Improved Sparrow Search Algorithm Based on Multistrategy Collaborative Optimization Performance and Path Planning Applications

Abstract

To address the problems of limited population diversity and a tendency to converge prematurely to local optima in the original sparrow search algorithm (SSA), an improved sparrow search algorithm (ISSA) based on multi-strategy collaborative optimization is proposed. ISSA employs three strategies to enhance performance: introducing one-dimensional composite chaotic mapping SPM to generate the initial sparrow population, thus enriching population diversity; introducing the dung beetle dancing search behavior strategy to strengthen the algorithm’s ability to jump out of local optima; integrating the adaptive t-variation improvement strategy to balance global exploration and local exploitation capabilities. Through experiments with 23 benchmark test functions and comparison with algorithms such as PSO, GWO, WOA, and SSA, the advantages of ISSA in convergence speed and optimization accuracy are verified. In the application of robot path planning, compared with SSA, ISSA exhibits shorter path lengths, fewer turnings, and higher planning efficiency in both single-target point and multi-target point path planning. Especially in multi-target point path planning, as the obstacle rate increases, ISSA can more effectively find the shortest path. Its traversal order is different from that of SSA, making the planned path smoother and with fewer intersections. The results show that ISSA has significant superiority in both algorithm performance and path planning applications.

1. Introduction

In the backdrop of rapid technological advancements and industrial transformations, mobile robots have permeated diverse facets of production and daily life. Path planning, being a linchpin in mobile robotics, dictates the efficacy of robotic operations. Central to this is the optimization of paths considering factors like operational dynamics and environmental constraints, with path smoothness, travel distance, and energy consumption serving as pivotal evaluation metrics [1,2,3].

In the past, robots have been navigated using traditional methods such as pre-programmed paths or simple sensor-based obstacle avoidance. However, these methods often lack flexibility and adaptability in complex and dynamic environments. With the increasing complexity of tasks and environments that robots are required to operate in, there is a growing need for more efficient and intelligent path planning algorithms. This paper focuses on improving the sparrow search algorithm (SSA) to address the limitations of traditional methods and existing algorithms. Multi-strategy collaborative optimization, including one-dimensional composite chaotic mapping, dung beetle dancing search behavior, and adaptive t-variation improvement, is introduced. The aim is to enhance the performance of the algorithm in aspects such as population diversity, convergence speed, and the ability to avoid local optima. This improvement is of crucial importance for enabling robots to navigate more effectively in various scenarios, especially in the presence of obstacles and multiple target points. The detailed focus of the research is on developing an improved SSA (ISSA) that can offer shorter, smoother paths with fewer turnings and higher planning efficiency, ultimately resulting in better overall performance in robot navigation tasks.

Traditional path planning algorithms, namely the A* algorithm, Rapidly-exploring Random Trees (RRT) algorithm, Artificial Potential Field (APF) algorithm, and Dynamic Window Approach (DWA), have paved the way [4,5,6,7,8,9,10,11,12]. Researchers have continuously strived to improve these algorithms. For instance, Li and colleagues [13] introduced enhancements to the A* algorithm to augment path smoothness. Sreesruthi et al. [14] compared multiple algorithms, validating the effectiveness of the A* algorithm. Kumar et al. [15] devised a hybrid algorithm by integrating concepts from the A* algorithm and the visibility graph algorithm. Wang et al. [16] proposed the EBHSA algorithm, fortifying it with diverse strategies to enhance its robustness and efficiency. Dewangan et al. [17] presented an algorithm that amalgamated centralized and decentralized approaches to optimize system performance.

In recent years, swarm intelligence optimization algorithms have emerged as a favored choice among scholars. These algorithms, including Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Grey Wolf Optimization (GWO), and Sparrow Search Algorithm (SSA), possess unique characteristics [18]. Xu et al. [19] proposed a novel path planning method for mobile robots by integrating a new quartic Bezier transition curve with an improved PSO algorithm. Gao et al. [20] developed an enhanced PSO algorithm for planning obstacle avoidance trajectories in robot spaces. Tian et al. [21] introduced an improved PSO variant based on leader-individual interaction for global path planning. Huo et al. [22] combined an improved ACO with B-spline curves for Ackerman mobile robot path planning. Hou et al. [23] proposed an improved GWO algorithm to address instability and convergence accuracy issues in mobile robot path planning. Zhang et al. [24] introduced the Turning Point Cuckoo Search Algorithm Grey Wolf Optimization (TPGWO) algorithm to overcome the limitations of GWO in path planning applications.

The SSA, with its simplicity and effectiveness in emulating sparrow feeding behavior for optimization, has nonetheless faced challenges. It exhibits a heavy reliance on the initial population and is prone to getting trapped in local optima during later iterations, accompanied by insufficient population diversity [25,26,27]. In response, numerous scholars have proposed improvements. Zhang et al. [28] introduced a novel sparrow search algorithm (NSSA) integrated with spawning technology. Ouyang et al. [27] incorporated reverse learning mechanisms. Tang et al. [29] proposed a chaotic sparrow search algorithm (CLSSA). Chen et al. [30] enhanced the SSA to improve its ability to escape local optima and accelerate convergence. Yang et al. [31] proposed improved versions of the sparrow search algorithm to enhance initial population quality and search capabilities. Liu et al. [32] proposed an improved SSA with multiple strategies to avoid premature convergence. Zhu et al. [33] and Ma et al. [34] proposed an enhanced SSA with hybrid strategies. Wang et al. [35] proposed an improved SSA to optimize population distribution and initial solution quality. Hu et al. [36] introduced an improved SSA with adaptive learning factors and crossover mutation strategies. Abbas et al. [37] improved the GWO algorithm for path planning and obstacle avoidance. Li et al. [38] proposed a novel SSA with sequential enhancement strategies. Wei et al. [39] proposed evacuation and path planning methods based on improved SSA. Geng et al. [40] devised an enhanced adaptive sparrow search algorithm. He et al. [41] proposed an improved chaos sparrow search algorithm for drone flight path planning.

Aiming to solve the issues of insufficient population diversity and the tendency to get trapped in local optima within the original sparrow search algorithm (SSA), this study has three distinct goals. Firstly, the one-dimensional composite chaotic mapping SPM is introduced. By combining Sine mapping and PWLCM mapping, a more diverse initial sparrow population is generated. Consequently, the population diversity is enriched, and the ergodicity and optimization ability of the algorithm are augmented. Secondly, the dung beetle dancing search behavior strategy is incorporated. The likelihood of the algorithm falling into local optima during iteration is mitigated, and its global search capabilities are bolstered. Thirdly, an improved adaptive t-variation strategy is integrated. By dynamically adjusting the variation strategy according to different search stages, a balance between global exploration and local exploitation is struck. As a result, an overall improvement in the algorithm’s performance and solution quality is achieved. In essence, the research gap is bridged by enhancing the SSA, providing a more effective solution for path planning in mobile robotics and potentially enhancing the efficiency and reliability of robotic applications in real-world scenarios. A workflow diagram is presented in Figure 1.

The significance of this study is profound. In the domain of mobile robotics, efficient path planning is essential for optimizing resource utilization and enhancing operational efficiency. The SSA algorithm is enhanced to offer a more dependable path planning solution. In industrial applications such as material transport robots in automated manufacturing and automatic guided vehicles in logistics warehousing, optimized paths can be achieved, which leads to reduced energy consumption and heightened work efficiency. Thus, enterprises are enabled to optimize resource allocation, curtail operating costs, and boost production benefits, thereby fortifying their market competitiveness.

The primary goal of this research is to develop an enhanced sparrow search algorithm (ISSA) through the integration of multi-strategy collaborative optimization. Specifically, the objectives are to improve population diversity and convergence speed, and to enhance the ability to avoid local optima. By introducing one-dimensional composite chaotic mapping, dung beetle dancing search behavior strategy, and an adaptive t-variation improvement strategy, the ISSA is expected to outperform existing algorithms in handling complex optimization problems. In the context of robot path planning, the ISSA aims to provide more efficient and accurate solutions, especially in scenarios with obstacles and multiple target points, ultimately leading to improved overall performance and reliability in robotic applications.

2. Methods

2.1. Sparrow Search Algorithm (SSA)

The Sparrow Search Algorithm (SSA) is a swarm intelligence optimization algorithm modeled after the foraging and anti-predation behaviors of sparrow groups. It involves three key roles: discoverers, followers, and sentinels [42].

Discoverers are tasked with finding food-rich areas in the entire search space and guiding the foraging direction for followers. The formula for discoverer position update is as follows:

$$X_{ij}^{t+1}=\left\{\begin{array}{l}{X}_{i,j}^t\times \exp ({\displaystyle \frac{-i}{\alpha \times T}}) R\le ST\\ \\ X_{i,j}^t+Q\times L R\ge ST\end{array}\right.$$

(1)

where t represents the current iteration number; T represents the maximum number of iterations; ST∈[0.5, 1] is the population’s safety value; R∈[0, 1] is a random number; $\alpha$ is a random number uniformly distributed in [0, 1]; Q follows the standard normal distribution; L represents a 1 × d matrix with all elements being 1.

Followers adjust their positions based on the behavior of discoverers. Their position update formula is as Equation (2):

$$X_{i,j}^{t+1}=\left\{\begin{array}{l} Q\times \exp ({\displaystyle \frac{X_{wex}^t-X_{i,j}^t}{i^2}}) ,i>n/2\\ \\ X_{i,j}^t+|X_{i,j}^t-X_p^{t+1}|\times A^T{(A{A}^T)}^{-1}L ,otherwise\end{array}\right.$$

(2)

where $X_P^{t+1}$ is the best position found by the follower; X_worst is the global worst position. When i > n/2, it indicates that the i-th follower has a lower fitness and needs to migrate to other locations for food.

Sentinels, accounting for 10–20% of the population, are randomly generated and responsible for sensing danger and adjusting the population’s position. Their position update method is as Equation (3):

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{bet}^t+\beta \times \left|X_i^t-X_{best}^t\right| f_i>f_g\\ \\ X_{i,j}^t+K\cdot {\displaystyle \frac{\left|X_i^t-X_{worst}^t\right|}{\left(f_i-f_w\right)+\epsilon }} f_i=f_g\end{array}\right.$$

(3)

where X_best is the current global optimal position; β is a standard normal distribution step control; K∈[−1, 1] is a random movement direction; f_i is the sparrow’s fitness; f_g and f_w are the global optimal and worst fitness values respectively; $\epsilon$ is a small positive number to avoid division by zero.

2.2. Proposed Algorithm

2.2.1. SPM Chaotic Mapping

The one-dimensional composite chaotic mapping SPM combines Sine and PWLCM mappings, expanding the chaotic range and ensuring ergodicity. It improves the generation efficiency of chaotic sequences without compromising security. To address the issue of insufficient population diversity in the later iteration of the sparrow population, this study employs SPM to generate the initial population, enhancing diversity and the quality of the initial solution. The chaotic mapping equation SPM is defined as in Equation (4):

$$x_i=\left\{\begin{array}{l} \mathrm{mod}(x_{i-1}/\eta +\mu \sin (\pi x_{i-1})+tur,1) 0\le x_{i-1}<\eta \\ \mathrm{mod}((x_{i-1}/\eta )/(0.5-\eta )+\mu \sin (\pi x_{i-1})+tur,1) \eta \le x_{i-1}<0.5\\ F(1-x_{i-1},\eta ,\mu ) 0.5\le x_{i-1}<1\end{array}\right.$$

(4)

When η∈(0, 1) and μ∈(0, 1), the system is chaotic. The perturbation parameter tur helps solve the zero-point problem of PWLCM at x = 0.5 by combining with the Sine mapping and modular operation, maintaining chaos at x = 0.5, thus expanding the chaotic range and enhancing the mapping’s chaotic performance.

The Sine mapping can be expressed as Equation (5):

$$x_{i+1}=\alpha \sin (\mathit{\pi }{x}_i)$$

(5)

where α∈[0, 1] and x_i∈[0, 1]. Compared to high-dimensional mappings, the Sine mapping has a simpler structure and faster chaotic sequence generation. But as computing power grows, the cost of cracking it by exhaustive attack decreases, somewhat limiting its security application.

The piecewise linear chaotic mapping (PWLCM) can be expressed as Equation (6):

$$x_i=\left\{\begin{array}{l} x_{i-1}/\beta 0\le x_{i-1}<\beta \\ (x_{i-1}-\beta )/(0.5-\beta ) \beta \le x_{i-1}<0.5\\ 0 x_{i-1}=0.5\\ F(1-x_{i-1},\beta ) 0.5<x_{i-1}<1.0\end{array}\right.$$

(6)

When the control parameter is in (0, 0.5), is in (0, 1) and the system is chaotic. The PWLCM mapping has consistent variable distribution and good ergodicity, suitable for encryption algorithms. But its zero-point problem makes it lose chaos, reducing security in practice, thus requiring improvement.

The SPM Chaotic Mapping combines the Sine mapping, which has a simple structure and fast chaotic sequence generation, and the PWLCM mapping, which has consistent variable distribution and good ergodicity. The goal is to address the issue of insufficient population diversity in the later iteration of the sparrow population. It generates a more diverse initial sparrow population, enriches the population diversity, augments the ergodicity and optimization ability of the algorithm, and improves the quality of the initial solution.

2.2.2. Introduction of Dancing Search Behavior (DBO)

Inspired by the behavior of dung beetles in nature, the Dancing Search Behavior (DBO) strategy is introduced into the algorithm [8,43,44,45,46]. When dung beetles encounter environmental disturbances such as wind or uneven ground, they often climb on the dung ball and dance (involving rotations and pauses) to determine a new movement direction. The aim is to assist the algorithm in exploring new regions and avoid being trapped in local optima. It enhances the global search ability of the algorithm, allowing it to jump out of local optima during the iteration process and find better solutions. In the proposed algorithm, the tangent function is utilized to model this behavior and acquire a new rolling direction. The formula for position update is as follows Equation (7):

$$X_i(t+1)=X_i(t)+\tan \beta |X_i(t)-X_i(t-1)|$$

(7)

where t represents the current iteration number, x_i(t) is the position of the i-th dung beetle in the t-th iteration, and β∈[0, π] is the deflection angle. It should be noted that when β = 0, π/2 or π, the position of the dung beetle remains unchanged. This strategy is expected to assist the algorithm in exploring new regions, avoiding being trapped in local optima, thereby enhancing the global search ability.

2.2.3. Integration of Adaptive t-Variation Improvement Strategy

The probability density function of the t-distribution is Equation (8):

$$P_{,}(x)={\displaystyle \frac{\Gamma \left(\frac{m+1}{2}\right)}{\sqrt{m\pi }\Gamma \left(\frac{m}{2}\right)}}{(1+{\displaystyle \frac{x^2}{m}})}^{-{\displaystyle \frac{m+1}{2}}}-\mathit{\infty }<x<\mathit{\infty }$$

(8)

where $\Gamma ({\displaystyle \frac{m+1}{2}})={\int }_0^{+\omega }{x}^{{\displaystyle \frac{m+1}{2}}1}{e}^{-x}dx$ is the second type of Euler integral.

This paper mutates the original position $X_i^t$ of the i-th individual sparrow as follows Equation (9):

$$X_i^{t+1}=X_i^t+X_i^t\cdot t(m)$$

(9)

where $X_i^{t+1}$ represents the position of the sparrow individual after t-distribution mutation, and t(m) represents the t-distribution with the iteration number m as the parameter.

Adaptive t-distribution probability density plots are presented in Figure 2.

This strategy is based on the probability density function of the t-distribution. In the early iteration stage, it enables a large jump to explore the global optimal solution, and in the later stage, it conducts a fine local search. The objective is to balance the global exploration and local exploitation capabilities, improve the overall performance and solution quality of the algorithm, and make the sparrow population finally gather near the global optimal solution.

The flowchart of the proposed algorithm is presented in Figure 3.

3. Results

3.1. Algorithm Performance Testing

In this section, the process of evaluating the performance of the proposed ISSA is described. To ensure the reliability and validity of the results, established experimental procedures and benchmarks commonly utilized in the field of optimization algorithm research are examined. The experimental environment is configured to be consistent for all algorithms. MATLAB R2022a is used as the simulation software, running on a Microsoft Win 11 (64-bit) operating system. The computer is equipped with an AMD Ryzen 5 4600U processor operating at 2.10 GHz and 16 GB of memory. The parameters for all algorithms are set uniformly. The number of runs is set at 100. The maximum number of iterations, set as 300, was ascertained through a series of convergence tests. As the improvement in the objective function value diminished and computational cost rose after a certain number of iterations, this value was selected to balance accuracy and computational efficiency. The number of search agents, set to 30, was determined after a parameter tuning process where different numbers were tested and the algorithm’s performance in solution quality and convergence speed was evaluated.

To further assess the ISSA, it is compared with four prominent swarm intelligence algorithms: Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), Whale Optimization Algorithm (WOA), and the original Sparrow Search Algorithm (SSA). The parameter settings for each algorithm are determined based on the recommendations in their respective reference papers. These settings are summarized in

Table 1. Parameter settings for each algorithm.

PSO	Learning factor $C_1=C_2=2$; initial inertia weight $W_{\max }=0.9$; inertia weight at maximum evolution generation $W_{\min }=0.6$.
GWO	Convergence factor a linearly decreases from 2 to 0 during the iteration process.
WOA	Convergence factor a linearly decreases from 2 to 0 during the iteration process, b is a constant representing the shape of the spiral, b = 1.
SSA	The proportion of discoverers in the population is set to about 20%. Safety value ST = 0.8.

.

3.1.1. Benchmark Test Functions

Twenty-three benchmark test functions, widely acknowledged in the literature for assessing optimization algorithms, are selected. These functions are categorized into three groups: high-dimensional unimodal (f₁–f₇), high-dimensional multimodal (f₈–f₁₃), and fixed-dimensional multimodal benchmark functions (f₁₄–f₂₃). Such a diverse set is employed to comprehensively test the ISSA’s performance in various scenarios, including handling functions with single global optima and those with multiple local optima. The expressions and details of these functions are presented in Table S1 and Figure S1, respectively.

3.1.2. Comparative Analysis with Other Swarm Intelligence Algorithms

The ISSA proposed in this paper is independently run 100 times on 23 benchmark functions together with Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA), Grey Wolf Optimization (GWO), and Sparrow Search Algorithm (SSA). In order to analyze the superiority of the proposed algorithm in search performance and stability compared to other swarm intelligence algorithms, the worst values, best values, average values, and standard deviations in Dim (where Dim is the abbreviation of Dimensions) of 20, 60, and 100 are calculated, respectively. The selection of Dim values (20, 60, and 100) is designed to provide a comprehensive assessment of the ISSA algorithm. The value of 20 is used as a lower-complexity baseline, while 60 and 100 are chosen to represent increasingly complex scenarios. This approach allows us to observe the algorithm’s performance and adaptation as the problem dimension grows, providing insights into its scalability and effectiveness in handling diverse optimization tasks. It should be noted that while these values are chosen somewhat arbitrarily for the purpose of a thorough evaluation, they are commonly used in the field to benchmark algorithms across different complexity levels.

The experimental results at different dimensions (20, 60, and 100) are presented in Tables S2, S3 and

Table A1. Experimental results (Dim = 100).

Function Name	Metric	PSO	GWO	WOA	SSA	ISSA
f1	Worst	6.73 × 10−2	1.62 × 10−26	5.45 × 10−71	3.48 × 10−45	0
	Best	1.73 × 10−4	1.43 × 10−29	1.02 × 10−90	0	0
	Average	7.07 × 10−3	1.13 × 10−27	7.42 × 10−73	3.48 × 10−47	0
	STD	9.77 × 10−3	2.48 × 10−27	5.59 × 10−72	3.48 × 10−46	0
f2	Worst	20	8.72 × 10−16	3.22 × 10−49	1.23 × 10−22	5.56 × 10−269
	Best	3.36 × 10−3	6.65 × 10−18	2.26 × 10−57	0	0
	Average	1.62	1.04 × 10−16	8.19 × 10−51	1.24 × 10−24	5.56 × 10−271
	STD	4.20	1.12 × 10−16	3.96 × 10−50	1.23 × 10−23	0
f3	Worst	1.62 × 104	2.79 × 10−4	8.8 × 104	4.71 × 10−50	0
	Best	509.214	8.43 × 10−9	3926.1054	0	0
	Average	2510.0837	1.25 × 10−5	45,513.8954	7.15 × 10−52	0
	STD	2577.5337	3.94 × 10−5	14,720.657	5.28 × 10−51	0
f4	Worst	11.5859	5.38 × 10−6	91.3319	5.64 × 10−31	9.25 × 10−285
	Best	3.4959	7.05 × 10−8	5.0264	0	0
	Average	7.0503	9.21 × 10−7	53.5678	6.78 × 10−33	9.33 × 10−287
	STD	1.6733	1.10 × 10−6	28.4516	5.74 × 10−32	0
f5	Worst	9.01 × 104	28.7558	28.7825	8.03 × 10−2	28.7644
	Best	20.0117	25. 2684	26.8084	1.77 × 10−4	27.7019
	Average	3.01 × 103	27.0547	27.9666	2.42 × 10−2	28.3511
	STD	1.54 × 104	7.30 × 10−1	4.81 × 10−1	1.64 × 10−2	3.06 × 10−1
f6	Worst	0.050133	1.7605	1.4544	1.5722 × 10−3	0.60586
	Best	1.11762 × 10−4	7.9901 × 10−5	0.085123	2.0102 × 10−5	0.023603
	Average	8.5796 × 10−3	0.75119	0.40004	1.3649 × 10−4	0.15755
	STD	0.011305	0.40083	0.21512	1.7587 × 10−4	0.09308
f7	Worst	5.4406	4.69 × 10−3	4.24 × 10−2	2.71 × 10−3	3.32 × 10−4
	Best	1.88 × 10−2	4.10 × 10−4	6.80 × 10−5	1.86 × 10−6	8.29 × 10−7
	Average	1.28 × 10−1	2.00 × 10−3	3.92 × 10−3	7.01 × 10−4	8.41 × 10−5
	STD	5.99 × 10−1	9.56 × 10−4	5.42 × 10−3	5.26 × 10−4	7.95 × 10−5
f8	Worst	−7057.4627	−3004.6203	−6769.8771	−4853.5867	−9392.9792
	Best	−9801.355	−7540.6924	−1.26 × 104	−1.26 × 104	−1.26 × 104
	Average	−8604.4849	−5763.6179	−10,500.6792	−8522.0113	−12,219.0844
	STD	606.3484	775. 7291	1800.849	2408.7056	708.0306
f9	Worst	108.5314	20.0383	1.14 × 10−13	0	0
	Best	29.4845	5.68 × 10−14	0	0	0
	Average	58.9817	2.9108	2.27 × 10−15	0	0
	STD	16.6001	3.6237	1.38 × 10−14	0	0
f10	Worst	2.123	1.47 × 10−13	7.99 × 10−15	8.88 × 10−16	8.88 × 10−16
	Best	4.43 × 10−3	7.55 × 10−14	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	Average	0.56194	1.01 × 10−13	4.70 × 10−15	8.88 × 10−16	8.88 × 10−16
	STD	0.65389	1.57 × 10−14	2.49 × 10−15	0	0
f11	Worst	0.21726	0.033534	0.21418	0	0
	Best	0.00144	0	0	0	0
	Average	2.97 × 10−2	5.27 × 10−3	5.16 × 10−3	0	0
	STD	3.24 × 10−2	9.03 × 10−3	3.01 × 10−2	0	0
f12	Worst	1.3007	0.13785	7.82 × 10−2	0.1127	1.67 × 10−2
	Best	2.65 × 10−5	6.57 × 10−3	3.89 × 10−3	5.71 × 10−8	1.30 × 10−3
	Average	0.1627	4.67 × 10−2	2.26 × 10−2	1.15 × 10−3	6.09 × 10−3
	STD	0.24573	2.34 × 10−2	1.23 × 10−2	1.13 × 10−2	2.71 × 10−3
f13	Worst	4.0927	1.3696	1.3048	0.023044	0.83937
	Best	2.77 × 10−4	1.02 × 10−1	9.94 × 10−2	1.60 × 10−6	1.04 × 10−1
	Average	0.22425	0.63757	0.53971	4.74 × 10−3	0.38609
	STD	0.55636	0.2552	0.26339	7.09 × 10−3	0.16402
f14	Worst	0.998	12.6705	10.7632	12.6705	12.6705
	Best	0.998	0.998	0.998	0.998	0.998
	Average	0.998	4.0933	2.8645	10.1041	8.2394
	STD	2.24 × 10−15	4.0065	3.0543	4.295	4.7131
f15	Worst	2.04 × 10−2	2.04 × 10−2	2.25 × 10−3	1.39 × 10−3	1.60 × 10−3
	Best	3.08 × 10−4	3.07 × 10−4	3.08 × 10−4	3.08 × 10−4	3.08 × 10−4
	Average	3.17 × 10−3	5.85 × 10−3	7.48 × 10−4	3.87 × 10−4	4.47 × 10−4
	STD	6.68 × 10−3	8.87 × 10−3	4.79 × 10−4	1.92 × 10−4	2.56 × 10−4
f16	Worst	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	Best	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	Average	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	STD	1.53 × 10−15	3.78 × 10−8	1.42 × 10−9	4.44 × 10−8	9.87 × 10−12
f17	Worst	0.39789	0.39791	0.39796	0.39789	0.39789
	Best	0.39789	0.39789	0.39789	0.39789	0.39789
	Average	0.39789	0.39789	0.39789	0.39789	0.39789
	STD	1.06 × 10−15	4.03 × 10−6	1.36 × 10−5	6.58 × 10−7	2.83 × 10−9
f18	Worst	84	84	3.0005	3	3
	Best	3	3	3	3	3
	Average	3.81	3.81	3	3	3
	STD	8.1	8.1	8.06 × 10−5	1.52 × 10−6	4.81 × 10−10
f19	Worst	−3.8628	−3.8549	−3.8252	−3.8618	−3.8628
	Best	−3.8628	−3.8628	−3.8628	−3.8628	−3.8628
	Average	−3.8628	−3.8615	−3.856	−3.8627	−3.8628
	STD	6.30 × 10−15	2.41 × 10−3	8.30 × 10−3	1.11 × 10−4	1.50 × 10−6
f20	Worst	−3.0839	−3.0222	−2.0612	−3.121	−3.151
	Best	−3.322	−3.322	−3.3219	−3.322	−3.322
	Average	−3.2647	−3.253	−3.2134	−3.2639	−3.2715
	STD	7.23 × 10−2	8.29 × 10−2	1.72 × 10−1	6.96 × 10−2	6.25 × 10−2
f21	Worst	−2.6305	−2.6302	−0.88098	−9.235	−10.1531
	Best	−10.1532	−10.1531	−10.153	−10.1532	−10.1532
	Average	−5.8962	−9.0744	−8.012	−10.1432	−10.1532
	STD	3.4536	2.2781	2.7952	0.091737	1.64 × 10−5
f22	Worst	−1.8376	−5.0876	−1.8332	−5.0877	−10.4029
	Best	−10.4029	−10.4027	−10.4018	−10.4029	−10.4029
	Average	−7.5483	−10.2061	−7.3723	−10.3493	−10.4029
	STD	3.5718	0.97141	3.0248	0.53148	1.46 × 10−5
f23	Worst	−2.4217	−2.4216	−1.6764	−10.5254	−10.5364
	Best	−10.5364	−10.5363	−10.5362	−10.5364	−10.5364
	Average	−8.0599	−10.1839	−7.1367	−10.5361	−10.5364
	STD	3.5863	1.5655	3.2759	1.11 × 10−3	1.06 × 10−5

. At 20 dimensions, for high-dimensional unimodal functions f₁–f₇, ISSA demonstrated remarkable performance. It had a standard deviation and best value of 0 in f₁–f₄ and outperformed other algorithms in other respects. On f₇, its minimal standard deviation indicated high stability and accuracy. ISSA could stably and efficiently obtain the optimal values for f₁ and f₃. For f₂, f₄, and f₇, while no algorithm found the optimal solution, ISSA had the lowest average value and higher optimization accuracy than its counterparts, except being second to SSA on f₅ and to PSO and SSA on f₆. For multimodal functions f₈–f₁₃, ISSA outperformed the other four algorithms on f₈–f₁₁. Specifically, on f₈, while none of the algorithms found the optimal value, ISSA had the closest approximation. On f₁₀, ISSA’s performance was on par with SSA and superior to the others. Additionally, ISSA found the optimal values of f₉ and f₁₁ with high convergence accuracy. ISSA was only inferior to SSA on f₁₂ and to SSA and PSO on f₁₃. By observing the standard deviation data, it was evident that ISSA had the smallest standard deviation on f₈–f₁₁, with values of 0 on f₉, f₁₀, and f₁₁, indicating good stability. For f₁₂ and f₁₃, ISSA’s stability was second only to SSA and surpassed that of the other algorithms in the comparison. For fixed-dimensional multimodal benchmark functions f₁₄–f₂₃, due to their relatively simple function constructions, each algorithm could achieve satisfactory search results.

At 60 dimensions, similar trends were observed. For unimodal functions f₁–f₇, ISSA performed better than the other four algorithms on f₁, f₂, f₃, f₄, and f₇. It obtained the optimal results on f₁ and f₃. For f₂, f₄, and f₇, although no algorithm obtained the optimal value, ISSA was the closest to the optimal solution. On f₅, ISSA was second only to SSA. On f₆, ISSA was inferior to PSO and SSA. Additionally, ISSA had the smallest standard deviation on f₁–f₄ and f₇, with 0 on f₁–f₄, showing strong stability. For multimodal functions f₈–f₁₃, ISSA performed well on f₈–f₁₁. It found the optimal values of f₉ and f₁₁. On f₈, none of the five algorithms found the optimal value, but ISSA had the highest degree of approximation. On f₁₀, ISSA had the same performance as SSA and was superior to other algorithms. ISSA was only inferior to SSA on f₁₂ and to SSA and PSO on f₁₃. Observing the standard deviation data, it could be seen that the standard deviation of ISSA on f₉, f₁₀, and f₁₁ was 0, indicating good stability. For f₈ and f₁₃, the stability of ISSA was second only to SSA and PSO. For f₁₂, the stability of ISSA was second only to SSA and was superior to the other algorithms in the comparison. For fixed-dimensional multimodal benchmark functions f₁₄–f₂₃, due to their relatively simple function constructions, the numerical differences among the algorithms were not significant.

At 100 dimensions, for high-dimensional unimodal functions f₁–f₇, ISSA again performed well on f₁, f₂, f₃, f₄, and f₇. It obtained the optimal results on f₁ and f₃. For f₂, f₄, and f₇, although no algorithm obtained the optimal value, ISSA was the closest to the optimal solution. Especially on f₂ and f₇, its average value was close to 0, and the optimization accuracy was high. ISSA was only inferior to other algorithms on f₅ and f₆. Additionally, ISSA had the smallest standard deviation on f₁–f₄ and f₇. The standard deviation on f₁–f₄ was 0, showing strong stability. For multimodal functions f₈–f₁₃, ISSA performed well on f₈–f₁₁. It found the optimal values of f₉ and f₁₁. On f₈, none of the five algorithms found the optimal value. On f₁₀, ISSA had the same performance as SSA and was superior to other algorithms. ISSA was only inferior to SSA on f₁₂ and was close to GWO on f₁₃. Observing the standard deviation data, it could be seen that the standard deviation of ISSA on f₉, f₁₀, and f₁₁ was 0, and the standard deviation on f₁₂ was the lowest, indicating good stability. For f₁₃, the stability of ISSA was second only to SSA and was superior to the other algorithms in the comparison. For multimodal benchmark functions f₁₄–f₂₃, the numerical differences among the algorithms were not significant.

ISSA consistently demonstrates excellent performance. It shows a rapid convergence speed in the early stages of iteration, which is mainly attributed to the use of the one-dimensional chaotic mapping SPM for generating the initial sparrow population. This approach enriches the population diversity and improves the quality of the initial solution. In the later stages of iteration, when other algorithms tend to stagnate, ISSA continues to search downward and achieves better convergence accuracy. This is due to the introduction of the adaptive t-variation strategy, which enables the algorithm to have good global exploration in the early stage and excellent local exploitation in the later stage. ISSA outperforms the other algorithms in terms of both convergence speed and accuracy on these unimodal functions, highlighting its superiority in handling functions with a single global optimal solution.

To more intuitively compare the convergence accuracy and convergence speed of the algorithms, the convergence curves of the 23 test functions at a dimension of 20 were plotted according to the number of iterations and fitness values (Figure 4, Figure 5 and Figure 6). Figure S1 shows the two-dimensional shapes of some test functions used to evaluate the ISSA algorithm. Figure 4 shows the convergence process of different algorithms on unimodal functions f₁-f₇. In f₁ and f₃, only ISSA reached the theoretical optimal value and found the optimal solution after about 300 iterations, with a significantly faster convergence speed than other comparison algorithms. For f₂, f₄, and f₇, algorithms other than ISSA did not find the optimal value. ISSA found a relatively good position in the early stage of iteration. In the early stage of the search, its convergence curve showed a rapid decline, indicating that the strategy of generating the initial sparrow population using SPM chaotic mapping could effectively improve the convergence rate of the algorithm in the early stage of iteration. In the middle and late stages of search, ISSA could more accurately lock the global optimal solution and then conduct local development near the optimal solution. The four algorithms involved in the comparison fell into local optimum prematurely. Only ISSA continuously approached the optimal value, and its approach speed to the optimal value was the fastest, indicating that introducing the dung beetle dancing search behavior strategy enhanced its global search ability and avoided falling into local optimum. For f₅ and f₆, the performance of ISSA was second only to SSA. ISSA had achieved excellent performance in unimodal test functions.

Figure 5 shows the convergence process of different algorithms on multimodal functions f₈-f₁₃. For f₉ and f₁₁, except that PSO did not reach the theoretical optimal value in f₉, the other algorithms reached the theoretical optimal value. However, the convergence speed of ISSA was significantly faster than other comparison algorithms. At the same time, the performance effects of SSA and WOA were similar to those of ISSA. For f₈ and f₁₀, none of the five algorithms found the optimal value. GWO and PSO fell into stagnation very early. SSA and WOA performed slightly better but also fell into local optimum, and the convergence speed was slow. In contrast, especially for f₁₀, the convergence curve of ISSA had the largest slope and the fastest approach speed to the optimal solution. Compared with the other four algorithms, the optimization effect was more ideal. This was due to ISSA’s fusion of adaptive t-variation swarm strategy, which enabled the algorithm to make a large jump in the early stage to explore the global optimal solution, and conduct fine local search in the later stage to improve the accuracy of the solution. For f₁₂ and f₁₃, the performance of ISSA was second only to SSA and was better than other algorithms involved in the comparison. In conclusion, ISSA had also achieved excellent performance in multi-peak test functions. Figure 6 shows the convergence process of different algorithms on fixed-dimensional multi-peak functions. It could be found that the convergence curves of each algorithm were similar. Although the optimal values found by each algorithm were close, ISSA was still slightly better than other algorithms in convergence accuracy and convergence speed. In terms of exploration ability, the convergence speed and convergence accuracy of the ISSA algorithm were generally better than other algorithms, indicating that this algorithm had better global search ability and the ability to jump out of local optimum, and could make the sparrow population finally gather near the global optimal solution. This phenomenon was fully demonstrated in the experiment.

3.1.3. Analysis of the Effectiveness of Improvement Strategies

f₁–f₇ are unimodal functions with only one global optimal solution, mainly used to test an algorithm’s convergence speed and accuracy. In the early iteration stage, ISSA converges faster than other algorithms. When others stagnate later, ISSA keeps searching and obtains better convergence accuracy. This is because ISSA uses one-dimensional chaotic mapping SPM to generate the initial sparrow population, enhancing population diversity and initial solution quality. Also, the adaptive t-variation strategy gives it good global exploration early and excellent local exploitation later, improving its optimization ability. Thus, ISSA with SPM and adaptive t-variation strategy has significantly better convergence speed and accuracy.

f₈–f₁₃ are high-dimensional multimodal functions, and f₁₄–f₂₃ are fixed-dimensional multimodal benchmark functions. These functions have many local extreme points, easily leading algorithms to stagnation, and are used to test the global search and local optima escape abilities. From the data and average convergence curve, ISSA is significantly better. This is due to the added dung beetle dancing search behavior strategy, which effectively improves the ability to jump out of local optima, reduces the probability of getting trapped, and enhances the algorithm’s robustness.

Therefore, regardless of function type or dimension, based on its improvement strategies, ISSA has higher performance advantages than other algorithms in global search, local development, and convergence speed.

3.2. Simulation and Verification of Path Planning

3.2.1. Single-Target Point Path Planning

In this subsection, the performance of the ISSA in single-target point path planning is evaluated. Grid maps of different sizes (10 × 10, 20 × 20, 30 × 30, and 40 × 40) with varying obstacle rates are created to simulate different levels of environmental complexity. In these maps, each grid represents a 1 m × 1 m area, with black grids indicating obstacles and white grids representing free space. The start and end points of the path are located at the lower left and upper right corners of the map, respectively. Sparrows are modeled as points with perception capabilities to detect and avoid obstacles on their way to the target point.

The ISSA is compared with the SSA in these path planning scenarios. Both algorithms are run independently with the same initial population size and a maximum iteration number of 100. Each algorithm is executed 50 times, and the average value is taken as the final result, following the standard experimental procedures in algorithm comparison for path planning. The performance of the algorithms is evaluated based on several key metrics, including path length, number of turns, and number of iterations. The path trajectories and corresponding measurement results are presented in Figure 7, Figure 8, Figure 9 and Figure 10, Figures S2–S9 and

Table 2. Performance indicators of different algorithms in different scenarios.

	Obstacle Rate	Metric	SSA	ISSA	Decrease Percentage (%)
10 × 10	10%	Path Length (m)	14.4853	13.8998	4.04%
		Number of Turns	5	4	20.00%
		Iteration	4	17	−352.00%
	15%	Path Length (m)	14.4853	13.8995	4.04%
		Number of Turns	7	7	0%
		Iteration	20	32	−60.00%
	20%	Path Length (m)	13.3137	12.7279	4.40%
		Number of Turns	3	0	100.00%
		Iteration	73	5	93.15%
20 × 20	10%	Path Length (m)	28.6274	28.0416	2.05%
		Number of Turns	6	5	16.67%
		Iteration	4	42	−950.00%
	15%	Path Length (m)	29.2132	28.0416	4.01%
		Number of Turns	9	5	44.45%
		Iteration	3	1	66.67%
	20%	Path Length (m)	29.2132	28.6274	2.00%
		Number of Turns	12	5	58.33%
		Iteration	47	7	85.11%
30 × 30	10%	Path Length (m)	42.7696	42.1838	1.37%
		Number of Turns	9	8	11.11%
		Iteration	14	27	−92.86%
	15%	Path Length (m)	43.3553	42.7696	1.35%
		Number of Turns	8	11	−37.50%
		Iteration	72	51	29.17%
	20%	Path Length (m)	44.5269	43.9411	1.32%
		Number of Turns	15	10	33.33%
		Iteration	2	63	−3050.00%
40 × 40	10%	Path Length (m)	59.2548	58.0833	1.98%
		Number of Turns	12	12	0%
		Iteration	24	93	−287.50%
	15%	Path Length (m)	59.2548	58.0833	1.98%
		Number of Turns	20	18	10.00%
		Iteration	300	50	83.33%
	20%	Path Length (m)	59.2548	58.699	0.94%
		Number of Turns	14	15	−7.14%
		Iteration	3	11	−266.67%

, providing a visual and quantitative assessment of the algorithms.

Observing the paths planned by ISSA and SSA, it can be found that they have similarities. This is because ISSA is obtained by optimizing on the basis of SSA. But in general, the path planned by ISSA is simpler, smoother, and more stable than that of SSA. In grid maps of different sizes, as the obstacle rate increases, the advantages of ISSA become more and more obvious. The path planned by SSA has more turning points and is not smooth; while the path of ISSA is smoother. Especially in some cases, the path of ISSA can be a straight line with better quality. This is likely because the three improvement strategies adopted by ISSA enrich the diversity of the sparrow population, enhance the ability of fast iterative optimization, and avoid local optimal solutions at the same time, so that the algorithm can effectively jump out of local optima. Therefore, the route planned by ISSA is smoother, has fewer turning points, is shorter in path, takes less search time, can effectively avoid obstacles, realizes orderly access to all points, and finds the optimal solution with fewer iterations. In contrast to ISSA, the path planned by SSA has more inflection points and a longer path length, lacks smoothness, and has insufficient local exploration ability during subsequent iterations and is easily trapped in local optima.

Observing the convergence curve, it can be seen that whether on a simple map of 10 × 10 or a complex map of 40 × 40, the convergence curve of the ISSA algorithm is better than that of the SSA algorithm. The convergence curve of the ISSA algorithm is smoother, drops faster and stabilizes earlier. At the same time, the curve of the ISSA algorithm does not fall into local optimum after a rapid drop. There are still broken lines in the later stage of iteration, which shows the effectiveness of the three improvement strategies adopted by ISSA.

Table 2 clearly shows the path length, number of turns, number of iterations of the two algorithms and their comparison. In general, the path length, number of turns, and number of iterations of ISSA are all better than those of SSA. Whether on a simple map of 10 × 10 or a complex map of 40 × 40, the path length of ISSA is shorter than that of SSA, showing the good search ability of ISSA. Especially in some cases, the path length of ISSA is much less than that of SSA, the number of turns is greatly reduced, and the number of iterations is also significantly reduced, realizing convergence to the shortest and smoothest path at the fastest speed. On maps of different sizes, as the obstacle rate increases, the performance of ISSA is even more excellent, indicating that the three proposed improvement strategies can effectively improve the algorithm performance. In some cases, although the number of turns of ISSA may be higher than that of SSA, the path length of ISSA is lower than that of SSA, and the number of iterations is less than that of SSA. Therefore, the performance of ISSA is still better than that of SSA. From this, it can be more clearly seen that ISSA has greater optimization than SSA in all aspects. The planned path length is shorter, the stability is good, it is not easily affected by the map size, there are fewer inflection points, the algorithm runs more smoothly, and it is more suitable for finding the optimal path in an environment with complex obstacles.

3.2.2. Multi-Target Point Path Planning

The multi-target point path planning problem can be described as follows: The mobile robot needs to traverse several target points, and each target point can only be traversed once. The purpose is to find the shortest path passing through all target points. The path traversing n target points is defined as a specific form. At this time, the total distance of the traversed path can be expressed as Equation (10):

$$S={\displaystyle \sum _{i=1,j=1}^{n}x\left(d_i,d_j\right)}$$

(10)

Among them, d(x(i), x(i + 1)) is the shortest distance between two adjacent target points in the traversal list. In the planning process, it is necessary to fully consider the movement constraints of the mobile robot in an obstacle environment. By using the shortest obstacle avoidance distance calculated by the A* algorithm to replace the Euclidean distance that ignores obstacles, it is ensured that the path planned for the mobile robot in a scene with obstacles is consistent with the actual optimal path. This paper is dedicated to combining the single-target point path planning of the A* algorithm and the multi-target point optimization of the improved sparrow optimization algorithm to minimize the value of S.

To verify the accuracy and convergence speed of the ISSA algorithm in multi-target point path planning in complex scenarios, as the obstacle rate in the scenario increases, 4~6 target nodes are selected for testing, respectively (Figure 11).

Table 3. Comparison of path lengths of SSA and ISSA algorithms in different scenarios.

Map	Obstacle Rate	Index	SSA	ISSA
10 × 10	0.1	Minimum path length/m	17.89949	17.89949
		Maximum path length/m	19.89949	17.89949
		Average path length/m	18.0003	17.89949
	0.15	Minimum path length/m	23.48528	23.48528
		Maximum path length/m	28.48528	25.89949
		Average path length/m	23.68922	23.49094
	0.2	Minimum path length/m	28.38478	27.79899
		Maximum path length/m	28.38478	28.38478
		Average path length/m	28.38478	27.80485
20 × 20	0.1	Minimum path length/m	39.45584	39.45584
		Maximum path length/m	43.69848	39.79899
		Average path length/m	39.47799	39.46545
	0.15	Minimum path length/m	40.04163	40.04163
		Maximum path length/m	49.35534	42.45584
		Average path length/m	40.06543	40.05612
	0.2	Minimum path length/m	49.35534	45.69848
		Maximum path length/m	56.76955	58.94113
		Average path length/m	49.39388	46.19976
30 × 30	0.1	Minimum path length/m	84.35534	84.35534
		Maximum path length/m	85.76955	86.18377
		Average path length/m	84.41756	84.37533
	0.15	Minimum path length/m	106.8406	96.3259
		Maximum path length/m	116.5685	114.4975
		Average path length/m	106.8795	98.13832
	0.2	Minimum path length/m	110.669	99.25483
		Maximum path length/m	130.9828	132.5685
		Average path length/m	111.9271	100.2721
40 × 40	0.1	Minimum path length/m	165.1249	165.1249
		Maximum path length/m	178.6102	165.1249
		Average path length/m	165.1788	165.1249
	0.15	Minimum path length/m	161.196	141.2965
		Maximum path length/m	161.196	144.468
		Average path length/m	161.196	141.5692
	0.2	Minimum path length/m	226.5807	219.8823
		Maximum path length/m	252.066	231.5097
		Average path length/m	227.6108	223.412

shows the comparison of algorithm performance,

Table 4. Comparison of traversal orders of SSA and ISSA in different scenarios.

Map	Obstacle Rate	Traversal Order		Number of Turning Points
		SSA	ISSA	SSA	ISSA
10 × 10	0.1	[1, 2, 3, 4, 5, 6]	[1, 2, 3, 4, 5, 6]	9	9
	0.15	[1, 2, 4, 3, 5, 7, 6]	[1, 2, 4, 3, 5, 7, 6]	12	12
	0.2	[1, 5, 7, 6, 3, 2, 4, 8]	[1, 2, 5, 7, 6, 3, 4, 8]	13	12
20 × 20	0.1	[1, 2, 3, 4, 5, 6]	[1, 2, 3, 4, 5, 6]	13	13
	0.15	[1, 2, 3, 4, 5, 6, 7]	[1, 2, 3, 4, 5, 6, 7]	18	18
	0.2	[1, 3, 5, 2, 4, 6, 7, 8]	[1, 5, 3, 2, 4, 6, 7, 8]	20	18
30 × 30	0.1	[5, 6, 3, 2, 1, 4]	[4, 1, 2, 3, 6, 5]	18	13
	0.15	[1, 2, 3, 5, 6, 7, 4]	[1, 2, 3, 4, 7, 6, 5]	24	22
	0.2	[1, 2, 3, 6, 7, 8, 5, 4]	[1, 3, 2, 4, 5, 8, 7, 6]	29	26
40 × 40	0.1	[4, 1, 5, 3, 2, 6]	[2, 3, 6, 5, 1, 4]	26	25
	0.15	[1, 3, 2, 5, 4, 6, 7]	[1, 4, 2, 3, 5, 6, 7]	41	37
	0.2	[3, 1, 7, 4, 2, 5, 6, 8]	[3, 5, 4, 6, 8, 7, 1, 2]	52	37

is the comparison of traversal orders, and Figure 12 and Figures S10–S12 show the results of path planning of different sizes and the corresponding fitness value iteration curves of multiple path plans of different quantities selected under different obstacle rates.

According to the data in Table 3, in the 10 × 10 path planning, when the obstacle rate is 0.1, compared with the SSA algorithm, the minimum path length of the ISSA algorithm is basically unchanged, and the average path length is reduced by 0.56%; when the obstacle rate is 0.15, the minimum path length is basically unchanged, and the average path length is reduced by 0.84%; when the obstacle rate is 0.2, the minimum path length is reduced by 2.06%, and the average path length is reduced by 2.04%. In the 20 × 20 path planning, when the obstacle rate is 0.1, the minimum path length is basically unchanged, and the average path length is reduced by 0.03%; when the obstacle rate is 0.15, the minimum path length is basically unchanged, and the average path length is reduced by 0.02%; when the obstacle rate is 0.2, the minimum path length is reduced by 7.41%, and the average path length is reduced by 6.47%. In the 30 × 30 path planning, when the obstacle rate is 0.1, the minimum path length is basically the same, and the average path length is reduced by 0.05%; when the obstacle rate is 0.15, the minimum path length is reduced by 9.84%, and the average path length is reduced by 10.41%; when the obstacle rate is 0.2, the minimum path length is reduced by 10.31%, and the average path length is reduced by 10.41%. In the 40 × 40 path planning, when the obstacle rate is 0.1, the minimum path length is basically the same, and the average path length is reduced by 0.03%; when the obstacle rate is 0.15, the minimum path length is reduced by 12.34%, and the average path length is reduced by 12.18%; when the obstacle rate is 0.2, the minimum path length is reduced by 2.96%, and the average path length is reduced by 1.84%.

From the fitness value iteration curve of Figure S10, in Figure S10c,f, the SSA algorithm falls into a local optimal solution at the beginning. Although it finds the optimal solution quickly, the ISSA algorithm finds the optimal solution at the beginning, indicating that the spatial search ability of the ISSA algorithm is better than that of the SSA algorithm. In Figure S10i, the SSA algorithm falls into a local optimal solution at the beginning and fails to jump out, resulting in cross paths in the generated path, and the minimum path length is not as expected. However, the ISSA algorithm finds the optimal solution and the convergence speed is very fast. From the fitness value iteration curve of Figure S11, in Figure S11c,f, ISSA and SSA both find the optimal solution, and the convergence speed is very fast. In Figure S11i, both the SSA and ISSA algorithms fall into local optimal solutions. The SSA algorithm fails to jump out of the local optimal solution due to its inferior search ability compared to the ISSA algorithm, resulting in cross paths in the generated path and the minimum path length not as expected. However, the ISSA algorithm adopts the dung beetle dancing search behavior strategy and jumps out of the local optimal solution and finds the optimal solution. From the fitness value iteration curve of Figure S12, in Figure S12c, ISSA and SSA both find the optimal solution. Since the ISSA algorithm introduces a one-dimensional composite chaotic mapping SPM, which improves the quality of the initial solution of the algorithm, the ISSA algorithm is the first to find the optimal solution and the convergence speed is better than that of the SSA algorithm. Moreover, the ISSA algorithm does not fall into a local optimal solution, indicating that the search and solution ability of the ISSA algorithm is better than that of the SSA algorithm. In Figure S12f, the ISSA algorithm and the SSA algorithm fall into different local optimal solutions at the beginning. The SSA algorithm fails to jump out of the local optimal solution due to its inferior search ability compared to the ISSA algorithm, resulting in the minimum path length not as expected. However, the ISSA algorithm finds the optimal solution after 320 iterations. In Figure S12i, the ISSA algorithm and the SSA algorithm both fall into different local optimal solutions at the beginning. However, the SSA algorithm falls into the local optimal solution multiple times, resulting in a slower convergence speed of the algorithm and failing to jump out of the local optimal solution, resulting in the minimum path length not as expected. However, the ISSA algorithm falls into the local optimal solution only once and quickly jumps out of the local optimal solution to find the optimal solution. From the fitness value iteration curve of Figure 12, in Figure 12c, ISSA and SSA both find the optimal solution, and the curve convergence speed is very fast. In Figure 12f, both the SSA and ISSA algorithms fall into local optimal solutions. However, the ISSA algorithm jumps out of the local optimal solution at 50 iterations and finds the optimal solution, while the SSA algorithm fails to jump out of the local optimal solution. In Figure 12i, both the SSA and ISSA algorithms fall into local optimal solutions. However, the ISSA algorithm jumps out of the local optimal solution and finds the optimal solution, while the SSA algorithm fails to jump out of the local optimal solution.

Table 4 shows that in the 10 × 10 path planning with an obstacle rate of 0.2, the 20 × 20 path planning with an obstacle rate of 0.2, and all obstacle-rate scenarios in the 30 × 30 and 40 × 40 path planning, the ISSA algorithm employs a traversal order distinct from that of the SSA algorithm. As a result, when the obstacle rate is 0.2 in the 10 × 10 case, the ISSA algorithm has one fewer turn than the SSA algorithm; when the obstacle rate is 0.2 in the 20 × 20 case, the ISSA algorithm has two fewer turns; in the 30 × 30 case with an obstacle rate of 0.1, the ISSA algorithm has five fewer turns, with an obstacle rate of 0.15 it has two fewer turns, and with an obstacle rate of 0.2 it has three fewer turns; in the 40 × 40 case, when the obstacle rate is 0.1, the ISSA algorithm has one fewer turn, when the obstacle rate is 0.15 it has four fewer turns, and when the obstacle rate is 0.2 it has fifteen fewer turns. Consequently, the ISSA algorithm generates a smoother path, reduces the path-planning length, and, to a certain extent, decreases the number of crossings in the generated path.

4. Discussion

4.1. Comparison with the Prior Literature

In the context of prior research on optimization algorithms and path planning, the performance of the proposed ISSA algorithm was evaluated through a series of experiments and comparisons with existing algorithms, showing significant improvements.

When contrasted with traditional path planning algorithms including the A* algorithm, rapidly exploring random trees (RRT) algorithm, artificial potential field (APF) algorithm, and dynamic window approach (DWA), ISSA presents a notable enhancement. Previous studies on these traditional algorithms mainly centered on improving path smoothness, effectiveness, and system performance via diverse modifications [8,11,46,47,48]. However, with its multi-strategy collaborative optimization, ISSA outperforms them in convergence speed and optimization accuracy. In addition, in the experiments with 23 benchmark test functions, ISSA demonstrates better capabilities in handling functions with single global optima and those with multiple local optima compared to algorithms like PSO, GWO, and WOA. In contrast to previous studies where these algorithms might have encountered challenges in either convergence speed or getting trapped in local optima, ISSA shows faster convergence in the early stages and better accuracy in the later stages. Especially when dealing with high-dimensional multimodal functions (f₈–f₁₃) and fixed-dimensional multimodal benchmark functions (f₁₄–f₂₃), which have many local extreme points and are prone to leading algorithms to stagnation, ISSA’s dung beetle dancing search behavior strategy effectively improves its ability to jump out of local optima, reduces the probability of getting trapped, and enhances the algorithm’s robustness. This gives ISSA a significant advantage over other algorithms in handling complex multi-modal functions.

Among the swarm intelligence optimization algorithms such as PSO, GWO, and WOA, ISSA also exhibits its superiority. Previous works on enhancing these algorithms mainly focused on adjusting parameters or combining different strategies. In contrast, ISSA’s unique combination of one-dimensional composite chaotic mapping SPM, dung beetle dancing search behavior strategy, and adaptive t-variation improvement strategy offers a novel perspective for algorithm optimization. In the experiments with 23 benchmark test functions, ISSA shows faster convergence in the early stages and better accuracy in the later stages compared to these algorithms.

In the comparison with the original SSA, ISSA displays better performance in handling functions with different characteristics and in path planning applications. The improvement strategies adopted by ISSA effectively tackle the issues of limited population diversity and premature convergence of SSA, which were the primary concerns in previous research. In the area of path planning, previous research has emphasized improving path smoothness and reducing travel distance. ISSA not only attains shorter path lengths and fewer turnings in both single-target and multi-target point path planning but also exhibits better performance as the obstacle rate rises, which is an improvement over existing methods where the performance may deteriorate in complex environments.

4.2. Implications of the Research Results

The results of this research have important implications in both practical applications and theoretical research. In practical application scenarios, such as automated material transport robots in industrial production and automatic guided vehicles in logistics warehousing, the better paths planned by the ISSA algorithm can be directly converted into lower energy consumption and higher work efficiency. This helps enterprises optimize resource allocation strategies, reduce operating costs, and improve production benefits, thereby enhancing market competitiveness. For example, in a large warehouse, robots using the ISSA algorithm to plan paths can reduce unnecessary travel distances and turns, save time and energy, and improve the efficiency of goods handling.

From the perspective of theoretical research, the success of the ISSA algorithm provides new ideas and methods in the field of swarm intelligence optimization algorithms. It verifies the effectiveness of multi-strategy collaborative optimization (such as introducing one-dimensional composite chaotic mapping, dung beetle dancing search behavior strategy, and adaptive t-variation improvement strategy) in improving algorithm performance. This will inspire subsequent research to further explore the impact of different strategy combinations on optimization algorithms, promote the development of this field in a more efficient and intelligent direction, and provide theoretical support for solving more complex practical problems.

5. Conclusions

This study successfully proposes an improved sparrow search algorithm (ISSA) based on multi-strategy collaborative optimization. By introducing the one-dimensional composite chaotic mapping SPM, the dung beetle dancing search behavior strategy, and the adaptive t-distribution variation improvement strategy, it effectively overcomes the shortcomings of limited population diversity and a tendency to converge prematurely to local optima of the original sparrow search algorithm (SSA). Through 23 benchmark test function experiments and application comparisons in robot path planning, it can be seen that ISSA exhibits excellent performance in convergence speed and optimization accuracy, and is significantly superior to SSA in both single-target point and multi-target point path planning. Especially in the situation where the obstacle rate increases in multi-target point path planning, its advantage is more prominent.

From a practical perspective, in many robot application scenarios, such as material transport robots in the automated manufacturing process in industrial production and automatic guided vehicles in the logistics warehousing link, the better paths planned by the ISSA algorithm can be directly transformed into lower energy consumption and higher work efficiency. This achievement will strongly assist enterprises in optimizing resource allocation strategies, reducing operating costs, and improving production benefits, thereby enhancing the competitiveness of enterprises in the market.

For future decision-making in related fields, the results of this study provide extremely valuable reference bases. It will guide decision-makers to preferentially consider adopting efficient algorithms such as ISSA when facing path planning needs, thereby promoting the continuous development of the entire industry at the technical application level, prompting the evolution of relevant technical standards and specifications towards a more optimized and efficient direction, and providing solid theoretical and practical support for the precise path planning of robot technology in more complex environments.