In recent years, quadrotors have been applied in various fields owing to their size, low costs, and capability of VTOL(vertical takeoff and landing) [1,2,3]. These considerable attributes and widespread applications propel significant focus on quadrotors [4,5,6,7]. In reality, external disturbances and measurement errors are inevitably present [8,9,10,11,12]. The key to complete tasks, e.g., power line inspection [13], logistics transportation [14], and pesticide spraying [15] depends on the performance of trajectory tracking. As a result, improving the effectiveness of trajectory tracking control under the influence of external disturbances holds important theoretical significance and practical value, which ensures that quadrotors can meet the flight requirements and avoid uncontrollable crashes.

Nair et al. achieved high-performance tracking for a satellite launch rocket system with time-varying uncertainties by designing an adaptive PID control in reference [16]. In [17], an online identification of aerodynamic parameters using the Kalman filter was utilized, followed by compensating the rotor speed using LQR strategy, resulting in improved control performance for heading and altitude. Such works generally linearize the nonlinear dynamics of quadrotors, which leads to approximation errors and performance limitations. In [18], a fuzzy logic system was used to assess the unmodeled dynamics of the quadrotor. A nonlinear disturbance observer was applied to compensate for external disturbances and assessment error. In reference [19], the authors proposed a radical basis function neural network (RBFNN) to fix the unknown dynamic problem. To increase the convergence speed and control precision, a fractional-order backstepping control was introduced. Sliding mode control is widely applied in practical systems, e.g., robot manipulation [20], linear motor positioning [21], and underwater robots [22], because of its strong robustness, fast response, and low sensitivity to uncertainties and disturbances. Conventional sliding mode (CSM) cannot guarantee the fast convergence of control system [23,24,25]. Terminal sliding mode (TSM) was designed to deal with this situation [26]. Besides, CSM may cause chattering. In [27], a modified second-order sliding mode control was introduced to address the chatting phenomenon. Omid Mofid et al. planed a super-twisting terminal sliding mode control considering input-delay, model uncertainty and wind disturbance for fast response. In [28], the dynamic model was divided into inner loop and outer loop. Nonsingular terminal sliding mode control (NTSM) was utilized to converge to the desired position and attitude in finite time. In [29], Hamid Ghadiri et al. proposed a novel adaptive nonsingular terminal sliding mode control (ANTSM). The control signals’ chattering was eliminated. Adaptive estimation was applied to compensate for unknown disturbance. In [30], the implementation of neural network (NN) approximator was to estimate the model uncertainty. Adaptive algorithms were designed to make up for the approximation error and update the NN weight matrix. Nonsingular fast terminal sliding mode control (NFTSMC) was presented to guarantee the finite-time convergence of the quadrotor to its desired trajectory. An NN-NFTSMC algorithm was formulated to provide the system with robustness in the presence of the model uncertainty and external disturbance. In [31], Mo H et al. took efforts to make comprehensive survey of control techniques for quadrotors. In this study, adaptive terminal sliding mode is the most appropriate method for quadrotor tracking control.

To develop a control scheme for full trajectory tracking of 6 DOF quadrotor’s position and attitude, an adaptive backstepping fast nonsingular integral terminal sliding mode (ABFNITSM) control is proposed to track desired attitude. The position loop is based on adaptive integral backstepping control. The main contributions of this article can be summarized as follows:

This article is organized as follows: the derivation of the quadrotor dynamic model is introduced in Section 2. The adaptive backstepping fast nonsingular integral terminal sliding mode for attitude control and the adaptive integral backstepping for position control are designated in Section 3. Meanwhile, the stability of the control system is proven using Lyapunov theory. In Section 4, the simulation result is presented, followed by the conclusion.

The quadrotor is an underactuated system. The vector $\mathrm{P}={(\mathrm{x},\mathrm{y},\mathrm{z})}^{\mathrm{T}}$ represents the translational motion and the vector $\mathsf{\Theta }={(\mathsf{\varphi },\mathsf{\theta },\mathsf{\psi })}^{\mathrm{T}}$ denotes the Euler angle. Figure 1 shows a schematic diagram of the quadrotor.

This paper adopts the right-hand rule to define body reference frame (BRF) $\mathbb{B}=\{{\mathrm{O}}_{\mathrm{B}},{\mathrm{x}}_{\mathrm{b}},{\mathrm{y}}_{\mathrm{b}},{\mathrm{z}}_{\mathrm{b}}\}$ and inertial reference frame (IRF) $\mathbb{E}=\{{\mathrm{O}}_{\mathrm{E}},{\mathrm{x}}_{\mathrm{e}},{\mathrm{y}}_{\mathrm{e}},{\mathrm{z}}_{\mathrm{e}}\}$ to describe the 6DOF rigid-body motion of the quadrotor. x, y, and z represent longitudinal, lateral, and vertical translational motion respectively. $\mathsf{\varphi }$, $\mathsf{\theta }$, and $\mathsf{\psi }$ are roll, pitch, and yaw angle respectively.

According to Newton-Euler principle, the system model can be written:

${\mathrm{F}}_{\mathrm{A}}$ is the aerodynamic force in the inertial frame.

${\mathrm{F}}_{\mathrm{i}}$ is the thrust produced by the i-th motor. ${\mathrm{R}}_{\mathrm{i}}$ describes the rotational speed of the motor. ${\mathrm{k}}_{\mathrm{L}}$ represents the lift coefficient of the motor.

${\mathrm{R}}_{\mathrm{B}}^{\mathrm{E}}$ is the transformation matrix from the BRF to the IRF.

${\mathrm{F}}_{\mathrm{G}}$ is the gravitational force. ${\mathrm{F}}_{\mathrm{D}}$ represents the force due to air resistance.

$\mathrm{I}$ is the inertia matrix of the quadrotor. Based on the assumptions in Appendix A, $\mathrm{I}$ is defined as follows:

The notation ${\left[\right]}_{\times }$ means the cross-product operator. Based on the vector $\mathsf{\Omega }$, it defines a skew-symmetric matrix.

${\mathsf{\tau }}_{\mathrm{A}}$ denotes the aerodynamic torque produced by motors. $\mathrm{d}$ is the distance between the rotation axis of one motor and the center of gravity. ${\mathrm{k}}_{\mathrm{M}}$ is the torque coefficient of one motor during rotation.

${\mathsf{\tau }}_{\mathrm{G}}$ stands for the torque caused by the gyroscopic effect. ${\mathsf{\tau }}_{\mathrm{D}}$ reflects the torque owing to air friction.

Actually, a quadrotor undergoes various problems such as inaccurate parametric (mass and inertia) measurement, control effectiveness and external disturbances. These throw obstacles in control design. In order to formulate the control problem, equation (2) is rewritten incorporating matched/unmatched uncertainties as where $∆\mathrm{m}\in \mathcal{R},∆\mathrm{I}\in {\mathcal{R}}^{3\times 3},∆{\mathrm{F}}_{\mathrm{A}}\in {\mathcal{R}}^3$ and $∆{\mathsf{\tau }}_{\mathrm{A}}\in {\mathcal{R}}^3$represent uncertainty in mass, inertia and control effectiveness, respectively. According to lemma1, the above equation can be re-written as: where ${\mathrm{D}}_{\mathrm{P}}$ and ${\mathrm{D}}_{\mathrm{A}}$ are lumped uncertainties.

where $\mathrm{A}\in {\mathcal{R}}^{\mathrm{m}\times \mathrm{m}}$, $\mathrm{B}\in {\mathcal{R}}^{\mathrm{m}\times \mathrm{n}}$, $\mathrm{C}\in {\mathcal{R}}^{\mathrm{n}\times \mathrm{p}}$, $\mathrm{D}\in {\mathcal{R}}^{\mathrm{p}\times \mathrm{m}}$, and $\mathrm{E}$ is unit diagonal matrix.

The relationships between the angular velocity of a quadrotor in BRF and the angular velocity in IRF are represented as follows:

According the small angle approximation principle, the transformation matrix can be approximated as the identity matrix.

The vector $\mathrm{X}=\left[\mathsf{\varphi },\dot{\mathsf{\varphi }},\mathsf{\theta },\dot{\mathsf{\theta }},\mathsf{\psi },\dot{\mathsf{\psi }},\mathrm{x},\dot{\mathrm{x}},\mathrm{y},\dot{\mathrm{y}},\mathrm{z},\dot{\mathrm{z}}\right]\in {\mathcal{R}}^{12}$ denotes the state variables.

The virtual position control inputs are defined as follows:

The equation (13) can be rewritten as follows: where:

This section is committed to the design of trajectory tracking control and stability analysis. Taking parametric uncertainties and external disturbances into consideration, a trajectory tracking control strategy is formulated to ensure quick and accurate tracking of reference commands. To achieve this goal, a new control is proposed integrating terminal sliding mode, integral element, backstepping technique and dead zone technique for attitude trajectory tracking. For the sake of chattering elimination, a new saturation function is designed. Additionally, an online adaptive estimation technique is introduced to compensate for parametric uncertainties and external disturbances. Adaptive integral backstepping control is introduced in position loop. Figure 2 illustrates the entire control flow diagram.