The coordinated flight of multiple unmanned aerial vehicles, known as drone formation flight, represents a significant area of research with applications spanning aerial displays, surveillance, and cooperative payload transport. The core challenge lies in designing control systems that enable a group of drones to not only maintain a specific geometric configuration, or formation, during complex maneuvers but also to reconfigure this formation seamlessly and safely in response to mission demands or environmental changes. This work delves into the detailed design and simulation of such controllers, focusing on the critical aspects of three-dimensional formation keeping and transformation for a two-vehicle, leader-follower drone formation.
The fundamental structure in many drone formation studies is the “leader-wingman” paradigm. In this setup, one designated drone, the leader, follows a predefined trajectory or is manually piloted. The other drones, the wingmen, are tasked with autonomously maintaining prescribed relative positions to the leader. This paper addresses the control problem for a wingman drone in a two-drone formation, initially configured in a left-diamond pattern relative to the leader. The objective is to develop a control system that ensures stable formation keeping during leader maneuvers and enables collision-free transitions to other standard formations, such as a trail formation or a right-diamond pattern.
Mathematical Modeling of the Drone Formation
Accurate modeling is the cornerstone of effective controller design for any drone formation. The model must capture both the individual vehicle dynamics and the kinematic relationships that bind the vehicles together in formation. We begin by assuming each drone in the formation is equipped with standard autopilot loops for speed, heading, and altitude hold. These are commonly modeled as first-order lags for speed and heading, and a second-order system for altitude.
The autopilot dynamics for any drone \( i \) (where \( i = L, W \) for Leader and Wingman) are given by:
$$ \dot{V}_i = -\frac{1}{\tau_V} V_i + \frac{1}{\tau_V} V_{ic} $$
$$ \dot{\psi}_i = -\frac{1}{\tau_{\psi}} \psi_i + \frac{1}{\tau_{\psi}} \psi_{ic} $$
$$ \ddot{h}_i = -\left( \frac{1}{\tau_{ha}} + \frac{1}{\tau_{hb}} \right) \dot{h}_i – \frac{1}{\tau_{ha}\tau_{hb}} h_i + \frac{1}{\tau_{ha}\tau_{hb}} h_{ic} $$
Here, \( V \), \( \psi \), and \( h \) represent velocity, heading angle, and altitude, respectively. The variables with subscript \( c \) denote command inputs to the autopilots. The time constants \( \tau_V \), \( \tau_{\psi} \), \( \tau_{ha} \), and \( \tau_{hb} \) define the response characteristics of each channel and are summarized in Table 1.
| Parameter | Description | Typical Value / Symbol |
|---|---|---|
| \( \tau_V \) | Velocity autopilot time constant | Assumed known |
| \( \tau_{\psi} \) | Heading autopilot time constant | Assumed known |
| \( \tau_{ha}, \tau_{hb} \) | Altitude autopilot time constants | Assumed known |
| \( V_i, \psi_i, h_i \) | Drone states: Velocity, Heading, Altitude | State Variables |
| \( V_{ic}, \psi_{ic}, h_{ic} \) | Autopilot command inputs | Control Inputs (for Wingman) |
The essence of drone formation modeling lies in defining the relative motion between the leader and the wingman. We derive this not through complex Coriolis equations but via straightforward geometric relationships. Consider the wingman’s body-fixed coordinate frame \( (x_W, y_W, z_W) \), where \( x_W \) points forward along the wingman’s velocity vector, \( y_W \) points to its right, and \( z_W \) points down. In this frame, the desired relative position of the leader is defined by the constant offsets \( (x_0, y_0, z_0) \). For a left-diamond formation, \( x_0 > 0 \) and \( y_0 > 0 \). The actual relative position vector is \( (x, y, z) \).
By analyzing the kinematics of both vehicles in an inertial frame and projecting the leader’s velocity onto the wingman’s rotating frame, we arrive at the fundamental nonlinear relative kinematics for the horizontal plane:
$$ \dot{x} = V_L \cos(\psi_L – \psi_W) + \dot{\psi}_W y – V_W $$
$$ \dot{y} = V_L \sin(\psi_L – \psi_W) – \dot{\psi}_W x $$
For the vertical channel, the relative motion is simply the difference in their altitudes and its derivative:
$$ z = h_W – h_L $$
$$ \dot{z} = \xi $$
$$ \dot{\xi} = \ddot{h}_W – \ddot{h}_L $$
To design a linear controller, we linearize these equations around a nominal flight condition: constant velocity \( V_0 \), constant heading \( \psi_0 = 0 \), and constant altitude \( h_0 \). The linearization point for the relative position is the desired formation spacing \( (x_0, y_0, z_0=0) \). Substituting the linear autopilot models and performing small-perturbation analysis yields a linear state-space model for the drone formation error dynamics. The state vector is chosen as \( \mathbf{X} = [x, V_W, y, \psi_W, z, \xi]^T \). The wingman’s autopilot commands \( V_{Wc}, \psi_{Wc}, h_{Wc} \) are the control inputs. The leader’s states \( V_L, \psi_L, \) and its altitude command \( h_{Lc} \) are treated as measurable disturbances or reference commands to be tracked.
The resulting linear state-space model is:
$$ \dot{\mathbf{X}} = \mathbf{A} \mathbf{X} + \mathbf{B} \mathbf{u} + \mathbf{\Gamma} \mathbf{d} $$
Where:
$$ \mathbf{u} = \begin{bmatrix} V_{Wc} \\ \psi_{Wc} \\ h_{Wc} \end{bmatrix}, \quad \mathbf{d} = \begin{bmatrix} V_L \\ \psi_L \\ h_{Lc} \end{bmatrix} $$
$$ \mathbf{A} = \begin{bmatrix}
0 & -1 & 0 & -y_0/\tau_{\psi} & 0 & 0 \\
0 & -1/\tau_V & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & x_0/\tau_{\psi} – V_0 & 0 & 0 \\
0 & 0 & 0 & -1/\tau_{\psi} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & -1/(\tau_{ha}\tau_{hb}) & -(\frac{1}{\tau_{ha}}+\frac{1}{\tau_{hb}})
\end{bmatrix} $$
$$ \mathbf{B} = \begin{bmatrix}
0 & y_0/\tau_{\psi} & 0 \\
1/\tau_V & 0 & 0 \\
0 & -x_0/\tau_{\psi} & 0 \\
0 & 1/\tau_{\psi} & 0 \\
0 & 0 & 0 \\
0 & 0 & 1/(\tau_{ha}\tau_{hb})
\end{bmatrix}, \quad \mathbf{\Gamma} = \begin{bmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & V_0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1/(\tau_{ha}\tau_{hb})
\end{bmatrix} $$
This model elegantly captures the coupling within the drone formation. Notably, the vertical channel \( (z, \xi) \) is decoupled from the horizontal channels. The horizontal channels are coupled through the states \( x \) and \( \psi_W \). This model forms the basis for our controller design and analysis.
Controller Design for Formation Keeping and Transformation
The primary goal for the wingman in a drone formation is twofold: 1) to track the leader’s maneuvers (changes in \( V_L, \psi_L, h_{Lc} \)) while maintaining the prescribed separation \( (x_0, y_0, z_0) \), and 2) to change the formation geometry by adopting new separation commands \( (x_c, y_c, z_c) \) on demand. We propose a control architecture based on Proportional-Integral-Derivative (PID) control, renowned for its simplicity, robustness, and clear physical interpretation. The overall control structure is illustrated in Figure 1.
| Channel | Error Signal (\( e \)) Composition | Control Law (\( u(t) \)) |
|---|---|---|
| Velocity / X | \( e_x = k_v(V_L – V_W) + k_x(x_c – x) \) | \( V_{Wc}(t) = K_{Px} e_x + K_{Ix} \int e_x dt + K_{Dx} \dot{e}_x \) |
| Heading / Y | \( e_y = k_{\psi}(\psi_L – \psi_W) + k_y(y_c – y) \) | \( \psi_{Wc}(t) = K_{Py} e_y + K_{Iy} \int e_y dt + K_{Dy} \dot{e}_y \) |
| Altitude / Z | \( e_z = k_z(z_c – z) \) | \( h_{Wc}(t) = K_{Pz} e_z + K_{Iz} \int e_z dt + K_{Dz} \dot{e}_z \) |
The error signal for each channel is a linear combination of a “maneuver tracking error” and a “formation spacing error”. For example, in the velocity (x-direction) channel, the error \( e_x \) blends the velocity difference between leader and wingman with the error in the longitudinal separation. The gains \( k_v, k_x, k_{\psi}, k_y, k_z \) are mixing gains that weight the importance of tracking versus spacing. This structure is key to the dual functionality of the controller. When the formation is to be kept, the spacing commands \( (x_c, y_c, z_c) \) are set to the nominal values \( (x_0, y_0, z_0) \). When a formation transformation is ordered, these commands are switched to new target values (e.g., \( y_c = 0 \) to transition from diamond to trail formation). The PID controller then acts on the newly defined error to drive the wingman to the new relative position.
Parameter Tuning and Simulation Analysis
Tuning the numerous PID and mixing gains is critical for the performance of the drone formation controller. We employed a time-domain optimization tool, Simulink Response Optimization (SRO), which utilizes sequential quadratic programming. Performance specifications such as settling time, overshoot, and steady-state error for step responses in tracking and spacing changes were defined as constraints. The SRO tool iteratively adjusted the gains to meet these specifications, resulting in an optimized set of parameters, as summarized in Table 3.
| Gain | Value | Gain | Value | Gain | Value |
|---|---|---|---|---|---|
| \( k_x \) | -7.54 | \( k_y \) | -0.85 | \( k_z \) | 18.98 |
| \( k_v \) | 10.34 | \( k_{\psi} \) | 5.45 | ||
| \( K_{Px} \) | 5.86 | \( K_{Py} \) | 9.85 | \( K_{Pz} \) | 3.29 |
| \( K_{Ix} \) | 0.39 | \( K_{Iy} \) | 0.88 | \( K_{Iz} \) | 0.46 |
| \( K_{Dx} \) | 0.01 | \( K_{Dy} \) | 0.002 | \( K_{Dz} \) | 0.04 |
To validate the controller, extensive simulations were conducted over a 60-second duration. The nominal flight condition was: altitude \( h_0 = 3000 \) ft, velocity \( V_0 = 450 \) ft/s, and heading \( \psi_0 = 0^\circ \). Three key scenarios were tested to demonstrate both formation keeping and transformation capabilities of the drone formation control system.
Scenario 1: Formation Keeping During Leader Maneuver
In this test, the leader executes a +30° heading change for the first 30 seconds, then returns to its original heading. The wingman’s task is to maintain the left-diamond formation (\( x_0=300ft, y_0=100ft \)) throughout. The simulation results show excellent tracking performance. The wingman’s heading accurately follows the leader’s maneuver with zero steady-state error in both phases. The velocity response shows a characteristic transient: to maintain the lateral (\( y \)) separation during the turn, the wingman must adjust its speed, initially increasing and then decreasing back to the nominal value. The \( x \) and \( y \) separation distances remain very close to their commanded values, with symmetric deviations during the transient periods. This confirms the controller’s ability to maintain a tight drone formation during aggressive leader maneuvers.
Scenario 2: Formation Transformation: Left-Diamond to Trail
This scenario tests the transformation capability. Initially in a left-diamond, the wingman receives a new spacing command at \( t=0 \) s to transition to a trail formation (\( x_c=300ft, y_c=0ft \)). The leader continues straight and level flight. Analysis of the response is revealing. The wingman must move from a position to the left and behind the leader to a position directly behind. To achieve this, it commands a positive (right) heading rate, causing a small increase in heading angle. Simultaneously, it increases thrust to cover the extra relative distance involved in the curved path. The \( x \)-separation shows a transient increase before settling to 300 ft, while the \( y \)-separation smoothly converges from 100 ft to 0 ft. Crucially, the relative distance between the two drones never approaches zero, indicating a collision-free transformation. The inertial flight paths clearly show the wingman’s curved trajectory to assume the new position directly behind the leader, successfully reconfiguring the drone formation.
Scenario 3: Formation Transformation: Left-Diamond to Right-Diamond
A more demanding transformation is commanded at \( t=20 \) s: switch from left-diamond (\( y_0=+100ft \)) to right-diamond (\( y_c=-100ft \)). This requires the wingman to cross the leader’s flight path. The controller manages this complex maneuver effectively. The wingman initiates a right turn. The velocity and heading responses show larger transients compared to the previous case, reflecting the more significant positional change required. The \( y \)-separation error drives the response, moving from +100 ft through 0 ft and finally settling at -100 ft. The \( x \)-separation is maintained around its nominal value with an acceptable transient. Most importantly, the minimum recorded distance between the drones remains safe, proving that the controller can execute even crossing-maneuver transformations within the drone formation without collision risk. The trajectory plot visually confirms the wingman’s path crossing behind the leader to settle into the new, mirrored formation geometry.
| Scenario | Initial Formation | Command / Disturbance | Key Controller Performance Observation |
|---|---|---|---|
| 1. Keeping | Left-Diamond | Leader: +30° heading step (0-30s), then return. | Accurate heading tracking; maintained x,y spacing; stable velocity transients. |
| 2. Transformation | Left-Diamond | Wingman: Change \( y_c \) from +100ft to 0ft (Trail). | Smooth, collision-free transition; wingman executes curved path to new position. |
| 3. Transformation | Left-Diamond | Wingman: Change \( y_c \) from +100ft to -100ft (Right-Diamond). | Successful crossing maneuver; stable settling to new offset; safe minimum separation. |
Conclusion and Discussion
This work has presented a complete methodology for the design and analysis of control systems for drone formation flight. By deriving a linearized model from first geometric principles and combining it with standard autopilot dynamics, we established a tractable model for controller design. The proposed PID-based control law, featuring blended error signals for maneuver tracking and spacing control, proved highly effective. The controller successfully demonstrated the two fundamental requirements for autonomous drone formation flight: robust formation keeping in the presence of leader maneuvers and safe, collision-free transformation between different standard formation geometries.
The simulation results for all three scenarios—formation keeping, left-diamond to trail transformation, and left-diamond to right-diamond transformation—validate the design. The wingman accurately tracked leader commands, maintained precise relative positioning, and executed transformation maneuvers with stable, well-damped responses. The use of modern tuning tools like SRO ensured an optimized set of control parameters. The architecture’s strength lies in its simplicity and extensibility. While demonstrated for a two-vehicle formation, the “leader-wingman” building block can be replicated and extended to control larger, multi-vehicle drone formations by defining a hierarchy or acyclic communication graph. Future work may explore integrating this approach with robust or adaptive control techniques to handle model uncertainties and external disturbances, further enhancing the reliability of autonomous drone formation systems in complex operational environments.