Reactive trajectory optimization for robotics presents formidable challenges, demanding the rapid generation of purposeful robot motion in complex and swiftly changing dynamic environments. While much existing research predominantly addresses robotic motion planning with predefined objectives, emerging problems in robotic trajectory optimization frequently involve dynamically evolving objectives and stochastic motion dynamics. However, effectively addressing such reactive trajectory optimization challenges for robot manipulators proves difficult due to inefficient, high-dimensional trajectory representations and a lack of consideration for time optimization. In response, we introduce a novel trajectory optimization framework called RETRO. RETRO employs adaptive optimization techniques that span both spatial and temporal dimensions. As a result, it achieves a remarkable computing complexity of $O(T^{2.4}) + O(Tn^{2})$, a significant improvement over the traditional application of DDP, which leads to a complexity of $O(n^{4})$ when reasonable time step sizes are used. To evaluate RETRO's performance in terms of error, we conducted a comprehensive analysis of its regret bounds, comparing it to an Oracle value function obtained through an Oracle trajectory optimization algorithm. Our analytical findings demonstrate that RETRO's total regret can be upper-bounded by a function of the chosen time step size. Moreover, our approach delivers smoothly optimized robot trajectories within the joint space, offering flexibility and adaptability for various tasks. It can seamlessly integrate task-specific requirements such as collision avoidance while maintaining real-time control rates. We validate the effectiveness of our framework through extensive simulations and real-world robot experiments in closed-loop manipulation scenarios.

翻译：响应式轨迹优化对机器人领域提出了严峻挑战，要求在高动态、快速变化的复杂环境中快速生成具有目的性的机器人运动。尽管现有研究主要针对具有预定义目标的机器人运动规划，但新兴的机器人轨迹优化问题往往涉及动态演化的目标与随机运动动力学。然而，由于低效的高维轨迹表征方式以及对时间优化的忽视，现有方法难以有效应对机械臂的响应式轨迹优化挑战。为此，我们提出了一种名为RETRO的新型轨迹优化框架。RETRO采用自适应优化技术，同时覆盖空间与时间维度，实现了显著的计算复杂度：$O(T^{2.4}) + O(Tn^{2})$，相较于传统微分动态规划（DDP）方法在采用合理时间步长时所产生的$O(n^{4})$复杂度，取得了显著提升。为评估RETRO的误差性能，我们通过对比Oracle轨迹优化算法获得的Oracle值函数，系统分析了其遗憾界。理论分析表明，RETRO的总遗憾值可被所选时间步长的函数上界所约束。此外，我们的方法能够在关节空间中生成平滑优化的机器人轨迹，具有针对不同任务的灵活性与适应性，可在维持实时控制频率的同时无缝集成避碰等任务特定要求。通过闭环操作场景下的广泛仿真与真实机器人实验，我们验证了该框架的有效性。