Real solutions of univariate polynomials

2011 ◽  
pp. 13-23
Author(s):  
Frank Sottile
Keyword(s):  
Real Solutions ◽  
Univariate Polynomials

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>

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>

scholarly journals On the Effect of Control-Point Spacing on the Multisolution Phenomenon in the P3P Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
M. Vynnycky ◽  
G. M. M. Reddy
Keyword(s):  
Image Analysis ◽  
Camera Calibration ◽  
Pose Estimation ◽  
Computer Animation ◽  
Solution Structure ◽  
Control Point ◽  
Quartic Equation ◽  
Real Solutions ◽  
Algebraic Analysis ◽  
And Robotics

The perspective 3-point (P3P) problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis, and robotics. One possibility is to formulate it mathematically in terms of finding the solution to a quartic equation. However, there is yet no quantitative knowledge as to how control-point spacing affects the solution structure—in particular, the multisolution phenomenon. Here, we consider this problem through an algebraic analysis of the quartic’s coefficients and its discriminant and find that there are significant variations in the likelihood of two or four solutions, depending on how the spacing is chosen. The analysis indicates that although it is never possible to remove the occurrence of the four-solution case completely, it could be possible to choose spacings that would maximize the occurrence of two real solutions. Moreover, control-point spacing is found to impact significantly on the reality conditions for the solution of the quartic equation.

Geometric Design of Symmetric 3-RRS Constrained Parallel Platforms

2004 ◽  
Author(s):  
Eric Wolbrecht ◽  
Hai-Jun Su ◽  
Alba Perez ◽  
J. Michael McCarthy
Keyword(s):  
Geometric Design ◽  
Homotopy Continuation ◽  
Total Degree ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Kinematic Synthesis ◽  
Real Solutions ◽  
Parallel Platform ◽  
Three Degree Of Freedom ◽  
Serial Chains

The paper presents the kinematic synthesis of a symmetric parallel platform supported by three RRS serial chains. The dimensional synthesis of this three degree-of-freedom system is obtained using design equations for each of three RRS chains obtained by requiring that they reach a specified set of task positions. The result is 10 polynomial equations in 10 unknowns, which is solved using polynomial homotopy continuation. An example is provided in which the direction of the first revolute joint (2 parameters) and the z component of the base and platform are specified as well as the two task positions. The system of polynomials has a total degree of 4096 which means that in theory it can have as many solutions. Our example has 70 real solutions that define 70 different symmetric platforms that can reach the specified positions.

An Inverse Force Analysis of a Planar Two-Spring System

1994 ◽  
Author(s):  
Thomas M. Pigoski ◽  
Joseph Duffy
Keyword(s):  
The Other ◽  
Spring Stiffness ◽  
Force Analysis ◽  
Degree Polynomial ◽  
Real Solutions ◽  
Variable Elimination ◽  
Spring System ◽  
The Common ◽  
Resultant Length ◽  
Inverse Force

Abstract A closed-form inverse force analysis was performed on a planar two-spring system. The two springs were grounded to pivots at one end and attached to a common pivot at the other. A known force was applied to the common pivot of the system, and it was required to determine all of the assembly configurations. By variable elimination, a sixth degree polynomial in the resultant length of one spring was derived, and from this, six real solutions of the point of application of force were obtained. Following this, the applied force was incremented along a line and the six paths of the moving pivot were tracked starting from the zero-load configurations. An analysis of these results showed stability phenomena indicating the workspace of this system contained regions of negative spring stiffness and points of catastrophe.

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