Path Planning of a Mobile Robot Based on the Improved Rapidly Exploring Random Trees Star Algorithm

Abstract

With the increasing utilization of sampling-based path planning methods in the field of mobile robots, the RRT* algorithm faces challenges in complex indoor scenes, including high sampling randomness and slow convergence speed. To tackle these issues, this paper presents an improved RRT* path-planning algorithm based on the generalized Voronoi diagram with an adaptive bias strategy. Firstly, the algorithm leverages the properties of the generalized Voronoi diagram (GVD) to obtain heuristic paths, and a sampling region with target bias is constructed, increasing the purposefulness of the sampling process. Secondly, the node expansion process incorporates an adaptive bias strategy, dynamically adjusting the step size and expanding direction. This strategy allows the algorithm to adapt to the local environment leading to improved convergence speed. To ensure the generation of smooth paths, the paper employs the cubic spline curve interpolation algorithm for trajectory optimization to ensure that the mobile robotic can obtain the best trajectory. Finally, the proposed algorithm is experimentally compared with existing algorithms, including the RRT* and Informed-RRT* algorithms, to verify the feasibility and stability.

1. Introduction

With the rapid advancement of robotics, mobile robots have garnered significant attention in the industrial, agricultural, medical, and military sectors due to their high mobility, flexibility, and operational safety [1,2,3,4,5,6]. However, the complexity of tasks undertaken by mobile robots, particularly in intricate indoor environments with irregular obstacles [7], presents a significant challenge in finding feasible paths that meet the robots’ requirements [8]. Path planning, an essential field in mobile robotics, focuses on planning a collision-free path from start to end locations that are either optimal or suboptimal, ensuring the robots’ safety during their operations [9,10].

Currently, common path-planning algorithms can be categorized into three groups: search-based algorithms, sampling-based algorithms, and intelligent bionic algorithms [11]. Intelligent bionic algorithms are path-planning methods that mimic the evolutionary processes of organisms in nature or the collective behavior of swarming animals. Among them, ant colony algorithms [12] and genetic algorithms [13] are widely recognized. While intelligent bionic algorithms can be combined with other algorithms to solve optimization problems such as vehicle path planning and travel agency problems [14,15], their efficiency in solving robot path-planning problems is relatively low [16]. Search-based algorithms, such as A* [17] and D* [18], are known for finding optimal solutions. These algorithms guarantee finding a globally optimal solution under certain conditions, such as no time limitations. However, as the size of the map increases, the computational complexity of search-based algorithms also increases significantly. Therefore, while search-based algorithms have the advantage of finding globally optimal solutions, their computational efficiency can be a limitation. Sampling-based algorithms, such as rapidly exploring random trees (RRT) [19] and the probabilistic road map (PRM) [20], offer advantages in handling large-scale maps and high-dimensional problems. They exhibit strong flexibility and require low map accuracy, allowing for rapid path-planning solutions. Nonetheless, due to the random sampling approach, the paths obtained are not necessarily optimal. To address this issue, Karaman et al. proposed the rapidly exploring random trees star (RRT*) algorithm [21]. By relaxing the iteration limit, RRT* can ensure asymptotic optimality. Indeed, the RRT* algorithm still has limitations in both the sampling and expansion stages [22].

In recent years, researchers have introduced several improvement measures to enhance the performance of sampling-based algorithms. Kuffner et al. proposed RRT-connect [23], which entails creating two trees in the environment. This technique effectively speeds up the random tree search process by performing a bidirectional search from the starting point and the goal point, but the final path is not optimal. Wang et al. proposed target-biased bidirectional RRT* [24], which optimizes the selection of sampling points through a target-biased strategy. Nasir et al. introduced RRT*-smart [25], an algorithm that accelerates convergence speed by incorporating intelligent sampling and path optimization techniques. The abundance of obstacles in indoor environments often requires extensive map-wide sampling to obtain a high-quality path. Reducing unnecessary sampling regions is crucial for improving the efficiency of path-planning algorithms. Gammell et al. proposed informed RRT* [26], which accelerates convergence by constraining the sampling region through the construction of elliptical sampling regions after obtaining an initial path. Ding et al. proposed an improved RRT* algorithm [27] by heuristic search and greedy sampling strategy, but this initiative does not guarantee the optimality of the heuristic path. The GVD [28] method is capable of effectively restricting the sampling region. Wang et al. proposed combining GVD with multiple potential functions to achieve non-uniform sampling [29]. Zhang et al. introduced a novel target area adaptive rapidly exploring random tree algorithm (TAA-RRT*) [30], which utilizes environmental information to achieve non-uniform sampling. In indoor environments, where obstacles have varying shapes and sizes, the step size of an algorithm has a significant impact on its convergence speed. Employing an appropriate expansion strategy is beneficial for achieving fast convergence of the path. P-RRT* [31] and PQ-RRT* [32] utilize artificial potential fields [33] to generate new nodes that are closer to the desired values, thereby reducing the time required to expand into the target region. Fan et al. proposed a target-biased bidirectional APF-RRT* algorithm [34], which incorporates a greedy strategy to bias the sampling nodes towards the target point, guiding the growth of the tree.

Inspired by the aforementioned literature, this paper proposes an improved RRT* algorithm to enhance its efficiency during the sampling and expansion stages. Firstly, a target-biased sampling region is constructed utilizing the properties of a generalized Voronoi diagram, and subsequent sampling will be carried out in this region, which effectively improves the purpose and efficiency of sampling. Secondly, an adaptive bias strategy is incorporated to control the expansion length and direction of new nodes, enhancing the effectiveness of the expansion process and facilitating rapid path convergence. Finally, a cubic spline interpolation algorithm is used to smooth the path. Extensive simulation experiments are conducted to compare the proposed improvements, demonstrating that the improved RRT* algorithm effectively addresses the limitations of the original RRT* algorithm in the sampling and expansion stages.

The structure of this paper is as follows: Section 2 discusses the definition of mobile robot path-planning problem. Section 3 provides an overview of RRT and RRT*, highlighting the limitations of the original RRT* algorithm in Section 3.3. Section 4 presents a detailed description of the improved RRT* algorithm, including the construction of a target-biased sampling region using a generalized Voronoi diagram in Section 4.2 and the utilization of an adaptive bias strategy to adjust node expansions in Section 4.3. Section 5 conducts comprehensive experimental comparisons and applies a cubic spline curve to optimize the paths generated by the improved RRT* algorithm, demonstrating its feasibility. Section 6 concludes the paper and provides prospects for future research.

2. Problem Definition

This segment provides the foundation for understanding the core concepts of path planning problems and introduces the notation used, which is essential for comprehending the enhancements proposed in Section 4. The objective of path planning in mobile robotics is to find a collision-free path from an initial starting position to a predetermined goal location. This article focuses on path-planning algorithms for mobile robotics operating in complex indoor environments. To accurately define the path planning problem, the environment is abstracted into a mathematical representation known as the state space. The state space, denoted as X, represents a 2D space that encompasses the entire indoor environment. $X_{obs}$ represents a subset of this space that corresponds to areas where the mobile robot cannot traverse, forming the obstacle space. Conversely, the free exploration area for the mobile robot referred to as $X_{free}$, represents the space available for exploration and path planning.

$$X_{obs}\subseteq X$$

(1)

$$X_{free}=X/X_{obs}$$

(2)

The path-planning problem is formulated with $(X,x_{start},x_{goal})$ as its input parameters, wherein $x_{start}$ denotes the start or root node within the randomly generated tree, while $x_{goal}$ signifies the goal node of the tree. In the randomized tree, if there exists a continuous function $\sigma :[0,1]\to X$, the bounded function is represented as a path in state space. Assuming $v\in [0,1]$ and $\sigma \left(v\right)\in X_{free}$, $\sigma$ is said to be a collision-free path from the root node to the goal node.

Definition 1.

Feasible Path

Given a path-planning problem formulated as $(X,x_{start},x_{goal})$, it theoretically always finds any solution to a feasible path $\sigma \left(v\right)$. If no viable solutions are identified, the outcome is reported as a failure, indicative of the algorithm’s inability to ascertain a feasible path within the prescribed parameters.

$$\sigma \left(v\right)\to X_{free},\sigma \left(0\right)=x_{start},\sigma \left(1\right)=x_{goal}$$

(3)

Definition 2.

Optimal Path

Given a path-planning issue represented by $(X,x_{start},x_{goal})$, let Σ represent the set of all possible pathways. The path cost function $Cost\left(\sigma \right)$ calculates the Euclidean distance between the starting node and the goal node for a given path σ. If ${\sigma }^{\prime }$ and the path cost $Cost\left({\sigma }^{\prime }\right)$ are minimized in $Cost\left(\sigma \right)$, then ${\sigma }^{\prime }$ is considered the optimum path.

$${\sigma }^{\prime }=\underset{\sigma \in \Sigma }{argmin}Cost\left(\sigma \right)$$

(4)

Definition 3.

Minimum Time Cost

Assume the optimal path constraint $Cost\left({\sigma }^{\prime }\right)<L$, where L is a program setup value corresponding to the minimum time cost $t_{min}\in R$ to check the fastness of the algorithm.

3. RRT Algorithm and RRT* Algorithm

In this section, we introduced the rapidly exploring random tree algorithm and its variant, RRT*, followed by a discourse on the limitations inherent to the RRT* algorithm.

3.1. RRT Algorithm

The RRT algorithm stands out as one of the most prominent approaches in sampling-based path planning. It generates a random search tree in the workspace, with the root node representing the starting point. In the state space, random sampling is employed to generate sample nodes, and the nearest node in the tree is selected to extend a new node using a predetermined step size. As the number of nodes increases, the search tree progressively covers the entire state space. The path search process concludes when a path connecting the start node to the goal node is found. The algorithm is based on whether the link between the new node and the nearest node collided or not. Then it determines whether the connection can be built to expedite the search process and overcome the difficulties of huge computation and high dimensionality in the traditional path planning algorithm. Figure 1 illustrates the principle of the classic RRT algorithm, depicting the generation and expansion of the search tree.

In Figure 1, $x_{start}$ denotes the root node of the random search tree T. The symbol $x_{goal}$ represents the goal node, while $x_{rand}$ represents a random node for sampling. $x_{near}$ represents the node in the search tree that is nearest to the random node. $x_{new}$ represents a new node and takes $x_{near}$ as the parent node to grow a fixed step size D in the direction from $x_{near}$ to $x_{rand}$ to obtain it. $X_{goal}$ denotes the goal region in the state space with radius $Thr$. $X_{obs}$ indicates the obstacle space.

3.2. RRT* Algorithm

The RRT* algorithm is an improvement of the RRT algorithm that focuses on optimizing the quality of the generated paths. It incorporates two key improvements, outlined in Algorithm 1. The first enhancement is the reselection of the parent node, as shown in Algorithm 2. After generating a new node, the algorithm reselects the closest node in the search tree as the parent of the new node. The second enhancement is the rewrite strategy, as depicted in Algorithm 3. This strategy involves measuring and updating the path cost for the nodes surrounding the new node. With the same sampling sites, the RRT* method yields a substantial reduction in path cost compared to the RRT algorithm.

Algorithm 1: RRT*
	Input   : $Map,x_{start},x_{goal}$
	Output: $Path$
1	$V\leftarrow \left\{x_{start}\right\},E\leftarrow ⌀,T=(V,E)$;
2	for $iter\in [1,Maxiter]$ do
3		$x_{rand}\leftarrow RandSample\left(X\right)$;
4		$x_{near}\leftarrow Nearst(V,x_{rand})$;
5		$x_{new}\leftarrow Steer(x_{near},x_{rand},D)$;
6			if $CollisionChecking(Map,x_{nera},x_{new})$ then
7				$X_{near}\leftarrow NearCricle(V,x_{new})$;
8				$x_{min}\leftarrow ChooseParent(X_{near},x_{new},x_{near}$);
9				$x_{nearest}\leftarrow x_{min}$;
10				$V\leftarrow V\cup \left\{x_{new}\right\}$;
11				$E\leftarrow E\cup \left\{(x_{nearest},x_{new})\right\}$;
12				$T\leftarrow Rewrite(T,X_{near},x_{new})$;
13			end
14			if $norm(x_{new}-x_{goal})<Thr$ then
15				$Path\left(1\right)\leftarrow x_{goal}$;
16				$Path(2,end)=GetPath(T,x_{start},x_{new})$;
17				break;
18			end
19	end

Algorithm 2: ChooseParent
	Input   : $X_{near},x_{new},x_{near}$
	Output: $x_{min}$
1	$x_{min}\leftarrow x_{near}$;
2	$T_{mincost}\leftarrow T\left(x_{near}\right)+dist(x_{new},x_{near})$;
3	for $x_{nearest}\in X_{near}$ do
4			$T_{cost}\leftarrow T\left(x_{nearest}\right)+dist(x_{new},x_{nearest})$;
5			if $T_{mincost}>T_{cost}$ then
6				$T_{mincost}=T_{cost}$; $x_{min}\leftarrow x_{nearest}$;
7			end
8	end

Algorithm 3: Rewrite
	Input   : $T,X_{near},x_{new}$
	Output: T
1	for $x_{near}\in X_{near}$ do
2			if $T\left(x_{new}\right)+dist(x_{new},x_{near})<T\left(x_{near}\right)$ then
3				$E\leftarrow ChangeParent\left(x_{near}\right)$;
4				$T\leftarrow UpdataCost\left(x_{near}\right)$;
5			end
6	end

3.3. Limitations of RRT* Algorithm

The RRT* algorithm has been proven to provide asymptotically optimal paths [21], which means that as the number of nodes approaches infinity, the generated paths will converge to the optimal solution. This property makes RRT* an improvement over the basic RRT algorithm. Despite its advantages, the RRT* algorithm still has certain limitations that should be addressed. These limitations include the following:

(1)

The generation of high-quality paths necessitates a substantial number of samples, resulting in increased computational complexity. This limitation arises from the necessity of thoroughly exploring the configuration space, which can be time-consuming.

(2)

The conventional RRT* algorithm suffers from a lack of purposeful sampling, resulting in excessive sampling in certain areas and slowing down the convergence speed. This issue arises because the random sampling strategy fails to prioritize regions that are more likely to contribute to the optimal path.

(3)

The fixed step sizes employed in the RRT* algorithm impede its adaptability to diverse environments. In confined spaces, larger step sizes constrain the expansion of the tree, limiting exploration capabilities. Conversely, smaller step sizes are inefficient in open areas as they lead to excessive tree growth without substantial progress towards the goal closest. This limitation arises from the inability of fixed step sizes to dynamically adjust according to the characteristics of the environment.

To address and refine these limitations, this paper proposes two improvements. The first improvement introduces non-uniform sampling techniques to enhance the efficacy and purposefulness of sampling. By assigning higher sampling densities to regions more likely to yield optimal paths, this approach effectively mitigates the issues of high computational complexity and inefficient convergence. The second improvement focuses on implementing an adaptive bias strategy to improve the algorithm’s adaptability. By dynamically adjusting the step sizes and expanding direction based on the environment’s characteristics, this strategy enables more efficient exploration in confined spaces and facilitates faster progress in open areas. These methods collectively aim to overcome the limitations of the RRT* algorithm, enabling the generation of high-quality paths with improved computational efficiency, faster convergence, and enhanced adaptability to varying environmental conditions.

4. The Improved RRT* Algorithm

In this section, we describe in detail the fundamental principles of each component module of the improved RRT* algorithm. Initially, the generalized Voronoi diagram generation for environment maps is presented. Subsequently, the acquisition of heuristic paths and the construction of sampled regions are illustrated. Furthermore, the implementation of the adaptive bias expansion strategy is described. Finally, our proposed improved RRT* algorithm is analyzed in general.

4.1. Generalized Voronoi Diagram

Let’s assume that the workspace of the mobile robot is denoted as X which consists of convex obstacles $C_1,\dots ,C_n$. The set of convex obstacles encompasses the boundaries of X, while the non-convex obstacles are represented as concatenated sets of convex sets, denoted as $C_i$. In Equation (5), the distance from any point in X to any point in p can be defined as The distance from any point p in X to $C_i$ can be defined.

$$d_i\left(p\right)=\underset{c_0\in C_i}{min}\left|\right|p-c_0\left|\right|$$

(5)

In this context, the symbol $\left|\right|·\left|\right|$ represents the norm operation between two points, while $c_0$ represents the closest obstacle point to point p on the segment $ci$. The gradient of $d_i$ is determined by Equation (6).

$$▿d_i\left(p\right)=\frac{p-c_0}{\left|\right|p-c_0\left|\right|}$$

(6)

Furthermore, Equation (7) defines the distance function $D\left(p\right)$ as the measure of proximity between point p and the nearest obstacle.

$$D\left(p\right)=\underset{i\in 1,\dots ,n}{min}{d}_i\left(p\right)$$

(7)

In a space X containing only disjoint obstacles, it is possible to define the two-equidistant surface using Equation (8).

$$F_{ij}=\{p\in X|d_i\left(p\right)=d_j\left(p\right)and▿d_i\left(p\right)\ne ▿d_j\left(p\right)\}$$

(8)

$$GVD=\{F_{ij},F_{ij}\in X\}$$

(9)

The generalized Voronoi diagram consists of edges and vertices, where $F_{ij}$ represents its edges. When visualizing the edges that do not intersect with obstacles, the resulting diagram is known as the Voronoi diagram. In Figure 2a, the solid green line represents the Voronoi diagram of the environment. For path planning in the same scenario, the topology of the Voronoi diagram can be reused by constructing it only once, which significantly simplifies the planning complexity and speeds up the search time.

4.2. Construction of Heuristic Paths and Sample Regions

Traditional RRT* path-planning algorithms use global random sampling to select sampling nodes, which results in a uniform distribution of sampling nodes over the entire map. However, this approach is time-consuming and inefficient in terms of computational resources. In order to improve the purposefulness of sampling, this paper proposes a target bias sampling strategy.

This strategy constructs a topological map of the environment through the properties of a generalized Voronoi diagram. Then, a heuristic path is searched on this topological map using the A* algorithm. The heuristic path acquisition process is shown in lines 3–4 of Algorithm 4: Given starting node $x_{start}$ and goal node $x_{goal}$, the nearest vertices $p_{start}$ and $p_{goal}$ on the GVD are identified for each. The A* search algorithm is then quickly executed on the topological map to derive heuristic paths ${\sigma }_{heuri}$. In Figure 2b, the blue path denotes the heuristic path.

Finally, a series of sampling circles are generated with the heuristic path point as the center, and the distance from the path point to the nearest obstacle as the radius. This region is referred to as the target bias sampling region $X_{tbs}$, and lines 5–9 of Algorithm 4 describe the construction of this region in detail. In Figure 2c, $X_{tbs}$ is visually depicted as light blue-filled areas.

$$X_{tbs}=\left\{Cir\left(p_i\right)\right|p_i\in {\sigma }_{heuri}\}$$

(10)

In Equation (10), ${\sigma }_{heuri}$ represents the heuristic path, $p_i$ denotes the heuristic path point, and $Cir\left(p_i\right)$ denotes the sampling circle. To achieve non-uniform sampling, a target bias sampling region, denoted as $X_{tbs}$, is constructed using the heuristic path point as the circle’s center and the distance to the nearest obstacle as the radius. As a result, conducting uniform sampling within $X_{tbs}$ ensures the identification of at least one feasible path. The main objective of this study is to replace the global sampling region X of the original RRT* path planning algorithm with a goal-biased sampling region $X_{tbs}$. This strategy improves the sampling efficiency by focusing the sampling region on the region that is more likely to lead to the optimal path, thus avoiding the blindness caused by random sampling.

Algorithm 4: $Construction{X}_{tbs}$
	Input   : $Map,x_{stat},x_{goal}$
	Output: $X_{tbs}$
1	$ObsInformed=ObstacleCheck\left(Map\right)$;
2	$GVD=BuildGVD(Map,ObsInformed)$;
3	$p_{stat},p_{goal}=FindNearestVertices\left(GVD\right)$;
4	${\sigma }_{heuri}=A^{\ast }(GVD,p_{stat},p_{goal})$;
5	for $p\in {\sigma }_{heuri}$ do
6		$Rad=MinDistanceToObs\left(p\right)$;
7		$Cir=Circle(p,Rad)$;
8		$X_{tbs}=[X_{tbs},Cir]$;
9	end

4.3. Adaptive Bias Expansion Strategy

In the traditional RRT* algorithm, the extended step size is typically predetermined as a fixed parameter, and then the iterative process of constructing the random tree is executed. However, the choice of step size significantly impacts both the quality of the generated path and the efficiency of the search. A larger step size is generally more effective in wide regions, while a smaller step size is more suitable for navigating through narrow regions. This limitation restricts the flexibility of the RRT* algorithm when dealing with complex environments.

An adaptive bias expansion strategy is introduced to adjust the step size and node growth direction dynamically to address this challenge. This strategy considers two crucial factors. One is the information about the density of obstacles around the nearest node, and the other is whether the nearest node’s connection to the goal point is in collision or not. After finding the nearest node, the program does not execute the expansion process immediately. First, the distribution of obstacles around the nearest node is analyzed with the nearest point as the center of the circle and R as the radius to provide information about the density of obstacles. Figure 3 illustrates the implementation of the dynamic step-size process for the dispersion obstacle and dense obstacle cases.

$$D_{adaptive}=\left\{\begin{array}{cc}\hfill \alpha D,& \hfill R>D,\\ \hfill \alpha R,& \hfill else.\end{array}\right.$$

(11)

$$\alpha =e^{-\rho }$$

(12)

The adjustment of the step size $D_{adaptive}$ is mathematically expressed in Equation (11). R denotes the radius of $Cir\left(p_{near}\right)$ where the nearest heuristic path point to $x_{near}$ is located. D represents the preset step size. The weight coefficient $\alpha$ takes values between 0 and 1, where a smaller value indicates denser obstacles (closer to 0), and a larger value indicates sparser obstacles (closer to 1). In Equation (12), $\rho$ denotes the obstacle density coefficient, which comes from the area share of obstacles within the circle. By incorporating these factors, the step size can be adjusted dynamically to adapt to the different obstacle configurations in the environment.

Second, the strategy connects $x_{near}$ and $x_{goal}$ for collision detection as a way to determine whether to grow directly toward $x_{goal}$. Two cases of growth towards random and goal points are shown in Figure 4. This operation resembles a greedy decision-making process but with a distinction. While greedy decision-making relies on a set threshold to determine whether to designate the sampling point as the goal point, it does not guarantee the effectiveness of node expansion. In contrast, the adaptive bias expansion strategy ensures the effectiveness of expansion while addressing the limitations of the RRT* algorithm in terms of convergence speed.

$$x_{new}=\left\{\begin{array}{cc}\hfill x_{near}+D_{adaptive}\ast (x_{goal}-x_{near}),& \hfill CollisionChecking(Map,x_{near},x_{goal}),\\ \hfill x_{near}+D_{adaptive}\ast (x_{rand}-x_{near}),& \hfill else.\end{array}\right.$$

(13)

Combining the dynamic adjustment step and node growth direction yields all the elements needed to expand a new node. As shown in Equation (13), $x_{new}$ is generated based on $D_{adaptive}$ and $CollisionChecking\left(\right)$ results.

Algorithm 5: $AdaptiveSteer$
	Input   : $x_{near},x_{rand},D$
	Output: $x_{new}$
1	$p_{near}=FindNearestPoint\left({\sigma }_{heuri}\right)$;
2	$R=GetRadius\left(Cir\left(p_{near}\right)\right)$;
3	$\rho =GetObstacleDensity(x_{near},R,ObsInformed)$;
4	$\alpha =e^{-\rho }$;
5	if $R>D$ then
6		$D_{adaptive}=\alpha D$;
7	else
8		$D_{adaptive}=\alpha R$;
9	end
10	if $CollisionChecking(x_{nera},x_{goal},Map)$ then
11		$x_{new}=x_{near}+D_{adaptive}\ast (x_{goal}-x_{near})$;
12	else
13		$x_{new}=x_{near}+D_{adaptive}\ast (x_{rans}-x_{near})$;
14	end

To further analyze the overall framework for improving RRT*, we give the adaptive bias strategy implemented as an algorithmic pseudo-code as shown in Algorithm 5. The algorithm performs parameter initialization in lines 1–4, dynamic adjustment of the step size in lines 5–9, and adjustment of the node growth direction and generation of new nodes in lines 10–14.

4.4. The Improved RRT* Algorithm

This study introduces an improvement to the RRT* algorithm by addressing three key aspects. The pseudo-code of the algorithm is shown in Algorithm 6. Line 2 of the algorithm is to generate the sampling region $X_{tbs}$ with target bias based on the Voronoi diagram. Once $X_{tbs}$ is constructed, line 4 of the algorithm $x_{rand}$ will be generated in this region. Line 6 of the algorithm implements the implementation of the goal bias strategy to assist in the generation of $x_{new}$.

• Efficient sampling strategy: In order to improve the purposefulness of sampling, this paper proposes a target bias sampling strategy. This strategy constructs a topological map of the environment through the properties of a generalized Voronoi diagram. Then, a heuristic path is searched on this topological map using the A* algorithm. Finally, the heuristic path point is the center of the circle to the distance to the nearest obstacle as the radius to construct a set of consecutive sampling circles. This set of sampling circles is called the target-biased sampling region $X_{tbs}$, and it will restrict the original global sampling region of the RRT* algorithm to that region. The detailed implementation of this part is described in Section 4.1 and Section 4.2.
• High-quality expansion strategy: To speed up the path convergence, this paper proposes an adaptive expansion strategy. After this strategy finds $x_{n}ear$, it does not run the expansion process directly. First, it analyzes the density of obstacles near $x_{n}ear$ to develop a dynamic step size adjustment strategy. Then, it connects the nearest node with the goal node for collision detection to decide the node growth direction. The specific implementation of this part is described in Section 4.3.
• Optimization algorithm: The presence of too many inflection points in the path planning result makes it difficult to implement the mobile robot. In order to make the path smoother, this paper uses the third spline interpolation algorithm to optimize the final path. The improved effect of this part is shown in Section 5.4.

The improved algorithm effectively addresses the challenges of high sampling randomness and slow convergence speed encountered in the RRT* algorithm by optimizing the sampling and expansion processes. During the preprocessing phase, the algorithm creates a map topology using a generalized Voronoi diagram and utilizes the A* algorithm to obtain heuristic paths on this topology. To optimize the sampling process, a target-biased sampling region, denoted $X_{tbs}$, is constructed with the heuristic path points as the center of the circle and the distance to the nearest obstacle as the radius. Algorithm 4 provides a detailed implementation of this process. In addition, to optimize the expansion process an adaptive bias expansion strategy is proposed to guide the generation of new nodes. This strategy dynamically adjusts the length and direction of generating new nodes, facilitating in-depth exploration of the environment. Algorithm 5 provides a detailed procedure for guiding the generation of new nodes. By combining these optimization methods, the improved algorithm overcomes the limitations of high sampling randomness and slow convergence, thereby enhancing the performance of path planning.   

Algorithm 6: The Improved RRT*
	Input   : $Map,x_{start},x_{goal}$
	Output: $Path$
1	$V\leftarrow \left\{x_{start}\right\}, E\leftarrow ⌀,T=(V,E)$;
2	$X_{tbs}=Construction{X}_{tbs}(Map,x_{start},x_{goal})$;
3	for $iter\in [1,n]$ do
4			$x_{rand}\leftarrow RandSample\left(X_{tbs}\right)$;
5			$x_{near}\leftarrow Nearst(V,x_{rand)}$;
6			$x_{new}\leftarrow AdaptiveSteer(x_{near},x_{rand},D)$;
7			if $CollisionChecking(Map,x_{nera},x_{new})$ then
8				$X_{near}\leftarrow NearCricle(V,x_{new})$;
9				$x_{min}\leftarrow ChooseParent(X_{near},x_{new},x_{near}$);
10				$x_{nearest}\leftarrow x_{min}$;
11				$V\leftarrow V\cup \left\{x_{new}\right\}$;
12				$E\leftarrow E\cup \left\{(x_{nearest},x_{new})\right\}$;
13				$T\leftarrow Rewrite(T,X_{near},x_{new})$;
14			end
15			if $norm(x_{new}-x_{goal})<Thr$ then
16				$Path\left(1\right)\leftarrow x_{goal}$;
17				$Path(2,end)=GetPath(T,x_{start},x_{new})$;
18				break;
19			end
20	end

5. Simulation Results

In this section, a comprehensive set of comparative experiments is conducted in diverse environments to demonstrate the superiority of the proposed improvement scheme. To validate the feasibility of the proposed method, the trajectory of the resulting path is optimized to meet the motion requirements of the mobile robot.

The experiments in this paper were conducted on a Windows 10 64-bit operating system using an Intel i5-12490f processor running at 3.0 GHz and equipped with 16 GB of RAM. The simulation software utilized for the experiments was MATLAB R2018a. The algorithms were evaluated through simulations in three distinct environments: a wide environment, a narrow environment, and a complex environment, as depicted in Figure 5. The simulation maps were produced with reference to [29,30]. This study employed the same subroutines for all algorithms to ensure a fair comparison and verify performance differences between the algorithms, including collision detection, neighborhood node traversal, and path cost updating. The algorithms compared in this section include RRT*, informed-RRT*, and improved RRT* algorithms.

The size of the simulation environment is 800 × 800 a 2D grid map. Blue dots indicate starting points and red dots indicate goal points. The blue line segment is the connecting line between $x_{new}$ and $x_{nearest}$ for each iteration of the random tree, the green solid line is the result of the path planning, the red solid line indicates the result of the optimized path, and the area where the mobile robot cannot move is the obstacle indicated by the black color block. In order to highlight the growth process of the random tree of different algorithms, the result when the number of samples is 1000 is chosen as the simulation result. A measurement of 1.05 times the optimal path length is used as the termination condition, and each algorithm is simulated 50 times in the environment to obtain the experimental data table.

5.1. Wide Environment

In the wide environment, the start point’s coordinates are (20, 20) and the goal point’s coordinates are (700, 700). The simulation results are illustrated in Figure 6.

Figure 6a showcases the RRT* algorithm, which employs uniform sampling throughout the space. However, this approach leads to a large number of redundant nodes during the growth of the random tree. In Figure 6b, the informed-RRT* algorithm utilizes an elliptical sampling method to narrow down the sampling area. Despite this improvement, the elliptical sampling area still includes obstacles, resulting in less effective iterations that slow down the convergence speed during random tree growth. Finally, Figure 6c illustrates the improved algorithm proposed in this paper. This algorithm improves sampling efficiency by constructing the $X_{tbs}$ sampling region. Combined with the adaptive bias strategy, it further accelerates the convergence speed of the algorithm. As a result, the algorithm achieves path planning in the environment using fewer sampling points while obtaining high-quality planning results.

Analyzing the experimental results presented in

Table 1. Wide environment simulation data.

Algorithm	Samples (n)	Initial Length	Optimal Length	Search Time (s)
RRT*	1154	1121.57	1026.86	6.27
Informed-RRT*	1024	1115.26	1023.90	4.46
Improved RRT*	395	1021.17	1021.17	0.94

reveals significant improvements achieved by the improved algorithm proposed in this paper. Compared to informed-RRT*, the search time required to obtain the optimal solution is reduced by 78.82%. Furthermore, in comparison to the RRT* algorithm, the improved algorithm showcases remarkable performance. It reduces the number of samples by 65.77% and decreases the search time by 85.00%. These results demonstrate the algorithm’s advantages in terms of fast convergence and stability within the wide environment.

5.2. Narrow Environment

In the narrow environment, the start point’s coordinates are (80, 420) and the goal point’s coordinates are (720, 420). The simulation results are illustrated in Figure 7.

The simulation results presented in Figure 7 demonstrate that as the density of obstacles increases in the environment, particularly in narrow channels, the effectiveness of sampling is significantly diminished. In the narrow channel, both the RRT* algorithm and the informed-RRT* algorithm show a concentration of samples near the start and end points in the simulation results. This concentration arises from the scarcity of effective sampling within the narrow channel. Consequently, a significant number of ineffective iterations are generated during the process, leading to a substantial decrease in search efficiency. In contrast, the improved RRT* algorithm proposed in this paper exhibits remarkable adaptability to sampling within narrow channels, thanks to the combination of an adaptive bias strategy. Consequently, it can swiftly generate high-quality path results, offering a substantial improvement over the existing approaches.

In

Table 2. Narrow environment simulation data.

Algorithm	Samples (n)	Initial Length	Optimal Length	Search Time (s)
RRT*	1513	801.74	679.53	2.72
Informed-RRT*	1309	794.98	681.16	2.05
Improved RRT*	278	664.97	662.49	0.89

, the informed-RRT* algorithm, when compared to the RRT* algorithm, exhibits a gradually diminishing advantage in sampling, which is also a drawback of the elliptical sampling region. In this paper, the improved RRT* algorithm demonstrates a 56.58% reduction in search time compared to the Informed-RRT* algorithm, along with a 78.76% decrease in the number of samples. These improvements represent a 67.27% and 81.62% decrease, respectively, compared to the RRT* algorithm. These findings further emphasize the necessity and effectiveness of the proposed adaptive bias strategy.

5.3. Complex Environment

In the complex environment, the start point is located at (222, 133) and the goal point is located at (230, 550). As the complexity of the environment increases, the advantage of Informed-RRT* over RRT* becomes less apparent. The simulation results in Figure 8 illustrate that the improved RRT* algorithm proposed in this paper exhibits a clearer sampling objective, resulting in shorter search times for high-quality initial and optimal solutions.

Table 3. Complex environment simulation data.

Algorithm	Samples (n)	Initial Length	Optimal Length	Search Time (s)
RRT*	1291	1100.59	1042.44	4.07
Informed-RRT*	1190	1073.65	1046.66	2.73
Improved RRT*	370	1040.14	1014.88	0.83

presents the experimental results, which demonstrate that the improved RRT* algorithm not only reduces the number of samples by 68.90% and the search time for the optimal solution by 69.59% when compared to the informed-RRT* algorithm but also achieves higher quality in terms of the initial and optimal solutions. When compared to the RRT* algorithm, the improved algorithm reduces the number of samples and search time by 71.34% and 79.60%, respectively.

Each algorithm was run 50 times, and their box plots were generated as shown in Figure 9. The black line in the box plot represents the median, while the red dots indicate outliers. Figure 9 compares the experimental data of all algorithms in three different environments, including the number of samples, the cost of the optimal path, and the search time. The box plot for the number of samples, depicted in Figure 9a, reveals that our proposed improved RRT* algorithm requires the fewest samples, indicating its superior sampling strategy compared to other algorithms. The box plot for the length of the optimal path, shown in Figure 9b, demonstrates that our algorithm consistently achieves shorter path costs compared to other algorithms, even when the cost is used as a termination condition in all three environments. The comparison of search time data in Figure 9c shows that our proposed algorithm is more efficient, particularly in narrow or complex environments. This validates the enhanced adaptability and stability of our proposed algorithm.

5.4. Path Optimization

During the simulations conducted in three different environments (Figure 6, Figure 7 and Figure 8), the path generated for the mobile robot exhibits high quality but falls short of being optimal. The presence of excessive turning points and large curvatures in the paths makes it impractical for the mobile robot to directly adopt them. To address this issue, the improved RRT* algorithm proposed in this paper incorporates the cubic spline interpolation algorithm to optimize the generated path. The optimization results are illustrated in Figure 10.

The primary objective of this optimization approach is to eliminate inflection and redundant points in the path, thereby obtaining a smooth path that aligns with the dynamic constraints of the mobile robot. The red path shown in Figure 11 represents the smoothed path after the optimization process. It is evident that the optimized path contains fewer redundant and inflection points. Through the reduction of redundant points, the path becomes smoother and adheres more closely to the motion rules of the mobile robot.

5.5. Dynamic Environment

In the dynamic environment, we add moving obstacles. Once the algorithm finds the initial solution, the mobile robot will move along the path and continuously optimize the feasible path to the goal point as it moves forward. In this process, the sampling strategy and the quality of the path solution are particularly important. As shown in

Table 4. Dynamic environment experiment date.

Algorithm	Total Time (s)	Standard Deviation (s)
RRT*	24.11	1.53
Improved RRT*	13.51	0.96

, we adopt the total time taken by the mobile robot to reach the goal point as the performance index for evaluating the algorithm. The dynamic environment is shown in Figure 11, where it is experimentally demonstrated that the improved RRT* algorithm takes less time to reach the goal point, and the standard deviation is also smaller compared to RRT, which further illustrates the stability of the proposed sampling strategy. These results fully demonstrate that the improved RRT* algorithm proposed in this paper still holds in dynamic environments.

6. Discussion

Sampling-based path planning for mobile robots encompasses performance metrics such as sample count, path cost, search time, and smoothness. High-quality paths for mobile robots exhibit efficient sampling, short path lengths, fast search times, and smooth trajectories. While the traditional RRT* algorithm is structurally simple and adaptable, it suffers from low search efficiency and stability in complex environments. This paper presents an improved RRT* algorithm that enhances sampling efficiency by constraining the sampling space to subregions with a target bias. The proposed adaptive bias strategy determines the step size and direction of new nodes based on obstacle density in the environment and the connectivity between the nearest point and the goal point. This strategy effectively guides the expansion of the random tree towards suitable regions, reducing path costs. Finally, the generated paths undergo the cubic spline curve interpolation algorithm optimization, resulting in smoother trajectories.

In the simulation, careful consideration is given to various indoor environmental conditions. The black regions represent physical obstacles such as tables, walls, and boxes. These regions are considered impassable for the mobile robot to avoid any potential damage. Figure 6c, Figure 7c, and Figure 8c depict the path-planning results of the improved RRT* algorithm in three different environments. In the spacious environment, as shown in Figure 6c, the algorithm in this paper demonstrates significant advantages compared to other algorithms, with fewer and smoother path waypoints. From Figure 9 and Table 1, it is evident that this algorithm achieves the minimum number of samples, path length, and search time, indicating its superior performance. As depicted in Figure 10a, the path optimized using the three-time spline interpolation method is smoother and more aligned with the navigation requirements of the mobile robot. In narrow environments, compared to other algorithms, the paths planned by the new algorithm are shown in Figure 7c. The minimal number of nodes in the random tree indicates an improvement in search efficiency. From Figure 9 and Table 2, it can be observed that the performance of the proposed algorithm is superior. Figure 10b illustrates the path after optimization using three-time spline interpolation in a narrow environment, which is smoother compared to other paths. In complex environments with intricate obstacles, as shown in Figure 8c, the path-planning results of the new algorithm are presented. Intuitively, its path exhibits less oscillation compared to other algorithms. From Figure 9 and Table 3, it is evident that the performance metrics for path planning have significantly improved. The results show that compared with the RRT* algorithm, the improved RRT* algorithm reduces the number of samples and search time by 71.34% and 79.60%, respectively, in complex indoor environments. The experimental results in a dynamic environment further demonstrate the general applicability of the improved RRT* algorithm proposed in this paper.

7. Conclusions

The RRT* algorithm possesses the characteristics of probabilistic completeness and asymptotic optimality. However, it suffers from slow convergence and low search efficiency in narrow passages within complex indoor environments. To address these issues, this paper proposes an improved RRT* algorithm that incorporates target bias sampling regions, adaptive deviation strategies, and trajectory optimization based on cubic spline curves. Experimental analysis conducted in this study demonstrates that the proposed algorithm effectively mitigates the limitations of RRT* in indoor complex environments, such as significant sampling randomness, weak expansion adaptability, and slow convergence, as mentioned in Section 3.3. Furthermore, the proposed algorithm exhibits higher stability. Nevertheless, there are certain limitations to our algorithm: (1) The parameters used in the proposed algorithm need to be adjusted based on specific circumstances, and parameter self-tuning is the focus of future research. (2) In dynamic scenarios, the algorithm exhibits poor real-time performance and average obstacle avoidance capabilities, which will be the main focus of our future investigations.