Some New Results on the Kinematics of 3R Serial Robots Using Nested Determinants

2017 ◽  
Author(s):  
Federico Thomas ◽  
M. Alba Perez-Gracia
Keyword(s):  
Computer Algebra ◽  
Inverse Kinematics ◽  
High Order ◽  
Computer Algebra Systems ◽  
Quadratic Polynomials ◽  
Serial Robots ◽  
Singularity Locus

It has been recently shown that the singularity locus of a 3R robot, and in particular its nodes and cusps, can be algebraically characterized in terms of nested determinants. This neat and structured formulation contrasts with the huge and often meaningless formulas generated using computer algebra systems. In this paper we explore further this kind of formulation. We present two new results which we think are of interest by themselves. First, it is shown how Chrystal’s method, used to obtain the resultant of two quadratic polynomials, can be formulated as nested determinants. Second, it is also shown how the coefficients of the harmonic conic of two given conics, can also be expressed in the same form. These results lead to new formulations for the inverse kinematics of 3R robots, their singularity loci, their nodes, and some of their high-order singularities.

A New Approach to Obtaining Closed-Form Solutions for Higher Order Tetrahedral Finite Elements Using Modern Computer Algebra Systems

2011 ◽  
Author(s):  
Sara E. McCaslin ◽  
Panos S. Shiakolas ◽  
Brian H. Dennis ◽  
Kent L. Lawrence
Keyword(s):  
Parallel Processing ◽  
Closed Form ◽  
Computer Algebra ◽  
High Order ◽  
Computer Algebra Systems ◽  
Element Analysis ◽  
New Approach ◽  
Closed Form Solutions ◽  
Modern Computer ◽  
Stiffness Matrices

Closed-form solutions for straight-sided tetrahedral element stiffness matrices used in finite element analysis have been proven more efficient than numerically integrated solutions. These closed-form solutions are symbolically integrated using computer algebra systems such as Mathematica or Maple. However, even with memory and processing speed available on desktop computers today, major hindrances exist when attempting to symbolically evaluate the stiffness matrices for high order elements. This research proposes a new approach to obtaining closed-form solutions. Results are presented that demonstrate the feasibility of obtaining the stiffness matrices for high order tetrahedral elements through p-level 9 by use of parallel processing tools in Mathematica 7. Comparisons are made between serial and parallel approaches based on memory required to generate a solution. The serial approach requires more memory and can only generate closed-form solutions up to 7th order. The parallel processing approach presented requires less memory and can generate solutions up to 9th order.

scholarly journals New ideas for handling of loop and angular integrals in D-dimensions in QCD

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Valery E. Lyubovitskij ◽  
Fabian Wunder ◽  
Alexey S. Zhevlakov
Keyword(s):  
Computer Algebra ◽  
Cross Sections ◽  
Orthogonal Basis ◽  
Computer Algebra Systems ◽  
Covariant Formalism ◽  
Linear Combinations ◽  
Recursion Relations ◽  
New Ideas ◽  
Rotational Invariant ◽  
Geometric Picture

Abstract We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of Fi24

2010 ◽  
Vol 17 (03) ◽  
pp. 389-414 ◽  
Author(s):  
Faryad Ali ◽  
Jamshid Moori
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Maximal Subgroup ◽  
Computer Algebra ◽  
Character Table ◽  
Computer Algebra Systems ◽  
Split Extension ◽  
Sporadic Group ◽  
Local Subgroup ◽  
Fischer Group

The Fischer group [Formula: see text] is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA.

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