The coordinated flight of multiple unmanned aerial vehicles, commonly referred to as a drone formation, represents a paradigm shift in aerial operations, enabling capabilities far beyond those of a single platform. From precision agriculture and infrastructure inspection to complex search-and-rescue missions and aerial light shows, the applications are vast and growing. The core challenge in maintaining an effective drone formation lies not just in stable flight but in the ability to dynamically adapt—to reconfigure. Drone formation reconfiguration is the process by which a group of UAVs intelligently changes its spatial arrangement from an initial pattern to a target pattern, often in response to changing environmental threats, mission objectives, or unforeseen obstacles. This capability is paramount for operational resilience and efficiency.
The essence of a successful drone formation reconfiguration strategy is safe, collision-free, and timely trajectory generation for each member of the group. The problem is inherently complex, involving multi-agent path planning under stringent kinematic, dynamic, and cooperative constraints. Traditional methods, such as potential field approaches, often suffer from local minima. Graph-based searches can be computationally expensive in large, continuous spaces, while bio-inspired algorithms may lack guarantees on solution quality or real-time performance.
In recent years, sampling-based planning algorithms, particularly the Rapidly-exploring Random Tree (RRT), have shown significant promise for solving high-dimensional motion planning problems. Their strength lies in their ability to quickly explore the configuration space without the need for explicit geometric modeling of free space. This makes them exceptionally suitable for planning in complex, obstacle-rich environments—precisely the scenario for a drone formation needing to navigate unknown or hostile airspace. This article delves into a comprehensive methodology for drone formation reconfiguration built upon a modified RRT framework. We will explore the kinematic modeling of formations, the adaptation of the core RRT algorithm to meet the unique demands of multi-UAV coordination, and the critical constraints that ensure safety and feasibility.
Kinematic Modeling of a Drone Formation
To effectively control and plan for a drone formation, a precise mathematical model of the relative motion between UAVs is essential. Among various control architectures, the leader-follower mode offers a balance between simplicity and effectiveness, making it a widely adopted scheme for drone formation control. In this model, one UAV is designated as the leader, which follows a predefined mission path. The follower UAVs are tasked with maintaining specific relative positions and orientations with respect to this leader.
We establish a formation-fixed coordinate frame \(\{O_l-X_lY_lZ_l\}\) attached to the leader drone. The origin \(O_l\) is at the leader’s center of mass. The \(O_lX_l\) axis is aligned with the leader’s velocity vector, \(O_lZ_l\) points downwards within the leader’s symmetry plane, and \(O_lY_l\) completes the right-handed system. Consider a follower drone \(F\) in a 2D planar motion scenario (assuming level flight, negligible pitch). Its position relative to the leader \(L\) is given by coordinates \([x, y]^T\) in this frame. The kinematic equations governing the follower’s relative motion are derived from the Coriolis equations of relative velocity:
$$
\dot{\mathbf{R}}^l_{LF} = \dot{\mathbf{R}}^l_F – \dot{\mathbf{R}}^l_L + \boldsymbol{\omega}^l_L \times (\mathbf{R}^l_F – \mathbf{R}^l_L)
$$
Where \(\mathbf{R}\) denotes position vector and \(\boldsymbol{\omega}^l_L = [\dot{\mu}_L, 0, \dot{\psi}_L]^T\) is the angular velocity vector of the leader frame, with \(\psi\) being the flight path angle and \(\mu\) the flight path bank angle. Expanding and simplifying for 2D planar motion yields the relative motion model for the follower in the formation frame:
$$
\begin{aligned}
\dot{x} &= V_F \cos\psi_E \cos\mu_E + \dot{\psi}_L y – V_L \\
\dot{y} &= V_F \sin\psi_E \cos\mu_E – \dot{\psi}_L x \\
\psi_E &= \psi_F – \psi_L \\
\mu_E &= \mu_F – \mu_L
\end{aligned}
$$
Here, \(V\) denotes velocity. The subscripts \(L\) and \(F\) denote leader and follower, respectively, while the subscript \(E\) denotes error difference. For steady, coordinated turns, the turn rate \(\dot{\psi}\) is constrained by the maximum bank angle \(\mu_{max}\) and the velocity \(V\), defining a minimum turn radius \(R_{min}\), a critical parameter for path planning:
$$
R_{min} = \frac{V^2}{g \cdot \tan(\mu_{max})}
$$
This model provides the foundation for describing the state of any member within a drone formation and is vital for formulating the reconfiguration problem as a constrained trajectory planning task for each UAV.
The RRT Algorithm: Foundation and Adaptation for Drone Formation
The basic Rapidly-exploring Random Tree algorithm is a single-query, probabilistic motion planner. It incrementally builds a search tree \(T\) rooted at the initial state \(Z_I\) by randomly sampling the free configuration space \(\bar{S}_T\) (the complement of the threat/obstacle space \(S_T\)). Its operation in each iteration can be summarized in three key steps, which we will later adapt for drone formation planning:
- Sampling: Generate a random sample \(Z_{rand}\). With a probability \(P\), \(Z_{rand}\) is drawn uniformly from \(\bar{S}_T\); with probability \(1-P\), \(Z_{rand}\) is set to the goal state \(Z_G\) to bias the search.
- Nearest Neighbor Search: Find the node \(Z_{near}\) in the current tree \(T\) that is closest to \(Z_{rand}\) according to a chosen distance metric (e.g., Euclidean distance).
- Extension: From \(Z_{near}\), extend a branch towards \(Z_{rand}\) by a predefined step size \(L\) to generate a new candidate node \(Z_{new}\). If the path segment between \(Z_{near}\) and \(Z_{new}\) is collision-free, \(Z_{new}\) is added to \(T\) as a child of \(Z_{near}\).
This process repeats until a node is generated within a threshold distance \(L\) of the goal \(Z_G\). The path is then extracted by tracing parent pointers from this final node back to the root. While elegant, the vanilla RRT algorithm produces paths that are inherently unsuited for direct drone formation tracking due to two primary issues:
- Excessive Nodes and Jagged Paths: The random expansion creates unnecessary nodes and sharp turns, leading to oscillatory and inefficient flight.
- Kinematic Infeasibility at Nodes: The instantaneous direction changes at nodes require infinite turn rates, violating the drone’s minimum turn radius constraint \(R_{min}\).
To render RRT viable for drone formation reconfiguration, two crucial post-processing and in-processing strategies are employed: Path Pruning and Transition Trajectory Insertion.
Path Pruning: Removing Redundant Nodes
Given the set of waypoints \(\text{Root} = \{Z_I, Z_1, Z_2, …, Z_n, Z_G\}\) generated by the basic RRT, the pruning algorithm seeks a minimal subset \(\text{Refined} \subseteq \text{Root}\) that defines a shorter, smoother, yet still collision-free path. The algorithm works iteratively:
- Start with \(\text{Refined} = \{Z_I\}\). Set the current start point \(Z_s = Z_I\).
- For the current \(Z_s\), check connectivity (collision-free and kinematically feasible) directly to the goal \(Z_G\). If feasible, add \(Z_G\) to \(\text{Refined}\) and terminate.
- If not, iterate backward from the last node before \(Z_G\) (i.e., \(Z_n, Z_{n-1}, …\)). For each node \(Z_i\), check direct connectivity from \(Z_s\) to \(Z_i\).
- Upon finding the first \(Z_i\) that is directly reachable from \(Z_s\), add \(Z_i\) to \(\text{Refined}\), set \(Z_s = Z_i\), and return to step 2.
This greedy shortening significantly reduces path length and the number of turns, directly benefiting the stability of the drone formation during reconfiguration.
Transition Trajectory Insertion: Ensuring Kinematic Feasibility
Pruning reduces nodes, but the remaining turns may still be kinematically infeasible. To address this, we insert transition trajectories between path segments, inspired by Dubins path concepts. Consider a drone traveling along segment \(\overrightarrow{Z_{i-1}Z_i}\) needing to turn towards the next segment \(\overrightarrow{Z_{i}Z_{i+1}}\). A turn is executed by following a circular arc of radius \(R \geq R_{min}\). The objective is to find the shortest such transition.
Let the direction of turn (clockwise/CW or counter-clockwise/CCW) be determined by the relative geometry. We define a condition for the existence of a feasible transition arc tangent to both segments. For a CCW turn, we compute the center \(O_c\) of the transition circle by offsetting a distance \(R_{min}\) perpendicularly from the line \(\overrightarrow{Z_{i-1}Z_i}\). The entry point \(O_e\) and exit point \(O_v\) are the points of tangency. A transition exists if \(O_v\) lies on the segment \(\overrightarrow{Z_{i}Z_{i+1}}\) and the distance from \(O_c\) to the line containing \(\overrightarrow{Z_{i}Z_{i+1}}\) equals \(R_{min}\). The transition is then the arc from \(O_e\) to \(O_v\) on the circle centered at \(O_c\). The swept angle \(\theta_{out}\) is given by the change in heading between the two path segments.
The geometry can be solved directly. Let \(\mathbf{d}_1\) and \(\mathbf{d}_2\) be unit direction vectors of the incoming and outgoing segments. The center \(O_c\) lies at the intersection of two lines, each parallel to \(\mathbf{d}_1\) and \(\mathbf{d}_2\) respectively, and offset by a distance \(R_{min}\) in the appropriate direction (left for CCW, right for CW). The condition for a valid transition is:
$$
\begin{aligned}
\text{Let } L_1 &\parallel \overrightarrow{Z_{i-1}Z_i}, \quad \text{dist}(L_1, \overrightarrow{Z_{i-1}Z_i}) = R_{min} \\
\text{Let } L_2 &\parallel \overrightarrow{Z_{i}Z_{i+1}}, \quad \text{dist}(L_2, \overrightarrow{Z_{i}Z_{i+1}}) = R_{min} \\
\text{If } O_c &= L_1 \cap L_2 \text{ exists, and } O_v = \text{proj}_{O_c \rightarrow \text{line}(Z_i Z_{i+1})} \text{ lies on the segment,} \\
\text{then a feasible transition arc exists.}
\end{aligned}
$$
If the geometry does not satisfy this condition (e.g., segments are too close), alternative strategies like adaptive guidance laws must be employed at that node to ensure stable tracking without violating constraints. The integration of pruning and transition arcs transforms the erratic RRT output into a flyable path, making it a suitable core planner for a drone formation reconfiguration system.
Integrating RRT into Drone Formation Reconfiguration: Constraints and Strategy
Planning for a single drone is challenging; planning for an entire drone formation adds layers of complexity. The reconfiguration problem is not merely \(N\) independent RRT plans. It is a coupled problem where the trajectory of each drone must satisfy individual kinematic limits, avoid static and dynamic threats (including other members of the formation), and meet cooperative timing objectives. The following constraints are fundamental to a safe and effective drone formation reconfiguration strategy.
| Constraint Category | Description | Mathematical / Implementation Formulation |
|---|---|---|
| 1. Drone Kinematics | Minimum turn radius and minimum straight segment length. | $$R_{planned} \geq R_{min} = \frac{V^2}{g \tan(\mu_{max})}$$ Step size \(L \geq l_{min}\). |
| 2. Formation Geometry | Initial and target relative positions of all drones must form specific patterns (line, wedge, diamond). | Defined by sets \(\mathbf{P}_{init} = \{\mathbf{p}_{1,init}, …, \mathbf{p}_{N,init}\}\) and \(\mathbf{P}_{goal} = \{\mathbf{p}_{1,goal}, …, \mathbf{p}_{N,goal}\}\) relative to leader or virtual structure. |
| 3. Collision Avoidance (Internal) | Maintain a safe separation \(d_{min}\) between all drones at all times during reconfiguration. | For any two drones \(i\) and \(j\) at time \(t\): $$\|\mathbf{r}_i(t) – \mathbf{r}_j(t)\|_2 > d_{min}$$ |
| 4. Collision Avoidance (External) | Avoid all static obstacles/threat zones modeled as circular regions. | For drone \(i\) and obstacle \(k\) with center \((a_k, b_k)\) and radius \(r_k\): $$(x_i(t)-a_k)^2 + (y_i(t)-b_k)^2 \geq (\lambda \cdot r_k)^2$$ where \(\lambda \geq 1\) is a safety margin. |
| 5. Coordinated Timing | Drones should arrive at their target positions simultaneously or in a specified sequence. | Simultaneous arrival: For planned path lengths \(l_i\) and speed bounds \([v_{min}, v_{max}]\), find \(t_f\) such that \(t_f = l_i / v_i\) with \(v_i \in [v_{min}, v_{max}]\) for all \(i\). |
Sequential Constrained Planning Strategy
A practical strategy for integrating RRT into drone formation reconfiguration is a sequential, constraint-aware approach:
- Leader Path Planning: First, an optimal or feasible path for the leader drone from its start to its goal is computed using the modified RRT algorithm (with pruning and transition arcs). This path must satisfy all external obstacle constraints.
- Follower Path Planning with Coupling: The paths for follower drones are then planned sequentially or in a coupled manner. Crucially, the already-planned leader’s trajectory is treated as a dynamic obstacle for the followers. During the RRT’s random sampling step for a follower, a probabilistic bias is introduced: a sample \(Z_{rand}\) is accepted with probability \(\hat{P}\) that decays as the sample’s predicted position at a corresponding time gets closer to the leader’s position at that time.
$$
\hat{P} = \begin{cases}
1, & \text{if } d_r > d_{min} \\
\exp\left(\frac{d_r – d_{min}}{D}\right), & \text{if } d_r \le d_{min}
\end{cases}
$$
Here, \(d_r\) is the distance between the sample and the leader’s position at a correlated time, and \(D\) is a scaling factor. This discourages the tree from expanding into regions where the follower would be too close to the leader. - Velocity Scheduling for Coordination: After all geometric paths \(\{l_1, l_2, …, l_N\}\) are planned, the simultaneous arrival constraint is enforced by solving for the individual drone speeds \(v_i\). Given the leader’s desired speed \(\bar{v}_L\), the nominal formation reconfiguration time is \(t_f = l_L / \bar{v}_L\). The speed for follower \(i\) is then set to \(v_i = l_i / t_f\), provided \(v_i \in [v_{min}, v_{max}]\). If not, \(t_f\) or the leader’s path may need adjustment.
- Collision Validation: Finally, the complete spatiotemporal plan is discretized and checked at fine time intervals \(\Delta t\) to verify both internal (\(d_{ij}(t) > d_{min}\)) and external collision constraints. If violations are found, local replanning or speed adjustments are performed.
Simulation Analysis and Experimental Insights
To validate the proposed RRT-based drone formation reconfiguration framework, comprehensive simulations and flight tests are conducted. A typical scenario involves a three-drone formation in a 1000m x 1000m area with multiple circular threat zones. The initial formation is a horizontal line (“line abreast” for search), and the target formation is a vertical trail (“line astern” for transit or engagement). The key simulation parameters are summarized below:
| Parameter | Value |
|---|---|
| Number of Drones (\(N\)) | 3 (1 Leader, 2 Followers) |
| Drone Speed Range | 20 m/s to 25 m/s |
| Max Bank Angle (\(\mu_{max}\)) | 0.6 rad |
| Min Turn Radius (\(R_{min}\)) | ~120 m (at ~23 m/s) |
| Min Safe Separation (\(d_{min}\)) | 40 m |
| Obstacle Safety Margin (\(\lambda\)) | 1.2 |
| RRT Step Size (\(L\)) | 20 m |
The modified RRT planner generates successful reconfiguration paths. A comparative analysis highlights the efficiency of the approach. The table below contrasts the performance of the RRT-based method with a Particle Swarm Optimization (PSO) with Dubins paths method and a Pseudospectral optimal control method, averaged over multiple runs.
| Performance Metric | RRT-Based Method | PSO-Dubins Method | Pseudospectral Method |
|---|---|---|---|
| Average Reconfiguration Time (\(t_f\)) | 45.56 s | 118.56 s | 37.13 s |
| Min Inter-Drone Distance | 49.87 m | 51.38 m | 45.68 m |
| Min Obstacle Distance | 5.24 m | 15.52 m | 7.12 m |
| Average Computation Time | 2.48 s | 20.52 s | 34.83 s |
The results are illuminating. The Pseudospectral method generates the most time-optimal paths (shortest \(t_f\)) but at the highest computational cost, making online replanning difficult. The PSO-Dubins method, while generating smooth paths, suffers from long convergence times and less optimal path lengths. The RRT-based method strikes an excellent balance: it achieves a reconfiguration time close to the optimal pseudospectral result, maintains all safety margins, and does so with a computation time nearly an order of magnitude faster. This makes it highly suitable for real-time or near-real-time drone formation reconfiguration in dynamic environments.
Flight Test Validation of Transition Trajectories
The critical role of transition trajectories for stable tracking is validated through physical flight tests. A fixed-wing UAV is tasked with tracking a rhombus-shaped path defined by waypoints, simulating the output of a pruned RRT. Two flight modes are compared: one using only a standard nonlinear guidance law at waypoints, and another where pre-computed transition arcs (with \(R = 70m > R_{min}\)) are inserted at the turns.
The flight data clearly demonstrates the superiority of the transition arc method. Without transition arcs, the aircraft exhibits significant oscillatory overshoot when negotiating acute-angle turns, requiring extended stabilization periods on the subsequent leg. The bank angle \(\mu\) commands are aggressive and prolonged. In contrast, with transition arcs, the UAV enters and exits turns smoothly, tracking the prescribed circular arc. The bank angle profile is smoother and more moderate, and the aircraft seamlessly transitions onto the next path segment without oscillatory behavior. This confirms that the path modification strategies are not just theoretical enhancements but are essential for translating RRT-generated plans into safe, stable flight paths for a drone formation.
Future Directions and Concluding Perspectives
The exploration of RRT-based strategies for drone formation reconfiguration opens several avenues for future research. Extending the framework from 2D to 3D space is a natural and necessary progression, requiring the consideration of climb/descent rates and energy constraints. The current method assumes a static environment; integrating real-time sensor data for dynamic obstacle avoidance and online replanning is crucial for field deployment. This could involve a rolling-horizon RRT or dynamic RRT* (RRT-star) approach. Furthermore, investigating decentralized versions of the algorithm, where each drone in the drone formation runs its own coupled planner with communicated intent, would enhance robustness and scalability. Finally, the integration of higher-fidelity dynamic models and wind disturbance rejection into the planning loop will bridge the gap between geometric planning and robust flight control.
In conclusion, the Rapidly-exploring Random Tree algorithm, when thoughtfully adapted and constrained, provides a powerful and efficient foundation for solving the complex problem of drone formation reconfiguration. By addressing the kinematic infeasibility of raw RRT outputs through path pruning and transition trajectory insertion, and by rigorously enforcing formation-specific constraints like internal collision avoidance and coordinated timing, the method generates safe, flyable, and efficient reconfiguration plans. The synergy between the rapid exploration capability of RRT and the stringent requirements of multi-agent coordination makes this a highly promising approach. As drone swarms move from demonstration to widespread application, such reliable and computationally tractable reconfiguration strategies will be indispensable for unlocking their full potential in an ever-changing operational airspace.