Non-singular Predefined-Time Backstepping Sliding Mode Trajectory Tracking Control for Quadrotor UAVs

In recent years, quadrotor unmanned aerial vehicles (UAVs) have garnered significant attention in industrial and academic communities due to their simple structure, high maneuverability, compact size, low cost, and stealth capabilities. The flight control system is the core of a quadrotor, typically divided into an inner attitude loop and an outer position loop. However, quadrotors often operate in outdoor and complex environments, necessitating robust disturbance rejection capabilities to handle actuator faults and unknown external disturbances. This paper addresses the trajectory tracking control problem for quadrotors under dynamic uncertainties and unknown disturbances by designing a non-singular predefined-time backstepping sliding mode controller (NPBSSM). The proposed controller ensures that the quadrotor achieves high-precision trajectory tracking within a predefined time frame, while maintaining robustness against disturbances.

The quadrotor dynamics are derived using the Newton-Euler formulation, with simplifications assuming a rigid, symmetric structure, constant mass and inertia, and coincidence of the geometric center and center of gravity. The system is subject to bounded disturbances that vary slowly. The dynamic model under disturbances is expressed as:

$$ \begin{aligned}
\dot{x} &= \frac{U_1}{m} (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi) + d_1 \\
\dot{y} &= \frac{U_1}{m} (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi) + d_2 \\
\dot{z} &= \frac{U_1}{m} (\cos\phi \cos\theta) – g + d_3 \\
\ddot{\phi} &= \frac{U_2}{I_x} + \frac{I_y – I_z}{I_x} \dot{\theta} \dot{\psi} + d_4 \\
\ddot{\theta} &= \frac{U_3}{I_y} + \frac{I_z – I_x}{I_y} \dot{\phi} \dot{\psi} + d_5 \\
\ddot{\psi} &= \frac{U_4}{I_z} + \frac{I_x – I_y}{I_z} \dot{\phi} \dot{\theta} + d_6
\end{aligned} $$

where $ m $ is the mass, $ g $ is gravity, $ [x, y, z]^T $ represents position, $ [\phi, \theta, \psi]^T $ denotes attitude angles, $ I_x, I_y, I_z $ are moments of inertia, $ U_1, U_2, U_3, U_4 $ are control inputs, and $ d_i $ ($ i = 1, \dots, 6 $) are lumped disturbances. The control inputs relate to rotor thrusts $ F_1, F_2, F_3, F_4 $ as:

$$ \begin{aligned}
U_1 &= F_1 + F_2 + F_3 + F_4 \\
U_2 &= (F_4 – F_2) L \\
U_3 &= (F_3 – F_1) L \\
U_4 &= (F_1 + F_3 – F_2 – F_4) L_\tau
\end{aligned} $$

where $ L $ is the arm length. Virtual control inputs $ u_x, u_y, u_z $ are introduced to decouple position and attitude control:

$$ \begin{aligned}
u_x &= U_1 (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi) \\
u_y &= U_1 (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi) \\
u_z &= U_1 (\cos\phi \cos\theta)
\end{aligned} $$

The desired attitude angles $ \phi_d, \theta_d $ and control $ U_1 $ are derived from $ u_x, u_y, u_z $ and the desired yaw $ \psi_d $:

$$ \begin{aligned}
\theta_d &= \arctan\left( \frac{u_x \cos\psi_d + u_y \sin\psi_d}{u_z} \right) \\
\phi_d &= \arctan\left( \frac{\cos\theta_d (u_x \sin\psi_d – u_y \cos\psi_d)}{u_z} \right) \\
U_1 &= \frac{u_z}{\cos\phi_d \cos\theta_d}
\end{aligned} $$

The control system employs a dual-loop structure, where the outer position loop generates desired attitude commands for the inner loop. The NPBSSM controller ensures predefined-time convergence and non-singularity. The system error is defined as $ e = x_1 – x_d $, where $ x_1 = [x, y, z, \phi, \theta, \psi]^T $ and $ x_d $ is the desired trajectory. An integral terminal sliding surface is designed as:

$$ s = k_1 e + k_2 \int_0^t \text{sign}(e) |e|^\alpha \, d\tau $$

where $ k_1, k_2 > 0 $, $ 1 < \alpha < 2 $, and $ \text{sign}(\cdot) $ is the signum function. A second-order auxiliary differential equation is constructed:

$$ \begin{aligned}
\dot{s}_1 &= s_2 \\
\dot{s}_2 &= -k_1 e – f(e)
\end{aligned} $$

with $ f(e) = \alpha k_2 |e|^{\alpha-1} \text{sign}(e) $. The predefined-time function is integrated into the controller using Lyapunov analysis. A quadratic continuous function is employed to design virtual controllers, ensuring non-singularity without filters or piecewise functions. The virtual controller $ s_{2d} $ is given by:

$$ s_{2d} = -\frac{s_1^T s_{2d}}{s_1^T s_1 + v_1} \left( s_1 + \frac{\pi}{2\gamma T_c} \left( \lambda_1 \text{sig}^{\gamma+1}(s_1) + (3-2\gamma) \text{sig}^{\gamma-1}(s_1) \right) \right) $$

where $ 0 < \gamma < 1 $, $ \lambda_1 \geq 0.5 $, $ v_1 \geq 0 $, and $ T_c $ is the predefined time. The actual controller $ u $ is derived as:

$$ u = -\frac{\tau^T u_1}{\tau^T \tau + v_2} \left( k G^{-1} \left( u_1 + k D + k f(\Theta) + k \dot{x}_d – f(e) – s_{2d} \right) \right) $$

with $ u_1 $ containing disturbance compensation terms. A disturbance observer is designed to estimate lumped disturbances $ D $:

$$ \begin{aligned}
\sigma &= \int_0^t \left( \Gamma_1 \Xi_1 + \Gamma_2 \Xi_2 \right) d\tau \\
\dot{\sigma} &= -\iota \sigma – \zeta \tanh\left( \frac{\eta \sigma}{|\sigma| + \beta} \right) + \text{sig}^l(\sigma) + \text{sig}^r(\sigma) + G u + f(\Theta)
\end{aligned} $$

where $ \iota, \eta, \beta, \zeta, \Xi_1, \Xi_2 > 0 $, $ 0 < l, r < 1 $. The observer ensures finite-time convergence of estimation errors.

Stability is proven using Lyapunov functions. For the sliding surface, $ V_1 = \frac{1}{2} s_1^T s_1 $ yields $ \dot{V}_1 \leq -\frac{\pi}{2\gamma T_c} V_1^{\frac{\gamma+1}{2}} + \varsigma_1 $, ensuring predefined-time convergence. Similarly, for the controller, $ V_2 = V_1 + \frac{1}{2} \tau^T \tau $ leads to $ \dot{V}_2 \leq -\frac{\pi}{2\gamma T_c} V_2^{\frac{\gamma+1}{2}} + \varsigma_2 $, with $ \varsigma_1, \varsigma_2 $ bounded. The total convergence time $ T_c’ $ satisfies $ T_c’ < 2T_c $.

Simulations validate the controller’s performance. The quadrotor parameters are: $ m = 2 \, \text{kg} $, $ L = 0.2 \, \text{m} $, $ g = 9.8 \, \text{m/s}^2 $, $ I_x = 0.0021 \, \text{kg·m}^2 $, $ I_y = 0.0036 \, \text{kg·m}^2 $, $ I_z = 0.0021 \, \text{kg·m}^2 $. Controller parameters include $ k_1 = 1 $, $ k_2 = 1 $, $ \alpha = 1.5 $, $ \gamma = 0.6 $, $ T_c = 3 \, \text{s} $ for position loop. The desired trajectory is $ x_d = \sin(2t) $, $ y_d = \cos(2t) $, $ z_d = 0.2t + 5 $, $ \psi_d = 0 $. Initial conditions are $ [0.5, 1.4, -0.4, 0, 0, 10\pi]^T $.

Comparative studies with integral backstepping sliding mode (ITBSSM), adaptive backstepping (ABS), variable-gain backstepping sliding mode (EVBSSM), and prescribed performance backstepping (CLBS) controllers demonstrate the superiority of NPBSSM. Under normal conditions, NPBSSM achieves convergence within 2 s without overshoot, while others exhibit slower convergence or overshoot. Under lumped disturbances $ D = 0.5\sin(t) + 0.5\sin^3(t) + 0.5\cos^3(t) + 0.001e^{-t} $, NPBSSM maintains robust tracking, whereas other controllers show significant deviations. The disturbance observer accurately estimates disturbances, enhancing robustness. Evaluation metrics include root mean square error (RMSE) and standard deviation of tracking error (SDTE):

$$ \begin{aligned}
\text{RMSE}_i &= \sqrt{\frac{1}{n} \sum_{k=1}^n e_i(k)^2}, \quad i = x, y, z, \psi \\
\text{SDTE} &= \sqrt{\frac{1}{n} \sum_{k=1}^n \left( T(k) – \bar{T} \right)^2 }, \quad T = \sqrt{e_x^2 + e_y^2 + e_z^2}
\end{aligned} $$

Table 1 summarizes the performance under disturbances:

Control Algorithm	RMSE_x	RMSE_y	RMSE_z	RMSE_ψ	SDTE
ITBSSM	0.0815	0.0742	0.0658	0.0151	0.1027
ABS	0.3311	0.2648	0.2372	0.0232	0.2789
EVBSSM	0.2072	0.1986	0.1989	0.0280	0.2109
CLBS	0.0649	0.0549	0.0676	0.0297	0.1053
NPBSSM	0.0498	0.0420	0.0498	0.0135	0.0807

NPBSSM reduces RMSE by 10.0–84.9% and SDTE by 21.4–71.1% compared to other methods, confirming its precision and robustness. Control inputs remain bounded and smooth, facilitating practical implementation.

In conclusion, the NPBSSM controller for quadrotor UAVs ensures predefined-time convergence, non-singularity, and disturbance rejection. The integration of a sliding surface, predefined-time function, and disturbance observer achieves high-precision trajectory tracking under uncertainties. Future work will focus on experimental validation and extension to multi-quadrotor systems.