Enhanced Robot Motion Block of A-Star Algorithm for Robotic Path Planning

An optimized robot path-planning algorithm is required for various aspects of robot movements in applications. The efficacy of the robot path-planning model is vulnerable to the number of search nodes, path cost, and time complexity. The conventional A-star (A*) algorithm outperforms other grid-based algorithms because of its heuristic approach. However, the performance of the conventional A* algorithm is suboptimal for the time, space, and number of search nodes, depending on the robot motion block (RMB). To address these challenges, this paper proposes an optimal RMB with an adaptive cost function to improve performance. The proposed adaptive cost function keeps track of the goal node and adaptively calculates the movement costs for quickly arriving at the goal node. Incorporating the adaptive cost function with a selected optimal RMB significantly reduces the searches of less impactful and redundant nodes, which improves the performance of the A* algorithm in terms of the number of search nodes and time complexity. To validate the performance and robustness of the proposed model, an extensive experiment was conducted. In the experiment, an open-source dataset featuring various types of grid maps was customized to incorporate the multiple map sizes and sets of source-to-destination nodes. According to the experiments, the proposed method demonstrated a remarkable improvement of 93.98% in the number of search nodes and 98.94% in time complexity compared to the conventional A* algorithm. The proposed model outperforms other state-of-the-art algorithms by keeping the path cost largely comparable. Additionally, an ROS experiment using a robot and lidar sensor data shows the improvement of the proposed method in a simulated laboratory environment.


I. INTRODUCTION
In the field of robotics, there are several challenging tasks, and path planning is one of the most important tasks among them to find a way to obtain the desired goal position.In this research area, the main objective is to develop an efficient path-planning algorithm by enhancing its performance in terms of searching time complexity, path cost, and search area [1].As the application area of robot usage is increasing rapidly, the number of ongoing studies is also increasing in this area [2].Generally, path-planning algorithms always search for the target position from the current position of the robot by keeping the overall path as shortest as possible.There are many path-planning algorithms for minimizing the path cost and finding the shortest path; however, for a path-planning algorithm, the importance of the three main factors, time complexity, path cost, and search area, mainly depends on the robot application [1].Nevertheless, considering these factors, a general and efficient path-planning algorithm fits most robot applications.
Generally, path-planning algorithms are categorized into two groups based on the surroundings of the known degree of environmental information global path planning and local path planning of the robot [3,4].A robot with global path planning needs to know the environmental information partially, such as a map of the environment required to calculate the optimal route from the start position to the goal position.The algorithm partially knew the obstacle map and the effectiveness of these algorithms is evaluated by their ability to find a valid path if one exists [5,6].In contrast, local path planning requires the robot to search its surroundings using sensors and find a path to reach the goal in an environment with unknown information [5].As a result, local path-planning algorithms are typically more time-consuming and complex and may require the use of techniques such as segmentation and object detection to avoid obstacles [7].Some commonly used local pathplanning algorithms include the artificial potential field method, fuzzy logic method, and dynamic window method.In some cases, an ArUco markers-based searching system can make the process easier and more efficient [8].In the global path-planning field, various algorithms are available, such as rapidly exploring random trees (RRTs), graph search, tangent graph-based planning, and predefined path optimization.Our research focus is on the area of graph-based searching algorithms.
Graph-based searching algorithms are extensively used and considered highly effective in the field of global path planning.Some popular examples of graph search algorithms include A*, Dijkstra, depth-first search (DFS), and breadth-first search (BFS).Among these algorithms, A* is particularly notable for its completeness, efficient search area, and ability to use heuristics [9].A* is a pathfinding algorithm developed on the basis of the Dijkstra algorithm and is designed for use on weighted graphs, such as grid maps with obstacles.The algorithm works by constructing a tree of paths from the starting node and adding nodes as it searches the graph until it reaches the goal node or exhausts all options [9].The final result is the shortest path found in the tree of searched paths.To search the graph or grid map, the algorithm requires a RMB to navigate the surrounding environment.The research goal is to develop an optimal RMB to improve the overall performance of the A* algorithm.
The robot motion block is a crucial element in pathplanning algorithms such as A*, as it determines how the robot will search the cells surrounding its current position as it moves toward its destination [10].The RMB is a matrix that specifies the cells that the robot should search in each iteration to find the goal node starting from the start node.The performance and efficiency of the algorithm are significantly affected by the size of the motion block matrix, as it determines the search area of the robot in each iteration [11].An inefficient motion block results in the robot searching for a larger number of cells to find the goal, increasing the time and space complexity.In the conventional approach, the motion block matrix comprises eight surrounding cell coordinates for the robot to search [10].In each iteration, the A* algorithm adds the information about these eight cells to its search queue and continues to add cells proportionally until it reaches the goal or exhausts all options.This approach necessitates a significant amount of searching time complexity and many search nodes.Thus, keeping these limitations in mind, this study proposes an optimal RMB with an adaptive cost function for robot movement to improve the performance.The proposed method considers eight neighboring coordinates with different searching distances and puts only the end node in the search queue, significantly reducing the number of search nodes and time complexity with a small path cost increment.As the path cost increases and the number of search nodes and time complexity decrease with different sizes of motion block, a formula is also proposed to decide the optimal one among different distances.Using the proposed method, the A* algorithm can find the goal position by avoiding obstacles with significantly improved overall performance.A comparison study with the conventional A* and different similar stateof-the-art path-planning algorithms validates the significant improvement of the proposed model.In addition, this study discusses the effectiveness, drawbacks, and applicability of the proposed approach in different application areas.
The remainder of this paper is structured as follows: Section 2 provides an overview of related works, and Section 3 presents the details of the proposed model.Section 4 offers the discussion and results of the experiments conducted, and Section 5 concludes the study.

II. RELATED WORKS
In the field of robotics, path-planning algorithms have been extensively investigated by researchers to improve their efficiency and reduce search costs.Despite considerable progress in this area, the performance of these algorithms still has room for improvement.Researchers are still exploring new ways to enhance their performance to make them more reliable and efficient.This is particularly important in robotics applications, where fast and accurate path planning is essential for the safe and effective operation of robots that can handle increasingly complex environments and tasks while maintaining a high-performance level.
In one study, Li Changgeng et al. proposed a new global pathplanning algorithm called bidirectional alternating search A* (BAS-A*) for mobile robots [12].Their algorithm combines the best features of traditional A* and bidirectional search algorithms to generate efficient and optimal paths.By utilizing a bidirectional alternating search strategy, the algorithm improves search efficiency by searching forward and backward path lists until they intersect.The algorithm addresses issues such as long calculation times, large turning angles, and unsmoothed paths in large task spaces.They introduced weighted heuristic functions, a filtering function for path nodes, and Bézier curves to ensure smooth and optimized path planning [13].Simulation results demonstrated the algorithm's efficiency and smoothness compared to other algorithms.Practical validation was also attempted on the TurtleBot3 Waffle Pi mobile robot [14].
Meanwhile, Szczepanski et al. presented a hybrid approach for the global path planning of mobile robots in variable workspaces [15].Their approach comprised an offline global path optimization algorithm using the artificial bee colony algorithm and an online path-planning scheme that uses a graph created from the control path points generated in the offline part [16].The online part uses the Dijkstra algorithm to find the shortest path [17].Their proposed approach aimed to generate optimal paths for mobile robots in variable workspaces feasible even with new obstacles.Their approach was tested in the Matlab/Simulink environment and achieved a good result.
In another study, Sánchez-Ibáñez et al. conducted a comprehensive review article that provided an overview of path-planning algorithms for autonomous mobile robots [18].They highlighted the importance of autonomous capabilities in enhancing economic and safety aspects.With numerous path-planning algorithms available, the authors aimed to classify and analyze these algorithms, particularly for autonomous ground vehicles [19].Their review covered environmental representation, robot mobility, dynamics, and various path-planning categories.The study served as a valuable resource for understanding the research conducted on path-planning algorithms, aiding in the selection of appropriate algorithms for specific requirements.
Meanwhile, Tripathy et al. proposed a collision-free navigation scheme for mobile robots in a grid environment [20].This research scheme combines a radio frequency identification method for robot localization, a hybrid approach for path planning, and a predefined decision table for navigation [21,22].The algorithm has two stages: construction of the virtual world and generation of the optimal shortest path.The performance of the algorithm was evaluated in different grid-based environments with and without obstacles [23].The results showed that the robot explored fewer cells to find the shortest path when there were no obstacles.However, in environments with obstacles, the number of turns in the shortest path was always less than that when exploring the entire virtual world.
Separately, Chen et al. introduced a three-neighbor search A* algorithm combined with the artificial potential field method to overcome irregular forward direction obstacles and guide robot movement [24,25].Their method demonstrated significant improvements by reducing the number of search nodes, search time, and path length by 88.85%, 77.05%, and 5.58%, respectively.Saeed et al. presented the boundary node method, an offline path-planning approach that generated collision-free paths for mobile robots [26].Their method utilized a nine-node quadrilateral element and a potential function to guide robot movement, generating initial collisionfree paths quickly and safely.They also employed a path enhancement method to further optimize the path length.The simulation results validated the effectiveness of their proposed methods.In a different approach, Ichter et al. introduced latent sampling-based motion planning, combining recent control advances with techniques from sampling-based motion planning (SBMP) [27,28].Their methodology involved learning a plannable latent representation using autoencoding, dynamics, and collision-checking networks.The learned latent RRT algorithm demonstrated global exploration capabilities and generalization to new environments, as showcased in visual planning and humanoid robot path-planning problems [29].These studies provide valuable insights into different path-planning approaches that helped us to address the limitations of traditional A* algorithm and offer improved performance and adaptability for various robotic systems.

III. PROPOSED MODEL
The A* algorithm is most effective in finding the shortest path due to its heuristic search technique.The conventional A* algorithm takes a long searching time and checks a large number of cells for searching the goal because of an inefficient RMB.For a robot to reach its destination, it needs to find the shortest path from the starting position to the goal by checking a minimum number of search nodes and small time complexity.The proposed RMB helps to significantly reduce those two criteria and improves the overall performance of the A* path-planning algorithm while avoiding obstacles.The block diagram in Fig. 1 illustrates the proposed model, which includes a robot path-planning algorithm with an efficient RMB to search for the optimal path.In this model, the robot examines the neighboring grid cells using the proposed optimal RMB, which significantly improves the performance of the path-planning algorithm and increases its usability in real time.The proposed model was experimented using various grid maps that contain different obstacle environments, starting and goal positions.

A. A* PATH PLANNING ALGORITHM WITH PROPOSED RMB
Currently, there exist numerous algorithms for searching paths and traversing graphs, one of which is A*.It is extensively recognized for its efficiency in being both optimal and complete.A* is classified as a best-first search or informed search algorithm that operates on a weighted graph.Its objective is to find the goal node, beginning from a specified starting node with the smallest amount of travel cost and time.The algorithm creates a tree of search nodes, beginning from the starting node, and continues until the search terminates, which can either be the discovery of the goal node or no discovery at all.The A* algorithm uses a heuristic function ℎ() to estimate the cost (,  ′ ) to reach the goal node from the current node [30], where ℎ() ≤ (,  ′ ) + ℎ(′), and (,  ′ ) is the cost of an individual node  ≠   and any successor ′.This heuristic should be both admissible (it should never overestimate the actual cost to reach the goal) and consistent (the cost from one node to another should not be more than the cost to move directly from the first node to the goal node).The heuristic function for the grid map is the Euclidean distance between the current node and the goal node.The algorithmic workflow of the A* algorithm, using the proposed model, shown in Fig. 2, is as follows: 1) To begin the A* algorithm with the proposed robot motion block, initialize the start node (  ) with a cost of 0, and add it to an  list of nodes to be considered for expansion.
2) Check the neighboring grid nodes for the goal node (  ) with the proposed robot motion block explained in subsection 3.2.If the current node is an obstacle node (  ), ignore it, and add only the end nodes in the  list.While the  list is not empty, select the node with the lowest total cost () and remove it from the  list.

𝐶𝑜𝑠𝑡(𝐶) = 𝑚𝑖𝑛{𝑂𝑃𝐸𝑁, ℎ(𝑆 goal , 𝑂𝑃𝐸𝑁 ) }
Here, ℎ( goal , ) denotes a heuristic function that uses Euclidean distance, 3) If the selected node is the goal node, stop the search and return the optimal path.Generate all possible successors of the selected node and calculate their costs and heuristic values.
4) For each successor node, update its cost and heuristic value if it has not already been visited or has a higher cost.Add each updated successor node to the  list if it is not already in the list.
5) If a successor node is already in the  list, update its cost and heuristic value if the new values are lower than the previous values.If a successor node is already in the visited list (i.e., has already been expanded), ignore it.
6) If all nodes in the motion block cells have been checked, return to step ІІ and repeat the steps until every node on the grid map has been checked.

7)
Terminate the algorithm when the goal node is found or the  list is empty.The proposed RMB is represented by a matrix as Equation (1), where  is the size of the RMB, which indicates the distance between the robot and the cells being searched, and  denotes a natural number.The first column represents the x-axis, the second column represents the y-axis, and the third column represents the robot movement cost of each point in a two-dimensional grid space.When  increases, the proposed method checks the nodes till the  th node but only adds the  th node to the  list.Fig. 3 provides a visual representation of the proposed RMBs.For achieving the costs  and  of the nodes of the motion matrix, an adaptive cost function  is formulated, which is multiplied by the costs associated with cardinal  and diagonal  ′ movements as Equation ( 4).In this adaptive cost function, first, the current node's cost  is summed with the Euclidian distance between the current node  and the node of motion matrix  to get the absolute differences in the x and y coordinates.The cost  ′ is achieved as in Equation ( 2).Next,  ′ summed with the Euclidean distance between the node of motion matrix  and goal node  by multiplying with a constant of  in Equation ( 3).Here,  is used to increase the importance of the distance value between each of the motion matrix nodes and the goal node.This helps the adaptive cost function to keep track of the goal node with each node of the motion matrix and reduce the number of searches to find the goal node.The optimal range of  is between 0.001 and 0.009 after testing on a large number of maps.However,  = 0.007 is used in this paper which provided the best results.

C. OPTIMAL ROBOT MOTION BLOCK SELECTION
The proposed robot motion block has different sizes ().The output results for each motion block differ.However, a general optimal RMB is essential for any robotic application.
To obtain the optimal one from different RMB sizes, a selection formula is proposed as Equations ( 5), ( 6) and ( 7).
Here,  denotes the array of  elements, each element contains the mean of output data.Which is experimented on the dataset containing {1, … , } data points, based on the RMB size. denotes paramiters and (  )denotes individual output array eliments for each RMB size; where {  ,  = 1, … , }.To optimize the efficiency of the path-planning algorithm, this research considered three paramiters , those are path costs, the number of search cells in the grid map, and time complexity.Next, Each data array is brought into a range of  = 1000 using Equation ( 5) and achieves a ranged array .After that, each indexed data from all resultant data arrays is summed and subscribed by the total number of arrays, which in this research is three, to obtain  ′ .Equation ( 6) was used to calculate  ′ .Finally, the optimal RMB is determined by selecting the minimum value of the final resultant data array for each RMB.Equation ( 7) was used to select the optimal RMB .

IV. EXPERIMENTAL RESULTS AND DISCUSSION
The experimental outcomes justify the validity and demonstrate the effectiveness of the proposed model.The following subsections describe the experimental results of the (2) proposed optimal RMB for the A* algorithm.First, the preprocessing steps of the dataset with different types of grid maps are described.After that, the experimental results using the prepared dataset show the effectiveness of the proposed model.

1) DATASET
To validate the proposed approach, a collection of grid maps is essential for the experiment on the proposed RMB of the A* path-planning algorithm.This experiment uses a benchmarked online public repository dataset named "motion_planning_datasets" [31].There are eight types of grid maps in this dataset.Each grid map type contains thousands of PNG format images with a resolution of 201 x 201.Five types of grid maps of planning environments are used among these eight types: alternating_gaps, forest, bugtraq_forest, gaps_and_forest, and mazes.For validation purposes, 4000 grid maps data are used in this research, with 800 grid maps data for each type.Fig. 4 shows some sample images of these grid maps from the dataset.

1) GRID-MAP PREPARATION
The dataset described in the previous subsection has some limitations, such as all grid map data points being images in PNG format rather than the matrix format required for the experiment.For the robot not to leave the grid map, a border surrounding the grid map is essential, but there is no border in the dataset images, and it also has no start and endpoint information.To make the proposed method's efficiency more concrete and generalize the method, the proposed method needs to test on different dimensional grid maps, but the dataset images have a diminution of 201 x 201.To overcome these limitations and fit the dataset with the experiment, the dataset images are preprocessed and prepared the grid maps for the experimental format, as depicted in Fig. 5. Where, white = free cell and black = obstacle cell.
Next, two borders surrounding the dataset grid map images with 15 pixels each are added, making the grid map image dimension 261 x 261, 462 x 261, and 462 x 462.Among these two borders, the outer border is black obstacle pixels, and the inner border is white obstacle-free pixels.Next, these modified map images are split into grid cells containing information about each cell with and without obstacles to make it a matrix format.Finally, the grid maps with the required format are achieved to validate the proposed model.

2) SOURCE AND DESTINATION SELECTION
For the experiment, four sets of different sources and destinations have been selected for each of the threedimensional maps, as shown in Fig. 6.The red arrow represents the direction from the source (depicted as the green circle) to the destination (represented by the blue circle).In addition besides the blue and green circles, source and destination coordinates are provided.For the map with dimensions 261 x 261, 200 maps experimented with for each direction, as 800 maps are prepared for this dimensional map.Similarly, 100 maps are used for each direction in the case of the 462 x 261 dimensional map, and 50 maps are employed for the 462 x 462 dimensional maps.

B. EXPERIMENTAL RESULTS OF THE PROPOSED MODEL
This paper shows the experimental results using the preprocessed 4000 grid maps of five different types for the proposed optimal RMB of the A* algorithm.First, the experiment for the proposed model demonstrates the effectiveness of the proposed RMB by minimizing the number of search cells and time complexity while maintaining the minimum path cost from the start position to the target position.Then, the selection of the optimal RMB ensures algorithmic optimality by considering the three major criteria: the number of search cells, time complexity, and path cost.Finally, a comparison between the proposed model and other similar algorithms shows the effective performance of the proposed optimal RMB.

1) COMPARISON BETWEEN CONVENTIONAL A* AND THE PROPOSED METHOD
The proposed method performs better than the conventional A* algorithm because of the adaptive cost function of the robot movement costs, which keeps track of the goal position.When the search distance of the motion block  increases, the performance also increases even more.This is because, in that case, only the end node of this search distance is placed in the next search list.In Table I, the comparison shows the improvement for RMB  = 1.However, this paper shows further experiments for other values of RMB.  Figure 7 shows improved performance and differences of the proposed model compared to the conventional A* algorithm.
It indicates that the proposed model maintains a safe distance from the obstacles where the conventional algorithm goes through close to the obstacles.This is because of its consideration of an offset during the final path calculation.

2) EXPERIMENTAL RESULTS USING DIFFERENT SIZES (n) OF THE ROBOT MOTION BLOCK
The simulated outcomes using the proposed with different sizes ( = 1-6) for the A* path-planning algorithm are depicted in Fig. 8.The RMB size is assumed ( = 1-6) for the experiment because if  increases more, it provides mostly similar results as  = 6, which does not affect the results.The experimental results for three types of map dimensions and three of the five map types are shown in Fig. 8. (a1-6), (b1-6), and (c1-6) show the result of the bugtraq_forest, gaps_and_forest, and mazes grid maps.Where (a1-6) has a map dimension of 261 x 261, (b1-6) has 462 x 261, and (c1-6) has 462 x 462.To find the target point, the number of search cells drastically decreases with an increase in the size of the proposed RMB.The results for the RMB size  = 1 are shown in (a1, b1, c1), those for  = 2 in (a2, b2, c2), those for  = 3 in (a3, b3, c3), and so on.However, the path cost increased by a tolerable amount.
( In Fig. 8, the red line indicates the path that the proposed approach found toward the target node.Furthermore, the searched grid cells are marked by a cyan cross, of which the number decreases when  increases.For the proposed RMB  = (1-6), three types of data were collected from the experimental results: path cost, number of search cells, and the time required (seconds) to find the path.
Table ІІ shows the resultant data for each RMB size and map dimension, which is a mean of all grid maps for each type of map among five types.In addition, a comparison of the outcome results is shown in Fig. 9     The results in Table II and Fig. 9, 10, and 11 show that, with the increase of distance for the RMB, the time complexity and number of search nodes to find the target decreased, whereas the path cost increased.Therefore, an optimal RMB selection is essential and is performed using the proposed formula in Equations ( 5), (6), and (7), as described in subsection III (C).

3) RESULTS FOR OPTIMAL ROBOT MOTION BLOCK SELECTION
Table III presents the range data for each criterion: the number of search cells, path cost, and time complexity.The mean of each map type and map dimension shows that in most cases, the size of RMB  = 3 has the minimum numeric value, and it provides the optimal performance of the A* algorithm.Fig. 12 also shows the result for the proposed optimal RMB (which is  = 3).This is because it has the lowest value for both the number of search nodes and time complexity while still having a low value for the path cost.Therefore, it balances the three criteria well and provides an optimal overall performance.It also indicates that for the map type, gaps_and_forest  = 2 is optimal because of its complex and dense obstacle environments.But the differences between the value of  = 2 and  = 3 is not much, such as for the map dimension 462 x 462 the values are 94.02 and 105.11 respectively.In the performance calculation, the considered range for the number of search cells is 0 -133654.33,as the maximum number of cells in the dataset is 68121 for the map dimension 261 x 261, 120321 for 462 x 261, and 212521 for 462 x 462.The average of these three values is 133654.33.Similarly, The considered ranges for the path cost and time complexity are assumed as follows: 0 -4500 and 0 -20 seconds, respectively.In this graph, the lowest performance percentage indicates good performance as the lowest values represent the lowest number of search cells, less amount of path cost, and less time complexity to find the target.This table shows that the proposed model outperforms the other algorithms in terms of the number of search cells and time complexity.Its path cost is mostly similar to that of the other algorithms except for DFS, which has the highest path cost.

C. DISCUSSION
This research aims to enhance the efficiency of the A* pathplanning algorithm by minimizing the time complexity and the number of search nodes while maintaining a minimum path cost.The experimental results presented in Fig. 7 and Table I show improved performance compared to the conventional A* algorithm because of its adaptive cost function for the robot movement, which keeps track of the goal position.The reduction in time complexity, the number of search nodes, and path cost is around 29.05 %, 32.58 %, and 0.04 %.According to Table II and Fig. 9, 10, and 11, increasing the size of the proposed RMB reduces the time complexity and the number of search nodes significantly.Fig. 8 samples the simulation results.The path costs increased slightly for the increased size of RMB; however, the time complexity and the number of search nodes decreased.Thus, the proposed model selects the optimal RMB to apply in different applications.The experimental results for the optimal RMB selection are presented in Table III and Fig. 12.According to the experimental results, the RMB  = 3 is optimal as it provides a balanced improved results in most cases, as shown in Fig.
12.An additional experiment was conducted, where the proposed model was compared with other similar algorithms, as shown in Table IV, Fig. 13 and 14.The experimental results show that the proposed model outperforms the comparison algorithms.The conventional A* algorithm outperformed the other algorithms.Whereas the performance of the proposed model with optimal RMB ( = 3) improved by decreasing the number of search cells 93.98 % and the time complexity 98.94 % compared to the conventional A* algorithm, calculated using Equation (8).The path cost of the proposed model was mostly similar to that of the other algorithms except for DFS, which had the highest path cost.

V. CONCLUSION
Path planning is crucial for autonomous robot navigation, allowing robots to find optimal routes while avoiding obstacles.A* is a popular global path-planning algorithm due to its heuristic approach.However, conventional A* has limitations, such as computational complexity and extensive cell searching in the grid spaces.This can be problematic for autonomous robots operating in real-time and dynamic environments.To address these limitations, this study proposes an optimal RMB for the A* algorithm that significantly improves its overall performance.In this approach, robot movement costs are calculated by the proposed adaptive cost function that keeps track of the goal to reduce the number of searches to find the goal.Also, the robot checks a certain distance surrounding eight directions of it each time, puts only the end node in the  list of the A* algorithm, and applies a heuristic approach to select the next movement from the  list.These two approaches significantly reduce the number of search cells and time complexity while achieving an acceptable path cost.To evaluate the proposed approach, a benchmarked public dataset with five types of grid maps is used, containing nearly thousands of maps for each type.To generalize the outcome, three different dimensional maps are prepared using the dataset.First, the comparison with the conventional A* shows the improvement of the proposed model.Next, the experimental results show that increasing the proposed RMB size further reduces time complexity and the number of search nodes.Also, the results from the proposed optimal RMB selection method show that the RMB size  = 3 is the optimal one, considering the number of search cells, time complexity, and path cost.Finally, the comparison with the different stateof-the-art algorithms shows the effectiveness of the proposed model in improving the performance of the A* algorithm for autonomous robot navigation.
= Maximum value of each parameter.

FIGURE 1 .
FIGURE 1. Block diagram of our proposed model.

FIGURE 2 .
FIGURE 2. Block diagram of A* path planning algorithm with proposed optimal RMB.B. PROPOSED ROBOT MOTION BLOCK WITH AN ADAPTIVE COST-FUNCTIONIn a grid-based search, robots must search the cells around them to find the goal node in the smallest number of cells possible.The robot uses a block of cells, called robot motion block or motion kernel, to search for its neighboring nodes.However, the conventional RMB used by A* leads to a higher number of search cells and a longer search time.The proposed motion block, which surrounds the robot with eight neighboring cells, aims to minimize the number of search cells and the time taken to find the goal position while maintaining a tolerable path cost.

3 …FIGURE 3 .
FIGURE 3. Block diagram of A* path planning algorithm with proposed optimal RMB.The RMB has eight neighboring cell directions, including up, down, left, right, up-right, up-left, down-right, and downleft.The costs of the four cardinal and diagonal directed points are represented by (, ), and ( ′ , ) respectively.These costs are calculated by the following Equations (2), (3), and (4):

FIGURE 6 .
FIGURE 6. (a), (b), (c) shows the source to destination direction for three dimension maps'.Where, blue circle = destination, green circle = source, and red arrow = path direction.

FIGURE 8 .
FIGURE 8. Simulated experimental outcomes of the proposed RMBs (n = 1-6) for three types of maps.Here, white = free cell, black = obstacle cell, green & blue circle = start & goal point, red line = found path, and cyan cross = cell searched.

( a )
Number of search nodes.(b) Path cost.(c) Time required for searching the goal node.

FIGURE 12 . 4 )FIGURE 13 .
FIGURE 12. (a), (b) and (c) are the experimental results for the proposed optimal RMB based on the experimental data of Table 3.

Figure 14
Figure 14 shows the performance comparison graph of the different algorithms and the proposed model considering the data in Table IV and Equation (9).

FIGURE 14 .
FIGURE 14. Performance of different algorithms compared with the proposed model with optimal RMB.

Average 666.67 167.35 132.70 199.42 325.67 333.33
In terms of the number of search cells and time complexity, the proposed model outperformed the other algorithms.Specifically, first, the number of search cells for the proposed model is 2824.45,whereas that for the other algorithms ranges from 46878.88 to 84677.83.Second, regarding time complexity, the proposed model requires 0.1406 seconds, whereas the other algorithms require 1.6375 to 16.5898 seconds.Finally, regarding the path cost, DFS has a value of 4115.53, which is significantly higher than that of the others, and the path cost of the proposed model is mostly similar to that of the remaining algorithms.
Table ІV represents a comprehensive comparison between the proposed model and the different algorithms, Dijkstra, DFS, BFS, and conventional A*.The values from this comparison are the cumulative results collected from the experiment conducted on the considered dataset; thus, these values have fraction numbers.

TABLE IV EXPERIMENTAL
RESULTS FOR THE COMPARISON BETWEEN DIFFERENT STATE-OF-THE-ART ALGORITHMS AND THE PROPOSED MODEL WITH OPTIMAL RMB.