Nowadays, with the increasing popularity of quadcopter unmanned aerial vehicles in several real-world applications, achieving a fully autonomous quadcopter flight has become an imperative topic investigated in many studies. One of the most pressing issues in such a topic is the precision landing task, which always is devastatingly influenced by the ground effect and external disturbances. In this paper, we present an autonomous quadcopter landing algorithm allowing the vehicle to land robustly and precisely onto a heaving platform. Firstly, a robust control algorithm addressing the altitude flight under the ground effect and external disturbances is derived. We strictly prove the closed-loop system stability by using the Lyapunov theory. Secondly, a landing target state estimator is proposed to provide state estimations of the moving landing target. In addition, we propose a landing procedure to ensure the landing task is achieved safely and reliably. Finally, we use a DJI-F450 drone equipped with an infrared sensor and a laser ranging sensor as the experimental quadcopter platform and conduct experiments to evaluate the performance of our new algorithm in real flight conditions. The experimental results demonstrate the effectiveness of the proposed method. INDEX TERMS Autonomous landing, precision landing, moving target, quadcopter, heaving platform, ship deck, robust control, sliding mode control, disturbance observer.