Application of Lissajous curves in trajectory planning of multiple agents

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.