Dynamic model and ADRC of a novel water-air unmanned vehicle for water entry with in-ground effect

2.1. Structural description of the water-air unmanned vehicle

The water-air unmanned vehicle with double quadrotor structure mainly consists of double-layer body frame, control box, motor, air power mechanism, and water power mechanism. The core component is the control box, inside of which is installed with control system, batteries, and other components. The control system is a dual processor structure. One of them is for management mission, and the other one is to calculate the attitude and control law. The hardware of the control system is shown in Fig. 1.

Carbon fiber is selected for the body frame to improve the intensity and reduce the weight. When the vehicle works, the motor is driven by a 12 V direct current battery found inside the control box and the maximum rotation speed is approximately 9,000 r/min. The 1050 real carbon fiber propeller and two-stroke D50 underwater propeller are selected as the air power mechanism and water power mechanism, respectively. The control box is cuboid, which is simply processed and installed but increases the drag to a certain extent when the vehicle moves.

When the water-air unmanned vehicle flies in the air, only the air power mechanism works. By contrast, when water-air unmanned vehicle moves underwater, only the water power mechanism works. On the upper surface of the body frame, a turntable is installed. The propeller arms are folded at ordinary time to save space for easy storage. When the vehicle starts to fly, the propeller arms are unfolded and the air power mechanism starts to work. When the vehicle crosses the water surface, the switch between the air and water power mechanisms is achieved by the cooperation of pressure sensor and operator observation. The water entry process of the vehicle should meet certain conditions. The vehicle enters the water vertically because its attitudes in air and underwater are almost the same. In other words, the body of the vehicle should be kept stable in the water entry process (the attitude angle is 0). The design sketch and motion principle of the water-air unmanned vehicle are shown in Fig. 2.

Fig. 1The hardware of control system

Fig. 2The design sketch and motion principle of the water-air unmanned vehicle

2.2. Dynamic model

The major focus of this study is the force of the water-air unmanned vehicle before it enters the water, and the dynamic model of the vehicle flying in air is established. The dynamic model of the normal quadrotor aircraft is given in the literature, and the following is a simple description:

Establish the ground coordinate system $E:\left\{x_e,y_e,z_e\right\}$ and the body coordinate system $B:\left\{x_b,y_b,z_b\right\}$ based on the right-hand principle, as shown in Fig. 3.

Fig. 3Body coordinate system and ground coordinate system

$\left(x,y,z\right)$ are the location coordinates of the body coordinate system, and $\left(\varphi ,\theta ,\psi \right)$ are three Euler angles (roll angle, pitch angle, and yaw angle). The model can be divided into the linear motion subsystem and angular motion subsystem, which are shown as follows:

The transformation matrix from the body coordinate system to the ground coordinate system is shown as follows:

Linear motion dynamic model can be expressed as follows:

where, $\dot{\xi }$ and $V$ are the velocity vector in the ground and body coordinate systems, respectively.

Angular motion dynamic model can be expressed as follows:

where $\dot{\eta }$ and $\omega$ are the angular velocity vector in the ground and body coordinate systems, respectively.

The six degrees of freedom movement of the normal quadrotor aircraft includes three directions of angular movement and three directions of linear movement. When the aircraft flies in high height, the formula of the linear motion is based on the Newton-Euler formula, as follows:

where $m$ is the mass of the quadrotor aircraft, $g$ is the acceleration due to gravity, $e_z={\left(\mathrm{0,0},1\right)}^T$ is the unit vector in the ground coordinate system, and $u_T$ is the total lift produced by four rotors, as follows:

where ${\mathrm{\Omega }}_i$ and $F_i$ are the rotating speed and the lift produced by the $i$th rotor, respectively, and $b$ is the lift coefficient.

The dynamic model of the angular motion can be expressed as follows:

where $I$ is inertia matrix, $-\omega \times I\omega$ and $G_a$are the gyroscopic effect produced by rigid body rotation and the direction change of the propeller, respectively, and τ is the control moment, as follows:

where, $J_r$ is the rotational inertia, $l$ is the distance from motor to the gravity center and $d$ is the drag coefficient.

According to Eqs. (7) and (9), the dynamic model of linear motion and angular motion are as follows respectively:

Therefore, quadrotor aircraft is typically under actuated system with 4 input of $\left(u_T,{\tau }_{\varphi },{\tau }_{\theta },{\tau }_{\Psi }\right)$ and 6 output of $\left(x,y,z,\varphi ,\theta ,\mathrm{\Psi }\right)$. The complete dynamic model is shown as follows:

where:

The expression of control input is as follows:

2.3. IGE

I. C. Cheeseman [5] proposed a general model of the IGE of helicopters, which is shown as follows:

where $r$ is the radius of the propeller, $z$ is the distance from the propeller to the ground, $T_{OGE}$ is the thrust produced by the propeller without IGE, and $T_{IGE}$ is the total thrust IGE. The previously presented theoretical formula is applied to a single-rotor helicopter. For the large difference between helicopter and quadrotor aircraft, the formula cannot be used directly in calculating the IGE of the quadrotor. According to the IGE formula of a normal quadrotor [16], we introduce a correct coefficient based on Eq. (17) that can be applied to the water-air unmanned vehicle with double quadrotor structure, as shown in Eq. (18):

where $T_{INPUT}$ is the required thrust, $T_{OUTPUT}$ is the real thrust produced by the vehicle, $r_r$ is the propeller radius of the water–air unmanned vehicle, $z_r$ is the distance from the vehicle to the water surface, and $\alpha$ is a coefficient that must be determined. $\alpha$ is determined by experiments. The experimental setup is shown in Fig. 4, and mainly functions of each part are introduced.

Fig. 4IGE experimental setup

(1) Power system.

The power system provides the motor power, which consists of DC power supply, brushless motor and speed governor. The rated voltage of the supply is 15 V and the rotor is inner rotor brushless motor.

(2) Operation system.

The operation system adjusts the input current of the bruthless DC motor by changing the duty cycle of PWM control pulse, which can realise the control of motor rotor speed.

(3) Measurement system.

The measurement system is used to measure the speed, thrust and power consumption of the rotor wings. Tachometer is used to measure the speed. Press sensor is applied to measure the thrust and its precision is –0.019 V/10 g. The power consumption is obtained by processing the recorded data of voltage and current.

(4) Fixing equipments.

The fixing equipments are fixed at the bottom of the tank which can support the power system. The trestle has a scale which is convenient for reading out the hovering height of the water-air vehicle.

The calculation steps are as follows:

Step 1. Determine the input thrust $T_{INPUT}$ of the vehicle at different heights from the water surface. Typically, these values can be obtained directly. Table 1. shows the relationship between the height where the water-air unmanned vehicle hovers and the thrust it requires.

Step 2. Determine the real thrust $T_{OUTPUT}$ at different heights from the water surface. Under the condition that the vehicle is steadily hovering, the thrust it produces is equal to its gravity, that is, $T_{OUTPUT}=mg$, and $T_{OUTPUT}$ can be regarded as a constant.

Step 3. Fit the value of $\alpha$ through the least square method.

The $r_r$ and $m$ of the water-air unmanned vehicle designed in this study are 0.15 m and 2.57 kg, respectively. In the range of 0.15 m $<z_r<$ 0.5 m, $\alpha$ should be a fitting formula, which can be expressed as:

Fig. 5IGE curve of water-air vehicle

Fig. 6Change of water surface

The experiments are conducted in an indoor pool, which can reduce the influence of the wind and waves in natural waters. We assume that the dimensional factors are $T=T_{INPUT}/T_{OUTPUT}$ and $r=z_r/r_r$, and a smaller $T$ indicates a more obvious IGE. When $T=$ 1, the IGE can be ignored. Fig. 5. shows that, when $r=$ 3.3, the IGE of the water-air unmanned vehicle is still obvious, and thereafter, the thrust changes gradually. I. C. Cheeseman [5] pointed out that the significant interval of the IGE of a single rotor is 0.5 $<r<$2. Li [16] pointed out that the significant interval of the IGE of the normal quadrotor aircraft is 0.5 $<r<$ 4. In this study, the significant interval of the water-air unmanned vehicle is 0.5 $<r<$ 3.3 in the water entry process because, when the vehicle approaches the water surface gradually, it will cause a series of complex disturbances on the still water surface. Part of the air flow between air propellers and the water surface is lost in random directions. As a result, the significant interval of the IGE of the water-air unmanned vehicle in the water entry process is larger than that of the single-rotor helicopter but smaller than that of the normal quadrotor aircraft. Fig. 6. shows the changes of the water surface when $z_r=$ 0.25 m. It can be seen that when the water-air unmanned vehicle approaches to the water surface, below it, the water sags. At the same time, waves spread around.

3.1. Tracking differentiator

where, defined $fst\left(\right)$ as follows [17]:

where:

${\theta }_d$ is the target pitch angle, $v\left(k\right)$ is the input signal at time $k$, $T$ is the integral step of the tracking differentiator, $r$ is the velocity factor, and $h$ is the filtering factor. According to engineering practice, a larger $r$ indicates a shorter transient process and a closer distance to the input signal. A smaller $r$ indicates a longer transient process, which influences the response rate of the system. A larger $h$ indicates a better filtering result, and the phase loss of the tracking signal is larger. In practical application, $h=3T$-$7T$.

3.2. Extended state observer

The definition of $fal\left(\right)$ is as follows:

The extended state observer is the core of the ADRC, which directly affects the control result. The nonlinear shape of $fal\left(\right)$ is determined by $\alpha$. The linear interval width of $fal\left(\right)$ near the origin is determined by $\delta$. If $\delta$ is excessively large, then the approximation capability of the nonlinear signal will weaken. If $\delta$ is excessively small, then it will cause high-frequency chattering. ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are feedback gains of state error, which affect the convergence velocity of ESO.

3.3. Nonlinear state error feedback

$k_1$ and $k_2$ are controller gains. The larger the value of $k_1$, the sooner the system responds. The largeness of $k_1$ may cause divergence of the output. The appropriate increase of $k_2$ can improve the dynamic performance of the system and restrain overshoots in the transient process. $b$ is the compensation factor, which determines the variation range of the estimated value of the disturbance.

3.4. Stability condition of ADRC

In an ADRC system, the role of TD and NLSEF is to accelerate response, and the system stability mainly depends upon ESO [18, 19].

For the second order discrete-time ESO proposed in this paper, when ${\alpha }_1={\alpha }_2=1$, the ESO is linear. That is:

Characteristic equation of the second order discrete-time ESO is:

Assume $z=\omega +1/\omega -1$, and transform $D\left(z\right)$ to $D\left(\omega \right)$. According to the Routh stability criterion, the stability condition of a linear second order discrete-time ESO is:

For a nonlinear second order discrete-time ESO, if $({\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2)$ is determined:

That is the stability condition of ESO.

4.1. Experiment on attitude control

The chief flow chart of attitude control system and program flow chart of attitude control algorithm are shown in Fig. 9. and Fig. 10., respectively.

The initial roll angle of the water-air unmanned vehicle is ${\varphi }_0=$−0.13 rad, the pitch angle is ${\theta }_0=$ 0.038 rad, the yaw angle is ${\psi }_0=$0.074 rad, and all desired attitude angles are 0 rad, which is the stable hovering state. Under the control of the ADRC and the normal PID controller, the change curves of the attitude angle are shown as follows.

Fig. 9Chief flow chart of attitude control system

Fig. 10Program flow chart of attitude control algorithm

Fig. 11. shows that the attitude response curve of the ADRC can track the desired value well. Although the PID controller can also track the desired value, a large oscillation exists and a long setting time is needed. The performance indices of the ADRC and the normal PID controller are shown in Table 4. With the roll angle $\varphi$ control as an example, the overshoot of the PID controller is 35 %, approximately four times more than that of the ADRC, which is 7.2 %. The advantage of ADRC in setting time is obvious in the water entry process of the water-air unmanned vehicle. Similar conclusions on the control of pitch angle $\theta$ and yaw $\psi$ angle are obtained. The main reason for the aforementioned results is that a coupling interference exists in the attitude angle control channels, and this interference can be regarded as internal disturbance. ESO has the abilities to estimate and compensate, which enhance the robustness of the system significantly.

Fig. 11Attitude angle change curves of PID controller and ADRC

a) Change curve of roll angle

b) Change curve of pitch angle

c) Change curve of yaw angle

4.2. Experiment on height control

The initial height of the water-air unmanned vehicle is 2 m. When the vehicle hovers steadily at such height, the water entry order starts. Given the double-layer structure and body frame of the water-air unmanned vehicle, when its bottom touches the water, the air rotor is 0.15 m high from the water surface. The curves of the UAV height changing under the control of the PID controller and the ADRC are shown in Figs. 12. and 13.

Fig. 12. shows that, at the early stage of the descent of the vehicle, the tracking trajectory of the PID controller coincides well with the reference trajectory. However, when the height is lower than 0.5 m, the PID controller cannot track it effectively because of the increase of IGE. The maximum error is more than 0.3 m, and the root mean square error is 0.14 m. Fig. 13. shows that the tracking trajectory of the ADRC coincides well with the reference trajectory at the early stage of the descent or approach of the UAV to the water surface. In the experiment, the real descending height is 0.17 m, which is close to the theoretical value. The maximum error is 0.06 m, and the root mean square error is 0.03 m. The IGE can be regarded as an external interference, and the ADRC method can deal with such a problem.

Fig. 12Change curves of height under PID controller

Fig. 13Change curves of height under ADRC controller