In modern battlefield environments, a drone formation consisting of multiple unmanned aerial vehicles (UAVs) offers significant advantages over a single drone. By employing drone formations for hazardous missions, task success rates can be improved, and close formation flying can save energy. According to the basic requirements of formation flight, each drone should maintain a specific formation shape during flight or adjust its position promptly based on external environments and mission demands to transform into other formation patterns. This paper focuses on the design of a sliding mode controller for drone formation keeping and transformation, addressing the challenges of robust control in dynamic scenarios.
The control of drone formations has become a hotspot in UAV research. Internationally, numerous studies have been conducted, such as the MICA project funded by DARPA, which emphasizes autonomous and cooperative control. Researchers have used Lyapunov stability theory for adaptive formation controllers and continuous high-order sliding mode methods to account for vortex effects. Domestically, achievements include robust decentralized model predictive control, fuzzy PID controllers, and neural network-based adaptive inverse controllers. Building on these ideas, this paper introduces sliding mode variable structure control theory into drone formation control for the first time, offering a novel approach to formation keeping and transformation.
We consider a drone formation consisting of two UAVs in a right-diamond formation, with a leader (L) and a wingman (W). Both drones employ identical autopilot models, including first-order speed hold, second-order heading hold, and second-order altitude hold. The models are given by:
$$ \dot{V}_i = -\frac{1}{\tau_V} V_i + \frac{1}{\tau_V} V_{ic} $$
$$ \ddot{\psi}_i = -\frac{1}{\tau_{\psi a} \tau_{\psi b}} \psi_i – \left( \frac{1}{\tau_{\psi a}} + \frac{1}{\tau_{\psi b}} \right) \dot{\psi}_i + \frac{1}{\tau_{\psi a} \tau_{\psi b}} \psi_{ic} $$
$$ \ddot{h}_i = -\frac{1}{\tau_{ha} \tau_{hb}} h_i – \left( \frac{1}{\tau_{ha}} + \frac{1}{\tau_{hb}} \right) \dot{h}_i + \frac{1}{\tau_{ha} \tau_{hb}} h_{ic} $$
where \( i = L, W \) for leader and wingman, respectively; \( V \) is velocity, \( \psi \) is heading angle, \( h \) is altitude; and \( \tau_V \), \( \tau_{\psi a} \), \( \tau_{\psi b} \), \( \tau_{ha} \), \( \tau_{hb} \) are time constants. The formation geometry is described using inertial and wingman rotating reference frames. In the wingman rotating frame, the relative dynamics between leader and wingman are derived. Let \( x \), \( y \), and \( z \) be the relative coordinates of the leader with respect to the wingman, with \( z = h_L – h_W \). The kinematic relationships are:
$$ \dot{x} = V_L \cos(\psi_L – \psi_W) – V_W + y \dot{\psi}_W $$
$$ \dot{y} = V_L \sin(\psi_L – \psi_W) – x \dot{\psi}_W $$
$$ \dot{z} = \dot{h}_L – \dot{h}_W $$
By combining these with the autopilot models, a three-dimensional state-space model for drone formation is established. The state vector is defined as \( \mathbf{X} = [x, V_W, y, \psi_W, \dot{\psi}_W, z, \xi]^T \), where \( \xi = \dot{z} \). The control inputs are \( V_{Wc} \), \( \psi_{Wc} \), and \( h_{Wc} \) for the wingman, while leader variables \( V_L \), \( \psi_L \), and \( h_{Lc} \) are treated as disturbances. The state-space equation is:
$$ \dot{\mathbf{X}} = A \mathbf{X} + B \mathbf{u} + \Gamma \mathbf{d} $$
with matrices defined as:
$$ A = \begin{bmatrix}
0 & 1 & 0 & 0 & \sin \alpha & 0 & 0 \\
-\frac{1}{\tau_V} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & \cos \alpha & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & -\frac{1}{\tau_{\psi a} \tau_{\psi b}} & -\left( \frac{1}{\tau_{\psi a}} + \frac{1}{\tau_{\psi b}} \right) & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & -\frac{1}{\tau_{ha} \tau_{hb}} & -\left( \frac{1}{\tau_{ha}} + \frac{1}{\tau_{hb}} \right)
\end{bmatrix} $$
$$ B = \begin{bmatrix}
0 & 0 & 0 \\
\frac{1}{\tau_V} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & \frac{1}{\tau_{\psi a} \tau_{\psi b}} & 0 \\
0 & 0 & 0 \\
0 & 0 & \frac{1}{\tau_{ha} \tau_{hb}}
\end{bmatrix}, \quad \Gamma = \begin{bmatrix}
0 & 0 & 0 \\
\frac{1}{\tau_V} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -\frac{1}{\tau_{ha} \tau_{hb}}
\end{bmatrix} $$
where \( \alpha \) is the formation angle, calculated from desired spacing \( x_0 \) and \( y_0 \) as \( \alpha = \arcsin(y_0 / \sqrt{x_0^2 + y_0^2}) \). This model serves as the basis for controller design.
The sliding mode variable structure control is introduced due to its robustness against disturbances and parameter variations. We decompose the drone formation model into three channels: lateral (Y-direction), vertical (Z-direction), and longitudinal (X-direction). This decoupling simplifies controller design and reduces tuning difficulty. For each channel, a sliding mode controller is designed separately, leveraging different reaching laws to ensure stability and performance.
In the Y-channel, the objective is to maintain lateral spacing \( y \) and synchronize heading with the leader. The decoupled state equation is:
$$ \begin{bmatrix} \dot{y} \\ \ddot{y} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} y \\ \dot{y} \end{bmatrix} + \begin{bmatrix} 0 \\ \cos \alpha \end{bmatrix} \dot{\psi}_W + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \psi_{Wc} $$
After deriving the relationship between \( y \) and control input \( \psi_{Wc} \), we define the error \( e = y – y_d \), where \( y_d \) is the desired spacing. Using feedback linearization, the control law is designed as:
$$ \psi_{Wc} = \tau_{\psi a} \tau_{\psi b} \left( f(x) – v \right) / \cos \alpha $$
with \( v = \ddot{y}_d + k_1 \dot{e} + k_2 e \), where \( k_1 \) and \( k_2 \) are positive constants. This ensures the error dynamics \( \ddot{e} + k_1 \dot{e} + k_2 e = 0 \), driving errors to zero exponentially.
For the Z-channel, altitude control is critical to maintain vertical spacing. The sliding surface is defined as \( s = c e + \dot{e} \), with \( e = z – z_d \) and \( c > 0 \). Using an equal-rate reaching law \( \dot{s} = -\eta \text{sign}(s) \), the control law for altitude is:
$$ h_{Wc} = \tau_{ha} \tau_{hb} \left( \dot{h}_W + \frac{1}{\tau_{ha} \tau_{hb}} h_W – c \dot{e} – \ddot{h}_{Ld} + \eta \text{sign}(s) \right) $$
This guarantees \( s \dot{s} < 0 \), ensuring sliding mode existence and robustness.
In the X-channel, longitudinal spacing \( x \) is controlled while accounting for coupling with the Y-channel. The state equation is:
$$ \begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & -\frac{1}{\tau_V} \end{bmatrix} \begin{bmatrix} x \\ V_W \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{1}{\tau_V} \end{bmatrix} V_{Wc} + \begin{bmatrix} \sin \alpha \\ 0 \end{bmatrix} \dot{\psi}_L $$
Similarly, by defining error \( e_x = x – x_d \) and using feedback linearization, the control law for velocity is:
$$ V_{Wc} = \tau_V \left( f(x) – v_x \right) $$
with \( v_x = \ddot{x}_d + k_{x1} \dot{e}_x + k_{x2} e_x \), leading to stable error dynamics. This three-channel approach effectively manages the drone formation with reduced complexity.
To validate the controller, simulations were conducted over 30 seconds. Initial conditions: altitude \( h_0 = 3000 \) ft, heading \( \psi_0 = 0^\circ \), velocity \( V_0 = 450 \) ft/s. Desired spacings for formations are set as: left-diamond with \( x_0 = 500 \) ft, \( y_0 = 500 \) ft; right-diamond with \( x_0 = 500 \) ft, \( y_0 = -500 \) ft; and trail formation with \( x_0 = 500 \) ft, \( y_0 = 0 \) ft. Time constants are \( \tau_{\psi a} = 1.838 \), \( \tau_{\psi b} = 1.838 \), \( \tau_V = 10 \), \( \tau_{ha} = 0.615 \), \( \tau_{hb} = 7.692 \). Controller parameters: Y-channel \( k_1 = 10 \), \( k_2 = 10 \); Z-channel \( c = 1 \); X-channel \( k_{x1} = 10 \), \( k_{x2} = 10 \).
For formation keeping, the leader maneuvers at 2 seconds, changing heading from 0° to 30°, holding until 17 seconds, then returning to 0°. Altitude increases by 300 ft. The results demonstrate that the wingman accurately tracks the leader’s motion. Key performance metrics are summarized in Table 1, showing small errors and fast convergence.
| Metric | Value | Error Bound | Convergence Time |
|---|---|---|---|
| Y-spacing (ft) | 500 | ±1 ft | < 8 s |
| X-spacing (ft) | 500 | ±6 ft | < 8 s |
| Velocity (ft/s) | 450 | ±6 ft/s | < 5 s |
| Altitude (ft) | 3000 | ±9 ft | < 3 s |
| Heading (°) | 30 | ±0.5° | < 2 s |
Compared to a traditional PID controller, the sliding mode controller offers superior performance with smaller overshoot, faster response, and better robustness. For formation transformation, the drone formation switches from left-diamond to right-diamond at 5 seconds, then to trail formation at 20 seconds. The real-time spacing changes smoothly, as shown in Table 2, with no collisions ensured by maintaining safe distances.
| Time (s) | Formation | Y-spacing (ft) | X-spacing (ft) |
|---|---|---|---|
| 0-5 | Left-diamond | 500 | 500 |
| 5-20 | Right-diamond | -500 | 500 |
| 20-30 | Trail | 0 | 500 |
The simulation results confirm that the sliding mode controller effectively maintains and transforms drone formations. The Y-channel and X-channel controllers use exponential reaching laws for fast convergence, while the Z-channel employs an equal-rate law for simplicity. This multi-channel design reduces parameter tuning difficulty and enhances system reliability. The drone formation remains stable even during aggressive maneuvers, with spacing errors tightly bounded.
In conclusion, we have developed a sliding mode controller for drone formation keeping and transformation based on a three-dimensional mathematical model. By decoupling the system into lateral, vertical, and longitudinal channels, we simplified the controller design and achieved robust performance. The controller ensures rapid tracking of leader motions, minimal spacing errors, and collision-free formation transformations. Compared to PID control, it offers improved response speed and adaptability, making it suitable for multi-drone applications in dynamic environments. Future work may extend this approach to larger drone formations or integrate it with obstacle avoidance algorithms for enhanced autonomy. This research contributes a novel method to the field of drone formation control, leveraging sliding mode theory for practical implementation.
The effectiveness of this controller underscores the potential of variable structure systems in aerospace applications. As drone formations become increasingly prevalent for military and civilian purposes, robust control strategies like this will be essential. We encourage further exploration of adaptive sliding mode techniques to handle uncertainties and nonlinearities in real-world drone operations. Overall, this study demonstrates a viable solution for maintaining and transforming drone formations with high precision and reliability.