Path Planning of Robot Based on Improved Multi-Strategy Fusion Whale Algorithm

Abstract

In logistics and manufacturing, smart technologies are increasingly used, and warehouse logistics robots (WLR) have thus become key automation tools. Nonetheless, the path planning of mobile robots in complex environments still faces the challenges of excessively long paths and high energy consumption. To this end, this study proposes an innovative optimization algorithm, IWOA-WLR, which aims to optimize path planning and improve the shortest route and smoothness of paths. The algorithm is based on the Whale Algorithm with Multiple Strategies Fusion (IWOA), which significantly improves the obstacle avoidance ability and path optimization of mobile robots in global path planning. First, improved Tent chaotic mapping and differential dynamic weights are used to enhance the algorithm’s optimization-seeking ability and improve the diversity of the population. In the late stage of the optimization search, the positive cosine inertia threshold and the golden sine are used to perform adaptive position updating during the search strategy to enhance the global optimal search capability. Secondly, the fitness function of the path planning problem is designed, and the path length is taken as the objective function, the path smoothness as the evaluation index, and the multi-objective optimization is realized through the hierarchical adjustment strategy and is applied to the global path planning of WLR. Finally, simulation experiments on raster maps with grid sizes of 15 × 15 and 20 × 20 compare the IWOA algorithm with the WOA, GWO, MAACO, RRT, and A* algorithms. On the 15 × 15 maps, the IWOA algorithm reduces path lengths by 3.61%, 5.90%, 1.27%, 15.79%, and 5.26%, respectively. On the 20 × 20 maps, the reductions are 4.56%, 5.83%, 3.95%, 19.57%, and 1.59%, respectively. These results indicate that the improved algorithm efficiently and reliably finds the global optimal path, significantly reduces path length, and enhances the smoothness and stability of the path’s inflection points.

1. Introduction

With the rapid development of the manufacturing and logistics industries, there is a growing need to reduce labor costs, improve efficiency, and enhance operational safety [1]. Against this backdrop, intelligent handling robots are beginning to play a vital role in production lines. These robots can not only automate and prioritize tasks such as parts supply, finished product handling, and material transportation but also adapt to different handling needs by being equipped with different types of handling devices such as pallets, carriages, or robotic arms. The flexibility and versatility of these robots allow them to adapt to changing production line environments, thus improving overall productivity and logistics efficiency [2]. The robot’s design encompasses its components, structural framework, and algorithms. In terms of components, the robot is constructed from modular units that operate collaboratively to perform various functions. Structurally, it features a robust base and a flexible arm, enabling adaptation to different handling requirements. Algorithmically, the robot employs advanced path planning and navigation technologies to ensure efficient operation within complex environments, as illustrated in Figure 1. These components interact within the system, working in unison to automate the robot’s tasks. The structural design incorporates essential elements such as the overall layout, base, wheels, and manipulator, all contributing to the robot’s stability and adaptability. The algorithms, including those for path planning, autonomous navigation, and obstacle avoidance, are central to optimizing the robot’s performance in intricate environments [3,4,5].

On the production line, the navigation process of mobile robots includes three stages: environment sensing, path planning, and motion control. Among them, path planning is the key link [6], which directly affects the robot’s navigation effect and overall operational efficiency. Although the genetic algorithm [7], Dijkstra’s algorithm [8], and A-algorithm [9] perform well on specific problems, they also reveal some limitations when dealing with complex problems. For example, although the genetic algorithm performs well in global search, it tends to converge slowly and is prone to fall into local optimal solutions; Dijkstra’s algorithm’s efficiency drops significantly when the graph size is large; and the performance of an algorithm is highly dependent on the choice of heuristic functions.

To overcome these limitations, swarm intelligence algorithms have emerged. These algorithms are able to balance solution completeness and timeliness in solving path-planning problems by modeling the behavior of groups in nature [10]. For example, the African Vulture Optimization Algorithm (AVOA) [11], Snake Optimization Algorithm (SO) [12], Ant Colony Optimization Algorithm (ACO) [13], and Gray Wolf Optimization Algorithm (GWO) [14] are successful applications based on bio-inspired learning. These algorithms are known for their versatility, ease of implementation, and high optimization efficiency and are well-suited for solving numerous practical problems [15,16,17].

Whale Optimization Algorithm (WOA) [18,19] is a relatively new metaheuristic algorithm. Compared with other algorithms, the WOA algorithm is characterized by a simple structure, easy implementation, and good parallel performance. It has been widely used in practical applications such as data prediction [20], path planning [21], and fault diagnosis [22]. However, WOA also faces challenges during the iterative process, including low solution accuracy, slow convergence speed, and susceptibility to local optima. Qing et al. [23] proposed an improved version of WOA, which utilizes Logistic Chaos Mapping to generate chaotic sequences to initialize the population location and enhance the initial population diversity. Despite these efforts, the diversity of the original population is still insufficient, thus limiting exploration and equilibrium. Pengju et al. [24] enhanced the global optimization capability of the algorithm by designing an inertia weighting strategy and combining it with a simulated annealing strategy. Although these improvements improved the performance of the algorithm to some extent, it still suffers from weak global search capability and poor convergence accuracy. Zhang et al. [25] introduced the golden sine strategy in the global search phase to better coordinate the local and global search capabilities. However, the lack of comprehensive consideration of the advantages and disadvantages of the algorithm before and after the integration of the golden sine strategy resulted in suboptimal convergence accuracy.

On the basis of previous research, this study aims to further explore how to effectively solve the problems of excessively long planning path generation and low efficiency in the path planning algorithm of warehouse logistics robots. For the complex multi-objective path planning problem, we propose a multi-strategy fusion algorithm based on the whale optimization algorithm (WOA)-IWOA. For the Whale Algorithm, in the three aspects of the population initialization, the search strategy, and the position update place to improve, in order to verify the effectiveness of the IWOA algorithm, we carried out a path planning simulation experiment. In the experiment, we take the path length and obstacle avoidance performance as the adaptation function and the smoothness of the path as the evaluation index. The experimental results show that the IWOA algorithm performs well in terms of path length, obstacle avoidance performance, and path smoothness and is able to effectively solve complex multi-objective path planning problems. These results not only verify the practicality of the IWOA algorithm but also provide the possibility of further optimization and improvement in a wider range of application scenarios in the future.

The contributions of this paper are summarized as follows:

Improved Initialization and Optimization: Addressing the issue of maladaptive initialization in the standard Whale Optimization Algorithm (WOA) for path planning, the proposed Improved Whale Optimization Algorithm (IWOA) incorporates a population initialization method based on an enhanced Tent Chaos Theory. This method significantly improves the quality of initial solutions. To tackle the inefficiency of the WOA’s optimization search, the paper introduces differentiated dynamic weights into the search strategy, which better balances convergence performance during the early and late stages of exploration and development. Additionally, an Adaptive Golden Sine Algorithm is proposed for the position update phase, greatly enhancing the convergence speed and local search capability of the algorithm.

Hierarchical Multi-Objective Optimization: The paper implements a hierarchical adjustment strategy for multi-objective optimization, effectively enhancing path shortening, smoothing, and reducing the number of path inflection points.

Comparative Simulation Experiments: Two sets of simulation experiments were conducted on maps of varying sizes. The performance of the proposed IWOA algorithm was compared with several swarm intelligence algorithms and classical path planning algorithms. The results demonstrate that the IWOA algorithm outperforms others in terms of optimization results, computational efficiency, stability, and robustness.

2. WOA Algorithm

2.1. Encircling Prey

The WOA algorithm simulates the contraction encirclement mechanism of whales, where the population converges to the position of the optimal individual. The position update formula is as follows:

$$\overrightarrow{D_1}=\left|\overrightarrow{C}\cdot \overrightarrow{X^{*}}(t)-\overrightarrow{X}(t)\right|,$$

(1)

$$\overrightarrow{X}(t+1)=\overrightarrow{X^{*}}(t)-\overrightarrow{A}\cdot \overrightarrow{D_1},$$

(2)

$$\overrightarrow{A}=2\overrightarrow{a}\cdot \overrightarrow{r}-\overrightarrow{a},$$

(3)

$$\overrightarrow{C}=2\cdot \overrightarrow{r},$$

(4)

where t is the current number of iterations and $\overrightarrow{X}(t)$ is the position vector of an individual whale. $\overrightarrow{X^{*}}(t)$ is the best position of the individual so far. || represents the absolute value and is an element-by-element multiplication. $\overrightarrow{D_1}$ is the distance between individuals. $\overrightarrow{A}$ and $\overrightarrow{C}$ are coefficient vectors, $\overrightarrow{a}$ is a parameter decreasing from 2 to 0, $\overrightarrow{r}$ is a random number between [0, 1].

2.2. Spiral Search

The spiral position update is to create a spiral formula, as in (6), between the whale and prey positions to simulate the whale’s spiral motion.

$$\overrightarrow{D_2}=\left|\overrightarrow{X^{*}}(t)-\overrightarrow{X}(t)\right|,$$

(5)

$$\overrightarrow{X}(t+1)=\overrightarrow{D_2}\cdot e^{bl}\cdot cos(2pl)+\overrightarrow{X^{*}}(t),$$

(6)

where $\overrightarrow{D_2}$ is the distance from the individual to the optimal position. The parameter b is used to define the shape of the spiral search. l is a random number in the range [−1, 1]. When a whale captures prey, both its contraction envelopment and spiral position updates are performed simultaneously. We can represent this with the mathematical model in (9). In the formula, P is a random number in [0, 1].

$$\overrightarrow{X}(t+1)=\{\begin{array}{l}\overrightarrow{X^{*}}(t)-\overrightarrow{A}\cdot {\overrightarrow{D}}_1,\\ \overrightarrow{D_2}\cdot e^{bl}\cdot \cos (2\pi l)+\overrightarrow{X^{*}}(t),\end{array}\begin{array}{c}p<0.5\\ p\ge 0.5\end{array}.$$

(7)

2.3. Search for Prey

In this phase, whales perform global exploration. When |$\overrightarrow{A}$| ≥ 1, in contrast to the contraction–enclosure phase, a whale is randomly selected in the population, and its position is updated. At this time, the behavior of humpback whales in searching for prey corresponds to the development stage in the algorithm, that is, global optimization, as shown in Equation (9):

$${\overrightarrow{D}}_{rand}=\left|\overrightarrow{C}\cdot {\overrightarrow{X}}_{rand}-\overrightarrow{X}(t)\right|,$$

(8)

$$\overrightarrow{X}(t+1)={\overrightarrow{X}}_{rand}-\overrightarrow{A}\cdot {\overrightarrow{D}}_{rand},$$

(9)

where ${\overrightarrow{X}}_{rand}$ is the position vector of a random individual and ${\overrightarrow{D}}_{rand}$ is the distance of a randomly selected individual from the prey.

3. IWOA Algorithm

3.1. Population Initialization with Improved Tent Mapping

In the realm of WOA population optimization, the initial quality of individuals is pivotal as it directly affects the species diversity and the overall performance of the algorithm. Logistic mapping has been a prevalent choice for scholars due to its extensive use in generating chaotic sequences, which serve to initialize population positions and, in turn, enhance the diversity of the initial population.

This paper delves into a comparative analysis between the traditional Logistic mapping and the refined Tent mapping in the context of chaotic mapping. The construction of the initial population is demonstrated under the premise of a population size of 500 and a dimensionality of 1, with values spanning the range from 0 to 1. The comparative assessment, as depicted in Figure 2, indicates that the Improved Tent chaotic mapping manifests a more uniform distribution and superior uniformity in comparison to the Logistic mapping, where the orange bars represent the logistic population distribution and the blue bars represent the tent population distribution. The mathematical expression for the Tent mapping is as follows:

$$Z(t+1)=\{\begin{array}{c}Z(t)/\alpha ,\\ (1-Z(t)/\alpha )/(1-\alpha )),\end{array}\begin{array}{c}0\le Z(t)<\alpha \\ \alpha \le Z(t)\le 1\end{array}$$

(10)

where $\alpha$ is a parameter in the range of (0, 1).

3.2. Differentiated Dynamic Weights

To elevate the algorithm’s overall proficiency concerning convergence precision and the quality of solutions, a novel approach for the distribution of weight coefficients among individual whales within the population is introduced. This method categorizes the whale individuals into three distinct classes predicated on their fitness value ratios after sorting them in ascending order of fitness: the elite individual class, the average individual class, and the subpar individual class. The mathematical framework for this stratification is encapsulated within Equation (11):

$$W=\{\begin{array}{l}{w}_1=0.1+0.2\times \mathrm{rand},\\ w_2=1.0+0.2\times \mathrm{rand},\\ w_3=\left(w_1+w_2\right)/2,\end{array}\begin{array}{c}0.9\le r\le 1\\ 0\le r\le 0.1\\ 0.1<r<0.9\end{array}.$$

(11)

In the pursuit of a globally optimal solution, our algorithm effectively leverages the optimal individuals to enhance local search optimization. This is achieved through a series of meticulously designed optimization operations, ensuring a thorough exploration of the optimal solution within a constrained search space. Simultaneously, other individuals in the algorithm dynamically update the set of optimal individuals by adjusting their positions based on assigned weights. This mechanism not only maintains the breadth of the population’s search range but also significantly enhances the capability to identify the global optimum. As illustrated in Figure 3, this updating strategy successfully broadens the search horizon while preserving detailed exploration within the optimal region.

$$\overrightarrow{X}(t+1)=\{\begin{array}{l}W\cdot \overrightarrow{X}*(t)-\overrightarrow{A}\cdot \overrightarrow{D_1},\\ \overrightarrow{D_2}\cdot e^{bl}\cdot \cos (2\pi l)+W\cdot \overrightarrow{X}*(t),\\ W\cdot \overrightarrow{X}{(t)}_{rand}-\overrightarrow{A}\cdot \overrightarrow{D_{rand}},\end{array}\begin{array}{c}p<Q\\ p\ge Q,\left|\overrightarrow{A}\right|<1\\ p\ge Q,\left|\overrightarrow{A}\right|\ge 1\end{array},$$

(12)

where in $\overrightarrow{X}(t+1)$, a weighted learning factor W is introduced into the three position updating mechanisms outlined in Section 2 in order to modulate the influence of the target whales.

3.3. Sine-Cosine Inertial Threshold

In the traditional WOA, the hunting strategy is governed by a static probability threshold of 0.5, which is uniformly applied across iterations. However, this can lead to suboptimal performance, particularly in terms of avoiding local optima as the algorithm iterates. To counteract this, a dynamic probability threshold, denoted as q, has been introduced. This threshold adjusts the hunting strategy by comparing a randomly generated p-value against a nonlinear probability threshold that evolves during the optimization process. This dynamic mechanism is crafted to reduce the likelihood of the algorithm becoming ensnared in local optima. The mathematical formulation for the dynamic thresholding is presented in Equation (13):

$$Q=\{\begin{array}{l}\cos ({\lambda }_1\times (t-250)/T_{\max }\times pi),\\ {\lambda }_2\cdot \sin ({\lambda }_3\times (t-250)/T_{\max }\times pi)+\beta ,\end{array}\begin{array}{c}t/T_{\max }\le 0.6\\ t/T_{\max }\ge 0.6\end{array},$$

(13)

where T_max denotes the maximum number of iterations for which the algorithm runs. We use the cosine function in the pre-iteration period to update the threshold; in the pre-iteration period, we increase the spiral position update and increase the ability of the algorithm’s global search; in the late iteration period, the algorithm is basically converged, and the use of the positive selection on the threshold of the random perturbation, to help the algorithm remove the local optimum so that the whale group is constantly close to the optimal solution, which in turn improves the convergence accuracy of the algorithm. The threshold image is shown in Figure 4:

The sine–cosine inertial threshold of Q is pivotal as it empowers the algorithm to navigate away from local optima while steering towards the global optimum. By intelligently modulating the search behavior based on the stage of the algorithm, this approach imbues the WOA with a robust framework that harmonizes exploration and exploitation, ultimately refining the convergence characteristics toward the optimal solution.

3.4. Adaptive Golden Sine Strategy

In this study, the Golden Sine Algorithm (GSA) [26] is adeptly utilized for spiral position perturbation, offering a more efficient reduction of the search space compared to conventional optimization algorithms. The GSA updates the whale’s position in a quadratic manner, which is designed to expedite the convergence process, where the red line represents the image of the sine function and the black line is the search space for the golden sine function. The location update strategy is shown in Figure 5.

Moreover, a dynamic threshold, as delineated in Section 3.3, modulates the likelihood of position updates, integrating the golden sine strategy for adaptive position adjustments of the whales. Given that the efficacy of the perturbed position post-GSA application cannot be guaranteed to surpass the initial position, a greedy selection strategy is implemented. This strategy facilitates an adaptive update of the spiral position by juxtaposing the fitness of the current whale’s position with the newly perturbed one. The specific update formula is encapsulated within the equation provided above, ensuring that only improvements are accepted, thereby driving the optimization process towards more promising solutions. The formula is shown in (15), (19), (20):

$$\overrightarrow{D_2}=\left|\overrightarrow{X^{*}}(t)-\overrightarrow{X}(t)\right|,$$

(14)

$$\overrightarrow{X}(t)=\overrightarrow{D_2}\cdot e^{bl}\cdot \cos (2\pi l)+W\cdot \overrightarrow{X^{*}}(t),$$

(15)

$$\tau =\left(\sqrt{5}-1\right)/2,$$

(16)

$${\theta }_1=-\pi +2\pi (1-\tau ),$$

(17)

$${\theta }_2=-\pi +2\pi \tau ,$$

(18)

$$\overrightarrow{X^{\&}}(t+1)=\overrightarrow{X}(t)\cdot \left|\sin R_1\right|+R_2\cdot \sin R_1\cdot \left|{\theta }_1\cdot \overrightarrow{X^{*}}(t)-{\theta }_2\cdot \overrightarrow{X}(t)\right|,$$

(19)

$$\overrightarrow{X}(t+1)=\{\begin{array}{c}\overrightarrow{X}(t),f(X^{\&}(t+1))>f(X(t))\\ {\overrightarrow{X}}^{\&}(t+1),f(X^{\&}(t+1))\le f(X(t))\end{array}$$

(20)

where $f(X(t))$ is the adaptation value of position t. If the generated position is better than the original position, it is replaced with the original position to make it globally optimal. R₁ and R₂ are random numbers in the range [0, 2], and θ₁ and θ₂ are coefficients obtained by introducing the golden ratio constant. $\overrightarrow{X^{\&}}(t+1)$ is the secondary update position of the whale spiral using the golden sine algorithm.

3.5. Flowchart of IWOA Algorithm

The flowchart of the IWOA algorithm is shown in Figure 6.

The steps of the proposed multi-strategy fusion IWOA in this paper are outlined as follows:

Step 1: Parameter Initialization: Establish the size of the whale population N, the maximum number of iterations T_max, and initialize the parameters for the algorithm;

Step 2: Population Initialization: Utilize the improved Tent chaos theory to initialize the positions of the whale population, ensuring a diverse initial distribution;

Step 3: Fitness Calculation: Compute the individual fitness value for each whale based on their current position in the search space;

Step 4: Prey Identification: Identify the whale with the smallest fitness value, denoting this position as the “prey”;

Step 5: Parameter Update: Update the parameters a, A, C, and Q, which are crucial for the algorithm’s dynamic behavior and search capability;

Step 6: Global Exploitation and Local Optimization: In early iterations, the algorithm’s spiral position updating capability is enhanced using a threshold q. The algorithm is then optimized for the local search. In later iterations, improve the local search optimization capability. If the stochastic parameter p < q, the whale position is updated according to Equation (12);

Step 7: Adaptive Golden Sine Algorithm with Dynamic W-strategy: When p > q, the adaptive golden sine algorithm is used, supplemented by an adaptive dynamic W-strategy, to update the whale position according to Equations (15), (19) and (20);

Step 8: Iterative update and termination check: Increase the number of iterations and evaluate whether the maximum number of iterations, T_max, has been reached; if the termination criterion is met, output the best fitness value. Otherwise, return to Step 3 to continue the optimization process.

4. Comparison of Simulation Experiments and Result Data

4.1. Algorithm Experiment Parameters and Test Function

All algorithm experiments are run independently 30 times, with a total of 500 iterations each time, and the population size N is set to 30. To verify the effectiveness of the IWOA algorithm, this paper adopts eight classical benchmark test functions, as shown in

Table 1. Benchmarking functions.

Index	Function	Dimension	Search Range	Optimal Solution
$f_1$	$Sphere$	$30$	$\left[-100,100\right]$	$0$
$f_2$	$Schwefel2.22$	$30$	$\left[-10,10\right]$	$0$
$f_3$	$Schwefel1.2$	$30$	$\left[-100,100\right]$	$0$
$f_4$	$Schwefel2.21$	$30$	$\left[-100,100\right]$	$0$
$f_7$	$Rosenbrock$	$30$	$\left[-30,30\right]$	$0$
$f_9$	$Rastrigin$	$30$	$\left[-5.12,5.12\right]$	$0$
$f_{10}$	$Ackley$	$30$	$\left[-32,32\right]$	$0$
$f_{11}$	$Griewank$	$30$	$\left[-600,600\right]$	$0$

. Functions F1–F4 and F7 are single-peak functions, while F9–F11 are complex multi-peak functions.

4.2. Impact of Improved Strategies on Algorithm Optimization Performance

To verify the role of each strategy and its impact on algorithm performance, this paper evaluates the Whale Algorithm (WOA), Improved Tent Algorithm (WOA1), the Whale Algorithm with Differential Dynamic Weighting Strategy (WOA2), Whale Algorithm with Golden Orthogonal (WOA3), Whale Algorithm with Improved Dynamic Threshold Strategy (WOA4), and the Whale Algorithm improved in this paper (IWOA) on the eight benchmark functions listed in Table 1. The eight benchmark functions listed in Table 1 are run independently 30 times. The optimal value, worst value, and average value of each algorithm are used as the final evaluation indices. Simultaneously, the algorithms are ranked based on the mean value of the results in each test function (if the mean values are equal, the standard deviation is compared). The Rank indicates the ranking of each improved strategy in different functions, while the Total rank represents the overall ranking. The detailed results are presented in

Table 2. Comparison of the five improved strategies.

Function	Result	WOA	WOA1	WOA2	WOA3	WOA4	IWOA
$f_1$	optimal value	7.20 × 10−81	4.33 × 10−82	1.28 × 10−205	0	3.25 × 10−84	0
	worst value	6.72 × 10−81	9.95 × 10−80	5.06 × 10−202	0	4.70 × 10−82	0
	Mean	6.49 × 10−81	9.62 × 10−80	1.70 × 10−203	0	3.13 × 10−81	0
	standard deviation	1.22 × 10−81	1.80 × 10−80	0	0	1.19 × 10−82	0
	Rank	6	5	3	1	4	1
$f_2$	optimal value	3.71 × 10−50	1.22 × 10−52	6.12 × 10−112	1.30 × 10−173	2.84 × 10−50	0
	worst value	1.84 × 10−49	4.19 × 10−52	1.22 × 10−110	7.39 × 10−170	3.52 × 10−50	0
	Mean	4.71 × 10−57	1.32 × 10−52	1.00 × 10−111	7.14 × 10−170	2.35 × 10−51	0
	standard deviation	1.00 × 10−48	5.42 × 10−53	2.13 × 10−111	0	8.95 × 10−51	0
	Rank	4	5	3	2	6	1
$f_3$	optimal value	2.69 × 104	1.19 × 102	1.04 × 10−1	1.16 × 10−251	1.26 × 103	0
	worst value	3.34 × 104	2.59 × 103	7.89 × 100	2.13 × 10−249	3.14 × 104	0
	Mean	2.71 × 104	2.54 × 103	7.63 × 100	1.83 × 10−250	2.08 × 103	0
	standard deviation	1.17 × 103	2.55 × 102	1.42 × 100	0	7.93 × 103	0
	Rank	6	5	3	2	4	1
$f_4$	optimal value	2.41 × 10−2	9.73 × 10−2	6.08 × 10−30	1.92 × 10−148	7.96 × 100	0
	worst value	2.21 × 100	1.63 × 10−1	6.46 × 10−29	2.05 × 10−142	7.19 × 101	0
	Mean	2.15 × 100	9.95 × 10−2	8.03 × 10−30	6.85 × 10−144	4.78 × 100	0
	standard deviation	4.00 × 100	0.12 × 10−1	1.06 × 10−29	3.75 × 10−143	1.82 × 101	0
	Rank	5	4	3	2	6	1
$f_7$	optimal value	1.15 × 10−4	2.23 × 10−4	1.66 × 10−4	4.25 × 10−6	1.44 × 10−6	2.48 × 10−7
	worst value	3.07 × 10−2	2.80 × 101	1.74 × 10−2	5.70 × 10−4	9.33 × 10−5	1.79 × 10−4
	Mean	5.76 × 10−3	3.59 × 10−3	3.76 × 10−3	1.11 × 10−4	6.22 × 10−6	4.71 × 10−5
	standard deviation	6.41 × 10−3	4.22 × 10−3	4.14 × 10−3	1.18 × 10−4	2.36 × 10−5	4.21 × 10−5
	Rank	6	4	5	3	1	2
$f_9$	optimal value	0	0	0	0	0	0
	worst value	5.68 × 10−14	5.68 × 10−14	0	0	0	0
	Mean	1.89 × 10−15	3.79 × 10−15	0	0	0	0
	standard deviation	1.04 × 10−14	1.44 × 10−14	0	0	0	0
	Rank	5	6	1	1	1	1
$f_{10}$	optimal value	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	worst value	7.99 × 10−15	7.99 × 10−15	4.44 × 10−15	8.88 × 10−16	2.26 × 10−15	8.88 × 10−16
	Mean	4.20 × 10−15	3.97 × 10−15	4.32 × 10−15	8.88 × 10−16	5.92 × 10−16	8.88 × 10−16
	standard deviation	2.45 × 10−15	2.23 × 10−15	6.48 × 10−16	0	2.25 × 10−16	0
	Rank	5	4	6	1	3	1
$f_{11}$	optimal value	0	0	0	0	0	0
	worst value	1.11 × 10−16	1.61 × 10−1	0	0	0	0
	Mean	7.40 × 10−18	1.38 × 10−2	0	0	0	0
	standard deviation	2.81 × 10−17	4.31 × 10−2	0	0	0	0
	Rank	5	6	1	1	1	1
Total rank		6	5	3	2	4	1

.

Firstly, the theoretical analysis demonstrates IWOA’s ability to attain the theoretical optimum in functions F1, F2, F3, and F4 while achieving superior accuracy in optimizing other functions. This underscores IWOA’s robust performance after comprehensive enhancements. In function F7, although the optimal value is not achieved, IWOA demonstrates a superior search capability compared to other enhanced algorithms.

To visualize the mean rankings of the five improvement strategies, a radar plot is drawn (Figure 7), with IWOA represented in blue. The smallest enclosed area in the plot indicates IWOA’s superior overall performance, followed in descending order by WOA3, WOA2, WOA4, WOA1, and WOA.

Secondly, upon observing the mean values of the eight benchmark functions and their total rankings, the Whale Algorithm (WOA3) augmented with the golden sine occupies a dominant position among the five strategies. This highlights the importance of enhancing the algorithm’s global optimal search capability iteratively. Additionally, the Whale Algorithm with the second-highest impact is WOA2, which utilizes differentiated dynamic weights to eliminate local optimal solutions and continuously approach the global optimum.

In addition, the positive cosine inertia threshold Whale Algorithm (WOA4) is compared to the WOA. This algorithm adds a dynamic threshold for adjusting the probability of position update, thus realizing adaptive updating of whale position. Although the effect of the WOA4 algorithm on the ability of the optimal solution of the single-peak function than the original algorithm is not a significant improvement, in the multi-peak function, it can be clearly seen that the strategy can improve the overall algorithm to find the optimal and enhance the whale optimization algorithm to remove the local optimal ability.

Finally, in the population initialization stage, the improved Tent Chaos Theory Whale Algorithm (WOA1) is added. It improves the uneven distribution in the pre-population stage and enhances the convergence speed of the algorithm in the pre-population stage. The ability to find the optimal solution is improved in the initial stage.

Therefore, the core of the IWOA algorithm in this paper is to add the Golden Sine Whale Algorithm (WOA3) based on the Whale Algorithm with differential dynamic weights (WOA2) and adaptive perturbation through the strategy of the Whale Algorithm with positive cosine inertia threshold (WOA4). And the improved Tent Chaos Theory Algorithm (WOA1) was used for uniform initialization of the population, which comprehensively improved the optimization ability of the Whale Algorithm.

4.3. Main Parameters of the Algorithm

To validate the performance of this algorithm in solving optimization problems, this paper compares the proposed improved Whale Optimization Algorithm (IWOA) against several established methods: the classical Gray Wolf Optimization (GWO) [14] Algorithm for its global search capability, the African Vulture Optimization (AVOA) [11] Algorithm for its local search ability, the Snake Optimization (SO) [12] Algorithm, the classical improved Whale Optimization Algorithm (NGS-WOA) [25], the latest improved Whale Optimization Algorithm (WOA-LFGA) [27], and the classical improved Whale Optimization Algorithm (CASAWOA) [24]. This comparative analysis aims to evaluate the efficacy of the IWOA algorithm relative to these seven benchmark algorithms. The algorithm information is shown in

Table 3. Principal parameter.

Algorithm	Main Parameter
GWO	The value of A ranges from 2 to 0.
AVOA	C1 = C2 = 2, a = 6, R = 1.
SO	C1 = 0.5, C2 = 0.05, C3 = 2.
NGS-WOA	b = 1, x1 = −π + (1 − τ) × 2π, x2 = −π + τ × 2π.
WOA-LFGA	beta = 1.5.
CASAWOA	w = 2[sin(π × t)/(2T) + π) +1] × n.
IWOA	P = rand(), b = 1, x1 = −π + (1 − τ) × 2π, x2 = −π + τ·2π.

.

4.4. Comparison among Different Algorithms

To compare the improvement effect of IWOA with other improved algorithms, the experimental results are obtained by solving the benchmark test functions listed in Table 1, which replicate the experiments from the literature [11,12,14,24,25,27]. The specific results are shown in

Table 4. Experimental comparison of different improvement strategies.

Function	Result	GWO	AVOA	SO	NGS-WOA	WOA-LFGA	CASAWOA	IWOA
$f_1$	optimal value	3.71 × 10−29	0	5.31 × 10−99	0	0	0	0
	worst value	2.12 × 10−26	1.55 × 10−285	5.71 × 10−94	0	2.41 × 10−310	2.49 × 10−311	0
	Mean	1.65 × 10−27	5.16 × 10−287	8.51 × 10−95	0	8.28 × 10−312	8.35 × 10−313	0
	standard deviation	3.96 × 10−27	1.75 × 10−285	1.54 × 10−94	0	3.51 × 10−290	0	0
	Rank	7	5	6	1	4	3	1
$f_2$	optimal value	1.47 × 10−17	9.76 × 10−175	3.82 × 10−44	2.29 × 10−220	6.67 × 10−160	3.45 × 10−161	0
	worst value	5.20 × 10−16	5.47 × 10−148	2.51 × 10−43	1.53 × 10−208	4.86 × 10−159	4.70 × 10−154	0
	Mean	9.61 × 10−17	2.95 × 10−149	2.43 × 10−43	1.06 × 10−209	8.07 × 10−160	2.39 × 10−155	0
	standard deviation	9.98 × 10−17	1.15 × 10−148	3.98 × 10−44	1.28 × 10−210	7.67 × 10−160	9.05 × 10−155	0
	Rank	7	5	6	2	3	4	1
$f_3$	optimal value	3.15 × 10−8	8.86 × 10−288	1.43 × 10−69	0	2.12 × 10−313	1.79 × 10−257	0
	worst value	1.72 × 10−4	1.12 × 10−207	3.95 × 10−57	0	1.31 × 10−225	5.82 × 10−254	0
	Mean	1.68 × 10−5	3.74 × 10−209	2.30 × 10−58	0	7.99 × 10−277	5.45 × 10−254	0
	standard deviation	3.67 × 10−5	1.72 × 10−237	7.44 × 10−58	0	1.51 × 10−235	5.24 × 10−251	0
	Rank	7	5	6	1	4	3	1
$f_4$	optimal value	4.61 × 10−8	5.16 × 10−175	3.27 × 10−43	5.46 × 10−203	1.19 × 10−162	3.61 × 10−152	0
	worst value	2.31 × 10−6	4.99 × 10−140	9.50 × 10−43	1.02 × 10−187	4.87 × 10−161	1.27 × 10−147	0
	Mean	6.23 × 10−7	1.66 × 10−141	3.47 × 10−43	3.45 × 10−189	2.78 × 10−162	1.22 × 10−148	0
	standard deviation	6.30 × 10−7	9.11 × 10−141	1.14 × 10−43	1.65 × 10−189	8.81 × 10−162	2.73 × 10−148	0
	Rank	7	4	6	2	3	5	1
$f_7$	optimal value	3.56 × 10−4	1.62 × 10−6	3.06 × 10−5	4.55 × 10−6	2.21 × 10−6	2.09 × 10−6	2.48 × 10−7
	worst value	6.33 × 10−3	3.52 × 10−4	7.53 × 10−4	1.75 × 10−4	1.57 × 10−4	1.81 × 10−4	1.79 × 10−4
	Mean	2.06 × 10−3	6.25 × 10−5	2.05 × 10−4	6.51 × 10−5	5.19 × 10−5	6.87 × 10−5	4.71 × 10−5
	standard deviation	1.25 × 10−3	8.74 × 10−5	1.93 × 10−4	3.86 × 10−5	5.17 × 10−5	4.51 × 10−5	4.21 × 10−5
	Rank	7	3	6	4	2	5	1
$f_9$	optimal value	5.68 × 10−14	0	0	0	0	0	0
	worst value	1.24 × 101	0	3.58 × 101	0	0	0	0
	Mean	2.62 × 100	0	7.09 × 100	0	0	0	0
	standard deviation	3.59 × 100	0	1.19 × 101	0	0	0	0
	Rank	6	1	7	1	1	1	1
$f_{10}$	optimal value	7.55 × 10−14	8.88 × 10−16	4.44 × 10−15	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	worst value	1.36 × 10−13	8.88 × 10−16	2.18 × 100	8.88 × 10−16	8.88 × 10−16	4.44 × 10−15	8.88 × 10−16
	Mean	9.99 × 10−14	8.88 × 10−16	7.29 × 10−2	8.88 × 10−16	8.88 × 10−16	2.66 × 10−15	8.88 × 10−16
	standard deviation	1.69 × 10−14	0	4.00 × 10−1	0	0	1.80 × 10−15	0
	Rank	6	1	7	1	1	5	1
$f_{11}$	optimal value	0	0	0	0	0	0	0
	worst value	2.25 × 10−2	0	7.45 × 10−1	0	0	0	0
	Mean	3.46 × 10−3	0	9.91 × 10−2	0	0	0	0
	standard deviation	6.19 × 10−3	0	2.05 × 10−1	0	0	0	0
	Rank	6	1	7	1	1	1	1
Total rank		7	4	6	2	3	5	1

.

Based on the data in Table 4, the convergence curves of IWOA and various improvement strategies are depicted in Figure 8a–f. These curves reflect the convergence speed, accuracy, stability of the algorithm, and its ability to escape local optima.

The CASAWOA employs a Tent chaotic map for designing an inertia weight simulated annealing strategy during the initial population stage, thereby enhancing the convergence performance of the original algorithm. However, this random search characteristic necessitates numerous iterations before convergence to the optimal solution is achieved, resulting in poor convergence speed. Uniform convergence is exhibited by the improved whale algorithm WOA-LFGA when the levy flight strategy is utilized for position updates. However, large levy flight steps may result in the local optimal solution being overlooked. The golden sine strategy is also introduced by the classical improved whale optimization algorithm NGS-WOA, but its performance is hindered by not considering the advantages and disadvantages before and after the improvement. In contrast, the process is optimized by the golden sine algorithm proposed in this paper using the Greedy choice, followed by the selection of the best solution through a greedy selection strategy. The fitness function is used to compare and select the optimal whale position. Significant improvements in the convergence speed and accuracy of the IWOA algorithm are demonstrated by the experimental results.

Compared to other algorithms, the capability to escape local optima and achieve the theoretical optimum is demonstrated by the IWOA, which also completes convergence within 50 generations; this reflects the excellent exploration ability of the IWOA in multimodal functions F9, F10, and F11.

Figure 8 not only illustrates the convergence aspects but also highlights the good balance and stability of the IWOA algorithm, indicating that it is not only fast and accurate but also reliable and consistent in its performance across different types of optimization problems.

The paper’s findings suggest that the IWOA algorithm is a robust optimization tool that effectively balances exploration and exploitation, leading to improved performance across a variety of benchmark functions. The visual representation of the convergence curves in Figure 4 serves as a testament to the algorithm’s capabilities and the impact of the various improvement strategies on its performance.

4.5. IWOA Algorithm Time Complexity Calculation

For the native WOA algorithm, let N represent the population size, T_max the maximum number of iterations, and d the dimension of each individual. The time complexity for the random initialization of the population phase is O₁ (N × d), and the time complexity for the main iteration phase is O₂ (N × T_max × d). Consequently, the overall time complexity is O = O₁ + O₂ = (N × d + N × T_max × d).

In the IWOA algorithm, the time complexity for initializing the population using the improved Tent map is O₃ (N × d). During the main iteration phase, updating the position of any dimension in the native algorithm takes time t₁. The primary contributor to the increased time complexity in IWOA is the improved iterative local search, which involves differential dynamic weighting and adaptive golden sine operations on two adjacent neighborhoods, executed in times t₂ and t₃, respectively. Additionally, the improved Sine–Cosine Inertial Threshold search performs fitness comparisons and position updates for each whale individual, taking time t₄. Consequently, the time complexity for this stage is O₄ (N × T_max × d × (t₁ + t₂ + t₃) + N × T_max × t₄). For high-dimensional complex problems, this can be approximated as O₄ (N × T_max × d).

Therefore, the total time complexity of the IWOA algorithm is O = O₃ + O₄ = (N × d + N × T_max × d). This indicates that the IWOA algorithm maintains the same time complexity as the original WOA algorithm and does not adversely affect its operational efficiency.

5. Path Planning Based on IWOA Algorithm

5.1. Problem Description and Environment Modeling

The mobile robot path planning problem in a static environment is addressed in this paper [28]. A simple and effective grid method is employed to model the environment. In this model, collisions between the mobile robot and obstacles are avoided. The robot is required to move from the starting point S = (1,1) to the endpoint D = (20,20). In this model, black areas represent obstacles, while white areas represent free space. The red positions are the start and end points of the robot. Figure 9 presents an example of a simple 15 × 15 and a complex 20 × 20 working environment modeled using the raster method [29].

5.2. Adaptability Functions

In the process of robotic path planning, the primary focus is on obstacle avoidance while addressing multiple factors such as path length, smoothness, stability, energy consumption, and time. In this study, we employed a tiered adjustment strategy. Initially, we preserved the classic obstacle avoidance and shortest path as the first reference element to achieve the ultimate goal of the vehicle. Subsequently, we reconsidered the smoothness and stability of the path generated in the first phase through re-planning. By employing curve optimization and smoothing strategies, we further enhanced the vehicle’s smoothness. Adjusting the turning angles improved the vehicle’s stability and addressed energy consumption issues. The formula is shown below:

$$F=g(i)+G(i),$$

(21)

$$g(i)=\sqrt{{(x_i-x_{i-1})}^2+{(y_i-y_{i-1})}^2},$$

(22)

$$G(i)=k{\displaystyle \sum map(i)},$$

(23)

$$q=arccos((b_2+c_2-a_2)/2bc),$$

(24)

$$S={\displaystyle \sum _{j=1}^{end}1/q_j},$$

(25)

where i is the current node, g(i) is the actual cost spent by the node before and after the iteration, G(i) is the collision function, where K is a large constant, and map(i) is the map information of the current node. $\theta$is the angle corresponding to the inflection point, and S is the curve smoothness.

5.3. Tiered Adjustment Strategy

In this study, a tiered approach is implemented for multi-objective optimization in robotic path planning. The initial phase utilizes an enhanced whale optimization algorithm for rapid route generation. Subsequent phases refine the path through node validation and simplification, followed by dynamic tangent adjustments to smooth the trajectory, thereby enhancing stability and energy efficiency.

The flow of the robot path planning algorithm based on smooth curves is shown in Figure 10.

The steps for curve optimization are as follows:

The first stage is IWOA algorithm path optimization.

Step 1: Environment and Parameter Initialization: Configure the raster map path planning environment and establish the initial parameters for the IWOA algorithm. This includes defining the robot’s starting and ending positions, the dimensions of the map, the population size, and the maximum number of optimization iterations;

Step 2: Initialize the whale positions in the population to establish the shortest initial path, and then update these positions using the IWOA algorithm with a defined fitness function to evaluate their performance in the current environment.

The second stage is Path Smoothing and Optimization.

Step 3: Path Smoothing and Optimization: It must be determined whether there is an obstacle in 3 consecutive nodes. If no obstacle exists, the intermediate invalid nodes are deleted; otherwise, they are retained. All points are traversed according to the following procedure:

$$a_i=\{{\displaystyle \frac{\left[\right],[a_{i-1},a_{i+1}]\times [b_{i-1},b_{i+1}]=0}{a_i,[a_{i-1},a_{i+1}]\times [b_{i-1},b_{i+1}]=1}}$$

(26)

The third stage is the Dynamic tangential method.

Step 4: Apply the dynamic cut point adjustment method to ensure the curve is smooth and efficient. as illustrated in Figure 11.

Step 4 Select the shorter edges of A_i−1, A_i, A_i+1. and Take the endpoint of the shorter edge as the initial point P(x_p, y_p) to create a perpendicular line intersecting with the angular bisector of ∠A_i−1A_iA_i+1, A_i−1Q_i (I = 1, 2, 3, …, n − 1), Denote the intersection point as O_i(x_o, y_o), which is given by the following formula:

$$x_0=(x_P+k_1{y}_p+k_1{x}_2-y_2)/(1+k_0{k}_1)$$

(27)

$$y_0=k_0(x_o-x_2)+y_2$$

(28)

The radius and equation of the tangent garden are as follows:

$$R=\sqrt{x_0^2-2x_0{x}_p+y_0^2-2y_0{y}_p+x_p^2+y_p^2}$$

(29)

$${(x-x_0)}^2+{(y-y_0)}^2=R^2$$

(30)

Step 5: Determine whether there is an intersection point S between the tangent circle and the long side. If such a point exists, proceed to step 6; otherwise, proceed to step 7;

Step 6: Determine whether there is an obstacle on the circular arc PS. If an obstacle exists, proceed to step 7; otherwise, replace the corner with the circular arc PS and proceed to step 6;

Step 7: Use the gradient descent method, where the tangent point P(x_p, y_p) moves to P₂(x_p2, y_p2) along the line segment. The new position x_p2 can be expressed as follows:

$$x_{p_2}=x_p+\lambda \left|x_2-x_p\right|,\lambda \in (0,1)$$

(31)

where λ is set based on the actual situation. Meanwhile, set P₂ as the initial tangent point and return to Step 6;

Step 8: Check: Evaluate whether the algorithm has met the maximum number of iterations. If the maximum has been reached, output the shortest path length and the optimal path planning information. If not, revert to Step 4 to resume the iterative calculation process.

5.4. Experimental Analysis of Raster Method Simulation

To assess the viability and efficacy of the SSWOA in the domain of path planning, the algorithm was subjected to simulation on raster maps, with each trial iterating 50 times. For the 15 × 15 and 20 × 20 raster maps, the population size was configured at 50. The IWOA’s performance was juxtaposed with that of the Whale Optimization Algorithm (WOA), Grey Wolf Optimization Algorithm (GWO), and Ant Colony Optimization (MAACO) [30]. A strategy for curve simplification was implemented to attenuate the complexity of the path within the fundamental ant colony algorithm. The parameters for MAACO were delineated as: M = 50, α = 1, β = 7, Q = 1, ρ = 0.3, and q₀ = 0.3. The parameters for the remaining algorithms were in accordance with those enumerated in Table 3. Each algorithm underwent independent testing 15 times.

Figure 12 and Figure 13 elucidate the simulation outcomes of the algorithms on 15 × 15 and 20 × 20 raster maps, respectively. Visual inspection of these Figures reveals that the path rendered by the IWOA is both more concise and more fluid. The convergence graphs indicate that the IWOA outperforms its counterparts on both the rudimentary 15 × 15 map and the intricate 20 × 20 map. In the initial iterations, the IWOA demonstrated rapid convergence with high precision, underscoring the significance of population initialization. The IWOA’s convergence trajectory also descends more swiftly, which mirrors the potency of the refined adaptive golden sinusoidal perturbation strategy and the differentiated dynamic weighting strategy. The evenness of the iterative trajectory suggests that the IWOA maintains an optimal state.

Table 5. Comparison of 15 ×15 map path planning results.

Algorithm	Shortest Path	Inflection Points Count	Smoothness	Average Path Length	Path Standard Deviation	Time-Consuming
WOA	21.86	5	2.122	22.41	0.41	2.75
GWO	22.39	8	3.545	22.96	0.39	2.64
MAACO	21.34	9	3.821	22.06	0.31	16.45
IWOA	21.07	5	1.389	21.35	0.29	3.41

and

Table 6. Comparison of 20 ×20 map path planning results.

Algorithm	Shortest Path	Inflection Points Count	Smoothness	Average Path Length	Path Standard Deviation	Time-Consuming
WOA	31.82	5	2.619	23.12	1.24	6.23
GWO	32.25	11	4.071	24.05	1.62	5.78
MAACO	31.62	7	3.125	22.89	1.42	46.79
IWOA	30.37	5	1.992	22.24	0.96	6.89

present the mean outcomes from 15 simulations conducted using various algorithms on raster graphs of distinct dimensions. On the 15 × 15 and 20 × 20 grid graphs, the optimal paths yielded by the IWOA algorithm were found to be 1.27% and 3.95% superior, respectively, to those obtained by the best-performing MAACO algorithm. Furthermore, the smoothness of the IWOA algorithm was observed to be 34.54% and 23.94% greater than that of the WOA algorithm. In the context of mobile robot path planning, the IWOA algorithm notably enhanced the convergence velocity and precision through the implementation of a curve smoothing strategy and a dynamic tangential approach, thereby augmenting the algorithm’s reliability.

In terms of average time consumption, the ACO algorithm exhibits the longest average duration across all map environments. The other algorithms show similar time consumption. However, as the map size increases, the time required by all algorithms also rises. Notably, the IWOA algorithm demonstrates a relatively smaller increase in time consumption. This improvement is attributed to the four enhanced strategies implemented, which increase the optimization accuracy and convergence speed without significantly adding to the time complexity.

5.5. Feasibility Analysis of Path Planning Algorithms

To further evaluate the effectiveness of the algorithms proposed in this paper for path planning, two widely used traditional algorithms are selected for comparative analysis: the A* [31] algorithm and the RRT [32] algorithm. In the experimental setup for the A* algorithm, its search neighborhood is extended from 8 to 48 to enhance search capability. Additionally, Floyd’s algorithm is employed to smooth the generated paths, improving their overall smoothness. For the RRT algorithm, a strategy prioritizing target points and search nodes is implemented, optimizing both node selection and some of the planned paths. These enhancements are designed to improve the algorithm’s performance and efficiency in complex environments. The results of all simulation experiments are illustrated in Figure 14, providing a visual comparison of the performance and differences among the algorithms.

As depicted in Figure 14, the RRT algorithm effectively identifies the optimal path early in the iteration when there are fewer obstacles. However, with an increase in obstacles, the RRT algorithm generates numerous redundant points, which adversely affects path smoothness. Additionally, the RRT algorithm employs a randomized search tree strategy, introducing a high level of randomness that can result in longer algorithm generation times.

In contrast, the A* algorithm primarily relies on Euclidean distance for updating location information. As shown in Figure 14b, the paths generated by the A* algorithm are close to the optimal solution. However, despite employing Floyd’s algorithm for path smoothing, its smoothness still lags behind that of the algorithm proposed in this paper. Specifically, in Figure 14a, the A* algorithm’s performance is significantly compromised when the critical path is obstructed by numerous obstacles, making it challenging to generate the optimal path. Overall, the IWOA algorithm presented in this paper demonstrates superior performance in terms of both smoothness and stability.

From

Table 7. Comparison of raster map path planning results.

Map Size	Evaluation Indicators	RRT	A*	IWOA
15 × 15	path length	25.02	22.24	21.07
	path smoothness	5.901	1.589	1.389
	time-consuming	15.23	3.84	3.41
20 × 20	path length	37.76	30.86	30.37
	path smoothness	11.698	2.951	1.992
	time-consuming	33.25	7.12	6.89

, it is evident that the optimal paths generated by the IWOA algorithm outperform those produced by the RRT and A* algorithms, with improvements of 15.78% and 5.26% on 15 × 15 maps, and 19.57% and 1.59% on 20 × 20 maps, respectively. Additionally, regarding path smoothness, the IWOA algorithm achieves improvements of 76.46% and 12.59% on 15 × 15 maps and 82.97% and 32.50% on 20 × 20 maps, compared to the RRT and A* algorithms, respectively. Finally, in terms of efficiency, the IWOA algorithm demonstrates the lowest energy consumption time, affirming its superior performance. These simulation results indicate that the IWOA algorithm is well-suited for mobile robot path-planning applications.

6. Robot Path Planning Simulation and Physical Verification

6.1. System Components

The robot platform system is composed of a server layer, an operating system layer, and a driver layer. The overall architecture and connection mode of the system are shown in Figure 15.

(1) Server Layer

The primary function of the server layer in the service robot system is to facilitate human-computer interaction and to monitor map information and platform status in real-time during the robot’s navigation. This layer employs a Samsung laptop equipped with an Intel i5 processor, 16 GB of RAM, and running the Ubuntu 16.04 operating system.

(2) Operating System Layer

The robot software system operating within the operating system layer serves three primary functions: First, it receives instructions from the host computer and executes the corresponding actions promptly. Second, it processes feedback from the driver layer, overseeing the control of the mobile robot’s movements and monitoring its current state. Third, it utilizes LiDAR data to construct an environmental map. Compared to units like the Raspberry Pi, an Industrial PC (IPC) offers superior computational speed, multi-threading capabilities, and enhanced image processing. Consequently, the main controller for this system is the Jimmei GK3000 industrial controller, which is equipped with the Ubuntu operating system and ROS Kinetic version. The manufacturer is China’s Shenzhen XDTech Electronics Co. The physical layout is illustrated in Figure 16, with performance specifications detailed in

Table 8. Performance parameters of the industrial controller.

Module	Parameter Description
CPU	Intel I5-10310U, 4 cores, 8 threads, 1.8 G main frequency
Video card	Intel HD Ultra Core Graphics
RAM	8 G RAM + 256 G solid state disk
USB	2 × USB2.0. 2 × USB3.0
Sizes	234 mm × 150 mm × 52 mm
Working environment	Temperature −20–60 °C, humidity 0–90% non-condensing

.

The LiDAR, depicted in Figure 17, is the YueDeng Intelligent EAI G4 model. It utilizes the laser triangulation method and operates at a maximum frequency of 9000 Hz. This LiDAR is capable of omnidirectional scanning. The manufacturer is China’s Shenzhen YueDeng Intelligent Technology Co. Detailed technical parameters are provided in

Table 9. EAI G4 performance parameters.

EAI G4	Numerical Value	Note
Ranging frequency	4000–9000 Hz	Typical 4000 Hz
Scanning frequency	5–12 Hz	Typical 7 Hz
Scanning angle	0–360°	/
Measuring range	0.1–16 m	/

.

(3) Driver Layer

The driver layer is responsible for controlling the service robot’s motors and processing sensor data. It regulates motor speed and position through pulse signal generation and utilizes I2C communication to collect odometer information, which is then relayed to the operating system layer via a serial port. The control chip used is the STM32F103ZET6, known for its robustness and extensive peripheral interfaces, including PWM control, RS232, Ethernet, and I2C. The manufacturer of the STM32F103ZET6 is STMicroelectronics, a company located in Switzerland. Programming is performed primarily in C.

The system employs a DC brushless hub motor, which offers advantages over traditional brushed motors, such as greater efficiency, enhanced reliability, and reduced noise. This motor has a rated power of 250 W and an encoder resolution of 4096.

For the Inertial Measurement Unit (IMU), the GY-85 nine-axis sensor is utilized. This sensor integrates a three-axis gyroscope, a three-axis accelerometer, and a magnetometer, all interfaced via the I2C bus. The I2C protocol is used to retrieve data from these sensors, with the system relying primarily on the yaw angle data from the ITG3205 three-axis gyroscope and the acceleration data from the ADXL345 three-axis accelerometer.

6.2. Simplicity of the ROS System

The Robot Operating System (ROS) is an open-source meta-operating system that provides essential services such as hardware abstraction, device control, common functions, inter-process messaging, and package management. It also offers a suite of tools and libraries for code acquisition, compilation, development, and execution across multiple computers, enhancing code reuse and simplifying robotics development. ROS’s flexible communication architecture integrates loosely coupled components, streamlining the development process.

Despite its designation as an “operating system”, ROS functions as a sub-operating system or software framework that operates on top of a primary operating system, such as Linux. It does not handle core tasks such as scheduling, compilation, or device drivers, which are managed by the host operating system (e.g., Ubuntu/Linux). Thus, ROS is more accurately described as a distributed communication framework that facilitates seamless inter-process communication.

The ROS architecture is organized into three core tiers: the file system tier, the computational graph tier, and the open-source community tier. Each tier has distinct functions and components within the ROS ecosystem. Figure 18 depicts the file system structure. The file system tier includes essential elements such as function packages, package lists, message type definitions, service descriptions, and source code. A function package, the basic unit of software organization in ROS, contains related code, configuration files, and additional resources. The function package manifest provides metadata about the package, including its name, version, and dependencies. Additionally, messages and services define data structures and interaction methods for inter-node communication within ROS, playing a crucial role in the communication framework.

The ROS computational graph is a network consisting of multiple processing units, known as nodes, and the communication links that connect them. Each node represents a distinct process within ROS, responsible for tasks such as environmental sensing, robot control, or decision-making algorithms. Nodes interact through predefined interfaces, including Topics and Services, among others. Figure 19 illustrates the structure of this computational graph.

In ROS, topic communication allows a node to publish a message to a topic while other nodes can subscribe to that topic to receive the message. Publishers and subscribers are decoupled from each other; they only need to reference the topic name to facilitate communication. When a publisher sends a message to a topic, all subscribers to that topic receive the message. This communication is asynchronous, meaning publishers and subscribers operate independently without waiting for responses from each other, thus enabling real-time data transmission. Furthermore, ROS topic communication supports multiple message types, which can be customized by developers to fit specific application needs. Figure 20 provides a schematic diagram of this communication process.

6.3. Driver Layer Programming

Given the STM32’s requirement to manage multiple sensor inputs and respond to control commands from the industrial control machine, achieving real-time performance with a traditional sequential structure is challenging. To overcome this, the FreeRTOS operating system is utilized, and four tasks are created: motion control task, IMU information acquisition task, handle control task, and ROS control task. The program structure is illustrated in Figure 21.

The RTOS task scheduler organizes the execution sequence of tasks based on their priority. Each task is run for a short duration, effectively making it appear as though all tasks are executed simultaneously. During this time, the system can handle any interrupts that may occur. Specifically, Serial Port 3 interrupts are used to receive data from the Robot Operating System (ROS).

6.4. Operating System Layer Program Design

In the driver layer software design, to ensure effective reception and processing of relevant data by the operating system, it is essential to target the appropriate topic nodes. These nodes act as critical components for data reception. The implementation of control nodes and LIDAR nodes must leverage the ROS framework, utilizing its extensive functions and interfaces. Additionally, the design of internal sensor nodes is crucial, as illustrated in Figure 22.

In this diagram, ellipses represent node names, and boxes denote topics that facilitate communication between nodes. To enable communication between the ROS system and STM32, the /stm32_serial_node communication framework is utilized. Given that the IMU, odometer, and mobile platforms operate within different coordinate systems, it is essential to first establish their spatial relationships and create a unified reference frame using the TF coordinate transformation method for subsequent data fusion and positioning calculations. The Figure shows /base_footprint_to_base_link and /base_footprint_to_imu_link. The /raw_odom topic represents odometer data, which can be displayed using the rostopic echo command in the ROS system.

6.5. Simulation and Physical Verification

Initially, a personal computer (PC) connects to the Wi-Fi signal emitted by the mobile robot. Upon successful communication, the relevant software package is executed to construct a map of the environment using Light Detection and Ranging (LiDAR) technology and to perform navigation experiments. Figure 23b illustrates the generated map, where the dark red areas represent obstacles, and the black square denotes the robot’s position. The corresponding real-world scene is depicted in Figure 23.

The path planning results in the real-world scenario are depicted in Figure 24.

Figure 25 illustrates the robot successfully navigating around all obstacles and generating an optimal path. The red curve represents the trajectory of the mobile robot, while the yellow curve denotes the trajectory generated by the MMACO algorithm.

Following testing in the real environment, the mobile robot demonstrated successful path planning from the starting point to the target location while ensuring safety. The path was both short and smooth. These experimental results confirm the reliability of the constructed platform and robustly validate the effectiveness and practicality of the proposed algorithm.

7. Conclusions

In this study, a multi-strategy fusion whale-based optimization algorithm is proposed, which aims to optimize the shortest route and smoothness of WLR to solve the global path planning problem. Through simulation experiments on IWOA and global path planning, the following conclusions are obtained:

(1)

Improved Multi-Strategy Whale Algorithm: to address the problem that the standard WOA is not applicable for initialization in path planning, this study introduces the improved Tent Chaos Mapping Theory to generate the initial population. This approach significantly enhances the diversity and global search capability of the algorithm. Meanwhile, in order to adapt to the search demand during the iteration process of the algorithm, the IWOA algorithm adopts a differentiated dynamic weighting strategy, which balances the global search and the local search, thus enhancing the algorithm’s optimization seeking ability. In terms of population position update, the probability of position update before and after whale iteration is adaptively adjusted through the combination of heuristic dynamic thresholding and golden sine strategy, and an improved golden sine strategy is introduced to perturb the algorithm’s position update, which further enhances the searchability of global optimal solution;

(2)

Multi-objective optimization for path planning: in this study, an adaptation function is designed to take the path length and obstacle avoidance performance as the main objectives, while the curve smoothness is taken as the evaluation index. Through path simplification and path smoothing strategies, the generated paths are adjusted hierarchically to achieve the multi-objective optimization of path planning. This method not only improves the productivity of WLR but also reduces the production costs;

(3)

Simulation experiments and performance comparison: Comparative analyses of simulation results across various map environments, generated using the raster method, indicate that the IWOA algorithm shows enhanced environmental adaptability compared to WOA, GWO, MAACO, RRT, and A*. On a 15 × 15 grid, the IWOA algorithm demonstrates performance improvements of 1.27% over the shortest-path MAACO algorithm and 12.59% over the smoothest-path optimal A* algorithm. On a 20 × 20 grid, IWOA achieves performance enhancements of 1.59% compared to A* and 23.94% compared to WOA optimized for smoothness.

In summary, the IWOA algorithm demonstrates notable optimization benefits in WLR path planning, offering innovative approaches and methodologies for research and application in mobile robot path planning. As a leading example of autonomous handling tools, mobile robots are driving the logistics and warehousing industry into a new era of smart automation. This advancement will help enterprises enhance efficiency, reduce operational costs, increase competitiveness, and create greater value for society. Future research will explore further integration of robotic arms with mobile platforms to enable more sophisticated interactions and operations. In logistics and warehousing applications, this integration could allow robots to execute complex tasks autonomously, such as moving to specific locations within a warehouse and performing actions like sorting, picking, and packing items, thereby significantly improving operational efficiency and service quality.