Quadcopters, as unmanned aerial vehicles (UAVs), have great potential in civil applications such as surveying, building monitoring, and infrastructure condition assessment. Quadcopters however are relatively sensitive to rotational noises so that their performance may be quickly downgraded in the case of inadequate control, system uncertainties and/or disturbances. In this study, we deal with the quadrotor control problem by proposing a new control scheme named the adaptive second-order quasi-continuous sliding mode control (adaptive 2-QCSM). The goal is to achieve robust attitude control for the task of monitoring and inspection of built infrastructure. We first formulate the mathematical model of the quadcopter, incorporating nonlinearity, strong coupling, uncertain dynamics and external disturbances. We then use a Lyapunov function to derive the adaptive 2-QCSM control laws. We further tune it to ensure finite-time reaching under these control laws. Extensive simulations have been carried out for evaluation. Results show that the proposed control scheme can attenuate chattering to obtain stable operations under large parameter variations and strong disturbance conditions.