# Optimization of UAV Cooperative Path Planning Mathematical Model Based on Personalized Multigroup Sparrow Search Algorithm in Complex Environment

Authors: Shaoqiang Yan; Weidong Liu; Ping Yang; Fengxuan Wu; Donglin Zhu; Gang Chen
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Sparrow search algorithm has the problem of redundancy of convergence speed due to its fast convergence speed, and it is easy to fall into local optimum in multimodal environment. To solve the above problem, this paper presents a personalized multipopulation sparrow search algorithm (MPSSA). By introducing multiple population mechanisms to reduce the probability of falling into the local optimum due to single-population search, by using a personalized subpopulation strategy to improve the personalized differences of subpopulations and balance the exploratory ability of algorithm development, then by using weighted center-of-gravity communication strategy to improve the quality of communication between populations, and finally by using dimension by dimension dynamic reverse learning to improve the accuracy of search. The superiority of MPSSA is validated by comparing the benchmark function and CEC2017. Finally, the algorithm solves the problem of poor quality due to the dimension increase of the UAV cooperative track. MPSSA helps the UAV to quickly plan a better and stable track group to ensure the UAV to complete the cooperative task safely and stably.


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## Body

## 1. Introduction

With the development of the times, science and technology is increasingly showing the characteristics of cross-cutting and penetration, the complexity of various engineering problems is increasing, and the calculation dimension is increasing, so the efficiency of optimization has become a major difficulty in soft computing [1–5]. As the core means to solve optimization problems, algorithm is the focus of many scholars’ attention and research. For common optimization problems, there are often discontinuity, nonlinearity, nonunique variables, and constraints, and modeling is not easy. Traditional optimization algorithms such as exhaustion, integer programming, constraint planning, graph theory, etc. need to traverse the entire space to solve such problems, resulting in high time and space complexity, space-occupying and low efficiency of the algorithm. At the same time, traditional algorithms require high objective functions and constraints and can only be used in differentiable cases. Therefore, the solution obtained by the algorithm is generally only a local optimal solution, and the accuracy of the solution cannot reach the actual requirements. As a new evolutionary computing technology, the swarm intelligence algorithm overcomes the drawbacks of the traditional optimization algorithm perfectly and enters a new stage for solving optimization problems with its own self-organization and adaptive characteristics.The swarm intelligence optimization algorithm is a random optimization algorithm (also known as probability search algorithm) constructed to simulate the group behavior of natural organisms [6–10]. Compared with most gradient-based optimization algorithms and traditional algorithms, the intelligence of swarm intelligence optimization algorithm is mainly because the algorithm is independent of the optimization problem itself, insensitive to initial conditions, self-organizing, and adaptive. The swarm intelligence optimization algorithm is simple in overall design, requires fewer parameters, is easy to implement, and can be processed in parallel, so it has the advantages of good fault tolerance, strong robustness and stability.With the development of the whole intelligent algorithm system, more and more classical swarm intelligence optimization algorithms appear, such as grey wolf optimization (GWO) [11], whale optimization algorithm (WOA) [12], Harris hawks optimization (HHO) [13], salp swarm algorithm (SSA) [14], artificial bee colony algorithm (ABC) [15], firefly algorithm (FA) [16], bat algorithm (BA) [17], chicken swarm optimization (CSO) [18], bird swarm algorithm (BSA) [19], and pigeon-inspired optimization (PIO) [20]. They have become brilliant stars in the study of swarm intelligence algorithms.The sparrow search algorithm (SSA) is a new population intelligence optimization algorithm proposed by Xue et al. based on the foraging, predatory, and antipredatory behaviors of sparrow population [21]. Like other traditional swarm intelligence optimization algorithms, the sparrow search algorithm has some drawbacks such as uneven initial population distribution, insufficient convergence ability at the end of iteration, easy to fall into local optimum, and premature stagnation [22]. Compared with the research history of classical algorithms such as the particle swarm optimization (PSO) [23] and genetic algorithm (GA) [24], the research and application of this algorithm is still in the initial stage and to be developed. With the advantages of SSA, compared with other swarm intelligence algorithms, the algorithm has faster convergence speed, stronger stability, and higher search accuracy, and it has great research potential and development prospects. At the same time, the improvement of the sparrow search algorithm’s own defects, whether in theoretical research, algorithm design, or engineering practice, has the necessity and value of further research [25, 26].With the rapid development of modern warfare and the more complex operational tasks, the cooperative operation of multiple unmanned aerial vehicles has become an inevitable trend. Collaborative track planning is based on the path planning of multiple UAVs and further considers the collaborative constraints to make the path shortest. It also brings new problems, such as the difficulty of fast convergence with increasing dimension of solution, the difficulty of high-precision time coordination of tracks, and the resolution of complex spatial conflicts. As a result, the solution of this kind of more complex optimization problem has higher requirements on the performance of the algorithm. Literature [27] proposed a parallel genetic algorithm to obtain a stable collaborative planning path through parallel computing of genetic algorithm. Literature [28] proposed an improved pigeon swarm algorithm based on social class, which uses social class strategy to accelerate the convergence of path planning. Literature [29] proposed an improved sparrow search algorithm based on logarithmic spiral strategy and adaptive step size strategy, which improved the optimization quality of collaborative path planning.In this paper, a personalized multipopulation sparrow search algorithm (MPSSA) is proposed and applied to UAV cooperative track planning. The results show that MPSSA algorithm has better optimization performance than the comparison algorithm and can help UAV safely and steadily plan better cooperative path. The main contributions of this paper are as follows:(1)
In Section1, the development of swarm intelligence algorithm in the context of big data is briefly introduced, and the main contributions of the article are introduced(2)
In Section2, the paper briefly introduces the principle and mathematical model of sparrow search algorithm(3)
In Section3, a variety of population mechanisms are applied to SSA, and a personalized subpopulation strategy is proposed to make different subpopulations have different parameter settings, so that each subpopulation has a different exploration ability(4)
In Section3, a weighted center-of-gravity communication strategy is proposed to reduce the disturbance of one population, reduce the risk that one subpopulation will fall into local optimum, which will result in all subpopulations falling into local optimum, and prevent other high-quality solutions from being ignored because of the fast convergence to the optimal solution(5)
In Section3, dimensional dynamic reverse learning is introduced to update the scouters’ position in SSA, enhance their feeding back behavior, and reduce the shortcomings of slow convergence speed and inadequate convergence accuracy caused by the reduction of factor population;(6)
In Section4, MPSSA is tested and compared with six other algorithms in standard test function and CE2017 [30, 31], and the performance of these algorithms is analyzed by ranking of algorithms, Wilcoxon rank sum test [32](7)
In Section5, the above algorithm is applied to the UAV track planning, further verifying the applicability of the algorithm, and MPSSA is applied to the UAV cooperative track planning to help improve the quality of the UAV cooperative track planning(8)
In Section6, the experimental results are summarized, and the next research content is pointed out


## 2. Sparrow Search Algorithm

As a group bird, sparrows are active in places where humans live. They are very active, intelligent, and have a good memory. They are bold and approachable, but they are very vigilant. During the sparrows’ feeding process, individual populations have a clear division of labor and can be divided into discoverers and participants according to their suitability for the environment. Discoverers have a high degree of environmental adaptability and need to search extensively to discover food, guide individuals to obtain food, and master the search direction of the entire population. Participants were less adaptable to the environment than the discoverers, and to improve their own fitness, they followed the discoverers to obtain food. At the same time, the sparrow population is bound to encounter various threats from the external natural environment such as natural enemies during feeding. In order to improve the survival probability, the sparrow population will randomly allocate a part of the individual as scouters, keep alert to the surrounding environment, and alert the population to flee whenever a threat is found.In the case of sparrow populations searching for lost grains in harvested fields, the discoverer is responsible for skipping across a wide range of fields to find lost grains and stopping eating when they discover them. When the participants notice that the discoverers have found food, they jump straight to the location of the grain, grab the food with the discoverer, and eat it in competition. However, due to the limited number of grains, there is no guarantee that each sparrow will be free from hunger among the participants, so sparrows farther away from the grain (i.e., less adaptable to the environment) will give up competing with the population for food and choose to fly elsewhere to feed alone. At the same time, in order to ensure the safe feeding of the population, the sparrow population will randomly arrange a certain number of sparrows for sentinel investigation in the periphery and interior of the population, so that the whole population can escape in time in response to emergencies. This is how the sparrow population feeds.In the process of sparrows’ foraging, the discoverer-participant model was used, and the two behavioral strategies were predatory behavior and anti-predatory behavior. Individuals with better locations in the sparrow population were considered discoverers, while the remaining individuals are participants, and 20% of the individuals were randomly assigned as scouters. Predatory behavior means that the discoverer is responsible for leading the population in search direction and discovering food, while the participant follows the discoverer to seize food. Antipredatory behavior means that scouters are always vigilant against environmental threats, mainly natural enemies, and sends timely dangerous signals to alert sparrow populations to move closer to safe areas in order to prevent them from being preyed.Set the current iteration numberT and the maximum iteration number M. The current position of the ith sparrow in the jth dimension is Xi,j.When the discoverer did not find the threat (R2<ST), they were responsible for guiding the population to forage and conduct extensive search. When individuals in the population have found predators (natural enemies) and issued an alarm (R2≥ST), guide the population to the location of the safe area. The location update is described as follows:
(1)Xi,jt+1=Xi,jt·exp−iα·M,R2<ST,Xi,jt+Q·L,R2≥ST,whereα is a random number belonging to [0, 1]. R2∈0,1 represents the early warning value. ST∈0.5,1 represents the security threshold of the current environment. Q is responsible for controlling the step size, which is a random number subject to normal distribution. L is a matrix of 1×d, and all elements are 1, and d represents dimension.In order to obtain food, the participants follow and supervise the discoverer to grab food (i≤N/2) or look for food alone (i>N/2). Therefore, the location update description of the participants is as follows:
(2)Xi,jt+1=Q·expXworstt−Xi,jti2,i>n2,XPt+1+Xi,jt−XPt+1·A+·L,otherwise,whereXworst represents the worst position of the current population and Xp is the best position currently occupied by the discoverer. A is responsible for controlling the direction 1×d matrix, the element is only 1 or -1, and A+=ATAAT−1.When aware of the danger, the sparrow population will make anti predation behavior. Whenfi≠fg, it means that the current sparrow is on the edge of the population and aware of the danger and needs to move closer to the population center to reduce the risk of predation. When fi=fg, it indicates that the sparrow in the center of the population is aware of the danger and needs to escape from its current position. The location update description of the scouters is as follows:
(3)Xi,jt+1=Xbestt+β·Xi,jt−Xbestt,fi≠fg,Xi,jt+K·Xi,jt−Xworsttfi−fw+ε,fi=fg,whereXbest represents the optimal location of the current population. β is responsible for controlling the step size, which is a random number subject to standard normal distribution. K controls the direction of sparrow movement and the moving step length. It is a random number belonging to [-1, 1]. fi, fg, and fw represent the fitness value of the ith individual and the best and worst fitness values of the current population, respectively. To prevent the denominator from being 0, ε, take a minimal positive real number.


## 3. Improved Sparrow Search Algorithm

### 3.1. Multigroup Search Mechanism

The current SSA algorithm adopts the single-population mode to search, and all the search processes are limited to one population. Once the population falls into the local optimization and cannot escape the local optimization trap, the optimization performance of the whole algorithm will be reduced. For the SSA algorithm, the convergence speed is very fast due to its discoverer-participant model. The participants will gather to the position of the discoverer for food, and a large number of individuals will converge to the optimal position currently occupied by the discoverer (formula (2)), which will increase the risk of falling into local optimization. Moreover, for the single-population sparrow algorithm and multipopulation sparrow algorithm, the number of individuals in a single population is multiple times that of multiple populations, which will lead to the redundancy of convergence ability of the single-population sparrow algorithm and premature stagnation.The efficiency of traditional single-population search is low. It is easy to fall into the trap of local optimization in the search space with more local optimization traps or multipeaks, resulting in the reduction of the overall performance of the algorithm. Compared with single-population search, multipopulation search has the following advantages [33–36]:
(1)
Different parameters can be set for each population. Take the DE algorithm as an example, such as the setting of crossover probability and mutation probability, which can make each population evolve in different directions and comprehensively enhance the search ability(2)
Each subpopulation can communicate and share. For example, the DE algorithm can set immigration operator to introduce the optimal individual into other subpopulations, so as to realize the learning from the current subpopulation to other subpopulations, which is conducive to the convergence of the algorithm(3)
The population is not easy to fall into the local optimum. Different subpopulations search separately, and the search method is more flexible. When a population falls into the local optimum, it can escape the attraction of the local optimum by communicating with other subpopulationsThe multipopulation search mechanism is to divide the population individualN into m subpopulations, then the individual of each subpopulation is round N/m=n, and round is a downward integer operation. If modN,m≠0 and mod is the remainder operation, the remainder C is evenly divided into the first C subpopulation. In the search process, the jth dimension of the ith individual of the kth subpopulation in the tth iteration is expressed as Xk,i,jt, at this time, i≤NGk, NGk is the number of individuals of the kth subpopulation, and ∑k=1mNGk=N, the details are shown in Figure 1.Figure 1
Individual distribution in multiple populations.


### 3.2. Personalized Subpopulation Strategy

At the same time, the discoverer in the SSA algorithm is responsible for guiding the direction of the population and conducting extensive search. The participants directly jump to the current global optimal position occupied by the discoverer and conduct detailed search, so the discoverer and participant control the global search and local search of the population, respectively. At present, the general SSA algorithm sets the proportion of discoverer to individual population to be 20%, that is, the proportion of discoverer to entrant is 1 : 4, so SSA has strong local search ability and high search accuracy.For multigroup SSA, we can set different discoverer proportions PD according to different populations and set the discoverer proportion of thekth population as PDk, so that different populations have different development and exploration capabilities, which can further balance the development and exploration stages. The subpopulation with strong global search ability can more easily find the approximate location of the global optimal solution. At this time, information sharing can be carried out according to the exchange between populations, so that other subpopulations can converge, and the subpopulation with strong local search ability can be used to search the location of the global optimal solution more carefully, so as to enter the development stage and further improve the optimization accuracy.The principle is shown in Figure2. In (a), sparrow individuals have less proportion of discoverers and weak global search ability and can only find local optima, but the convergence speed is fast. In (b), sparrow individuals have a high proportion of discoverers and high global search ability and can find the global optimum, but the convergence speed is slow.Figure 2
Personalized subpopulation.
(a)(b)


### 3.3. Weighted Center of Gravity Communication Strategy

The population is divided into several subpopulations, and each individual searches in their own subpopulation. This search method can reduce the risk of the whole population falling into local optimization. However, when the whole population size remains unchanged, the number of individuals in the subpopulation decreases and searches independently and converges to the optimal solution found by each subpopulation. Once the optimal solutions in each subpopulation are inconsistent, the solution efficiency will be reduced. Therefore, the subpopulations communicate with each other, that is, the elite individuals among the subpopulations share information, which helps the subpopulations converge to the global optimal solution.The general population exchange mechanism is to compare the best individuals in each subpopulation [37], select the best individual X, and replace the best or worst individual in each subpopulation with this individual, so as to improve the population quality. However, once the selected optimal individual X is local optimal, all subpopulations will fall into local optimal. Another common population exchange mechanism is the random recombination strategy [38], which means that the whole population will be randomly reorganized every certain number of iterations, and each individual in the population will be divided into a new subpopulation and start a new search. This strategy greatly improves the communication between populations, but the existence probability causes a chaotic state. For example, the subpopulation individuals who have fallen into the local optimum may lead to more new subpopulations falling into the local optimum. The individuals with poor quality are divided into the same subpopulation, resulting in the low quality of the subpopulation. Therefore, this kind of strategy has the defect of instability.Therefore, this paper proposes a weighted center of gravity communication strategy, that is, the worst position of each subpopulation is replaced by the weighted center of gravity of the optimal position in all other subpopulations, and the weight is the proportion of the fitness of the optimal position of each subpopulation to the total fitness. It is worth noting that the subpopulation of the worst position replaced does not participate in the selection of the center of gravity. The role of this strategy is as follows:(1)
The worst position is replaced by the weighted center of gravity of the best position of other subpopulations. First, the worst position is replaced. If the quality of the subpopulation is poor, it will help to improve the population. If the quality of the subpopulation is good, it is helpful to help the population test whether it falls into local optimization(2)
Selecting and replacing the weighted center of gravity of the optimal position in all other subpopulations that do not include the subpopulation can reduce the interference of its own population, reduce the risk that a subpopulation falls into the local optimum and all subpopulations fall into the local optimum, and prevent the situation that the convergence speed to the optimal solution is too fast and other high-quality solutions are ignoredTaking the four subpopulations as an example,G1, G2, G3, and G4 are the four populations, respectively; W1, W2, W3, and W4 are the positions of the worst individuals of the four subpopulations after weighted center of gravity communication; and the black circle is the global optimum, as shown in Figure 3. In (a), because G1 and G4 are close to the global optimum and their weights are high, W3 is closer to G1 and G4. In (b), W3 is located closer to G2 because G2 is closer to the global optimum with its higher weight. In both cases, the individual after the weighted center of gravity exchange will be closer to the global optimum, but W2 (b) is far from the global optimum, which helps to test whether it is a local optimum.Figure 3
Weighted center of gravity exchange.
(a)(b)The worst position formula for thekth subpopulation is updated as follows:
(4)Xworstk=∑j=1nfgj∑i=1nfgi−fgk·Xbestj−fgj∑i=1nfgi−fgk·Xbestk.The above superscript represents theith, jth, and kth subpopulations, which can be further simplified as
(5)Xworstk=∑j=1nfgj∑i=1nfgi·Xbestj,i,j≠k.


### 3.4. Dimension by Dimension Dynamic Reverse Learning

As a result of the multiple population mechanisms dividing the population into multiple subpopulations, the number of individuals in the subpopulation is smaller than that in the original population, which leads to a decrease in convergence rate and optimization accuracy, which is also a disadvantage of the multiple population mechanisms. The reverse learning strategy helps to accelerate the convergence speed and improve the optimization accuracy, and the solution after reverse learning can be closer to the optimal solution. General reverse learning can only find the optimal solution [39–41] in a fixed spatial search, while dimension by dimension dynamic reverse learning has better search ability than general direction learning, reduces the interference between dimensions, and can continuously converge to the optimal solution in dynamic space. At the same time, the alert has antipredatory behavior to help the population jump out of local optimum and accelerate convergence (Formula (3)). In this paper, dimension by dimension dynamic reverse learning is added to the scouters’ location update to enhance the antipredatory behavior and reduce the drawbacks of slow convergence speed and inadequate convergence accuracy caused by the reduction of the number of factor population.The principle is as follows:(6)X∗=a+b−X.Set the original position toX and the reverse learning position toX∗. a and b are upper and lower boundaries, respectively, which is the general reverse learning strategy.This paper expands reverse learning to each dimension and reverse learning to each dimension. The formula is expanded as follows:(7)X:,j∗=aj+bj−X:,j,wherej is the dimension, aj is the lower boundary of the jth dimension, and bj is the upper boundary of the jth dimension.At the same time, the dynamic boundary is adopted in this paper:(8)aj=minX:,j.bj=maxx:,j,where minX:,j is the minimum of the jth dimension in all individuals and maxX:,j is the maximum of the jth dimension in all individuals.Figure4 is a dimension by dimension dynamic reverse learning diagram for the ith individual. The black circle is the global optimal solution, and the squares are the largest and smallest values in each dimension of the current population, that is, the upper and lower boundaries of each dimension. a+b/2 is the midpoint of each dimension boundary, and it is worth noting that in practice it is not a straight line, which is abstracted as a straight line. X is the original location and X∗ is the inverse solution of X under one-dimensional dynamic reverse learning. It can be seen that the dimension by dimension dynamic reverse learning, like general reverse learning, can approach the optimal solution, accelerate the convergence process, and further eliminate the interference between dimensions so that each dimension has no influence on each other.Figure 4
Dynamic per dimension reverse learning.


### 3.5. MPSSA Algorithm Flow

The MPSSA algorithm adopts a variety of population search mechanisms and divides the population into equal sized subpopulations as evenly as possible after initialization. We set different “discoverer-participant” ratios in different subpopulations to ensure that each algorithm has different development and exploration capabilities, increase the personalized differences among populations, and make the search mode of populations more flexible. Then, dimension by dimension dynamic reverse learning is added when the location of the scouter is updated, which can accelerate the convergence speed of the subpopulation and have more detailed search accuracy. Finally, after all subpopulations are updated, the weighted center of gravity communication strategy is adopted to exchange among populations and transfer high-quality solutions, so as to help improve the quality of populations and jump out of local optima. The specific implementation steps are shown in Algorithm1.Algorithm 1: The framework of the MPSSA.
Set populationsNSet the number of subpopulationsnSet maximum number of iterationsMSet alert value STSet the proportionPDk of discoverers in the kth subpopulationSet proportion of scouters in the population SDt=1;Initialize population.Randomize the population inton equal-sized subpopulations.While(t≤M)Fork=1: nIn thekth subpopulation, sort fitness values and mark the positions of the best and worst sparrows.R2=rand (1).Fori=1:PDk∗N/nUsing formula (1) to update the location of the discoverers;End forFori=PDk∗N/n+1:NUsing formula (2) to update the location of the participants;End forForl=1: SDUsing formula (3) to update the location of the scouters;Update the location of the sparrows according to formula (7) and (8);End forGet the new optimal individual;If the new location is better than the original location, update the location;End forUpdate the worst position of each subpopulation according to the formula (5);The optimal solution of each subpopulation is compared, and the best one is selected as the optimal solution and optimal location of the whole population;t=t+1;End whileReturn:Xbest,fg


### 3.6. Algorithm Complexity Analysis

As shown in the above algorithm steps, MPSSA mainly performs the following operations more than SSA:(1)
The population numbern is divided into n subpopulations, and the time complexity of this task is On(2)
Although the location update is carried out according ton subpopulations, the time complexity of individual update is ON×M×D as the original SSA because the total number remains unchanged, D is the dimension. At the same time, due to the increase of subpopulation, the sorting times before individual location update will be increased, but the same number of population remains unchanged, and the computational complexity remains unchanged(3)
Personalized multigroup strategy only changes the initial parameters and the proportion of discoverers and participants and does not increase the complexity of the algorithm(4)
The weighted center of gravity AC strategy updates only the worst individuals of then subpopulations, and its algorithm complexity is On×M(5)
The dimension by dimension dynamic reverse learning only operates when the scouter locations in the population are updated with a time complexity ofOSD∗N×M×D, SD as a proportion of the population of scouters, did not add an order of magnitude to the temporal complexity of the algorithm, despite the added level of complexityIn summary, MPSSA has a computational complexity ofON×D×M.


## 3.1. Multigroup Search Mechanism

The current SSA algorithm adopts the single-population mode to search, and all the search processes are limited to one population. Once the population falls into the local optimization and cannot escape the local optimization trap, the optimization performance of the whole algorithm will be reduced. For the SSA algorithm, the convergence speed is very fast due to its discoverer-participant model. The participants will gather to the position of the discoverer for food, and a large number of individuals will converge to the optimal position currently occupied by the discoverer (formula (2)), which will increase the risk of falling into local optimization. Moreover, for the single-population sparrow algorithm and multipopulation sparrow algorithm, the number of individuals in a single population is multiple times that of multiple populations, which will lead to the redundancy of convergence ability of the single-population sparrow algorithm and premature stagnation.The efficiency of traditional single-population search is low. It is easy to fall into the trap of local optimization in the search space with more local optimization traps or multipeaks, resulting in the reduction of the overall performance of the algorithm. Compared with single-population search, multipopulation search has the following advantages [33–36]:
(1)
Different parameters can be set for each population. Take the DE algorithm as an example, such as the setting of crossover probability and mutation probability, which can make each population evolve in different directions and comprehensively enhance the search ability(2)
Each subpopulation can communicate and share. For example, the DE algorithm can set immigration operator to introduce the optimal individual into other subpopulations, so as to realize the learning from the current subpopulation to other subpopulations, which is conducive to the convergence of the algorithm(3)
The population is not easy to fall into the local optimum. Different subpopulations search separately, and the search method is more flexible. When a population falls into the local optimum, it can escape the attraction of the local optimum by communicating with other subpopulationsThe multipopulation search mechanism is to divide the population individualN into m subpopulations, then the individual of each subpopulation is round N/m=n, and round is a downward integer operation. If modN,m≠0 and mod is the remainder operation, the remainder C is evenly divided into the first C subpopulation. In the search process, the jth dimension of the ith individual of the kth subpopulation in the tth iteration is expressed as Xk,i,jt, at this time, i≤NGk, NGk is the number of individuals of the kth subpopulation, and ∑k=1mNGk=N, the details are shown in Figure 1.Figure 1
Individual distribution in multiple populations.


## 3.2. Personalized Subpopulation Strategy

At the same time, the discoverer in the SSA algorithm is responsible for guiding the direction of the population and conducting extensive search. The participants directly jump to the current global optimal position occupied by the discoverer and conduct detailed search, so the discoverer and participant control the global search and local search of the population, respectively. At present, the general SSA algorithm sets the proportion of discoverer to individual population to be 20%, that is, the proportion of discoverer to entrant is 1 : 4, so SSA has strong local search ability and high search accuracy.For multigroup SSA, we can set different discoverer proportions PD according to different populations and set the discoverer proportion of thekth population as PDk, so that different populations have different development and exploration capabilities, which can further balance the development and exploration stages. The subpopulation with strong global search ability can more easily find the approximate location of the global optimal solution. At this time, information sharing can be carried out according to the exchange between populations, so that other subpopulations can converge, and the subpopulation with strong local search ability can be used to search the location of the global optimal solution more carefully, so as to enter the development stage and further improve the optimization accuracy.The principle is shown in Figure2. In (a), sparrow individuals have less proportion of discoverers and weak global search ability and can only find local optima, but the convergence speed is fast. In (b), sparrow individuals have a high proportion of discoverers and high global search ability and can find the global optimum, but the convergence speed is slow.Figure 2
Personalized subpopulation.
(a)(b)


## 3.3. Weighted Center of Gravity Communication Strategy

The population is divided into several subpopulations, and each individual searches in their own subpopulation. This search method can reduce the risk of the whole population falling into local optimization. However, when the whole population size remains unchanged, the number of individuals in the subpopulation decreases and searches independently and converges to the optimal solution found by each subpopulation. Once the optimal solutions in each subpopulation are inconsistent, the solution efficiency will be reduced. Therefore, the subpopulations communicate with each other, that is, the elite individuals among the subpopulations share information, which helps the subpopulations converge to the global optimal solution.The general population exchange mechanism is to compare the best individuals in each subpopulation [37], select the best individual X, and replace the best or worst individual in each subpopulation with this individual, so as to improve the population quality. However, once the selected optimal individual X is local optimal, all subpopulations will fall into local optimal. Another common population exchange mechanism is the random recombination strategy [38], which means that the whole population will be randomly reorganized every certain number of iterations, and each individual in the population will be divided into a new subpopulation and start a new search. This strategy greatly improves the communication between populations, but the existence probability causes a chaotic state. For example, the subpopulation individuals who have fallen into the local optimum may lead to more new subpopulations falling into the local optimum. The individuals with poor quality are divided into the same subpopulation, resulting in the low quality of the subpopulation. Therefore, this kind of strategy has the defect of instability.Therefore, this paper proposes a weighted center of gravity communication strategy, that is, the worst position of each subpopulation is replaced by the weighted center of gravity of the optimal position in all other subpopulations, and the weight is the proportion of the fitness of the optimal position of each subpopulation to the total fitness. It is worth noting that the subpopulation of the worst position replaced does not participate in the selection of the center of gravity. The role of this strategy is as follows:(1)
The worst position is replaced by the weighted center of gravity of the best position of other subpopulations. First, the worst position is replaced. If the quality of the subpopulation is poor, it will help to improve the population. If the quality of the subpopulation is good, it is helpful to help the population test whether it falls into local optimization(2)
Selecting and replacing the weighted center of gravity of the optimal position in all other subpopulations that do not include the subpopulation can reduce the interference of its own population, reduce the risk that a subpopulation falls into the local optimum and all subpopulations fall into the local optimum, and prevent the situation that the convergence speed to the optimal solution is too fast and other high-quality solutions are ignoredTaking the four subpopulations as an example,G1, G2, G3, and G4 are the four populations, respectively; W1, W2, W3, and W4 are the positions of the worst individuals of the four subpopulations after weighted center of gravity communication; and the black circle is the global optimum, as shown in Figure 3. In (a), because G1 and G4 are close to the global optimum and their weights are high, W3 is closer to G1 and G4. In (b), W3 is located closer to G2 because G2 is closer to the global optimum with its higher weight. In both cases, the individual after the weighted center of gravity exchange will be closer to the global optimum, but W2 (b) is far from the global optimum, which helps to test whether it is a local optimum.Figure 3
Weighted center of gravity exchange.
(a)(b)The worst position formula for thekth subpopulation is updated as follows:
(4)Xworstk=∑j=1nfgj∑i=1nfgi−fgk·Xbestj−fgj∑i=1nfgi−fgk·Xbestk.The above superscript represents theith, jth, and kth subpopulations, which can be further simplified as
(5)Xworstk=∑j=1nfgj∑i=1nfgi·Xbestj,i,j≠k.


## 3.4. Dimension by Dimension Dynamic Reverse Learning

As a result of the multiple population mechanisms dividing the population into multiple subpopulations, the number of individuals in the subpopulation is smaller than that in the original population, which leads to a decrease in convergence rate and optimization accuracy, which is also a disadvantage of the multiple population mechanisms. The reverse learning strategy helps to accelerate the convergence speed and improve the optimization accuracy, and the solution after reverse learning can be closer to the optimal solution. General reverse learning can only find the optimal solution [39–41] in a fixed spatial search, while dimension by dimension dynamic reverse learning has better search ability than general direction learning, reduces the interference between dimensions, and can continuously converge to the optimal solution in dynamic space. At the same time, the alert has antipredatory behavior to help the population jump out of local optimum and accelerate convergence (Formula (3)). In this paper, dimension by dimension dynamic reverse learning is added to the scouters’ location update to enhance the antipredatory behavior and reduce the drawbacks of slow convergence speed and inadequate convergence accuracy caused by the reduction of the number of factor population.The principle is as follows:(6)X∗=a+b−X.Set the original position toX and the reverse learning position toX∗. a and b are upper and lower boundaries, respectively, which is the general reverse learning strategy.This paper expands reverse learning to each dimension and reverse learning to each dimension. The formula is expanded as follows:(7)X:,j∗=aj+bj−X:,j,wherej is the dimension, aj is the lower boundary of the jth dimension, and bj is the upper boundary of the jth dimension.At the same time, the dynamic boundary is adopted in this paper:(8)aj=minX:,j.bj=maxx:,j,where minX:,j is the minimum of the jth dimension in all individuals and maxX:,j is the maximum of the jth dimension in all individuals.Figure4 is a dimension by dimension dynamic reverse learning diagram for the ith individual. The black circle is the global optimal solution, and the squares are the largest and smallest values in each dimension of the current population, that is, the upper and lower boundaries of each dimension. a+b/2 is the midpoint of each dimension boundary, and it is worth noting that in practice it is not a straight line, which is abstracted as a straight line. X is the original location and X∗ is the inverse solution of X under one-dimensional dynamic reverse learning. It can be seen that the dimension by dimension dynamic reverse learning, like general reverse learning, can approach the optimal solution, accelerate the convergence process, and further eliminate the interference between dimensions so that each dimension has no influence on each other.Figure 4
Dynamic per dimension reverse learning.


## 3.5. MPSSA Algorithm Flow

The MPSSA algorithm adopts a variety of population search mechanisms and divides the population into equal sized subpopulations as evenly as possible after initialization. We set different “discoverer-participant” ratios in different subpopulations to ensure that each algorithm has different development and exploration capabilities, increase the personalized differences among populations, and make the search mode of populations more flexible. Then, dimension by dimension dynamic reverse learning is added when the location of the scouter is updated, which can accelerate the convergence speed of the subpopulation and have more detailed search accuracy. Finally, after all subpopulations are updated, the weighted center of gravity communication strategy is adopted to exchange among populations and transfer high-quality solutions, so as to help improve the quality of populations and jump out of local optima. The specific implementation steps are shown in Algorithm1.Algorithm 1: The framework of the MPSSA.
Set populationsNSet the number of subpopulationsnSet maximum number of iterationsMSet alert value STSet the proportionPDk of discoverers in the kth subpopulationSet proportion of scouters in the population SDt=1;Initialize population.Randomize the population inton equal-sized subpopulations.While(t≤M)Fork=1: nIn thekth subpopulation, sort fitness values and mark the positions of the best and worst sparrows.R2=rand (1).Fori=1:PDk∗N/nUsing formula (1) to update the location of the discoverers;End forFori=PDk∗N/n+1:NUsing formula (2) to update the location of the participants;End forForl=1: SDUsing formula (3) to update the location of the scouters;Update the location of the sparrows according to formula (7) and (8);End forGet the new optimal individual;If the new location is better than the original location, update the location;End forUpdate the worst position of each subpopulation according to the formula (5);The optimal solution of each subpopulation is compared, and the best one is selected as the optimal solution and optimal location of the whole population;t=t+1;End whileReturn:Xbest,fg


## 3.6. Algorithm Complexity Analysis

As shown in the above algorithm steps, MPSSA mainly performs the following operations more than SSA:(1)
The population numbern is divided into n subpopulations, and the time complexity of this task is On(2)
Although the location update is carried out according ton subpopulations, the time complexity of individual update is ON×M×D as the original SSA because the total number remains unchanged, D is the dimension. At the same time, due to the increase of subpopulation, the sorting times before individual location update will be increased, but the same number of population remains unchanged, and the computational complexity remains unchanged(3)
Personalized multigroup strategy only changes the initial parameters and the proportion of discoverers and participants and does not increase the complexity of the algorithm(4)
The weighted center of gravity AC strategy updates only the worst individuals of then subpopulations, and its algorithm complexity is On×M(5)
The dimension by dimension dynamic reverse learning only operates when the scouter locations in the population are updated with a time complexity ofOSD∗N×M×D, SD as a proportion of the population of scouters, did not add an order of magnitude to the temporal complexity of the algorithm, despite the added level of complexityIn summary, MPSSA has a computational complexity ofON×D×M.


## 4. Simulation Experiment

### 4.1. The Basic Test Function

In this paper, nine kinds of test functions were selected to carry out simulation experiments to verify the performance of MPSSA, among which F1-F3 are high-dimensional unimodal functions, F4-F6 are high-dimensional multimodal functions, F7-F9 are fixed dimensional functions, their dimensions, boundaries, theoretical minima, etc. are shown in Table1, and the function space is shown in Figure 5. Meanwhile, PSO [23], DE [42], GWO [1], SSA [22], BSSA [25], and CSSA [26] were selected as the comparison algorithms in this paper, and the parameters of the algorithms were set as shown in Table 2.Table 1
Test function [22].
FunctionDimensionsBoundaryMinHigh-dimensional single peakF1x=∑i=1nxi230[-100, 100]0F2x=∑i=1n∑j=1ixj230[-100, 100]0F3x=∑i=1nxi+0.5230[-100, 100]0High-dimensional multipeakF4x=∑i=1n−xisinxi30[-500, 500]−418.98nF5x=πn10sinπy1+∑i=1n−1yi−121+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4yi=1+xi+14uxi,a,k,m=kxi−amxi>a0−a<xi<ak−xi−amxi<−a30[-50, 50]0F6x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,430[-50, 50]0Fixed dimensionF7x=∑i=111ai−x1bi2+b1x2/bi2+b1x3+x424[-5, 5]0.0003F8x=∑i=17X−aiX−aiT+ci−14[0, 10]-10.4029F9x=∑i=110X−aiX−aiT+ci−14[0, 10]-10.5364Figure 5
Function diagram.Table 2
Algorithm parameter setting.
AlgorithmPSODEGWOSSABSSACSSAMPSSAParameterc1=2c2=2Wmin=0.2Wmax=0.9CR=0.2Fmin=0.2Fmax=0.8a=2⟶0ST=0.8PD=0.2SD=0.2ST=0.8PD=0.2SD=0.2ST=0.8PD=0.2SD=0.2n=4ST=0.8PDk=0.1,0.2,0.3,0.4SD=0.2Setting the number of populations in the experimentN to 100 and the maximum number of iterations M to 500, each algorithm was run independently 30 times, and the optimal, worst, mean, standard deviation 4-term outcomes were recorded, in which the best outcome in each indicator was coarsened; then, each algorithm was ranked in different functions (depending on the average, or the standard deviation if the average is equal), and finally, the ranking results were averaged to give a total ranking, as detailed in Table 3. “A/I” means “ Algorithm/Index” in Table 3. The average convergence of each algorithm in each function is shown in Figure 6.Table 3
Function experimental results.
FA/IPSODEGWOSSABSSACSSAMPSSAF1Best5.95E−093.81E−048.30E−410000Worst1.69E−069.87E−041.19E−370000Ave1.78E−076.30E−041.16E−380000Std3.24E−071.55E−042.18E−380000Rank6751111F2Best5.48E−052.02E−035.69E−230000Worst1.21E−033.73E−035.58E−227.14E−243000Ave4.02E−042.78E−031.92E−222.38E−244000Std2.96E−044.18E−041.18E−220000Rank6754111F3Best3.98E−093.88E−042.97E−083.70E−321.35E−131.35E−130Worst9.02E−071.20E−036.96E−015.83E−096.87E−106.87E−104.12E−16Ave1.27E−076.73E−041.73E−013.93E−104.84E−114.84E−111.98E−16Std1.77E−071.68E−042.03E−011.16E−091.31E−101.31E−101.50E−16Rank5674231F4Best−8.32E+03−9.69E+03−8.12E+03−1.26E+04−1.26E+04−1.26E+04−1.26E+04Worst−4.33E+03−8.78E+03−3.53E+03−7.25E+03−8.41E+03−8.41E+03−1.07E+04Ave−6.72E+03−9.21E+03−6.38E+03−1.09E+04−1.15E+04−1.15E+04−1.21E+04Std8.75E+022.23E+028.32E+021.72E+038.14E+028.14E+025.15E+02Rank6574231F5Best1.55E−103.65E−051.20E−081.57E−322.48E−152.48E−151.57E−32Worst1.04E−011.83E−044.67E−022.95E−107.65E−127.65E−124.71E−11Ave3.46E−038.86E−051.57E−023.45E−111.46E−121.46E−121.57E−12Std1.89E−023.57E−051.20E−026.69E−112.14E−122.14E−128.60E−12Rank6574231F6Best1.60E−092.66E−041.25E−072.56E−141.60E−141.60E−141.35E−32Worst1.10E−026.02E−044.91E−015.55E−097.01E−107.01E−106.55E−17Ave7.91E−044.01E−041.37E−019.16E−106.23E−116.23E−111.09E−17Std2.79E−038.12E−051.29E−011.58E−091.38E−101.38E−102.48E−17Rank6574231F7Best3.61E−043.56E−043.07E−043.07E−043.07E−043.07E−043.07E−04Worst1.07E−037.80E−042.04E−023.32E−043.07E−043.07E−043.07E−04Ave7.78E−046.51E−042.40E−033.08E−043.07E−043.07E−043.07E−04Std2.05E−048.89E−056.09E−034.39E−061.31E−101.31E−106.59E−11Rank6574231F8Best−1.04E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01Worst−5.09E+00−1.04E+01−5.09E+00−1.04E+01−1.04E+01−1.04E+01−1.04E+01Ave−9.17E+00−1.04E+01−1.00E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01Std2.27E+006.71E−041.35E+001.96E−059.33E−169.33E−161.87E−15Rank7564123F9Best−1.05E+01−1.05E+01−1.05E+01−1.05E+01−1.05E+01−1.05E+01−1.05E+01Worst−5.13E+00−1.05E+01−1.05E+01−5.13E+00−1.05E+01−1.05E+01−1.05E+01Ave−1.02E+01−1.05E+01−1.05E+01−1.04E+01−1.05E+01−1.05E+01−1.05E+01Std1.37E+001.54E−076.24E−089.87E−012.29E−152.29E−155.14E−15Rank7456231Total rank6.115.446.223.891.672.441.22Figure 6
Comparison of algorithm convergence.In order to test the role of each strategy, the sparrow algorithm (SSA), the sparrow algorithm with multi group mechanism (SSA1), the sparrow algorithm with personalized multigroup strategy (SSA2), the sparrow algorithm with weighted barycenter exchange mechanism based on SSA2 (Ssa3), and the sparrow algorithm with dynamic reverse learning based on SSA3 (MPSSA) are tested and simulated inF1, as shown in the first figure in Figure 6. The two algorithms can be used as the control group to test the necessity of the newly added mechanism or strategy. It can be seen that the performance of SSA, SSA1, SSA2, SSA3, and MPSSA shows an increasing trend, indicating that the addition of each mechanism or strategy has a certain improvement on the performance of the previous algorithm.It can be seen from the experimental results in Table3 that MPSSA has found the theoretical optimal solution in F1,F2,F3,F4,F7,F8,andF9, and in addition to the above functions, the optimal solution in the comparison algorithm is found inF5 and F6, indicating that MPSSA algorithm has strong optimization performance. In terms of function types, in high-dimensional unimodal functions, MPSSA has strong search ability with other algorithms such as SSA, BSSA, and CSSA in F1 and F2, and only MPSSA has found the theoretical optimal solution in F3. In the high-dimensional multimodal function, MPSSA ranks first in all indicators, and its overall performance is better than other algorithms, which is also the advantage of multiple swarm algorithms in the case of multimodal. In the fixed dimension function, MPSSA has little difference from other SSA improved algorithms, only a small difference in standard deviation.From the convergence of each algorithm in Figure6, in the high-dimensional unimodal function, the convergence speed of MPSSA is not fast, but due to its superior optimization performance, it can directly find the optimal value and end the convergence. In the high-dimensional multimodal function, the advantage of MPSSA is highlighted. The convergence speed is significantly faster than all comparison algorithms, and it does not fall into local optimization due to the attraction of multimodal.To sum up, PSO, DE, and GWO perform poorly in the standard test function, which is significantly worse than the SSA and SSA improved algorithms. Moreover, MPSSA has a strong search ability in the basic test function, ranking first, followed by BSSA, CSSA, and SSA. Especially in the case of high-dimensional and multi peak, MPSSA is outstanding.


### 4.2. CEC 2017

In order to test the general adaptability of the algorithm and prevent the randomness of the selected function in the above experiment, it is necessary to select a more complex test function, that is, the CEC 2017 test function used in the international algorithm competition, to test the above algorithm. Due to the defect of F2 [43], the article will not adopt it. The evaluation times are set to 1000 ∗ dim, and other parameters are consistent with the above experiments. The experimental results after 30 times of operation are shown in Table 4.Table 4
CEC 2017 [30] experimental results
(a)FF1F3F4F5A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.79E+024.44E+032.55E+0363.00E+023.00E+025.62E−0224.13E+024.72E+021.88E+0136.05E+026.49E+022.18E+014DE1.28E+029.52E+027.60E+0216.60E+048.11E+047.30E+0374.87E+024.89E+021.19E+0066.05E+026.31E+028.97E+003GWO4.89E+071.06E+098.07E+0878.00E+032.73E+049.92E+0354.94E+025.31E+023.32E+0175.46E+025.83E+022.03E+011SSA1.12E+023.17E+034.07E+0332.25E+043.84E+041.43E+0464.71E+024.89E+021.71E+0147.36E+027.97E+022.93E+017BSSA2.25E+023.96E+035.44E+0353.00E+023.03E+022.70E+0034.63E+024.89E+022.14E+0156.91E+027.83E+023.66E+016DSSA1.07E+023.54E+033.86E+0343.00E+024.67E+021.57E+0244.00E+024.53E+024.04E+0125.82E+026.95E+024.70E+015MPSSA1.02E+021.40E+031.75E+0323.00E+023.00E+022.60E−0214.00E+024.33E+023.58E+0115.70E+026.30E+022.72E+012(b)FF6F7F8F9A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO6.19E+026.31E+025.36E+0047.72E+028.39E+022.98E+0118.72E+029.05E+021.93E+0131.79E+033.03E+035.33E+024DE6.00E+026.00E+021.06E−1318.42E+028.66E+026.98E+0039.10E+029.31E+028.38E+0059.00E+029.00E+022.16E−101GWO6.01E+026.03E+021.26E+0027.83E+028.50E+025.34E+0128.55E+028.81E+022.13E+0119.66E+021.34E+034.27E+022SSA6.50E+026.64E+027.56E+0071.21E+031.32E+034.10E+0179.48E+021.01E+032.61E+0175.36E+035.41E+033.02E+017BSSA6.46E+026.55E+024.79E+0061.23E+031.32E+033.03E+0169.19E+029.91E+022.00E+0164.90E+035.40E+032.00E+026DSSA6.20E+026.35E+021.27E+0159.50E+021.11E+038.63E+0158.80E+029.25E+022.66E+0142.06E+034.84E+038.76E+025MPSSA6.00E+026.06E+023.43E+0038.19E+028.94E+028.63E+0148.65E+029.03E+022.70E+0121.04E+032.67E+031.53E+033(c)FF10F11F12F13A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.60E+034.46E+037.00E+0211.15E+031.19E+032.27E+0111.17E+044.89E+043.51E+0421.45E+031.52E+041.70E+041DE5.73E+036.31E+032.65E+0261.20E+031.21E+036.61E+0021.55E+065.86E+062.05E+0664.44E+041.24E+058.05E+045GWO3.22E+034.53E+031.17E+0321.21E+031.32E+031.06E+0279.91E+052.30E+072.51E+0772.53E+047.73E+043.66E+044SSA5.04E+036.38E+031.13E+0371.15E+031.24E+033.69E+0143.11E+041.10E+061.18E+0654.44E+033.15E+061.31E+077BSSA4.40E+035.56E+034.94E+0241.16E+031.25E+036.11E+0152.76E+045.92E+055.12E+0542.96E+032.40E+061.31E+076DSSA4.31E+035.60E+037.00E+0251.18E+031.26E+036.49E+0161.56E+041.41E+057.75E+0431.46E+031.76E+041.63E+042MPSSA3.70E+034.57E+035.91E+0231.15E+031.22E+034.17E+0136.84E+033.36E+041.16E+0411.50E+033.10E+042.79E+043(d)FF14F15F16F17A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO1.50E+036.20E+033.59E+0321.72E+038.20E+038.88E+0332.13E+032.63E+032.42E+0241.79E+032.23E+032.26E+024DE1.75E+046.05E+042.88E+0455.18E+031.92E+041.18E+0461.79E+032.15E+031.78E+0211.80E+031.87E+033.51E+011GWO2.69E+031.26E+052.56E+0571.37E+045.07E+043.26E+0471.73E+032.16E+032.86E+0221.80E+031.94E+031.05E+022SSA4.71E+037.26E+041.82E+0562.02E+037.11E+035.15E+0312.87E+033.65E+037.78E+0272.25E+032.70E+033.10E+027BSSA3.46E+031.27E+045.00E+0341.82E+038.63E+038.59E+0342.39E+032.90E+032.64E+0262.23E+032.63E+031.82E+026DSSA1.82E+036.59E+035.09E+0332.00E+037.38E+039.86E+0322.07E+032.84E+033.20E+0252.05E+032.47E+032.81E+025MPSSA1.76E+035.13E+032.62E+0311.61E+038.84E+037.48E+0352.13E+032.56E+033.29E+0231.85E+032.15E+031.45E+023(e)FF18F19F20F21A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO4.41E+041.51E+051.56E+0542.11E+031.34E+049.97E+0352.28E+032.60E+031.46E+0252.37E+032.43E+032.53E+014DE1.15E+053.98E+051.94E+0551.25E+042.04E+047.83E+0362.07E+032.17E+036.17E+0112.42E+032.44E+039.74E+005GWO4.47E+044.79E+056.51E+0561.06E+046.04E+056.97E+0572.10E+032.30E+031.33E+0222.34E+032.38E+032.52E+012SSA2.34E+048.34E+053.96E+0672.22E+031.12E+048.28E+0342.33E+032.87E+032.53E+0272.49E+032.56E+034.69E+017BSSA1.26E+048.34E+046.04E+0422.12E+039.39E+031.27E+0432.44E+032.69E+031.27E+0262.46E+032.52E+033.60E+016DSSA2.90E+041.06E+055.57E+0432.07E+035.81E+034.91E+0312.17E+032.34E+031.59E+0232.20E+032.40E+031.43E+023MPSSA1.13E+044.68E+043.06E+0412.84E+037.96E+038.10E+0322.09E+032.44E+032.12E+0242.35E+032.38E+031.75E+011(f)FF22F23F24F25A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.30E+033.35E+031.77E+0342.87E+033.05E+039.78E+0152.95E+033.15E+037.74E+0152.88E+032.89E+031.39E+011DE2.82E+033.24E+032.96E+0232.76E+032.78E+036.60E+0032.95E+032.98E+031.04E+0132.89E+032.89E+039.86E−022GWO2.35E+033.48E+031.50E+0352.69E+032.76E+035.73E+0122.86E+032.91E+036.06E+0112.92E+032.95E+031.61E+017SSA2.31E+037.28E+031.18E+0372.96E+033.21E+031.68E+0273.11E+033.40E+031.20E+0272.89E+032.91E+032.00E+016BSSA2.30E+035.91E+032.44E+0362.91E+033.09E+037.70E+0163.08E+033.26E+039.03E+0162.88E+032.91E+032.07E+015DSSA2.30E+032.51E+037.99E+0222.80E+032.89E+035.85E+0142.96E+033.05E+038.57E+0142.88E+032.89E+031.42E+013MPSSA2.30E+032.46E+038.66E+0212.71E+032.75E+033.17E+0112.88E+032.95E+033.55E+0122.88E+032.91E+032.55E+014(g)FF26F27F28F29A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.80E+035.66E+031.78E+0343.15E+033.34E+031.88E+0253.10E+033.21E+035.50E+0143.49E+033.86E+032.05E+024DE4.74E+034.88E+039.30E+0133.20E+033.21E+031.79E+0013.22E+033.23E+031.08E+0153.53E+033.64E+036.84E+011GWO4.06E+034.45E+031.67E+0213.21E+033.23E+031.41E+0123.28E+033.34E+034.10E+0173.39E+033.65E+038.43E+012SSA7.49E+038.83E+039.13E+0273.39E+033.63E+031.83E+0273.20E+033.23E+033.03E+0164.08E+035.58E+038.03E+027BSSA4.01E+037.63E+039.11E+0263.27E+033.49E+031.46E+0263.10E+033.16E+035.63E+0133.99E+034.48E+034.84E+026DSSA2.90E+035.75E+036.14E+0253.22E+033.30E+038.42E+0143.10E+033.13E+035.74E+0113.56E+034.15E+033.09E+025MPSSA2.90E+034.69E+037.16E+0223.21E+033.24E+032.04E+0133.10E+033.14E+036.68E+0123.49E+033.79E+032.23E+023(h)FF30Total rankA/IBestAveStdRankPSO3.38E+035.64E+032.50E+0313.43DE1.44E+042.63E+048.94E+0333.4GWO1.05E+063.28E+061.75E+0674.13SSA1.25E+043.28E+041.52E+0455.97BSSA7.21E+033.48E+041.09E+0564.97DSSA5.29E+039.78E+033.60E+0323.67MPSSA5.93E+032.76E+041.07E+0542.43From the results in Table4, MPSSA finds the optimal value in all algorithms in F1,F3,F4,F12,F15,F18,F22,andF28, and the average value is the closest to the theoretical optimal value in F3,F4,F12,F14,F18,F22,F23,andF28. Among the 29 test functions, MPSSA ranks in the top three except F7,F15,F20,F25,andF30. In the comprehensive ranking, MPSSA ranks first, followed by De, PSO, GWO, CSSA, BSSA, and SSA.The values in Table5 represent the p value of Wilcoxon rank sum test for the results of 30 operations. When p<0.05, it can be considered that MPSSA is significantly different from the algorithm, “+”, “=”, and “-”, respectively, represent that MPSSA is superior, equal, and inferior to the comparison algorithm [44]. It can be seen that most algorithms are inferior to the MPSSA algorithm, which shows that MPSSA is significantly different from other algorithms, and the performance is better than other algorithms, which shows that MPSSA has better advantages than other algorithms.Table 5
Wilcoxon rank sum test.
PSODEGWOSSABSSACSSAF19.99E−078.42E−012.93E−115.29E−033.50E−021.07E−02F39.59E−092.71E−112.79E−112.83E−114.70E−112.38E−11F44.93E−056.44E−112.42E−116.65E−086.92E−074.54E−01F51.00E−035.38E−011.52E−072.37E−112.51E−119.10E−07F62.61E−114.55E−124.80E−062.62E−112.56E−112.60E−11F71.76E−042.83E−012.33E−043.86E−113.46E−111.32E−09F88.18E−011.20E−057.84E−032.75E−117.89E−113.97E−03F93.75E−022.15E−111.93E−061.34E−103.59E−105.43E−07F107.96E−012.55E−111.11E−012.61E−105.98E−093.94E−07F111.97E−057.05E−011.67E−062.63E−021.99E−012.38E−01F122.57E−012.69E−112.66E−111.89E−101.95E−086.33E−09F131.45E−017.87E−094.50E−064.82E−013.62E−013.54E−01F142.21E−012.78E−112.36E−083.26E−093.15E−083.70E−01F156.84E−019.69E−051.62E−096.20E−019.23E−012.05E−01F161.26E−017.32E−071.94E−055.05E−091.14E−044.36E−04F178.19E−029.19E−101.22E−061.77E−107.57E−111.73E−06F182.25E−073.01E−114.71E−097.33E−062.58E−022.18E−06F194.01E−022.88E−089.17E−113.24E−012.50E−011.44E−01F201.15E−022.47E−077.22E−034.84E−087.04E−066.10E−02F216.18E−102.77E−116.09E−012.80E−112.69E−118.12E−03F223.78E−034.87E−104.81E−106.36E−113.40E−094.29E−01F232.72E−115.17E−067.28E−012.63E−112.74E−114.80E−11F242.43E−101.61E−054.86E−052.88E−112.65E−112.51E−07F256.05E−051.61E−062.81E−051.08E−015.58E−012.54E−05F264.79E−032.79E−025.05E−042.79E−117.52E−105.99E−09F279.94E−014.40E−095.68E−022.59E−113.29E−111.75E−05F289.85E−046.54E−052.83E−112.38E−051.14E−039.73E−03F291.70E−019.68E−031.25E−014.04E−111.05E−092.60E−05F301.40E−051.46E−092.84E−111.00E−093.63E−071.42E−02+/=/-10/0/194/0/255/0/244/0/255/0/249/0/21


## 4.1. The Basic Test Function

In this paper, nine kinds of test functions were selected to carry out simulation experiments to verify the performance of MPSSA, among which F1-F3 are high-dimensional unimodal functions, F4-F6 are high-dimensional multimodal functions, F7-F9 are fixed dimensional functions, their dimensions, boundaries, theoretical minima, etc. are shown in Table1, and the function space is shown in Figure 5. Meanwhile, PSO [23], DE [42], GWO [1], SSA [22], BSSA [25], and CSSA [26] were selected as the comparison algorithms in this paper, and the parameters of the algorithms were set as shown in Table 2.Table 1
Test function [22].
FunctionDimensionsBoundaryMinHigh-dimensional single peakF1x=∑i=1nxi230[-100, 100]0F2x=∑i=1n∑j=1ixj230[-100, 100]0F3x=∑i=1nxi+0.5230[-100, 100]0High-dimensional multipeakF4x=∑i=1n−xisinxi30[-500, 500]−418.98nF5x=πn10sinπy1+∑i=1n−1yi−121+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4yi=1+xi+14uxi,a,k,m=kxi−amxi>a0−a<xi<ak−xi−amxi<−a30[-50, 50]0F6x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,430[-50, 50]0Fixed dimensionF7x=∑i=111ai−x1bi2+b1x2/bi2+b1x3+x424[-5, 5]0.0003F8x=∑i=17X−aiX−aiT+ci−14[0, 10]-10.4029F9x=∑i=110X−aiX−aiT+ci−14[0, 10]-10.5364Figure 5
Function diagram.Table 2
Algorithm parameter setting.
AlgorithmPSODEGWOSSABSSACSSAMPSSAParameterc1=2c2=2Wmin=0.2Wmax=0.9CR=0.2Fmin=0.2Fmax=0.8a=2⟶0ST=0.8PD=0.2SD=0.2ST=0.8PD=0.2SD=0.2ST=0.8PD=0.2SD=0.2n=4ST=0.8PDk=0.1,0.2,0.3,0.4SD=0.2Setting the number of populations in the experimentN to 100 and the maximum number of iterations M to 500, each algorithm was run independently 30 times, and the optimal, worst, mean, standard deviation 4-term outcomes were recorded, in which the best outcome in each indicator was coarsened; then, each algorithm was ranked in different functions (depending on the average, or the standard deviation if the average is equal), and finally, the ranking results were averaged to give a total ranking, as detailed in Table 3. “A/I” means “ Algorithm/Index” in Table 3. The average convergence of each algorithm in each function is shown in Figure 6.Table 3
Function experimental results.
FA/IPSODEGWOSSABSSACSSAMPSSAF1Best5.95E−093.81E−048.30E−410000Worst1.69E−069.87E−041.19E−370000Ave1.78E−076.30E−041.16E−380000Std3.24E−071.55E−042.18E−380000Rank6751111F2Best5.48E−052.02E−035.69E−230000Worst1.21E−033.73E−035.58E−227.14E−243000Ave4.02E−042.78E−031.92E−222.38E−244000Std2.96E−044.18E−041.18E−220000Rank6754111F3Best3.98E−093.88E−042.97E−083.70E−321.35E−131.35E−130Worst9.02E−071.20E−036.96E−015.83E−096.87E−106.87E−104.12E−16Ave1.27E−076.73E−041.73E−013.93E−104.84E−114.84E−111.98E−16Std1.77E−071.68E−042.03E−011.16E−091.31E−101.31E−101.50E−16Rank5674231F4Best−8.32E+03−9.69E+03−8.12E+03−1.26E+04−1.26E+04−1.26E+04−1.26E+04Worst−4.33E+03−8.78E+03−3.53E+03−7.25E+03−8.41E+03−8.41E+03−1.07E+04Ave−6.72E+03−9.21E+03−6.38E+03−1.09E+04−1.15E+04−1.15E+04−1.21E+04Std8.75E+022.23E+028.32E+021.72E+038.14E+028.14E+025.15E+02Rank6574231F5Best1.55E−103.65E−051.20E−081.57E−322.48E−152.48E−151.57E−32Worst1.04E−011.83E−044.67E−022.95E−107.65E−127.65E−124.71E−11Ave3.46E−038.86E−051.57E−023.45E−111.46E−121.46E−121.57E−12Std1.89E−023.57E−051.20E−026.69E−112.14E−122.14E−128.60E−12Rank6574231F6Best1.60E−092.66E−041.25E−072.56E−141.60E−141.60E−141.35E−32Worst1.10E−026.02E−044.91E−015.55E−097.01E−107.01E−106.55E−17Ave7.91E−044.01E−041.37E−019.16E−106.23E−116.23E−111.09E−17Std2.79E−038.12E−051.29E−011.58E−091.38E−101.38E−102.48E−17Rank6574231F7Best3.61E−043.56E−043.07E−043.07E−043.07E−043.07E−043.07E−04Worst1.07E−037.80E−042.04E−023.32E−043.07E−043.07E−043.07E−04Ave7.78E−046.51E−042.40E−033.08E−043.07E−043.07E−043.07E−04Std2.05E−048.89E−056.09E−034.39E−061.31E−101.31E−106.59E−11Rank6574231F8Best−1.04E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01Worst−5.09E+00−1.04E+01−5.09E+00−1.04E+01−1.04E+01−1.04E+01−1.04E+01Ave−9.17E+00−1.04E+01−1.00E+01−1.04E+01−1.04E+01−1.04E+01−1.04E+01Std2.27E+006.71E−041.35E+001.96E−059.33E−169.33E−161.87E−15Rank7564123F9Best−1.05E+01−1.05E+01−1.05E+01−1.05E+01−1.05E+01−1.05E+01−1.05E+01Worst−5.13E+00−1.05E+01−1.05E+01−5.13E+00−1.05E+01−1.05E+01−1.05E+01Ave−1.02E+01−1.05E+01−1.05E+01−1.04E+01−1.05E+01−1.05E+01−1.05E+01Std1.37E+001.54E−076.24E−089.87E−012.29E−152.29E−155.14E−15Rank7456231Total rank6.115.446.223.891.672.441.22Figure 6
Comparison of algorithm convergence.In order to test the role of each strategy, the sparrow algorithm (SSA), the sparrow algorithm with multi group mechanism (SSA1), the sparrow algorithm with personalized multigroup strategy (SSA2), the sparrow algorithm with weighted barycenter exchange mechanism based on SSA2 (Ssa3), and the sparrow algorithm with dynamic reverse learning based on SSA3 (MPSSA) are tested and simulated inF1, as shown in the first figure in Figure 6. The two algorithms can be used as the control group to test the necessity of the newly added mechanism or strategy. It can be seen that the performance of SSA, SSA1, SSA2, SSA3, and MPSSA shows an increasing trend, indicating that the addition of each mechanism or strategy has a certain improvement on the performance of the previous algorithm.It can be seen from the experimental results in Table3 that MPSSA has found the theoretical optimal solution in F1,F2,F3,F4,F7,F8,andF9, and in addition to the above functions, the optimal solution in the comparison algorithm is found inF5 and F6, indicating that MPSSA algorithm has strong optimization performance. In terms of function types, in high-dimensional unimodal functions, MPSSA has strong search ability with other algorithms such as SSA, BSSA, and CSSA in F1 and F2, and only MPSSA has found the theoretical optimal solution in F3. In the high-dimensional multimodal function, MPSSA ranks first in all indicators, and its overall performance is better than other algorithms, which is also the advantage of multiple swarm algorithms in the case of multimodal. In the fixed dimension function, MPSSA has little difference from other SSA improved algorithms, only a small difference in standard deviation.From the convergence of each algorithm in Figure6, in the high-dimensional unimodal function, the convergence speed of MPSSA is not fast, but due to its superior optimization performance, it can directly find the optimal value and end the convergence. In the high-dimensional multimodal function, the advantage of MPSSA is highlighted. The convergence speed is significantly faster than all comparison algorithms, and it does not fall into local optimization due to the attraction of multimodal.To sum up, PSO, DE, and GWO perform poorly in the standard test function, which is significantly worse than the SSA and SSA improved algorithms. Moreover, MPSSA has a strong search ability in the basic test function, ranking first, followed by BSSA, CSSA, and SSA. Especially in the case of high-dimensional and multi peak, MPSSA is outstanding.


## 4.2. CEC 2017

In order to test the general adaptability of the algorithm and prevent the randomness of the selected function in the above experiment, it is necessary to select a more complex test function, that is, the CEC 2017 test function used in the international algorithm competition, to test the above algorithm. Due to the defect of F2 [43], the article will not adopt it. The evaluation times are set to 1000 ∗ dim, and other parameters are consistent with the above experiments. The experimental results after 30 times of operation are shown in Table 4.Table 4
CEC 2017 [30] experimental results
(a)FF1F3F4F5A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.79E+024.44E+032.55E+0363.00E+023.00E+025.62E−0224.13E+024.72E+021.88E+0136.05E+026.49E+022.18E+014DE1.28E+029.52E+027.60E+0216.60E+048.11E+047.30E+0374.87E+024.89E+021.19E+0066.05E+026.31E+028.97E+003GWO4.89E+071.06E+098.07E+0878.00E+032.73E+049.92E+0354.94E+025.31E+023.32E+0175.46E+025.83E+022.03E+011SSA1.12E+023.17E+034.07E+0332.25E+043.84E+041.43E+0464.71E+024.89E+021.71E+0147.36E+027.97E+022.93E+017BSSA2.25E+023.96E+035.44E+0353.00E+023.03E+022.70E+0034.63E+024.89E+022.14E+0156.91E+027.83E+023.66E+016DSSA1.07E+023.54E+033.86E+0343.00E+024.67E+021.57E+0244.00E+024.53E+024.04E+0125.82E+026.95E+024.70E+015MPSSA1.02E+021.40E+031.75E+0323.00E+023.00E+022.60E−0214.00E+024.33E+023.58E+0115.70E+026.30E+022.72E+012(b)FF6F7F8F9A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO6.19E+026.31E+025.36E+0047.72E+028.39E+022.98E+0118.72E+029.05E+021.93E+0131.79E+033.03E+035.33E+024DE6.00E+026.00E+021.06E−1318.42E+028.66E+026.98E+0039.10E+029.31E+028.38E+0059.00E+029.00E+022.16E−101GWO6.01E+026.03E+021.26E+0027.83E+028.50E+025.34E+0128.55E+028.81E+022.13E+0119.66E+021.34E+034.27E+022SSA6.50E+026.64E+027.56E+0071.21E+031.32E+034.10E+0179.48E+021.01E+032.61E+0175.36E+035.41E+033.02E+017BSSA6.46E+026.55E+024.79E+0061.23E+031.32E+033.03E+0169.19E+029.91E+022.00E+0164.90E+035.40E+032.00E+026DSSA6.20E+026.35E+021.27E+0159.50E+021.11E+038.63E+0158.80E+029.25E+022.66E+0142.06E+034.84E+038.76E+025MPSSA6.00E+026.06E+023.43E+0038.19E+028.94E+028.63E+0148.65E+029.03E+022.70E+0121.04E+032.67E+031.53E+033(c)FF10F11F12F13A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.60E+034.46E+037.00E+0211.15E+031.19E+032.27E+0111.17E+044.89E+043.51E+0421.45E+031.52E+041.70E+041DE5.73E+036.31E+032.65E+0261.20E+031.21E+036.61E+0021.55E+065.86E+062.05E+0664.44E+041.24E+058.05E+045GWO3.22E+034.53E+031.17E+0321.21E+031.32E+031.06E+0279.91E+052.30E+072.51E+0772.53E+047.73E+043.66E+044SSA5.04E+036.38E+031.13E+0371.15E+031.24E+033.69E+0143.11E+041.10E+061.18E+0654.44E+033.15E+061.31E+077BSSA4.40E+035.56E+034.94E+0241.16E+031.25E+036.11E+0152.76E+045.92E+055.12E+0542.96E+032.40E+061.31E+076DSSA4.31E+035.60E+037.00E+0251.18E+031.26E+036.49E+0161.56E+041.41E+057.75E+0431.46E+031.76E+041.63E+042MPSSA3.70E+034.57E+035.91E+0231.15E+031.22E+034.17E+0136.84E+033.36E+041.16E+0411.50E+033.10E+042.79E+043(d)FF14F15F16F17A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO1.50E+036.20E+033.59E+0321.72E+038.20E+038.88E+0332.13E+032.63E+032.42E+0241.79E+032.23E+032.26E+024DE1.75E+046.05E+042.88E+0455.18E+031.92E+041.18E+0461.79E+032.15E+031.78E+0211.80E+031.87E+033.51E+011GWO2.69E+031.26E+052.56E+0571.37E+045.07E+043.26E+0471.73E+032.16E+032.86E+0221.80E+031.94E+031.05E+022SSA4.71E+037.26E+041.82E+0562.02E+037.11E+035.15E+0312.87E+033.65E+037.78E+0272.25E+032.70E+033.10E+027BSSA3.46E+031.27E+045.00E+0341.82E+038.63E+038.59E+0342.39E+032.90E+032.64E+0262.23E+032.63E+031.82E+026DSSA1.82E+036.59E+035.09E+0332.00E+037.38E+039.86E+0322.07E+032.84E+033.20E+0252.05E+032.47E+032.81E+025MPSSA1.76E+035.13E+032.62E+0311.61E+038.84E+037.48E+0352.13E+032.56E+033.29E+0231.85E+032.15E+031.45E+023(e)FF18F19F20F21A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO4.41E+041.51E+051.56E+0542.11E+031.34E+049.97E+0352.28E+032.60E+031.46E+0252.37E+032.43E+032.53E+014DE1.15E+053.98E+051.94E+0551.25E+042.04E+047.83E+0362.07E+032.17E+036.17E+0112.42E+032.44E+039.74E+005GWO4.47E+044.79E+056.51E+0561.06E+046.04E+056.97E+0572.10E+032.30E+031.33E+0222.34E+032.38E+032.52E+012SSA2.34E+048.34E+053.96E+0672.22E+031.12E+048.28E+0342.33E+032.87E+032.53E+0272.49E+032.56E+034.69E+017BSSA1.26E+048.34E+046.04E+0422.12E+039.39E+031.27E+0432.44E+032.69E+031.27E+0262.46E+032.52E+033.60E+016DSSA2.90E+041.06E+055.57E+0432.07E+035.81E+034.91E+0312.17E+032.34E+031.59E+0232.20E+032.40E+031.43E+023MPSSA1.13E+044.68E+043.06E+0412.84E+037.96E+038.10E+0322.09E+032.44E+032.12E+0242.35E+032.38E+031.75E+011(f)FF22F23F24F25A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.30E+033.35E+031.77E+0342.87E+033.05E+039.78E+0152.95E+033.15E+037.74E+0152.88E+032.89E+031.39E+011DE2.82E+033.24E+032.96E+0232.76E+032.78E+036.60E+0032.95E+032.98E+031.04E+0132.89E+032.89E+039.86E−022GWO2.35E+033.48E+031.50E+0352.69E+032.76E+035.73E+0122.86E+032.91E+036.06E+0112.92E+032.95E+031.61E+017SSA2.31E+037.28E+031.18E+0372.96E+033.21E+031.68E+0273.11E+033.40E+031.20E+0272.89E+032.91E+032.00E+016BSSA2.30E+035.91E+032.44E+0362.91E+033.09E+037.70E+0163.08E+033.26E+039.03E+0162.88E+032.91E+032.07E+015DSSA2.30E+032.51E+037.99E+0222.80E+032.89E+035.85E+0142.96E+033.05E+038.57E+0142.88E+032.89E+031.42E+013MPSSA2.30E+032.46E+038.66E+0212.71E+032.75E+033.17E+0112.88E+032.95E+033.55E+0122.88E+032.91E+032.55E+014(g)FF26F27F28F29A/IBestAveStdRankBestAveStdRankBestAveStdRankBestAveStdRankPSO2.80E+035.66E+031.78E+0343.15E+033.34E+031.88E+0253.10E+033.21E+035.50E+0143.49E+033.86E+032.05E+024DE4.74E+034.88E+039.30E+0133.20E+033.21E+031.79E+0013.22E+033.23E+031.08E+0153.53E+033.64E+036.84E+011GWO4.06E+034.45E+031.67E+0213.21E+033.23E+031.41E+0123.28E+033.34E+034.10E+0173.39E+033.65E+038.43E+012SSA7.49E+038.83E+039.13E+0273.39E+033.63E+031.83E+0273.20E+033.23E+033.03E+0164.08E+035.58E+038.03E+027BSSA4.01E+037.63E+039.11E+0263.27E+033.49E+031.46E+0263.10E+033.16E+035.63E+0133.99E+034.48E+034.84E+026DSSA2.90E+035.75E+036.14E+0253.22E+033.30E+038.42E+0143.10E+033.13E+035.74E+0113.56E+034.15E+033.09E+025MPSSA2.90E+034.69E+037.16E+0223.21E+033.24E+032.04E+0133.10E+033.14E+036.68E+0123.49E+033.79E+032.23E+023(h)FF30Total rankA/IBestAveStdRankPSO3.38E+035.64E+032.50E+0313.43DE1.44E+042.63E+048.94E+0333.4GWO1.05E+063.28E+061.75E+0674.13SSA1.25E+043.28E+041.52E+0455.97BSSA7.21E+033.48E+041.09E+0564.97DSSA5.29E+039.78E+033.60E+0323.67MPSSA5.93E+032.76E+041.07E+0542.43From the results in Table4, MPSSA finds the optimal value in all algorithms in F1,F3,F4,F12,F15,F18,F22,andF28, and the average value is the closest to the theoretical optimal value in F3,F4,F12,F14,F18,F22,F23,andF28. Among the 29 test functions, MPSSA ranks in the top three except F7,F15,F20,F25,andF30. In the comprehensive ranking, MPSSA ranks first, followed by De, PSO, GWO, CSSA, BSSA, and SSA.The values in Table5 represent the p value of Wilcoxon rank sum test for the results of 30 operations. When p<0.05, it can be considered that MPSSA is significantly different from the algorithm, “+”, “=”, and “-”, respectively, represent that MPSSA is superior, equal, and inferior to the comparison algorithm [44]. It can be seen that most algorithms are inferior to the MPSSA algorithm, which shows that MPSSA is significantly different from other algorithms, and the performance is better than other algorithms, which shows that MPSSA has better advantages than other algorithms.Table 5
Wilcoxon rank sum test.
PSODEGWOSSABSSACSSAF19.99E−078.42E−012.93E−115.29E−033.50E−021.07E−02F39.59E−092.71E−112.79E−112.83E−114.70E−112.38E−11F44.93E−056.44E−112.42E−116.65E−086.92E−074.54E−01F51.00E−035.38E−011.52E−072.37E−112.51E−119.10E−07F62.61E−114.55E−124.80E−062.62E−112.56E−112.60E−11F71.76E−042.83E−012.33E−043.86E−113.46E−111.32E−09F88.18E−011.20E−057.84E−032.75E−117.89E−113.97E−03F93.75E−022.15E−111.93E−061.34E−103.59E−105.43E−07F107.96E−012.55E−111.11E−012.61E−105.98E−093.94E−07F111.97E−057.05E−011.67E−062.63E−021.99E−012.38E−01F122.57E−012.69E−112.66E−111.89E−101.95E−086.33E−09F131.45E−017.87E−094.50E−064.82E−013.62E−013.54E−01F142.21E−012.78E−112.36E−083.26E−093.15E−083.70E−01F156.84E−019.69E−051.62E−096.20E−019.23E−012.05E−01F161.26E−017.32E−071.94E−055.05E−091.14E−044.36E−04F178.19E−029.19E−101.22E−061.77E−107.57E−111.73E−06F182.25E−073.01E−114.71E−097.33E−062.58E−022.18E−06F194.01E−022.88E−089.17E−113.24E−012.50E−011.44E−01F201.15E−022.47E−077.22E−034.84E−087.04E−066.10E−02F216.18E−102.77E−116.09E−012.80E−112.69E−118.12E−03F223.78E−034.87E−104.81E−106.36E−113.40E−094.29E−01F232.72E−115.17E−067.28E−012.63E−112.74E−114.80E−11F242.43E−101.61E−054.86E−052.88E−112.65E−112.51E−07F256.05E−051.61E−062.81E−051.08E−015.58E−012.54E−05F264.79E−032.79E−025.05E−042.79E−117.52E−105.99E−09F279.94E−014.40E−095.68E−022.59E−113.29E−111.75E−05F289.85E−046.54E−052.83E−112.38E−051.14E−039.73E−03F291.70E−019.68E−031.25E−014.04E−111.05E−092.60E−05F301.40E−051.46E−092.84E−111.00E−093.63E−071.42E−02+/=/-10/0/194/0/255/0/244/0/255/0/249/0/21


## 5. UAV Cooperative Track Planning

Track planning is a problem of path optimization in continuous space, which is mainly divided into single track planning [45] and cooperative track planning [46].


### 5.1. Single UAV Track Planning Model

#### 5.1.1. Flight Fuel Cost

In the actual combat mission, the track length can reflect the fuel consumption. LetLi be the track length of ith segment and N be the number of track points [47]. That is, the fuel consumption cost of UAV flight can be expressed as the track length:
(9)fpath=∑i=1NLi.


#### 5.1.2. Flight Altitude Change Cost

In order to avoid radar search and prevent collision with mountains, the UAV must adjust the flight altitude in time. A stable flight altitude can reduce the burden of the control system and save more fuel. The variance of track altitude change can describe the stability of flight altitude, which can be expressed as(10)fheight=∑i=1Nzi−1/N∑i=1Nzi2N,wherezi is the Z coordinate of the ith track point.


#### 5.1.3. Smoothing Cost

The smaller the deflection angle during flight, the more stable the flight state of UAV and the smoother the flight path. The smoothing cost can be expressed by the change degree of deflection angleδ, and the function is set as follows:
(11)fsmooth=∑i=1Nδi−δi−1.


#### 5.1.4. Comprehensive Threat Constraint

When passing through the enemy area, UAV will encounter the enemy’s air defense system, including detection radar, air defense antiaircraft gun, ground to air missile, and other threats. The above threats are approximately regarded as a cylindrical area on the three-dimensional plane, and the detection range or attack range is taken as its radiusR. The current track segment Li is divided into five segments, M represents the comprehensive threat, a total of nM comprehensive threats are set, kM represents the current kth comprehensive threat, RkM represents the radius of the current threat, and dk,i represents the distance from the current threat point to the five equally divided segments. The threat constraint principle of the current comprehensive threat point to track segment Li is shown in Figure 7, and its threat constraint function is set as follows:
(12)Constraint=∑i=15N∑kM=1nMmaxRkM−dkM,i,0.Figure 7
Comprehensive threat constraint.Therefore, the cost function of UAV with single track is(13)minfitness=ω1·fpath+ω2·fheight+ω3·fsmooth+η·Constraint.


### 5.2. Multi-UAV Collaborative Planning Model

In the process of collaborative track planning, it is necessary to plan multiple candidate tracks that meet the flight constraints of single UAV for each UAV at the single-plane planning level in advance and then establish the cooperative constraint relationship to plan the cooperative track of multiple UAVs. Cooperative constraints mainly include spatial cooperative constraints and temporal cooperative constraints among multiple UAVs. Spatial cooperative constraint means to avoid collision between multiple UAVs while meeting the track planning of a single UAV. Time cooperative constraint means that multiple UAVs can reach the specified target point at the same time or within a certain time difference.The specific operations of multi-UAV cooperative track planning are as follows: firstly, generate multiple candidate tracks of each UAV, then calculate the time range of reaching the target point according to the minimum and maximum flight speed and track length of each UAV, and calculate the time intersection of reaching the target area between different UAVs, then determine the cooperative time range, and finally select the flight scheme corresponding to the minimum cooperative cost that meets the constraints.


#### 5.2.1. Spatial Cooperative Constraint

When large UAVs perform combat tasks in the modern battlefield, it is necessary to specify that the distance of each UAV at the same time should be greater than the safe distanceD, so as to avoid the damage caused by the collision between UAVs and ensure the UAVs to complete the combat tasks safely. Assuming that the European distance between the ith UAV and the jth UAV at a certain time is Di,j, where i,j∈1,n and n are the number of UAVs, the space constraints that multiple UAVs should meet at this time are as follows.For∀i,j∈1,n, there is D−Di,j≤0. Then, set the spatial collaboration constraint function as follows:
(14)Constraintspace=∑i=1n∑j=1nmaxD−Di,j,0.


#### 5.2.2. Time Cooperative Constraint

When multiple UAVs perform tasks cooperatively, if they cannot reach the task target area at the same time, it will not only reduce the cooperative work efficiency of UAVs but also increase the probability that a single UAV will be destroyed, resulting in the reduction of task completion rate. Therefore, it is necessary to restrict the time when UAVs arrive at the target area, that is, set the cooperative time, that is, meet the intersection of the time intervals when all UAVs arrive at the target area. When the intersection is not empty, it indicates that there is a period of time to enable the UAV to reach the target point at the same time by adjusting the flight speed [28].There aren UAVs in total, and each UAV has m alternative tracks. The jth flight path of the ith UAV is Li,j, and the speed of each UAV is vi∈vmin,vmax; then, the flight time range of the jth flight path of the ith UAV reaching the target area is Ti,j=Li,j/vmax,Li,j/vmin=Ti,jmin,Ti,jmax, Ti,jmin, and Ti,jmax respectively, represent the shortest and longest time of the jth flight path of the ith UAV reaching the target area, then the time range of the ith UAV is Ti=Ti,1∪Ti,1∪⋯∪Ti,m; then, the requirement is T=∩i=1NTi≠∅; then, there is a feasible solution in the track alternative group of the current UAV.LetTimin and Timax, respectively, be the shortest and longest time for a track selected from the candidate track group of the I UAV to reach the target area. When the intersection of the time intervals for all UAVs to reach the target area is not empty, there is a feasible solution, that is, the time cooperation constraints of multiple UAVs are
(15)maxT1min,T2min,⋯,Tmmin≤minT1max,T2max,⋯,Tmmax.Then, set the time collaboration constraint function as(16)Constrainttime=maxmaxT1min,T2min,⋯,Tmmin−minT1max,T2max,⋯,Tmmax,0.Figure8 is the schematic diagram of time cooperative constraint of three UAVs, in which each UAV has three alternative tracks. In the figure, only the same order track of each UAV is used to explain. It can be seen that the blue and purple lines have time intersection, that is, they meet the cooperative time constraint, while the red line does not meet the cooperative time constraint.Figure 8
Time collaboration constraint.


#### 5.2.3. Cooperative Cost Function

The cooperative objective function includes two parts. One part is the sum of the cost∑i=1nfitnessi of the track objective function of a flight track selected from the alternative track group by each UAV, where fitnessi is the cost of the track objective function selected by the ith aircraft. The other part is the planned coordination time cost Tarrival, in order to minimize the coordination cost of UAV, the minimum value in the intersection of coordination time is taken as Tarrival, as shown in Figure 7. Therefore, the coordination cost function is as follows.
(17)minF=∑i=1nfitnessi+Tarrival+ηConstraintspace+Constrainttime


### 5.3. Experimental Setup

The map environment adopts the establishment of three-dimensional topographic map within the range of100km×150km×3km, and the track is smoothed by cubic B-spline curve.An individualSi in the population is defined as a path connected by multiple track points, Si=sx1,sy1,sz1,sx2,⋯⋯sxn,syn,szn, in which every three constitute the three-dimensional coordinates of a track point, a total of n track points, and the dimension of each individual is 3∗n. If the starting point coordinate S is xs,ys,zs and the ending point coordinate E is xE,yE,zE, the three-dimensional coordinates of the track point di are
(18)xdi=xS+sxi·xS−xE,ydi=yS+syi·yS−yE,zdi=zdi∗+szi·zmax−zdi∗,wherezdi∗ can be obtained in the elevation map according to xdi and ydi and zmax refers to the upper bound of coordinate Z in the elevation map.


### 5.4. Experimental Simulation of Track Planning

In single UAV track planning, the coordinates of starting point and ending point are set as (10, 90, 1.1) and (130, 10, 1.1). There are five nonflying areas with comprehensive threats. The coordinates of the central points are (20, 60), (40, 80), (60, 40), (80, 60), and (100, 30), and the radius is 10 km,η=107, ω1=0.5, ω2=0.3, and ω3=0.2. The algorithm parameters, iteration times, and population number are consistent with the above experiments. The track point is set to 10 and runs independently for 20 times. Figure 9(a) is the original topographic map, (b) is the optimal three-dimensional track map of each algorithm, (c) is the contour top view, and (d) is the objective function convergence map. The objective function results after the simulation experiment are shown in Figure 10.Figure 9
Single UAV track planning.
(a)
3-D topographic map(b)
Track map(c)
Top view(d)
Cost function convergenceFigure 10
Single UAV planning results comparison.As can be seen from Figure9, MPSSA can cling to the ground, perfectly avoid the threat area, quickly get rid of the constraints, and the planned route is smoother. PSO, De, GWO, SSA, BSSA, and CSSA fall into local optimization after a certain number of iterations and cannot jump out. The planned line is obviously not a good line, while MPSSA converges faster and finds a better solution than other algorithms.Figure10 shows the results of the objective function after the above seven algorithms independently conduct track planning for 20 times. It can be seen that the four indexes of MPSSA are better than other algorithms, indicating that it has strong optimization ability and stability. Although the time consumed by the algorithm is not the shortest, it gains better performance at the cost of time.To sum up, MPSSA can quickly avoid the constraints of threat areas and complete the track planning, and the planned route is shorter and smoother than other comparison algorithms. At the same time, the convergence speed, optimization accuracy, and robustness of the algorithm are better than other algorithms.


### 5.5. Experimental Simulation of Cooperative Track Planning

In the collaborative track planning, three UAVs are selected for experiments, and there are 10 alternative tracks in each alternative track group. The coordinates of starting point 1, starting point 2, starting point 3, and ending point are set as (10, 90, 1.1), (10, 70, 2.2), (10, 90, 1.2), and (130, 10, 1.1), respectively. The settings of other parameters such as comprehensive threat are the same as above, and the safety distanceD is set as 0.2 km, Vmax=1km/s, and Vmin=0.5km/s. The objective function results after the simulation experiment are shown in Table 6. Figure 11(a) is the original topographic map, (b) is the three-dimensional route map of cooperative track planning, (c) is the top view of contour line, and (d) is the convergence map of cooperative objective function. Table 6 shows the results of UAV objective function of single UAV track planning. Table 7 shows the results of multi-UAV cooperative track planning cooperative objective function.Table 6
Track planning for each UAV.
UAV/costBestWorstAveStdUAV-174.3177.6175.961.55UAV-269.4376.1972.813.10UAV-369.1072.8770.991.75Figure 11
Collaborative track planning.
(a)
3-D topographic map(b)
Track map(c)
Top view(d)
Cost function convergenceTable 7
Collaborative track planning results.
UAV/indexCostTrack length (km)Arrival time (s)Collaborative arrival time (s)Collaborative costUAV-174.31148.05[148.05, 296.10][148.05, 275.94]360.99UAV-269.43137.99[137.99, 275.99]UAV-369.21137.97[137.97, 275.94]As can be seen from (b) and (c), MPSSA can still avoid the threat area and get rid of constraints in the case of collaborative planning of multiple UAVs to fly smoothly at a lower altitude relative to the terrain. Although UAV1 and UAV3 seem to coincide from the top view, it can be seen from (b) that the two UAVs are still outside the safe distance and have a time difference. It can be seen that due to the constraints of cooperative space and time, the tracks of UAVs are not always optimal. It can be seen from the graph that the optimal solution (SSD) can still converge faster and jump out of the optimal solution.It can be seen from Table6 that the standard deviation of the cost function value of the alternative track under MPSSA is small, and the alternative track group meeting the constraints can be planned stably for each UAV at different starting points. Table 7 shows the multi-UAV collaborative planning scheme, which shows that each UAV can reach the end point within the time range of [148.05 s, 275.94 S], and the collaborative cost is 360.99. It can be seen that the track selected by each UAV is not the best in the alternative track group, indicating that the cooperative track planning scheme is not necessarily composed of the best track in the alternative track group of each UAV but selects the relatively better track that meets the coordination requirements to perform the task.In conclusion, the MPSSA algorithm adds a variety of population search mechanisms and sets different “discoverer enrollee” ratios in different subpopulations, ensuring that each algorithm has different development and exploration capabilities. By adding dynamic dimension by dimension dynamic reverse learning, it accelerates the convergence speed of subpopulations and has more detailed search accuracy. After all subpopulations are updated, it adopts the weighted center of gravity exchange strategy, The exchange between populations and the transfer of high-quality solutions help to improve the quality of the population and jump out of the local optimum. The above strategies make MPSSA’s search mode more flexible and its comprehensive search capability enhanced, so that MPSSA can help UAVs quickly bypass threat areas and obtain better track routes under complex environments and mathematical models, and help multiple UAVs to plan better, stable, and smooth track schemes, ensure that UAVs can complete cooperative tasks safely and stably, and further verify MPSSA’s good algorithm performance.


## 5.1. Single UAV Track Planning Model

### 5.1.1. Flight Fuel Cost

In the actual combat mission, the track length can reflect the fuel consumption. LetLi be the track length of ith segment and N be the number of track points [47]. That is, the fuel consumption cost of UAV flight can be expressed as the track length:
(9)fpath=∑i=1NLi.


### 5.1.2. Flight Altitude Change Cost

In order to avoid radar search and prevent collision with mountains, the UAV must adjust the flight altitude in time. A stable flight altitude can reduce the burden of the control system and save more fuel. The variance of track altitude change can describe the stability of flight altitude, which can be expressed as(10)fheight=∑i=1Nzi−1/N∑i=1Nzi2N,wherezi is the Z coordinate of the ith track point.


### 5.1.3. Smoothing Cost

The smaller the deflection angle during flight, the more stable the flight state of UAV and the smoother the flight path. The smoothing cost can be expressed by the change degree of deflection angleδ, and the function is set as follows:
(11)fsmooth=∑i=1Nδi−δi−1.


### 5.1.4. Comprehensive Threat Constraint

When passing through the enemy area, UAV will encounter the enemy’s air defense system, including detection radar, air defense antiaircraft gun, ground to air missile, and other threats. The above threats are approximately regarded as a cylindrical area on the three-dimensional plane, and the detection range or attack range is taken as its radiusR. The current track segment Li is divided into five segments, M represents the comprehensive threat, a total of nM comprehensive threats are set, kM represents the current kth comprehensive threat, RkM represents the radius of the current threat, and dk,i represents the distance from the current threat point to the five equally divided segments. The threat constraint principle of the current comprehensive threat point to track segment Li is shown in Figure 7, and its threat constraint function is set as follows:
(12)Constraint=∑i=15N∑kM=1nMmaxRkM−dkM,i,0.Figure 7
Comprehensive threat constraint.Therefore, the cost function of UAV with single track is(13)minfitness=ω1·fpath+ω2·fheight+ω3·fsmooth+η·Constraint.


## 5.1.1. Flight Fuel Cost

In the actual combat mission, the track length can reflect the fuel consumption. LetLi be the track length of ith segment and N be the number of track points [47]. That is, the fuel consumption cost of UAV flight can be expressed as the track length:
(9)fpath=∑i=1NLi.


## 5.1.2. Flight Altitude Change Cost

In order to avoid radar search and prevent collision with mountains, the UAV must adjust the flight altitude in time. A stable flight altitude can reduce the burden of the control system and save more fuel. The variance of track altitude change can describe the stability of flight altitude, which can be expressed as(10)fheight=∑i=1Nzi−1/N∑i=1Nzi2N,wherezi is the Z coordinate of the ith track point.


## 5.1.3. Smoothing Cost

The smaller the deflection angle during flight, the more stable the flight state of UAV and the smoother the flight path. The smoothing cost can be expressed by the change degree of deflection angleδ, and the function is set as follows:
(11)fsmooth=∑i=1Nδi−δi−1.


## 5.1.4. Comprehensive Threat Constraint

When passing through the enemy area, UAV will encounter the enemy’s air defense system, including detection radar, air defense antiaircraft gun, ground to air missile, and other threats. The above threats are approximately regarded as a cylindrical area on the three-dimensional plane, and the detection range or attack range is taken as its radiusR. The current track segment Li is divided into five segments, M represents the comprehensive threat, a total of nM comprehensive threats are set, kM represents the current kth comprehensive threat, RkM represents the radius of the current threat, and dk,i represents the distance from the current threat point to the five equally divided segments. The threat constraint principle of the current comprehensive threat point to track segment Li is shown in Figure 7, and its threat constraint function is set as follows:
(12)Constraint=∑i=15N∑kM=1nMmaxRkM−dkM,i,0.Figure 7
Comprehensive threat constraint.Therefore, the cost function of UAV with single track is(13)minfitness=ω1·fpath+ω2·fheight+ω3·fsmooth+η·Constraint.


## 5.2. Multi-UAV Collaborative Planning Model

In the process of collaborative track planning, it is necessary to plan multiple candidate tracks that meet the flight constraints of single UAV for each UAV at the single-plane planning level in advance and then establish the cooperative constraint relationship to plan the cooperative track of multiple UAVs. Cooperative constraints mainly include spatial cooperative constraints and temporal cooperative constraints among multiple UAVs. Spatial cooperative constraint means to avoid collision between multiple UAVs while meeting the track planning of a single UAV. Time cooperative constraint means that multiple UAVs can reach the specified target point at the same time or within a certain time difference.The specific operations of multi-UAV cooperative track planning are as follows: firstly, generate multiple candidate tracks of each UAV, then calculate the time range of reaching the target point according to the minimum and maximum flight speed and track length of each UAV, and calculate the time intersection of reaching the target area between different UAVs, then determine the cooperative time range, and finally select the flight scheme corresponding to the minimum cooperative cost that meets the constraints.


### 5.2.1. Spatial Cooperative Constraint

When large UAVs perform combat tasks in the modern battlefield, it is necessary to specify that the distance of each UAV at the same time should be greater than the safe distanceD, so as to avoid the damage caused by the collision between UAVs and ensure the UAVs to complete the combat tasks safely. Assuming that the European distance between the ith UAV and the jth UAV at a certain time is Di,j, where i,j∈1,n and n are the number of UAVs, the space constraints that multiple UAVs should meet at this time are as follows.For∀i,j∈1,n, there is D−Di,j≤0. Then, set the spatial collaboration constraint function as follows:
(14)Constraintspace=∑i=1n∑j=1nmaxD−Di,j,0.


### 5.2.2. Time Cooperative Constraint

When multiple UAVs perform tasks cooperatively, if they cannot reach the task target area at the same time, it will not only reduce the cooperative work efficiency of UAVs but also increase the probability that a single UAV will be destroyed, resulting in the reduction of task completion rate. Therefore, it is necessary to restrict the time when UAVs arrive at the target area, that is, set the cooperative time, that is, meet the intersection of the time intervals when all UAVs arrive at the target area. When the intersection is not empty, it indicates that there is a period of time to enable the UAV to reach the target point at the same time by adjusting the flight speed [28].There aren UAVs in total, and each UAV has m alternative tracks. The jth flight path of the ith UAV is Li,j, and the speed of each UAV is vi∈vmin,vmax; then, the flight time range of the jth flight path of the ith UAV reaching the target area is Ti,j=Li,j/vmax,Li,j/vmin=Ti,jmin,Ti,jmax, Ti,jmin, and Ti,jmax respectively, represent the shortest and longest time of the jth flight path of the ith UAV reaching the target area, then the time range of the ith UAV is Ti=Ti,1∪Ti,1∪⋯∪Ti,m; then, the requirement is T=∩i=1NTi≠∅; then, there is a feasible solution in the track alternative group of the current UAV.LetTimin and Timax, respectively, be the shortest and longest time for a track selected from the candidate track group of the I UAV to reach the target area. When the intersection of the time intervals for all UAVs to reach the target area is not empty, there is a feasible solution, that is, the time cooperation constraints of multiple UAVs are
(15)maxT1min,T2min,⋯,Tmmin≤minT1max,T2max,⋯,Tmmax.Then, set the time collaboration constraint function as(16)Constrainttime=maxmaxT1min,T2min,⋯,Tmmin−minT1max,T2max,⋯,Tmmax,0.Figure8 is the schematic diagram of time cooperative constraint of three UAVs, in which each UAV has three alternative tracks. In the figure, only the same order track of each UAV is used to explain. It can be seen that the blue and purple lines have time intersection, that is, they meet the cooperative time constraint, while the red line does not meet the cooperative time constraint.Figure 8
Time collaboration constraint.


### 5.2.3. Cooperative Cost Function

The cooperative objective function includes two parts. One part is the sum of the cost∑i=1nfitnessi of the track objective function of a flight track selected from the alternative track group by each UAV, where fitnessi is the cost of the track objective function selected by the ith aircraft. The other part is the planned coordination time cost Tarrival, in order to minimize the coordination cost of UAV, the minimum value in the intersection of coordination time is taken as Tarrival, as shown in Figure 7. Therefore, the coordination cost function is as follows.
(17)minF=∑i=1nfitnessi+Tarrival+ηConstraintspace+Constrainttime


## 5.2.1. Spatial Cooperative Constraint

When large UAVs perform combat tasks in the modern battlefield, it is necessary to specify that the distance of each UAV at the same time should be greater than the safe distanceD, so as to avoid the damage caused by the collision between UAVs and ensure the UAVs to complete the combat tasks safely. Assuming that the European distance between the ith UAV and the jth UAV at a certain time is Di,j, where i,j∈1,n and n are the number of UAVs, the space constraints that multiple UAVs should meet at this time are as follows.For∀i,j∈1,n, there is D−Di,j≤0. Then, set the spatial collaboration constraint function as follows:
(14)Constraintspace=∑i=1n∑j=1nmaxD−Di,j,0.


## 5.2.2. Time Cooperative Constraint

When multiple UAVs perform tasks cooperatively, if they cannot reach the task target area at the same time, it will not only reduce the cooperative work efficiency of UAVs but also increase the probability that a single UAV will be destroyed, resulting in the reduction of task completion rate. Therefore, it is necessary to restrict the time when UAVs arrive at the target area, that is, set the cooperative time, that is, meet the intersection of the time intervals when all UAVs arrive at the target area. When the intersection is not empty, it indicates that there is a period of time to enable the UAV to reach the target point at the same time by adjusting the flight speed [28].There aren UAVs in total, and each UAV has m alternative tracks. The jth flight path of the ith UAV is Li,j, and the speed of each UAV is vi∈vmin,vmax; then, the flight time range of the jth flight path of the ith UAV reaching the target area is Ti,j=Li,j/vmax,Li,j/vmin=Ti,jmin,Ti,jmax, Ti,jmin, and Ti,jmax respectively, represent the shortest and longest time of the jth flight path of the ith UAV reaching the target area, then the time range of the ith UAV is Ti=Ti,1∪Ti,1∪⋯∪Ti,m; then, the requirement is T=∩i=1NTi≠∅; then, there is a feasible solution in the track alternative group of the current UAV.LetTimin and Timax, respectively, be the shortest and longest time for a track selected from the candidate track group of the I UAV to reach the target area. When the intersection of the time intervals for all UAVs to reach the target area is not empty, there is a feasible solution, that is, the time cooperation constraints of multiple UAVs are
(15)maxT1min,T2min,⋯,Tmmin≤minT1max,T2max,⋯,Tmmax.Then, set the time collaboration constraint function as(16)Constrainttime=maxmaxT1min,T2min,⋯,Tmmin−minT1max,T2max,⋯,Tmmax,0.Figure8 is the schematic diagram of time cooperative constraint of three UAVs, in which each UAV has three alternative tracks. In the figure, only the same order track of each UAV is used to explain. It can be seen that the blue and purple lines have time intersection, that is, they meet the cooperative time constraint, while the red line does not meet the cooperative time constraint.Figure 8
Time collaboration constraint.


## 5.2.3. Cooperative Cost Function

The cooperative objective function includes two parts. One part is the sum of the cost∑i=1nfitnessi of the track objective function of a flight track selected from the alternative track group by each UAV, where fitnessi is the cost of the track objective function selected by the ith aircraft. The other part is the planned coordination time cost Tarrival, in order to minimize the coordination cost of UAV, the minimum value in the intersection of coordination time is taken as Tarrival, as shown in Figure 7. Therefore, the coordination cost function is as follows.
(17)minF=∑i=1nfitnessi+Tarrival+ηConstraintspace+Constrainttime


## 5.3. Experimental Setup

The map environment adopts the establishment of three-dimensional topographic map within the range of100km×150km×3km, and the track is smoothed by cubic B-spline curve.An individualSi in the population is defined as a path connected by multiple track points, Si=sx1,sy1,sz1,sx2,⋯⋯sxn,syn,szn, in which every three constitute the three-dimensional coordinates of a track point, a total of n track points, and the dimension of each individual is 3∗n. If the starting point coordinate S is xs,ys,zs and the ending point coordinate E is xE,yE,zE, the three-dimensional coordinates of the track point di are
(18)xdi=xS+sxi·xS−xE,ydi=yS+syi·yS−yE,zdi=zdi∗+szi·zmax−zdi∗,wherezdi∗ can be obtained in the elevation map according to xdi and ydi and zmax refers to the upper bound of coordinate Z in the elevation map.


## 5.4. Experimental Simulation of Track Planning

In single UAV track planning, the coordinates of starting point and ending point are set as (10, 90, 1.1) and (130, 10, 1.1). There are five nonflying areas with comprehensive threats. The coordinates of the central points are (20, 60), (40, 80), (60, 40), (80, 60), and (100, 30), and the radius is 10 km,η=107, ω1=0.5, ω2=0.3, and ω3=0.2. The algorithm parameters, iteration times, and population number are consistent with the above experiments. The track point is set to 10 and runs independently for 20 times. Figure 9(a) is the original topographic map, (b) is the optimal three-dimensional track map of each algorithm, (c) is the contour top view, and (d) is the objective function convergence map. The objective function results after the simulation experiment are shown in Figure 10.Figure 9
Single UAV track planning.
(a)
3-D topographic map(b)
Track map(c)
Top view(d)
Cost function convergenceFigure 10
Single UAV planning results comparison.As can be seen from Figure9, MPSSA can cling to the ground, perfectly avoid the threat area, quickly get rid of the constraints, and the planned route is smoother. PSO, De, GWO, SSA, BSSA, and CSSA fall into local optimization after a certain number of iterations and cannot jump out. The planned line is obviously not a good line, while MPSSA converges faster and finds a better solution than other algorithms.Figure10 shows the results of the objective function after the above seven algorithms independently conduct track planning for 20 times. It can be seen that the four indexes of MPSSA are better than other algorithms, indicating that it has strong optimization ability and stability. Although the time consumed by the algorithm is not the shortest, it gains better performance at the cost of time.To sum up, MPSSA can quickly avoid the constraints of threat areas and complete the track planning, and the planned route is shorter and smoother than other comparison algorithms. At the same time, the convergence speed, optimization accuracy, and robustness of the algorithm are better than other algorithms.


## 5.5. Experimental Simulation of Cooperative Track Planning

In the collaborative track planning, three UAVs are selected for experiments, and there are 10 alternative tracks in each alternative track group. The coordinates of starting point 1, starting point 2, starting point 3, and ending point are set as (10, 90, 1.1), (10, 70, 2.2), (10, 90, 1.2), and (130, 10, 1.1), respectively. The settings of other parameters such as comprehensive threat are the same as above, and the safety distanceD is set as 0.2 km, Vmax=1km/s, and Vmin=0.5km/s. The objective function results after the simulation experiment are shown in Table 6. Figure 11(a) is the original topographic map, (b) is the three-dimensional route map of cooperative track planning, (c) is the top view of contour line, and (d) is the convergence map of cooperative objective function. Table 6 shows the results of UAV objective function of single UAV track planning. Table 7 shows the results of multi-UAV cooperative track planning cooperative objective function.Table 6
Track planning for each UAV.
UAV/costBestWorstAveStdUAV-174.3177.6175.961.55UAV-269.4376.1972.813.10UAV-369.1072.8770.991.75Figure 11
Collaborative track planning.
(a)
3-D topographic map(b)
Track map(c)
Top view(d)
Cost function convergenceTable 7
Collaborative track planning results.
UAV/indexCostTrack length (km)Arrival time (s)Collaborative arrival time (s)Collaborative costUAV-174.31148.05[148.05, 296.10][148.05, 275.94]360.99UAV-269.43137.99[137.99, 275.99]UAV-369.21137.97[137.97, 275.94]As can be seen from (b) and (c), MPSSA can still avoid the threat area and get rid of constraints in the case of collaborative planning of multiple UAVs to fly smoothly at a lower altitude relative to the terrain. Although UAV1 and UAV3 seem to coincide from the top view, it can be seen from (b) that the two UAVs are still outside the safe distance and have a time difference. It can be seen that due to the constraints of cooperative space and time, the tracks of UAVs are not always optimal. It can be seen from the graph that the optimal solution (SSD) can still converge faster and jump out of the optimal solution.It can be seen from Table6 that the standard deviation of the cost function value of the alternative track under MPSSA is small, and the alternative track group meeting the constraints can be planned stably for each UAV at different starting points. Table 7 shows the multi-UAV collaborative planning scheme, which shows that each UAV can reach the end point within the time range of [148.05 s, 275.94 S], and the collaborative cost is 360.99. It can be seen that the track selected by each UAV is not the best in the alternative track group, indicating that the cooperative track planning scheme is not necessarily composed of the best track in the alternative track group of each UAV but selects the relatively better track that meets the coordination requirements to perform the task.In conclusion, the MPSSA algorithm adds a variety of population search mechanisms and sets different “discoverer enrollee” ratios in different subpopulations, ensuring that each algorithm has different development and exploration capabilities. By adding dynamic dimension by dimension dynamic reverse learning, it accelerates the convergence speed of subpopulations and has more detailed search accuracy. After all subpopulations are updated, it adopts the weighted center of gravity exchange strategy, The exchange between populations and the transfer of high-quality solutions help to improve the quality of the population and jump out of the local optimum. The above strategies make MPSSA’s search mode more flexible and its comprehensive search capability enhanced, so that MPSSA can help UAVs quickly bypass threat areas and obtain better track routes under complex environments and mathematical models, and help multiple UAVs to plan better, stable, and smooth track schemes, ensure that UAVs can complete cooperative tasks safely and stably, and further verify MPSSA’s good algorithm performance.


## 6. Conclusion

In this paper, a sparrow search algorithm based on personalized multipopulation is proposed to solve the original defects of SSA and help UAV complete complex cooperative track planning. Through the introduction of multipopulation mechanism, personalized subpopulation strategy, weighted center of gravity communication strategy, and dimension by dimension dynamic reverse learning, it can prevent falling into local optimization, enhance the personalized difference of subpopulation, improve the quality of population communication, accelerate convergence and improve optimization accuracy, and effectively improve the performance of the original SSA algorithm. The experimental results of benchmark function and CEC2017 show that MPSSA can stably find better solutions than PSO, GWO, DE, SSA, and SSA improved algorithms, has good advantages, and can adapt to more complex optimization problems. Through the simulation experiment of UAV cooperative track, the results show that MPSSA is helpful to the static planning of UAV cooperative track, with shorter time, better planning path, and better stability.At the same time, there is still a lot of work to be done in the future, such as checking the impact on the performance of the algorithm when more constraints are added. The algorithm is applied to more complex dynamic path planning and real-time obstacle avoidance. The algorithm is applied to multiobjective programming problem and so on.



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*Source: 2521737-2022-06-29.xml*