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.

翻译：在自动驾驶车辆的生态驾驶和四旋翼飞行器导航等应用中，快速且精确地求解在时空约束下自主智能体的最优控制问题至关重要。然而，由于动力学与事件时序之间的耦合，近似求解最优控制问题的非线性规划本质上是非凸的，因此求解具有挑战性。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，但往往只能得到次优解。为了显著改善数值特性，我们提出了一种采用时间缩放直接多重打靶法的公式，该方法将预测时域划分为与特征时间约束对齐的多个分段。此外，我们开发了一种基于半定规划的快速凸松弛方法，该方法利用了提升公式的稀疏结构。全面的仿真研究证明了该方法的解最优性和计算效率。此外，在具有约束开放时间窗口的四旋翼飞行器航点飞行任务上进行的真实世界实验，验证了该方法在复杂环境中的实际适用性。