Compared with traditional vehicle transportation methods, unmanned aerial vehicles (UAVs) have many advantages in transporting cargo because they are not limited by factors such as road traffic. To address practical applications, this paper presents a local obstacle avoidance control scheme for the quadrotor with suspended payload, which is suitable for complex environments with dense obstacles and external disturbances. Firstly, the overall dynamic model of the quadrotor-payload system is established using Euler-Lagrange equations. Secondly, considering the envelope circle radius switching problem caused by changes in the amplitude of the swing angles of the payload, an obstacle avoidance constraint of the quadrotor-payload system is presented, and an R-function is introduced to improve the traditional artificial potential field method, so as to achieve a less conservative obstacle avoidance. Thirdly, a cascade control scheme based on sliding mode algorithm is designed for the quadrotor-payload system trajectory tracking under external disturbances. Moreover, to avoid payload oscillations during flight, a feed-forward controller is designed to suppress the swing of the payload. Finally, simulation results verify the effectiveness and superiority of the proposed local obstacle avoidance strategy and control scheme.