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Given the dynamic model of a UAV, a set of virtual geometric constraints is imposed to the system here. Once these constraints are reached through a feedback control law, it is possible to show that the system exhibits a limit cycle that is the periodic trajectory to track. The aim of the work is to keep the UAV on such periodic trajectory. It is possible to stabilize the aerial vehicle on this oscillating motion through a modified LQR controller, which is designed on the basis of a linear periodic system approximating the system dynamics around the desired orbit.