1. Introduction
Using tugboats has been an effective method for positioning and transporting marine vessels in various scenarios, especially those with limited or no maneuverability. Berthing and unberthing of large ships, towing disabled vessels, navigating cargo barges through narrow channels, etc., can be named as a few of those scenarios. A tug assists by towing or pushing the vessels, thus adding maneuverability to the whole system. By cooperative control of several tugs, complex movements can be achieved while ensuring safe operations. Furthermore, the deployment of oversized or unwieldy offshore platforms like heavy cargo ships, floating cranes, offshore wind turbines, etc., is becoming popular thanks to the growth of offshore energy exploitation and maritime trade. Hence, transportation systems using multiple tugboats are becoming more and more relevant in marine mobility. A recent study forecasts that the tugboat service market will grow at a compound annual growth rate (CAGR) of 14.33% between 2023 and 2028 and the market size will increase by USD 3.15 billion [
1]. Even so, automated control problems of these transportation systems are still wide-open.
Firstly, most of these systems rely on manually controlled tugboats. That is, each tugboat is controlled by a skilled pilot, managing its thrust, position, heading angle, etc. With the involvement of multiple tugs, their pilots also need to communicate and cooperate. This is not an easy task, as it requires significant time and effort and also puts involved people at risk. In recent years, there has been a noticeable increase in the commercialization of autonomous tugboats [
2,
3,
4]. This opens the door for automated control of the offshore platform transportation system with better performance and without human intervention.
Secondly, the configuration of tugboats, i.e., the number of boats and their role in transportation, is considered. The tugs can contact and push the platform to be transported or tow it over towing lines. Existing proposals vary from a single tug to a maximum of six cooperating, as shown in
Figure 1. Lee et al. [
5] proposed the use of a single tugboat to tow a vessel via a towing line and move it in a tight space. The vessel has no propulsion but is equipped with passive actuation mechanisms like rudders. In this configuration, not only is the tug controlled, but also the vessel’s actuators have to actuate to be able to ensure the desired trajectory of the vessel and the stability of the whole transportation system. An offshore platform without any actuators requires at least two tugboats to assist. The popular arrangement is that both tugboats tow the platform, where one of them plays the guiding role at the front and the other plays the following row at the back [
6,
7]. Even so, the system dynamics are still nonholonomic. Therefore, the configurations of four tugboats are usually used in practical applications, and in research, the combinations of six tugs are proposed in order to provide sufficient maneuverability in complex operating situations. In [
8,
9,
10], the tugs attach and provide propelling to the platform from different directions. Towing a platform by towing lines provides more freedom to the tugboats so a system of four towing tugboats is preferred in [
7,
11,
12]. Regardless of pushing or towing, the tugs are usually positioned evenly around the transported platform to keep the balance for the platform, as well as provide comparable controllability in all directions on the horizontal plane. However, this arrangement takes up space, so in crowded and complicated work areas, they are difficult to implement. To overcome this, we previously proposed in [
13] the novel configuration of tugboats for the offshore transportation system. It consists of two pushing tugboats and two other tugs to tow the platform. All are positioned on one side of the platform so that they use the space more effectively. By coordinating opposing pushing and towing forces from the tugs, the maneuverability of the system is still comparable with conventional configurations. Unfortunately, the system models in [
13] do not fully reflect the interaction between the platform and the tugs in this configuration.
The control system for this transportation configuration then needs designing. The objective is to govern the tugs such that the desired movement of the transported platform is achieved and the stability of the entire system is preserved. The control flow starts by determining the force and torque required to act on the platform to achieve this desired movement. It is then distributed across the tugboats from where the control action for each tug is calculated and acted upon. [
13] introduced the first control attempt for our proposed tugboat configuration by a centralized
H∞ control with pseudo-inverse-based allocation of the platform control inputs. However, the resulting outputs can be unusable due to this allocation method. For example, force amplitude is negative, or the impact angle violates practical geometry constraints. In addition, the motion references for the tugs have not been defined in this study due to the shortcomings in the system model. Therefore, the feasibility of the proposed control cannot be verified. Our latter study [
14] solved these issues by (1) considering the allocation design as a constrained optimization problem and (2) designing the
H∞ control based on a modified system model with the state variables of the towing tugboats containing the towing force and its impact angle, which correspond with the output of the allocation. Simulation results have validated the proper control performances of this transportation control system but also hinted that they can be improved in terms of positioning accuracy, control-distribution efficiency, and control-effort consumption. For other conventional tugboat configurations, decentralized architecture has also been adopted besides our centralized control. An optimal-based allocation is combined with the sliding mode control in [
10] and with feedback control in [
9] for the offshore platform-transportation system with attached tugboats. In the case of four tugboats towing the platform, model predictive control with a dynamic coordination-decision mechanism was introduced in [
11], consensus control was designed in [
15], and dynamic surface control was combined with optimal-based and robust control in [
12]. The decentralized control has flexibility in the coordination of different control techniques that suit the characteristics of each vessel in the system. However, most decentralized control systems adopt the cascade scheme, where only the input-to-state stability can be preserved.
Therefore, we propose a novel control system for offshore platform transportation with the mentioned arrangement of pushing and towing tugboats in this paper. The control aims for precise positioning and tracking of the platform and preserves the stability of the entire system. In such, challenging tasks like berthing and navigating through a busy workspace can be conducted safely and successfully, even with the presence of ocean disturbances. For this, the mathematical models of the entire system are first derived, in which, the tugboat models are written with new state variables. Then, the control strategy is proposed, consisting of (1) an observer estimating the environmental disturbances, (2) a platform motion controller acting as the supervisory control, (3) an optimal constrained allocation distributing the required actions over the tugboats, and (4) the tugboats’ controller deriving the control inputs of the tugs. The boundedness of the transportation is proved via the Lyapunov stability theory, and comparative simulations are conducted in challenging scenarios for evaluating the control performance. The problem formulation, control-system design, stability analysis, and simulation studies are respectively organized from the following
Section 2,
Section 3,
Section 4 and
Section 5. In
Section 6, conclusions are drawn. Hence, the contribution of the paper can be summarized as follows:
A complete model of the offshore platform-transportation system using two pushing tugboats and two towing tugboats is first introduced with new state variables.
A novel control scheme is proposed for the offshore floating platform transportation system utilizing two pushing and two towing tugboats.
Proof of the boundedness of the entire system is provided. Simulation results evaluate the effectiveness of the proposed system.
2. Problem Formulation
The floating platform to be transported has neither propulsion nor actuators to regulate its movement. The proposed system uses two tugboats towing the platform via towing lines and two other tugs pushing it in the opposite direction. All are positioned on one side of the platform; for example, the front of the platform in a route following the task or the outside of the platform in the berthing operation. Hence, the longitudinal distances from their contact/connection points on the platform to the center of the platform can be considered equal. The other connection points of the ropes on the towing tugs are assumed to be at the tugs’ center. Among the tugs, the towing ones are on the outside and the pushing ones are on the inside. The arrangement of the floating platform and the tugboats, collectively referred to as vessels, are illustrated in
Figure 2a. By varying the amplitude and angle of impact of the towing and pushing forces, the maneuverability of the platform can be sufficiently fulfilled. This is achieved by controlling the motion of the corresponding tugboats.
2.1. Kinematic and Dynamic Equations
Figure 2b depicts the schematic drawing with notations for system parameters and motion variables. The motions are realized in a north–east–down reference frame
xyz and the standard body-fixed frame of each vessel. One can see that the whole system has 30 degrees of freedom (DoF), with the 6-DoF motions for each vessel. However, in a vessel dynamic-positioning problem, we can assume that the heave, pitch, and roll motion are sufficiently small and, therefore, neglected. Only the planar motions, i.e., surge, sway, and yaw, are worth considering [
16]. The planar kinematics and dynamics of these vessels can be generally written as follows [
17]:
where the subscript
refers to the transported platform, the towing tugboats on the left and the right of the platform, and the pushing tugboats on the left and the right of the platform, respectively. For each vessel, the inertia matrix
consists of its mass, the moment of inertia about its vertical axis, the added mass, and added inertia due to hydrodynamic forces and moments.
is the total damping matrix. Assume that both are diagonal matrices. The vessel’s horizontal position and the heading angle in the reference frame are provided by
, while the vector
indicates the velocities of the corresponding surge, sway, and yaw motions in the body-fixed frames. The kinematic relationship between frames is provided by the rotation matrix
which is:
In the body-fixed frame, the total forces and torque acting on a vessel comprise the controllable one
and the unpredictable disturbances
. The disturbances are majorly contributed by the wind and wave forces whose models have been thoroughly presented, such as in [
11]. For the transported platform, Vector
represents the surge and sway forces and yaw moment resultant of the towing and pushing forces from the tugboats. From the schematic drawing in
Figure 2b,
is computed as follows:
In Equation (3), is the amplitude of the towing/pushing force from the corresponding tugboat, and is the angle of impact of the force on the transported platform. Apparently, . is the distance from the mass center of the platform to the contacting/connection point with the tugboats in the longitudinal direction of the platform. on the other hand, is the corresponding distance in the platform’s lateral direction. Additionally, and for simplification.
In the case of the towing tugboats, the input
combines the forces and torques generated by the onboard actuators
and the towing force
. It is noteworthy that the direction of the towing force is along the towing line. That is, its impact angle on the tug depends on the relative angle
and the yaw angle
of the tug itself:
Similarly,
of the pushing tug
p is the result of actuation
and the contact force
along the tug’s longitudinal axis. They are written as follows:
From the theory of elasticity, the contact forces and towing forces are modeled in the same form [
18]:
In the case of the towing forces, is the stiffness of the rope, is the difference between the rope’s length and the distance between two connecting points, and is a smooth function to define that the force only appears when the rope is tensioned. For the contact force, is the stiffness of the contact patch, is the difference between the longitudinal distance from the tug’s center to its front end and the distance from the tug’s center to the contact point on the platform, and indicates the force only appears when the tug contacts with the platform.
Moreover, the motion control problems of pushing-type tugboats have been well established, especially in comparison to the towing-type ones. Therefore, their dynamics are dismissed, and it is assumed that their responses perfectly meet any movement requirements. That leaves the motions of the platform and the towing tugboats to be examined in the remainder of the paper.
2.2. Transformed System Model
The motion equation of the transported platform and the towing tugboats can be written solely in the reference frame as follows:
where:
Additionally, from the relative position between the platform and the towing tugs, the horizontal position of the tugs can be written as follows:
Taking the second-time derivatives from both sides of Equation (9) and substitute
and
from Equation (7) results in:
where the details of the coefficient matrices are provided in
Appendix A. From Equation (10), one can see that the dynamics of the towing tugboats are now represented through the relative motion with the platform they convey. The state variables contain the impact angle
of the towing force on the platform and the distance
, which, from Equation (6), corresponds to the amplitude of the force.
4. Stability Analysis
To analyze the stability of the entire system and derive the choice of control gains, the following Lyapunov function candidate is considered:
with:
being the diagonal matrix with the diagonal line provided by the vector in
.
indicates scalars that satisfy the following conditions such that
is positive definite [
20].
Taking the time derivative of
results in:
with:
One can see that
is a positive semi-definite matrix. Hence, the second term in Equation (30) is negative semi-definite. In addition with the Theorem 1 in [
20], by choosing the observer gains
,
, and
as:
there exists some positive
such that:
In other words, the proposed observer is finite-time stable. On the other hand, taking the time derivative of
provides:
where:
With
being bounded, the proposed control ensures the system to be bounded if the controllers’ gains are chosen such that the matrix
S is positive definite [
21]. It is noted that the function
models the smooth transition between two discrete states: tension or intension (in the case of towing lines), and contact or non-contact (in the case of the pushing tugboats). Then, during the effective operation of the tugs, we can simply replace
. Additionally,
S contains nonlinear elements. By considering the nonlinear system as a polytopic system, the controllers’ gains can be optimally derived using linear matrix inequality techniques [
22]. Then, the stability of the whole system is preserved.