The latest advances in mobile platforms, such as robots, have enabled the automatic acquisition of full coverage point cloud data from large areas with terrestrial laser scanning. Despite this progress, the crucial post-processing step of registration, which aligns raw point cloud data from separate local coordinate systems into a unified coordinate system, still relies on manual intervention. To address this practical issue, this study presents an automated point cloud registration approach optimized for a stop-and-go scanning system based on a quadruped walking robot. The proposed approach comprises three main phases: perpendicular constrained wall-plane extraction; coarse registration with plane matching using point-to-point displacement calculation; and fine registration with horizontality constrained iterative closest point (ICP). Experimental results indicate that the proposed method successfully achieved automated registration with an accuracy of 0.044 m and a successful scan rate (SSR) of 100% within a time frame of 424.2 s with 18 sets of scan data acquired from the stop-and-go scanning system in a real-world indoor environment. Furthermore, it surpasses conventional approaches, ensuring reliable registration for point cloud pairs with low overlap in specific indoor environmental conditions.