The localization problem in robotics has been widely studied both for indoor and outdoor applications, but is still open for improvements. In indoor environments, GPS-based methods are not preferred due to reflections, and the pose of the robot is determined according to the measurements taken around with its sensors. One of them is iterative closest point (ICP)-based localization method. ICP is a point set registration method, the essence of which is to iteratively compute the transformation between two point sets. However, it is also utilized to solve the localization problem thanks to its high precision in registration. Precise localization is important for applications that require highly accurate pose estimation, such as for smart-AGVs to be used in smart factories to reach a station at industrial standards. Traditional ICP finds transformation in terms of a rotation and translation, and thus can be directly applied to the localization problem. On the other hand, the affine variant of ICP is not adapted to solve the localization problem. In this study, the necessary arrangements to make affine ICP suitable for precise localization are given as a procedure such that the transformation between point sets is found by affine ICP, the resulting transformation is projected to rotation plane by polar decomposition and then the pose is estimated. The enhancements achieved with the usage of affine ICP in precise localization problems are demonstrated in simulation by comparing localization performance of affine ICP with that of traditional ICP. For this purpose, in a factory environment, a scenario where a smart-AGV approaching the target autonomously to carry out an operation has been prepared. The performances of the algorithms have been evaluated for five different docking stations with 30 separate experiments. Moreover, the challenges related to the affine ICP-based fine localization, in particular about finding projection of affine transformation to rotation plane, are highlighted in this study.