Research on Target Allocation for Hard-Kill Swarm Anti-Unmanned Aerial Vehicle Swarm Systems

Abstract

In response to the saturated attacks by low, slow, and small UAV swarms, there is currently a lack of effective countermeasures. Counter-UAV swarm technology is an important issue that urgently requires breakthroughs. This paper conducts research on a mid–short-range hard-kill counter-swarm scenario where fewer swarms confront multiple swarms and stronger swarms confront weaker swarms. The requirement is for counter-swarm UAVs to quickly penetrate the swarm at mid–short range and collide with as many incoming UAVs as possible to destroy them. To address the sparse solution space problem, an improved genetic algorithm that integrates multiple strategies is adopted to calculate the spatial density distribution of the incoming swarm. A baseline is identified through gradient descent that maximizes the density integral in a straight-line direction. Based on this baseline, the solution space for single strikes on the swarm is filtered. During the solution process, an elite strategy is introduced to prevent the overall degradation of the population performance. Additionally, the feasibility of the flight trajectory needs to be assessed. A piecewise cubic spline interpolation method is used to optimize the flight trajectory, minimizing the maximum curvature. Ultimately, multiple counter-swarm UAV targets within the swarm and their corresponding trajectories are obtained.

1. Introduction

With the development of unmanned aerial vehicles (UAVs) towards intelligence, autonomy, and miniaturization, the types of UAVs are increasing, and their mission fields are continuously expanding, with a wider range of mission types [1]. In recent years, driven by the military competition among major countries and the surge of artificial intelligence technology, swarm combat has received widespread attention and has excelled on the battlefield. Swarms are typically low-cost and have a single function, and can be obtained through appropriate modifications based on existing mature technologies. In combat, they can rely on the advantages of quantity to achieve victory in local conflicts. At the same time, the system’s robustness is very strong. Even if some UAVs are shot down, the system can still be reconfigured to continue executing combat missions. This subversive combat model has drawn the attention of various countries, with many nations initiating projects and research on swarm combat [2,3,4].

Faced with the offensive of a large number of low-cost swarms, there is no very effective solution. Even “low, slow, and small” swarms without intelligence pose significant challenges to the existing air defense system when conducting counter-swarm operations. Currently, the research on anti-swarm technology significantly is lagging behind the development of swarm technology, and there is an urgent need to make breakthroughs to address the subversive combat method posed by UAV swarm operations. Anti-swarm technologies are primarily categorized into three areas: detection and early warning, jamming and deception, and destructive interception [4]. Detection and early warning primarily involve technologies such as radar, electro-optical, infrared, radio, and acoustic detection and tracking systems. Jamming and deception are considered soft-kill methods and include techniques such as electromagnetic interference, cyber-attacks, information deception, and camouflage deception. Hard-kill methods generally employ air defense missiles, laser weapons, high-power microwave weapons, artillery shells, and swarm-anti-swarm tactics. The use of air defense missiles for interception is too costly. Meanwhile, laser weapons, high-power microwave weapons, and artillery shells are short-range destructive means with a limited strike capacity and suboptimal effectiveness. They are also affected by the weight of the platforms, which can make deployment difficult. Additionally, there is the risk of further damage caused by the crash and fall of the UAVs. The swarm-anti-swarm method is a novel approach that encompasses many advanced technologies, including intelligent decision-making and control [5]. It has a natural advantage in anti-swarm combat. This method can engage incoming swarms with as many strikes as possible at medium to short range, using either suicidal or recoverable UAVs [6]. For the remaining swarms, short-range artillery, directed energy weapons, and other means can be employed for attack. This approach allows for a more efficient and targeted response to the threat posed by UAV swarms.

For the swarm-anti-swarm approach, systematic research is currently lacking. One line of research concentrates on multi-agent systems. Simonjan et al. [7] have employed the Multi-Agent Deep Deterministic Policy Gradient (MADDPG) algorithm to model two multi-agent deep reinforcement learning strategies, but the number of agents is restricted to seven due to the suboptimal performance of multi-agent learning in large groups. SEO et al. [8] have based their threat modeling on Partially Observable Markov Decision Processes (POMDPs), aiming to asymmetrically maximize defensive advantage and attack complexity while minimizing the success rate of network attacks. These studies, similar to intelligent air combat, mainly focus on the action confrontation between UAVs [9,10,11]. Another area of research is more focused on the construction of swarm defense systems, encompassing reconnaissance, interception, and striking capabilities. Brust et al. [12] have proposed a drone defense system for intercepting and escorting malicious drones outside the flight zone. The defense drone swarm can self-organize defense formations upon detecting intruders and hunt down malicious drones in the form of a network swarm. This is a system where multiple UAVs surround and capture one UAV. Li et al. [13] have investigated the optimization problem of deploying air defense systems against reconnaissance swarms. Shi et al. [3] at Zhejiang University have developed an anti-UAV system, which integrates various passive surveillance technologies for the detection, localization, and radio frequency interference of UAVs.

In real scenarios of swarm attacks on targets, the offensive actions of UAV swarms differ from those in air combat or missile attacks. The actual attack process often relies on a large number of small, slow, and low-flying UAVs that directly assault the target, with the swarm not demonstrating a high degree of intelligence. For the effective method of swarm-anti-swarm, there is a lack of research that closely mirrors real scenarios. Previous research on air combat is not applicable because the small, slow, and low-flying UAVs are not equivalent to the anti-swarm UAVs, including aspects such as the performance, cost, and level of intelligence. The scenario of high-speed anti-swarm UAVs hard-killing small, slow, and low-flying UAVs is akin to a “cavalry assault.” In this scenario, the anti-swarm UAVs rapidly penetrate the swarm at mid-short range and, while satisfying constraints, aim to strike as many UAVs as possible. However, research on this scenario is limited. Studies that are relevant include UAV inspection scheduling, task allocation, and weapon–target assignment [14], which provide a reference but still have significant differences and require a reassessment of the issue.

This paper conducts research on the target allocation for high-speed, recoverable anti-swarm UAVs during a single penetration of an approaching swarm. The anti-swarm UAV is a tail-sitter vertical takeoff and landing (VTOL) configuration, with a speed greater than that of the approaching swarm UAVs, subject to the constraint of turning radius R. The difficulties of this problem include the following: (1) the difficulty of optimized modeling: the maximum number of feasible solutions is unknown, and it is unknown which solutions are feasible, with the two being coupled; (2) the coupling of constraints and the problem process: when searching for the optimal solution, it is also necessary to simultaneously judge whether the trajectory satisfies the constraint of the turning radius; (3) the sparsity of solutions: due to the constraints, the optimal feasible solutions are sparse, as they consist of a set of points whose coordinates satisfy the constraints simultaneously and in the largest number; (4) the difficulty of constraint modeling: trajectory optimization is necessary, otherwise, feasible optimal solutions might be discarded.

Addressing the aforementioned challenges, this paper first analyzes the scenario of a single anti-swarm UAV penetrating a stationary two-dimensional distributed swarm and then extends the study to three-dimensional cases, multiple anti-swarm UAVs penetrating the swarm, and the impact of the movement of the swarm. Finally, a qualitative discussion on solutions for the movement of swarming drones is provided. By analyzing the scale between the turning radius and the spatial distribution of the swarm, a baseline for a single anti-swarm UAV to penetrate is solved using the gradient descent method, filtering out a small number of feasible solution spaces, which greatly reduces the sparsity problem. During the solution process, diversity is designed into the initial population, and an elite strategy is introduced to prevent the overall degradation of the population performance. Each time a feasible point is selected, a judgment must be made on whether the flight trajectory satisfies the constraint of the turning radius. To address this issue, this paper employs a piecewise cubic spline interpolation method to optimize the flight trajectory, minimizing the maximum curvature. The reciprocal of the optimized maximum curvature is used as the criterion for judging whether the turning radius constraint is met. Ultimately, the sequence of swarm UAVs to be engaged by the anti-swarm UAVs and their corresponding trajectories can be obtained.

The outline of this paper is as follows: Section 2 describes the hard-kill anti-swarm UAVs and their operational styles. Section 3 presents the process of the improved optimization genetic algorithm and the preparation conditions for optimization, including the calculation of the baseline and the optimization of trajectories. Section 4 provides the optimization results, ranging from single to multiple anti-swarm UAVs striking, and on this basis, analyzes the impact of the strike frequency, swarm density, and turning radius constraints on the strike effectiveness. Section 5 qualitatively discusses the solutions for dealing with the movement of the swarm.

2. Problem Description

The anti-swarm UAV is designed with a tail-sitter configuration, capable of vertical takeoff and landing, and is recoverable, which facilitates flexible deployment, a low cost, and superior performance compared to the incoming low, slow, and small swarm UAVs, as illustrated in Figure 1. We assume that the maximum speed of the anti-swarm UAV is significantly greater than that of the incoming low, slow, and small UAVs, with a constant rate of speed, and that the coordinates of the incoming swarm are known. Due to the higher speed of the anti-swarm UAVs, collisions are intended to effectively destroy incoming drones. Consequently, the flight path is significantly influenced by the turning radius constraints.

Figure 2 illustrates the coordinate distribution of the incoming swarm with the blue dots (in a two-dimensional scenario), while the red circles depict the scale of the turning radius for our anti-swarm UAVs. The larger red circle represents the correct scale, and the smaller, semi-transparent circle indicates an incorrect scale. Given the spacing relative to the incoming swarm, the anti-swarm UAVs focus on “stabs”; hence, the minimum turning radius is relatively large. If there is no turning radius limit, the problem will resemble the Traveling Salesman Problem.

The problem description and optimization process for the three-dimensional scenario are similar to those for the two-dimensional scenario, but the mathematical models for the calculation of the baseline and trajectory optimization are slightly different and require further extension. The schematic diagram for the three-dimensional case is shown in Figure 3.

After analyzing the scale, we can obtain a basic description of the problem: N anti-swarm UAVs set off from the starting point to penetrate the incoming swarm. The objective is to strike as many drones as possible in a single attack while satisfying the constraint of the turning radius.

3. Modeling and Optimization Method

This section addresses the challenges posed by hard-kill anti-swarm UAVs and provides an overall framework for an improved genetic algorithm that integrates various strategies, along with the necessary preparatory work.

3.1. Improved Genetic Algorithm with Multi-Strategy Integration

Genetic algorithms are heuristic search algorithms that simulate the principles of natural selection and genetics. They are known for their strong global search capabilities, flexible pattern operations, high robustness, and natural parallelism. However, they also have drawbacks such as slow convergence, complex parameter settings, genetic drift, and a tendency to become stuck in local optima. To enhance the efficiency of genetic algorithms and adapt them to specific problems, it is often necessary to improve or integrate them with other strategies.

Given the assumption that the coordinates of the swarm UAVs are known, in order to determine the number of points that the anti-swarm UAV can strike with a single penetration of the swarm and the corresponding coordinates, Equation (1) is established in the form of the following solution:

$$X=[x_1,x_2,x_3,\dots ,x_n],\hspace{1em}\begin{array}{c}{x}_i\in \{0,1\},n=numbe{r}_{attact}\end{array}$$

(1)

The form of the population is then as follows:

$$Population=\left[\begin{array}{cccc}{x}_{1,1}& x_{1,j}& \dots & x_{1,n}\\ x_{2,1}& x_{2,2}& \dots & x_{2,n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m,1}& x_{m,j}& \dots & x_{m,n}\end{array}\right],\hspace{1em}\begin{array}{c}\begin{array}{l}{x}_{i,j}\in \{0,1\},n=numbe{r}_{attact},\\ m=numbe{r}_{pop}\end{array}\end{array}$$

(2)

The matrix of position coordinates for the two-dimensional swarm is defined as P:

$$P=\left[\begin{array}{cc}{p}_{x1}& p_{y1}\\ p_{x2}& p_{y2}\\ ⋮& ⋮\\ p_{xn}& p_{yn}\end{array}\right]$$

(3)

Each row of $(P\cdot Population)$ represents a possible combination of swarm coordinates during the search process, which is the combination that the anti-swarm UAV can strike in a single penetration. These combinations need to be interpolated using interpolation methods, and the curvature of the curve is judged. If the reciprocal of the curvature, which is the turning radius, is smaller than the minimum value, it indicates that the constraint is not satisfied, and the solution needs to be discarded. In summary, the design of the fitness function is as follows:

$$Fit=Cons\cdot {{\displaystyle {\sum }_{i}x}}_i,\begin{array}{cc}& Cons=\{0,1\}\end{array}$$

(4)

where Cons represents the constraint on the curvature radius, and if it meets the constraint, it is set to 1; otherwise, it is set to 0. ${{\displaystyle {\sum }_{i}x}}_i$ represents the number of strike points, and it is desired to have as many as possible while meeting the constraints. Therefore, we need to process the population further. As shown in the first row of Figure 4, the population requires a random distribution of fewer individuals, as having too many is likely to make it difficult to meet the constraint of the turning radius, and the iterative convergence process should proceed from few to many, rather than from many to few. Secondly, as shown in the second row of Figure 4, there are a large number of completely random 0–1 sequences to increase the diversity of the population. Finally, as shown in the third row of Figure 4, we also need to force the number of solutions to be no less than 2, because a straight line always meets the curvature condition. To further solve the problem of sparse solutions converging slowly, an elite strategy is introduced. As shown in Figure 4, in each iteration, the population with the $Max(Fit)$ is replicated in proportion to the initial population to prevent the optimization process from diverging. This method can effectively solve the problem of sparse solutions being difficult to converge.

The overall framework of the improved genetic algorithm is depicted in Figure 5. We first significantly reduce the solution space through the calculation of the baseline, then generate the initial population based on the principles shown in Figure 4, and proceed into the iterative process. For each iteration, possible combinations are interpolated using piecewise cubic spline interpolation, and the spline curves are optimized. The constraints of the turning radius are then checked to determine the value of the fitness function. Following this, the selection, crossover, and mutation stages are performed. Finally, an elite strategy is employed to forcefully retain the population with the highest fitness, and the process enters the next iteration until the specified number of iterations is reached.

3.2. Baseline Calculation

Due to the low cost of UAVs, incoming swarms are often numerous, as illustrated in Figure 2, with an assumed number of 300. The problem of maximizing the number of drones that an anti-swarm UAV can strike in a single penetration of the swarm while satisfying the constraint of the turning radius is a complex optimization problem with a vast solution space, and the solutions are extremely sparse. Assuming that a single UAV can strike a maximum of 10 drones, the possible choices would be from a pool of $C_{300}^{10}$ drones, but whether striking these 10 drones would satisfy the constraint of the turning radius is unknown. In reality, we do not yet know the maximum number of drones that can be struck in a single strike.

To reduce the complexity of the solution, we need to screen the points. From the scale analysis in the problem description, it is known that the anti-swarm UAV penetrates the swarm at a high speed and is constrained by the turning radius, which does not allow it to continuously circle within the swarm. Instead, it “stabs” through, although not necessarily in a straight line, and the degree of curvature is limited. Therefore, we can draw a baseline with screening the swarm coordinates near the baseline as the solution space. Intuitively, it is felt that in areas where the point set has a high distribution density, passing through approximately in a straight line can traverse more points. Another important factor is the depth, because even if the density is not high, if the depth is sufficient, many points can still be traversed. Combining these two points, the two criteria for selecting the baseline are density D and depth L. These two factors can be further integrated in an integral form, which can be expressed as ${\displaystyle \int D}dL$.

3.2.1. Baseline for Two-Dimensional Scenario

For a two-dimensional distribution of points, the baseline can be obtained using the gradient descent method, with the specific calculation steps as follows:

(1)

Divide the grid and calculate the density distribution of the point set, interpolating to obtain the function $\min {\displaystyle \int F_D(x_i,a{x}_i+b)}dx$.

(2)

Define the straight line, and use Monte Carlo sampling to generate points in the range [x, y]. The coefficients [a, b] are treated as optimization variables.

(3)

Apply the gradient descent method to find a set of [a, b] that maximizes the integral of the objective function $\min {\displaystyle \int F_D(x_i,a{x}_i+b)}dx$ along the straight line.

The grid division is determined by the interquartile range (IQR) criterion for histogram width, which is a method that can better extract the characteristics of the data. First, the IQR is calculated based on the distribution of the point set spacing, as follows:

$$IQR=Q_3-Q_1$$

(5)

Among them, Q₁ is the first quartile, meaning that 25% of the data in the dataset is less than or equal to this value; Q₃ is the third quartile, indicating that 75% of the data in the dataset is less than or equal to this value. The grid width (GW) is as follows:

$$GW={\displaystyle \frac{2IQR}{n^{\frac{1}{3}}}}$$

(6)

The final solution is [a, b] = [0.51, 5.34]. The density distribution and the baseline are shown in Figure 6.

After obtaining the baseline, we select points close to the baseline, and points far from the baseline do not enter the single optimization process. With the coefficients of the baseline known, we calculate the distance from each point to the line using the point-to-line distance formula. The number of points selected to enter the optimization process should be several times the estimated impact value, but it should not be too large, otherwise, the solution space will still be huge. This article takes a value of 80. We then choose 80 of these points for single optimization, reducing the solution space from 300 to 80, as shown in Figure 7.

3.2.2. Baseline for Three-Dimensional Scene

In three-dimensional cases, the calculation of the baseline is similar to the process in two-dimensional cases, but the mathematical computations are more complex. The parametric equation of a line in space is as follows:

$$\left\{\begin{array}{c}x(t)=x_0+lt\\ y(t)=y_0+mt\\ z(t)=z_0+nt\end{array}\right.$$

(7)

For a plane line, the variables to be optimized are the coefficients [a, b], while for a space line, there are six variables to be optimized, specifically, [x₀, y₀, z₀] and [l, m, n]. The remaining process is similar to the two-dimensional procedure, and the final solution obtained is [x₀, y₀, z₀] = [3.34, 6.18, 15.85] m and [l, m, n] = [51.45, 4.47, 22.65]. The spatial density distribution and the baseline are shown in Figure 8.

Similarly, 80 points are selected for single optimization, reducing the solution space from 300 to 80, as shown in Figure 9.

In both the two-dimensional and three-dimensional cases, the benefit of filtering the point set through the baseline is not only in the significant reduction in the solution space but also in making the optimization process more likely to converge. This is because when searching for feasible solutions within the point set, it is necessary to judge the curvature of the trajectory. After filtering, the point set is distributed around the baseline, and when searching for feasible solutions, the likelihood of satisfying the curvature constraints is much greater than searching within the initial point set, making it easier to find solutions. In the initial point set, the set of feasible solutions that satisfy the curvature constraints is extremely sparse, which can easily lead to divergence in the solution process.

3.3. Trajectory Optimization

An important difficulty of the problem lies in the coupling of constraints and the solution process; i.e., when searching for the optimal solution, it is also necessary to simultaneously judge whether the trajectory satisfies the curvature constraints. Suppose a sequence of 10 points is selected and after curve interpolation, it is found that these 10 points do not meet the curvature constraints; then, this solution will be discarded. However, it is also possible that the interpolation curve is not suitable and the curve has not been optimized. By using an appropriate curve and optimizing the curvature of the curve, it may be possible to satisfy the curvature constraints. Therefore, we need to adopt suitable interpolation methods and necessary curvature optimization to ensure that we can obtain a better flight trajectory while also ensuring that feasible solutions are not eliminated.

There has been extensive research on trajectory optimization, with fitting methods based on B-splines, Bezier curves, Dubins curves, etc., but there is less research on interpolation. In this study, the anti-swarm drone needs to collide with the swarm drones, which is an interpolation problem rather than a fitting problem. It has been assumed that the anti-swarm drone moves at a constant speed; therefore, the first and second derivatives of the velocity are continuous. Considering this, we plan to use piecewise cubic spline interpolation to handle the trajectory optimization.

A piecewise cubic spline interpolation function is given on the interval [m, n] with a set of node divisions, $m=x_0<x_1<\cdots x_{n-1}<x_n=n$, satisfying the following conditions: (1) The function has continuous first and second derivatives on [m, n]; (2) The function is a polynomial of degree no greater than three on each subinterval. Then, the piecewise function $s(x)$ is called the cubic spline function for the given set of nodes. If the spline function satisfies the interpolation conditions

$$s(x_i)=y_i,\begin{array}{cc}& i=0,1,\cdots ,n\end{array},$$

(8)

then the cubic spline function $s(x)$ is referred to as the cubic spline interpolation function. It must also satisfy the following conditions:

$$s(x_i-0)=s(x_i+0),$$

(9)

$$s\prime (x_i-0)=s\prime (x_i+0),$$

(10)

$$s″(x_i-0)=s″(x_i+0).$$

(11)

In the formula, $i=1,2,\cdots ,n-1$. Equations (9)–(11) are the continuity conditions that the cubic spline function must satisfy. $s(x)$ is an interpolation polynomial of degree no greater than three in each subinterval, so there are four unknown parameters in each subinterval. The number of subintervals is n, which means the total number of unknown parameters is 4n. From Formulas (8) to (11), it can be seen that the interpolation conditions and spline conditions provide only 4n − 2 conditions. Therefore, it is necessary to add two known conditions to solve for the unknown parameters. Usually, the known conditions are increased by imposing endpoint constraints, which are the boundary conditions. Commonly used boundary conditions include natural boundaries, fixed boundaries, and clamped boundaries, among others, or custom boundaries can be defined to adjust the curve. In this problem, we need to find a set of boundary conditions $[m_0,m_n]$ such that the maximum curvature of the spline curve over all the intervals is minimized.

3.3.1. Trajectory for Two-Dimensional Scene

The formula for calculating the curvature of a two-dimensional plane curve is as follows:

$$\kappa ={\displaystyle \frac{\left|y″(x)\right|}{{\left(1+{\left|y\prime (x)\right|}^2\right)}^{\frac{3}{2}}}}$$

(12)

Taking a two-dimensional space curve as an example, we are given a set of random points. We use the particle swarm optimization algorithm to optimize this problem. The specific optimization process is similar to the general particle swarm optimization process. Referring to reference [15], after obtaining the optimized boundary conditions, we compare the resulting curve with the curves of two conventional boundary conditions, as well as the comparison of the curvatures of each curve, as shown in Figure 10.

After particle swarm optimization, it can be ensured that the maximum curvature is minimized, which means the minimum turning radius of the trajectory is maximized. When assessing the feasibility of the curves generated during the search process, using the results after optimizing the curvature can prevent the search process from discarding feasible optimization solutions.

3.3.2. Trajectory for Three-Dimensional Scene

The expression of a spatial curve is much more complex than that of a plane curve. A spatial line is obtained by the intersection of two surfaces in space. The general equation of a spatial curve is

$$\left\{\begin{array}{c}F(x,y,z)=0\\ G(x,y,z)=0\end{array}\right.$$

(13)

Another way to express this is by using parametric equations in the following form:

$$\left\{\begin{array}{c}x=\phi (t)\\ y=\psi (t)\\ z=\omega (t)\end{array}\right.$$

(14)

The expression forms of Equations (13) and (14) are not easily extended from the optimization method we use for the two-dimensional problem. In the two-dimensional case, by optimizing the endpoint states of the piecewise cubic spline function, the maximum curvature of the spline curve is minimized. The expression of the two-dimensional plane curve is $y=f(x)$. If one wishes to extend this to a three-dimensional space curve, the expression of the space curve needs to be improved. As shown in Equation (14), t can be represented as $t=\phi \prime (x)$ through the inverse function. Therefore, Equation (13) can be converted into a parametric equation similar to Equation (15):

$$\left\{\begin{array}{c}t=\phi \prime (x)\\ y=\psi (\phi \prime (x))\\ z=\omega (\phi \prime (x))\end{array}\right.$$

(15)

By further setting $p=\phi \prime (x)$, $t=x$, Equation (15) can be represented as a set of two plane curves and a line connected by a parameter in the form of a parametric equation:

$$\left\{\begin{array}{c}x=p\\ y=f(p)\\ z=g(p)\end{array}\right.$$

(16)

The curvature of the space curve can be calculated using Equation (17):

$$\kappa ={\displaystyle \frac{{‖{\overrightarrow{T}}^{\prime }(t)\times \overrightarrow{T}(t)‖}_2}{{\left|{\overrightarrow{T}}^{\prime }(t)\right|}^3}}$$

(17)

Wherein t and T are the tangent vector and acceleration vector of the curve, respectively. $\times$ denotes the vector cross product, and $\left|·\right|$ represents the magnitude of a vector. In Equation (16), $y=f(p)$ and $z=g(p)$ are the two-dimensional planar expressions, which can be controlled by the states of the starting and ending points. After the transformation by Equations (14)–(17), the spatial curve can be controlled by two sets of starting and ending points, i.e., four parameters $\left[m_{0y},m_{ny},m_{0z},m_{nz}\right]$. By substituting the parameters into Equation (16) and then using Equation (17), the curvature of the spatial curve can be calculated. By adopting an optimization method similar to the two-dimensional process, an optimized spatial piecewise cubic spline interpolation curve can be obtained, with the minimum maximum curvature in space. Given a set of random points, the curve with optimized boundary conditions is compared with two conventional boundary condition curves, along with the comparison of their curvatures, as shown in Figure 11. This result can be used to determine whether the turning radius meets the constraints in the optimization process of three-dimensional trajectory problems.

After the extension of the optimization of the planar curve curvature, the optimization results of the spatial curve curvature were obtained. Its function is similar to that in the two-dimensional case, both aiming to ensure the trajectory with the minimum maximum curvature.

4. Optimization Results and Discussion

In this section, the striking results optimized by a genetic algorithm that integrates multiple strategies will be presented, including two-dimensional and three-dimensional scenarios. For each scenario, the striking effects of a single anti-swarm UAV and a fleet of 20 anti-swarm UAVs will be demonstrated, respectively, with different turning radius.

4.1. Optimization Results for Two-Dimensional Scenarios

In the original two-dimensional plane, there are 300 drones forming a swarm with known coordinates, and the swarm is assumed to be stationary. After undergoing a variety of strategic treatments, the problem of striking is amenable to traditional genetic algorithms, with the application of general parameter setting principles being sufficient. Genetic algorithm parameters: the population size is 500, maximum number of iterations is 500, crossover rate is 0.8, and mutation rate is 0.1. A single anti-swarm drone passes through the swarm once, and under the condition of satisfying different turning radius constraints, the optimization results of the maximum number of drones in the swarm that a single anti-swarm drone can target are shown in Figure 12.

As R_min varies from 10 to 50 m, the maximum number of strikes in a single pass is 16, 14, 12, 11, and 10, respectively. With the increase in the minimum turning radius constraint, the optimized curve tends to be more linear, and the number of strikes decreases accordingly, but the overall reduction is not significant. On the other hand, within the interval where x, y $\in$ [0, 30], the point set is more densely populated. In this interval, the number of strikes in a single pass under different turning radius constraints are 9, 8, 8, 6, and 6, respectively, and their proportions of the total number of strikes are 56.25%, 57.14%, 66.67%, 54.55%, and 60%, all of which are higher than 50%. This indicates that densely distributed swarms are more likely to be struck. Since the problem assumes the swarms are stationary, in reality, densely packed swarms would be penetrated and struck in a shorter time. Therefore, for the attacking swarms, a sparse distribution would be more difficult to target. The situation of multiple anti-swarm drones intercepting together is shown in Figure 13, where we assume the number of anti-swarm drones is 20.

For the issue involving multiple drones, this paper does not consider cooperative effects but instead adopts a cycling strategy. Namely, after calculating the strikes of the first anti-swarm drone on the swarms, the points that have been struck are removed, and the calculation steps are repeated until the number of cycles reaches the number of deployed anti-swarm drones. To further analyze the effect of multiple drones striking together and the impact of different turning radius constraints on the strike effectiveness, statistical analyses on the number of strikes for each cycle under different turning radius constraints are conducted, as shown in Figure 14a–e. From the statistical results, it can be observed that with more relaxed turning radius constraints, the average number of strikes is higher. In scenarios with the same turning radius constraints, there is randomness in the number of strikes per cycle, which is due to local optima. However, overall, the more anti-swarm drones deployed, the relatively fewer the number of strikes. The statistics on the total number of strikes for each scenario are presented in Figure 14f, which shows that the constraint of the turning radius has a significant impact on the total number of strikes. Each reduction in the minimum turning radius constraint results in a significant increase in the overall number of strikes.

4.2. Optimization Results for Three-Dimensional Scenarios

In a three-dimensional space of 100 × 100 × 100, there are 300 swarms with known coordinates, assuming the swarms are stationary. Other parameter settings are the same as in the two-dimensional process. The optimization results for the maximum number of swarms a single anti-swarm drone can strike, given that it passes through a swarm once and satisfies different turning radius constraints, are shown in Figure 15.

As R_min varies from 10 to 50 m, the maximum number of strikes in a single operation is, successively, 9, 7, 7, 7, and 6. Within the interval where x, y belong to [0, 30], the point set is relatively dense. In this interval, under different turning radius constraints, the numbers of strikes in a single operation are 9, 8, 8, 6, 6, respectively, and their proportions in the total number of strikes are 66.67%, 71.43%, 71.43%, 71.43%, and 66.67%, all of which are higher than 50%. By comparing with the two-dimensional results, it can be observed that in the three-dimensional case, on one hand, the number of strikes is fewer, and on the other hand, the proportion of strike points distributed in dense areas is larger. The situation of multiple anti-swarm drones intercepting together is shown in Figure 16, where we also assume that the number of anti-swarm drones is 20.

In the three-dimensional scenario, for the issue involving multiple drones, the collaborative effects are once again not considered. By statistically analyzing the number of strikes under various turning radius constraints, as depicted in Figure 17a–e, it can be observed that when the turning radius constraints are relaxed, the average number of strikes increases. Upon statistically compiling the total number of strikes for each scenario, as presented in Figure 17f, it becomes apparent that the constraint on the turning radius has a significant influence on the total number of strikes. With each reduction in the constraint of the minimum turning radius, there is a marked increase in the overall number of strikes. This observation is consistent with the findings in the two-dimensional scenario.

4.3. Qualitative Discussion of Swarm Dynamics

For a large number of small, slow-moving swarms, we can approximately consider that the direction of the swarm’s velocity is consistent. From the optimization results, it can be seen that within the framework of this paper, in terms of the process of using anti-swarm drones to strike swarms, the main impact of speed is on the constraint of the turning radius. For general fixed-wing aircraft, the relationship between the speed and turning radius is as follows:

$$R={\displaystyle \frac{V^2}{g\tan \varphi }}$$

(18)

Among the elements of this formula, V is the speed of the fixed-wing UAV, g is the acceleration due to gravity, and $\varphi$ is the bank angle. From Equation (18), it can be observed that the turning radius constraint for a fixed-wing aircraft is significantly influenced by speed. On one hand, this means that when anti-swarm drones are flying at a higher speed, the minimum turning radius becomes large, which will result in a significant reduction in the number of strikes. On the other hand, if the anti-swarm drone’s speed is insufficient, it will not be able to achieve the desired destructive effect. There exists a certain threshold for the striking speed, which is the speed that satisfies the destructive effect. Exceeding this speed will negatively impact the number of strikes. The selection of the threshold speed needs to be determined through further simulation experiments, which is beyond the scope of this paper’s research.

5. Conclusions

Anti-swarm technology, as a critical method to counter the threat posed by drone swarms, is of paramount importance in the current international security landscape. This paper presents an optimized algorithm that merges various strategies and takes into account the issue of trajectory planning, addressing for the first time in a quantitative manner the effectiveness of hard-kill anti-swarm measures involving piercing impacts. On this basis, this paper further analyzes the influence of the turning radius and the number of strikes on the quantity of strikes, and concludes with a qualitative discussion on solutions for managing swarm movement. The paper’s emphasis is on systematically providing optimized methods for target allocation in this previously unexplored area, illustrating the striking effects of hard-kill anti-swarm tactics where weaker or fewer swarms confront stronger or larger swarms. In the practical application of anti-swarm measures, many issues remain unresolved. This paper does not examine the formation configurations of incoming swarms, and the dynamic changes in formation during actual swarm attacks require additional study. A more systematic analysis is needed on the impact of the swarm speed on anti-swarm technology, including aspects such as the speed magnitude and direction, while also considering factors like the swarm formation topology and communication. There is also a lack of comprehensive research on real-time dynamic decision-making for anti-swarm operations. In summary, this paper offers an optimized framework for target allocation in hard-kill anti-swarm tactics against swarms, which can provide a more thorough understanding of the destructive potential of such anti-swarm methods based on this framework. The method presented in this paper can also be extended to maximum traversal problems such as robot capture.