In this paper, we present a method for calculating reaction forces for a crane mounted on a ship moving in waves. The method is used to calculate the reaction forces between the crane base and the vessel deck. This includes the case where the crane is mounted on the platform that keeps the base of the crane horizontal when the vessel is moving in roll and pitch. The wave motion of the ship is modeled with force response amplitude operators (RAOs) based on the JONSWAP wave spectrum. The combined equations of motion for a vessel and a crane are derived using Kane’s equations of motion, where velocities and angular velocities are formulated in terms of twists, and the associated partial velocities and partial angular velocities are given as lines in Plücker coordinates. The unknown reaction forces are represented as wrenches and are determined using screw transformations. The method is used to study the effect of the roll and pitch compensation platform in numerical simulations. The efficiency of the platform is evaluated in terms of the magnitude of reaction forces and crane payload sway angles.
Crane operations are important in the offshore industry as a vital element in the supply chain of offshore installations. Offshore cranes may be large and heavy structures, which apply significant dynamic reaction forces and moments to the structure of the vessel. For small vessels, the mass of a crane may be significant compared to the mass of the vessel, which means that the motion of the crane will influence the motion of the ship. In such cases, it will be useful to determine the reaction forces from a dynamic analysis, where the equations of motion for the combined crane and vessel system are used. The determined reaction forces will be needed in the design of the connection between the crane and the deck. Alternatively, the reaction forces can be used to determine the design specifications for a roll and pitch compensation platform, when it is used to keep the base of the crane horizontal, while the vessel is moving in waves. The dynamic analysis based on the combined equations of motion can be used to determine the reaction forces for the cases with and without the motion compensation platform, which may be useful for the evaluation of the platform efficiency.
The dynamic coupling between crane load and vessel motion was studied in Ref.  for a floating crane barge. The response of the barge was found when the barge and the crane are modeled as one rigid body. A simulator for a vessel with a crane was presented in Ref. , where the authors used an object-oriented modeling approach and co-simulation with the functional mock-up interface .
The combined dynamics of a ship and a deck crane were derived in Ref. , where Lie Groups and Lagrangian mechanics were used to derive the equations of motion for a general vehicle-manipulator system where the vehicle has six degrees-of-freedom (DOFs). Earlier work on the same topic was presented in Ref. , where the equations of motion of a spacecraft-manipulator system were derived using Kane’s equation of motion. Kane’s equations of motion  for multibody systems are based on the Newton–Euler formulation, where the forces of constraints are eliminated using the principle of virtual work. In Kane’s method, the principle of virtual work is formulated in terms of partial velocities and partial angular velocities, which serve as projection operators that project inertial and external forces onto the directions associated with the generalized speeds. In this way, a minimal set of ordinary differential equations is obtained. Kane’s method was originally formulated at the component level using coordinate-free vectors , and it was used in a very detailed derivation of the equations of motion for a Stanford manipulator in Ref. . An alternative formulation of Kane’s equations of motion for serial-link mechanisms was achieved by Angeles and Ma , where a matrix formulation was introduced using the link twists to determine the projection operators in the form of a natural orthogonal complement. This was further developed in Ref. , where the partial velocities and the partial angular velocities used in the derivation of Kane’s equations of motion were represented by lines in Plücker coordinates , and this was demonstrated in the development of the equations of motion for the combined dynamics of a ship and a crane mounted on the ship.
The forces of constraint (i.e., the reaction forces) in a multibody system can be determined using Lagrange multipliers, which results in a differential–algebraic system of equations, which can be solved for the reaction forces [11,12]. This modeling approach might require stabilization , and the obtained reaction forces are related to the constraint equations and may need conversion to the defined coordinates. Another approach is to use a minimal set of ordinary differential equations as in Kane’s formulation, where the constraint forces are eliminated from the equations. The constraint forces can then be brought to evidence using auxiliary generalized speeds, which define fictitious velocities and partial velocities in the directions of the unknown reaction forces . This was investigated for a knuckleboom crane in Refs. [14,15], where the method of auxiliary generalized speeds was formulated in terms of twists and screws, which led to the systematic derivation procedure of the auxiliary partial velocities used in the determination of the constraint forces.
In this work, we extend the results of Refs. [9,14], first by deriving the equations of motion for a crane and a vessel system including a roll and pitch compensation platform between the ship deck and the crane base. Next, a procedure is developed for the determination of the reaction forces between the crane and the platform and between the platform and the vessel. A new feature of the proposed method is that the auxiliary partial velocities are given by lines in Plücker coordinates, and the reaction forces are given in terms of wrenches that are transformed with screw transformations. This gives a formulation with a clear geometric interpretation that is helpful in the development of the dynamical model. The performance of the derived model is studied by numerical simulations where the ship motion in waves is simulated using the JONSWAP spectrum [16–18] in combination with force response amplitude operators (RAOs) [19,20]. In addition, we investigate the efficiency of the motion compensation platform. Efficiency is evaluated in terms of the magnitude of the reaction forces at the deck/platform and platform/crane interfaces and in terms of the payload sway angles.
The rest of this article is organized as follows: Section 2 presents the theoretical preliminaries. In Sec. 3, we show the detailed derivation of the dynamical model, while in Sec. 4, we provide the procedure for the determination of reaction forces. In Secs. 5 and 6, we present the simulation results and provide the discussion of the results. Section 7 presents conclusions.
In this section, we present the theoretical preliminaries of this work. First, we present the procedure for modeling a marine vessel in wave motion. Then, we introduce twists, which are screw representation of linear and angular velocities of rigid bodies, and wrenches, which are screw representation of forces and torques on rigid bodies. Then, we present a general method for modeling dynamics of open-chain manipulators, which is based on Kane’s method, where partial angular velocities and partial linear velocities are represented as lines or screws. The modeling procedures given in the preliminaries will later be used to model a coupled crane/vessel system.
Equations of Motion of a Vessel in Waves.
Definition and Properties of Screws.
In this section, the necessary background in screw theory is presented [10,24,25]. Screws are used in the following sections to formulate the equations of motion with Kane’s method and to calculate the reaction forces and moments. The use of screws gives a clear geometric interpretation and transformation rules that are useful in the derivation.
Lines and Screws.
Equation of Motion of a Serial Manipulator.
Kinematics of a Crane/Vessel System.
We consider a mechanical system given in Fig. 1. The system is a knuckleboom crane with a pitch and roll compensation platform mounted on a vessel in wave motion. The system is described as a serial-link mechanism with eight links, representing the rigid bodies of the system. Body 0 is the vessel, bodies 1 and 2 describe the motion compensation platform, body 3 is the crane king, body 4 is the first boom, body 5 is the second boom, and bodies 6 and 7 describe the payload. Relative motion between bodies i and i + 1 is modeled with single degree-of-freedom joints. Each body i has a body-fixed frame i and a mass mi. The COG of body i for i = 1…7 is located at a distance di from the origin of frame i.
Equations of Motion of a Crane/Vessel System.
In this section, we present the procedure for determination of reaction forces (we will, in general, refer to both forces and moments by just writing forces) between the vessel and the motion compensation platform (i.e., in joint 1), as well as between the platform and the crane king (i.e., in joint 3).
The dynamical model of the marine vessel with the deck crane and the procedure for the determination of reaction forces are implemented numerically in this section. The lengths of the crane bodies are given in Table 1. The COG distances for the crane bodies are defines as di = li/2 for i = 1…6 and d7 = l7. The masses of the crane bodies are given in Table 2.
The implemented vessel model is the same supply vessel model as used in Ref. . The main vessel dimensions are as follows: the length between perpendiculars is 82.5 m, the breadth is 8.0 m, and the draught is 6.0 m. The mass of the vessel is 6362 mT. The hydrodynamic coefficients, force RAOs, parameters of the radiation force model, and rigid body mass matrix of the vessel were taken from the marine systems simulator . The wave parameters considered in the simulations are significant wave height Hs = 5 m and peak frequency ωp = 1.26 rad/s.
The vessel is initialized at , and a proportional-derivative (PD) controller is implemented for surge, sway, and yaw control, where the control task is to keep the controlled DOFs at zero. The crane is initialized at q3 = −90 deg, q4 = −45 deg, q5 = −90 deg, q6 = −45 deg, and q7 = 0 deg. We have implemented a PD controller with gravity compensation to keep qi for i = 3, 4, 5 at its initial values, and the pendulum DOFs q6, q7 are unactuated.
The roll and pitch compensation platform is initialized at q1 = 0 deg and q2 = 0 deg. Two control strategies are assumed in this paper, where in both, a PD controller is implemented. The first strategy is when the motion platform is controlled to stay horizontal (mode “MC on”). The second strategy is when the motion platform is controlled to stay parallel to the vessel deck (mode “MC off”). The second control strategy is used to simulated the absence of motion compensation for the comparison purposes.
Two simulations are run with two different crane locations on the vessel deck; see Fig. 2. In location I, the crane is placed in the origin of frame 0, which is in the middle of the deck seen in the transverse section. In location II, the crane is moved along the y0 axis by a distance of ec = 3.0 m.
Crane Location I.
The results of the numerical simulation of the system with the crane in location I (see Fig. 2) are presented in this section. The results in time series with an active roll and pitch compensation platform (mode “MC on”) are shown as solid lines, while the results with a locked roll and pitch compensation platform (mode “MC off”) are shown as dashed lines. Roll, pitch, and heave time histories are shown in Figs. 3 and 4, while surge, sway, and yaw are close to zero throughout the simulation, and those graphs are omitted. The time histories for DOFs of the roll and pitch compensation platform are shown in Fig. 5. The time histories for the payload angles θ1 and θ2 (see Fig. 6) are shown in Figs. 7 and 8. The reaction moments and reaction forces on platform from the deck given in the coordinates of frame 1 are shown in Figs. 9–11.
Crane Location II.
The results of the numerical simulation of the system with the crane in location II (see Fig. 2) are presented in this section. The results in time series with an active roll and pitch compensation platform (mode “MC on”) are shown as solid lines, while the results with a locked roll and pitch compensation platform (mode “MC off”) are shown as dashed lines. Roll, pitch, and heave time histories are shown in Figs. 15 and 16, while surge, sway, and yaw are close to zero throughout the simulation, and those graphs are omitted. The time histories for DOFs of the roll and pitch compensation platform are shown in Fig. 17. The time histories for the payload angles θ1 and θ2 (see Fig. 6) are shown in Figs. 18 and 19. The reaction moments and reaction forces on platform from the deck are shown in Figs. 20–22.
Discussion of Results
In this section, we discuss the results that were presented in Sec. 5.
The comparison of the maximum values of payload angles throughout the simulation for crane locations I and II are given in Figs. 26 and 27. The active roll and pitch motion compensation platform resulted in reduction of the pendulum sway of 66.5% for θ1 and 99.7% for θ2 in crane location I. Reduction of the crane pendulum sway in location II for θ1 and θ2 is accordingly 68.5% and 97.7%.
The relative comparison of the normalized maximum values of the reaction forces and moments on the platform from the vessel deck for crane locations I and II are given in Figs. 28 and 29. The relative comparison of the normalized maximum values of the reaction forces and moments on the crane king from the platform for crane locations I and II are given in Figs. 30 and 31. Reduction of the reaction moment and force components due to the active roll and pitch compensation platform is summed up in Tables 3 and 4.
The active roll and pitch compensation platform provided significant reduction in the maximum magnitudes of the payload sway angles, reaction forces, and reaction moments. The reduction of the payload sway angles improves the operational weather window of the cranes, while the reduction of the reaction forces and moments leads to the benefits in structural design and fatigue lifetime.
We have presented a procedure for dynamic modeling of a coupled crane and vessel system when the vessel is moving in waves. We have included the case when a roll and pitch compensation platform is installed between the vessel and the crane. The kinematics of the model were derived by representing velocities and angular velocities of the bodies as twists and by representing partial velocities and partial angular velocities as lines in Plücker coordinates. In addition, we have presented a procedure for the determination of reaction forces in the deck/platform and platform/crane interfaces. The reaction forces were determined from algebraic relations, which were conveniently derived by representing the unknown reaction forces as wrenches. Since wrenches are screws, screw transformations were also used in the derivations.
The presented model was implemented, and the results of numerical simulations were provided. The analysis was carried out for two crane locations: in the middle of the deck and closer to the starboard. The simulation results were used to study the efficiency of a roll and pitch compensation platform installed between a crane and a vessel. The efficiency was evaluated in terms of the magnitude of the determined reaction forces and in terms of the magnitude of the payload sway angles. It was shown that the compensation of roll and pitch angles, such that the base of the crane stays horizontal, leads to significant reduction of both reaction forces and payload sway angles. The simulation results demonstrated that the proposed model can be used to determine dynamical forces for structural and fatigue analyses of the crane/vessel interface.
Further development of this work could be implementation of a crane/platform control system for damping out the payload oscillations. In addition, the crane model can be extended with a payload hoisting degree-of-freedom.
Norwegian Research Council, SFI Offshore Mechatronics, Project No. 237896