<p>This paper investigates the tracking control problem of a quadrotor equipped with a flexible robotic arm (QeRA), where the arm can move quickly under a smooth-changing control command reflecting the propeller speed variations. Considering the effects of the moving robotic arm on the center of mass and moment of inertia, the dynamic model of QeRA is derived by incorporating the forces and torques generated by the arm, upon which a triple-loop control framework is used to solve the tracking problem with position tracking achieved through rapid angle velocity tracking. Unlike previous studies that treat arm motion as a disturbance and use robust or adaptive controllers, a gains-scheduling controller based on the multi-mode switched linear parameter varying (sLPV) approach is designed to tackle the large-range variations in the moment of inertia in QeRA caused by arm motion, which leads to significant nonlinearity and strong coupling in the angle velocity dynamics. By inserting a transition interval after the mode switching, the bump of the multi-mode control command at the switching instant is alleviated in the continuous-time domain via gains co-design of transition and non-transition intervals, rather than ignoring the smoothness constraint of the actuator (i.e., the propeller of the QeRA) or discretizing the dynamic model in the past literature. The effectiveness and advantages of the designed smooth multi-mode gains-scheduling controller based on the smooth sLPV approach are demonstrated through simulations in representation scenarios for QeRA, including position and attitude tracking.</p>