Table of values for <i>p.</i>

Ensuring that a robot employing demonstration learning models can simultaneously achieve accurate trajectory tracking of demonstrated paths and effective avoidance of moving obstacles in dynamic environments remains a critical research challenge. This paper proposes a real-time trajectory planning framework based on an enhanced artificial potential field (APF) approach to address this dual-objective problem. Specifically, the proposed method deploys a sequence of virtual target points along the demonstrated trajectory to guarantee both path-following precision and goal convergence for robotic systems. A dynamic obstacle repulsion model is developed by integrating velocity-coupled and acceleration-associated force components, enabling proactive obstacle motion anticipation and adaptive trajectory reconfiguration. Furthermore, a probabilistic obstacle motion prediction framework is established through motion pattern analysis to actively optimize the robot’s motion strategy and reduce tracking errors. Simulation-based experimental results demonstrate that, under complex obstacle motion scenarios, the proposed method achieves a 55.8% reduction in trajectory tracking error compared with recently proposed improved APF methods and a 41.5% decrease relative to Dynamic Movement Primitives (DMP) baselines. These quantitative improvements validate the framework’s superior robustness and safety performance in unstructured environments, with all evaluations systematically conducted in simulated settings.确保采用演示学习模型的机器人在动态环境中同时实现对演示路径的精准轨迹跟踪与移动障碍物的有效规避，仍是一项关键的科研挑战。本文提出一种基于增强型人工势场（Artificial Potential Field, APF）方法的实时轨迹规划框架，以解决该双目标问题。具体而言，所提方法沿演示轨迹部署一系列虚拟目标点，以保障机器人系统的路径跟踪精度与目标收敛性。本文通过整合速度耦合与加速度关联的力分量，构建了动态障碍物斥力模型，可实现对障碍物运动的主动预判与轨迹的自适应重构。此外，本文通过运动模式分析构建了概率型障碍物运动预测框架，以主动优化机器人运动策略并降低跟踪误差。基于仿真的实验结果表明，在复杂障碍物运动场景下，与近期提出的改进型APF方法相比，所提方法的轨迹跟踪误差降低了55.8%；相较于动态运动基元（Dynamic Movement Primitives, DMP）基准模型，误差降幅达41.5%。上述量化优化结果验证了该框架在非结构化环境中具备更优异的鲁棒性与安全性能，所有评估均在仿真环境中系统化开展。