Modern unmanned aerial vehicles (UAVs) rely on constant periodic sensor measurements to help avoid obstacles, deal with system disturbances and correct uncertainties. However, constant checking is time and energy consuming and is often not necessary especially in situations in which the UAV can safely fly using open loop control without entering unsafe states. Thus, in this paper we focus on two main problems: i) how to predict the possible reachable states of the UAV considering disturbances and system noise and ii) when to schedule the next state monitoring and replanning time to avoid unnecessary executions. To this end, we propose a self-triggered framework in which we consider realistic UAV dynamics and reachability analysis to predict future events while scheduling replanning times to guarantee safe high performance behavior under various conditions. Finally we validate the proposed approach with simulations and experiments in which a quadrotor UAV is tasked to navigate to a goal while avoiding obstacles in the environment.