Four degrees of freedom (4DoF) registration is a class of point cloud registration problems for finding a rigid transformation to align two point clouds under the constraint that the rigid transformation is composed of a three-dimensional (3D) translation and 1D rotation. This constraint is suitable to align scan pairs acquired using modern terrestrial Light Detection and Ranging (LiDAR) scanners, the scans of which can share the direction of gravity as the Z-axis due to such scanners using tripods or internal inclinometers. We propose a fast convergence method for global optimal 4DoF registration. The proposed method consists of (i) our newly developed 4DoF registration model formulated as an optimization problem involving the <i>cylindrical norm</i> to measure the distance between two points, and (ii) a fast convergence algorithm to find a global optimal solution of the model. We experimentally demonstrated that the proposed method reduced the number of iterations to convergence and computation time compared with a current 4DoF registration method, especially when the given scan pairs are similar but cannot be aligned, which often appears in registration of multiple point clouds.