scholarly journals Advances in Polynomial Continuation for Solving Problems in Kinematics

2004 ◽  
Vol 126 (2) ◽  
pp. 262-268 ◽  
Author(s):  
Andrew J. Sommese ◽  
Jan Verschelde ◽  
Charles W. Wampler
Keyword(s):  
Mechanical Systems ◽  
Robotic Manipulators ◽  
Degree Of Freedom ◽  
Polynomial Equations ◽  
Dimensional Solution ◽  
Solution Sets ◽  
The Past ◽  
Higher Dimensional ◽  
Isolated Roots ◽  
System Of Polynomial Equations

For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.

scholarly journals Multiprojective witness sets and a trace test

2020 ◽  
Vol 20 (3) ◽  
pp. 297-318 ◽  
Author(s):  
Jonathan D. Hauenstein ◽  
Jose Israel Rodriguez
Keyword(s):  
Algebraic Geometry ◽  
Polynomial Equations ◽  
Numerical Algebraic Geometry ◽  
Dimensional Solution ◽  
New Approach ◽  
Solution Sets ◽  
Irreducible Decomposition ◽  
Case Examples ◽  
Trace Test ◽  
Systems Of Polynomial Equations

AbstractIn the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalise the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.

A Method for Identifying Parameters of Mechanical Joints

1990 ◽  
Vol 57 (2) ◽  
pp. 337-342 ◽  
Author(s):  
J. Wang ◽  
P. Sas
Keyword(s):  
Physical Parameter ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Modal Testing ◽  
Physical Parameters ◽  
Mechanical Joints ◽  
Joint Parameters

A method for identifying the physical parameters of joints in mechanical systems is presented. In the method, a multi-d.o.f. (degree-of-freedom) system is transformed into several single d.o.f. systems using selected eigenvectors. With the result from modal testing, each single d.o.f. system is used to solve for a pair of unknown physical parameters. For complicated cases where the exact eigenvector cannot be obtained, it will be proven that a particular physical parameter has a stationary value in the neighborhood of an eigenvector. Therefore, a good approximation for a joint physical parameter can be obtained by using an approximate eigenvector and the exact value for the joint parameters can be reached by carrying out this process in an iterative way.

scholarly journals A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

2009 ◽  
Vol 47 (5) ◽  
pp. 3608-3623 ◽  
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Solution Set ◽  
Polynomial Equations ◽  
Local Dimension ◽  
System Of Polynomial Equations

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

scholarly journals Where Was Past Low-Entropy?

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 466
Author(s):  
Carlo Rovelli
Keyword(s):  
Early Universe ◽  
Scale Factor ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Dominant Source ◽  
Single Degree ◽  
The Past ◽  
Low Entropy

Where was the past low-entropy of the early universe located? Contrary to some popular answers, I argue that that the dominant source of low-entropy is the fact that a single degree of freedom, the scale factor, was not at equilibrium. I also discuss possible interpretations of the “improbability” of this early low-entropy.

scholarly journals Smart and Modular

2011 ◽  
Vol 133 (09) ◽  
pp. 48-51
Author(s):  
Harry H. Cheng ◽  
Graham Ryland ◽  
David Ko ◽  
Kevin Gucwa ◽  
Stephen Nestinger
Keyword(s):  
Degrees Of Freedom ◽  
Robotic System ◽  
Search And Rescue ◽  
Degree Of Freedom ◽  
Robotic Systems ◽  
Modular Robots ◽  
Modular Robot ◽  
Modular Systems ◽  
The Past ◽  
Three Degrees Of Freedom

This article discusses the advantages of a modular robot that can reassemble itself for different tasks. Modular robots are composed of multiple, linked modules. Although individual modules can move on their own, the greatest advantage of modular systems is their structural reconfigurability. Modules can be combined and assembled to form configurations for specific tasks and then reassembled to suit other tasks. Modular robotic systems are also very well suited for dynamic and unpredictable application areas such as search and rescue operations. Modular robots can be reconfigured to suit various situations. Quite a number of modular robotic system prototypes have been developed and studied in the past, each containing unique geometries and capabilities. In some systems, a module only has one degree of freedom. In order to exhibit practical functionality, multiple interconnected modules are required. Other modular robotic systems use more complicated modules with two or three degrees of freedom. However, in most of these systems, a single module is incapable of certain fundamental locomotive behaviors, such as turning.

Cayley kinematics and the Cayley form of dynamic equations

2005 ◽  
Vol 461 (2055) ◽  
pp. 761-781 ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Matrix Form ◽  
Dynamic Equations ◽  
Cayley Transform ◽  
Higher Dimensional ◽  
The Matrix ◽  
General Motion

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

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