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.

Characterization of Instantaneous Point-Line Motions

2004 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Rigid Bodies ◽  
Dual Quaternion ◽  
Third Order ◽  
Displacement Operator ◽  
Instantaneous Point ◽  
Rigid Body Motions ◽  
Point Line ◽  
Derivatives Of

This article discusses systematically the characterization of instantaneous point-line motions, and the higher-order relationship between a point-line motion and the associated rigid body motions. The transformation of a point-line between two positions is depicted as a pure translation along the point-line followed by a screw displacement about their common normal and expressed with a unit dual quaternion referred to as the point-line displacement operator. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. Such a treatment leads to a consistent expression or unified treatment for the transformation of lines, point-lines, and rigid bodies. The relationships between point-line motions and rigid body motions are addressed in detail up to the third order.

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 Research on the rigid body pose estimation using dual quaternions

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401882311
Author(s):  
Jing Li
Keyword(s):  
Rigid Body ◽  
Rigid Body Motion ◽  
Rotation Vector ◽  
Body Motion ◽  
Mathematical Tool ◽  
Dual Quaternion ◽  
Information Need ◽  
Traditional Algorithm ◽  
Relative Pose ◽  
Body Pose Estimation

The dual quaternion is the simplest and most effective mathematical tool to describe the translational and rotational motion of a general rigid body. Its computation and updating require screw vector. The relative pose information need to be updated when measuring the pose of the rigid body. In the traditional algorithm, it generally focuses on attitude updating, but less research on position updating. The rotation vector algorithm is used to represent the quaternion to update the attitude of the rigid body, but it cannot update the position. Because any general rigid body motion can be realized by rotation about a certain axis and translation along this axis, this article proposes an algorithm to update the position and attitude of the rigid body’s relative motion based on the screw vector. The rotation vector and screw vector are introduced in the rigid body motion and update the quaternion and dual quaternion, respectively; then, the relative pose information of the leader–follower rigid body based on the screw vector algorithm is deduced. The single-sample, two-sample, and three-sample algorithms are compared and simulated, and the simulation results show that this method not only overcomes the deficiencies associated with the separate updating of position and attitude using a traditional algorithm but also has higher precision than the traditional algorithm.

Paradoxes in Rigid-Body Kinematics of Structures

1998 ◽  
Vol 65 (1) ◽  
pp. 218-222
Author(s):  
L. Mentrasti
Keyword(s):  
Rigid Body ◽  
Kinematic Analysis ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Kinematic Chains ◽  
Straight Line ◽  
Two Bodies

The paper discusses two paradoxes appearing in the kinematic analysis of interconnected rigid bodies: there are structures that formally satisfy the classical First and Second Theorem on kinematic chains, but do not have any motion. This can arise when some centers of instantaneous rotation (CIR) relevant to two bodies coincide with each other (first kind paradox) or when the CIRs relevant to three bodies lie on a straight line (second kind paradox). In these cases two sets of new theorems on the CIRs can be applied, pointing out sufficient conditions for the nonexistence of a rigid-body motion. The question is clarified by applying the presented theory to several examples.

The Design of an Augmented Reality Based Rigid Body Motion Experiment System

2015 ◽  
Vol 719-720 ◽  
pp. 275-278
Author(s):  
Yang Xiang Zhang ◽  
Yun Zhu ◽  
Hui Ye
Keyword(s):  
Augmented Reality ◽  
Rigid Body ◽  
Real Time ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Motion Information ◽  
Experiment System ◽  
The Real ◽  
Physical Motion

The authors have designed an augmented reality based rigid body motion experiment system (ARRBMES), which could capture the physical motion information from the interaction between the users and virtual rigid bodies in real-time. The launcher and the container of the rigid bodies in this system are all positioned by tag cards, and the initial physical quantities of the rigid bodies are captured and analyzed through the motion information of the launcher tag card. Then ARRBMES will realize the real-time display of rigid body motion and collision events in a virtual-real fusion environment. ARRBMES can simulate the motion of rigid bodies in ideal state which cannot be achieved in the real world. As a result, the users can obtain realistic experience and the system can increase their physical intuition and cognitive experience. Moreover, ARRBMES can obtain physical information from the interaction in real-time between users and the system, which makes it a special Cyber-Physical System.

On the geometry of rigid-body motions: The relation between Lie groups and screws

2002 ◽  
Vol 216 (1) ◽  
pp. 13-23 ◽  
Author(s):  
S Stramigioli ◽  
B Maschke ◽  
C Bidard
Keyword(s):  
Rigid Body ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Approach ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Group Approach ◽  
Rigid Body Motions

This paper gives a synthetic presentation of the geometry of rigid-body motion in a projective geometrical framework. An important issue is the geometric approach to the identification of twists and wrenches in a Lie group approach and their relation to screws. The paper presents a novel formal way to describe the spaces of lines, axials, polars and screws as subsets or subspaces of Lie algebras in order to make clear the relation between screw concepts and Lie group concepts.

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