The Position Analysis of Spherical Seven-Bar Mechanism

2011 ◽  
Vol 101-102 ◽  
pp. 193-196
Author(s):  
Zhao Feng Zhang ◽  
Zhi Huan Zhang
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Polynomial Equation ◽  
Constraint Equations ◽  
Position Analysis ◽  
Numerical Example ◽  
Location Parameters ◽  
Sylvester Resultant ◽  
Univariate Polynomial

In this paper, we turn plane seven-bar mechanism into spherical seven-bar mechanism, using quaternion to construct mathematical model for spherical seven-bar mechanism. Three constraint equations are obtained according to the angles constraint. Using Sylvester resultant elimination by two steps, a 32 degree univariate polynomial equation can be obtained. A numerical example confirms that analytical solutions of spherical seven-bar mechanism are 32 and with the help of Mathematic software to solve the location parameters.

Research on Position Analysis of a Kind of Nine-Link Barranov Truss

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Pin Wang ◽  
Qizheng Liao ◽  
Yufeng Zhuang ◽  
Shimin Wei
Keyword(s):  
Gaussian Elimination ◽  
Constraint Equation ◽  
Polynomial Equation ◽  
Vector Method ◽  
Constraint Equations ◽  
Position Analysis ◽  
Sylvester Resultant ◽  
Univariate Polynomial ◽  
New Equation ◽  
First Time

The position analysis of a nine-link Barranov truss is finished by using Dixon resultants together with Sylvester resultants. Above all, using vector method in complex plane to construct four constraint equations and transform them into complex exponential form, then three constraint equations are used to construct a 6×6 Dixon matrix, which contains two variables to be eliminated. We extract the greatest common divisor (GCD) of two columns of Dixon matrix and compute its determinant to obtain a new equation. This equation together with the fourth constraint equation can be used to construct a Sylvester resultant. A 50deg univariate polynomial equation is obtained from the determinant of Sylvester resultant. Other variables can be computed by Euclidean algorithm and Gaussian elimination. Lastly, a numerical example confirms that the analytical solution number of the Barranov truss is 50. It is the first time to complete analytical solutions of this kind of Barranov truss.

Polynomial Solution to the Five-Position Synthesis of Spatial CC Dyads via Dialytic Elimination

2000 ◽  
Author(s):  
Chintien Huang ◽  
Yu-Jui Chang
Keyword(s):  
Polynomial Solution ◽  
Polynomial Equation ◽  
Elimination Method ◽  
Real Solutions ◽  
Numerical Example ◽  
Univariate Polynomial ◽  
Position Synthesis

Abstract This paper presents a polynomial solution to the five-position synthesis of spatial cylindrical-cylindrical dyads. The solution procedures start with the simplification of the synthesis equations derived by Tsai and Roth. The simplified equations are solved by Sylvester’s dialytic elimination method to obtain a univariate polynomial equation of degree six, which gives at most 6 CC dyads for the five-position synthesis. A numerical example with six real solutions is provided.

Forward Kinematics in Polynomial Form of the General Stewart Platform1

1998 ◽  
Vol 123 (2) ◽  
pp. 254-260 ◽  
Author(s):  
Carlo Innocenti
Keyword(s):  
Parallel Manipulator ◽  
Polynomial Equation ◽  
Polynomial Form ◽  
Floating Point ◽  
Forward Kinematics ◽  
Numerical Example ◽  
Univariate Polynomial ◽  
General Geometry ◽  
Floating Point Computation

The paper presents a new algorithm to solve, in polynomial form, the forward kinematics of the general-geometry 6-6 fully-parallel manipulator. The forty solutions that the problem at hand admits in the complex domain are found by determining the roots of a 40th-order univariate polynomial equation. Unlike the existing algorithm, the proposed one is suitable for implementation in a standard floating-point computation environment. A numerical example shows application of the new algorithm to a case study.

Forward Position Analysis of Nearly General Stewart Platforms

1992 ◽  
Author(s):  
Change-de Zhang ◽  
Shin-Min Song
Keyword(s):  
Closed Form Solution ◽  
Polynomial Equation ◽  
Stewart Platform ◽  
Form Solution ◽  
Constraint Equations ◽  
Position Analysis ◽  
Order Polynomial ◽  
Moving Platform ◽  
Order Polynomial Equation ◽  
Simultaneous Elimination

Abstract This paper presents the closed-form solution of forward position analysis of the nearly general stewart platform, which consists of a base and a moving planar platforms connected by six extensible limbs through spherical joints in the two planar platforms. It becomes a general stewart platform if the centers are not constrained to those two planes. In this study, transformation matrix is used to represent the position of the moving platform. Based on the six dependency equations of the rotation matrix and the six constraint equations related to the six link lengths, a set of six 4-th degree equations in three unknowns are derived. Further derivations produce twenty-one dependent constraint equations. By simultaneous elimination of two unknowns a 20-th order polynomial equation in one unknown is obtained. Due to dual solutions of other unknowns, this indicates a maximum of forty possible solutions. The roots of this polynomial are solved numerically and the realistic solutions are constructed using computer graphics.

Direct position analysis of parallel manipulators which generate SP-2PS structures

Robotica ◽  
2005 ◽  
Vol 23 (4) ◽  
pp. 521-526 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Polynomial Equation ◽  
Analytic Form ◽  
Parallel Manipulators ◽  
Real Case ◽  
End Effector ◽  
Position Analysis ◽  
Parallel Structures ◽  
Univariate Polynomial

The determination of the assembly modes of the parallel structures with three legs of type PS or SP (P and S stand for prismatic pair and spherical pair, respectively) consists of solving the direct position analysis of all the three-legged parallel manipulators which have, in each leg, one not actuated prismatic pair, one not actuated spherical pair and one or two one-dof actuated pairs of any type, placed along the leg in any order. There are two types of such structures: (i) 3PS structures and (ii) SP-2PS structures. The procedure to determine the assembly modes of the SP-2PS structures has not been presented yet, in the literature. This paper presents the analytic form determination of the assembly modes of the SP-2PS structures. In particular, the closure equations of a generic SP-2PS structure will be written and their solution will be reduced to the solution of an eight-degree univariate polynomial equation with real coefficients. Finally, the proposed algorithm will be applied to a real case. The result of this study is that the assembly modes of any SP-2PS structure are at most eight, and the end-effector poses, which solve the direct position analysis of the parallel manipulators that generate those structures, are also eight.

Forward Position Analysis of Nearly General Stewart Platforms

1994 ◽  
Vol 116 (1) ◽  
pp. 54-60 ◽  
Author(s):  
Chang-de Zhang ◽  
Shin-Min Song
Keyword(s):  
Closed Form Solution ◽  
Polynomial Equation ◽  
Stewart Platform ◽  
Form Solution ◽  
Constraint Equations ◽  
Position Analysis ◽  
Order Polynomial ◽  
Moving Platform ◽  
Order Polynomial Equation ◽  
Simultaneous Elimination

This paper presents the closed-form solution of the forward position analysis of the nearly general Stewart platform, which consists of a base and a moving planar platform connected by six extensible limbs through spherical joints in the two planar platforms. It becomes a general Stewart platform if the centers are not constrained to those two planes. In this study, the coordinate transformation matrix is used to represent the position of the moving platform. Based on the six dependency equations of the rotation matrix and the six constraint equations related to the six link lengths, a set of six 4th degree equations in three unknowns are derived. Further derivations produce 21 dependent constraint equations. By simultaneous elimination of two unknowns a 20th order polynomial equation in one unknown is obtained. Due to dual solutions of other unknowns, this indicates a maximum of 40 possible solutions. The roots of this polynomial are then solved numerically and the realistic solutions are constructed using computer graphics.

Forward Kinematics in Polynomial Form of the General Stewart Platform

1998 ◽  
Author(s):  
Carlo Innocenti
Keyword(s):  
Parallel Manipulator ◽  
Polynomial Equation ◽  
Polynomial Form ◽  
Floating Point ◽  
Forward Kinematics ◽  
Numerical Example ◽  
Univariate Polynomial ◽  
General Geometry ◽  
Floating Point Computation

Abstract The paper presents a new algorithm to solve, in polynomial form, the forward kinematics of the general-geometry 6-6 fully-parallel manipulator. The forty solutions that the problem at hand admits in the complex domain are found by determining the roots of a 40th-order univariate polynomial equation. Unlike the existing algorithm, the proposed one is suitable for implementation in a standard floating-point computation environment. A numerical example shows application of the new algorithm to a case study.

scholarly journals Forward Displacement Analysis of a General 6-3 Stewart Platform Using Conformal Geometric Algebra

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Feng Wei ◽  
Shimin Wei ◽  
Ying Zhang ◽  
Qizheng Liao
Keyword(s):  
Geometric Algebra ◽  
Polynomial Equation ◽  
Stewart Platform ◽  
Conformal Geometric Algebra ◽  
Kinematic Constraint ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Sylvester Resultant ◽  
Univariate Polynomial ◽  
Forward Displacement Analysis

In this paper, a new algorithm for the forward displacement analysis of a general 6-3 Stewart platform (6-3SPS) based on conformal geometric algebra (CGA) is presented. First, a 6-3SPS structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on the geometric characteristics, one of which is built according to the point characteristic four-ball intersection in CGA. A 16th-degree univariate polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination. Finally, a numerical example is given to verify the algorithm.

scholarly journals Modelling time efficiency of cobot-supported kit preparation

2019 ◽  
Vol 106 (5-6) ◽  
pp. 2227-2241 ◽  
Author(s):  
Patrik Fager ◽  
Martina Calzavara ◽  
Fabio Sgarbossa
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Cycle Time ◽  
Laboratory Experiments ◽  
Spare Parts ◽  
Batch Preparation ◽  
Assembly Quality ◽  
Numerical Example ◽  
Business To Business ◽  
Support Time

AbstractKitting – meaning to supply assembly with components in presorted kits – is widely seen as beneficial for assembly quality and efficiency when there is a multitude of component variants. However, the process by which kits are prepared – the kit preparation – is labour-intensive, and kit errors are problematic at assembly processes. The use of robotics to support kit preparation has received some attention by researchers, but literature is lacking with respect to how collaborative robots – cobots – can support kit preparation activities. The purpose of this paper is to identify the potential of a cobot to support time-efficient batch preparation of kits. To address the purpose, the paper presents a mathematical model for estimation of the cycle time associated with cobot-supported kit preparation. The model is applied in a numerical example with experimental data from laboratory experiments, and cobot-supported kit preparation is compared with manual kit preparation. The findings suggest that cobot-supported kit preparation is beneficial with diverse kits and smaller components quantities per SKU (Stock Keeping Unit) and provides less variability of the outcome, when compared to manual kit preparation. The paper reveals several insights about cobot-supported kit preparation that can be valuable for both academics and practitioners. The model developed can be used by practitioners to assess the potential of cobots to support kit-batch preparation in association with assembly, spare parts, repair and maintenance, or business to business industry.

Closed-Form Displacement Relations of a Five-Link R-R-C-C-R Spatial Mechanism

1971 ◽  
Vol 93 (1) ◽  
pp. 221-226 ◽  
Author(s):  
A. H. Soni ◽  
P. R. Pamidi
Keyword(s):  
Closed Form ◽  
Polynomial Equation ◽  
Input Output ◽  
Degree Polynomial ◽  
Spatial Mechanism ◽  
Dual Number ◽  
Numerical Example

Using (3 × 3) matrices with dual-number elements, closed form displacement relationships are derived for a spatial five-link R-R-C-C-R mechanism. The input-output closed form displacement relationship is an eighth degree polynomial equation. A numerical example is presented.

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