For the cable-suspended aerial transportation system (ATS), double pendulum effects caused by the hook and payload is unignorable. This paper presents a nonlinear controller incorporating a model predictive control (MPC)-based gains auto-tuning scheme designed specifically for the ATS with double-pendulum swing (DPS) effects. With the gains auto-tuning method, control gains can be tuned effectively and optimally compared with manual tuning. Specifically, an energy-based controller is firstly presented. Subsequently, to improve the control performance, a gains auto-tuning scheme is developed for optimizing the control gains. Ultimately, simulation results demonstrate that the proposed controller can precisely fulfill quadrotor positioning and effectively suppress payload and hook swing angles.