Dynamic obstacle avoidance is a challenging topic for optimal control and optimization-based trajectory planning problems. Many existing works use Control Barrier Functions (CBFs) to enforce safety constraints for control systems. CBFs are typically formulated based on the distance to obstacles, or integrated with path planning algorithms as a safety enhancement tool. However, these approaches usually require knowledge of the obstacle boundary equations or have very slow computational efficiency. In this paper, we propose a framework based on model predictive control (MPC) with discrete-time high-order CBFs (DHOCBFs) to generate a collision-free trajectory. The DHOCBFs are first obtained from convex polytopes generated through grid mapping, without the need to know the boundary equations of obstacles. Additionally, a path planning algorithm is incorporated into this framework to ensure the global optimality of the generated trajectory. We demonstrate through numerical examples that our framework allows a unicycle robot to safely and efficiently navigate tight, dynamically changing environments with both convex and nonconvex obstacles. By comparing our method to established CBF-based benchmarks, we demonstrate superior computing efficiency, length optimality, and feasibility in trajectory generation and obstacle avoidance.

翻译：动态避障是最优控制及基于优化的轨迹规划领域中的一个具有挑战性的课题。现有许多研究利用控制屏障函数（CBFs）来强制控制系统的安全约束。CBFs通常基于到障碍物的距离来构建，或与路径规划算法集成作为安全增强工具。然而，这些方法通常需要已知障碍物的边界方程，或计算效率非常低下。本文提出一个基于模型预测控制（MPC）与离散时间高阶控制屏障函数（DHOCBFs）的框架，用于生成无碰撞轨迹。DHOCBFs首先通过网格映射生成的凸多面体获得，无需已知障碍物的边界方程。此外，该框架中集成了一个路径规划算法，以确保生成轨迹的全局最优性。我们通过数值算例证明，该框架能使一个独轮车机器人在紧凑、动态变化且包含凸与非凸障碍物的环境中安全高效地导航。通过将我们的方法与已有的基于CBF的基准方法进行比较，我们证明了本方法在轨迹生成与避障方面具有更优的计算效率、长度最优性及可行性。