Model, Control, and Realistic Visual 3D Simulation of VTOL Fixed-Wing Transition Flight Considering Ground Effect

Abstract

The research topic of VTOL (vertical take-off and landing) fixed wing (VFW) is gaining significant attention, particularly in the transition phase from VTOL to fixed wing and vice versa. One of the latest and most challenging transition strategies is the bird take-off mode, where vertical and horizontal take-off is carried out simultaneously, mimicking the behavior of birds. The condition that is rarely considered when taking off is the ground effect. Under natural conditions, a ground effect is bound to occur, which can significantly impact the stability of the transition when the VFW is close to the ground. This paper addresses this issue by proposing a model and control strategy and conducting realistic visual 3D simulations of the VFW transition that incorporates ground effect using full complex aerodynamic parameters. This research represents a novel approach, using the robot operating system (ROS) and Gazebo to conduct realistic visual 3D simulations for VFW transition. The linear quadratic regulator (LQR) control method is used to manage the transitions and compensate for any disturbances. The flight tests demonstrate the effectiveness of the proposed model and controller in executing flight missions using the bird take-off mode transition. Moreover, the controller has demonstrated reliability and robustness in compensating for attitude errors induced by ground effects and external disturbances.

1. Introduction

Unmanned aerial vehicles (UAVs), also known as drones, have become increasingly prevalent in military and civilian applications due to their numerous advantages. These advantages include low risk to human life, relatively inexpensive operational costs, and ease of use. UAVs come in various categories, such as multirotor and fixed-wing. Multirotor UAVs have vertical take-off and landing (VTOL) capabilities and can hover in place, but their cruising range is relatively short [1,2]. In contrast, fixed-wing UAVs can travel long distances at high speeds but require a long runway for take-off and landing [3,4].

Hybrid UAVs combine the advantages of both multirotor and fixed-wing types. They can take off and land vertically like multirotor and travel long distances like fixed-wing UAVs. Depending on their propulsion type, hybrid UAVs can be further divided into two categories: those with the same propulsion for vertical and horizontal flight and those with separate propulsion for each flight mode. The former includes tilt-wing, tilt-rotor, and tail sitter UAVs, while the latter includes VTOL fixed wing (VFW). There is a complicated mechanism for the same propulsion and a complex control system, so it is not used as a design reference. Unfortunately, the VFW is simple to implement and can be further expanded with quadrotors and fixed-wing configuration [5,6].

The advantage of VFW is that their geometry remains unchanged during the transition from VTOL flight mode to fixed-wing mode and vice versa, making their mechanical motions simple and straightforward [5,6,7,8]. However, the weakness of the VFW is that it has more propulsion than the same propulsion. This condition causes the VFW to be heavier than the same propulsion type. In addition, the VFW requires control of two propulsion systems for vertical take-off and cruising simultaneously during the transition process.

In general, VFW has four stages of flight; take-off, transition, cruising, and landing. Figure 1 shows the conventional transition stages. The movement from point A to point B is a take-off process. Meanwhile, moving from point B to point C is a transition process. The transition is the most critical process [5] in VFW. Point C is the end of the transition process and the beginning of cruising. The transition ends when the airspeed has exceeded the fixed wing’s stall speed [9].

The strategy above is safe enough because the transition process begins when the VFW is at cruising altitude [9]. However, vertical take-off to reach cruising altitude requires significant power, so it will shorten the cruising range. Based on the flight stages, power consumption from the largest to the smallest is taking off, landing, climbing, and cruising, respectively. Vertical take-off on a VFW requires much power because it rotates the four rotors [10,11].

To improve power efficiency, developing effective transition strategies for VFW UAVs is crucial. One promising approach is to emulate a bird’s take-off, as demonstrated in Figure 2. This innovative strategy involves a simultaneous vertical take-off and transition from point A by activating all four quadrotor propulsions to generate lift and fixed-wing propulsion to provide forward thrust. By combining these actions, the UAVs can achieve a more efficient and effective transition, reducing power consumption and improving flight performance [12]. These findings have important implications for the future of VFW technology, as power efficiency is an essential aspect to consider in the design and development of VFW. With this promising approach to improve power efficiency, VFW can continue to push the boundaries of aerial capabilities and contribute to the advancement of the field of unmanned aerial vehicles. However, one important factor to consider is the ground effect because the transition starts when VFW is still close to the ground.

In the context of UAVs, a phenomenon known as the ground effect occurs when the rotor UAVs operate at low altitudes. This phenomenon is characterized by a downwash propeller effect reflected by the ground, resulting in increased lift force but compromised flight stability. Prior research [13] has shown that the ground effect affects the flight stability of rotor UAVs during take-off when they are close to the ground. As the UAV moves further away from the ground, the ground effect diminishes rapidly, eventually disappearing altogether, thus destabilizing flight. It is a challenge for the bird take-off mode transition to carry out the stable transition when the ground effect occurs.

The ground effect on the VFW has not yet received attention from previous studies, both at the simulation and experimental stages. The ground effect in helicopters has been well-researched and studied for take-off, landing, and hovering near the ground [14,15,16]. In contrast, the ground effect in multirotor has less attention research [17,18]. Furthermore, while recent research has evaluated ground effects in VTOL planes, these studies have only focused on conventional take-off and have not explored the impact on flight stability [19]. This gap in research motivates this study, which aims to explore and apply the ground effect as a critical factor influencing the stability of VFW systems during bird take-off mode transition. By exploring this previously unexplored area of research, it is possible to advance the understanding of the complex interplay between flight dynamics and environmental factors and contribute to the development of more robust and adaptable flight control systems.

During the transition of bird take-off mode, the VFW needs a controller to maintain flight stability. Even more, there is a ground effect that affects flight stability. In autonomous VFW, a controller is at least needed to control attitude and trajectory to achieve stable flight [20,21]. Based on several studies [22,23,24], linear quadratic regulator (LQR) control has robustness for multirotor control. In addition, LQR also has satisfactory performance for fixed-wing control [25,26,27]. In this context, LQR control has the advantage of being able to handle nonlinear dynamics and uncertainties. Moreover, the feedback loop inherent in LQR control can help stabilize the VFW during the bird take-off transition mode, where the VFW experiences a significant change in its flight dynamics. Therefore, the application of LQR control for VFW, particularly during the bird take-off transition mode, is a promising solution to address the challenges of maintaining stability in the presence of ground effect and external disturbance.

The application of aerodynamics parameters in Gazebo 3D visual simulation for flapping wings has been discussed [28]. However, it is just a brief analysis, not the simulation application stage, because the flapping and morphing wings have airfoils that move relative to global inertia and body inertia to produce aerodynamic forces. This phenomenon is the same as in the multirotor. Furthermore, previous studies [29,30,31,32] simulated UAVs using Gazebo but did not apply full aerodynamic parameters and did not yield satisfactory outcomes. Different from previous studies, this research applies complete aerodynamic parameters to generate force and torque so that the simulation is like real conditions by considering the ground effect.

Overall, the contribution of this paper is summarized as follows:

• The paper presents a dynamic model of VFW using the Newton–Euler approach and bird take-off mode transition trajectory.
• The paper also proposes a ground effect model that influences flight stability during the transition. This model is an important consideration that was not previously addressed in similar research.
• To handle the bird take-off mode transition and compensate for disturbances, this study applies a robust LQR controller.
• Finally, the paper conducts a realistic 3D simulation of VFW’s bird take-off mode transition using Gazebo/ROS. This simulation is an essential step in verifying the model and robustness of the controller in the presence of external disturbances and ground effects.

This paper is organized into five sections, as follows; Section 2 describes VFW mathematical model, the ground effect model, and the system dynamic and aerodynamic parameters; Section 3 presents the state space model and the controller; Section 4 presents the visual 3D simulation using bird take off mode transition in any conditions. Finally, the conclusions are given in Section 5.

2. System Modeling and Parameters

Section 2.1 presents VFW modeling using the Newton–Euler approach, Section 2.2 explains the ground effect model, and Section 2.3 describes the aerodynamics parameters for realistic 3D visual simulation in ROS/Gazebo.

2.1. Mathematical Model of VTOL Fixed Wing

VFW modeling refers to the Earth’s inertial frame G: {$O_G;X_G,Y_G,Z_G$} and the body’s inertial frame B: {$O_B;X_B,Y_B,Z_B$}. According to the rotation matrix equation, ^GR_B will represent the orientation of the body frame B to the earth frame G. ^GR_B ∈ SO(3) is the rotation matrix representing the orientation of the body frame on the earth frame is presented as

$${{}_{}{}^G\mathbf{R}}_B=\left[\begin{array}{ccc}c\psi c\theta & -s\psi c\varphi +c\psi s\theta s\varphi & s\psi s\varphi -c\psi s\theta c\varphi \\ s\psi c\theta & c\psi c\varphi +s\psi s\theta s\varphi & -c\psi s\varphi +s\psi s\theta c\varphi \\ -s\theta & c\theta s\varphi & c\varphi c\theta \end{array}\right]$$

(1)

where c is cos, s is sin, $\theta$ is pitch angle, $\varphi$ is roll angle and $\psi$ is yaw angle.

As shown in Figure 3, the VFW was built using a Skysurfer airframe and attaching an H-type quadrotor frame. Skysurfer is a slow flayer fixed wing with a 1400 mm wing span and 915 mm length. Skysurfer has ailerons, an elevator, a rudder, and one puller propulsion. The force acting on the VFW during the transition involved two propulsion systems:

$$F=F_Q+F_{E.}$$

(2)

The subscript Q denotes the quadrotor, and the subscript E denotes the fixed wing. The resultant force F is the sum of forces from the quadrotor and fixed wing, while the force on each axis is

$$\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]=\left[\begin{array}{c}{F_Q}_x\\ {F_Q}_y\\ {F_Q}_z\end{array}\right]+\left[\begin{array}{c}{F_E}_x\\ {F_E}_y\\ {F_E}_z\end{array}\right]$$

(3)

Therefore, the working torque is the sum of two torques from the quadrotor and fixed-wing torque is defined as

$$M=M_Q+M_E$$

(4)

and the torque on each axis is

$$\left[\begin{array}{c}{M}_x\\ M_y\\ M_z\end{array}\right]=\left[\begin{array}{c}{M_Q}_x\\ {M_Q}_y\\ {M_Q}_z\end{array}\right]+\left[\begin{array}{c}{M_E}_x\\ {M_E}_y\\ {M_E}_z\end{array}\right]$$

(5)

The VFW dynamics can be expressed in the following form [6]

$$F_x={F_Q}_x+{F_E}_x-mg \mathrm{s}\mathrm{i}\mathrm{n} \theta =m\left(\dot{u}+qw-rv\right)$$

(6)

$$F_y={F_Q}_y+{F_E}_y-mg \mathrm{c}\mathrm{o}\mathrm{s} \theta .\cos \varphi =m\left(\dot{v}+ru-pw\right)$$

(7)

$$F_z={F_Q}_z+{F_E}_z+mg \mathrm{c}\mathrm{o}\mathrm{s} \theta .\sin \varphi =m\left(\dot{w}+pv-qu\right)$$

(8)

$$M_x={M_Q}_x+{M_E}_x=I_{xx}\dot{r}-I_{xy}\left(\dot{q}-pr\right)+\left(I_{zz}-I_{yy}\right)qr$$

(9)

$$M_y={M_Q}_y+{M_E}_y=I_{yy}\dot{q}-I_{xy}\left(\dot{p}-qr\right)+\left(I_{xx}-I_{zz}\right)pr$$

(10)

$$M_z={M_Q}_z+{M_E}_z=I_{zz}\dot{r}-I_{xy}\left(q^2-p^2\right)+\left(I_{yy}-I_{xx}\right)pq$$

(11)

where F, M, m, and g denote force, torque, mass, and gravity. While u, v, and w represent angular velocity on the x-axis, y-axis, and z-axis, translation velocity along the x-axis, y-axis, and z-axis are p, q, and r. The inertial matrix of the vehicle’s rigid body is described as

$$I=\left[\begin{array}{ccc}{I}_{xx}& I_{yx}& I_{xz}\\ I_{xy}& I_{yy}& I_{yz}\\ I_{xz}& I_{yz}& I_{zz}\end{array}\right]$$

(12)

When VFW is maneuvering, there will be a change in angular and translational positions. The rate of change of angular position is defined as [7]

$$\frac{d\varphi }{dt}=p+\left(r \cos \varphi +r \mathrm{s}\mathrm{i}\mathrm{n} \varphi \right)\mathrm{t}\mathrm{a}\mathrm{n} \theta$$

(13)

$$\frac{d\theta }{dt}=q \mathrm{c}\mathrm{o}\mathrm{s} \varphi -r \mathrm{s}\mathrm{i}\mathrm{n} \varphi$$

(14)

$$\frac{d\psi }{dt}=\left(q \mathrm{s}\mathrm{i}\mathrm{n} \varphi +r \mathrm{c}\mathrm{o}\mathrm{s} \varphi \right) \mathrm{s}\mathrm{e}\mathrm{c} \theta$$

(15)

$$\begin{gathered} \frac{d{X}_G}{dt}=u \mathrm{c}\mathrm{o}\mathrm{s} \theta .\mathrm{c}\mathrm{o}\mathrm{s} \psi +v\left(\mathrm{c}\mathrm{o}\mathrm{s} \varphi .\mathrm{s}\mathrm{i}\mathrm{n} \theta -\mathrm{s}\mathrm{i}\mathrm{n} \varphi .\mathrm{s}\mathrm{i}\mathrm{n} \psi \right) \\ +w\left(\mathrm{c}\mathrm{o}\mathrm{s} \varphi .\mathrm{s}\mathrm{i}\mathrm{n} \psi +\mathrm{s}\mathrm{i}\mathrm{n} \varphi .\mathrm{c}\mathrm{o}\mathrm{s} \psi \right) \end{gathered}$$

(16)

$$\frac{d{Y}_G}{dt}=u \mathrm{s}\mathrm{i}\mathrm{n} \theta .\mathrm{c}\mathrm{o}\mathrm{s} \psi +v \mathrm{c}\mathrm{o}\mathrm{s} \varphi .\mathrm{c}\mathrm{o}\mathrm{s} \psi -w \mathrm{s}\mathrm{i}\mathrm{n} \varphi .\mathrm{c}\mathrm{o}\mathrm{s} \theta$$

(17)

$$\begin{gathered} \frac{d{Z}_G}{dt}=-u \mathrm{c}\mathrm{o}\mathrm{s} \theta .\mathrm{c}\mathrm{o}\mathrm{s} \psi +v\left(\mathrm{c}\mathrm{o}\mathrm{s} \gamma .\mathrm{s}\mathrm{i}\mathrm{n} \theta .\mathrm{s}\mathrm{i}\mathrm{n} \psi -\mathrm{s}\mathrm{i}\mathrm{n} \varphi .\mathrm{c}\mathrm{o}\mathrm{s} \psi \right) \\ +w\left(\mathrm{c}\mathrm{o}\mathrm{s} \varphi .\mathrm{c}\mathrm{o}\mathrm{s} \psi -\mathrm{s}\mathrm{i}\mathrm{n} \varphi .\mathrm{s}\mathrm{i}\mathrm{n} \theta .\mathrm{s}\mathrm{i}\mathrm{n} \psi \right) \end{gathered}$$

(18)

where $X_G$, $Y_G$, and $Z_G$ represent its position on the global axis. The total airspeed of VFW:

$$V=\sqrt{u^2+v^2+w^2}$$

(19)

is the resultant of the aircraft’s speed on the Earth’s x-axis, y-axis, and z-axis. $X_G$, $Y_G$, and $Z_G$ represent its position on the global axis, while α is the angle of attack and β is the sideslip angle of the fixed wing given by

$$\beta =\arcsin \left(\frac{w}{V}\right)$$

(20)

$$\alpha =\arctan \left(\frac{-v}{w}\right)$$

(21)

To generate lift, the aircraft needs to maintain a positive pitch and angle of attack relative to the wind axis. The origin of the wind axis is at the vehicle’s center of mass. The direction cosine matrix transforms the inertial frame into a wind frame as follows:

$$\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]={\mathbf{D}\mathbf{C}\mathbf{M}}_{w\to b}\left[\begin{array}{c}{F}_{x_a}\\ F_{y_a}\\ F_{z_a}\end{array}\right], \left[\begin{array}{c}{M}_x\\ M_y\\ M_z\end{array}\right]={\mathbf{D}\mathbf{C}\mathbf{M}}_{w\to b}\left[\begin{array}{c}{M_x}_a\\ {M_y}_a\\ {M_z}_a\end{array}\right]$$

(22)

$${\mathbf{D}\mathbf{C}\mathbf{M}}_{w\to b}=\left[\begin{array}{c}\cos \alpha .\cos \beta \\ \sin \alpha \\ -\cos \alpha .\sin \beta \end{array}\begin{array}{c}-\sin \alpha .\cos \beta \\ \cos \alpha \\ \sin \alpha .\sin \beta \end{array}\begin{array}{c}\sin \beta \\ 0\\ \cos \beta \end{array}\right]$$

(23)

where $F_{x_a}$, $F_{y_a}$, $F_{z_a}$, $M_{x_a}$, $M_{y_a},$ and $M_{z_a}$ are the forces and moments in the wind frame.

At the beginning of the bird take-off mode transition, the quadrotor propulsion will dominate in producing lift. As airspeed increases, the fixed wing will produce more lift until the end of the transition. The thrust of each quadrotor propeller $F_i$ is calculated as

$$F_i=b{\omega }_i^2, i=1, 2, 3, 4$$

(24)

where the thrust coefficient of each propeller $b$ can be obtained as

$$b=C_T\rho {AR}^2$$

(25)

$\omega ,C_T,\rho ,A\mathrm{a}\mathrm{n}\mathrm{d}R$ represent the thrust coefficient propeller, angular velocity, dimensionless thrust constant, air density, propeller area, and propeller radius, respectively. The total thrust of the quadrotor is calculated as

$$F_Q=F_1+F_2+F_3+F_4, i=1, 2, 3, 4$$

(26)

The aerodynamic moment resulting from the rotation of each propeller on the quadrotor is described as

$$M_i=k{\omega }_i^2, i=1, 2, 3, 4$$

(27)

This moment is characterized by the aerodynamic moment coefficient, denoted as k and defined as

$$k=C_Q\rho {AR}^3$$

(28)

which is utilized for yaw control. Furthermore, the differential thrust generated by the propellers, measured from the center of gravity (CoG), induces roll and pitch moments. These moments can be expressed as

$$\left[\begin{array}{c}{M_Q}_x\\ {M_Q}_y\\ {M_Q}_z\end{array}\right]=\left[\begin{array}{c}-l\\ -l\\ -k\end{array}\begin{array}{c}l\\ -l\\ k\end{array}\begin{array}{c}-l\\ l\\ -k\end{array}\begin{array}{c}l\\ l\\ k\end{array}\right]\left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\end{array}\right]$$

(29)

which quantifies the roll, pitch, and yaw moments acting on the center of mass. In these equations, $k$, $C_Q$, and l denote the moment coefficient for each propeller, the dimensionless moment constant, and the distance from the center of mass to each propeller, respectively.

In fixed-wing mode, the rotation of the puller propulsion generates forward speed, as described by

$$F_E=b{\omega }_E^2$$

(30)

When the aircraft is flying forward, the lift and drag forces acting on the wing can be expressed using

$$F_L=C_l\frac{\rho }{2}A{v}^2$$

(31)

$$F_D=C_d\frac{\rho }{2}A{v}^2.$$

(32)

The deflection of the aileron, elevator, and rudder generates aerodynamic forces, which can be quantified as

$$F_{csA}=j \mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{csA}$$

(33)

$$F_{csE}=j \mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{csE}$$

(34)

$$F_{csR}=j \mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{csR}$$

(35)

The aerodynamic forces coefficient for the aileron, elevator, and rudder is represented by

$$j=C_l\frac{\rho }{2}A{v}^2$$

(36)

These forces induce roll, pitch, and yaw moments, which can be calculated by multiplying each force by its distance from the CoG as follows:

$${M_E}_x={2F}_{csA}{l}_x$$

(37)

$${M_E}_y=F_{csE}{l}_y$$

(38)

$${M_E}_z=F_{csR}{l}_z$$

(39)

By incorporating the quadrotor and fixed-wing models discussed earlier, the total forces acting on each axis are expressed as

$$\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ -mg\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ F_Q\end{array}\right]+\left[\begin{array}{c}-F_D+F_E\\ 0\\ F_L\end{array}\right]$$

(40)

2.2. Ground Effect

The thrust required for the quadrotor to hover in the ground effect and out-of-ground effect is depicted in Figure 4. In the ground effect, the quadrotor needs less thrust than the out-of-ground effect due to the downwash reflections, as illustrated in Figure 5b. In the presence of a ground effect, the lift generated will increase while the propeller angular velocity remains constant. However, when almost leaving the ground effect there will be a sudden drop in a lift. Consequently, the flight control of the VFW in the ground effect became more intricate [13]. Figure 5a demonstrates that elevating the aircraft height resulted in a more uniform flow field.

The ground effect calculation for the rotor is defined as [33]

$$\frac{F_{IGE}}{F_{OGE}}=\frac{1}{1-{\left(\frac{r}{{4h}_r}\right)}^2}, maka F_{IGE}=F_{OGE}\frac{1}{1-{\left(\frac{r}{{4h}_r}\right)}^2}$$

(41)

and measurable until the distance to the ground $\frac{h_r}{r}=6$ [34,35], where $F_{IGE}$ is the thrust in-ground effect, $F_{OGE}$ is the thrust without out-of-ground effect, r is the propeller radius, $h_r$ is the distance from the ground, and $K_b$ is coefficient lift body. In the quadrotor, the resultant ground effect will be affected by four rotors is expressed as

$$\frac{F_{IGE}}{F_{OGE}}={\left(1-{\left(\frac{r}{{4h}_r}\right)}^2-r^2\left(\frac{h_r}{\sqrt{{\left(d^2-{{4h}_r}^2\right)}^3}}\right)-\left(\frac{r^2}{2}\right){\left(\frac{h_r}{\sqrt{{\left(d^2-{{4h}_r}^2\right)}^3}}\right)}^3-2r^2\left(\frac{h_r}{\sqrt{{\left(b^2-{{4h}_r}^2\right)}^3{K}_b}}\right)\right)}^{-1}$$

(42)

After applying the ground effect to Equation (40), the force acting on the aircraft becomes

$$\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ -mg\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ F_{IGE}\end{array}\right]+\left[\begin{array}{c}-F_D+T_E\\ 0\\ F_L\end{array}\right]$$

(43)

2.3. VFW Simulation Parameters

The VFW 3D model incorporates a fixed-wing airframe based on the Skysurfer and a quadrotor frame utilizing the H550. Isometric front, top, and side views of the model are depicted in Figure 6a, Figure 6b and Figure 6c, respectively, with dimensions expressed in millimeters. The Skysurfer was selected as the fixed-wing airframe due to its low stall speed, which is crucial for the VFW’s intended application. This characteristic also contributes to a shorter and simpler transition process, as documented in reference [36]. The physical and aerodynamic parameters for the model are presented in

Table 1. VFW parameters.

Parameters	Value
wing span	1.470 m
aircraft length	0.924 m
mass	1.9835 kg
airfoil wing	NACA 2412
airfoil propeller	NACA 2412
airfoil aileron, elevator dan rudder	NACA 0012
initial AoA fixed wing	5°
fixed wing and quadrotor propeller diameter	0.254 m (10 inches)
area formed propeller quadrotor rotate	5.06 × 10−2 m2
distance center of mass and motor quadrotor	0.38891 m
center of mass	X = 0.34, Y = 0.00, Z = 0.036
gravity	9.81 kg/m2
inertia about x-axis (Ix)	0.023426 kg/m2
inertia about y-axis (Iy)	0.044475 kg/m2
inertia about z-axis (Iz)	0.066685 kg/m2
air density ($\mathit{\rho }$)	1.225 kg/m3
coefficient torque (CT)	1.3602 × 10−1
coefficient thrust (CQ)	5.5555 × 10−2

.

Proper component placement is critical in achieving an optimal center of mass, which is a crucial factor in vehicle stability. The law of leverage can be used to calculate the center of mass (CoM) based on the principle that the weight on one side of a fulcrum multiplied by its distance from the fulcrum is equal to the weight on the opposite side multiplied by its distance from the fulcrum [37]. To ensure adequate thrust, propeller selection for both fixed-wing and quadrotor configurations should be based on the thrust requirements, as detailed in the performance analysis [38,39,40].

The URDF (unified robot description format) in Algorithm 1 declares the robot parameters necessary for visual simulations of the VFW in ROS. As illustrated in Figure 7, the VFW comprises ten links, namely base_link (wing and fuselage), link_1 (front right propeller), link_2 (front left propeller), link_3 (rear left propeller), link_4 (rear right propeller), link_5 (fixed wing propeller), link_6 (right aileron), link_7 (left aileron), link_8 (elevator), and link_9 (rudder). These links are connected to base_link via joints to form the VFW structure. Each link is defined by four properties, namely inertial body properties, visual properties, collision body properties, and aerodynamic parameters. The inertial body properties include the link’s mass, inertial matrix, and center of mass. Collision body properties are defined by geometry functions with subfunctions meshes, while visual properties are the visible parts of the links that are rendered in the simulator. Finally, the aerodynamic parameters are the parameters that affect the aerodynamic forces acting on each link.

Algorithm 1: URDF Configuration.

<robot name=“quad-plane”> <link name=“base_link”> <inertial> <origin … /> <mass … /> <inertia … /> </inertial> <visual> <origin … /> <geometry> … </geometry> <material … /> </material> </visual> <collision> <origin … /> <geometry> … </geometry> </collision> </link> <joint name … type … > <origin… /> <parent link …/> <child link… /> <axis … /> <limit … /> </joint> <plugin>

The parameters in Algorithm 2 obtained the aerodynamic force based on the airfoil geometry and aerodynamic curves, as shown in Figure 4. The aerodynamic parameters of each link are presented in detail in

Table 2. Aerodynamic parameters of each link.

Aerodynamics Parameter	Base_Link	Link_1, Link_2, Link_3, Link_4, Link_5	Link_6, Link 7	Link_8	Link_9
air_density	1.225	1.225	1.225	1.225	1.225
cla	4.75	4.75	5.9	5.9	5.9
cla_stall	−3.85	−3.85	−3.85	−3.85	−3.85
cda	−0.4	−0.4	−0.4	−0.4	−0.4
cda_stall	−0.92	−0.92	−0.92	−0.92	−0.92
alpha_stall	0.339	1.5	0.339	0.339	0.339
a0	0.05	0.4	0.0	0.0	0.0
area	0.2	0.02	0.04	0.05	0.04
upward	0 0 1	0 0 1 (link_1, link_2, link_3, link_4) 1 0 0 (link_5)	0 0 1	0 0 1	0 1 0
forward	−1 0 0	0 −1 0 (link_1, link_4) 0 1 0 (link_2, link_3, link_5)	−1 0 0	−1 0 0	−1 0 0
cp	0.340 0.000 0.030	0.065 0.275 0.036 (link_1) 0.065 −0.275 0.036 (link_2) 0.615 −0.275 0.036 (link_3) 0.615 0.275 0.036 (link_4) 0 0 0 (link_5)	0.394 0.312 0.046 (link_6) 0.394 −0.312 0.046 (link_7)	0.900 0.000 0.015	1.571 0.000 0.010

.

Algorithm 2: Aerodynamic Plugin Parameters.

<gazebo> <plugin name=“aero_fly” filename=“libLiftDragPlugin.so”> <air_density> … </air_density> # Density of the fluid this model is suspended in (kg/m3) <cla> … </cla> #The ratio of the coefficient of lift and alpha slope before stall <cla_stall> … </cla_stall> #The ratio of coefficient of lift and alpha slope after stall <cda> … </cda> #The ratio of the coefficient of drag and alpha slope before stall <cda_stall> … </cda_stall> #The ratio of coefficient of drag and alpha slope after stall <alpha_stall> … </alpha_stall> #Angle of attack at stall point (radians) <a0> … </a0> #initial angle of attack (radians) <area> … </area> #Surface area of the link (m2) <upward> x y z</upward> # 3-vector representing the upward direction of motion in the link frame <forward> x y z</forward> # 3-vector representing the forward direction of motion in the link frame <link_name>base_link</link_name> #Name of the link affected by the group of lift/drag properties <cp> x y z</cp> #Center of pressure in link frame. The forces due to lift and drag will be applied here </plugin> </gazebo>

The lift and drag forces acting on the wings, propellers, ailerons, elevator, and rudder are depicted in Figure 8a by the red lines. For generating lift, draft, and moment, the propeller must rotate, whereas the wings, ailerons, elevators, and rudder require relative airspeed to generate aerodynamic forces. The top view of the chord line wing is shown in Figure 8b and Figure 8c displays the aerodynamic parameter curves for the wing, propeller, aileron, elevator, and rudder, which correspond to the airfoil’s shape specified by the NACA number. The NACA 2412 airfoil is used for the wings and propellers, whereas the NACA 0012 airfoil is employed for the ailerons, elevators, and rudder.

3. State Space Model and Controller

Considering the dynamic equations described in Section 2.1, a state space controller is proposed to maintain the stability of VFW and track the trajectory. Figure 9 depicts the block diagram of the LQR (linear quadratic regulator) controller based on state space representation. The diagram illustrates how system state is obtained from various sensors, including those for position, orientation, linear velocity, and angular velocity, resulting in twelve state data. This state is then compared to the desired state, and the comparison results are used as input for a full state feedback controller, which generates an input for the control signal (u). The control signal is calculated by multiplying the full-state feedback controller input by a negative feedback gain K. The control signal is first interpreted by the control element as the value of the motor rotational speed and servo position for the VFW actuator before being forwarded to the process plant. It iterates continuously as long as the VFW is carrying out its mission.

The LQR controller’s ability to provide feedback on the system’s entire state, rather than just partial feedback, is crucial. This ensures that the VFW moves precisely as desired. The output of the entire system (y) is in the form of six states that describe the VFW’s position and orientation. This level of detail enables close monitoring of the VFW’s movement, ensuring that it performs its task with high accuracy and reliability.

The VFW is characterized by twelve states, namely ($X,Y,Z,V_x,V_y,V_{z,}\varphi ,\theta ,\psi ,{\omega }_{\varphi },{\omega }_{\theta },$ and ${\omega }_{\theta })$, representing the position, translation velocity, orientation, and angular velocity. Utilizing the parameters listed in Table 1, the state space fixed wing model is derived using Matlab to obtain matrices A, B, C, D, Q, R, and K. Equation (44) as follows:

$$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$$

(44)

describes the relationship between the system’s present state, input, and future state, and for VFW, detailed as

$$\left[\begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{Z}\\ \dot{V_x}\\ \dot{V_y}\\ \dot{V_z}\\ \dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\\ \dot{{\omega }_{\varphi }}\\ \dot{{\omega }_{\theta }}\\ \dot{{\omega }_{\psi }}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ g\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ -g\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\end{array}\right]\left[\begin{array}{c}X\\ Y\\ Z\\ V_x\\ V_y\\ V_z\\ \varphi \\ \theta \\ \psi \\ {\omega }_{\varphi }\\ {\omega }_{\theta }\\ {\omega }_{\theta }\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1/m\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1/I_x\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1/I_y\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1/I_z\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right]$$

(45)

$$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$$

The relationship among the system’s state, input, and output id defined as

$$\mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}$$

(46)

and elaborated in

$$\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]\left[\begin{array}{c}X\\ Y\\ Z\\ V_x\\ V_y\\ V_z\\ \varphi \\ \theta \\ \psi \\ {\omega }_{\varphi }\\ {\omega }_{\theta }\\ {\omega }_{\psi }\end{array}\right]+\left[0\right] \left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right]$$

(47)

$$\mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}$$

The control signal

$$\mathbf{u}=-\mathbf{K}\mathbf{x}$$

(48)

is obtained by multiplying the negative feedback gain K with the state control. A, B, C, D, x, u, y, and K denote the state matrix, input matrix, output matrix, feed-forward matrix, current state, control signal, output, and feedback gain, respectively. Feedback gained K as follows:

$$\mathbf{K}=\left[\begin{array}{c}0.0000\\ 0.0000\\ -1.0000\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 1.0000\\ 0.0000\\ 0.0000\end{array}\begin{array}{c}1.0000\\ 0.0000\\ 0.0000\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 0.0000\\ -1.4515\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 1.4515\\ 0.0000\\ 0.0000\end{array}\begin{array}{c}1.0000\\ 0.0000\\ 0.0000\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 5.7562\\ 0.0000\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 0.0000\\ 6.0383\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 0.0000\\ 0.0000\\ 1.0000\end{array}\begin{array}{c}0.0000\\ 1.1268\\ 0.0000\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 0.0000\\ 1.2939\\ 0.0000\end{array}\begin{array}{c}0.0000\\ 0.0000\\ 0.0000\\ 1.0646\end{array}\right]$$

(49)

which depends on Q and R matrix. Q matrix defines the weights on the states, and the R matrix defines the weights on the control input in the cost function. The Q and R matrices are given as identity matrices below.

$$\mathbf{Q}=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\mathbf{R}=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$$

(50)

The control signals are defined as

$$u_2=K_{2.7 }\varphi +K_{2.10 }{\omega }_{\varphi }$$

(51)

$$u_3=K_{3.8 }\theta +K_{3.11 }{\omega }_{\theta }$$

(52)

$$u_4=K_{4.9 }\psi +K_{4.12 }{\omega }_{\psi }$$

(53)

where $u_2$ is the control signal for roll, $u_3$ is the control signal for pitch, and $u_4$ is the control signal for yaw. For $K_{\mathrm{a}.\mathrm{b}}$ in Equations (51)–(53), a is the number of rows, and b is the number of columns in K. The value of u is the multiplication of the negative feedback gain K and the current state, as in Equation (46). The signal control for the quadrotor’s motor R is described as

$$R_{front\_right}={u_1+u}_{2}db+u_3\ast db+u_4\ast k$$

(54)

$$R_{front\_left}={u_1+u}_2(-db)+u_{3}db+u_4(-k)$$

(55)

$$R_{rear\_left}={u_1+u}_2(-db)+u_3(-db)+u_4\ast k$$

(56)

$$R_{rear\_right}={u_1+u}_{2}db+u_3(-db)+u_4(-k)$$

(57)

Meanwhile, the control signals for fixed-wing flight control surfaces E are described as

$$E_{aileron\_left}=u_2\ast {j\ast l}_x$$

(58)

$$E_{aileron\_right}={-u}_2\ast {j\ast l}_x$$

(59)

$$E_{elevator}=u_3\ast {j\ast l}_y$$

(60)

$$E_{rudder}=u_4\ast {j\ast l}_z$$

(61)

4. Simulation

This section presents the simulation of VFW bird take of mode transition that applied ground effect and external disturbance. Section 4.1 presents the transition trajectory of bird take-off mode, Section 4.2 presents the simulation setup, and Section 4.3 discusses the simulation result.

4.1. Bird Take-Off Mode Transition Trajectory

The transition scenario in Figure 10 imitates how a bird takes off. VFW starts the transition from the ground and gains altitude and airspeed simultaneously. The quadrotor and fixed-wing propulsion turn on together. VFW flies up and gets forward speed with a specific rate of climb until it reaches the desired altitude. After reaching the desired altitude, VFW maintains its altitude by adjusting the rate of climb value to 0 m/s until it reaches the desired airspeed. When VFW exceeds the desired airspeed, the fixed wing can fly using the lift from the wing. So the quadrotor can be slowed down and turned off. It indicates that the transition is successful.

4.2. Simulation Setup

Several key components must be set up to carry out a realistic visual 3D simulation using ROS and the Gazebo simulator. The first step is to define the physical properties of the robot, which include inertial body properties, visual properties, collision body properties, and aerodynamic parameters. These parameters are described using URDF configuration, as outlined in Algorithm 1 and Algorithm 2 in Section 2.3. Additionally, the physical parameters of the VFW and environmental parameters, such as gravity and air density, are described in detail in Table 1. The full aerodynamic parameters of each link of the VFW are described in Table 2 and implemented using a modified LiftDragPlugin. These parameters play a crucial role in determining how closely the simulation will resemble the real world.

Once the physical properties have been defined, the VFW’s sensors, actuators, and controllers must be added to the robot model using ROS packages. The robot’s position, orientation, linear velocity, and angular velocity relative to the global inertial frame are represented by twelve states (X, Y, Z, V_x, V_y, V_z, ϕ, θ, ψ, ω_ϕ, ω_θ, and ω_ψ) acquired using rosservices GetModelState as a sensor. The actuators are represented by joint_state_publisher, which enables movement between connected links.

The controller in Gazebo consists of two types: a physical controller and a dynamic controller. The physical controller uses the joint controller package to set the maximum and minimum joint angles, joint velocity limits, and actuator response. On the other hand, dynamic controllers use rostopic to implement the LQR controller. These controllers play an essential role in governing the VFW’s behavior in the simulation.

4.3. Simulation Result

In this research, a visually realistic VFW simulation has been successfully conducted using Gazebo and ROS. This simulator integrates physical parameters, aerodynamics parameters, visual parameters, sensors, and actuator control. Figure 11a shows VFW visual 3D simulation in Gazebo, while Figure 11b the inertial of each link. Each link has a center of mass, as shown in Figure 11c, where the center of pressure for aerodynamic force is applied. Figure 11d shows the joint position of each child’s link with its parent, represented by three arrows (red, green, and blue).

To evaluate the performance of the proposed system, a fully autonomous transition was tested under three flight conditions. The first test was conducted under normal conditions, without applying ground effect and external disturbance. The second test was carried out in ground effect but without external disturbance. The third test included both ground effect and external disturbance. For a successful transition, predetermined parameters were set, including an altitude of more than 5 m and airspeeds of more than 5 m/s. Additionally, the attitude stability indicators were monitored, with steady-state error roll and pitch less than 5 degrees and yaw rate less than 3 degrees/s.

Figure 12 presents a real-time visualization of the simulated transition process, beginning with the activation of both quadrotor and fixed-wing propulsions in Figure 12a. Due to the low airspeed during this phase, the quadrotor is primarily responsible for controlling the vehicle’s flight attitude, while deflections of the ailerons, elevator, and rudder have negligible effects on the vehicle’s roll, pitch, and yaw moments. As shown in Figure 12b, the vehicle aims to attain the predetermined altitude and airspeed in the middle of the transition. Figure 12c displays the end of the transition, with the vehicle cruising at its designated altitude and minimum airspeed.

Figure 13 shows the flight test result in normal conditions. In this flight test, the ground effect was not applied without disturbance, as shown in Figure 13d. Figure 13a shows the flight phase; there are preparation flight, transition, and cruising. Figure 13b,e show attitude response with a maximum roll error value is 3.87° and a maximum pitch error value is 3.66°. The maximum yaw rate is 0.3 degree/s, as shown in Figure 13g. It indicates that the controller can maintain flight stability during the transition. The transition was completed at a predetermined altitude and airspeed in 4.71 s, as shown in Figure 13a. Compared to previous research [10,12], this uninterrupted test result has better control results seen from the maximum attitude error during the transition process.

The influence of the ground effect on VFW performance can be seen in the test results presented in Figure 14a. This effect occurs up to a height of 0.762 m ($h_r=6r$ [34,35]) when t = 51 to t = 210, as shown in Figure 14d. The test results indicate that aircraft stability is more challenging to maintain in ground effect than in normal conditions. The maximum roll error is 4.3 when t = 187, and the maximum error pitch is 8.17 when t = 183 due to the momentary lack of lift when leaving the ground effect. Despite these challenges, the controller was able to maintain flight stability with a settling time roll of 0.45 s and a settling time pitch of 0.60 s. It is noteworthy that the yaw rate is not disturbed and still maintains stability, with a maximum value of 0.2 degree/s, as shown in Figure 14g.

Moreover, the ground effect provides the advantage of additional lift that increases the vertical speed of the aircraft. This effect enables the predetermined altitude to be reached more quickly, resulting in a faster transition time of 4.53 s, as shown in Figure 14c. In comparison to previous research on handling ground effect [41,42,43], the proposed controller in this study demonstrated a good attitude response. Despite the challenges posed by the ground effect, the results of the test suggest that with the right controller, it is possible to maintain flight stability and take advantage of the increased lift provided by this phenomenon.

The flight test results with external disturbances are presented in Figure 15. At time t = 358, external disturbances were applied at body coordinates 0.6, −0.3, and 0 with a force of 600 N to each of the x, y, and z axes, as shown in Figure 15d. Despite being disturbed, the controller was able to maintain stability with a settling time of 0.93 s for roll and 0.90 s for pitch, while the maximum roll and pitch errors were 9.43° and 9.40°, respectively. The maximum yaw rate was 1.1 degrees/s. The results indicate that the proposed controller can quickly handle external disturbances. The transition was completed in 5.15 s, as Figure 15a shows.

In short, the simulation results showed that the proposed controller was able to maintain flight stability during the transition of the proposed VFW model and trajectory, even in the presence of ground effect and external disturbance, which caused greater instability. The ground effect provided additional lift, which increased the vertical speed and allowed for a quicker transition to the predetermined altitude. Despite external disturbances, the proposed controller could handle them quickly and maintain stability during the transition. Overall, the study emphasized the importance of considering the ground effect in transitioning from vertical take-off and landing to fixed-wing flight and the effectiveness of the proposed controller in maintaining flight stability during the transition under various flight conditions.

However, it is important to note that there may be some differences between the simulation and the real-world application of the ground effect on VTOL. These differences could arise from various factors, such as variations in atmospheric conditions, surface properties, and other environmental factors that may not be accurately represented in the simulation. So it is necessary to conduct experiments to validate the influence of ground effect on bird take-off mode transition of VFW in the real world.

5. Conclusions

This paper concentrated on modeling, controlling, and conducting a realistic 3D simulation of the VFW transition considered ground effect. The mathematical and 3D models were obtained with fully dynamic and aerodynamic properties. Ground effect in VFW also had been successfully modeled as a considerable influence in transition flight. Furthermore, a realistic 3D simulation of the bird take-off mode transition was conducted on Gazebo and ROS to validate the proposed model, trajectory, and controller performance. Various flight tests have been performed to verify the robustness of the LQR controller in ground effect and external disturbance. The results show that the proposed model and controller carry out flight missions successfully with the bird take-off mode transition in any conditions. The controller has reliability and robustness in compensating for attitude errors caused by ground effects and external disturbance.

In the future, there are several areas of research that could build on the findings of this study. One promising direction is to optimize the flight performance of the system by implementing adaptive control techniques. Adaptive control can enhance the system’s ability to operate effectively even in the face of significant uncertainties, such as those caused by ground effects and external disturbances. Additionally, it would be valuable to conduct a bird takeoff mode transition experiment to further validate the effectiveness of the proposed approach in real-world scenarios.