As we move to increasingly complex cyber-physical systems (CPS), new approaches are needed to plan efficient state trajectories in real-time. In this paper, we propose an approach to significantly reduce the complexity of solving optimal control problems for a class of CPS with nonlinear dynamics. We exploit the property of differential flatness to simplify the Euler-Lagrange equations that arise during optimization, and this simplification eliminates the numerical instabilities that plague optimal control in general. We also present an explicit differential equation that describes the evolution of the optimal state trajectory, and we extend our results to consider both the unconstrained and constrained cases. Furthermore, we demonstrate the performance of our approach by generating the optimal trajectory for a planar manipulator with two revolute joints. We show in simulation that our approach is able to generate the constrained optimal trajectory in $4.5$ ms while respecting workspace constraints and switching between a `left' and `right' bend in the elbow joint.

翻译：随着我们走向日益复杂的网络-物理系统（CPS），实时规划高效状态轨迹的新方法变得不可或缺。本文提出了一种方法，能够显著降低具有非线性动力学的一类CPS最优控制问题的求解复杂度。我们利用微分平坦性特性简化了优化过程中出现的欧拉-拉格朗日方程，这一简化消除了普遍困扰最优控制的数值不稳定性。我们还给出了描述最优状态轨迹演化的显式微分方程，并将结果扩展至无约束和有约束两种情况。此外，我们通过为具有两个旋转关节的平面机械臂生成最优轨迹，验证了该方法的表现。仿真结果表明，我们的方法能够在$4.5$毫秒内生成受约束最优轨迹，同时满足工作空间约束并在肘关节的“左”弯和“右”弯之间进行切换。