###### Abstract
Determining the failure probability of a system, due to uncertain external factors, is referred to as the reliability design problem [23]. For reliability-based design optimization (RBDO) problem solving, two calculation cycles are required: the optimization cycle and the reliability evaluation cycle. Therefore, the computational cost is one of the major concerns. The most commonly used methods to solve reliability analysis problems are the first-order reliability method, second-order reliability method [24], Monte Carlo simulation method [25], double loop method, decouple double loop method, and single loop deterministic method [26]. While reliability analysis optimization for CMs has also recently received interest, there is still little research on this topic [27; 28; 29]. In these studies, problems are usually treated via explicit mathematical functions [30]. The Hasofer-Lind approximation method is appropriate for linear functions, but not suitable for high-order functions. The Monte Carlo simulation (MCS) method [31], which evaluates the limit state function of the data samples, is one of the most popular reliability analysis methods. It is simple to use, and the results are easy to analyze. However, it often requires many samples to ensure accuracy, leading to high computational costs. The first-order reliability method (FORM) is based on the Taylor series expansion of the limit state function, whose approximation may not be accurate for complex nonlinear limit state functions. Its applicability may also not be appropriate for discontinuous or multiple failure modes limit state functions. However, it can be integrated into a FEM tool to quickly evaluate any high-order nonlinear limit state functions, regardless of the need for an explicit mathematical model. Therefore, the FORM can overcome the drawbacks of the previous methods, and still effectively solve any complicated reliability-based problems. RBDO of implicit CM design problems that are difficult to mathematically formulate has not yet been extensively investigated. Thus, this paper aims to fill this gap. Regarding the optimal design algorithm used for synthesizing positioning CMs, in previous research, there has often been a distinction between the study of rigid links and the design of the elastic components. This paper proposes a reliability-based optimization method for their design that first optimizes a pseudo-rigid-body model (PRBM) via an analytical approach. The 2D model is then built in the Ansys ADPL language for the analysis of its behaviors by the finite element method. Subsequently, a multi-objective design optimization problem is solved using the nondominated sorting genetic algorithm II (NSGA-II). This research also integrates a reliability analysis task using the FORM into the design process to assess the impact of uncertainty in the practical input parameters, and implements this as an RBDO problem. In this stage, the constraints of the multi-objective design optimization problem are treated as limit state functions to evaluate the reliability of the obtained optimal solutions. The above combined design method is applied to design a compliant micropositioning feed drive mechanism (CFDM). In this paper, its natural frequency and displacement functions are also analyzed using the PRBM. Eventually, the optimal design is machined by wire EDM, with experiments conducted to evaluate the fabricated prototype. The output displacement is tested in a LabVIEW 2014 closed-loop control system, whose PID parameters have previously been determined using a single-objective genetic algorithm (GA) in MATLAB 2014a.