A decline in oil-related revenues challenges Norway to focus on new types of offshore installations. Often, ship-mounted crane systems transfer cargo or crew onto offshore installations such as floating windmills. This project analyzes the motion of a ship induced by an onboard crane in operation using a new theoretical approach to dynamics: the moving frame method (MFM). The MFM draws upon Lie group theory and Cartan's moving frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. While others have applied aspects of these mathematical tools, the notation presented here brings these methods together; it is accessible, programmable, and simple. In the MFM, the notation for multibody dynamics and single body dynamics is the same; for two-dimensional (2D) and three-dimensional (3D), the same. Most importantly, this paper presents a restricted variation of the angular velocity to use in Hamilton's principle. This work accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass. This research solves the equations numerically using a relatively simple numerical integration scheme. Then, the Cayley–Hamilton theorem and Rodriguez's formula reconstruct the rotation matrix for the ship. Furthermore, this work displays the rotating ship in 3D, viewable on mobile devices. This paper presents the results qualitatively as a 3D simulation. This research demonstrates that the MFM is suitable for the analysis of “smart ships,” as the next step in this work.

## References

References
1.
MacGregor, 2018, “
MacGregor 3-Axis Motion Compensated Cranes
,” MacGregor Finland Oy, Kaarina, Finland, accessed Sept. 24, 2018, https://www.macgregor.com/Products-solutions/products/offshore-and-subsea-load-handling/3-axis-motion-compensation-cranes/
2.
Rolls-Royce, 2018, “
Rolls Royce Marine, Dual Draglink Crane
,” Rolls-Royce plc, London, accessed Sept. 24, 2018, https://www.rolls-royce.com/products-and-services/marine/product-finder/cranes/subsea-crane/dual-drag-link-crane-subsea.aspx#/
3.
Maczyński
,
A. A.
, and
Wojciech
,
S. S.
,
2011
, “
Stabilization of Load's Position in Offshore Cranes
,”
ASME. J. Offshore Mech. Artic Eng.
,
134
(
2
), p.
021101
.
4.
Xu
,
J.
,
Ren
,
Z.
,
Li
,
Y.
,
Skjetne
,
R.
, and
Halse
,
K.
,
2018
, “
Dynamic Simulation and Control of an Active Roll Reduction System Using Free-Flooding Tanks With Vacuum Pumps
,”
ASME J. Offshore Mech. Arct. Eng.
,
140
(
6
), p.
061302
.
5.
Kongsberg Gruppen ASA, 2018, “
Autonomous Ship Project, Key Facts About YARA Birkeland
,” Kongsberg Gruppen ASA, Kongsberg, Norway, accessed Sept. 24, 2018, https://www.km.kongsberg.com/ks/web/nokbg0240.nsf/AllWeb/4B8113B707A50A4FC125811D00407045?
6.
Nordvik
,
A.
,
Khan
,
N.
,
Burcă
,
R. A.
, and
Impelluso
,
T. J.
,
2017
, “
A Study of Roll Induced by Crane Motion on Ships: A Case Study of the Use of the Moving Frame Method
,”
ASME
Paper No. IMECE2017-70111
.
7.
Impelluso
,
T.
,
2016
, “
Rigid Body Dynamics: A New Philosophy, Math and Pedagogy
,”
ASME
Paper No. IMECE2016-65970
.
8.
Impelluso
,
T.
,
2017
, “
The Moving Frame Method in Dynamics: Reforming a Curriculum and Assessment
,”
Int. J. Mech. Eng. Educ.
,
46
(2), pp. 158–191.
9.
Cartan
,
É.
,
1986
,
On Manifolds With an Affine Connection and the Theory of General Relativity
,
Bibiliopolis
,
Napoli, Italy
.
10.
Frankel
,
T.
,
2012
,
The Geometry of Physics, an Introduction
,
3rd ed.
,
Cambridge University Press
,
New York
.
11.
Murray
,
R. M.
,
Li
,
Z.
, and
Sastry
,
S. S.
,
1994
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
,
Boca Raton, FL
.
12.
Murakami
,
H.
,
2013
, “
A Moving Frame Method for Multibody Dynamics
,”
ASME
Paper No. IMECE2013-62833
.
13.
Denavit
,
J.
, and
Hartenberg
,
R.
,
1955
, “
A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices
,”
ASME J. Appl. Mech.
,
22
, pp.
215
221
.
14.
Murakami
,
H.
,
2015
, “
A Moving Frame Method for Multi-Body Dynamics Using SE(3)
,”
ASME
Paper No. IMECE2015-51192
.
15.
Wittenburg
,
J.
,
1977
,
Dynamics of Rigid Bodies
,
B. G. Teubner
, ed.,
ZAMM
,
Weinheim, Germany
.
16.
Wittenburg
,
J.
,
2008
,
Dynamics of Multibody Systems
,
2nd ed.
,
Springer
,
Berlin
.
17.
Holm
,
D. D.
,
2008
,
Geometric Mechanics, Part II: Rotating, Translating and Rolling
,
World Scientific
,
Hackensack, NJ
.
18.
Rios
,
O.
,
Murakami
,
H.
, and
Impelluso
,
T.
,
2017
, “
A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena
,”
ASME J. Appl. Mech.
,
83
(
11
), p.
111006
.
19.
Khronos Group, 2018, “
OpenGL ES for the Web
,” The Khronos Group Inc., Beaverton, OR, accessed Sept. 24, 2018, https://www.khronos.org/webgl/
20.
Norbach
,
A.
,
Fjetland
,
K.
,
Hestetun
,
G.
, and
Impelluso
,
T.
,
2018
, “
Gyroscopic Wave Energy Converter for Fish Farms
,”
ASME
Paper No. IMECE2018-86188
.
21.
Austefjord
,
K.
,
Hestvik
,
M.
, and
Laresen
,
L.
,
2018
, “
Modeling Subsea ROV Motion Using the Moving Frame Method
,”
Int. J. Dyn. Control
(in press).
22.
Flatlandsmo
,
J.
,
Torbjørn
,
S.
,
Halvorsen
,
Ø.
, and
Impelluso
,
T.
,
2018
, “
Modeling Stabilization of Crane and Ship by Gyroscopic Control Using the Moving Frame Method
,”
ASME J. Comput. Non-Linear Dyn.
,
14
(3), p. 031006.
23.
Rios
,
O.
,
2016
, “
Development of Active Mechanical Models for Flexible Robots to Duplicate the Motion of Inch Works and Snakes
,”
ASME
Paper No. IMECE2016-65550
.
24.
Murakami
,
H.
,
2016
, “
Modeling and Experimentation of a Ribbed Caudal Fin for Aquatic Robots
,”
ASME
Paper No. IMECE2017-71235
.