Introduction
With the rapid improvement of functionality and performance, drones, i.e., Unmanned Aerial Vehicles (UAVs), have been widely used in various applications, especially in areas that are too dangerous or inaccessible to humans[1], e.g., disaster sites and battlefields. By dynamically grouping a number of drones that interact with one another to complete common tasks cooperatively and autonomously, a drone swarm can have significantly higher flexibility, scalability, survivability, and cost-effectiveness than the simple accumulation of individual drones[2]. In recent years, drone swarms have demonstrated great superiority in modern battlefields, such as in Syria, Nagorno-Karabakh, and Ukraine.
The development of military attack and defense technologies is similar to the relationship between a spear and a shield. The aforementioned advantages of the “spears” of drone swarms have posed great challenges to the “shields” of their opponents, who need to protect against the invasion of drone swarms by reducing their mission capabilities as much as possible. However, current research on countermeasures against drone swarms, such as Global Positioning System (GPS) spoofing, electromagnetic interference, false target jamming[3], [4], and anti-aircraft weapons[5], is still limited and relatively simple. Compared to these spoofing and interference measures, anti-aircraft weapons are more active and aggressive in causing damage to drones and are, therefore, more appealing to the opponents of drones. However, the flexibility and survivability of drone swarms make it difficult or impossible for a single air defense system to bring substantial damage to a swarm at a time. Typically, if one or several drones are damaged by an attack from the air defense system, the remaining drones can quickly escape from the attack area and reform the swarm that still has a majority of the original mission capability. Therefore, how to deploy available air defense systems to effectively defend against drone swarms is a challenging problem, which is important for the safety and security of the defending side.
In this paper, we study an optimization problem for deploying air defense systems against enemy reconnaissance drone swarms. Given a set of available air defense systems, the problem determines the location of each air defense system in a predetermined region where a reconnaissance drone swarm will pass through. Whenever the swarm is threatened/attacked by an air defense system, it replans the path to the target, as illustrated by the flowchart in Fig. 1. Therefore, we need to optimize the deployment of air defense systems, such that the cost for the enemy drone swarm to pass through the region will be maximized. We calculate the cost based on a counterpart drone path planning problem. To solve this adversarial problem, we first present an iterative search algorithm, which can produce exact optimal solutions to small-size problem instances; then, we propose an evolutionary framework for obtaining optimal or near-optimal solutions to large-size instances. We implement the evolutionary framework with six popular evolutionary algorithms, namely Genetic Algorithm (GA) with variable mutation rate[6], adaptive Particle Swarm Optimization (PSO) using comprehensive learning and self-adaptive parameters[7], Ecogeography-Based Optimization (EBO)[8], Quantum-inspired Tabu Search (QTS)[9], dual-strategy Differential Evolution (DE)[10], and Water Wave Optimization (WWO)[11]. We conducted computational test on a set of problem instances, the results of which validate the effectiveness of the method for defending against reconnaissance drone swarms. Among the six algorithms, EBO and WWO exhibit more competitive performance than the other ones. The main contributions of this work can be stated as follows:
It presents a problem of deploying air defense systems against drone swarms, which, to our best knowledge, has not been addressed before.
It proposes an exact iterative algorithm and an evolutionary framework with operators adapted for the problem, which are efficient for solving small- and large-size problem instances, respectively.
It conducts tests to validate the effectiveness of the proposed approach for improving the defensive abilities against reconnaissance drone swarms.
In the remainder of this paper, Section 2 reviews related work, Section 3 formulates the air defense system deployment optimization problem based on a basic drone path planning problem, Section 4 presents the exact iterative search algorithm, Section 5 describes the evolutionary framework and its typical implementations, Section 6 presents the computational experiments, and finally Section 7 concludes this paper.
Basic flow of the invading drone swarm and the opponent for defending against the swarm.
Related Work
The considered air defense system deployment problem can be classified as a facility location problem which is known to be NP-hard[12]. Exact algorithms are only applicable to small-size instances[13], [14]. Many recent studies have been conducted on heuristic and evolutionary algorithms for finding near-optimal solutions. Teran-Somohano and Smith[15] used a bi-objective evolutionary strategy algorithm to solve a semi-obnoxious and multiple capacitated facility location problem that often arises in public planning. Zheng et al.[16] presented a master-slave evolutionary algorithm for a problem of integrated civilian-military emergency supply pre-positioning. Wang et al.[17] presented a dual-population evolutionary algorithm to solve a facility location problem with two objectives on reliability and coverage under the uncertainty of facilities. Vansia and Dhodiya[18] utilized non-dominated sorting genetic algorithm and modified self-adaptive multi-population elitism Jaya algorithm for a multi-objective transportation-p-facility location problem that minimizes the overall transportation time, cost of transportation, and carbon emission. Zhang et al.[19] proposed an enhanced group theory based evolutionary algorithm to solve the uncapacitated facility location problem. Eriskin et al.[20] studied a robust multi-objective model for the location of hospitals during pandemics. Karatas et al.[21] surveyed the facility location models and solution techniques for military organizations.
The present problem aims to maximize the effectiveness of defense against enemy targets. In this sense, the problem is similar to a subclass of Weapon Target Assignment (WTA) problems[22] that aim to maximize the total expected damage or total cost of the targets. Observing that exact algorithms[23], [24] are only applicable to small-size WTA instances, most recent research efforts have been devoted to heuristic and metaheuristic algorithms, including very large neighborhood search[23], [25], ant colony optimization[26], [27], GA[28], PSO[29], eminent domain[30], etc., to efficiently solve medium- and large-size WTA instances. Moreover, our problem considers that a drone swarm can dynamically reform and replan after being attacked and hence is closer to dynamic WTA[31] which uses a sequence of decisions to tackle the dynamic change of targets and is much more complex than its static countcrpart[32], [33].
Although there are many research work on general WTA problems, studies focusing on air defense system deployment are relatively limited. Han and Shi[34] proposed a simulated annealing algorithm for optimizing the deployment of air defense missile systems, where the combat effectiveness of defense systems is evaluated based on the stochastic service system theory. Wang and GuO[35] studied the disposition of air defense systems at the company level in two scenarios: one using a single system to protect multiple targets, and the other using multiple systems to protect a single target. They developed linear programming and dynamic programming methods to solve the problems. Yu et al.[36] proposed an artificial neural network approach to weapon system configuration, which first uses a backtracking network to approximate the system effectiveness, and then translates the problem to the traveling salesman problem which is solved by Hopfield network. Wang and Pan[37] considered an air defense disposition problem with uncertainties and risks measured by fuzzy entropy, and they presented a hybrid intelligent algorithm for the problem. Han et al.[38] proposed two air defense system configuration models based on integer programming, one for firing range covering and the other for firing angle covering, and they proved that the layout solution obtained with the firing angle concept is more efficient.
Currently, few studies have been conducted on air defense system deployment against drone swarms, which is much more difficult than traditional anti-aircraft air defense system deployment because drone swarms are much more flexible in reforming and replanning than traditional aircraft. To explore drone vulnerability to deceptive GPS signals, Kerns et al.[39] established necessary conditions for drone capture via GPS spoofing and explored the spoofer's range of possible post-capture control over drones. Considering a problem of using a team of mini drones acting as a cooperative defensive system against enemy unmanned aerial systems, Castrillo et al.[5] studied sensing, mitigation, and command and control technologies to realize such a defense system. Su et al. [40] studied a problem of false target jamming against UAVs, where each false target jamming solution is evaluated based on its adversarial effects on possible VA V detection solutions. To the best of our knowledge, the problem of air defense system deployment against drone swarms has not been addressed before.
Problem Formulation
In this section, we formulate the air defense system deployment optimization problem, which aims to maximize the cost for enemy reconnaissance drones to pass through a predetermined region. We first describe a basic drone path planning problem from the viewpoint of the holders of drones, and then describe the adversarial defense problem from the viewpoint of the opponents of drones.
3.1 Drone Path Planning
First consider the basic drone path planning problem in three-dimensional space[41], [42], where a swarm of drones needs to traverse a predetermined region. According to the terrain of the region, a total number
• Length of the Path
The longer the path, the higher the cost. We calculate the length cost as the ratio of the total length to the Euclidean distance between \begin{equation*}
L(\boldsymbol{P})=\frac{1}{\vert S,T\vert }\sum_{i=0}^{n_{P}-1}\vert P_{i},P_{i+1}\vert
\tag{1}
\end{equation*}
• Height of the Path
The higher the path, the higher the cost. This is because flying at a low flight height can improve the reconnaissance efficiency and reduce the risk of being discovered. We calculate the height cost as the ratio of the total height to \begin{equation*}
H(\boldsymbol{P})=\frac{1}{z_{\max}-z_{\max}}\sum_{i=0}^{n_{P}-1}(\overline{z}(P_{i},P_{i+1})-z_{\min})
\tag{2}
\end{equation*}
• Excessive Horizontal Rotation of the Path
If a horizontal rotation angle is larger than a predefined threshold \begin{equation*}
R(\boldsymbol{P})=\frac{1}{n_{P}\widehat{\omega}}\sum_{i=0}^{n_{P}-2}\max(\omega(P_{i},P_{i+1},P_{i+2})-\widehat{\omega},0)
\tag{3}
\end{equation*}
• Excessive Vertical Inclination Angle of the Path
If an inclination angle is larger than a predefined threshold \begin{equation*}
S(\boldsymbol{P})=\frac{1}{n_{P}\widehat{\theta}}\sum_{i=0}^{n_{P}-2}\max(\theta(P_{i},P_{i+1},P_{i+2})-\widehat{\theta,}0)
\tag{4}
\end{equation*}
The objective function of drone path planning is defined as the weighted sum of the above costs,\begin{equation*}
\min C(\boldsymbol{P})=L(\boldsymbol{P})+w_{H}H(\boldsymbol{P})+w_{R}R(\boldsymbol{P})+w_{S}S(\boldsymbol{P})
\tag{5}
\end{equation*}
3.2 Air Defense System Deployment Against Reconnaissance Drone Swarms
First, we assume that the holder of reconnaissance drones can always obtain the optimal path
Let \begin{equation*}
E_{1}(\boldsymbol{P})=\frac{1}{\vert S,T\vert }\sum_{i=0}^{n_{P}-1}\Lambda_{1}(P_{i},P_{i+1})
\tag{6}
\end{equation*}
Illustration of the path replanning of a drone swarm after being intercepted by air defense systems. Solid (black) line: original path; dash (green) line: first replanned path; dotted (blue) line: second replanned path; red circle: defense range.
Consequently, the objective function of drone path replanning after being intercepted by the first air defense system should incur the fire threat cost in the following:\begin{equation*}
\min C_{1}(\boldsymbol{P})=C(\boldsymbol{P})+w_{E}E_{1}(\boldsymbol{P})
\tag{7}
\end{equation*}
Let
By analog, after placing the j-th air defense system on the path \begin{equation*}
E_{j}(\boldsymbol{P})=\frac{1}{\vert S,T\vert }\sum_{i=0}^{n_{P}-1}\Lambda_{j}(P_{i},P_{i+1})
\tag{8}
\end{equation*}
\begin{equation*}
\min C_{j}(\boldsymbol{P})=C(\boldsymbol{P})+w_{E}E_{j}(\boldsymbol{P})
\tag{9}
\end{equation*}
From the viewpoint of the opponent of the drone swarm, the air defense system deployment optimization problem is to determine \begin{align*}
\max f(A)= & C(\boldsymbol{P}^*[S, S_1])+ \\
& \sum_{j=1}^{m-1} C_j(\boldsymbol{P}^{(j)}[S_j, S_{j+1}])+C(\boldsymbol{P}^{(m)}[S_m, T])
\tag{10}
\end{align*}
In practice, some areas are not suitable for deploying air defense systems due to terrain constraints. We use
Exact Iterative Search Algorithm
To solve the above air defense system deployment problem, we first propose an iterative search algorithm for finding the exact optimal solution to the problem. Suppose that the previous \begin{align*}
\max f(A_m)= & C_{m-1}(\boldsymbol{P}^{(m-1)}[S_{m-1}, S_m])+ \\
& C_m(\boldsymbol{P}^{(m)}[S_m, T])
\tag{11}
\end{align*}
The position
Next, suppose that the previous \begin{align*}
& \max f(A_{m-1})=C_{m-2}(\boldsymbol{P}^{(m-2)}[S_{m-2}, S_{m-1}])+ \\
& C_{m-1}(\boldsymbol{P}^{(m-1)}[S_{m-1}, S_m])+C_m(\boldsymbol{P}^{(m)}[S_m, T])
\tag{12}
\end{align*}
The procedure FindA
By analogy, we can derive the procedures for determining the positions
As illustrated in Fig. 3, FindA \begin{align*}
& O(C(N, m) O(A^{\star}))= \\
& O(N(N-1)(N-2) \cdots(N-m+1) O(A^{\star})),
\end{align*}
Evolutionary Algorithm
As the running time of the above exact optimization algorithm increases exponentially with
To efficiently solve the air defense system deployment problem, we design a specific encoding and decoding scheme. Each solution to the problem is encoded as an m-dimensional real-valued vector
Step 1:
Find out the set of all candidate positions whose defense range intersects with the path
, and sort these positions in order of their intersections with the path from front to back (i.e., the closer the intersection to the starting point\boldsymbol{P}^{\ast} , the higher the rank;S Step 2:
Let
be the number of candidate positions andN_{1} be the integer closest tok ; take the k-th candidate position to place the first air defense system;N_{1}x_{1} Step 3:
Let
;j=1 Step 4:
Replan the drone path
based on\boldsymbol{P}^{(j)} ;A_{j} Step 5:
Find out the set of all candidate positions whose defense range intersects with the path
, and sort these positions in order of their intersections with the path from front to back;\boldsymbol{P}^{(j)} Step 6:
Let
be the number of candidate positions andN_{j+1} be the integer closest tok ; take the k-th candidate position to place theN_{j+1}x_{j+1} -th air defense system;j Step 7:
Set
ifj=j+;1 , then stop; otherwise, go to Step 4.j > m
The above scheme encodes the position of each air defense system as the rank of its intersection with the path of the drone swarm to relate the position to the anti-drone task. The use of real-valued encoding makes it easy to adapt existing evolutionary algorithms for continuous optimization to our problem.
In the following subsections, we use GA[6], PSO[7], EBO[8], QTS[9], DE[10], and WWO[11] to implement the framework.
5.1 Ga With Variable Mutation Rate
GA performs a stochastic search based on the principle of genetic crossover and mutation. On the basis of the above encoding/decoding scheme, we use the breeder crossover operation[43] that generates offsprings \begin{align*}
& x_j^c=r x_j^a+(1-r) x_j^b, \\
&\ x_j^d=(1-r) x_j^a+r x_j^b
\tag{13}
\end{align*}
Another key difference of our algorithm from classical GA is that it uses a variable mutation rate \begin{equation*}
r_{\boldsymbol{m}}(\boldsymbol{x})=\frac{f(\boldsymbol{x})}{f_{\max}}r_{m}^{\max}
\tag{14}
\end{equation*}
Algorithm 4 presents the pseudo-code of the GA, where
5.2 Adaptive Comprehensive Learning PSO
PSO[44] associates each solution (particle) \begin{equation*}
x_{i,j}=x_{i,j}+v_{i,j}
\tag{15}
\end{equation*}
In the classical PSO, each velocity vector is adjusted by learning from both the personal historical best \begin{equation*}
v_{i,j}=wv_{i,j}+c\times rand(0,1)(pbest_{i^{\prime},j}-x_{i,j})
\tag{16}
\end{equation*}
The exemplar solution
Algorithm 5 presents the pseudo-code of the adaptive comprehensive learning PSO algorithm for the problem. The average time complexity of the algorithm is
5.3 Ebo
EBO is an extended version of Biogeography-Based Optimization (BBO)[46], which associates each solution \begin{align*}
& r_\mu(\boldsymbol{x})=\frac{f_{\max }-f(\boldsymbol{x})+\epsilon}{f_{\max }-f_{\min }+\epsilon}, \\
&\ \ r_v(\boldsymbol{x})=\frac{f(\boldsymbol{x})-f_{\min }+\epsilon}{f_{\max }-f_{\min }+\epsilon}
\tag{17}
\end{align*}
Compared to the basic BBO, EBO employs a local neighborhood structure[47] and differentiates local migration and global migration. Local migration sets the current dimension of the immigrating solution \begin{equation*}
x_{j}=x_{j}+rand(0,1)(x_{j}^{\prime}-x_{j})
\tag{18}
\end{equation*}
Global migration selects two emigrating solutions \begin{equation*}
x_j=\begin{cases}
x_j^{\prime}+rand(0,1)(x_j^{\prime \prime}-x_j), & f(\boldsymbol{x}^{\prime}) > f(\boldsymbol{x}^{\prime \prime}); \\
x_j^{\prime \prime}+rand(0,1)(x_j^{\prime}-x_j), & f(\boldsymbol{x}^{\prime}) < f(\boldsymbol{x}^{\prime \prime})
\end{cases}
\tag{19}
\end{equation*}
Whether a migration operation is a local or global migration is determined by a parameter
Algorithm 6 presents the pseudo-code of the EBO algorithm, the time complexity of which is
5.4 Qts
Tabu search[48] is an extension of basic local search using short-term memory to save recently obtained local optimal solutions to avoid repeated searches. QTS[9] introduces quantum-inspired bits and gates into the algorithm to suppress premature convergence and better balance exploration and exploitation. The QTS algorithm first initializes a quantum matrix
Algorithm 7 presents the pseudo-code of the QTS algorithm, the time complexity of which is
5.5 Dual-Strategy Differential Evolution
DE is a simple but fast evolutionary algorithm that evolves solutions according to the difference between them[49]. At each iteration, a mutation vector is produced for each solution by adding the difference between two randomly selected solutions to a third one; a trial vector is then produced by the crossover of the solution and its mutation vector; finally, the better one between the solution and the trial vector is chosen for the next generation.
Dual-strategy DE[10] uses multiple sub-populations of solution. In each sub-population, the solutions are sorted in decreasing order of fitness, and the following DE/rand/1 mutation scheme is applied to the first half of the solutions:\begin{equation*}
\boldsymbol{v}_{j}=\boldsymbol{x}_{r_{1}}+F(\boldsymbol{x}_{r_{2}}-\boldsymbol{x}_{r_{3}})
\tag{20}
\end{equation*}
The following DE/lbest/1 mutation scheme is applied to the second half of the solutions:\begin{equation*}
\boldsymbol{v}_{i}=\boldsymbol{x}_{lbest}+F(\boldsymbol{x}_{r_{1}}-\boldsymbol{x}_{r_{2}})
\tag{21}
\end{equation*}
Algorithm 8 presents the pseudo-code of the dual-strategy DE algorithm, the time complexity of which is
5.6 Wwo
Inspired by the shallow water wave theory, WWO assigns each solution \begin{equation*}
\lambda(\boldsymbol{x})=\lambda(\boldsymbol{x}) \alpha^{-(f(x)-f_{\min }+\epsilon) /(f_{\max }-f_{\min }+\epsilon)}
\tag{22}
\end{equation*}
At each iteration, WWO uses a propagation operation to produce an offspring \begin{equation*}
x_{j}^{\prime}=x_{j}+rand(-1,1)\lambda(\boldsymbol{x})L_{j}
\tag{23}
\end{equation*}
WWO also performs a breaking operation on each newly found best solution \begin{equation*}
x_{j}^{\prime}=x_{j}^{\ast}+N(0,1)\beta L_{j}
\tag{24}
\end{equation*}
We use a variable population size[50], which linearly decreases from
Algorithm 9 presents the pseudo-code of the WWO algorithm. The time complexity of the algorithm is
Computational Experiment
The experiments were conducted on eight test instances, which were constructed based on four selected maps with different topographic features in South East China. Table 1 presents the basic features of the instances, including the regional area (2nd column), number of waypoints (3rd column), number of available air defense systems (4th column), and number of drones in the swarm (5th column). The important weights
For each test instance, we first use the
Figures 4–11 present the experimental results on the eight test instances, respectively. In each figure, the dashed line denotes the path cost
On smallest-size Instance 1, five evolutionary algorithms (except GA) obtain the same median value, which is the exact optimal objective value obtained by IS (but the computation time of the evolutionary algorithms is much shorter than that of IS). EBO, DE, and WWO always obtain the optimal solution, whereas the minimum values of PSO and QTS are smaller than the optimal value, i.e., the two algorithms occasionally fail to obtain the optimal solution. The median value of GA is smaller than the optimal value, although its maximum value is the optimal, i.e., GA occasionally obtains the optimal solution, but fails to do so in most cases. On Instance 2 where the region is the same as that of Instance 1 but the numbers of waypoints, air defense systems, and drones are all larger than those of Instance 1, WWO obtains the best median value, EBO, DE, and WWO occasionally obtain the optimal solution as IS, but GA, PSO, and QTS never do so.
On the remaining larger-size instances, IS cannot obtain the optimal solution within 12 hours, indicating that the exact algorithm can only solve small-size instances. DE obtains the best median value on Instance 4, and EBO obtains the best median values on the other five instances.
We conducted a nonparametric Wilcoxon rank sum test on the results of the six evolutionary algorithms to evaluate their differences on each instance. The results (all at a confidence level of 95%) show that, on either Instance 1 or Instance 2, there is no significant difference among the results of PSO, EBO, QTS, DE, and WWO, and they are all significantly better than the result of GA. On Instances 3 and 6, the result of EBO is significantly better than those of GA, PSO, and QTS, but it is not significantly different than those of DE and WWO. On Instance 4, the result of DE is significantly better than those of GA, EBO, and QTS, but it is not significantly different than those of PSO and WWO. On Instances 5, 7, and 8, the result of EBO is significantly better than those of the other five algorithms. Tables 2–7 present the statistical test results on Instances 3–8, respectively, where “>” denotes that the result of the algorithm in the current row is significantly better than that in the current column.
Table 8 presents the rank of the median value of each algorithm on each test instance, and Table 9 presents the total rank number (row 2), the number of times of obtaining the best median value (row 3, abbreviated to “best”), and the number of times of being significantly better than other algorithms (row 4, abbreviated to “significantly better”) over all eight instances. EBO exhibits the best overall performance, WWO performs the second best, and GA performs the worst because it is easily trapped in local optima, given that its crossover operator is relatively weak in global search. QTS only performs better than GA, as its neighborhood search mechanism is not also inefficient in searching a large-size solution space. PSO's particle motion mechanism makes it converge fast to some good solutions on small-size instances, but often leads to premature convergence on large-size instances. The differential crossover operator and multi-population strategy make DE more suitable for searching large-size solution spaces than small-size ones. The integration of global and local migrations in EBO and the wavelength-based propagation in WWO can achieve a quite good balance between global search and local search, which is the main reason why the two algorithms exhibit good performance on the test instances. Comparatively, the migration mechanism endows EBO with a stronger global search ability to solve large-size instances.
The computational results also validate that increasing the number of air defense systems can effectively increase the cost of enemy drones. For example, the ratio of the path cost after the deployment of the air defense systems (in terms of the maximum cost achieved by the algorithms) to the path cost before the deployment is around 500% on Instance 1 with five air defense systems, but the ratio increases to over 580% on Instance 2 with six systems. From the viewpoint of the drone holder, under the threat of air defense systems, increasing the number of drones can effectively decrease its total path cost. Facing the invasion of a large swarm of drones, it is crucial to obtain high-quality air defense system deployment solutions to defend against reconnaissance drones.
Conclusion
This paper presents an optimization problem of air defense system deployment against enemy reconnaissance drone swarms, which determines the locations of air defense systems to intercept drones and force them to continually change their paths, such that the total cost of replanned paths of the drone swarm is maximized. We propose an exact iterative search algorithm and an evolutionary framework implemented using six concrete algorithms to solve the problem. The computational experimental results validate the performance of air defense system deployment solutions obtained by the algorithms for defending against reconnaissance drones.
This study only considers defense against reconnaissance drones that cannot attack our defense systems. In our ongoing study, we consider that the target drone swarm can have ground attack drones[51], and the problem should take the possible damage caused by the drones to the air defense systems into consideration, aiming at maximizing the cost of drones and minimizing the loss of air defense systems simultaneously. In this study, we only implemented the evolutionary framework with six popular individual algorithms, and the ongoing study also includes implementing more hybrid algorithms for further possible performance improvement.
Although the countermeasure of air defense is more active than spoofing and interference, it is not sufficiently active because air defense systems are relatively static after deployment. Currently, we are also studying more comprehensive air defense systems that incorporate the cooperation of ground weapons, attacking drones, as well as human soldiers to fight against enemy droncs[52]. Future studies will consider air-ground cooperative defense against enemy drones, which could be significantly more complex and require algorithms to be more efficient and intelligent in predicting drone behaviors and utilizing problem-solving knowledge [53].
ACKNOWLEDGMENT
This work was supported by the National Natural Science Foundation of China (No. 61872123) and the Natural Science Foundation of Zhejiang Province (No. LR20F030002).