2. Related works

Route planning techniques can be categorized as either global or local. Global path planning takes into consideration the provided map and searches for the overall transportation route, which is crucial for determining the best path rather than the amount of time needed for path calculation [Reference Pande, Parhi and Pratihar1]. On the other hand, a local path establishes a route by identifying obstacles in real time using a variety of sensors (such as cameras, LiDAR sensors, laser sensors, ultrasonic sensors, and sensors for sound and heat). According to Li et al. [Reference Li, Cao, Li, Huang, Wang and Liu2], a robot going through an unfamiliar ecosystem can estimate its own pose using position and map data, and it can even create a progressive map as it moves. This allows the robot to execute autonomous obstacle avoidance and navigation. The motion trajectory of the vehicle is composed of the moment coordinates x ₁, x ₂, and x_r that correspond to the discrete moments t = 1, 2, and r that make up the SLAM issue [Reference Parker3]. For the nonlinear system in the actual motion process, the motion model and observation model are as follows:

(1) \begin{equation}\begin{cases} x_{r}=f(x_{r-1}, a_{r},c_{r})\\ J_{r}={k}(x_{r}, b_{r}) \end{cases}\end{equation}

where $x_{r}$ stands for robot posture, j_r for the system’s observed value, f(x) for the robot motion equation and k(x) for the robot observation equation, $a_{r}$ for the robot control value, $c_{r}$ for the system’s process noise, and $b_{r}$ for the system’s observation noise. Elnabarawy et al. provide an overview of the SLAM algorithms for autonomous mobile robots and its various algorithms such as EKF-SLAM [Reference Elnabarawy, Elhafsi and Elhawary4], Hector SLAM, Fast SLAM, and Graph SLAM [Reference Niaz, Lee and Kim5]. Genetic algorithms were suggested as a path planning strategy by Pande et al. for mobile robots [Reference Pande, Parhi and Pratihar1, Reference Goudar and Deshpande6]. The algorithm generates a set of feasible paths and selects the best one based on their fitness function. Parker presented a survey of various path planning algorithms for mobile robotics, including grid-based, potential field, heuristic search, and machine learning-based methods [Reference Parker3]. The paper provides a comprehensive review of the state-of-the-art in the field. Peng et al. [Reference Peng, Li and Zhang7] proposed a path planning approach based on the artificial potential field method, which simulates attractive and repulsive forces acting on the robot to guide it toward the goal while avoiding obstacles. A B Wahab et al. [Reference Wahab, Nadhir, Lee, Akbar and Hassan8] presented a path planning approach for mobile robots in unknown environments using PSO. The algorithm optimizes a cost function based on the distance to the goal and the obstacle avoidance criteria. Arifin et al. [Reference Arifin, Matsui and Sugimoto9] proposed a path planning approach using ant colony optimization and fuzzy logic. The algorithm generates a set of paths based on the pheromone trail left by the ants and selects the best one using fuzzy logic.

A comparative analysis of evolutionary algorithms, such as genetic algorithm, PSO, and ant colony optimization, for path planning of mobile robots and analyze the performance of each algorithm based on various metrics [Reference Sathiya and Chinnadurai10]. Sharma and Singh [Reference Sharma and Singh11] proposed a heuristic-based navigation method for mobile robots using a hybrid fuzzy logic and artificial potential field approach. The proposed method combines the advantages of both methods to improve the efficiency and robustness of mobile robot navigation [Reference Sharma and Singh11]. Hao, He, and Sun [Reference Hao, He and Sun12] presented a new hybrid heuristic method for mobile robot path planning based on an improved artificial potential field algorithm. The proposed method integrates a virtual attractive force and a repulsive force based on an improved potential field algorithm to achieve efficient path planning [Reference Hao, He and Sun12]. Wang, Zhao, and Zeng [Reference Wang, Zhao and Zeng13] proposed an adaptive artificial potential field algorithm for mobile robot path planning. The proposed method adapts to the environment and the robot’s motion characteristics to achieve efficient path planning in complex environments [Reference Wang, Zhao and Zeng13]. Gao, Xu, Zhang, and Huang [Reference Gao, Xu, Zhang and Huang14] presented a path planning method for mobile robots based on an improved ant colony optimization algorithm. The proposed method enhances the ant colony optimization algorithm by introducing a local search strategy and pheromone evaporation mechanism to achieve efficient path planning [Reference Gao, Xu, Zhang and Huang14]. Wang and Wu [Reference Wang and Wu15] proposed a novel path planning method for mobile robots based on the A^* algorithm and artificial potential field. Fusic et al. proposed the classical algorithms like RRT^* [Reference Julius Fusic and Sitharthan16] are widely used in 3D environment navigation approach [Reference Wang and Wu15].

Liu and Zhang [Reference Liu and Zhang17] propose an improved potential field algorithm for mobile robot path planning. The algorithm includes a repulsion field generated by obstacles and an attraction field generated by the goal, and the weights of these two fields are adjusted using a variable gain factor to improve the path planning performance [Reference Liu and Zhang17]. Hussain et al. [Reference Hussain, Ahmad and Khan18] present a hybrid path planning algorithm based on bat-inspired algorithm and ant colony optimization for mobile robots. The algorithm generates a set of potential paths using the bat-inspired algorithm and then optimizes them using ant colony optimization to obtain the optimal path. Wu et al. [Reference Wu, Wu and Wei19] propose an improved artificial potential field algorithm for mobile robot path planning. The algorithm uses a Gaussian function to model the repulsion field generated by obstacles and a sigmoid function to model the attraction field generated by the goal, which results in smoother and more efficient path planning [Reference Wu, Wu and Wei19].

Ren et al. [Reference Ren, Liu and Zhang20] introduce a novel artificial potential field method for mobile robot path planning. The method includes a repulsion field generated by obstacles and an attraction field generated by the goal and introduces a curvature correction factor to adjust the direction of the robot to avoid local minima [Reference Ren, Liu and Zhang20]. Al-Atabany et al. [Reference Al-Atabany, Gao and Liu21] propose a hybrid heuristic algorithm for mobile robot path planning in dynamic environments. The algorithm combines the artificial potential field method with the genetic algorithm to generate an optimal path and uses a dynamic obstacle avoidance strategy to handle changes in the environment [Reference Al-Atabany, Gao and Liu21]. This section thoroughly explains the concept of multiple route planning strategies for mobile robots. These approaches are classified into numerous algorithm ways in order to enhance the optimization time, path length, path smoothness, and adaptation of feasibility across varied contexts. These methods include classical, sampling-based, map-based, and heuristic strategies for identifying optimal path. The PSO and its variants have a substantial advantage over other algorithms in terms of adapting to complex situations because the majority of impediments in the real world are tangible [Reference Krell, Sheta, Balasubramanian and King22]. It is evident that the proposed SALPSO method did not need to take a longer computing time or path to complete the task. Figure 1 depicts the pipeline flow diagram for the suggested navigation system.

Figure 1. Pipeline diagram for Lidar-based navigation system.

3. Environment modeling algorithm

The construction of an environment model is the initial stage in determining a mobile robot’s likely path. The proposed SALPSO and other PSO variants are used in logistic mobile applications to determine the best path for loading and unloading operations. For the proposed work, three scenarios with four routes are taken into account. The Lidar-mapped image is then generated into 2D occupancy grids using MATLAB function Binary occupancy grid. The modified grids allow for the interpretation of objects (obstacle) on the map as 1 and spaces as 0. The flow diagram in Fig. 2 shows in detail how the input Lidar map image is converted step-by-step into an occupancy grid. The grid generation technique and its pseudo code are displayed in Table I for the environment algorithm.

Figure 2. Flow diagram for Lidar map conversion.

The provided pseudocode details the steps required to transform a LiDAR image into an occupancy grid. Using binary Occupancy Map [Reference Fusic, Kanagaraj, Hariharan and Karthikeyan38], the occupancy grid is prepared, an RGB image is converted to a grayscale image for pixel identification, and thresholds are defined for the range of LiDAR data. The final product is a set of occupancy grid statistics for four distinct environmental configurations. Here, a range threshold is set to classify LiDAR points. Points with a range between 90 and 100 are likely to be treated differently in the occupancy grid generation. To make identifying individual pixels easier, you’ve apparently taken the next step of transforming a color (RGB) image into a grayscale image. This nesting loop traverses the Y axis of the occupancy matrix, where j is the row index. This line determines whether the given grid cell [i, j] is occupied or not. It assigns the values i and j to x _obs and y _obs and 0.8 to r _obs.

4. Improved self-adaptive learning (SALPSO) methodology

The population-based stochastic search method known as PSO is used to solve optimization problems in multidimensional space and has been widely used in scientific and technical fields [Reference Wang, Li, Weise, Wang, Yuan and Tian23]. The existing PSO variant universality, or their capacity to perform well across a range of various fitness environments, remains an issue [Reference Zhou and Duan24, Reference Li, Li, Li and Li39]. The state-of-the-art PSO variants, including Chaotic Cooperative Particle Swarm Optimization (CCPSO) proposed by Liu et al. [Reference Liu, Sun and Lu25], are an advanced variant of the PSO algorithm that incorporates chaotic dynamics to enhance its performance, Comprehensive Learning Particle Swarm Optimization (CLPSO) proposed by Zhao et al. [Reference Hu, Wen and Zhang26] is developed to enhance the search and optimization capabilities of PSO, particularly in complex and high-dimensional problem spaces, Local Search Strategy Particle Swarm Optimization (LPSO) proposed by Jiang et al. [Reference Jiang, Wang and Zhang27, Reference Liu and Zhang28] enhances the exploration and exploitation capabilities of PSO by introducing a local search mechanism. LPSO retains the fundamental components of PSO, including the concept of particles, their positions, velocities, and the social learning process. Each particle in the swarm maintains its position and velocity, updates them based on its own experience, and communicates with neighboring particles to share information about the best solutions found, Velocity Particle Swarm Optimization (VPSO) implemented by Fusic, S. Julius [Reference Zhang, Zhang and Xiong29, Reference Fusic, Sitharthan, Masthan and Hariharan30] stated the primary distinction of VPSO is its central focus on controlling and optimizing particle velocities. The adjustment of velocities helps particles navigate through the search space effectively, allowing them to escape local optima and converge toward global optima, and Binary Particle Swarm Optimization (BPSO) tailored for discrete optimization problems where variables can take binary values, typically 0 or 1 [Reference Gad31, Reference Zhao and Liu32]. BPSO is widely used in various domains, including machine learning, feature selection, and combinatorial optimization to name a few. Even though numerous PSO variants have been put forth, one computation algorithm with resilience and universality, that is, its ability to handle a variety of problems with various features, remains unsatisfactory. In general, the selection of appropriate strategies, especially the choice of the velocity update mechanism, plays a critical role in the success of PSO in solving a particular problem [Reference Srijaroon, Sethanan, Jamrus and Chien33–Reference Zhang, Ning, Li, Pan, Gao and Zhang35].

Table I. Pseudocode Lidar map into occupancy grid.

Table II. Pseudocode for proposed ISALPSO algorithm.

The objective of employing a self-adaptive learning framework is to autonomously select appropriate strategies on various optimization problems and at multiple objective process phases depending on feedback on the fineness of solutions generated via SALPSO proposed by Tao et al. In the proposed improved SALPSO derived from the standard PSO algorithm where PSO has two models: G _best and P _best. The G _best model is more oriented toward exploration, while the P _best model is more oriented toward utilization. The P _best and G _best depend on the $X_{K}^{i} \,\text{and}\, X_{K}^{j}$ for the learning process but majority of PSO algorithms fall short of efficiently obtaining the optimal solutions, particularly when attempting to solve difficult multi-objective or complex environment optimization issues that exhibit multimodality. The single search approach that obtains learning information only from $X_{K}^{i} \, \text{and} \, X_{K}^{j}$ reduces the performance of path planning prediction in real-time data from sensors like Lidar and Kinect sensors. The proposed Improved SALPSO algorithm has a fast convergence rate and avoids being caught by local optima locations. To solve this challenge, the performance of the Pbest and Gbest prediction must be balanced. Instead of immediately updating the current learning information with the monotonic learning strategy in SLPSO, each particle adaptively selects the best appropriate search behavior from a collection of various learning strategies based on its selection ratio of the related operator. In the proposed work, the introduction of improved SALPSO has four operator which is the common difference from the standard PSO algorithm. The proposed SALPSO stochastically decides which approach is used to update the current particle using execution probability. Also, during the entire optimization procedure, avoid employing execution probabilities that are comparatively fixed. In order to adaptively update the execution probabilities depending on the feedback of prior optimization methods, SALPSO employs a gradual updating technique based on a given learning rate (α). Improved SALPSO (ISALPSO) uses a learning rate to regulate the execution probability learning speed during the optimization process. By using a gradual learning mechanism, SALPSO can minimize the influence of fitness attribution of various learning landscapes. The proposed SALPSO-based path planning algorithm consists of the following steps: (1) defining the problem, including the start and goal locations, obstacles in the environment, and robot’s dynamics constraints; (2) designing the SALPSO algorithm that generates a feasible path while optimizing the performance criteria and avoiding obstacles; (3) implementing the SALPSO algorithm on the robot’s hardware or software and collecting sensor data in real time; (4) generating a path using the SALPSO algorithm based on the real-time sensor data; (5) evaluating the generated path based on the performance criteria and obstacle avoidance; (6) adjusting the SALPSO algorithm’s parameters based on the performance of the generated path; and (7) converting the generated path into commands that can be sent to the robot’s actuators to move the robot along the path. The psedocode for the proposed ISALPSO algorithm is shown in the Table II.

4.1. Step 1: Initialization

The population or particle count is initially produced at random with n (n = 1000). Each particle’s initial position ( $X_{k}^{0}$ ) is chosen at random for the environment’s coordinate range of ( $X_{a},Y_{a}$ ) to ( $X_{b},Y_{b}$ ). The ( $X_{a},Y_{a}$ ) to ( $X_{b},Y_{b}$ ) is vary with the hector SLAM generated map images which are bound and converted as (0,0) to (100,200), respectively. Every particle in the population possesses $\langle \textit{Position of particle}\, X_{k}^{i}|\textit{Velocity of particle}\, V_{k}^{i} \rangle$ the listed properties are possessed by every particle in the population. In the proposed constrain, the position $\boldsymbol{X}_{\boldsymbol{k}}^{\boldsymbol{i}}$ and velocity $\boldsymbol{V}_{\boldsymbol{k}}^{\boldsymbol{i}}$ are similar interpretation as in the other PSO algorithm proposed by Tao et al. [Reference Tao, Li, Chen, Liang, Li, Guo and Qi34]. The learning rate α can be selected as less than 1 with the particle weight assumed to be $W_{k}$ for the kth best particle assumed to be k = 1, 2 … ps.

(2) \begin{equation}W_{k} = \frac{\log (ps-j+1)}{\log \left(1\right)+\ldots +\log (ps)}\end{equation}

4.2. Step 2: Fitness function initialization

In order to find the shortest route, the fitness value is essential. Once the particle begins to accelerate and travel at a certain velocity $\boldsymbol{V}_{\boldsymbol{k}}^{\boldsymbol{i}}$ , the probability of violation is presumed to be zero [Reference Srijaroon, Sethanan, Jamrus and Chien33]. Because it provides the best fitness value, the proposed condition is considered to be the overall optimal option. The best local solution produced by each particle is compared to the best global solution while taking the violation value into account. The execution probability of the kth updating strategy is updated according to the following rule after a predetermined number of generations Gs.

(3) \begin{equation}Tp_{k}^{\prime}=\left(1-\alpha \right)Tp_{k}+\alpha \frac{S_{k}^{i}}{Gs}\end{equation}

(4) \begin{equation}Tp_{k}=Tp_{k}^{\prime}/(Tp_{1}^{\prime}+I....+Tp_{n}^{\prime})\end{equation}

(5) \begin{equation}S_{k}^{i}=\frac{R_{k}^{i}}{\sum _{j}^{n}R_{k}^{i}}*\left((1-n\right)* S_{k}^{i}\max )+S_{k}^{i}\min\end{equation}

where Tpk is the probability of temporal execution and α represents the learning rate, which regulates the repeating proportion. $\frac{S_{k}^{i}}{Gs}$ is assumed to be less than 1 based on the normalized execution probabilities. It is to be assumed that techniques that have produced higher-quality solutions will gradually raise their implementation probabilities. Further the selection ratio ( $S_{k}^{i}$ ) for each learning operator is selected to determine the probability of selecting the updated particle. The ( $S_{k}^{i}$ ) value incremented or decremented for successive iteration k of previous performance results in higher fitness value. The selection ratios of all operators are all set to 0.25 for each particle. The following modified equation will be used to update the selection ratio of the operator of particle k at iteration i + 1 within the framework of the self-adaptive learning mechanism.

4.4. Step 4: Completion of iteration

Between the starting point and the finishing point, the global best position is determined and navigated ( $x_{a},y_{a},x_{b},y_{b}$ ) for iteration i = 1. The solution is excluded while determining the optimum path if the violation $\varnothing _{i}$ happens during particle movement. Once the optimal approach to getting there is identified, steps 2–4 are repeated, and pseudocode is shown in Tables II and as the result. Table III displays the ISALPSO method and its variation parameters. The entire simulation framework utilized in the suggested study is shown as a flowchart in Fig. 2. The particle, that represents a significant response to the optimization process, may begin to move around in order to find the best answer in a given search space. The velocity and position of each and every swarm particle at the ith iteration will be modified as follows during the next iteration based on the four learning operators:

(6) \begin{equation} \text{Operator 1: } V_{k}^{i+1}= W_{k}*V_{k}^{i}+c_{\mathrm{c}}\mathrm{ran}_{1}(P_{\text{best}}-X_{k}^{i}) + c_{\mathrm{s}}\mathrm{ran}_{1}(P_{\text{best}}-X_{k}^{i})\end{equation}

(7) \begin{equation}\text{Operator 2: } V_{k}^{i+1}= W_{k}*V_{k}^{i}+c_{\mathrm{c}}\mathrm{ran}_{1}(P_{\text{best}\_ \text{nearest}}-X_{k}^{i}) + c_{\mathrm{s}}\mathrm{ran}_{1}\left(P_{\text{best}}-X_{k}^{i}\right) \end{equation}

(8) \begin{equation} \text{Operator 3: } X_{k}^{i+1}= X_{k}^{i}+V_{\text{average}}^{i}\text{ * }\mathrm{N}(0,1)\end{equation}

(9) \begin{equation}\text{Operator 4: } V_{k}^{i+1}= W_{k}*V_{k}^{i}+c_{\mathrm{c}}\mathrm{ran}_{1}\left(G_{\text{best}}-X_{k}^{i}\right) +c_{\mathrm{c}}\mathrm{ran}_{1}\left(G_{\text{best}}-X_{k}^{i}\right)\end{equation}

where $X_{k}^{i+1}, P_{\text{best}\_ \text{neares}t}$ and $V_{\text{average}}^{i}$ are the personal best position of the particle closest to particle k at iteration i and the average velocity of all particles at iteration i, respectively. N (0,1) is a random number taken from a uniform distribution ranging from 0 to 1. δ and C_c are weight and acceleration coefficients that can be used to balance the effect of exploration and exploitation during the search process. In general, w and 3 are set depending on the designer’s practical expertise. Based on the working environment, the search behaviors are adjusted and learning strategy locate the nearer local minima point. The $X_{k}^{i}$ and $V_{k}^{i}$ signify the kth position and velocity at the ith iteration, respectively; $P_{\text{best}}$ and $G_{\text{best}}$ symbolize the kth particle’s optimal positioning (Local best) and the ideal position of the global swarm (Global best) up to iteration i, where P _best signifies the best individual particle position, G _best denotes the best global particle position, $C_{C}^{i}$ and $C_{S}^{i }$ are ISALPSO’s cognitive and social constant factors, alpha represents the learning rate 0–1 and ran1, ran2 are random numbers ranging from 0 to 1 are created. The C_c , C_s , and number of particles (nP) values for the proposed work are chosen based on variable values ranging from 20 to 100 for particles and 0 to 2 for learning factor, and the maze environment is run for 20 iterations. Based on the optimization, the nP = 100, C_c = 1.1, and C_s = 1.7 the best violation-free path is found.

Table III. Constriction factors of proposed SALPSO.

Table IV. PSO variants comparison for four environments with different routes.

Considering the violation $\varnothing _{k+1}$ represents the plotted point of particle is out of the using the Eq. (11) to forecast the path,

(10) \begin{equation}X_{k}^{i}=\sqrt{{(x_{a}}-{x_{b}})^{2}+\left(y_{a}-y_{b}\right)^{2}}\end{equation}

(11) \begin{equation}\varnothing _{k+1}=\varnothing _{k}+\text{mean}\left(1-\sqrt{{(x_{k}}-{x_{\mathrm{obs}}})^{2}+\left(y_{k}-y_{\mathrm{obs}}\right)^{2}}\right)/r_{\mathrm{obs}}\end{equation}

where $x_{\text{obs}}$ represents the environment obstacle in x coordinate, $y_{\text{obs}}$ represents the environment obstacle in y coordinate. The radius of the various structural grid obstacles is represented by $r_{\text{obs}}$ . All of the proposed obstacles are shaped using cubic spline interpolation, and the path prediction additionally employs cubic spline interpolation in proposed algorithm.

The proposed SALPSO algorithm constriction factor parameters for proposed environment simulations are listed in Table III. The Learning coefficient and learning rate parameters detailed in the Eq. (3) are used to tune the proposed SALPSO algorithm. In addition to this, the population size of the particles (n) is selected based on 20 experimental simulation results implemented using environment E3 and considered the constriction factor value for proposed SALPSO as shown in Table III for proposed application. These above-mentioned parameters are tuned in the simulated robot to achieve violation-free optimal path to reach the destination. In this model, the violation as ( $\mathrm{\varnothing }_{i}$ ) and the violation-free distance (D) need to achieve this shortest path as shown in environment 3. The values of the input and their response that are used for the experimental design as mentioned in Table III. From the optimal solution in Table III, the proposed SALPSO parameters are fixed as C_c = 1.1; C_s = 1.7; α = 0.16; n = 1000, and iteration as 100. The inertial weight was selected based on the equation w _k .