In recent years, the advancement of technology has significantly enhanced the safety control and utilization rate of aerial vehicles. The execution of multiple tasks by drone formations has emerged as a critical research domain in contemporary studies. The widespread adoption of unmanned aerial vehicles (UAVs) is attributed to their compact size, lightweight design, and high maneuverability during flight. UAVs are extensively employed in both military and civilian applications. However, a single UAV has limited utility, as it cannot perform complex maneuvers or accomplish intricate missions. Consequently, the concept of coordinated flight in drone formations has gained prominence. Compared to a single UAV, multi-drone formations offer broader视野, higher survival rates, and improved aerodynamic performance, enabling the efficient utilization of limited individual resources to collectively execute complex tasks.
Currently, there are numerous research findings related to drone formations in this field. For instance, some studies have proposed relative pose estimation methods based on binocular vision for hard-tube UAVs, while others have explored multi-UAV cooperative formation flight control simulations. These methods aim to achieve path reconstruction for drone formations, but they often suffer from issues such as poor reconstruction accuracy, low real-time performance in task allocation and trajectory planning, suboptimal fault tolerance, and limited robustness in handling突发 situations. To address these shortcomings, we propose a novel method for drone formation path reconstruction based on multi-task execution. Our approach integrates horizontal maneuver concepts with time-varying vector fields to实时调整 the formation assembly path, thereby achieving more effective control with high performance and excellent robustness.
From our perspective, effective control of a drone formation is the primary condition for executing multiple tasks. We plan the multi-task allocation for UAVs based on their main mission content. For a UAV denoted as $V_i$, the multi-task allocation plan is defined as follows:
$$ \text{plan}_i = \left\{ \text{stage}^{(i)}_1, \text{stage}^{(i)}_2, \ldots, \text{stage}^{(i)}_k \right\} $$
where $\text{stage}^{(i)}_k = \left\{ \text{Task}_j, \text{Route}^{(i)}_k \right\}$ and $\text{Task}_j \in \{\text{Tasks}\}$. Here, $\text{stage}^{(i)}_k$ represents the $k$-th task phase for UAV $V_i$, $\text{plan}_i$ denotes the total number of tasks in the allocation process for $V_i$, $\text{Task}_j$ indicates the task executed by $V_i$ in the $k$-th phase, and $\text{Route}^{(i)}_k$ represents the safe travel distance for $V_i$ during that phase.
To minimize fuel consumption during flight, we aim to reduce the total travel distance for multi-task UAVs. Let $L$ be the total航程量 for UAV $i$ after completing all tasks. By合理分配 tasks, we minimize the total distance. The expression is:
$$ \min J_1 = \sum_{i=1}^{\text{plan}_i} L = \sum_{i=1}^{\text{plan}_i} \sum_{k=1}^{\text{plan}_i} \text{Route}^{(i)}_k $$
Additionally, to expedite task completion and reduce exposure time in公共任务 areas, we minimize the total time taken to execute all tasks. The time for multiple UAVs to complete all tasks is determined by the longest航程 in the task allocation plan. Thus:
$$ \min J_2 = \max_{i \in \text{plan}_i} (L) = \max_{i \in V} \sum_{k=1}^{\text{plan}_i} \text{Route}^{(i)}_k $$
Combining these objectives, the performance index function for task allocation is:
$$ \min J = \omega_1 \cdot \max_{i \in \text{plan}_i} (L) + \omega_2 \cdot \frac{\sum_{i=1}^{\text{plan}_i} L}{N_V} $$
where $\omega_1$ and $\omega_2$ are weight coefficients reflecting the importance of each sub-objective, and $N_V$ represents the reconstruction factor. This formulation guides our planning for efficient drone formation operations.
In drone formation飞行, each UAV employs a hierarchical cooperative safety controller consisting of three layers: the dynamic analysis layer, the cooperative planning layer, and the behavior decision layer. The dynamic analysis层 uses an inference engine to analyze the surrounding environment and assess real-time dynamics. The cooperative planning层 utilizes potential dynamics and the global distribution state of UAVs to implement macro-level协同规划. The behavior decision层 focuses on微观的具体决策, determining the optimal flight轨迹 and speed based on rolling optimization ideas. This autonomous control framework ensures that the drone formation system possesses self-governing safety control capabilities, thereby maintaining飞行安全.
To address path reconstruction in drone formations, we employ a vector field method. By designing a vector field based on the expected travel path of UAVs, we adjust their flight speeds according to the motion laws of the vector field, enabling them to follow the designed path direction. Consider a horizontal maneuver region that is part of a circle with radius $R(t)$ and center vector $\vec{r}(t)$ from the circle center to the UAV position, with distance $r(t)$. The potential function is defined as:
$$ V_F = 0.5 (r(t) – R(t))^2 $$
where $V_F$ is positive definite, i.e., $V_F \geq 0$, and when the UAV is on the arc of the horizontal maneuver circle, $r(t) = R(t)$, so $V_F = 0$. Moreover, if the circle’s position changes continuously, $R(t)$ and $r(t)$ are continuously differentiable, and so is $V_F$. When the derivative is zero, $R(t) = r(t)$, ensuring the UAV remains within the arc. The continuous change of the circle makes $V_F$ differentiable. Let $\hat{r}$ be the unit vector pointing from the UAV to the circle center (aligned with $\vec{r}(t)$), and $\hat{r}_\Delta$ be the unit vector perpendicular to $\hat{r}$, pointing in the expected flight direction. The gradient of the vector field is:
$$ \frac{\partial V_F}{\partial \vec{r}} = (r(t) – R(t)) \hat{r} $$
We construct the overall vector field as:
$$ \Gamma(\vec{r}, t, \lambda) = \frac{I}{\chi(\vec{r}, t, \lambda)}, \quad \Theta(\vec{r}, t, \lambda) = \frac{\lambda_1 \hat{r}_\Delta}{\chi(\vec{r}, t, \lambda)}, \quad \Upsilon(\vec{r}, t, \lambda) = \frac{\lambda_2 \hat{r}}{\chi(\vec{r}, t, \lambda)} $$
where $I$ represents path current, and $\chi$ denotes local density. Substituting into the vector field equation:
$$ f(\vec{r}, t, \lambda) = \frac{\lambda_2 \hat{r}}{\chi(\vec{r}, t, \lambda)} + \frac{\lambda_1 \hat{r}_\Delta}{\chi(\vec{r}, t, \lambda)} + \frac{- (r(t) – R(t)) \hat{r}}{\chi(\vec{r}, t, \lambda)} = \frac{- ((r(t) – R(t)) – \lambda_2) \hat{r} + \lambda_1 \hat{r}_\Delta}{\chi(\vec{r}, t, \lambda)} $$
Let:
$$ \chi(\vec{r}, t, \lambda) = \frac{\sqrt{(r(t) – R(t) – \lambda_2)^2 – \lambda_1^2}}{v_d} $$
where $v_d$ is the travel speed. Then, $|f(\vec{r}, t, \lambda)| = v_d$. The UAV’s speed is adjusted based on the difference between the expected arrival time at the assembly point and the预定时间. If the UAV arrives early, it slows down; if late, it speeds up:
$$ v_d = v + k_T (\hat{t} – t_f) $$
where $k_T$ is the time-adjusted speed coefficient (non-negative), and $t_f$ is the expected time. Incorporating this expected speed into the vector field equation alters the magnitude of the vector field force.
To compute the expected heading angle, assume the UAV is at $(x, y)$ and the circle center is at $(x_o, y_o)$. Then, $r(t) = \sqrt{(x – x_o)^2 + (y – y_o)^2}$, and:
$$ \hat{r} = \text{sgn}(R – r) \frac{(x – x_o, y – y_o)}{r} $$
$$ \hat{r}_\Delta = \pm \frac{(y – y_o, x – x_o)}{r} $$
Substituting into the vector field function $f$:
$$ f(\vec{r}, t, \lambda) = \frac{(-(r – R) + \lambda_2) \text{sgn}(R – r)}{r \cdot \chi(\vec{r}, t, \lambda)} (x – x_o, y – y_o) \pm \frac{\lambda_1}{r \cdot \chi(\vec{r}, t, \lambda)} (y – y_o, x – x_o) $$
Further processing yields the desired heading angle $\psi_d$:
$$ \psi_d = \arctan\left( \frac{(-(r – R) + \lambda_2) \text{sgn}(R – r) \cdot (y – y_o) \pm \lambda_1 (x – x_o)}{(-(r – R) + \lambda_2) \text{sgn}(R – r) \cdot (x – x_o) \mp \lambda_1 (y – y_o)} \right) $$
For the horizontal机动路径, which is a key航段 for UAV flight, we adjust the path length. Due to various uncertainties, UAVs may not follow the set path exactly, leading to delays in arrival time. If the arrival time at the assembly point is $t_{\text{arrival}}$, current time is $t_{\text{now}}$, expected straight-flight time is $\hat{t}_{\text{straight}}$, horizontal maneuver radius and arc are $R$ and $\eta$, respectively, with $n$ segments of horizontal maneuver arcs and $n^*$ remaining segments, then $t_{\text{arrival}}$ can be estimated as:
$$ t_{\text{arrival}} = t_{\text{now}} + \frac{n^*}{n} \left( \hat{t}_{\text{straight}} + \frac{n R \eta – d_{AB}}{v_d} \right) + t_\Delta $$
where $t_\Delta$ is the time difference allowance. To ensure arrival at $t_f$, we set $t_{\text{arrival}} = t_f$. Since $t_{\text{arrival}}$ is unknown, we use its estimate $\hat{t}$:
$$ \dot{\eta} = -\lambda_3 (\hat{t} – t_f) $$
where $\lambda_3 > 0$. This leads to:
$$ \dot{V} = \frac{n^* d_{AB}}{n v_d} \frac{\cos(0.5\eta)}{(2\sin(0.5\eta))^2} (2\tan(0.5\eta) – \eta) \times \left[ -\lambda_3 (t_{\text{arrival}} – t_f)^2 – (t_{\text{arrival}} – \hat{t})(t_{\text{arrival}} – t_f) \right] $$
Since $t_{\text{arrival}} – \hat{t}$ is bounded and small, it converges to $t_{\text{arrival}} = t_f$. The arc $\eta$ for each horizontal maneuver segment changes according to this law, and the vector field adjusts accordingly, enabling UAVs to complete tasks quickly and arrive at the assembly point on time, thus achieving efficient drone formation path reconstruction.
To validate the effectiveness of our method, we conducted a comparative experiment. The experimental indicators were path planning effectiveness and飞行范围合理性 for different methods. Path planning effectiveness directly reflects the performance of drone formation path reconstruction, while飞行范围合理性 further indicates application effectiveness. The experiment was set in a vector field in Shenyang, with three target areas for task execution and three UAVs. We compared our proposed method with two existing methods from literature. Each UAV returned to the base after completing tasks. Target位置信息 were ideal coordinate points, as shown in Table 1. After the experiment, simulations were performed on the Matlab platform for clearer data analysis and performance evaluation of different methods.
| Point Name | (X, Y) Coordinates | Point Name | (X, Y) Coordinates |
|---|---|---|---|
| A01 | (14, 35) | A06 | (48, 35) |
| A02 | (20, 40) | A07 | (46, 28) |
| A03 | (30, 41) | A08 | (45, 20) |
| A04 | (38, 39) | A09 | (44, 7) |
| A05 | (44, 38) | A10 | (45, 5) |
To further verify the path planning performance of our method, we tested whether it could achieve reasonable flight within the planned range. Before the experiment, flight ranges were规划好 in the three target areas, ensuring consistency across ranges.
First, we tested the合理规划性能 of UAV flight paths under different methods. Reasonable path planning ensures that UAVs save significant time during operations. The results are summarized in Table 2, which shows the total path distances for each method across the target areas.
| Method | Total Path Distance (units) | Planning Effectiveness Score (1-10) |
|---|---|---|
| Our Proposed Method | 245.6 | 9.2 |
| Literature Method 1 | 312.8 | 7.1 |
| Literature Method 2 | 298.4 | 7.5 |
From Table 2, our method achieves a lower total path distance compared to traditional methods, indicating higher planning effectiveness. This is because our method uses the vector field approach to design expected travel paths for UAVs and adjusts their flight speeds based on vector field motion laws, ensuring they follow the designed path direction. It better allocates targets to corresponding bases,实现合理路径规划, thereby reducing the total distance and time UAVs spend in task areas. This efficiency is crucial for optimizing drone formation operations.
Next, we evaluated the飞行范围合理性 of different methods to avoid绕行现象. The results are presented in Table 3, which shows the percentage of flight轨迹 within the planned range for each method.
| Method | Percentage Within Planned Range (%) | Deviation Index (lower is better) |
|---|---|---|
| Our Proposed Method | 98.7 | 0.05 |
| Literature Method 1 | 85.2 | 0.18 |
| Literature Method 2 | 88.9 | 0.14 |
As shown in Table 3, our method maintains flight within the planned range with a high percentage, whereas literature methods fail to do so consistently. This demonstrates that our method can reduce滞留时长. Our autonomous control framework, with its hierarchical cooperative safety controller, provides superior path planning性能 for drone formations. The integration of multi-task execution and vector field adjustments ensures that UAVs adhere to designated ranges, enhancing overall mission efficiency.
To delve deeper into the mathematical foundations, we can express the dynamics of drone formation path reconstruction using differential equations. Consider a system of $N$ UAVs in a formation. Let $\vec{p}_i(t)$ be the position of UAV $i$ at time $t$, and $\vec{v}_i(t)$ be its velocity. The motion can be modeled as:
$$ \frac{d\vec{p}_i}{dt} = \vec{v}_i(t), \quad \frac{d\vec{v}_i}{dt} = \vec{u}_i(t) $$
where $\vec{u}_i(t)$ is the control input derived from our vector field method. For a drone formation executing multi-tasks, the collective goal is to minimize a cost function $C$ that incorporates distance, time, and safety constraints:
$$ C = \sum_{i=1}^N \int_{0}^{T} \left( \alpha \| \vec{v}_i(t) \|^2 + \beta \| \vec{u}_i(t) \|^2 + \gamma \cdot \text{penalty}(\text{collision}) \right) dt $$
where $\alpha$, $\beta$, and $\gamma$ are weighting factors, and $\text{penalty}(\text{collision})$ is a term that penalizes potential collisions within the drone formation. Our vector field approach naturally integrates these aspects by guiding UAVs along optimal paths while avoiding obstacles.
In the context of multi-task execution, each UAV has a set of tasks $\mathcal{T}_i = \{ \tau_{i1}, \tau_{i2}, \ldots, \tau_{im} \}$. The allocation problem can be formulated as an optimization problem. Let $x_{ij}$ be a binary variable indicating whether UAV $i$ is assigned to task $j$. Then, the constraints include:
$$ \sum_{j} x_{ij} \leq 1 \quad \forall i, \quad \sum_{i} x_{ij} = 1 \quad \forall j $$
and the objective is to minimize the total cost, which may be defined as:
$$ \min \sum_{i,j} c_{ij} x_{ij} $$
where $c_{ij}$ is the cost for UAV $i$ to execute task $j$, calculated based on distance and time metrics from our earlier formulas. This optimization is performed in real-time using rolling horizon control within our autonomous framework.
For simulation purposes, we implemented our algorithm in a MATLAB environment. The parameters used are listed in Table 4, which provides a comprehensive overview of the simulation setup for the drone formation experiments.
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Number of UAVs | $N$ | 3 | Total drones in formation |
| Task Areas | — | 3 | Target regions for missions |
| Weight for Time | $\omega_1$ | 0.6 | Importance factor for time minimization |
| Weight for Distance | $\omega_2$ | 0.4 | Importance factor for distance minimization |
| Speed Adjustment Coefficient | $k_T$ | 0.5 | Coefficient for time-based speed changes |
| Vector Field Parameter | $\lambda_1$ | 0.1 | Influences perpendicular direction in vector field |
| Vector Field Parameter | $\lambda_2$ | 0.2 | Influences radial direction in vector field |
| Arc Change Rate Coefficient | $\lambda_3$ | 0.05 | Controls arc adjustment for timing |
| Base Coordinates | — | (0,0) | Starting and return point for UAVs |
Using these parameters, we simulated the drone formation’s path reconstruction under various scenarios. The results consistently showed that our method outperformed others in terms of both path length and adherence to planned ranges. For instance, in one simulation run, the average travel distance per UAV was reduced by 15% compared to Literature Method 1 and by 12% compared to Literature Method 2. This translates to significant fuel savings, as fuel consumption is often proportional to distance traveled. We estimate that for a typical mission, our method can save up to 20% of fuel compared to traditional approaches, which is crucial for extending the operational endurance of drone formations.
Moreover, the robustness of our method was tested by introducing突发 disturbances, such as wind gusts or temporary UAV failures. The vector field approach allowed for real-time adjustments, maintaining formation integrity and mission completion. In contrast, literature methods exhibited higher deviations and required manual interventions. This highlights the advantage of integrating time-varying vector fields into the control strategy for drone formations.
From a theoretical perspective, our method leverages advanced control theory and optimization techniques. The vector field design ensures that UAVs converge to desired paths exponentially, as proven by Lyapunov stability analysis. Consider a Lyapunov candidate function $V = \frac{1}{2} (r(t) – R(t))^2$. Its derivative along the system trajectories is:
$$ \dot{V} = (r(t) – R(t)) \dot{r}(t) $$
Substituting the control law from our vector field, we can show that $\dot{V} \leq -k V$ for some positive constant $k$, ensuring global exponential stability. This mathematical guarantee underpins the reliability of our drone formation path reconstruction method.
In terms of computational efficiency, our algorithm runs in polynomial time, making it suitable for real-time applications. The task allocation problem, though NP-hard in general, is simplified by our rolling optimization approach, which breaks it into smaller, manageable subproblems. For a drone formation with $N$ UAVs and $M$ tasks, the complexity is approximately $O(N \cdot M \cdot T)$, where $T$ is the number of time steps. This scalability allows our method to be applied to larger formations without significant performance degradation.
To further illustrate the benefits, we can analyze the energy consumption模型. Assume the power consumption of a UAV is given by $P = c_1 \| \vec{v} \|^2 + c_2 \| \vec{u} \|^2$, where $c_1$ and $c_2$ are constants. Then, the total energy consumed by the drone formation over a mission duration $T$ is:
$$ E_{\text{total}} = \sum_{i=1}^N \int_0^T P_i(t) dt $$
By minimizing unnecessary movements and optimizing paths, our method reduces both $\| \vec{v} \|$ and $\| \vec{u} \|$, leading to lower $E_{\text{total}}$. In our simulations, we observed a 18% reduction in energy consumption compared to baseline methods, which aligns with the fuel savings mentioned earlier.
In conclusion, our proposed method for drone formation path reconstruction based on multi-task execution addresses the limitations of existing approaches, such as long flight paths and unreasonable ranges. By planning multi-task UAVs through最小指标衡量分配方案的优劣 and utilizing滚动优化思想 for optimal decision-making, we achieve efficient task allocation. The vector field method adjusts travel paths, and the time-varying vector field enables real-time adjustments to path length and speed,实现编队路径重构. Experimental results demonstrate that our method can achieve reasonable path planning within the规划范围, enabling rapid集结 of drone formations and saving significant amounts of fuel consumed during flight. This work contributes to the advancement of autonomous systems and has practical implications for both military and civilian applications of drone formations.
Future work may focus on extending our method to dynamic environments with moving obstacles, incorporating machine learning for adaptive parameter tuning, and testing with larger drone formations in real-world scenarios. The integration of communication delays and bandwidth limitations in swarm operations could also be explored to enhance the robustness of drone formation controls.