Lie Group Periodic Structures Generation and Control in Robotic Swarm

Abstract

This thesis explores target-surrounding techniques for swarms of UAVs. Most traditional target-surrounding approaches limit the swarm to circular trajectories because it is easier to maintain the agents in such trajectories and apply collision avoidance techniques. This thesis provides experimental validation, in real quadcopters, of a novel target-surrounding technique from the literature, which introduces the concept of circular embedding as a virtual circular trajectory where the swarm is stabilized. This virtual circle is distorted, via quaternion rotation, to form a 3D geometry that is topologically homeomorphic to the circle. This thesis also provides an improved version of this technique, using the Lie group SO(3), expanding the number of 3D shapes that can be generated via circle distortion. Additionally, this thesis provides experimental validation of a novel low-level LQR controller for quadcopters, defined using the Lie group SE_2(3). This group enables the controller to stabilize all the quadcopters' states (orientation, position, and velocity), which is efficient when the vehicle follows more complex 3D trajectories, such as those provided by the target-surrounding previously mentioned. The LQR is replaced by an MPC, which was necessary to combine this controller with the swarm techniques studied in this work, forming a Lie group cascade swarm controller for quadcopters. Finally, this low-level controller was adapted to the dynamics of a fixed-wing, and the cascade swarm controller was successfully tested for this type of UAVs. The swarm technique proposed here was tested with the dynamics of double integrators, quadcopters, and fixed wings, proving that it is agnostic of the agents in the swarm.