Special Unitary Matrices in the Analysis and Synthesis of Spherical Linkages

2018 ◽  
Vol 11 (1) ◽  
Author(s):  
Saleh M. Almestiri ◽  
Andrew P. Murray ◽  
David H. Myszka
Keyword(s):  
Design Problem ◽  
Homotopy Continuation ◽  
Unitary Matrices ◽  
Spherical Mechanisms ◽  
Analysis And Synthesis ◽  
Spherical Linkages

This work seeks to systematically model and solve the equations associated with the kinematics of spherical mechanisms. The group of special unitary matrices, SU(2), is utilized throughout. Elements of SU(2) are employed here to analyze the three-roll wrist and the spherical Watt I linkage. Additionally, the five orientation synthesis of a spherical four-bar mechanism is solved, and solutions are found for the eight orientation synthesis of the Watt I linkage. Using SU(2) readily allows for the use of a homotopy-continuation-based solver, in this case Bertini. The use of Bertini is motivated by its capacity to calculate every solution to a design problem.

Spherical Linkages Analysis and Synthesis by Special Unitary Matrices for Solution via Numerical Algebraic Geometry

2017 ◽  
Author(s):  
Saleh M. Almestiri ◽  
Andrew P. Murray ◽  
David H. Myszka
Keyword(s):  
Algebraic Geometry ◽  
Complex Numbers ◽  
Numerical Algebraic Geometry ◽  
Unknown Parameters ◽  
Unitary Matrices ◽  
Analysis And Design ◽  
Analysis And Synthesis ◽  
Spherical Linkages ◽  
Four Bar Linkage ◽  
Systems Of Polynomial Equations

Numerical algebraic geometry is the field that studies the computation and manipulation of the solution sets of systems of polynomial equations. The goal of this paper is to formulate spherical linkages analysis and design problems via a method suited to employ the tools of numerical algebraic geometry. Specifically, equations are developed using special unitary matrices that naturally use complex numbers to express physical and joint parameters in a mechanical system. Unknown parameters expressed as complex numbers readily admit solution by the methods of numerical algebraic geometry. This work illustrates their use by analyzing the spherical four-bar and Watt I linkages. In addition, special unitary matrices are utilized to solve the five orientation synthesis of a spherical four-bar linkage. Moreover, synthesis equations were formulated for the Watt I linkage and implemented for an eight orientation task. Results obtained from this method are validated by comparison to other published work.

Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Homotopy Continuation

2002 ◽  
Vol 124 (4) ◽  
pp. 652-661 ◽  
Author(s):  
Eric Lee ◽  
Constantinos Mavroidis
Keyword(s):  
Design Problem ◽  
Free Choice ◽  
Robot Manipulators ◽  
Continuation Method ◽  
Geometric Design ◽  
Design Parameters ◽  
Homotopy Continuation ◽  
Serial Link ◽  
End Effector ◽  
Homotopy Continuation Method

In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints is solved using a polynomial homotopy continuation method. Three spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3-R manipulator are computed so that the manipulator will be able to place its end-effector at these three pre-specified locations. Denavit and Hartenberg parameters and 4×4 homogeneous matrices are used to formulate the problem and obtain eighteen design equations in twenty-four design unknowns. Six of the design parameters are set as free choices and their values are selected arbitrarily. Two different cases for selecting the free choices are considered and their design equations are solved using polynomial homotopy continuation. In both cases for free choice selection, eight distinct manipulators are found that will be able to place their end-effector at the three specified spatial positions and orientations.

scholarly journals Displacement Analysis of Spherical Mechanisms Having Three or Fewer Loops

2004 ◽  
Vol 126 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Inverse Kinematics ◽  
Solution Method ◽  
Degree Of Freedom ◽  
Serial Link ◽  
Single Degree Of Freedom ◽  
Displacement Analysis ◽  
Spherical Mechanisms ◽  
Common Center ◽  
Parallel Link ◽  
Spherical Linkages

Spherical linkages, having rotational joints whose axes coincide in a common center point, are sometimes used in multi-degree-of-freedom robot manipulators and in one-degree-of-freedom mechanisms. The forward kinematics of parallel-link robots, the inverse kinematics of serial-link robots and the input/output motion of single-degree-of-freedom mechanisms are all problems in displacement analysis. In this article, loop equations are formulated and solved for the displacement analysis of all spherical mechanisms up to three loops. We show how to solve each mechanism type using either a formulation in terms of rotation matrices or quaternions. In either formulation, the solution method is a modification of Sylvester’s elimination method, leading directly to numerical calculation via standard eigenvalue routines.

Position and Orientation Analyses of Mechanism by PHCpack Solver of Homotopy Continuation

2012 ◽  
Vol 152-154 ◽  
pp. 1779-1784
Author(s):  
Hong Bing Xin ◽  
Qiang Huang ◽  
Yue Qing Yu
Keyword(s):  
High Efficiency ◽  
Graphical Method ◽  
Continuation Method ◽  
Sufficient Accuracy ◽  
Homotopy Continuation ◽  
Analytic Method ◽  
Mechanism Analysis ◽  
Homotopy Continuation Method ◽  
Analysis And Synthesis ◽  
Position And Orientation

Mechanism analysis and synthesis commonly used analytic method and graphical method, etc., but the homotopy method has high efficiency in nonlinear analysis of mechanism. This paper applies the homotopy continuation method PHCpack solver to solve mechanism position and orientation, and draws the mechanism configuration according to the results obtained, the relative error of numerical results to exact solutions shows that the homotopy continuation method PHCpack solver can provide sufficient accuracy of mechanism position and orientation analysis, and meet the engineering needs.

Design Problem Solving

2006 ◽  
Vol 214 (2) ◽  
pp. 73-86 ◽  
Author(s):  
Constance Winkelmann ◽  
Winfried Hacker
Keyword(s):  
Problem Solving ◽  
Design Problem

Zusammenfassung. In zwei experimentellen Studien werden mögliche alternative Bedingungen der lösungsgüteverbessernden Wirkung einer Frage-Antwort-Technik im Entwurfsdenken (Design Problem Solving) geprüft. In der ersten Studie (N = 84) wurden differential- und kognitionspsychologische Bedingungen geprüft. Eine Aussagenanalyse in einer zweiten Studie (N = 94) sollte die nachgewiesenen Ergebnisse vertiefen. Für die in beiden Studien ermittelten Lösungsgüteverbesserungen konnten keine differentialpsychologischen Abhängigkeiten von Vorgehensstilen nachgewiesen werden. In kognitionspsychologischer Hinsicht zeigte sich, dass sowohl Fragen, die die Forderungen der Aufgabe in Erinnerung bringen, als auch Fragen, deren Beantwortung die Auslösung von Denkprozessen fördern, zur Lösungsgüteverbesserung beitragen. Größere Effekte werden bei der Kombination beider Fragenarten nahe gelegt. Die Ergebnisse der Aussagenanalyse weisen auf eine sowohl beschreibende als auch begründende und bewertende Vorgehensweise der Untersuchungsteilnehmer hin, die Verbesserungen erzielten. Fragen für weiterführende Untersuchungserfordernisse werden abgeleitet.

НАРАТИВ ЯК КОНТЕНТНА ОСНОВА МІЖНАРОДНИХ СТРАТЕГІЧНИХ КОМУНІКАЦІЙ

2020 ◽  
pp. 4
Author(s):  
N. Zykun ◽  
A. Bessarab ◽  
L. Ponomarenko
Keyword(s):  
Selection Method ◽  
Historical Narrative ◽  
Neutral Position ◽  
Main Task ◽  
Grand Narrative ◽  
National Narrative ◽  
Strategic Communications ◽  
Analysis And Synthesis ◽  
The Media ◽  
International Interaction

<p><em>The article, basing on the analysis of selected media texts with reference to narrative from the leading Ukrainian newspapers «Dzerkalo Tyzhnia» (Weekly Mirror), «Den» (Day), «Ukraina Moloda» (Young Ukraine) for 2016–2020, the semantic and content characteristics of the «narrative», «strategic narrative», «small narratives» nominations has established; the directions of the semantic realization of the meaning of the narrative and its possibilities in the process of international strategic communications aimed at both external and internal audience, are outlined. It is proved that the main task of a strategic, or national, narrative is a reasoned explanation to the state population and interested audiences of specific realities, intentions, plans; justification of certain directions of state activity aimed at partners, at opponents and those occupying a neutral position.</em></p><p><em>There are divided the spheres of use of different narratological nominations: in international communications and in scientific discourse, the conceptual foundations of state identity and international interaction are referred to as strategic narrative or grand narrative, in publicistic discourse the narrative nomination is used, more rarely – historical narrative, national narrative.</em></p><p><em>The scientific novelty of the research is that the focus is on the media aspect of the use of one of the key concepts of strategic communications and the role of the media in its implementation.</em></p><p><em>The main general scientific methods used in this article are descriptive and comparative ones, as well as analysis and synthesis. The following empirical methods were also used: solid selection method (solid selection method for allocation texts with the «narrative» lexeme; quantitative method of content analysis with elements of qualitative one – for characterizing the semantic of the «narrative» term).</em></p><p><em>The results of the study can be used in the complex research of the technology of international strategic communications and in the practical activity of specialists in international strategic communications, a new trend in Ukraine, which is currently under active institutionalization.</em></p><strong><em>Key words:</em></strong><em> international strategic communications, propaganda, narrative, strategic narrative, grand narrative, «small narratives».</em>

Sign in / Sign up

Export Citation Format

Share Document