# Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage
## 1 Introduction
The first one is caused by the randomness of a physical variation. On the other hand, the second one is due to the lack and incomplete of knowledge [16]. In general, the aleatory uncertainty is handled with the probabilistic model, but epistemic uncertainty is modeled with a convex set, a fuzzy set method, and anti-optimization [15]. Next, state of the art steering linkage synthesis is introduced. A planar steering linkage works by applying a single input at a rack and pinion to drive two steering arms and tie rods as shown in Figure 1. It is a commonly used steering system in passenger cars, called a central take-off (CTO) type [2]. The design objective of this kind of mechanism is to avoid skidding and wear, which can be achieved by following the Ackermann principle. According to this principle, all lines passing the front wheel axle should meet at the line passing the rear axle, causing an instantaneous center, as shown in Figure 2. By using trigonometric relations, the outer wheel angle, \(\theta_{OA}\), can be expressed as a function of the inner front wheel angle, \(\theta_{I}\). Figure 1: Central take-off planar steering linkage schemes (reproduced from [17]). \[\theta_{OA}(\theta_{I})=\tan^{-1}\frac{1}{\cot\theta_{I}+W_{t}/W_{b}} \tag{1}\]  
where \(W_{t}\) is the wheel track, and \(W_{b}\) is a wheelbase. In principal, there is no mechanism that can totally follow the Ackerman principle at every angle including the four-bar and six-bar linkages. Many researchers have paid attention to finding the solution for minimizing the steering error (STE) between the desired and practical angles of the outer front wheels during turning. The objective of optimum synthesis of a steering linkage using STE has been proposed in [18]. The definition of STE was defined as  
\[\delta\theta_{O}(\theta_{I})=\left|\theta_{O}(\theta_{I})-\theta_{OA}(\theta_ {I})\right| \tag{2}\]  
where \(\theta_{O}\) and \(\theta_{OA}\) are the actual and desired angles made by the same outer front wheel during turning, respectively. In this work, it is called angle distance (AD). The function was a traditional or reference objective function for the steering linkage synthesis until the work of [PERSON] and [PERSON] [19], who proposed the steering error in the form of the distance deviation from the instantaneous center. This is a concept of Euclidean distance (EUD). For more than a decade, researchers have tried to improve the steering mechanism by applying optimization techniques with the traditional objective function rather than reconsidering other functions that may result in a better design. The traditional function has been used in several works for single-objective optimization of the link-length sensitivity [2; 18; 20] or steering error [18; 21; 22; 23] or six-bar steering linkage [22; 23; 24; 25; 26; 27; 28; 29; 30] or topology optimization [31]. Then, the work by [PERSON] et al. [30] proposes the weighted sum method to combine STE and toe-in angle deviation during a wheel jumping. As previously mentioned, only one work by [PERSON] and [PERSON] [19] has defined the steering error in the form called a cornering radius function [32]. Later, a new steering error was compared with the traditional one [17; 20], providing better optimization results [33]. Due to the robustness of multi-objective evolutionary algorithms (MOEAs) in solving the steering linkage design, which can perform only once in finding a solution set [17], they can be an alternative choice to gradient-based optimizers. It was shown that the performance of such a method for solving minimization of steering error and turning radius is acceptable [33]. A steering error in form of Euclidean distance (EUD) can be defined by a distance deviation rather than angle distance (AD) that is defined in (2).