Mobile Robot Path Planning Based on Improved Particle Swarm Optimization and Improved Dynamic Window Approach

. To enable mobile robots to efectively complete path planning in dynamic environments, a hybrid path planning method based on particle swarm optimization (PSO) and dynamic window approach (DWA) is proposed in this paper. First, an improved particle swarm optimization (IPSO) is proposed to enhance the exploration capability and search accuracy of the algorithm by improving the velocity update method and inertia weight. Secondly, a particle initialization strategy is used to increase population diversity, and an addressing local optimum strategy is used to make the algorithm overcome the local optimum. Tirdly, a method of selecting navigation points is proposed to guide local path planning. Te robot selects the appropriate navigation points as the target points for local path planning based on the position of the robot and the risk of collision with dynamic obstacles. Finally, an improved dynamic window approach (IDWA) is proposed by combining the velocity obstacle (VO) with the DWA, and the evaluation function of the DWA is improved to enhance trajectory tracking and dynamic obstacle avoidance capabilities. Te simulation and experimental results show that IPSO has greater exploration capability and search accuracy; IDWA is more efective in trajectory tracking and dynamic obstacle avoidance; and the hybrid algorithm enables the robot to efciently complete path planning in dynamic environments.


Introduction
In recent years, mobile robots have been applied in a wide range of felds.As a key technology of robotics, path planning has become a research hotspot in robotics [1].Tere are two types of path planning methods for robots: global path planning and local path planning.Global path planning is based on the entire environmental information, such as the shape and size of obstacles, and their distribution on the map.On the other hand, local path planning has some unknown environmental information, such as the size and position of dynamic obstacles [2].
Currently, several algorithms have been applied to the global path planning of robots.Te A * algorithm [3] is a heuristic approach capable of theoretically determining the global optimal path.However, its computational complexity scales with the complexity of the environment, which can result in a considerable increase in execution time.Te ant colony optimization (ACO) [4] is a heuristic algorithm that can efectively solve path planning problems.However, it sufers from limitations such as slow convergence speed and a tendency to converge towards the local optimum, which can impair its ability to fnd the global optimal solution.Te genetic algorithm (GA) [5] is an intelligent algorithm known for its robustness and parallelism, and it is commonly used in path planning.However, it is susceptible to limitations such as early convergence, poor convergence path quality, and poor population diversity, which can restrict its efectiveness in fnding the global optimal path.Te slime mould optimization algorithm (SMOA) [6] is an intelligent algorithm that mimics the behavioral and morphological changes observed during the foraging process of slime mould.Although it can be used to solve path planning problems, it sufers from limitations such as slow convergence speed, and a tendency to converge towards the local optimum.Te salp swarm algorithm (SSA) [7] is a recently developed intelligent algorithm that draws inspiration from the foraging behavior of the salp.While it exhibits strong performance in solving path planning problems, it sufers from drawbacks such as slow convergence speed, convergence towards local optimum, and poor population diversity.Te particle swarm optimization (PSO) [8] is an intelligent algorithm that boasts several advantages, including a simple structure, few parameters, easy implementation, and fast convergence.However, the algorithm also has limitations such as low search accuracy, poor population diversity, and convergence towards the local optimum [9].Existing literature shows that intelligent algorithms used for path planning often sufer from limitations such as low accuracy, poor diversity, and the local optimum.Tese limitations can restrict the ability of the algorithm in meeting the demands of global path planning.Consequently, many scholars have focused on improving these algorithms.Ji et al. [10] proposed improvements to the pheromone update strategy and state transfer strategy of the ACO to increase the randomness of the path selection by the ant colony, thereby enhancing the global exploration ability and reducing the probability of the algorithm trapping in the local optimum.Additionally, they introduced a local optimal escape strategy to efectively address the problem of local optimum in the ACO.Zhai and Feng [11] proposed a method to integrate simulated annealing into the GA, which can efectively mitigate the issue of the search process getting stuck in the local optimum.Te approach also includes adaptive operators for crossover and mutation probabilities, which help to maintain population diversity and enhance the algorithm's global exploration capability.Wang et al. [12] proposed a learning technique based on orthogonal lensing opposition in SSA to enhance the algorithm's ability to escape the local optimum and fnd higher quality paths.Additionally, they 4introduced a scheme to adaptively adjust the number of leaders and a dynamic learning strategy to improve the global exploration capability and convergence speed.Wang et al. [13] enhanced the traditional SSA algorithm by incorporating inertia weight, designing a velocity clamping strategy, and designing an adaptive transformation parameter strategy.Tese modifcations were aimed at achieving a better balance between the global exploration capability and local search accuracy of the algorithm.Agarwal and Bharti [14] proposed an improved version of the SMOA with improved slime mould weights.Te proposed method maintains a specifc perturbation rate in the early stage of the algorithm to avoid premature convergence towards the local optimum and a reduced perturbation rate in the later stage of the algorithm to accelerate convergence.Additionally, the authors designed a multiobjective ftness function that considers both path minimization and obstacle avoidance.Tis approach improves the performance of the algorithm and enables it to achieve better results.Zhang et al. [15] proposed improvements to the linearly decreasing inertia weight and learning factors in PSO to balance the exploration capability and search accuracy of the algorithm.Te authors also introduced a measure of population diversity using ftness variance.However, the issue of premature and local optimum in the algorithm was not addressed.Zhong et al. [16] proposed an improved PSO that enhances the global exploration capability and diversity of the population.In the initialization phase, mutations were introduced to the particle positions to increase the search space coverage.Furthermore, a multi-particle competition mechanism was incorporated to promote population diversity in later stages of the algorithm.Wang et al. [17] proposed a more efective quantum swarm optimization algorithm that enhances both the population diversity and global exploration capabilities of the algorithm.Li et al. [18] proposed a modifed PSO algorithm that utilizes hierarchical random disturbance to increase population diversity and improve the algorithm's local search capability by modifying particle positions.Additionally, a random positive feedback factor is included to further enhance the algorithm's performance.Tan et al. [19] proposed a method to determine the excellence of particles and populations by calculating their evolutionary capabilities.Tey then adjusted the inertia weight and learning factors based on the evolutionary ratio to improve the global exploration capability and convergence speed of the algorithm.Te aforementioned improvements in PSO aim to enhance the exploration capability and search accuracy of the algorithm, yet they fail to address the issue of local optimum in PSO, leading to longer convergence path lengths.Terefore, this paper proposes an improved particle swarm optimization (IPSO) to address the limitations of PSO, including premature and local optimum, which can lead to long convergence path lengths.IPSO overcomes these drawbacks by improving the velocity update and inertia weight of the algorithm, as well as using an initialization strategy and addressing the local optimum strategy, resulting in a shorter convergence path length.
Teoretically, the aforementioned algorithms can obtain the shortest global path, but in practical environments, dynamic obstacles may render the obtained global path useless.In the context of dynamic environments, local path planning has good adaptability and can better handle dynamic obstacles.Some of the well-known local path planning algorithms include the BUG algorithm [20], the artifcial potential feld algorithm [21], the dynamic window approach (DWA) [22], etc. Te DWA algorithm, which takes into account the dynamic constraints of the robot, is considered a practical local path planning method [23].However, due to the absence of global path guidance, DWA is susceptible to getting trapped in the local optimum and is unable to avoid fast-moving obstacles [24].
Many scholars divide the path planning of robots into two phases: global path planning and local path planning.Te global path planning phase aims to obtain a collisionfree global path, which is used to guide local path planning.Te local path planning phase is guided by the global path to obtain a path that avoids dynamic obstacles and satisfes the robot dynamics constraints [25].Liu et al. [26] proposed a fusion algorithm that combines Jump A * and DWA to improve the smoothness of the path and reduce the number of turn points.By improving the DWA, the efciency of local path planning and obstacle avoidance is increased.Ji et al. [27] proposed a fusion algorithm that can be applied in complicated environments.Te algorithm incorporates the weight information of the road surface into the heuristic Te main contributions of this paper are as follows: (1) an adaptive selection strategy is proposed to enhance the global exploration ability and search accuracy of PSO by enabling some particles to select the corresponding inertia weight and velocity update methods based on their performance.(2) A positive feedback factor is added to PSO to adjust the particle positions, enhance the search accuracy, and accelerate the convergence speed of the algorithm.(3) A particle initialization strategy is proposed to increase the diversity of the population and use the strategies of adding random perturbation and random crossover to make the algorithm more capable of addressing the local optimum, thereby enhancing the exploration ability of PSO.(4) IDWA is proposed to enhance global path guidance and obstacle avoidance by combining with VO and improving the evaluation function.(5) A hybrid path planning method is proposed to complete the path planning task for mobile robots in dynamic environments.
Te structure of the remaining part of this paper is as follows: Section 2 describes the path planning problem and models the path planning problem and obstacles.Section 3 introduces the traditional PSO.Sections 4 and 5, respectively, introduce IPSO and IDWA.Section 6 introduces the process of algorithm fusion.Section 7 verifes the superiority and efectiveness of IPSO, IDWA, and the fusion algorithm through simulation experiments.Section 8 summarizes and concludes this study.

Modelling of Path Planning.
To ensure the safety of robots and reduce energy consumption, the goal of global path planning for mobile robots is to fnd a collision-free path with the shortest length in the environment [31].Robot global path planning is usually seen as fnding the set P of points generated by the robot moving through the environment, and the connection between adjacent points does not cross static obstacles.Te set of points P can be expressed as the following equation: where is the sequence of points in the environment, S is the starting point, T is the target point, L is the length of the path, and L is calculated by the following equation: where l (i,i+1) represents the distance from point i to point i + 1 and n represents the number of elements of P.
To make the global path more suitable for mobile robots, this paper uses the third spline interpolation method to optimize the global path, so that the global path is smoother and the turning angle is smaller [32].Te method for generating a cubic spline curve path between points p i and p i+1 is as follows.
(1) Generate m interpolation points between points p i to p i+1 using cubic spline interpolation a continuous path is called a cubic spline interpolation path.
Te length Cubl (i,i+1) of the cubic spline interpolation path between points p i to p i+1 can be calculated from the following equation: Te cubic spline interpolation path length L can be calculated from the following equation: (4)

Modelling of Obstacle.
To simplify the obstacle avoidance problem in the robot's working environment, it is common to consider the infuence range of obstacles as the outer circle of the obstacle.Here the infuence radius is R obs .
To ensure the safety of the robot, the safety distance d is added to the radius of infuence of the obstacle, which depends on the size of the robot.d is added to the radius of infuence of the obstacle, as shown in Figure 1.
Te obstacle's real radius of infuence R can be calculated by the following equation: (5)

PSO
Te fundamental concept of PSO is based on the cooperative learning of particles to achieve optimal solutions.Te particle has two properties: velocity and position.Te velocity determines the direction and size of the particle movement, and the position represents the solution to the problem.Te particle updates its velocity by learning from the position of the global optimal particle and its historical optimal position and fnds the optimal solution by updating the particle's velocity and position for multiple iterations.Te velocity and position update equations of the particle i at the t + 1 th iteration are calculated by equations ( 6) and (7), respectively.
where ω is the inertia weight, C 1 is the self-learning factor, which denotes the weight coefcient for learning towards its historical optimal position; C 2 is the global learning factor, which denotes the weight coefcient for learning toward the global optimal position; R 1 and R 2 are random numbers from 0 to 1; P i best is the historical optimal position of particle i; G best is the current global optimal position; V i (t) denotes the velocity of particle i at moment t, and X i (t) denotes the position of the particle i at moment t.
To improve the search accuracy and global exploration ability of PSO, an improved IPSO is proposed in this paper.

Adaptive Selection Strategy.
Te concept of inertia plays a crucial role in PSO, as it refects a particle's ability to maintain its previous momentum.A higher inertia weight can enhance the global exploration capability of PSO, allowing particles to discover high-quality solution regions.Conversely, reducing the inertia weight can increase the local search ability of PSO, enabling particles to improve their search accuracy in a specifc region [33].However, the global and local search capabilities of PSO are contradictory due to this property.
To solve the above problems, many scholars have proposed adaptive inertia weights to balance the exploration ability and search accuracy of the algorithm.For instance, Wang et al. [34] and Ding et al. [35] introduced adaptive inertia weight to adjust the position of followers, thereby enhancing the exploration ability and search accuracy of the SSA.
We proposed an adaptive inertia weight.Some particles will select the corresponding inertia weight based on their performance, calculated as in the following equation: where ω min is 0.4, ω max is 0.9, ω i is the inertia weight of the particle i, F i (t) is the ftness of the particle i at time t, F ave is the average ftness of the population, and F Gbest is the ftness of the current global optimal particle.When the F i (t) is larger than F ave , the particle i is considered to perform poorly, and the inertia weight coefcient of the particle is increased to fnd more high-quality solution regions.When the F i (t) is smaller than F ave , the particle is considered to perform better, and its inertia weight coefcient is reduced to enhance the local search accuracy of the particle.
Traditional PSO solely computes the particle's velocity based on its historical optimal position and global optimal position, which cannot exploit the potential of each particle in the population.To address this issue, we proposed a velocity update strategy with adaptive selection, in which some particles can select the velocity update method based on their performance, which can be calculated by the following equation: where F bad is the ftness threshold of the poorly performing particles and F good is the ftness threshold of the wellperforming particles, η is a random number from 0 to 1.
Te poorly performing particles tend to search around their 4 Journal of Robotics historical best position to enhance their global exploration ability.Te well-performing particles tend to search around the global best position to enhance local search accuracy and the algorithm's convergence speed.Particles with average performance learn from the historical optimal position and the global optimal position to fnd more high-quality solution regions.

Positive Feedback
Factor.Li et al. [18] adds a positive feedback factor to the traditional PSO, which increases the local search capability of PSO by adjusting the position of the particles based on their ftness after iteration compared with that before iteration.In this paper, we improve the positive feedback factor proposed by [18], as in the following equation: where θ i is the positive feedback factor of the particle i, Rand is a random number from 0 to 1, and t is the number of current iterations.When F i (t + 1) is lower than F i (t) but higher than F ave , the particle i moves θ i length to speed up its entry into the high-quality solution region.When F i (t + 1) is lower than F i (t) and lower than F ave , particle i moves 1/2 * Rand * θ i length increasing the local search accuracy by reducing the extra step added due to the positive feedback factor while accelerating the entry into the high-quality solution region.

Initialization Strategy.
In PSO, the diversity of the population plays a crucial role in determining the quality of the solution obtained.However, as the algorithm runs, the diversity of the population tends to decrease, especially in the later stages of the algorithm when particles cluster together, leading to premature convergence.To improve the diversity of the population, this paper uses an initialization strategy that classifes particles into excellent-performing, averageperforming, and poor-performing based on their ftness values.At the beginning of each iteration, the initialization operation is performed on some of the poorly performing particles to increase the diversity of the population and prevent the algorithm from maturing prematurely.

Addressing the Local Optimum Strategy.
Trapping in the local optimum has been a shortcoming of PSO that is diffcult to overcome.Terefore, this paper proposes a method to address local optimum by adding a random perturbation and random crossover.If the global optimal particle stays the same for a long period, the algorithm is judged to be in the local optimum, and a random perturbation of length δ is added to the global optimal particle.Ten, check whether the ftness after adding the perturbation is lower than the ftness before adding the perturbation.If the ftness is lower, the perturbed particle is saved as a new global optimal particle.Otherwise, the perturbation is discarded, and the same operation is repeated in the next iteration.Te δ can be calculated by the following equation: where β is the value assigned based on the F Gbest .β is 0.2 if the current global optimal particle is in the high-quality solution region.Te β is 0.5 if the current global optimal particle is in the inferior solution region.Te process of adding random perturbations to make the algorithm address the local optimum is shown in Figure 2.
Besides adding random perturbations, a random crossover strategy is designed to make the algorithm address the local optimum.
After judging that the algorithm is trapped in the local optimum, replace a point in the global optimal particle with a point at the corresponding position of a random particle.Te random crossover process is shown in Figure 3.

Fusion with VO.
VO has the ability to predict whether a robot will collide with dynamic obstacles and has been widely used in recent years for robot avoidance of dynamic obstacles [36].Te principle of VO prediction collision is shown in Figure 4 where V rob and V obs are the velocities of the robot and the dynamic obstacle, respectively.V r is the relative velocity of the robot and the dynamic obstacle, and R rob and R obs are the radius of the robot and the dynamic obstacle, respectively.
Te relative velocity V r of the robot to the obstacle can be calculated by the following equation: where ⊕ is the Minkowski sum.Defne a ray starting from the robot center P r along the V r as the following equation: where t denotes time, so ray λ(P r , V r ) represents the realtime relative position of the robot and the obstacle if the robot and the obstacle maintain their motion without changing.Defne the set V r as the relative collision cone (RCC), the mathematical representation of the RCC is as in the following equation: where Obstacle is the infuence range after the obstacle radius R obs expansion R rob .
If λ(P r , V r ) is within the RCC, it indicates the risk of collision between the robot and the dynamic obstacle.In this paper, we use a fusion of VO and IDWA for dynamic obstacle avoidance.

Improving the heading (v, ω) Function.
Te heading(v, ω) function of conventional DWA is calculated by the following equation: where α is the angular deviation of the robot orientation at the end of the trajectory from the target point.Te traditional DWA heading(v, ω) function may cause the situation in Figure 5.It can be seen that the robot is in a better state after one unit of time following trajectory 1 than after one unit of time following trajectory 2. However, the calculation results based on the heading(v, ω) function shows that trajectory 2 has a higher score, which is unreasonable.
To solve this problem, we proposed an improved Heading(v, ω) function.When the robot approaches the target point, the heading(v, ω) function no longer calculates the angle between the robot's orientation at the end of the trajectory and the target point but instead calculates the angle between the robot's orientation after moving one unit of time and the target point, as shown in Figure 6.

Improving the dist(v, ω) Function.
When modelling for the static obstacles, the infuence range of the static obstacles is increased based on the size of the robot, so they do not collide when the global path is close to the obstacles.Te dist(v, ω) function causes the robot to select the trajectory movement away from the static obstacles, which not only weakens the guidance efect of the global path on local path planning but also increases the path length of local path planning, as shown in Figure 7.
To address these problems, the dist(v, ω) function has been improved as the following equation:  6 Journal of Robotics where x rob , x obs , y rob , and y obs represent the horizontal and vertical coordinates of the robot and the static obstacle, respectively, and R is the radius of infuence of the obstacle.Te improved Dist(v, ω) function allows the robot to select a trajectory that is close to but without collision with static obstacles, which enhances the guidance efect of the global path on local path planning.

Adding the vh(v, ω) Function.
To make IDWA more efective in avoiding dynamic obstacles, the vh(v, ω) function is added to the evaluation function to calculate the change in angle diference with the obstacle before and after robot movement.Te vh(v, ω) function can be calculated by the following equation: where VA(t) and VA(t + 1) denote the angular diference between the velocity of the robot and the velocity of the dynamic obstacle before and after motion, respectively.After adding the vh(v, ω) function, the algorithm will select trajectories for the robot that increase the angular diference between the velocity of the robot and the velocity of the dynamic obstacle, as shown in Figure 8. Te evaluation function of IDWA is stated in the following equation: where σ is a normalization operation.Te α, β, c, and λ are the weighting factors of Heading(v, ω), Dist(v, ω), vel(v, ω), and vh(v, ω), respectively.Te λ is calculated as in the following equation: where is collision is the judgment result of VO, and the value of is collision is 1 if the robot will collide with the dynamic obstacle, otherwise is collision is 0.

Algorithm Fusion
To complete the path planning of mobile robots in dynamic environments, this paper proposes a hybrid algorithm combining IPSO and IDWA.In the proposed hybrid algorithm, multiple navigation points are identifed in the Journal of Robotics global path generated by IPSO.Te robot selects the most suitable navigation point as the target point for IDWA based on its current position and the risk of collision with obstacles and iteratively updates its target point until reaching the global target point.Selecting navigation points is crucial for IDWA.If the selected navigation point is too far away from the robot, it may not efectively guide the robot, while if the selected navigation point is too close to the robot, it may not efectively avoid dynamic obstacles.For this reason, we proposed a method for selecting navigation points on global paths.Tis paper uses the cubic spline interpolation to optimize the global path, assuming there are Num points in the global path, and the robot's location is the i th point.If the distance of the robot from dynamic obstacles does not reach the obstacle avoidance distance or reaches the obstacle avoidance distance but the robot will not collide with dynamic obstacles, the i + n th point will be chosen as the navigation point for IDWA.If the distance of the robot from dynamic obstacles is less than the obstacle avoidance distance and the robot will collide with the dynamic obstacle, the i + kn th point will be chosen as the navigation point for IDWA.Where the n depends on the velocity of the robot, and the k depends on the obstacle avoidance distance and the size of the dynamic obstacle.
Te steps of the hybrid algorithm are as follows.
Step 1: Modelling of the robot's working environment.
Step 2: Get the global path by using IPSO.
Step 3: Divides the global path into multiple navigation points.
Step 4: Use the above method to select the navigation points in turn as the target points for local path planning, and use IDWA to complete the local path planning.
Step 5: Te robot reaches the global target point, and the path planning ends.
Te fow chart of the fusion path planning method is shown in Figure 9.

Simulation Experimental Analysis
To determine the efectiveness of the IPSO, IDWA, and hybrid algorithm, we use the MATLAB 2021a software to conduct simulation experiments on a computer with Win 10 and 8GB RAM.

Simulation Experimental of IPSO.
To determine that IPSO has strong exploration ability and search accuracy, this paper conducts simulation experiments and data comparisons with traditional PSO, [18] improved particle swarm optimization (IPSO1), and [19] improved particle swarm optimization (IPSO2) in 10 * 10 and 15 * 15 environments, respectively.
To minimize the efect of the randomness of the intelligent algorithm, 50 simulation experiments are conducted in two environments, and the average path length and shortest path length of the four algorithms are compared.
Te average path length and shortest path length can refect the exploration ability and search accuracy of the algorithm, respectively.Te coordinates of the start and target points of Environment 1 and Environment 2 are (0, 0), (10,10) and (0, 0), (15,15), respectively.Te start and target points are represented by yellow squares and green squares, respectively.Te relevant parameters set in this paper are as follows: the number of populations is 50, the maximum number of iterations is 300, and the learning factor C 1 � C 2 � 1.5, the inertia weight ω min � 0.4, and ω max � 0.9.
Te simulation experimental data of the four algorithms in environment 1 and environment 2 are described in Tables 1 and 2, respectively.
Te shortest paths of the four algorithms in Environment 1 and Environment 2 are shown in Figures 10 and 11, respectively.
Te simulation experimental data demonstrate that the average path length and shortest path length obtained by IPSO are superior to the other three algorithms in both environments.Tis is attributed to the superior exploration ability and search accuracy of IPSO.
To determine the efectiveness of the address local optimum strategy and the faster convergence speed of IPSO, the convergence curves in the two environments are shown in Figure 12(a) and Figure 12(b), respectively.In this experiment, the number of populations is 150, the maximum number of iterations is 500, and the learning factor C 1 � C 2 � 1.5, the inertia weight ω min � 0.4, and ω max � 0.9.8

Journal of Robotics
Te convergence curves in Figure 12 demonstrate that IPSO outperforms PSO, IPSO1, and IPSO2 in terms of convergence speed and convergence path length in environments 1 and 2. Te convergence path length of IPSO in environment 1 and environment 2 are reduced at the 343 th and 387 th iteration, respectively, proved the efectiveness of the address local optimum strategy.Te convergence path length in two environments is shown in Table 3.

Simulation Experimental of IDWA.
To determine the obstacle avoidance capability of IDWA, the simulation experiments of avoiding dynamic obstacles were compared using DWA and IDWA, respectively.
Te radius of the robot in this experiment is 0.1 m, the maximum linear velocity is 1 m/s, the maximum acceleration is 0.2 m/s 2 , the maximum rotational velocity is 40/s, the maximum rotational acceleration is 40/s 2 , the initial   Journal of Robotics 9   Journal of Robotics orientation angle is 90 °, the radius of the dynamic obstacle is 0.6 m, and the velocity is 0.6 m/s.Te results of the simulation experiments are shown in Figures 13 and 14, respectively.Te dist(v, ω) function of conventional DWA drives the robot to turn to the side with the same direction of motion as the dynamic obstacle, and the distance of the robot from the dynamic obstacle is too close, leading to an empty set of velocity sampling space, resulting in obstacle avoidance failure, as shown in Figure 13.
If the robot is judged to be at risk of collision with the dynamic obstacle by using VO, the vh(v, ω) function in the IDWA evaluation function drives the robot to turn in the direction opposite to the dynamic obstacle's motion direction, which not only successfully avoids the dynamic obstacle but also shortens the path length increased by avoiding the dynamic obstacle, as shown in Figure 14.

Algorithm Fusion.
Te above experiments demonstrated the efectiveness of IPSO and IDWA.To further assess whether the fusion algorithm is efective for robot path planning in the dynamic environment, it will be verifed in a dynamic environment of 25 * 25.
In this experiment, the robot has a radius of 0.1 m, a maximum linear velocity of 1 m/s, a maximum acceleration of 0.2 m/s 2 , the maximum rotational velocity of 40/s, a rotational acceleration of 40/s 2 , and an initial orientation angle of 36 °.Te coordinates of the starting point are (0, 0) and the coordinates of the global target point are (23,23), which are represented as yellow and green squares, respectively, the red * is navigation point for IDWA.Te radius of both dynamic obstacles is 0.6 m and the velocity is 0.8 m/s.Te process and results of the fusion algorithm for robot path planning in dynamic environments are shown in Figure 15. 12 Journal of Robotics

Journal of Robotics 13
Te black line is the global path planned by IPSO, the green line is the trajectory of IDWA, and the red line is the trajectory of two dynamic obstacles.
From the experimental results, it can be seen that the global path and the local path ft closely when there are no dynamic obstacles.Under the direction of the global path, IDWA successfully completes the dynamic obstacle avoidance and path planning tasks.

Conclusion
Tis paper proposes a hybrid algorithm for the path planning of robots in dynamic environments.
To improve the traditional PSO, IPSO is proposed to enhance the exploration ability, search accuracy, and address the local optimum to obtain a shorter global path.Firstly, the exploration capability and search accuracy of PSO are increased using an adaptive selection strategy and a positive feedback factor.Secondly, the initialized particle strategy is used to increase population diversity.Finally, the strategy of adding random perturbation and random crossover is used to help the algorithm address the local optimum.In addition, the global path is optimized using cubic spline interpolation to make the global path smoother and have smaller turning angles.Simulation experiments prove that the IPSO has better exploration ability and search accuracy.
To improve the disadvantages of DWA trapped in local optimums and insufcient obstacle avoidance, an IDWA is proposed.Firstly, the global path is used to provide global guidance for the robot, which improves the problem of DWA getting stuck at the local optimum due to the lack of global guidance.Secondly, the Heading(v, ω) and Dist(v, ω) functions are improved to enhance the guidance of the global path to the local path planning.Finally, VO is used to determine whether the robot will collide with dynamic obstacles, and the vh(v, ω) function is added to the evaluation function to improve the ability of the algorithm to avoid dynamic obstacles.Te simulation experiments prove that IDWA can efectively complete the path planning of the robot in a dynamic environment with the guidance of the global path.

Figure 1 :
Figure 1: Te actual radius of infuence of the obstacle R.

Figure 2 :
Figure 2: Te process of adding random perturbation.

Figure 9 :
Figure 9: Flow chart of the fusion algorithm.

Figure 12 :
Figure 12: Te convergence curves in two environments.(a) Convergence curve of environment 1.(b) Convergence curve of environment 2.

Figure 15 :
Figure 15: Fusion algorithm in the dynamic environment.(a) Te global paths in the environment obtained using IPSO.(b) Te fusion algorithm for the avoidance of dynamic obstacle 1. (c) Te fusion algorithm for the avoidance of dynamic obstacle 2. (d) Te fusion algorithm completes path planning in the dynamic environment.

Table 1 :
Simulation experimental data of environment 1.Te bold values are the smallest among the four groups, representing the average path length and the shortest path length, respectively.

Table 2 :
Simulation experimental data of environment 2.Te bold values are the smallest among the four groups, representing the average path length and the shortest path length, respectively.

Table 3 :
Convergence path length and number of convergence iterations in environment 1 and environment 2.
Te values in bold represent the best of the four sets of values.