Path Planning of Mobile Robots with an Improved Grey Wolf Optimizer and Dynamic Window Approach

Abstract

To address the critical limitations of conventional Grey Wolf Optimization (GWO) in path planning scenarios—including insufficient exploration capability during the initial phase, proneness to local optima entrapment, and inherent deficiency in dynamic obstacle avoidance—this paper proposes a multi-strategy enhanced GWO algorithm. Firstly, the Piecewise chaotic mapping is applied to initialize the Grey Wolf population, enhancing the initial population quality. Secondly, the linear convergence factor is modified to a nonlinear one to balance the algorithm’s global and local search capabilities. Thirdly, Evolutionary Population Dynamics (EPD) is incorporated to enhance the algorithm’s ability to escape local optima, and dynamic weights are used to improve convergence speed and accuracy. Finally, the algorithm is integrated with the Improved Dynamic Window Approach (IDWA) to enhance path smoothness and perform dynamic obstacle avoidance. The proposed algorithm is named PAGWO-IDWA. The results demonstrate that, compared to traditional GWO, PAGWO-IDWA reduces the path length, number of turns, and running time by 9.58%, 33.16%, and 30.31%, respectively. PAGWO-IDWA not only overcomes the limitations of traditional GWO but also enables effective path planning in dynamic environments, generating paths that are both safe and smooth, thus validating the effectiveness of the algorithm.

1. Introduction

With the rapid development of robotics, robots are increasingly deployed in various fields, and path planning, one of the fundamental challenges for mobile robots, has gradually attracted widespread attention [1]. Path planning refers to the process by which mobile robots use algorithms to plan paths that satisfy task requirements, with evaluation metrics such as path distance, number of turns, and running time [2]. Path planning can typically be divided into two categories: global and local path planning. Traditional approaches in global path planning often involve algorithms such as Dijkstra’s algorithms [3], the A* algorithms [4], and the RRT algorithms [5], and local path planning includes the dynamic window approach (DWA) [6] and the artificial potential field (APF) algorithm [7]. In path planning, researchers worldwide have proposed many algorithms and made significant contributions. Feng et al. [8] proposed an autonomous path planning method capable of dynamically adjusting coverage paths and priorities, by introducing a dynamic weight matrix to Dijkstra’s algorithm, thereby generating cost-optimal paths rather than merely the shortest paths. To address vehicle path safety in uncertain dynamic obstacle environments, Lathrop et al. [9] calculated the Wasserstein distance between the distributions of vehicle and obstacle states, improved the RRT algorithm through the incorporation of a dynamic distribution model, and provided safe paths that are probabilistically guaranteed. Jiang et al. [10] added a risk penalty to the evaluation function of the A* algorithm, making the improved algorithm more suitable for practical situations, and combined the A* algorithm with the DWA, which allows the DWA to traverse the A* path points, thus endowing the algorithm with dynamic path planning capability. Ding et al. [11] employed a beetle antennae search algorithm to address the optimization issues of the APF algorithm. They dynamically adjusted the search step size based on the distance between the robot and the target point, thereby achieving a balance between path generation speed and planning accuracy. To address the suboptimal path problem of the DWA algorithm due to fixed weights in complex environments, Gong et al. [12] improved the velocity cost function. They input the distances between the robot, obstacles, and the target point into a fuzzy control module, thus enabling the robot to adapt to environmental changes. Although traditional global path algorithms and local path algorithms perform well in many applications, they also have limitations in addressing optimization problems. For example, the A* algorithm is subject to computational complexity in high-dimensional environments, which results in poor dynamic capability. The APF can result in path planning failures due to local minima problems and is computationally intensive, requiring considerable time.

In recent years, with the development of intelligent optimization algorithms, such as ant colony optimization (ACO) [13], genetic algorithm (GA) [14], sine cosine algorithm (SCA) [15], and particle swarm optimization (PSO) [16], these algorithms have also been widely applied to path planning problems. Wang et al. [17] enhanced the ACO by employing several heterogeneous ant colonies to enhance solution diversity and optimize UAV task allocation through task insertion boundaries and random interruption mechanisms. Akay et al. [18] introduced a differential update strategy into the SCA, where different strategies are chosen to update candidate solutions, thereby enhancing the diversity of the algorithm. Xu et al. [19] incorporated adaptive weighted delayed velocity in the PSO algorithm to improve convergence stability and proposed quartic Bézier curves for smoothing the paths, thereby reducing abrupt turns in robot motion. Pan et al. [20] combined GA with deep learning, and the trained deep neural networks can quickly generate planning results, which significantly reduces the running time.

Gray Wolf Optimization (GWO) algorithm [21] is a swarm intelligence algorithm proposed in 2014, widely used in various fields due to its strong global search capability. It primarily finds the global optimal solution through the natural predatory behavior of wolves. Compared with other intelligent algorithms, GWO offers advantages such as fewer parameters, faster convergence speed, and improved stability. However, in complex environments, GWO may exhibit reduced convergence accuracy and is prone to falling into local optima. Therefore, improving GWO has become an active area of research in recent years. To address the issue of GWO falling into local optima in complex multi-modal problems, Yu et al. [22] proposed HGWODE, which introduces a ranking-based mutation strategy and enhances the exploratory capability using the Differential Evolution algorithm, thereby enabling the algorithm to escape local optima. Ding et al. [23] introduced a probabilistic jump mechanism into GWO to help it escape local optima and improve its the convergence speed by replacing the linear convergence factor with an exponentially decaying one. Liu et al. [24] enhanced GWO by using a Gaussian map to improve the quality of the initial gray wolf population, incorporating the Lévy flight strategy to increase the population’s diversity, and integrating the Golden Sine function to improve the convergence accuracy. Jiang et al. [25] introduced the $\epsilon -level$ comparison method in GWO to enhance the algorithm’s exploratory capability by dynamically adjusting the $\epsilon$-value. Additionally, they incorporated a communication mechanism that randomly selects individuals to exchange information with the optimal individual, thereby improving the population’s diversity. Zhang et al. [26] introduced the centrifugal distance rate of change to calculate the population distribution and dynamically assigned weights based on the centrifugal distance rate of change between individuals and leaders, thereby enhancing the algorithm’s optimization performance and convergence speed. Qu et al. [27] simplified the position update formula of GWO, accelerating convergence while retaining global search capability, and modified the symbiotic phase of the Symbiotic Organisms Search (SOS) algorithm, enhancing information exchange among individuals and avoiding local optima. They combined the improved SOS algorithm with the simplified GWO algorithm, proposing a new algorithm, HSGWO-MSOS, which balances global exploration and local exploitation, thereby improving the effectiveness of path planning. Zhang et al. [28] significantly enhanced the search efficiency and stability of GWO path planning through the introduction of a discrete search mechanism, adaptive parameter adjustment, and local optimization strategies, demonstrating superior performance in path planning for porous materials. Liu et al. [29] integrated the Gaussian mutation strategy and spiral function perturbation to enhance the algorithm’s ability to escape local optima by randomly perturbing the position of the optimal individual. Additionally, they enhanced the global search ability of the algorithm by dynamically adjusting the weights, thereby increasing the influence of a wolf via Gaussian distribution. They applied the proposed NAS-GWO to trajectory planning in agricultural drone applications. Wang et al. [30] developed a poor solution repair strategy, thereby improving the population quality, and incorporated a multi-constraint path cost function to optimize the efficiency and safety of UAV path planning. Although the existing improvement methods have improved the performance of GWO, the algorithm remains constrained in complex environments. For example, the algorithm tends to fall into local optima and exhibits suboptimal path smoothness. Moreover, in dynamic environments, traditional GWO cannot handle dynamic obstacle avoidance in dynamic settings. Furthermore, some improved algorithms have enhanced performance but introduced excessive tuning parameters, which may increase algorithmic complexity, reduce generalization ability, and hinder adaptation to diverse application scenarios.

Based on previous research, this paper conducts an in-depth study of GWO and applies comprehensive improvements to overcome its performance limitations. It proposes the PAGWO algorithm and combines it with an enhanced version of the DWA, forming the PAGWO-IDWA approach for mobile robot path planning and dynamic obstacle avoidance.

The structure of the paper is outlined as follows: Section 2 provides an overview of the standard GWO algorithm; Section 3 improves the GWO algorithm with various strategies to address its shortcomings; Section 4 describes the improvements to the Dynamic Window Approach (DWA); Section 5 integrates the PAGWO algorithm with the Improved Dynamic Window Approach (IDWA) to address the limitations in dynamic path planning; Section 6 simulates and analyzes the proposed algorithm to validate the performance of PAGWO-IDWA in path planning; and finally, Section 7 provides a summary and concludes the paper.

2. Standard Gray Wolf Optimization Algorithm and Fitness Function

2.1. Standard Gray Wolf Optimization Algorithm

The GWO algorithm is a simulation-based optimization algorithm that mimics social hierarchy and group hunting behaviors observed in gray wolf packs. The algorithm has the advantages of few parameters, simple principles, and strong global search capability [31].

The social hierarchy of gray wolves is a pyramid, divided into four levels: the alpha wolf is the population leader, responsible for decision-making, representing the algorithm’s optimal solution; the beta wolf assists the alpha wolf in managing the rest of the pack, representing the suboptimal solution; the delta wolf occupies the third level, responsible for reconnaissance and auxiliary tasks; and omega wolves are the lowest-ranked and most numerous wolves, following the commands of others. The GWO algorithm is divided into three phases: the prey encirclement phase, the prey pursuit phase, and the prey attack phase [32].

The gray wolf pack, as a cohesive unit during the prey encirclement phase, collectively updates its position based on the relative distance to the prey.

$$D=\left|C{X}_p(t)-X(t)\right|$$

(1)

$$X(t+1)=X_p(t)+AD$$

(2)

where $D$ is the distance between the prey and the individual gray wolf’s current position, $t$ is the current iteration, $X_p$ is the prey position, and $X$ is the individual gray wolf’s current position. The coefficient vectors $A$ and $C$ are expressed as follows:

$$A=2a{r}_1-a$$

(3)

$$C=2r_2$$

(4)

where $a$ decreases linearly from 2 to 0 as the iterations progress, while parameters $r_1$ and $r_2$ are random values within the range [0, 1].

During the prey pursuit phase, each gray wolf calculates the distances to the alpha, beta, and delta wolves based on their current positions. Subsequently, individual gray wolves move closer to the prey using these distance vectors. The position update formula integrates the influence of the leaders with stochastic exploration components, ensuring that the pack remains within the potential vicinity of the globally optimal solution. The position update formula is as follows:

$$D_{\alpha }=\left|\begin{array}{l}{C}_1{X}_{\alpha }-X\end{array}\right|\begin{array}{ll}\begin{array}{ll}& \end{array}& X_1=X_{\alpha }-A_1{D}_{\alpha }\end{array}$$

(5)

$$D_{\beta }=|C_2{X}_{\beta }-X|\begin{array}{lll}& & X_2=X_{\beta }-A_2{D}_{\beta }\end{array}$$

(6)

$$D_{\delta }=\left|\begin{array}{l}{C}_3{X}_{\delta }-X\end{array}\right|\begin{array}{lll}& & X_3=X_{\delta }-A_3{D}_{\delta }\end{array}$$

(7)

$$X(t+1)={\displaystyle \frac{X_1+X_2+X_3}{3}}$$

(8)

where $D_{\alpha }$, $D_{\beta }$, and $D_{\delta }$ represent the relative position difference between the current positions of individual gray wolves and those of alpha, beta, and delta wolves, respectively; $X_{\alpha }$, $X_{\beta }$, and $X_{\delta }$ represent the current positions of alpha, beta, and delta wolves, respectively; $X_1$, $X_2$, and $X_3$ are the updated positions of omega wolves based on alpha, beta, and delta wolves, respectively. Finally, omega wolves recalculate the new mean position based on the updated positions.

The attacking prey phase is a critical stage in the optimization process, during which the algorithm approaches the global optimal solution with greater precision. As $a$ decreases, the value of $A$ changes accordingly. When $\left|A\right|<1$, the gray wolf performs a local search and attacks the prey, and when $\left|A\right|>1$, gray wolves perform a global search and gradually disperse.

In summary, the gray wolf optimization algorithm has the following drawbacks:

(1)

The initial population of gray wolves in the encirclement phase is randomly generated, and the initial population is unevenly distributed, which negatively affects the exploration ability in the initial phase;

(2)

$A$ is an important parameter controlling global and local search, and its value varies with $a$. The linear decay of $a$ has limitations;

(3)

The position update strategy lacks flexibility when addressing complex problems and is prone to becoming trapped in local optima.

(4)

The standard GWO exhibits limited adaptability in dynamic path planning tasks.

2.2. Fitness Function

Path planning is a crucial step in achieving autonomous navigation for mobile robots, entailing the discovery of a collision-free route from a start point to a destination while meeting specific requirements. In the path optimization process, two core issues are considered: path length, which reflects the algorithm’s optimization capability, and obstacle avoidance, which ensures safety. Therefore, this paper designs the fitness function of the proposed algorithm to incorporate these two aspects: path length and obstacle avoidance.

Path length is a key metric for evaluating the quality of a route. Suppose a route is represented by a series of points, denoted as $P=\{P_1,P_2,P_3\cdots ,P_n\}$, where each point’s coordinates are given by $(x_i,y_i)$. The path length can be obtained by summing the Euclidean distances between consecutive points, as in Formula (9):

$$F_1={\displaystyle \sum _{i=1}^{n-1}\sqrt{{(x_{i+1}-x_i)}^2+{(y_{i+1}-y_i)}^2}}$$

(9)

In order to ensure the safety of the path, obstacle avoidance must be considered in the fitness function. The safety of path planning is enforced by introducing a penalty value, which depends on the number of obstacles traversed by the path, as in Formula (10):

$$F_2=\kappa O_b$$

(10)

where $\kappa$ is the penalty factor, which is used to regulate the weight of obstacle avoidance, and $O_b$ is the number of obstacles that the path crosses.

Combining the path length with the avoidance penalty value, the integrated fitness function can be expressed as follows:

$$F={\displaystyle \sum _{i=1}^{n-1}\sqrt{{(x_{i+1}-x_i)}^2+{(y_{i+1}-y_i)}^2}}+\kappa O_b$$

(11)

This design of the fitness function has wide applicability in the field of path planning. By comprehensively considering the path length and obstacle conflicts, the robot is able to plan efficient and safe moving routes in complex environments and improve its navigation capability.

Such a fitness function design is widely applicable in path planning. By simultaneously considering path length and obstacle conflicts, the robot can plan efficient and safe trajectories in complex environments, thereby enhancing its navigation capabilities.

3. Improved Gray Wolf Optimization Algorithm

3.1. Piecewise Chaotic Mapping

This study employs a piecewise chaotic mapping strategy to enhance population diversity and augment the optimization process’s early-stage exploration capability, mitigating the issues of insufficient population diversity and low-quality initialization in conventional GWO algorithms. Chaotic mapping exhibits ergodicity and regularity [33], producing a random sequence that can cover a more expansive search space, allowing the initial population to search from a broader range of positions. Standard chaotic mappings include the Logistic, Tent, Sinusoidal, and Piecewise chaotic mappings. The piecewise chaotic mapping is simple and efficient, resulting in a more uniform distribution of the initial gray wolf population, as shown in Formula (12):

$$x(t+1)=\left\{\begin{array}{ll}{\displaystyle \frac{x(t)}{p}},\hfill & 0\le x(t)<p\hfill \\ {\displaystyle \frac{x(t)-p}{0.5-p}},\hfill & p\le x(t)<0.5\hfill \\ {\displaystyle \frac{1-p-x(t)}{0.5-p}},\hfill & 0.5\le x(t)<1-p\hfill \\ {\displaystyle \frac{1-x(t)}{p}},\hfill & 1-p\le x(t)<1\hfill \end{array}\right.$$

(12)

where $x(t)$ represents a random iteration value, $x(1)$ is a random number in the range [0, 1]; $p$ is a control parameter with values within the open interval (0, 0.5), meaning that $p$ cannot be 0 or 0.5, as this would lead to undefined results in the piecewise chaotic mapping function, and in this paper, the value of $p$ is set to 0.3.

Piecewise chaotic mapping operates by defining different functional relationships across distinct segments, ensuring that each generated initial position is determined by more than just a single random distribution. This leads to a more uniformly distributed initial population, effectively improving the quality of the initial gray wolf population and enhancing the algorithm’s early-stage exploration capability.

3.2. Nonlinear Convergence Factor

In the gray wolf optimization algorithm, the parameter $A$ is the key factor that controls the balance between global and local search. When $\left|A\right|>1$, the wolf pack searches in distant regions to enhance its global search capability, and when $\left|A\right|<1$, the wolf pack gradually converges towards the prey and performs local search [34].

From Formula (3), $a$ is linearly reduced from 2 to 0 during the iteration process, and the value of $A$ changes accordingly; this linear decay often leads to local optima in practical applications. In this paper, we introduce a nonlinear convergence factor based on the adjustment factor $n$, which allows for changing the decay of $a$ by adjusting the size of $n$ based on practical application requirements.

$$a=\left\{\begin{array}{ll}-{\displaystyle \frac{t^2}{n^2}}+2,\hfill & t<{\displaystyle \frac{2n^2}{t_m}}\hfill \\ {\displaystyle \frac{2{(t-t_m)}^2}{t_m^2-2n^2}},\hfill & {\displaystyle \frac{2n^2}{t_m}}\le t\le t_m\hfill \end{array}\right.$$

(13)

where $t_m$ is the maximum number of iterations, and $n$ is in the range $(0,t_m)$.

Figure 1 shows the effect of different values of $n$ on the decay of the convergence factor a when the number of iterations is 1000. The dashed horizontal line labeled “baseline” corresponds to $a=1$, serving as a key reference point for the GWO. As discussed earlier, in GWO the parameter $A$ is given by $A=2a{r}_1$, and the threshold $\left|A\right|=1$ marks the transition between global and local search. Consequently, when $a$ crosses 1, the search behavior shifts from primarily global to more local exploitation.

As can be seen in Figure 1, smaller $n$ values cause $a$ to approach 1 more quickly, thereby triggering local search earlier and reducing the duration of the global search phase. Conversely, larger $n$ values slow the convergence to 1, extending the global search phase. By selecting an appropriate $n$, one can, thus, balance the algorithm’s global and local exploration more effectively according to the specific optimization problem.

3.3. Evolutionary Population Dynamics

Evolutionary population dynamics (EPD) is inspired by self-organizing critical theory (SOC), which posits that dynamic systems can adaptively adjust to maintain an efficient and stable dynamic equilibrium. EPD takes advantage of this idea by eliminating less-adapted individuals and repositioning them to enhance population diversity [35].

During iterations of the gray wolf algorithm, the population gradually converges toward the top three individuals, which aids local exploration but also reduces overall population diversity, increasing the risk of becoming trapped in local optima. In later stages, a decline in population diversity may negatively impact the global optimization performance. In this paper, EPD is used to enhance the diversity of the population.

The specific implementation of EPD is divided into the following two steps. Firstly, in each iteration, the bottom 20% of individuals with the lowest fitness values are removed. Then, the removed individuals are regenerated in four ways. The first three methods involve generating new individuals near the elite individuals, while the fourth method generates individuals randomly; each of these four methods is applied with a probability of 25%. The mathematical formula is as follows:

$$X(t+1)=\left\{\begin{array}{cc}{X}_{\alpha }-r_4(X_{\alpha }-l_b),& r_3\le 0.5\\ X_{\alpha }+r_4(u_b-X_{\alpha }),& r_3>0.5\end{array}\right.$$

(14)

$$X(t+1)=\left\{\begin{array}{cc}{X}_{\beta }-r_4(X_{\beta }-l_b),& r_3\le 0.5\\ X_{\beta }+r_4(u_b-X_{\beta }),& r_3>0.5\end{array}\right.$$

(15)

$$X(t+1)=\left\{\begin{array}{cc}{X}_{\delta }-r_4(X_{\beta }-l_b),& r_3\le 0.5\\ X_{\delta }+r_4(u_b-X_{\beta }),& r_3>0.5\end{array}\right.$$

(16)

$$X(t+1)=l_b+r_5(u_b-l_b)$$

(17)

where $r_3$ and $r_5$ are random numbers between $[0,1]$, $r_4$ is a random number between $[0,0.5]$, and $u_b$ and $l_b$ are the upper and lower bounds of the search space, respectively.

EPD updates the less-fit wolves in various ways, including generating new individuals around elite wolves to improve local search ability and employing random generation to improve population diversity. This mechanism enhances the algorithm’s ability to escape local optima while maintaining its optimization capability.

3.4. Dynamic Weighting

In the standard GWO, the gray wolf individuals are mainly updated based on the positions of the alpha, beta, and delta wolves. The three wolves play the same role; this approach fails to emphasize the superior importance of the alpha wolf. Such an averaging process adversely affects both the convergence speed and accuracy. To overcome this limitation and accelerate convergence, we introduce dynamic weights based on fitness changes to amplify the role of the alpha wolf. The mathematical expression is as follows:

$$w_1={\displaystyle \frac{\left|f_{\delta }\right|}{\left|f_{\alpha }\right|+\left|f_{\beta }\right|+\left|f_{\delta }\right|}}$$

(18)

$$w_2={\displaystyle \frac{\left|f_{\beta }\right|}{\left|f_{\alpha }\right|+\left|f_{\beta }\right|+\left|f_{\delta }\right|}}$$

(19)

$$w_3={\displaystyle \frac{\left|f_{\alpha }\right|}{\left|f_{\alpha }\right|+\left|f_{\beta }\right|+\left|f_{\delta }\right|}}$$

(20)

$$X(t+1)=w_1{X}_1+w_2{X}_2+w_3{X}_3$$

(21)

where $f_{\alpha }$, $f_{\beta }$, and $f_{\delta }$ represent the current fitness values of the alpha, beta, and delta wolves, respectively.

4. Improved Dynamic Window Approach

4.1. Standard Dynamic Window Approach

Dynamic Window Approach (DWA) is a local path planning algorithm. Its core idea is to assess the robot’s surroundings and sample candidate linear and angular velocities, simulate the resulting trajectory for the next time step, and subsequently select a safe and efficient path by evaluating these trajectories using an evaluation function [36].

The mobile robot kinematic model is shown in Formula (22):

$$\begin{array}{l}x(t)=x(t-1)+v(t)\Delta t\cos \left(\theta (t-1)\right)\hfill \\ y(t)=y(t-1)+v(t)\Delta t\sin \left(\theta (t-1)\right)\hfill \\ \theta (t)=\theta (t-1)+\omega (t)\Delta t\hfill \end{array}$$

(22)

where $x(t)$, $y(t)$, and $\theta (t)$ are the current pose information of the mobile robot, $v(t)$ and $\omega (t)$ are the current linear and angular velocities of the mobile robot, and $\Delta t$ is the time interval. The mobile robot is constrained by linear and angular velocities as shown in Formula (23):

$$V_T=\{(v,w)|v_{\min }\le v\le v_{\max },w_{\min }\le w\le w_{\max }\}$$

(23)

Due to the limited motor torque, the robot is constrained by its capability, as shown in Formula (24):

$$V_D=\{(v,w)|v_c-\dot{v}\Delta t\le v\le v_c+\dot{v}\Delta t,w_c-\dot{w}\Delta t\le w\le w_c+\dot{w}\Delta t\}$$

(24)

where $v_c$ and $w_c$ are the linear and angular velocities of the mobile robot at the current time, and $\dot{v}$ and $\dot{w}$ are the maximum linear acceleration and maximum angular acceleration due to the motor. To ensure collision-free path planning, the mobile robot must maintain a predefined safety margin when navigating near obstacles, with the constraint relationship expressed in the following Formula (25):

$$V_A=\left\{(v,w),\left|v\le \sqrt{2\mathrm{dist}(v,w)\dot{v}},w\le \sqrt{2\mathrm{dist}(v,w)\dot{w}}\right.\right\}$$

(25)

where $\mathrm{dist}(v,w)$ is the distance evaluation function, which calculates the distance between the mobile robot and the obstacle. The mobile robot must satisfy the aforementioned three constraints, and after sampling, the motion trajectory is computed. Subsequently, the evaluation function is applied to the motion trajectory to determine the optimal path, and the standard evaluation function is shown in Formula (26):

$$G(v,w)={\lambda }_1\mathrm{heading}(v,w)+{\lambda }_2\mathrm{dist}(v,w)+{\lambda }_3\mathrm{velocity}(v,w)$$

(26)

where $\mathrm{heading}(v,w)$ is the azimuth evaluation function, denoting the angle difference between the mobile robot’s heading and the target point at the current time; $\mathrm{velocity}(v,w)$ is the velocity evaluation sub-function, denoting the speed of the current robot trajectory; and the weights for the three evaluation criteria are represented by ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, respectively.

We found that the conventional DWA algorithm still exhibits some shortcomings, and we have introduced the following improvements to address them:

(1)

Global Optimization Strategy

Instead of using only the endpoint as the goal—which can cause the robot to become trapped in local optima—we combine DWA with PAGWO. The globally optimal path generated by PAGWO is decomposed into several local goal points, guiding the robot step-by-step toward the final destination.

(2)

Dynamic Heading Angle

Rather than relying on a fixed heading angle based solely on the start-to-target direction, we now adjust the robot’s heading based on the next waypoint provided by PAGWO. This dynamic adjustment enables the robot to respond more effectively to environmental changes and to follow the optimal path.

(3)

Conflict Avoidance Strategy

We have implemented a new conflict avoidance mechanism that assesses the relative positions and angles between the robot and dynamic obstacles. If a frontal conflict is detected, the robot accelerates to bypass the obstacle; if a non-frontal conflict is identified, it stops and waits. This tailored response improves overall navigation efficiency by better handling dynamic obstacles.

4.2. Key Improvements to the Dynamic Window Approach

4.2.1. Global Optimization Strategy

Standard DWA only takes the endpoint as the goal point in path planning. However, it is easy for this approach to get stuck in local optima or deviate from the optimal path when encountering situations such as U-shaped obstacles. To address this shortcoming, we combine DWA with PAGWO. The globally optimal path generated by PAGWO is decomposed into individual local goal points, which guide the robot via DWA toward these local goal points, thereby ensuring that the robot successfully traverses these points to reach the endpoint.

4.2.2. Adaptive Heading Angle

The path quality of DWA is closely related to the initial heading angle. Traditional DWA usually selects a fixed value or uses the line between the start and end points as the initial heading angle. However, this approach is prone to detours, thereby adversely affecting path quality in complex environments. In this paper, we propose a novel initial heading angle by incorporating local target points. The optimized initial heading angle calculation is shown in Formula (27):

$$\theta =\arctan \left({\displaystyle \frac{y_1-y_0}{x_1-x_0}}\right)$$

(27)

where $(x_0,y_o)$ is the starting point, and $(x_1,y_1)$ is the first local target point.

The method incorporates the relationship between the robot and the local target points when calculating the initial heading angle, and by adaptively adjusting the initial heading angle, the robot can reduce unnecessary detours during the initial local path planning and bypass obstacles more safely, thereby enhancing the overall quality of the path.

4.2.3. Conflict Avoidance Strategies

In dynamic environments, mobile robots need to avoid dynamic obstacles to ensure the safety and efficiency of their tasks. The traditional DWA sometimes fails to take effective action in time, resulting in collisions when facing head-on conflict with dynamic obstacles. Additionally, when facing a dynamic obstacle approaching from the side, it often employs a non-discriminatory avoidance strategy, which may lead to unnecessary detours and increase the probability of collisions. To address these issues, this paper proposes a new obstacle avoidance strategy that utilizes the distance and angle relationship between the dynamic obstacle and the robot, assesses the obstacle’s behavior, and adopts corresponding avoidance maneuvers. The specific strategies are as follows.

First, the Euclidean distance between the robot’s current position and the dynamic obstacle’s current position is calculated. In this formula, the robot’s current position is at $(x_r,y_r)$, and the dynamic obstacle’s current position is at $(x_{obs},y_{obs})$.

$$d=\sqrt{{(x_{obs}-x_r)}^2+{(y_{obs}-y_r)}^2}$$

(28)

As shown in Figure 2, set the distance threshold $d_{\min }$ to 3 and the angle threshold ${\theta }_{\min }$ to 30°, then proceed to the following judgment when $d\le d_{\min }$ and the dynamic obstacle enters the mobile robot’s angle detection range.

In this context, the heading angle of the mobile robot, $\theta$, is initially defined with a specific value, representing the robot’s actual orientation. For simplicity in the obstacle avoidance strategy, we redefine the robot’s heading angle as ${\theta }_r$ and consider it as 0°. This simplification allows for easier judgment of the relative positions and angles of the robot and dynamic obstacle during the avoidance process. When the heading angle of the dynamic obstacle, ${\theta }_{obs}$, falls within the range of $[160°,200°]$, it is considered a head-on conflict; otherwise, it is considered a non-head-on conflict. When judged as a head-on conflict, the mobile robot quickly avoids the obstacle interference area by adjusting its angular velocity and accelerating to avoid collision. Specifically, if ${\theta }_{obs}\in (180°,200°]$, it indicates that the dynamic obstacle is moving to the left relative to the mobile robot, and the mobile robot accelerates to the left to bypass it. If ${\theta }_{obs}\in [160°,180°]$, it indicates that the dynamic obstacle is traveling to the right relative to the mobile robot or moving face-to-face, the mobile robot accelerates to the right to bypass it. If the conflict is determined to be non-head-on, the mobile robot stops and waits for the dynamic obstacle to leave the interference area before continuing path planning.

With this strategy, the mobile robot can minimize the risk of collisions in complex environments, thereby ensuring the safety of its movement while simultaneously reducing unnecessary detours and enhancing the efficiency of path planning.

5. PAGWO-IDWA Algorithm

5.1. Overall Framework

Figure 3 presents the main steps of the proposed path planning framework, from environment initialization to generating a safe and smooth path. First, the environment is represented as raster maps and relevant parameters are defined. The mobile robot’s start and goal positions, as well as any obstacles, are then specified. Next, a fitness function that combines Euclidean distance with an obstacle avoidance penalty is established to evaluate candidate paths. The improved PAGWO is subsequently used to determine the globally optimal path, and the resulting waypoints are provided to the improved DWA for local trajectory planning. By integrating PAGWO’s global guidance with DWA’s local maneuvering, the method ensures efficient, collision-free navigation in dynamic environments.

In the following section, we present the detailed algorithm flowchart and pseudo-code for the PAGWO-IDWA framework. This comprehensive algorithmic description elucidates the step-by-step procedures, key components, and decision-making logic that underpin the proposed method, thereby providing a clear pathway for both implementation and further analysis.

5.2. Algorithm Flowchart and Pseudo-Code

In this paper, the improved DWA is integrated with PAGWO (termed PAGWO-IDWA) to plan paths for mobile robots and to enable them to avoid obstacles in dynamic environments, thereby enhancing the algorithm’s dynamic performance and robustness. The algorithm’s flowchart is presented in Figure 4:

In this framework, PAGWO is first utilized to generate a global collision-free path, incorporating four key enhancements: (1) piecewise chaotic mapping to diversify the search space; (2) an EDP strategy to prevent premature convergence; (3) an improved nonlinear convergence factor to balance exploration and exploitation; and (4) a dynamic weighting scheme in the position update formula. Once the global path points are obtained, the improved IDWA uses these points to guide the heading angle and optimize local obstacle avoidance. Specifically, the conflict avoidance strategy is based on the relative angle and position between the mobile robot and dynamic obstacles. When a direct conflict is detected—where the robot and obstacle are aligned in a manner that poses a high risk of collision—the robot accelerates to quickly bypass the obstacle. Conversely, if a non-direct conflict is identified, the robot decelerates and stops, waiting for a safer opportunity to proceed. Moreover, similar to the conventional Dynamic Window Approach (DWA), IDWA employs an evaluation function that considers factors such as heading alignment, velocity, and obstacle clearance to determine the optimal local trajectory. By combining PAGWO’s global guidance with IDWA’s adaptive local maneuvering, the proposed method achieves efficient and safe navigation in dynamic environments. The pseudo-code of the PAGWO-IDWA is presented in Algorithm 1.

Algorithm 1: PAGWO-IDWA pseudo-code
Input:
Number of grey wolves: n.
Maximum number of iterations: Max_iter.
Dimension of the problem: dim.
Grid map: G.
DWA parameters: ${\lambda }_1,{\lambda }_1,{\lambda }_1$.
Output:
The smooth optimal trajectory for the robot.
1.	Initialize the Grey Wolf population using Piecewise Chaotic Mapping.
2.	Initialize Alpha, Beta, and Delta wolves.
3.	While iter < Max_iter
4.	for i = 1: n
5.	Evaluate fitness and update leaders;
6.	end for
7.	for i = 1: round(n/4)
8.	Apply Evolutionary Population Dynamics (EPD);
9.	end for
10.	Update nonlinear convergence factor;
11.	For i = 1: n
12.	Update positions with dynamic weighting;
13.	end for
14.	Obtain the current new location;
15.	Update the optimal location of the current individual
16.	iter = iter + 1;
17.	end while
18.	Acquire global optimal path point information of the PAGWO;
19.	Fuse the point information and put it into local planner DWA;
20.	Control the robot to move according to the global optimal with Conflict avoidance strategies;
21.	Return the smooth optimal trajectory.

5.3. Complexity Analysis

In optimization problems, computational complexity analysis is an important metric for evaluating the performance of an algorithm, and the computational cost of an algorithm is quantified using Big O notation. When applying the proposed PAGWO-IDWA algorithm to the path planning problem, the complexity mainly depends on these key factors: the population size, $n$, the dimensionality, $d$, of the path planning problem, the maximum number of iterations, $t_m$, and the evaluation complexity of the objective function, $c$. The computational complexity of the proposed PAGWO-IDWA is specified as follows:

$$\begin{array}{cc}\hfill O(\mathrm{PAGWO}\text{-}\mathrm{IDWA})& =O(\mathrm{path} \mathrm{planning} \mathrm{problem})+O(\mathrm{init}.)+O(\mathrm{cost} \mathrm{function})\hfill \\ \hfill & +O(\mathrm{solution} \mathrm{update})+O\left(\mathrm{EPD}\right)\hfill \\ \hfill & =O(1+nd+t_{m}cn+t_{m}nd+n\log (n)+nd)\hfill \\ \hfill & \cong O(t_{m}cn+t_{m}nd)\hfill \end{array}$$

(29)

6. Experiments Simulation and Analysis

6.1. Algorithm Testing and Analysis

To validate the performance of the PAGWO algorithm proposed in this study, we systematically selected a diverse set of benchmark functions commonly employed in optimization, including different problem types (as shown in

Table 1. International standard test function.

F	Function	Dim	Range	Best
$f_1$	Sphere Model	30/100	[−100, 100]	0
$f_2$	Schwefel’s problem 2.22	30/100	[−10, 10]	0
$f_3$	Schwefel’s problem 1.2	30/100	[−10, 10]	0
$f_4$	Schwefel’s problem 2.21	30/100	[−100, 100]	0
$f_5$	Quartic Function	30/100	[−1.28, 1.28]	0
$f_6$	Shifted Zakharov Function	10	[−100, 100]	300
$f_7$	Generalized Rastrigin’s Function	30/100	[−5.12, 5.12]	0
$f_8$	Ackley’s Function	30/100	[−32, 32]	0
$f_9$	Generalized Griewank Function	30/100	[−600, 600]	0
$f_{10}$	Schaffer F7 Function	10	[−100, 100]	600
$f_{11}$	Shifted Levy Function	10	[−100, 100]	900
$f_{12}$	Kowalik’s Function	4	[−5, 5]	0.0003
$f_{13}$	Goldstein-Price Function	2	[−2, 2]	3
$f_{14}$	Composition Function 2	10	[−100, 100]	2400
$f_{15}$	Composition Function 7	10	[−100, 100]	2700

), to ensure comprehensive evaluation. $f_1$–$f_6$ are single-peak functions that test the algorithm’s essential optimization ability and convergence speed; $f_7$–$f_{11}$ are multimodal functions to evaluate the algorithm’s global search capability, particularly its ability to find the global optimum in complex search spaces; $f_{12}$ and $f_{13}$ are low-dimensional fixed functions to ensure that the algorithm can balance global and local search; $f_{14}$ and $f_{15}$ are composite functions used to assess the robustness of algorithms in handling complex, multimodal problems.

Subsequently, the algorithms are tested independently. The selected algorithms and parameters, as shown in

Table 2. Algorithmic parameter.

Algorithms	GA	SCA	PSO	GWO	PAGWO
Parameter configuration	$\begin{array}{l}{p}_c=0.8;\\ p_m=0.05;\\ Pop=50;\\ t_m=1000\end{array}$	$\begin{array}{l}a=2;\\ Pop=50;\\ t_m=1000\end{array}$	$\begin{array}{l}w=0.8;\\ c_1=1.5;\\ c_2=1.5;\\ Pop=50;\\ t_m=1000\end{array}$	$\begin{array}{l}Pop=50;\\ t_m=1000\end{array}$	$\begin{array}{l}p=0.3;\\ n=600;\\ Pop=50;\\ t_m=1000\end{array}$

, are tested using a population size of 50 and 1000 iterations. To ensure the validity of the test and maintain objectivity, each algorithm is independently tested 30 times, using the average, standard deviation, and optimal value as evaluation metrics. The test results are shown in

Table 3. Comparison of different algorithms.

Function	Metric	GA	SCA	PSO	GWO	PAGWO
	Mean	$4.52\times {10}^3$	$4.19\times {10}^{-3}$	$4.46\times {10}^{-6}$	$9.31\times {10}^{-71}$	$4.22\times {10}^{-156}$
$f_1$	Std	$3.47\times {10}^3$	$9.05\times {10}^{-3}$	$4.27\times {10}^{-6}$	$1.57\times {10}^{-70}$	$2.31\times {10}^{-155}$
	Best	$4.58\times {10}^2$	$6.76\times {10}^{-7}$	$3.19\times {10}^{-7}$	$9.39\times {10}^{-73}$	$3.75\times {10}^{-165}$
	Mean	$3.62\times {10}^1$	$5.70\times {10}^{-6}$	$1.25\times {10}^{-4}$	$4.71\times {10}^{-41}$	$5.57\times {10}^{-90}$
$f_2$	Std	$1.14\times {10}^1$	$1.42\times {10}^{-5}$	$7.70\times {10}^{-5}$	$4.97\times {10}^{-41}$	$1.80\times {10}^{-89}$
	Best	$1.61\times {10}^1$	$3.34\times {10}^{-8}$	$2.53\times {10}^{-5}$	$4.09\times {10}^{-42}$	$1.39\times {10}^{-92}$
	Mean	$3.76\times {10}^4$	$2.24\times {10}^3$	$4.46\times {10}^2$	$1.12\times {10}^{-20}$	$2.10\times {10}^{-51}$
$f_3$	Std	$1.19\times {10}^4$	$1.89\times {10}^3$	$9.19\times {10}^2$	$3.09\times {10}^{-20}$	$9.83\times {10}^{-51}$
	Best	$1.20\times {10}^4$	$1.24\times {10}^2$	$1.11\times {10}^2$	$9.16\times {10}^{-26}$	$2.45\times {10}^{-62}$
	Mean	$5.56\times {10}^1$	$1.23\times {10}^1$	$3.33\times {10}^0$	$2.12\times {10}^{-17}$	$3.41\times {10}^{-50}$
$f_4$	Std	$9.78\times {10}^0$	$7.05\times {10}^0$	$7.41\times {10}^{-1}$	$2.72\times {10}^{-17}$	$1.01\times {10}^{-49}$
	Best	$4.25\times {10}^1$	$1.15\times {10}^0$	$2.16\times {10}^0$	$3.03\times {10}^{-19}$	$6.97\times {10}^{-54}$
	Mean	$1.25\times {10}^0$	$1.55\times {10}^{-2}$	$1.71\times {10}^{-2}$	$5.13\times {10}^{-4}$	$1.91\times {10}^{-4}$
$f_5$	Std	$1.31\times {10}^0$	$8.88\times {10}^{-3}$	$6.14\times {10}^{-3}$	$2.49\times {10}^{-4}$	$1.15\times {10}^{-4}$
	Best	$1.21\times {10}^{-1}$	$1.17\times {10}^{-3}$	$6.91\times {10}^{-3}$	$1.55\times {10}^{-4}$	$1.69\times {10}^{-5}$
	Mean	$9.93\times {10}^{12}$	$1.22\times {10}^3$	$3.00\times {10}^2$	$1.49\times {10}^3$	$3.00\times {10}^2$
$f_6$	Std	$2.00\times {10}^{-3}$	$6.35\times {10}^2$	$1.69\times {10}^{-7}$	$1.53\times {10}^3$	$5.31\times {10}^{-2}$
	Best	$9.93\times {10}^{12}$	$4.36\times {10}^2$	$3.00\times {10}^2$	$3.27\times {10}^2$	$3.00\times {10}^2$
	Mean	$1.45\times {10}^2$	$2.38\times {10}^1$	$3.76\times {10}^1$	$1.91\times {10}^{-1}$	$0$
$f_7$	Std	$4.53\times {10}^1$	$3.74\times {10}^1$	$1.12\times {10}^1$	$1.04\times {10}^0$	$0$
	Best	$7.32\times {10}^1$	$1.68\times {10}^{-7}$	$1.85\times {10}^1$	$0$	$0$
	Mean	$1.90\times {10}^1$	$1.13\times {10}^1$	$8.29\times {10}^{-4}$	$1.36\times {10}^{-14}$	$5.39\times {10}^{-15}$
$f_8$	Std	$7.85\times {10}^{-1}$	$9.62\times {10}^0$	$6.98\times {10}^{-4}$	$2.22\times {10}^{-15}$	$1.60\times {10}^{-15}$
	Best	$1.67\times {10}^1$	$6.41\times {10}^{-6}$	$1.38\times {10}^{-4}$	$7.99\times {10}^{-15}$	$4.44\times {10}^{-15}$
	Mean	$2.93\times {10}^1$	$1.24\times {10}^{-1}$	$1.48\times {10}^{-2}$	$1.65\times {10}^{-3}$	$0$
$f_9$	Std	$1.75\times {10}^1$	$1.75\times {10}^{-1}$	$1.84\times {10}^{-2}$	$5.70\times {10}^{-3}$	$0$
	Best	$5.58\times {10}^0$	$1.79\times {10}^{-6}$	$2.02\times {10}^{-6}$	$0$	$0$
	Mean	$9.34\times {10}^2$	$6.18\times {10}^2$	$6.00\times {10}^2$	$6.01\times {10}^2$	$6.00\times {10}^2$
$f_{10}$	Std	$1.15\times {10}^{-13}$	$3.63\times {10}^0$	$5.82\times {10}^{-1}$	$8.64\times {10}^{-1}$	$3.58\times {10}^{-1}$
	Best	$9.34\times {10}^2$	$6.12\times {10}^2$	$6.00\times {10}^2$	$6.00\times {10}^2$	$6.00\times {10}^2$
	Mean	$2.59\times {10}^4$	$1.01\times {10}^3$	$9.01\times {10}^2$	$9.06\times {10}^2$	$9.00\times {10}^2$
$f_{11}$	Std	$0$	$4.32\times {10}^1$	$2.35\times {10}^0$	$1.15\times {10}^1$	$2.67\times {10}^{-1}$
	Best	$2.59\times {10}^4$	$9.51\times {10}^2$	$9.00\times {10}^2$	$9.00\times {10}^2$	$9.00\times {10}^2$
	Mean	$4.96\times {10}^{-3}$	$8.62\times {10}^{-4}$	$1.81\times {10}^{-3}$	$3.71\times {10}^{-3}$	$3.56\times {10}^{-4}$
$f_{12}$	Std	$6.49\times {10}^{-3}$	$3.98\times {10}^{-4}$	$5.06\times {10}^{-3}$	$7.58\times {10}^{-3}$	$1.68\times {10}^{-4}$
	Best	$6.83\times {10}^{-4}$	$3.15\times {10}^{-4}$	$3.07\times {10}^{-4}$	$3.07\times {10}^{-4}$	$3.07\times {10}^{-4}$
	Mean	$8.08\times {10}^0$	$3.00\times {10}^0$	$3.00\times {10}^0$	$3.00\times {10}^0$	$3.00\times {10}^0$
$f_{13}$	Std	$1.61\times {10}^1$	$1.82\times {10}^{-5}$	$1.39\times {10}^{-15}$	$5.84\times {10}^{-6}$	$7.55\times {10}^{-7}$
	Best	$3.00\times {10}^0$	$3.00\times {10}^0$	$3.00\times {10}^0$	$3.00\times {10}^0$	$3.00\times {10}^0$
	Mean	$5.56\times {10}^3$	$2.50\times {10}^3$	$2.55\times {10}^3$	$2.55\times {10}^3$	$2.53\times {10}^3$
$f_{14}$	Std	$0$	$4.69\times {10}^{-1}$	$5.68\times {10}^1$	$6.48\times {10}^1$	$5.08\times {10}^1$
	Best	$5.56\times {10}^3$	$2.50\times {10}^3$	$2.50\times {10}^3$	$2.50\times {10}^3$	$2.50\times {10}^3$
	Mean	$1.27\times {10}^4$	$2.87\times {10}^3$	$2.87\times {10}^3$	$2.87\times {10}^3$	$2.86\times {10}^3$
$f_{15}$	Std	$0$	$1.46\times {10}^0$	$9.55\times {10}^0$	$6.70\times {10}^0$	$1.75\times {10}^0$
	Best	$1.27\times {10}^4$	$2.87\times {10}^3$	$2.86\times {10}^3$	$2.86\times {10}^3$	$2.86\times {10}^3$

.

The proposed PAGWO algorithm demonstrates superior convergence accuracy compared to conventional algorithms in single-peak functions $f_1$–$f_6$ while maintaining a low standard deviation, indicating its excellent fundamental optimization capability and rapid convergence speed. Notably, in function $f_5$ with introduced random noise, PAGWO exhibits robust performance, further verifying its capability to handle interference environments. For multimodal functions, PAGWO not only successfully attains the global optimum of 0 in both $f_7$ and $f_9$ but also outperforms other algorithms on $f_8$, $f_{10}$, and $f_{11}$, thereby confirming its robust global search capability and effective mechanism for escaping local optima. In low-dimensional fixed functions, PAGWO outperforms baseline methods in $f_{12}$, and the average solution quality in $f_{13}$ also reaches the theoretical optimal value, demonstrating its consistent superiority in high-dimensional scenarios. In the combined functions $f_{14}$ and $f_{15}$, the average performance of PAGWO surpasses that of the other algorithms, demonstrating that it is more robust when tackling complex, multimodal optimization problems. These experimental results collectively validate PAGWO’s state-of-the-art performance across diverse benchmarks. The algorithmic improvements can be specifically attributed to four key strategies: enhanced initial exploration through piecewise chaotic mapping, balanced global–local search dynamics via a nonlinear convergence factor, accelerated convergence through optimized position updating, and robustness improvement via EPD for local optima avoidance.

6.2. Experimental Content of Path Planning

To evaluate the effectiveness of the PAGWO-IDWA algorithm, the following three sets of experiments are designed in this paper:

• Global path planning experiments: the GA, the SCA, the PSO, and the GWO algorithm are used as references in three raster maps of varying complexity to verify the performance of the PAGWO algorithm in global path planning.
• PAGWO-IDWA and DWA Comparison Experiment: in the DWA, two enhancements are based on PAGWO, so the improved version can essentially be regarded as PAGWO-IDWA. To evaluate the performance of the improved algorithm relative to the original DWA, we conducted two sub-experiments. The first validates that using PAGWO-generated waypoints significantly enhances DWA’s global path planning capability, while the second demonstrates that employing PAGWO-generated waypoints as orientation angles improves its path efficiency. The dynamic obstacle avoidance performance of DWA will be validated in a subsequent experiment;
• Dynamic obstacle avoidance experiment: dynamic obstacles are added to the map to assess the PAGWO-IDWA algorithm’s obstacle avoidance performance in dynamic environments and evaluate its safety and generalizability. Figure 5 shows the three raster maps used for testing in this paper. The environment complexity increases with the number of obstacles. Figure 5a shows a simple 20 m × 20 m map, Figure 5b shows a moderately complex 30 m × 30 m map, and Figure 5c shows a highly complex map of 30 m × 30 m.

The experimental hardware consisted of a laptop with an AMD R9-7940H processor at 4 GHz (AMD, Santa Clara, CA, USA), 16 GB of RAM, running Windows 11 64-bit with MATLAB R2021b software.

6.3. Global Path Planning Experiment

To validate the efficacy of the PAGWO algorithm, the mobile robot is simplified as a point on a two-dimensional network and compared with four algorithms, GA, SCA, PSO, and GWO, in the three simulation maps described above. The black squares in the figure represent obstacles, and the red dots and squares indicate the start and goal positions, respectively (the start is located at the bottom-left corner, and the goal is at the top-right corner). The evaluation metrics are path length, number of turns, and running time. In the experiment, the population size for all algorithms is set to 50, and the number of iterations is set to 1000.

The simulation results for map 1 are presented in Figure 6, which illustrates the paths generated by each algorithm and their convergence behaviors. Among the five algorithms compared—namely, GA, SCA, PSO, GWO, and PAGWO, PAGWO produces the shortest path with the fewest turns, indicating that PAGWO approaches the target location faster while avoiding static obstacles. This is due to incorporating strategies such as Piecewise chaotic mapping and nonlinear convergence factors, which make PAGWO more effective in path planning algorithms. Specifically, in Figure 6a, GA exhibits redundant path segments in the early stage of the route. In Figure 6b, SCA generates the highest number of path deviations. In Figure 6c,d, PSO and GWO perform well, demonstrating fewer deviations and shorter path lengths. However, further improvements are needed to achieve optimality. In Figure 6f, PAGWO is observed to converge faster toward the minimum, quickly finding the optimal solution during the iterative process, indicating that it outperforms other algorithms in global planning and convergence efficiency.

For further verification of the performance of the PAGWO algorithm in this study, the second set of experiments was conducted in simulation environment 2. Compared with the first set of experimental maps, the obstacle distribution in the second set is more complex, demanding higher path planning capabilities from the algorithm. The comparison of path planning results is shown in Figure 7. For example, both GA and PSO exhibited moderate performance by navigating into the U-shaped obstacle twice in succession. However, SCA and GWO performed poorly in this path planning, generating paths with a large number of redundant segments. In contrast, PAGWO not only avoids the U-shaped obstacles but also performs exceptionally well in terms of path smoothness, with a notable reduction in the number of turns, indicating its ability to effectively avoid redundant steering and improve the smoothness of the path. From Figure 7f, PAGWO is shown to be the most effective in both convergence accuracy and speed, having converged after 200 iterations, outperforming other algorithms. The computational efficiency and path planning stability of PAGWO in complex environments are further verified.

To further verify the generalizability of the PAGWO algorithm, path planning simulation experiments were performed in simulation environment 3, which has the most complex obstacle distribution. The results of the experiment are presented in Figure 8. Both GA and SCA exhibit significant path redundancy in the early stages of their routes, characterized by unnecessary detours into the interior U-shaped obstacles. Additionally, PSO becomes trapped in a local optimum, resulting in a suboptimal path, while GWO produces a path with multiple turns in its middle section, compromising overall path quality. In contrast, the PAGWO algorithm generates a path with significantly fewer turns and a smoother trajectory, demonstrating a practical improvement in path smoothing and efficiency when handling complex obstacle distributions. The convergence curve shows that PAGWO converges quickly within 200 iterations and significantly outperforms other algorithms in terms of both convergence speed and stability. The other algorithms exhibit slower convergence speeds and higher final stabilization values, further indicating that PAGWO excels in global optimization.

In summary, across three simulation environments with varying obstacle distributions and map sizes, PAGWO outperforms other algorithms in terms of path length, number of turns, and other metrics. To verify the robustness of PAGWO, the proposed algorithm was run 30 times in three different environments and was compared with other algorithms. The evaluation metrics are path length, number of turns, and running time, as shown in

Table 4. Comparison of algorithm performance metrics in path planning.

ENV. Model		Algorithm	GA	SCA	PSO	GWO	PAGWO
	Index
ENV. 1	Path Length (m)		30.65	30.82	30.75	31.13	28.21
	Number of turns		6.50	6.40	6.60	7.10	6.10
	Running time (s)		6.39	5.20	4.06	2.39	2.06
ENV. 2	Path Length (m)		50.18	51.56	50.49	49.56	44.30
	Number of turns		14.90	14.73	15.83	15.30	7.53
	Running time (s)		14.96	3.10	6.69	4.98	3.06
ENV. 3	Path Length (m)		52.68	52.57	53.56	51.30	46.82
	Number of turns		18.50	18.27	19.57	18.40	12.03
	Running time (s)		13.49	0.84	4.13	4.20	2.58

. In the three simulation map environments, the PAGWO algorithm shows significant advantages in path optimization. In terms of path length, the average shortest path lengths for PAGWO in the three maps are 28.21m, 44.30m, and 46.82m, respectively, outperforming other algorithms. In terms of the number of turns, PAGWO also performs better. In map 3, which is the most complex in terms of obstacles, the average number of turns is reduced by 34.62%, 34.97%, 34.15%, and 38.53% compared to GWO, GA, SCA, and PSO, respectively. In terms of running time, PAGWO performs exceptionally well in maps 2 and 3, demonstrating its higher computational efficiency in complex environments. In summary, the PAGWO algorithm not only improves the global optimization performance of the paths but also achieves a good balance between path smoothness and computational efficiency, making it an effective method for solving path planning problems in complex environments.

6.4. PAGWO-IDWA and DWA Comparison Experiment

To evaluate the impact of incorporating PAGWO-generated global waypoints into DWA, the first experiment was designed with the mobile robot’s starting point at (1, 1) and its endpoint at (10, 10). In the simulation environment, the triangle denotes the start point, the circle denotes the goal, the asterisks represent local goal points generated by PAGWO, and the green area indicates the detection range of the DWA algorithm. The experimental results are presented in Figure 9. In Figure 9a, the robot is trapped in the local optimum and cannot reach the end point successfully, while in Figure 9b, the robot is guided to the endpoint by the global path of PAGWO and successfully arrives at the endpoint. This method effectively reduces the limitation imposed by a single global goal point on path planning, allowing the robot to complete the path planning task successfully.

To evaluate the effect of the adaptive facing angle on DWA, a second set of experiments was conducted. In these experiments, the mobile robot’s starting point was set at (1, 1) and the endpoint at (20, 20). Path planning was performed using both the DWA algorithm and the PAGWO-IDWA algorithm on the same raster map, and the results are shown in Figure 10. In Figure 10a, the initial facing angle of DWA is directed toward the endpoint, causing the robot to deviate from the optimal path at the outset and resulting in redundant path segments. Furthermore, after encountering an obstacle, DWA continues to follow the wall because its facing angle remains fixed toward the endpoint, lacking adaptive guidance. In contrast, Figure 10b shows that during the initial stage of path planning with PAGWO-IDWA, the robot’s facing angle is oriented toward the PAGWO-generated path point. Under the correct guidance of this path point, the robot follows the optimal path to the endpoint, thereby verifying that the improved algorithm outperforms DWA in path optimization. The data for these experiments are presented in

Table 5. PAGWO-IDWA and DWA comparison experiment.

	GWO Path Length (m)	PAGWO-IDWA Path Length (m)
Comparison experiment 1	Inf	14.29
Comparison experiment 2	34.52	27.25

.

6.5. Dynamic Obstacle Avoidance Experiment

To validate the dynamic obstacle avoidance capability of the PAGWO-IDWA algorithm, a red dynamic obstacle is introduced into the static map 1. The dynamic obstacle starts at (5, 4) and ends at (2, 2), while the mobile robot starts at (1, 1) and aims to reach (20, 20). Figure 11a shows the globally optimal path planned by PAGWO, which maintains the shortest distance from start to end and avoids static obstacles, providing a basis for subsequent local planning. In Figure 11b, the robot starts from the initial position and moves along the global path toward the local goal point. Upon encountering the dynamic obstacle, it adjusts its direction to the right and accelerates to avoid it. After safely bypassing the obstacle, the robot decelerates and readjusts its angle to the left, resuming its movement toward the local goal point. Finally, as shown in Figure 11c, the robot successfully reaches the endpoint, guided by the local goal points.

To further validate the path planning capability of the PAGWO-IDWA algorithm in dynamic environments, this study introduces dynamic obstacles approaching from the side in simulation map 2. The dynamic obstacle’s starting point is (6, 6) and the endpoint is (6, 1), while the mobile robot’s starting point is (1, 1) and the endpoint is (30, 30). The dynamic obstacle avoidance process is shown in Figure 12. Figure 12a shows the optimal path found by the PAGWO algorithm in simulation map 2. In Figure 12b, the mobile robot detects the dynamic obstacle approaching from the side and stops to avoid the collision. In Figure 12c, after the dynamic obstacle leaves the robot’s detection range and safety is confirmed, the robot continues toward the local target point. Finally, in Figure 12d, the robot returns to the global optimal path and successfully completes the path planning task. The experimental results show that in more complex map environments, when facing dynamic obstacles approaching from the side, the robot can safely stop during local planning, continue with the path planning task after the obstacle passes, and maintain stability of the path planning, which demonstrates the adaptive ability of PAGWO-IDWA.

To further validate the generalization capability of PAGWO-IDWA, two dynamic obstacles are added to map 3. The first dynamic obstacle has start and end points at (7, 1) and (7, 5), respectively, while the second dynamic obstacle starts at (13, 9) and ends at (7, 4). The mobile robot starts at (1, 1) and ends at (30, 30). The experimental results are shown in Figure 13. In Figure 13a, the global optimal path planned by the PAGWO algorithm is presented, where the mobile robot safely avoids static obstacles and finds the shortest path. In Figure 13b, the robot approaches the first dynamic obstacle from the side and decelerates to avoid it. In Figure 13c, after confirming the obstacle has passed, the robot returns to the global path and continues forward. In Figure 13d, the robot encounters the second dynamic obstacle, and in Figure 13e, it adjusts its direction and safely avoids it. Finally, the trajectory in Figure 13f shows the robot successfully reaching the endpoint, avoiding two successive dynamic obstacles under the guidance of the PAGWO-IDWA algorithm, thus completing the complex dynamic path obstacle avoidance task. The experimental results demonstrate that PAGWO-IDWA efficiently avoids dynamic obstacles in complex environments, further validating the algorithm’s generalization capability and practicality.

To further investigate the impact of additional obstacles on dynamic obstacle avoidance performance, we introduced five dynamic obstacles in map 2. Specifically, the first obstacle has start and end points at (6, 6) and (6, 1), the second at (7, 1) and (5, 7), the third at (14, 7) and (7, 5), the fourth at (15, 7) and (8, 5), and the fifth at (26, 23) and (17, 13). The mobile robot’s start and end points are (1, 1) and (30, 30), respectively. The experimental results are presented in Figure 14, where Figure 14a shows the globally optimal path identified by the PAGWO algorithm.

In Figure 14b, the mobile robot begins local path planning and detects two dynamic obstacles at (3, 2), approaching from the left and right, respectively. As a result, the robot halts and performs obstacle avoidance. In Figure 14c, after ensuring safety, the robot resumes movement. After a short distance, it detects two oncoming dynamic obstacles. In Figure 14d, the robot is seen avoiding the dynamic obstacles by skirting to the right and quickly returning to its original global path. Finally, in Figure 14e, the mobile robot encounters the last dynamic obstacle, evading it to the left. As shown in Figure 14f, the robot successfully avoids all dynamic obstacles and reaches the endpoint along the preferred path.

The current experimental results clearly demonstrate that our improved PAGWO-IDWA algorithm significantly enhances obstacle avoidance compared to traditional methods, effectively handling scenarios where dynamic obstacles appear consecutively. This reactive strategy ensures safe navigation and validates the algorithm’s practical utility in complex dynamic environments. However, we also recognize that when multiple dynamic obstacles simultaneously enter the detection range, the current approach may encounter processing limitations. Future work will address these challenges by incorporating predictive models—such as Reinforcement Learning—to anticipate obstacle trajectories and further improve avoidance performance in high-density dynamic settings.

7. Conclusions

This paper proposes a new algorithm, PAGWO-IDWA, to solve the path planning problem. First, to improve the quality of the initial population, the Piecewise chaotic mapping is used to initialize the Grey Wolf population. Second, a nonlinear convergence factor is introduced to balance global and local search capabilities. Then, the EPD strategy is applied, and dynamic weights are introduced, enhancing the algorithm’s ability to escape local optima and accelerate the convergence speed. Finally, combining PAGWO with IDWA addresses the issues of path smoothness and dynamic obstacle avoidance in the PAGWO algorithm.

The algorithm’s performance is verified through both static and dynamic obstacle avoidance experiments. PAGWO outperforms GA, SCA, PSO, and GWO in path planning performance across three static maps with varying complexity. Compared with the standard GWO, the improved PAGWO reduces the path length, number of turns, and running time by 9.39%, 14.08%, and 13.81%, respectively, in map 1. In map 2, the improvements are 10.61%, 50.78%, and 38.55%, respectively, and in map 3, they are 8.73%, 34.62%, and 38.57%. Dynamic obstacle avoidance experiments further demonstrate that PAGWO-IDWA generates efficient, smooth paths in complex dynamic environments, safely bypassing dynamic obstacles or waiting for them to pass, according to the obstacle avoidance strategy. This progression verifies its strong global path planning and dynamic obstacle avoidance capabilities.

While our current conflict avoidance strategy has enhanced the mobile robot’s obstacle avoidance performance compared to the original approach, limitations still remain. In the future, we plan to explore predictive techniques, such as Reinforcement Learning, to allow the robot to anticipate obstacle trajectories and proactively adjust its path. We expect that these improvements will further enhance both the safety and efficiency of path planning in highly dynamic environments. Additionally, our simulations presently model the mobile robot as a point, with a fitness function that primarily minimizes path length while applying a basic obstacle-avoidance penalty. However, practical applications necessitate a more comprehensive consideration of factors such as the robot’s physical dimensions, obstacle sizes, and necessary safety margins to account for localization errors. In subsequent studies, we intend to integrate these real-world constraints into our optimization framework. By combining predictive techniques with a more realistic model, we aim to further improve the robustness and applicability of the PAGWO-IDWA algorithm for mobile robot path planning.