This paper is concerned with characterizing the shortest path of a Dubins vehicle from a position with a prescribed heading angle to a target circle with the final heading tangential to the target circle. Such a shortest path is of significant importance as it is usually required in real-world scenarios, such as taking a snapshot of a forbidden region or loitering above a ground sensor to collect data by a fixed-wing unmanned aerial vehicle in a minimum time. By applying Pontryagin's maximum principle, some geometric properties for the shortest path are established {without considering any assumption on the relationship between the minimum turning radius and radius of the target circle}, showing that the shortest path must lie in a sufficient family of 12 types. By employing those geometric properties, the analytical solution to each type is devised so that the length of each type can be computed in a constant time. In addition, some properties depending on problem parameters are found so that the shortest path can be computed without checking all the 12 types. Finally, some numerical simulations are presented, illustrating and validating the developments of the paper.
Dubins路径是具有有界曲率的最短路径。非完整运动规划的开创性结果是（在没有障碍物的情况下）Dubins路径由一个圆弧、一个线段和另一个圆弧组成，或者由三个圆弧组成[Dubins，1957]。Dubins最初的证明使用了高等微积分;后来，Dubins的结果被使用控制理论技术重新证明[Reeds and Shepp，1990]，[Sussmann and Tang，1991]，[Boissonnat，Cérézo，and Leblond，1994]。我们介绍和研究的曲率约束运动的离散模拟。我们发现，最短的“有界曲率”的多边形路径具有相同的结构Dubins路径。Dubins路径的性质遵循我们的结果作为一个限制的情况下-这给出了一个新的，“离散”的证明Dubins结果。
An Efficient Method for Extracting the Shortest Path from the Dubins Set for Short Distances Between Initial and Final Positions
State Key Laboratory for Manufacturing Systems Engineering（制造系统工程国家重点实验室）；School of Automation Science and Engineering（自动化科学与工程学院）；Faculty of Electronic and Information Engineering（电子与信息工程学院）
A Novel Model for 3D Motion Planning for a Generalized Dubins Vehicle with Pitch and Yaw Rate Constraints
Department of Electrical Engineering and Computer Science, University of California, Irvine（加州大学 Irvine 分校 电气工程与计算机科学系）；Department of Mechanical Engineering, Texas A&M University（德克萨斯大学 A&M 学院 机械工程系）；DCS Corporation（DCS 公司）；Control Science Center, Air Force Research Laboratory, Wright-Patterson Air Force Base（空军研究实验室 威格伍德-帕特森空军基地 控制科学中心）
