For mobile robots, localization is essential for navigation and spatial correlation of its collected data. However, localization in Global Positioning System-denied environments such as underwater has been challenging. Light-emitting diode (LED)-based optical localization has been proposed in the literature, where the bearing angles extracted from the line-of-sight of the robot viewed from a pair of base nodes (also known as beacon nodes) are used to triangulate the position of the robot. The state-of-the-art in this approach uses a stop-and-go motion for the robot in order to ensure an accurate position measurement, which severely limits the mobility of the robot. This work presents an LED-based optical localization scheme for a mobile robot undergoing continuous motion, despite the two angles in each measurement cycle being captured at different locations of the robot. In particular, the bearing angle measurements are captured by the robot one at a time and are properly correlated with respect to the base nodes by utilizing the velocity prediction from Kalman filtering. The proposed system is evaluated in simulation and experiments, with its performance compared to the traditional state-of-the-art approach where the two angle measurements in each cycle are used directly to compute the position of the robot. In particular, the experimental results show that the average position and velocity estimation errors are reduced by 55% and 38%, respectively, when comparing the proposed method to the state-of-the-art.