A Displacement Metric for Finite Sets of Rigid Body Displacements

2008 ◽  
Author(s):  
Venkatesh Venkataramanujam ◽  
Pierre Larochelle
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Orthogonal Group ◽  
Polar Decomposition ◽  
Coordinate Frame ◽  
Invariant Metric ◽  
Motion Generation ◽  
Euclidean Group ◽  
Finite Set ◽  
Potential Applications

There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the principal frame (PF) associated with the finite set of displacements and the polar decomposition to map the homogenous transform representation of elements of the special Euclidean group SE(N-1) onto the special orthogonal group SO(N). Once the elements are mapped to SO(N) a bi-invariant metric can then be used. The metric obtained is thus independent of the choice of fixed coordinate frame i.e. it is left invariant. This metric has potential applications in motion synthesis, motion generation and interpolation. Three examples are presented to illustrate the usefulness of this methodology.

A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition

2006 ◽  
Vol 129 (8) ◽  
pp. 883-886 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray ◽  
Jorge Angeles
Keyword(s):  
Singular Value Decomposition ◽  
Rigid Body ◽  
Polar Decomposition ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Coordinate Frames ◽  
Open Research ◽  
Value Decomposition

An open research question is how to define a useful metric on the special Euclidean group SE(n) with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of SE(n) with elements of the special orthogonal group SO(n+1). This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of SE(n). The embedding of the elements of SE(n) into SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n).

A Coordinate Frame Useful for Rigid-Body Displacement Metrics

2010 ◽  
Vol 2 (4) ◽  
Author(s):  
Venkatesh Venkataramanujam ◽  
Pierre M. Larochelle
Keyword(s):  
Rigid Body ◽  
Point Mass ◽  
Coordinate Frame ◽  
Invariant Metric ◽  
Mass Model ◽  
Principal Axes ◽  
System Of Units ◽  
Rigid Body Displacement ◽  
Definition Of

This paper presents the definition of a coordinate frame, entitled the principal frame (PF), that is useful for metric calculations on spatial and planar rigid-body displacements. Given a set of displacements and using a point mass model for the moving rigid-body, the PF is determined from the associated centroid and principal axes. It is shown that the PF is invariant with respect to the choice of fixed coordinate frame as well as the system of units used. Hence, the PF is useful for left invariant metric computations. Three examples are presented to demonstrate the utility of the PF.

Projection Metrics for Rigid-Body Displacements

2005 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray
Keyword(s):  
Rigid Body ◽  
Robot Control ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Robot Calibration ◽  
Coordinate Frames ◽  
Open Research ◽  
Spatial Displacements

An open research question is how to define a useful metric on SE(n) with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. We present two techniques for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group SO(n+1). These techniques are based on the singular value and polar decompositions (denoted as SVD and PD respectively) of the homogeneous transform representation of the elements of SE(n). The projection of the elements of SE(n) onto SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements (hence the term projection metric. A bi-invariant metric on SO(n+1) may then be used to measure the distance between any two spatial displacements. The results are PD and SVD based projection metrics on SE(n). These metrics have applications in motion synthesis, robot calibration. motion interpolation, and hybrid robot control.

Synthesis of Planar, Compliant Four-Bar Mechanisms for Compliant-Segment Motion Generation

1999 ◽  
Vol 123 (4) ◽  
pp. 535-541 ◽  
Author(s):  
L. Saggere ◽  
S. Kota
Keyword(s):  
Compliant Mechanisms ◽  
Shape Change ◽  
Synthesis Method ◽  
Elastic Equilibrium ◽  
Body Motion ◽  
Motion Generation ◽  
Generation Task ◽  
Potential Applications ◽  
Flexible Segment ◽  
Definition Of

Compliant four-bar mechanisms treated in previous works consisted of at least one rigid moving link, and such mechanisms synthesized for motion generation tasks have always comprised a rigid coupler link, bearing with the conventional definition of motion generation for rigid-link mechanisms. This paper introduces a new task called compliant-segment motion generation where the coupler is a flexible segment and requires a prescribed shape change along with a rigid-body motion. The paper presents a systematic procedure for synthesis of single-loop compliant mechanisms with no moving rigid-links for compliant-segment motion generation task. Such compliant mechanisms have potential applications in adaptive structures. The synthesis method presented involves an atypical inverse elastica problem that is not reported in the literature. This inverse problem is solved by extending the loop-closure equation used in the synthesis of rigid-links to the flexible segments, and then combining it with elastic equilibrium equation in an optimization scheme. The method is illustrated by a numerical example.

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.

scholarly journals Intermolecular correlations are necessary to explain diffuse scattering from protein crystals

IUCrJ ◽  
2018 ◽  
Vol 5 (2) ◽  
pp. 211-222 ◽  
Author(s):  
Ariana Peck ◽  
Frédéric Poitevin ◽  
Thomas J. Lane
Keyword(s):  
Rigid Body ◽  
Conformational Changes ◽  
Protein Function ◽  
Diffuse Scattering ◽  
Conformational Dynamics ◽  
Protein Crystals ◽  
Biological Unit ◽  
Long Range Correlations ◽  
Correlated Motions ◽  
Potential Applications

Conformational changes drive protein function, including catalysis, allostery and signaling. X-ray diffuse scattering from protein crystals has frequently been cited as a probe of these correlated motions, with significant potential to advance our understanding of biological dynamics. However, recent work has challenged this prevailing view, suggesting instead that diffuse scattering primarily originates from rigid-body motions and could therefore be applied to improve structure determination. To investigate the nature of the disorder giving rise to diffuse scattering, and thus the potential applications of this signal, a diverse repertoire of disorder models was assessed for its ability to reproduce the diffuse signal reconstructed from three protein crystals. This comparison revealed that multiple models of intramolecular conformational dynamics, including ensemble models inferred from the Bragg data, could not explain the signal. Models of rigid-body or short-range liquid-like motions, in which dynamics are confined to the biological unit, showed modest agreement with the diffuse maps, but were unable to reproduce experimental features indicative of long-range correlations. Extending a model of liquid-like motions to include disorder across neighboring proteins in the crystal significantly improved agreement with all three systems and highlighted the contribution of intermolecular correlations to the observed signal. These findings anticipate a need to account for intermolecular disorder in order to advance the interpretation of diffuse scattering to either extract biological motions or aid structural inference.

Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)

1994 ◽  
Author(s):  
Pierre Larochelle ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Numerical Results ◽  
Image Space ◽  
Dimensional Synthesis ◽  
Invariant Metric ◽  
Planar Mechanisms ◽  
Position And Orientation ◽  
Position Synthesis

Abstract In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.

Task Analysis and Motion Generation for Service Robots

2012 ◽  
pp. 30-50 ◽  
Author(s):  
Jian S. Dai
Keyword(s):  
Mathematical Models ◽  
Path Analysis ◽  
Point Of View ◽  
Domestic Service ◽  
Service Robots ◽  
Motion Generation ◽  
Orientation Analysis ◽  
Potential Applications ◽  
Position Representation ◽  
Robotic Automation

This chapter is to summarise research in the direction of domestic service robots particularly with reference to robotic implementation of ironing process. The chapter presents the garment handling and ironing from a procedural point of view and discusses the devices for handling. The handling is categorised into several steps with common handling operations, resulting in categorisation of gripping and handling devices with potential applications to domestic automation. Based on this, ironing paths are explored with an orientation-position representation. This is followed by the introduction of development of folding and unfolding and by the region segregation based garment folding. This involves path analysis, folding algorithms, and mechanisms review for ironing. The paths produced from the ironing process are presented with mathematical models to be possibly implemented in robotic automation and their orientation is presented, dependent on the regions of garment. The orientation analysis is useful in finding the similarity in motion to determine the effective and efficient way of ironing a garment with orientation region diagrams and workspace presentation.

Optimal Motion Generation for Groups of Robots: A Geometric Approach

2004 ◽  
Vol 126 (1) ◽  
pp. 63-70 ◽  
Author(s):  
Calin Belta ◽  
Vijay Kumar
Keyword(s):  
Kinetic Energy ◽  
Rigid Body ◽  
Mobile Robots ◽  
Configuration Space ◽  
Geodesic Flow ◽  
Geometric Approach ◽  
General Treatment ◽  
Rectilinear Motion ◽  
Motion Generation ◽  
Optimal Motion

In this paper we generate optimal smooth trajectories for a set of fully-actuated mobile robots. Given two end configurations, by tuning one parameter, the user can choose an interpolating trajectory from a continuum of curves varying from that corresponding to maintaining a rigid formation to motion of the robots toward each other. The idea behind our method is to change the original constant kinetic energy metric in the configuration space and can be summarized into three steps. First, the energy of the motion as a rigid structure is decoupled from the energy of motion along directions that violate the rigid constraints. Second, the metric is “shaped” by assigning different weights to each term. Third, geodesic flow is constructed for the modified metric. The optimal motions generated on the manifolds of rigid body displacements in 3-D space SE3 or in plane SE2 and the uniform rectilinear motion of each robot corresponding to a totally uncorrelated approach are particular cases of our general treatment.

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