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

Abstract

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.

Kinematics and Dynamics of Flexible Robotic Manipulators Using Dual Screws

“…where p j rs is a vector from r to s given in the coordinates of Frame j. It is straightforward to verify that the twists of the composite displacement T a c = T a b T b c can be added according to [25] t i ac/r = t i ab/r + t i bc/r (5) under the condition that the twists are referenced to the same point r, and are given in the coordinates of the same Frame i.…”

“…Consider the system given in [25] to bet where u j is the generalized speed of Joint j and the dual line of Joint j isL…”

“…The time derivative of a dual line is given as an extension of the time derivative of a real line [25]. The geometric differentiation procedure given in this section is more computationally efficient for symbolic expressions than using the build-in differentiation commands in Matlab and the chain rule.…”

“…This was used for rigid mechanisms in [20], [21], and [22], and for flexible systems in [23] and [24]. The fact that a twist is a screw given in Plücker coordinates allows for an alternative formulation in terms of screw theory [25], [26], where the geometrical interpretation of the derivations may be more evident.…”

“…The kinematic framework of the method is based on the Lie groups of dual rotation and dual homogeneous transformation matrices, which are the basis for the derivation of dual twists. The method is an extension of the previous work [25], [28], [29] and is based on the linearization procedure proposed in [12] and [15]. The advantage of the proposed method is that we propose to use screw theory and dual linear algebra for the derivations, which is easier to implement computationally than the original formulation in [12] and [15] where coordinate-free vectors and dyadics were used.…”