In modern electronic warfare, networked radar systems pose significant challenges due to their robust multi-target detection and anti-jamming capabilities. Traditional electronic countermeasures often fail to maintain spatiotemporal consistency across multiple radar perspectives, limiting their effectiveness. To address this, we explore the use of Unmanned Aerial Vehicle (UAV) swarms for coordinated deception. Drone technology enables multi-dimensional协同干扰, which can effectively penetrate enemy radar networks. This paper focuses on phantom track deception, where UAVs generate false trajectories to mislead networked radars. By leveraging advanced drone technology, we aim to disrupt enemy situational awareness and deplete strategic resources.
Phantom track deception involves creating虚假目标 through signal delay and forwarding via Digital Radio Frequency Memory (DRFM) systems. The Line of Sight (LOS) criterion must be satisfied, meaning the radar’s main beam, UAV, and phantom target must align. Previous studies have addressed this using proportional guidance laws and协同 control methods. However, these approaches often rely on centralized pre-planning, which becomes inadequate when UAVs experience failures or damage during missions. To overcome this, we propose a decentralized cooperative game model for UAV swarms. Our approach optimizes task allocation and flight paths in real-time, ensuring continuous deception even under platform losses.
The core of our method lies in modeling the UAV swarm’s deception as a cooperative game. Each UAV acts as a player, optimizing its utility based on task choices and trajectories. We define the utility function to maximize the number of phantom tracks that pass the radar network’s homology test. This test checks if multiple radars detect the same phantom target, with success determined by the spatial resolution cell (SRC). Our model incorporates constraints from UAV kinematics and task allocation limits. We prove the existence and convergence of Nash equilibrium using exact potential game theory. The solution is achieved through a distributed coalition game combined with a genetic algorithm.
In the following sections, we detail the system model, game formulation, optimization algorithm, and simulation results. We demonstrate that our approach maintains phantom track continuity under UAV failures, outperforming existing methods. The integration of drone technology and game theory provides a robust framework for dynamic mission re-planning in contested environments.
System Model for UAV Swarm Deception
Consider a swarm of $N_u$ UAVs deceiving a networked radar system comprising $N_r$ monostatic radars. The swarm generates $N_p$ phantom tracks over time steps $k = 1, 2, \ldots, K$. Let $u_{n,k}$ denote the 3D Cartesian coordinates of UAV $n$ at time $k$, $\tilde{p}_{m,k}$ represent the preset phantom target $m$, and $r_q$ be the fixed coordinates of radar $q$. The coordinates are given by:
$$ u_{n,k} = [x^u_{n,k}, y^u_{n,k}, z^u_{n,k}]^T, \quad \tilde{p}_{m,k} = [\tilde{x}^p_{m,k}, \tilde{y}^p_{m,k}, \tilde{z}^p_{m,k}]^T, \quad r_q = [x^r_q, y^r_q, z^r_q]^T $$
To describe task allocation, we introduce binary variables $\mu^q_{n,m,k} \in \{0,1\}$, where $\mu^q_{n,m,k} = 1$ indicates that UAV $n$ deceives radar $q$ to generate phantom target $m$ at time $k$. Additionally, $s_{n,k} \in \{0,1\}$ represents the operational status of UAV $n$, with $s_{n,k} = 1$ meaning it is functional. The set of operational UAVs at time $k$ is $W_k = \{ n \mid s_{n,k} = 1 \}$.
Under the LOS criterion, the actual phantom target position $p^q_{m,k}$ generated by UAV $n$ for radar $q$ is derived from similar triangles:
$$ p^q_{m,k} = \begin{cases}
\frac{\tilde{z}^p_{m,k}}{z^u_{n,k}} (u_{n,k} – r_q) + r_q, & \text{if } s_{n,k} \mu^q_{n,m,k} = 1 \\
\emptyset, & \text{otherwise}
\end{cases} $$
The deception signal model involves DRFM delay and Doppler modulation. The time delay $\Delta t$ is calculated as:
$$ \Delta t = \frac{2 \| p^q_{m,k} – u_{n,k} \|_2}{c} $$
where $c$ is the speed of light. The Doppler frequency shift $f_d$ compensates for the UAV’s motion:
$$ f_d = \frac{2 (v^b_{q,k} \cdot v^p_{m,q,k} – v^b_{q,k} \cdot v^u_{n,k})}{\lambda_q} $$
Here, $v^u_{n,k}$ and $v^p_{m,q,k}$ are velocity vectors of the UAV and phantom target, $v^b_{q,k}$ is the radar’s signal direction vector, and $\lambda_q$ is the wavelength. The transmitted deception signal $J(t)$ is:
$$ J(t) = \rho \frac{G_t G_j \lambda_q^2}{L (4\pi R_{n,q,k})^2} s(t – \Delta t) e^{j2\pi f_d (t – \Delta t)} $$
where $G_t$ and $G_j$ are antenna gains, $L$ is path loss, $R_{n,q,k}$ is the distance between UAV $n$ and radar $q$, and $\rho$ is the amplitude modulation parameter:
$$ \rho = \frac{R_{n,q,k}^2}{R_{n,m,q,k}^2 \lambda_q} \sqrt{\frac{4\pi \sigma_{m,q,k}}{G_j}} $$
$R_{n,m,q,k}$ is the distance between the phantom target and radar, and $\sigma_{m,q,k}$ is the radar cross-section.
Cooperative Game Model for UAV Swarm Deception
We formulate the UAV swarm deception as a cooperative game to enable decentralized decision-making. The game is defined by the tuple $\Gamma = \langle W_k, A, U_k, u_{n,k} \rangle$, where:
- $W_k$ is the set of players (operational UAVs).
- $A$ is the strategy set, encompassing task choices and flight parameters.
- $U_k(C_m)$ is the characteristic function for coalition $C_m$, defined as the homology test outcome for phantom track $m$.
- $u_{n,k}(a_{n,k}, A_{-n,k})$ is the utility of UAV $n$, depending on its own strategy $a_{n,k}$ and others’ strategies $A_{-n,k}$.
The homology test checks if phantom tracks are consistent across radar pairs. For radars $q_1$ and $q_2$, the test passes if:
$$ \| p^{q_1}_{m,k} – p^{q_2}_{m,k} \|_2 < \delta_{\min} $$
where $\delta_{\min} = \min(\delta_{q_1}, \delta_{q_2})$ is the minimum distance resolution. The binary outcome $h^{q_1,q_2}_{m,k}$ is:
$$ h^{q_1,q_2}_{m,k} = u\left( \delta^{q_1,q_2}_{\min} – \| p^{q_1}_{m,k} – p^{q_2}_{m,k} \|_2 \right) $$
where $u(\cdot)$ is the unit step function. The overall homology test result $g_{m,k}$ for phantom track $m$ is:
$$ g_{m,k} = u\left( \sum_{q_1=1}^{N_r} \sum_{q_2=q_1+1}^{N_r} h^{q_1,q_2}_{m,k} – N_K \right) $$
Here, $N_K$ is the threshold number of radar pairs required to pass the test. The characteristic function for coalition $C_m$ is $U_k(C_m) = g_{m,k}$, and the utility of UAV $n$ is:
$$ u_{n,k}(a_{n,k}, A_{-n,k}) = \sum_{m \in G_{n,k}} g_{m,k} $$
where $G_{n,k} = \{ m \mid \mu^q_{n,m,k} = 1 \}$ is the set of phantom tracks generated by UAV $n$.
Joint Optimization Algorithm Based on Coalition Game
We aim to maximize the total utility of the UAV swarm, defined as:
$$ U_{\text{sum}}(A_k) = \sum_{m \in HG_{k-1}} U_k(C_m) $$
where $HG_{k-1} = \{ m \mid g_{m,k} = 1 \}$ is the set of phantom tracks passing the homology test. The optimization problem is:
$$ \begin{aligned}
\max_{v_k, \Delta\theta_k, \beta_k, \mu_k} & \quad U_{\text{sum}}(A_k) \\
\text{s.t.} & \quad \text{C1: } v_{\min} \leq v_{n,k} \leq v_{\max} \\
& \quad \text{C2: } \alpha_{\min} \leq \alpha_{n,k} \leq \alpha_{\max} \\
& \quad \text{C3: } \Delta\theta_{\min} \leq \Delta\theta_{n,k} \leq \Delta\theta_{\max} \\
& \quad \text{C4: } \beta_{\min} \leq \beta_{n,k} \leq \beta_{\max} \\
& \quad \text{C5: } \mu^q_{n,m,k} \in \{0,1\} \\
& \quad \text{C6: } 1 \leq \sum_{m \in HG_{k-1}} \mu^q_{n,m,k} \leq \mu_{\max} \\
& \quad \text{C7: } \sum_{n \in W_k} \mu^q_{n,m,k} \leq 1 \\
& \quad \text{C8: } n \in W_k
\end{aligned} $$
Here, $v_k$, $\Delta\theta_k$, and $\beta_k$ are vectors of UAV speeds, yaw angles, and pitch angles, respectively. Constraints C1–C4 ensure kinematic feasibility, while C5–C8 handle task allocation limits.
We prove that this game is an exact potential game, guaranteeing convergence to a Nash equilibrium. The potential function $\psi(a_{n,k}, A_{-n,k})$ is set equal to $U_{\text{sum}}(A_k)$. For any UAV $n$ changing strategy from $a_{n,k}$ to $\tilde{a}_{n,k}$, the change in individual utility matches the change in the potential function:
$$ u_{n,k}(a_{n,k}, A_{-n,k}) – u_{n,k}(\tilde{a}_{n,k}, A_{-n,k}) = \psi(a_{n,k}, A_{-n,k}) – \psi(\tilde{a}_{n,k}, A_{-n,k}) $$
This ensures that the game has at least one pure strategy Nash equilibrium, and the maximum total utility is achieved at equilibrium.
The distributed optimization for each UAV is:
$$ \begin{aligned}
\max_{v_{n,k}, \Delta\theta_{n,k}, \beta_{n,k}, \mu_{n,k}} & \quad u_{n,k}(a_{n,k}, A_{-n,k}) \\
\text{s.t.} & \quad \text{C1–C8}
\end{aligned} $$
We solve this using an iterative algorithm combining coalition game theory and a genetic algorithm. The algorithm proceeds as follows:
- Initialize task allocation based on previous time step.
- Each UAV computes its utility under current strategies.
- For a fixed number of iterations, randomly select a UAV to optimize its task and trajectory using GA, while others hold strategies fixed.
- If the new strategy improves utility, update the UAV’s strategy and coalition structure.
- Remove phantom tracks that fail the homology test.
The computational complexity is $O(G \times P \times (M_n N_r^2 + N))$ for distributed optimization, where $G$ is GA iterations, $P$ is population size, $M_n$ is the number of phantom tracks per UAV, and $N$ is optimization dimensions. This scales linearly with UAV count, making it suitable for large swarms.
Simulation Results and Analysis
We evaluate our algorithm in three scenarios, considering UAV failures and varying radar configurations. The homology test threshold is $D_\gamma = \chi^2_n(1-\gamma)$ with $\gamma = 10^{-6}$ and $n=4$. The association distance between radars $i$ and $j$ is:
$$ D^{i,j}_k = (\tilde{X}^i_k – \tilde{X}^j_k) (P^i_k + P^j_k)^{-1} (\tilde{X}^i_k – \tilde{X}^j_k)^T $$
where $\tilde{X}^i_k$ is the state estimate and $P^i_k$ is the error covariance. If $D^{i,j}_k < D_\gamma$, the measurements are considered homologous.
UAV parameters: speed 50–100 m/s, acceleration ±5 m/s², yaw angle ±0.5 rad, pitch angle ±0.1 rad. DRFM delay error is 100 ns, and UAV jitter error is 5 m. Each UAV can generate up to $\mu_{\max} = 3$ phantom tracks per radar.
Scenario 1: Baseline Performance
12 UAVs deceive 3 radars located at (50,25,0) km, (52,20,0) km, and (47,15,0) km. Initial task allocation is shown in Table 1.
| Phantom Track | Radar 1 | Radar 2 | Radar 3 |
|---|---|---|---|
| Track 1 | UAV 1 | UAV 2 | UAV 3 |
| Track 2 | UAV 1 | UAV 4 | UAV 5 |
| Track 3 | UAV 6 | UAV 2 | UAV 7 |
| Track 4 | UAV 8 | UAV 9 | UAV 3 |
| Track 5 | UAV 10 | UAV 11 | UAV 12 |
Under errors, phantom tracks deviate by 20–50 m randomly. Our algorithm maintains spatial consistency, with all tracks passing the homology test. Association distances remain below the threshold, confirming effective deception.
Scenario 2: UAV Failures
UAVs 2, 6, and 12 fail during mission. Without re-planning, phantom tracks 1, 3, and 5中断. With our algorithm, UAV 10 takes over track 2 for radar 1, and UAV 11 handles track 1 for radar 2. Only track 3 interrupts. The utility function ensures maximal track continuity despite losses.
Scenario 3: Curved Phantom Tracks
16 UAVs deceive 4 radars at (1,10,0) km, (21,10,0) km, (1,2,0) km, and (21,2,0) km. Phantom tracks follow S-maneuvers. UAVs 2,4,11,16 are destroyed. Our algorithm recovers 3 interrupted tracks, though deviations increase due to complex kinematics. Tracks with similar motion patterns form deception clusters, masking real targets.
Comparative analysis with centralized optimization and random allocation shows our approach balances track count and success rate. Centralized methods yield more tracks but lower precision due to conflicting path constraints. Random allocation achieves higher success per track but fewer overall tracks. Our distributed method closely matches centralized performance while enhancing robustness.
| Algorithm | Phantom Tracks Generated | Deception Success Rate (%) |
|---|---|---|
| Proposed Distributed | 4 | 85 |
| Centralized Optimization | 5 | 83 |
| Random Allocation | 1 | 98 |
Figure 1 illustrates the impact of UAV losses on track generation. As failures increase, track count drops non-linearly. When only one UAV remains per radar, up to 2–3 tracks can be maintained. If all UAVs for a radar are lost, no tracks are generated for that radar.
Conclusion
We presented a game theory-based joint optimization algorithm for UAV swarm task allocation and trajectory planning in phantom track deception. Our decentralized approach leverages coalition games to dynamically re-plan tasks and paths under UAV failures. The formulation as an exact potential game ensures convergence to Nash equilibrium, maximizing deception utility. Simulations confirm that our method maintains track continuity and effectiveness against networked radars, outperforming existing techniques. Future work will integrate model predictive control and dynamic spectrum allocation to handle communication constraints and unknown radar locations. Advances in drone technology will further enhance the robustness of Unmanned Aerial Vehicle swarms in electronic warfare.