Research on Autonomous Mobile Robot Path Planning Based on M-RRT Algorithm

The Rapidly-Exploring Random Tree (RRT) algorithm has demonstrated proficiency in adapting to path search challenges within high-dimensional dynamic environments. However, a notable limitation of the RRT algorithm lies in its inability to fulfill the criteria for achieving the shortest and smoothest path for mobile sensing nodes. To address the limitations of the conventional RRT algorithm and enhance the path planning for mobile robots, this paper proposed an innovative approach named M-RRT, designed to overcome the aforementioned shortcomings and optimize the path planning process for mobile sensing nodes. First, the search area is constructed according to the defined coverage density. After searching the path in the search area, the RRT algorithm uses the greedy method to delete the intermediate nodes in the path, and obtains the uniquly optimal path. Finally, the Bezier curve is used to optimize the path, which makes the path shortest and meets the dynamic requirements of the mobile node. Simulation results show that M-RRT has better path and faster convergence speed than traditional RRT, which can better meet the planning requirements of mobile nodes.


Introduction
In the rapidly evolving field of wireless sensor networks, the path planning problem of mobile nodes stands out as a fundamental challenge.The path planning problem of mobile node is to search an optimal, safe (collision-free) and complete path from the starting point to the end point in the robot configuration space according to a certain optimization criterion.Researchers have proposed that deploying a small number of redundant mobile nodes in wireless sensor networks can solve the coverage and node failure problems [12][13].This scheme not only improves the network performance but also controls the network overhead.However, the process of moving requires nodes to navigate obstacles efficiently, follow the shortest possible route, and maintain a continuous smooth path, aligning with the dynamics and energy efficiency needs of mobile nodes [6].
The strategic deployment of redundant mobile nodes in wireless sensor networks represents a pivotal enhancement, addressing coverage and node failure issues effectively.This approach not only boosts network performance but also manages network overhead, fostering a resilient and efficient system.Nevertheless, the principal challenge lies in the motion of these nodes.They must skillfully negotiate obstacles, minimize travel distance, and ensure a smooth, uninterrupted path, a trio of requirements crucial for their operational utility.
Researchers have extensively explored various algorithms to tackle these complex requirements.Classical pathfinding algorithms like Dijkstra and Floyd have been foundational, offering reliable solutions that necessitate a thorough understanding of the entire environment and necessitate subsequent path smoothing [7].On the other hand, the A* and D* algorithms represent a leap forward, capable of rapidly deriving optimal paths [10].Their efficiency, however, is contingent on factors like grid size and the choice of heuristic methods, which can be limiting in more dynamic or constrained environments.
The advent of evolutionary intelligence algorithms -including genetic algorithms, ant colony optimization, simulated annealing, and particle swarm optimization -marked a significant milestone in global path optimization [9].These methods excel in accuracy and have found widespread application.Nevertheless, these methods require prior global environmental information, and their time and space complexity are high, limiting their adaptability to edge devices like mobile robots.Moreover, in most instances, wireless sensor networks can only provide limited environmental data, making the application of such algorithms challenging due to the extensive preprocessing needed, especially in dynamic unknown environments [3].
It is found that planning algorithms based on random sampling can adapt to more complex environments with lower time and space complexity and can effectively adapt to the path planning problem of mobile sensor nodes [16].such as the Rapidly-exploring Random Tree (RRT) and its derivatives, including RRT-CONNECT, RRT*, RRT*-FN, Informed RRT* and other algorithms [1][2]7], which can well adapt to the motion dynamics requirements of mobile nodes However, their reliance on random node generation introduces a significant drawback: the paths generated are often longer than optimal, lacking in efficiency and practicality for real-world applications.In order to obtain smooth, shortest and obstacle-avoiding paths, researchers combined Bezier, Benstine and other curves with heuristic and artificial intelligence methods, and proposed many excellent results [15,18].For example, the artificial potential field method based on Bezier curve adopted the quadratic Bezier method and artificial potential field method to avoid obstacles.Better optimization results can be obtained in sparse environment.However, for dense wireless sensor networks, performance degrades significantly.Therefore, in dense infinite sensor networks, a reasonable path planning algorithm is needed to obtain the path.
In recent developments, methods like the Dynamic Window Approach combined with deep reinforcement learning have shown improvements in dynamic obstacle avoidance, yet they face limitations in movement options and environmental adaptability [8].Bi-layer hybrid algorithms involving ACO, PSO, and A* have been explored for multi-task path planning, but their complexity and interaction requirements limit their efficiency in dynamic environments [14].
The Modified Adaptive Ant Colony Optimization (MAACO) algorithm offers faster convergence and reduced path lengths but struggles in dynamic scenarios requiring real-time adaptability [17].The Expanding Path RRT (EP-RRT) has improved efficiency but faces challenges in diverse environments and dynamic changes [4].The Probability Smoothing Bi-RRT (PSBi-RRT) algorithm enhances convergence speed and collision reduction but does not guarantee initial solution quality [11].A continuous RRT-based method using B-spline curves for non-holonomic mobile robots showed promise but needed additional collision avoidance mechanisms [5].
In response to these challenges, this paper introduces the Modified Rapidly-exploring Random Tree (M-RRT) search strategy, an innovative approach designed for efficient path planning in dense, dynamic wireless sensor networks.M-RRT represents a novel solution that adeptly combines the strengths of various existing algorithms while addressing their primary limitations.This strategy is especially suited for real-time applications where rapid adaptation to changing environments is paramount.The main contributions are: 1 This paper introduces the M-RRT search strategy to achieve coverage and connectivity in wireless sensor networks.The strategy accounts for the distribution of initial access points, obstacles, and static nodes, employing a density set to define the search area.
2 Enhanced path optimization is proposed, utilizing a greedy algorithm to eliminate intermediate nodes for a more efficient path.The optimization aims to achieve a path that meets mobile nodes' needs, including shortest distance, obstacle avoidance, and smooth trajectory.
Path smoothing with quadratic Bezier curves is utilized to refine the solution path obtained through the M-RRT strategy.This step aims to minimize angles between adjacent segments, ensuring the final path is not only optimal but also smooth for mobile nodes.

RRT Algorithm
The RRT algorithm initiates with the root node set at the initial point and proceeds to construct a random expansion tree by iteratively adding leaf nodes through random sampling.The expansion process encompasses four essential steps: (1) the random generation of a node; (2) identification of the nearest point in the tree to the randomly generated node and connecting them; (3) generation of a node along the line based on a predetermined value; and (4) insertion of the node into the tree if it satisfies certain criteria.Illustrated in Figure 1, the process commences with the starting point, represented as , and a randomly selected point .Subsequently, the nearest node is determined, and a point on the line between and is chosen, with the condition that the distance, D( , ), satisfies the prescribed step size .If this condition is met, the newly generated node is inserted into the tree; otherwise, the process is repeated until a suitable node is found.This iterative procedure continues until the destination point is reached.Ultimately, the constructed random tree provides a path from the initial point to the destination point, as depicted in Figure 1 and Figure 2.  2) Enhanced path optimization is proposed, utilizing a greedy algorithm to eliminate intermediate nodes for a more efficient path.The optimization aims to achieve a path that meets mobile nodes' needs, including shortest distance, obstacle avoidance, and smooth trajectory.
Path smoothing with quadratic Bezier curves is utilized to refine the solution path obtained through the M-RRT strategy.This step aims to minimize angles between adjacent segments, ensuring the final path is not only optimal but also smooth for mobile nodes.

RRT Algorithm
The RRT algorithm initiates with the root node set at the initial point and proceeds to construct a random expansion tree by iteratively adding leaf nodes through random sampling.The expansion process encompasses four essential steps: (1) the random generation of a node; (2) identification of the nearest point in the tree to the randomly generated node and connecting 2) Enhanced path optimization is proposed, utilizing a greedy algorithm to eliminate intermediate nodes for a more efficient path.The optimization aims to achieve a path that meets mobile nodes' needs, including shortest distance, obstacle avoidance, and smooth trajectory.
Path smoothing with quadratic Bezier curves is utilized to refine the solution path obtained through the M-RRT strategy.This step aims to minimize angles between adjacent segments, ensuring the final path is not only optimal but also smooth for mobile nodes.

RRT Algorithm
The RRT algorithm initiates with the root node set at the initial point and proceeds to construct a random expansion tree by iteratively adding leaf nodes through random sampling.The expansion process encompasses four essential steps: (1) the random generation of a node; (2) identification of the nearest point in the tree to the randomly generated node and connecting them; (3) generation of a node along the line

M-RRT Algorithm Modeling
After considering the assumptions, M-RRT is elaborated as follows: (1) All mobile nodes, except for themselves, are considered as obstacles; (2) The static sensing node has a radius of R static and the mobile node has a radius of R move ; (3) Let R obsi represent the projected radius of the i th obstacle in the forward direction; (4) By increasing the size of all obstacles by R move, the algorithm simplifies the search process by not needing to account for its own size.

Initial Region Setting Based On Coverage Density
The performance of the algorithm is directly influenced by the configuration of the initial region, which is conducted in two steps.
1 Establishing the baseline for the initial region: In a wireless sensor network, the baseline line refers to the line connecting the starting point and ending point.The search area extends on both sides of this line, with a length denoted as and .
2 Determining the size of the initial area, as depicted in Figures 3 and Figure 4, referred to as .
Based on the connectivity within a wireless sensor network, the number and distribution of nodes within a specific area can be deduced.Despite potential local inconsistencies, a generally uniform distribution is observable.To derive a more accurate initial value, this paper introduces the concept of obstacle coverage density.The initial area calculated by Equation ( 2) mi include significant obstacles within the ini area, rendering the RRT algorithm unable find a path from the initial region, as depicted Figure 5.

Figure 5
The Obstacle initial region Definition 1: Obstacle Coverage Density.This metric, defined within wireless sensor networks, quantifies the ratio of the combined projected area of all obstacles and static nodes to the total area covered by the wireless sensor network, as expressed in Formula (1).In this formula, represents the transmission area occupied by obstacles, denotes the individual area of a static node, signifies the overall area covered by the wireless sensor network, and indicates the quantity of static nodes. ( The calculation of the initial coverage area width, considering the coverage and connectivity characteristics of nodes, allows for the determination of the width of the initial coverage area ( ) for a given number of static nodes (M), as shown in Equation (2).The symbol [ ] denotes the operation of taking the integer part. (2) The initial area calculated by Equation (2) might include significant obstacles within the initial area, rendering the RRT algorithm unable to find a path from the initial region, as depicted in Figure 5.
However, at this time, it is found that the size of the initial area can be modified according to the obstacle projection, and the size to be adjusted can be calculated according to the obstacle projection, as shown in The initial area calculated by Equation ( 2) might include significant obstacles within the initial area, rendering the RRT algorithm unable to find a path from the initial region, as depicted in Figure 5.However, at this time, it is found that the size of the initial area can be modified according to the obstacle projection, and the size to be adjusted can be calculated according to the obstacle projection, as shown in Equations ( 3)-( 4).Formula ( 3) is to calculate the minimum distance from each obstacle to the baseline projection, where is the projection length of the obstacle on the right side of the baseline, is the projection length of the obstacle on the left side of the baseline, and is to find their minimum value.Formula (4) means to find the maximum value after the obstacle projection is solved, so that the value to be adjusted can be obtained. ( In accordance with the formulations presented in Equations ( 4) and ( 2), the derivation of Equation ( 5) is achievable through the process of unification, wherein the equation is consolidated to yield the essential initial region, denoted as .
The initial area calculated by Equation ( 2) might include significant obstacles within the initial area, rendering the RRT algorithm unable to find a path from the initial region, as depicted in Figure 5.However, at this time, it is found that the size of the initial area can be modified according to the obstacle projection, and the size to be adjusted can be calculated according to the obstacle projection, as shown in Equations ( 3)-( 4).Formula ( 3) is to calculate the minimum distance from each obstacle to the baseline projection, where is the projection length of the obstacle on the right side of the baseline, is the projection length of the obstacle on the left side of the baseline, and is to find their minimum value.Formula (4) means to find the maximum value after the obstacle projection is solved, so that the value to be adjusted can be obtained. ( In accordance with the formulations presented in Equations ( 4) and ( 2), the derivation of Equation ( 5) is achievable through the process of unification, wherein the equation is consolidated to yield the essential initial region, denoted as .
(1) Obstacle traversing the initial region (2) Obstacle forward direction projection Equations ( 3)-( 4).Formula ( 3) is to calculate the minimum distance from each obstacle to the baseline projection, where is the projection length of the obstacle on the right side of the baseline, is the projection length of the obstacle on the left side of the baseline, and is to find their minimum value.Formula (4) means to find the maximum value after the obstacle projection is solved, so that the value to be adjusted can be obtained. ( In accordance with the formulations presented in Equations ( 4) and ( 2), the derivation of Equation ( 5) is achievable through the process of unification, wherein the equation is consolidated to yield the essential initial region, denoted as . (5)

Extended Search Area Setting Based on Coverage Density
Due to the influence of the shape, size and position of obstacles, the RRT algorithm cannot find the path in the set area.In this case, the search area needs to be expanded, as shown in Equation ( 6).In Formula ( 6), the step size is expanded after each search failure.Repeat the preceding steps until the search path is found, as shown in Figures 6-7.

Figure 6
Region expansion

Figure 7
Extension length and collision detection position of obstacles, the RRT algorithm cannot find the path in the set area.In this case, the search area needs to be expanded, as shown in Equation ( 6). ( 6) In Formula ( 6), the step size is expanded after each search failure.Repeat the preceding steps until the search path is found, as shown in Figures 6-7.

Figure 6
Region expansion Extension length and collision detection

Op
The cons nece requ algo search area needs to be expanded, as shown in Equation ( 6).
In Formula ( 6), the step size is expanded after each search failure.Repeat the preceding steps until the search path is found, as shown in Figures 6-7.

Figure 6
Region expansion Extension length and collision detection

Op
The cons nece requ algo search area needs to be expanded, as shown in Equation ( 6).
In Formula ( 6), the step size is expanded after each search failure.Repeat the preceding steps until the search path is found, as shown in Figures 6-7.

Figure 6
Region expansion Extension length and collision detection (

1) Extension length (2) Collision and region detection
Normally, it can be set according to the initial value, but during the expansion process, the value depends on: (1) the coverage density, (2) whether the obstacle crosses the extended area; At the s taking Formula area, so area can (7), and Formula It can b after gr initial v natural formula become satisfyin

R
RRT ad the sear area lim not in again.F random Extende

G Optim
The pa consist necessa required algorith

(1) Extension length (2) Collision and region detection
Normally, it can be set according to the initial value, but during the expansion process, the value depends on: (1) the coverage density, (2) whether the obstacle crosses the extended area; (3) , the initial value of , as shown in Formula (7), k represents the number of extensions, indicating that the more the number of extensions, the smaller the area to be expanded; represents the coefficient, indicating the size of the initial area in the default state.(7) At the same time, if the obstacle is too large after taking into account the extended area, the Formula ( 6) is used to calculate the adjustment area, so that the width of the expanded search area can be obtained according to the Formula (7), and can be obtained, as shown in the Formula (8).It can be seen from the Formulas ( 7)-( 8) that after gradual expansion, is the sum of the initial value of the region multiplied by the natural reciprocal; from the limit summation formula.After infinite expansion, will become infinite to cover the entire region, thus satisfying the probability completeness.

Random Expansion Tree
RRT adds the boundary detection function in the search process after increasing the search area limit, as shown in Figure 7(2).When is not in the search area, nodes are generated again.Figure 8 shows the flow of M-RRT's random search tree algorithm (Extend_Tree).

Figure 8
Extended search tree of M-RRT tion area can be obtained according to the Formula (7), and can be obtained, as shown in the Formula ( 8).(8) It can be seen from the Formulas ( 7)-( 8) that after gradual expansion, is the sum of the initial value of the region multiplied by the natural reciprocal; from the limit summation formula.After infinite expansion, will become infinite to cover the entire region, thus satisfying the probability completeness.

Greedily Delete Node Optimization Path
The paths obtained through tree expansion consist of a series of nodes, which are not necessarily the shortest.Further optimization is required, and this paper employs a greedy algorithm to prune nodes from the expanded

Greedily Delete Node Optimization Path
The paths obtained through tree expansion consist of a series of nodes, which are not necessarily the shortest.Further optimization is required, and this paper employs a greedy algorithm to prune nodes from the expanded tree, resulting in a shorter path.The key steps of the proposed greedy algorithm are as follows: 1 Initialization of the original path , where represents the path between and obtained from a random tree .
2 Initialization of the sequence of optimized path nodes , with an initial empty value.The starting point from is inserted into as the first anchor point.
3 Iterative selection of anchor points and insertion into .Nodes are continuously selected from , and it is determined whether the line connecting the selected point and the anchor point intersects with obstacles.If no intersection occurs, the selection process continues until an intersection is detected.At this point, the node selected just before the intersection becomes the next anchor point, and it is inserted into Sk.The search continues until the endpoint becomes the last anchor point and is inserted into Sk.This process effectively removes many intermediate nodes.In Figure 9, the sequence of points represents the anchor points selected from .After this step, in wireless sensor networks, situations similar to Figure 10 often arise, where obstacles are contained between three anchor points.In such cases, the intermediate anchor points need to be removed, constituting step (4).The obstacle is in the middle of the anchors

Figure 10
The obstacle is in the middle of the three anchors tree, resulting in a shorter path.The key steps of the proposed greedy algorithm are as follows: (1) Initialization of the original path , where represents the path between and obtained from a random tree .
(2) Initialization of the sequence of optimized path nodes , with an initial empty value.The starting point from is inserted into as the first anchor point.
(3) Iterative selection of anchor points and insertion into . Nodes are continuously selected from , and it is determined whether the line connecting the selected point and the anchor point intersects with obstacles.If no

Shortest Smooth Path
The path formed by the optimized node sequence cannot meet the demand of stable movement of nodes, mainly because the node

Shortest Smooth Path
The path formed by the optimized node sequence cannot meet the demand of stable movement of nodes, mainly because the node sequence cannot meet the kinematic mechanical demand of mobile nodes at the connection point.For mobile nodes, their dynamic equation can be expressed by Formula ( 9).where, v represents the moving speed of the node, represents the angular speed, and represents the maximum angular speed acceptable to the mobile node.In this way, the minimum radius of the mobile node when turning can be obtained, as shown in Formula (10).(10) The necessary condition for a mobile node to move smoothly is that the curvature at each point of the path is less than the value that the mobile node can bear.Therefore, the average probability condition [8] is adopted here, which satisfies Equation (11), where a and b represent the slope of any two points on the curve and the change point of P(a).According to the greedy optimization path, there must be obstacles between the two-line segments.In special cases involving a static sensor node, the radius length is: . If an arc is arbitrarily taken from a circle with radius and length R, the appropriate moving time can be found to enable smooth movement of the mobile sensing node.This arc can be replaced by a quadratic Bezier curve.Therefore, the process of searching for a smooth path involves search control points in , ensuring that the path is both smooth and smooth, and the path is shortest.Given n+1 points, the smooth path is solved by determining the junction points a 1 , a 2 , ... ,a n , ... ,a 2n on the two adjacent line segments, satisfying Formula (11) and collision avoidance.This makes the path shortest, that is, satisfying the solution of the Equation ( 12).(12) In the Equation ( 12), B i represents the length of the second Bezier curve.The integral method is employed for solving, where the a 2i subscript in B i is an even number to start counting.The control points of the second quadratic curve, a 2i and a 2i+1 , indicate that the horizontal coordinate of the previous control point is less than the horizontal coordinate of the next curve li represents the remaining line segment on S i S i+1 , implying that, after smoothing, some line segments need to be traversed on S i S i+1 .L represents the sum of the length of the conic and the remaining line segment.To solve for L is to find an optimal conic so that the path is the shortest.
During the solution process, if only the quadratic curve between each two adjacent line segments meets the conditions for the shortest path.(that is, the quadratic curve does not intersect with obstacles, the curvature meets the Formula (11)), the path between the two-line segments is the shortest.The sequence of control points may cross, that is, the horizontal coordinate of the i control point is greater than the horizontal coordinate of the i+1 control point, resulting in a conflict, and the required path is not obtained, as shown in Figure 11, a i+2 >a i+3 .This situation, depicted in Figure 11 (a i+2 > a i+3 ), implies that when the sensor node moves to a i+2 , it needs to fall back to a i+3 , which doesn't align with the requirements.
It is evident from Formula (12) that the process of solving smooth curves is an NP-difficult problem.However, it has been observed that minimizing the angle between two-line segments fulfills the requirements of smooth processing without control point conflicts.Figure 12 illustrates this scenario with a i+2 =a i+3 , and

Mobile Node Planning Strategy
M-RRT includes five key processing processes: region initialization, random search, region extension, path optimization, and smoothing, as shown in Figure 14.The find extend Tree function calls Extend Tree(tree) and iterates step

Mobile Node Planning Strategy
RRT includes five key processing processes: region initialization, random search, region extension, path optimization, and smoothing, as shown in Figure 14.
The find extend Tree function calls Extend Tree(tree) and iterates step by step to determine whether the path can be searched for a finite number of times.If the path is found, Greedy Optimal(Tree) is called for optimization.Then call the Bezier Smooth(Path) function to smooth and get the required path.

Experimental Results and Analysis
This study conducts simulations in a rectangular coordinate system [0, 100] × [0, 100], deploying 120 static sensing nodes and 5 redundant mobile nodes, where Nodes=125, k=10, S=100, M=1, featuring low-density circular obstacles.The radius of static nodes is denoted as r = 0.1, while the radius of mobile nodes is represented by R=0.2.For the R-RRT algorithm, the step size is set to 3, is 10, is 2, and N is 200.The outcomes of the M-RRT algorithm, as depicted in Figure 15, and the RRT algorithm, as illustrated in Figure 16, are compared.The results reveal that the enhanced M-RRT algorithm confines the search region to a smaller space, yielding paths significantly superior to those generated by the RRT algorithm.

RRT algorithm after adding obstacles
To further verify the improved strategy proposed in this paper, the number of obstacles was increased to 100, with other conditions unchanged.The results, shown in Figure 17 for M-RRT and Figure 18 for RRT, clearly demonstrate M-RRT's superior performance.
To further verify the improved strategy proposed in this paper, the number of obstacles was increased to 100, with other conditions unchanged.The results, shown in Figure 17 for M-RRT and Figure 18 for RRT, clearly demonstrate M-RRT's superior performance.

RRT algorithm after adding obstacles
In order to further verify the superiority of M-RRT algorithm, a case study involving autonomous vehicle navigation with dynamic obstacles (e.g., moving vehicles and pedestrians) was examined.five maps were randomly generated by changing the number of obstacles, and M-RRT and RRT algorithms were used to In order to further verify the superiority of M-RRT algorithm, a case study involving autonomous vehicle navigation with dynamic obstacles (e.g., moving vehicles and pedestrians) was examined.five maps were randomly generated by changing the number of obstacles, and M-RRT and RRT algorithms were used to search the path.The experiments were repeated for 100 times, and their average convergence time and average path length were obtained, as shown in Figure 19 and 20, respectively.It can be seen from the results that the search time of M-RRT is much shorter than RRT, and the path length of M-RRT is also much shorter than RRT.The M-RRT algorithm facilitated efficient path planning, optimizing the route in real-time as the environmental conditions changed.This application highlighted the algorithm's potential in industries such as autonomous driving and drone navigation, where adaptability and real-time processing are critical.
search the path.The experiments were repeated for 100 times, and their average convergence time and average path length were obtained, as shown in Figure 19 and 20, respectively.It can be seen from the results that the search time of M-RRT is much shorter than RRT, and the path length of M-RRT is also much shorter than RRT.The M-RRT algorithm facilitated efficient path planning, optimizing the route in real-time as the environmental conditions changed.This application highlighted the algorithm's potential in industries such as autonomous driving and drone navigation, where adaptability and real-time processing are critical.

Conclusion
By leveraging the coverage and connectivity of wireless sensor networks, this paper introduces the M-RRT strategy for efficient path planning of mobile nodes.From setting the initial search area based on coverage density to optimizing paths using a greedy algorithm and smoothing with quadratic Bezier curves, the M-RRT strategy not only reduces search Comparison of average path length

Conclusion
By leveraging the coverage and connectivity of wireless sensor networks, this paper introduces the M-RRT strategy for efficient path planning of mobile nodes.From setting the initial search area based on coverage density to optimizing paths using a greedy algorithm and smoothing with quadratic Bezier curves, the M-RRT strategy not only reduces search time and path length but also provides a smooth and optimal path for mobile sensor nodes.Compared to the traditional RRT algorithm, the M-RRT algorithm demonstrates superior performance, especially in dense sensor networks, underscoring its advantages in terms of search efficiency and path quality.
time and path length but also provides a smooth and optimal path for mobile sensor nodes.Compared to the traditional RRT algorithm, the M-RRT algorithm demonstrates superior performance, especially in dense sensor networks, underscoring its advantages in terms of search efficiency and path quality.Future work will focus on expanding the algorithm's applicability to a wider range of real-world scenarios, further refining its computational efficiency, and exploring its integration into larger-scale autonomous systems.

Figure 1
Figure 1 Schematic diagram of RRT algorithm random tree growth

Figure 2
Figure 2 RRT planning path

Figure 3 Figure 4 Figure 4
Figure 3 Schematic diagram of the initial expanded area

Figure 4
Figure 4Initial area

Figure 5
Figure 5The Obstacle initial region

Figure 9
Figure 9Anchor sequence after iterative selection

( 4 )
Verification of the equality of le between and sequences.If they ar equal, is cleared, is copied to , cleared, and step (3) is repeated.If the equal, the optimization process is conclud

Figure 9
Figure 9Anchor sequence after iterative selection

4
Verification of the equality of lengths between and sequences.If they are not equal, is cleared, is copied to , is cleared, and step (3) is repeated.If they are equal, the optimization process is concluded.

Figure 11 Cross conflict of control points Figure 12
Figure 11Cross conflict of control points

Figure 13
Figure 13Smooth Optimization Path

Figure 11 Cross conflict of control points Figure 12
Figure 11Cross conflict of control points Figure 13Smooth Optimization Path

Figure 11 Cross conflict of control points Figure 12
Figure 11Cross conflict of control points

Figure 13
Figure 13Smooth Optimization Path

Figure 14 M
Figure 14 M-RRT path planning strategy

Figure 15 Path
Figure 15Path planning of M-RRT

Figure 18 RRT
Figure 18 RRT algorithm after adding obstacles

Figure 16
Figure 16 Path planning of RRT

Figure 17 M
Figure 17 M-RRT planning after adding obstacles

Figure 17 M 18 RRT
Figure 17 M-RRT planning after adding obstacles Figure 19Comparison of average search time

Figure 20
Figure 20 Comparison of average path length