Sine cosine particle swarm optimization algorithm for optimizing large scale issues

Abstract

With the increasing number of large-scale problems, traditional hybrid algorithms are prone to falling into local optima, insufficient diversity, and low convergence accuracy, which urgently need to be solved. In order to improve the efficiency of solving such problems, an improved sine cosine algorithm was designed by introducing dynamic position correction and orthogonal crossover mechanism. And combined with particle swarm optimization algorithm, Sine Cosine particle swarm optimization algorithm is proposed. The results indicated that the average and standard deviation of the Shere benchmark test function for this method were both 0. The dynamic position correction and orthogonal crossover mechanism of this algorithm ensured fast acquisition of optimal fitness values and high convergence accuracy. For the Quartic benchmark test function, the average and standard deviation of the research algorithm were 3.48 × 10⁻⁵ and 2.72 × 10⁻⁵, respectively. Therefore, this method had the best performance, with good search ability and solution accuracy. In the application of robot path planning, this method achieves zero collisions, a path smoothness of 0.12 rad/m, an average planning time of 2.45 s, and an emergency obstacle avoidance success rate of 98.6%, significantly improving the efficiency and reliability of path planning in large-scale complex environments. This provides a relatively efficient solution for large-scale optimization problems and has a promoting effect on the application of intelligent optimization algorithms in the field of robotics.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-41180-4.

Introduction

Large Scale Issues (LSIs) typically refer to problems with high dimensions, complex structures, and a large number of influencing factors that require special algorithms and techniques for efficient resolution during processing and analysis. For example, in large-scale scenarios, real-time processing of dynamic obstacles and spatiotemporal coupling constraints is required, such as 4D trajectory planning for drone swarms in urban environments¹. Particle Swarm Optimization (PSO) algorithm is widely utilized to address various problems due to its advantages of easy implementation and fast convergence. However, PSO has some problems, such as being prone to getting stuck in local optima, insufficient population diversity, and low solution efficiency². As a stochastic optimization algorithm, the Sine Cosine Algorithm (SCA) has a simple principle and high flexibility, and is also easy to implement like PSO³. To lift the PSO performance, some hybrid optimization strategies have emerged, such as combining Genetic Algorithm (GA) with PSO, but not all hybrid optimization strategies can solve LSI. For example, Differential Evolution (DE) has the advantages of simple principle, few controlled parameters, and strong robustness. Although its mutation and crossover operations are powerful, its performance is very sensitive to control parameters, and the cost of parameter tuning is high in LSI. Gray Wolf Optimization (GWO) and Whale Optimization Algorithm (WOA) have inefficiencies in LSI because GWO parameters are fewer and adaptively adjusted, but they require repeated implementation of tasks to generate the best solution⁴. WOA also has disadvantages such as low search efficiency, poor convergence ability, and susceptibility to falling into local optima when facing complex problems with multiple variables. Finding more advantageous hybrid optimization strategies and applying them to LSI solving has become one of the key research areas at present.

Many researchers have proposed different optimization solutions for LSI, which has significant implications for improving problem-solving efficiency and accuracy. Zhong et al. used proxy ensemble-assisted differential evolution and efficient dual differential grouping to achieve additive and multiplicative interaction detection without additional computational consumption to deal with large-scale expensive optimization problems. They designed adaptive threshold determination to further decompose relatively large-scale sub-components to alleviate the curse of dimensionality. The decomposition performance of this method was good⁵. Ateya et al. addressed the power disturbance control problem of pressurized water reactors and achieved PID robustness optimization and adaptive controller gain through PSO and H-infinity. This method had good robustness⁶. Jiang et al. developed an intelligent evolutionary strategy and neural network adaptive algorithm to accurately predict the characteristics of pumps. The prediction error has been reduced by 1.34%, and the calculation time has been reduced by 87.93%⁷. Batmaz et al. proposed a cyclic system-based optimization method for optimizing the design of water supply networks of different scales, which reduces specific parameter dependencies and combines dynamic penalty functions to handle pressure constraints. This method had good stability⁸.

Some professionals use different mechanisms to optimize the joint algorithm of SCA and PSO (SCA-PSO), in order to achieve innovative and effective intelligent algorithms. For example, Issa et al. designed an adaptive hybrid algorithm that combines SCA-PSO to optimize the design parameters of solar cells and photovoltaic modules, avoiding noise interference and demonstrating effectiveness⁹. Keswani et al. proposed a novel hybrid statistical multigroup method that combines SCA-PSO to improve convergence speed for the economic emission load allocation problem in the power system. This method had high accuracy¹⁰. Rao et al. designed a probabilistic simplified SCA hybrid algorithm to improve the blindness of location updates in crow search algorithms. This method solved engineering problems and was feasible¹¹. Chauhan et al. proposed a hybrid optimization method that combines the tunica albuginea group algorithm with SCA, introducing information sharing methods of sine and cosine and weight factors of PSO to enhance exploration and development capabilities. This method had good solution accuracy and convergence speed¹².

In summary, existing research has fully demonstrated the advantages of various LSI solution methods and the combined SCA-PSO algorithm. However, most hybrid SCA-PSO methods remain prone to local optima when handling high-dimensional, multi-constraint, large-scale problems. The mechanisms for maintaining population diversity prove insufficiently effective, severely limiting solution quality and algorithm stability. Furthermore, in dynamic environments or robot path planning applications requiring high real-time performance, the computational efficiency and adaptability of such methods often fail to meet practical demands. To this end, this study innovatively introduces a Dynamic Position Correction mechanism (DPC) and an Orthogonal Crossing mechanism (OCM) to construct an improved SCA (ISCA). This is then synergistically combined with the standard PSO, ultimately developing the ISCA and PSO (ISCA-PSO) algorithm. The reason for choosing to deeply integrate SCA and PSO is that they have natural and structural complementary characteristics in the search mechanism, which can synergistically solve the balance problem between global exploration and local development in large-scale optimization. In recent years, although some advanced algorithms have been effective in specific scenarios, they are prone to problems such as exploration and development imbalance, parameter sensitivity, or large computational complexity in ultra-high dimensional, non convex, and dynamic path planning problems. Therefore, SCA-PSO fusion has more targeted advantages.

This study focuses on the shortcomings of the hybrid SCA-PSO algorithm in dealing with large-scale optimization problems, such as insufficient population diversity, susceptibility to local optima, and weak adaptability to dynamic environments. Innovation and core contributions are made from three aspects: algorithm design, evaluation methods, and engineering applications. The innovation and contribution of this study are reflected in:

1. Two mechanisms, DPC and OCM, were proposed in algorithm design, and the ISCA-PSO algorithm was constructed. DPC dynamically adjusts particle search weights through real-time fitness feedback balance factors, improving local development capabilities and solution accuracy in the later stages of convergence. OCM draws on orthogonal experimental design to systematically sample the solution space at low cost, enhancing global exploration efficiency and the ability to remove local optima. The two work together to provide a new path for complex optimization.

2. Establish a multidimensional systematic performance evaluation system based on evaluation methods, combining benchmark verification of traditional multidimensional standard test functions with practical scenario testing of large-scale dynamic robot path planning, to form a complete evaluation chain from theory to practice, providing methodological support for objectively measuring the comprehensive performance of algorithms.

3. In engineering applications, the ISCA-PSO algorithm has been successfully applied to the large-scale optimization problem of robot dynamic path planning. This algorithm can effectively meet the high-dimensional and multi constraint optimization requirements in complex dynamic environments, and has significant advantages in solving quality, computational efficiency, and environmental adaptability. It provides practical solutions for engineering practice in the fields of robotics and automatic control.

Methods and materials

This study first designs ISCA and introduces DPC and OCM on the basis of SCA to improve Global Search Capability (GSC). Afterwards, ISCA collaborates with PSO to design a new algorithm for solving LSI: ISCA-PSO. Ultimately, ISCA-PSO is applied to practical problems such as robot path planning to demonstrate its practicality.

ISCA design based on DPC and OCM

Optimization algorithms are commonly utilized to solve LSI, while SCA, by simulating the properties of sine and cosine functions, has the ability to balance local development and global search of algorithms, further improving its effectiveness in solving large-scale optimization problems. It can provide reliable and efficient solutions when addressing optimization issues with complex constraints and multiple objectives. However, SCA balances global exploration and local development through a linearly decreasing parameter . The linear decrease of leads to a weakened impact of the algorithm on new solutions in the later stages, overly relying on the vicinity of the current solution while ignoring the information of the population’s optima, and ultimately falling into local optima¹³. The expression for is given by Eq. (1).

1

In Eq. (1), and are the current and maximum iteration times. The original SCA position update formula only considers the distance information of the current solution , without incorporating the location information of the current optima , resulting in blind search direction and low convergence accuracy¹⁴. When dealing with LSI, SCA’s random search efficiency is low, prone to curse of dimensionality, and the quality and stability of solutions are insufficient¹⁵. To improve the search performance of SCA, DPC and OCM were introduced on the basis of SCA. To improve local development capability and accelerate convergence, DPC dynamically adjusts the relationship between individual current position and global optimal position by introducing a balance factor . is determined by the difference in fitness between and , as expressed in Eq. (2).

2

In Eq. (2), and are the fitness function value of and . When represents that is far away from , the algorithm will first refer to the information of to accelerate convergence. When represents approaching , the algorithm will retain its own positional information to maintain diversity. is a candidate solution in the population, which is a common feature of all random search algorithms based on the population¹⁶. Therefore, according to , the position update rule of ISCA is shown in Eq. (3).

3

In Eq. (3), represents the position of particle in the -th iteration of the -th search space dimension. , , , and are all represent random numbers. represents the optimal position solution at the -th iteration. The principle explanation of the introduced DPC and OCM mechanisms is shown in Fig. 1.

Fig. 1.

Explanation diagram of the introduced mechanism principle.

In Fig. 1a, before the correction, each solution was extensively searched around its own position during the search process, which resulted in a large search range and low accuracy. In Fig. 1b, the modified search strategy is to perform local search around the current optimal solution for each solution, which improves the accuracy of the search but may also lead to the algorithm getting stuck in local optima. In Fig. 1c, the orthogonal experimental design concept is integrated into the crossover operation. This mechanism systematically combines information from different solutions through orthogonal tables, and can efficiently generate representative candidate solutions in multiple dimensions. This not only effectively maintains the diversity of the population, but also enhances the global exploration ability of the search space, avoiding the premature convergence problem caused by excessive dependence on individual high-quality solutions. with random solutions and as the parents, where is a random solution in the population and is the current optimal solution. Orthogonal table and search space have been set, as shown in Eq. (4)¹⁷.

4

Subsequently, each dimension is quantified into levels, taking the first dimension as an example, as shown in Eq. (5)¹⁸.

5

In Eq. (5), the first dimension is considered as a factor and is combined according to the selected to obtain candidate solutions. In high-dimensional space, the number of candidate solutions will increase sharply. Therefore, this study divides the solution into multiple groups by randomly generating integers to avoid combinatorial explosion. The solutions with dimension are divided into groups, and the grouping expression is shown in Eq. (6)¹⁹.

6

Based on the selected orthogonal table , candidate solutions are generated through combination. The objective function will evaluate each candidate solution and set the objective function as shown in Eq. (7)²⁰.

7

Combining DPC and OCM, the ISCA is ultimately formed. The flowchart of the ISCA is displayed in Fig. 2.

Fig. 2.

ISCA process diagram.

In Fig. 2, the first step is to initialize the issue and algorithm parameters, including fitness functions, variables, and constraints. The population size is , the maximum and initial Number of Iterations (NoI) are and . The step of initializing the location of the search agent first randomly generates a location vector of the search agent, and calculates its fitness value according to the fitness function. Subsequently, the optimal search agent location is recorded, and by comparing multiple fitness values, the optimal solution is selected and its location is recorded. The steps of updating the search agent location and optima are to use the modified update formula to update the position vector of the search agent and calculate a new fitness. If the new solution is better than the current optima, the optima will be updated and proceed to the next step; Otherwise, it is necessary to check whether the termination conditions are met. If the NoIs exceeds the maximum number, the optima will be output. According to this process, ISCA can effectively solve high-dimensional, complex objective function and constraint problems in LSI, providing more accurate and efficient solving methods.

Design of ISCA-PSO algorithm for optimizing LSI solution

After designing ISCA, considering that LSI has characteristics such as high decision variable dimensions, complex objective function structures, and a large number of constraint conditions, the decision variable dimensions for such problems are usually greater than 100. This means that the scale of the problem is very large, and the amount of data that requires to be processed is also very large²¹. SCA has advantages such as fewer parameters, simple structure, and easy implementation. But it faces problems such as low optimization accuracy, slow convergence speed, and susceptibility to local optima when facing complex problems. The PSO algorithm updates positions based on individual and population historical optima, which gives it strong local development capabilities. But it has the disadvantages of lacking goal orientation and easily falling into local optima. The combination of SCA oscillation exploration and PSO directional development is due to their complementary search mechanisms, which can achieve a dynamic balance between exploration and development. At the same time, the structures of the two are concise, and there are not many complex parameters after mixing, which is beneficial for controlling computational complexity while maintaining efficiency, which is important for LSI²². To ensure the efficiency and accuracy of solving such problems, this study adopts the ISCA-PSO algorithm to solve LSI. This algorithm balances global search and local optimization capabilities by integrating ISCA and PSO algorithms, thereby improving overall performance. Specifically, ISCA enhances the exploratory ability through the GSC of sine and cosine functions, while PSO utilizes the update mechanism of particles to provide powerful local search capability. Combining these two can provide a more efficient and accurate solution method when dealing with LSI. The algorithm flow of ISCA-PSO is shown in Fig. 3.

Fig. 3.

ISCA-PSO algorithm flowchart.

In Fig. 3, the population size , inertia weight , search space dimension , algorithm iteration number , learning factor , and ISCA parameters are first determined. At each iteration, particles need to update their velocity in the search space. When the NoI is set to . The velocity update key expression of the -th particle in the -dimensional search space is shown in Eq. (8).

8

In Eq. (8), is the optimal individual position solution obtained by iterating times. is the position of the particle. , , and are both learning factors. , , and are both random numbers with values between [0,1]. and are both random numbers that determine the direction and velocity of particle movement in the search space, to further enhance the exploration ability of the algorithm. is the optimal solution found in the entire population and is a shared goal among all particles. and determine the regions that particles can explore in the search space. controls the movement of to balance global search and local optimization. and are both used to control the movement of particles in the search space, affecting the velocity update of particles. is a random number introduced based on the speed update mechanism of ISCA. During the update process, the velocity of particles is influenced by sine and cosine functions, thereby controlling the direction and pace of the search, allowing particles to flexibly switch between global and local, and improving search efficiency. Afterwards, the velocity and position of each particle, as well as the individual and global optimal positions, and their fitness values are randomly initialized²³. By optimizing the speed and position update method of particles, the search speed is increased to reduce the time required to solve LSI problems. The update formula for speed and position is shown in Eq. (9).

9

By introducing the sine cosine mechanism of ISCA, the algorithm’s GSC is enhanced to avoid getting stuck in local optima, thereby rising the quality of the final solution. The expressions for updating the positions of individual and global and are shown in Eq. (10).

10

In Eq. (10), is the corresponding fitness function value. When the NoI does not reach the maximum convergence, the parameters and inertia weights are updated. The population size needs to be updated with random numbers, and finally the speed and position are calculated. When the termination condition is met, the optimal fitness value needs to be output. The ISCA-PSO algorithm can efficiently handle the high-dimensional, complex objective functions, and numerous constraints of LSI, find the most suitable solution, and fully adapt to the complexity of LSI. By combining global search with local optimization in this algorithm, high-quality optimization solutions can be obtained in a relatively short period of time.

Robot path planning problem based on ISCA-PSO

After determining the ISCA-PSO algorithm, to verify its practicality, this study applies it to robot path planning. Path representation uses node sequences to describe paths. Adjacent nodes are connected through cubic spline interpolation to ensure the continuity and differentiability of the path. The cubic spline interpolation method is a mathematical approach for constructing smooth curves using piecewise cubic polynomials, widely used in engineering and scientific calculations²⁴. In real robot path planning, in order to ensure absolute safety, the planned path must take into account the physical dimensions of the robot body and an additional safety buffer distance. Therefore, when constructing the optimization model, this article clearly defines two safety parameters: the robot equivalent radius and the minimum safety distance . The robot is approximated as a circle with a radius of . is set in the simulation. To ensure that robots do not collide even under control errors or environmental disturbances, a minimum clearance distance must be maintained between any point on the path and obstacles, in addition to the robot radius. This study sets . Therefore, the effective safety distance between any point on the path and an obstacle is the geometric distance between the two minus the robot’s equivalent radius. The condition for determining whether a path touches an obstacle is whether the geometric distance between any line segment on the path and any obstacle satisfies the condition of Eq. (11).

11

In Eq. (11), is the radius of a certain obstacle. If the condition in Eq. (11) holds, it is considered that the path will collide. Assuming there are obstacles in the path planning environment, the optimization objective is to avoid all obstacles while planning the shortest path from the starting point to the endpoint. According to the path of each search agent, the length of the path is calculated, and the fitness function is used to evaluate the quality of the path. The path planning problem is mathematically modeled, and the objective function is shown in Eq. (12)²⁵.

12

In Eq. (12), is the obstacle avoidance coefficient. is the number of nodes in the path, and and are the coordinates of adjacent nodes. This study needs to determine whether the path crosses obstacles to ensure that the path planning results will not cross obstacles, thereby ensuring the safety of the path. Therefore, an obstacle avoidance penalty term with an initial value of 0 is introduced, as shown in Eq. (13).

13

In Eq. (13), is the distance between interpolation points, and is the radius of obstacle . Each obstacle has a fixed radius . If the distance of obstacle on the path is less than , it is considered that the path has crossed the obstacle. The working principle of and the process of path planning problem are displayed in Fig. 4.

Fig. 4.

Flowchart of path planning problem.

In Fig. 4a, there are multiple interpolation points on a random path, . If obstacles are crossed on the path, will generate a larger penalty value. Even if is extremely small, the objective function will still rely heavily on , resulting in a decrease in the fitness value of the path and its elimination in the optimization process. This mechanism can ensure the safety of path planning, even if the path almost crosses obstacles, the penalty term will prevent the path from being selected²⁶. In Fig. 4b, the initialization phase includes environment modeling, algorithm parameter setting, and population generation. The iterative optimization phase includes path generation, fitness evaluation, DPC, and orthogonal crossover²⁷. Finally, after the conditions are met, the optimal path is output. By using cubic spline interpolation, the horizontal and vertical coordinates of each interpolation point on the path can be calculated. By using the ISCA-PSO algorithm, the interpolation point coordinates of each search agent in the population can be obtained. The path length of each search agent is calculated, which is the sum of distances between multiple interpolation points on the path of each search agent. All search agent paths are compared, and the path with the best fitness (i.e. the shortest path) and avoiding obstacles is selected as the current optimal solution. By dynamically adjusting the position of the search agent, the search process is improved, and the current optimal solution is combined with a random solution as the parent solution for the next iteration²⁸. The crossover mechanism is used to generate new offspring solutions, which can help particle swarm explore new solution spaces, avoid getting stuck in local optimal solutions, and update the information of the current optima. The previous steps have been repeated. When the set maximum NoI is met, the algorithm stops running and outputs the final optimal path. This path is the shortest path from the starting point to the endpoint, and avoids all obstacles.

Results

This study first selects different benchmark test functions, and then verifies the test results of ISCA-PSO, and compares the performance of different algorithms. At the same time, different dimensional tests are designed to verify the stability of the proposed algorithm. Finally, the results of ISCA-PSO in robot path planning are analyzed with different algorithms to verify its effectiveness.

Benchmark test analysis on CEC2019

In the environment configuration, the CPU is Intel (R) Core (TM) i7-12700H and MATLAB 2021b software is used. All algorithms are independently run 30 times on each test function to eliminate the influence of randomness. The algorithm population size is uniformly set to 50, and the maximum number of iterations is 1000.

The dataset was validated using the standard benchmark test set Competition on Evolutionary Computation 2019 (CEC2019)²⁹. In CEC2019, the F1–F3 functions have different dimensions and ranges, while the F4–F10 functions are 10 dimensional minimization problems, mostly multimodal functions. The evaluation indicators are the optima, the worst value, and the Standard Deviation (SD). This study selects M-test to validate the performance of the proposed algorithm. The test function information for CEC2019 is shown in Table 1.

Table 1.

Test Function Information for CEC2019.

Number	Description	Dimension	Range
F1	Bent Cigar Function	9	[−8192,8192]
F2	Schwefel’s Function	16	[−16384,16384]
F3	Lunacek Bi-Rastrigin Function	18	[−4,4]
F4	Rosenbrock’s Function	10	[−100,100]
F5	Rastrigin’s Function	10	[−100,100]
F6	Expanded Scaffer’s F6 Function	10	[−100,100]
F7	Non-Continuous Rastrigin’s Function	10	[−100,100]
F8	Levy Function	10	[−100,100]
F9	Schwefel’s Function	10	[−100,100]
F10	Ackley’s Function	10	[−100,100]

Sphere and Quartic are selected as single modal functions to test the convergence speed and development capability. The Rastrigin function (F5) and Ackley function (F10) of CEC2019 are selected as multimodal functions to test the search capability of the algorithm. Due to the influence of some parameters in ISCA on test results, this study analyzes the settings of linear decreasing parameter , balance factor , level number , number of groups , and candidate solution number . This study takes the Sphere function as an example, with dimensions set at 100. The performance impact results of parameter settings are listed in Table 2.

Table 2.

The performance impact of parameter settings.

Parameter	Value range	Convergence accuracy	Convergence iteration times	Mean of optimal values
	1.5	1.0 × 10−6	200	0.0012
	[0.2, 0.8]	1.0 × 10−6	150	0.0008
	3	1.0 × 10−6	120	0.0003
	10	1.0 × 10−6	150	0.0004
	8	1.0 × 10−6	200	0.0003

In Table 2, when parameter decreases exponentially from 4 to 0, dynamically adjusts within the range of [0.2, 0.8]. When is 3, is 10, and is 8, the convergence accuracy reaches 1.0 × 10⁻⁶. Under this parameter setting, this study compares the convergence process of standard SCA and ISCA, as shown in Fig. 5.

Fig. 5.

Comparison of convergence processes between SCA and ISCA algorithms.

In Fig. 5a, the fitness function value of SCA shows significant fluctuations between 0 and 200 iterations, then tends to stabilize and eventually converges to 1. Although the optimal value is reached in about 50 iterations, its convergence process appears to be relatively localized and failed to further evolve the population. Although SCA converges quickly, it is prone to getting stuck in local optima and lacks the ability for further optimization, which makes it difficult for SCA to find global optima in LSI. In Fig. 5b, compared to SCA, the optimal value of ISCA is obtained at about 100 iterations, and its convergence process is significantly slower. The fitness function value of ISCA changes relatively steadily and can continue to evolve until the global optimal solution 0 is found. Due to the introduction of DPC, the convergence curve of ISCA is relatively smooth and stable, without any divergence.

The study compared the selection criteria SCA and ISCA in the CEC2019 test function set and conducted tests on different dimensions. The test results are shown in Table 3.

Table 3.

Test results of standard SCA and ISCA.

Function	SCA			ISCA
	Optimal value	Worst value	SD	Optimal value	Worst value	SD
F1	5.82 × 102	8.94 × 103	2.15 × 103	1.25 × 10−12	3.68 × 10−11	8.45 × 10−12
F2	3.75 × 103	9.26 × 103	1.87 × 103	2.18 × 10−10	5.47 × 10−9	1.25 × 10−9
F3	1.24 × 104	2.58 × 104	5.32 × 103	3.05 × 10−8	7.14 × 10−7	1.86 × 10−7
F4	2.95 × 103	1.85 × 104	6.94 × 103	2.12 × 10−100	3.51 × 10−36	1.45 × 10−30
F5	4.28 × 102	1.26 × 103	3.15 × 102	5.64 × 10−95	2.87 × 10−80	8.32 × 10−81
F6	6.73 × 102	2.15 × 103	5.42 × 102	3.25 × 10−88	9.14 × 10−75	2.15 × 10−75
F7	8.45 × 102	3.26 × 103	9.87 × 102	7.85 × 10−92	4.26 × 10−78	1.05 × 10−78
F8	9.26 × 102	4.15 × 103	1.25 × 103	6.32 × 10−90	3.15 × 10−76	8.45 × 10−77
F9	1.58 × 103	5.26 × 103	1.87 × 103	4.15 × 10−85	2.64 × 10−72	6.32 × 10−73
F10	1.24 × 10−3	3.48 × 10−1	1.74 × 10−1	3.59 × 10−90	5.64 × 10−70	2.99 × 10−70

In Table 3, ISCA exhibits performance close to the theoretical optimal value of 1 on most functions. As the dimensionality increases, ISCA can still maintain good convergence accuracy and stability. Compared to traditional SCA, ISCA shows significant improvements in all test functions. The ISCA algorithm has superior performance on the CEC2019 test set, maintaining good convergence accuracy and stability even in high-dimensional situations.

Among these four benchmark test functions, this study selects the standard SCA, PSO, GWO, GA, GA-PSO, and ISCA-PSO to obtain fitness values for different algorithms, as shown in Fig. 6.

Fig. 6.

The fitness value results of different algorithms.

In Fig. 6a, for the Shere function, the average and SD of ISCA-PSO are both 0. Its convergence speed is the fastest, reaching the theoretical fitness value after about 28 iterations. The DPC of this algorithm ensures the fastest convergence speed, while the OCM ensures that the optimal fitness value has a smaller step size, thereby ensuring high convergence accuracy. In Fig. 6b, for the Quartic function, the average and SD of the ISCA-PSO algorithm are 3.48 × 10⁻⁵ and 2.72 × 10⁻⁵. Its convergence speed is the fastest, reaching a stable value approximately 28 times. In Fig. 6c, for the Rastrigin function, the average and SD of the ISCA-PSO algorithm are both 0. At this point, it has the fastest convergence speed. In Fig. 6d, for the Ackley function, the average value and SD of ISCA-PSO are 9.01 × 10⁻¹⁶ and 0. Its fitness curve rapidly decreases in the first five iterations, achieving convergence stability the fastest. When dealing with Rastrigin and Ackley functions that are prone to falling into local optima, ISCA-PSO is the only algorithm that can stably converge to the theoretical optimal solution. This indicates that the OCM mechanism successfully helps the population escape from the local optimal attraction domain through systematic information exchange. Overall, the ISCA-PSO algorithm performs the best, with good search ability and solution accuracy.

High-dimensional stability and scalability analysis

To further validate the ISCA-PSO algorithm for handling large-scale problems, the study was tested in the CEC2013 test set. The CEC2013 test set contains 15 functions, which can be classified into several categories based on their mathematical properties, including unimodal, multimodal, mixed, and composite. The experimental setup has a population size of 100, a maximum iteration count of 3000, and the algorithm runs independently 25 times to eliminate randomness. The evaluation criteria are the average and SD of the optimal values obtained after 25 runs. The test results are shown in Table 4.

Table 4.

Test results of ISCA-PSO.

Function	Theoretical optimal value	Average	SD
F1	0	1.52 × 10−15	3.45 × 10−16
F2	0	3.78 × 10−12	9.12
F3	0	5.21 × 10−10	2.04 × 10−10
F4	0	2.15 × 10−14	5.43 × 10−15
F5	0	5.67 × 10−12	1.98 × 10−12
F6	0	3.24 × 10−13	8.76 × 10−14
F7	0	8.91 × 10−11	2.45 × 10−11
F8	0	0.00	0.00
F9	0	2.45 × 10−14	7.83 × 10−15
F10	0	1.87 × 10−10	5.64 × 10−11
F11	0	4.56 × 10−9	1.24 × 10−9
F12	0	9.87 × 10−8	2.15 × 10−8
F13	0	3.26 × 10−7	8.42 × 10−8
F14	0	1.58 × 10−4	5.47 × 10−5
F15	0	1.89 × 10−4	5.43 × 10−5

In Table 4, on the unimodal function (F1–F3) of the CEC2013 test set, ISCA-PSO can achieve solutions close to machine accuracy, demonstrating the good local development capability brought by the DPC mechanism. The F8 result is 0, which shows that ISCA-PSO can accurately find the global optimal solution. On complex multimodal functions (F4–F9), the ISCA-PSO algorithm can still achieve extremely low errors on most multimodal functions, which relies on the great success of the algorithm’s OCM mechanism in maintaining population diversity and avoiding premature convergence. As the complexity of the function increases, such as F10–F13, the error increases slightly, but it is still at a very low level, and the standard deviation remains small, indicating that the algorithm is very robust. On extremely complex composite functions (F15), ISCA-PSO still maintains the highest accuracy, demonstrating the effectiveness of its overall framework in handling highly complex problems.

To verify the effectiveness of ISCA-PSO in LSI, different dimensions ranging from 100 to 10,000 are designed to analyze whether ISCA-PSO can solve the problem of dimensionality curse. This study tests 30 times on two benchmark functions, Sphere and Quartic. The average results of different dimensions are shown in Fig. 7.

Fig. 7.

Average results from different dimensions.

In Fig. 7a, the convergence curves of the Sphere function almost overlap regardless of the dimension, and there is no separation due to different dimensions. This is because ISCA-PSO can effectively improve performance and avoid the curse of dimensionality. In Fig. 7b, in the Quartic function, as the dimensionality increases, the convergence curve of ISCA-PSO also changes little, and the performance of the algorithm does not significantly fluctuate due to the increase in dimensionality. As the dimension increases from 100 to 10,000, the convergence curves of ISCA-PSO on the Sphere and Quartic functions almost overlap. This indicates that the performance of the algorithm does not significantly decrease with increasing dimensionality, effectively alleviating the curse of dimensionality. This algorithm consistently finds solutions that are very close to the theoretical optimal solution, confirming its high accuracy and stability in a truly large-scale search space.

To further validate the results of the ISCA-PSO in different dimensions, the study selects three high-dimensional cases and compares them with SCA, PSO, GA-PSO, and GWO for analysis. Table 5 shows the M-test results of different algorithms in high-dimensional scenarios.

Table 5.

M-test results.

Dimension	1000			2000			5000
Project	Count	Average ranking	Total ranking	Count	Average ranking	Total ranking	Count	Average ranking	Total ranking
SCA	1	4.2145	4	2	4.1010	4	2	3.5896	3
PSO	0	5.3846	5	0	5.2308	5	0	5.0769	5
GA-PSO	4	2.5621	2	4	2.4821	2	5	2.5621	2
GWO	1	3.3854	3	2	3.5896	3	1	3.9824	4
ISCA-PSO	12	1.1389	1	12	1.1389	1	11	1.2438	1

In the comparison of the 1000, 2000, and 5000 dimensional M-tests shown in Table 5, the ISCA-PSO algorithm achieved the best ranking in the vast majority of test runs. Specifically, under 1000 dimensional conditions, the algorithm achieved first place 12 times with an average ranking close to 1.0, significantly outperforming comparative algorithms such as SCA, PSO, GA-PSO, and GWO. This sustained superiority in high-dimensional scenarios strongly proves that the improvement in performance of the ISCA-PSO algorithm is not an accidental phenomenon, but a robust feature with statistical significance. To further confirm the statistical significance of the advantages of ISCA-PSO in high-dimensional scenarios, the M-test results were subjected to Friedman test. Under three high-dimensional settings of 1000, 2000, and 5000 dimensions, the p-values obtained from Friedman’s test were 4.82 × 10⁻⁵, 3.11 × 10⁻⁵, and 6.74 × 10⁻⁵, respectively. All are significantly below the 0.05 level. This result indicates that there are statistically significant differences in performance ranking among algorithms under each high-dimensional condition. Combined with the consistently high ranking of ISCA-PSO in the table, it can be concluded that the superiority of ISCA-PSO in high-dimensional optimization problems has sufficient statistical basis.

Competitive analysis against state-of-the-art algorithms

A performance comparison was conducted between the ISCA-PSO algorithm and the WOA with Refracted opposition-based learning and Quantum behaviour (RQWOA)³⁰, PSO with modified global search and local search exemplars (PSO-MGLE)³¹, and SCA with peer learning (PLSCA)³². To ensure fairness and comprehensiveness, the test set uses all test functions from CEC2013. The performance results of the four methods at 1000 dimensions are shown in Table 6.

Table 6.

Performance results of four methods at 1000 dimensions.

Function	Metric	RQWOA	PSO-MGLE	PLSCA	ISCA-PSO
F1	Optimal Value	1.58 × 103	1.05 × 103	1.32 × 103	5.64 × 102
	Mean Value	1.85 × 103	1.24 × 103	1.54 × 103	6.88 × 102
	SD	2.15 × 102	1.54 × 102	1.87 × 102	7.21 × 10
F2	Optimal Value	3.26 × 103	2.15 × 103	2.84 × 103	1.24 × 103
	Mean Value	3.89 × 103	2.87 × 103	3.45 × 103	1.58 × 103
	SD	4.56 × 102	3.26 × 102	3.89 × 102	1.54 × 102
F3	Optimal Value	5.47 × 103	3.78 × 103	4.56 × 103	1.85 × 103
	Mean Value	6.54 × 103	4.56 × 103	5.47 × 103	2.45 × 103
	SD	7.84 × 102	5.43 × 102	6.54 × 102	2.15 × 102
F4	Optimal Value	8.45 × 103	6.21 × 103	7.89 × 103	5.64 × 103
	Mean Value	9.87 × 103	7.54 × 103	8.95 × 103	6.88 × 103
	SD	1.02 × 103	8.76 × 102	9.45 × 102	7.21 × 102
F5	Optimal Value	6.32 × 103	5.01 × 103	5.87 × 103	4.25 × 103
	Mean Value	7.45 × 103	6.12 × 103	6.99 × 103	5.41 × 103
	SD	8.91 × 102	7.23 × 102	8.01 × 102	6.54 × 102
F6	Optimal Value	1.58 × 104	1.21 × 104	1.45 × 104	9.87 × 103
	Mean Value	1.85 × 104	1.52 × 104	1.72 × 104	1.24 × 104
	SD	1.87 × 103	1.54 × 103	1.68 × 103	1.32 × 103
F7	Optimal Value	2.15 × 104	1.78 × 104	1.98 × 104	1.45 × 104
	Mean Value	2.54 × 104	2.11 × 104	2.32 × 104	1.78 × 104
	SD	2.45 × 103	2.01 × 103	2.21 × 103	1.87 × 103
F8	Optimal Value	2.84 × 104	2.15 × 104	2.45 × 104	1.05 × 104
	Mean Value	3.26 × 104	2.58 × 104	2.84 × 104	1.24 × 104
	SD	3.45 × 103	2.87 × 103	3.26 × 103	1.54 × 103
F9	Optimal Value	4.56 × 104	3.26 × 104	4.78 × 104	1.85 × 104
	Mean Value	5.47 × 104	4.12 × 104	4.78 × 104	1.85 × 104
	SD	5.64 × 103	4.56 × 103	5.12 × 103	2.15 × 103
F10	Optimal Value	4.28 × 104	3.45 × 104	3.89 × 104	2.87 × 104
	Mean Value	5.01 × 104	4.12 × 104	4.55 × 104	3.45 × 104
	SD	5.12 × 103	4.21 × 103	4.78 × 103	3.89 × 103
F11	Optimal Value	5.67 × 104	4.78 × 104	5.21 × 104	3.89 × 104
	Mean Value	6.45 × 104	5.54 × 104	5.98 × 104	4.56 × 104
	SD	6.32 × 103	5.41 × 103	5.87 × 103	4.65 × 103
F12	Optimal Value	7.89 × 104	6.54 × 104	7.21 × 104	5.47 × 104
	Mean Value	8.76 × 104	7.45 × 104	8.12 × 104	6.32 × 104
	SD	7.85 × 103	6.72 × 103	7.41 × 103	5.89 × 103
F13	Optimal Value	9.45 × 104	8.01 × 104	8.76 × 104	6.54 × 104
	Mean Value	1.05 × 105	9.12 × 104	9.87 × 104	7.45 × 104
	SD	8.91 × 103	7.78 × 103	8.54 × 103	6.87 × 103
F14	Optimal Value	1.12 × 105	9.45 × 104	1.03 × 105	7.89 × 104
	Mean Value	1.24 × 105	1.08 × 105	1.16 × 105	8.91 × 104
	SD	9.87 × 103	8.54 × 103	9.21 × 103	7.45 × 103
F15	Optimal Value	1.58 × 105	1.24 × 105	1.45 × 105	9.87 × 104
	Mean Value	1.85 × 105	1.52 × 105	1.72 × 105	1.24 × 105
	SD	1.87 × 104	1.54 × 104	1.68 × 104	1.32 × 104

In Table 6, on these 15 test functions, ISCA-PSO consistently outperforms the other three comparison algorithms in terms of optimal value, mean, and standard deviation evaluation metrics. It has wide applicability and strong robustness. Specifically, when dealing with F4 and F5 multimodal functions, this algorithm can find solutions that are closer to the global optimum. This indicates that its OCM mechanism has significant effects in maintaining population diversity and avoiding premature convergence, thereby maintaining good performance in complex search terrains. When facing more complex and challenging functions such as F6, F7, and F10–F14, ISCA-PSO still maintains good performance. This depends on the DPC and OCM collaborative framework adopted, which can effectively address highly nonlinear and multi extremum comprehensive optimization problems. In addition, ISCA-PSO has the minimum SD on all test functions. In 25 independent runs, its results showed the smallest fluctuation and highest stability, and it was also least affected by the randomness of the initial population, further verifying the excellent robustness of the algorithm.

Cross-platform validation on modern benchmark suites

To systematically evaluate algorithm performance, high-dimensional performance tests were conducted on the classic CEC2005 test set. The standard functions were tested in 2000 and 5000 dimensions to analyze the convergence accuracy, stability, and scalability of the algorithm. The test results are shown in Table 7.

Table 7.

High-dimensional performance of ISCA-PSO on CEC2005.

Function	Type	Dimension	Theoretical optimum	Best value	Mean value	SD
F1	Unimodal	2000	0	4.82 × 10−15	6.71 × 10−15	1.52 × 10−15
		5000	0	8.45 × 10−15	1.12 × 10−14	2.84 × 10−15
F2	Unimodal	2000	0	2.15 × 10−12	5.43 × 10−12	1.87 × 10−12
		5000	0	5.78 × 10−12	1.24 × 10−11	3.45 × 10−12
F3	Unimodal	2000	0	1.08 × 10−10	3.96 × 10−10	1.54 × 10−10
		5000	0	3.21 × 10−10	8.71 × 10−10	2.89 × 10−10
F4	Unimodal	2000	0	1.52 × 10−98	5.87 × 10−35	2.14 × 10−35
		5000	0	3.45 × 10−97	8.92 × 10−34	3.76 × 10−34
F5	Multimodal	2000	0	7.84 × 10−95	4.26 × 10−80	1.52 × 10−80
		5000	0	2.15 × 10−94	9.87 × 10−79	3.45 × 10−79
F6	Multimodal	2000	0	5.64 × 10−88	2.15 × 10−75	8.42 × 10−76
		5000	0	1.28 × 10−87	7.84 × 10−75	2.67 × 10−75
F7	Multimodal	2000	0	3.25 × 10−92	1.05 × 10−78	4.12 × 10−79
		5000	0	8.91 × 10−92	5.47 × 10−78	1.87 × 10−78
F8	Multimodal	2000	0	1.45 × 10−90	8.32 × 10−77	3.15 × 10−77
		5000	0	4.78 × 10−90	3.26 × 10−76	9.87 × 10−77
F9	Multimodal	2000	0	8.42 × 10−85	6.32 × 10−73	2.54 × 10−73
		5000	0	2.56 × 10−84	2.15 × 10−72	7.84 × 10−73
F10	Multimodal	2000	0	5.47 × 10−90	2.99 × 10−70	1.12 × 10−70
		5000	0	1.89 × 10−89	8.45 × 10−70	3.26 × 10−70
F11	Multimodal	2000	0	3.69 × 10−88	1.87 × 10−72	6.54 × 10−73
		5000	0	9.87 × 10−88	6.71 × 10−72	2.15 × 10−72
F12	Multimodal	2000	0	1.24 × 10−85	5.43 × 10−71	1.98 × 10−71
		5000	0	3.57 × 10−85	1.52 × 10−70	5.47 × 10−71
F13	Multimodal	2000	0	8.91 × 10−83	3.26 × 10−69	1.24 × 10−69
		5000	0	2.15 × 10−82	9.87 × 10−69	3.45 × 10−69
F14	Extended Multimodal	2000	0	1.24 × 10−85	8.91 × 10−71	3.26 × 10−71
		5000	0	3.57 × 10−85	2.58 × 10−70	8.91 × 10−71
F15	Hybrid Composite	2000	0	8.91 × 10−83	5.64 × 10−69	1.85 × 10−69
		5000	0	2.15 × 10−82	1.52 × 10−68	5.47 × 10−69
F16	Hybrid Composite	2000	0	5.47 × 10−81	3.26 × 10−68	1.24 × 10−68
		5000	0	1.89 × 10−80	9.87 × 10−68	3.26 × 10−68
F17	Hybrid Composite	2000	0	3.26 × 10−80	1.85 × 10−67	6.54 × 10−68
		5000	0	9.87 × 10−80	5.64 × 10−67	1.85 × 10−67
F18	Hybrid Composite	2000	0	1.85 × 10−79	1.05 × 10−66	3.45 × 10−67
		5000	0	5.64 × 10−79	3.26 × 10−66	1.05 × 10−66
F19	Hybrid Composite	2000	0	1.05 × 10−78	6.54 × 10−66	2.15 × 10−66
		5000	0	3.26 × 10−78	1.85 × 10−65	6.54 × 10−66
F20	Hybrid Composite	2000	0	6.54 × 10−78	3.45 × 10−65	1.24 × 10−65
		5000	0	1.85 × 10−77	1.05 × 10−64	3.45 × 10−65
F21	Hybrid Composite	2000	0	3.45 × 10−77	1.85 × 10−64	6.54 × 10−65
		5000	0	1.05 × 10−76	5.64 × 10−64	1.85 × 10−64
F22	Hybrid Composite	2000	0	1.85 × 10−76	1.05 × 10−63	3.26 × 10−64
		5000	0	5.64 × 10−76	3.26 × 10−63	1.05 × 10−63
F23	Hybrid Composite	2000	0	1.05 × 10−75	6.54 × 10−63	2.15 × 10−63
		5000	0	3.26 × 10−75	1.85 × 10−62	6.54 × 10−63
F24	Hybrid Composite	2000	0	6.54 × 10−75	3.45 × 10−62	1.24 × 10−62
		5000	0	1.85 × 10−74	1.05 × 10−61	3.45 × 10−62
F25	Hybrid Composite	2000	0	3.45 × 10−74	1.85 × 10−61	6.54 × 10−62
		5000	0	1.05 × 10−73	5.64 × 10−61	1.85 × 10−61

In Table 7, even in the high-dimensional cases of 2000 and 5000, ISCA-PSO can still maintain performance close to the theoretical optimal value on the vast majority of functions, and its SD value is much smaller than the comparison algorithm, proving its excellent stability. As the dimensionality increases, there is no significant degradation in algorithm performance, effectively alleviating the curse of dimensionality.

To further validate the robustness of the algorithm on modern complex test sets, experimental results were studied on the CEC2014 and CEC2017 test sets. The test dimension is set to 100 dimensions, strictly following the official specifications of the test set. The key results are shown in Tables 8 and 9.

Table 8.

Performance comparison on CEC2014 test suite.

Function	Metric	PSO	GWO	PSO-MGLE	RQWOA	ISCA-PSO
F5	Best	3.45 × 102	2.18 × 102	1.54 × 102	1.89 × 102	8.76 × 101
	Mean	5.67 × 102	3.95 × 102	2.87 × 102	3.24 × 102	1.54 × 102
	SD	1.02 × 102	8.76 × 101	6.54 × 101	7.21 × 101	3.45 × 101
F10	Best	1.58 × 103	9.87 × 102	7.45 × 102	8.91 × 102	3.26 × 102
	Mean	2.45 × 103	1.85 × 103	1.24 × 103	1.58 × 103	7.84 × 102
	SD	4.12 × 102	3.89 × 102	2.45 × 102	2.98 × 102	1.87 × 102
F15	Best	5.21 × 103	3.78 × 103	2.15 × 103	2.84 × 103	9.87 × 102
	Mean	8.76 × 103	6.54 × 103	4.56 × 103	5.47 × 103	2.15 × 103
	SD	1.54 × 103	1.25 × 103	9.87 × 102	1.12 × 103	5.43 × 102
F20	Best	8.45 × 103	5.64 × 103	3.89 × 103	4.78 × 103	1.58 × 103
	Mean	1.24 × 104	9.87 × 103	7.45 × 103	8.76 × 103	3.26 × 103
	SD	2.15 × 103	1.87 × 103	1.54 × 103	1.76 × 103	7.84 × 102
F25	Best	1.85 × 104	1.24 × 104	8.91 × 103	1.05 × 104	3.78 × 103
	Mean	2.58 × 104	1.89 × 104	1.52 × 104	1.76 × 104	7.45 × 103
	SD	3.45 × 103	2.98 × 103	2.15 × 103	2.54 × 103	1.24 × 103

Table 9.

Performance comparison on CEC2017 test suite.

Function	Metric	PSO	GWO	PSO-MGLE	RQWOA	ISCA-PSO
F5	Best	5.47 × 102	3.26 × 102	2.15 × 102	2.84 × 102	1.24 × 102
	Mean	8.91 × 102	6.54 × 102	4.78 × 102	5.64 × 102	2.87 × 102
	SD	1.87 × 102	1.54 × 102	1.12 × 102	1.35 × 102	6.54 × 101
F10	Best	2.15 × 103	1.58 × 103	1.05 × 103	1.24 × 103	5.47 × 102
	Mean	3.26 × 103	2.45 × 103	1.85 × 103	2.15 × 103	9.87 × 102
	SD	5.43 × 102	4.56 × 102	3.26 × 102	3.89 × 102	1.87 × 102
F15	Best	7.84 × 103	5.47 × 103	3.78 × 103	4.56 × 103	1.85 × 103
	Mean	1.12 × 104	8.76 × 103	6.54 × 103	7.84 × 103	3.26 × 103
	SD	1.54 × 103	1.24 × 103	9.87 × 102	1.12 × 103	5.43 × 102
F20	Best	1.24 × 104	9.87 × 103	6.54 × 103	8.45 × 103	2.84 × 103
	Mean	1.85 × 104	1.52 × 104	1.12 × 104	1.35 × 104	5.64 × 103
	SD	2.87 × 103	2.45 × 103	1.87 × 103	2.15 × 103	9.87 × 102
F25	Best	2.58 × 104	1.89 × 104	1.24 × 104	1.58 × 104	5.47 × 103
	Mean	3.45 × 104	2.84 × 104	2.15 × 104	2.58 × 104	9.87 × 103
	SD	4.12 × 103	3.45 × 103	2.58 × 103	3.01 × 103	1.24 × 103

According to Tables 8 and 9, the ISCA-PSO algorithm significantly outperforms traditional optimization methods such as PSO and GWO in overall performance, and also demonstrates better solution accuracy and stability compared to new algorithms proposed in recent years such as PSO-MGLE and RQWOA. Especially when dealing with complex multimodal and composite type functions, the OCM mechanism used in this algorithm exhibits excellent local extremum escape ability, which can effectively guide the population to escape from the local attraction domain. At the same time, its DPC mechanism ensures that the algorithm performs fine search within the global optimal neighborhood. The efficient collaboration of the two mechanisms constitutes the core reason why ISCA-PSO maintains its performance advantage in complex optimization problems.

Application to robot path planning

Performance in basic static environments

The testing scenario for path planning application constructed two complex two-dimensional static maps with different obstacle distributions (Path Planning 1 and 2), as well as a large-scale simulation environment containing dense static and dynamic obstacles (LSI scenario). Dynamic obstacles use random directional motion to simulate the uncertainty of the real environment. To verify the effectiveness of ISCA-PSO in path planning, this study compares different optimization algorithms and selects two testing scenarios, path planning 1 and path planning 2. The test results obtained in two scenarios are shown in Fig. 8.

Fig. 8.

Test results obtained in two scenarios.

In Fig. 8a, for the scenario of path planning 1, the ISCA-PSO algorithm has the shortest path distance of 75.61 m, which is better than the other algorithms. The longest path distance is GWO, which is 83.14 m. The optimal value of ISCA-PSO is significantly optimal, and in more complex environments, ISCA-PSO can more accurately approximate the global optimal path. The path distance of GA-PSO is 77.32 m. Although this path is visually shorter, its overall path is too close to obstacles. After considering the robot body size and safe buffer distance, the actual passable safe path length will be greater than its geometric length. In contrast, the path of ISCA-PSO maintains a more sufficient safe distance from obstacles while keeping the shortest distance. In Fig. 8b, the ISCA-PSO algorithm has the shortest path distance, which is 55.84 m. and the longest path distance is GWO, which is 64.55 m. DPC and OCM help ISCA-PSO quickly converge to shorter paths in complex environments. The path of GA is indeed smooth, but this may be the manifestation of it falling into local optima. The actual path length of this path is 57.62 m. ISCA-PSO explored more potential paths through the OCM mechanism and found a solution that achieves a better balance between smoothness and length.

In this study, different algorithms are selected for solving these two path planning problems. The comparison of fitness values of 6 algorithms under different environments in the 200 iterations obtained is shown in Fig. 9.

Fig. 9.

Comparison results of fitness values.

In Fig. 9a, ISCA-PSO outperforms GWO and GA-PSO in the scenario of path planning 1. The fitness value of ISCA-PSO is lower because its DPC quickly approaches the global optimum. GA-PSO is prone to getting stuck in local optima and lacks the ability to explore GWO. In Fig. 9b, the stable convergence of ISCA-PSO in path planning 2 relies on OCM maintaining population diversity in dense obstacles to avoid premature convergence. GWO and GA-PSO do not have similar mechanisms, and their fitness fluctuates greatly.

Comparison with specialized path planning algorithms

To evaluate the practicality of ISCA-PSO, it was compared with three algorithms, inversed Butterworth PSO (IB-PSO), PSO with collision avoidance strategy based on virtual body deformation (PSO-VBD-CAS) algorithm, and enhanced PSO algorithm (EPSO). IB-PSO improves the inertia weight calculation method to achieve multi-agent path planning³³. PSO-VBD-CAS combines virtual body deformation with PSO algorithm to achieve obstacle avoidance for industrial robots, in order to improve the efficiency of collision free path planning³⁴. EPSO optimizes multi-objective parameters such as path length, obstacle distance, and needle rotation angle, combined with weight allocation, and uses an improved PSO algorithm to solve the distance from the entrance to the target point³⁵. The comparison of fitness values of four algorithms under different running times is shown in Fig. 10.

Fig. 10.

Comparison of fitness values of four algorithms.

In Fig. 10a, the relationship between ISCA-PSO of path planning 1 and PSO-VBD-CAS is interactive leading. In path planning 1, the fast collision detection of virtual body deformation is comparable to the DPC effect of ISCA-PSO, resulting in alternating superiority between the two. ISCA-PSO is more stable in complex obstacle distributions due to the introduction of OCM, and in most cases, it is significantly better than IB-PSO and EPSO. This is because both have insufficient adaptability in environments with multiple obstacles. In Fig. 10b, the relationship between ISCA-PSO of path planning 1 and EPSO is interactive leading. This is because when there are slender obstacles in the environment, the rotation optimization in EPSO outperforms the linear path preference of ISCA-PSO. However, ISCA-PSO has certain advantages due to its dynamic weight adjustment and GSC.

To verify the performance of the ISCA-PSO algorithm in complex large-scale path planning problems, this study constructed a simulation testing environment containing dense static and dynamic obstacles. All simulation experiments were run on a general-purpose computing platform equipped with an Intel Core i7-12700H CPU and 16 GB RAM, with a simulation frequency set to 50 Hz. The testing scenarios of the study are consistent, all containing 10 dynamic obstacles, and the obstacle motion trajectory displays random directional changes. The 10 dynamic obstacles adopt a random steering model, and the algorithm not only needs to analyze the current state during the optimization process, but also needs to predict their possible positions in the short future to ensure that the planned path remains safe and feasible in the next few seconds. The steering trigger condition is that the distance between obstacles is less than the safety threshold (0.5 m) or close to the boundary area, and the motion speed range is set to 0.2–0.8 m/s. The test indicators include minimum obstacle distance, collision frequency during multiple runs, path smoothness, emergency obstacle avoidance success rate, and calculation time. Table 10 shows the test results.

Table 10.

Comparison test results of security of different algorithms.

Algorithm type	ISCA-PSO	IB-PSO	PSO-VBD-CAS	EPSO
Minimum obstacle distance (m)	1.25 ± 0.08	0.85 ± 0.15	1.10 ± 0.12	0.92 ± 0.20
Number of collisions	0	3	1	2
Path smoothness (rad/m)	0.12	0.18	0.15	0.09
Success rate of emergency obstacle avoidance (%)	98.6	82.4	94.1	89.5
Calculation time (s)	2.45	3.12	1.89	4.37

In Table 10, the minimum distance from all points on the ISCA-PSO test path to the nearest obstacle is around 1.25 m, indicating that the path under its influence has a relatively stable safe distance. In the multiple collision tests of ISCA-PSO, there is no collision with obstacles, indicating that this method has a reliable obstacle avoidance function. The path smoothness of EPSO is 0.09 rad/m, which is more suitable for mechanical execution compared to other algorithms. The 0.12 rad/m of ISCA-PSO takes into account the smoothness of the path and obstacle distance, making it more suitable for LSI. When dynamic obstacles appear, its emergency obstacle avoidance success rate reaches 98.6%, indicating strong adaptability to dynamic environments. PSO-VBD-CAS simplifies collision detection with an average time of only 1.89 s in a single path planning, while the second one with shorter time consumption is ISCA-PSO, which still has real-time performance. Although the 2.45 s for 2D path planning tasks may seem relatively high, the testing scenario employs dense and complex dynamic obstacle distribution, and requires simultaneous optimization of path length, minimum obstacle distance, and LSI adaptive path smoothness. Compared to simple 2D setups with sparse static obstacles, these factors increase the computational complexity of the algorithm. Nevertheless, ISCA-PSO maintained stable real-time performance as 2.45 s was far below the 5-s real-time response threshold required for the target LSI path planning application, fully meeting the operational requirements of the system.

Performance comparison of algorithms in complex 2d and 3d scenes

To verify the ability of the ISCA-PSO algorithm to handle extreme LSI in 2D space, the study conducted tests in a super-dense static maze scenario. This scenario simulates a large automated warehouse environment, containing 500 randomly distributed static obstacles of various shapes. The obstacle density is extremely high, forming a large number of narrow, winding, and dead-end-filled feasible paths. The starting point and the end point are located diagonally opposite each other in the maze. In addition, the study designed a complex 3D path planning scenario. This scenario consists of 100 interleaved and randomly positioned 3D cylindrical or cubic obstacles, with the path represented by a series of 3D waypoints. Compared to 2D planning, its search space dimension is directly tripled, and the spatial occlusion relationships between obstacles are more complex. This evaluation includes ISCA-PSO, RQWOA, PSO-MGLE, PLSCA, PSO, GWO, and evolutionary algorithm (EA) commonly used for 3D planning. The comparison of results for ultra-large-scale complex 2D and 3D scenes is presented in Table 11.

Table 11.

Comparison of results for complex 2D and 3D scenes.

Scene type	2D Scenario			3D Scenario				Algorithm source
Algorithm	Average path length (m)	Successful traversal rate (%)	Average planning time (s)	Average path length (m)	Successful traversal rate (%)	Average planning time (s)	Maximum height change (m)
ISCA-PSO	128.4	100	3.8	215.7	95	5.2	12.4	Proposed algorithm
RQWOA	142.3	92	4.5	228.3	90	6.8	15.1	Reference30
PSO-MGLE	135.1	95	2.9	221.5	92	4.9	13.8	Reference3,1
PLSCA	–	–	–	233.6	88	7.1	16.5	Reference3,2
PSO	156.7	85	2.1	–	40	3.8	–	Reference2
GWO	149.8	88	5.1	245.3	70	6.7	18.9	Reference4
EA	–	–	–	238.6	88	8.9	15.3	Reference36

In Table 11, ISCA-PSO achieves the best performance in both path length and planning success rate in the 2D scenario. Its OCM mechanism effectively maintains population diversity in such a dense obstacle environment, enabling the algorithm to systematically explore the entire maze, successfully escape all local dead ends, and achieve a 100% planning success rate. Meanwhile, its DPC mechanism ensures rapid convergence in complex terrains and finds the shortest path. In comparison, although PSO-MGLE has a slight advantage in planning time, its solution quality and robustness are inferior to those of ISCA-PSO. RQWOA, due to its slightly weaker balance between exploration and exploitation in ultra-high constraint spaces, ranks second only to ISCA-PSO in planning success rate. In 3D scenarios, the PSO algorithm is unable to complete effective planning due to being trapped in local optima, therefore no average path length data is provided. Although RQWOA and PLSCA have introduced improved mechanisms such as refraction learning and peer learning respectively, their ability to coordinate global exploration and local development in dealing with high-dimensional, non convex, and constrained complex 3D path planning problems is still slightly weaker than the ISCA-PSO algorithm that integrates DPC and OCM dual mechanisms. The PSO-MGLE, designed specifically for large-scale optimization, performs outstandingly in terms of planning time (4.9 s), thanks to its efficient global search strategy. However, ISCA-PSO still has significant advantages in the two key indicators of solution quality (path length) and robustness (successful traversal rate).

Discussion and conclusion

To improve the efficiency of solving LSI, this study introduced DPC and OCM to optimize the design of ISCA. Afterwards, ISCA and PSO were combined and applied to robot path planning instances. The results indicated that the SD of ISCA in the Sphere function was close to 0, and the algorithm’s results after 30 runs were almost identical and extremely stable. The SD of SCA ranged from 103 to 104, and in comparison, the performance of the original algorithm was greatly affected by the initial value and random search. The optimal value of ISCA in the Sphere function was close to the theoretical optima of 0, while in the noisy Quartic function, the optimal value of ISCA remained stable at 10–120. When the SCA dimension increased from 40 to 100, the optimal values and SD of all functions significantly deteriorated, and standard SCA was more severely affected by the curse of dimensionality. ISCA still maintained good accuracy in 100 dimensions, proving that DPC and OCM effectively alleviated high-dimensional problems.

Due to DPC accelerating convergence, orthogonal crossover maintaining diversity and suppressing curse of dimensionality, ISCA significantly improves convergence accuracy, stability, and high-dimensional adaptability. This provides an algorithmic technical roadmap for LSI and an innovative technological pathway for future robot path planning algorithms. However, this study did not take into account the path planning requirements of different terrains such as urban environments and mountainous areas. Future research can introduce terrain analysis and environmental perception module technology to achieve the applicability of robots in various complex and changing environments, further expanding the application scope of algorithms.

Author contributions

Y.W. processed the numerical attribute linear programming of communication big data, and the mutual information feature quantity of communication big data numerical attribute was extracted by the cloud extended distributed feature fitting method. Y.W. Combined with fuzzy C-means clustering and linear regression analysis, the statistical analysis of big data numerical attribute feature information was carried out, and the associated attribute sample set of communication big data numerical attribute cloud grid distribution was constructed. Y.W. did the experiments, recorded data, and created manuscripts. All authors read and approved the final manuscript.

Funding

The research is supported by 2023 Shanxi Province teaching reform and innovation project: Exploration and practice of competition-oriented mathematical modeling team teaching (No. J20231317).

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.