## 1 Introduction  
Accurate mathematical models are essential for research on submarine maneuvering and control. Currently, the dynamic mathematical model of submarines relies on the standard equations of submarine motion published by the United States Navy Ship Research and Development Center in 1967 [1]. However, these equations involve more than 100 hydrodynamic coefficients that must be estimated and obtained for accurate dynamic modeling. Although free-running model tests are good for studying the maneuverability characteristics of submarines [2], they are costly and easily influenced by the experimental environment and measuring instruments. The restrained model tests of submarine models, including Planar Motion Mechanism (PMM) [3] and Rotation Arm (RA) [4], are widely used to obtain the hydrodynamic coefficients. RA can be used to obtain the cross coefficients related to angular velocities, but it is unable to determine acceleration derivatives and accurately measure position derivatives. Additionally, the experimental facilities for RA impose high costs. On the other hand, PMM is less expensive and can obtain more hydrodynamic coefficients by simulating the translational and rotational motions of the test model, thereby obtaining the six-degree-of-freedom hydrodynamic coefficients of the submarine hull, but the motion cannot be large in the PMM test, so the nonlinear coefficients are not reliable. Although most hydrodynamic coefficients can be obtained through these test methods, their high costs and long cycles remain practical problems. With the development of computer technology, computational fluid dynamics (CFD) methods for simulating and calculating hydrodynamic coefficients have become more prevalent. For instance, [PERSON] et al. [5] used CFD to simulate the PMM test of the KCS model and obtained partial hydrodynamic coefficients. [PERSON] et al. [6] separated the linear and nonlinear components of the total resistance coefficient using CFD numerical methods, fitted the resistance coefficients for translational motion, and calculated the hydrodynamic coefficients for rotational motion using surface fitting. [PERSON] et al. [7] combined the inertial and rotating reference frames, along with the added momentum source method, to simulate the RA test. By varying the angular velocity and drift angle, they obtained stability derivatives of yawing moment and yawing force that aligned well with the experimental results. Their study also explored the impact of the rotating arm radius. [PERSON] et al. [8] extracted all hydrodynamic coefficients of the forced motion of a submarine hull by conducting only one test using CFD simulation. Parameter identification methods are powerful technical means for establishing accurate system-dynamic mathematical models using experimental or numerical data. Many researchers have studied model identification and parameter estimation methods for autonomous underwater vehicles. For example, [PERSON] [9] first applied the Kalman filter for ship hydrodynamic parameter identification in the 1990s. Since then, numerous scholars have extended the research of [9], such as [PERSON] et al. [10], who successfully determined the hydrodynamic coefficients of the NPS AUV II and ISIMI by employing velocity and displacement measurements in conjunction with the implementation of an extended Kalman filter (EKF) estimator. These hydrodynamic coefficients were then incorporated into the augmented state vector of a nonlinear model with six degrees of freedom. To enhance the accuracy and convergence speed of the algorithm, a suitable covariance matrix was selected based on the ARMA process model. [PERSON] and [PERSON] [11] utilized a combination of an analytical and semi-empirical estimation (ASE) method that integrated the hydrodynamics and geometric characteristics of submarine vehicles. This approach was further enhanced by incorporating a parameter estimator based on the extended Kalman filter.