Operation of a ship in harbors, including the docking operation, is a tense task for onboard members because of the risk of collisions with other ships and harbor facilities. In particular, a large ship typically has powerful propellers that have poor maneuverability at a low speed; thus, it becomes difficult for the ship to move with a delicate motion, despite having a dynamic positioning system. Therefore, tugboats play an active role in maneuvering such large ships in harbors [1].

Meanwhile, pushing or pulling motion enables us to move a large object easily in our daily activities, and it can be more effective on the sea because the frictional force in sea is less than that on land. A tugboat utilizes this advantage to move large floating objects, such as (disabled) large ships, barges, oil platforms, caissons, etc. Moreover, multiple tugboats can cooperatively perform more delicate tasks like docking a large ship in a narrow space. However, the radio communications by humans between tugboats often interrupt the smooth flow of the operation, and operators are required to have considerable skills; this has encouraged researchers to automate the maneuvering operation of a large ship through multiple tugboats.

The research group in [2]-[5] investigated the manipulation problem of a large (disabled) ship using swarms of autonomous marine surface vehicles (tugboats). Each tugboat in contact with the ship was considered to be a three-degree-of-freedom (3-DOF) azimuth thruster of the ship, and its influence was treated as a pushing force to the ship. Ship dynamics were used to develop control strategies. Robust control methods based on the aforementioned concept have also been studied by other researchers [6][7].

The manipulation of a large ship by multiple surface vehicles is worth investigating as indicated in [2] and there are few related works. However, in robotics, there is a significant amount of research on the manipulation of a large object by multi-robot cooperation [8]-[16]. Related works were reviewed in [8]. Approaches using pushing [9]-[12], formation [13], grasping [14], caging [15], and object closure [16] have been investigated; these robotics technologies can be applied to marine robotics.

This study involves pose (position and orientation) control of a large ship by multiple tugboats with the aim of automating docking operation. To this end, we first challenge to incorporate the dynamics of both the ship and tugboats as a preliminary study; the previous research in [2]-[7] considered only the ship dynamics. The contact between the ship and each tugboat is assumed to be a point contact. The velocity constraints at the contact points yield a reduced dynamic equation that is expressed by independent variables, e.g., ship variables (surge, sway, and yaw rates), and excludes contact force terms; this allows us to treat the ship and multiple tugboats as a single system to control the independent variables. Section 2 of this paper formulates the reduced dynamic equation inspired by multi-fingered robot hand dynamics [17]. In Section 3, a controller to manipulate the ship to track a straight-line path while maintaining a desired heading angle is presented; a separate controller is, however, required for a single tugboat because a singular point appears when the contact point is aligned with two centers of mass of the ship and tugboat. Section 3 further considers a controller for this case. Lastly, Section 4 conducts simulations to verify the effectiveness of the presented control method based on the reduced dynamics.

Consider a (disabled) ship manipulated by n number of tugboats as shown in Figure 1. To formulate the reduced dynamic equation, we use the following notations:

Figure 1:  
Ship manipulated by n number of tugboats

• 𝑥_𝑜-𝑦_𝑜 coordinate system: the fixed frame
• 𝑥_𝑠-𝑦_𝑠 coordinate system: the ship frame
• 𝑢_𝑠: the surge rate of the ship
• 𝑢_𝑡𝑖 : the surge rate of the i-th tugboat
• 𝑣_𝑠: the sway rate of the ship
• 𝑣_𝑡𝑖 : the sway rate of the i-th tugboat
• 𝜔_𝑠 : the yaw rate of the ship
• 𝜔_𝑡𝑖 : the yaw rate of the i-th tugboat
• ^𝑠𝑢_𝑡𝑖, ^𝑠𝑣_𝑡𝑖, ^𝑠𝜔_𝑡𝑖 : the rates expressed with respect to (w.r.t.) the ship frame
• 𝑽_𝑠 = (𝑢𝑠 𝑣𝑠 𝜔𝑠)𝑇 ∈ ℜ³
• 𝑽_𝑡𝑖 = (𝑢𝑡𝑖 𝑣𝑡𝑖 𝜔𝑡𝑖 )𝑇 ∈ ℜ³
• ^𝑠𝑽_𝑡𝑖 = ( ^𝑠𝑢_𝑡𝑖^𝑠𝑣_𝑡𝑖^𝑠𝜔_𝑡𝑖 )^𝑇 ∈ ℜ³
• 𝜓_𝑠 : the yaw angle of the ship w.r.t. the fixed frame
• 𝜓_𝑡𝑖 : the yaw angle of the i-th tugboat w.r.t. the fixed frame
• ^𝑠𝜓_𝑡𝑖 : the yaw angle of the i-th tugboat w.r.t. the ship frame
• 𝝉_𝑠 ∈ ℜ³: the resultant forces and moment applied to the ship
• 𝝉_𝑡𝑖 ∈ ℜ³ : the forces and moment exerted by the i-th tugboat, which are expressed w.r.t. its body frame
• ^𝑠𝝉_𝑡𝑖 ∈ ℜ³ : 𝝉_𝑡𝑖 expressed w.r.t. the ship frame
• 𝒇_𝑖 ∈ ℜ²: the contact force at the contact point of the i-th tugboat
• 𝒓_𝑖 ∈ ℜ²: the position vector pointing from the center of mass of the ship to the contact point of the i-th tugboat
• 𝒓_𝑡𝑖 ∈ ℜ²: the position vector pointing from the center of mass of the i-th tugboat to the contact point with the ship, which is expressed w.r.t. its body frame
• ^𝑠𝒓_𝑡𝑖 ∈ ℜ²: 𝒓_𝑡𝑖 expressed w.r.t. the ship frame

From Figures 1 and 2, we obtain the dynamic equations of the ship and the i-th tugboat expressed w.r.t. the ship frame as follows:

Figure 2:  
Force and moment relationships between the ship and the i-th tugboat

where 𝑀_𝑠 = 𝑀_𝑠𝑅𝐵 + 𝑀_𝑠𝐴 ∈ ℜ^3×3, 𝑀_𝑠𝑅𝐵 and 𝑀_𝑠𝐴 are the rigid-body inertia matrix and the added inertia matrix, respectively, 𝐶_𝑠(𝑽_𝑠) = 𝐶_𝑠𝑅𝐵(𝑽_𝑠) + 𝐶_𝑠𝐴(𝑽_𝑠) ∈ ℜ^3×3, 𝐶_𝑠𝑅𝐵(𝑽_𝑠) and 𝐶_𝑠𝐴(𝑽_𝑠) are the rigid-body Coriolis and centripetal matrix, as well as the hydrodynamic Coriolis and centripetal matrix, respectively, and 𝐷_𝑠(𝑽_𝑠) ∈ ℜ^3×3 is the hydrodynamic damping matrix [18]; these are for the ship and 𝑀_𝑡𝑖, 𝐶_𝑡𝑖(𝑽_𝑡𝑖), and 𝐷_𝑡𝑖(𝑽_𝑡𝑖) represent the same things associated with the tugboat.

Assuming that the contact between the ship and each tugboat is a point contact, and considering the influence of the contact force to the ship and tugboat, as shown in Figure 2, Equations (1) and (2) can be rewritten as

where 𝐼₂ is a 2×2 identity matrix, and E is the orthogonal rotation matrix rotating an arbitrary vector 90^o counterclockwise in a plane.

Combining Equations (8) and (9) for a ship and n number of tugboats, we obtain the following:

Let 𝒗_𝑠 = (𝑢_𝑠 𝑣_𝑠)^𝑇, ^𝑠𝒗_𝑡𝑖 = ( ^𝑠𝑢_𝑡𝑖^𝑠𝑣_𝑡𝑖)𝑇 , and 𝒗_𝑖 be the velocity of the contact point with the i-th tugboat; moreover, let us assume that the contact point does not change during operation.

From Figure 3, the velocity constraint at the contact point can then be expressed as

Figure 3:  
Velocity constraint at the contact point between the ship and the i-th tugboat

Combining all the velocity constraints at n number of contact points, we have

Here, selecting 𝑽_𝑠 as independent variables, we obtain the following:

Substituting Equations (26) and (27) into Equation (14) and eliminating the contact force terms because 𝐵𝑁 = 𝑂_2𝑛×3 from Equation (25), the reduced dynamic equation is finally expressed as follows:

As a complex trajectory can be formed by combinations of straight lines, we discuss the straight-line path-tracking control of a ship using multiple tugboats while maintaining a certain yaw angle (heading angle) of the ship. Figure 4 shows notations for tracking a particular straight-line path. 𝒑_𝑠, 𝒑_𝑝, and ^𝑝𝒑_𝑠 = (𝜂_𝑥 𝜂_𝑦)^𝑇 are the position vectors of the ship, the origin of the 𝑥_𝑝-𝑦_𝑝 frame (path frame) w.r.t. the fixed frame, and the ship w.r.t. the path frame, respectively, and 𝜓_𝑝, α, and 𝜂_𝜓 are the angle of the 𝑥_𝑝-axis from the 𝑥_𝑜-axis, the angle from the front side of the ship to the moving direction, and the angle of the moving direction from the 𝑥_𝑝-axis, respectively.

Figure 4:  
Tracking a straight-line path. The tugboats are omitted from the figure

The control purpose ensures that 𝜂_𝑦 and 𝜂_𝜓 are zeros, and 𝜂_𝑦 and 𝜂_𝜓 are obtained by

Letting 𝑈_𝑑 and 𝑽_𝑠𝑑 be the desired moving speed and the desired value of 𝑽_𝑠, respectively, we can set 𝑽_𝑠𝑑 as

From the reduced dynamic equation in Equation (30), the forces and moments required to be exerted by the tugboats to achieve 𝑽_𝑠 → 𝑽_𝑠𝑑 are obtained by

where 𝐾 is a control gain matrix. Substituting Equation (37) into Equation (30) yields V˙s=K(Vsd-Vs), and 𝑽𝑠 ≈ 𝑽𝑠𝑑 if the elements of K are sufficiently large.

Here, we should note that 𝝉_𝑡 in Equation (37) can be calculated if 𝑁^𝑇𝑇 is invertible; however, when using a single tugboat, i.e., n = 1, it cannot be invertible when the contact point is aligned with the two centers of mass of the ship and tugboat as shown in Figure 5 (a) (singular configuration); this situation usually occurs when a single tugboat pushes the ship to move. To address this problem, a kinematic control approach is written as follows:

Figure 5:  
(a) Singular configuration. (b) Pushing the ship to track a given path by a single tugboat

where a control variable 𝜙 was added to the desired value of the yaw rate in Equation (36), 𝜙 = (𝜓_𝑠 − α) − 𝜓_𝑡1 is the angle from the line passing through the center of mass of the tugboat and the contact point to the moving direction as shown in Figure 5 (b), and α depends on the contact point here.

For more than two tugboats, singular configurations can also occur, e.g., when the lines passing through the contact points and the centers of mass of the tugboats intersect at a point; however, such singular configurations are difficult to occur in actual operations and thus this paper does not consider this issue.

To verify the effectiveness of the path-tracking control method based on the reduced dynamics, simulations were conducted. It was assumed that the ship did not use its propulsive forces for movement (we can consider another floating object, e.g., barge, instead of a ship) and the tugboat can exert forces and moment for 3-DOF motion on a horizontal plane. The contact points were chosen arbitrarily to prove the correct functioning of the reduced dynamics.

The parameter values of the ship and tugboats used for simulations are shown in Table 1, and they were determined for future lab-scale experiments. L, W, m and 𝐼_𝑧 are the length, width, mass, and moment of inertia, respectively. Xu˙, Yν˙, Yω˙, Nν˙, and Nω˙ are the added mass and inertia derivatives, 𝑋_𝑢, 𝑌_𝑣, 𝑌_𝜔, 𝑁_𝑣, and 𝑁_𝜔 are the coefficients of the linear skin friction, and 𝑋_|𝑢|𝑢, 𝑌_|𝑣|𝑣, and 𝑁_|𝜔|𝜔 are the coefficients of the quadratic drag (refer to [18]). 𝐼_𝑧, Xu˙, Yν˙, and Nω˙ were obtained through calculations, and others were chosen by referring to the ship parameters in [18] and through simulations. Three cases were examined: the use of two tugboats, three tugboats, and a single tugboat.

First, we tested the case when two tugboats were used, i.e., n = 2. The two tugboats were initially in contact with the ship as shown in Figure 6. 𝒓₁ = (−1, 0.5)^𝑇 , 𝒓₂ = (−1, −0.5)^𝑇 , 𝒓_𝑡1 = 𝒓_𝑡2 = (0.5, 0)^𝑇 , and the origin and angle of the path frame were set as (2, 0)^𝑇 and 𝜓_𝑝 = 30^o, respectively. The desired moving speed was 𝑈_𝑑 = 1 knot, α in Figure 4 and Equation (35) is set as 0^o, and the control gains are 𝑘_𝑦 = 1, 𝑘_𝜓 = 1.2, 𝑘_𝑢 = 𝑘_𝑣 = 𝑘_𝜔 = 1000, i.e., 𝐾 = diag(1000, 1000, 1000). As observed in the figure, the ship tracks the given path well while maintaining the heading angle specified (α = 0^o). Figure 7 presents the case when a given path is tracked while maintaining α = 45^o . The result proved the good performance in this case.

Figure 6:  
Tracking a given path using two tugboats while maintaining α = 0^o

Figure 7:  
Tracking a given path using two tugboats while maintaining α = 45^o

Figures 8 and 9 are the cases of using three tugboats, i.e., n = 3. First, the initial conditions for the case of Figure 8 are 𝒓₁ = (1, 0.5)^𝑇, 𝒓₂ = (0, 0.5)^𝑇, 𝒓₃ = (−1, 0.5)^𝑇, 𝒓_𝑡1 = 𝒓_𝑡2 = 𝒓_𝑡3 = (0.5, 0)^𝑇, 𝜓_𝑠 = 90^o, and α = 135^o. The origin and angle of the path frame, the desired moving speed, and the control gains are the same as those used in the previous simulations. For Figure 9, 𝒓₁ = (1, 0.5)^𝑇, 𝒓₂ = (0, 0.5)^𝑇, 𝒓₃ = (−1, 0)^𝑇, 𝒓_𝑡1 = 𝒓_𝑡2 = 𝒓_𝑡3 = (0.5, 0)^𝑇, 𝜓_𝑠 = 90^o , and α = 45^o . The two simulation results verified that the controller worked correctly. Figure 10 shows the changes of α in the simulations shown in Figures 7, 8, and 9.

Figure 8:  
Tracking a given path using three tugboats while maintaining α = 135^o

Figure 9:  
Tracking a given path using three tugboats while maintaining α = 45^o

Figure 10:  
Changes of α in Figures 7, 8, and 9

In addition, simulation was conducted using a single tugboat, as shown in Figure 11. 𝒓₁ = (−1, 0)^𝑇, 𝒓_𝑡1 = (0.5, 0)^𝑇, 𝜓_𝑝 = 10^o, and α = 0^o, and the control gains are 𝑘_𝑦 = 1, 𝑘_𝜓 = 1.92, 𝑘_𝜙 = 2.55 , and 𝐾 = diag(100, 100, 100) . The figure shows that the ship tracks the given path well.

Figure 11:  
Tracking a given path using a single tugboat

From these simulation results, it was confirmed that the reduced dynamics functioned correctly; however, this study did not consider the friction angle of the contact force that assures the pushing motion of the tugboat for manipulation of the ship. For future works, we aim to incorporate/combine the contact force constraints into/with the reduced dynamics.

In this study, the manipulation of a large ship was conducted using multiple tugboats. We formulated a reduced dynamic equation that incorporates the dynamics of the ship and n number of tugboats and excludes the contact force terms; this allowed us to treat the ship and n number of tugboats as a single system to manipulate the ship. A pose control method based on the reduced dynamics was presented, and its effectiveness was verified through simulations. Our future works will involve incorporating the contact force constraints, as mentioned in the last part of Section 4, and conducting lab-scale experiments.