Locating N Points of a Rigid Body on N Given Planes

2004 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Forward Kinematics ◽  
Linear Least Squares ◽  
Geometric Problem ◽  
Quadratic Equations ◽  
Linear Least Squares Problem ◽  
Minimum Number ◽  
Parallel Link ◽  
And Robotics

This paper describes a method for finding the location of a rigid body such that N specified points of the body lie on N given planes in space. Of special interest is the case N = 6, which is the minimum number to fully constrain the body. This geometric problem arises in two seemingly disparate contexts: metrology, as a generalization of so-called “3-2-1” locating schemes; and robotics, as the forward kinematics problem for 6ES or 6SE parallel-link platform robots. For N = 6, the geometric problem can be formulated algebraically as 3 quadratic equations having, in general, eight possible solutions. We give a method for finding all eight solutions via an 8 × 8 eigenvalue problem. We also show that for N ≥ 7, the solution can be found uniquely as a linear least squares problem.

scholarly journals On a Rigid Body Subject to Point-Plane Constraints

2005 ◽  
Vol 128 (1) ◽  
pp. 151-158 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Least Squares Method ◽  
Contact Problems ◽  
The Body ◽  
Eigenvalue Method ◽  
All Solutions ◽  
Minimum Number ◽  
Parallel Link ◽  
Three Degree Of Freedom

This paper investigates the location of a rigid body such that N specified points of the body lie on N given planes in space. Variants of this problem arise in kinematics, metrology, and computer vision, including some, such as the motion of a spherical four-bar, that are not at first glance point-plane contact problems. The case N=6, the minimum number to fully constrain the body, is of special interest: We give an eigenvalue method for finding all solutions, which may number up to eight. For N⩾7 there are, in general, no solutions, but if the constraints are compatible and not degenerate, we show how to find the unique solution by a linear least-squares method. For N⩽5, the body is underconstrained, having in general 6−N degrees of freedom; we determine the degree of the general motion for each case. We also examine the workspace of a particular three-degree-of-freedom parallel-link tripod mechanism.

Multiple‐source Werner deconvolution

Geophysics ◽  
1993 ◽  
Vol 58 (12) ◽  
pp. 1792-1800 ◽  
Author(s):  
Richard O. Hansen ◽  
Marc Simmonds
Keyword(s):  
Analytic Signal ◽  
The Body ◽  
Multiple Source ◽  
Juan De Fuca Ridge ◽  
Spreading Center ◽  
Complex Polynomial ◽  
Aeromagnetic Data ◽  
Linear Least Squares ◽  
Linear Least Squares Problem ◽  
Juan De Fuca

A reformulation of the Werner deconvolution algorithm using the analytic signal is extended to multiple source bodies. The extended algorithm involves solving a linear least‐squares problem; the coefficients so obtained determine a complex polynomial whose roots define the locations and depths of the body contacts. The extended algorithm has been used to map the structure of the Cobb offset zone of the Juan de Fuca Ridge from aeromagnetic data; both the top and bottom of the spreading center basalts can be delineated. Connections between the multiple‐source Werner technique and CompuDepth™ are discussed.

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.

Does a rigid body limit maneuverability?

2000 ◽  
Vol 203 (22) ◽  
pp. 3391-3396 ◽  
Author(s):  
J.A. Walker
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Alternative Measure ◽  
Pectoral Fin ◽  
Caudal Fin ◽  
Minimum Radius ◽  
Current Definition ◽  
Theoretical Minimum

Whether a rigid body limits maneuverability depends on how maneuverability is defined. By the current definition, the minimum radius of the turn, a rigid-bodied, spotted boxfish Ostracion meleagris approaches maximum maneuverability, i.e. it can spin around with minimum turning radii near zero. The radius of the minimum space required to turn is an alternative measure of maneuverability. By this definition, O. meleagris is not very maneuverable. The observed space required by O. meleagris to turn is slightly greater than its theoretical minimum but much greater than that of highly flexible fish. Agility, the rate of turning, is related to maneuverability. The median- and pectoral-fin-powered turns of O. meleagris are slow relative to the body- and caudal-fin-powered turns of more flexible fish.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

scholarly journals On weighted Strand’s iteration

2018 ◽  
Vol 34 (2) ◽  
pp. 183-190
Author(s):  
D. CARP ◽  
◽  
C. POPA ◽  
T. PRECLIK ◽  
U. RUDE ◽  
...  
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Numerical Approximation ◽  
Iterative Algorithms ◽  
Linear Least Squares ◽  
Minimal Norm ◽  
Least Squares Problem ◽  
Minimal Norm Solution ◽  
Linear Least Squares Problem

In this paper we present a generalization of Strand’s iterative method for numerical approximation of the weighted minimal norm solution of a linear least squares problem. We prove convergence of the extended algorithm, and show that previous iterative algorithms proposed by L. Landweber, J. D. Riley and G. H. Golub are particular cases of it.

The Forced Oscillations of Submerged Bodies

2003 ◽  
Vol 125 (4) ◽  
pp. 710-715
Author(s):  
Angel Sanz-Andre´s ◽  
Gonzalo Tevar ◽  
Francisco-Javier Rivas
Keyword(s):  
Rigid Body ◽  
Small Amplitude ◽  
Center Of Mass ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Central Point ◽  
Modal Testing ◽  
Forced Oscillations ◽  
Motion Equations ◽  
Marine Applications

The increasing use of very light structures in aerospace applications are given rise to the need of taking into account the effects of the surrounding media in the motion of a structure (as for instance, in modal testing of solar panels or antennae) as it is usually performed in the motion of bodies submerged in water in marine applications. New methods are in development aiming at to determine rigid-body properties (the center of mass position and inertia properties) from the results of oscillations tests (at low frequencies during modal testing, by exciting the rigid-body modes only) by using the equations of the rigid-body dynamics. As it is shown in this paper, the effect of the surrounding media significantly modifies the oscillation dynamics in the case of light structures and therefore this effect should be taken into account in the development of the above-mentioned methods. The aim of the paper is to show that, if a central point exists for the aerodynamic forces acting on the body, the motion equations for the small amplitude rotational and translational oscillations can be expressed in a form which is a generalization of the motion equations for a body in vacuum, thus allowing to obtain a physical idea of the motion and aerodynamic effects and also significantly simplifying the calculation of the solutions and the interpretation of the results. In the formulation developed here the translational oscillations and the rotational motion around the center of mass are decoupled, as is the case for the rigid-body motion in vacuum, whereas in the classical added mass formulation the six motion equations are coupled. Also in this paper the nonsteady motion of small amplitude of a rigid body submerged in an ideal, incompressible fluid is considered in order to define the conditions for the existence of the central point in the case of a three-dimensional body. The results here presented are also of interest in marine applications.

A Parameterization of the Central Axis Congruence Associated with Four Positions of a Rigid Body in Space

1993 ◽  
Vol 115 (3) ◽  
pp. 547-551 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Central Axis ◽  
Spatial Mechanisms ◽  
Center Point ◽  
Point Curve ◽  
Screw Surface ◽  
Computer Based

Given four positions of a rigid body in space, there is a congruence of lines that can be used as the central axes of cylindric cranks to guide the body through the four positions. This “central axis congruence” is a generalization of the center point curve of planar kinematics. It is known that this congruence is identical to the screw congruence which arises in the study of complementary screw quadrilateral. It is less well-known that the screw congruence is the “screw surface” of the 4C linkage formed by the complementary screw quadrilateral, and it is this relationship that we use to obtain a parameterization for the screw congruence and in turn, the central axis congruence. This parameterization should facilitate the use of this congruence in computer based design of spatial mechanisms.

scholarly journals Analysis of the three dimensional structure of the kidney glomerulus capillary network

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Mark Terasaki ◽  
Jason Cory Brunson ◽  
Justin Sardi
Keyword(s):  
3D Structure ◽  
Three Dimensional ◽  
The Body ◽  
Capillary Network ◽  
Dimensional Structure ◽  
Pixel Resolution ◽  
Longitudinal Splitting ◽  
Kidney Glomerulus ◽  
Minimum Number ◽  
Glomerular Size

AbstractThe capillary network of the kidney glomerulus filters small molecules from the blood. The glomerular 3D structure should help to understand its function, but it is poorly characterized. We therefore devised a new approach in which an automated tape collecting microtome (ATUM) was used to collect 0.5 μm thick serial sections from fixed mouse kidneys. The sections were imaged by scanning electron microscopy at ~ 50 nm/pixel resolution. With this approach, 12 glomeruli were reconstructed at an x–y–z resolution ~ 10 × higher than that of paraffin sections. We found a previously undescribed no-cross zone between afferent and efferent branches on the vascular pole side; connections here would allow blood to exit without being adequately filtered. The capillary diameters throughout the glomerulus appeared to correspond with the amount of blood flow within them. The shortest path (minimum number of branches to travel from afferent to efferent arterioles) is relatively independent of glomerular size and is present primarily on the vascular pole size. This suggests that new branches and longer paths form on the urinary pole side. Network analysis indicates that the glomerular network does not form by repetitive longitudinal splitting of capillaries. Thus the 3D structure of the glomerular capillary network provides useful information with which to understand glomerular function. Other tissue structures in the body may benefit from this new three dimensional approach.

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