Effect of dwell time,distance,and reflecting area on UAV swarm elimination in the presence of directed energy weapons

Abstract

The defeat of a swarm of uncrewed aerial systems is an important problem for current fighting forces. The purpose of this paper is to develop a simplified framework in which the defeat of such a swarm may be quantified. This is in the context where area surveillance is provided by an $X$ -band radar system and defense is through a directed energy weapon, hence providing an integrated solution. When specialized to high energy laser defense, it is necessary to develop a new modeling framework for performance prediction. Previous studies modeled threat arrivals through a renewal process. When threat arrivals are governed through a detection process, there is a significant increase in modeling complexity. Hence, this paper demonstrates how these difficulties can be avoided by developing a performance metric based upon order statistics. This metric provides a means of quantifying the probability of defeat of a swarm within a given time interval, as well as the probability that at least one swarm member reaches its intended target. This allows the operational effectiveness of the integrated sensing and effecting system to be determined under mathematical modeling assumptions, which can provide some important insights for military strategists and system designers.

Keywords

Counter UAV integrated sensing and effecting directed energy weapons operational effectiveness

1. Introduction

1.1. Context and objectives

This paper is concerned with the operational effectiveness of directed energy weapons (DEWs) applied for counter uncrewed aerial system (CUAS) defeat. Countering of swarms of uncrewed aerial vehicles (UAVs) is a problem of considerable importance in modern warfare.^1–4 The main focus here is on high energy laser (HEL) DEW performance when protecting a region against such threats.⁵ However, the analysis may be applied to other DEW effectors such as high power radio frequency (HPRF) systems. HEL DEWs are an effective solution to the CUAS problem, since they can provide precision strikes at considerable distances with near speed of light engagement times.^6–8

The approach adopted in this study is a simplified version of the kill chain of threat detection, classification, tracking, engagement decision making and effecting. It develops a mathematically driven model to quantify integrated sensing and effecting for CUAS effectiveness. This basic model yields some insights into integrated defensive system performance under a number of operational conditions and threat parameters.

From a sensing perspective, swarms of UAVs present a challenge because they may consist of low radar cross section (RCS) targets making them difficult to detect. Due to the fact that UAV propellers admit a micro-Doppler signature, Doppler processing, in addition to amplitude-based detection, has been shown to enhance radar performance in detecting UAVs.⁹ These micro-Doppler signatures have also been used as a basis for detection from a continuous-wave coherent lidar.¹⁰ In addition, there have been applications of acoustic systems for UAV detection, since the latter emits acoustic waves.¹¹ Although the focus in this paper is on amplitude-based radar detection, there is scope to extend the analysis to other sensor systems such as those outlined above.

HEL DEWs require a dwell time in which to induce thermal damage on targets, resulting in detected threats requiring a waiting time for processing by the effector. Given a swarm of UAVs is not likely remaining stationary, these UAVs may be converging onto their target while the HEL is processing the current threat. This potential delay in processing threats may result in at least one UAV reaching its intended target. Hence, it is important to understand a DEW system’s performance when it is facing a swarm of UAVs, especially the nature of its processing delays.

An integrated sensing and effecting system will have threats cued through the sensing process, which can be complex for DEWs. If the effecting mechanism does not introduce waiting times for processing, then threats will be acted on by the DEW once they are in its effective range. This is the case for HPRF DEWs and laser dazzlers. In the HEL, DEW situation threats will inevitably experience a waiting time. The approach developed in Weinberg,¹² and extended in Weinberg,¹³ assumes that threats arrive into the theater of operation through a renewal process. This then permits HEL performance to be modeled through a single-server queue, where the service times correspond to that taken to neutralize the current threat and delays experienced by waiting threats may be modeled through a recursive relationship.¹⁴ Although this is a valid approach, it becomes difficult to introduce an analytically tractable solution when the arrival process is determined through a sensing mechanism. Even if detection of threats is assumed independent, the interarrival times are not independent, resulting in complexity in adapting the queuing framework for delay processing as discussed in Ross.¹⁴ This is further complicated by the fact that it becomes necessary to consider the ranked order of threat arrivals, requiring the application of order statistics (OS). Although the joint density of such OS may be constructed, when combined with variations of the delay structure in Ross,¹⁴ the resulting expressions for performance prediction are not useful in practice. Hence, it became necessary to develop a new framework in which HEL DEW performance may be assessed when a sensing process is generating the arrival distribution.

Therefore, the main objective of this paper is to develop a preliminary model to account for integrated sensing and effecting without dealing with the complexity of queuing theory models. Under a restriction on the behavior of the UAVs, a new approach is developed to allow detection times to replace the assumption of arrivals into the theater of operation through a renewal process. Metrics for both BLUFOR and REDFOR mission success will be introduced, which will be based upon OS of detection times that will then permit some insights into expected system performance.

1.2. UAVs in modern warfare

In modern warfare the application and use of UAVs is becoming prevalent.^15,16 The initial use of UAVs in military settings was for intelligence, surveillance, and reconnaissance purposes; however, their utility as a weapon has also emerged.¹⁷ As reported in the latter, countries including the United States, United Kingdom, Israel and Italy were among the first to see the potential in developing weaponized drones. For an example of a large-scale UAV, General Atomics developed the MQ-9 Reaper, which can be equipped with Paveway laser guided bombs and AGM-114 Hellfire missiles. It has been deployed by the US Air Force for over a decade, and has been used extensively for precision strikes.¹⁸ Smaller UAVs have also been used to attack military targets, an example being the frequent attacks on the Russian Hmeymim Air Base in Syria. As a particular instance of this, on 6 January 2018, 13 combat fixed-wing UAVs attacked this base, launched by terrorists.¹⁷

The ongoing war in Ukraine has also demonstrated the utility of UAVs in contemporary conflicts and the need for effective countermeasures.¹⁹ What one observes from this war is that UAVs can be used to provide an asymmetric advantage. The conflict has also demonstrated the utility in utilizing commercial off-the-shelf (COTS) UAVs in warfare. As is discussed in Molloy,²⁰ small weaponized UAVs have provided Ukraine with the ability to deploy a cheap weapon to strike targets remotely while preserving the safety of the operator and providing the enemy with a difficult to detect threat. Ukraine has used small swarms of UAVs to destroy armored convoys in the Donetsk Region,²¹ while Russia has deployed Iranian Kamikaze drones (loitering munitions) to attack Ukrainian infrastructure.²² Both forces in this conflict have had to develop their tactics, techniques, and procedures (TTP) for UAV usage as the war evolves, with Ukraine taking the lead in utilizing COTS UAVs to defend against a stronger aggressor.

1.3. Modeling approach and assumptions

There are quite a number of studies investigating the problem of UAV swarms and their neutralization. Examinations of swarms and the interaction between members are considered in Tahira et al.,²³ Castrillo et al.,²⁴ and Chang et al.,²⁵ while Seidaliyev et al.²⁶ provide an overview of the issues concerning detection and classification. The way in which UAV swarms may be used to attack targets, and their neutralization, is overviewed in Chamola et al.¹⁷

As is discussed in Castrillo et al.,²⁴ there are a number of ways in which a group of UAVs may be configured, based upon collaboration between members. The four main formations can be described as follows:

Isolated individual, in which each UAV acts independently;

Group of isolated individuals, where there is no coordination between uncrewed aerial system (UAS);

Team, which is a networked group with a common mission, acting in a synchronous way; and

Swarm, which is a uniform mass of undifferentiated UAS with a single mission or task.

In this study, a combination of the latter two will be adopted in the analysis. In particular, it will be assumed that a collection of UAVs are converging on a target, such that they are traveling at the same speed but may differ in spatial position initially as well as their electromagnetic signature. The assumption of common speed results in the group/swarm remaining in a coherent pattern throughout. Otherwise, the collection of UAVs may be categorized as groups of isolated individuals as well as a core team/swarm.

This analysis will focus on small COTS UAVs resulting in smaller weight, speed, and RCS assumptions. These UAVs, classified as NATO class 1 and 2, have featured significantly in the Russian−Ukrainian War.²⁷ They tend to weigh up to 24 kg, and travel at maximum speeds ranging from 13 to 30 m/s.²⁸ As discussed in Gong et al.,⁹ group 1 UAVs, as classified by the US Department of Defense, generally have an RCS of between 0.01 and 0.1 m².

The tactics employed include coordinated strikes to targets, enhanced by a large number of swarm members. Specifically, a swarm may be directed toward an airfield to disrupt air traffic and to attack critical infrastructure. A series of low-flying UAVs may be used to strike a convoy of vehicles, as mentioned previously. The fact that they may be operated autonomously and can exhibit complex decentralized behavior can result in them becoming a difficult threat to counter.¹⁷ For the purposes of this study, it will be assumed that the swarm is converging onto its target without complex individual motion.

It is important to note that the solution proposed in this paper is contingent on a common speed assumption, otherwise it becomes necessary to re-prioritize detected UAVs for processing by the HEL DEW. For brevity, the term swarm will be used to encompass both teams/swarms as defined above.

From a sensing perspective, a number of modeling simplifications will be adopted. It will be assumed that the existence of the swarm has been established, that it has been classified as consisting of potential threats, and is being tracked by the sensor system which has passed onto the DEW the necessary engagement information to allow it to act. Since these threats will have been determined to be traveling at the same speed, the engagement decision process will be to neutralize threats in the order in which they have been detected. This simplified command and control mechanism will only work under the assumptions adopted in this paper. The conclusions will discuss future research efforts which will address this limitation.

2. Performance models

Throughout this study, units of distance and wavelength will be measured in meters (m), time in seconds (s) and speed in meters per second (m/s). RCS will be in square meters (m²). Irradiance is measured in Watts per square meter (W/m²) with the same units applied to thermal disruption thresholds.

2.1. Generic model development

Suppose that there are $M$ sensors available, one DEW effector and that there are a total of $N$ threats. It will be assumed throughout that these threats are converging on the DEW’s location, traveling at a constant speed. Hence, the distance between the DEW and threat $j$ at time $t$ will be $R_{j} (t) = ν_{j} (T_{j} - t)$ , for each $1 \leq j \leq N$ and $0 \leq t \leq T_{j}$ . Here, threat $j$ travels at $ν_{j}$ m/s and originates at a distance of $ν_{j} T_{j}$ m from the DEW.

It is necessary to distinguish between arbitrary labeling of threats and the order in which they are detected by the sensor network. Hence, the $N$ threats may be labeled arbitrarily as threat 1, threat 2, …, threat $N$ and this is not assumed to be a ranking of the order in which they are detected. Define $p_{j}^{n} (t)$ to be the probability that sensor $j$ detects threat $n$ by time $t$ . Suppose that $τ_{n}$ is the time of detection of threat $n$ , and let $τ_{(n)}$ be the detection time of the $n$ th detected threat. Set $L_{(n)}$ to be the lifetime of the $n$ th detected threat. While it is possible, in general terms, to construct statistical models for this general lifetime the resultant mathematical models are not easy to work with, and so one specific choice will be used as the basis for a performance metric. This is the lifetime of the last detected threat, namely $L_{(N)}$ . To explain this choice, note that the event ${L_{(N)} < t}$ is equivalent to the swarm being completely defeated by time $t$ . Its complement, namely ${L_{(N)} \geq t}$ , corresponds to the event that at least one UAV has survived to time $t$ . One may adopt the scenario where the integrated sensing and effecting network has been deployed in an operational setting where BLUFOR is protecting a region of importance while REDFOR is launching the UAV swarm attack on BLUFOR. From a BLUFOR perspective, the metric ${L_{(N)} < t}$ is a success measure, while for REDFOR a suitable success measure is the metric ${L_{(N)} \geq t}$ . One may also examine the event ${L_{(1)} > t}$ since it provides an indicator of whether the complete swarm has survived to time $t$ . However, from an operational perspective, the latter two metrics are able to quantify mission success and failure sufficiently.

Figure 1 provides an illustration of the combat scene, from a two-dimensional perspective. In this example, the one DEW (denoted $E$ ) is located at the origin and is flanked by two sensors ( $S_{1}$ and $S_{2}$ respectively). A swarm of five UAVs ( $U_{1}, \dots, U_{5}$ ) is converging onto the effectors location as illustrated. The legend relates the lifetime order statistic to the appropriate performance measure.

What is now required is a statistical model for the lifetime distribution of the last detected threat. It is worth noting that the following analysis may be used to construct a model for the lifetime of the first detected threat if of interest in numerical analysis.

Note that the event ${τ_{(N)} < t}$ implies that by time $t$ each of the $N$ threats has been detected by at least one member of the sensor network. Hence, it follows that

P (τ_{(N)} < t) = Π_{n = 1}^{N} [1 - Π_{j = 1}^{M} (1 - p_{j}^{n} (t))],

(1)

under the assumption that detection of threats is independent, where $P$ denotes probability. The modeling sine qua non adopted in the following is that any given threat will survive beyond a given time $t$ if either it is not detected by the sensor network or if it is detected by at least one sensor then it is not neutralized by time $t$ .

Consider the last detected threat, which is detected at time $τ_{(N)}$ , where the latter is given by Equation (1). Then by conditional probability

\begin{matrix} P (L_{(N)} > t) = P (τ_{(N)} > t) + P (τ_{(N)} \leq t) \times \\ P (The last detected threat is not neutraliszed in \\ [0, t] given that the threat is detected in this interval) . \end{matrix}

(2)

One may utilize Equation (1) to simplify Equation (2), and what is then required is the probability that this threat is not neutralized in the observational window $[0, t]$ . Suppose that this last detected threat has a disruption threshold $μ_{N}$ , irradiance $I_{N} (w)$ , and delay $D_{N}$ , where the latter is assumed to be a constant. Then the probability that the threat is neutralized in the time interval $[0, t]$ is given by

P [\int_{τ_{(N)} + D_{N}}^{t} I_{N} (w) dw \geq μ_{N}],

(3)

since this threat enters service at time $τ_{(N)} + D_{N}$ . If it is assumed that $I_{N} (t) = \frac{κ_{N}}{{(R_{N}^{*} (t))}^{2}}$ with $R_{N}^{*} (t) = ν_{N}^{*} (T_{N}^{*} - t)$ for $0 \leq t \leq T_{N}^{*}$ and $κ_{N}$ a constant, then by evaluating the integral in Equation (3), one may demonstrate that Equation (2) yields

\begin{matrix} P (Swarm defeated by time t) \\ = P (L_{(N)} < t) \\ = F_{τ_{(N)}} (t) \times \\ F_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1} - D_{N}), \end{matrix}

(4)

where Equation (1) may then be utilized in the above. In this analysis, the parameters $(T_{N}^{*}, ν_{N}^{*})$ correspond to those for the last detected threat in the swarm of $N$ UAVs.

It is informative to note that in order to ensure that the argument of the second distribution function in Equation (4) is non-negative one must impose the condition

D_{N} \leq T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1},

(5)

which provides a time-dependent upper bound on the maximal permissible delay.

In order to evaluate Equation (4), one requires the parameter set $(T_{N}^{*}, ν_{N}^{*}, μ_{N})$ , noting that these are associated with the last detected threat. If it is assumed that each threat travels at the same speed and each $T_{j}$ is the same then the corresponding parameters are immediate. In the situation where this is not the case, it may be necessary to consider carefully how these parameters are to be determined based upon physical considerations. To illustrate this, if all threats have the same RCS then the last detected threat will be the one furthest from the DEW initially, so may be determined from the maximum of $ν_{j} T_{j}$ for each $j \in {1, 2, \dots, N}$ . When the RCS also varies, it becomes necessary to utilize an algorithm to extract the desired information.

For the initial arbitrary ranking of threats, suppose that the sequence $τ_{1}, τ_{2}, \dots, τ_{N}$ are the detection times of the threats, where $τ_{(1)} < τ_{(2)} < \dots < τ_{(N)}$ are the corresponding ranked detection times. Since it is required to determine which will be the last threat detected, consider for some index $1 \leq j \leq N$ the probability that $τ_{j}$ is the last detected threat. Note that through an application of conditional probability that

\begin{matrix} P (τ_{(N)} = τ_{j}) = P (τ_{1} < τ_{j}, τ_{2} < τ_{j}, \dots, \\ τ_{j - 1} < τ_{j}, τ_{j + 1} < τ_{j}, \dots, τ_{N} < τ_{j}) \\ = \int_{0}^{\infty} f_{τ_{j}} (x) P (τ_{1} < x, τ_{2} < x, \dots, \\ τ_{j - 1} < x, τ_{j + 1} < x, \dots, τ_{N} < x) dx \\ = \int_{0}^{\infty} f_{τ_{j}} (x) \times \\ Π_{n = 1, n \neq j}^{N} [1 - Π_{k = 1}^{M} (1 - p_{k}^{n} (x))] dx . \end{matrix}

(6)

where $f_{τ_{j}}$ is the density of $τ_{j}$ .

Expression (6) enables one to produce a probability distribution of the last detected threat’s position in the sequence of detection times, and then permits the specification of relevant parameters in Equation (4). A useful way in which Equation (6) may be evaluated is through Monte Carlo simulation. Since $f_{τ_{j}} (x)$ is a density on the domain of integration one may average realizations of $τ_{j}$ and evaluate these over the corresponding integrands. In order to simulate $τ_{j}$ one requires its distribution function. It is not difficult to argue that

F_{τ_{j}} (x) = 1 - Π_{k = 1}^{M} (1 - p_{k}^{j} (x)),

(7)

and one may apply the inverse cumulative distribution function method to simulate $τ_{j}$ from Equation (7).

To summarize this section the modeling framework is illustrated in Figure 2, indicating how one may use it to inform system specifications to ensure a desired level of operational effectiveness against a given threat class. The probability of defeating the swarm is labeled metric 1, while its complement is metric 2.

2.2. Sensing through an X-band radar

In order to apply the above what is required is a model for sensor performance. Hence, an example is adapted from Mahafza,²⁹ specifically example 3.1 together with the signal-to-noise ratio (SNR) given by (3.16) in the latter but with no radar losses. Consequently, it will be supposed that each radar is $X$ -band operating with a frequency of 10 GHz (wavelength 0.029979 m), 5 kHz bandwidth with a power level of 1 kW and antenna gain of 10 dB. The SNR, in dB, can be shown to be

SNR (t) = 123.54755 + σ_{dB} - 40 lo g_{10} (R (t)),

(8)

where $σ_{dB}$ is the target RCS in dB and $R (t)$ is the distance to the target. It will be assumed that the operation is taking place in a land-based setting, so that one may consider Gaussian radar clutter statistics for simplicity. If it is supposed that detection decisions result from a cell-averaging constant false alarm rate detector then one may show that

p_{j}^{n} (t) = {[1 + \frac{({P_{FA}}^{- 1 / H} - 1)}{1 + {SNR}_{j}^{n} (t)}]}^{- H},

(9)

where $H$ is the number of averaged cells in the detector, $P_{FA}$ is the probability of false alarm (Pfa) and ${SNR}_{j}^{n}$ is the SNR (Equation (8)) where $R (t)$ is the distance from sensor $j$ to threat $n$ at time $t$ .³⁰ Expression (9) assumes that the threats RCS varies as a Swerling I fluctuation model.³¹ In the following the Pfa is set to $10^{- 3}$ and $H = 2$ .

2.3. Effecting through a HEL

For the numerical analysis to follow it will be assumed that the DEW is a HEL, which may be functioning in a scenario where threats may be delayed before processing. It will be supposed that the laser operates in an environment where atmospheric effects are minimized, so for example thermal blooming will be considered compensated for through adaptive optics. Under the assumption of a Gaussian beam profile, the irradiance (5.11) from Weinberg¹³ is adopted. This is given by

I (t) = \frac{P_{T}}{2 π R {(t)}^{2} (α_{D}^{2} + α_{J}^{2})},

(10)

where $P_{T}$ is the peak laser power, $R (t)$ is the distance between the target and the DEW, $α_{D} = \frac{2}{π} M^{2} \frac{λ}{D}$ is the angular standard deviation of the Gaussian diffraction pattern, where $M^{2}$ is the laser beam quality, $λ$ is the wavelength and $D$ is the laser aperture diameter and $α_{J}$ is the beam jitter.

Throughout the following specific parameterizations will be adopted: $P_{T} = 1$ kW, $λ = 1.045 μ$ m, $D = 0.8$ , $M^{2} = 1$ and $α_{J} = 0.001$ . These parameters have been selected from the examples in Weinberg.¹³ The disruption threshold will be assumed to be $μ = 10^{6}$ W/m². This threshold represents the typical melting benchmark for a number of materials which appear in UAV designs.¹³

Note that in view of Equation (10), it follows that $κ_{N} = \frac{P_{T}}{2 π (α_{D}^{2} + α_{J}^{2})}$ .

4. Performance examples

A series of examples is now considered to illustrate the application of the main results. Throughout the following, a two-dimensional setting is taken for the study, where the HEL DEW is located at (0, 0) and two sensors are positioned at (−10, 0) and (10, 0), respectively. It will be assumed that the threats are converging onto the DEW’s position and arrive at different angles of arrival, as illustrated in Figure 1. Each case considered will discuss the example’s variation in threat parameters.

In terms of the delay which the last detected threat may experience, the following choice is adopted. Based upon considerations in Burley³² and Puent,³³ the time it takes to cause a thermal disruption will be in the order of seconds depending on the UAV’s hardening to thermal effects. Hence, as an approximation, it will be assumed that it takes about 3 s to cause a thermal effect. Since we are concerned with the delay the last detected threat can experience, this will be maximal if all threats are detected simultaneously and all are placed in a queue for processing by the HEL DEW. Therefore, if there are $N$ threats in total, the last detected threat will have to wait roughly $D_{N} \approx 3 \times (N - 1)$ s to be processed. This choice for $D_{N}$ will be adopted throughout the following analyses.

4.1. Case 1

For this case, it is assumed that the swarm consists of five identical UAVs, originating at the same distance and traveling at the same speed, with common RCS. Hence, $D_{N} = 12$ will be utilized. However, they are assumed to travel on different paths with $θ_{1} = π / 6$ , $θ_{2} = π / 4$ , $θ_{3} = π / 8$ , $θ_{4} = π / 2$ and $θ_{5} = π / 3$ respectively. The first example corresponds to $T_{j} = 200$ , $ν_{j} = 15$ m/s and $σ_{j} = 1$ m². Figure 3 plots the probability that the entire swarm is defeated as a function of its spatial distance to the HEL DEW, which is metric 1. The swarm originates at a distance of 3 km from the DEW in this situation. In such situations the $T_{N}^{*}$ and $ν_{N}^{*}$ of the last detected threat are immediate. Figure 4 provides a plot of the complement of this, namely that at least one swarm member survives as a function of its distance to the DEW (metric 2). Both plots illustrate that the swarm will be defeated before any member is within 1 km of the DEW. In this example, it is the large RCS which is resulting in all threats being detected in sufficient time to be defeated by the HEL DEW.

In this same scenario, when the common RCS is reduced to 0.01 m², all threats will be defeated when the swarm is 500 m from the DEW, since the smaller RCS means that they take longer to detect but there is sufficient time to neutralize the swarm regardless of the delay (figure not include for brevity).

For the remaining examples, only plots of metric 1 will be included since that for metric 2 is simply the complement, as illustrated above.

Next consider the case where $T_{j} = 100$ , $ν_{j} = 15$ m/s and $σ_{j} = 0.01$ m², so that the swarm originates at a distance of 1.5 km from the DEW. Figure 5 shows that in this case all threats are eliminated before they reach 400 m from the DEW. The reduction in elimination range is due to the decrease in common RCS.

It is interesting to explore the impact of faster traveling threats in the situation of Figure 5. In particular, when the common UAV speed is increased beyond 30 m/s one begins to see the impact of the delay on performance. At $ν_{j} = 30$ m/s, there is only a marginal difference but when $ν_{j} = 45$ m/s one begins to see a clear difference as illustrated in Figure 6. Although the difference is small here, there are situations where this can become more significant. However, it is worth noting that a COTS UAV traveling at 45 m/s is unusual; this choice has been used to illustrate that the delay cases can result in different probabilities of defeat.

Obviously, the initial positioning of the swarm will have a significant impact on the DEW’s performance. To explore this further consider the case where $T_{j} = 30$ with $ν_{j} = 35$ m/s, with $σ_{j} = 0.01$ m². Hence, the swarm is located at a distance of 1.05 km from the DEW initially. Figure 7 plots the corresponding results, showing that the swarm is eliminated within 350 m of the DEW, but the delay case has a slight decrease in probability of defeating the swarm.

To explore this further, in Figure 8 the speed has been increased to 45 m/s, so that the threats are at a distance of 1.35 km from the DEW initially. Here there is a more severe difference between the delay and non-delay case. When the speed is increased to 50 m/s the probability of defeat of the swarm, when accounting for delays, reaches 0.7 at its maximum, as shown in Figure 9. In the latter case, the swarm is located at a distance of 1.8 km from the DEW initially.

It is important to note that the speeds used in Figures 7 –9 are a bit excessive for COTS UAVs but this has been an important exercise since it illustrates potential problems when swarm members are traveling somewhat fast, with small RCS. However, with larger RCS, it can be shown that this situation improves, since faster traveling threats will be detected sufficiently early. As an example, when the RCS is increased to unity with a speed of 50 m/s in this case all threats will be defeated before they are within 1.2 km of the DEW, regardless of the delay.

4.2. Case 2

In this subsection, some examples are considered where the $T_{j}$ varies slightly, noting that for a consistent swarm they should originate in close proximity, requiring small variation in these parameters. For illustration purposes it will be assumed that $T_{1} = 100$ , $T_{2} = 99$ , $T_{3} = 98$ , $T_{4} = 97$ , $T_{5} = 96$ , with a common $σ_{j} = 0.001$ m². Since all threats will be presumed to travel at the same speed, and have the same RCS, the last detected threat will be the one furthest from the DEW initially, hence $T_{1}$ will be used for the last detected threat’s $T_{N}^{*}$ parameter. Figure 10 is for the case where the common speed is 15 m/s. Here there is no difference between the non-delay and delay cases, and all threats will be defeated before the last swarm member is within 200 m of the DEW. When the speed is increased to 20 m/s there is a slight change between these two cases, which becomes clearer when $ν_{j} = 30$ m/s as illustrated in Figure 11. For the delay case there is an 80% chance that the entire swarm is eliminated before the last member reaches the DEW, while for the no delay case the conclusion is the same as for Figure 10.

The example of this section showed that if the threats originate at different positions, then in the delay case one might see REDFOR achieving its mission objective at considerably smaller speeds than that illustrated in the previous subsection.

4.3. Case 3

This section shows an example where variation in both the RCS and $T_{j}$ across threats is now considered, with parameterisations as in Table 1. In this situation, it is not so clear which will be the last detected threat, so the algorithm introduced in the previous section was employed to determine this, and using 10,000 Monte Carlo runs the distribution of times was produced. This is given by the vector

\begin{matrix} [0.00283458, 0.28224932, 0.29258738, 0.42176255, 0.00056617], \end{matrix}

whose $j$ th entry is the probability that threat $j$ is the last detected threat. Based upon this it is most likely that this is threat 4, and so its parameters are used to produce the probability of swarm disruption. Figure 12 shows the results when the common speed is 15 m/s. Here, one may conclude that all threats are eliminated within 100 m of the DEW, with only a small difference between the delay and non-delay case.

Table 1.

Adopted variation in each RCS and $T_{j}$ for case 3.

Threat	RCS	$T_{j}$
1	1	100
2	0.01	95
3	0.01	90
4	0.001	75
5	2	75

When all threat speeds are increased, one can see the impact of the delay. Figure 13 is when the common speed is increased to $ν_{j} = 20$ m/s. This results in a sudden drop of probability of detection for the delay case. When no delay is present, the swarm is likely defeated within 50 m of the DEW, but in the delay case, there is just over an 80% probability that the swarm is eliminated before it reaches the DEW. A more extreme result occurs when the speed is increased to 30 m/s, as illustrated in Figure 14. For the non-delay case, the probability that the swarm is completely destroyed is about 80%, while for the delay case this has reduced to about 20%.

In this example, there are two threats with large RCS and three with smaller RCS, so once they are traveling fast it is likely that the sensing network will not detect them with sufficient time to result in a successful disruption by the DEW, as the results have indicated. If a maximal delay is experienced then this results in REDFOR more likely to achieve its mission aims. The faster speeds utilized here are significantly slower than in the previous subsections, indicating variations in the RCS and points of threat origin can have a significant impact on BLUFOR mission success.

4.4. Case 4

As a final example, the case where there are 10 threats in the swarm is considered, such that their parameters are given in Table 2. Since there are 10 threats the maximum delay is $D_{N} = 27$ s. In this example the threat speed and $T_{j}$ are held fixed. Figure 15 shows the probability of defeating the swarm when $T_{j} = 50$ with a common speed of 15 m/s. The figure shows that in the non-delay case the swarm is likely defeated completely before it is within 200 m of the DEW. However, with the maximal delay it is clear that the probability of defeating the swarm completely is only just above 40%. In the latter the small RCS and increased maximal delay have severely limited the DEW’s mission effectiveness, noting that the speed is only 15 m/s. If this speed is increased to 20 m/s, then metric 1 is effectively zero (figure excluded for brevity).

Table 2.

Threat RCS and angle of arrival $θ_{j}$ for case 4.

Threat	RCS	$θ_{j}$
1	0.001	$π / 3$
2	0.001	$π / 6$
3	0.001	$π / 7$
4	0.001	$π / 8$
5	0.01	$π / 9$
6	0.01	$π / 10$
7	0.01	$π / 11$
8	0.1	$π / 12$
9	0.1	$π / 13$
10	0.1	$π / 14$

4.5. Some considerations

The previous example highlighted the impact that increased delay can have on expected performance. If instead of the maximum delay one assumed that $D_{N} = 13$ s, then there is no difference between the delay and non-delay cases. Hence, it is of interest to better understand permissible delays so that the integrated system’s performance is not compromised.

To ensure in practice that this delay is minimized, there are several potential solutions. In the first instance increasing the radar’s power, and the number of available sensors, will result in earlier detection of threats, which will then provide the DEW with more time to neutralize threats. Second, the DEW’s power may be increased to reduce the dwell time, provided there is compensation for atmospheric attenuation. This alone would be sufficient to ensure the delay of the last detected threat is below the limit required to ensure BLUFOR mission success.

Note that with reference to Equation (4), if one wanted to obtain bounds on the difference between this expression and the non-delay case, the key term is the second one in this expression, namely

F_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1} - D_{N}) .

(11)

The difference between Equation (11) and the non-delay case may be bounded by utilizing the mean value theorem. In particular, it can be shown that

\begin{matrix} F_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1}) - \\ F_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1} - D_{N}) \\ = f_{τ_{(N)}} (c (t)) D_{N} \end{matrix}

(12)

where the function $c (t)$ is between

T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1} - D_{N}

and

T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1},

and $f_{τ_{(N)}}$ is the density of $τ_{(N)}$ . In some cases of interest, it can be shown that this density is an increasing function, such as when a series of homogeneous threats is converging on a sensor colocated with the DEW. This then enables one to produce the bound

\begin{matrix} F_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1}) - \\ F_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1} - D_{N}) \\ \leq f_{τ_{(N)}} (T_{N}^{*} + {[{[t - T_{N}^{*}]}^{- 1} + \frac{{(ν_{N}^{*})}^{2} μ_{N}}{κ_{N}}]}^{- 1}) D_{N} . \end{matrix}

(13)

Suppose that it is desired to ensure that the difference between the delay and non-delay cases is bounded by an arbitrary amount. Then from Equation (13), one may deduce the bound required on the delay in order to ensure this requirement. Calculations for the example mentioned in this discussion showed that the delay is bounded by a constant multiple of a power of the ratio of the square of the threat speed to its RCS and a product of three functions of the SNR. When the laser power to square speed ratio is increased, this bound also increases, implying that increasing laser power for slower moving targets will result in larger permissible delays. The bound is also reduced for larger RCS and slower traveling UAVs, which is a result consistent with observations made in the numerical analysis. Further mathematical exploration is required to further quantify bounds on Equation (13) in terms of system parameters.

5. Conclusions and future work

The purpose of this paper was to demonstrate how performance metrics for CUAS may be constructed for the case where sensing is provided by a network of $X$ -band radars and effecting is through a HEL DEW. A simplified modeling framework was developed for analysis. The complexity of processing threats through a ranked detection process was addressed by utilizing metrics based upon OS. A necessary assumption adopted was that members of a swarm of UAVs traveled at a constant speed, so that engagement decisions could be based upon the order in which threats are detected. The HEL DEW delay of the last detected threat was modeled as a function of the average dwell time to achieve a thermal effect times the number of UAVs serviced prior to this threat. It was then possible to explore the role of threat parameters on BLUFOR mission success, quantified through metric 1. A secondary metric, measuring the probability that at least one UAV survives and reaches its target, was also proposed, which is directly related to metric 1. This permits one to quantify REDFOR likelihood of mission success.

The analysis presented in this paper has indicated the need for further exploration of the problem of assessing integrated defensive system performance through mathematical models. First, since the swarm may contain UAVs which deviate from the assumed uniform speed it is necessary to examine how this can be accounted for in a performance assessment model. One approach may be to introduce a time-dependent threat ranking process, which schedules threats based upon the expected times they are likely to reach their intended targets. This may require a re-examination of the queuing theory approach of Weinberg¹² with the introduction of priority queuing. A second important requirement is the validation of adopted models through analysis of trials data with real operational systems and with a variety of UAVs, operating under various conditions. These two principal lines of effort will form the basis for subsequent future work.

Footnotes

Acknowledgements

The author would like to thank the two reviewers for their comments which provided practical insights into the complexity of UAV swarm defeat. The reviewers’ comments not only improved the current paper but have provided the author with important future research directions.

Declaration of conflicting interests

The author has declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

ORCID iD

Graham V Weinberg

Author biography

Graham V Weinberg is a mathematician specializing in probability and stochastic processes, who completed a PhD degree at the University of Melbourne in 2001. He has been employed by the Australian Defence Science and Technology (DST) Group since 2002 and has worked across a number of different domains. Most of his career was spent in airborne maritime surveillance radar but over the last 5 years has shifted to land-based combat modeling and directed energy weapons performance prediction.

References

Xiaoning

. Analysis of military application of UAV swarm technology. In: 3rd international conference on unmanned systems, Harbin, China, 27-28 November 2020, pp. 1200-1204. New York: IEEE.

Stodola

Nohel

Rybansky

Modeling a reconnaissance operation in an urban environment using a swarm of UAVs. J Def Model Simul Appl Methodol Technol 2023; 22: 129–146.

Middeldorp

Malone

Noordkamp

Quantifying the robustness of a Bayesian belief network in the context of unmanned aerial system threat prediction. J Def Model Simul Appl Methodol Technol. Epub ahead of print 30 October 2023. DOI: 10.1177/15485129231206825.

Overview and key technology summary of drone swarm operation. Proceedings of the 2022 international conference on autonomous unmanned systems. Singapore: Springer, 2023.

Nielsen

PE.

Effects of directed energy weapons. Technical Report, National Defence University, Washington: DC, 1994.

Yun

Song

Pei

Modeling the impact of high energy laser weapon on the mission effectiveness of unmanned combat aerial vehicle. IEEE Access. 2020; 8:32246–32257.

Steinvall

. The potential role of laser in combating UAV: part 2; laser as a countermeasure and weapon. In: Proceedings SPIE, technologies for optical countermeasures XVIII and high-power lasers: technology and systems, platforms, effects V, Online, 12 September 2021, p. 1186707. https://www.spiedigitallibrary.org/conference-proceedings-of-spie/11867/1186707/The-potential-role-of-laser-in-combating-UAVs–part/10.1117/12.2601755.short

Kracman

MM.

A stochastic jump model applied to collaborative queue-based high energy laser defense. J Def Model Simul Appl Methodol Technol 2023; 22: 187–195.

Gong

Yan

, et al. Improved radar detection of small drones using Doppler signal-to-clutter (DSCR) detector. Drones 2023; 7: 316.

10.

Rodrigo

Larsen

Pedersen

CW coherent detection lidar for micro-Doppler sensing and raster-scan imaging of drones. Opt Express 2023; 31: 7389.

11.

Dumitrescu

Minea

Costea

, et al. Development of an acoustic system for UAV detection. Sensors 2020; 20: 4870.

12.

Weinberg

GV.

Performance prediction of directed energy weapons. Prog Electromagn Res 2022; 108: 79–88.

13.

Weinberg

GV.

Directed energy system performance prediction. Boston, MA: Artech House, 2023.

14.

Ross

Stochastic processes. New York, NY: Wiley, 1996.

15.

Jung

Park

Lee

, et al. Enhancing UAV swarm tactics with edge AI: adaptive decision making in changing environments. Drones 2024; 8: 582.

16.

Bakirci

Ozer

MM.

Involvement of unmanned aerial vehicles and swarm intelligence in future modern warfare: an overview. In: Computing, internet of things and data analytics. Vol 1145. Berlin, Germany: Springer, 2024.

17.

Chamola

Kotesh

Agarwal

, et al. A comprehensive review of unmanned aerial vehicle attacks and neutralization techniques. Ad Hoc Networks 2021; 111: 102324.

18.

Air Force. MQ-9 reaper. https://www.af.mil/About-Us/Fact-Sheets/Display/Article/104470/mq-9-reaper/ (accessed April 17, 2025)

19.

Gong

Yan

Kong

, et al. Introduction to drone detection radar with emphasis on automatic target recognition technology. arXiv. 2023; arXiv:2307.10326.

20.

Molloy

. Drones in modern warfare: lessons learnt from the war in Ukraine. Australian Army Occasional Paper No. 29 2024. https://researchcentre.army.gov.au/library/occasional-papers/drones-modern-warfare

21.

Kyivpost. Russian assaults fail in Donbas as Ukraine destroys tanks, wipes out troops. https://www.kyivpost.com/post/49140 (accessed April 17, 2025)

22.

Edmonds

Bendett

. Russia’s use of uncrewed systems in Ukraine. US Center for Naval Analyses Technical. Report DRM-2022-U-034223, 2023. https://www.cna.org/reports/2023/05/russias-use-of-drones-in-ukraine

23.

Tahira

Boling

Haghbayan

, et al. Swarms of unmanned aerial vehicles—a survey. J Ind Inf Integr 2019; 16: 100106.

24.

Castrillo

Manco

Pascarella

, et al. A review of counter-UAS technologies for cooperative defensive teams of drones. Drones 2022; 6: 65.

25.

Chang

Zhao

, et al. Overview of research on intelligent swarm munitions. Def Technol. Epub ahead of print 22 August 2024. DOI: 10.1016/j.dt.2024.08.017.

26.

Seidaliyeva

Ilipbayeva

Taissariyeva

, et al. Advances and challenges in drone detection and classification techniques: a state-of-the-art review. Sensors 2024; 24: 125.

27.

Australian Defence. Defence outlines C-UAS plans. https://www.australiandefence.com.au/news/news/defence-outlines-c-uas-plans (accessed April 21, 2025)

28.

Jouav. How fast can a drone fly? Top speeds of various types. https://www.jouav.com/blog/how-fast-can-a-drone-fly.html (accessed April 21, 2025)

29.

Mahafza

BR.

Introduction to radar analysis. Boca Raton, FL: CRC Press, 1998.

30.

Gandhi

Kassam

SA.

Optimality of the cell-averaging CFAR detector. IEEE Trans Inf Theory 1994; 40: 1226–1228.

31.

Levanon

Radar principles. New York: John Wiley and Sons, 1988.

32.

Burley

JL.

Comparison of high energy laser expected dwell times and probability of kill for mission planning scenarios in actual and standard atmospheres. Master of Science Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, USA, 2012.

33.

Puent

Integration of adaptive optics into high energy laser modelling and simulation. Master of Science Thesis, Naval Postgraduate School, Monterey, CA, USA, 2017.