The deployment of unmanned aerial vehicles (UAVs) has become ubiquitous across diverse sectors, from aerial photography and logistics to critical infrastructure inspection. In my work within the power grid sector, a persistent challenge has been the limitation of single-drone systems when tasked with inspecting extensive, high-density transmission networks. Single drones, constrained by battery life, sensor footprint, and flight regulations, often struggle with efficiency and the risk of missing defects between parallel lines. This operational gap highlights the compelling potential of drone formation strategies. Coordinated drone formation flight can significantly increase the effective scanning swath, thereby enhancing inspection throughput and reliability. However, the practical realization of such a drone formation, particularly one capable of performing coordinated planar maneuvers like following the turns of a power line corridor, presents a significant control and planning challenge. The core problem lies in maintaining a stable geometric configuration while optimizing the flight trajectory for the entire group under complex dynamic and environmental constraints.
Traditional methods for drone formation control, such as the leader-follower approach, offer simplicity but are fundamentally limited. In these schemes, follower drones rigidly track the leader’s position based on a pre-defined offset. This rigidity often ignores the individual dynamic limits (e.g., maximum turn rate, acceleration) of the followers, leading to instability, excessive control effort, or even loss of formation during aggressive maneuvers. My objective was to develop a more robust and flexible framework that enforces formation cohesion not as a rigid constraint but as an emergent property of inter-agent interactions, while simultaneously solving for a globally optimal flight path. This led to the integration of two powerful concepts: a virtual spring-based formation model and an hp-adaptive pseudospectral optimal control solver.
The foundation of any trajectory optimization problem is an accurate dynamic model. For a fixed-wing UAV performing coordinated turns in a horizontal plane (assuming zero sideslip), the equations of motion can be simplified. The key states are the position $(x, y)$, velocity $V$, and flight path angle $\chi$. The primary control inputs are the engine thrust $T$ and the bank angle $\gamma_V$. The fundamental dynamics governing the drone formation member are:
$$
\begin{aligned}
\dot{x} &= V \cos \chi \\
\dot{y} &= V \sin \chi \\
\dot{V} &= \frac{1}{m} (T – D) \\
\dot{\chi} &= \frac{g}{V} \tan \gamma_V
\end{aligned}
$$
Here, $m$ is the mass, $g$ is gravity, and $D$ is the aerodynamic drag, calculated as $D = \frac{1}{2} C_D \rho S V^2$, with $C_D$ as the drag coefficient, $\rho$ air density, and $S$ the wing reference area. Practical constraints must be imposed: $V_{\min} \leq V \leq V_{\max}$, $| \gamma_V | \leq \gamma_{V_{\max}}$, and $T_{\min} \leq T \leq T_{\max}$.
To manage the drone formation geometry, I moved away from rigid offsets. Instead, I modeled the formation as a network of virtual elastic connections. Each drone is connected to its neighbors by virtual springs. The desired inter-agent distance $d_{ij}^*$ serves as the spring’s resting length. If the actual distance $d_{ij}$ deviates from $d_{ij}^*$, a virtual force is applied to both drones, attracting or repelling them to restore the desired spacing. For a drone $i$ connected to neighbor $j$, this force $\vec{F}_{ij}^{vs}$ is modeled using a spring-damper system:
$$
\vec{F}_{ij}^{vs} = -k_{s} (d_{ij} – d_{ij}^*) \hat{\mathbf{e}}_{ij} – k_{d} (\vec{v}_{ij} \cdot \hat{\mathbf{e}}_{ij}) \hat{\mathbf{e}}_{ij}
$$
where $k_s$ is the spring stiffness, $k_d$ is the damping coefficient, $\hat{\mathbf{e}}_{ij}$ is the unit vector from drone $i$ to $j$, and $\vec{v}_{ij}$ is their relative velocity. The total virtual formation force on drone $i$ is the sum over all its neighbors: $\vec{F}_{i}^{formation} = \sum_{j \in \mathcal{N}_i} \vec{F}_{ij}^{vs}$. This approach creates a flexible yet cohesive drone formation where the geometry is maintained through local interactions, allowing for natural deformation and recovery during maneuvers.
The trajectory planning problem is to find the control histories $T(t)$ and $\gamma_V(t)$ for all drones in the drone formation that minimize a cost function (e.g., total flight time or energy) while satisfying the dynamics, constraints, and boundary conditions (start/end positions). This is a complex, high-dimensional optimal control problem (OCP). I employed the hp-adaptive pseudospectral method to transcribe this continuous-time OCP into a tractable Nonlinear Programming (NLP) problem.
The time interval $t \in [t_0, t_f]$ is mapped to $\tau \in [-1, 1]$. The state and control variables are discretized at a set of Legendre-Gauss (LG) collocation points $\{\tau_1, …, \tau_N\}$. The states are approximated using Lagrange interpolating polynomials $L_j(\tau)$:
$$
\mathbf{x}(\tau) \approx \sum_{j=0}^{N} \mathbf{X}_j L_j(\tau)
$$
The dynamics are then enforced as algebraic constraints at each collocation point by differentiating the polynomial approximation. The derivative matrix $\mathbf{D}$ allows us to write the dynamics discretely: $\sum_{j=0}^{N} D_{kj} \mathbf{X}_j – \frac{t_f-t_0}{2} f(\mathbf{X}_k, \mathbf{U}_k) = 0$. The “hp-adaptive” part is key: it automatically refines the solution by strategically increasing the number of mesh intervals ($h$-refinement) or the polynomial degree ($p$-refinement) in regions where the solution exhibits high curvature or discontinuities, leading to superior accuracy and efficiency compared to fixed-grid methods.
The complete OCP for a multi-drone drone formation can be formulated as follows. Let $N_d$ be the number of drones. We aim to minimize a composite cost function $J$ that includes the mission objective (e.g., final time) and penalties for formation error:
$$
\min_{\mathbf{X}, \mathbf{U}, t_f} J = t_f + w_f \sum_{k=1}^{N} \sum_{(i,j) \in \mathcal{E}} (d_{ij}[k] – d_{ij}^*)^2
$$
subject to:
1. Discretized Dynamics: $\sum_{j=0}^{N} D_{kj} \mathbf{X}_j^{(i)} – \frac{t_f-t_0}{2} f(\mathbf{X}_k^{(i)}, \mathbf{U}_k^{(i)}) = 0$ for $i=1,…,N_d$ and $k=1,…,N$.
2. Path Constraints: $V_{\min} \leq V^{(i)}[k] \leq V_{\max}$, $| \gamma_V^{(i)}[k] | \leq \gamma_{V_{\max}}$, $T_{\min} \leq T^{(i)}[k] \leq T_{\max}$.
3. Boundary Conditions: $\mathbf{x}^{(i)}(t_0) = \mathbf{x}_0^{(i)}$, $\mathbf{x}^{(i)}(t_f) = \mathbf{x}_f^{(i)}$.
4. Collision Avoidance: $d_{ij}[k] \geq d_{safe}$ for all $i \neq j$.
This large-scale NLP is then solved using a robust solver like IPOPT. The virtual spring forces are integrated by including the formation-keeping error directly in the cost function or as a soft constraint, guiding the optimizer to find trajectories that inherently maintain the drone formation structure.
To validate the proposed framework, I conducted extensive numerical simulations. A drone formation of four drones in a diamond configuration was tasked with executing a 90-degree coordinated turn while transitioning from an initial speed to a final cruise speed. The simulation parameters for a representative fixed-wing drone are summarized below.
| Parameter | Symbol | Value |
|---|---|---|
| Mass | $m$ | 15 kg |
| Wing Reference Area | $S$ | 0.4 m² |
| Drag Coefficient | $C_D$ | 0.024 |
| Min / Max Thrust | $T_{\min}, T_{\max}$ | 5 N, 150 N |
| Min / Max Velocity | $V_{\min}, V_{\max}$ | 10 m/s, 30 m/s |
| Max Bank Angle | $\gamma_{V_{\max}}$ | 45° |
| Virtual Spring Stiffness | $k_s$ | 15 N/m |
| Virtual Spring Damping | $k_d$ | 0.8 N·s/m |
The first simulation demonstrates the capability for a single drone. Using the hp-adaptive pseudospectral method alone, the drone successfully planned and executed a minimum-time 90-degree turn while satisfying all velocity and control constraints. The resulting trajectories for states and controls were smooth and feasible.
The critical test involved the four-drone drone formation. I compared three methods: (A) a basic leader-follower with rigid offsets, (B) an artificial potential field method, and (C) the proposed virtual spring + hp-adaptive pseudospectral (VS-HP) method. The primary performance metrics were formation geometric error and stability under wind disturbance.
| Method | Avg. Position Error (m) | Max Formation Deformation (%) | Stability Metric* |
|---|---|---|---|
| (A) Leader-Follower | 15.32 | > 60% | 15% |
| (B) Artificial Potential Field | 9.87 | ~40% | 40% |
| (C) Proposed VS-HP Method | 4.35 | < 25% | 70% |
*Stability Metric: Percentage of flight time where formation deformation is below a 15% threshold.
The leader-follower method (A) failed to maintain the diamond shape during the turn, as followers could not adhere to the rigid offsets without violating their own dynamic limits. The artificial potential field method (B) performed better but exhibited oscillatory behavior and larger steady-state errors. The proposed VS-HP method (C) produced a smooth, coordinated turn. The virtual springs allowed the formation to flex slightly during the maneuver’s high-curvature phase and then smoothly recover to the precise diamond shape, all while optimizing the overall flight path. The resulting control inputs were smooth and within limits for all drones in the drone formation.
To evaluate robustness, I introduced a challenging 10 m/s crosswind disturbance model during the turn. The results, included in Table 2, show the superior disturbance rejection of the VS-HP method. The virtual spring network provides a distributed “web” of restoring forces, allowing the drone formation to absorb and compensate for the wind as a cohesive unit, much like a physical net deforming in the wind but retaining its overall structure.
The framework’s adaptability was further tested in complex 3D terrain. The state model was extended to include altitude $z$ and flight path angle $\gamma$. Virtual springs were applied in 3D, and terrain obstacles were modeled as repulsive potential fields. The hp-adaptive solver efficiently managed the added complexity. The following table summarizes the drone formation‘s performance across different terrains, demonstrating consistent, robust behavior.
| Terrain Scenario | Avg. 3D Formation Error (m) | Altitude Tracking RMSE (m) | Energy Increase vs. Nominal |
|---|---|---|---|
| Flat Terrain (Baseline) | 3.12 | 0.87 | 0% |
| Rolling Hills | 4.35 | 2.24 | 12.4% |
| Mountainous Ridge | 5.72 | 5.82 | 28.6% |
| Canyon Transit | 8.63 | 8.74 | 46.9% |
The integration of virtual spring dynamics with hp-adaptive pseudospectral optimal control presents a powerful and versatile framework for drone formation trajectory planning. This methodology fundamentally addresses the shortcomings of rigid formation control schemes. The virtual spring model provides an elegant, physics-inspired mechanism for decentralized cohesion and collision avoidance, granting the drone formation inherent flexibility and robustness to disturbances. The hp-adaptive pseudospectral method offers a highly accurate and efficient means to solve the resulting complex, constrained multi-agent optimal control problem, ensuring globally optimal performance with respect to time, energy, or other mission-specific criteria.
Simulation results across various scenarios—from simple coordinated turns to complex 3D terrain navigation under wind disturbances—confirm the effectiveness of this approach. The drone formation maintains stable geometry, adheres to individual vehicle constraints, and executes smooth, feasible trajectories. This work provides a solid theoretical and computational foundation for advanced multi-UAV applications, particularly in demanding, structured environments like automated power line inspection, where precise, efficient, and robust drone formation flight is paramount. Future work will focus on real-time implementation, dynamic re-planning in response to unforeseen obstacles, and scaling the approach to manage very large-scale swarms.