###### Abstract  
In precision engineering, the use of compliant mechanisms (CMs) in positioning devices has recently bloomed. However, during the course of their development, beginning from conceptual design through to the finished instrument based on a regular optimization process, many obstacles still need to be overcome, since the optimal solutions often lie on constrained boundaries or at the margin of safe/unsafe domains. Accordingly, if uncertainty occurs during the fabrication or operation of the mechanism, it might lose its functions, rendering the design infeasible. This paper proposes a universal design process for positioning CMs, consisting of two steps: optimal design of the pseudo-rigid-body model, and reliability-integrated multi-objective optimization design using NSGA-II algorithms. This optimization algorithm is applied in the design of a feed drive mechanism for micromachining. The optimal design is also fabricated and tested. The results calculated for the displacement amplification ratio, natural frequency, and input/output stiffness using different approaches, including analytical methods, simulations, and experiments, were compared to evaluate the efficiency of the proposed synthesis method, and show discrepancies of less than 5%. Thus, the results convincingly support the applicability of the proposed optimization algorithm for the design of other precision-positioning CMs prone to failure in vulnerable conditions. capulant mechanism; feed drive mechanism; NSGA-II algorithm; reliability-based optimization; FEM; FORM +
Footnote †: journal: Applied Sciences  
Among the various applications of CMs, their use as high-precision positioning mechanisms has received the most interest [18]. Researchers in this field usually use bridge-type [19], lever-type, L-shape levers, and half-bridge [18] or their combination [20] to design the motion transmission platforms. Figure 1 illustrates two examples of positioning CMs using multistage displacement amplifiers, wherein rigid links and flexure hinges are integrated in their configurations. Shape and size optimization approaches are often used as part of the design methodology for this type of structure. Previous studies have usually focused on the dimension of the hinges to maximize the amplification ratio. Although the shape and size of the elastic joints are essential factors in assuring the deformation capacity, the length of the rigid links determines the displacement range, parasitic motion, and amplification magnitude. Selecting adequate design parameters for specific purposes is a persistently challenging task. Due to the multitude of parameters, in earlier studies the selection of design variables (DVs) to formulate the problem has often been dependent on the experience of the particular researchers involved. In the optimization process, it is also often assumed that the dimensions of the design parameters, as well as the operating conditions of the mechanism after fabrication, are ideal. For applications where high accuracy is required, any reckless variation in the fabricated structure, compared with the theoretical design, may result in an unignorable impact on the performance of the instrument. Prac  
Figure 1: Pseudo-rigid-body of (**a**) a compliant 2-DOF stage for locating biomaterial samples [21], and (**b**) a large-stroke nanopositioning stage [22]. ing also show that in many realistic input parameters, there is random fluctuation according to a specific probability distribution function. Examples of the problematic uncertainty in parameters include fluctuations in cutting forces, inhomogeneous material properties, tolerance of fabricated parts, assembly accuracy, as well as the imperfect constraints of the mechanism. This consequently leads to inconsistency in the output performance of the designed instrument. In some unexpected scenarios, this may exceed the predetermined allowable limit. The likelihood for this jeopardy to occur is called the unsafety or failure probability of the structure.