Considering a welding joints welding task C ¼ ðc1; c2; .. .cN Þ, the distance between two welding joints can be described as dðci; cjÞÇ 0, where ci; cj 2 C(1 ≤ i, j ≤ N), ci stands for a welding joint. The task in this paper is to find the best welding joints sequence p ¼ fc1; c2; .. .cN ; g to make sure the total path length of the welding robot is minimal.

N—1

f ðpÞ¼ X dðci; ciþ1Þ þ dðcN ; c1Þ ð4:3Þ

i¼1

where f ðpÞ denotes the distance when the welding path is p.

4.3.2 2D  Welding  Path Planning

Above algorithms are used for a two-dimensional path planning problem where 40 welding joints exist. Figure 4.2a shows the comparison of the convergence curves of GA, PSO, partition-PSO, and GA-PSO. It can be seen that the curve based on GA-PSO converges much faster than any other algorithm. Figure 4.2b is the path planning result of GA-SPO algorithm. The optimization results and convergence speeds of various algorithms are given in the Table 4.1. The search capability of basic PSO is better than basic GA in both best solution and convergence speed. The 40 welding joints are not a large-scale optimization problem; hence, basic PSO has a good effect, and the result of PSO is very close to the true optimal solution. Because the partition PSO is a local optimum algorithm designed to solve the large- scale problems, the result of partition PSO is worse than the basic PSO. GA-PSO combines the advantages of GA and PSO, so it converges faster than the basic PSO.

4.3.3 3D  Welding  Path Planning

Figure 4.3a is a car door with 115 welding joints, and Fig. 4.3b is the 3D coor- dinates of the welding joints. Figure 4.3c is the convergence curve comparison   of

Fig. 4.3 Simulation results display, a welding joints of a door, b 3D coordinates of the welding joints, c comparison of convergence rate curves, d path planning result of   GA-PSO

GA, PSO, partition PSO, and GA-PSO. Figure 4.3d shows the path optimized based on GA-PSO. Table 4.2 shows the comparison of path planning results, and GA- PSO is better than other three algorithms. 115 welding joints is a large-scale path planning problem, the shortcoming of prematurity of GA is completely revealed. The optimal solution of GA is the worst among all the algorithms, and the con- vergence speed is the slowest as well. PSO is also easy to fall into local optimum when it comes to more welding joints, but it is still better than GA. Partition PSO shows its advantage when there are many welding joints, all the welding joints are pided into four zones to search optimal path in this case, which increase its optimization capability greatly. GA-PSO remains its good characteristic, the persity of particles are greatly increased after the optimal solution of GA was used

Table 4.2  Path planning results of four algorithms

Optimal solution (cm) Convergence speed

GA 744.1097 2,000

PSO 512.4208 1,400

Partition PSO 479.8333 220

GA-PSO 467.0721 50

as another global optimal position. Hence, its global search capability is greatly improved, and the best result was obtained based on  GA-PSO.

4.3.4 Performance Comparison

Although GA and PSO are global convergence intelligent algorithms, both of them have the drawback of prematurity, which means they are easy to fall into local optimum, especially in a large search space. They can be used to solve small-scale path planning in a welding task. Partition PSO pides large search space into several small ones, and simplifies the complexity problem. However, when the partition is established, the exchange of two zones is limited to the two connection points, and points inside of each zone are separated. Hence, the partition PSO is a local convergence intelligent algorithm, the advantage of partition PSO can be shown in more welding joints path planning. The GA-PSO proposed in this article is a combination of GA and PSO, the persity of particles is greatly increased, and the global search capability is improved as well. It is suitable for both small welding joints case and more welding joints case. Besides, after effective mutation operator was applied in the partition PSO algorithm to make the algorithm jump out of local optimal solutions, the better optimization results were obtained  [11].