Study on the Turning Characteristics and Influencing Factors of the Unmanned Sailboat

Abstract

Unmanned sailboats can convert wind energy with sails to provide power for navigation, which can independently plan routes and collect data without human intervention. They have received increasing attention in recent years due to their low power consumption and strong self-sustainability. Due to the greater difficulty of manipulation, the unmanned sailboats have a weaker maneuverability than the propeller-driven vessels in the complex and variable marine environment. Typically, the turning motion is evaluated to characterize the maneuverability of a vessel, which has rarely been investigated in the existing research on unmanned sailboats. Therefore, this study builds a motion simulation platform for unmanned sailboats based on the 3 m class Petrel Sail to investigate the turning characteristics. The index of the approximate turning circle is introduced based on the turning motion trajectory, which is used to obtain the effect of rudder angle, wind angle, wind speed, and current speed on the turning performance of the sailboat in ideal hydrostatic conditions and under flow disturbance, respectively. Finally, a harbor pool test is conducted with an unmanned sailboat to verify the analysis results, and the errors in maximum transverse distance and maximum advance distance are in the reasonable range, proving the correctness of the theoretical results. The current study also provides theoretical guidance for subsequent research on sailboat manipulation and maneuverability.

1. Introduction

The ocean covers 71 percent of the Earth’s surface, and it plays an important role in the production and living practices of human beings. Various ocean observations have been made to understand and develop the ocean [1]. Traditional ocean observation methods include satellites, research vessels, and (subsurface) buoys. With the development of automatic control, navigation, communication, and other technologies, marine unmanned equipment has been widely used in ocean observation to make up for the shortcomings of traditional ocean observation. Autonomous underwater robots, underwater gliders, wave gliders, unmanned sailboats, and other marine unmanned equipment have their own advantages in terms of speed, working depth, endurance, and load [2]. Driven by the wind, unmanned sailboats can perform long-duration and long-distance navigation tasks without the limitation of fuel shortage. Therefore, unmanned sailboats can autonomously perform the observation of remote sea areas and conduct long-term monitoring of specific sea areas [3].

In recent years, a number of unmanned sailboats have been developed and put into practical applications. Autonomous Marine Systems Inc. (Somerville, MA, USA) developed an unmanned catamaran called Datamaran in 2013, which has a self-righting function [4]. Spain launched the A-Tirma G2 in 2015, which features a biplane sail layout and shows advantage in track keeping [5]. Offshore Sensing AS from Norway developed the Sailbuoy unmanned sailboat in 2018, which can carry equipment of up to 15 kg in weight for the measurement of marine resources and was the first unmanned surface vehicle to complete a transatlantic crossing [6]. Recently, Saildrone Inc. developed the 10 m Voyager in 2022, with a studdingsail and self-balancing sail turning function, which can sail continuously at sea for up to 180 days with low power consumption [7].

Maneuverability is an important index for the use of unmanned sailboats during sailing, as it ensures the stable operation of sailboats in the complex marine environment. In fact, the maneuverability of unmanned sailboats depends largely on the manipulation of sails and rudders, and great efforts have been made to optimize the manipulation methods. Meng et al. [8] proposes a novel method to achieve self-balancing sail rotation based on elastic rope for unmanned sailboats and introduces the sum of the angular thrust of spherical surface S_L as a performance evaluation index. The sailboat’s performance is increased by 24% after optimization. Yin et al. [9] carries out orthogonal analysis and CFD simulations to analyze the influence of coupling parameters on the aerodynamic performance of an arc sail and identifies an arc sail suitable for the existing Petrel unmanned sailboats. Kuang et al. [10] proposes to optimize the chord length ratio of a two-element wing sail to enhance the aerodynamic performance of a wind-powered unmanned sailboat. Liu et al. [11] dealt with the two scenarios of unmanned sailing virtual and fixed-speed cruising and proposed a corresponding optimal manipulation strategy for sail rotation by taking the minimum slewing diameter and the maximum speed as the evaluation indexes. Saoud et al. [12] proposed a sail control method to maximize the longitudinal speed by restricting the transverse rocking angle and reported that the method can improve the performance of the sailboat. Augenstein et al. [13] proposed to replace the water rudder with a tail blade rudder, which produces a stable windward angle and promotes the stability of the boat, greatly improving its sailing maneuverability. For a 12 m class unmanned sailboat, Zhang et al. [14] calculated the wind speed boundaries for the sailboat at different rudder angles to find the controllable zone under different angles of attack and analyzed the factors affecting the controllable zone to make suggestions for rudder maneuvering.

The turning performance of a vessel is one of the important indicators of its maneuverability, which is closely related to its safety and operational efficiency. Many previous studies have investigated the turning motion of various vessels. Jiang et al. [15] studied the turning characteristics of unmanned boats, determined the relevant parameters by building a model and performing real-boat tests, and verified the accuracy of the model with the least recursive squares method. Jiang et al. [16] determined maneuverability parameters of the trimaran and reported the influences of the lateral and longitudinal positions of the side hulls on the maneuverability of the trimaran. Wang et al. [17] simulated and analyzed the turning process of a hydrofoil catamaran under different sea conditions by establishing a slewing motion control system. Wang et al. [18] constructed a mathematical model of glide boat maneuvering motion under different wind and current conditions and simulated the turning characteristics under different rudder angles and water flow rates, providing a valuable reference for predicting glide boat maneuvering performance. Tavakoli et al. [19] proposed a mathematical model for the slewing maneuvering motion of a glide boat by coupling the motions of six degrees of freedom, experimentally predicted the radius and trajectory of the glide boat, and analyzed the effect of the rudder angle on the glide boat’s turning motion. Wu et al. [20] designed a berthing maneuvering algorithm for a catamaran unmanned surface vehicle and optimized speed control and the turning radius through simulation experiments during the berthing process, improving the berthing efficiency and safety in complex port environments.

Although these studies investigated the turning performances of different vessels under the interference of wind, waves, currents, and other complex environmental factors, few have dealt with the turning characteristics of unmanned sailboats, which are vital for their working performances. In this case, this study conducts theoretical and simulation analysis on the turning characteristics of unmanned sailboats, and the main contributions are summarized as follows:

• A motion simulation platform is built for the unmanned sailboat, which is used to investigate the turning characteristics under the two conditions of ideal static water and flow interference and obtain the turning law of the sailboat;
• The index of the approximate turning circle is proposed based on the turning trajectory, which provides a new quantitative tool for analyzing the turning performances of unmanned sailboats;
• A harbor pool test is carried out based on the 3 m class Petrel Sail unmanned sailboat to verify the correctness of the simulation results.

The rest of this paper is structured as follows. Section 2 introduces the structure, the control architecture, and the path-following strategy of the Petrel Sail. Section 3 describes the turning motion of the sailboat and its main evaluation indexes. Section 4 integrates the simulation model and studies the turning characteristics under the ideal hydrostatic and flow disturbance conditions. Section 5 introduces the harbor pool test for the turning characteristics. Finally, Section 6 summarizes the conclusions and gives an outlook for future works.

2. Petrel Sail Unmanned Sailboat

2.1. Configuration of Petrel Sail

The Petrel Sail unmanned sailboat is mainly composed of a hull, a sail, a rudder, and a keel, as shown in Figure 1. The hull is streamlined. The sail stands on the middle of the hull and helps the sailboat to obtain thrust, and its skeleton is composed of three parts: the mast, transverse rib, and frame. The rotary sail mechanism is installed at the bottom of the sail, involving a base, a main axis of revolution, a turntable, a sail motor, a pressure cabin, and various couplings. The turntable is fixed on the base and adopts a liquid-filled worm gear structure. The sail motor is installed in the pressure cabin. The upper end of the main rotary shaft is tightly connected to the bottom of the sail, and the lower end is connected to the output shaft of the turntable with an elastic coupling to transmit the sail torque. The weather station and anemometer are installed on top of the sail to obtain necessary field parameters during the voyage.

The keel is installed at the bottom the sailboat, opposite to the sail, which uses the lateral force of the water to balance the lateral force of the sail and prevent shifting and toppling, thus ensuring the stability of the unmanned sailboat. The rudder is installed under the stern of the hull to control the course of the ship with hydrodynamic force. The communication unit of the sailboat is located at the stern of the hull, including GPS and BDS modules for positioning, as well as the iridium module and wireless module for communication and data exchange.

Table 1. The main parameters of the Petrel Sail unmanned sailboat.

Configuration	Parameter	Value
Hull	Length (mm)	3100
	Width (mm)	650
	Aspect ratio	1.5
	Draft (mm)	250
	Heel angle (deg)	40
Sail	Height (mm)	1400
	Width (mm)	600
	Area (m2)	0.84
	Weight (kg)	8
Keel	Length (mm)	750
	Width (mm)	250
	Area (m2)	0.1875
	Weight (kg)	15

shows the main parameters of the Petrel Sail unmanned sailboat.

2.2. Dynamic Model

A dynamic model is established for analyzing the motion of the unmanned sailboat, which considers the surge, sway, heave, roll, pitch, and yaw. A six-degree-of-freedom model [21] is adopted, involving the geocentric coordinate system E-XYZ, the body coordinate system O-xyz, and the velocity coordinate system O-x_vy_vz_v, as shown in Figure 2.

The translational motion and rotational motion of the unmanned sailboat can be expressed as

$$\left\{\begin{array}{c}m[u-vr+wq-x_G(q^2+r^2)+y_G(pq-\dot{r})+z_G(pr+\dot{q})]=F_x\\ m[\dot{v}-wp+ur-y_G(p^2+r^2)+z_G(qr-\dot{p})+x_G(pq+\dot{r})]=F_y\\ m[\dot{w}-up+vp-z_G(p^2+q^2)+x_G(rp-\dot{q})+y_G(rq+\dot{p})]=F_z\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}\begin{array}{c}{I}_{Bx}\dot{p}+(I_{Bz}-I_{By})qr-(\dot{r}+pq)I_{Bzz}+(r^2-q^2)I_{Byz}+(pr-\dot{q})I_{Bxy}\\ +m[\left.y_G(\dot{w}-uq+vp)-z_G(\dot{\nu }-wp+ur)\right]=M_x\end{array}\\ \begin{array}{c}{I}_{By}\dot{q}+(I_{Bx}-I_{Bz})rp-(\dot{p}+qr)I_{Bxy}+(p^2-r^2)I_{Bzx}+(qp-\dot{r})I_{Byz}\\ +m[z_G(\dot{u}-\nu r+wq)-x_G(\dot{w}-qu+\nu p)]=M_y\end{array}\\ \begin{array}{c}{I}_{Bz}\dot{r}+(I_{By}-I_{Bx})pq-(\dot{p}-qr)I_{Bxz}+(q^2-p^2)I_{Bxy}+(rq-\dot{p})I_{Bzx}\\ +m[x_G(\dot{\nu }-wp+ur)-y_G(\dot{\nu }-vr+wq)]=M_z\end{array}\end{array}\right.$$

(2)

where (X, Y, Z) denotes the origin of the body coordinate system in the geodetic coordinate system, (u, v, w) indicates the moving speed, (p, q, r) denotes the angular velocity, (F_x, F_y, F_z) indicates the force on the unmanned sailboat in all directions, (M_x, M_y, M_z) shows the torque in all directions, and I_Bxy, I_Byz, and I_Bzx are the moments of inertia of the mass of the unmanned sailboat to the xOy, yOz, and xOz planes in the body coordinate system, respectively.

2.3. Control Architecture

The navigation control system of the unmanned sailboat needs to consider the current environment and the motion parameters and formulate reasonable strategies to control the actuators including the sail and rudder. Here, we adopt a three-layer control architecture, which is divided into the upper control layer (A), middle control layer (B), and lower control layer (C), as shown in Figure 3.

Control layer A makes the work decision of the sailboat. For the specific working scenario, multiple navigation points are set to divide the path into multiple segments. The next segment is started automatically after the completion of the current segment is detected until reaching the target position.

Control layer B is used for data collection during navigation. The navigation information, such as the surrounding environment, hull attitude, and navigation speed, are collected with the sensors onboard.

Control layer C controls the operation of the sail and rudder, including the sail controller D and the rudder controller E.

Based on the aerodynamic characteristics of the sail, the sail controller D adjusts the sail angle according to the relative wind direction angle of the target, thus allowing us to achieve optimal thrust and lateral force transfer performance, as well as speed maximization and auxiliary path-following. In practice, the relative wind direction angle changes with time and cannot be predicted due to the instability of the wind direction and the change in course. Therefore, the input (relative wind direction angle) is divided into multiple control intervals. The functions of the optimal sail angle and the relative wind direction angle can be obtained according to the aerodynamic characteristics of the sail, and the sail angle can be generated by taking the median value of each interval group as a reference. This control method is easy to implement in practice and has high stability, which avoids the disorder of the actuators caused by the disturbance of the input signals. Here, the sail rotary mechanism is regarded as a first-order inertia link, expressed as

$${\displaystyle \frac{\theta }{{\theta }_e}}={\displaystyle \frac{1}{T_{S}s+1}}$$

(3)

where θ_e is the target sail angle, and T_S is the working time of the sail rotary mechanism, which is a constant.

The steering controller E is a major component used for adjusting the course of the sailboat. The control of the rudder angle is realized based on the practical sailing experience. The traditional incremental PID algorithm is adopted, and the recursive principle is expressed as

$$\begin{array}{c}{\delta }_R\left(t\right)-{\delta }_R\left(t-1\right)=K_p\left[\beta \left(t\right)-\beta \left(t-1\right)\right]+K_i\beta \left(t\right)\\ +K_d\left[\beta \left(t\right)-2\beta \left(t-1\right)+\beta \left(t-2\right)\right]\end{array}$$

(4)

where K_p is the proportional coefficient, K_i is the integral coefficient, K_d is the differential coefficient, and β(t) is the heading deviation value collected at time t.

On this basis, the fuzzy PID control is established to realize the real-time adjustment of the parameters. The actual heading deviation angle β and its change rate β_C are taken as the input of the controller, and, thus, the three parameters of K_p, K_i, and K_d can be adjusted in real time. The output accurate value is transmitted to the steering gear to complete the heading control. Similar to the sail controller, the fuzzy PID control is considered to be a first-order inertial link.

2.4. Path-Following Strategy

The path of the unmanned sailboat can be roughly represented by a two-dimensional path composed of multiple straight lines and arcs. Therefore, the widely used line-of-sight [22] (LOS) method is adopted to update the heading deviation angle in real time and correct the yaw of the unmanned sailboat during the sailing in a timely manner.

A navigation diagram generated with the LOS method is illustrated in Figure 4. For the segment T_kT_k+1 that takes T_k+1 as the target, a vertical line is made from the current position S of the unmanned sailboat. The forward distance parameter Δ is introduced for navigation control, and the point T_LOS is obtained at Δ with T_k as the starting point and T_kT_k+1 as the direction. The LOS angle of the sailboat is the angle between the direction vector of point S to point T_LOS and the north direction, which is recorded as β_T. The heading deviation angle β is obtained with the difference between β_T and the heading angle β_O. Relative wind direction is the wind speed and wind direction measured by the anemometer on the sailing boat, which is related to the true wind direction and the sailing course angle. As the unmanned sailboat approaches the target T_k+1, β gradually tends toward 0° under the control of the layer C. In addition, when the unmanned sailboat enters the circular area of T_k+1 with R as the radius, control layer A will end the current navigation task and start navigation for the next target point, and control layer B will calculate the new β_T and β values and constantly perform navigation control.

3. Turning Motion

The turning motion of a surface vehicle is composed of the straight navigation stage with a constant speed and the turning stage, as does the turning motion of the unmanned sailboat, as shown in Figure 5. In the direct navigation stage, the unmanned sailboat relies on the small swing of the rudder angle δ_R to maintain the route. In the turning stage, the unmanned sailboat begins to turn due to the adjustment of δ_R. The trajectory of this stage, namely the shape of the turning circle, is affected by both δ_R and the environmental conditions, such as wind speed and flow velocity.

Based on the relationship between the turning direction and the wind direction, the turning motion of the unmanned sailboat can be divided into tacking and gybing motions. When the turning direction of the unmanned sailboat has a component in the direction of the wind, it is defined as tacking motion. On the other hand, when the turning direction has a component in the same direction as the wind direction, it is defined as a gybing motion.

The main characteristic parameters for describing the turning motion are listed in

Table 2. The main characteristic parameters of the turning motion.

Parameter	Definition
Advance Ad	The longitudinal distance traveled from the initiation of steering to the 90° change in course
Max. advance Admax	The maximum longitudinal distance traveled during the turning motion
Tactical diameter DT	The lateral distance traveled when the initial heading changes by 180°
Transfer Tr	The lateral distance traveled from the initiation of steering to the 90° change in course
Max. transfer Trmax	The maximum lateral distance traveled during the turning motion

.

Because the unmanned sailboat is only driven by the sail, the wind direction angle φ_w changes constantly when the sailboat travels in a stable wind field, and there will be a powered area and a powerless area.

When the sail can provide thrust, the unmanned sailboat is in the powered area. In this area, the wind direction angle φ_w is in the ranges of −180° to −45° and 45° to 180°. The maximum sail thrust strategy is adopted to obtain the best rotary angle of the sail and generate the maximum longitudinal thrust along the bow direction, so as to reach the maximum sailing speed.

When the unmanned sailboat cannot obtain forward thrust from the sail, it is in the powerless area. In this area, the wind direction angle φ_w is in the range of −45° to 45°. Under the aerodynamic and hydrodynamic resistance, the navigation speed of the unmanned sailboat gradually decreases, and the rudder efficiency also decreases, which, in turn, affects the turning characteristics. By utilizing the cambered shape of the arc sail, the sail is maintained at an aerodynamic attack angle of 0° during both tacking and gybing maneuvers in this area. At the same time, the sailboat generates a moment of turning to execute the turning movement.

4. Simulation Analysis

In this section, the three-layer control structure of the unmanned sailboat is integrated based on the motion mathematical model and corresponding control strategy design, and MATLAB R2022a software is adopted to build the motion simulation model for the unmanned sailboat, as shown in Figure 6. The simulation model mainly includes the control architecture, path-following strategy, motion model, sail controller, and rudder controller.

When performing turning motion on the water surface, the unmanned sailboat is affected by external factors, such as wind, waves, and currents. Among these factors, the wind is the only driving source that cannot be ignored in the whole simulation process. The turning motion is a typical motion of surface vessels, and there is currently limited research on the turning characteristics of unmanned sailboats. First, the turning motion of unmanned sailboat under the ideal condition is investigated to clarify the law of turning motion under a fixed wind direction. Further, the interference of waves and currents is considered to predict the turning motion trajectory of the unmanned sailboat.

4.1. Turning Characteristics under Ideal Hydrostatic Conditions

Based on the motion simulation model established, we simulated the turning motion scenario of the unmanned sailboat under a constant and uniform wind field in the hydrostatic condition. The Petrel Sail unmanned sailboat is small in size, and the designed prototype is suitable for coastal waters with calm wind and waves. Therefore, the Petrel Sail unmanned sailboat has strong stability. The influence of wind on the tilt of the sailboat was not considered for simplifying the complexity of the model. For the calculation, the rudder angle δ_R, the wind speed v_w, and the wind direction angle φ_w were taken as the inputs of the simulation model, while the maximum advance distance A_dmax and the maximum transfer distance T_rmax were taken as the evaluation indexes.

Given that the motion of the unmanned sailboat is nonlinear in nature, the ternary quadratic regression orthogonal experimental was adopted to investigate the turning motion, allowing for the simultaneous examination of three main factors and revealing the nonlinear effect. The regression equation was established with the inputs δ_R, v_w, and φ_w according to the principle of least squares [23], expressed as

$$y=a+b_1{\delta }_R+b_2{v}_w+b_3{\phi }_w+b_{12}{\delta }_R{v}_w+b_{13}{\delta }_R{\phi }_w+b_{23}{v}_w{\phi }_w+b_{11}{\delta }_R^2+b_{22}{v}_w^2+b_{33}{\phi }_w^2$$

(5)

where a is a constant; b₁, b₂, and b₃ are the linear regression coefficients; b₁₂, b₁₃, and b₂₃ are the interaction term partial regression coefficients; b₁₁, b₂₂, and b₃₃ are the quadratic term partial regression coefficients; and A_dmax and T_rmax are used as the evaluation indexes of the test, respectively.

The range of the input parameters was set as follows:

• The maximum rudder value that could be realized by the unmanned sailboat in practical work was set to 35°, and the minimum value was set to 5°, since a smaller rudder angle cannot produce enough torque to complete the turning motion. Therefore, the ranges of the rudder angle were set to −15° to −35° and 15° to 35°.
• The unmanned sailboat works on the sea surface in all weather conditions. The wind situation in the South China Sea region was taken as the reference for the current study, and it is affected by monsoon all year round. The wind is distributed in the range from level 3 to level 5 more than 90% of the time. In this case, the range of wind speed v_w was set from 4 m/s to 8 m/s.
• The wind angle was set to be less than 120°, thus ensuring the utilization of wind power in both tacking and gybing. If the wind angle was greater than 120° during tacking, the unmanned sailboat would enter the non-navigable area, which makes it difficult to effectively utilize the wind power. Therefore, the wind angle φ_w was set from −120° to 120°.

Each level of δ_R, v_w, and φ_w was coded, and the range of each input, as well as its coding, can be found in the Appendix A.

Let the quadratic term in the regression equation be z_j² and the centering of the corresponding coding can be expressed as

$$z_j^{\prime }=z_j^2-{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{z}_{ji}^2}$$

(6)

where z_j′ is the coding after centering, and n is the number of trials.

In the three-factor quadratic regression orthogonal test, altogether, 15 test combinations were generated, and the detailed schemes can be found in Appendix B. With the simulation model established, the 15 combinations were simulated for analyzing the turning motion of the unmanned sailboat in tacking and gybing, respectively. Finally, the corresponding A_dmax and T_rmax values were obtained. The test results are shown in

Table 3. The orthogonal test results for the turning motion of the unmanned sailboat in tacking and gybing.

No.	Admax (m)		Trmax (m)		No.	Admax (m)		Trmax (m)
	Tacking	Gybing	Tacking	Gybing		Tacking	Gybing	Tacking	Gybing
1	11.1	8.2	13.3	18.9	9	9.9	9.6	16.5	18.4
2	10	10.2	18.7	19	10	9.2	8.2	30.6	35
3	10.7	8.6	14.6	18.9	11	12.3	12	19.9	22.7
4	10	10.1	18.6	19.1	12	12.3	11.8	20.3	22.7
5	18.7	13.3	22.1	31.1	13	13.2	9.5	16.3	22.6
6	16.4	16.7	31.4	32.1	14	12.1	12.1	22.9	22.9
7	17.9	14.1	23.9	31.4	15	12.3	11.9	20.1	22.6
8	16.6	16.6	31.5	32.1

.

The simulation results for the second set of data are illustrated in Figure 7 as an example. The unmanned sailboat departed from point (0, 0) to carry out a steady speed sailing along the X-direction to point A, and then entered the tacking and gybing motion by applying the rudder angle δ_R in two different directions.

Further, the regression relationship between the input and the evaluation indicators was obtained through data analysis. The goodness of fit and analysis of variance for each regression equation are shown in

Table 4. Regression results.

Evaluation Index	Difference Source	Sum of Squares	Degree of Freedom	Mean Square	F	Significance	R2
Tacking Admax	Regression	152.983	3	50.994	414.118	**	0.991
	Residual	1.355	11	0.123
	Sum	154.337	14
Gybing Admax	Regression	130.098	4	32.524	158.539	**	0.984
	Residual	2.052	10	0.205
	Sum	132.149	14
Tacking Trmax	Regression	472.988	4	118.247	246.529	**	0.990
	Residual	4.796	10	0.480
	Sum	477.784	14
Gybing Trmax	Regression	491.411	3	163.804	2089.084	**	0.999
	Residual	0.863	11	0.078
	Sum	492.273	14

**: This indicates that the statistical test results corresponding to the effect or factor are very reliable.

.

Tacking:

$$\left\{\begin{array}{l}{A}_{dUp}=13.513-3.57z_1+0.615z_3+1.456z_1^{\prime }\\ T_{rUp}=21.379-5.554z_1-3.133z_3+2.17z_1^{\prime }+0.937z_1{z}_3\end{array}\right.$$

(7)

Gybing:

$$\left\{\begin{array}{l}{A}_{dDown}=12.193-3.109z_1-1.147z_3+1.214z_1^{\prime }-0.885z_3^{\prime }\\ T_{rDown}=24.633-6.48z_1-0.216z_3+2.667z_1^{\prime }\end{array}\right.$$

(8)

Analysis of the regression results shows that the regression equations established for the test evaluation indicators reached a significant level and fit well with the simulation data.

The actual regression equations of the evaluation indexes can be obtained as

Tacking:

$$\left\{\begin{array}{l}{A}_{dUp}=35.978-1.508{\delta }_R+0.0124{\phi }_w+0.0215{\delta }_R^2\\ T_{rUp}=63.947-2.415{\delta }_R-0.121{\phi }_w+0.032{\delta }_R^2+0.0023{\delta }_R{\phi }_w\end{array}\right.$$

(9)

Gybing:

$$\left\{\begin{array}{l}{A}_{dDown}=32.682-1.274{\delta }_R+0.0203{\phi }_w+0.0179{\delta }_R^2-0.00036{\phi }_w^2\\ T_{rDown}=66.709-2.756{\delta }_R-0.00437{\phi }_w+0.0394{\delta }_R^2\end{array}\right.$$

(10)

The relationship between the inputs δ_R and φ_w and the evaluation indexes A_dmax and T_rmax is visualized with three-dimensional response plots.

Figure 8a shows the three-dimensional response surfaces of the rudder angle δ_R and wind angle φ_w with respect to the maximum advance distance A_dmax values for tacking and gybing. For tacking, A_dmax decreased significantly with the increase in δ_R and showed an increasing trend with the increase in φ_w. For gybing, A_dmax also decreased significantly with the increase in δ_R but showed a decreasing trend with the increase in φ_w. When φ_w was in the range of 0° to 60°, the values of A_dmax generated by tacking and gybing under the same rudder angle were basically the same. When φ_w was in the range of 60° to 120°, the value of A_dmax corresponding to the gybing was lower than that of tacking, and, thus, the selection of gybing would generate shorter advance distances and better operation effect.

Figure 8b shows the three-dimensional response surfaces of the rudder angle δ_R and wind angle φ_w with respect to the maximum transverse distance T_rmax for tacking and gybing. For both tacking and gybing, T_rmax decreased significantly with the increase in δ_R. Moreover, it also showed a decreasing trend with the increase in φ_w, albeit only if the decrease for tacking was greater than that for gybing. Therefore, the selection of tacking would produce a shorter transfer distance and, thus, improve the operating effect.

In the hydrostatic condition, the maximum advance distance A_dmax and the maximum transfer distance T_rmax were mainly affected by the rudder angle δ_R and the wind angle φ_w, while the wind speed v_w did not have a significant effect. For A_dmax, the effect of tacking and gybing was comparable when φ_w was in the range from 0° to 60°, while gybing had a stronger effect when it was in the range from 60° to 120°. In addition, the value of T_rmax decreased with the increase in φ_w, and the decreasing trend was more obvious in tacking. In brief, choosing tacking will produce better turning performance of the unmanned sailboat.

4.2. Turning Characteristics under Flow Disturbance Conditions

Based on the constant wind field condition established, the wave and current interferences were further introduced to study the turning characteristics of unmanned sailboats. The ambient flow field was defined as a steady uniform flow field where the flow velocity and its direction do not change with time or spatial location. The ternary quadratic regression orthogonal experimental scheme in the hydrostatic condition was adopted, also taking rudder angle δ_R, wind speed v_w, and wind direction angle φ_w as the inputs.

During the turning motion in a constant uniform flow field, the size of the turning circle of the unmanned sailboat under flow disturbance was the same as that in the hydrostatic condition. The simulated trajectory of the unmanned sailboat obtained from the regression orthogonal test could not directly derive the maximum advance distance A_dmax and maximum transfer distance T_rmax because the turning circle was deformed along the flow direction according to the offset of the flow velocity in the geodetic coordinate system.

Therefore, we proposed an approximate turning circle index to describe the virtual circular path of a sailboat in its turning motion. We utilized the centroid coordinates of the approximated turning circle (x_S, y_S), the radius r_S, and the centroid angle coefficient K_C as the output quantities of the regression test and then compared them with the maximum advance distance A_dmax and maximum transfer distance T_rmax. Finally, it generated the variation in the turning characteristics for the flow velocity and wind speed under flow disturbance.

Through the orthogonal regression test, the centroid coordinates, radius, and angle coefficient of the approximate turning circle in the tacking and gybing states were obtained, and detailed information can be found in Appendix C.

Here, the simulation trajectory of the fourth set of data is analyzed as an example. The unmanned sailboat passes through five marker points A-E in one turning circle in the hydrostatic state, as shown in Figure 9. Point A denotes the beginning point of the turning, and the circle taking points B, C and D as the base points is considered to be the approximate turning circle of the unmanned sailboat, with the centroid denoted as O (x_S, y_S) and the radius as r_S. The parameters of the turning circle can be expressed as:

$$\left\{\begin{array}{l}x=x_S+r_S\sin \gamma \\ y=y_S+r_S\cos \gamma \\ \gamma \in [0,2\pi )\end{array}\right.$$

(11)

where (x_S, y_S) are the coordinates of the center of the circle, r_S is the radius of the circle, γ is the turning angle, and (x, y) are the coordinates of an arbitrary point on the trajectory.

The coordinate system for the centroid of the turning circle is shown in Figure 9. The unmanned sailboat is set to rotate around the center of the circle O, with the approximate turning circle apex A′ being the starting point. The center angle γ_S is approximated to be equivalent to the heading change angle Ψ_S.

The regression relationship between the input variables and the evaluation indexes is obtained as follows:

Tacking is defined as follows:

$$\left\{\begin{array}{l}{x}_S=30.474-0.598{\delta }_R+0.274v_w+0.0157{\phi }_w+0.00918{\delta }_R^2-0.000131{\phi }_w^2\\ y_S=39.641-1.511{\delta }_R-0.0893{\phi }_w+0.00303{\delta }_R{\phi }_w+0.0192{\delta }_R^2-0.000499{\phi }_w^2\\ r_S=25.537-0.941{\delta }_R-0.18v_w-0.00698{\phi }_w-0.000704{\delta }_R{\phi }_w+0.0135{\delta }_R^2+0.000287{\phi }_w^2\\ K_S=-1.36+0.191{\delta }_R+0.248v_w+0.00188{\phi }_w+0.00841{\delta }_R{v}_w-0.00298{\delta }_R^2-0.0000267{\phi }_w^2\end{array}\right.$$

(12)

Gybing is defined as follows:

$$\left\{\begin{array}{l}{x}_S=23.516-0.0525{\phi }_w\\ y_S=-26.414+0.469{\delta }_R+0.0322{\phi }_w\\ r_S=31.476-1.474{\delta }_R+0.0279{\phi }_w+0.0232{\delta }_R^2\\ K_S=0.298+0.144{\delta }_R+0.235v_w+0.000197{\phi }_w+0.00886{\delta }_R{v}_w+0.000615v_2{\phi }_w-0.00297{\delta }_R^2\end{array}\right.$$

(13)

The goodness of fit and analysis of variance of the regression equations can be found in Appendix D. The analysis of the regression results reveals the significant regression effects of each parameter for the approximate turning circle, as well as good fitting with the simulation data.

According to the turning process under the hydrostatic state, a wind speed of 6 m/s and wind angle of 60° were selected for gybing, and heading change angle with time for different rudder angles could be obtained. As shown in Figure 10, Ψ_S and the turning time have an approximately linear relationship.

Further, the center angle of the approximate turning circle can be considered a function of time t, expressed as:

$${\gamma }_s=K_{C}t$$

(14)

where K_C is the centroid angle coefficient.

The maximum advance distance A_dC and the maximum transfer distance T_rC under flow disturbance can be calculated by:

$$A_{dC}=Max(x_S+r_S\sin (K_{C}t)+v_C\cos {\phi }_{C}t)-x_A$$

(15)

$$y_s>0\hspace{1em}\hspace{1em}{T}_{rC}=Max(y_S-r_S\cos (K_{C}t)+v_C\sin {\phi }_{C}t)$$

(16)

$$y_s<0\hspace{1em}\hspace{1em}{T}_{rC}=\left|Min(y_S+r_S\cos (K_{C}t)+v_C\sin {\phi }_{C}t)\right|$$

(17)

where v_C is the absolute velocity of the flow, φ_C is the absolute direction, and the parameters x_S, y_S, r_S, and K_C can be calculated by substituting the rudder angle δ_R, the wind speed v_w, and the true wind angle φ_w into the approximate turning regression equations, respectively.

4.3. Effect of Flow Velocity under Flow Disturbance

Figure 11 shows the 3D surface response plots and contours for the maximum advance distance A_dC and maximum transfer distance T_rC with variations in flow velocity v_C and flow angle φ_C under flow disturbance. In this specific case, the unmanned sailboat was set to be subjected to a wind speed v_w of 6 m/s, a wind angle φ_W of 60°, and a rudder angle δ_R of 35° in tacking. The variables v_C and φ_C were set in the ranges from 0 m/s to 0.25 m/s and from 0° to 360°, respectively.

As shown in Figure 11a, a concave arc was observed in the surface diagram of the A_dC with respect to the maximum advance distance of the flow Angle φ_C, where the value of A_dC varied within the range 5 m to 18 m. The value of A_dC showed a decreasing trend, with an increase in v_C when the sailboat sailed against the flow and φ_C lay in the range of 90° to 270°. Notably, the smallest A_dC occurred when the sailing direction was opposite to the flow direction. In contrast, A_dC was positively proportional to v_C for the sailboat sails along the flow.

As shown in Figure 11b, a wave-like phenomenon was observed in the surface diagram of the T_rC with respect to the maximum transverse distance of the flow angle φ_C, where the value of T_rC varied within the range 5 m to 31 m. T_rC was susceptible to the flow interference than A_dC, and the impact was the greatest when the sailboat was subjected to crossflow and φ_C lay in the range of 90° and 270°. When the turning direction was consistent with the direction of the crossflow and φ_C was 90°, T_rC took the maximum value. On the contrary, T_rC took the minimum value when φ_C was 270°. In addition, T_rC was positively proportional to flow velocity v_C when φ_C is within the range of 0° to 180°, and inversely proportional to v_C when φ_C is within the range of 180° to 360°.

In the above analysis, A_dC varied significantly when the unmanned sailboat was subjected to different flow situations, which showed a decreasing trend when the unmanned sailboat sailed along the flow but presented an increasing trend when the sailboat sailed against the flow. In addition, T_rC varied significantly when the sailboat was subjected to crossflow, and the variation trend depended on the relationship between the turning direction and the flow direction. Notably, the increase in v_C enhanced the effect of flow interference on the turning motion of the unmanned sailboat.

4.4. Effect of Wind Speed under Flow Interference

Wind speed v_w is also a major element affecting A_dC and T_rC, since the parameter of time is also involved in the turning performance of the unmanned sailboat under flow disturbance. Figure 12 shows the variation in A_dC and T_rC with wind speed v_w and flow angle φ_C. Here, the unmanned sailboat was set to be subjected to a flow velocity v_C of 0.2 m/s, a wind angle φ_w of 60°, and a rudder angle δ_R of 35° in tacking. The variables v_w and φ_C were set in the ranges 4 m/s to 8 m/s and within 0° to 360°, respectively.

As shown in Figure 12a, the cross flow and longitudinal flow were distinguished with flow angles of 90° and 270°. When the sailboat sailed in the crossflow, namely when φ_C was located in the range of 90° to 270°, the maximum advance distance A_dC increased with the wind speed v_w. When the sailboat sailed in downstream, A_dC was inversely proportional to v_w.

As shown in Figure 12b, upstream sailing and downstream sailing were distinguished with a flow angle φ_C of 180°. During upstream sailing, when φ_C lay in the range of 0° to 180°, the maximum transfer distance T_rC decreased with the increase in the wind speed v_w. During downstream sailing, T_rC had a positively proportional relationship with v_w.

In the above analysis, an increase in wind speed v_w attenuated the effect of flow disturbance on the turning motion of the unmanned sailboat.

5. Harbor Pool Test

5.1. Test Conditions

In October 2023, a harbor pool test for the unmanned sailboat was conducted, and the 3 m class Petrel Sail unmanned sailboat was deployed to evaluate its turning characteristics. The prototype of Petrel Sail is shown in Figure 13, and the specific parameters follow those listed in Table 1.

The harbor pool test for the turning motion of Petrel Sail was carried out in the harbor pool (about 450 m long and 650 m wide) of Qingdao National Deep-Sea Base in Qingdao, China, as shown in Figure 14. In this harbor pool, the wind direction and wind speed are relatively stable, the ocean current is gentle, and the water depth is moderate, making it suitable for the test.

The main purpose of the harbor pool test was to test the turning characteristics of Petrel Sail in a real marine environment, thus verifying the previous analysis results.

The communication module in the stern of the hull could always measure the wind speed and direction of the sailboat and then receive the current environment through the wireless module of the shore station. By setting a different initial course, the unmanned sailboat could sail steadily for a period. When the heading angle entered the control range, the rudder turns to the fixed rudder angle and maintains it, and the actual turning motion route of the unmanned sailing prototype was collected. Four trials were designed for the tacking of the unmanned sailboat with different initial headings. For each trial, a fixed rudder angle was reached after a period of steady sailing and then maintained. During the harbor pool test, the true wind angle φ_w varied from −150° to −180° and the average wind speed was about 4 m/s in the natural wind field. The sailing trajectories for the four trials are shown in Figure 15. See also

Table 5. Different trial schemes.

Number	Relative Wind Angle (deg)	Rudder Angle (deg)	Wind Speed (m/s)
Trial 1	180	20	3.9
Trial 2	120	20	4.2
Trial 3	150	30	4.5
Trial 4	90	30	4.1

.

5.2. Test Results

The results of the maximum advance distance A_dmax and transverse distance T_rmax for the four trials obtained during the harbor pool test are shown in

Table 6. Results of turning motion test.

Number	Boat Speed (m/s)	Admax (m)	Deviation (%)	Trmax (m)	Deviation (%)
Trial 1	1.0	12.5	14.8	24.1	13.5
Trial 2	1.0	12.8	10.4	25.3	6.6
Trial 3	1.0	11.2	7.1	19.9	6.2
Trial 4	1.0	11.5	2.7	17.2	10.1

.

Trials 1 and 2 share the same rudder angle and a close average wind speed, with similar settings for trials 3 and 4. For all trials, the maximum advance distance A_dmax and transverse distance T_rmax decrease with the increase in φ_w, while trials 1 and 3 have larger advance distances and transverse distances than trials 2 and 4. In addition, the comparison between trials 1–2 and trials 3–4 reveals that the maximum advance distance A_dmax and transverse distance T_rmax decrease significantly with the increase in the rudder angle δ_R.

The sailing parameters for the four trials are substituted into the regression equation. The error between the test and simulation results for maximum advance distance A_dmax is within 2.7–14.8%, and that for the maximum transfer distance T_rmax is within 6.2–13.8%. The maximum errors for the two are observed in trial 1, which are 14.8% and 13.5%, respectively. In summary, the test results for A_dmax and T_rmax are in good agreement with the simulation results, and the errors of the harbor pool test are in a reasonable range, proving the correctness of the theoretical results.

6. Discussion

Our research has achieved certain results and provides research value for the design and operation of the unmanned sailboats. The prototype design of the Petrel Sail unmanned sailboat principle still has limitations. The prototype used is relatively small and used in relatively calm coastal waters with less wind and waves. Therefore, the lateral capsizing of the sailing vessel was not considered in this harbor pool test. In future studies, we plan to further explore the lateral stability of unmanned sailing vessels during long ocean voyages.

7. Conclusions

This study focuses on the turning characteristics of unmanned sailboats under complex environmental conditions. A simulation platform is built and the simulation experiment is designed with the regression orthogonal method. The turning characteristics of unmanned sailboat under hydrostatic condition and flow disturbance are obtained through the simulation analysis of the rudder angle, wind speed and direction, flow velocity, and other parameters. Finally, a harbor pool test for the unmanned sailboat is carried out.

The conclusions of this paper are as follows:

(1)

In the hydrostatic condition, the maximum advance distance A_dmax and transverse distance T_rmax are mainly affected by the rudder angle and wind angle, but not significantly affected by the wind speed. Better maneuverability of the unmanned sailboat can be obtained by selecting tacking.

(2)

Under flow disturbance, the maximum advance distance changes significantly when the unmanned sailboat is subjected to different flow situations (countercurrent or concurrent current). The maximum transverse distance varies significantly when sailboats are subjected to transverse currents, and the changing trend depends on the relationship between the turning direction and the flow direction. The increase in flow velocity will enhance the influence of flow interference on the maximum advance distance A_dmax and transverse distance T_rmax, while the opposite is true for wind speed.

(3)

The index of the approximate turning circle is a key metric that quantifies the maneuverability of a sailboat during its turning motion. This index is defined as the supposed circular path formed by the sailboat’s trajectory during turning maneuvers.

(4)

The error between the test results and simulation results for the maximum advance distance is within the range 2.7% to 14.8%, and that for the maximum transverse distance is within the range 6.2% to 13.8%. The errors are in a reasonable range, verifying the correctness of the simulation model established.

In future works, we will further investigate the boundaries for the turning performances of unmanned sailboats. A lateral comparison of different types of unmanned sailboats will be carried out to compare their turning performances. At the same time, the impacts of more practical factors such as attitude of the sailboat will be discussed to further improve the turning performances of unmanned sailboats, and long-voyage sea trials are expected for verification.