## 1 Introduction  
In recent years, various high-precision applications have been produced and applied to prompt the micro/nano-technology to develop rapidly, such as adjustable posture institution of microwave antenna, scanning tunnel microscopy, bionic mechanism, micro-gripper, etc. [1; 2; 3; 4; 5]. A compliant mechanism with flexure hinges is regarded as one of the best and has been implemented in these fields due to its incomparable merits including no friction, lubrication and backlash, reduced weight, low clearance, and being easy to manufacture, integrally. For specific micro-technology applications, the micro-manipulator should have the ability to amplify the output displacement of lead zirconate titanate (PZT) in order to meet the actual requirements [6; 7]. In the literature, research results have been published on every aspect of the compliant mechanism. Some works are dedicated to the design and verification of the mechanical configurations. A flexure-based compliant mechanism with three degrees of freedom (3-DOF) is presented in [8]. A parallel compliant mechanism with two axis linear motion, for ultra-precise motion, is designed in [9]. By employing flexure hinges, a micro manipulator with six degrees of freedom (6-DOF) as a fiber optics aligner is presented in [10]. The design of an XY novel flexure-based stage is proposed in [11] with high placement precision. A compliant stage with high scanning speed and a relatively large range is presented in [12]. A novel parallel-kinematic XY micro-positioning system is designed with the characteristics of a large range of motion in [13]. Another flexure-based parallel manipulator is designed with 3-DOF motion in [14]. Some researchers are focused on the quasi-static properties of the micro-manipulator. An analytical model for a classic compliant mechanism is proposed in [15]; by using this model the amplification ratio, input and output stiffness can be calculated. Another mathematical model is proposed in [16] to calculate the amplification ratio by using kinematic theory and elastic beam theory. A similar work is presented in [17]; the authors discuss several previously established models and set an improved analytical model based on Euler-Bernoulli beam theory, but the result does not match well with that of FEA. The matrix method for investigating the compliant mechanism is presented in [18; 19]. The authors in [20] designed a micro-manipulator, with multi-lever and flexure hinges, and proposed equations to assess the properties of the mechanism. A two-step amplifier is designed in [21], and [PERSON]'s equation is used to calculate its performance. On the other hand, researchers are also interested in the approaches to build the stiffness analytical equations of flexure hinges. The spatial stiffness matrix is built in [22] by using the finite element method. The regular circular hinge is regarded as a beam element with three nodes in [23] for the purpose of enhancing the calculation accuracy. In [24], the flexure hinge is taken as a varied cross-section in-plane beam element with two nodes. A similar work is presented in [25]; a high-order Hermite multinomial is utilized to build shape functions of three nodes of a non-prismatic beam element. The authors in [26] presented the exact shape functions through the governing differential equations, but this method has narrow application scope. To better acquire the satisfactory comprehensive performance of the mechanism, the studies of multi-objective optimization have been introduced to solve this problem. A hybrid technique based on differential evolution algorithms is introduced to solve multi-pass turning optimization problems in [27]. In [28], a new particle swarm-based optimization method is employed to optimize vehicle crashworthiness. A new hybrid optimization approach, based on the particle swarm optimization algorithm andreceptor-editing properties of immune systems, seeks to solve problems in both design and manufacturing areas in [29].