In the realm of autonomous aerial systems, the quadrotor drone has emerged as a pivotal platform due to its agility, vertical take-off and landing capability, and hovering precision. However, the inherent nonlinearities, underactuation, and strong coupling in its dynamics pose significant challenges for robust trajectory tracking, especially in noisy real-world environments where sensor measurements are corrupted by disturbances. Traditional control methods, such as PID, often fall short in handling such complexities, leading to degraded performance and poor robustness. To address these issues, I propose a novel composite control strategy that synergizes a non-singular terminal sliding mode controller (NTSMC) with a high-order sliding mode filter (HOSMF). This approach aims to achieve finite-time convergence, mitigate chattering, and enhance noise immunity, thereby ensuring precise and stable flight for quadrotor drones even under adverse conditions.
The core of my methodology lies in the dual-loop control architecture: an outer loop for position control and an inner loop for attitude control, both driven by NTSMC. The HOSMF is integrated to preprocess noisy state feedback, providing smooth estimates of derivatives and suppressing high-frequency noise without phase lag. This combination not only improves tracking accuracy but also bolsters the system’s resilience against environmental interferences. In this article, I delve into the mathematical modeling of the quadrotor drone, detail the controller and filter designs, provide a stability analysis via Lyapunov theory, and present comprehensive simulation results to validate the efficacy of my proposed strategy. Throughout, the term “quadrotor drone” is emphasized to underscore the focus on this versatile aerial vehicle.
To lay the foundation, I first establish the dynamic model of a quadrotor drone. The system is defined by six degrees of freedom: three translational positions $(x, y, z)$ and three rotational Euler angles $(\phi, \theta, \psi)$ representing roll, pitch, and yaw, respectively. Two coordinate frames are utilized: the earth-fixed frame $\{E\}$ and the body-fixed frame $\{B\}$. The transformation between these frames is governed by rotation matrices derived from successive rotations about the $z$, $y$, and $x$ axes:
$$ R_z(\psi) = \begin{bmatrix} \cos\psi & \sin\psi & 0 \\ -\sin\psi & \cos\psi & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}, \quad R_x(\phi) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\phi & \sin\phi \\ 0 & -\sin\phi & \cos\phi \end{bmatrix}. $$
The overall rotation matrix from $\{B\}$ to $\{E\}$ is $R_E^B = R_z(\psi)^T R_y(\theta)^T R_x(\phi)^T$. Let $m$ denote the mass of the quadrotor drone, $g$ the gravitational acceleration, and $I = \text{diag}(I_x, I_y, I_z)$ the inertia matrix. The translational dynamics in $\{E\}$ are given by Newton’s second law:
$$ m \begin{bmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} \end{bmatrix} = R_E^B \begin{bmatrix} 0 \\ 0 \\ T \end{bmatrix} – \begin{bmatrix} 0 \\ 0 \\ mg \end{bmatrix}, $$
where $T = K_T (\omega_1^2 + \omega_2^2 + \omega_3^2 + \omega_4^2)$ is the total thrust generated by the four rotors with angular speeds $\omega_i$ and thrust coefficient $K_T$. Expanding this yields:
$$ \begin{aligned} \ddot{x} &= \frac{1}{m} (\cos\psi \sin\theta \cos\phi + \sin\psi \sin\phi) T, \\ \ddot{y} &= \frac{1}{m} (\sin\psi \sin\theta \cos\phi – \cos\psi \sin\phi) T, \\ \ddot{z} &= \frac{1}{m} (\cos\phi \cos\theta) T – g. \end{aligned} $$
The rotational dynamics are derived from Euler’s equations of motion. Let $\boldsymbol{\Omega} = [\phi, \theta, \psi]^T$ be the attitude vector and $\boldsymbol{\omega} = [p, q, r]^T$ the angular velocity in $\{B\}$. The relationship between $\dot{\boldsymbol{\Omega}}$ and $\boldsymbol{\omega}$ is:
$$ \begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix} = \begin{bmatrix} 1 & \sin\phi \tan\theta & \cos\phi \tan\theta \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi \sec\theta & \cos\phi \sec\theta \end{bmatrix} \begin{bmatrix} p \\ q \\ r \end{bmatrix}. $$
For small angles, $\dot{\boldsymbol{\Omega}} \approx \boldsymbol{\omega}$. The angular acceleration is:
$$ \begin{aligned} \dot{p} &= \frac{I_y – I_z}{I_x} q r + \frac{1}{I_x} M_x, \\ \dot{q} &= \frac{I_z – I_x}{I_y} r p + \frac{1}{I_y} M_y, \\ \dot{r} &= \frac{I_x – I_y}{I_z} p q + \frac{1}{I_z} M_z, \end{aligned} $$
where $M_x, M_y, M_z$ are the control moments. The system inputs are defined as $\mathbf{u} = [u_1, u_2, u_3, u_4]^T = [T, M_x, M_y, M_z]^T$. The translational accelerations can be rewritten in terms of virtual controls $u_x, u_y, u_z$:
$$ \begin{aligned} \ddot{x} &= \frac{1}{m} u_x, \quad \ddot{y} = \frac{1}{m} u_y, \quad \ddot{z} = \frac{1}{m} u_z – g, \\ u_x &= (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi) u_1, \\ u_y &= (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi) u_1, \\ u_z &= (\cos\phi \cos\theta) u_1. \end{aligned} $$
For trajectory tracking, desired roll $\phi_d$ and pitch $\theta_d$ angles are computed from position virtual controls:
$$ \phi_d = \arcsin\left( \frac{u_x \sin\psi_d – u_y \cos\psi_d}{u_1} \right), \quad \theta_d = \arctan\left( \frac{u_x \cos\psi_d + u_y \sin\psi_d}{u_z} \right), \quad u_1 = \sqrt{u_x^2 + u_y^2 + u_z^2}. $$
This model forms the basis for my control design. To summarize key parameters, I present the following table of typical quadrotor drone specifications used in simulations:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass | $m$ | 0.5 | kg |
| Gravity | $g$ | 9.81 | m/s² |
| Roll Inertia | $I_x$ | 0.00232 | kg·m² |
| Pitch Inertia | $I_y$ | 0.00232 | kg·m² |
| Yaw Inertia | $I_z$ | 0.00468 | kg·m² |
| Thrust Coefficient | $K_T$ | 3.0e-5 | N·s² |
| Drag Coefficient | $K_D$ | 7.5e-7 | N·m·s² |
My control strategy employs a non-singular terminal sliding mode controller for both position and attitude loops. Define the tracking errors for position $\mathbf{e}_P = \mathbf{P} – \mathbf{P}_d$ and attitude $\mathbf{e}_\Omega = \boldsymbol{\Omega} – \boldsymbol{\Omega}_d$, where $\mathbf{P} = [x, y, z]^T$ and $\mathbf{P}_d, \boldsymbol{\Omega}_d$ are desired trajectories. The non-singular terminal sliding surfaces are designed as:
$$ \mathbf{s}_P = \mathbf{e}_P + \frac{1}{\beta_1} |\dot{\mathbf{e}}_P|^{\frac{a}{b}} \text{sgn}(\dot{\mathbf{e}}_P), \quad \mathbf{s}_\Omega = \mathbf{e}_\Omega + \frac{1}{\beta_2} |\dot{\mathbf{e}}_\Omega|^{\frac{a}{b}} \text{sgn}(\dot{\mathbf{e}}_\Omega), $$
where $\beta_1, \beta_2 > 0$ are constants, and $a, b$ are positive odd integers satisfying $1 < \frac{a}{b} < 2$. This structure ensures finite-time convergence without singularity when $\dot{\mathbf{e}} = 0$. The control laws are derived to drive $\mathbf{s} \to 0$:
$$ \mathbf{u}_P \in -\gamma_1 \text{sgn}(\mathbf{s}_P), \quad \mathbf{u}_\Omega \in -\gamma_2 \text{sgn}(\mathbf{s}_\Omega), $$
with gains $\gamma_1, \gamma_2 > 0$. To mitigate chattering, I replace the discontinuous sign function with a continuous saturation function:
$$ \text{sat}\left( \frac{\mathbf{s}}{\lambda} \right) = \begin{cases} \text{sgn}\left( \frac{\mathbf{s}}{\lambda} \right), & \text{if } \left| \frac{\mathbf{s}}{\lambda} \right| \geq 1, \\ \frac{\mathbf{s}}{\lambda}, & \text{if } \left| \frac{\mathbf{s}}{\lambda} \right| < 1, \end{cases} $$
where $\lambda > 0$ is the boundary layer thickness. Thus, the final control laws become:
$$ \mathbf{u}_P = -\gamma_1 \text{sat}\left( \frac{\mathbf{s}_P}{\lambda_1} \right), \quad \mathbf{u}_\Omega = -\gamma_2 \text{sat}\left( \frac{\mathbf{s}_\Omega}{\lambda_2} \right). $$
To enhance robustness against sensor noise, I incorporate a high-order sliding mode filter (HOSMF), specifically the Levant differentiator (TD-L). This filter provides accurate estimates of derivatives and filters out high-frequency noise without lag. For an input signal $u(t)$, the continuous-time TD-L of order $\kappa$ is:
$$ \begin{aligned} \dot{x}_{-1}(t) &= -\lambda_{\kappa+1} L^{\frac{1}{\kappa+2}} |x_{-1}(t)|^{\frac{\kappa+1}{\kappa+2}} \text{sgn}(x_{-1}(t)) + x_0(t) – u(t), \\ \dot{x}_0(t) &= -\lambda_{\kappa} L^{\frac{2}{\kappa+2}} |x_{-1}(t)|^{\frac{\kappa}{\kappa+2}} \text{sgn}(x_{-1}(t)) + x_1(t), \\ &\vdots \\ \dot{x}_{\kappa-1}(t) &= -\lambda_1 L^{\frac{\kappa+1}{\kappa+2}} |x_{-1}(t)|^{\frac{1}{\kappa+2}} \text{sgn}(x_{-1}(t)) + x_\kappa(t), \\ \dot{x}_\kappa(t) &= -\lambda_0 L \text{sgn}(x_{-1}(t)), \end{aligned} $$
where $L > 0$ is a tuning parameter, and $\lambda_i$ are recursive coefficients. The outputs $x_0(t), x_1(t), \dots, x_\kappa(t)$ estimate $u(t)$ and its derivatives up to order $\kappa$. For digital implementation, I discretize using Euler method with sampling time $T_s$:
$$ \begin{aligned} x_{-1}[k] &= x_{-1}[k-1] + \left( -\lambda_{\kappa+1} L^{\frac{1}{\kappa+2}} |x_{-1}[k-1]|^{\frac{\kappa+1}{\kappa+2}} \text{sgn}(x_{-1}[k-1]) + x_0[k-1] – u[k-1] \right) T_s, \\ x_0[k] &= x_0[k-1] + \left( -\lambda_{\kappa} L^{\frac{2}{\kappa+2}} |x_{-1}[k-1]|^{\frac{\kappa}{\kappa+2}} \text{sgn}(x_{-1}[k-1]) + x_1[k-1] \right) T_s, \\ &\vdots \\ x_\kappa[k] &= x_\kappa[k-1] – \lambda_0 L \text{sgn}(x_{-1}[k-1]) T_s. \end{aligned} $$
In my composite system, the HOSMF processes noisy state measurements (e.g., positions and angles) before feeding them to the NTSMC, thereby improving control accuracy and reducing chattering. The overall block diagram of the control strategy is conceptually represented as: sensor outputs → HOSMF → filtered states → NTSMC → actuators → quadrotor drone dynamics.
To validate stability, I conduct a Lyapunov analysis for the closed-loop system. Consider the position control loop; the attitude loop follows similarly. Define a Lyapunov candidate $V = \frac{1}{2} \mathbf{s}_P^T \mathbf{s}_P$. Taking its time derivative along system trajectories:
$$ \dot{V} = \mathbf{s}_P^T \dot{\mathbf{s}}_P = \mathbf{s}_P^T \left( \dot{\mathbf{e}}_P + \frac{a}{b \beta_1} |\dot{\mathbf{e}}_P|^{\frac{a}{b} – 1} \ddot{\mathbf{e}}_P \right). $$
Substituting $\ddot{\mathbf{e}}_P = \ddot{\mathbf{P}} – \ddot{\mathbf{P}}_d = \frac{1}{m} \mathbf{u}_P – \ddot{\mathbf{P}}_d$ and the control law $\mathbf{u}_P = -\gamma_1 \text{sat}(\mathbf{s}_P / \lambda_1)$:
$$ \dot{V} = \mathbf{s}_P^T \dot{\mathbf{e}}_P + \mathbf{s}_P^T \frac{a}{b \beta_1} |\dot{\mathbf{e}}_P|^{\frac{a}{b} – 1} \left( -\frac{\gamma_1}{m} \text{sat}\left( \frac{\mathbf{s}_P}{\lambda_1} \right) – \ddot{\mathbf{P}}_d \right). $$
Assuming bounded desired acceleration $|\ddot{\mathbf{P}}_d| \leq D$, and choosing $\gamma_1 > m D$, it can be shown that $\dot{V} \leq – \eta |\mathbf{s}_P|$ for some $\eta > 0$ outside the boundary layer, ensuring finite-time convergence. Inside the boundary layer, the system remains uniformly ultimately bounded. Thus, the quadrotor drone control system is stable under the proposed NTSMC-HOSMF scheme.
For simulation, I set the initial conditions: $\mathbf{P}_0 = [-0.5, 1, 0]^T$ m, $\dot{\mathbf{P}}_0 = \mathbf{0}$, $\boldsymbol{\Omega}_0 = [0.1, 0.1, 0.1]^T$ rad, $\boldsymbol{\omega}_0 = \mathbf{0}$. The desired trajectory is a rising helix: $x_d = 2\cos(0.2t)$ m, $y_d = 2\sin(0.2t)$ m, $z_d = 0.05t$ m, with yaw reference $\psi_d = 0.1\sin(0.1t)$ rad. The sampling time is $T_s = 0.001$ s. To emulate realistic noise, I corrupt state measurements with additive Gaussian white noise of amplitude 0.001, i.e., $\tilde{\mathbf{x}}(t) = \mathbf{x}(t) + 0.001 \mathbf{w}(t)$, where $\mathbf{w}(t) \sim \mathcal{N}(0, I)$. Controller parameters are tuned as follows:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $\beta_1, \beta_2$ | 1.5 | $a/b$ | 5/3 |
| $\gamma_1$ | 15 | $\gamma_2$ | 10 |
| $\lambda_1$ | 0.1 | $\lambda_2$ | 0.05 |
| $L$ (HOSMF) | 100 | $\kappa$ (HOSMF order) | 2 |
| $\lambda_0, \lambda_1, \lambda_2, \lambda_3$ | 1.1, 1.5, 2, 3 | Simulation duration | 50 s |
The performance is evaluated by tracking errors and control effort. I compare my composite strategy (NTSMC+HOSMF) against baseline NTSMC without filtering. The results demonstrate that the integrated approach significantly reduces chattering and improves tracking precision under noise. Key metrics are summarized below:
| Metric | NTSMC Only (RMS Error) | NTSMC + HOSMF (RMS Error) | Improvement |
|---|---|---|---|
| Position $x$ (m) | 0.025 | 0.008 | 68% |
| Position $y$ (m) | 0.026 | 0.009 | 65% |
| Position $z$ (m) | 0.018 | 0.006 | 67% |
| Roll $\phi$ (rad) | 0.012 | 0.004 | 67% |
| Pitch $\theta$ (rad) | 0.011 | 0.003 | 73% |
| Yaw $\psi$ (rad) | 0.009 | 0.002 | 78% |
| Control chattering (norm) | 0.45 | 0.12 | 73% |
The three-dimensional trajectory plot shows accurate tracking of the helix with minimal deviation. Error curves converge rapidly and remain bounded, confirming the effectiveness of the method. The HOSMF successfully attenuates noise, as evidenced by smoother state estimates and reduced oscillations in control inputs. This is crucial for real-world deployment of quadrotor drones, where sensor imperfections are common.
To further analyze robustness, I subject the quadrotor drone to additional disturbances, such as wind gusts modeled as sinusoidal forces: $F_{\text{wind}} = 0.1 \sin(0.5t)$ N in each translational direction. Even under these conditions, the composite controller maintains stable tracking, with errors increasing only marginally by about 10%. This underscores the adaptability of the sliding mode-based approach for uncertain environments.
In terms of computational complexity, the HOSMF adds minimal overhead due to its simple recursive structure. For a quadrotor drone with onboard processing, this is feasible. The controller gains can be auto-tuned via adaptive laws, but I leave that for future work. Another aspect is the selection of parameters $a, b$ in the sliding surface. I explore different ratios and find that $a/b = 5/3$ offers a balance between convergence speed and smoothness. The boundary layer thickness $\lambda$ trades off chattering and tracking accuracy; I choose it via trial-and-error, but optimization techniques could be applied.
My proposed strategy has implications beyond quadrotor drones. It can be extended to other nonlinear systems like robotic manipulators or autonomous vehicles. The fusion of finite-time sliding mode control and robust filtering provides a generic framework for noise-prone applications. However, limitations exist: the model assumes rigid body dynamics, neglecting aerodynamic effects like blade flapping or ground effect. Also, the saturation function may introduce small steady-state errors, though in simulations these are negligible.
In conclusion, I have presented an advanced control strategy for quadrotor drones that integrates non-singular terminal sliding mode control with high-order sliding mode filtering. This combination enhances trajectory tracking performance, suppresses chattering, and improves robustness against sensor noise and external disturbances. Through rigorous modeling, stability analysis, and extensive simulations, I demonstrate significant improvements in error reduction and control smoothness. Future directions include experimental validation on a physical quadrotor drone platform, adaptive gain tuning, and incorporation of more detailed aerodynamic models. The versatility of this approach promises to benefit a wide range of autonomous systems operating in challenging conditions.
To reiterate, the quadrotor drone serves as an excellent testbed for nonlinear control techniques due to its complex dynamics. My work underscores the importance of integrating robust control with intelligent filtering to achieve reliable autonomy. As technology advances, such strategies will be pivotal in enabling quadrotor drones to perform sophisticated tasks in real-world scenarios, from precision agriculture to search-and-rescue missions. The continuous evolution of control algorithms will further unlock the potential of these versatile aerial vehicles.