Research on Motion Control and Compensation of UAV Shipborne Autonomous Landing Platform

Abstract

As an important interface between unmanned aerial vehicles (UAVs) and ships, the stability and motion control compensation technology of the shipborne UAV landing platform are paramount for successful UAV landings. This paper has designed a new control compensation method for an autonomous UAV landing platform to address the impact of complex sea conditions on the stability of UAV landing platforms. Firstly, the parallel Stewart platform was introduced as the landing platform, and its structure was analyzed with forward and inverse kinematic calculations conducted in Matlab to verify its accuracy. Secondly, a least-squares recursive AR prediction algorithm was designed to predict the future attitudes of ships under varying sea conditions. Finally, the prediction algorithm was combined with the platform’s control strategy and a dual-sensor system was adopted to ensure the stability of the UAV landing process. The experimental results demonstrate that these innovative improvements enhanced the compensation accuracy by 59.6%, 60.3%, 48.4%, and 47.9% for the rolling angles of 5° and 10° and the pitching angles of 5° and 10°, respectively. Additionally, the compensation accuracy for the roll and pitch in sea states 2 and 5 improved by 51.2%, 59.4%, 58.7%, and 55.9%, respectively, providing technical support for UAV missions such as maritime rescue and exploration.

1. Introduction

In recent years, China’s marine resource development and maritime power strategy have propelled the advancement of intelligent equipment such as unmanned ships and shipborne unmanned aerial vehicles (UAVs). Leveraging the synergy between ships (including unmanned vessels) and UAVs can optimally exploit the long endurance capabilities of ships and the high maneuverability of UAVs. However, the limited takeoff and landing space provided by ships, coupled with the stringent requirements for landing control precision, underscores the pivotal role of autonomous landing technology for shipborne UAVs in offshore operations and mission execution.

Since the 1960s, scientists have been relentlessly pursuing the development of autonomous landing systems for shipborne UAVs. In 1958, the United States introduced the “Skyhook Recovery” system [1], albeit with stringent trajectory control accuracy demands and vulnerability to the wind and waves. Zhang Jianfeng [2] addressed these challenges by constructing linear and nonlinear models for Skyhook-recovered UAVs and designing optimized L1 trajectory control laws, enhancing recovery precision. Nevertheless, this approach failed to fully mitigate the influence of wind, waves, and currents. Yao Tiancheng from Shanghai Jiao Tong University [3] proposed a vision-based autonomous landing technology, enhancing landing accuracy and safety under complex sea conditions, and enabling collaborative control between the UAVs and unmanned ships [4,5,6]. Nevertheless, this technique is sensitive to lighting conditions, prone to interference in complex backgrounds, and demands high real-time performance with stringent hardware requirements, limiting its applicability in severe sea states. Sun Ruina at Taiyuan University of Technology [7] devised a polar ice-based buoy landing platform, providing a stable environment for UAVs; however, the buoy’s drifting with the sea ice and the potential swaying due to waves and winds necessitate precise landing in dynamic environments, posing severe challenges to the UAV navigation systems and control algorithms.

Concurrently, maritime self-stabilizing platform technology has significantly advanced in UAV landing applications. Juneyoung Kwak [8] introduced a novel two-degree-of-freedom leveling mobile platform that maintains a horizontal landing surface and assists quadcopter stabilization. However, its two-dimensional adjustment limits its adaptability to complex maritime landing environments. The T700 maritime stabilization platform by BARGEMASTER in the Netherlands employs a three-degree-of-freedom parallel mechanism with a 700-ton load capacity [9], but it is tailored for large-scale offshore operations, potentially compromising flexibility in specific scenarios. You Xuhan [10] addressed heavy load stability requirements at sea by adopting a Stewart parallel mechanism as a compensation system, utilizing inertial sensor technology to analyze ship disturbances in real time. This study integrated long-term, short-term, and instantaneous ship motion prediction models to enhance landing platform compensation accuracy. Nevertheless, shortcomings in the disturbance analysis algorithm, the lack of a comprehensive system, and insufficient simulation validation hinder robust performance under complex disturbances, limiting its widespread applicability. Hong Liang [11] designed a PID and linear active disturbance rejection control (LADRC)-based height–position–speed cascade control system for quadrotor UAVs. By incorporating LADRC as a secondary loop controller and utilizing PID for height–position tracking, it mitigated height control disturbances from ground airflows during landing. However, the system’s real-time performance and position control system’s anti-interference capability are inadequate, potentially delaying control commands and affecting mission execution.

Drawing upon prior research, this paper presents a novel control compensation method for autonomous landing platforms for UAVs. Firstly, a Stewart parallel platform [12], renowned for its structural stability and six degrees of freedom, is selected as the autonomous landing platform for shipborne UAVs to enhance landing precision and stability. Forward and inverse kinematics simulations validate the proposed kinematic model’s accuracy and effectiveness. Secondly, a limited-memory recursive least squares AR prediction algorithm is designed to forecast the ship’s future attitude by solving the influence of external disturbances on the lower platform, transmitting data to the host computer, and thereby improving the upper platform’s compensation precision. Finally, an experimental platform is established, with the lower platform simulating ship motion and the upper platform serving as the UAV landing platform. A dual-sensor system ensures real-time transmission of ship attitude and platform data, bolstering the landing platform’s resilience against disturbances in complex sea conditions.

2. Landing Platform Based on the Parallel Stewart Platform

The landing platform model suggested in this paper is according to the parallel Stewart platform model. The parallel Stewart platform has a symmetrical structure, the same legs, and free expansion, which are easy to control, and the maximum displacement of each degree of freedom is easy to calculate [13]. The drone shipborne autonomous landing platform consists of the upper and lower platform, six electric cylinders, and 12 Hooke hinges, which are driven by six degrees of freedom through a motor and a ball screw. Due to the small shape variable, Hooke hinge deformation is ignored during modeling. This platform has the characteristics of a high bearing capacity, strong load, high precision, fast response, and low error.

In order to accurately describe the parallel Stewart mechanism, three parts of the three-dimensional rectangular coordinate system were built, used to finely determine the specific location of Stewart platform in a three-dimensional space, as shown in Figure 1. The coordinate system includes the global coordinate system {D}, base frame {G}, and mobile platform coordinate system {N}, while i = 1, 2, …, 6 represents the Hooke joint position on the fixed and mobile platforms.

In the configuration presented in Figure 2, the center of the circular platform is used as the reference point, and a new coordinate system is established at this point. By considering the arrangement of the Hooke hinge on the platform, it is noted that the position of each hinge relative to the center of the platform can be described by the platform radius as well as specific offset angles, which are marked as ${\phi }_c$, ${\phi }_d$, respectively, in Figure 2. Using the offset angle of the platform and the known radius of the circular platform, it is possible to accurately calculate the position of each Hooke hinge within its corresponding coordinate system. In the polar coordinate system, the hinge position of the Hooke hinge can be indicated by its polar angle as follows:

$$angle{c}_i=\left\{\begin{array}{l}2\pi \left(i\mathit{mod}3\right)/3-{\phi }_c/2\\ 2\pi \left(\left(i+1\right)\mathit{mod}3\right)/3+{\phi }_c/2\end{array}\right\}\begin{array}{c}i\in \left[1, 3, 5\right]\\ i\in \left[2, 4, 6\right]\end{array}$$

(1)

$$angle{d}_i=\left\{\begin{array}{l}2\pi \left(\left(i-1\right)\mathit{mod}3\right)-{\phi }_d/2\\ 2\pi \left(\left(i+1\right)\mathit{mod}3\right)/3+{\phi }_d/2\end{array}\right\}\begin{array}{c}i\in \left[1, 3, 5\right]\\ i\in \left[2, 4, 6\right]\end{array}$$

(2)

The radii of the upper and lower platform are calculated as follows:

$$c_i=\left(r_c,angle{C}_i\right),i\in \left[1, 6\right]$$

(3)

$$d_i=\left(r_d,angle{D}_i\right),i\in \left[1, 6\right]$$

(4)

To verify the kinematic accuracy of the parallel platform, the model is examined in Matlab and the forward and inverse kinematics are calculated. The process of forward kinematics [14] calculation is to determine the exact position and attitude of the upper platform according to the known drive expansion length, and the calculation formula of the expansion amount of each drive is shown in Formula (5); while the inverse kinematics [15] is to calculate the necessary drive expansion length [16] by determining the desired position and attitude of the upper platform, and the reverse solution calculation formula of the platform is shown in (6) as follows:

$${\displaystyle \sum }_{i=0}^5{F}_i(\chi ,y,z,\alpha ,\beta ,\gamma )=\left({}_N{}^{G}R{q}_i+\tilde{q}-G_i\right){({}_N{}^{G}R{q}_i+\tilde{q}-G_i)}^T-L_i^2$$

(5)

$$I=\sqrt{{({}_N{}^{G}R{n}_i+{{}^{G}q}_{N_0}-g_i)}^T\left({}_N{}^{G}R{n}_i+{{}^{G}q}_{N_0}-g_i\right)}$$

(6)

where ${}_N{}^{G}R$ is the rotation matrix, $\tilde{q}$ is the displacement variable, $G_i$ is the center where the base is located, $L_i$ is the bar length, and I is the expansion of the drive rod. Platform simulations were performed in the Matlab software, and the simulation model is shown in Figure 3.

For the calculation of the forward and inverse kinematics, the position of the upper platform is first shown in

Table 1. Upper Platform pose data.

Serial No.	X (mm)	Y (mm)	Z (mm)	Roll (°)	Pitch (°)	Yaw (°)
1	0	0	0	10	10	0
2	20	0	0	15	0	10
3	15	30	30	−10	15	10
4	20	−20	−20	15	−8	−10

, where the expansion of each driving rod is solved according to the inverse kinematics calculation in Formula (6), and then the change value of the expansion of the driving rod is brought into the forward kinematics calculation as in Formula (5); the value of the upper platform is shown in

Table 2. Positive kinematics to solve the upper platform.

Serial No.	No. of Iterations	Duration	Solution Position x (x, y, z roll, pitcℎ, yaw, x, y, z)
1	5	15 ms	(0.902, 1.003, 101.002, 10.804, 10.905, 0.203)
2	5	15 ms	(19.698, 0.302, 120.605, 13.993, 0.303, 10.802)
3	6	20 ms	(17.003, 32.005, 148.956, −9.382, 15.804, 9.698)
4	6	20 ms	(23.051, −21.862, 98.589, 15.824, −8.275, 10.206)

.

Analysis of results: According to Table 2, the change value of each drive obtained by the inverse kinematics is calculated by forward kinematics using Newton’s iteration method, and the platform pose data are very close to the initial platform data, verifying the scientific nature of the kinematics of the platform.

3. AR Prediction Algorithm with Qualified Memory Recursive Least Squares

In order to overcome the complex sea conditions and their real-time influence on drone landing platform stability, we put forward a limited memory of the least squares method with the recurrence AR prediction algorithm to predict the future of a ship under different sea conditions, where we input the prediction value to the landing platform compensation hysteresis to adjust and improve the stability of the landing platform in real time. The main work of this section is to build the prediction algorithms and to prove the accuracy of the algorithms.

The mathematical expression formula of the AR model $x\left(n\right)$ is as follows:

$$x\left(n\right)=-{\displaystyle \sum }_{k=1}^p{a}_{k}x\left(n-k\right)+u\left(n\right)$$

(7)

where p, which is the order of the model, determines that the model uses p past data to predict the current moment; $a_k$ is the autoregressive coefficient of the model, indicating the influence of the historical data on the current moment; $u\left(n\right)$ is the random error term of the model, usually assuming a white noise sequence with the zero mean and the variance of ${\sigma }^2$.

Determination of AR model order: The AIC (A-Information Criterion) [17] criterion in the best criterion function fixed-order method is used to determine the order. Assuming that $\left\{x_n\right\}$ is a random time series, and the AIC criterion function is defined according to the fitted residual variance ${\stackrel{̯}{\sigma }}_a^2$ of the model and the order n in the model, the AIC criterion function is as follows:

$$AIC\left(n\right)={\stackrel{̯}{\mathit{ln}\sigma }}_a^2\left(n\right)+{\displaystyle \frac{2n}{N}}$$

(8)

When fitting the autoregressive model, the model order is gradually increased from one, and the residual variance of the fitted model and the $AIC\left(n\right)$ value are calculated. Increasing the order of the model is then brought into the formula, and the minimum value $AI{C}_{(\mathit{min})}$ and the corresponding model order are recorded until the order of the model reaches the preset maximum value M and the final calculation of $n_0$, the best order of the model.

$$AIC\left(n_0\right)=\underset{1\le n\le M}{min}AIC\left(n\right)$$

(9)

AR model since the regression coefficient: According to Formula (7), using the AR model under t, the t + 1 time optimization estimation is often not comprehensive behind a value in the actual application prediction, so there is a need to subsequently analyze the data at the time of $t+n$ in the future, based on the least squares method of prediction [18], to obtain the following autoregressive prediction model:

$$\widehat{x}\left(t+n\right)=\{\begin{array}{ll}{\displaystyle {\displaystyle \sum }_{p=1}^p}-a_{i}x\left(t+1-i\right),& n=1\\ {\displaystyle {\displaystyle \sum }_{p=1}^p}-a_{i}x\left(t+1-i\right)+{\displaystyle {\displaystyle \sum }_{i=n}^p}-a_{i}x\left(t+n-i\right),& 1<n<p\\ {\displaystyle {\displaystyle \sum }_{p=1}^p}-a_{i}x\left(t+n-i\right),& n>p\end{array}$$

(10)

To overcome the impact of complex sea conditions on the stability and real-time performance of the unmanned aerial vehicle (UAV) landing platform, it is necessary to predict the future attitude of the ship. Commonly used prediction algorithms include the AutoRegressive (AR) algorithm [19], the Moving Average (MA) algorithm [20], and the AutoRegressive Moving Average (ARMA) algorithm [21]. The AR algorithm is a method for processing time series, which predicts the current value by linearly combining the values of several past time points. It is used to capture trends and autocorrelation in time series, making it suitable for data with long-term dependencies and has been deeply applied in the prediction of natural phenomena. The MA algorithm, another method for processing time series, is based on the past white noise errors of the time series and predicts future values by linearly combining these errors. It is typically used to capture short-term fluctuations and noise in time series, suitable for data with oscillating characteristics. However, the MA algorithm primarily relies on the linear combination of past error terms to predict the future, which results in relatively weaker capture of long-term trends. Additionally, its parameters are complex and cannot intuitively reflect the dynamic characteristics of the time series. The ARMA algorithm combines the characteristics of both the AR and MA models as a time series prediction model. It aims to predict future values by fitting historical data while capturing both autocorrelation and short-term fluctuations in the time series. However, its model is complex, parameter estimation is difficult, and it is sensitive to outliers and missing values in the time series, which can lead to ineffective data fitting and reduced prediction accuracy. Therefore, to avoid the impact of continuously accumulating data and increasing time series data volume, this paper selects the AR algorithm combined with the recursive least squares (RLSs) method to improve prediction efficiency.

4. Experimental Verification and Analysis of the Results

The AR prediction algorithm is used to verify the limited memory of the recursive least squares method. First, the one-step, two-step, and five-step predictions of the ship’s rolling and pitching in the next 90 s under the conditions of secondary sea conditions and fifth-level sea conditions were obtained, respectively, and then the data were compared with actual values. RMSE was then used to calculate the root mean square error of each simulation. The hardware configuration of the experiment was as follows: 64-bit Windows 11 operating system, 16 GB of RAM, NVIDIA GeForce RTX 4060 GPU, and a 13th Gen Intel(R) Core(TM) i7-13700H processor with a base clock speed of 2.40 GHz. The software environment used was MATLAB 9.12.0 (R2022a).

The prediction results of a ship rolling in the second-level sea condition are as shown in Figure 4, with the root mean square error of the one-step prediction for second-level rolling.

The prediction results of ship pitching under second-level sea conditions are shown in Figure 5.

The prediction results of ship roll under fifth-level sea conditions are shown in Figure 6.

The prediction results of a ship rocking under fifth-level sea conditions are shown in Figure 7.

The prediction results show that although the sea condition level gradually increased, the prediction time gradually extended; however, the RMSE value is within the ideal range, where the limited memory least squares recursive AR ship motion prediction results and the actual values are smaller, so the algorithm in different sea condition levels and different prediction times have ideal results.

Movement control and compensation experiment of UAV shipborne autonomous landing platform: A double-layer parallel and stable physical platform is built as shown in Figure 8. The two platforms are fixed with rivets, and the lower platform simulates the movement of the ship, while the upper platform compensates. The control end uses two upper machines, where one sends simulated motion data to the lower platform, the other collects the motion signal of the lower platform and sends it to the upper platform for compensation after processing. The system is realized by C + + programming. Meanwhile, in order to meet the compensation accuracy of the landing platform in extreme sea conditions, a control compensation method with attitude prediction module is proposed and applied as shown in Figure 9. Compared with the traditional compensation method, this method introduces a real-time ship motion prediction module to improve the response speed of the platform.

First, the ship motion under the interference of a single degree of freedom is selected for motion compensation, and two sinusoidal functions with amplitudes of 5 and 10 are input as motion simulation. The experimental results of pose data collected from the upper and lower platforms are shown in Figure 10 and

Table 3. Sine compensation error of degree-of-freedom of parallel stable platform.

	Maximum Error |e|max (°)	Average Error μ|e| (°)	Performance Improvement (%)
Rolling angle (5°)	0.57	0.23	59.6
Pitching angle (5°)	0.63	0.25	60.3
Rolling angle (10°)	0.93	0.48	48.4
Pitching angle (10°)	0.98	0.51	47.9

.

The data from Table 3 show that, after the compensation mechanism, the landing platform attitude error significantly reduces, with a 5° rolling angle, 5° pitching angle, 10° rolling angle, and 10° pitching angle compensation accuracy increase to 59.6%, 6%, 60.3%, 48.4%, and 47.9%, respectively; thus, the results prove that the platform has excellent compensation ability and is able to compensate well for a single degree of freedom.

The following is the experimental verification at different sea condition levels. In the experiment, the simulated motion of the ship under the secondary and fifth sea conditions is taken as the simulated signal input of the lower platform. The compensation results are indicated in Figure 11, where the blue curve is the attitude signal curve of the lower platform, and the red curve is the attitude signal curve after the compensated motion of the upper platform.

Analysis of maximum error versus average error is shown in

Table 4. Compensation error of shipborne autonomous landing platform of shipborne UAV.

	Maximum error |e|max (°)	Average Error μ|e| (°)	Performance Improvement (%)
Rolling at level 2	0.41	0.20	51.2
Pitching at level 2	0.32	0.13	59.4
Rolling at level 5	1.67	0.69	58.7
Pitching at level 5	1.02	0.45	55.9

.

Analysis of results: The data presented in the table reveal notable improvements in the compensation accuracy under secondary roll, secondary pitch, fifth-grade roll, and fifth-grade pitch conditions, with respective enhancements of 51.2%, 59.4%, 58.7%, and 55.9%. When the shipborne unmanned aerial vehicle (UAV) autonomous landing platform compensates for ship disturbances across various sea states, despite the increase in platform compensation errors with higher sea state grades, the overall compensation errors remain relatively low. This indicates a robust compensation performance, demonstrating that the platform effectively mitigates disturbances induced by varying sea conditions. Its adaptability and stability have met the expected criteria.

Validation of Method Advancement: To verify the superiority of the proposed method, eight landing experiments were conducted. After each landing, the distance between the center of the UAV’s downward-facing camera and the center of the landing platform was measured. Comparative tests were also performed against methods proposed by References [22,23,24]. As shown in Figure 12, the compensation accuracy achieved by the shipborne UAV landing platform compensation method designed in this study surpasses most autonomous landing algorithms tailored for UAVs, highlighting its improved performance compared to existing research.

5. Conclusions

This paper delves into the challenges of stability and precision faced by shipborne unmanned aerial vehicle (UAV) landing platforms under complex sea conditions. It thoroughly summarizes previous research achievements and proposes a novel control compensation strategy for an autonomous UAV landing platform based on the Stewart parallel platform. This strategy aims to significantly enhance the landing accuracy and stability of UAVs on dynamic ship decks, thereby bolstering the capability of ship–UAV collaborative operations. Firstly, this research adopts the Stewart parallel mechanism, a celebrated choice for its unparalleled structural robustness and versatile six-degree-of-freedom control capabilities, as the cornerstone of the shipborne unmanned aerial vehicle (UAV) landing platform. Through meticulous forward and inverse kinematic analyses, corroborated by rigorous Matlab simulations, the theoretical soundness and practical feasibility of the platform design are meticulously established, guaranteeing exquisite control precision and exceptional stability throughout the UAV landing process. Secondly, to mitigate the deleterious effects of external disturbances on landing accuracy, an innovative control strategy is formulated, grounded in a constrained memory recursive least squares adaptive regression (AR) prediction algorithm. This algorithm adeptly interprets real-time ship motion data, anticipating the vessel’s imminent attitude fluctuations with unparalleled precision. The prompt dissemination of these predictions to the supervisory control system accelerates the platform’s response to dynamic ship motions, substantially amplifying the adaptability and accuracy of the compensation control. This advancement overcomes the limitations of conventional methods in terms of compensating for platform deviations, thereby ensuring unprecedented landing precision. Lastly, a dual-parallel platform experimental framework is constructed, wherein the lower platform meticulously emulates the diverse motion profiles of ships navigating through varying sea conditions, while the upper platform serves as the tangible interface for UAV landings. This sophisticated system integrates a dual-sensor array, facilitating seamless real-time acquisition and transmission of both ship attitude and platform status data. This enhancement fortifies the system’s resilience against interference and solidifies its overall stability, setting a new benchmark for precision and reliability in shipborne UAV landing operations. The results of the experiments showed that ship under the interference of a single degree-of-freedom, for instance, the compensation accuracy of 5° rolling angle, 5° pitching angle, 10° rolling angle, and 10° pitching angle increased by 59.6%, 60.3%, 48.4%, and 47.9%, respectively; the compensation accuracy of rolling and pitching in the second and fifth sea conditions increased by 51.2%, 59.4%, 58.7%, and 55.9%, respectively. Improvements in the above compensation accuracy achieves the accurate prediction of the ship’s future attitude and the smooth landing of the UAV on the ships, enhancing the accuracy of the collaborative control of ships and UAV. Also, it promotes the development of cooperative control technology between unmanned ships and UAV, laying a solid foundation for the development of intelligent ships in China.