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>

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

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.

On the Gröbner basis triangularization of constraint equations in natural coordinates

2013 ◽  
Vol 31 (3) ◽  
pp. 371-392 ◽  
Author(s):  
Thomas Uchida ◽  
Alfonso Callejo ◽  
Javier García de Jalón ◽  
John McPhee
Keyword(s):  
Gröbner Basis ◽  
Natural Coordinates ◽  
Grobner Basis ◽  
Constraint Equations

scholarly journals A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method

Information ◽  
2020 ◽  
Vol 11 (5) ◽  
pp. 246
Author(s):  
Honghui He ◽  
Jinzhao Wu
Keyword(s):  
Hybrid Systems ◽  
Gröbner Basis ◽  
State Variables ◽  
Grobner Basis ◽  
Constraint Equations ◽  
Free Variables ◽  
Special Cases ◽  
Finite Set ◽  
Algebraic Constraints ◽  
System Variables

In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and directly solve for the complete inductive conditions. An instance means the verification of a proposition after instantiating free variables to numbers. A lattice array is a key notion in this paper, which is essentially a finite set of instances. Verifying that a proposition holds over a Lattice Array suffices to prove that the proposition holds in general; this interesting feature inspires us to present CIM. On one hand, instead of computing a comprehensive Gröbner basis, CIM uses a Lattice Array to generate the constraints in parallel. On the other hand, we can make a clever use of the parallelism of CIM to start with some constraint equations which can be solved easily, in order to determine some parameters in an early state. These solved parameters benefit the solution of the rest of the constraint equations; this process is similar to the domino effect. Therefore, the constraint-solving tractability of the proposed method is strong. We show that some existing approaches are only special cases of our method. Moreover, it turns out CIM is more efficient than existing approaches under parallel circumstances. Some examples are presented to illustrate the practicality of our method.

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.

scholarly journals On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

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