An adaptive sliding mode controller based on global asymptotic convergent observer for attitude tracking of quadrotor unmanned aerial vehicles

Abstract

This article presents an adaptive sliding mode controller based on global asymptotic convergent observer for attitude tracking problem of a quadrotor unmanned aerial vehicle under modeling uncertainties and external disturbances. For avoiding the singularity problem, the attitude tracking error equations of a quadrotor with quaternion representation are introduced. For obtaining unmeasured states of the control system, a global asymptotic convergent observer without model-based is presented to estimate. To reject uncertainties and disturbances, a new observer-based controller for the quadrotor is designed by combining adaptive sliding mode control algorithm with time-delay estimation method. The stability analysis shows that the tracking errors of the proposed scheme are uniformly ultimately bounded. Simulation results are carried out to illustrate that compared with the existing controller, the proposed control scheme has better attitude tracking performance and higher robustness to reject the disturbances.

Introduction

As a kind of the micro helicopter, the quadrotor unmanned aerial vehicles (UAV) have some special advantages, such as vertical taking off and landing, hovering, and rapid maneuvering. As a typical underactuated system, the flight control scheme of the quadrotor UAV is always a focus of the research. ¹

Relying on the mathematical model of the quadrotor based on Euler angle representation to describe the rotational dynamics, various advanced control approaches have been presented for the quadrotor UAV to guarantee flight performance, such as proportional–integral–derivative (PID) control,^2,3 linear quadratic (LQ) control, ³ flatness-based control, ⁴ and model reference–based control. ⁵ Meanwhile, to reject the disturbances (including parameter perturbations, nonlinearities, and unknown external disturbances), several robust control schemes such as backstepping control,^6,7 switching model predictive control, ⁸ adaptive control, ⁹ sliding mode control,^7,10,11 robust motion control, ¹² fuzzy control, ¹³ and some other strategies have been developed for the quadrotor UAV. In Zhang et al., ¹⁰ a feedback sliding mode controller was proposed for stabilizing the attitude of a quadrotor aircraft under time-varying disturbances. In the study of Liu et al., ¹² without depending on the time scale separation assumption and tracking errors, a robust motion controller was designed for a quadrotor to cancel out internal and external disturbances. However, the rotational dynamics of the quadrotor UAV has singularity problem with Euler representation. For avoiding the problem, some quaternion-based attitude control schemes have been developed to ensure attitude tracking of the quadrotor UAV.^14–17 In particular, quaternion-based control has been addressed by resorting to the output feedback method for attitude tracking of the quadrotor UAV.^14–16 In the study of Djamel et al., ¹⁷ a quaternion-based attitude optimal controller was designed for attitude motion of a quadrotor using backstepping and nonlinear $H_{\infty }$ technology.

The above controllers have same problem depending on full state feedback. Actually, all state measurements of the quadrotor UAV are not always available. Therefore, some observers and filters have been designed to solve the drawback for quadrotor UAV.^18–23 Sliding mode observers have been provided for velocity and disturbances estimation of quadrotor UAV.^18–20 In the study of Boudjedir et al., ²¹ a neural network observer was presented to reject the measurement noise and estimate unavailable states of quadrotor. In the study of Lee et al., ²² a nonlinear disturbance observer was designed for dealing with unknown input gain matrix. In the study of Xiong et al., ²³ an optimal Kalman filter was employed to estimate state vector for rejecting white Gaussian process and measurement noises.

Based on the above literature reviews, most of the controllers were developed by imposing on full state feedback. Thus, it is significant to design the control scheme for a quadrotor UAV, which does not depend on all state measurements. This article provides an observer-based adaptive sliding mode control (ASMC) method for attitude tracking problem of the quadrotor UAV under modeling uncertainties and external disturbances. The proposed control scheme is constructed without full state feedback. The controller is well-effective to track reference and high robustness against disturbances. Since the Euler angle representation suffers from singularity problem, the attitude error dynamics model of the quadrotor UAV is constructed using unit quaternion. A global asymptotic convergent observer is employed to estimate the system states. The proposed observer is designed independent of model of system, and its gains are easily obtained. Based on the proposed observer, an adaptive sliding mode controller combined with time-delay estimation technique is designed. The stability analysis shows that the control system is globally asymptotic stable.

This article is organized as follows. In section “Attitude dynamics of a quadrotor with quaternion,” the quaternion-based attitude error dynamics model is introduced. In section “Attitude tracking control design,” the global asymptotic convergent observer is presented, and the observer-based adaptive sliding mode controller is developed. The stability analysis of the closed-loop system is provided. In section “Simulation results,” simulation results are presented. Finally, conclusions are included in section “Conclusion.”

Attitude dynamics of a quadrotor with quaternion

For obtaining the attitude dynamic of the quadrotor UAV, two right-hand frames denoted by $E_I$ and $E_B$ are utilized. $E_B=\{\begin{array}{ccc}{x}_B\hfill & y_B\hfill & z_B\hfill \end{array}\}$ denotes a body-fixed frame rigidly attached to the aircraft with origin in the mass center, where $z_B$ is the normal axis of the principal plane of the quadrotor directed from bottom to top, the axis $x_B$ is along with the forward flying direction of the quadrotor, and the axis $y_B$ is determined by the right-hand rule. $E_I=\{\begin{array}{ccc}{x}_I\hfill & y_I\hfill & z_I\hfill \end{array}\}$ denotes the inertial frame fixed at same position of the earth, where $z_I$ is the vertical direction upward into the sky, as shown in Figure 1. Then, the dynamics for rotational motion of the quadrotor UAV using Euler representation can be given by^14,17

$$\mathbf{J}\stackrel{·}{\mathbf{\omega }}=-\mathbf{S}(\mathbf{\omega })\mathbf{J}\mathbf{\omega }+\mathbf{\tau }\mathbf{+}\mathbf{D}$$

(1)

where $\mathbf{\omega }\in {\Re }^3$ is the angular velocity in the frame $E_B$ , $\mathbf{J}\in {\Re }^{3\times 3}$ denotes positive definite inertia matrix, $\mathbf{\tau }\mathbf{\in }{\mathbf{\Re }}^3$ is the control torque input, $\mathbf{D}\in {\Re }^3$ represents the disturbance including the modeling uncertainties and external disturbances; it is assumed that the first and second time derivatives of the disturbance $\mathbf{D}$ are bounded all time. $\mathbf{S}(·)$ in equation (1) denotes a general form of skew symmetric matrix as follows

$$\mathbf{S}(\mathbf{\xi }\mathbf{)}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}0\hfill & \mathbf{-}{\mathbf{\xi }}_3\hfill & {\mathbf{\xi }}_2\hfill \\ \multicolumn{1}{c}{{\mathbf{\xi }}_3}& 0\hfill & \mathbf{-}{\mathbf{\xi }}_1\hfill \\ \multicolumn{1}{c}{\mathbf{-}{\mathbf{\xi }}_2}& {\mathbf{\xi }}_1\hfill & 0\hfill \end{array}\mathbf{\right]}$$

(2)

for $\mathrm{\forall }$ , $\mathbf{\xi }=\mathbf{[}\begin{array}{ccc}{\mathbf{\xi }}_1\hfill & {\mathbf{\xi }}_2\hfill & {\mathbf{\xi }}_3\hfill \end{array}{\mathbf{]}}^{\mathbf{T}}\mathbf{\in }{\mathbf{\Re }}^3$ .

Figure 1.

Schematic of the quadrotor UAV.

For avoiding singularity problem with Euler angle–based representation to describe the rotational matrix $\mathbf{R}$ which represents the rotational matrix from the frame $E_B$ to the inertial frame $E_I$ , the rotational matrix $\mathbf{R}$ is described using unit quaternion representation $\mathbf{q}=[\begin{array}{cc}{q}_0\hfill & {\mathbf{q}}_v^T\hfill \end{array}{]}^T$ ,which satisfies $q_0^2+\Vert {\mathbf{q}}_v{\Vert }_2^2=1$ , as follows

$$\mathbf{R}=(q_0^2-{\mathbf{q}}_v^T{\mathbf{q}}_v){\mathbf{I}}_3+2{\mathbf{q}}_v{\mathbf{q}}_v^T-2q_0\mathbf{S}({\mathbf{q}}_v)$$

(3)

where ${\mathbf{I}}_3$ denotes a $3\times 3$ identity matrix.

The relation between the quaternion $\mathbf{q}$ and the angular velocity $\mathbf{\omega }$ can be expressed via the following differential equation as

$$\stackrel{·}{\mathbf{q}}=\frac{1}{2}\mathbf{B}(\mathbf{q})\mathbf{\omega }$$

(4)

where the auxiliary matrix $\mathbf{B}(\mathbf{q})=[\begin{array}{cc}{\mathbf{B}}_0^T\hfill & {\mathbf{B}}_v^T\hfill \end{array}]\in {\Re }^{4\times 3}$ is defined as ${\mathbf{B}}_0=-{\mathbf{q}}_v^T\in {\Re }^{1\times 3}$ and ${\mathbf{B}}_v=q_0{\mathbf{I}}_3+\mathbf{S}({\mathbf{q}}_v)\in {\Re }^{3\times 3}$ , respectively.

The desired quaternion and angular velocity are defined by ${\mathbf{q}}_d$ and ${\mathbf{\omega }}_d$ in a desired body-fixed frame $E_{\mathrm{Bd}}=\{\begin{array}{ccc}{x}_{\mathrm{Bd}}\hfill & y_{\mathrm{Bd}}\hfill & z_{\mathrm{Bd}}\hfill \end{array}\}$ , respectively. The rotational matrix from the frame $E_{\mathrm{Bd}}$ to the inertial frame $E_I$ is defined by ${\mathbf{R}}_d$ . The quaternion tracking error from the current orientation to the desired orientation is defined by ${\mathbf{q}}_e=[\begin{array}{cc}{q}_{0e}\hfill & {\mathbf{q}}_{\mathrm{ve}}^T\hfill \end{array}{]}^T\in {\Re }^4$ , where $q_{0e}=q_0{q}_{0d}+{\mathbf{q}}_v^T{\mathbf{q}}_{\mathrm{vd}}$ , ${\mathbf{q}}_{\mathrm{ve}}=\mathbf{S}({\mathbf{q}}_v){\mathbf{q}}_{\mathrm{vd}}+q_{0d}{\mathbf{q}}_v-q_0{\mathbf{q}}_{\mathrm{vd}}$ are the scalar part and the vector part of the quaternion ${\mathbf{q}}_e$ , respectively, and the quaternion ${\mathbf{q}}_e$ satisfies that ${\mathbf{q}}_e^T{\mathbf{q}}_e=1$ . The angular velocity of $E_B$ with respect to $E_{\mathrm{Bd}}$ expressed in $E_B$ is defined by ${\mathbf{\omega }}_e=\mathbf{\omega }-{\mathbf{R}}_e{\mathbf{\omega }}_d$ , where ${\mathbf{R}}_e$ represents the mismatch between the rotational matrix $\mathbf{R}$ to the rotational matrix ${\mathbf{R}}_d$ and is defined as ${\mathbf{R}}_e={\mathbf{RR}}_d^T=(q_{0e}^T-{\mathbf{q}}_{\mathrm{ve}}^T{\mathbf{q}}_{\mathrm{ve}}){\mathbf{I}}_3+2({\mathbf{q}}_{\mathrm{ve}}{\mathbf{q}}_{\mathrm{ve}}^T-q_{0e}\mathbf{S}({\mathbf{q}}_{\mathrm{ve}}))$ .

The quaternion error ${\mathbf{q}}_e$ can be related to the angular velocity ${\mathbf{\omega }}_e$ via the following differential equation as

$${\stackrel{·}{\mathbf{q}}}_e=\frac{1}{2}{\mathbf{B}}_e({\mathbf{q}}_e){\mathbf{\omega }}_e$$

(5)

where the auxiliary matrix ${\mathbf{B}}_e({\mathbf{q}}_e)=[\begin{array}{cc}{\mathbf{B}}_{0e}^T\hfill & {\mathbf{B}}_{\mathrm{ve}}^T\hfill \end{array}]\in {\Re }^{4\times 3}$ is defined as ${\mathbf{B}}_{0e}=-{\mathbf{q}}_{\mathrm{ve}}^T\in {\Re }^{1\times 3}$ and ${\mathbf{B}}_{\mathrm{ve}}=q_{0e}{\mathbf{I}}_3+\mathbf{S}({\mathbf{q}}_{\mathrm{ve}})\in {\Re }^{3\times 3}$ , respectively.

After taking the time derivative of ${\mathbf{\omega }}_e$ and using the part of equations (5) and (1), the attitude dynamics and kinematics of the quadrotor UAV can be rewritten by error quaternion as follows

$$\{\begin{array}{c}{\stackrel{·}{\mathbf{q}}}_{\mathrm{ve}}=\frac{1}{2}{\mathbf{B}}_{\mathrm{ve}}{\mathbf{\omega }}_e\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathbf{\omega }}}_e=-{\mathbf{J}}^{-1}\mathbf{S}({\mathbf{\omega }}_e+{\mathbf{R}}_e{\mathbf{\omega }}_d)\mathbf{J}({\mathbf{\omega }}_e+{\mathbf{R}}_e{\mathbf{\omega }}_d)-{\mathbf{R}}_e(\mathbf{S}({\mathbf{\omega }}_e){\mathbf{\omega }}_d+{\stackrel{·}{\mathbf{\omega }}}_d)+{\mathbf{J}}^{-1}\mathbf{\tau }\mathbf{+}{\mathbf{J}}^{\mathbf{-}1}\mathbf{D}}\end{array}$$

(6)

where the fact has been used that ${\stackrel{·}{\mathbf{R}}}_e={\mathbf{R}}_e\mathbf{S}({\mathbf{\omega }}_e)$ . ²⁴

Attitude tracking control design

The global asymptotic convergent observer design

Defining the estimates ${\hat{\mathbf{x}}}_1$ , ${\hat{\mathbf{x}}}_2$ of the states ${\mathbf{q}}_{\mathrm{ve}}$ , ${\mathbf{\omega }}_e$ , the estimate errors can be written as follows

$$\{\begin{array}{c}{\mathbf{e}}_1={\hat{\mathbf{x}}}_1-{\mathbf{q}}_{\mathrm{ve}}\hfill \\ \multicolumn{1}{c}{{\mathbf{e}}_2={\hat{\mathbf{x}}}_2-{\mathbf{\omega }}_e}\end{array}$$

(7)

The observer for the attitude of the quadrotor UAV is proposed as

$$\{\begin{array}{c}{\stackrel{·}{\hat{\mathbf{x}}}}_1=\frac{1}{2}{\mathbf{B}}_{\mathrm{ve}}{\hat{\mathbf{x}}}_2-{\mathbf{L}}_1{\left|{\mathbf{e}}_1\right|}^{{\alpha }_1}\mathrm{sgn}({\mathbf{e}}_1)-{\mathbf{e}}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\mathbf{x}}}}_2=-L\mathrm{sgn}({\mathbf{e}}_1)-{\mathbf{L}}_2{\left|{\mathbf{e}}_1\right|}^{{\alpha }_2}\mathrm{sgn}({\mathbf{e}}_1)-\frac{\mathrm{nL}{\mathbf{B}}_{\mathrm{ve}}}{2}{\mathbf{e}}_1+{\mathbf{J}}^{-1}\mathbf{\tau }}\end{array}$$

(8)

where $L>0$ , $0<{\alpha }_i(i=1,2)<1$ , $n$ is the degree of the attitude of the quadrotor UAV, and ${\mathbf{L}}_1,{\mathbf{L}}_2\in {\Re }^{3\times 3}$ denote diagonal constant positive matrices.

Taking the time derivative of equation (7), the equations of the estimation errors can be written as follows

$$\{\begin{array}{c}{\stackrel{·}{\mathbf{e}}}_1=\frac{1}{2}{\mathbf{B}}_{\mathrm{ve}}{\mathbf{e}}_2-{\mathbf{L}}_1{\left|{\mathbf{e}}_1\right|}^{{\alpha }_1}\mathrm{sgn}({\mathbf{e}}_1)-{\mathbf{e}}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathbf{e}}}_2=-L\mathrm{sgn}({\mathbf{e}}_1)-{\mathbf{L}}_2{\left|{\mathbf{e}}_1\right|}^{{\alpha }_2}\mathrm{sgn}({\mathbf{e}}_1)-\frac{\mathrm{nL}{\mathbf{B}}_{\mathrm{ve}}}{2}{\mathbf{e}}_1-\mathbf{G}}\end{array}$$

(9)

where $\mathbf{G}=-{\mathbf{J}}^{-1}\mathbf{S}({\mathbf{\omega }}_e+{\mathbf{R}}_e{\mathbf{\omega }}_d)\mathbf{J}({\mathbf{\omega }}_e+{\mathbf{R}}_e{\mathbf{\omega }}_d)-{\mathbf{R}}_e(\mathbf{S}({\mathbf{\omega }}_e){\mathbf{\omega }}_d+{\stackrel{·}{\mathbf{\omega }}}_d)+{\mathbf{J}}^{-1}\mathbf{D}$ .

Theorem 1

For the observer (8) and system (6) with the bounded uncertainties, if the gain $L$ in equation (9) satisfies the sufficient condition $L>|\mathbf{G}(t)|+|\stackrel{·}{\mathbf{G}}(t)|$ , then ${\mathbf{e}}_1(t)\to 0$ and ${\mathbf{e}}_2(t)\to 0$ as $t\to \infty$ . The proposed observer ensures that estimate errors are asymptotic convergence.

The proof is presented in “Appendix 1.”

The adaptive sliding mode controller based on the observer design

The control objective is of designing a controller to track the desired trajectory of attitude without measuring the angular velocity. The reference of the state ${\mathbf{q}}_{\mathrm{ve}}$ is denoted by ${\overline{\mathbf{q}}}_{\mathrm{ve}}$ ; its first and second derivatives of the desired reference ${\overline{\mathbf{q}}}_{\mathrm{ve}}$ are available and bounded.

In the controller design procedure, the states ${\mathbf{q}}_{\mathrm{ve}}$ , ${\mathbf{\omega }}_e$ are replaced by the observer estimation states ${\hat{\mathbf{x}}}_i(i=1,2)$ , respectively. The controller is designed step by step via the observer-based ASMC arithmetic. Define the tracking errors ${\mathbf{z}}_1={\overline{\mathbf{q}}}_{\mathrm{ve}}-{\hat{\mathbf{x}}}_1$ , ${\mathbf{z}}_2={\stackrel{·}{\overline{\mathbf{q}}}}_{\mathrm{ve}}-{\hat{\mathbf{x}}}_2$ . To achieve the control objective, the sliding variable is defined as follows

$$\mathbf{s}={\mathbf{z}}_2+{\mathbf{K}}_s{\mathbf{z}}_1$$

(10)

where ${\mathbf{K}}_s=\mathrm{diag}(k_{s1},k_{s2},\dots ,k_{\mathrm{sn}})$ . According to equation (10), for compensating the nonlinearities and disturbances, the controller is designed by combining sliding mode control technique with time-delay estimation method as follows ²⁵

$$\mathbf{\tau }\mathbf{=}\mathbf{-}\mathbf{J}\hat{\mathbf{G}}\mathbf{+}\mathbf{J}\mathbf{(}{\stackrel{··}{\overline{\mathbf{q}}}}_{\mathbf{ve}}\mathbf{+}{\mathbf{K}}_{\mathbf{s}}{\mathbf{z}}_2\mathbf{+}\mathbf{\beta }\mathbf{s}\mathbf{)}$$

(11)

where $\beta$ is a positive constant, $\hat{\mathbf{G}}(t)\approx \mathbf{G}(t-L)={\stackrel{·}{\hat{\mathbf{x}}}}_2(t-L)-{\mathbf{J}}^{-1}\mathbf{\tau }\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{L}\mathbf{)}$ , $L$ is a sampling time.

Substituting equations (10) and (11) into equation (6), the closed-loop system can be obtained as follows

$${\stackrel{·}{\mathbf{z}}}_2+({\mathbf{K}}_s+\beta \mathbf{I}){\mathbf{z}}_2+\beta {\mathbf{K}}_s{\mathbf{z}}_1+{\stackrel{·}{\mathbf{e}}}_2+\mathbf{G}-\hat{\mathbf{G}}=0$$

(12)

If the time-delay estimation error $\mathbf{G}-\hat{\mathbf{G}}$ is sufficiently small, the tracking error can converge to zero by adjusting ${\mathbf{K}}_s$ and $\beta$ . For enhancing the tracking accuracy and adaptation of equation (11), an ASMC scheme is added as follows

$$\mathbf{\tau }\mathbf{=}\mathbf{-}\mathbf{J}\hat{\mathbf{G}}\mathbf{+}\mathbf{J}\mathbf{(}{\stackrel{··}{\overline{\mathbf{q}}}}_{\mathbf{ve}}\mathbf{+}{\mathbf{K}}_{\mathbf{s}}{\mathbf{z}}_2\mathbf{+}\mathbf{\beta }\mathbf{s}\mathbf{)}\mathbf{+}\mathbf{J}\mathbf{(}\hat{\mathbf{K}}\mathbf{tanh}\mathbf{(}\mathbf{s}\mathbf{)}\mathbf{)}$$

(13)

where $\hat{\mathbf{K}}=\mathrm{diag}({\hat{k}}_1,{\hat{k}}_2,\dots ,{\hat{k}}_n)$ is a positive switching matrix, $\tanh (\mathbf{s})=(e^{\mu \mathbf{s}}-e^{-\mu \mathbf{s}})/(e^{\mu \mathbf{s}}+e^{-\mu \mathbf{s}})$ , $\mu$ is a positive gain.

The adaptive law is employed for the ASMC scheme as

$${\stackrel{·}{\hat{k}}}_i={\phi }_i{\left({\alpha }_i^{-1}\left|s_i\right|\right)}^{\tanh ({\Vert s_i\Vert }_{\infty }-\epsilon )}·\tanh ({\Vert s_i\Vert }_{\infty }-\epsilon )$$

(14)

where $\phi$ , $\alpha$ , and $\epsilon$ are positive gains, and $\epsilon$ is the adaptation gain.

Theorem 2

For the closed-loop system (6) controlled by equation (13), $\mathrm{\forall }$ a finite time ${\tau }_{\epsilon }>0$ , for $t>{\tau }_{\epsilon }$ , the tracking errors of the proposed control scheme are uniformly ultimately bounded in the sense that

$$\Vert \mathbf{s}{\Vert }_2<\sqrt{\sum _{i=1}^n{\epsilon }^2+{\Gamma }_M}$$

(15)

where ${\Gamma }_M$ is the maximum value of $\sum _{i=1}^n{\alpha }_i/{\phi }_i({\overline{G}}_i-{\hat{k}}_i{)}^2$

The proof is presented in “Appendix 1.”

Simulation results

In this section, some simulation results are presented to demonstrate the performance of the proposed control method. The mass of the quadrotor UAV is 2 kg, and the inertial matrix of the quadrotor UAV is $\mathbf{J}=\mathrm{diag}(\begin{array}{ccc}0.01567,\hfill & 0.01567,\hfill & 0.02834\hfill \end{array})N{s}^2/\mathrm{rad}$ . The gains for the presented observer are set as ${\mathbf{L}}_1=(\begin{array}{ccc}86\hfill & 86\hfill & 92\hfill \end{array}{)}^T$ , ${\mathbf{L}}_2=(\begin{array}{ccc}300\hfill & 300\hfill & 516\hfill \end{array}{)}^T$ , ${\mathbf{\alpha }}_1=(\begin{array}{ccc}0.75\hfill & 0.75\hfill & 0.8\hfill \end{array}{)}^T$ , ${\mathbf{\alpha }}_2=(\begin{array}{ccc}0.83\hfill & 0.83\hfill & 0.92\hfill \end{array}{)}^T$ , $L=12$ , $n=3$ . The initial values of the presented observer are set as ${\hat{\mathbf{x}}}_1=(\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}{)}^T$ , ${\hat{\mathbf{x}}}_2=(\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}{)}^T$ . The parameters of the controller are selected as ${\mathbf{K}}_s=[\begin{array}{ccc}6\hfill & 6\hfill & 5\hfill \end{array}]$ , $\beta =[101023]$ , $\phi =[0.250.250.5]$ .

The desired quaternion ${\mathbf{q}}_d$ is defined as ${\mathbf{q}}_d=(1\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}{)}^T$ . The quaternion $\mathbf{q}$ is denoted as $\mathbf{q}=(\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \end{array})$ , as $t=0$ ; $\mathbf{q}=(0.7-\begin{array}{ccc}0.358\hfill & 0.346\hfill & -0.512\hfill \end{array}{)}^T$ , as $t=0.01$ . The objective of the quadrotor attitude tracking is $\underset{t\to \infty }{lim}{\mathbf{q}}_{\mathrm{ve}}(t)=0$ ; therefore, the desired reference ${\overline{\mathbf{q}}}_{\mathrm{ve}}$ can be denoted as ${\overline{\mathbf{q}}}_{\mathrm{ve}}=(\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}{)}^T$ . The initial angular velocity $\mathbf{\omega }$ is set as $\mathbf{\omega }=[\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array}{]}^T\mathrm{rad}/\mathrm{s}$ . The desired angular velocity ${\mathbf{\omega }}_d$ is set as ${\mathbf{\omega }}_d=[\begin{array}{ccc}0.5\cos \mathrm{t}\hfill & 0.5\cos \mathrm{t}\hfill & 0.5\cos \mathrm{t}\hfill \end{array}{]}^T\mathrm{rad}/\mathrm{s}$ . The disturbances $\mathbf{D}$ are denoted as stochastic disturbance, and its amplitude is 5 N m.

For verifying the robustness of the proposed control scheme, starting from 2 s, and lasts 1 s, the continuous noises 15 N m is added to the first subsystem $q_{\mathrm{ve}1}$ , and starting from 3 s, and lasts 1 s, the time-varying disturbance 20cos(t) Nm is added to the second subsystem $q_{\mathrm{ve}2}$ . Meanwhile, for comparing the effectiveness of controller, the simulation of tracking results using the existing ASMC scheme in the study of Plestan et al. ²⁶ combined with the proposed observer are also presented.

Some simulation results are provided in Figures 2–8. Figures 2 and 3 show the estimations of ${\mathbf{q}}_{\mathrm{ve}}$ and ${\mathbf{\omega }}_e$ by employing the proposed observer. It is clearly seen that the proposed observer has well-performed in estimations. When the control subsystems are subject to disturbances seriously, the estimated states ${\mathbf{q}}_{\mathrm{ve}}$ and ${\mathbf{\omega }}_e$ of the proposed observer are still holded. From the results of the estimations, it is proved that the proposed observer can contribute reliable estimate values to design the controller. Figure 4 shows the estimation errors of the proposed observer. It can be obtained that the estimation errors can be converged to a small neighborhood of zero in t < 0.5 s. The estimation errors can be reduced to zero quickly while the control subsystems are subject to the disturbances. Figure 5 shows the control input of the proposed control system. It is observed that the control input $\mathbf{\tau }$ has a stable value in t < 0.5 s. When the disturbances are present, the values of the control inputs are changed adaptively to reject the disturbances.

Figure 2.

Estimation of the state ${\mathbf{q}}_{\mathrm{ve}}$ : (a) estimation of $q_{\mathrm{ve}1}$ , (b) estimation of $q_{\mathrm{ve}2}$ , and (c) estimation of $q_{\mathrm{ve}3}$ .

Figure 3.

Estimation of the state ${\mathbf{\omega }}_e$ : (a) estimation of ${\omega }_{e1}$ , (b) estimation ${\omega }_{e2}$ , and (c) estimation of ${\omega }_{e3}$ .

Figure 4.

Estimation errors of the control system states: (a) estimation errors of the state ${\mathbf{q}}_{\mathrm{ve}}$ and (b) estimation errors of the state ${\mathbf{\omega }}_e$ .

Figure 5.

Control input of the proposed scheme: (a) control input ${\tau }_1$ , (b) control input ${\tau }_2$ , and (c) control input ${\tau }_3$ .

Figure 6.

Comparison of tracking results for ${\mathbf{q}}_{\mathrm{ve}}$ with different control schemes: (a) comparison of tracking results for $q_{\mathrm{ve}1}$ , (b) comparison of tracking results for $q_{\mathrm{ve}2}$ , and (c) comparison of tracking results for $q_{\mathrm{ve}3}$ .

Figure 7.

Comparison of tracking results for ${\mathbf{\omega }}_e$ with different control schemes: (a) comparison of tracking results for ${\omega }_{e1}$ , (b) comparison of tracking results for ${\omega }_{e2}$ , and (c) comparison of tracking results for ${\omega }_{e3}$ .

Figure 8.

Comparison of tracking errors with different control schemes: (a) tracking errors of the proposed control scheme and (b) tracking errors of the existing control scheme.

Figures 6 and 7 show the comparison of the tracking results for ${\mathbf{q}}_{\mathrm{ve}}$ and ${\mathbf{\omega }}_e$ with different control schemes. The output values of the control system can track values of the reference trajectory about in t < 1 s with the proposed control scheme, and in t < 1.7 s with the existing control scheme. Figures 6(a), (b) and 7(a), (b) show that compared to the existing control scheme, the fluctuations of output values are smaller with the proposed control scheme, when the control subsystems are subject to the disturbances. Figure 8 shows the comparison of tracking errors with different control schemes. From Figure 8(a), it can be obtained that the tracking errors can be reduce to zero approximately in t < 1 s with the proposed control scheme. When the control subsystems are subject to the disturbances, the variation range of tracking errors are −0.005 to 0.005. From Figure 8(b), it can be obtained that the tracking errors can be reduce to zero approximately in t < 1.7 s with the existing control scheme. When the control subsystems are subject to the disturbances, the variation range of tracking errors are −0.01 to 0.01. Thus, it is easy to obtain that compared to the existing scheme, the proposed scheme has better effectiveness to track the desired reference and robust performance to reject the disturbances.

Conclusion

In this article, an observer-based adaptive sliding mode controller is developed for tracking the attitude of a quadrotor UAV under modeling uncertainties and external disturbances. Using unit quaternion, the attitude error dynamics model of the quadrotor UAV is introduced. A global asymptotic convergent observer which is not sensitive to system model is presented. Based on the proposed observer, compared with time-delay estimation method, an adaptive sliding mode controller is designed. The stability analysis shows that the estimate errors are asymptotic convergent, and the tracking errors are guaranteed to be uniformly ultimately bounded. Numerical simulation results show that the control system can ensure attitude tracking performance and has better robustness to reject disturbances.