## 1 Introduction  
A typical mechanism for the propulsion of underwater robots is a thruster with a propeller. The dynamic model of a tilting thruster is essential for designing a controller for underwater robots. Due to the complex effects of nonlinear fluid dynamics, studies have derived the dynamic model using experiments. [PERSON] suggested that the thrust force is proportional to the signed square of the propeller's rotational speed [1]. [PERSON] proposed a nonlinear model considering the fluid dynamics of the motor model and propeller blade [2]. [PERSON] presented a combination model consisting of an axial flow model and a rotational flow model. [3]. [PERSON] developed a three-state model based on dimensionless propeller parameters, thrust coefficients, and an advance ratio [4]. [PERSON] proposed an advanced three-state model considering the surrounding flow rate and inflow angle [5; 6; 7]. [PERSON] [8] and [PERSON] [9] numerically analyzed the effect of thrust ducts on different boundary conditions. [PERSON] conducted Bollard-pull experiments to find the relationship between axial thrust and input while considering the effect of thrust-thrust and thrust-hull interactions [10]. [PERSON] used a system identification process to derive an accurate dynamic model for commercially available fixed-pitch variable speed thrusters [11]. [PERSON] developed a novel thruster based on a pump-jet [12] and applied it to ASV (Autonomous Surface Vehicle) in shallow water [13]. Most underwater robots are driven by fixed thrusters, but tilting thrusters have recently been used in underwater robots [14]. A tilting thruster can reduce the number of thrusters in an underwater robot and contributes to compact design [15]. When an underwater robot is controlled with tilting thrusters, the influence of the rotational motion of the thruster has to be considered, unlike a fixed thruster. The rotating thrust force can be calculated by multiplying the steady-state thrust force by a trigonometric function of the rotation angle [16]. When a thruster is operated with rotation motion, additional forces and moments are generated, and they can have a complicated relationship. Studies have numerically analyzed dynamic models according to the rotation angle of the azimuth thrusters [17; 18], but a model that takes into account the effects that occur while the thruster is rotating is needed. In our previous research, dynamic models for rotation motion were derived by oscillating a 1-Degree-of-Freedom (DoF) tilting thruster using sine waves [19]. Modeling errors occurred as a result of relying on discrete data in the frequency domain, and it was necessary to design a model-free robust controller to overcome them [20]. This paper presents a 2-DoF azimuth thruster, as shown in Figure 1. Like fish fins, the thruster has a 2-DoF direction of propulsion. There are two rotational joints with the gimbal mechanism. It can produce effects similar to those of bioinspired robots that produce two-way propulsion through tail motion [21]. In a steady state, the direction of the thrust is the same as the direction of the thruster. However, the thrust is bent and dispersed during the rotational motion of the azimuth thruster, and a reaction moment also occurs. It is difficult to analytically describe this phenomenon, because the system model highly depends on complicated fluid dynamics. In this study, a signal compression method was applied to derive an empirical model of the azimuth thruster. Signal compression methods are often used for the empirical modeling of mechanical systems and can extract the frequency response of a system in the whole range of mechanically constrained frequencies using a pseudo-impulse signal [22]. Signal compression methods have been used for the cylinder modeling of an automatic excavator [23] and friction modeling of a robot manipulator [24].