Approximation of Finite Rigid Body Motions

2009 ◽  
Author(s):  
Andreas Mu¨ller
Keyword(s):  
Rigid Body ◽  
Euler Equations ◽  
Time Derivative ◽  
Time Integration ◽  
Body Motion ◽  
Motion Group ◽  
Time Stepping ◽  
Approximation Formulas ◽  
Rigid Body Motions ◽  
Local Parameters

It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion, since the angular velocity components are non-holonomic velocity coordinates. Moreover, the reconstruction of the body’s motion requires to solve a set of differential equations on the rigid body motion group. This is usually avoided by introducing local parameters (e.g. Euler angles) so that the problem becomes an ordinary differential equation on a vector space (e.g. kinematic Euler equations). In this paper the original problem on the motion group is treated. A family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order. It is shown that a k-th order accurate reconstruction requires the first k – 1 time derivative of the velocity. As an application the reconstruction formulas are used for the rotation update in a momentum preserving time stepping scheme for time integration of the dynamic Euler equations.

On Higher Order Point-Line and the Associated Rigid Body Motions

2006 ◽  
Vol 129 (2) ◽  
pp. 166-172 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Higher Order ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Dual Quaternion ◽  
Displacement Operator ◽  
Screw Motion ◽  
Rigid Body Motions ◽  
Point Line

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

A Computationally Efficient Planar Rigid Body Velocity Measure

2009 ◽  
Author(s):  
Luis E. Criales ◽  
Joseph M. Schimmels
Keyword(s):  
Rigid Body ◽  
Rigid Bodies ◽  
Body Motion ◽  
Computationally Efficient ◽  
Straight Line ◽  
Body Velocity ◽  
Velocity Measure ◽  
Rigid Body Motions ◽  
Line Path ◽  
Planar Motions

A planar rigid body velocity measure based on the instantaneous velocity of all particles that constitute a rigid body is developed. This measure compares the motion of each particle to an “ideal”, but usually unobtainable, motion. This ideal motion is one that would carry each particle from its current position to its desired position on a straight-line path. Although the ideal motion is not a valid rigid body motion, this does not preclude its use as a reference standard in evaluating valid rigid body motions. The optimal instantaneous planar motions for general rigid bodies in translation and rotation are characterized. Results for an example planar positioning problem are presented.

scholarly journals Rigid-body motion is the main source of diffuse scattering in protein crystallography

IUCrJ ◽  
2019 ◽  
Vol 6 (2) ◽  
pp. 277-289 ◽  
Author(s):  
T. de Klijn ◽  
A. M. M. Schreurs ◽  
L. M. J. Kroon-Batenburg
Keyword(s):  
Rigid Body ◽  
Diffuse Scattering ◽  
Body Motion ◽  
Maximum Information ◽  
Biologically Relevant ◽  
Internal Motions ◽  
X Ray Scattering ◽  
The Subject ◽  
Rigid Body Motions ◽  
A Minor

The origin of diffuse X-ray scattering from protein crystals has been the subject of debate over the past three decades regarding whether it arises from correlated atomic motions within the molecule or from rigid-body disorder. Here, a supercell approach to modelling diffuse scattering is presented that uses ensembles of molecular models representing rigid-body motions as well as internal motions as obtained from ensemble refinement. This approach allows oversampling of Miller indices and comparison with equally oversampled diffuse data, thus allowing the maximum information to be extracted from experiments. It is found that most of the diffuse scattering comes from correlated motions within the unit cell, with only a minor contribution from longer-range correlated displacements. Rigid-body motions, and in particular rigid-body translations, make by far the most dominant contribution to the diffuse scattering, and internal motions give only a modest addition. This suggests that modelling biologically relevant protein dynamics from diffuse scattering may present an even larger challenge than was thought.

scholarly journals A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 531
Author(s):  
Xiaomin Duan ◽  
Huafei Sun ◽  
Xinyu Zhao
Keyword(s):  
Rigid Body ◽  
Change Point ◽  
Order Approximation ◽  
Local Maximum ◽  
Change Point Detection ◽  
Body Motion ◽  
Geometric Method ◽  
Change Points ◽  
Gradient Descent Algorithm ◽  
Rigid Body Motions

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group S E ( 3 ) , so the Riemannian mean based on the Lie group structures of S E ( 3 ) reflects the characteristics of change-points. Once a change-point occurs, the distance between the current point and the Riemannian mean of its neighbor points should be a local maximum. A gradient descent algorithm is proposed to calculate the Riemannian mean. Using the Baker–Campbell–Hausdorff formula, the first-order approximation of the Riemannian mean is taken as the initial value of the iterative procedure. The performance of our method was evaluated by numerical examples and manipulator experiments.

Estimating the Orientation of a Game Controller Moving in the Vertical Plane Using Inertial Sensors

2012 ◽  
Author(s):  
Peng He ◽  
Philippe Cardou ◽  
André Desbiens
Keyword(s):  
Rigid Body ◽  
Inertial Sensors ◽  
Time Integration ◽  
Vertical Plane ◽  
Nonlinear Function ◽  
Body Motion ◽  
Human Hand ◽  
Game Controller ◽  
State Space Systems ◽  
Novel Method

This paper presents a novel method of estimating the orientation of a rigid body in the vertical plane from point-acceleration measurements, by discerning its gravitational and inertial components. In this method, a simple stochastic model of the human-hand motions is used in order to distinguish between the two types of acceleration. Two mathematical models of the rigid-body motion are formulated as distinct state-space systems, each corresponding to a proposed method. In both two cases, the output is a nonlinear function of the state, which calls for the application of the extended Kalman filter (EKF). The proposed filter is shown to work efficiently through two simulated trajectories, which are representative of human-hand motions. A comparison of the orientation estimates obtained from the proposed method shows that the filter offers more accuracy than a tilt sensor under high accelerations, and avoids the drift obtained by the time-integration of gyroscope measurements.

Dynamic Response of Composite Pressure Vessels to Inertia Loads

1991 ◽  
Vol 113 (1) ◽  
pp. 86-91
Author(s):  
J. C. Prucz ◽  
J. D’Acquisto ◽  
J. E. Smith
Keyword(s):  
Rigid Body ◽  
Combustion Engine ◽  
Pressure Vessels ◽  
Body Motion ◽  
Connecting Rod ◽  
Filament Wound ◽  
Potential Benefits ◽  
Rigid Body Motions ◽  
Crank Mechanism ◽  
Selection Of

A new analytical model has been developed in order to investigate the potential benefits of using fiber-reinforced composites in pressure vessels that undergo rigid-body motions. The model consists of a quasi-static lamination analysis of a cylindrical, filament-wound, pressure vessel, combined with an elastodynamic analysis that accounts for the coupling effects between its rigid-body motion and its elastic deformations. The particular type of motion investigated in this paper is that of an oil-pressurized, tubular connecting rod in a slider-crank mechanism of an internal combustion engine. A comprehensive parametric study has been focused on the maximum wall stresses induced in such a rod by the combined effect of internal pressure and inertia loads associated with its motion. The numerical results illustrate potential ways to reduce these stresses by appropriate selection of material systems, lay-up configurations and geometric parameters.

Coordinate Mappings for Rigid Body Motions

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Equations Of Motion ◽  
Time Integration ◽  
Exponential Mapping ◽  
Group Setting ◽  
Dual Quaternions ◽  
Integration Schemes ◽  
Rigid Body Motions ◽  
Lie Group Integration

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

scholarly journals Rigid Body Motions and Local Compliance Response during Impact of Two Deformable Spheres

2017 ◽  
Vol 8 (1) ◽  
pp. 1 ◽  
Author(s):  
Akuro Big-Alabo
Keyword(s):  
Rigid Body ◽  
Dynamic Models ◽  
Numerical Solutions ◽  
Simple Procedure ◽  
Body Motion ◽  
Impact System ◽  
Rigid Body Motions ◽  
Local Compliance ◽  
Nonlinear Dynamic Models ◽  
The Impact

The impact of two hard deformable spheres is revisited with the aim of investigating the constituent rigid body motions and indentation response of each sphere during collision. The latter are determined theoretically and the theoretical solutions are validated by comparing with numerical solutions of the coupled nonlinear dynamic models for impact of two hard deformable spheres. For elastic impact events, normalized tabulated solutions are derived using the Force Indentation Linearisation Method (FILM) and the tabulated solutions can be used to generate actual rigid body motions and indentation histories for each of the colliding spheres without need for numerical or finite element solutions. The analysis shows that the rigid body motion and local compliance response of each sphere depend on: (a) ratio of mass of sphere to effective mass of impact system, and (b) ratio of initial velocity of sphere to initial relative velocity of impact system. Finally, the 2-D collision problem is discussed and a simple procedure to determine the unique solution of all four unknowns is presented.

Partitioned Transient Analysis Procedures for Coupled-Field Problems: Accuracy Analysis

1980 ◽  
Vol 47 (4) ◽  
pp. 919-926 ◽  
Author(s):  
K. C. Park ◽  
C. A. Felippa
Keyword(s):  
Rigid Body ◽  
Time Integration ◽  
Governing Equations ◽  
Implicit Integration ◽  
Practical Applications ◽  
Coupled Field ◽  
Stable Partition ◽  
Rigid Body Motions ◽  
Coupled Field Problems ◽  
The Right

Partitioned solution procedures for direct time integration of second-order coupled-field systems are studied from the standpoint of accuracy. These procedures are derived by three formulation steps: implicit integration of coupled governing equations, partitioning of resulting algebraic systems and extrapolation on the right-hand partition. It is shown that the combined effect of partition, extrapolation, and computational paths governs the choice of stable extrapolators and preservation of rigid-body motions. Stable extrapolators for various computational paths are derived and implementation-extrapolator combinations which preserve constant-velocity and constant-acceleration rigid-body motions are identified. A spectral analysis shows that the primary error source introduced by a stable partition is frequency distortion. Finally, as a guide to practical applications, the advantages and shortcomings of five specific partitions are discussed.

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