International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1306–1320

# Modelling Subsea ROV robotics using the moving frame method

Article

## Abstract

This paper introduces a new method (notation and theory) in engineering dynamics. The goal of this paper is to apply this new method to the analysis of multi-body systems, specifically for the analysis of the motion of a Remotely Operated Vehicle (ROV). Norway conducts operations on a variety of structures in the North Sea. Research that can improve the efficiency of these operations are of high interest. This paper researches the motion of an ROV induced by the motion of the robotic manipulators, motor torques, and buoyancy. This research also introduces a new method in engineering 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 pieces and aspects of these mathematical tools, the notation presented here brings them together; it is accessible, programmable and simple. The notation for multi-body dynamics and single rigid body dynamics is the same. Most important, this paper presents a restricted variation of the angular velocity to use in Hamilton’s Principle to extract the equations of motion. This research solves the equations using a relatively simple numerical integration scheme. The Cayley-Hamilton theorem and Rodriguez’s formula reconstructs the rotation matrix for the ROV. 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 ROVs” as the next step in this work.

## Keywords

Moving Frame Method Dynamics ROV motion Subsea Engineering Robotics

## List of symbols

$$I_{3 } ,I_{d }$$

3 × 3 identity matrix

Jc( α)

3 × 3 mass moment of inertia matrix

K

Kinetic energy

L

Lagrangian

$$\left[ M \right]$$

Mass matrix

$$\left[ {{\text{M}}^{*} } \right]$$

Reduced mass matrix

$$\left[ {{\text{N}}^{*} } \right]$$

Reduced non-linear velocity matrix

q

Generalized coordinates

$$\dot{q}$$

Generalized velocity

$$\ddot{q}$$

Generalized acceleration

R

Rotation matrix

r

Absolute position vector

s

Relative position vector

U

Potential energy

$$\left\{ {\dot{X}} \right\}$$

List of velocities

$$\delta$$

Variation

$$\delta W$$

Virtual work

$$\delta \Pi$$

Variation of frame connection matrix

$$\delta \dot{\tilde{X}}$$

Variation of the generalized rates

$$\delta \tilde{X}$$

Virtual generalized displacement

$$\delta \dot{X}$$

Virtual generalized velocity

$$\delta q$$

Virtual essential generalized displacement

$$\tilde{\delta }\pi$$

Virtual rotational displacement

$$\Omega$$

Time rate of frame connection matrix

$$\omega$$

Angular velocity vector

$$\tilde{\omega }$$

Skew symmetric angular velocity matrix

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Katrine Oen Austefjord
• 1
• 2
• Linn-Kristin Skeide Larsen
• 1
• 3
• Martin Oddøy Hestvik
• 1
• 4
• Thomas J. Impelluso
• 1
• 5
1. 1.Mechanical and Marine EngineeringWestern Norway University of Applied Sciences (HVL)BergenNorway
2. 2.SandsliNorway
3. 3.LoddefjordNorway
4. 4.AverøyNorway
5. 5.BergenNorway

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