The working environment of the robot is established through a grid model, which can be divided into N × N squares, as seen in Figure 1. The black grid represents obstacles which are impassable, and the white grids represent feasible passing free areas, denoted as 0 and 1, respectively. The five-pointed star indicates the starting point and the green point is the target point. Then the grid model is placed in the coordinate system so as to establish the robot working environment.

It can be seen from Figure 1 that the model is easy to construct, represent, and store data for processing and it is convenient for computer processing.

The PSO algorithm is an intelligent optimization algorithm proposed by Kennedy and Eberhart (1995) based on the study of the living habits of animal flocks (Tang et al., 2020). Suppose the optimal solution of a certain problem exists in D dimensional space for a swarm with size m, and the population can be expressed as, Swarm = {x₁, x₂, …, x_m} where x_i(i = 1, ⋯, m) is the particle without mass, k represents the total number of the required iterations, and the position information of the ith particle in the kth iteration can be represented by a d-dimensional vector xik=(xi1k,xi2k,…,xidk),i=1,2,…m, the velocity of each particle can be represented as vik=(vi1k,vi2k,⋯,vidk), i = 1, 2, ···, m. In each iteration, the position and velocity of the particles are dynamically adjusted according to the historical optimal fitness values of each particle and the population. The calculation for the (k + 1)th iterations of the ith particle in d-dimensional space can be written as,

where c₁ and c₂ represent the learning factors. c₁ and c₂ are the control variables to control the step lengths of the individual particle flying toward the local optimal value and the swarm optimal value, respectively. pidk  is the historical optimal fitness value of each particle in the optimization process, pgdk is the optimal fitness value reached by all particles, that is, the optimal fitness value of the population; the rand() function is to generate a random number between (0,1) to differentiate particles. The subscript d(1 ≤ d ≤ D) represents the dimension of the searching space. In the above Equation (1) and (2), the speed of the PSO is composed of the local and global three parts: vidk represents the speed of the particle at the k^th iteration, {c1∗rand()∗(pidk−xidk)} represents the information of the particle itself, and {c2∗rand()∗(pgdk−xidk)} represents the part of the particle in the population for collaboration and information sharing.

The PSO-GWO algorithm is an improved PSO version incentive inspired by gray wolf predation (Narinder and Singh, 2017; Teng et al., 2019; Gul et al., 2021). Different from bird flocks, the gray wolf pack is quite a strict socially hierarchical organization; its hierarchical arrangement is illustrated in Figure 2.

The first layer α in Figure 2 represents the leader in the population, where the leader α is the core of the wolf pack, being mainly responsible for leading and assigning tasks.

The second layer β in Figure 2 represents the think tank team, which is used to assist the leader in management, i.e., when leader α is vacant, β will quickly take over the position of α. In the entire wolf pack, the status of β is only lower than that of α. If α is occupied, β is an advisor to α and discipliner for the group.

The third layer δ follows the command and management of leader α and think tank, and are mainly responsible for care and supervision.

The function of the fourth layer, ω, is to balance the membership within the population.

The essence of the GWO is that the particle with the highest fitness is taken as the leader α to manage other particles. The specific steps of the GWO are summarized as follows:

As seen in Equations (3)–(5), the dimension of the spatial solution is d, and the current number of the iteration is k. c₁, c₂, and c₃ represent the learning factors, rand() is a random number between (0, 1), and w represents the inertia weight coefficient. The larger w makes the algorithm better in global search, in contrast, the smaller w is more suitable for local search. The core idea of the PSO-GWO is to arrange the particles according to their fitness values during each iteration, and set the three particles with the best fitness values to α, β, and δ, while these three particles can predict the approximate range of the location where the optimal solution may exist, and the remaining particles can search the optimal solution within the predicted range. In such a way, the particles can find the optimal solution more quickly and effectively with the improved convergence performance, so the path planning ability of the PSO algorithm can be improved accordingly.

In the previous section, GWO was added to PSO to form the PSO-GWO algorithm, and the search ability of the PSO algorithm can be enhanced to improve its path planning ability, but the PSO algorithm still has the problem of premature convergence, and its convergence speed and global search ability can be further strengthened. Hence chaos and a new adaptive inertia weight are added to provide solutions for these problems.

The PSO can randomly distribute particles while the optimal solution is highly related to the particle initialization. The more uniform the initial particle distribution, the richer the diversity of the group, and the faster the optimal solution can be obtained.

Chaos (Demir et al., 2020; Lian et al., 2020; Lu et al., 2020; Wu et al., 2020; Guo et al., 2021; Ouertani et al., 2021) refers to a nonlinear motion that can traverse all situations within a specified range. A chaotic sequence can represent all states in a prescribed space, which is commonly generated by mapping. Many researchers have found that chaotic mapping has unpredictable characteristics when studying chaotic mapping relations. Although it is somewhat unpredictable, certain laws can still be used in the mapping process. The most commonly used form of chaotic mapping is logistic mapping, as shown in Equation (6):

In Equation (6), 0 ≤ Z₀ ≤ 1, Z_i is the value obtained by i times Logistic mapping of Z₀, and μ represents the control variable. When μ = 4, the system is within a completely chaotic state and the range of the chaotic space is [0, 1].

The steps of using chaos to initialize the PSO are summarized as follows. First, an n-dimensional vector Z₁ = (z₁, z₂, …, z_n) is randomly generated, and Equation (6) is used to map the other vectors so as to generate a chaotic sequence Z₁, Z₂, …, Z_N. Then the chaotic sequence z_i is inversely mapped from the chaotic space[0, 1] to the space [a, b] where the optimal solution is located, and the particle position is x_ij = a + (b − a)z_ij, j = 1, 2, …, n, i = 1, 2, …, N. Finally, the particles with higher fitness values are determined as the initial particles of the population.

When the PSO is trapped in the local optimum, the algorithm will select the historical optimal value of the particles in the iterative process and convert it into a chaotic sequence through inverse mapping to obtain the optimal position of the particle, then randomly replace a certain particle position in the current search space so that the local optimum can be jumped out by the algorithm. Whether particles fall into precocity is determined by the variance of the population fitness, calculated as,

where n is the size of the population, f_i represents the adaptability of the first particle, and f_avg represents the average adaptability of the current swarm particles. The population variance σ² reflects the precocious state of the particles. When σ² is less than a certain threshold, it is calculated that the particle algorithm will fall into precocity. Then the chaos is applied to process the optimal particles to increase the diversity of the population. The detailed steps are described as follows.

After the particle swarm performs the chaotic initialization operation, the particles are more evenly distributed in the search space, and the chaotic sequence can be used to reduce the prematurity, improve the diversity of particles, and enhance the convergence speed of the algorithm.

It is known that the quality of PSO is closely related to inertia weight where the local search ability of the algorithm is higher with smaller inertial weight and global search capability is stronger with larger inertial weight. To enable the algorithm maintaining higher search ability during the entire operation process, many methods have been proposed to adjust the inertia weight (Li et al., 2019a,b; Gopal et al., 2020; Wang et al., 2020; Wang, 2021; Zhang et al., 2021). However, the current inertia weight improvement methods have a close relationship with the iteration number and cannot adapt to the nonlinear variations well. For this reason, this paper deals with the inertia weight via the particle adjacent fitness values. The inertia weight can be updated and calculated as,

In Equation (8), the global optimal fitness of the kth iteration is pgbestk, and the global optimal fitness of the (k − 1)th iteration is pgbestk-1; the maximum ω_max and minimum ω_min of ω is set as 0.9 and 0.4, respectively. k is the current iteration number; MaxIter represents the maximum number of the iterations. It can be seen from Equation (8) that a is larger at the beginning of the iteration, so the algorithm has strong global searching ability, and a becomes smaller gradually at the later iteration stage, so the algorithm has strong local search ability. In summary, the inertia weight combined with the fitness ratio of the neighboring particles can adaptively adjust the size of w with the number of iterations so that the algorithm has a higher global search ability.

The steps of the IPSO-GWO algorithm for path planning are summarized as follows:

To verify the superiority of the IPSO-GWO algorithm, this paper uses MATLAB R2018b software to perform benchmark function tests on PSO and IPSO-GWO. The variables of the simulation experiments are set as follows: the population size is 50, the dimension of the optimization variable is 4, the learning factor c₁ = c₂ = 2.05, and the test functions of the simulation experiments are ten benchmark functions such as Drop Wave, Peaks, Rosenbrock, etc. For Rosenbrock function, the number of the iterations is 200, and the number of iterations of others is 100. Simulation experiments are performed on the above functions. The algorithm iteration curve is shown in Figure 3. The experimental results of the three test functions are analyzed and compared, as listed in Tables 1–10.

It can be seen from Figure 3 that, compared with the PSO and PSO-GWO algorithms, the IPSO-GWO algorithm converges the fastest. From Tables 1–10, it is seen that the results obtained from IPSO-GWO is closer to the global minimum, which verify that the performance of the IPSO-GWO algorithm is higher than those of the other algorithms.

To verify the superiority of the IPSO-GWO optimization in robot global path planning, we have carried out two kinds of path planning simulation tests: one is the test of IPSO with PSO, and the other is the test of IPSO with Genetic algorithm (GA) and Ant Colony Optimization (ACO). Both tests use 20 × 20 and 30 × 30 map environments. For IPSO-GWO and PSO, the population size is set to 50 and c₁ = c₂= 1.6. For GA, the crossover probability is set to 0.8, the mutation probability is set to 0.2, and the population size is set to 50. For ACO, the stimulating factor of the pheromone concentration α is set to 1, the stimulating factor of visibility β is set to 7, pheromone evaporation coefficient ρ is set to 0.3, pheromone intensity is set to 1, and the number of iterations of the four algorithms is set to 200. The experimental results are illustrated in Figures 4–7.

From Figure 4, IPSO-GWO and PSO simulate the path and iterative convergence curve of the path planning in a 20 × 20 map environment; it can be concluded that the PSO can obtain the optimal path at the 138th iteration with path length 30.36. Whereas the proposed IPSO-GWO can acquire the optimization at the 75th iteration with the obtained path length 28.63.

From Figure 5, IPSO and PSO simulate the path and iterative convergence curve of the path planning in a 30 × 30 map environment. The PSO algorithm searches for the optimal path at the 169th iteration, and the obtained path length is 45.36. The proposed IPSO-GWO can find the optimal path in the 86th iteration with acquired path length 42.77.

From Figure 6, IPSO, ACO, and GA are used to simulate the path and iterative convergence curve of the path planning in a 20 × 20 map environment. It can be seen that IPSO-GWO can acquire the optimal path length of 28.63 in the 11th iteration, the optimal path length found by ACO in the 62th iteration is 29.21, and the optimal path length found by GA in the 22nd iteration is 29.21. It can be concluded that the proposed IPSO-GWO algorithm converges faster in a 20 × 20 map environment with shortest path acquirement.

From Figure 7, IPSO, ACO, and GA are used to simulate the path and iterative convergence curve of the path planning in a 30 × 30 map environment. It can be seen that IPSO-GWO can find the optimal path length of 42.77 in the 86th iteration, the optimal path length found by ACO in the 166th iteration is 42.77, and the optimal path length found by GA in the 33rd iteration is 45.11. It can also be concluded that although GA finds the optimal path faster, the path length is longer, whilst ACO finds the same optimal path as IPSO-GWO, but it is slower than IPSO-GWO. In summary, IPSO-GWO algorithm has the highest performance efficiency.