稳定风场中 Dubins 飞机最短时间路径的闭式解
Closed-Form Solutions for Minimum-Time Paths of Dubins Airplane in Steady Wind
AI总结：
针对稳定风场中 Dubins 飞机最短时间路径规划问题，本文利用 Pontryagin 最大值原理划分四类候选路径，并通过二次方程、超越方程求根及改进二分法实现常数时间全局最优求解。
摘要：
本文研究稳定风场影响下 Dubins 飞机的最短时间路径规划问题。通过将路径规划问题转换到空速相对坐标系，该问题等价于寻找 Dubins 飞机拦截一个运动目标的最短时间控制策略。在空速相对坐标系中，应用 Pontryagin 最大值原理，最短时间解的候选路径被划分为四种类型：SC、CC、CCC 和 CSC，其中 S 表示直线段，C 表示圆弧段。此外，本文分析了每种类型的几何性质，指出 SC 和 CC 路径可以通过求解一个二次方程的根得到，而 CCC 和 CSC 路径则由若干非线性超越方程的根确定。文中提出了一种改进的二分法，使所有超越方程的根都能够在常数时间内计算完成。因此，通过比较四种类型的全部候选路径，即可在常数时间内获得全局最优路径。最后，数值算例表明，与文献中的现有方法相比，本文推导的闭式解能够保证找到全局最优解。
英文摘要：
This paper is concerned with the minimum-time path-planning problem for a Dubins airplane under the influence of steady wind. The path-planning problem, by transforming into the air-relative frame, is equivalent to finding the minimum-time control strategy for a Dubins airplane to intercept a moving target. In the air-relative frame, by applying Pontryagin's maximum principle, the candidates for the minimum-time solution are categorized into a family of four types: SC, CC, CCC, CSC, where S denotes a straight line segment and C denotes a circular segment. Furthermore, the geometric properties for each type are analyzed, indicating that the paths of SC and CC can be obtained by finding the roots of a quadratic equation, while the paths of CCC and CSC are determined by the roots of some nonlinear transcendental equations. An improved bisection method is presented in the paper so that all the roots of the transcendental equations can be computed within a constant time. As a consequence, the globally optimal path can be obtained within a constant time by comparing all the candidates of the four types. Finally, numerical examples are presented, showing that the closed-form solutions derived in the paper can ensure to find the globally optimal solution by comparing with existing methods in the literature.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
