Deceptive Waypoint Sequencing Based UAV–UAV Interception Control Using DBSCAN Learning Strategy

Abstract

Modern multi-Unmanned Aerial Vehicle (UAV) attacks pose significant challenges to existing counter-UAV frameworks due to their agility, irregular spatial formations, and increasing reliance on intelligent evasive behaviors. This paper proposes a unified interception architecture that integrates Density-Based Spatial Clustering of Applications with Noise (DBSCAN) for multi-target grouping, a deceptive waypoint sequencing (DWS) mechanism for adversarial evasion, and a robust sliding-mode backstepping controller augmented with extended state observers (ESOs) for precise tracking under disturbances. DBSCAN enables real-time clustering of attacking UAVs without prior knowledge of the number of formations, producing dynamic centroids that serve as tactical interception references. To counter risky attackers capable of predicting defender trajectories, a novel DWS strategy introduces centroid-relative waypoints that preserve mission objectives while reducing trajectory predictability. Lyapunov-based analysis is developed for stability, guaranteeing uniform ultimate boundedness of the tracking errors. The proposed approach achieves successful interception in both scenarios, with an interception time of 7 s and final interception error of $0.023$ m in the single-UAV case, and an interception time of 8 s with final interception error of $0.050$ m in the multiple-UAV case, whereas the PID baseline fails to achieve interception under the same conditions. Extensive simulations involving single and multi-cluster engagements demonstrate that the proposed strategy achieves fast, accurate, and deception-resilient interception, outperforming the conventional PID approach in the presence of disturbances, nonlinearities, and dynamic swarm configurations. The obtained results show the effectiveness of integrating adaptive clustering, deceptive planning, and robust nonlinear control for modern UAV–UAV defensive operations.

1. Introduction

Unmanned aerial vehicles (UAVs), commonly referred to as drones, have rapidly increased in both civilian and military sectors, enabling new applications while also raising additional security concerns. In 2020, the global drone market was valued at approximately $\$22.5$ billion, with projections expected to exceed $\$42.8$ billion by 2025 [1]. UAVs are used for infrastructure inspections, logistics (delivery), surveillance, search and rescue, and military reconnaissance and strike operations [2,3,4]. They range from those covered by the hobbyist UAV to those that are for long-range fixed-wing systems. However, this growth also brings threats: small drones can be weaponized, used for smuggling or espionage, or flown into restricted areas [5,6]. The increasing accessibility of commercial off-the-shelf UAV technology means that potential adversaries can deploy swarms of intelligent drones. Several studies show that these UAVs are of low cost and high maneuverability, and are therefore difficult to detect and track [7,8,9,10]. Authors highlight that their small size, agile flight, and ability to fly close to the ground allow UAVs to evade traditional surveillance methods [9,10]. This highlights the urgent need for advanced counter-UAV solutions to ensure public safety and protect assets.

Quadrotors and other micro and mini UAVs deserve special attention. These small multirotor drones, often weighing less than 20 kg, are inexpensive, highly maneuverable, and increasingly common [11]. Improvements in battery and motor technology enable small quadrotors to carry cameras, sensors, or even small payloads over practical distances. They can be operated by amateurs or programmed for autonomous missions [11,12]. The common use of this system among the civilian is because they are simple to use and economical [9]. While these features offer benefits, they also pose security risks. Even a hobbyist drone can be equipped with surveillance cameras or weapons. Cases regarding the UAVs being used for smuggling and spying have recently been published by researchers [13]. In a worst-case scenario, coordinated swarms of small drones could bypass defenses or exploit vulnerabilities. Ref. [10] caution that “significant security threats can emerge from UAVs approaching in swarms”, emphasizing the need for detection systems to be capable of tracking multiple targets at once [14]. Small quadrotors pose unique challenges due to their very low radar and visual signatures, ability to operate at low altitudes among obstacles, and potential for mass deployment by adversaries.

Due to these challenges, effective countermeasures depend on a range of detection and tracking technologies. Common counter-UAV sensor suites consist of radar, electro-optical/infrared (EO/IR) cameras with computer vision, acoustic microphones, and radio frequency (RF) receivers for monitoring known drone frequencies [15,16]. In practice, a network of diverse sensors is utilized: for instance, Ref. [10] discuss systems that combine data from distributed radar, camera, and microphone arrays to ensure comprehensive coverage [15]. Techniques such as multilateration and triangulation can use data from multiple stations to determine a drone’s position. Additionally, machine learning and artificial intelligence have been utilized; for example, neural networks can identify the optical or RF signatures of known drone models. Despite these advancements, significant challenges still persist. Small and fast-maneuvering UAVs are very difficult to detect in a harsh environment [10,17]. Line-of-sight issues can obstruct cameras and radars. Optical methods are affected by shadows, changes in lighting, and clutter from the sky. Additionally, small drones have low thermal and acoustic signatures, making them difficult to detect. Even active radar can be deceived by GPS spoofing, and clever drones may use navigation tricks to counteract jamming. Counter-UAV systems must address low radar cross-sections, limited signal-to-noise ratios, and adversarial countermeasures in complex environments [10,17].

Once a threat is detected, there are several available strategies for neutralization. Mechanical (kinetic) methods physically intercept UAVs, while electronic and cyber methods disrupt their systems [18]. Kinetic capture and interdiction techniques include nets, tethered projectiles, and hard-kill interceptors. For example, net guns or capture drones can ensnare small UAVs and bring them safely to the ground [19]. Net-based systems are particularly effective for small, slow drones that are hard to target with guns or missiles; nets can safely immobilize the drone with minimal collateral damage [20,21]. Alternative non-explosive munitions or frangible projectiles have been proposed. Directed-energy weapons, such as high-power microwave or laser systems, can disable UAV electronics at a distance without physical contact [22,23]. These methods can disable a drone without firing projectiles; however, they require a clear line of sight and considerable power. In contrast, electronic non-physical countermeasures disrupt a drone’s control and navigation signals. Common methods include RF or Global Positioning System (GPS) jamming and spoofing. By broadcasting noise or fake Global Navigation Satellite System (GNSS) signals, defenders can cause a drone to lose communication or deviate from its intended path. Authors in one study note that most counter-UAV systems currently utilize jamming and spoofing techniques [24,25,26]. Each neutralization tactic for drones comes with trade-offs. Kinetic methods may cause collateral damage or violate property laws. Jamming is limited by regulations on signal emissions, while directed energy weapons require specialized equipment. Because of safe and legal intent, non-lethal techniques used to capture nuisance drones are often preferred.

Classical guidance laws provide simple yet effective rules for directing an interceptor toward a moving target. Two well-known examples are pure pursuit (PP) and proportional navigation (PN). In PP guidance, the interceptor continuously points its velocity directly at the target’s current position [27,28]. This yields an intuitive “chase” behavior that is easy to implement, but it often leads to a lagging trajectory where the interceptor falls in behind the target (a tail-chase), especially if the target is fast or maneuvering. Pure pursuit can result in a longer path and potential overshoot of the target’s position [29]. By contrast, PN is a more sophisticated and widely used guidance law originally developed for homing missiles [30]. The core idea of PN is to steer the interceptor such that it maintains a constant bearing (constant line-of-sight angle) to the moving target, which is the condition for collision in a constant-velocity scenario [31].

Most of the research on onboard detection of non-cooperating UAVs focuses on visual methods based on deep learning [32,33,34]. However, detection and precise state estimation of small drones in cluttered environments proves to be a difficult problem, which several works have attempted to address by utilizing depth images instead of (or in addition to) RGB [35,36]. Flying objects are typically spatially distinctive, making them easily detectable in depth images, and the depth information provides a full 3D relative location of the target, which is otherwise problematic with monocular approaches [37].

Controlling defensive UAVs relies on an effective control architecture. Pursuing or intercepting target drones requires strong guidance [37]. Recent studies have emphasized nonlinear robust controllers capable of managing UAV dynamics uncertainties at low speeds and altitudes [38]. Sliding-mode control (SMC) is well-known for its robustness to disturbances and parameter variations, while backstepping control is appreciated for its systematic Lyapunov-based design [39,40]. These techniques have been utilized in quadrotors for tasks such as tracking and following paths [39]. For instance, Zhang et al. designed a disturbance observer-augmented sliding-mode backstepping controller for a quadrotor, demonstrating Lyapunov stability and exhibiting improved tracking compared to adaptive backstepping [14]. These schemes are designed to inherently compensate for errors caused by wind or modeling inaccuracies. Additionally, the two results are combined in sliding-mode backstepping controllers, which integrate the disturbance rejection of SMC with the recursive structure of backstepping. In [13], the authors developed a sliding-mode backstepping law for UAV interception that ensures chattering-free and asymptotically stable tracking of target trajectories. SMC and backstepping techniques are particularly effective for low-speed interception tasks, as they ensure both convergence and robustness, even in the presence of payload variations or wind gusts. These features make them ideal for use in defender UAVs tasked with fast and precise interception of adversaries.

Fully capable models enable rearranging when faced with multiple attacking UAVs to improve task efficiency. When confronted with multiple attackers, defenders must prioritize the threats they face. Unsupervised learning such as clustering offers a method to group attacker UAVs based on their spatial proximity. DBSCAN (Density-Based Spatial Clustering of Applications with Noise) is particularly effective for discovering clusters of arbitrary shapes while automatically categorizing outlier drones as noise [41,42]. DBSCAN does not require a pre-defined number of clusters and can handle irregular cluster shapes, unlike k-means [43,44]. In a nut shell, unsupervised clustering for UAV positioning improves resource allocation and multi-interception.

Defensive planning should also take into account the presence of intelligent adversaries. Modern attacking UAVs do not necessarily follow fixed, naive trajectories; rather, they can utilize onboard decision-making logic or learning algorithms [45,46]. An adversarial UAV can adjust its flight path to evade defenders and detect attempts at jamming [47]. Static intercept paths may not be adequate. One strategy to mitigate this issue is deception; defenders can create false cues or waypoints to mislead attackers. Spoofed navigation beacons or decoy routes can mislead an attacker about the defender’s intent. By adding deceptive waypoints to the defenders trajectory, we interrupt the attacker’s decision-making and mislead it from the actual trajectory. While this concept is not yet common in the UAV literature, similar ideas are present in electronic warfare: jamming and spoofing are established countermeasures [48,49,50]. Recent advancements in ground robotics and computer security have demonstrated that misleading an AI planning process can reduce adversarial effectiveness [51,52]. Incorporating deception into UAV defense can enhance its effectiveness against risky attackers.

Inspired by the discussions above, this research aims to integrate concepts from machine learning, specifically DBSCAN, SMC, backstepping, stability analysis, and waypoint sequencing to develop a UAV–UAV interception control strategy. In this research, the actual interception strategy involves clustering, transforming chaotic multi-agent environments into manageable sub-problems. The approach simplifies assignment by classifying attacking UAVs based on proximity and density, allowing for uncertainty in attacker trajectories. To provide precise and stable interceptions, a sliding-mode backstepping control technique combines the recursive stabilization of backstepping with the robustness of sliding-mode control. The stability of the quadrotor system is rigorously guaranteed by Lyapunov’s direct method, which ensures that the defenders’ centroids will converge to attackers’ notwithstanding dynamic uncertainties. Specifically, our contributions are as follows:

1.

This is the first research to integrate a multi-target grouping scheme based on DBSCAN for coordinating defenses against UAV attackers that are spatially distributed with sliding-mode control and backstepping for an UAV–UAV attacker–defender interception strategy and robust trajectory tracking for interception.

2.

In contrast to the research in [13], in this work a novel deceptive route point strategy is incorporated for defender paths that deceive attacking UAVs. The attackers are non-reactive and do not have access to the defender’s internal reference-switching and waypoint sequencing logic.

3.

A robust sliding-mode backstepping controller with an extended state observer (ESO) is incorporated for defender UAVs to accurately intercept cluster centroids under time-varying disturbances.

4.

This work employs the Lyapunov stability analysis method to guarantee the interception of attackers by UAV defenders and the stability of the system states.

The remainder of the paper is structured as follows. Section 2 provides the problem formulation of the UAV–UAV interception. Section 3 describes the UAV–UAV deceptive waypoint interception design using DBSCAN learning. Section 4 evaluates the approach in simulation. Finally, Section 5 concludes the work.

2. Problem Formulation

In Figure 1, the notations $A_i$ and $D_j$ refer to the attacking UAV i and the defending UAV j, respectively. Consider a swarm of M attackers $A_i\in {\mathbb{R}}^n$ with positions $P_A^i=[x_A^i,y_A^i,z_A^i]$ and velocities $V_A^i=[v_{A,x}^i,v_{A,y}^i,v_{A,z}^i]$, alongside N defending UAVs $D_j\in {\mathbb{R}}^n$ described by positions $P_D^j=[x_D^j,y_D^j,z_D^j]$ and velocities $V_D^j=[v_{D,x}^j,v_{D,y}^j,v_{D,z}^j]$. These states define the framework for UAV-to-UAV interception. The attacking swarm is clustered into N groups using the DBSCAN algorithm, after which each defender is tasked with intercepting the centroid of its allocated cluster. This technique reduces the complexity of interception and improves the efficiency of intercepting numerous attackers. To enhance robustness against risky attackers capable of anticipating defender motion, a conditional risk-activation strategy is introduced: when such behavior is detected, the defender switches from direct centroid pursuit to a waypoint sequencing strategy. The switching condition is defined by a binary risk flag

$$r_j\left(t\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{attackers}\mathrm{predict}\mathrm{the}\mathrm{path}\mathrm{of}\mathrm{defender}{D}_j,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

so that $r_j\left(t\right)=0$ corresponds to direct centroid pursuit and $r_j\left(t\right)=1$ activates deceptive waypoint sequencing. In the latter case, each defender $D_j$ generates a finite waypoint sequence $W_j=\{w_{j,1},w_{j,2},\dots ,w_{j,K}\}$, $w_{j,k}\in {\mathbb{R}}^3$, such that the final waypoint satisfies $w_{j,K}=c_{a\left(j\right)}\left(t_{\mathrm{term}}\right)$, where $c_{a\left(j\right)}$ is the centroid of the assigned attacker cluster. The risk flag $r_j\left(t\right)$ is implemented as a predefined piecewise-constant signal in the simulation studies in order to isolate and evaluate the effect of switching between direct centroid pursuit and the DWS waypoint-sequenced reference. Specifically, $r_j\left(t\right)$ is set to

$$r_j\left(t\right)=\left\{\begin{array}{cc}0,\hfill & t\in [0,t_1)\cup [t_2,t_3)\cup \cdots ,\hfill \\ 1,\hfill & t\in [t_1,t_2)\cup [t_3,t_4)\cup \cdots ,\hfill \end{array}\right.$$

where $\left\{t_k\right\}$ are prescribed switching instants. These intermediate waypoints ensure that the trajectory remains deceptive and less predictable, while guaranteeing that the ultimate mission objective arrival at the cluster centroid is achieved.

The defending UAVs are assigned to engage the centroids of the attacker clusters, which act as tactical reference points representing the geometric center of each concentrated attacker group. Once a defender arrives in the vicinity of its designated centroid, it initiates a neutralization procedure either through physical means or through non-kinetic actions such as electronic interference, kinetic measures, or net-based capture, targeting the nearby attackers within its operational interception radius. In this work, the centroid is not considered the terminal trajectory, but as a reference point that positions defenders advantageously for intercepting the attacker UAVs. This shifts the strategy from a mere rendezvous task to an active interception process, where reaching the centroid effectively triggers engagement at the swarm level.

Assumption 1: The positions and velocities of the attacking UAVs are assumed to be available through the radar sensing system at any given time.

Assumption 2: The defending quadrotors are considered to have the same dynamics, mass, drag characteristics, and inertia, and are modeled as rigid bodies with symmetric structures.

Assumption 3: The defenders operate independently, such that their trajectory and control inputs are not influenced by the states or actions of the other defenders.

3. UAV–UAV Deceptive Waypoint Interception Design Using DBSCAN Learning

3.1. Clustering Overview

Clustering is an essential, unsupervised learning approach used to group data points based on similarity or spatial closeness without the need for predefined labels [53]. In swarms, clustering is important for reducing the computational complexity and improving multiagent interception techniques by representing swarm UAVs through their centroids [54]. There are several clustering techniques, each providing different functional strength.

Uncertain, data-based techniques like Gaussian Mixture Models (GMMs) allow for flexible partition that can represent similar group structures; however, they need extra effort and rely on assumptions about the underlying data distribution and the number of clusters [55]. Clustering methods like k-means are simple and computationally efficient but require the number of clusters to be specified in advance and assume spherical cluster geometry, making them less suitable for dynamically evolving swarm patterns [56].

In contrast, density-based approaches, particularly the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm, are highly effective for identifying clusters of irregular shapes and handling noise in dynamic environments [41]. DBSCAN automatically determines the number of clusters by identifying dense regions within the data and isolating sparse or anomalous points as noise. This property makes DBSCAN particularly advantageous in swarm interception scenarios, where the number and spatial configuration of attacking UAVs can change unpredictably over time.

By relying on two essential factors, namely the neighborhood radius $\epsilon$ and the minimum number of surrounding points (MinPts), DBSCAN adaptively groups UAVs that are spatially close or moving cohesively, without requiring predefined cluster counts. This adaptability allows defensive systems to detect compact UAV formations, isolate outliers, and reassign interception resources dynamically. Given that the attacking UAVs in our system may demonstrate non-uniform movements behaviors and irregular location distributions, DBSCAN provides a robust and adaptable solution for real-time clustering, making it extremely appropriate for rapid interception tasks.

3.2. DBSCAN Clustering

DBSCAN is an unsupervised clustering algorithm that groups data points based on their spatial density rather than a predefined number of clusters. Unlike k-means, DBSCAN does not require the number of clusters N to be specified in advance and can effectively identify clusters of arbitrary shape, as well as isolate outliers [57]. This property makes DBSCAN suitable for dynamic multi-UAV environments where the number and formation of attacking quadrotors change over time.

In this work, DBSCAN is utilized to group attacking UAVs based on their spatial proximity and motion behavior. The clustering is performed using the instantaneous positions and velocities of the attackers, allowing defenders to identify dense formations of UAVs and detect outliers that may represent individual or evasive threats. The obtained clusters allow the system to assign appropriate defenders, predict attacker trajectory, and change the formation accordingly in real time. For simplicity and comprehensive understanding of UAV–UAV interception, we again consider the case when $M>N$.

The clusters are defined as

$$\begin{array}{c}\hfill C_{a\left(j\right)}=\{A_i\in E\mid A_i\mathrm{is}\mathrm{density-connected}\mathrm{to}\mathrm{any}{A}_j\in C_{a\left(j\right)}\},\\ \hfill a=1,2,\dots ,N\end{array}$$

(1)

where

• $E=\{A_1,A_2,\dots ,A_M\}$ represents the set of all attacking UAVs;
• $C_{a\left(j\right)}$ denotes the cluster a (i.e., a group of attackers identified as a dense region);
• Clustering is performed based on the attackers’ positions $P_A^i$ and velocities $V_A^i$.

In DBSCAN, two parameters govern the clustering behavior:

• $\epsilon$: the neighborhood radius defining spatial proximity;
• MinPts: the minimal quantity of adjacent points necessary to define a dense area.

For every attacker $A_i$, its $\epsilon$-neighborhood can be identified as

$$N_{\epsilon }\left(A_i\right)=\{A_j\in E\mid \parallel P_A^j-P_A^i\parallel \le \epsilon \}.$$

(2)

The point $A_i$ is categorized as:

• A core point if $|N_{\epsilon }\left(A_i\right)| \ge \mathrm{MinPts}$;
• A border point if it lies within $N_{\epsilon }$ of a core point but has fewer than MinPts neighbors;
• A noise point if it is neither a core nor a border point.

Each cluster $C_{a\left(j\right)}$ thus consists of a maximal set of density-connected points. To represent each cluster, we define a density-based centroid $c_{a\left(j\right)}$ as the mean position of all core points within the cluster:

$$c_{a\left(j\right)}={\displaystyle \frac{1}{|Q_a|}}\sum _{i\in Q_a}{P}_A^i,Q_a=\{A_i\in C_{a\left(j\right)}\mid A_i\mathrm{is}\mathrm{a}\mathrm{core}\mathrm{point}\}.$$

(3)

This centroid serves as the representative position of cluster $C_{a\left(j\right)}$, allowing defenders to coordinate interception based on the most stable (densest) regions of attacker formations.

Unlike k-means, DBSCAN does not minimize a global variance function but rather satisfies the density-connectivity condition. Formally, two attackers $A_i$ and $A_j$ belong to the same cluster if there exists a sequence of core points $(A_i=p_1,p_2,\dots ,p_k=A_j)$ such that

$$p_{t+1}\in N_{\epsilon }\left(p_t\right),\forall t=1,2,\dots ,k-1.$$

(4)

This ensures that attackers within each cluster are connected through dense neighborhoods, capturing natural formations without enforcing artificial cluster boundaries. Consequently, DBSCAN dynamically partitions the M attackers into N clusters (and possible noise points) suitable for cluster-to-defender assignments. DBSCAN uses two parameters: the neighborhood radius $\epsilon$ and the minimum number of neighbors $MinPts$ required to declare a core point. In the UAV interception setting, $\epsilon$ should be chosen to reflect the maximum separation at which attackers are considered part of the same locally dense formation, while remaining larger than sensing/measurement uncertainty and smaller than the typical separation between distinct formations. If $\epsilon$ is too small, a coherent formation can fragment into multiple clusters and more points can be labeled as noise; if $\epsilon$ is too large, nearby formations can be merged into a single cluster. The parameter $MinPts$ controls the required local density: larger values make clustering more conservative, whereas smaller values make cluster formation easier but may admit spurious clusters.

The analysis for the feasibility of the DBSCAN is done as follows.

3.2.1. Computational Complexity Analysis of DBSCAN Clustering

The complexity of DBSCAN clustering in this work depends primarily on the total number of attacking UAVs M, the spatial dimensionality $d=(x,y,z)$, and the neighborhood query mechanism determined by the radius $\epsilon$ and minimum point threshold $MinPts$. Unlike k-means, DBSCAN does not rely on iterative centroid updates; instead, its performance is dominated by the process of neighborhood searches required to identify dense regions [58].

For each UAV position, DBSCAN performs the following key operations:

• Neighborhood query: For every point, the algorithm retrieves all neighboring points within distance $\epsilon$. The computational cost of this step depends on the data structure used:

  -

  Using a naive linear search: $O\left(M^2\right)$;

  -

  Using an optimized spatial index (e.g., k-d tree or ball tree): $O(MlogM)$.

• Cluster expansion: Once a core point is identified ($|N_{\epsilon }| \ge \mathrm{MinPts}$), DBSCAN recursively expands its cluster by visiting all density-reachable points. This operation, in the worst-case scenario, requires visiting all M points once, resulting in $O\left(M\right)$ complexity.

Thus, the overall computational complexity of DBSCAN is expressed as

$$O(MlogM)\mathrm{or}O\left(M^2\right).$$

(5)

For three-dimensional UAV swarm data ($d=3$), the spatial indexing structure efficiently supports $\epsilon$-neighborhood queries in $O(\log M)$ time on average, resulting in near-linear scalability even for moderately large formations.

In standard UAV defense situations ($M\le 100$), the DBSCAN method attains clustering speeds of significantly less than ten milliseconds on an Intel i7 CPU with 16 GB of RAM, clearly confirmed by our simulations. This verifies that DBSCAN clustering is highly computational and appropriate for real-time use in unpredictable UAV interception scenarios.

Compared with k-means, DBSCAN eliminates the dependence on iteration count i and predefined cluster count N, while providing adaptive clustering performance and automatic noise handling. This is a significant advantage for online multi-UAV defense systems operating under uncertain swarm configurations.

3.2.2. Scalability Analysis of DBSCAN

The computational cost of DBSCAN primarily comes from neighborhood searches within the radius $\epsilon$ for each attacker point. A naive all-pairs implementation has worst-case complexity $\mathcal{O}\left(M^2\right)$, whereas practical implementations using spatial indexing typically scale closer to $\mathcal{O}(MlogM)$ for 3D data. Consistent with this, our measured clustering time is <10 ms for $M=100$ attackers. The clustering step remains within real-time limits at the tested update rate for the considered swarm sizes; large-scale swarms can be further supported by lowering the clustering update frequency and/or using incremental neighbor caching, which does not affect the controller update rate.

3.2.3. Sensitivity Analysis

To manage computational load during dynamic interception, clustering is performed at fixed intervals or whenever the spatial distribution of attackers exceeds a predefined deviation threshold. Since DBSCAN operates solely on attacker position data, the clustering step remains lightweight and scalable, even for large swarms. In MATLAB simulations (version 2023a) on a standard machine (Intel i7, 16 GB RAM), clustering between 50 and 100 attackers typically requires under 10 ms. The control loop runs at its own update rate, while cluster reassignment is triggered only when centroid movement surpasses a designated spatial limit, thereby avoiding unnecessary task reallocation. This approach preserves the real-time capability of the interception framework while keeping both communication overhead and computational effort low.

3.3. Deceptive Waypoint Sequencing (DWS) for UAV–UAV Interception

Deceptive Waypoint Sequencing (DWS) is introduced as a novel, mission-preserving maneuver-planning paradigm for defenders in centroid-based UAV interception. In scenarios where M attacking UAVs (with $M>N$) are clustered into N groups and defenders are assigned to intercept cluster centroids, straightforward pursuit of centroids can become predictable and thus exploitable by intelligent attackers that attempt to infer or anticipate defender motion. DWS combats this vulnerability by replacing a single direct centroid reference with a short, structured sequence of intermediate waypoints that are generated relative to the cluster centroid. The sequence deliberately introduces controlled, bounded deviations from the straight-line pursuit so as to increase the defender’s trajectory unpredictability, while preserving the primary mission requirement: arrival at the cluster centroid. DWS therefore augments centroid interception with a lightweight deception layer that is only activated when attacker predictability is detected to be high.

Within the centroid-based architecture, each defender $D_j$ is nominally tasked to reach the centroid $c_{a\left(j\right)}\left(t\right)$ of its assigned attacker cluster ${\mathcal{C}}_{a\left(j\right)}$. DWS modifies the centroid trajectory used by a defender: instead of commanding the defender directly to the centroid, DWS replaces the centroid trajectory by a finite sequence of centroid-relative waypoints $W_j$ that terminates at the centroid. Because DWS is defined relative to a moving centroid, it preserves scalability (cluster abstraction remains intact), simplifies waypoint parameterization, and ensures mission correctness (final waypoint equals the centroid).

Let $c\left(t\right)\in {\mathbb{R}}^3$ denote the current position of the centroid assigned to defender $D_j$ at planning time t. DWS produces a finite ordered waypoint sequence

$$W_j=\{w_{j,1},w_{j,2},\dots ,w_{j,K}\},w_{j,k}\in {\mathbb{R}}^3$$

where each waypoint is parametrically defined relative to $c\left(t\right)$ by three scalar parameters $(r_i,{\eta }_i,{\zeta }_i)$:

$$x_d^W\left(t\right)=w_{j,i}=c\left(t\right)+r_i\left[\begin{array}{c}cos{\eta }_i\\ sin{\eta }_i\\ {\zeta }_i\end{array}\right],i=1,\dots ,K.$$

(6)

Here, $r_i\ge 0$ is the radial offset from the centroid in the horizontal plane, ${\eta }_i\in (-\pi ,\pi ]$ is the planar bearing around the centroid, and ${\zeta }_i$ is the normalized vertical parameter. The final waypoint satisfies the mission constraint

$$w_{j,K}=c\left(t_{\mathrm{term}}\right),$$

which ensures that despite deceptive intermediate motion, the defender ultimately reaches the centroid.

3.3.1. Feasibility, Safety, and Smoothness Constraints

To keep DWS practical and compatible with the defender vehicle, the parameters $(r_i,{\eta }_i,{\zeta }_i)$ and derived waypoints $\left\{w_{j,i}\right\}$ must satisfy:

• Radial and vertical bounds:

  $$0\le r_i\le r_{max},\left|{\zeta }_i\right|\le {\zeta }_{max}.$$
• Inter-waypoint displacement and curvature: for $i=1,\dots ,K-1$,

  $$\parallel w_{i+1}-w_i\parallel \le d_{max},wrap\left(|{\eta }_{i+1}-{\eta }_i|\right)\le \Delta {\theta }_{max}.$$
• Safety:

  $$w_{j,i}\in \mathcal{F},\forall i,$$

  where $\mathcal{F}$ is the free-space set (no-fly zones excluded).
• Timing and derivative bounds: Given waypoint times $0={\tau }_1<{\tau }_2<\cdots <{\tau }_K$, the smoothed reference $x_d\left(\tau \right)$ must satisfy

  $$x_d\left({\tau }_i\right)=w_i,\parallel {\dot{x}}_d\left(\tau \right)\parallel \le v_{max},\parallel {\ddot{x}}_d\left(\tau \right)\parallel \le a_{max}.$$

3.3.2. Risk Activation and Switching

DWS is activated only when attackers demonstrate sufficient ability to predict attacker motion. A binary risk flag $r\left(t\right)\in \{0,1\}$ is used:

$$r\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{direct}\mathrm{centroid}\mathrm{pursuit},\hfill \\ 1,\hfill & \mathrm{DWS}\mathrm{mode}(\mathrm{deceptive}\mathrm{sequencing}).\hfill \end{array}\right.$$

(7)

When $r\left(t\right)=0$, the defender tracks $c\left(t\right)$ directly. When $r\left(t\right)=1$, the defender executes a DWS sequence W that terminates at $c\left(t_{\mathrm{term}}\right)$.

Remark 1. 

The signal $r_j\left(t\right)\in \{0,1\}$ switches the reference used by the defender between direct centroid pursuit and the DWS waypoint-sequenced reference. In this paper, $r_j\left(t\right)$ is selected as a predefined piecewise-constant signal in the simulation studies (with switching at prescribed time instants), and no autonomous supervisory logic is assumed that could induce arbitrarily fast switching. Since switching introduces no state reset and only modifies the bounded reference signal, and since the same stabilizing backstepping–sliding-mode structure is used in both modes, the closed-loop trajectories remain bounded in the considered cases.

3.3.3. Assigning UAV Defenders to Clusters of UAV Attackers Based on DWS

Each defender $D_j$ is mapped to a specific attacker cluster ${\mathcal{C}}_{a\left(j\right)}$. Under this assignment, the defender is guided toward the centroid $c_{a\left(j\right)}$ of its corresponding cluster, as defined in (2). Consequently, the interception behavior of the defender is oriented toward

$$P_D^j\to c_{a\left(j\right)},a=1,2,\dots ,N.$$

(8)

Therefore, to get an optimal attacker–defender assignment, we need to minimize the equation below:

$$\underset{\pi }{arg\min }\sum _{j=1}^N\parallel P_D^j-c_{a\left(j\right)}\parallel$$

(9)

where $\pi$ denotes the permutation function responsible for mapping defenders to their respective clusters.

This formulation guarantees that each defender is paired with the nearest cluster, thereby reducing the total travel distance between defenders and the corresponding centroids. As a result, the interception task can be accomplished in the shortest possible time. The flowchart in Figure 2 provides an overall summary of the DWS interception framework.

As illustrated in Figure 3, the base-station radar records the positions $P_A^i$ and velocities $V_A^i$ of all attacking UAVs at every time instant t. The DBSCAN algorithm is then applied to the acquired position data to generate clusters, with the corresponding centroids updated continuously as the attackers change location. This update process is performed by re-clustering the most recent measurements and recalculating the centroids $c_{a\left(j\right)}$ as defined in Equation (3). The resulting centroid positions trace trajectories $P_{c_{a\left(j\right)}}$, capturing the time-varying behavior of the attacker groups. In parallel, the base station continuously performs a risk assessment that monitors whether attackers display predictive or intelligent pursuit behaviors. If no elevated risk is detected, defenders pursue their assigned centroid trajectories directly. When the risk monitor indicates that defenders’ motions are predictable to the attackers, the base station activates the DWS strategy: rather than commanding the defender to track $c_{a\left(j\right)}$ directly, the base station generates a finite sequence of centroid-relative waypoints. Executing $W_j$ yields a deceptive, centroid-relative reference that reduces predictability to attackers while preserving the mission objective of intercepting the dynamically updated cluster centroid.

3.4. Defender UAV Model for DWS UAV Interception

In this work, the attacker UAVs are represented as time-varying sensor position points $P_A^i$. In contrast, the defender UAVs are modeled using a full nonlinear dynamic system to enable accurate maneuvering and effective interception. The quadrotor defender dynamics are governed by nonlinear differential equations that represent both its translational and rotational motion. Control is applied to regulate the defender’s $x,y,z$ coordinates so that it can track the evolving centroid of the attacker groups. The formulation and detailed modeling of quadrotor dynamics are well-established in the literature. Accordingly, the defending UAV’s six-degree-of-freedom state-space equations are given as in refs. [59,60].

Based on assumption 2, the dynamics of the defending UAVs are identical, and are given in Equation (10).

For $j=1,2\dots N$

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=a_1{x}_4{x}_6+a_2{x}_2^2+a_3\overline{\mathsf{\Omega }}{x}_4+b_1{U}_{j,2}+d_{j,2}\hfill \\ {\dot{x}}_3=x_4\hfill \\ {\dot{x}}_4=a_4{x}_2{x}_6+a_5{x}_4^2+a_6\overline{\mathsf{\Omega }}{x}_2+b_2{U}_{j,3}+d_{j,4}\hfill \\ {\dot{x}}_5=x_6\hfill \\ {\dot{x}}_6=a_7{x}_2{x}_4+a_8{x}_6^2+b_3{U}_{j,4}+d_{j,6}\hfill \\ {\dot{x}}_7=x_8\hfill \\ {\dot{x}}_8=a_9{x}_8+U_{j,x}{\displaystyle \frac{U_{j,1}}{m}}+d_{j,8}\hfill \\ {\dot{x}}_9=x_{10}\hfill \\ {\dot{x}}_{10}=a_{10}{x}_{10}+U_{j,y}{\displaystyle \frac{U_{j,1}}{m}}\hfill \\ {\dot{x}}_{11}=x_{12}\hfill \\ {\dot{x}}_{12}=a_{11}{x}_{12}+{\displaystyle \frac{C{x}_{1}C{x}_3}{m}}{U}_{j,1}-g\hfill \end{array}\right.$$

(10)

Assumption 4: The states of the defending UAVs in Equation (10) are all measurable.

where the state vector of the system is given as

$$\begin{array}{cc}X=& [{\varphi }_D^j,{\dot{\varphi }}_D^j,{\theta }_D^j,{\dot{\theta }}_D^j,{\psi }_D^j,{\dot{\psi }}_D^j,\hfill \\ \hfill & x_D^j,{\dot{x}}_D^j,y_D^j,{\dot{y}}_D^j,z_D^j,{\dot{z}}_D^j{]}^{\mathrm{T}}=[x_1,x_2,x_3,x_4,x_5,x_6,\hfill \\ \hfill & x_7,x_8,x_9,x_{10},x_{11},x_{12}{]}^{\mathrm{T}}\hfill \end{array}$$

$U_{j,1},U_{j,2},U_{j,3}$, and $U_{j,4}$ are the control inputs of the system, determined by the angular velocities of the four rotors. The angular state variables $\mathsf{\Phi },\theta ,\psi$ are the roll, pitch and yaw respectively. $d_2,d_4,d_6$ and $d_8$ are the additive disturbances to the state of the quadrotor.

The dynamics in Equation (10) can be interpreted as two coupled subsystems:

• Rotational subsystem $\left(x_1–x_6\right)$: the states $(x_1,x_3,x_5)$ correspond to the attitude angles $(\varphi ,\theta ,\psi )$ and $(x_2,x_4,x_6)$ correspond to the corresponding angular rates $(\dot{\varphi },\dot{\theta },\dot{\psi })$. The expressions for ${\dot{x}}_2$, ${\dot{x}}_4$, and ${\dot{x}}_6$ include the gyroscopic/cross-inertia coupling terms parameterized by $a_1–a_8$ and are driven by the torque inputs $U_{j,2}$, $U_{j,3}$, and $U_{j,4}$.
• Translational subsystem $\left(x_7–x_{12}\right)$: the states $(x_7,x_9,x_{11})$ correspond to inertial positions $(x,y,z)$ and $(x_8,x_{10},x_{12})$ correspond to the associated linear velocities $(\dot{x},\dot{y},\dot{z})$. The expressions for ${\dot{x}}_8$, ${\dot{x}}_{10}$, and ${\dot{x}}_{12}$ include drag terms parameterized by $a_9–a_{11}$ and are driven by the total thrust $U_{j,1}$ projected via the direction-cosine terms $U_{j,x}$ and $U_{j,y}$.

• The coefficients $a_i$ and $b_i$ (as defined in Equation (10) and the inertia-related expressions in Equation (11)) therefore map explicitly to the appropriate rotational/translational states.

  $$\begin{array}{cccccc}\hfill a_1& ={\displaystyle \frac{I_y-I_z}{I_x}},\hfill & \hfill a_2& ={\displaystyle \frac{-K_{fax}}{I_x}},\hfill & \hfill a_3& ={\displaystyle \frac{-J_r}{I_x}},\hfill \\ \hfill a_4& ={\displaystyle \frac{I_z-I_x}{I_y}},\hfill & \hfill a_5& ={\displaystyle \frac{-K_{fay}}{I_y}},\hfill & \hfill a_6& ={\displaystyle \frac{J_r}{I_y}},\hfill \\ \hfill a_7& ={\displaystyle \frac{I_x-I_y}{I_z}},\hfill & \hfill a_8& ={\displaystyle \frac{-K_{faz}}{I_z}},\hfill & \hfill a_9& ={\displaystyle \frac{-K_{ftx}}{m}},\hfill \\ \hfill a_{10}& ={\displaystyle \frac{-K_{fty}}{m}},\hfill & \hfill a_{11}& ={\displaystyle \frac{-K_{ftz}}{m}},\hfill \\ \hfill b_1& ={\displaystyle \frac{d}{I_x}},\hfill & \hfill b_2& ={\displaystyle \frac{d}{I_y}},\hfill & \hfill b_3& ={\displaystyle \frac{1}{I_z}},\hfill \end{array}$$

  $$\left\{\begin{array}{c}{U}_{j,x}=C{x}_{1}S{x}_{3}C{x}_5+S{x}_{1}S{x}_5,\hfill \\ U_{j,y}=C{x}_{1}S{x}_{3}S{x}_5-S{x}_{1}C{x}_5,\hfill \end{array}\right.$$

m is the total mass of the structure and $J\in R^{3\times 3}$ is a symmetric positive definite constant inertia matrix of the quadrotor given by

$$J=\left(\begin{array}{ccc}{I}_x& 0& 0\\ 0& I_y& 0\\ 0& 0& I_z\end{array}\right)$$

(11)

$\mathsf{\Omega }$ is the angular velocity of the airframe:

$$\mathsf{\Omega }=\left(\begin{array}{ccc}1& 0& -sin\theta \\ 0& cos\varphi & cos\theta sin\varphi \\ 0& -sin\varphi & cos\varphi cos\theta \end{array}\right)\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]$$

(12)

where $K_{ftx},K_{fty}$ and $K_{ftz}$ are the translation drag coefficients. $K_{fax},K_{fay}$ and $K_{faz}$ are the friction aerodynamic coefficients. d is the distance between the rotors. The rotor is a unit comprising a D.C-motor actuating a propeller via a reducer. The D.C-motor is governed by the following dynamic equations:

$$\left\{\begin{array}{c}V=ri+L{\displaystyle \frac{di}{dt}}+k_e\omega \hfill \\ k_{m}i=J_r{\displaystyle \frac{d\omega }{dt}}+C_s+k_r{\omega }^2\hfill \end{array}\right.$$

(13)

The motor parameters are defined as follows: V denotes the input voltage applied to the motor; $k_e$ and $k_m$ are the electrical and mechanical torque constants, respectively; $k_r$ represents the constant torque load; r is the internal resistance of the motor; and $J_r$ is the rotor’s moment of inertia.

Based on these definitions, the rotor dynamics are modeled as

$${\dot{\omega }}_i=b{V}_i-{\gamma }_0-{\gamma }_1{\omega }_i-{\gamma }_2{\omega }_i^2$$

(14)

$i\in [1,4]$

With

$${\gamma }_0={\displaystyle \frac{C_s}{J_r}},{\gamma }_1={\displaystyle \frac{k_e{k}_m}{r{J}_r}},{\gamma }_2={\displaystyle \frac{k_r}{J_r}}\mathrm{and}b={\displaystyle \frac{k_m}{r{J}_r}}$$

Therefore,

$$\overline{\mathsf{\Omega }}=\left({\omega }_1-{\omega }_2+{\omega }_3-{\omega }_4\right)$$

(15)

3.5. Control

This section introduces the proposed methodology for designing the DWS interception control strategy. As outlined in Algorithm 1, the control goal is to intercept the trajectories of attacking UAVs while misleading intelligent adversaries. This is accomplished by generating deceptive waypoints along the paths of the defending UAVs, which are activated when a high threat level is detected, i.e., when the base station identifies the attacking UAVs as intelligent. The overall interception framework is illustrated in the block diagram shown in Figure 4.

Algorithm 1: DWS Interception Algorithm using DBSCAN

To formalize the objective, let the positions of the attacking UAVs correspond to the measured and clustered trajectories, denoted by $x_{m,d}=P_{c_{a\left(j\right)}}$, where the desired defender positions $P_{c_{a\left(j\right)}}$ form a 12-dimensional state vector representing the position and orientation of the attacking UAVs. A backstepping-based sliding-mode control law is employed to carry out the interception, ensuring system stability, robustness, and accurate tracking of the desired dynamics while handling all system nonlinearities. Furthermore, an Extended State Observer (ESO) is utilized to estimate additive disturbances affecting the states of the defending quadrotor UAVs.

Let $P_{c_{a\left(j\right)}}\left(t\right)$ denote the nominal centroid reference. The active desired trajectory provided to the controller is

$$x_{m,d}\left(t\right)=\left(1-r_j\left(t\right)\right)P_{c_{a\left(j\right)}}\left(t\right)+r_j\left(t\right)x_d^W\left(t\right).$$

(16)

Accordingly, the desired signals $(x_{m,d},{\dot{x}}_{m,d},{\ddot{x}}_{m,d})$ are from $x_{m,d}$ and its time derivatives. Thus, when $r_j\left(t\right)=0$ the controller tracks the centroid trajectory; when $r_j\left(t\right)=1$ it tracks the DWS trajectory that ends at the centroid. $x_d^W\left(t\right)$ is the trajectory of the waypoint defined in (6).

Assumption 5: All waypoints $w_{j,i}$ lie in the free space and respect separation/no-fly constraints (safety).

Theorem 1. 

The stability of the closed-loop system described in (10) can be guaranteed by appropriately selecting the parameters of the proposed controllers given in (24), (27), (30), (33), (36), and (38), along with the ESOs in (25), (28), (31), and (34).

Proof. 

A backstepping control approach is applied recursively to construct the control laws, which reduces the computational effort involved in handling tracking errors and Lyapunov function evaluations. The procedure is applied for $j=1,2,\dots ,N$, where N represents the total number of defending UAVs:

$$\begin{array}{cccc}{z}_m\hfill & =x_{m,d}-x_m,\hfill & \hfill & m=1,3,\dots ,11\hfill \end{array}$$

(17a)

$$\begin{array}{cccc}{z}_m\hfill & =x_m-{\dot{x}}_{(m-1),d}-{\alpha }_{m-1}{e}_{m-1},\hfill & \hfill & m=2,4,\dots ,12\hfill \end{array}$$

(17b)

where $x_m$ denotes the actual position of the defending UAVs, $x_{m,d}$ specifies the desired position, and $z_m$ represents the tracking error between them.

$${\alpha }_m>0,\forall m=1,2,\dots ,12.$$

$$\begin{array}{cccc}\hfill V_m& =\frac{1}{2}{z}_m^2,\hfill & \hfill & m=1,3,\dots ,11\hfill \end{array}$$

(18a)

$$\begin{array}{cccc}\hfill V_m& =V_{m-1}+\frac{1}{2}{z}_m^2,\hfill & \hfill & m=2,4,\dots ,12\hfill \end{array}$$

(18b)

The coefficient ${\alpha }_m$ is a positive design gain chosen to stabilize the virtual control error using Lyapunov-based backstepping. The value influences the convergence rate of the tracking error $z_m$ and must meet ${\alpha }_m>0$. $V_m$ is the candidate for the Lyapunov function chosen. □

Chattering Suppression: To suppress high-frequency chattering caused by the discontinuous sign function, we define a continuous saturation function:

$${\mathrm{sat}}_f\left(S\right)=\left\{\begin{array}{cc}1,\hfill & S>f,\hfill \\ {\displaystyle \frac{S}{f}},\hfill & \left|S\right|\le f,\hfill \\ -1,\hfill & S<-f,\hfill \end{array}\right.f\in \{1,3,5,7,9,11\}.$$

(19)

where $f>0$ defines the boundary layer thickness.

• Step $m=2$ (surface $S_1$)

The choice of sliding surfaces depends on the derived tracking errors in (16). If the sliding surface is chosen to be

$$S_1=z_2=x_2-{\dot{x}}_{1,d}-{\alpha }_1{z}_1$$

(20)

From Equation (18), the Lyapunov function is chosen to be

$$V_2=\left({\displaystyle \frac{1}{2}}{z}_1^2+{\displaystyle \frac{1}{2}}{z}_2^2\right)$$

(21)

Combining (17), (19) and (20), the derivative of the Lyapunov is

$$\begin{array}{cc}\hfill {\dot{V}}_2& =z_1{\dot{z}}_1+S_1{\dot{S}}_1,\hfill \\ \hfill {\dot{V}}_2& =z_1{\dot{z}}_1+S_1(a_1{x}_4{x}_6+a_2{x}_2^2+a_3{x}_4\overline{\mathsf{\Omega }}+b_1{U}_{j,2}+d_{j,2}\hfill \\ \hfill & -{\ddot{x}}_{1,d}-{\alpha }_1\left({\dot{x}}_{1,d}-x_2\right)).\hfill \end{array}$$

(22)

The assumed law for the surface is the derivative of (19) and therefore satisfies $\left(S_1{\dot{S}}_1<0\right)$

$$\begin{array}{cc}\hfill {\dot{S}}_1& =-q_1{sat}_1\left(S_1\right)-k_1{S}_1\hfill \\ \hfill & ={\dot{x}}_2-{\ddot{x}}_{1,d}-{\alpha }_1{\dot{z}}_1\hfill \\ \hfill & =a_1{x}_4{x}_6+a_2{x}_2^2+a_3\overline{\mathsf{\Omega }}{x}_4+b_1{U}_{j,2}+d_{j,2}\hfill \\ \hfill & -{\ddot{x}}_{1,d}-{\alpha }_1\left({\dot{x}}_{1,d}-x_2\right)\hfill \end{array}$$

(23)

Using the backstepping approach, the control input $U_{j,2}$ is extracted as

$$\begin{array}{cc}{U}_{j,2}=& {\displaystyle \frac{1}{b_1}}(-q_1{\mathrm{sat}}_1\left(S_1\right)-k_1{S}_1-a_1{x}_4{x}_6-a_2{x}_2^2-a_3\overline{\mathsf{\Omega }}{x}_4-\widehat{d_{j,2}}\hfill \\ \hfill & +{\ddot{x}}_{1,d}+{\alpha }_1\left({\dot{x}}_{1,d}-x_2\right))\hfill \end{array}$$

(24)

Using the first order ESO, the disturbance $d_2$ can be estimated as

$$\begin{array}{cc}\hfill {\dot{\xi }}_2& =-h_2{\xi }_2-h_2^2{x}_2\hfill \\ \hfill & -h_2\left(a_1{x}_4{x}_6+a_2{x}_2^2+a_3\overline{\mathsf{\Omega }}{x}_4+b_1{U}_{j,2}\right),h_2>0,\hfill \\ \hfill {\widehat{d}}_{j,2}& ={\xi }_2+h_2{x}_2.\hfill \end{array}$$

(25)

By substituting the control input $U_{j,2}$ from (23) into (21), we obtain

$${\dot{V}}_2=z_1{\dot{z}}_1-k_1{S}_1^2-S_1({\widehat{d}}_{j,2}-d_{j,2})-q_1{S}_1{\mathrm{sat}}_1\left(S_1\right)\le 0$$

(26)

Given that $k_1>0$ and $q_1>0$, ${\dot{V}}_2$ is negative semi-definite. This ensures that the tracking errors $z_1$ and $z_2$ remain bounded and that the sliding surface $S_1$ converges to zero. Consequently, the control law defined in (24) guarantees the Lyapunov stability of this subsystem.

The remaining steps in the control design can be obtained recursively based on the tracking errors in (17a) and (17b), the Lyapunov candidates (18a) and (18b), the saturation law (19), the surface (20), the derivations (23)–(25), and the ESO (25); we repeat the same design for the even indices $m\in \{4,6,8,10,12\}$. For each case, define $S_{m-1}=z_m=x_m-{\dot{x}}_{(m-1),d}-{\alpha }_{m-1}{z}_{m-1}$ (as in (17b)) and use ${sat}_f(·)$ with the corresponding odd index $f\in \{3,5,7,9,11\}$.

Lemma 1. 

Consider the first-order ESO in Equation (25) used to estimate the additive disturbance $d_{j,i}\left(t\right)$ in the channel ${\dot{x}}_i$, where $i\in \{2,4,6,8\}$. Define the estimation error as ${\tilde{d}}_{j,i}\left(t\right)=d_{j,i}\left(t\right)-{\widehat{d}}_{j,i}\left(t\right)$, where ${\widehat{d}}_{j,i}\left(t\right)={\xi }_i\left(t\right)+h_i{x}_i\left(t\right)$ and $h_i>0$ (Equation (25)). Assume that $d_{j,i}\left(t\right)$ is differentiable and satisfies $|{\dot{d}}_{j,i}\left(t\right)|\le {\overline{\delta }}_i$ for all $t\ge 0$. Then, the estimation error satisfies

$${\dot{\tilde{d}}}_{j,i}\left(t\right)=-h_i{\tilde{d}}_{j,i}\left(t\right)+{\dot{d}}_{j,i}\left(t\right),i\in \{2,4,6,8\},$$

(27)

and is exponentially ultimately bounded as

$$|{\tilde{d}}_{j,i}\left(t\right)|\le e^{-h_{i}t}|{\tilde{d}}_{j,i}\left(0\right)|+{\displaystyle \frac{{\overline{\delta }}_i}{h_i}}\left(1-e^{-h_{i}t}\right),i\in \{2,4,6,8\}.$$

(28)

In particular, if $d_{j,i}\left(t\right)$ is constant (i.e., ${\dot{d}}_{j,i}\left(t\right)=0$), then ${\tilde{d}}_{j,i}\left(t\right)$ converges exponentially to zero.

Proof. 

For each $i\in \{2,4,6,8\}$, the plant channel is ${\dot{x}}_i=f_i+g_i{U}_{j,i}+d_{j,i}$, and the ESO structure in Equation (25) gives ${\widehat{d}}_{j,i}={\xi }_i+h_i{x}_i$ with ${\dot{\xi }}_i=-h_i({\xi }_i+h_i{x}_i)-h_i{f}_i-h_i{g}_i{U}_{j,i}$. Thus, ${\dot{\widehat{d}}}_{j,i}={\dot{\xi }}_i+h_i{\dot{x}}_i=-h_i({\xi }_i+h_i{x}_i)-h_i{f}_i-h_i{g}_i{U}_{j,i}+h_i(f_i+g_i{U}_{j,i}+d_{j,i})=-h_i{\widehat{d}}_{j,i}+h_i{d}_{j,i}$. Therefore, ${\dot{\tilde{d}}}_{j,i}={\dot{d}}_{j,i}-{\dot{\widehat{d}}}_{j,i}={\dot{d}}_{j,i}-(-h_i{\widehat{d}}_{j,i}+h_i{d}_{j,i})=-h_i(d_{j,i}-{\widehat{d}}_{j,i})+{\dot{d}}_{j,i}=-h_i{\tilde{d}}_{j,i}+{\dot{d}}_{j,i}$, which proves Equation (27). Solving the linear system yields $|{\tilde{d}}_{j,i}\left(t\right)|\le e^{-h_{i}t}|{\tilde{d}}_{j,i}\left(0\right)|+{\int }_0^t{e}^{-h_i(t-\tau )}|{\dot{d}}_{j,i}\left(\tau \right)|d\tau \le e^{-h_{i}t}\left|{\tilde{d}}_{j,i}\left(0\right)\right|+{\displaystyle \frac{{\overline{\delta }}_i}{h_i}}(1-e^{-h_{i}t})$, which proves Equation (28). □

• Step $m=4$ (surface $S_3$)

$$S_3=z_4=x_4-{\dot{x}}_{3,d}-{\alpha }_3{z}_3.$$

$$\begin{array}{cc}{U}_{j,3}=& {\displaystyle \frac{1}{b_2}}(-q_3{\mathrm{sat}}_3\left(S_3\right)-k_3{S}_3-(a_4{x}_2{x}_6+a_5{x}_4^2+a_6\overline{\mathsf{\Omega }}{x}_2)\hfill \\ \hfill & -\widehat{d_{j,4}}+{\ddot{x}}_{3,d}+{\alpha }_3\left({\dot{x}}_{3,d}-x_4\right))\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\dot{\xi }}_4& =-h_4{\xi }_4-h_4^2{x}_4\hfill \\ \hfill & -h_4\left(a_4{x}_2{x}_6+a_5{x}_4^2+a_6\overline{\mathsf{\Omega }}{x}_2+b_2{U}_{j,3}\right),h_4>0,\hfill \\ \hfill {\widehat{d}}_{j,4}& ={\xi }_4+h_4{x}_4.\hfill \end{array}$$

(30)

$${\dot{V}}_4=z_3{\dot{z}}_3-k_3{S}_3^2-S_3(\widehat{d_{j,4}}-d_{j,4})-q_3{S}_3{\mathrm{sat}}_3\left(S_3\right)\le 0.$$

(31)

• Step $m=6$ (surface $S_5$)

$$S_5=z_6=x_6-{\dot{x}}_{5,d}-{\alpha }_5{z}_5.$$

$$\begin{array}{cc}{U}_{j,4}=& {\displaystyle \frac{1}{b_3}}(-q_5{\mathrm{sat}}_5\left(S_5\right)-k_5{S}_5-(a_7{x}_2{x}_4+a_8{x}_6^2)\hfill \\ \hfill & -\widehat{d_{j,6}}+{\ddot{x}}_{5,d}+{\alpha }_5\left({\dot{x}}_{5,d}-x_6\right))\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\dot{\xi }}_6& =-h_6{\xi }_6-h_6^2{x}_6\hfill \\ \hfill & -h_6\left(a_7{x}_2{x}_4+a_8{x}_6^2+b_3{U}_{j,4}\right),h_6>0,\hfill \\ \hfill {\widehat{d}}_{j,6}& ={\xi }_6+h_6{x}_6.\hfill \end{array}$$

(33)

$${\dot{V}}_6=z_5{\dot{z}}_5-k_5{S}_5^2-S_5(\widehat{d_{j,6}}-d_{j,6})-q_5{S}_5{\mathrm{sat}}_5\left(S_5\right)\le 0.$$

(34)

• Step $m=8$ (surface $S_7$)

$$S_7=z_8=x_8-{\dot{x}}_{7,d}-{\alpha }_7{z}_7.$$

$$\begin{array}{cc}{U}_{j,x}=& {\displaystyle \frac{m}{U_{j,1}}}(-q_7{\mathrm{sat}}_7\left(S_7\right)-k_7{S}_7-a_9{x}_8\hfill \\ \hfill & -\widehat{d_{j,8}}+{\ddot{x}}_{7,d}+{\alpha }_7\left({\dot{x}}_{7,d}-x_8\right))\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\dot{\xi }}_8& =-h_8{\xi }_8-h_8^2{x}_8\hfill \\ \hfill & -h_8\left(a_9{x}_8+U_{j,x}{\displaystyle \frac{U_{j,1}}{m}}\right),h_8>0,\hfill \\ \hfill {\widehat{d}}_{j,8}& ={\xi }_8+h_8{x}_8.\hfill \end{array}$$

(36)

$${\dot{V}}_8=z_7{\dot{z}}_7-k_7{S}_7^2-S_7(\widehat{d_{j,8}}-d_{j,8})-q_7{S}_7{\mathrm{sat}}_7\left(S_7\right)\le 0.$$

(37)

• Step $m=10$ (surface $S_9$)

$$S_9=z_{10}=x_{10}-{\dot{x}}_{9,d}-{\alpha }_9{z}_9.$$

$$\begin{array}{cc}{U}_{j,y}=& {\displaystyle \frac{m}{U_{j,1}}}(-q_9{\mathrm{sat}}_9\left(S_9\right)-k_9{S}_9-a_{10}{x}_{10}\hfill \\ \hfill & +{\ddot{x}}_{9,d}+{\alpha }_9\left({\dot{x}}_{9,d}-x_{10}\right))\hfill \end{array}$$

(38)

$${\dot{V}}_{10}=z_9{\dot{z}}_9-k_9{S}_9^2-q_9{S}_9{\mathrm{sat}}_9\left(S_9\right)\le 0.$$

(39)

• Step $m=12$ (surface $S_{11}$)

$$S_{11}=z_{12}=x_{12}-{\dot{x}}_{11,d}-{\alpha }_{11}{z}_{11}.$$

$$\begin{array}{cc}{U}_{j,1}=& {\displaystyle \frac{m}{C{x}_{1}C{x}_3}}(-q_{11}{\mathrm{sat}}_{11}\left(S_{11}\right)-k_{11}{S}_{11}-a_{11}{x}_{12}+g\hfill \\ \hfill & +{\ddot{x}}_{11,d}+{\alpha }_{11}\left({\dot{x}}_{11,d}-x_{12}\right))\hfill \end{array}$$

(40)

$${\dot{V}}_{12}=z_{11}{\dot{z}}_{11}-k_{11}{S}_{11}^2-q_{11}{S}_{11}{\mathrm{sat}}_{11}\left(S_{11}\right)\le 0.$$

(41)

Hence, for $i\in \{3,5,7,9,11\}$, with $S_i$, $z_i$, $k_i>0$, and $q_i>0$, it follows that ${\dot{V}}_i$ for $i\in \{4,6,8,10,12\}$ is negative semi-definite. This ensures that the tracking errors $z_i$ remain bounded and that the sliding surfaces $S_i$ converge to zero. As a result, the selected control inputs guarantee the Lyapunov stability of the entire system.

Remark 2. 

Under the gain conditions of Theorem 1 and assumptions (5) and (6), the Lyapunov inequalities (26), (31), (34), (37), (39), (41) hold on each mode ($r_j=0$ or $r_j=1$), guaranteeing bounded tracking errors and convergence of each sliding surface to zero on any interval of constant mode. With continuity at switching, standard multiple-Lyapunov functions imply uniform ultimate boundedness of the switched closed loop. If the mode remains fixed over a nonzero interval (e.g., during an engagement), asymptotic convergence follows, as in the baseline.

Remark 3. 

We assume $U_{j,1}>0$ in flight so that $\left(35\right)$–$\left(38\right)$ are well-defined.

4. Simulation Results

Simulations were performed to evaluate the performance and accuracy of the proposed interception control strategy implemented on a quadrotor UAV. The controller gains were set as $k1=k2=k3=k4=k5=k6=1$, $q1=q2=q3=q4=q5=q6=1$, and $h_2=h_4=h_6=h_8=10$. These gains were chosen empirically to obtain stable closed-loop behavior and smooth control signals, and they were kept fixed across all simulation cases to ensure a consistent comparison. The parameters of the defending quadrotor UAVs employed in this study are summarized in

Table 1. Parameters of the UAVs (defender).

Parameter	Value	Parameter	Value
${\gamma }_0$	$189.63$	${\gamma }_1$	$6.0612$
${\gamma }_2$	$0.0122$	J	$\mathrm{diag}(3.8278,3.8288,7.6566)\times {10}^{-3}\mathrm{N}·\mathrm{m}/(\mathrm{rad}/{\mathrm{s}}^2)$
m	$486\mathrm{g}$	$K_{fa}$	$\mathrm{diag}(5.5670,5.5670,6.3540)\times {10}^{-4}\mathrm{N}/(\mathrm{rad}/\mathrm{s})$
d	$25\mathrm{cm}$	b	$280.19$
$J_r$	$2.8385\times {10}^{-5}\mathrm{N}·\mathrm{m}/(\mathrm{rad}/{\mathrm{s}}^2)$	$K_{ft}$	$\mathrm{diag}(5.5670,5.5670,6.3540)\times {10}^{-4}\mathrm{N}/(\mathrm{m}/\mathrm{s})$

[9]. The DWS values $w_{j,i}$ are determined by selecting appropriate parameters $r_i$, ${\eta }_i$, and ${\zeta }_i$. The centroid-relative waypoint parameters were selected empirically (trial-and-error) within conservative bounds consistent with the defender UAVs nominal maneuverability and safety margins, ensuring smooth tracking without excessive control effort, while remaining sufficiently nontrivial to realize the intended deception effect. To illustrate the effectiveness of the proposed interception framework, simulations were carried out for both single and multiple UAV-to-UAV interception scenarios.

The simulation focuses on intercepting the centroids of attacker clusters rather than pursuing individual UAVs. It is assumed (Assumption 1) that the initial positions of the attackers are obtained via sensing. DBSCAN clustering is applied to compute the cluster centroids based on spatial proximity. Each of the trajectories of the centroid are then regarded as the reference position for the defending UAVs.

To evaluate the robustness and effectiveness of the proposed sliding-mode backstepping interception strategy, a baseline comparison is conducted against a conventional Proportional–Integral–Derivative (PID) controller. The PID control input along each axis is formulated as

$$u_{\mathrm{PID}}\left(t\right)=K_{p}e\left(t\right)+K_i{\int }_0^{t}e\left(\tau \right)d\tau +K_d{\displaystyle \frac{de\left(t\right)}{dt}}$$

(42)

In this setup, $e\left(t\right)$ denotes the tracking error between the actual position of the defending UAV and the assigned centroid trajectory. The PID gains $K_p$, $K_i$, and $K_d$ were tuned empirically for each axis to achieve accurate and stable tracking under ideal operating conditions.

To assess robustness under practical uncertainties, bounded additive disturbances were introduced into the defender UAV dynamics. In particular, disturbances denoted by $d_2,d_4,d_6,d_8$ were incorporated into Equation (10), which describe ${\dot{x}}_2,{\dot{x}}_4,{\dot{x}}_6,{\dot{x}}_8$. These disturbances simulate unmodeled system dynamics, as well as environmental factors such as wind gusts and sensor measurement noise. The disturbances are defined as $d_{1,2}=4sin\left(0.5t\right)$, $d_{1,4}=sin\left(2t\right)$, $d_{1,6}=7sin\left(5t\right)$ and $d_{1,8}=sin\left(3t\right)$. for $j=2$$d_{2,2}=4sin\left(0.5t\right)$, $d_{2,4}=sin\left(2t\right)$, $d_{2,6}=7sin\left(5t\right)$ and $d_{2,8}=sin\left(3t\right)$.

In this work, the attacker is modeled as non-reactive and does not have access to the defender’s internal reference-switching and waypoint sequencing logic. Therefore, the purpose of the simulations is to validate the feasibility and robustness of the proposed interception framework when DWS generates the reference trajectory, rather than to model a full adversarial decision-making process. Evaluating DWS against specific intelligent attacker predictors (e.g., learning-based anticipation or game-theoretic avoidance) is an important direction for future work.

Case 1: 

This scenario involves a single defending quadrotor UAV and 10 attacking UAVs i.e., $N=1$ and $M=10$. Here, $N=1$, as there is only one cluster output by DBSCAN. Unlike partition-based methods such as k-means, the DBSCAN algorithm does not require a predefined number of clusters. Instead, it autonomously identifies the most suitable cluster structure based on the spatial distribution and density of quadrotor UAVs. This capability makes DBSCAN highly effective for detecting clusters of arbitrary shapes and distinguishing noise or outlier UAVs in dynamic environments. Clustering is conducted to determine the centroid of the 10 quadrotors. In Case 1, which involves one defender and one centroid, the trajectory is described by $x=sin\left(t\right),y=cos\left(t\right)$, and $z=t$. The DWS is introduced at $(T/3)$ s and $(2T/3)$ s, which is done by varying the risk $r_j\left(t\right)$ in Equation (16), and the waypoints introduced are $w_{1,1},$ and $w_{1,2},$ where the last waypoint $w_{1,k},$ is the actual centroid equation. In this case, the waypoints parameters are $r_{1,1}=2$, ${\eta }_{1,1}=20$, $r_{1,2}=2$, ${\eta }_{1,2}=20$, ${\zeta }_{1,1}=10$ and ${\zeta }_{1,2}=10$. At each time step, the positions of the UAVs and their corresponding centroids are recorded, as shown in Figure 5. The motion of the attacking UAVs produces dynamically updated centroids $c_a$, forming trajectories $P_{c_a\left(j\right)}$. Figure 6 depicts the interception of the attacking UAV by the defending UAV using the DWS at 3 s and 6 s, with the final interception occurring at 10 s, as determined by the ode45 solver in the MATLAB simulation. In contrast, Figure 7 indicates that the PID-based approach fails in achieving interception. Figure 8 shows the interception tracking error for the proposed approach, which converges to nearly zero in under 4 s. The control input evolution for this interception is illustrated in Figure 9. The control signals remain smooth and free from chattering, ensuring that the defending quadrotor effectively tracks and intercepts the attacking UAV without overshoot during the transient phase. Additionally, Figure 10 presents the disturbances and their corresponding estimates obtained via the Extended State Observer (ESO). As shown, the estimation error converges to zero quickly. This shows that the proposed control technique is robust despite disturbances in the dynamics of the defending UAVs. To ensure robustness, Figure 6 and Figure 7 demonstrate that the PID controller fails to accurately intercept the attacker centroid when disturbances are present. This leads to substantial interception errors and can ultimately cause mission failure. In comparison, the proposed interception strategy demonstrates reliable performance, successfully accomplishing the interception while showing strong robustness against external disturbances and modeling uncertainties.

Case 2: 

In this case, we use two defending quadrotor UAVs and 20 attacking UAVs, i.e., $N=2$ and $M=20$. Here, $N=2$ as there are only two clusters output by DBSCAN. Clustering is conducted to determine the centroids of the 20 quadrotors. In this case, each centroid takes a different, pre-defined 3D path. It is thought that the attackers follow the motion of their cluster centroid. To keep things simple and to highlight control performance, only the centroid trajectories are presented in the trajectory plots. The DWS is introduced at $(T/3)$ s and $(2T/3)$ s, which is done by varying $r_j\left(t\right)$ in Equation (16) for $j\in \{1,2\}$, and the waypoints introduced for each defender are $w_{j,1},$ and $w_{j,2},$, where the last waypoint $w_{1,k},$ is the actual centroid equation. In this case, the waypoints parameters are $r_{1,1}=2$, $r_{1,2}=2$, ${\eta }_{1,1}=20$, ${\eta }_{1,2}=20$, $r_{2,1}=2$, $r_{2,2}=2$, ${\eta }_{2,1}=20$, ${\eta }_{2,2}=20$, ${\zeta }_{1,1}=10$, ${\zeta }_{1,2}=10$, ${\zeta }_{2,1}=10$ and ${\zeta }_{2,2}=10$. At each time instant, the positions of the UAVs and their corresponding centroids are recorded, as shown in Figure 11. The motion of the attacking UAVs generates dynamically updated centroids $c_a$, forming the trajectories $P_{c_a\left(j\right)}$. Figure 12 illustrates the interception of the attacking UAVs by two defending UAVs for $j\in \{1,2\}$ using the proposed strategy in the presence of disturbances, with successful interception achieved in under 8 s for both centroids. For comparison, Figure 13 shows the results using PID control, where interception fails due to the lack of a mechanism to compensate for DWS and external disturbances. Figure 14 presents the interception tracking error for the proposed strategy, which converges to nearly zero in less than 4 s. Figure 15 depicts the disturbances and their estimates obtained via the ESO, showing that the estimation error quickly approaches zero. The disturbance estimation plots show that ${\widehat{d}}_{j,i}\left(t\right)$ closely follows $d_{j,i}\left(t\right)$, with a short transient and small steady-state mismatch, indicating accurate online reconstruction of the lumped uncertainties. Consequently, the ESO-based compensation effectively attenuates disturbance effects in the tracking channels and contributes to the small interception errors. These results demonstrate that the proposed control approach is robust against disturbances affecting the defending UAVs. The control efforts are smooth and free from chattering, ensuring that the defending quadrotor effectively intercepts the attacking UAVs, as in Case 1. Overall, the control strategy successfully guides the defending quadrotors to track and intercept the attacking UAVs without overshoot during the transient response.

Table 2. Interception performance comparison between PID baseline and the proposed approach (single- and multiple-UAV scenarios).

Approach (Scenario)	Interception (Yes/No)	Interception Time (s)	Average Interception Error (m)
PID-based (Single UAV)	No	–	High
Proposed approach (Single UAV)	Yes	7	$0.023$
PID-based (Multiple UAVs)	No	–	High
Proposed approach (Multiple UAVs)	Yes	8	$0.050$

below compare the proposed interception control strategy against the PID controller.

5. Conclusions

This paper introduced a comprehensive UAV–UAV interception framework that jointly exploits density-based clustering, deceptive maneuver generation, and robust nonlinear control to address the complexities of modern multi-UAV threats. The DBSCAN clustering module enables real-time grouping of irregular attacker formations while eliminating the need for fixed cluster assumptions, thereby improving scalability in dynamic and uncertain environments. In order to counter intelligent attackers that can predict the defenders’ trajectory, the proposed DWS algorithm generates an unpredictable defender trajectory which significantly improves the robustness in interception. Coupled with a sliding-mode backstepping controller supported by extended state observers, the overall architecture guarantees reliable convergence to attacker centroids under disturbances and modeling uncertainties, as verified through Lyapunov stability analysis.

Simulation studies across single-cluster and multi-cluster scenarios confirm that the proposed approach consistently achieves faster convergence, lower tracking errors, and superior disturbance rejection compared to baseline PID strategies. The result of this mechanism highlight the significance of integrating learning-based position detection with deception planning and robust control for future counter-UAV systems. We will conduct hardware-in-the-loop validation using an autopilot-in-the-loop setup (e.g., Pixhawk running PX4/ArduPilot) connected to a real-time simulator (e.g., Gazebo/JSBSim) via MAVLink/ROS to assess performance under sensing noise, actuator limits, and communication delays. We will also investigate extending the framework to underwater vehicles by adopting a 6-DoF hydrodynamic model with added-mass and nonlinear damping and adapting sensing/communication to acoustic positioning/links, while preserving DBSCAN clustering, DWS reference generation, and ESO-based robust tracking. Moreover, in a subsequent work, we will address extending the comparison set and incorporating additional real-world effects (varying disturbances, formation changes, and network-induced delays).