: Control Lyapunov function ames2019control
• Theorem 1
• Definition 3: Forward invariance set
• Definition 4: Control barrier function ames2019control
• Theorem 2
• Remark 1
• Remark 2
• Remark 3
• Theorem 3