A Control Strategy for Autonomous Approaching and Coordinated Landing of UAV and USV

Abstract

Unmanned aerial vehicles (UAVs) autonomous landing plays a key role in cooperative work with other heterogeneous agents. A neglected aspect of UAV autonomous landing on a moving platform is addressed in this study. The landing process is divided into three stages: positioning, tracking, and landing. In the tracking phase, MPCs are designed to implement tracking of the target landing platform. In the landing phase, we adopt a nested Apriltags collaboration identifier combined with the Aprilatags algorithm to design a PID speed controller, thereby improving the dynamic tracking accuracy of UAVs and completing the landing. The experimental data suggested that the method enables the UAV to perform dynamic tracking and autonomous landing on a moving platform. The experimental results show that the success rate of UAV autonomous landing is as high as 90%, providing a highly feasible solution for UAV autonomous landing.

1. Introduction

With the continuous advancement of unmanned aerial vehicle (UAV) technology, UAVs have been increasingly applied in both civilian [1,2,3] and military fields [4,5,6] due to their advantages of high flexibility, fast speed, and low cost. However, the limited payload capacity and short endurance time of consumer-level UAVs (typically 20–30 min) make it difficult for them to complete long-duration tasks. Consequently, UAVs may face an urgent need for emergency landing after extended operation.

Autonomous landing is a key challenge in UAV self-flight. In recent years, a lot of research has been conducted on the autonomous landing of UAVs. From the target platform’s state, drone autonomous landing is primarily divided into two scenarios, static scenes and dynamic scenes [7,8]:

(1)

Autonomous Landing of UAVs in Static Scenes

Compared with the landing of UAVs in dynamic scenes, it is more stable and easier for UAVs to land in static scenes. Therefore, static target landing-related technologies have relatively matured. According to the sensors carried on the UAVs, there are different ways to realize autonomous landing [9,10,11,12,13]. Vision-guided UAV landing is a primary method to realize UAVs landing in static scenes. Many artificial markers have been designed to assist in completing the landing [14,15,16], such as: “T”, “H”, circular, rectangular, and combination marks. Some markers are shown in Figure 1.

Although the technology of autonomous landing for UAVs is relatively mature, in practical applications the UAVs need to work with a moving platform.

(2)

Autonomous Landing of UAVs in Dynamic Scenes

While performing complex tasks, UAVs need to collaborate with moving platforms. Compared with static scenes, landing in dynamic scenes is harder and the complexity of the technology is much higher, which necessitates the need to improve the control and navigation systems of UAVs. Meanwhile, the mission objectives of the two types of landing are almost the same, which means the autonomous landing methods mentioned above in static scenes are also widely used in dynamic scenes. Li proposed a Bi-directional Long and Short-Term Memory (Bi-KSTM) neural network for predicting the attitude of an unmanned boat, designed a PID controller, and achieved attitude synchronization between the UAV and the USV during autonomous landing [17]. Khazetdinov uses visual sensors to design embedded ArUco markers (e-ArUco), enabling landing using only the ArUco algorithm in a Gazebo simulation platform. The experimental results showed an average landing precision of 2.03 cm with a standard deviation of 1.53 cm [18]. In Lin’s study, Lin proposed a novel vision system that integrates infrared imaging technology and deep learning algorithms, aiming to enhance both accuracy and robustness. The experimental results demonstrate the method’s reliability [19]. Zhang designed an adaptive learning navigation rule with multiple layers, determining the position of the unmanned boat and achieving horizontal tracking and vertical descent of the UAV based on position feedback [20].

Table 1. The relevant work over the past five years.

	Method	Result
[21]	1 Reinforcement learning framework 2 The continuous action space algorithms (DDPG, TD3, SAC) 3 The reward function design based on reward shaping	The framework successfully achieves a 100% autonomous landing success rate for UAVs
[22]	1 Multimodal detector 2 Reinforcement learning decision model 3 Embedded deployment framework	1 The DQN model achieves an average landing accuracy of 0.25 m 2 The multimodal fusion maintains high reliability even when sensors fail
[23]	1 The fusion of sensor data 2 The dynamic adjustment of LiDAR accumulation time and self-assessment of depth map accuracy 3 Multifeature Fusion	1 The accuracy of landing point selection is increased to 98% after the fusion of multiple features
[24]	1 Multi-Stage Fusion Framework 2 Visual Enhancement Design 3 Dynamic Switching Strategy	1 The dynamic switching strategy ensures safe approach even when vision is lost, with a success rate of 100% 2 The visual system has an attitude estimation error of less than 2° in the horizontal direction, and the final landing error is less than 10 cm

is a further organization of the relevant work over the past five years. From the experimental results, although Jiang and his team [21] achieved 100% success, they did not validate the experimental algorithm on a real UAV. Moreover, their simulation environment was overly simple, giving the algorithm a higher fault-tolerance. While Neves et al. [22] demonstrated strong innovation and achieved good results, their method relies on high-performance GPUs. This dependency may limit the method’s application in resource-constrained environments. Chen’s team [23] used a complex model to achieve up to 98% accuracy. However, this boosted performance also raised computational demands, requiring high-end hardware and making it unsuitable for lightweight UAVs. In Dong et al.’s study [24], the UAV integrated multiple sensors, which increased its payload. The UAV achieved 100% success only on a slow-moving (0.5 m/s) target platform, as stated in the paper.

While the technology for autonomous landings is relatively mature, there is still a lot of room for development in the dynamic landing of UAVs. In this paper, we divided the process of autonomous landing into three main parts: target platform localization, trajectory tracking in fixed height, and final autonomous landing. The first part proposed a generic positioning method for the UAV to achieve real-time localization of the target platform. The second part simplified the Motion Planning and Control Controller (MPC) to enable the UAV to track the target landing platform on a horizontal plane while maintaining a constant altitude. The third part designed nested Apriltags cooperative markers and combined them with a PID speed controller using a Intel RealSense Depth Camera D435 (Intel, Santa Clara, CA, USA) for real-time tracking of the cooperative markers, enabling the UAV to complete autonomous landing. This method is not constrained by UAV computing resources and can be deployed in diverse real-world scenarios. The comprehensive technical flowchart for the UAV’s autonomous landing process is presented in Figure 2.

2. Modeling and Experimental Platform Specifications

In this section, we introduce the mathematical model of the UAV, the simulation platform, and physical platform specifications.

2.1. UAV Modeling

For the study of collaborative landing between UAVs and USVs, first, a mathematical model for the UAV must be established [25]. This study employs an X-type UAV as the research subject. The UAV is assumed to be a rigid and symmetric structure with its center of mass located at its geometric center. To describe the UAV’s position and attitude, we define the earth-centered inertial reference frame O_E-X_EY_EZ_E and the body-fixed frame O_B-X_BY_BZ_B, as shown in Figure 3.

The lift of the four-rotor UAV is generated by the rapid rotation of its four propellers. By adjusting the rotational speeds of the four motors, the position and attitude of the four-rotor UAV can be controlled. Based on the illustration provided in Figure 3, the mathematical model of the UAV is as follows:

$$\left\{\begin{array}{c}\ddot{\mathit{\varphi }}=\dot{\mathit{\theta }}\dot{\mathit{\psi }}\left({\displaystyle \frac{{\mathit{I}}_{\mathit{y}}-{\mathit{I}}_{\mathit{z}}}{{\mathit{I}}_{\mathit{x}}}}\right)-{\displaystyle \frac{{\mathit{J}}_{\mathit{r}}}{{\mathit{I}}_{\mathit{x}}}}\dot{\mathit{\theta }}\mathbf{\Omega }+{\displaystyle \frac{\mathit{l}}{{\mathit{I}}_{\mathit{x}}}}{\mathit{U}}_2-{\displaystyle \frac{\mathit{l}{\mathit{K}}_4}{\mathit{m}}}\dot{\mathit{\varphi }}\\ \ddot{\mathit{\theta }}=\dot{\mathit{\varphi }}\dot{\mathit{\psi }}\left({\displaystyle \frac{{\mathit{I}}_{\mathit{z}}-{\mathit{I}}_{\mathit{x}}}{{\mathit{I}}_{\mathit{y}}}}\right)+{\displaystyle \frac{{\mathit{J}}_{\mathit{r}}}{{\mathit{I}}_{\mathit{y}}}}\dot{\mathit{\varphi }}\mathbf{\Omega }+{\displaystyle \frac{\mathit{l}}{{\mathit{I}}_{\mathit{y}}}}{\mathit{U}}_3-{\displaystyle \frac{\mathit{l}{\mathit{K}}_5}{\mathit{m}}}\dot{\mathit{\theta }}\\ \ddot{\mathit{\psi }}=\dot{\mathit{\varphi }}\dot{\mathit{\theta }}\left({\displaystyle \frac{{\mathit{I}}_{\mathit{x}}-{\mathit{I}}_{\mathit{y}}}{{\mathit{I}}_{\mathit{z}}}}\right)+{\displaystyle \frac{\mathit{l}}{{\mathit{I}}_{\mathit{z}}}}{\mathit{U}}_4-{\displaystyle \frac{{\mathit{K}}_6}{\mathit{m}}}\dot{\mathit{\psi }}\\ \ddot{\mathit{x}}=(\mathbf{cos} \mathit{\varphi }\mathbf{sin} \mathit{\theta }\mathbf{cos} \mathit{\psi }+\mathbf{sin} \mathit{\varphi }\mathbf{sin} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_1}{\mathit{m}}}\dot{\mathit{x}}\\ \ddot{\mathit{y}}=(\mathbf{cos} \mathit{\varphi }\mathbf{sin} \mathit{\theta }\mathbf{sin} \mathit{\psi }-\mathbf{sin} \mathit{\varphi }\mathbf{cos} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_2}{\mathit{m}}}\dot{\mathit{y}}\\ \ddot{\mathit{z}}=(\mathbf{cos} \mathit{\varphi }\mathbf{cos} \mathit{\theta }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-\mathit{g}-{\displaystyle \frac{{\mathit{K}}_3}{\mathit{m}}}\dot{\mathit{z}}\end{array}\right.$$

(1)

In the above equations: $\mathit{\theta }, \mathit{\psi }, \mathit{\varphi }$ are the rotational angles of the body around the y-axis, x-axis, and z-axis. ${\mathit{I}}_{\mathit{x}}={\mathit{I}}_{\mathit{y}}=0.0745 \mathrm{kg}\cdot {\mathrm{m}}^2$ and ${\mathit{I}}_{\mathit{z}}=0.03175 \mathrm{kg}\cdot {\mathrm{m}}^2$ are the rotational inertia of the body in three axes (x, y, z). ${\mathit{K}}_1–{\mathit{K}}_6$ are the coefficients of air resistance.

${\mathit{U}}_1$,${\mathit{U}}_2$, ${\mathit{U}}_3$, ${\mathit{U}}_4$, and $\mathit{\Omega }$ are given by the following equation:

$$\left\{\begin{array}{c}{\mathit{U}}_1=\mathit{b}({\mathit{\Omega }}_1^2+{\mathit{\Omega }}_2^2+{\mathit{\Omega }}_3^2+{\mathit{\Omega }}_4^2)\\ {\mathit{U}}_2=\mathit{b}({\mathit{\Omega }}_1^2-{\mathit{\Omega }}_2^2-{\mathit{\Omega }}_3^2+{\mathit{\Omega }}_4^2)\\ {\mathit{U}}_3=\mathit{b}(-{\mathit{\Omega }}_1^2-{\mathit{\Omega }}_2^2+{\mathit{\Omega }}_3^2+{\mathit{\Omega }}_4^2)\\ {\mathit{U}}_4=\mathit{d}({\mathit{\Omega }}_2^2+{\mathit{\Omega }}_4^2-{\mathit{\Omega }}_1^2-{\mathit{\Omega }}_3^2)\\ \mathit{\Omega }={\mathit{\Omega }}_2+{\mathit{\Omega }}_4-{\mathit{\Omega }}_1-{\mathit{\Omega }}_3\end{array}\right.$$

(2)

${\mathit{\Omega }}_1, {\mathit{\Omega }}_2, {\mathit{\Omega }}_3, {\mathit{\Omega }}_4$ are the speeds of the four motors.

2.2. Relative Motion

It is necessary to consider the kinematic issues of the relative motion between the UAV and the USV. We define ${\mathit{r}}_{\mathit{u}}$ as the position vector of the drone in the inertial coordinate system and ${\mathit{r}}_{\mathit{s}}$ as the position vector of the USV in the inertial coordinate system.

$${\mathit{r}}_{\mathit{u}/\mathit{s}}={\left[\begin{array}{c}{\mathit{x}}_{\mathit{u}/\mathit{s}},{\mathit{y}}_{\mathit{u}/\mathit{s}},{\mathit{z}}_{\mathit{u}/\mathit{s}}\end{array}\right]}^{\mathit{T}}={\mathit{r}}_{\mathit{u}}-{\mathit{r}}_{\mathit{s}}$$

(3)

${\mathit{r}}_{\mathit{u}/\mathit{s}}$ is the relative position of the drone and the USV.

UAV translational dynamics:

$$\left\{\begin{array}{c}\ddot{\mathit{x}}=(\mathit{cos} \mathit{\varphi }\mathit{sin} \mathit{\theta }\mathit{cos} \mathit{\psi }+\mathit{sin} \mathit{\varphi }\mathit{sin} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_1}{\mathit{m}}}\dot{\mathit{x}}\\ \ddot{\mathit{y}}=(\mathit{cos} \mathit{\varphi }\mathit{sin} \mathit{\theta }\mathit{sin} \mathit{\psi }-\mathit{sin} \mathit{\varphi }\mathit{cos} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_2}{\mathit{m}}}\dot{\mathit{y}}\\ \ddot{\mathit{z}}=(\mathit{cos} \mathit{\varphi }\mathit{cos} \mathit{\theta }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-\mathit{g}-{\displaystyle \frac{{\mathit{K}}_3}{\mathit{m}}}\dot{\mathit{z}}\end{array}\right.$$

(4)

USV translational dynamics:

$$\left\{\begin{array}{c}{\ddot{\mathit{x}}}_{\mathit{s}}={\displaystyle \frac{{\mathit{F}}_{\mathit{s}}}{{\mathit{m}}_{\mathit{s}}}}\mathit{cos} {\mathit{\psi }}_{\mathit{s}}-{\mathit{d}}_{\mathit{x}}{\dot{\mathit{x}}}_{\mathit{s}}\\ {\ddot{\mathit{y}}}_{\mathit{s}}={\displaystyle \frac{{\mathit{F}}_{\mathit{s}}}{{\mathit{m}}_{\mathit{s}}}}\mathit{sin} {\mathit{\psi }}_{\mathit{s}}-{\mathit{d}}_{\mathit{y}}{\dot{\mathit{y}}}_{\mathit{s}}\\ {\ddot{\mathit{z}}}_{\mathit{s}}=-{\mathit{A}}_{\mathit{w}}{\mathit{\omega }}_{\mathit{w}}^2\mathit{sin}\left({\mathit{\omega }}_{\mathit{w}}t\right)\end{array}\right.$$

(5)

In formula (5), ${\mathit{F}}_{\mathit{s}}$ is the thrust of the unmanned ship. ${\mathit{\psi }}_{\mathit{s}}$ is the steering angle. ${\mathit{d}}_{\mathit{x}} \mathbf{a}\mathbf{n}\mathbf{d} {\mathit{d}}_{\mathit{y}}$ are linear damping factors. According to the above formula, we can get the relative dynamics formula of UAV and USV.

$$\left\{\begin{array}{c}{\ddot{\mathit{x}}}_{\mathit{u}/\mathit{s}}=(\mathit{cos} \mathit{\varphi }\mathit{sin} \mathit{\theta }\mathit{cos} \mathit{\psi }+\mathit{sin} \mathit{\varphi }\mathit{sin} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_1}{\mathit{m}}}\dot{\mathit{x}}-{\displaystyle \frac{{\mathit{F}}_{\mathit{s}}}{{\mathit{m}}_{\mathit{s}}}}\mathit{cos}{\mathit{\psi }}_{\mathit{s}}+{\mathit{d}}_{\mathit{x}}{\dot{\mathit{x}}}_{\mathit{s}}\\ {\ddot{\mathit{y}}}_{\mathit{u}/\mathit{s}}=(\mathit{cos} \mathit{\varphi }\mathit{sin} \mathit{\theta }\mathit{sin} \mathit{\psi }-\mathit{sin} \mathit{\varphi }\mathit{cos} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_2}{\mathit{m}}}\dot{\mathit{y}}-{\displaystyle \frac{{\mathit{F}}_{\mathit{s}}}{{\mathit{m}}_{\mathit{s}}}}\mathit{sin} {\mathit{\psi }}_{\mathit{s}}+{\mathit{d}}_{\mathit{y}}{\dot{\mathit{y}}}_{\mathit{s}}\\ {\ddot{\mathit{z}}}_{\mathit{u}/\mathit{s}}=(\mathit{cos} \mathit{\varphi }\mathit{cos} \mathit{\theta }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-\mathit{g}-{\displaystyle \frac{{\mathit{K}}_3}{\mathit{m}}}\dot{\mathit{z}}+{\mathit{A}}_{\mathit{w}}{\mathit{\omega }}_{\mathit{w}}^2\mathit{sin}\left({\mathit{\omega }}_{\mathit{w}}t\right)\end{array}\right.$$

(6)

2.3. Simulation Environment

ROS1 (Robot Operating System) under Ubuntu 20.04 is an open-source framework designed for robot operation systems, providing developers with a robust set of tools and APIs to build and control automated robotic systems. At its core, ROS incorporates three fundamental concepts: nodes, services, and topics. Nodes operate similarly to system processes, enabling developers to publish or subscribe to information flows; services, on the other hand, provide asynchronous functionality by allowing clients to request responses from remote or local servers; topics represent real-time data streams, facilitating dynamic event notifications in complex environments. These architectural components collectively enable robots to interact efficiently and perform operations within their operational environment.

The experiment utilized MAVROS and PX4 for joint simulation on the Gazebo platform. MAVROS is an officially provided tool under ROS that receives and sends MAVLink messages. Using MAVROS allows sending MAVLink messages to the flight control system and receiving data from it. The communication relationship between ROS and PX4 is illustrated in Figure 4.

By writing code in ROS, subscribing to or publishing MAVROS topics, subscribing to the drone’s state, and publishing the expected position or expected speed of the drone, PX4 flight control can track the expected motion.

2.4. Physical Platform Specifications

To verify the autonomous landing algorithm of the UAV, we assembled an F450 UAV, as shown in Figure 5. The UAV employs an Orangepi5b as the onboard computer, which is equipped with the Ubuntu 20.04 system. By configuring ROS, PX4, and MAVROS, we can conduct secondary development on the UAV. We use the D435 camera as the downward camera to recognize Apriltags identifiers.

3. Research Methods

3.1. Positioning Phase

This experiment is based on ROS + PX4 + MAVROS + Gazebo joint simulation, and the UAV control node needs to subscribe to the position topic of the mobile platform to obtain the position of the mobile platform as shown in Figure 6. The node diagram is illustrated in Figure 7.

In Figure 6, the apriltag_landing node represents the UAV’s motion control node. It subscribes to the odometry topic ugv_0/odom from the USV, enabling the UAV to receive real-time position and velocity information of the USV. The information is subsequently published through mavros-related publisher topics to facilitate position and attitude control of the UAV, thereby achieving localization and tracking of the USV by the UAV.

3.2. The Tracking Phase

After the UAV acquires the position information of the USV, it employs an MPC to enable the UAV to track the USV. The UAV first flies to a fixed altitude, and then tracks the trajectory of the unmanned boat in the XY plane. This enables the UAV to quickly fly to the vicinity of the USV, ensuring that the downward-facing camera on the UAV can identify the Apriltags markers on the USV. Tracking the unmanned boat in the XY plane at a fixed altitude simplifies the complexity of the MPC, transforming the UAV’s motion control in three-dimensional space into motion control in two-dimensional space. Figure 8 illustrates this configuration.

The predictive model forms the basis of MPC, enabling it to predict future system responses based on control actions. This capability enables the controller to utilize real-time states and operational constraints to forecast system dynamics over a specified prediction horizon, thereby optimizing control inputs to achieve desired system performance.

The UAV dynamics model is based on rigid body kinematics and Newton–Euler equations, and the state variables are defined as:

$$\mathit{X} \mathbf{=} {\mathbf{[}\mathbf{x}\mathbf{,} \mathit{y}\mathbf{,} \mathit{z}\mathbf{,} {\mathit{v}}_{\mathit{x}}\mathbf{,}{ \mathit{v}}_{\mathit{y}}\mathbf{,} {\mathit{v}}_{\mathit{z}}\mathbf{,} \mathit{r}\mathbf{,} \mathit{p}\mathbf{,} \mathit{y}\mathit{a}\mathbf{,} {\mathit{w}}_{\mathit{x}}\mathbf{,} {\mathit{w}}_{\mathit{y}},{\mathit{w}}_{\mathit{z}}\mathbf{]}}^{\mathit{T}}$$

Among them, x, y, z represent the UAV’s position, ${\mathit{v}}_{\mathit{x}}$, ${\mathit{v}}_{\mathit{y}}$, ${\mathit{v}}_{\mathit{z}}$ represents the UAV’s velocity, r, p, $\mathit{y}\mathit{a}$ represent the roll, pitch, and yaw angles of the UAV, respectively, and ${\mathit{w}}_{\mathit{x}}$, ${\mathit{w}}_{\mathit{y}}$, ${\mathit{w}}_{\mathit{z}}$ represent the angular velocity.

The nonlinear dynamics equations are:

$$\dot{\mathit{X}}=\left[\begin{array}{c}{\mathit{v}}_{\mathit{x}}\\ {\mathit{v}}_{\mathit{y}}\\ {\mathit{v}}_{\mathit{z}}\\ {\displaystyle \frac{{\mathit{u}}_1}{\mathit{m}}}\left(\mathit{c}\mathit{o}\mathit{s}\left(\mathit{y}\mathit{a}\right)\mathit{s}\mathit{i}\mathit{n}\left(\mathit{p}\right)\mathit{c}\mathit{o}\mathit{s}\left(\mathit{r}\right)+\mathit{s}\mathit{i}\mathit{n}\left(\mathit{y}\mathit{a}\right)\mathit{s}\mathit{i}\mathit{n}\left(\mathit{r}\right)\right)-{\displaystyle \frac{{\mathit{k}}_{\mathit{x}}{\mathit{v}}_{\mathit{x}}}{\mathit{m}}}+{\displaystyle \frac{{\mathit{d}}_{\mathit{z}}}{\mathit{m}}}\\ {\displaystyle \frac{{\mathit{u}}_1}{\mathit{m}}}\left(\mathit{s}\mathit{i}\mathit{n}\left(\mathit{y}\mathit{a}\right)\mathit{s}\mathit{i}\mathit{n}\left(\mathit{p}\right)\mathit{c}\mathit{o}\mathit{s}\left(\mathit{r}\right)-\mathit{c}\mathit{o}\mathit{s}\left(\mathit{y}\mathit{a}\right)\mathit{s}\mathit{i}\mathit{n}\left(\mathit{r}\right)\right)-{\displaystyle \frac{{\mathit{k}}_{\mathit{y}}{\mathit{v}}_{\mathit{y}}}{\mathit{m}}}+{\displaystyle \frac{{\mathit{d}}_{\mathit{y}}}{\mathit{m}}}\\ {\displaystyle \frac{{\mathit{u}}_1}{\mathit{m}}}\mathit{cos}(\mathit{p})\mathit{cos}(\mathit{r})-\mathit{g}-{\displaystyle \frac{{\mathit{k}}_{\mathit{z}}{\mathit{v}}_{\mathit{z}}}{\mathit{m}}}+{\displaystyle \frac{{\mathit{d}}_{\mathit{z}}}{\mathit{m}}}\\ {\mathit{w}}_{\mathit{x}}+{\mathit{w}}_{\mathit{y}}\mathit{s}\mathit{i}\mathit{n}\left(\mathit{r}\right)\mathit{t}\mathit{a}\mathit{n}\left(\mathit{p}\right)+{\mathit{w}}_{\mathit{z}}\mathit{cos}\left(\mathit{r}\right)\mathit{tan}\left(\mathit{p}\right)\\ {\mathit{w}}_{\mathit{y}}\mathit{cos}\left(\mathit{r}\right)-{\mathit{w}}_{\mathit{z}}\mathit{sin}\left(\mathit{r}\right)\\ {\displaystyle \frac{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{r}\right){\mathit{w}}_{\mathit{y}}}{\mathit{c}\mathit{o}\mathit{s}\left(\mathit{p}\right)}}+{\displaystyle \frac{\mathit{c}\mathit{o}\mathit{s}\left(\mathit{r}\right){\mathit{w}}_{\mathit{z}}}{\mathit{c}\mathit{o}\mathit{s}\left(\mathit{p}\right)}}\\ {\displaystyle \frac{{\mathit{u}}_2}{{\mathit{I}}_{\mathit{x}}}}-{\displaystyle \frac{\left({\mathit{I}}_{\mathit{y}}-{\mathit{I}}_{\mathit{z}}\right){\mathit{w}}_{\mathit{y}}{\mathit{w}}_{\mathit{z}}}{{\mathit{I}}_{\mathit{x}}}}\\ {\displaystyle \frac{{\mathit{u}}_3}{{\mathit{I}}_{\mathit{y}}}}-{\displaystyle \frac{\left({\mathit{I}}_{\mathit{z}}-{\mathit{I}}_{\mathit{x}}\right){\mathit{w}}_{\mathit{x}}{\mathit{w}}_{\mathit{z}}}{{\mathit{I}}_{\mathit{y}}}}\\ {\displaystyle \frac{{\mathit{u}}_4}{{\mathit{I}}_{\mathit{z}}}}-{\displaystyle \frac{\left({\mathit{I}}_{\mathit{x}}-{\mathit{I}}_{\mathit{y}}\right){\mathit{w}}_{\mathit{x}}{\mathit{w}}_{\mathit{y}}}{{\mathit{I}}_{\mathit{z}}}}\end{array}\right]$$

(7)

When the UAV is hovering or moving at low speed, the attitude angles involved in the UAV’s motion can be linearized: $\mathit{s}\mathit{i}\mathit{n}\left(\mathit{r}\right)\approx \mathit{r}$, $\mathit{c}\mathit{o}\mathit{s}\left(\mathit{r}\right)\approx 1$, $\mathit{t}\mathit{a}\mathit{n}\left(\mathit{r}\right)\approx \mathit{r}$, $\mathit{s}\mathit{i}\mathit{n}\left(\mathit{p}\right)\approx \mathit{p}$, $\mathit{c}\mathit{o}\mathit{s}\left(\mathit{p}\right)\approx 1$, $\mathit{t}\mathit{a}\mathit{n}\left(\mathit{p}\right)\approx \mathit{p}$, $\mathit{s}\mathit{i}\mathit{n}\left(\mathit{y}\mathit{a}\right)\approx \mathit{y}\mathit{a}$, $\mathit{c}\mathit{o}\mathit{s}\left(\mathit{y}\mathit{a}\right)\approx 1$. By removing the disturbance terms, the predictive model is further simplified to obtain Equation (8).

$$\dot{\mathit{X}}=\left[\begin{array}{c}{\mathit{v}}_{\mathit{x}}\\ {\mathit{v}}_{\mathit{y}}\\ {\mathit{v}}_{\mathit{z}}\\ {\displaystyle \frac{{\mathit{u}}_1}{\mathit{m}}}\left(\mathit{p}+\mathit{y}\mathit{a}\times \mathit{r}\right)-{\displaystyle \frac{{\mathit{k}}_{\mathit{x}}{\mathit{v}}_{\mathit{x}}}{\mathit{m}}}\\ {\displaystyle \frac{{\mathit{u}}_1}{\mathit{m}}}\left(\mathit{y}\mathit{a}\times \mathit{p}-\mathit{r}\right)-{\displaystyle \frac{{\mathit{k}}_{\mathit{y}}{\mathit{v}}_{\mathit{y}}}{\mathit{m}}}\\ {\displaystyle \frac{{\mathit{u}}_1}{\mathit{m}}}-\mathit{g}-{\displaystyle \frac{{\mathit{k}}_{\mathit{z}}{\mathit{v}}_{\mathit{z}}}{\mathit{m}}}\\ {\mathit{w}}_{\mathit{x}}+{\mathit{w}}_{\mathit{y}}\mathit{r}\mathit{p}+{\mathit{w}}_{\mathit{z}}\mathit{p}\\ {\mathit{w}}_{\mathit{y}}-{\mathit{w}}_{\mathit{z}}\mathit{r}\\ \mathit{r}{\mathit{w}}_{\mathit{y}}+{\mathit{w}}_{\mathit{z}}\\ {\displaystyle \frac{{\mathit{u}}_2}{{\mathit{I}}_{\mathit{x}}}}-{\displaystyle \frac{\left({\mathit{I}}_{\mathit{y}}-{\mathit{I}}_{\mathit{z}}\right){\mathit{w}}_{\mathit{y}}{\mathit{w}}_{\mathit{z}}}{{\mathit{I}}_{\mathit{x}}}}\\ {\displaystyle \frac{{\mathit{u}}_3}{{\mathit{I}}_{\mathit{y}}}}-{\displaystyle \frac{\left({\mathit{I}}_{\mathit{z}}-{\mathit{I}}_{\mathit{x}}\right){\mathit{w}}_{\mathit{x}}{\mathit{w}}_{\mathit{z}}}{{\mathit{I}}_{\mathit{y}}}}\\ {\displaystyle \frac{{\mathit{u}}_4}{{\mathit{I}}_{\mathit{z}}}}-{\displaystyle \frac{\left({\mathit{I}}_{\mathit{x}}-{\mathit{I}}_{\mathit{y}}\right){\mathit{w}}_{\mathit{x}}{\mathit{w}}_{\mathit{y}}}{{\mathit{I}}_{\mathit{z}}}}\end{array}\right]$$

(8)

Linearization using the Jacobian matrix near the hover equilibrium point:

$$\mathit{A}={\left.{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{x}}}\right|}_{{\mathit{X}}_{\mathit{e}},{\mathit{u}}_{\mathit{e}}},\mathit{B}={\left.{\displaystyle \frac{\partial \mathit{f}}{\partial \mathit{u}}}\right|}_{{\mathit{X}}_{\mathit{e}},{\mathit{u}}_{\mathit{e}}}$$

(9)

In the above formula, ${\mathit{X}}_{\mathit{e}}={\left[\begin{array}{c}{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,0,0,0,0,0,0,0\end{array}\right]}^{\mathit{\top }}$, ${\mathit{u}}_{\mathit{e}}={\left[\begin{array}{c}\mathit{m}\mathit{g},0,0,0\end{array}\right]}^{\mathit{\top }}$. The continuous state–space model obtained after linearization is:

$$\dot{\mathit{X}}=\mathit{A}\mathit{X}+\mathit{B}\mathit{u},\mathit{Y}=\mathit{C}\mathit{X}$$

(10)

The positions and yaw angles are extracted through the output matrix $\mathit{C}$. So $\mathit{Y}=[ \mathit{x}, \mathit{y}, \mathit{z}, \mathit{y}\mathit{a}]$. We use a zero-order hold (ZOH) for discretization with a sampling period of ${\mathit{T}}_{\mathit{s}}=1/{\mathit{f}}_{\mathit{s}}$. The discrete equation is as follows:

$$\begin{array}{c}\mathit{X}(\mathit{k}+1)={\mathit{A}}_{\mathit{d}}\mathit{X}\left(\mathit{k}\right)+{\mathit{B}}_{\mathit{d}}\mathit{u}\left(\mathit{k}\right)\\ \mathit{Y}\left(\mathit{k}\right)={\mathit{C}}_{\mathit{d}}\mathit{X}\left(\mathit{k}\right)\end{array}$$

(11)

In the prediction horizon ${\mathit{N}}_{\mathit{p}}$, the state prediction equation is:

$${\mathit{X}}_{\mathit{p}}=\mathbf{\Theta }\mathit{U}+\mathit{M}$$

(12)

${\mathit{X}}_{\mathit{p}}={\left[\begin{array}{c}\mathit{X}(\mathit{k}+1|\mathit{k}{)}^{\mathit{\top }},\dots ,\mathit{X}(\mathit{k}+{\mathit{N}}_{\mathit{p}}|\mathit{k}{)}^{\mathit{\top }}\end{array}\right]}^{\mathit{\top }}$ is the state prediction matrix. $\mathit{U}={\left[\begin{array}{c}\mathit{u}\left(\mathit{k}\right|\mathit{k}{)}^{\mathit{\top }},\dots ,\mathit{u}(\mathit{k}+{\mathit{N}}_{\mathit{c}}-1|\mathit{k}{)}^{\mathit{\top }}\end{array}\right]}^{\mathit{\top }}$ is the control input matrix. $\mathit{\Theta }$ is the state transition matrix.

$$\mathit{\Theta }=\left[\begin{array}{cccc}{\mathit{B}}_{\mathit{d}}& 0& \dots & 0\\ {\mathit{A}}_{\mathit{d}}{\mathit{B}}_{\mathit{d}}& {\mathit{B}}_{\mathit{d}}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ {\mathit{A}}_{\mathit{d}}^{{\mathit{N}}_{\mathit{p}}-1}{\mathit{B}}_{\mathit{d}}& {\mathit{A}}_{\mathit{d}}^{{\mathit{N}}_{\mathit{p}}-2}{\mathit{B}}_{\mathit{d}}& \dots & {\mathit{B}}_{\mathit{d}}\end{array}\right]$$

(13)

$\mathit{M}={\left[\begin{array}{c}{\mathit{A}}_{\mathit{d}}\mathit{x}\left(\mathit{k}\right),{\mathit{A}}_{\mathit{d}}^2\mathit{x}\left(\mathit{k}\right),\dots ,{\mathit{A}}_{\mathit{d}}^{{\mathit{N}}_{\mathit{p}}}\mathit{x}\left(\mathit{k}\right)\end{array}\right]}^{\mathit{\top }}$ is the free response matrix.

To minimize the tracking error and the cost of control inputs, the objective function $\mathit{J}$ is constructed

$$\mathit{J}={\mathbf{\sum }}_{\mathit{i}=1}^{{\mathit{N}}_{\mathit{p}}} \parallel \mathit{Y}(\mathit{k}+\mathit{i}|\mathit{k})-\mathit{r}(\mathit{k}+\mathit{i}){\parallel }_{\mathit{Q}}^2+{\mathbf{\sum }}_{\mathit{i}=0}^{{\mathit{N}}_{\mathit{c}}-1} \parallel \mathit{u}(\mathit{k}+\mathit{i}){\parallel }_{\mathit{R}}^2$$

(14)

In the above formula, $\mathit{Q}=\mathit{diag}(150,150,150,1)$ and $\mathit{R}=\mathit{diag}(1,1,1,1)$ are the weight matrices. Since the control input simulates the analog signal of a remote controller, the control input has the following constraints:

$$\left\{\begin{array}{c}{\mathit{u}}_{\mathbf{min}}\le \mathit{u}(\mathit{k}+i)\le {\mathit{u}}_{\mathbf{max}},i=0,1,\dots ,{\mathit{N}}_{\mathit{c}}-1,\\ {\mathit{u}}_{\mathbf{min}}=[-1,-1,-1,-1{]}^{\mathbf{\top }},\\ {\mathit{u}}_{\mathbf{max}}=[1,1,1,1{]}^{\mathbf{\top }}.\end{array}\right.$$

(15)

After transforming the objective function $\mathit{J}$ into the standard quadratic form, the following equation is obtained:

$$\underset{\mathit{U}}{\mathit{m}\mathit{i}\mathit{n}} {\displaystyle \frac{1}{2}}{\mathit{U}}^{\mathbf{\top }}\mathit{H}\mathit{U}+{\mathit{f}}^{\mathbf{\top }}\mathit{U} \mathrm{s}.\mathrm{t}. {\mathit{U}}_{\mathbf{min}}\le \mathit{U}\le {\mathit{U}}_{\mathbf{max}}$$

(16)

In the above formula, $\mathit{H}=2({\mathit{\Theta }}^{\mathit{\top }}\mathit{Q}\mathit{\Theta }+\mathit{R}),\mathit{f}={\mathit{\Theta }}^{\mathit{\top }}\mathit{Q}(\mathit{M}-{\mathit{R}}_{\mathit{ref}}).$ By solving the above QP problem, the first control input $\mathit{u}\left(\mathit{k}\right)$ obtained from the solution is applied to the UAV, and the state is updated.

3.3. Landing Phase

During the UAV landing phase, we designed a UAV PID speed controller based on the position error obtained by the downward-facing camera. Meanwhile, to ensure that the UAV can land stably in a windy environment, we conducted wind resistance tests on the UAV. First, we tested the algorithm in a simulation environment, as shown in Figure 9, and then validated the algorithm again by deploying it on a real UAV.

3.3.1. Multi-Apriltag Marker-Based Design

During the UAV landing process, as the altitude gradually decreases, the field of view of the camera continuously shrinks. When the UAV descends to a certain altitude, the camera can no longer identify the complete identifier. As shown in Figure 10, a single marker cannot meet the precision requirements for the unmanned aircraft’s accurate landing.

To address the problem mentioned above, we designed a cooperative identifier [26,27]. As shown in Figure 11. When the UAV is at a higher altitude, it recognizes the larger external marker. When the altitude of the UAV gradually decreases and it can no longer recognize the complete external marker, the UAV begins to recognize the internal marker. To achieve precise landing, a large empty space should be maintained in the center of the external marker. In this experiment, the TAG36H11 family member with ID 236 was selected as the large-sized Apriltag marker, while the ID 2 Apriltag marker was used for small-sized object marker.

3.3.2. Design of the PID Speed Controller

We designed a PID speed controller based on the position error obtained from the Apriltags algorithm under the ROS platform to enable the UAV to track the marker. The Apriltags algorithm’s subscription and publication of messages is illustrated in Figure 12.

At the UAV control node, the position of the camera relative to the center of the identifier is obtained by subscribing to the /tag_detections topic.

The relative position obtained is used as the error input for the PID controller to obtain the control input of the UAV speed controller, as shown in Figure 13. When the UAV is tracking, a permissible landing error range is set. When the relative position between the UAV and the identifier is within the error range, the UAV completes the autonomous landing by gradually reducing its altitude.

3.4. Aerodynamic Disturbances

During the flight process, the UAV is aways affected by wind. In order to enable the UAV to complete autonomous landing under the influence of wind, we conducted simulation experiments on the wind resistance capability of UAVs. Based on the existing wind field model [28] and in combination with the UAV dynamics model, we design a composite controller to allow the UAV to complete the landing task.

3.4.1. Wind Field Model

The wind speed model proposed in this paper is based on the following assumptions: The wind speed consists of a constant wind speed part and a dynamically changing gust part. The constant wind speed part represents the basic level of wind speed, while the gust part reflects the random fluctuations of wind speed. By introducing a time decay factor and random disturbances, the model can more accurately simulate the dynamic changes in wind speed.

The mathematical expression of the model is as follows:

$${\mathit{V}}_{\mathit{total}}={\mathit{V}}_{\mathit{c}}+\mathrm{G}$$

(17)

${\mathit{V}}_{\mathit{c}}$ represents the constant wind speed, and the $\mathrm{G}$ is the gust. The gust is composed of the decaying gust and the random gust. The expression is as follows:

$$\mathit{G}\mathbf{=} \mathit{Attenuation} \mathit{gust}+\mathit{random} \mathit{gust}$$

(18)

$$\mathit{Attenuation} \mathit{gust}=\mathit{last}\mathit{\text{\_}}\mathit{gust}\times {\mathit{e}}^{-\mathit{t}/\mathit{\tau }}$$

(19)

$$\mathit{random} \mathit{gust}=\mathit{turbulence}\mathit{\text{\_}}\mathit{intensity}\times \mathit{rand}\left(\mathbf{3}\right)\times (1-{\mathit{e}}^{-\mathit{t}/\mathit{\tau }})$$

(20)

The $\mathit{last}\mathit{\text{\_}}\mathit{gust}$ refers to the last gust speed, and the exponential decay function ${\mathit{e}}^{-\mathit{t}/\mathit{\tau }}$ is introduced to simulate the attenuation of the wind. In Equation (20), the $\mathit{turbulence}\mathit{\text{\_}}\mathit{intensity}$ represents the strength of turbulence. The function $\mathit{rand}\left(3\right)$ generates a random vector of length 3, which simulates the random fluctuations of wind speed in three directions. The variable $(1-{\mathit{e}}^{-\mathit{t}/\mathit{\tau }})$ is the time decay factor, used to simulate the decaying characteristics of random gusts over time.

3.4.2. Wind Resistance Control System

Based on the wind force model and the UAV dynamics model, we designed the UAV control system to achieve wind resistance. The controller included two main parts: the PID controller and the DOB (Disturbance Observer).

The PID controller is used to realize position tracking and the discrete PID controller is as follows:

$$\left\{\begin{array}{c}{\mathit{u}}_{\mathit{p}\mathit{i}\mathit{d}}\left(\mathit{k}\right)={\mathit{K}}_{\mathit{p}}\mathit{e}\left(\mathit{k}\right)+{\mathit{K}}_{\mathit{d}}\dot{\mathit{e}}\left(\mathit{k}\right)+{\mathit{K}}_{\mathit{i}}\mathbf{\int }\mathit{e}\left(\mathit{k}\right)\\ \mathit{e}\left(\mathit{k}\right)={\mathit{p}}_{\mathit{r}\mathit{e}\mathit{f}}-\mathit{p}\left(\mathit{k}\right)\\ \dot{\mathit{e}}\left(\mathit{k}\right)\approx {\displaystyle \frac{\mathit{e}\left(\mathit{k}\right)-\mathit{e}(\mathit{k}-1)}{\mathit{\Delta }\mathit{t}}}\\ \mathbf{\int }\mathit{e}\left(\mathit{k}\right)\approx \mathit{clip}\left({\mathbf{\sum }}_{\mathit{i}=0}^{\mathit{k}} \mathit{e}\left(\mathit{i}\right)\mathit{\Delta }\mathit{t},-5,+5\right)\end{array}\right.$$

(21)

${\mathit{u}}_{\mathit{p}\mathit{i}\mathit{d}}\left(\mathit{k}\right)\in {\mathbb{R}}^3$ is the input vector and $\mathit{e}\left(\mathit{k}\right)$ is the position error vector caused by wind; the proportional part can generate a control force to counteract wind force in real time. The differential part can predict the trend of error change and provide damping in advance to counteract the oscillations caused by wind. The integral part can accumulate deviations caused by wind. To prevent integral windup, the integral term is limited to the range (−5, +5). Due to the limited thrust of the UAV, the output of the PID controller is constrained within the range of (−15, +15).

The DOB can estimate and provide feedforward compensation for external disturbances in real time. The mathematical expression of the disturbance observer is as follows:

$${\displaystyle \frac{\mathit{d}\dot{\mathit{d}}}{\mathit{d}\mathit{t}}}={\mathit{K}}_{\mathit{d}\mathit{o}\mathit{b}}\left(\dot{\mathit{v}}-{\displaystyle \frac{\mathit{u}}{\mathit{m}}}-\dot{\mathit{d}}\right)$$

(22)

The ${\mathit{K}}_{\mathit{d}\mathit{o}\mathit{b}}$ is the sensitivity of the control observer to the error. The $\dot{\mathit{v}}$ is the actual acceleration of the UAV. The u is the control force currently output by the PID controller, so it can be inferred that ${\displaystyle \frac{\mathit{u}}{\mathit{m}}}$ is the theoretical acceleration. $\dot{\mathit{d}}$ is the currently estimated disturbance wind force.

The mathematical formula of this control system is as follows:

$$\mathit{u}={\mathit{u}}_{\mathit{p}\mathit{i}\mathit{d}}-\dot{\mathit{d}}$$

(23)

In this way, we can maintain the simplicity of PID while significantly enhancing wind resistance performance through the disturbance observer.

4. Results

In this section, we discuss the experimental results of the simulation and the real UAV.

4.1. MPC Tracking Controller Results Display

In this section, we use MATLAB R2023a to evaluate the results of the MPC trajectory tracking controller. Since the USV is affected by waves during its movement, it often has difficulty moving in a straight line. Therefore, a circular trajectory is adopted for the USV. The tracking performance is shown in Figure 14, where the red solid line represents the trajectory of the USV and the blue solid line represents the trajectory of the UAV.

Figure 15 and Figure 16 illustrate the UAV’s trajectory tracking in X and Y directions. Figure 17 and Figure 18 depict the tracking errors in X and Y directions, respectively. The results indicate that the errors in both X and Y directions remain within the range of −0.8 to 0.8 during the tracking process. Figure 19 shows the ability of the UAV to track the yaw angle of the USV, and the UAV can quickly perform this tracking task.

Based on the results presented, the tracking phase demonstrates that the UAV can effectively track the autonomous vessel and meet the requirements for the landing stage.

4.2. Landing Phase PID Controller Analysis and Results Demonstration

In the landing phase, we verified the effectiveness of the PID speed controller by having the UAV track both straight and circular trajectories.

The general form of the PID controller is as follows:

$$\begin{array}{c}\mathit{u}(\mathit{t})={\mathit{K}}_{\mathit{p}}\mathit{e}(\mathit{t})+{\mathit{K}}_{\mathit{i}}\mathbf{\int }\mathit{e}(\mathit{t})\mathit{d}\mathit{t}+{\mathit{K}}_{\mathit{d}}{\displaystyle \frac{\mathit{d}\mathit{e}(\mathit{t})}{\mathit{d}\mathit{t}}}\end{array}$$

(24)

After obtaining the relative position of the onboard camera with respect to the barcode, the camera coordinate system follows the right-hand coordinate system. Therefore, the X direction of the downward-looking camera corresponds to the Y direction of the UAV, and the Y direction corresponds to the X direction of the UAV. An error identified in the X direction is noted as ${\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)$. The PID controller for the X direction is as follows:

$${\mathit{u}}_{\mathit{x}}\left(\mathit{t}\right)={\mathit{K}}_{\mathit{p}}{\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)+{\mathit{K}}_{\mathit{i}}\mathbf{\int }{\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)\mathit{d}\mathit{t}+{\mathit{K}}_{\mathit{d}}{\displaystyle \frac{\mathit{d}{\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}}$$

(25)

Similarly, the PID controller for the Y direction is as follows:

$${\mathit{u}}_{\mathit{y}}\left(\mathit{t}\right)={\mathit{K}}_{\mathit{p}}{\mathit{e}}_{\mathit{y}}\left(\mathit{t}\right)+{\mathit{K}}_{\mathit{i}}\mathbf{\int }{\mathit{e}}_{\mathit{y}}\left(\mathit{t}\right)\mathit{d}\mathit{t}+{\mathit{K}}_{\mathit{d}}{\displaystyle \frac{\mathit{d}{\mathit{e}}_{\mathit{y}}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}}$$

(26)

We have tuned the controller parameters using the Ziegler–Nichols method. The PID controller parameters for the X and Y directions are as shown in

Table 2. PID controller parameters table for X and Y directions.

Parameter	Kp	Ki	Kd
x	0.8	0.02	0.1
y	0.6	0.02	0.1

.

According to Equation (1), the horizontal position dynamics of the UAV can be expressed as:

$$\ddot{\mathit{x}}=(\mathbf{cos} \mathit{\varphi }\mathbf{sin} \mathit{\theta }\mathbf{cos} \mathit{\psi }+\mathbf{sin} \mathit{\varphi }\mathbf{sin} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_1}{\mathit{m}}}\dot{\mathit{x}}$$

(27)

$$\ddot{\mathit{y}}=(\mathbf{cos} \mathit{\varphi }\mathbf{sin} \mathit{\theta }\mathbf{sin} \mathit{\psi }-\mathbf{sin} \mathit{\varphi }\mathbf{cos} \mathit{\psi }){\displaystyle \frac{1}{\mathit{m}}}{\mathit{U}}_1-{\displaystyle \frac{{\mathit{K}}_2}{\mathit{m}}}\dot{\mathit{y}}$$

(28)

Since the UAV does not consider air resistance in the simulation environment and the changes in the UAV’s body angle during the tracking process are small, the above formula can be simplified as:

$$\ddot{\mathit{x}}={\displaystyle \frac{1}{\mathit{m}}}{(\mathit{F}}_{\mathbf{x}}-\dot{\mathit{x})}$$

(29)

$$\ddot{\mathit{y}}={\displaystyle \frac{1}{\mathit{m}}}{(\mathit{F}}_{\mathbf{y}}-\dot{\mathit{y})}$$

(30)

Taking the Laplace transform of the above formula yields:

$${\mathit{G}}_{\mathbf{open},\mathit{x}}(\mathit{s})={\displaystyle \frac{\mathit{X}\left(\mathit{s}\right)}{{\mathit{F}}_{\mathit{x}}\left(\mathit{s}\right)}}={\displaystyle \frac{1}{\mathit{m}{\mathit{s}}^2+\mathit{s}}}$$

(31)

The transfer function in the Y direction has the same form as that in the X direction.

The transfer function of the PID controller is:

$$\mathit{C}(\mathit{s})={\mathit{K}}_{\mathit{p}}+{\displaystyle \frac{{\mathit{K}}_{\mathit{i}}}{\mathit{s}}}+{\mathit{K}}_{\mathit{d}}\mathit{s}={\displaystyle \frac{{\mathit{K}}_{\mathit{d}}{\mathit{s}}^2+{\mathit{K}}_{\mathit{p}}\mathit{s}+{\mathit{K}}_{\mathit{i}}}{\mathit{s}}}$$

(32)

The closed-loop transfer function of the system is:

$${\mathit{G}}_{\mathbf{closed}}(\mathit{s})={\displaystyle \frac{\mathit{C}\left(\mathit{s}\right){\mathit{G}}_{\mathbf{open}}\left(\mathit{s}\right)}{1+\mathit{C}\left(\mathit{s}\right){\mathit{G}}_{\mathbf{open}}\left(\mathit{s}\right)}}$$

(33)

$${\mathit{G}}_{\mathbf{closed}}(\mathit{s})={\displaystyle \frac{{\mathit{K}}_{\mathit{d}}{\mathit{s}}^2+{\mathit{K}}_{\mathit{p}}\mathit{s}+{\mathit{K}}_{\mathit{i}}}{\mathit{m}{\mathit{s}}^3+(1+{\mathit{K}}_{\mathit{d}}){\mathit{s}}^2+{\mathit{K}}_{\mathit{p}}\mathit{s}+{\mathit{K}}_{\mathit{i}}}}$$

(34)

According to the Routh stability criterion, the system is stable.

Figure 20 illustrates the line tracking error graph of the UAV controlled by the PID controller for the USV. The Y-direction error is within the range of −0.8 to 0.8. The X-direction error increases with the speed of the USV. As shown, when the USV moves at 1.8 m/s, the error between the UAV and the Apriltag marker is approximately 2 m. This fails to meet the landing conditions, hence an improvement of the PID controller is required.

This experiment improves the precision of autonomous landing of the UAV by adding bias terms to the PID controller. The PID controller with added bias terms is as follows:

$$\begin{array}{c}{\mathit{u}}_{\mathit{x}}\left(\mathit{t}\right)={\mathit{K}}_{\mathit{p}}{\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)+{\mathit{K}}_{\mathit{i}}\mathbf{\int }{\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)\mathit{d}\mathit{t}+{\mathit{K}}_{\mathit{d}}{\displaystyle \frac{\mathit{d}{\mathit{e}}_{\mathit{x}}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}}+\mathit{v}\mathit{e}{\mathit{l}}_{\mathit{x}}\end{array}$$

(35)

This bias term, $\mathit{v}\mathit{e}{\mathit{l}}_{\mathit{x}}$, represents the speed of the moving platform in the X-direction. By increasing the bias term $\mathit{v}\mathit{e}{\mathit{l}}_{\mathit{x}}$, we can reduce the tracking error for the target markers.

Similarly, the PID controller for the Y direction is as follows:

$$\begin{array}{c}{\mathit{u}}_{\mathit{y}}(\mathit{t})={\mathit{K}}_{\mathit{p}}{\mathit{e}}_{\mathit{y}}(\mathit{t})+{\mathit{K}}_{\mathit{i}}\mathbf{\int }{\mathit{e}}_{\mathit{y}}(\mathit{t})\mathit{d}\mathit{t}+{\mathit{K}}_{\mathit{d}}{\displaystyle \frac{\mathit{d}{\mathit{e}}_{\mathit{y}}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}}}+\mathit{v}\mathit{e}{\mathit{l}}_{\mathit{y}}\end{array}$$

(36)

The trajectory of the line tracking, as shown in Figure 21, illustrates the drone’s movement along the x-axis. The target moving platform moves at a speed of 3 m/s along the x-axis. The UAV’s y-axis error ranges from −0.8 m to 0.8 m, while the x-axis error ranges from −0.4 m to 0.4 m. When the errors in both the X and Y directions are within the range of (−0.3, 0.3) meters, the UAV will gradually reduce its altitude to complete the autonomous landing. Otherwise, it will continue to track at the current altitude until the autonomous landing is accomplished. The error illustration is shown in Figure 22.

Due to the influence of wind and waves on the USV, it is typically difficult for the USV to maintain straight-line movement, necessitating experiments on nonlinear trajectories. Figure 23 illustrates the trajectory tracking illustration of the drone when following a circular path by the USV moving at a speed of 2 m/s with angular velocity of 0.15 rad/s. When the drone identifies Apriltag markers, it performs height reduction if both X and Y directional errors are within −0.3 to 0.3 m. As shown in Figure 24, the X-direction error is between −0.4 to 0.4 m and the Y-direction error is between −0.4 to −0.2 m. Once the drone’s landing conditions are met, it lowers its height at a rate of −0.2 m/s in the Z-axis to complete the landing.

The experimental results indicate that when the USV moves at a speed of 3 m/s, the UAV’s X-direction error ranges between −0.5 and 0.5 m, while the Y-direction error ranges between −0.8 and 0.8 m, satisfying the landing conditions. Additionally, during the circular trajectory tracking process, when the USV moves at a speed of 2 m/s (v) and an angular speed of 0.2 rad/s (ω), the UAV’s X-direction error is within the range of −0.4 to 0.4 m, while the Y-direction error remains between −0.4 and −0.2 m. The experimental results demonstrate that the proposed method can satisfy the requirement for UAV landing on a dynamic platform.

4.3. The Results of Wind Force Simulation

In this section, we will analyze the simulation results of the controller based on the data obtained from the simulation.

To begin with, we will provide the initial values of the PID controller parameters and the tuning criteria.

• Proportional gain ${\mathit{K}}_{\mathit{p}}$:

Based on empirical formulas, the initial value of ${\mathit{K}}_{\mathit{p}}$ is

$${\mathit{K}}_{\mathit{p},\mathit{initial}}={\displaystyle \frac{16\mathit{m}}{{\mathit{T}}_{\mathit{settle}}^2}}$$

(37)

${\mathit{T}}_{\mathit{settle}}$ is the desired settling time. The target position is (10, 15, 5), and the desired time is set to 5. The mass of the UAV is 1.5 kg. We can get the ${\mathit{K}}_{\mathit{p},\mathit{initial}}$ = 0.96. A too-high ${\mathit{K}}_{\mathit{p}}$ can cause overshoot or oscillation, but a too-low ${\mathit{K}}_{\mathit{p}}$ will make the UAV respond untimely. The final experimental results show that ${\mathit{K}}_{\mathit{p}}$ = [3, 3, 4].

• Differential gain ${\mathit{K}}_{\mathit{d}}$:

We set the system damping ratio $\mathit{\zeta }$ to 0.7. The initial value of ${\mathit{K}}_{\mathit{d}}$ is

$${\mathit{K}}_{\mathit{d},\mathbf{initial}}=2\mathit{\zeta }\sqrt{\mathit{m}{\mathit{K}}_{\mathit{p}}}-{\mathit{K}}_{\mathit{drag}}$$

(38)

${\mathit{K}}_{\mathit{drag}}$ is the drag coefficient, which is used to quantify the effect of air resistance on the UAV during its motion. Since the current experiment uses a small quadrotor UAV, ${\mathit{K}}_{\mathit{drag}}$ is set to 0.2. We can get the ${\mathit{K}}_{\mathit{d},\mathit{initial}}$ = 1.48. The final experimental results show that ${\mathit{K}}_{\mathit{d}}$ = [1.5, 1.5, 2].

• Integral gain ${\mathit{K}}_{\mathit{i}}$:

The initial value of the integral gain ${\mathit{K}}_{\mathit{i}}$ should be much smaller than ${\mathit{K}}_{\mathit{p}}$.

$${\mathit{K}}_{\mathit{i},\mathit{initial}}={\displaystyle \frac{{\mathit{K}}_{\mathit{p}}}{10\mathit{\tau }}}$$

(39)

$\mathit{\tau }$ is the dominant time constant of the system. The initial value of $\mathit{\tau }$ is usually between one-third and one-half of the settling time, which is considered optimal. Therefore, we take $\tau$ as 2. We can get the ${\mathit{K}}_{\mathit{i},\mathit{initial}}=0.048$. The final experimental results show that ${\mathit{K}}_{\mathit{i}}$ = [0.1, 0.1, 0.2].

The UAV dynamics model is a second-order system, and the open-loop transfer function is:

$$\mathit{G}(\mathit{s})={\displaystyle \frac{1}{\mathit{m}{\mathit{s}}^2+{\mathit{K}}_{\mathbf{drag}}\mathit{s}}}$$

(40)

The closed-loop transfer function after introducing proportional, integral, and differential terms is:

$$\mathit{T}(\mathit{s})={\displaystyle \frac{{\mathit{K}}_{\mathit{p}}+{\mathit{K}}_{\mathit{d}}\mathit{s}+\frac{{\mathit{K}}_{\mathit{i}}}{\mathit{s}}}{\mathit{m}{\mathit{s}}^2+({\mathit{K}}_{\mathbf{drag}}+{\mathit{K}}_{\mathit{d}})\mathit{s}+{\mathit{K}}_{\mathit{p}}+\frac{{\mathit{K}}_{\mathit{i}}}{\mathit{s}}}}$$

(41)

The closed-loop system is proven to be stable using the Routh stability criterion.

We conducted experiments under the following wind field parameters: (X:2, Y:1.5, Z:1); (X:6, Y:7, Z:8); (X:11.5, Y:12.5, Z:12). The simulation results are shown in Figure 25, Figure 26 and Figure 27. When the wind force is relatively small, the UAV can respond quickly and reach the target position in a short time. However, as the wind force gradually increases, the overshoot of the UAV in the X and Y directions will continue to increase. But according to the results, the UAV can still fly to the designated position. Therefore, the controller can enable the UAV to have certain wind resistance capabilities.

5. Conclusions and Future Works

This paper divides the UAV landing process into three stages: positioning, tracking, and landing. Each stage is validated in the paper, which enhances the efficiency of UAV autonomous landing. Simultaneously, the UAV is tested for wind resistance through simulation.

Here it needs to be explained that since the MPC controller in the experiment requires control of the UAV’s torque, and MAVROS has a very poor effect on torque control, we used MATLAB for simulation during the tracking phase. Based on the simulation results, the MPC performs as expected during the tracking phase, but it still needs to be verified in real UAVs.

In the final landing phase, the algorithm from the simulation was transferred to a real UAV for algorithm validation. The experimental results are similar to the simulation results, capable of completing the landing task with a success rate of over 90%. It needs to be explained that, due to the limitations of the experimental conditions, we used manual dragging of a homemade identifier to simulate the movement of the unmanned boat. Therefore, it is not possible to fully replicate the motion of the unmanned boat on the water.

At this stage, we have achieved UAV autonomous landing on dynamic platforms, but some issues remain to be resolved. In our experiments, the UAV and USV must share position information over a local area network. Network instability can lead to data loss, thereby delaying the UAV’s response. Compared to other teams [22,23,24], we may need to add sensors like LiDAR to improve the final landing accuracy. When conducting the wind resistance experiments for UAVs, there are still some shortcomings when compared with those of other researchers [29].

In future work, to prevent UAVs from being affected by network instability, we may improve the UAV’s positioning method for target moving platforms. Additionally, to ensure final landing accuracy, we will consider adding sensors to the UAV. However, the main focus will remain on optimizing the UAV’s control algorithm.