In recent years, the rapid advancement of unmanned drone technology has significantly enhanced capabilities in both military and civilian domains. Unmanned drone swarms are increasingly deployed for tasks such as reconnaissance, surveillance, and target tracking, where passive positioning—locating radiation sources without emitting signals—is crucial for maintaining stealth and efficiency. However, most existing studies on path optimization for unmanned drone swarms in passive positioning scenarios are confined to two-dimensional planes, neglecting the complexities of three-dimensional space. This limitation often results in suboptimal positioning accuracy and increased travel distances for unmanned drones. In this paper, we address this gap by proposing a comprehensive framework for three-dimensional path optimization and passive positioning of unmanned drone swarms. Our approach integrates Time Difference of Arrival (TDOA) modeling, Cramér–Rao Lower Bound (CRLB)-based optimization, and a Particle Swarm Optimization (PSO) algorithm to enhance the precision and efficiency of unmanned drone operations. Additionally, we introduce mechanisms such as dynamic reference station selection, target coordinate stability detection, and a direct flight mode to reduce computational complexity and improve energy utilization. Through extensive simulations, we validate the effectiveness of our method, demonstrating superior performance in both positioning accuracy and path planning for unmanned drone swarms.
The core of our research lies in optimizing the paths of unmanned drone swarms to improve passive positioning accuracy in three-dimensional environments. We begin by establishing a TDOA-based signal model for passive positioning. Let the target coordinates be denoted as \( u = (x, y, z) \), and the estimated coordinates as \( u_0 = (x_0, y_0, z_0) \). For an unmanned drone swarm with \( n \) nodes, the main station (reference) coordinates are \( S_1 = (x_1, y_1, z_1) \), and the auxiliary station coordinates are \( S_i = (x_i, y_i, z_i) \) for \( i = 2, 3, \dots, n \). The distance from the target to the main station is given by:
$$ \gamma_1 = |u – S_1| = \sqrt{(x – x_1)^2 + (y – y_1)^2 + (z – z_1)^2} $$
Similarly, the distance to any auxiliary station is:
$$ \gamma_i = |u – S_i| = \sqrt{(x – x_i)^2 + (y – y_i)^2 + (z – z_i)^2} $$
The TDOA measurement leads to distance differences:
$$ h_i(u) = \gamma_i – \gamma_1 $$
and the actual measured distance difference, including noise \( \epsilon_i \), is:
$$ V_i = c \Delta t_i = h_i(u) + \epsilon_i $$
where \( c \) is the speed of light. To handle reference station selection dynamically, we compute a reference point \( u_{\text{ref}} \) at each iteration \( k \):
$$ u_{\text{ref}} = \begin{cases}
\frac{1}{N} \sum_{i=1}^{N} S_i^{(k)} & \text{if } k = 1 \\
u^{(k-1)} & \text{if } k \geq 2
\end{cases} $$
The main station is then selected as the unmanned drone closest to this reference point:
$$ r = \min_{i \in \{1, \dots, N\}} \| S_i^{(k)} – u_{\text{ref}} \| $$
This dynamic approach adapts to the evolving geometry of the unmanned drone swarm, improving positioning robustness.
For target localization, we employ a Chan-Taylor joint algorithm. The Chan algorithm provides an initial estimate through two-stage weighted least squares, while the Taylor algorithm refines this estimate iteratively. The Chan algorithm solves for the position variable matrix \( P = [x_0, y_0, z_0, \gamma_1]^T \) using:
$$ \hat{p} = (D^T \Psi^{-1} D)^{-1} D^T \Psi^{-1} \eta $$
where \( D \) is an intermediate matrix, \( \eta \) is a constant vector derived from measurements, and \( \Psi \) is the error covariance matrix. The Taylor algorithm then updates the estimate by solving:
$$ \Delta u = (G^T Q^{-1} G)^{-1} G^T Q^{-1} m_i $$
with \( u = u_0 + \Delta u \), where \( G \) is the Jacobian matrix and \( Q \) is the distance covariance matrix. This combination ensures high accuracy for unmanned drone-based positioning.
To optimize the paths of unmanned drone swarms, we formulate an objective function based on the CRLB, which represents the theoretical lower bound for estimation error. The Fisher information matrix \( F \) is computed as:
$$ F = \frac{1}{c^2} \sum_{i=2}^{n} \frac{1}{\sigma_i^2} \frac{\partial h_i}{\partial u} \left( \frac{\partial h_i}{\partial u} \right)^T $$
where \( \sigma_i^2 \) is the variance of time difference measurements. The CRLB is then:
$$ \text{CRLB} = F^{-1} $$
Our goal is to minimize the trace of the CRLB by optimizing the heading angles \( \theta_i \) and pitch angles \( \phi_i \) of each unmanned drone. The objective function is:
$$ J(\Theta) = \min_{\Theta} \text{trace}(\text{CRLB}(\Theta)) $$
where \( \Theta = [\theta_1, \dots, \theta_N, \phi_1, \dots, \phi_N]^T \). We incorporate constraints such as smooth motion and altitude limits into a composite fitness function for the PSO algorithm:
$$ f(\Theta) = J(\Theta) + \lambda \sum_{i=1}^{N} (\Delta \theta_i^2 + \Delta \phi_i^2) + \beta \left[ \sum_{i=1}^{N} g_\theta(\theta_i) + \sum_{i=1}^{N} g_\phi(\phi_i) \right] + \gamma \sum_{i=1}^{N} h(z_i) $$
Here, \( \lambda \), \( \beta \), and \( \gamma \) are weights for smoothness, angle constraints, and altitude constraints, respectively. The PSO algorithm updates angles iteratively:
$$ \Delta \theta_i^n(d+1) = \omega(d) \Delta \theta_i^n(d) + a_1 \gamma_1 \Delta p_{\theta,i}^n + a_2 \gamma_2 \Delta g_\theta^n $$
$$ \Delta \phi_i^n(d+1) = \omega(d) \Delta \phi_i^n(d) + a_1 \gamma_1 \Delta p_{\phi,i}^n + a_2 \gamma_2 \Delta g_\phi^n $$
where \( \omega(d) \) is the inertia weight, \( a_1 \) and \( a_2 \) are acceleration constants, and \( \gamma_1, \gamma_2 \) are random factors. The three-dimensional motion of each unmanned drone is modeled as:
$$ \begin{aligned}
V_{ix} &= V \cos \phi_i^{g(d)} \cos \theta_i^{g(d)} \\
V_{iy} &= V \cos \phi_i^{g(d)} \sin \theta_i^{g(d)} \\
V_{iz} &= V \sin \phi_i^{g(d)}
\end{aligned} $$
with position update:
$$ S_i^{(k+1)} = S_i^{(k)} + \Delta t \cdot [V_{ix}, V_{iy}, V_{iz}]^T $$
where \( V = 20 \, \text{m/s} \) is the constant speed of the unmanned drones.
To enhance efficiency, we introduce a target stability detection mechanism. If the target coordinate estimate remains stable for \( a \) consecutive iterations, with change less than a threshold \( b \), i.e., \( \| u^{(k)} – u^{(k-1)} \| < b \), we halt further updates and switch to a direct flight mode. In this mode, the unmanned drone closest to the target executes a direct path, while others hover to conserve energy. The direct flight objective is:
$$ J_{\text{direct}}(\theta, \phi) = \left\| S_N^{(k)} + v \Delta t \begin{bmatrix} \cos \phi \cos \theta \\ \cos \phi \sin \theta \\ \sin \phi \end{bmatrix} – u_s \right\| $$
with fitness function:
$$ f^*(\theta, \phi) = J_{\text{direct}}(\theta, \phi) + \lambda (\Delta \theta^2 + \Delta \phi^2) + \gamma h(u_n) $$
This approach reduces computational load and improves timeliness for unmanned drone swarms.
We conducted simulations to evaluate our method, comparing it with a classical two-dimensional approach. The unmanned drone swarm consisted of 6 nodes, with initial coordinates as shown in Table 1. The target was placed at varying heights from 100 m to 1000 m to assess three-dimensional performance. Key parameters included a minimum safe distance of 12000 m, PSO with 200 particles and 500 iterations, and constraints such as heading angles \( \theta \in [-45^\circ, 45^\circ] \), pitch angles \( \phi \in [-30^\circ, 30^\circ] \), and altitudes \( h \in [100, 1000] \, \text{m} \).
| Unmanned Drone | X | Y | Z |
|---|---|---|---|
| 1 | 0 | 0 | 200 |
| 2 | 8660 | 0 | 400 |
| 3 | 4660 | 7500 | 600 |
| 4 | 0 | 8660 | 750 |
| 5 | 4660 | 4660 | 500 |
| 6 | 0 | 4660 | 300 |
The positioning accuracy was measured using horizontal error \( e_h \), height error \( e_z \), and total three-dimensional error \( e_{3d} \). Results for our three-dimensional system (with and without direct flight) and the two-dimensional system are summarized in Table 2. The data clearly shows that the three-dimensional approach significantly reduces errors compared to the two-dimensional method, especially in height estimation. For instance, at a target height of 1000 m, the three-dimensional system achieved a total error of 2.4 m, while the two-dimensional system suffered from 950 m height error due to its lack of vertical modeling. The direct flight mode slightly increased errors but offered efficiency gains, balancing accuracy and timeliness for unmanned drone operations.
| Target Height (m) | 3D System (Height Error) | 3D System (Horizontal Error) | 3D System (Total Error) | 3D Direct Flight (Height Error) | 3D Direct Flight (Horizontal Error) | 3D Direct Flight (Total Error) | 2D System (Height Error) | 2D System (Horizontal Error) | 2D System (Total Error) |
|---|---|---|---|---|---|---|---|---|---|
| 100 | 1.5 | 0.626 | 1.8 | 2.0 | 5.148 | 5.8 | 50 | 0.604 | 50 |
| 200 | 1.8 | 1.326 | 2.4 | 2.4 | 4.504 | 5.6 | 150 | 0.402 | 150 |
| 300 | 1.0 | 6.34 | 1.8 | 3.1 | 3.624 | 4.9 | 250 | 0.488 | 250 |
| 400 | 1.9 | 1.188 | 2.5 | 3.2 | 2.630 | 4.4 | 350 | 0.680 | 350 |
| 500 | 1.9 | 1.986 | 2.8 | 2.2 | 3.022 | 4.1 | 450 | 0.254 | 450 |
| 600 | 1.8 | 0.722 | 2.2 | 2.1 | 4.116 | 4.9 | 550 | 0.810 | 550 |
| 700 | 1.8 | 0.882 | 2.2 | 2.5 | 2.086 | 3.6 | 650 | 0.438 | 650 |
| 800 | 1.8 | 1.144 | 2.2 | 3.3 | 2.214 | 4.2 | 750 | 0.264 | 750 |
| 900 | 2.6 | 1.038 | 2.9 | 3.1 | 3.572 | 4.9 | 850 | 0.468 | 850 |
| 1000 | 1.2 | 1.864 | 2.4 | 1.0 | 3.642 | 3.9 | 950 | 0.442 | 950 |
| Average | 1.7 | 1.712 | 2.3 | 2.5 | 3.456 | 4.6 | 500 | 0.485 | 500 |
In terms of path planning efficiency, we measured total flight time and total travel distance for the unmanned drone swarm. As shown in Table 3, our three-dimensional system reduced both metrics compared to the two-dimensional approach. For example, at a target height of 100 m, the three-dimensional system achieved a total flight time of 433.2 seconds and a total distance of 49787.2 meters, whereas the two-dimensional system required 712.0 seconds and 80627.9 meters. The direct flight mode further improved efficiency, with average reductions of 4.93% in flight time and significant savings in travel distance, demonstrating the benefits of our stability detection and optimization mechanisms for unmanned drone swarms.
| Target Height (m) | 3D System (Total Time, s) | 3D System (Total Distance, m) | 3D Direct Flight (Total Time, s) | 3D Direct Flight (Total Distance, m) | 2D System (Total Time, s) | 2D System (Total Distance, m) |
|---|---|---|---|---|---|---|
| 100 | 433.2 | 49787.2 | 409.2 | 19653.0 | 712.0 | 80627.9 |
| 200 | 433.2 | 50394.5 | 414.8 | 21979.8 | 717.6 | 76048.3 |
| 300 | 433.2 | 50412.8 | 417.6 | 22868.2 | 712.8 | 73521.2 |
| 400 | 436.8 | 50850.1 | 417.6 | 23908.4 | 713.2 | 80292.2 |
| 500 | 437.2 | 51038.4 | 422.4 | 24538.4 | 709.6 | 74719.8 |
| 600 | 441.2 | 51628.7 | 410.0 | 20099.5 | 710.4 | 73890.5 |
| 700 | 445.6 | 52225.3 | 418.8 | 23077.2 | 708.4 | 75348.5 |
| 800 | 452.8 | 53077.4 | 428.8 | 26373.8 | 708.4 | 77013.9 |
| 900 | 433.2 | 50378.3 | 416.8 | 23331.8 | 704.0 | 77434.5 |
| 1000 | 432.0 | 49934.8 | 406.4 | 19649.2 | 707.6 | 74393.9 |
| Average | 437.8 | 50972.7 | 416.2 | 22547.9 | 710.4 | 76329.1 |
Our simulations also revealed insights into error dynamics. The total positioning error decreased rapidly in early iterations but then plateaued, indicating diminishing returns from further optimization. This justifies our stability detection mechanism, which halts updates when changes fall below a threshold, saving computational resources for unmanned drone swarms. Additionally, the direct flight mode optimized paths by reducing unnecessary maneuvers, enhancing energy efficiency for unmanned drones in long-duration missions.
In conclusion, this paper presents a novel framework for three-dimensional path optimization and passive positioning of unmanned drone swarms. By integrating TDOA modeling, CRLB-based optimization, and PSO algorithms, we achieve higher accuracy and efficiency compared to traditional two-dimensional methods. The introduction of dynamic reference station selection, target stability detection, and direct flight mode further enhances performance, making our approach suitable for real-world applications where unmanned drones operate in complex three-dimensional environments. Future work could explore adaptive parameter tuning for stability thresholds or extend the method to dynamic targets. Overall, our research contributes to advancing the capabilities of unmanned drone swarms in passive positioning scenarios, paving the way for more autonomous and efficient systems.