A General Methodology for the Forward Kinematic Problem of Symmetrical Parallel Mechanisms and Application to 5-PRUR Parallel Mechanisms (3T2R)

2010 ◽  
Author(s):  
Mehdi Tale Masouleh ◽  
Manfred Husty ◽  
Cle´ment Gosselin
Keyword(s):  
Dimensional Space ◽  
Large Set ◽  
Parallel Mechanisms ◽  
General Methodology ◽  
Higher Dimensional ◽  
Kinematic Mapping ◽  
Rigid Body Displacement ◽  
Forward Kinematic Problem ◽  
Dimensional Projective Space ◽  
Forward Kinematic

In this paper, a general methodology is introduced in order to formulate the FKP of symmetrical parallel mechanisms in a 7-dimensional projective space by the means of the so-called Study’s parameters. The main objective is to consider rigid-body displacement, and consequently the FKP, based on algebraic geometry, rather than rely on classical recipes, such as Euler angles, to assist in problem-solving. The state of the art presented in this paper is general and can be extended to other types of symmetrical mechanisms. In this paper, we limit the concept of kinematic mapping to topologically symmetrical mechanisms, i.e., mechanisms with limbs having identical kinematic arrangement. Exploring the FKP in a higher dimensional space is more challenging since it requires the use of a larger number of coordinates. There are, however, advantages in adopting a large set of coordinates, since this approach leads to expressions with lower degree that do not involve trigonometric functions.

Forward kinematic problem of 5-RPUR parallel mechanisms (3T2R) with identical limb structures

2011 ◽  
Vol 46 (7) ◽  
pp. 945-959 ◽  
Author(s):  
Mehdi Tale Masouleh ◽  
Clément Gosselin ◽  
Manfred Husty ◽  
Dominic R. Walter
Keyword(s):  
Parallel Mechanisms ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

scholarly journals Constraint Equations for a Planar Parallel Platform

2021 ◽  
Author(s):  
◽  
Amani Ahmed Otaif
Keyword(s):  
Gröbner Basis ◽  
Linear Constraints ◽  
Grobner Basis ◽  
Real Solutions ◽  
Constraint Equations ◽  
Kinematic Mapping ◽  
Univariate Polynomials ◽  
Forward Kinematic Problem ◽  
Bézout’S Theorem ◽  
Forward Kinematic

<p>The aim of this thesis is to apply the Grünwald–Blaschke kinematic mapping to standard types of parallel general planar three-legged platforms in order to obtain the univariate polynomials which provide the solution of the forward kinematic problem. We rely on the method of Gröbner basis to reach these univariate polynomials. The Gröbner basis is determined from the constraint equations of the three legs of the platforms. The degrees of these polynomials are examined geometrically based on Bezout’s Theorem. The principle conclusion is that the univariate polynomials for the symmetric platforms under circular constraints are of degree six, which describe the maximum number of real solutions. The univariate polynomials for the symmetric platforms under linear constraints are of degree two, that describe the maximum number of real solutions.</p>

Solving the Forward Kinematic Problem of 4-DOF Parallel Mechanisms (3T1R) With Identical Limb Structures and Revolute Actuators Using the Linear Implicitization Algorithm

2011 ◽  
Author(s):  
Mehdi Tale Masouleh ◽  
Dominic R. Walter ◽  
Manfred Husty ◽  
Cle´ment Gosselin
Keyword(s):  
Linear Equations ◽  
Linear Constraint ◽  
Mathematical Framework ◽  
Parallel Mechanisms ◽  
Novel Approach ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Rotation Motion ◽  
Forward Kinematic ◽  
General Architecture

This paper investigates the forward kinematic problem of 4-DOF parallel mechanisms with revolute actuators and identical limb structures and performing a three translations and one rotation motion pattern. The general architecture of all the mechanisms under study in this paper originates from the type synthesis performed for 4-DOF parallel mechanisms with identical limb structures. The mathematical framework used in this paper is based on algebraic geometry where the forward kinematics and constraint expressions are explored in a seven-dimensional kinematic space by means of the so-called Study parameters (dual quaternions). In this paper, the algorithm applied for obtaining the forward kinematic and constraint expressions is based on a recent and novel approach, called linear implicitization algorithm, which is based on solving systematically a system of linear equations to determine the coefficients of the non-linear constraint equations. This paper presents also an example of a 4-DOF parallel mechanism.

scholarly journals Constraint Equations for a Planar Parallel Platform

2021 ◽  
Author(s):  
◽  
Amani Ahmed Otaif
Keyword(s):  
Gröbner Basis ◽  
Linear Constraints ◽  
Grobner Basis ◽  
Real Solutions ◽  
Constraint Equations ◽  
Kinematic Mapping ◽  
Univariate Polynomials ◽  
Forward Kinematic Problem ◽  
Bézout’S Theorem ◽  
Forward Kinematic

<p>The aim of this thesis is to apply the Grünwald–Blaschke kinematic mapping to standard types of parallel general planar three-legged platforms in order to obtain the univariate polynomials which provide the solution of the forward kinematic problem. We rely on the method of Gröbner basis to reach these univariate polynomials. The Gröbner basis is determined from the constraint equations of the three legs of the platforms. The degrees of these polynomials are examined geometrically based on Bezout’s Theorem. The principle conclusion is that the univariate polynomials for the symmetric platforms under circular constraints are of degree six, which describe the maximum number of real solutions. The univariate polynomials for the symmetric platforms under linear constraints are of degree two, that describe the maximum number of real solutions.</p>

Unified Kinematic Analysis of General Planar Parallel Manipulators

2004 ◽  
Vol 126 (5) ◽  
pp. 866-874 ◽  
Author(s):  
M. J. D. Hayes ◽  
P. J. Zsombor-Murray ◽  
C. Chen
Keyword(s):  
Parallel Manipulators ◽  
Inverse Kinematic ◽  
Parallel Robots ◽  
Kinematic Mapping ◽  
Forward Kinematic Problem ◽  
Forward Kinematic ◽  
Three Degree Of Freedom ◽  
Generalized Constraint ◽  
First Time ◽  
Planar Parallel Manipulators

A kinematic mapping of planar displacements is used to derive generalized constraint equations having the form of ruled quadric surfaces in the image space. The forward kinematic problem for all three-legged, three-degree-of-freedom planar parallel manipulators thus reduces to determining the points of intersection of three of these constraint surfaces, one corresponding to each leg. The inverse kinematic solutions, though trivial, are implicit in the formulation of the constraint surface equations. Herein the forward kinematic solutions of planar parallel robots with arbitrary, mixed leg architecture are exposed completely, and in a unified way, for the first time.

Kinematic effects of number of legs in 6-DOF UPS parallel mechanisms

Robotica ◽  
2017 ◽  
Vol 35 (12) ◽  
pp. 2257-2277 ◽  
Author(s):  
Mohammad H. Abedinnasab ◽  
Farzam Farahmand ◽  
Bahram Tarvirdizadeh ◽  
Hassan Zohoor ◽  
Jaime Gallardo-Alvarado
Keyword(s):  
Parallel Manipulators ◽  
Optimum Number ◽  
Parallel Mechanisms ◽  
Performance Indices ◽  
Singular Configurations ◽  
Redundantly Actuated ◽  
Forward Kinematic Problem ◽  
Kinematic Measures ◽  
Forward Kinematic ◽  
Legged Mechanism

SUMMARYIn this paper, we study the kinematic effects of number of legs in 6-DOF UPS parallel manipulators. A group of 3-, 4-, and 6-legged mechanisms are evaluated in terms of the kinematic performance indices, workspace, singular configurations, and forward kinematic solutions. Results show that the optimum number of legs varies due to priorities in kinematic measures in different applications. The non-symmetric Wide-Open mechanism enjoys the largest workspace, while the well-known Gough–Stewart (3–3) platform retains the highest dexterity. Especially, the redundantly actuated 4-legged mechanism has several important advantages over its non-redundant counterparts and different architectures of Gough–Stewart platform. It has dramatically less singular configurations, a higher manipulability, and at the same time less sensitivity. It is also shown that the forward kinematic problem has 40, 16, and 1 solution(s), respectively for the 6-, 3-, and the 4-legged mechanisms. Superior capabilities of the 4-legged mechanism make it a perfect candidate to be used in more challenging 6-DOF applications in assembly, manufacturing, biomedical, and space technologies.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

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