# Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage
## 1 Introduction  
Uncertainties such as tolerance, backlash, and clearance can affect kinematic and dynamic characteristics of a mechanism, especially for a steering linkage [1]. In previous studies, the uncertain parameters affecting steering linkages have been quantified [1], while the robust design was an alternative for synthesis of the mechanism [2]. The first study presented an effective surrogate model for reliability analysis wherein it was extended to time-dependent reliability analysis [1]. Previous work applied the first-passage approach to handle the time-dependent reliability analysis [3]. In reliability analysis of mechanisms, it is difficult to collect the uncertainty data due to geometrical properties [4], flexible components [5], and joint clearance within the model [6]. The complete evaluation of uncertainty for all possible realizations of mechanism motion within a specific condition caused by the useable mechanism is needed [1]. Kinematic reliability analysis of steering mechanisms has been proposed for sensitivity analysis and optimization design [2]. An alternative method of reliability analysis is MonteCarlo simulation (MCS), which is computationally time-consuming. This kind of technique is called the probabilistic analysis. This technique is straightforward but requires thousands of analyses for an optimization run, which is impractical in most cases. As a result, to improve computational burden, the surrogate models as mentioned in many research papers [2, 7] have been used to estimate function evaluations. Meanwhile, research in the field of topology optimization has been presented in [8], which depends on uncertainty of material properties, external loads, and other parameters. The uncertain parameters can result in impractical optimum design [9]. Recently, there are twomain strategies to address the problem while accounting for uncertainties in topology optimization. The first technique is robust topology optimization (RTO) [10] and the second is reliability-based topology optimization (RBTO) [11]. The robustness or reliability can be based on the probabilistic model or non-probabilistic model [10]. The probabilistic model is a more popular choice than the non-probabilistic model due to its fast progress. Unfortunately, this model requires more precision of objective information to make the statistical distribution of uncertainties in the primary design stage similar with the robust design of mechanisms. On the other hand, the well-known examples of the second model are the convex set [12], the fuzzy set method [13], and anti-optimization with fuzzy set [14]. The fuzzy set model is a proven alternative selection method for collection of the uncertainties in optimization. This is because it provides moderate conservative results and it is acceptable and simple to combine with other techniques. Such techniques include anti-optimization [14] and a target performance-based design approach [9]. The disadvantage of the anti-optimization and fuzzy method is complexity in analysis due to the double-loop nested problem where optimization and anti-optimization are solved at the same time [14]. Similar to RBTO with a fuzzy set model, it causes triple-loop nested problems due to the additional topology optimization design stage. The complexity has been reduced by using the target performance-based design approach, which can reduce the triple-loop nested problem to a double-loop nested or single-loop problem [9; 15]. Thus far, no technique can be applied to solve the nested problem of the reliability analysis optimization design of a steering linkage, and therefore our present work aimed to reduce the complexity of the double-loop nested problem in the reliability-based design optimization (RBDO) using a multi-objective optimization technique with the worst-case scenario and fuzzy uncertainties. Conventionally, RBDO and uncertainties can be categorized into two groups depending on uncertainty classification. They are aleatory uncertainty and epistemic uncertainty.