A Velocity Metric for Rigid-Body Planar Motion

2010 ◽  
Author(s):  
Joseph M. Schimmels ◽  
Luis E. Criales
Keyword(s):  
Rigid Body ◽  
Average Distance ◽  
The Body ◽  
Body Motion ◽  
Length Scale ◽  
Translational Velocity ◽  
Body Velocity ◽  
Assembly Problem ◽  
Instantaneous Center ◽  
Selection Of

A planar rigid-body velocity metric based on the instantaneous velocity of all particles that constitute a rigid body is developed. A measure based on the discrepancy in the translational velocity at each particle for two different planar twists is introduced. The calculation of the measure is simplified to the calculation of the product of: 1) the discrepancy in angular velocity, and 2) the average distance of the body from the instantaneous center associated with the twist discrepancy. It is shown that this measure satisfies the mathematical requirements of a metric and is physically consistent. It does not depend on either the selection of length scale or the frames used to describe the body motion. Although the metric does depend on body geometry, it can be calculated efficiently using body decomposition. An example demonstrating the application of the metric to an assembly problem is presented.

A Linear Algebra Approach to the Analysis of Rigid Body Velocity From Position and Velocity Data

1983 ◽  
Vol 105 (2) ◽  
pp. 92-95 ◽  
Author(s):  
A. J. Laub ◽  
G. R. Shiflett
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Matrix Theory ◽  
The Body ◽  
Instantaneous Velocity ◽  
Body Motion ◽  
Translational Velocity ◽  
Velocity Data ◽  
Algebra Approach ◽  
Body Velocity

The instantaneous velocity of a rigid body in space is characterized by an angular and translational velocity. By representing the angular velocity as a matrix and the translational component as a vector the velocity of any point in the rigid body may be found if the position of the point and the parameters of the angular and translational velocities are known. Alternatively, the parameters of the rigid body velocity may be determined if the velocity and position of three points fixed in the body are known. In this paper, a new matrix-theory-based method is derived for determining the instantaneous velocity parameters of rigid body motion in terms of the velocity and position of three noncollinear points fixed in the body. The method is shown to possess certain advantages over traditional vectoral solutions to the same problem.

scholarly journals The equations of motion for a rigid body using non-redundant unified local velocity coordinates

2019 ◽  
Vol 48 (3) ◽  
pp. 283-309 ◽  
Author(s):  
Stefan Holzinger ◽  
Joachim Schöberl ◽  
Johannes Gerstmayr
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Motion Vector ◽  
The Body ◽  
Body Motion ◽  
Integration Scheme ◽  
Exponential Map ◽  
Local Velocity ◽  
Translational Velocity ◽  
Angular Velocity Vector

Abstract A novel formulation for the description of spatial rigid body motion using six non-redundant, homogeneous local velocity coordinates is presented. In contrast to common practice, the formulation proposed here does not distinguish between a translational and rotational motion in the sense that only translational velocity coordinates are used to describe the spatial motion of a rigid body. We obtain these new velocity coordinates by using the body-fixed translational velocity vectors of six properly selected points on the rigid body. These vectors are projected into six local directions and thus give six scalar velocities. Importantly, the equations of motion are derived without the aid of the rotation matrix or the angular velocity vector. The position coordinates and orientation of the body are obtained using the exponential map on the special Euclidean group $\mathit{SE}(3)$SE(3). Furthermore, we introduce the appropriate inverse tangent operator on $\mathit{SE}(3)$SE(3) in order to be able to solve the incremental motion vector differential equation. In addition, we present a modified version of a recently introduced a fourth-order Runge–Kutta Lie-group time integration scheme such that it can be used directly in our formulation. To demonstrate the applicability of our approach, we simulate the unstable rotation of a rigid body.

Instantaneous Center of Rotation in Pitch Response of a FPSO Submitted to Head Waves

2018 ◽  
Author(s):  
Daniel de Oliveira Costa ◽  
Antonio Carlos Fernandes ◽  
Joel Sena Sales Junior ◽  
Peyman Asgari
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Tracking System ◽  
The Body ◽  
Body Motion ◽  
Floating Body ◽  
Wave System ◽  
Center Of Rotation ◽  
Head Waves ◽  
Instantaneous Center

When under influence of an incident wave system, any floating body presents a general motion with all six degrees of freedom, unless it presents some kind of restrains on it. For a free moving body, the center of rotation will depend on the force distribution and might not coincide with its center of gravity. For long and slender floating structures, such as FPSO platforms, a small change in the center of Pitch rotation would result in significant change in the overall motions in its fore and aft regions. Therefore, it is of high importance to obtain a better understating of the instantaneous position of the body center of rotation in Heave and Pitch response. This paper investigates the position of the Instantaneous Center of Rotation in Pitch Response of a scaled down model of a FPSO platform under different regular wave conditions. The investigation uses basic kinematics equations for rigid body, defining the 6 degrees of freedom of the rigid body motion from a finite number of markers installed in the model. A high quality tracking system captures the markers positions in order to define the rigid body at each instant of time. For an initial approach, the study considers the response due to head waves seas with experimental validation.

Instantaneous center/axis of rotation for planar and three-dimensional motion

2021 ◽  
pp. 030641902110511
Author(s):  
Donald L Kunz
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
The Body ◽  
Body Motion ◽  
Planar Motion ◽  
Instantaneous Axis Of Rotation ◽  
Axis Of Rotation ◽  
Instantaneous Center ◽  
Instantaneous Axis

This article discusses a direct analytical method for calculating the instantaneous center of rotation and the instantaneous axis of rotation for the two-dimensional and three-dimensional motion, respectively, of rigid bodies. In the case of planar motion, this method produces a closed-form expression for the instantaneous center of rotation based on a single point located on the rigid body. It can also be used to derive closed-form expressions for the body and space centrodes. For three-dimensional, rigid body motion, an extension of the technique used for planar motion locates a point on the instantaneous axis of rotation, which is parallel to the body angular velocity vector. In addition, methods are demonstrated that can be used to map the body and space cones for general rigid body motion, and locate the fixed point for the body.

scholarly journals Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed
Keyword(s):  
Rigid Body ◽  
Sobolev Spaces ◽  
Periodic Motion ◽  
Viscous Incompressible Fluid ◽  
Body Motion ◽  
Translational Velocity ◽  
Ill Posed ◽  
Homogeneous Sobolev Spaces ◽  
The Mean ◽  
Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

Using Dual Cam Tracks for High Speed Rigid Body Motion

1998 ◽  
Author(s):  
Clay Cooper ◽  
Stephen Derby
Keyword(s):  
Rigid Body ◽  
High Speed ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Transfer Rates ◽  
Track System ◽  
Selection Of

Abstract Rigid Body Motion has long been one of the standard problems for kinematicians. For high speed transfer rates, an industrial example of using a dual cam track system to achieve better performance is documented. The dual track establishes both a positional and orientational location of the followers. The selection of this mechanism type is discussed.

A New Theoretical and Numerical Approach to Finite Rigid Body Rotations Based on Lie Group Theory

2013 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Elias Paraskevopoulos ◽  
Nikolaos Potosakis
Keyword(s):  
Rigid Body ◽  
Configuration Space ◽  
Differentiable Manifold ◽  
Numerical Approach ◽  
Body Motion ◽  
Strong Basis ◽  
Lie Group Theory ◽  
Large Rotations ◽  
Basic Characteristics ◽  
Selection Of

A systematic theoretical approach is presented first, in an effort to provide a complete and illuminating study on motion of a rigid body rotating about a fixed point. Since the configuration space of this motion is a differentiable manifold possessing group properties, this approach is based on some fundamental concepts of differential geometry. A key idea is the introduction of a canonical connection, matching the manifold and group properties of the configuration space. Next, following the selection of an appropriate metric, the dynamics is also carried over. The present approach is theoretically more demanding than the traditional treatments but brings substantial benefits. In particular, an elegant interpretation can be provided for all the quantities with fundamental importance in rigid body motion. It also leads to a correction of some misconceptions and geometrical inconsistencies in the field. Finally, it provides powerful insight and a strong basis for the development of efficient numerical techniques in problems involving large rotations. This is demonstrated by an example, including the basic characteristics of the class of systems examined.

Control of Flexible Payloads Grasped by Actuated Grippers Undergoing Large Rigid-Body Motions: Part II — Controllability and Observability

2002 ◽  
Author(s):  
Edward J. Park ◽  
James K. Mills
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Time Scale ◽  
Body Motion ◽  
Deformation Control ◽  
Scale Modeling ◽  
Fixed Interface ◽  
Controllability And Observability ◽  
Scale Behavior ◽  
Selection Of

Part I of this work models the dynamics of a flexible payload grasped by an actuated gripper undergoing large rigid body motion by a robotic manipulator. In Part II, the controllability and observability conditions of the system are discussed. In Part I, the dynamic model of the actuated flexible payload is derived using the component mode synthesis (CMS) method with addition of actuator constraint, fixed-interface vibration and quasi-static modes. Here, the two-time scale modeling (TSM) technique is employed taking advantage of the two-time scale behavior between the quasi-static modes and vibration modes in the dynamic model. Due to the complexity of the resulting system, the controllability and observability conditions are not trivial. Hence, the controllability and observability study addressed herein becomes essential in showing the advantages of using the CMS and TSM techniques in control system design for the problem. A simulation example demonstrates that simultaneous vibration and quasi-static deformation control is achievable by proper selection of each type of modes.

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.

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