Determination of Reaction Forces of a Deck Crane in Wave Motion Using Screw Theory

Cibicik, Andrej; Tysse, Geir O.; Egeland, Olav

doi:10.1115/1.4043701

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.

Introduction

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. [1] 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. [2], where the authors used an object-oriented modeling approach and co-simulation with the functional mock-up interface [3].

The combined dynamics of a ship and a deck crane were derived in Ref. [4], 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. [5], where the equations of motion of a spacecraft-manipulator system were derived using Kane’s equation of motion. Kane’s equations of motion [6] 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 [6], and it was used in a very detailed derivation of the equations of motion for a Stanford manipulator in Ref. [7]. An alternative formulation of Kane’s equations of motion for serial-link mechanisms was achieved by Angeles and Ma [8], 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. [9], 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 [10], 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 [13], 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 [6]. 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.

Preliminaries

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.

In this section, the mathematical model of a ship moving in waves is presented. This is done by the well-established method where force RAOs are used to calculate the wave forces on the ship and the waves are described with the JONSWAP wave spectrum [9,17]. The JONSWAP wave spectrum describes the distribution of waves in the North Sea and is given by

S (ω) = 0.2053 {H_{s}}^{2} \frac{{ω_{p}}^{4}}{ω^{5}} \exp [- \frac{5}{4} {(\frac{ω_{p}}{ω})}^{4}] γ^{Y}

(1)

where H_s is the significant wave height, which is an approximation of the mean peak-to-peak wave height of the highest one-third of the waves, ω_p is the peak wave frequency, γ = 3.3 and

Y = \exp [- \frac{(ω - ω_{p})^{2}}{2 σ^{2} ω_{p}^{2}}], σ = {\begin{cases} 0.07, & if ω \leq ω_{p} \\ 0.09, & if ω > ω_{p} \end{cases}

(2)

The wave spectrum S(ω) is approximated with a discrete wave spectrum S(ω_i) where ω_i, i = 1… N, is selected randomly in the interval

[{\bar{ω}}_{i} - Δ ω / 2, {\bar{ω}}_{i} + Δ ω / 2]

⁠, where the frequency range is divided into N equal intervals Δω with a mean value

{\bar{ω}}_{i}

in each interval.

Short crested irregular waves are modeled by discretizing the direction χ of the waves into M equal intervals Δχ with a mean value

{\bar{χ}}_{j}

of each interval. Then, the discrete angles χ_j, j = 1… M, are selected randomly in each interval

[{\bar{χ}}_{j} - Δ χ / 2, {\bar{χ}}_{j} + Δ χ / 2]

⁠, and the spreading function is set to

D (χ_{j}) = {\begin{matrix} \frac{2}{π} \cos^{2} (χ_{j}), & - \frac{π}{2} < χ_{j} - χ_{0} < \frac{π}{2} \\ 0, & otherwise \end{matrix}

(3)

where χ₀ is the dominant wave direction. The resultant wave elevation for each frequency ω_i is

ζ_{i} (t) = \sum_{j = 1}^{M} Z_{i j} \cos (ω_{i} t + ϵ_{i j})

(4)

where

ϵ_{i j}

is a random phase angle and

Z_{i j} (t) = \sqrt{2 D (χ_{j}) S (ω_{i}) Δ ω Δ χ}

(5)

The wave forces acting on the vessel can then be calculated using the force RAOs. The force RAO F_k(ω, χ) for degree-of-freedom k is a transfer function, which is given in terms of its amplitude

| F (ω, χ) |

and phase

∠ F (ω, χ)

⁠, which are calculated from the geometry of the hull. The wave forces in the degree-of-freedom k are then determined by

τ_{w, k} = \sum_{i = 1}^{N} \sum_{j = 1}^{M} | F_{k} (ω_{i}, χ_{j}) | Z_{i j} \cos [ω_{i} t + μ_{i j}]

(6)

where

μ_{i j} = ϵ_{i j} + ∠ F_{k} (ω_{i}, χ_{j})

⁠. The equation of motion of a vessel in waves expressed in the coordinates of the vessel-fixed frame (frame 0) is given by

\begin{aligned} M_{0, A} {\dot{ν}}_{0}^{0} + D ν_{0}^{0} + {\bar{C}}_{r} x + G η_{0}^{n} & = τ_{w} + τ_{t h r} \\ \dot{x} & = {\bar{A}}_{r} x + {\bar{B}}_{r} ν_{0}^{0} \end{aligned}

(7)

where

τ_{w} = [τ_{w, 1} \dots τ_{w, 6}]^{T}

is the vector of forces due to the force RAOs, M_0,A = M₀ + A(∞), and D = B(∞). The matrices

{\bar{A}}_{r}

⁠,

{\bar{B}}_{r}

⁠,

{\bar{C}}_{r}

and the vector x are the terms from the state-space model of the fluid memory effect [21]. The term M₀ is the associated mass matrix, A(ω) is the frequency-dependent added mass matrix, B(ω) is the frequency-dependent damping matrix, and G is the restoring force matrix [22,23]. The vector

ν_{0}^{0}

is a six-dimensional vector including the linear velocity of the origin of frame 0 and the angular velocity of the vessel expressed in the vessel-fixed frame

ν_{0}^{0} = [\begin{matrix} v_{0}^{0} \\ ω_{0}^{0} \end{matrix}]

(8)

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.

Consider two coordinate frames i and j, and let

R_{j}^{i}

be the rotation matrix from frame i to frame j, which satisfies

(R_{j}^{i})^{- 1} = (R_{j}^{i})^{T} = R_{i}^{j}

⁠. Let u^j be a vector given in the coordinates of j. Then, the same vector given in the coordinates of frame i will be

u^{i} = R_{j}^{i} u^{j}

⁠. Let

\hat{a}

denote the skew-symmetric matrix form of a vector a = [a₁ a₂ a₃]^T, so that

\hat{a} = [\begin{matrix} 0 & - a_{3} & a_{2} \\ a_{3} & 0 & - a_{1} \\ - a_{2} & a_{1} & 0 \end{matrix}]

(9)

Then,

\hat{a} b = a \times b

for any vector b. It follows that

{\hat{u}}^{i} = R_{j}^{i} {\hat{u}}^{j} R_{i}^{j}

[25]. Consider the screw

s_{/ B}^{j} = [\begin{matrix} u^{j} \\ w^{j} \end{matrix}]

(10)

which is an ordered pair of vectors u^j and w^j given in the coordinates of j and referenced to the point B. The screw will satisfy the screw transformation

s_{/ A}^{i} = U_{A B}^{i} {\bar{R}}_{j}^{i} s_{/ B}^{j} = V_{A B}^{i j} s_{/ B}^{j}

(11)

where

s_{/ A}^{i}

is given in the coordinates of i and is referenced to the point A. The screw transformation

V_{A B}^{i j}

is given in terms of screw reference transformation matrix

U_{A B}^{i}

from B to A and the screw coordinate transformation matrix

{\bar{R}}_{j}^{i}

⁠, where

U_{A B}^{i} = [\begin{matrix} I & 0 \\ {\hat{p}}_{A B}^{i} & I \end{matrix}], {\bar{R}}_{j}^{i} = [\begin{matrix} R_{j}^{i} & 0 \\ 0 & R_{j}^{i} \end{matrix}]

(12)

Here,

p_{A B}^{i}

is the vector from A to B. It is noted that

U_{A B}^{i} {\bar{R}}_{j}^{i} = {\bar{R}}_{j}^{i} U_{A B}^{j}

⁠, which gives the alternative expressions

V_{A B}^{i j} = [\begin{matrix} R_{j}^{i} & 0 \\ {\hat{p}}_{A B}^{i} R_{j}^{i} & R_{j}^{i} \end{matrix}] = [\begin{matrix} R_{j}^{i} & 0 \\ R_{j}^{i} {\hat{p}}_{A B}^{j} & R_{j}^{i} \end{matrix}]

(13)

Lines and Screws.

A line can be described by the screw

L_{/ A}^{i} = [\begin{matrix} a^{i} \\ m_{/ A}^{i} \end{matrix}]

(14)

which is given in the coordinates of i and referenced to the point A. Here, aⁱ is the unit direction vector of the line in the coordinates of i and

m_{/ A}^{i} = {\hat{p}}_{A}^{i} a^{i}

is the moment of the direction vector aⁱ about the reference point A, where

p_{A}^{i}

is the vector from A to a point on the line. The elements of the line

L_{/ A}^{i}

are the Plücker coordinates of the line [10]. It is noted that the line

L_{/ A}^{i}

is a screw, and it satisfies the screw transformations . Moreover, if the line is through the reference point A, then

p_{A}^{i} = 0

and

L_{/ A}^{i} = [(a^{i})^{T} 0^{T}]^{T}

⁠.

Consider the screw

s_{/ A}^{i} = σ L_{/ A}^{i} + σ [\begin{matrix} 0 \\ h a^{i} \end{matrix}] = σ [\begin{matrix} a^{i} \\ m_{/ A}^{i} + h a^{i} \end{matrix}]

(15)

where σ is the magnitude of the screw and h is the pitch. The line

L_{/ A}^{i}

is said to be the screw axis of

s_{/ A}^{i}

⁠. It is seen that if σ = 1 and h = 0, then

s_{/ A}^{i} = L_{/ A}^{i}

⁠. A line is therefore said to be a screw with zero pitch. Moreover, if h → ∞ and σ = 1/h, then

s_{/ A}^{i} \to N_{/ A}^{i}

⁠, where

N_{/ A}^{i} = [\begin{matrix} 0 \\ a^{i} \end{matrix}]

(16)

The screw

N_{/ A}^{i}

is therefore called a screw with infinite pitch along the unit vector aⁱ.

Twists.

Let the displacement from frame i to frame j be described by a translation of the origin given by a vector

p_{i j}^{i}

followed by a rotation

R_{j}^{i}

about the translated origin. This means that the displacement is given by the homogeneous transformation matrix

T_{j}^{i} = [\begin{matrix} R_{j}^{i} & p_{i j}^{i} \\ 0^{T} & 1 \end{matrix}]

(17)

The time derivative of the homogeneous transformation matrix is

{\dot{T}}_{j}^{i} = [\begin{matrix} {\hat{ω}}_{i j}^{i} R_{j}^{i} & v_{i j}^{i} \\ 0^{T} & 0 \end{matrix}]

(18)

where

v_{i j}^{i}

is the velocity of frame j relative to frame i in the coordinates of i and

ω_{i j}^{i}

is the angular velocity of j relative to i in the coordinates of i. The time derivative can be written as follows [25]:

{\dot{T}}_{j}^{i} = {\hat{t}}_{i j / i}^{i} T_{j}^{i} = T_{j}^{i} {\hat{t}}_{i j / j}^{j}

(19)

where

{\hat{t}}_{i j / i}^{i} = [\begin{matrix} {\hat{ω}}_{i j}^{i} & v_{i j}^{i} + {\hat{p}}_{i j}^{i} ω_{i j}^{i} \\ 0^{T} & 0 \end{matrix}] and {\hat{t}}_{i j / j}^{j} = [\begin{matrix} {\hat{ω}}_{i j}^{j} & v_{i j}^{j} \\ 0^{T} & 0 \end{matrix}]

(20)

are the matrix forms of the screws

t_{i j / i}^{i} = [\begin{matrix} ω_{i j}^{i} \\ v_{i j}^{i} + {\hat{p}}_{i j}^{i} ω_{i j}^{i} \end{matrix}] and t_{i j / j}^{j} = [\begin{matrix} ω_{i j}^{j} \\ v_{i j}^{j} \end{matrix}]

(21)

The term

t_{i j / i}^{i}

is called the twist of frame j relative to frame i referenced to the origin of frame i, and given in the coordinates of i, while

t_{i j / j}^{j}

is the same twist referenced to the origin of frame j and given in the coordinates of j. It is noted that a twist is a screw, and it satisfies the screw transformation . From Eq. , it is seen that twists in the matrix form are transformed according to

{\hat{t}}_{i j / j}^{j} = (T_{j}^{i})^{- 1} {\hat{t}}_{i j / i}^{i} T_{j}^{i}

(22)

Note that this transformation is only valid when the twist is given in the coordinates of the reference frame. Consider the composite displacement

T_{k}^{i} = T_{j}^{i} T_{k}^{j}

⁠. It follows from Eq. that

{\hat{t}}_{i j / j}^{j} = (T_{j}^{i})^{- 1} {\dot{T}}_{j}^{i} and {\hat{t}}_{j k / k}^{k} = (T_{k}^{j})^{- 1} {\dot{T}}_{k}^{j}

(23)

which are the matrix forms of the twists

t_{i j / j}^{j} = [\begin{matrix} ω_{i j}^{j} \\ v_{i j / j}^{j} \end{matrix}] and t_{j k / k}^{k} = [\begin{matrix} ω_{j k}^{k} \\ v_{j k / k}^{k} \end{matrix}]

(24)

The twist of the composite displacement

t_{i k / k}^{k} = [\begin{matrix} ω_{i k}^{k} \\ v_{i k / k}^{k} \end{matrix}]

(25)

is given in the matrix form as

{\hat{t}}_{i k / k}^{k} = (T_{k}^{i})^{- 1} {\dot{T}}_{k}^{i}

⁠, which gives

{\hat{t}}_{i k / k}^{k} = (T_{k}^{j})^{- 1} (T_{j}^{i})^{- 1} {\dot{T}}_{j}^{i} {\dot{T}}_{k}^{j} + (T_{k}^{j})^{- 1} {\dot{T}}_{k}^{j} = {\hat{t}}_{i j / k}^{k} + {\hat{t}}_{j k / k}^{k}

(26)

where Eqs. and are used. It follows that

t_{i k / k}^{k} = t_{i j / k}^{k} + t_{j k / k}^{k}

(27)

which means that the twist of a composite displacement is the sum of the twists of the individual displacements when the twists are referenced to the same point and are given in the coordinates of the same frame.

Wrenches.

A wrench is a screw representation of forces and torques. The wrench with a line of action through the origin of frame i referenced to i and given in the coordinates of i is given as follows:

w_{i / i}^{i} = [\begin{matrix} f_{i}^{i} \\ n_{i / i}^{i} \end{matrix}]

(28)

where

f_{i}^{i}

is a force and

n_{i / i}^{i}

is a torque. Consider the lines

\begin{aligned} L_{x_{i} / i}^{i} & = {[\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} \\ L_{y_{i} / i}^{i} & = {[\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} \\ L_{z_{i} / i}^{i} & = {[\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 \end{matrix}]}^{T} \end{aligned}

(29)

referenced to the origin of frame i and given in the coordinates of i. The lines are through the origin of frame i,

L_{x_{i} / i}^{i}

is along the x_i axis,

L_{y_{i} / i}^{i}

is along the y_i axis, and

L_{z_{i} / i}^{i}

is along the z_i axis. Next, consider the screws with infinite pitch

\begin{aligned} N_{x_{i} / i}^{i} & = {[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]}^{T} \\ N_{y_{i} / i}^{i} & = {[\begin{matrix} 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}]}^{T} \\ N_{z_{i} / i}^{i} & = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]}^{T} \end{aligned}

(30)

referenced to the origin of frame i and given in the coordinates of i. Then, the wrench associated with the three orthogonal forces and the three orthogonal torques can be written as a sum as follows:

w_{i / i}^{i} = \sum_{j = 1}^{3} (L_{a_{j} / i}^{i} ρ_{f_{j}} + N_{a_{j} / i}^{i} ρ_{m_{j}})

(31)

where a_j for j = 1, 2, 3 stands for the axes x_i, y_i, z_i,

ρ_{f_{j}}

are the magnitudes of the forces and

ρ_{m_{j}}

are the magnitudes of the torques in x_i, y_i, z_i directions. The expression can alternatively be written in the matrix form as follows:

w_{i / i}^{i} = L_{i / i}^{i} ρ

(32)

where the columns of the matrix

L_{i / i}^{i}

are the screws in Eq. , so that

L_{i / i}^{i} = [L_{x_{i} / i}^{i} L_{y_{i} / i}^{i} L_{z_{i} / i}^{i} N_{x_{i} / i}^{i} N_{y_{i} / i}^{i} N_{z_{i} / i}^{i}]

(33)

and ρ is the vector of the magnitudes of forces and torques

ρ = {[ρ_{f_{1}} ρ_{f_{2}} ρ_{f_{3}} ρ_{m_{1}} ρ_{m_{2}} ρ_{m_{3}}]}^{T}

(34)

Since all the columns of Eq. are screws, it follows that the matrix

L_{i / i}^{i}

will satisfy the screw transformation. The matrix

L_{i / i}^{i}

can be referenced to the center of gravity (COG) of body j and expressed in the coordinates of j by

L_{i / m_{j}}^{j} = V_{m_{j}, i}^{j i} L_{i / i}^{i}

(35)

where

V_{m_{j}, i}^{j i}

is a screw transformation matrix. The wrench referenced to the COG of body j and given in the coordinates of j can then be found by

w_{i / m_{j}}^{j} = L_{i / m_{j}}^{j} ρ

(36)

Suppose that the wrench

w_{i / m_{j}}^{j}

is acting on a rigid body i that is moving with a twist

t_{n i / m_{j}}^{j}

⁠. Then, the power of the wrench on the twist is

P = {(w_{i / m_{j}}^{j})}^{T} Π t_{n i / m_{j}}^{j}

(37)

where Π is the interchange operator [10] defined by

Π = [\begin{matrix} 0 & I \\ I & 0 \end{matrix}]

(38)

If

{(w_{i / m_{j}}^{j})}^{T} Π t_{n i / m_{j}}^{j} = 0

⁠, the twist and the wrench are said to be reciprocal.

Dynamic Modeling

Equation of Motion of a Serial Manipulator.

Dynamics of a crane can be modeled in the same manner as dynamics of a robotic manipulator arm. Therefore, in this section, we present the detailed procedure for deriving the equations of motion for robotic manipulator arms, which is based on Ref. [14]. The equation of motion of a rigid link i is derived using the Newton–Euler approach. We describe dynamics about the COG and formulate equations in a convenient matrix form

[\begin{matrix} M_{m_{i}}^{i} {\dot{ω}}_{n i}^{i} + {\hat{ω}}_{n i}^{i} M_{m_{i}}^{i} ω_{n i}^{i} - n_{i / m_{i}}^{i} - n_{i / m_{i}}^{i (c)} \\ m_{i} ({\dot{v}}_{n i / m_{i}}^{i} + {\hat{ω}}_{n i}^{i} v_{n i / m_{i}}^{i}) - f_{m_{i}}^{i} - f_{m_{i}}^{i (c)} \end{matrix}] = 0

(39)

where

v_{n i / m_{i}}^{i}

is the linear velocity of the COG of link i relative to the inertial frame,

ω_{n i}^{i}

is the angular velocity of link i relative to the inertial frame, and

M_{m_{i}}^{i}

is the inertia tensor of link i. The external force

f_{m_{i}}^{i}

and external moment

n_{i / m_{i}}^{i}

are applied at the COG of link i,

f_{m_{i}}^{i (c)}

⁠, and

n_{i / m_{i}}^{i (c)}

are the forces and moments of constraint. All terms are given in the coordinates of frame i, which is fixed in link i.

The twist of link i relative to the inertial frame n and expressed in the coordinates of i is given by

t_{n i / i}^{i} = [\begin{matrix} ω_{n i}^{i} \\ v_{n i / i}^{i} \end{matrix}]

(40)

where the twist is referenced to the origin of frame i and is given in the coordinates of i. This means that

v_{n i / i}^{i}

is the velocity of the origin of frame i. This twist can be transformed, so that it is referenced to the COG point m_i of link i with the transformation

t_{n i / m_{i}}^{i} = U_{m_{i}, i}^{i} t_{n i / i}^{i} = [\begin{matrix} ω_{n i}^{i} \\ v_{n i / m_{i}}^{i} \end{matrix}]

(41)

where

v_{n i / m_{i}}^{i}

is the velocity of the COG of link i. The matrix

U_{m_{i}, i}^{i}

is the screw reference transformation matrix

U_{m_{i}, i}^{i} = [\begin{matrix} I & 0 \\ {\hat{p}}_{m_{i}, i}^{i} & I \end{matrix}]

(42)

where

p_{m_{i}, i}^{i}

is the vector from the COG of link i to the origin of frame i in the coordinates of i. In the same way, the twist can be referenced to the origin of frame i + 1, which is fixed in link i + 1, through the transformation

t_{n i / i + 1}^{i} = U_{i + 1, i}^{i} t_{n i / i}^{i}

(43)

The twist can be expressed in the coordinates of frame i + 1 by the screw coordinate transformation

t_{n i / i + 1}^{i + 1} = {\bar{R}}_{i}^{i + 1} t_{n i / i + 1}^{i}

(44)

where the screw coordinate transformation matrix is

{\bar{R}}_{i}^{i + 1} = [\begin{matrix} R_{i}^{i + 1} & 0 \\ 0 & R_{i}^{i + 1} \end{matrix}]

(45)

and

R_{i}^{i + 1}

is the orthogonal rotation matrix from i + 1 to i. The screw transformation for the twist to be referenced to the COG of link i + 1 and to be given in the coordinates of i + 1 can then be written as follows:

t_{n i / m_{i + 1}}^{i + 1} = U_{m_{i + 1}, i + 1}^{i + 1} {\bar{R}}_{i}^{i + 1} U_{i + 1, i}^{i} t_{n i / i}^{i} = V_{m_{i + 1}, i}^{i + 1, i} t_{n i / i}^{i}

(46)

where the screw transformation matrix is

V_{m_{i + 1}, i}^{i + 1, i} = [\begin{matrix} R_{i}^{i + 1} & 0 \\ {\hat{p}}_{m_{i + 1}, i + 1}^{i + 1} R_{i}^{i + 1} + R_{i}^{i + 1} {\hat{p}}_{i + 1, i}^{i} & R_{i}^{i + 1} \end{matrix}]

(47)

The twist of link i + 1 relative to frame n can be calculated recursively from the twist of link i according to

t_{n, i + 1 / m_{i + 1}}^{i + 1} = t_{n i / m_{i + 1}}^{i + 1} + t_{i, i + 1 / m_{i + 1}}^{i + 1}

(48)

Note that the twists in the recursive calculation scheme must be referenced to the same point and expressed in the coordinates of the same frame.

The links are connected with joints of a single rotational degree-of-freedom. The axis of rotation of the joint between link i − 1 and link i passes through the origin of frame i. This means that the twist of link i relative to link i − 1 will be given as a rotation about a line that passes through the origin of frame i. Then, the twist of frame i relative to i − 1 with the reference to i and given in the coordinates of i is

t_{i - 1, i / i}^{i} = u_{i} L_{i / i}^{i}

(49)

where the line

L_{i / i}^{i}

is the line of joint i and u_i is the generalized speed of joint i. The line of the joint j

L_{j / m_{i}}^{i}

referenced to the COG of link i and expressed in the coordinates of i is

L_{j / m_{i}}^{i} = V_{m_{i}, j}^{i j} L_{j / j}^{j}

(50)

The projection matrix of each link in the system is defined as follows:

P_{i} = [\begin{matrix} L_{1 / m_{i}}^{i} & \dots & L_{i / m_{i}}^{i} & 0_{6 \times (n_{q} - i)} \end{matrix}]

(51)

where n_q is a number of links. The twist of link i relative to the inertial frame, referenced to the COG of link i and expressed in the coordinates of i, can be given as a sum as follows:

t_{n i / m_{i}}^{i} = \sum_{j = 1}^{i} u_{j} L_{j / m_{i}}^{i}

(52)

The twist can be expressed by the projection matrix as

t_{n i / m_{i}}^{i} = P_{i} u

(53)

where u is a vector of generalized speeds, where the elements u_i are given in Eq. . The derivative of is given by

{\dot{t}}_{n i / m_{i}}^{i} = P_{i} \dot{u} + {\dot{P}}_{i} u

(54)

where the derivative of projection matrix

{\dot{P}}_{i}

with respect to time is defined as

{\dot{P}}_{i} = [{\begin{matrix} \dot{L} \end{matrix}}_{1 / m_{i}}^{i} \dots {\dot{L}}_{i / m_{i}}^{i} 0_{6 \times (n_{q} - i)}]

(55)

The derivative of joint j line referenced to the COG of link i and expressed in the coordinates of i is defined as

{\dot{L}}_{j / m_{i}}^{i} = {\bar{d}}_{j i}^{i} L_{j / m_{i}}^{i}

(56)

where

{\bar{d}}_{j i}^{i}

is the differentiation operator defined as

{\bar{d}}_{j i}^{i} = [\begin{matrix} {\hat{ω}}_{i j}^{i} & 0 \\ - {\hat{υ}}_{m_{i}, j}^{i} & {\hat{ω}}_{i j}^{i} \end{matrix}]

(57)

The skew-symmetric auxiliary velocity term

{\hat{υ}}_{m_{i}, j}^{i}

is defined in a vector form as

υ_{m_{i}, j}^{i} = {\hat{ω}}_{i j}^{i} p_{m_{i}, i}^{i} + \sum_{k = 1}^{i - j - 1} {\hat{ω}}_{i - k, j}^{i} p_{i - k + 1, i - k}^{i}

(58)

where

p_{m_{i}, i}^{i}

is a distance vector from the link COG point m_i to the origin of frame i,

p_{i - k + 1, i - k}^{i}

is a distance vector from frame i − k + 1 to frame i − k, and

{\hat{ω}}_{i j}^{i}

is a skew-symmetric form of the angular velocity of frame j with respect to frame i, all expressed in the coordinates of i. The equations of motion for link i can be written as follows:

D_{i} {\dot{t}}_{n i / m_{i}}^{i} + W_{i} D_{i} t_{n i / m_{i}}^{i} = Π w_{i / m_{i}}^{i (c)} + Π w_{i / m_{i}}^{i}

(59)

where

w_{i / m_{i}}^{i (c)}

and

w_{i / m_{i}}^{i}

are the wrenches of constraint and external forces, respectively. The mass matrix of link i and the auxiliary skew-symmetric matrix W_i are given by

D_{i} = [\begin{matrix} M_{m_{i}}^{i} & 0 \\ 0 & m_{i} I \end{matrix}] and W_{i} = [\begin{matrix} {\hat{ω}}_{n i}^{i} & 0 \\ 0 & {\hat{ω}}_{n i}^{i} \end{matrix}]

(60)

where

M_{m_{i}}^{i}

is the inertia tensor at the COG of link i, m_i is the mass of link i, and

{\hat{ω}}_{n i}^{i}

is the skew-symmetric form of the angular velocity vector taken from the first three rows of

t_{n i / m_{i}}^{i}

⁠. According to the D’Alembert’s principle and the principle of virtual work, constraint forces are cancelled from the equation of motion of the whole multibody system

\sum_{i} P_{i}^{T} Π w_{i / m_{i}}^{i (c)} = 0

(61)

The vector of generalized external forces is given by

τ_{s} = \sum_{i} P_{i}^{T} Π w_{i / m_{i}}^{i}

(62)

The equation of motion of the entire multibody system is then formulated as

\sum_{i} P_{i}^{T} (D_{i} {\dot{t}}_{n i / m_{i}}^{i} + W_{i} D_{i} t_{n i / m_{i}}^{i}) = τ_{s}

(63)

where the substitution of Eqs. and leads to the new formulation in a matrix form

M_{s} \dot{u} + C_{s} u = τ_{s}

(64)

where the mass and Coriolis force matrices are

\begin{aligned} M_{s} & = & \sum_{i} P_{i}^{T} D_{i} P_{i} \\ C_{s} & = & \sum_{i} P_{i}^{T} [D_{i} {\dot{P}}_{i} + W_{i} D_{i} P_{i}] \end{aligned}

(65)

It is noted that the mass and Coriolis force matrices have the property that

(\dot{M} - 2 C)

is a skew-symmetric matrix [26].

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 m_i. The COG of body i for i = 1…7 is located at a distance d_i from the origin of frame i.

Fig. 1

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A knuckleboom crane with a base motion compensation platform mounted on a vessel

The position and orientation of frame 0 relative to frame n (the inertial frame) is given by the vector

η_{0}^{n} = [x_{0} y_{0} z_{0} ϕ θ ψ]^{T}

(66)

where [x₀y₀z₀]^T is the position of the origin of frame 0, ϕ is the roll angle, θ is the pitch angle, and ψ is the yaw angle. The rotation matrix from frame n to frame 0 is

R_{0}^{n} = R_{z} (ψ) R_{y} (θ) R_{x} (ϕ)

(67)

where R_z(ψ) is the rotation matrix for rotation ψ about the z axis, R_y(θ) is the rotation matrix for rotation θ about the y axis, and R_x(ϕ) is the rotation matrix for rotation ϕ about the x axis. The rotation matrix from frame 0 to frame 1 is

R_{1}^{0} = R_{x} (π) R_{x} (q_{1})

⁠, where q₁ is the rotation angle about the x₁ axis. The rotation matrix from frame 1 to frame 2 is

R_{2}^{1} = R_{y} (q_{2})

⁠, where q₂ is the rotation angle about the y₂ axis. The rotation matrix from frame 2 to frame 3 is

R_{3}^{2} = R_{z} (q_{3})

⁠, where q₃ is the rotation angle about the z₃ axis. The rotation matrix from frame 3 to frame 4 is

R_{4}^{3} = R_{x} (\frac{π}{2}) R_{z} (\frac{π}{2}) R_{z} (q_{4})

⁠, where q₄ is the rotation angle about the z₄ axis. The rotation matrix from frame 4 to frame 5 is

R_{5}^{4} = R_{z} (q_{5})

⁠, where q₅ is the rotation angle about the z₅ axis. The rotation matrix from frame 5 to frame 6 is

R_{6}^{5} = R_{z} (q_{6})

⁠, where q₆ is the rotation angle about the z₆ axis. The rotation matrix from frame 6 to frame 7 is

R_{7}^{6} = R_{y} (q_{7})

⁠, where q₇ is the rotation angle about the y₇ axis.

The configuration of the system is given by the vector of generalized coordinates

q = {[\begin{matrix} (η_{0}^{n})^{T} & q_{1} & \dots & q_{7} \end{matrix}]}^{T}

(68)

and the vector of generalized speeds [6] is given by

u = {[\begin{matrix} (t_{n 0 / 0}^{0})^{T} & u_{1} & \dots & u_{7} \end{matrix}]}^{T}

(69)

where

t_{n 0 / 0}^{0}

is the twist of frame 0 relative to frame n referenced to the origin of frame 0 and expressed in the coordinates of frame 0 and

u_{i} = {\dot{q}}_{i}

⁠. The lines of the crane joints in the system are given as

\begin{aligned} L_{1 / 1}^{1} & = & {[\begin{matrix} 1 & 0 & 0 & 0_{1 \times 3} \end{matrix}]}^{T}, L_{2 / 2}^{2} & = & {[\begin{matrix} 0 & 1 & 0 & 0_{1 \times 3} \end{matrix}]}^{T}, \\ L_{3 / 3}^{3} & = & {[\begin{matrix} 0 & 0 & 1 & 0_{1 \times 3} \end{matrix}]}^{T}, L_{4 / 4}^{4} & = & {[\begin{matrix} 0 & 0 & 1 & 0_{1 \times 3} \end{matrix}]}^{T}, \\ L_{5 / 5}^{5} & = & {[\begin{matrix} 0 & 0 & 1 & 0_{1 \times 3} \end{matrix}]}^{T}, L_{6 / 6}^{6} & = & {[\begin{matrix} 0 & 0 & 1 & 0_{1 \times 3} \end{matrix}]}^{T}, \\ L_{7 / 7}^{7} & = & {[\begin{matrix} 0 & 1 & 0 & 0_{1 \times 3} \end{matrix}]}^{T} \end{aligned}

(70)

The twist

t_{n 0 / 0}^{0}

referenced to the origin of frame 0, relative to frame n, and expressed in the coordinates of frame 0 is

t_{n 0 / 0}^{0} = [\begin{matrix} ω_{n 0}^{0} \\ v_{n 0 / 0}^{0} \end{matrix}]

(71)

The twist of body 0 can be written as a sum as follows:

t_{n 0 / 0}^{0} = \sum_{j = 1}^{3} [L_{a_{j} / 0}^{0} (ω_{n 0}^{0})_{a_{j}} + N_{a_{j} / 0}^{0} (v_{n 0 / 0}^{0})_{a_{j}}]

(72)

where a_j for j = 1, 2, 3 stands for the axes x₀, y₀, and z₀, the terms

(ω_{n 0}^{0})_{a_{j}}

and

(v_{n 0 / 0}^{0})_{a_{j}}

are the projections of

ω_{n 0}^{0}

and

v_{n 0 / 0}^{0}

⁠, respectively, on a_j axis, while

L_{a_{j} / 0}^{0}

and

N_{a_{j} / 0}^{0}

are defined in Eqs. and . The sum can be written in a matrix form as follows:

t_{n 0 / 0}^{0} = L_{0 / 0}^{0} t_{n 0 / 0}^{0}

(73)

where the matrix

L_{0 / 0}^{0}

is

L_{0 / 0}^{0} = [\begin{matrix} L_{x_{0} / 0}^{0} & L_{y_{0} / 0}^{0} & L_{z_{0} / 0}^{0} & N_{x_{0} / 0}^{0} & N_{y_{0} / 0}^{0} & N_{z_{0} / 0}^{0} \end{matrix}]

(74)

Since the columns of the matrix are screws, then the matrix satisfies screw transformations. The matrix

L_{0 / 0}^{0}

can be referenced to the COG of body i and expressed in the coordinates of i by the screw transformation

L_{0 / m_{i}}^{i} = V_{m_{i}, 0}^{i 0} L_{0 / 0}^{0}

(75)

Similarly, the line of joint j

L_{j / m_{i}}^{i}

referenced to the COG of body i and expressed in the coordinates of i can be obtained by

L_{j / m_{i}}^{i} = V_{m_{i}, j}^{i j} L_{j / j}^{j}

as in Eq. . The projection matrix of body 0 is defined as

P_{0} = [\begin{matrix} L_{0 / 0}^{0} & 0_{6 \times n_{q}} \end{matrix}]

(76)

and the projection matrix of each body i for i = 1…7 is defined as

P_{i} = [\begin{matrix} L_{0 / m_{i}}^{i} & L_{1 / m_{i}}^{i} & \dots & L_{i / m_{i}}^{i} & 0_{6 \times (n_{q} - i)} \end{matrix}]

(77)

where n_q = 7 is the number of the crane and platform DOFs. The twist of body i referenced to the COG of body i and expressed in the coordinates of i can be given as a sum

t_{n i / m_{i}}^{i} = L_{0 / m_{i}}^{i} t_{n 0 / 0}^{0} + \sum_{j = 1}^{i} u_{j} L_{j / m_{i}}^{i}

(78)

The twist can be expressed by the projection matrix and the vector of generalized speeds as

t_{n i / m_{i}}^{i} = P_{i} u

⁠, as in Eq. . The derivative of Eq. with respect to time is then

{\dot{t}}_{n i / m_{i}}^{i} = P_{i} \dot{u} + {\dot{P}}_{i} u

⁠, as in Eq. , where the derivative of the projection matrix

{\dot{P}}_{0} = 0

and

{\dot{P}}_{i}

for i = 1…7 is defined as

{\dot{P}}_{i} = [\begin{matrix} {\dot{L}}_{0 / m_{i}}^{i} & {\dot{L}}_{1 / m_{i}}^{i} & \dots & {\dot{L}}_{i / m_{i}}^{i} & 0_{6 \times (n_{q} - i)} \end{matrix}]

(79)

The derivative of the matrix

L_{0 / m_{i}}^{i}

with respect to time is

{\dot{L}}_{0 / m_{i}}^{i} = {\bar{d}}_{0 i}^{i} L_{0 / m_{i}}^{i}

(80)

where

{\bar{d}}_{0 i}^{i}

is a differentiation operator . The derivative of the line

{\dot{L}}_{j / m_{i}}^{i}

is defined in Eq. .

Equations of Motion of a Crane/Vessel System.

We formulate the equation of motion for the whole system by summing up the equations of motion of separate bodies premultiplied with the transpose of projection matrices. The forces and moments of constraint are cancelled from the equation of motion for the whole system by D’Alambert’s principle and the principle of virtual work as shown in Eq. . The equations of motion of the vessel are defined in terms of

ν_{0}^{0}

; however, in this work, we use the twist

t_{n 0 / 0}^{0}

to represent velocities and angular velocities of the vessel. Therefore, some of the matrices in Eq. need to be rearranged before the vessel and crane models can be coupled into one system. The mass, Coriolis, and gravity force matrices are transformed as

D_{0} = Π M_{0, A} Π, W_{0} = Π D Π, G_{0} = Π G

(81)

where M_0,A, D, and G are defined in Eq. and Π is the interchange operator . The terms from the state-space model of the fluid memory effect are transformed as

A_{r} = {\bar{A}}_{r}, B_{r} = {\bar{B}}_{r} Π, C_{r} = Π {\bar{C}}_{r}

(82)

where

{\bar{A}}_{r}

⁠,

{\bar{B}}_{r}

⁠, and

{\bar{C}}_{r}

are also defined in Eq. .

The mass matrix of the whole system can be found by

M = \sum_{i = 0}^{7} P_{i}^{T} D_{i} P_{i}

(83)

where D₀ is given in and D_i for i = 1…7 is defined in Eq. . The matrix of centrifugal and Coriolis forces is

C = P_{0}^{T} [D_{0} {\dot{P}}_{0} + W_{0} P_{0}] + \sum_{i = 1}^{7} P_{i}^{T} [D_{i} {\dot{P}}_{i} + W_{i} D_{i} P_{i}]

(84)

where W₀ is given in Eq. and W_i for i = 1…7 is defined in Eq. . The vector of generalized gravitational and buoyancy forces is given by

τ_{g} = - P_{0}^{T} G_{0} η_{0}^{n} + \sum_{i = 1}^{7} P_{i}^{T} Π w_{i / m_{i}}^{i (g)}

(85)

where G₀ is the linearized gravitational and buoyancy force matrix given in Eq. , the wrench

w_{i / m_{i}}^{i (g)}

for i = 1…7 is generally defined as

w_{i / m_{i}}^{i (g)} = [\begin{matrix} R_{n}^{i} [\begin{matrix} 0 \\ 0 \\ m_{i} g \end{matrix}] \\ 0_{3 x 1} \end{matrix}]

(86)

The vector of generalized crane and motion compensation platform control forces is given by

τ_{c o n t} = \sum_{i = 0}^{5} P_{i}^{T} Π w_{i / m_{i}}^{i (c o n t)}

(87)

where

w_{i / m_{i}}^{i (c o n t)}

is a wrench of control inputs given in the coordinates of frame i and referenced to the origin of frame i for i = 0 and referenced to the COG of body i for i = 1…5. The equations of motion for the whole system can now be written as follows:

\begin{aligned} {\dot{η}}_{0}^{n} = J (η) u \\ M \dot{u} + C u + P_{0}^{T} C_{r} x = τ_{g} + P_{0}^{T} Π τ_{t h r} + P_{0}^{T} Π τ_{w} + τ_{c o n t} \\ \dot{x} = A_{r} x + B_{r} P_{0} u \end{aligned}

(88)

where A_r, B_r, and C_r are the auxiliary matrices from the state-space model of the fluid memory effect , τ_thr is the wrench of thruster control inputs, and τ_w is the wrench of wave forces estimated by Eq. . The term J(η) is the velocity transformation matrix [17].

Reaction Forces

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).

Define a vector of unknown magnitudes of the reaction forces

ρ_{c} = [\begin{matrix} Π ρ_{1} \\ Π ρ_{3} \end{matrix}]

(89)

where ρ₁ = [ρ₁₁…ρ₁₆]^T is a vector of magnitudes of the reaction forces in joint 1, ρ₃ = [ρ₃₁…ρ₃₆]^T is a vector of magnitudes of the reaction forces in joint 3, and Π is given in . In both cases, first three magnitudes are forces and last three magnitudes are moments. Auxiliary velocities and angular velocities can be expressed in terms of auxiliary twists, and then, partial auxiliary velocities and partial auxiliary angular velocities can be expressed as lines or screws. In the original formulation [6,14], the auxiliary velocities are first formulated and then partial differentiation is performed with respect to auxiliary generalized speeds. In the procedure presented in this paper, we do the derivation using lines and screws directly.

The wrench at the origin of frame 1 associated with the reaction forces and moments in joint 1 is given as

w_{1 / 1}^{1 (c)} = L_{c 1 / 1}^{1} ρ_{1}

(90)

where

L_{c 1 / 1}^{1}

is a set of joint lines and screws [27] given as follows:

L_{c 1 / 1}^{1} = [\begin{matrix} L_{x_{1} / 1}^{1} & L_{y_{1} / 1}^{1} & L_{z_{1} / 1}^{1} & N_{x_{1} / 1}^{1} & N_{y_{1} / 1}^{1} & N_{z_{1} / 1}^{1} \end{matrix}]

(91)

and

L_{/ 1}^{1}, N_{/ 1}^{1}

are defined in Eqs. and . Since columns of the matrix are screws, then the matrix satisfies screw transformations. The matrix

L_{c 1 / 1}^{1}

can be referenced to the COG of body i and expressed in the coordinates of frame i

L_{c 1 / m_{i}}^{i} = V_{m_{i}, 1}^{i 1} L_{c 1 / 1}^{1}

(92)

Analogically, the wrench at the origin of frame 3 associated with all the reaction forces and moments in joint 3 is given as

w_{3 / 3}^{3 (c)} = L_{c 3 / 3}^{3} ρ_{3}

(93)

where

L_{c 3 / 3}^{3}

is a set of joint lines and screws [27] given as

L_{c 3 / 3}^{3} = [\begin{matrix} L_{x_{3} / 3}^{3} & L_{y_{3} / 3}^{3} & L_{z_{3} / 3}^{3} & N_{x_{3} / 3}^{3} & N_{y_{3} / 3}^{3} & N_{z_{3} / 3}^{3} \end{matrix}]

(94)

and

L_{/ 3}^{3}

and

N_{/ 3}^{3}

are also defined in Eqs. and . The matrix

L_{c 3 / 3}^{3}

can be referenced to the COG of body i and expressed in the coordinates of frame i

L_{c 3 / m_{i}}^{i} = V_{m_{i}, 3}^{i 3} L_{c 3 / 3}^{3}

(95)

The auxiliary projection matrix of each body i in the system is defined as

\begin{aligned} {P_{c}}_{0} & = 0 \\ {P_{c}}_{i} & = [\begin{matrix} L_{c 1 / m_{i}}^{i} & 0_{6 \times 6} \end{matrix}], for i = 1, 2 \\ {P_{c}}_{i} & = [\begin{matrix} L_{c 1 / m_{i}}^{i} & L_{c 3 / m_{i}}^{i} \end{matrix}], for i \geq 3 \end{aligned}

(96)

The auxiliary vector of generalized gravitational and buoyancy forces is given by

τ_{f g} = - {P_{c}}_{0}^{T} G_{0} η_{0}^{n} + \sum_{i = 1}^{7} {P_{c}}_{i}^{T} Π w_{i / m_{i}}^{i (g)}

(97)

where the terms G₀,

η_{0}^{n}

⁠, and

w_{i / m_{i}}^{i (g)}

are the same as in Eq. . The auxiliary vector of generalized crane and motion compensation platform control forces is given by

τ_{f c o n t} = \sum_{i = 0}^{5} {P_{c}}_{i}^{T} Π w_{i / m_{i}}^{i (c o n t)}

(98)

where

w_{i / m_{i}}^{i (c o n t)}

is the same as in Eq. . The vector of the unknown magnitudes of the reaction forces can now be found by

\begin{aligned} ρ_{c} & = M_{f} \dot{u} + C_{f} u + {P_{c}}_{0}^{T} C_{r} x \\ - τ_{f g} - {P_{c}}_{0}^{T} τ_{t h r} - {P_{c}}_{0}^{T} τ_{w} - τ_{f c o n t} \end{aligned}

(99)

where the matrices M_f and C_f are defined as

M_{f} = \sum_{i = 0}^{7} {P_{c}}_{i}^{T} D_{i} P_{i}

(100)

and

C_{f} = {P_{c}}_{0}^{T} [D_{0} {\dot{P}}_{0} + W_{0} P_{0}] + \sum_{i = 1}^{7} {P_{c}}_{i}^{T} [D_{i} {\dot{P}}_{i} + W_{i} D_{i} P_{i}]

(101)

where D₀, D_i, W₀, and W_i are the same as in Eqs. and . Provided that P_c₀ = 0, then Eq. is simplified to

ρ_{c} = M_{f} \dot{u} + C_{f} u - τ_{f g} - τ_{f c o n t}

(102)

where

\dot{u}

and u are obtained from the simulation of Eq. .

Simulation Results

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 d_i = l_i/2 for i = 1…6 and d₇ = l₇. The masses of the crane bodies are given in Table 2.

Table 1

Crane dimensions (m)

Term	l₁	l₂	l₃	l₄	l₅	l₆	l₇
Value	0.01	3.0	10.0	10.0	8.0	0.01	5.0

Table 2

Crane masses (t)

Term	m₁	m₂	m₃	m₄	m₅	m₆	m₇
Value	0.01	40	70	40	30	0.01	20

The implemented vessel model is the same supply vessel model as used in Ref. [9]. 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 [19]. The wave parameters considered in the simulations are significant wave height H_s = 5 m and peak frequency ω_p = 1.26 rad/s.

The vessel is initialized at $η_{0}^{n} = [000000]^{T}$ ⁠, 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 q_i 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 e_c = 3.0 m.

Fig. 2

Two locations of the crane considered in simulations, the offset ec = 3.0 m along the y0 axis

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Two locations of the crane considered in simulations, the offset e_c = 3.0 m along the y₀ axis

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.

Fig. 3

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Roll and pitch angles for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode. The pitch data are superposed.

Fig. 4

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Heave of the vessel for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode. The lines are superposed.

Fig. 5

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DOFs of the motion compensation platform for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 6

Orientation of the payload is described by the rotation Rxn(θ1)Ry′(θ2) from the inertial frame

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Orientation of the payload is described by the rotation R_{x_n}(θ₁)R_y′(θ₂) from the inertial frame

Fig. 7

Payload orientation angle θ1 for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

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Payload orientation angle θ₁ for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 8

Payload orientation angle θ2 for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

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Payload orientation angle θ₂ for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 9

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Reaction moments on platform for crane location I. Solid lines show results in “MC on” mode dashed lines in “MC off” mode.

Fig. 10

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Reaction forces on platform for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 11

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Reaction force on platform for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

The reaction moments and reaction forces on the crane king from the platform given in the coordinates of frame 3 are shown in Figs. 12–14.

Fig. 12

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Reaction moments on crane for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 13

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Reaction forces on crane for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 14

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Reaction force on crane for crane location I. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

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.

Fig. 15

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Roll and pitch angles for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode. The pitch data are superposed.

Fig. 16

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Heave of the vessel for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode. The lines are superposed.

Fig. 17

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DOFs of the motion compensation platform for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 18

Payload orientation angle θ1 for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

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Payload orientation angle θ₁ for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 19

Payload orientation angle θ2 for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

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Payload orientation angle θ₂ for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 20

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Reaction moments on platform for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 21

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Reaction forces on platform for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 22

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Reaction force on platform for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

The reaction moments and reaction forces on the crane king from the platform are shown in Figs. 23–25.

Fig. 23

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Reaction moments on crane for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 24

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Reaction forces on crane for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

Fig. 25

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Reaction force on crane for crane location II. Solid lines show results in “MC on” mode and dashed lines in “MC off” mode.

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%.

Fig. 26

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Comparison of maximum values of payload orientation angles for crane location I

Fig. 27

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Comparison of maximum values of payload orientation angles for crane location II

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.

Fig. 28

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Relative comparison of reaction forces and moments on the platform for crane location I

Fig. 29

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Relative comparison of reaction forces and moments on the platform for crane location II

Fig. 30

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Relative comparison of reaction forces and moments on the crane for crane location I

Fig. 31

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Relative comparison of reaction forces and moments on the crane for crane location II

Table 3

Reduction of maximum values of reaction forces and moments on the platform due to active motion compensation, in (%)

Reaction	τ₇	$n_{y_{1}}$	$n_{z_{1}}$	$f_{x_{1}}$	$f_{y_{1}}$	$f_{z_{1}}$
Location I	29.3	99.7	88.6	76.5	77.4	0.0
Location II	42.2	97.7	90.0	81.5	84.2	0.0

Table 4

Reduction of maximum values of reaction forces and moments on the crane king due to active motion compensation, in (%)

Reaction	$n_{x_{3}}$	$n_{y_{3}}$	τ₉	$f_{x_{3}}$	$f_{y_{3}}$	$f_{z_{3}}$
Location I	99.6	24.5	99.8	76.3	99.2	0.0
Location II	97.8	36.3	97.4	83.3	97.8	0.0

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.

Conclusions

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.

Funding Data

Norwegian Research Council, SFI Offshore Mechatronics, Project No. 237896

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Determination of Reaction Forces of a Deck Crane in Wave Motion Using Screw Theory

Introduction

Preliminaries

Equations of Motion of a Vessel in Waves.

Definition and Properties of Screws.

Lines and Screws.

Twists.

Wrenches.

Dynamic Modeling

Equation of Motion of a Serial Manipulator.

Kinematics of a Crane/Vessel System.

Equations of Motion of a Crane/Vessel System.

Reaction Forces

Simulation Results

Crane Location I.

Crane Location II.

Discussion of Results

Conclusions

Funding Data

References

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Determination of Reaction Forces of a Deck Crane in Wave Motion Using Screw Theory

Introduction

Preliminaries

Equations of Motion of a Vessel in Waves.

Definition and Properties of Screws.

Lines and Screws.

Twists.

Wrenches.

Dynamic Modeling

Equation of Motion of a Serial Manipulator.

Kinematics of a Crane/Vessel System.

Equations of Motion of a Crane/Vessel System.

Reaction Forces

Simulation Results

Crane Location I.

Crane Location II.

Discussion of Results

Conclusions

Funding Data

References

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