Kinematic Registration Using Line Geometry

2004 ◽  
Author(s):  
Johannes K. Eberharter ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Biomedical Imaging ◽  
Geometric Method ◽  
Line Geometry ◽  
Spatial Displacement ◽  
Line Complex ◽  
Kinematic Registration ◽  
Point Data ◽  
Elegant Solution

This paper uses line geometry to find an elegant solution to the kinematic registration problem involving reconstruction of a spatial displacement from data on three homologous points at two finitely separated positions of a rigid body. The bisecting linear line complex of two position theory in kinematics is used in combination with recent results from computational line geometry to present an elegant computational geometric method for the solution of this old problem. The results have applications in robotics, manufacturing, and biomedical imaging. The paper considers when minimal, over-determined, and perturbed sets of point data are given.

Kinematic Registration in 3D Using the 2D Reuleaux Method

2005 ◽  
Vol 128 (2) ◽  
pp. 349-355 ◽  
Author(s):  
Johannes K. Eberharter ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Approximation Problem ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Spatial Displacement ◽  
Linear Solution ◽  
Classical Technique ◽  
Minimum Number ◽  
Kinematic Registration ◽  
Point Data

This paper presents a method for kinematic registration in three dimensions using a classical technique from two-dimensional kinematics, namely the Reuleaux method. In three dimensions the kinematic registration problem involves reconstruction of a spatial displacement from data on a minimum of three homologous points at two finitely separated positions of a rigid body. When more than the minimum number of homologous points are specified or when errors in specification of these points are considered, the problem becomes an over determined approximation problem. A computational geometric method is presented, resulting in a linear solution of the over determined system. The results have applications in robotics, manufacturing, and biomedical imaging. The paper considers the kinematic registration when minimal, over-determined, infinitesimal, and perturbed sets of homologous point data are given.

A Line Geometric Approach to Determining Displacements Based on Line Specifications

2008 ◽  
Author(s):  
Wuchang Kuo ◽  
Chintien Huang
Keyword(s):  
Rigid Body ◽  
Measurement Errors ◽  
Systematic Approach ◽  
Screw Theory ◽  
Geometric Approach ◽  
Line Geometry ◽  
Line Complex ◽  
Numerical Example ◽  
On Line ◽  
Best Fit

This paper presents a systematic approach to determining the displacements of a rigid body from line specifications. The underlying concept of the proposed approach pertains to screw theory and line geometry. We utilize the correspondence between a pair of homologous lines and a regulus and that between a screw and a linear line complex. In this paper, a displacement screw is obtained by fitting a linear line complex to two or more line reguli. When two exact pairs of homologous line are specified, we obtain a unique linear line complex, which determines the displacement screw correspondingly. When more than two pairs of homologous lines with measurement errors are specified, it becomes a redundantly specified problem, and a linear line complex that has the best fit to more than two reguli is determined. A numerical example with the specification of four pairs of homologous lines is provided.

Three-Dimensional Generalizations of Reuleaux’s and Instant Center Methods Based on Line Geometry

2010 ◽  
Vol 2 (4) ◽  
Author(s):  
Jasem Baroon ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Geometric Method ◽  
Two Dimensional ◽  
Line Geometry ◽  
Basic Form ◽  
Motion Reconstruction ◽  
Moving Body ◽  
On Line

In kinematics, the problem of motion reconstruction involves generation of a motion from the specification of distinct positions of a rigid body. In its most basic form, this problem involves determination of a screw displacement that would move a rigid body from one position to the next. Much, if not all of the previous work in this area, has been based on point geometry. In this paper, we develop a method for motion reconstruction based on line geometry. A geometric method is developed based on line geometry that can be considered a generalization of the classical Reuleaux method used in two-dimensional kinematics. In two-dimensional kinematics, the well-known method of finding the instant center of rotation from the directions of the velocities of two points of the moving body can be considered an instantaneous case of Reuleaux’s method. This paper will also present a three-dimensional generalization for the instant center method or the instantaneous case of Reuleaux’s method using line geometry.

A Three-Dimensional Generalization of Reuleaux’s Method Based on Line Geometry

2006 ◽  
Author(s):  
Jasem Baroon ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Least Squares ◽  
Least Squares Method ◽  
Three Dimensional ◽  
Line Geometry ◽  
Basic Form ◽  
Motion Reconstruction ◽  
Linear Solution ◽  
On Line

In kinematics, the problem of motion reconstruction involves generation of a motion from the specification of distinct positions of a rigid body. In its most basic form, this problem involves determination of a screw displacement that would move a rigid body from one position to the next. Much if not all of the previous work in this area has been based on point geometry. In this paper, we develop a method for motion reconstruction based on line geometry. An elegant geometric method is developed based on line geometry that can be considered as a generalization of the classical Reuleaux’s method used in 2D kinematics. The case of over determined system is also considered a linear solution is presented based on least squares method.

On the Regulus Associated With the General Displacement of a Line and Its Application in Determining Displacement Screws

2010 ◽  
Vol 2 (4) ◽  
Author(s):  
Chintien Huang ◽  
Wuchang Kuo ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Measurement Errors ◽  
The Body ◽  
Hyperbolic Paraboloid ◽  
Line Complex ◽  
General Displacement ◽  
Basic Entity ◽  
Best Fit

In the two position theory of finite kinematics, we are concerned with not only the displacement of a rigid body, but also with the displacement of a certain element of the body. This paper deals with the displacement of a line and unveils the regulus that corresponds to such a displacement. The regulus is then used as a basic entity to determine the displacements of a rigid body from line specifications. Residing on a special hyperbolic paraboloid, the regulus is obtained by the intersection of three linear line complexes corresponding to a specific set of basis screws of a three-system. When determining the displacements of a rigid body from line specifications, a displacement screw is obtained by fitting a linear line complex to two or more line reguli. When two exact pairs of homologous lines are specified, we obtain a unique linear line complex, which determines the corresponding displacement screw. When more than two pairs of homologous lines with measurement errors are specified, it becomes a redundantly specified problem, and a linear line complex that has the best fit to more than two reguli is determined.

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.

Modeling of Elastically Coupled Bodies: Part II—Exponential and Generalized Coordinate Methods

1998 ◽  
Vol 120 (4) ◽  
pp. 501-506 ◽  
Author(s):  
Ernest D. Fasse ◽  
Peter C. Breedveld
Keyword(s):  
Rigid Body ◽  
Constitutive Equations ◽  
Companion Paper ◽  
Rigid Bodies ◽  
Geometric Modelling ◽  
Geometric Method ◽  
Generalized Coordinate

This paper looks at spatio-geometric modelling of elastically coupled rigid bodies. Two methods are presented. In the first method constitutive equations are derived by associating rigid body displacements with twist displacements and then generating wrenches proportional to the twist displacements. In the second method consistitutive equations are derived by associating rigid body displacements with generalized coordinate displacements, generating generalized forces proportional to the displacements, and then computing corresponding wrenches. The application of these methods and the geometric method presented in the companion paper are illustrated in a nontrivial example.

scholarly journals On the line geometry of rigid-body inertia

2014 ◽  
Vol 225 (11) ◽  
pp. 3073-3101 ◽  
Author(s):  
J. M. Selig ◽  
D. Martins
Keyword(s):  
Rigid Body ◽  
Line Geometry ◽  
Body Inertia
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