## 1 Introduction  
The manoeuvring performance of the submarine is an important parameter in the initial design, especially for the geometry of the appendages, which in turn is important for the submarine resistance and hence total performance of the submarine. Conventionally, captive experiments, i.e., rotating arm (RA) and planar motion mechanism (PMM), and numerical simulations, i.e., computerised planar motion mechanism (CPMM), are adopted for in the early design stage until a reasonable maturity of the design is reached. For an alternative option of matured designs, free-running manoeuvring models have further become available for validation of the manoeuvring performance. The so-called submarine free-running model (SFRM) is a remotely controlled scale model replicating submarine manoeuvres. These models should be equipped with instrumentation to measure all the state variables required to describe its 6-DOF motion responses, including trajectories, attitudes, linear and angular velocities, and accelerations [1]. Over the last few decades, there have been numerous studies [2; 3; 4; 5] investigating SFRMs in controlled environments. Although the SFRM can evaluate the manoeuvring performances of the full-scale model accurately, there are still some differences between the scale model and the full-scale model due to scale effect. In order to keep the desired trajectories dynamically during the test, a robust control system is required for adjusting the control planes of the SFRM. Definitely, the input and output signals demand for the control planes are greatly affected by the manoeuvring characteristics. The CFD-based and coefficient-based methods are the two major mathematical models in exploring the hydrodynamic forces and moments exerted on the SFRM. As the name implies, the coefficient-based method uses the equation of motion (EOM) combined with the hydrodynamic coefficient as a mathematical model [6; 7]. Taking advantage of the predefined coefficients in the mathematical model, it usually has less computation time than the CFD-based method. Furthermore, the coefficient-based method can approximate the hydrodynamic interaction effect through semi-empirical correction when the positions of the appendages are rearranged at a preliminary design stage [8]. However, the limitation of the coefficient-based method is that the unsteady viscous effect cannot be captured in the linear model, such as cross flow and vorticity [1]. In order to overcome the limitations of the coefficient-based method and improve the accuracy of manoeuvring characteristics, the CFD-based method provides another option, including the resistance test [9], self-propulsion test, and free-running test simulations [10]. Since the estimation of hydrodynamic coefficients has uncertainty and variation in different environmental conditions [11], hydrodynamic forces and moments exerted on the SFRM are changed accordingly. Therefore, an applicable controller needs to be a self-tuning and robust counter to the variation of SFRM parameters and also unpredictable environmental disturbances. In recent years, numerous control models have been proposed, e.g., linear controllers [12; 13], sliding-mode controller (SMC) [14], adaptive control [15; 16], FLC (Fuzzy Logic Control) [17], predictive control [18], static feedback control [19], neural-network-based control [20], and PID (proportional-integral-derivative) control [21]. The advantage of using a PID controller is that it is simple to implement and maintain, but it is primarily applicable for linear time-invariant systems. The well-known classic equations of motion (EOMs) are those of [PERSON] [6], [PERSON] [7], and [PERSON] and Fjellstad [22]. They were derived by adopting Newton-Euler equations or Lagrangian methods, which can be calculated in matrix forms. For example, [PERSON], et al.