The dynamic Moore-Penrose inverse solution has attracted increasing attention because of its wide range of applications. The use of zeroing neural networks to solve the inverse problem of dynamic matrices has become a popular topic in recent years. However, existing studies have established zeroing neural network (ZNN) models for the left Moore-Penrose inverse and right Moore-Penrose inverse based on the different dimensions of the matrix, respectively. There is an urgent need for a new framework that can handle both the right inverse and the left inverse simultaneously. In addition, convergence performance is a critical metric when using neural networks to solve problems, regardless of whether the environment has measurement noise or not. Therefore, this article proposes a unified solution framework that can be used to simultaneously solve for the left Moore-Penrose inverse and the right Moore-Penrose inverse. Two novel ZNN models that converge in finite time are obtained by using the evolution formula based on finite-time convergence. The convergence and robustness of the models are analyzed using finite-time theory. The performance of the proposed method is compared to that of existing methods using numerical examples. Furthermore, the proposed neural network models for calibration have been implemented in a photoelectric tracking system that utilizes a redundant manipulator. The feasibility and effectiveness of the proposed model in the actual system have been verified through simulations and experiments. It has been observed that the root mean square value of tracking error is as low as 0.032 mm in the experiment.