Applications of the 4D Geometric Algebra to Dimensional Mobility Criteria of Delassus-Parallelogram and Bennett Paradoxical Linkages

2015 ◽  
Author(s):  
Chung-Ching Lee
Keyword(s):  
Rigid Body ◽  
Geometric Algebra ◽  
Three Dimensional ◽  
Grassmann Algebra ◽  
Body Motion ◽  
Finite Rotation ◽  
Finite Rotations ◽  
Hypercomplex Number ◽  
Geometric Problems ◽  
Methodical Approach

Geometric algebra is also termed Clifford-Grassmann algebra or hypercomplex number. It allows studying space geometric problems in an easy and compact way. Transforming three-dimensional (3D) Euclidean geometric entities to actual elements of four-dimensional (4D) geometric algebra (abbreviated to g4) through a methodical approach of geometric algebra, one can describe motion displacements as even elements of g4. This article relies on the combined rotation and translation in g4 to establish the dimensional constraints of two non-exceptional overconstrained paradoxical linkages. Firstly, fundamentals of geometric algebra are recalled. Then, the single finite rotation and the composition of two successive finite rotations are introduced. After that, a general rigid-body motion in g4 is revealed for a possible application in exploring paradoxical chains using the geometric algebra. Finally, the metric or dimensional mobility criteria of Delassus-parallelogram four-screw and Bennett four-revolute paradoxical linkages are algebraically verified.

Segmental motion and deformation of transmurally infarcted myocardium in acute postinfarct period

1995 ◽  
Vol 268 (3) ◽  
pp. H1304-H1312 ◽  
Author(s):  
J. A. Lima ◽  
V. A. Ferrari ◽  
N. Reichek ◽  
C. M. Kramer ◽  
L. Palmon ◽  
...  
Keyword(s):  
Rigid Body ◽  
Coronary Occlusion ◽  
Three Dimensional ◽  
Magnetic Resonance Images ◽  
Body Motion ◽  
Segmental Motion ◽  
Infarcted Myocardium ◽  
Tissue Tagging ◽  
Coronary Ligation ◽  
Principal Strains

Mechanical behavior of infarcted myocardium in the first week following coronary occlusion has not been well characterized. Prior unidimensional studies failed to account for perpendicular deformation or shearing. This study characterizes three-dimensional motion and deformation of transmural infarcts 1 wk after coronary ligation in seven sheep. Principal strains and systolic in-plane translation and rotation were calculated for triangular elements defined by tissue tagging in short- and long-axis magnetic resonance images. The magnitudes of the first and second principal strains were reduced in both the short- and long-axis planes 1 wk after infarction. In addition, the absolute angular difference between the direction of the first principal strain and the radial direction increased from 14.7 +/- 1.9 to 43.5 +/- 2.7 degrees in the short-axis plane and from 19.6 +/- 7.3 to 43.9 +/- 10.0 degrees (P < 0.05) in the long-axis plane. In-plane rigid-body translation and rotation were also reduced in both planes. In conclusion, marked reduction and reorientation of principal strains and reduction in segmental rigid-body motion characterize nonreperfused transmural myocardial infarctions 1 wk after coronary occlusion.

Active Control of Elastic and Rigid Body Response of a Three Dimensional Underwater Structure

1995 ◽  
Vol 117 (1) ◽  
pp. 30-37 ◽  
Author(s):  
H. Suzuki ◽  
K. Yoshida ◽  
K. Watanabe
Keyword(s):  
Numerical Model ◽  
Rigid Body ◽  
Active Control ◽  
Structural Model ◽  
Scale Up ◽  
Three Dimensional ◽  
Basic Research ◽  
Weather Conditions ◽  
Body Motion ◽  
Elastic Response

One key technology for the offshore development of the increasing water depth will be remotely operated installation and construction of flexible structure in the deep water or on the seabed. The flexibility comes from scale-up or weight reduction of the structure. Conventional operation from the sea surface is affected by the weather conditions, and, therefore, not so efficient. This paper presents basic research on active control of elastic response and rigid body motion of an underwater elastic structure toward the remotely operated installation technique. The numerical model of the dynamics of the structural model is formulated, and based on the numerical model the control is formulated. The formulated control is tested by computer simulations and model experiments. The structural model is propelled by thrusters and taken from initial position to another position, while the elastic responses are controlled by variable buoyancy-type actuators.

Position and Attitude Tracking Control Law Design Using Geometric Algebra

2014 ◽  
Vol 602-605 ◽  
pp. 1113-1116
Author(s):  
Di Min Wu ◽  
Zhen Jing Li ◽  
Bin Li ◽  
Yu Xia Chen ◽  
Li Li
Keyword(s):  
Rigid Body ◽  
Geometric Algebra ◽  
Tracking Control ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Control Law ◽  
Proportional Control ◽  
Attitude Tracking ◽  
Attitude Tracking Control ◽  
The Stability

A position and attitude tracking control law is developed using geometric algebra (GA). The rigid body motion can be represented by the screw versor (or motor) in GA. Using the kinematics of the motor, the tracking control law of the rigid body motion can be formulated similar to the proportional control law. This paper provides a GA-based position and attitude tracking control law by using the negative feedback of the motor logarithm. The stability of the control law is validated by the numerical simulation.

Optimizing spherical navigator echoes for three-dimensional rigid-body motion detection

2005 ◽  
Vol 53 (5) ◽  
pp. 1080-1087 ◽  
Author(s):  
Daniel W. Petrie ◽  
Andreu F. Costa ◽  
Atsushi Takahashi ◽  
Yi-Fen Yen ◽  
Maria Drangova
Keyword(s):  
Rigid Body ◽  
Motion Detection ◽  
Three Dimensional ◽  
Rigid Body Motion ◽  
Body Motion

scholarly journals Three-dimensional phase field sintering simulations accounting for the rigid-body motion of individual grains

2021 ◽  
Vol 186 ◽  
pp. 109963
Author(s):  
Robert Termuhlen ◽  
Xanthippi Chatzistavrou ◽  
Jason D. Nicholas ◽  
Hui-Chia Yu
Keyword(s):  
Rigid Body ◽  
Phase Field ◽  
Three Dimensional ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Individual Grains

Control Law Design Based on Geometric Algebra

2013 ◽  
Vol 347-350 ◽  
pp. 496-500 ◽  
Author(s):  
Bin Li ◽  
Qian Shou Liu ◽  
Di Min Wu ◽  
Zi Hui Zhang ◽  
Yang Ke Zhou
Keyword(s):  
Numerical Simulation ◽  
Rigid Body ◽  
Geometric Algebra ◽  
Attitude Control ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Control Law ◽  
Lyapunov Theorem ◽  
Computational System ◽  
The Stability

A position and attitude control law is developed using geometric algebra (GA). GA is a powerful representational and computational system for geometry. The rigid body motion can be represented by the versor product in GA framework. Using the kinematics of the motor (the versor which represents the rigid body motion in GA), the control law of the rigid body motion can be developed. This paper provides a GA-based position and attitude control law by using the negative feedback of the motor. The stability of the control law is validated by the Lyapunov theorem and the numerical simulation.

scholarly journals Estimation of the Rigid-Body Motion From Three-Dimensional Images Using a Generalized Center-of-Mass Points Approach

2006 ◽  
Vol 53 (5) ◽  
pp. 2712-2718 ◽  
Author(s):  
B. Feng ◽  
P.P. Bruyant ◽  
P.H. Pretorius ◽  
R.D. Beach ◽  
H.C. Gifford ◽  
...  
Keyword(s):  
Rigid Body ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Three Dimensional Images ◽  
Generalized Center
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