Double global optimum GA-PSO algorithm has two global optimal solutions: one is the best particle of PSO (gbest) and the other is from GA (GAbest). GA is

used to search another global optimal solution, which is used for particle update. Hence, the update of particles will be influenced by three parameters: pbest, gbest, and GAbest. It will increase the persity of particles and enhance the ability to find the optimal solution because gbest and GAbest are not  relevant.

The Eqs. (4.1) and (4.2) is the particle update equation for improved GA-PSO. Compared with the basic PSO, GAbest from GA is added in the Eq. (4.1). Thus, the particles are always updated according to the gbest, and GAbest. Mutation operator in GA will help the algorithm obtain global optimal solution, because it can help particle jump out of local optimal solution  greatly.

4.2.2 Partition PSO

The basic discrete PSO has a good ability to find optimal solution for small-scale problems. But usually there are a lot of welding joints in the welding task, and it is hard to get the optimal solution using basic discrete PSO. Hence, the partition-PSO will be used to solve large-scale welding joints path planning problem   here.

For N-dimensional welding path optimization problem, the number of the pos- sible solutions is ðN — 1Þ!=2, when N = 20, there will be more than 1016 possible solutions. For larger scale, the number will grow explosively. If the N-dimensional problem  is  pided  into  several  small-scale  problems,  which  are  described as

n1; n2; .. .nk , the number of the possible solutions will be  ðn1 —1Þ! þ  ðn2 —1Þ! þ ···þ 

2 2

 ðnk —1Þ!

2     , the search space will be reduced greatly. Hence, partitioning is an  effective

way for large welding joints path planning  problem.

Partitioning principle for particles is based on the distance between different particles. Particle Xi ¼ ðxi1; xi2; .. .xiDÞ will be pided into k zones as Xi ¼ ð½xi11 ; xi12 ; .. .]; ½xi21 ; xi22 ; .. .]; .. .; ½xik1 ; xik2 ; .. .]Þ according to the partition principle. The first step of partition is to find the boundary points, and define k reference points from the boundary points. In order to ensure that the welding joint can get a reasonable allocation, these k reference points should be defined as decentralized as possible. Based on the above operation, all the elements in particle i can be pided into k zones. Then, PSO algorithm is used to obtain optimal welding joints sequence for different zones separately. In order to ensure the global optimal, the distance between adjacent areas in the zones should be shortest; it means that the last element in a zone and the first element in the next zone should be the nearest. Figure 4.1 is the schematic of partition-PSO  algorithm.

Fig. 4.1 Schematic of partition-PSO algorithm, a welding joints distribution, b result of partition- PSO algorithm

4.3 Welding  Path Optimization

4.3.1 Definition of Welding Robot Path Planning

Welding joints sequence planning is an important issue of welding robot produc- tion, and how to find an optimal path to go through all the welding joints is the most concern. In general, there are many solutions for welding joints sequence planning, and one optimal path will be selected based on certain criteria. Hence, welding robot path planning problem can be described as obtaining a reasonable welding joints sequence for welding robot under some criteria. The criteria may be the shortest path, the least time consuming, the minimum welding deformation, or the minimum energy consumption, and so on. Therefore, the robot path planning is a constrained optimization problem. In this paper, the shortest path is used as the criterion to optimize the welding path for verifying the effectiveness of the opti- mization algorithms.