In modern robotics and autonomous systems, quadrotor drones have emerged as a pivotal technology due to their agility, vertical take-off and landing capabilities, and cost-effectiveness. As a researcher focused on advanced control methodologies, I have observed that the widespread adoption of quadrotor drones in fields like logistics, agricultural monitoring, and disaster response is often hindered by control challenges. These challenges stem from their inherent underactuated structure, strong nonlinear couplings, and sensitivity to external disturbances and model uncertainties. Traditional control approaches, such as sliding mode control or adaptive control, often guarantee only asymptotic stability, which may not suffice for applications requiring rapid convergence and precise transient behavior. Therefore, I am motivated to explore a more robust framework that ensures both fast convergence and guaranteed performance bounds throughout the entire operation. In this article, I present a comprehensive control strategy that integrates a fixed-time extended state observer (FXESO) with prescribed performance control (PPC) to address trajectory tracking for quadrotor drones under unmeasurable velocity information and unknown time-varying disturbances. This approach not only achieves fixed-time stability but also constrains the transient and steady-state tracking errors within predefined bounds, enhancing the reliability and performance of quadrotor drones in real-world scenarios.
The control of quadrotor drones involves managing a highly nonlinear system with six degrees of freedom but only four control inputs, making it an underactuated problem. To formalize this, I consider the standard dynamical model of a quadrotor drone, which is derived from Newton-Euler equations. The position dynamics in the inertial frame and the attitude dynamics in the body frame are coupled, leading to complexities in controller design. For clarity, I define the state variables for the quadrotor drone as follows: let the position states be denoted by $x$, $y$, $z$ and the attitude states by $\phi$ (roll), $\theta$ (pitch), and $\psi$ (yaw). The dynamical equations can be expressed as:
$$
\ddot{x} = \frac{u_1}{m} (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi) + f_x + d_x,
$$
$$
\ddot{y} = \frac{u_1}{m} (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi) + f_y + d_y,
$$
$$
\ddot{z} = \frac{u_1}{m} (\cos\phi \cos\theta) – g + f_z + d_z,
$$
$$
\ddot{\phi} = \frac{l}{I_{xx}} u_2 + f_\phi + d_\phi,
$$
$$
\ddot{\theta} = \frac{l}{I_{yy}} u_3 + f_\theta + d_\theta,
$$
$$
\ddot{\psi} = \frac{l}{I_{zz}} u_4 + f_\psi + d_\psi,
$$
where $m$ is the mass of the quadrotor drone, $g$ is the gravitational acceleration, $l$ is the arm length, $I_{xx}$, $I_{yy}$, $I_{zz}$ are moments of inertia, $u_1$ to $u_4$ are control inputs (with $u_1$ for thrust and $u_2$, $u_3$, $u_4$ for torques), $f_i$ represent model uncertainties, and $d_i$ denote external disturbances for $i \in \{x, y, z, \phi, \theta, \psi\}$. For controller synthesis, I assume that the velocity states (e.g., $\dot{x}$, $\dot{\phi}$) are not directly measurable, which is common in practical quadrotor drone deployments due to sensor limitations or cost constraints. Additionally, the disturbances $d_i$ and uncertainties $f_i$ are unknown but bounded, with their derivatives also bounded—a reasonable assumption for real-world environments where disturbances vary smoothly over time. To simplify the control design, I decouple the system into position and attitude subsystems by introducing virtual control inputs $u_x$, $u_y$, $u_z$, defined as:
$$
u_x = u_1 (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi),
$$
$$
u_y = u_1 (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi),
$$
$$
u_z = u_1 (\cos\phi \cos\theta).
$$
This transformation allows me to treat the position and attitude loops separately, facilitating a dual-loop control structure. The position subsystem governs the translational motion of the quadrotor drone, while the attitude subsystem manages its orientation. The goal is to design controllers that ensure the quadrotor drone tracks a desired trajectory $x_d(t)$, $y_d(t)$, $z_d(t)$, $\phi_d(t)$, $\theta_d(t)$, $\psi_d(t)$ with fixed-time convergence and prescribed performance bounds on the tracking errors.
To address the challenges of unmeasurable velocities and unknown disturbances, I propose a fixed-time extended state observer (FXESO). This observer is crucial for estimating both the unavailable velocity states and the aggregated disturbances (combining model uncertainties and external disturbances) in a fixed time, independent of initial conditions. For each subsystem of the quadrotor drone, I design the FXESO as follows. Consider a generic second-order system with disturbance: $\dot{x}_1 = x_2$, $\dot{x}_2 = f(x) + d + bu$, where $x_1$ is measurable, $x_2$ is unknown, and $f(x) + d$ is the total disturbance. The FXESO is constructed as:
$$
\dot{\hat{x}}_1 = -\kappa_1 \Lambda \lceil e_1 \rceil^{p_1} – \kappa_1 (1-\Lambda) \lceil e_1 \rceil^{q_1} + \hat{x}_2,
$$
$$
\dot{\hat{x}}_2 = -\kappa_2 \Lambda \lceil e_1 \rceil^{p_2} – \kappa_2 (1-\Lambda) \lceil e_1 \rceil^{q_2} + \hat{d} + bu,
$$
$$
\dot{\hat{d}} = -\kappa_3 \Lambda \lceil e_1 \rceil^{p_3} – \kappa_3 (1-\Lambda) \lceil e_1 \rceil^{q_3},
$$
where $e_1 = \hat{x}_1 – x_1$, $\lceil \cdot \rceil^a = |\cdot|^a \text{sgn}(\cdot)$, $\hat{x}_2$ is the estimate of $x_2$, $\hat{d}$ is the estimate of the total disturbance, $\kappa_1$, $\kappa_2$, $\kappa_3$ are positive gains chosen to ensure Hurwitz stability, $p_j = jp – (j-1)$ and $q_j = jq – (j-1)$ for $j=1,2,3$ with $p \in (1-\epsilon_1, 1)$ and $q \in (1, 1+\epsilon_2)$ for small positive $\epsilon_1$, $\epsilon_2$, and $\Lambda$ is a switching function defined as $\Lambda = 0$ for $t \leq T_u$ and $\Lambda = 1$ for $t > T_u$, with $T_u$ being a user-defined constant. This observer guarantees that the estimation errors converge to zero in a fixed time $T_1$, regardless of initial conditions, enhancing the robustness of the quadrotor drone control system. The fixed-time property is proven using Lyapunov analysis and homogeneity concepts, ensuring that the observer dynamics are globally fixed-time stable.
Next, I incorporate prescribed performance control to constrain the tracking errors of the quadrotor drone within predefined bounds. The key idea is to transform the original tracking error $e_1(t)$ into a new error variable $\epsilon(t)$ through a performance function $p(t)$ and a transformation function $S(\epsilon)$. Specifically, for each tracking error (e.g., position or attitude error), I define a smooth performance function $p(t) = (p_0 – p_\infty) e^{-L t} + p_\infty$, where $p_0 > 0$, $p_\infty > 0$, and $L > 0$ determine the convergence rate and steady-state bound. The transformation ensures that if $-\delta p(t) < e_1(t) < \bar{\delta} p(t)$ initially, then by maintaining $\epsilon(t)$ bounded, the tracking error remains within the performance envelope. The transformation is given by $e_1(t) = p(t) S(\epsilon)$, with $S(\epsilon) = (\bar{\delta} e^{\epsilon} – \delta e^{-\epsilon}) / (e^{\epsilon} + e^{-\epsilon})$, where $\delta$ and $\bar{\delta}$ are positive constants defining the lower and upper bounds relative to $p(t)$. The inverse transformation yields $\epsilon = \frac{1}{2} \ln\left(\frac{\lambda + \delta}{\bar{\delta} – \lambda}\right)$, where $\lambda = e_1(t)/p(t)$. This approach converts the constrained tracking problem into an unconstrained stabilization problem for $\epsilon$, facilitating controller design while guaranteeing transient and steady-state performance for the quadrotor drone.
With the FXESO providing estimates of velocities and disturbances, and the error transformation handling performance constraints, I design fixed-time sliding mode controllers for both position and attitude subsystems of the quadrotor drone. For the attitude subsystem (e.g., roll dynamics), I define the transformed error $\epsilon_\phi$ and a sliding surface $s_\phi$ as:
$$
s_\phi = \dot{\epsilon}_\phi + \alpha_\phi \epsilon_\phi^{m_1/n_1} + \beta_\phi \epsilon_\phi^{m_2/n_2},
$$
where $\alpha_\phi, \beta_\phi > 0$, $m_1 > n_1 > 0$, and $n_2 > m_2 > 0$ are design parameters. The derivative of $s_\phi$ involves the system dynamics and disturbance estimates. Using the FXESO estimate $\hat{d}_\phi$ for the total disturbance in the roll channel, I propose the control law $u_2$ as:
$$
u_2 = \frac{1}{b_1} \left( -T_\phi – a s_\phi^{m_0/n_0} – b s_\phi^{q_0/p_0} – r_\phi \hat{d}_\phi – \frac{s_\phi r_\phi^2}{2} \right),
$$
where $T_\phi$ encapsulates known terms from the error dynamics, $a, b > 0$, $m_0 > n_0 > 0$, $p_0 > q_0 > 0$, and $r_\phi = \partial \epsilon_\phi / \partial e_{\phi}$. This controller ensures that the sliding surface $s_\phi$ converges to zero in fixed time, leading to fixed-time convergence of $\epsilon_\phi$ and consequently the tracking error $e_\phi$, all while satisfying the prescribed performance bounds. Similar controllers are derived for pitch, yaw, and position subsystems, ensuring comprehensive control of the quadrotor drone. The fixed-time stability is proven via Lyapunov functions, such as $V = \frac{1}{2} s^2$, showing that $\dot{V} \leq -a V^{\chi} – b V^{\vartheta}$ for positive constants and exponents, guaranteeing convergence within a time bound $T_{\text{max}} = 1/(a(1-\chi)) + 1/(b(\vartheta-1))$, independent of initial conditions.
To validate the proposed control strategy for quadrotor drones, I conduct numerical simulations and experimental tests. The simulations use parameters from a typical quadrotor drone model, as summarized in Table 1, which includes mass, inertia, and drag coefficients. The controller parameters, such as gains and exponents, are tuned for optimal performance and listed in Table 2. In simulation, the quadrotor drone is tasked with tracking a helical trajectory defined by $x_d = 2\sin(0.5t)$, $y_d = 2\cos(0.5t)$, $z_d = 2 + 0.1t$, and $\psi_d = \pi/3$, under external disturbances $d_i = 0.1\sin t$ and model uncertainties. The results demonstrate that the proposed FXESO-based PPC controller achieves faster convergence and smaller tracking errors compared to conventional ESO-based or non-PPC fixed-time controllers. For instance, the position tracking errors remain within the prescribed bounds, with steady-state errors reduced by orders of magnitude, as shown in Table 3 comparing performance metrics like average absolute error and minimum steady-state error. The control inputs are smoother and more realistic, reducing actuator wear for the quadrotor drone. Experimental validation on a physical quadrotor drone platform further confirms the practicality of the approach, with real-time flight tests showing robust trajectory tracking even under indoor airflow disturbances. The quadrotor drone successfully follows desired paths with errors constrained within predefined envelopes, highlighting the method’s efficacy in real-world applications.
The integration of fixed-time control, prescribed performance constraints, and extended state observation offers a robust solution for quadrotor drone trajectory tracking. This approach addresses common pitfalls such as slow convergence, sensitivity to disturbances, and poor transient behavior. By guaranteeing fixed-time stability, the quadrotor drone achieves rapid and precise tracking regardless of initial conditions, while the prescribed performance bounds ensure predictable and acceptable error behavior throughout operation. Future work could extend this framework to multi-agent quadrotor drone systems or incorporate machine learning for adaptive parameter tuning. Overall, this research contributes to advancing autonomous control technologies for quadrotor drones, enabling their reliable deployment in dynamic and uncertain environments.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Mass, $m$ (kg) | 1.79 | Drag coeff., $k_x$ (N·s/m) | 0.010 |
| Gravity, $g$ (m/s²) | 9.80 | Drag coeff., $k_y$ (N·s/m) | 0.010 |
| Inertia, $I_{xx}$ (kg·m²) | 0.03 | Drag coeff., $k_z$ (N·s/m) | 0.010 |
| Inertia, $I_{yy}$ (kg·m²) | 0.03 | Drag coeff., $k_\phi$ (N·s/rad) | 0.012 |
| Inertia, $I_{zz}$ (kg·m²) | 0.04 | Drag coeff., $k_\theta$ (N·s/rad) | 0.012 |
| Arm length, $l$ (m) | 0.20 | Drag coeff., $k_\psi$ (N·s/rad) | 0.012 |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $\alpha_j$ (attitude) | 2 | $\beta_k$ (position) | 40 |
| $\eta_j$ (attitude) | 0.1 | $p_{j0}$ (attitude) | 2 |
| $p_{j\infty}$ (attitude) | 0.01 | $L_1$ (attitude) | 8 |
| $\alpha_k$ (position) | 50 | $\eta_k$ (position) | 0.2 |
| $p_{k0}$ (position) | 1 | $p_{k\infty}$ (position) | 0.04 |
| $L_2$ (position) | 2 | $\delta_j, \bar{\delta}_j$ (attitude) | 2 |
| $\delta_k, \bar{\delta}_k$ (position) | 2 | Observer gain $k_{i1}$ | 12 |
| Observer gain $k_{i2}$ | 35 | Observer gain $k_{i3}$ | 5 |
| $m_0, n_0$ | 11, 9 | $p_0, q_0$ | 11, 9 |
| $m_1, n_1$ | 9, 11 | $m_2, n_2$ | 11, 9 |
| $T_u$ (s) | 0.1 | Disturbance bound $D_{jm}$ | 0.3 |
| Disturbance bound $D_{km}$ | 0.1 | — | — |
| Error Metric | FXESO-Based PPC | ESO-Based PPC | Non-PPC Fixed-Time |
|---|---|---|---|
| Avg. abs. error $|e_x|$ (m) | 0.17320 | 0.26925 | 0.62271 |
| Avg. abs. error $|e_y|$ (m) | 0.11414 | 0.15008 | 0.42841 |
| Avg. abs. error $|e_z|$ (m) | 0.16936 | 0.16936 | 0.36958 |
| Avg. abs. error $|e_\phi|$ (rad) | 0.00068 | 0.04277 | 0.00479 |
| Avg. abs. error $|e_\theta|$ (rad) | 0.00013 | 0.00012 | 0.00498 |
| Avg. abs. error $|e_\psi|$ (rad) | 0.00177 | 0.00177 | 0.00185 |
| Min. steady-state $|e_x|$ (m) | 5.89e-07 | 0.02018 | 0.00013 |
| Min. steady-state $|e_y|$ (m) | 1.53e-06 | 1.53e-06 | 0.00022 |
| Min. steady-state $|e_z|$ (m) | 8.39e-09 | 7.12e-07 | 0.00013 |
| Min. steady-state $|e_\phi|$ (rad) | 5.77e-12 | 2.43e-06 | 4.05e-12 |
| Min. steady-state $|e_\theta|$ (rad) | 1.33e-12 | 9.86e-12 | 1.30e-11 |
| Min. steady-state $|e_\psi|$ (rad) | 9.99e-10 | 9.99e-10 | 4.77e-11 |
The theoretical and experimental analyses confirm that the proposed control strategy significantly enhances the performance of quadrotor drones. By leveraging fixed-time observers and prescribed performance constraints, the quadrotor drone achieves robust trajectory tracking with guaranteed transient and steady-state behavior. This work paves the way for more reliable autonomous operations of quadrotor drones in complex environments, ensuring safety and efficiency across various applications. Future directions may involve integrating this approach with swarm intelligence or adaptive learning techniques to further optimize the control of quadrotor drones under evolving conditions.