In this paper, a novel foldable quadrotor (FQR) inspired by an origami mechanism is designed. The FQR can fold its arms during flight to enable aggressive turning maneuvers and operations in cluttered environments. A dynamic model of folding is built for this system with the collected data, and a feedback controller is designed to control the position and orientation of the FQR. Lyapunov stability analysis is conducted to show that the system is stable during arm folding and extension, and motion planning of the FQR is achieved based on a modified minimum-snap trajectory generation method. Simulation results are provided to demonstrate the advantage of this design over the conventional quad-rotor in obstacle avoidance during flight.