Enforcing safety is critical for aerial robotics. In this paper we consider the safety control problem for a 3D quadrotor with limited sensing range subject to avoiding collisions with time-varying obstacles. By using the concepts of Control Lyapunov Functions (CLFs) and Control Barrier functions (CBFs), we propose a control algorithm that explicitly considers the nonlinear and underactuated dynamics of a quadrotor to strictly guarantee time-varying safety-critical constraints. We demonstrate the feasibility of our proposed control design through numerical validation of (a) aerial flight through a region of dense cluttered obstacles, and (b) aerial flight through a dense time-varying obstacle field.

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