## 1 Introduction  
Micro/nanopositioning technology is essential for various applications in many research and technical fields, such as microelectromechanical systems (MEMS), ultra-precision machining and measurement, micro-assembly, biological processes, manipulation, and other applications [1; 2; 3; 4; 5; 6; 7; 8]. For a scanning system, a parallel kinematic piezoelectric actuator (PZT) XYZ scanner was proposed, which could achieve high bandwidth and low coupling errors [9]. A new two-degree-of-freedom compliant parallel micromanipulator (CPM) utilizing flexure joints was proposed for two-dimensional nanomanipulation [10]. The XYZ micromanipulator has applications that include moving and positioning microcomponents, such as mirrors, lenses, and gratings. It can position microcomponents in the X-, Y-, and Z-directions by using three independent linear inputs and has achieved out-of-plane displacements [11]. In consideration of the wide application demands in many areas, micropositioning stages are the core equipment for realizing micropositioning. In recent decades, it has achieved rapid development and plays an important role in today's engineering applications. For example, the parallel kinematic XY flexure mechanism was designed based on systematic constraint patterns for realizing large ranges of motion without causing over-constraint or significant error motions [12]. A novel flexure-based mechanism with three degrees of freedom driven by three piezoelectric actuators focused on achieving kinematics with X-direction motions decoupled from those in the Y- and \(\theta\)-directions [13]. The piezo-actuated XY stage with integrated a parallel, decoupled, and stacked kinematics structure was used for micro/nanopositioning applications and a novel flexure-based XY micropositioning stage driven by electromagnetic actuators with high moving range and resolution was also produced [14; 15]. Based on the above analysis, numerous positioning stages applied in various fields with various types and functions have received wide attention and can be found in recent research [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. In particular, two-degree-of-freedom (2-DOF) micropositioning stages meet most requirements of practical applications. A micropositioning stage based on flexure hinges can provide an output displacement with desirable properties for realizing micro/nanometer resolutions. Compared with conventional linkage mechanisms, flexure hinges possess inherent advantages, such as no friction, no backlash, repeatable motion, no need for lubrication, and a low cost [27; 28; 29]. Flexure hinges are utilized as elastically deformable rotation/translation joints for precision motion in micropositioning stages. It is important to achieve both a high natural frequency and motion accuracy. So far, there have been a large number of studies that have explored the performance of various flexure hinges with different structures [30; 31; 32; 33; 34; 35; 36], such as the leaf-spring hinge, cross-axis flexural pivots, and carstwheel hinges. Different flexure hinges have their own characteristics. The various flexure hinges are most often used in the positioning stage as a guide mechanism to make them ideal for wide use in scientific instruments. In the proposed structure of the stage, to simplify the design, the rectangular beam with different geometric parameters was combined to form different flexure hinges to reduce the complexity due to the excessive number of structural parameters. Meanwhile, utilizing multiple instances of the same parallel flexure hinge instead of a single hinge to connect two rigid links can enhance the performance of the stage [37]. Except for the factors in terms of the stage structure, actuators and materials, fabrication accuracy, and assembly errors, sensors and unknown disturbances will likely reduce the positioning precision of the micropositioning stage [38].