Container ship model experiment of autonomous course-keeping ability at the beach

2.1 Container Ship Model

In the recent trend of a VLCS, the length overall (LOA) does not make it longer than 399.9 m. Inspired by COSCO Shipping Gemini having the length [19], we adjusted the scale ratio of 1/320 based on a 20,000 TEU ship. Considering the portable size, the LOA and beam of our model are selected as 1,250 and 185 mm, respectively. Table 1 lists the main parameters of the ship model. The draft and hull weights make us anticipate the stability of the roll motions of the model. It is a bottom-flat type and a heavy bottom model. We did not show the hydrodynamic information of this model in detail. It will be the next topic with the numerical simulations. See reference in [2].

Table 1:

Main parameters of the ship model

At first, we prepared the Iso-pink on the desk (Figure 1a). With a saw, we roughly cut the material. Then we carefully polished the Iso-pink (Figure 1b). Then it was taped to protect before rubbing the fiberglass (Figure 1c).

Figure 1:

Model construction process

The fiberglass and synthetic resins are put into the hull (Figure 2a). The hull of fiberglass is widely used because of its various merits, such as strength, lightness, low cost, and elasticity. Both the inner space and strength within the portable size make this material recommended. Basic painting with white color and the work of grinding was done several times to smooth the surface of hull (Figure 2b). Then, the painting with blue and red colors, which separates the load lines and preserves the preliminary buoyancy by the height of a freeboard, was finished (Figure 2c). The maximum draft of the VLCS model should be kept within the boundary of the blue and red colors of the hull. This point should be checked before going out to sea.

Figure 2:

Hull process of fiberglass and painting

Let’s see the form of the bow and stern. The bulbous bow was not fully constructed according to the author’s opinion, rather, it seems to be closer to an ax bow. There are tiny holes in the bow thrusters below the VLCS model (Figure 3a). The transversal beam of the position of the bow thrusters is not sufficient. Because the thrusters were not so effective, we decided not to use them. How can we balance the ship if we use the different weights of cargo and ballast tanks, except for the propulsion system, in this size of hull space? So, the too small size of the model of USVs may lose the beauty of the ship itself because its purpose is far from transportation and operation in ocean environments. Even though it is not so meaningful, the stern skeg is made as slim as possible against the wave breaking and viscous resistance (Figure 3b). The single rudder and propeller can be seen as well. A rubber tube was used to cover the propeller shaft. The long and narrow stern tube should be protected from the inflow of seawater.

Figure 3:

Snapshots of bow and stern

Figure 4 shows the equipment of the autonomous tracking system. We arranged the table to announce the simple structures of actuation. We employed various products, including the Holybro Pixhawk 4, Holybro SiK telemetry radio V3, Raspberry Pi 4 Model B, FLYSKY FS-i6 receiver, SeaKing-60A-V3.1 speed controller, 3680-880 KV brushless motors, and a laptop.

Figure 4:

Equipment for an autonomous tracking system

Holybro SiK telemetry can work well with Micro Air Vehicle Link (MAVLink) packets and be integrated with the Mission Planner and QGroundControl [20]. Even if the distance between the model ship and the computer is relatively far, information exchange is possible through two telemetries, which are connected to the Pixhawk 4 and a laptop. There are major sensors, such as GPS, Inertial Measurement Units (IMU), which include accelerometers and gyroscopes, a barometer, and a magnetometer. MAVLink is a serial protocol that is the primary means of transmitting sensor data [21]. ArduPilot adapts its telemetry rates using MAVLink Radio.

In advance, we put the deck containers (white color in Figure 5a), which are made of Styrofoam and Formax, on the hull. Then we checked the trim, draft, heeling, waterproof conditions, and actuation of the rudder and propeller in an indoor water tank. Because the leads were laid on the bottom of the hull and inside the deck containers, the stability of the VLCS model could not be better. This point is far from reality. The draft, trim, and heel must be trimmed to the cargoes, ballast, and bunkers. The deck containers were fixed to prevent collapse. One can see the single box of autonomous systems in the middle of the deck containers (Figure 5b). The edge of the transom touches the still water surface. The VLCS model was prepared at last.

Figure 5:

The tests of heeling, trim, and waterproof conditions

2.2 Guidance law

The guidance law in ArduPilot originated from the principles of an Unmanned Aerial Vehicle (UAV) system [22]. We illustrated the marine situation of autonomous tracking tasks in Figure 6. The container ship has a position of course track errors (d). The center of the frame (o_b) is fixed on the center of gravity (CG). The local position may have the cross-track errors from the tangential line (CD), which is parallel to the lines of AB || EF.

Figure 6:

Guidance law

The model is on a North-East-Down (NED) frame, where the north axis is directed to the true north (x_N). The pink lines mean the same direction. The yaw angle (ψ) is from the x_N to the direction of x_b axis. The ship plans to navigate to the next waypoint (WP) 1 autonomously. The red thick line means the well-known L₁ distance (See Table 2), which is denoted as:

Table 2:

Main parameters of the L1 Controller

where D_L1 is the damping, named as NAVL1_DAMPING in the ArduPilot; T_L1 is the period, which is one key parameter, as named NAVL1_PERIOD; V_f is the forward speed (See Table 2) [23]. The η denotes the angle between the V_f and L₁ distance. This angle is the same as the sum of η₁,η₂, and η₃. According to the definition of radian, the arc length (s) is Rη, where R is the radius of circle; η is in radians. The maximum distance from a WP is denoted as WP_RADIUS [23]. We can assume such that η₁≈d/L₁ and $$ {\eta }_2\approx \dot{d}/V_f$$. When the USV is controlled to reach the WP 1, the yaw angle (ψ) is adjusted, of course, as with the existing marine vessels. There may be ocean disturbances, so the actual direction normally can be varied as the course angle (χ). Both the angles (ψ, χ) are measured as the clockwise direction from the x-axis (x_N), which indicates the true north. The crab angle (β_c) is not very important in the study. We can see the two dashed lines in Figure 6, where α_⊥LOS denotes the acceleration perpendicular to the L₁; a_ccmdmeans the lateral (centripetal) acceleration command, which is calculated by the PD controller such that:

where k_d,k_s (>0) are the damping and restoring constants, respectively [24]. The key parameters of the L₁ controller are listed in Table 2.

In addition, the α_scmd is approximately equal to -$$ \ddot{d}$$, a second-order Ordinary Differential Equation (ODE) of course track errors can be calculated as:

With $$ {\omega }_n=\sqrt{\frac{k_x}{m}}=\sqrt{2}Vc/L_1$$, c≡cosη₃,

and $$ \zeta =\frac{k_d}{2\sqrt{m{k}_s}}=\frac{1}{\sqrt{2}}=0.707$$,

where ζ is the small damping coefficient; ω_n is the natural frequency of a resonator (rad/s), which is dependent on the speed and the L₁ distance, like the concept of Line of sight (LOS); mis the mass (11,480 g) of the typical Mass-Damper-Spring system.

3.1 Experimental place

Actually, we did not perform the collision avoidance, zig-zag test, berthing, or unberthing in this time, rather, focus on course-keeping ability, the lower vibrations, and rotational stability at sea. Figure 7 shows the place of test. Yeosu City is one of the marine cities in South Korea. The test date was 1 March, 2025. This coastal sea looks so clear; however, it represents dynamic scenes whenever we test. There are tourists on the beach and rocks at close distances. Sometimes we have to encounter low tides, seaweed, aquatic plants, and fishing nets. More interestingly, divers are working on the spot because it is the coast of a fishing village. After the rainy season, large amounts of waste from the town and the mountain prevent the test because of concerns about the damage to actuators. Outside of the yellow line, frequent traffic or berthing ships exist. Can we establish these things in each simulation?

Figure 7:

Experimental place: Yeosu, S. Korea

Information on waves and winds is listed in Tables 3 and 4, where taken from the National Climate Data Center [25]. The nearest official data on the wave of 1 March, 2025, is observed at the Nam-Hae in South Korea 26km off from the test place. Because the beach of experiment is closer to the Yeosu city, the wave height is less than that in Table 3. We started the tests in the mornings between 0600 and 0800 Local Time (LT). The wind speed was weak. We can see that the maximum wind speed and directions of air flow are from the sea.

Table 3:

Information on the wave

Table 4:

Information on the wind speed

3.2 Results of trajectories with waypoints

Figure 8 shows the snapshots of routes on the Mission Planner (MP) screen. The red line shows the results of the trajectories. We introduced this one result in the study, considering the number of test results. The corner of the right-down explicitly stated the date of the test. Our scenario is the star-shaped routes because we want to see the course track errors and performance after arriving at the WPs and the sharp turning of the USV. The advantage of the USV systems is that the results show a constant pattern. It cannot get out of its performance. By humans, one may mistake their duty watch because of many human factors.

Figure 8:

trajectories on the Mission Planner

The course track errors, which were explained in the previous section, are illustrated in Figure 9. Over 2 m results are the 6 edges of routes in Figure 8. The remaining courses are recorded with an error of less than 1 m. This model features a single rudder, allowing the ship to navigate with a slight zig-zag motion. On that day, there are no terrible wind gusts and rough waves. Of course, only the surface wave will be a burden on this size of model.

Figure 9:

track errors

Although the sea test, the model shows a certain level of speed (Figure 10). We set the speed to 1 m/s with a Froude number (Fn, $$ U/\sqrt{gL}$$) of 0.286. The design speed of a container ship and Fn are 24 to 26 knots and 0.22 to 0.25 [26], respectively. Thus, the actual speed of 0.8 m/s is more appropriate considering surge resistance. This cannot be changed during the navigation. It is a different point with a car driving. When we drive the car at a steep curve on the road, the speed can decline by deceleration. This model cannot be changed in its speed. Only the rudder action controls the position of the ship. So, we said that the yaw angle of the guidance law is one of the important keys. The result of a constant level of speed means that the ship overcomes the wave resistance. About half of the speed decreased at the six edges. The final peak of results was not the route. At that time, we recovered the ship from the sea. Then we have to check the ship's condition.

Figure 10:

speeds

The WPs and trajectories are compared in Figure 11. Red circles are WPs on the MP screen. Green dashed lines are connections between the WPs. The ship started west of Figure 11. It goes to the east side of the map, then comes back to the start position after navigating with a star-shaped route. Because the ship has a constant speed, it deviates at the edges of the routes. We think about the speed reduction of the corners in future tests.

Figure 11:

Waypoints (WPs) and results of trajectories

3.3 Results of roll, pitch, and yaw responses

In this section, the responses of the main states are represented. The roll motions (Figures 12, 13) are one of the important considerations. Table 5 presents the basic statistical results of the rotational responses, with a population of 949. The abnormal values of roll motions are frequent than those of pitch motion, resulting in larger standard deviations compared to the pitch angle.

Figure 12:

roll responses

Figure 13:

roll rate responses

Table 5:

Results of rotational responses

The concerns of capsizing, we do not have to worry about because the model has a flat-type of bottom and the lead on the inner space of the hull, and the weights of the deck container are not so heavy. This point is far from reality, such that the ability of maximum cargoes is not emphasized in the model. The hull of a container ship is designed in such a form, even though the resonance roll may be anticipated owing to the vertical height of deck containers and high propulsion for achieving the on-time voyages. Roll angles are not exceeded by 10 °, and their rate responses are higher than the roll angle. It means that the ship recovers to an upright position shortly. It has good stability performance in a transversal direction. Pitch responses (Figures 14, 15) show no special features. The bow shape of the model cuts through the sea surface very well. Considering the model has a full draft, the pitch responses are not bad. In addition, the rates of pitch are higher than the angles. In some intervals, the pitch rate was varied. This point is not so important because the ship moves while floating on the sea, and the weather conditions. We thought the routes were not straight or circular paths.

Figure 14:

pitch responses

Figure 15:

pitch rate responses

Yaw angles and rates (Figures 16, 18) recorded the general courses of the ship. We can know the ship's states and the routes from the information on yaw angles only. It did not overstep the 180 °. The headings (Figure 17) provide the information by comparison. The actual course angle (χ) of the model is different from these two angles. The forward speed (V_f ) does not coincide with the ship’s heading. Ocean environments affected the test. That’s why the guidance law and tuning parameters in Table 2 are necessary.

Figure 16:

yaw responses

Figure 17:

headings

Figure 18:

yaw rate responses

3.4 Results of Vibrations

The IMU Batch Sampler gathers high-frequency data for spectral analysis from the IMU sensors. Accelerometer data recorded is converted into the frequencies of the vibration [27]. The vibrations for each axis resulted in a comparatively low level, especially compared to UAV cases, where the vibrations are acceptable if they represent less than 30 m/s/s. These points are strong for marine vessels when we think about the origin of guidance law and its many UAV users. The wide transom with flat buttock shape helps to reduce resistance and vibrations [28]. These results are one of the satisfactory outcomes. Especially, the surge vibrations with low magnitude stipulated to us that the model can navigate the beach.

Figure 19:

vibrations of the x-axis

Figure 20:

vibrations of the y-axis

Figure 21:

vibrations of the z-axis