## 1 Introduction  
Hydraulic systems have the advantages of high output power density and precision control effects and have been widely used in production and engineering [1]. With the advancement of science and technology, the development of hydraulic systems also tends to be highly integrated. Especially in the field of hydraulic control valves, high-degree integration imposes strict requirements for the size, performance, and layout method of the electro-mechanical converters in control valves. The electro-mechanical converters employed in control valves primarily include voice coil motors (VCMs) [2], proportional electromagnets [3], piezoelectric ceramics [4], and magnetostrictions [5]. Although the piezoelectric ceramics and the magnetostrictions have the advantages of large force, fast response, and high control accuracy, their limited displacements bring challenges in actual products and applications [6]. Consequently, they are often utilized with displacement amplification mechanisms [7, 8]. Proportional electromagnets are the most widely used electromechanical converter in the field of hydraulic values, with the advantages of large output force, low cost, and stable operation. Compared to proportional electromagnets, VCMs have a simpler structure, smaller volume, lighter weight, and faster response speed [9]. Meanwhile, under the same input power, VCMs generally demonstrate greater displacements than proportional electromagnets, but their output forces are relatively low [10]. It is known that although VCMs have some advantages in the field of valve control, but may not meet the requirements of hydraulic valves for the output force of electromechanical converters, especially in the direct-driven valves. Consequently, a kind of force amplification mechanism with a compact structure and good amplification effect is needed to enhance the VCMs' output force. Amplification mechanisms can be divided into displacement amplification mechanisms and force amplification mechanisms based on the variation in input and output directions. Common amplification methods include lever amplification mechanisms [11], bridge amplification mechanisms [12], Scott-Russell amplification mechanisms [13], and hydraulic amplification mechanisms [14]. Although lever amplification mechanisms, bridge amplification mechanisms, and Scott-Russell amplification mechanisms all have good amplification effects, limited by their structural forms and amplification principles, it is difficult to achieve a compact and flexible layout, which even affects the bandwidth of the dynamic systems [8; 15]. In contrast, the hydraulic amplification mechanisms realize the amplification function based on PASCAL's principle [16] and have lower requirements for the spatial layout between the input and output connecting parts, which are more suitable for the working requirements of the hydraulic control valves. As shown in Figure 1, by adjusting the structure form and layout scheme, the hydraulic amplification mechanisms can realize the function of non-coaxial and orthogonal force amplification and transmission between the actuator and the load. In Figure 1a, the directions of the input force and the load force are parallel and non-coaxial. In Figure 1b, the directions of the input force and the load force are orthogonal. In recent years, extensive research has been carried out on hydraulic amplification mechanisms. Traditional hydraulic amplification mechanisms include flat diaphragm amplification mechanisms [7] and piston hydraulic amplification mechanisms [17]. [PERSON] et al. designed a piezoelectric pump utilizing a flat diaphragm hydraulic displacement amplification mechanism and conducted theoretical modeling and experimental analysis, as shown in Figure 2[18]. In the system, the deformation of piezoelectric vibrator is transferred to the flexible diaphragm through the driving liquid.