Quasi-Velocity Approach Toward Normalization Constraint for Euler Parameters

2021 ◽  
Author(s):  
Vatsal Joshi ◽  
Ashley Guy ◽  
Alan Bowling
Keyword(s):  
Euler Parameters ◽  
Normalization Constraint

STABILITY OF SPATIAL ELASTICA IN A GRAVITATIONAL FIELD

2012 ◽  
Vol 12 (02) ◽  
pp. 403-421 ◽  
Author(s):  
BOONCHAI PHUNGPAINGAM ◽  
LAWRENCE N. VIRGIN ◽  
SOMCHAI CHUCHEEPSAKUL
Keyword(s):  
Gravitational Field ◽  
Elastic Equilibrium ◽  
Experimental System ◽  
Primary Control ◽  
Arc Length ◽  
Euler Parameters ◽  
Reference Analysis ◽  
Horizontal Orientation ◽  
Equilibrium Configurations ◽  
Out Of Plane

This paper considers the behavior of a spatial elastica in a gravitational field. The slenderness of the system considered is such that the weight becomes an important consideration in determining elastic equilibrium configurations. Both ends of the elastica are clamped in an initially (planar) horizontal orientation at a fixed distance apart. However, one of the ends allows an increase in arc-length, that is, it is a sleeve joint. Thus, the total arc-length is the primary control parameter. This kind of elastica typically loses stability, resulting in out-of-plane deflections, when the total arc-length is increased beyond a critical value. A small mid-length torque can used to perturb a planar equilibrium configuration in order to test for stability. The aim of this study is to assess the effect of self-weight of the elastica (which is typically ignored) on promoting or delaying the loss of stability. To this end, it is useful to compare and contrast the results of orientation, that is, the system is configured in both an initial "upright" orientation and then in an "upside-down" orientation to highlight the influence of gravity. The results of the weightless elastica are used as a reference. Analysis is based on Kirchhoff's rod theory and Euler parameters, and the resulting set of governing differential equations are solved using a shooting method. The results from an experimental system using a slender superelastic wire made from Nitinol (Nickel Titanium Naval Ordnance Laboratory) exhibit close agreement with the analytical results.

Mechanical Power Losses of Ball Bearings: Model and Experimental Validation

2021 ◽  
pp. 1-29
Author(s):  
Ahmet Dindar ◽  
Amit Chimanpure ◽  
Ahmet Kahraman
Keyword(s):  
Dynamic Model ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Mechanical Power ◽  
Ball Bearings ◽  
Power Losses ◽  
Euler Parameters ◽  
Surface Shear ◽  
Axial Forces ◽  
Set Up

Abstract A tribo-dynamic model of ball bearings is proposed to predict their load-dependent (mechanical) power losses. The model combines (i) a transient, point contact mixed elastohydrodynamic lubrication (EHL) formulation to simulate the mechanics of the load carrying lubricated ball-race interfaces, and (ii) a singularity-free dynamics model, and establishes the two-way coupling between them that dictates power losses. The dynamic model employs a vectoral formulation with Euler parameters. The EHL model is capable of capturing two-dimensional contact kinematics, velocity variations across the contact as well as asperity interactions of rough contact surfaces. Resultant contact surface shear distributions are processed to predict mechanical power losses of example ball bearings operating under combined radial and axial forces. An experimental set-up is introduced for measurement of the power losses of rolling-element bearings. Sets of measurements taken by using the same example ball bearings are compared to those predicted by the model to assess its accuracy in predicting mechanical power loss of a ball bearing within wide ranges of axial and radial forces.

scholarly journals A Survey on the Computation of Quaternions From Rotation Matrices

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Soheil Sarabandi ◽  
Federico Thomas
Keyword(s):  
Three Dimensional ◽  
Rotation Matrix ◽  
Three Dimensions ◽  
Attitude Dynamics ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Rotation Matrices ◽  
Spacecraft Attitude Dynamics ◽  
Applied Fields ◽  
Quaternion Representation

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, three-dimensional image processing, and computer graphics. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in ℝ3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3 × 3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

Cyclic Constraint Analysis for Attitude Synchronization of Networked Spacecraft Agents

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Hanlei Wang ◽  
Yongchun Xie
Keyword(s):  
Multiple Equilibria ◽  
Closed Loop ◽  
Euler Parameters ◽  
Networked System ◽  
Synchronization Errors ◽  
Synchronization Problem ◽  
Constraint Analysis ◽  
The Stability ◽  
Error Orientation ◽  
Attitude Synchronization

In this paper, we investigate the attitude synchronization problem for multiple networked spacecraft, and the spacecraft agents are assumed to interact on an undirected and connected graph. We adopt a physically motivated PD-like attitude consensus scheme which takes Euler parameters or quaternions of the error orientation matrix between the spacecraft agents as the attitude deviation, resulting in nonlinear attitude coupling among the networked spacecraft agents and additionally multiple equilibria of the closed-loop networked system. The stability of the closed-loop networked system is shown by the Lyapunov stability analysis. To show the convergence of the attitude synchronization errors, we develop a new tool called cyclic constraint analysis. With this synthesis tool, we show that attitude synchronization is achieved without relying on any assumptions of the spacecraft orientations. Simulation study is presented to shed some light on the obtained results.

On the Dynamics of a Weighted Bowling Ball

1979 ◽  
Vol 46 (4) ◽  
pp. 937-943 ◽  
Author(s):  
R. L. Huston ◽  
C. Passerello ◽  
J. M. Winget ◽  
J. Sears
Keyword(s):  
Angular Velocity ◽  
Velocity Component ◽  
Equations Of Motion ◽  
Mass Center ◽  
Governing Equations ◽  
Euler Parameters ◽  
Factors Affecting ◽  
Dependent Variables ◽  
And Performance ◽  
Component Parallel

An analysis of the dynamics and performance of a weighted, slipping/rolling bowling ball is presented. The analysis uses Euler parameters and angular velocity components as dependent variables. The governing equations of motion are integrated using standard digital/numerical procedures. Particular attention is given to factors affecting ball performance (“hook”) and the lane oil tracing on the ball. It is found that factors most affecting hook are the mass-center location, the lane conditions (friction), and the initial angular velocity component parallel to the lane.

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