3。4。 Optimization algorithm: hybrid Taguchi-random coordinate search algorithm

To find the optimal value of each step, an optimization algorithm is required。 The objective function used in this paper is com- posed of an obvious equation, but it is too complex to analyze the function analytically。 This type of function is treated as a black- box function [20]。 To optimize the black-box function, derivative-free optimization (DFO) algorithms such as genetic algorithms (GA), simulated annealing (SA), or pattern search optimization (PSO) are applied [21]。 However, these methodologies are limited in that they sometimes cannot find global optimal points。 Rios and Sahinidis [21] suggest that a better solution to this kind of problem may be a combination of two algorithms。 Thus, the hybrid Taguchi-random coordinate search algorithm (HTRCA) is ap- plied [22]。 HTRCA combines the Taguchi method (TM) with the random coordinate search algorithm (RCA) to find the optimal solution  of  the  objective functions。

4。 Case studies

This section analyzes a number of cases that were investigated with the developed algorithm to validate the method。 For a performance comparison, the conventional method that minimizes  tracking error  is used。 The input  data set of the  given  points and their intervals is controlled for unprejudiced comparison。 The method minimizes the tracking error of the given data set and finds the optimal mechanism。 Moreover, the method developed in this paper derives the optimal mechanism to minimize the objective function with the reference slope and its change in angle of slope, which are estimated from the given data set。

4。1。 Case 1–1: type-I (equal intervals)

The  desired trajectory  of  the first  case is a mathematical  ellipse described  as follows:

1000 coupler points were chosen to identify the optimal solution that can trace an ellipse with a = 100 and b = 50。 Because the purpose of this case study is to find the shape and size of the trajectory, the position variables x0 and y0 are not needed。 Therefore, for this problem,

Target curve:

Cx;D  ¼ 100 cos θ ð11Þ

Cy;D ¼ 50 sin θ ð12Þ

θ ¼ i=1000; i ¼ 1; …; 1000

Constraint conditions: Desired shape: Type-I

Xsize ∈ ð 0; 500 ]

½ l2      l3      l4 ]=l1 ∈ ð 0; 10 ]

。 lcx       lcy 。=l1  ∈ ½—10; 10]

4。2。 Case 1–2: type-I (variable intervals)

This case is also an ellipse described by Eq。 (10)。 In some real situations, the velocity of each section needs to be considered。 Therefore, the velocity profile of the coupler points is modified from the first case。 The other conditions are the same。 For this problem,

Target curve:

Constraint conditions: Desired shape: Type-I

Xsize ∈ ð 0; 500 ]

½ l2      l3      l4 ]=l1 ∈ ð 0; 10 ]

。 lcx       lcy 。=l1  ∈ ½—10; 10]

4。3。 Case 2–1: type-III (equal interval)

The desired trajectory of case 2 looks like a crescent, as shown in Fig。 7 (A)。 The desired velocity in all sectors is constant。 For this  problem,  the target  curve  and constraint conditions  are  as follows。

Target curve:

The value of parameter composing the shape of the desired trajectory is demonstrated in Fig。 7(A)。 1000 coupler points  in  total are  positioned equidistantly。