Three-dimensional point cloud registration is a fundamental task for localization and mapping in autonomous navigation applications. Over the years, registration algorithms have evolved; nevertheless, several challenges still remain. Recently, non-repetitive scanning LiDAR sensors have emerged as a promising 3D data acquisition tool. However, the feasibility of this type of sensor to leverage robust point cloud registration still needs to be ascertained. In this paper, we explore the feasibility of one such LiDAR sensor with a Spirograph-type non-repetitive scanning pattern for robust 3D point cloud registration. We first characterize the data of this unique sensor; then, utilizing these results, we propose a new 3D point cloud registration method that exploits the unique scanning pattern of the sensor to register successive 3D scans. The characteristic equations of the unique scanning pattern, determined during the characterization phase, are used to reconstruct a perfect scan at the target distance. The real scan is then compared with this reconstructed scan to extract objects in the scene. The displacement of these extracted objects with respect to the center of the unique scanning pattern is compared in successive scans to determine the transformations that are then used to register these scans. The proposed method is evaluated on two real and different datasets and compared with other state-of-the-art registration methods. After analysis, the performance (localization and mapping results) of the proposed method is further improved by adding constraints like loop closure and employing a Curve Fitting Derivative Filter (CFDT) to better estimate the trajectory. The results clearly demonstrate the suitability of the sensor for such applications. The proposed method is found to be comparable with other methods in terms of accuracy but surpasses them in performance in terms of processing time.