# Optimization on Linkage System for Vehicle Wipers by the Method of Differential Evolution
## 1 Introduction  
Linkage systems are widely used on vehicles. Research on the design of linkage systems is important for vehicle industry development. In this article, we consider the linkage systems for vehicle wipers. Our goal is to improve the _steadiness_ of a linkage system for vehicle wipers without weakening its normal function. This improvement on the _steadiness_ of a linkage system for vehicle wipers could reduce the possible material fatigue of a linkage system. Besides, this improvement could also be helpful for reducing the unpleasant rubber wiper noise on vehicle windshield. The linkage systems considered in this article are the center driven linkage systems (CDLS). A center driven linkage system is a composition of two crank-rocker linkage systems driven by a single central input-link. A crank-rocker linkage system is shown in the following Figure 1.  
In Figure 1, the link OB is a fixed link with length \(q\). The link OA, with length \(s\), may rotates full 360 degrees and is called the \"crank\" of this linkage system. Usually the crank is driven by a motor. The link BC, with length \(p\), may only moves on a limited circular sector and is called the \"rocker\" of this linkage system. The link AC usually has the longest length \(l\) and acts as the transfer link of this linkage system. Generally the Grashof law  
\[s+l<p+q\]  
for crank-rocker system must be satisfied. See [1] for more discussions on linkage systems. To improve the _steadiness_ of a linkage system without weakening its normal function, we will consider an optimization problem on the maximal magnitude of _angular acceleration_ of the output-links of a CDLS under the assumptions that the frame of the fixed links of linkage system is unchanged and that the input link rotates at the same (constant) angular speed with its length unchanged. To meet the usual requirements for vehicle wipers on windshield, this optimization problem must be solved subject to 10 specific constraints. To tackle this optimization problem, we will use a Differential Evolution type search method. The usual stages of a DE search algorithm is shown in the following Figure 2.  
The method of Differential Evolution (DE) was introduced by [PERSON] and [PERSON] in 1995 [2; 3; 4]. In this DE search method, two important parameters \(F\) (the _scale factor_) and \(CR\) (the _crossover ratio_), for the mutation and crossover operations of DE, can be selected respectively from the interval \([0,1]\). These parameters are important both for controlling the _diversity_ of \"generation population\" and for controlling the _convergence rate_ of DE. The space complexity of DE is usually relatively low, when compared with the highly efficient \"restart covariance matrix adaptation evolution strategies (restart CMA-ES)\". Besides, the control parameters \(F\) (the _scale factor_) and \(CR\) (the _crossover ratio_) can be adjusted to improve the performance of the DE algorithm easily _without_ causing serious _computational burden_. Moreover, the introduction of mutation operator by \"difference vectors\" in DE obviously speed up the computational convergence of DE. Because of its simplicity and its experimental efficiency, this DE search method soon has become very popular. Since then, various variants of DE were presented. For examples, there are DE using \"Trigonometric Mutation\" [5]; \"opposition-based learning\" DE [6; 7; 8; 9]; DEGL with Neighborhood-Based Mutation [10]; SADE (DE with parameters adjusted through self-adaptive learning) [11; 12; 13; 14]. See [15; 16; 17] for more information on the various variants of DE. Though various complicated types of DE methods were presented, research articles on the theoretical analysis of DE are rare. Most of the variants of DE mentioned above were presented only with experimental studies. For classical DE algorithms, \"weak convergence\" for most objective functions were proved under reasonable assumptions in [18; 19; 20].