In recent years, quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their versatility in applications such as surveillance, transportation, and disaster response. However, ensuring safe and stable tracking control in complex environments with external disturbances and spatial constraints remains a critical challenge. This paper addresses the adaptive safe tracking control problem for a class of quadrotor systems, leveraging a secure boundary protection method (SBPM) and prescribed performance finite-time control. The proposed approach enables quadrotors to adaptively generate safe trajectories when spatial boundaries conflict with desired paths, while maintaining robust performance against disturbances via H∞ control. We present a comprehensive controller design, simulation results, and analysis to validate the effectiveness of our method.
Introduction
Quadrotor UAVs are characterized by their agility, vertical take-off and landing capabilities, and simple structure, making them ideal for various missions. However, in dynamic environments, these systems often encounter spatial boundary changes and external disturbances that compromise tracking safety. Traditional control methods, such as control barrier functions (CBFs), may suffer from poor tracking performance or computational complexity. To overcome these limitations, we propose an adaptive H∞ safe tracking control strategy that integrates SBPM with prescribed performance constraints. This approach ensures that the quadrotor output remains within safe boundaries while achieving finite-time convergence and disturbance rejection. Our contributions include a novel controller design for multi-output quadrotor systems and extensive simulations under boundary突变 scenarios.
System Description
Consider the following mathematical model of a quadrotor UAV:
$$
\begin{align*}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= \frac{u_x}{m} u_1 – b_1 x_2 + d_1 \\
\dot{x}_3 &= x_4 \\
\dot{x}_4 &= \frac{u_y}{m} u_1 – b_2 x_4 + d_2 \\
\dot{x}_5 &= x_6 \\
\dot{x}_6 &= \frac{\cos x_7 \cos x_9}{m} u_1 – g – b_3 x_6 + d_3 \\
\dot{x}_7 &= x_8 \\
\dot{x}_8 &= r_1 u_2 + a_1 x_{10} x_{12} – b_4 x_8 + d_4 \\
\dot{x}_9 &= x_{10} \\
\dot{x}_{10} &= r_2 u_3 + a_2 x_8 x_{12} – b_5 x_{10} + d_5 \\
\dot{x}_{11} &= x_{12} \\
\dot{x}_{12} &= r_3 u_4 + a_3 x_8 x_{10} – b_6 x_{12} + d_6
\end{align*}
$$
where $X = [x_1, x_2, \dots, x_{12}]^T = [x, \dot{x}, y, \dot{y}, z, \dot{z}, \phi, \dot{\phi}, \theta, \dot{\theta}, \psi, \dot{\psi}]^T$ represents the state vector, with $x$, $y$, $z$ denoting position coordinates and $\phi$, $\theta$, $\psi$ representing roll, pitch, and yaw angles, respectively. The control inputs $u_1$, $u_2$, $u_3$, $u_4$ correspond to thrust, roll, pitch, and yaw controls. External disturbances $d_i \in L_2[0,T]$ are bounded, and system parameters include mass $m$, gravity $g$, and inertia terms $J_x$, $J_y$, $J_z$. The coefficients $a_i$, $b_i$, and $r_i$ are derived from physical properties. Assumption 1 states that desired trajectories $x_{id}(t)$ and their derivatives are continuous and bounded. The control objectives are to ensure bounded stability, adaptive safe tracking under boundary conflicts, and H∞ disturbance attenuation.
Secure Boundary Protection Method for Quadrotor
The SBPM framework enables the quadrotor to handle sudden spatial boundary changes, such as obstacles appearing in the flight path. For each axis ($x$, $y$, $z$), safe boundaries are generated dynamically. Consider the $x$-direction as an example. The actual spatial boundaries $\rho_{x1}(t)$ and $\rho_{x2}(t)$ may突变 at times $T_{xA}$ and $T_{xB}$:
$$
\rho_{x1}(t) =
\begin{cases}
\rho_{x11}(t) & t < T_{xA} \\
\rho_{x12}(t) & t \geq T_{xA}
\end{cases}, \quad
\rho_{x2}(t) =
\begin{cases}
\rho_{x21}(t) & t < T_{xB} \\
\rho_{x22}(t) & t \geq T_{xB}
\end{cases}
$$
The desired boundaries $\bar{k}_{xup}(t)$ and $\bar{k}_{xdown}(t)$ are defined using a prescribed time performance function (PTPF) $\phi_x(t)$:
$$
\bar{k}_{xup}(t) = x_d(t) + \phi_x(t), \quad \bar{k}_{xdown}(t) = x_d(t) – \phi_x(t)
$$
where $\phi_x(t) = \left( \frac{T_{xs} – t}{T_{xs}} \right)^{q_x} \phi_{x0} + \phi_{xf}$ for $0 \leq t < T_{xs}$, and $\phi_x(t) = \phi_{xf}$ for $t \geq T_{xs}$, with design parameters $T_{xs} > 0$, $\phi_{x0} > 0$, $q_x \geq 3$, and $\phi_{xf} > 0$. The bidirectional filtering smoothing mechanism (BFSM) applies reverse filtering to create virtual boundaries and forward filtering to obtain smooth safe boundaries $k_{xup}(t)$ and $k_{xdown}(t)$. The self-adjustment law (SAL) ensures virtual boundaries do not violate spatial constraints, adapting to three cases of boundary conflicts. Similar procedures apply to $y$ and $z$ directions, ensuring the quadrotor output remains within safe limits while meeting performance constraints.
Adaptive H∞ Safe Tracking Controller Design
We design the controller using backstepping, divided into position and attitude control subsystems. Define error variables $z_i = x_i – x_{id}$ and $z_{i+1} = x_{i+1} – \alpha_i$ for $i = 1, 3, 5, 7, 9, 11$, where $\alpha_i$ are virtual control laws. First-order filters are introduced to avoid differentiation explosion: $\jmath_i \dot{\bar{\alpha}}_i + \bar{\alpha}_i = \alpha_i$, with filter error $e_i = \bar{\alpha}_i – \alpha_i$.
Position Controller Design
For the $x$-direction subsystem, define a barrier variable $\varepsilon_1 = 0.5 \ln(-k_{xdown} + x_1) – 0.5 \ln(k_{xup} – x_1)$. The Lyapunov function $V_1 = \frac{1}{2} \varepsilon_1^2$ leads to the virtual control law:
$$
\alpha_1 = -\frac{1}{M_1} (\beta_1 + c_1 \varepsilon_1 + \varepsilon_1)
$$
where $M_1 = 0.5 \left( \frac{1}{-k_{xdown} + x_1} + \frac{1}{k_{xup} – x_1} \right)$, $\beta_1 = \frac{\partial \varepsilon_1}{\partial k_{xup}} \dot{k}_{xup} + \frac{\partial \varepsilon_1}{\partial k_{xdown}} \dot{k}_{xdown}$, and $c_1 > 0$ is a design parameter. Proceeding to Step 2, the Lyapunov function $V_2 = V_1 + \frac{1}{2} z_2^2 + \frac{1}{2} e_1^2 + \frac{1}{2} \tilde{\theta}_2^2$ incorporates neural network (NN) estimation for unknown dynamics, with $\tilde{\theta}_2 = \theta_2 – \hat{\theta}_2$ representing the estimation error. The actual control law and adaptation law are:
$$
u_x = \frac{m}{u_1} \left( -\frac{1}{2\lambda_2} z_2 S_2^T(Z_2) S_2(Z_2) \hat{\theta}_2 – \frac{1}{2\gamma^2} z_2 – \frac{(1 + \jmath_1) z_2}{\jmath_1} + \dot{\bar{\alpha}}_1 – \varepsilon_1 M_1 – c_2 z_2 \right)
$$
$$
\dot{\hat{\theta}}_2 = \frac{1}{2\lambda_2} z_2^2 S_2^T(Z_2) S_2(Z_2) – h_2 \hat{\theta}_2
$$
where $Z_2 = [x_1, x_2, k_{xup}, k_{xdown}, \dot{k}_{xup}, \dot{k}_{xdown}]^T$, and $c_2, h_2, \lambda_2 > 0$ are design parameters. Similar designs apply to $y$ and $z$ directions, with control laws $u_y$ and $u_1$ derived accordingly. The desired roll and pitch angles $\phi_d$ and $\theta_d$ are resolved from $u_x$ and $u_y$ using inverse kinematics:
$$
\phi_d = \arcsin(u_x \sin(-\psi) – u_y \cos(-\psi)), \quad \theta_d = \arcsin\left( \frac{u_x \cos(-\psi) + u_y \sin(-\psi)}{\cos \phi_d} \right)
$$
Attitude Controller Design
For the roll subsystem, the virtual control law is $\alpha_7 = -c_7 z_7 – z_7 + \dot{x}_{7d}$, and the actual control law $u_2$ is:
$$
u_2 = \frac{1}{r_1} \left( -\frac{1}{2\lambda_8} z_8 S_8^T(Z_8) S_8(Z_8) \hat{\theta}_8 – \frac{1}{2\gamma^2} z_8 – \frac{(1 + \jmath_7) z_8}{\jmath_7} + \dot{\bar{\alpha}}_7 – z_7 – c_8 z_8 \right)
$$
with adaptation law $\dot{\hat{\theta}}_8 = \frac{1}{2\lambda_8} z_8^2 S_8^T(Z_8) S_8(Z_8) – h_8 \hat{\theta}_8$. Similar steps yield controllers for pitch and yaw subsystems. The overall Lyapunov function $V = \sum V_i$ ensures bounded stability and H∞ performance, satisfying $\int_0^t \| z(s) \|^2 ds < \gamma^2 \int_0^t \| d(s) \|^2 ds + \bar{V}(0)$.
Simulation Studies
We validate the proposed controller using a quadrotor model with parameters: $m = 2.5\, \text{kg}$, $g = 9.8\, \text{m/s}^2$, $l = 0.325\, \text{m}$, $J_x = J_y = 0.082\, \text{kg·m}^2$, $J_z = 0.149\, \text{kg·m}^2$, and damping coefficients $G_x = G_y = G_z = 0.6\, \text{kg/s}$. External disturbances $d_i$ are injected at $t = 10\, \text{s}$. Initial states are $X(0) = [0.55, 0.4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]^T$, and adaptation gains are initialized to zero. PTPF parameters are set as $T_{xs} = T_{ys} = T_{zs} = 2.5\, \text{s}$, $q_x = q_y = q_z = 3$, $\phi_{x0} = \phi_{y0} = 0.5$, $\phi_{z0} = 2.2$, $\phi_{xf} = \phi_{yf} = \phi_{zf} = 0.02$. Control parameters include $c_i > 0$, $h_i = 1$, $\lambda_i = 0.1$, $\gamma = 0.25$, and $\vartheta = 0.18$. Reference trajectories are $x_{1d} = \cos(t)$, $x_{3d} = \sin(t)$, $x_{5d} = 2$, and $x_{11d} = 0$.
Two cases are simulated: Case 1 without boundary mutations and Case 2 with mutations in $x$ and $y$ directions at $t = 6.5\, \text{s}$. In Case 1, the quadrotor tracks desired trajectories within $2.5\, \text{s}$, achieving prescribed performance. In Case 2, SBPM generates safe boundaries to avoid collisions, as shown in the tracking plots. Control inputs $u_1$ to $u_4$ exhibit adjustments under disturbances, confirming H∞ robustness. The table below summarizes key parameters used in simulations.
| Parameter | Value | Description |
|---|---|---|
| $m$ | 2.5 kg | Mass of quadrotor |
| $g$ | 9.8 m/s² | Gravity acceleration |
| 0.082 kg·m² | Roll and pitch inertia | |
| $J_z$ | 0.149 kg·m² | Yaw inertia |
| $T_{xs}, T_{ys}, T_{zs}$ | 2.5 s | Prescribed time constants |
| $c_1$ to $c_{12}$ | 1 to 5 | Control gains |
| $\gamma$ | 0.25 | H∞ performance level |
The results demonstrate that the quadrotor maintains safe tracking under boundary mutations and disturbances, with all signals remaining bounded. The controllers ensure rapid adaptation and convergence, highlighting the efficacy of the proposed method for real-world quadrotor applications.
Conclusion
This paper presents an adaptive H∞ safe tracking control strategy for quadrotor UAVs, integrating secure boundary protection and prescribed performance constraints. The controller design addresses multi-output safety in spatial flight, enabling adaptive trajectory generation when boundaries conflict. Through Lyapunov analysis and simulations, we prove bounded stability, H∞ disturbance attenuation, and adherence to safety boundaries. Future work will extend this approach to multi-quadrotor systems and real-time implementations. The proposed method offers a robust solution for enhancing quadrotor autonomy in complex environments.