一种快速的半正定凸松弛方法用于具有时空约束的最优控制问题
A Fast Semidefinite Convex Relaxation for Optimal Control Problems With Spatio-Temporal Constraints
AI总结：
本文提出了一种快速半正定凸松弛方法，用于解决具有时空约束的最优控制问题，通过时间缩放直接多重射击方案和稀疏模式利用，提升计算效率与解的最优性。
摘要：
解决在空间和时间约束下自主体的最优控制问题（OCPs）快速且准确，对于从自动驾驶车辆的生态驾驶到四旋翼导航等应用至关重要。然而，近似OCPs的非线性规划本质上非凸，因为动态和事件时间之间存在耦合，因此解决起来具有挑战性。大多数方法通过预定义航点时间或仅使用非凸轨迹优化来应对这一挑战，这虽然简化了问题，但往往导致次优解。为了显著提高数值特性，我们提出了一种具有时间缩放直接多重射击方案的公式，将预测时间轴划分为与特征时间约束对齐的段。此外，我们开发了一种快速的基于半正定规划的凸松弛方法，利用提升后的公式稀疏模式。全面的仿真研究证明了解决方案的最优性和计算效率。此外，针对具有受限开放时间窗口的四旋翼航点飞行任务的现实世界实验验证了该方法在复杂环境中的实际适用性。
英文摘要：
Solving optimal control problems (OCPs) of autonomous agents operating under spatial and temporal constraints fast and accurately is essential in applications ranging from eco-driving of autonomous vehicles to quadrotor navigation. However, the nonlinear programs approximating the OCPs are inherently nonconvex due to the coupling between the dynamics and the event timing, and therefore, they are challenging to solve. Most approaches address this challenge by predefining waypoint times or just using nonconvex trajectory optimization, which simplifies the problem but often yields suboptimal solutions. To significantly improve the numerical properties, we propose a formulation with a time-scaling direct multiple shooting scheme that partitions the prediction horizon into segments aligned with characteristic time constraints. Moreover, we develop a fast semidefinite-programming-based convex relaxation that exploits the sparsity pattern of the lifted formulation. Comprehensive simulation studies demonstrate the solution optimality and computational efficiency. Furthermore, real-world experiments on a quadrotor waypoint flight task with constrained open time windows validate the practical applicability of the approach in complex environments.
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