Use of the Moving Frame Method in Dynamics to Model Gyroscopic Control of Small Crafts at Sea: Theory — Part 1

2017 ◽  
Author(s):  
Joakim Nyland ◽  
Håkon Teigland ◽  
Thomas J. Impelluso
Keyword(s):  
Numerical Simulation ◽  
Large Scale ◽  
Equations Of Motion ◽  
Companion Paper ◽  
Cross Product ◽  
Moving Frame ◽  
3D Analysis ◽  
Active System ◽  
Moving Frame Method ◽  
Frame Method

This paper presents new method in dynamics — the Moving Frame Method (MFM) — and uses it to address a challenge faceing Norwegian shipping. Large offshore renewable energy investments require the use of maintenance boats to keep them in operable conditions. Unfortunately, due to rough seas in some project locations, the transferring of crew members from vessel to turbine or platform is fraught with safety concerns. These concerns can be alleviated by controlling the motion of the transfer vessel. This research studies an add-on stability system for marine vessels to ease the process of offshore platform maintenance and crew member safety. Specifically, this research concerns an internal active system — an active gyroscopic stabilizer — and a more powerful method of theoretical and computational mechanics. This paper derives the equations of motion of a model system equipped with dual gyroscopic stabilizers, using the MFM. The equations of motion are numerically solved to obtain a numerical simulation. The method exploits a variational principle with a restricted variation of the angular velocity. The MFM simplifies dynamics, enables a consistent notation, from 2D to 3D analysis and exploits matrix algebra in lieu of the vector cross product. Finally, in a companion paper to this one, the mathematical model and the numerical simulation is verified with experiments conducted in a large-scale wave tank.

Use of the Moving Frame Method in Dynamics to Model Gyroscopic Control of Small Crafts at Sea: Experiment — Part 2

2017 ◽  
Author(s):  
Håkon Teigland ◽  
Andreas Flåten ◽  
Martin R. Lied ◽  
Caspar C. Smith ◽  
Joakim F. Nyland ◽  
...  
Keyword(s):  
Experimental Validation ◽  
Wind Farms ◽  
Companion Paper ◽  
Moving Frame ◽  
Measurement Unit ◽  
3D Analysis ◽  
Gyroscopic Control ◽  
Moving Frame Method ◽  
Frame Method ◽  
Validation Tests

This companion paper represents the experimental validation of a new mathematical method that models the gyroscopic control of small craft at sea. In light of the development of offshore wind farms, there is a need for stabilizing smaller vessels that transfer crew to and from the wind farms. One way of stabilizing floating vessels is by the use of gyroscopes (inertial spinning disks) mounted on the vessels. The research presented in this paper uses the Moving Frame Method (MFM) as the underlying analytical method. The companion paper introduces the method and provides the mathematical analysis. This paper presents the experimental validation. Tests performed in a wave tank are used to verify the equations obtained by using the MFM. The tests are performed using a scaled model of a floating vessel. The vessel motions are obtained by an inertial measurement unit (IMU). The data received from the IMU and equations obtained by using the MFM calculates appropriate gyroscopic nutation rates to reduce the vessels roll motions. The rates are then applied to the gyroscopes using servo motors. This research demonstrates the power of the MFM. More importantly, it shows how the MFM invites experimental validation tests, as it is fundamentally a 3D analysis, yet open to understanding and use by undergraduates. The authors request that reader review the companion paper previous to this one.

scholarly journals Stabilizing Ship Motion With a Dual System Inertial Disk

2019 ◽  
Author(s):  
Torstein R. Storaas ◽  
Kasper Virkesdal ◽  
Gitle S. Brekke ◽  
Thorstein Rykkje ◽  
Thomas Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Moving Frames ◽  
New Approach ◽  
Lie Group Theory ◽  
Moving Frame Method ◽  
Frame Method ◽  
Modern Mathematics

Abstract Norwegian industries are constantly assessing new technologies and methods for more efficient and safer maintenance in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze ship stability moderated by a dual gyroscopic inertial device. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of two inertial disk devices, it accounts for the prescribed spin of the disks. It separates out the prescribed variables. This work displays the results in 3D on cell phones. It represents a prelude to testing in a wave tank.

scholarly journals Modeling a Knuckle-Boom Crane Control to Reduce Pendulum Motion Using the Moving Frame Method

2019 ◽  
Author(s):  
Maren Eriksen Eia ◽  
Elise Mari Vigre ◽  
Thorstein Ravneberg Rykkje
Keyword(s):  
Equations Of Motion ◽  
Moving Frame ◽  
Web Pages ◽  
Moving Frames ◽  
Pendulum Motion ◽  
Moving Frame Method ◽  
Frame Method ◽  
The Masses ◽  
And Control ◽  
Boom Crane

Abstract A Knuckle Boom Crane is a pedestal-mounted, slew-bearing crane with a joint in the middle of the distal arm; i.e. boom. This distal boom articulates at the ‘knuckle (i.e.: joint)’ and that allows it to fold back like a finger. This is an ideal configuration for a crane on a ship where storage space is a premium. This project researches the motion and control of a ship mounted knuckle boom crane to minimize the pendulum motion of a hanging load. To do this, the project leverages the Moving Frame Method (MFM). The MFM draws upon Lie group theory — SO(3) and SE(3) — and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. The work reported here accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass on the ship. The equations of motion are solved numerically using a 4th order Runge Kutta (RK4), while solving for the rotation matrix for the ship using the Cayley-Hamilton theorem and Rodriguez’s formula for each timestep. This work displays the motion on 3D web pages, viewable on mobile devices.

Modeling Stablization of Crane and Ship by Gyroscopic Control Using the Moving Frame Method

2018 ◽  
Author(s):  
Josef Flatlandsmo ◽  
Torbjørn Smith ◽  
Ørjan O. Halvorsen ◽  
Johnny Vinje ◽  
Thomas J. Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Long Term Results ◽  
Moving Frame ◽  
Moving Frames ◽  
Learning Modules ◽  
Moving Frame Method ◽  
Frame Method

Norwegian industries are constantly assessing new technologies and methods for more efficient and safer production in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze the motion induced by a crane and controlled by a gyroscopic inertial device mounted on a ship. The crane is a simple two-link system that transfers produce and equipment to and from barges. An inertial flywheel — a gyroscope — is used to stabilize the barge during transfer. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of both the crane and the stabilizing inertial device. Furthermore, this work allows for buoyancy and motor induced torques. Furthermore, this work displays the results in 3D on cell phones. The long-term results of this work leads to a robust 3D active compensation method for loading/unloading operations offshore. Finally, the interactivity between the crane and the stabilizing gyro anticipates the impending time of artificial intelligence when machines, equipped with on-board CPU’s and IP addresses, are empowered with learning modules to conduct their operations.

Modelling Buoy Motion at Sea

2019 ◽  
Author(s):  
Thorstein R. Rykkje ◽  
Tord Tørressen ◽  
Håvard Løkkebø
Keyword(s):  
Equations Of Motion ◽  
Moving Frame ◽  
Web Pages ◽  
Moving Frames ◽  
Lie Group Theory ◽  
Moving Frame Method ◽  
Frame Method ◽  
And Mathematics ◽  
The Masses ◽  
Theoretical Results

Abstract This project creates a model to assess the motion induced on a buoy at sea, under wave conditions. We use the Moving Frame Method (MFM) to conduct the analysis. The MFM draws upon concepts and mathematics from Lie group theory — SO(3) and SE(3) — and Cartan’s notion of Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. This work accounts for the masses and geometry of all components and for buoyancy forces and added mass. The resulting movement will be displayed on 3D web pages using WebGL. Finally, the theoretical results will be compared with experimental data obtained from a previous project done in the wave tank at HVL.

Production and Analytics of a Multi-Linked Robotic System

2019 ◽  
Author(s):  
Thorstein R. Rykkje ◽  
Eystein Gulbrandsen ◽  
Andreas Fosså Hettervik ◽  
Morten Kvalvik ◽  
Daniel Gangstad ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Robotic System ◽  
Moving Frame ◽  
Moving Frames ◽  
New Approach ◽  
Lie Group Theory ◽  
Senior Design ◽  
Moving Frame Method ◽  
Frame Method

Abstract This paper extends research into flexible robotics through a collaborative, interdisciplinary senior design project. This paper deploys the Moving Frame Method (MFM) to analyze the motion of a relatively high multi-link system, driven by internal servo engines. The MFM describes the dynamics of the system and enables the construction of a general algorithm for the equations of motion. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. The result is a dynamic 3D analytical model for the motion of a snake-like robotic system, that can take the physical sizes of the system and return the dynamic behavior. Furthermore, this project builds a snake-like robot driven by internal servo engines. The multi-linked robot will have a servo in each joint, enabling a three-dimensional movement. Finally, a test is performed to compare if the theory and the measurable real-time results match.

Modeling Crane Induced Ship Motion Using the Moving Frame Method

2018 ◽  
Author(s):  
Paulo Alexander Jacobsen Jardim ◽  
Jan Tore Rein ◽  
Øystein Haveland ◽  
Thomas J. Impelluso
Keyword(s):  
Equations Of Motion ◽  
Current Approach ◽  
Moving Frame ◽  
Current Phase ◽  
Integration Scheme ◽  
Moving Frames ◽  
Moving Frame Method ◽  
Frame Method ◽  
Interactive 3D ◽  
The Masses

A decline in oil-related revenues challenges Norway to focus on new types of offshore installations and their maintenance. Often, ship-mounted crane systems transfer cargo or crew onto marine structures such as floating windmills. This project analyzes the motion of a ship induced by an onboard crane in operation. It analyzes the motion of a crane mounted on a ship using 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. This work extends a previous project that assumed many simplifications. It accounts for the masses and geometry of all components. This current approach also accounts interactive motor couples and prepares for buoyancy forces and added mass. The previous work used a symbolic manipulator, resulting in unwieldy equations. In this current phase, this research solves the equations numerically using a relatively simple numerical integration scheme. Then, the Cayley-Hamilton theorem and Rodriguez’s formula reconstructs the rotation matrix for the ship. Furthermore, this work displays the rotating ship in 3D, viewable on mobile devices. WebGL is a JavaScript API for rendering interactive 3D and 2D graphics within any compatible web browser without the use of plug-ins. This paper presents the results qualitatively as a 3D simulation. This research proves that the MFM is suitable for the analysis of “smart ships,” as the next step in this work.

A Moving Frame Method for Multi-Body Dynamics

2013 ◽  
Author(s):  
H. Murakami
Keyword(s):  
Equations Of Motion ◽  
Rigid Bodies ◽  
Explicit Representation ◽  
Moving Frame ◽  
Moving Frames ◽  
Body Dynamics ◽  
Moving Frame Method ◽  
Frame Method ◽  
Multi Body ◽  
Kinematics And Kinetics

Élie Cartan’s moving frame method, developed in differential geometry, has been applied to multi-body dynamics to derive equations of motion. The explicit representation of a body-attached orthonormal coordinate basis and its origin, referred to as a moving frame, enables the usage of the special orthogonal group, SO(3), and the special Euclidean group, SE(3), to describe kinematics and kinetics of interconnected bodies by joints and force elements. The moving frame representation using Theodore Frankel’s compact notation is adopted to alleviate theoretical complexities of the Lie group theory to which SO(3) and SE(3) belong. For the variational formulation, the restricted variation of angular velocity is derived for the moving frame method. Starting from two connected rigid bodies, it will be demonstrated that the explicit representation of moving frames renders straight-forward symbolic computations of three-dimensional kinematics and kinetics. This simplicity eliminates errors in computing analytical expressions for kinematic and kinetic variables and streamlines the coding effort for numerical solution. For controller design, if the degrees-of-freedom is small, the moving frame method allows a straight-forward derivation of equations of motion in analytical form.

Modeling Crane-Induced Ship Motion Using the Moving Frame Method

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Paulo Alexander Jacobsen Jardim ◽  
Jan Tore Rein ◽  
Øystein Haveland ◽  
Thorstein R. Rykkje ◽  
Thomas J. Impelluso
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Moving Frame ◽  
Integration Scheme ◽  
Moving Frames ◽  
Offshore Installations ◽  
Numerical Integration Scheme ◽  
Moving Frame Method ◽  
Frame Method ◽  
The Masses

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.

A Moving Frame Method for Multi-Body Dynamics Using SE(3)

2015 ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Group Theory ◽  
Lie Group ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Lie Group Theory ◽  
Angular Velocities ◽  
Moving Frame Method ◽  
Frame Method ◽  
Cartesian Coordinate ◽  
Multi Body

To describe the configuration of a multi-body system, Cartesian coordinate systems are attached to all bodies comprising the system. Their connections through joints and force elements are efficiently expressed by using 4×4 matrices of the homogeneous transformation, presented by Denavit and Hartenberg in 1955. However, at this time, there is no systematic method to compute velocities and angular velocities using the matrices of such homogeneous transformations. In this paper, homogeneous transformation matrices are identified as a subset of a Lie group, called the special Euclidean group denoted by SE(3). This observation enables the usage of the Lie group theory in multibody kinematics. The effective use of the theory is built upon a platform of a moving frame method as presented in this paper. In this method, for each body-attached Cartesian coordinate system, the coordinate vector basis is written explicitly following Élie Cartan. This moving frame notation enables us to use the Lie algebra of SE(3), denoted by se(3), to compute velocities and angular velocities by minimizing the complexities of the Lie group theory. For kinetics, a variational method is established in se(3) by deriving a relationship between a virtual angular velocities and the corresponding virtual rotational displacements. This constrained variation of virtual angular velocities allows the derivation of the d’Alembert principle of virtual work from Hamilton’s principle for multibody systems. Utilizing this variational tool, we present a systematic computation of equations of motion from Hamilton’s principle. Finally, we reduce the spatial dynamics to planar dynamics and list the simplifications achieved in the two-dimensional problems using SE(2). Then, for a two-degree-of-freedom manipulator the analytical equations of motion are obtained to demonstrate the power of the moving frame method.

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