Dynamic Landing Control of a Quadrotor on the Wave Glider

Abstract

Aiming at the problems of difficult attitude stabilization, low landing accuracy, large external disturbance and slow dynamic response during the quadrotor dynamic landing on the wave glider, an improved series active disturbance rejection control method for the quadrotor is proposed. The quadrotor controller with inner-loop attitude angular velocity control and outer-loop position control based on the active disturbance rejection controller (ADRC) is designed by analyzing the dynamic model of the quadrotor. A tracking differentiator (TD) is adopted to track the input signal, and an expansive state observer (ESO) is used to estimate the total disturbance. Moreover, a nonlinear law state error feedback (NLSEF) is used to generate the virtual control volume of the system to realize the control of the quadrotor, and the stability of the cascaded self-turbulent controller is verified by Lyapunov’s theory. The simulation verifies that the proposed controller can accurately control the attitude and the position with better anti-interference capability and faster tracking speed. According to the final sea trial, a combination of an active disturbance rejection controller optimized with improved crow search algorithm (ICADRC) and April Tag visual reference system is used to land the quadrotor efficiently and successfully even under the surface float attitude uncertainty.

1. Introduction

Wave gliders can achieve long-term endurance without manual maintenance and provide real-time continuous observations of the air–sea interface in the open ocean by being equipped with ocean science sensors [1,2,3]. In addition, the wave glider can carry quadrotors for vertical observation of the atmosphere. The combined application of the wave glider and the quadrotor compensates for their own shortcomings and enhances their own functions [4,5]. During the landing of a quadrotor on a surface float of the wave glider, the quadrotor goes through several processes.

(1)

The quadrotor visually recognizes the guiding signs of the surface float of the wave glider.

(2)

The quadrotor dynamically follows the landing guidance signs on the surface float.

(3)

The quadrotor dynamically adjusts its attitude, gradually descends and achieves landing.

Among the mentioned processes, stable control algorithm, accurate dynamic target identification, real-time information fusion, precise attitude coupling and reliable landing strategy are all crucial to ensure successful landing. This paper primarily analyzes the control system of a quadrotor to improve the stability and anti-interference capability of the control system during the quadrotor landing on the surface float of the wave glider.

Currently, quadrotors have been made great progress in automatic landing on land and relatively stable water surfaces, which is shown in

Table 1. Aircraft landing methods.

Method	Target	Status	Scenes	Mode	Reference
T-MPSP	UAV	Static	Land	Simulation	Amit K et al. (2016)
Sensor Fusion	UAV	Mobile	Land	Experiment	Francisco et al. (2015)
Polytopic Observer	UAV	Mobile	Land	Simulation	Souad et al. (2017)
April-Tag	MAV	Mobile	Land	Experiment	Borowczyk et al. (2015)
April-Tag	Quadrotor	Mobile	Land	Experiment	Aleix et al. (2020)
Sensor Fusion	Quadrotor	Mobile	Aquatic	Experiment	Linnea et al. (2019)
Visual servo	Quadrotor	Mobile	Aquatic	Experiment	Wynn et al. (2019)

. The tracking model predictive static programming (T-MPSP) guidance and dynamic reverse autopilot were used to guide the unmanned aerial vehicle (UAV) for autonomous landing (Amit K et al., 2016) [6]. The devices, altimeter and inertial measurement unit (IMU) were used to obtain the relative estimation, and the UAV helicopter was guided to land on the field mobile platform by multi-sensor fusion strategy (Francisco et al., 2015) [7]. A functional unknown input observer in the polygenic Takagi Sugeno framework can make the quadrotor land on a mobile platform (Souad et al., 2017) [8]. By integrating Navigation Satellite Timing and Ranging Global Position System (GPS) and IMU information based on an April Tag visual reference system, the micro aerial vehicles (MAVs) was guided to land on the roof at 50 km/h (Borowczyk et al., 2017) [9]. The April Tag visual reference system was used to guide the UAV to land on a moving Automated Guided Vehicle (AGV) (Aleix et al., 2020) [10]. Based on the information sharing between the UAV and the unmanned surface vehicle (USV), the quadrotor could land on the sea (Linnea et al., 2019) [11]. The status of the landing platform was estimated by using GPS and visual measurement, and the quadrotor was able to land on the water canoe platform (Wynn et al., 2019) [12].

The state of the surface float of the wave glider is changed more frequently and significantly than that of landing platform on the land. The dual-body structure of the wave glider and its own underactuated characteristics make the attitude of the surface float become more unstable. Moreover, a flight controller with a strong anti-interference ability, fast response and a prominent signal tracking ability are required, when the quadrotor lands on the surface boat of the moving wave glider.

The control systems of the quadrotor have been extensively studied, consisting of nonlinear PID (NLPID) controller [13], adaptive neuro-fuzzy inference system [14], backstepping and dynamic inversion control [15], and model predictive control (MPC) [11].

To solve the control problems of nonlinear uncertain system and anti-disturbance capability, a lumped perturbation observer-based control method was presented by using a novel ex-tended multiple sliding surface for matched and unmatched uncertain nonlinear systems [16]. A novel fast terminal sliding mode control technique was presented based on the disturbance observer, which is recommended for the stabilization of underactuated robotic systems [17]. Although the sliding mode control has achieved good control effect, for the quadrotor control system, the input saturation phenomenon exists in the system under the sliding mode control, which will cause some impact on the actual control effect. Therefore, based on the above references, anti-saturation sliding mode control is introduced to achieve good control effect in the actual control of quadrotor. The ADRC has been widely applied for the attitude control of the UAV automatic landing on aircraft carrier [18], the fault-tolerant control of the hypersonic vehicle under multiple disturbances and actuator failures [19], helicopter trajectory tracking control [20], as well as quadrotor control. A double closed-loop ADRC was proposed to control the quadrotor, which could track the desired trajectory of the quadrotor efficiently and accurately [21]. Based on an ADRC and an embedded model control (EMC), a digital attitude control method for the quadrotor was proposed and the effectiveness was verified by simulation [22]. A minimum nominal model was adopted to predict and compensate for internal uncertainty and external disturbances, and a ADRC and a disturbance observer-based control (DOBC) were designed. As revealed from the simulation results, the proposed controller successfully remedied disturbances [20]. The finite-time convergent extended state observer and the chaotic grey wolf optimization algorithm were used to optimize the parameters of the ADRC. Furthermore, the effectiveness and robustness of the algorithm were verified through simulation [23].

This paper develops a quadrotor landing control method based on a series of ADRC. Moreover, an adaptive step size crow search algorithm is proposed to optimize the attitude parameters of the quadrotor when disturbed by the outside environment. The main contributions of this paper are as follows: (1) A series active disturbance rejection controller is designed, and the stability of the system is proofed by Lyapunov theorem. (2) The adaptive step size crow search algorithm is used to optimize the parameters of the active disturbance rejection controller, and the disturbance attributed to the unsteady winds and sea surface effect is compensated to improve the robustness of the control system.

This paper is organized as follows: In Section 2, the mathematical model of the quadrotor system is presented, and a series ADRC is designed. The stability of the cascade ADRC is verified by Lyapunov theorem. In Section 3 an active disturbance rejection controller is introduced, which is optimized by adopting an improved crow search algorithm. In Section 4, the stability and robustness of series ICADRC system for the quadrotor are verified by simulation and experiment. The conclusions are drawn in Section 5.

2. Control Models of the Quadrotor

2.1. Dynamic Model of Quadrotor

The attitude and motion of the quadrotor can be regulated by altering the rotation speeds of different rotors. To gain more insights into the control effect of the ICADRC in the presence of winds and sea waves, the full dynamic model of the quadrotor is built by referencing existing research results [24,25,26]. Figure 1 gives the coordinate frames of the quadrotor.

The centroid of the quadrotor is set as the origin of the body coordinate system, and the body coordinate frame (system frame, B) and the geodetic coordinate frame (E) are built. It is assumed that the attitude vector $E_o={\left(\xi ,\eta ,\zeta \right)}^{\mathrm{T}}$ of the quadrotor in the navigation coordinate system is used for the conversion from the system coordinate system to the inertial coordinate system. For the convenience, $c()$ represents the cosine function and $s()$ represents the sine function. The position and the attitude of the quadrotor can be described by ${\rm M}={\left(\varphi ,\theta ,\psi \right)}^{\mathrm{T}}$. The body coordinate system of the quadrotor is set as the starting coordinate system.

The coordinate transformation relationship between the navigation coordinate system $O{X}_e{Y}_e{Z}_e$ and the body coordinate system $O{X}_b{Y}_b{Z}_b$ can be determined by the rotation matrix of the body as

$$\left(\begin{array}{c}{\mathit{X}}_b\\ {\mathit{Y}}_b\\ {\mathit{Z}}_b\end{array}\right)=R\left(\varphi ,\theta ,\psi \right)\left(\begin{array}{c}{\mathit{X}}_e\\ {\mathit{Y}}_e\\ {\mathit{Z}}_e\end{array}\right)$$

(1)

$$R\left(\varphi ,\theta ,\psi \right)=R_z\left(\psi \right)R_y\left(\theta \right)R_x\left(\varphi \right)$$

(2)

$$R\left(\varphi ,\theta ,\psi \right)=\left(\begin{array}{ccc}c\left(\theta \right)c\left(\psi \right)& s\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)-c\left(\varphi \right)s\left(\psi \right)& c\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)+s\left(\varphi \right)s\left(\psi \right)\\ c\left(\theta \right)s\left(\psi \right)& s\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)+c\left(\varphi \right)s\left(\psi \right)& c\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)-s\left(\varphi \right)c\left(\psi \right)\\ -s\left(\theta \right)& s\left(\varphi \right)c\left(\theta \right)& c\left(\varphi \right)c\left(\theta \right)\end{array}\right)$$

(3)

The rotation matrix of the quadrotor in the inertial coordinate system can be determined as

$$R_{b/w}=R\left(\varphi ,\theta ,\psi \right)=\left(\begin{array}{ccc}c\left(\theta \right)c\left(\psi \right)& c\left(\theta \right)s\left(\psi \right)& -s\left(\theta \right)\\ s\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)-c\left(\varphi \right)s\left(\psi \right)& s\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)+c\left(\varphi \right)s\left(\psi \right)& s\left(\varphi \right)c\left(\theta \right)\\ c\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)+s\left(\varphi \right)s\left(\psi \right)& c\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)-s\left(\varphi \right)c\left(\psi \right)& c\left(\varphi \right)c\left(\theta \right)\end{array}\right)$$

(4)

where $R_{b/w}$ represents the rotation matrix from the body coordinate system to the navigation coordinate system, and the body coordinate system is used as the starting coordinate system.

In the quadrotor coordinate system, the three-axis accelerations $a_x$, $a_y$ and $a_z$ can be determined by the acceleration sensor. During the variable speed motion of the quadrotor, the three-axis acceleration vector is the vector of vehicle acceleration and gravity as well as the component of R on the three axes satisfies $R=\sqrt{a_x{}^2+a_y{}^2+a_z{}^2}$, the acceleration is normalized

$$\{\begin{array}{c}{R}_x=a_x/\left|R\right|\\ R_y=a_y/\left|R\right|\\ R_z=a_z/\left|R\right|\end{array}$$

(5)

The normalized acceleration vector $R_a$ at the sampling moment can be expressed as

$$\left|R_a\right|=\sqrt{R_x{}^2+R_y{}^2+R_z{}^2}=1$$

(6)

Combining Equation (4) with Equation (6), Equation (1) can be rewritten as

$$\left(\begin{array}{c}{R}_x\\ R_y\\ R_z\end{array}\right)=\left(\begin{array}{ccc}c\left(\theta \right)c\left(\psi \right)& s\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)-c\left(\varphi \right)s\left(\psi \right)& c\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)+s\left(\varphi \right)s\left(\psi \right)\\ c\left(\theta \right)s\left(\psi \right)& s\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)+c\left(\varphi \right)s\left(\psi \right)& c\left(\varphi \right)s\left(\theta \right)c\left(\psi \right)-s\left(\varphi \right)c\left(\psi \right)\\ -s\left(\theta \right)& s\left(\varphi \right)c\left(\theta \right)& c\left(\varphi \right)c\left(\theta \right)\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$$

(7)

The relationship between the components of the gravity vector and the corresponding attitude angles can be simplified as

$$\{\begin{array}{l}{R}_x=-s\left(\theta \right)\hfill \\ R_y=s\left(\varphi \right)c\left(\theta \right)\hfill \\ R_z=c\left(\varphi \right)c\left(\theta \right)\hfill \end{array}$$

(8)

Inverse transformation of Equation (8) can be expressed as

$$\{\begin{array}{l}\theta =\arcsin \left(-R_x\right)\hfill \\ \varphi =\arcsin \left(R_{\mathrm{y}}/c\left(\theta \right)\right)\hfill \end{array}$$

(9)

To avoid singularity when $\theta \in \left(-\mathsf{\pi }/2,\mathsf{\pi }/2\right)$, it should be ensured that $\varphi =0$°, while the inverse sine function takes a high linearity among $\left(-\mathsf{\pi }/4,\mathsf{\pi }/4\right)$ and the attitude angle should also be restricted in the control system when the quadrotor is controlled.

The six-degree-of-freedom dynamic model of the quadrotor can be expressed as [26]

$$\{\begin{array}{l}\begin{array}{c}\{\begin{array}{l}\dot{x}=c(\psi )c(\theta )u+[c(\psi )s(\varphi )s(\theta )-c(\varphi )s(\psi )]v+[s(\varphi )s(\psi )+c(\varphi )c(\psi )s(\theta )]w\hfill \\ \dot{u}=rv-qw-gs(\theta )+f_{wx}/m\hfill \end{array}\\ \{\begin{array}{l}\dot{y}=c(\theta )s(\psi )u+[c(\varphi )c(\psi )+s(\varphi )s(\psi )s(\theta )]v+[c(\varphi )s(\psi )s(\theta )-c(\psi )s(\varphi )]w\hfill \\ \dot{v}=-ru+pw+gs(\varphi )c(\theta )+f_{wy}/m\hfill \end{array}\end{array}\hfill \\ \{\begin{array}{l}\dot{z}=-s(\theta )u+c(\theta )s(\varphi )v+c(\varphi )c(\theta )w\hfill \\ \dot{w}=qu-pv+gc(\theta )c(\varphi )+(f_{wz}-f_t)/m\hfill \end{array}\hfill \\ \{\begin{array}{l}\dot{\varphi }=p+[s(\varphi )q+rc(\varphi )]t(\theta )\hfill \\ \dot{p}=[(I_{yy}-I_{zz})rq+({\tau }_x+{\tau }_{wx})]/I_{xx}\hfill \end{array}\hfill \\ \{\begin{array}{l}\dot{\theta }=c(\varphi )q-s(\varphi )r\hfill \\ \dot{q}=[(I_{zz}-I_{xx})pr+({\tau }_y+{\tau }_{wy})]/I_{yy}\hfill \end{array}\hfill \\ \{\begin{array}{l}\dot{\psi }=[s(\varphi )q+c(\varphi )r]/c(\theta )\hfill \\ \dot{r}=[(I_{xx}-I_{yy})pq+({\tau }_z+{\tau }_{wz})]/I_{zz}\hfill \end{array}\hfill \end{array}$$

(10)

The input of the system is defined as [26]

$$\{\begin{array}{l}{f}_t=b({\Omega }_1^2+{\Omega }_2^2+{\Omega }_3^2+{\Omega }_4^2)\hfill \\ {\tau }_x=bl({\Omega }_3^2-{\Omega }_1^2)\hfill \\ {\tau }_y=bl({\Omega }_4^2-{\Omega }_2^2)\hfill \\ {\tau }_z=d({\Omega }_2^2+{\Omega }_4^2-{\Omega }_1^2-{\Omega }_3^2)\hfill \end{array}$$

(11)

The parameters in Equations (10) and (11) are given in

Table 2. Parameters description.

Parameters	Description	Units
[x y z]	Linear position vector	m
$[\varphi \theta \psi$]	Angular position vector	rad
[u v w]	Linear velocity vector	m/s
[p q r]	Angular velocity vector	rad/s
[IxxIyyIzz]	Moment of inertia vector	kg∙m2
Wt	Total thrust generated by rotors	N
$[T_{\varphi }{T}_{\theta }{T}_{\psi }$]	Control torques	N∙m
[WwxWwyWwz]	Wind force vector	N
g	Gravitational force	m/s2
m	Total mass	kg
[Ω1 Ω2 Ω3 Ω4]	Rotors speeds vector	rad/s
b	Thrust coefficient	N∙s2
l	Motor to center length	m
d	Drag coefficient	N∙m∙s2

.

Due to the disturbance of the quadrotor during landing, the second-order system equation of the quadrotor with disturbance can be expressed as

$$\{\begin{array}{l}\dot{u}=rv-qw-gs(\theta )+f_{wx}/m+\Delta x\hfill \\ \dot{v}=-ru+pw+gs(\varphi )c(\theta )+f_{wy}/m+\Delta y\hfill \\ \dot{w}=qu-pv+gc(\theta )c(\varphi )+(f_{wz}-f_t)/m+\Delta z\hfill \\ \dot{p}=\left[\right(I_{yy}-I_{zz})rq+({\tau }_x+{\tau }_{wx}\left)\right]/I_{xx}+\Delta \varphi \hfill \\ \dot{q}=\left[\right(I_{zz}-I_{xx})pr+({\tau }_y+{\tau }_{wy}\left)\right]/I_{yy}+\Delta \theta \hfill \\ \dot{r}=\left[\right(I_{xx}-I_{yy})pq+({\tau }_z+{\tau }_{wz}\left)\right]/I_{zz}+\Delta \psi \hfill \end{array}$$

(12)

where $\Delta i(i=x,y,z,\varphi ,\theta ,\psi )$ denotes the nonlinear partial disturbance and external disturbance of gyro effect and coupling term in the quadrotor system.

2.2. Control System of the Quadrotor

The dynamic landing of the quadrotor on the moving surface float of the wave glider at sea should located by using the April Tag visual reference system. To ensure the stability and accuracy of landing, the control system of quadrotor needs both position control and attitude control. According to Equation (10), the quadrotor is a second-order nonlinear system with 6 degrees of freedom and 4 control inputs, whereas it is difficult for general controllers to estimate the unknown disturbance of the nonlinear system. However, an ADRC can effectively solve the mentioned problem, and the series ADRC model of the quadrotor is designed, which is shown in Figure 2.

In the cascaded self-tampering system, the input of the inner-loop attitude control is the desired attitude angle $\left({\varphi }_d,{\theta }_d,{\psi }_d\right)$ and the practical attitude angle $\left(\varphi ,\theta ,\psi \right)$. The control torques of the system can be adjusted by the altitude values obtained by the April Tag visual reference system. The input of the outer-loop position control is the desired position information $\left(x_d,y_d,z_d\right)$ and the real position information $\left(x,y,z\right)$, which can be solved by the April Tag visual reference system.

The state equation of the third-order linear system after the expansion of the system with internal and external perturbations can be expressed as

$$\{\begin{array}{l}{\dot{x}}_1=x_2\hfill \\ \begin{array}{l}{\dot{x}}_2=x_3+bu\left(t\right)\\ {\dot{x}}_3=\dot{f}\left(\right)\\ y=x_1\end{array}\hfill \end{array}$$

(13)

where $f\left(\right)=\left[\begin{array}{c}\left[\right(I_{yy}-I_{zz})rq+{\tau }_{wx}]/I_{xx}+\Delta \varphi \\ \left[\right(I_{zz}-I_{xx})pr+{\tau }_{wy}]/I_{yy}+\Delta \theta \\ \left[\right(I_{xx}-I_{yy})pq+{\tau }_{wz}]/I_{zz}+\Delta \psi \end{array}\right]$, $x_1=[\varphi ,\theta ,\psi {]}^{\mathrm{T}}$, $x_2=[\dot{\varphi },\dot{\theta },\dot{\psi }{]}^{\mathrm{T}}$, $b=\left[\begin{array}{ccc}\frac{l}{I_{xx}}& & \\ & \frac{l}{I_{yy}}& \\ & & \frac{1}{I_{xx}}\end{array}\right]$.

$f\left(\right)$ is the sum of the all perturbations, b, u(t) and $\Delta i(i=\varphi ,\theta ,\psi )$ are the gain, the control input and the perturbation of the system, respectively.

The design principles of the controller for six channels of the quadrotor are similar in that the control amount of the target can be determined based on the desired input signals $\left(x_d,y_d,z_d,{\varphi }_d,{\theta }_d,{\psi }_d\right)$. The ${\theta }_d$ acts as an example to illustrate the design process of the controller in this paper.

(1) Fast overshoot-free tracking of a given input signal x while obtaining its differential signal can be achieved using the TD. The TD takes the form of

$$\{\begin{array}{l}{\dot{x}}_1=v_2\hfill \\ {\dot{v}}_2=fhan\left(v_1-v_2,r_0,h_0\right)\hfill \end{array}$$

(14)

where $v_{\theta 1}$ is the tracking signal of the input signal ${\theta }_d$; $r_0$ is the speed factor, which determines the speed of convergence; h₀ is the filtering factor, the larger h is, the better the filtering effect of TD, but it will bring greater phase delay; $fhan\left(v_{\theta 1}-{\varphi }_d,v_{\theta 2},r_0,h_0\right)$ is the speed control synthesis function.

(2) The ESO is adopted to estimate the total perturbation of the system. The ESO can estimate the state variables of the system from the input and output of the system, while obtaining an estimate of the total perturbation of the model due to internal and external perturbations. The third-order nonlinear ESO can be expressed as

$$\{\begin{array}{l}{\epsilon }_1=z_1-y\hfill \\ {\dot{z}}_1=z_2-{\beta }_1{\epsilon }_1\hfill \\ {\dot{z}}_2=z_3-{\beta }_{2}fal({\epsilon }_1,{\alpha }_1,\delta )+bu\hfill \\ {\dot{z}}_3=-{\beta }_{3}fal({\epsilon }_1,{\alpha }_2,\delta )\hfill \end{array}$$

(15)

where $z_{\theta 1}$ is the observed value of the $\theta$-channel practical output $y_{\theta }$. $z_{\theta 2}$ is the first-order differentiation of the tracking $y_{\theta }$. $z_{\theta 3}$ is the total perturbation estimate, i.e., the expansion state. ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$ represent the observer gain parameters, all greater than zero, which mainly reduce the convergence rate of the ESO. e is the tracking error of the ESO on the output signal. ${\alpha }_1$, ${\alpha }_2$ and $\delta$ are the nonlinear parameters. $fal()$ is a nonlinear function, which can be expressed as

$$fal(e,\alpha ,\delta )=\{\begin{array}{l}{\left|e\right|}^{\alpha }\mathrm{sgn}[e],\left|e\right|>\delta \hfill \\ e/{\delta }^{1-\alpha },\left|e\right|\le \delta \hfill \end{array}$$

(16)

Since the segment function is not derivable, to facilitate analysis, let $\delta =0$. According to the limit theorem, it can be deduced that $fal(e,\alpha ,\delta )$ is continuous in the domain of definition, which can expressed as

$$fal(\epsilon ,\alpha ,\delta )={\left|\epsilon \right|}^{\alpha }\mathrm{sgn}\left[\epsilon \right]$$

(17)

(3) The NLSEF is based on the system tracking error generated by the TD and the ESO, a virtual control quantity $u_0$ is generated by nonlinear combination, and the disturbance estimation $z_{\theta 3}$ is compensated to obtain the final output quantity. The NLSEF can be expressed as

$$\{\begin{array}{l}{e}_1=v_{\theta 1}-z_{\theta 1}\hfill \\ e_2=x_{\theta 2}-z_{\theta 2}\hfill \\ u_0=k_{1}fal(e_1,{\alpha }_1,{\delta }_0)+k_{2}fal(e_1,{\alpha }_2,{\delta }_0)\hfill \\ u=u_0-z_{\theta 3}/b\hfill \end{array}$$

(18)

where $e_1$ is the deviation error between $v_{\theta 1}$ and $z_{\theta 1}$, $e_2$ is the differential error between $v_{\theta 1}$ and $z_{\theta 1}$; $u_0$ is the uncompensated control quantity, u is the control quantity after perturbation compensation; and $k_1$ and $k_2$ are the proportional gain and differential gain of the controller, respectively.

2.3. Stability Analysis

The ESO is the core of the ADRC. The ADRC only needs the input and output information of the system. While getting the estimation of each state variable of the system, it can estimate the internal and external disturbances of the system and perform state compensation in the feedback by ESO. The stability of the ADRC controller can be proved by the stability of the ESO.

Let ${\dot{e}}_1=z_{\theta 1}-y_{\theta }$, ${\dot{e}}_2=z_{\theta 2}-v_{\theta 2}$, ${\dot{e}}_2=z_{\theta 3}$, according to Equation (13) to Equation (15).

The observation error of the system can be expressed as

$$\{\begin{array}{l}{\dot{e}}_1=e_2-{\beta }_1{e}_1\hfill \\ {\dot{e}}_2=e_3-{\beta }_{2}fal(e_1,{\alpha }_1,\delta )\hfill \\ {\dot{e}}_3=f\left(\right)-{\beta }_{3}fal(e_1,{\alpha }_2,\delta )\hfill \end{array}$$

(19)

When the perturbation of the system is zero, the error of the third-order dilated state observer can be expressed as

$$\{\begin{array}{l}{\dot{e}}_1=e_2-{\beta }_1{e}_1\hfill \\ {\dot{e}}_2=e_3-{\beta }_{2}fal(e_1,{\alpha }_1,\delta )\hfill \\ {\dot{e}}_3=-{\beta }_{3}fal(e_1,{\alpha }_2,\delta )\hfill \end{array}$$

(20)

Equation (20) can be rewritten as

$$\dot{e}=-\mathit{A}e$$

(21)

where $e=\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right]$ and $\mathit{A}=\left[\begin{array}{ccc}{\beta }_1& -1& 0\\ {\beta }_2\frac{fal(e_1,{\alpha }_1,\delta )}{e_1}& 0& -1\\ {\beta }_3\frac{fal(e_1,{\alpha }_2,\delta )}{e_1}& 0& 0\end{array}\right]$.

Let the antisymmetric matrix L be

$$\mathit{L}=\left[\begin{array}{ccc}{l}_{11}& l_{12}& l_{13}\\ -l_{12}& l_{22}& l_{23}\\ -l_{13}& -l_{23}& l_{33}\end{array}\right]$$

(22)

If L is symmetric with respect to the product of A, the Lyapunov function of the simplified system with Equation (21) can be chosen as

$$V={\displaystyle {\int }_0^t(\mathit{L}\mathit{A}}e,\dot{e})d\tau$$

(23)

Theorem: If the principal diagonal elements of the matrix L are positive and the symmetric matrix $LA$ satisfies the positive definiteness condition of the matrix. The zero solution of the simplified system with Equation (7) is Lyapunov stable and thus the system will converge to the equilibrium point and is ultimately Lyapunov asymptotically stable.

Proof. 

For the matrix $LA$ there exists:

$$\mathit{L}\mathit{A}=\left[\begin{array}{ccc}{L}_{11}& -l_{11}& -l_{12}\\ L_{21}& l_{12}& -l_{22}\\ L_{31}& l_{13}& l_{23}\end{array}\right]$$

(24)

where $\left[\begin{array}{c}{L}_{11}\\ L_{21}\\ L_{31}\end{array}\right]=\left[\begin{array}{c}{l}_{11}{\beta }_1+l_{12}{\beta }_2\frac{fal(e_1,{\alpha }_1,\delta )}{e_1}+l_{13}{\beta }_3\frac{fal(e_1,{\alpha }_2,\delta )}{e_1}\\ -l_{12}{\beta }_1+l_{22}{\beta }_2\frac{fal(e_1,{\alpha }_1,\delta )}{e_1}+l_{23}{\beta }_3\frac{fal(e_1,{\alpha }_2,\delta )}{e_1}\\ -l_{13}{\beta }_1-l_{23}{\beta }_2\frac{fal(e_1,{\alpha }_1,\delta )}{e_1}+l_{33}{\beta }_3\frac{fal(e_1,{\alpha }_2,\delta )}{e_1}\end{array}\right]$ and $L_{11}>0$. □

The conditions for the symmetry matrix $LA$ are

$$\{\begin{array}{c}{L}_{21}=-l_{11}\\ L_{31}=-l_{12}\\ l_{13}=-l_{22}\end{array}$$

(25)

The matrix $LA$ are positive definite, then

$$\left|\begin{array}{cc}{L}_{11}& -l_{11}\\ L_{21}& l_{12}\end{array}\right|>0$$

(26)

$$\left|\begin{array}{ccc}{L}_{11}& -l_{11}& -l_{12}\\ L_{21}& l_{12}& -l_{22}\\ L_{31}& l_{13}& l_{23}\end{array}\right|>0$$

(27)

Assumption 1.

The expression M satisfies$M=\frac{fal\left(e_1,\alpha ,\delta \right)}{e_1}$and the constant α satisfies$\alpha ={\alpha }_1={\alpha }_2$.

According to Equation (16), M should be bounded.

According to Equation (25), it can be determined

$$l_{12}=l_{13}{\beta }_1+l_{23}{\beta }_{2}M+l_{13}{\beta }_{3}M$$

(28)

Combining Equation (25) with Equation (28) yields

$$l_{23}=\frac{l_{11}-l_{13}{\beta }_1^2-l_{13}{\beta }_1{\beta }_{3}N+l_{22}{\beta }_{2}M}{M({\beta }_1{\beta }_2-{\beta }_3)}$$

(29)

For the convenient analysis, $l_{13}$ is taken to be a negative number that tends to 0 infinitely. Due to $L_{11}>0$, substituting Equation (28) into Equation (29) yields

$$L_{11}\approx {\beta }_1+\frac{{\beta }_2{}^{2}M}{{\beta }_1{\beta }_2-{\beta }_3}>0$$

(30)

Assumption 2.

The parameters${\beta }_1$,${\beta }_2$, and${\beta }_3$are constants and meet the condition

$${\beta }_1{\beta }_2-{\beta }_3<0$$

(31)

It is possible to get

$${\beta }_1({\beta }_1{\beta }_2-{\beta }_3)+{\beta }_2{}^{2}M<0$$

(32)

Taking $l_{13}$ as a negative number that tends to 0 infinitely, Equations (28) and (29) can be simplified as

$$l_{12}\approx l_{23}{\beta }_{2}M$$

(33)

$$l_{23}\approx \frac{1}{({\beta }_1{\beta }_2-{\beta }_3)M}$$

(34)

and

$$\{\begin{array}{c}{l}_{12}<0\\ l_{23}<0\end{array}$$

(35)

This contradicts the nature of the symmetric matrix and eliminates the condition.

Assumption 3.

The parameters${\beta }_1$,${\beta }_2$, and${\beta }_3$are constants and meet the condition

$${\beta }_1{\beta }_2-{\beta }_3>0$$

(36)

According to Equation (26), it can be determined as

$$L_{11}{l}_{12}>l_{11}{}^2$$

(37)

Taking $l_{11}=1$, according to Equations (28)–(30), Equation (37) can be simplified as

$$1+\frac{M{\beta }_2{}^3}{{({\beta }_1{\beta }_2-{\beta }_3)}^2}+\frac{{\beta }_3}{{\beta }_1{\beta }_2-{\beta }_3}>0$$

(38)

where ${\beta }_1{\beta }_2-{\beta }_3>0$.

Likewise, Equation (27) can be satisfied under ${\beta }_1{\beta }_2-{\beta }_3>0$.

According to the mentioned conditions, the third-order ESO is asymptotically stable when perturbation is 0 and ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are positive and ${\beta }_1{\beta }_2-{\beta }_3>0$.

3. Parameter Optimization of the Series ADRC

Although the ADRC can significantly suppress disturbances of the system and improve the stability of the controlled object, some parameters need be adjusted, which is difficult. The selection of parameters directly affects the performance of the ADRC, so various methods such as empirical method, neural network tuning method and swarm intelligence algorithm were proposed to optimize the parameters of the ADRC. The empirical tuning method varies from person to person and has great randomness. The neural network method is computationally intensive and the activation function is not easy to choose, and parameter adjustment is difficult. The swarm intelligence algorithm is simple and robust and can maximally approximate the optimal solution of the controller parameters. Thus, it is suitable to adjust the parameters of the controller.

3.1. Improved Crow Search Algorithm

Crow search algorithm (CSA) is a novel swarm intelligence algorithm proposed by Askarzadeh [27]. Compared with conventional intelligent algorithms, crow search algorithm is faster and more effective in multi-parameter optimization than conventional swarm intelligence algorithms such as particle swarm algorithm (PSO) and genetic algorithm (GA). As indicated from considerable experiments, the CSA algorithm can achieve effective results when solving parameters in engineering applications. However, conventional crow search algorithm is easy to fall into the local optimal solution as impacted by the fixed search step length. Therefore, the crow flight distance is adaptively adjusted and the adaptive adjustment of the step length is used to replace the random step length to reduce the possibility of the algorithm falling into the local optimal and improve the search effect in this paper.

The adaptive step size adjustment strategy can be expressed as

$$f{l}_i=f{l}_{\min }+(f{l}_{\max }-f{l}_{\min })\cdot s_i$$

(39)

where $f{l}_{\max }$ and $f{l}_{\min }$, respectively, represent the maximum and minimum flight lengths of the crow, respectively.

$s_i$ is defined as

$$s_i=\frac{\Vert x^{i,iter}-m^{j,iter}\Vert }{s_{\max }}$$

(40)

where $x^{i,iter}$ represents the location of the crow i, $m^{j,iter}$ expresses the location with crow j hiding food. $s_{\max }$ represents the maximum distance between the crow i and the location where food may be hidden. iter is the ith iteration. The improved CSA (ICSA) can be expressed as

$$x_i^{i,t+1}=\{\begin{array}{l}{x}^{i,iter}+r_i\times f{l}^{i,iter}\times (m^{j,iter}-x^{i,iter})r_j\ge A{P}^{j,iter}\hfill \\ x^{i,iter}+f{l}_{\min }^{i,iter}+(f{l}_{\max }^{i,iter}-f{l}_{\min }^{i,iter})\times s_i{r}_j<A{P}^{j,iter}\hfill \end{array}$$

(41)

where $r_j$ represents a random number between 0 and 1. $A{P}^{j,iter}$ represents the perception probability of the crow j.

With the adaptive step size instead of randomly selecting positions, the ICSA is capable of improving the convergence at the early stage, reducing the limitation of falling into the local optimum, accelerating the convergence, as well as improving the adaptability.

3.2. Parameter Tuning Optimization of the ADRC

The main parameters of the second-order self-tampering controller for the θ-channel include the tracking differentiator (TD) of $(r_0,h_0)$, the ESO $({\alpha }_{01},{\alpha }_{02},{\beta }_1,{\beta }_2,{\beta }_3,\delta )$ and the NLSEF $(k_1,k_2,{\alpha }_1,{\alpha }_2,{\delta }_0)$. The five parameters $({\beta }_1,{\beta }_2,{\beta }_3,k_1,k_2)$ should be optimized and tuned, and the remaining parameters can be determined with empirical estimation and will not be changed during the optimization.

The population search space of the ICSA is set to five dimensions, and the respective position of the crow can be represented as a set of parameters (${\beta }_1,{\beta }_2,{\beta }_3,k_1,k_2$).

$$x^{i,iter}=({\beta }_1,{\beta }_2,{\beta }_3,k_1,k_2)$$

(42)

The values of the fitness function are used to assess the merit of the optimal position of individual crows in the crow search algorithm, and the choice of the fitness function will directly affect the convergence speed of the algorithm and the selection of the optimal solution.

With the pitch angle of the quadrotor as an example, the following requirements are raised for the control system of the quadrotor from the consideration of the practical landing requirements.

(1) The time-integrated performance index for reducing the absolute value of the pitch angle error is used to obtain a higher braking accuracy that allows the quadrotor to remain stable during the landing process.

(2) The change rate of pitch angle to avoid the quadrotor from swinging back and forth significantly in a short time.

Referring to the Integrated Time and Absolute Error (ITAE) criterion, the adaptation function is designed by combining the practical requirements of the quadrotor control system, which can be expressed as

$$J={\displaystyle \int \left\{w_1\cdot [t\cdot ({\theta }_d-\theta \left(k\right))]+w_2\cdot |\theta (k)-\theta (k-1)|\right\}}dt$$

(43)

where J is the current fitness function value and $\varphi \left(k\right)$ is the value of the practical controlled quantity at the kth iteration. $w_i(i=1,2)$ is the weight value of each indicator item, which can be selected according to the actual control objectives.

4. Simulations and Results

Under the MATLAB/Simulink environment, the quadrotor model with the ICADRC is introduced. Figure 3 illustrates the schematic diagram of the improved CSA with optimized θ-channel active disturbance rejection controller. The performance of the proposed ICADRC and landing control on the surface boat of the wave glider are evaluated and verified.

4.1. The Improved CSA Simulation

The control experiments are performed with the GA, the PSO, the CSA and improved CSA algorithm (ICSA) with dimension 5, population size 100 and iteration number 50. The results of iterations of fitness values of the four optimization algorithms are shown in Figure 4. As indicated from the figure, after 200 iterations, all four algorithms will converge to their respective optimal values. Among these algorithms, the ICSA algorithm converges relatively fast, and the results after 5 iterations are significantly better than those of the GA and the PSO after 50 iterations. It is shown that the CSA has better performance than conventional search algorithms when finding the optimal solution, and it can improve the search capability after adopting the adaptive step.

To assess and verify the tracking performance of the proposed algorithm, the simulation and experiment are conducted under conventional PID control, the PID controller optimized by improved crow search algorithm controller (ICPID), the ADRC and the ICADRC.

Table 3. Parameters of the test quadrotor.

Parameter	Description	Value	Unit
Ixx	Moment of inertia vector of x	5.794 × 10−3	kg m2
Iyy	Moment of inertia vector of y	5.794 × 10−3	kg m2
Izz	Moment of inertia vector of z	9.64 × 10−3	kg m2
g	Gravitational force	9.80 × 100	m/s2
b	Thrust coefficient	5.19 × 10−5	N s2
d	Drag coefficient	3.81 × 10−6	N m s2

gives the parameters of the quadrotor used in the simulation environment.

4.2. The Attitude Control Simulation of the Quadrotor

The PID controller, the ICPID controller, the ADRC controller, and the ICADRC controller are used to track and control the same input signals under the identical quadrotor dynamic model, respectively. As shown in Figure 5, a rectangular wave signal with variable amplitude is used as the control command of the desired pitch angle, and the control effects of the PID controller and the ADRC control are similar when the pitch angle remains unchanged. When the desired angle changes suddenly, the angle tracking errors of the ADRC and the ICADRC are small and do not show large overshoot and oscillation, while the angle tracking errors of the PID controller and the ICPID controller show larger overshoot and oscillation. However, the ICPID controller can suppress overshoot and oscillation to some extent.

Since the improved CSA-optimized ADRC shows the advantages of fast convergence, small overshoot and no steady-state error, it can track the upper desired signal smoothly and quickly. More specifically, angle tracking control has significant anti-interference capability, thus the ICADRC can track the input signal more quickly.

Figure 6 shows the tracking control effects of several controllers under the identical on a sinusoidal signal. Compared with the ADRC, the PID controller has a larger angle tracking error and slower tracking speed. Although the ICPID controller can improve the tracking speed and reduce the angle tracking control error, its tracking effect remains weaker than that of the ADRC. Compared with conventional ADRC, the ICADRC has the advantages of smaller tracking error, faster convergence and stable angle change.

According to Figure 5 and Figure 6, the angle tracking performance exhibited by the ADRC is higher than that of conventional PID controller. The ADRC exhibits more significant expected angle tracking performance, smaller tracking error and stronger ability to inhibit oscillation, which shows that the ADRC exhibits better robustness and stability. The ADRC controller after ICSA optimized parameters has better tracking performance and anti-disturbance capability than the conventional ADRC.

Although the ICPID controller can elevate the tracking speed and reduce the angle tracking error, its tracking effect is weaker than that of the ADRC. The ICADRC outperforms the ADRC for its advantages of small tracking error, fast convergence speed and stable angle changes.

Compared with the conventional ADRC, the ICSA can improve the tracking speed and the signal response ability of the ADRC, while improving the robustness and stability of the system in depth. To simulate the effect of winds on the landing of quadrotor and verify the anti-interference ability of the ICADRC, winds as disturbance is introduced into the quadrotor in the simulation. The ICPID controller and the ICADRC are embedded the control system of the quadrotor, respectively.

The responses of the quadrotor without disturbance under different controllers are shown in Figure 7. It can be seen from the simulation results that both the ICPID controller and the ICADRC are capable of maintaining the desired height without disturbance, whereas the ICADRC can track the desired signal smoothly. The ICPID controller has more significant oscillations and longer recovery time after external disturbance, while the ICADRC has smaller disturbance and can recover more efficiently.

To assess the performance of attitude tracking under rigorous disturbance conditions, Gaussian white noise, ranging from −0.2 rad to 0.2 rad, is introduced to the attitude tracking control of the quadrotor. The tracking responses of the quadrotor with disturbance under different controllers are illustrated in Figure 8. It can be seen that the control performance of the ICADRC is better than that of the ICPID controller, and the attitude error is lower. The ICADRC can suppress external disturbance and track the desired signal more smoothly in a short time without overshoot.

4.3. The Position Control Simulation of the Quadrotor

The desired trajectory tracking capability of the quadrotor is simulated with the same dynamic model with different controllers. The starting point of the desired trajectory is set to [0, 0, 0], the starting point of the quadrotor is set to [0.5, 0.5, 0], and the simulation time is set to 16 s. The PID controller, the ICPID controller, the ADRC controller, and the ICADRC controller are employed to track and control the identical desired trajectory, and the space trajectories of the quadrotor under different controllers are illustrated in Figure 9. The position trajectory tracking and trajectory tracking errors of the quadrotor under different controllers are shown in Figure 10, which indicate that the ADRC and the ICADRC track better, respond faster and have less overshoot compared with the ICPID and the PID controllers under the same conditions. The control effect of the ICADRC can be improved over conventional ADRC controller. The trajectory tracking errors of the ADRC and the ICADRC are among [−0.2 m, 0.2 m], while the trajectory tracking errors of the PID and the ICPID are relatively large. According to four controllers, the ICADRC controller exhibits the optimal position control capability, which is more consistent with the practical engineering requirements.

4.4. Sea Trials

The “Black Pearl” wave glider as a platform for quadrotor landing is used in the sea trial as shown in Figure 11. A protective net around the landing platform is adopted to prevent the quadrotor from crashing into the sea when landing failure. The porous pearl cotton and the matte finish of the April Tag visual reference system are used to minimize the impact of sunlight reflection on the quadrotor landing. High friction rubber pads installed on the bottom of the quadrotor can increase the frictional resistance between the quadrotor and the landing platform and prevent the quadrotor from sliding attributed to the shanking of the surface float. During the experiment, GPS positioning and WiFi communication are used to guide the quadrotor to fly around the wave glider, and the April Tag visual reference system is used to guide the quadrotor for further landing. The landing speed and the attitude of the quadrotor during the landing are used as indicators of the superior performance of the controller.

The experiment conditions are wave height of 0.2 m, wind speed of 0.3 m/s, solar radiation intensity of 947 W/m² and sampling frequency of 10 Hz for the quadrotor. Five experiments with different controllers are performed to verify the control performance. During the experiments, if the landing time of the quadrotor is longer than 30 s or the landing platform is failed, the experimental is regarded as a failure. The experimental results are given in

Table 4. Landing results on the sea under different controllers.

Controller	Test Number	Minimum Time	Success Rate
PID	5	15.87 s	20%
ICPID	5	14.93 s	60%
ADRC	5	12.45 s	80%
ICADRC	5	11.97 s	100%

.

The shortest landing time of the quadrotor under different controllers acts as an indicator for comparative analysis. The position changes of the quadrotor during landing are shown in Figure 12. Compared with the PID controller, the altitude change of the quadrotor under the ADRC controller is smoother and shorter, which demonstrates that the ADRC can effectively suppress the influence of external disturbances. After the ADRC controller parameters optimized by the ICSA, the external disturbances can be further suppressed, and the altitude change of the quadrotor is smoother and the landing time can be further shortened.

For the analysis, heat maps are drawn based on the probability distributions of the cross-roll and pitch angles of the quadrotor under different controllers, which is shown in Figure 13. According to the figure, the attitude of the quadrotor under the ADRC controller is more stable than that of the quadrotor under the PID controller. The distribution of the points with a high probability of occurrence is more concentrated in the range of [−4°,4°]. The attitude probability distributions of the ICADRC controller and the ICPID controller are more convergent than those of the ADRC controller and the PID controller, and the ICADRC shows a more concentrated attitude probability distribution.

Table 5. Landing results on the sea.

		PID	ICPID	ADRC	ICADRC
Pitch	SD	6.478	5.914	4.921	4.401
	RMSE	0.024	0.022	0.019	0.015
Roll	SD	3.934	3.341	1.765	1.486
	RMSE	0.015	0.012	0.007	0.006

lists the standard deviation (SD) and the root mean square error (RMSE) of the cross-roll as well as the pitch attitude of the quadrotor under different controllers. Compared with the PID, the ICPID, and the ADRC, the SD of pitch angle of the quadrotor under the ICADRC can be reduced by 2.077°, 1.513° and 0.52°, respectively, and the RMSE can be reduced by 37.5%, 31.8%, and 21.1%, respectively. The SD of traverse angle of the quadrotor under the ICADRC can be reduced by 2.448°, 1.855°, and 0.279°, respectively, and the RMSE can be reduced by 60.0%, 50.0%, and 14.3%, respectively. It is demonstrated that the ICADRC controller is capable of effectively improving the attitude stability of the quadrotor during practical landing on sea.

Figure 14a shows the dynamic landing of the quadrotor on the surface float of the wave glider. According to the figure, the quadrotor with the ICADRC controller can achieve a successful landing on the surface float of the wave glider wobbling in the sea by the guidance of the April Tag visual reference system.

Figure 14b gives the recovery of the quadrotor. The high friction rubber pad can ensure that the quadrotor still rests on the landing platform stably even if there is a certain wobble of the surface float. Meanwhile, the experiments show that the porous pearl cotton and the matte treated April Tag visual reference system can effectively reduce the adverse effect of sunlight reflection on the landing of the quadrotor.

5. Conclusions

The dynamic landing of a quadrotor on the surface float of a wave glider is recognized as a complex and demanding control process with high accuracy. The excellent anti-disturbance controller and tracking of the April Tag visual reference system are crucial. To deal with the weak anti-disturbance capability of the conventional controller in the sea landing of the quadrotor, an improved second-order tandem self-anti-disturbance controller is proposed to realize the inner-loop attitude angular velocity control and outer-loop position control of the quadrotor. The TD can enhance the quadrotor to track the position of the April Tag visual reference system, and the ESO can estimate the total disturbances of the quadrotor at sea. The NLSEF can compensate for the system tracking errors attributed to the TD and the ESO to finally obtain the practical control amount of the quadrotor. The parameters of the second-order ADRC controller are optimized to further enhance its stability and disturbance immunity. To verify the effectiveness and correctness of the proposed controller, simulations and experiments are performed. The results show that the proposed ICADRC has high stability, fast tracking speed, great robustness and prominent anti-interference capability. The combination of the ICADRC controller and the April Tag visual reference system for landing makes it possible to guarantee the success rate of the quadrotor landing on the surface float of the wave glider even in the case of uncertainty float attitude. In the follow-up study, authors consider adding the float attitude prediction of the wave glider to provide more accurate platform attitude information for quadrotor landing and further shorten the landing time of the quadrotor on wave glider float.