Lissajous curves have been used in various engineering applications such as optics, imaging, antenna scan, machining, as well as mobile robotics. In this article, we propose and analytically justify a Lissajous curve based trajectory planning strategy for aerial multi-agent systems to achieve the following objectives simultaneously: (i) Collision free paths for repeated coverage of a region while maintaining a closed sensor ring around a specified center for all time. (ii) Guaranteed detection of any stationary or moving object enclosed within the ring in finite time without the possibility of undetected escape. This leverages known and some novel properties of Lissajous curves that we establish as a part of this work. This has several potential applications in civil and military missions such as search and surveillance, repeated patrolling, target detection and capture, and the proposed strategy meets all these objectives simultaneously. We validate the proposed strategy through simulations and experiments using differential drive ground robots. We also demonstrate the applicability of this strategy for aerial surveillance through a Software-In-Loop-Simulation for quadrotors.

中文翻译：

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李沙育曲线在多主体轨迹规划中的应用

Lissajous曲线已用于各种工程应用中，例如光学，成像，天线扫描，机械加工以及移动机器人。在本文中，我们提出并分析性地证明了基于Lissajous曲线的空中多智能体系统轨迹规划策略，以同时实现以下目标：（i）无碰撞路径，可重复覆盖区域，同时在指定区域周围保持闭合的传感器环始终处于中心位置。（ii）确保在有限的时间内检测到圈内封闭的任何静止或活动物体，而不会发生未发现的逃逸的情况。这利用了我们在这项工作中建立的Lissajous曲线的已知特性和一些新颖特性。这在民用和军事任务中具有多种潜在应用，例如搜索和监视，反复巡逻，目标检测和捕获，并且所提出的策略可以同时满足所有这些目标。我们通过使用差动驱动地面机器人的仿真和实验来验证所提出的策略。我们还通过四旋翼的环内仿真软件演示了该策略在空中监视中的适用性。