scholarly journals A Geometric Modeling and Computing Method for Direct Kinematic Analysis of 6-4 Stewart Platforms

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ying Zhang ◽  
Xin Liu ◽  
Shimin Wei ◽  
Yaobing Wang ◽  
Xiaodong Zhang ◽  
...  
Keyword(s):  
Kinematic Analysis ◽  
Geometric Modeling ◽  
Polynomial Equation ◽  
Solution Procedure ◽  
Geometric Constraints ◽  
The Other ◽  
Univariate Polynomial ◽  
Stewart Platforms ◽  
Resultant Matrix ◽  
Computing Method

A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Gröbner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure. The novelty of the paper lies in that (1) the formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA and (2) the size of the resultant matrix is smaller than those existed.

A New Algebraic Solution to Inverse Static Force Analysis of a Special Planar Three-Spring System

2016 ◽  
Author(s):  
Ying Zhang ◽  
Qizheng Liao ◽  
Shimin Wei ◽  
Feng Wei ◽  
Duanling Li
Keyword(s):  
Polynomial Equation ◽  
Algebraic Solution ◽  
Static Force ◽  
Force Analysis ◽  
Elimination Algorithm ◽  
Equilibrium Configurations ◽  
Spring System ◽  
Univariate Polynomial ◽  
Variable Substitution ◽  
Resultant Matrix

In this paper, we present a new algebraic elimination algorithm for the inverse static force analysis of a special planar three-spring system. The system consists of three linear springs joined to the ground at the two fixed pivots and connected to the two moving pivots at the platform. When exerted by specified static force, the goal of inverse static analysis is to determine all the equilibrium configurations. First of all, a system of seven polynomial equations in seven variables is established based on the geometric constraint and static force balancing. Then, four basic constraint equations in four variables are obtained by variable substitution. Next, a 20 by 20 resultant matrix is reduced by means of three consecutive Sylvester elimination process. Finally, a 54th-degree univariate polynomial equation is directly derived without extraneous roots in the computer algebra system Mathematica 9.0. At last, a numerical example is given to verify the elimination procedure.

On the Solution of Five-Precision-Points Path Synthesis of Planar Four-Bar Linkages Using Algebraic Method

2016 ◽  
Author(s):  
Feng Wei ◽  
Shimin Wei ◽  
Ying Zhang ◽  
Qizheng Liao
Keyword(s):  
Algebraic Method ◽  
Polynomial Equation ◽  
Type I ◽  
Type Iv ◽  
Univariate Polynomial ◽  
Degenerate Solutions ◽  
Path Synthesis ◽  
Four Bar Linkage ◽  
High Degree ◽  
Resultant Matrix

The problem of five precision points path synthesis of planar four-bar linkage can be divided into four types in term of the input parameters. A unified formulation for the four types is built based on the planar displacement matrix. Next, the corresponding resultant matrix is constructed based on Groebner bases generated by applying the new term ordering (the groups graded reverse lexicographic ordering, <ggrevlex) for four types. Then, a high-degree univariate polynomial equation is accordingly obtained. At last, several examples are provided to validate the algorithm and the solutions are verified in the software SAM. And it is concluded that type I has 36 solutions, type II has 64 including 16 degenerate solutions, type III has 92 solutions and type IV has 82 solutions including 16 degenerate solutions.

scholarly journals A Novel Geometric Modeling and Calculation Method for Forward Displacement Analysis of 6-3 Stewart Platforms

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 442
Author(s):  
Ganmin Zhu ◽  
Shimin Wei ◽  
Ying Zhang ◽  
Qizheng Liao
Keyword(s):  
Calculation Method ◽  
Geometric Modeling ◽  
Constraint Equation ◽  
Polynomial Equation ◽  
Inner Product ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Moving Platform ◽  
Stewart Platforms ◽  
Forward Displacement Analysis

A novel geometric modeling and calculation method for forward displacement analysis of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework. Firstly, two formulas between 2-blade and 1-blade are formulated. Secondly, the expressions for two spherical joints of the moving platform are given via CGA operation. Thirdly, a coordinate-invariant geometric constraint equation is deduced. Fourthly, a 16-degree univariate polynomial equation without algebraic elimination by using the Euler angle substitution is presented. Fifthly, the coordinates of three spherical joints on the moving platform are calculated without judging the radical symbols. Finally, two numerical examples are used to verify the method. The highlight of this paper is that a new geometric modeling and calculation method without algebraic elimination is obtained by using the determinant form of the CGA inner product algorithm, which provides a new idea to solve a more complex spatial parallel mechanism in the future.

Dynamic Binding of Features and Solid Modelers

1993 ◽  
Author(s):  
Jami J. Shah ◽  
Viren Pherwani
Keyword(s):  
Geometric Modeling ◽  
Object Oriented ◽  
The Other ◽  
Dynamic Binding ◽  
Communication Architecture ◽  
X Window ◽  
Feature Based

Abstract The work described in this paper investigates the feasibility of standardizing communications between geometric modeling core systems and generic feature-based applications. Since geometric modelers differ in the functionality they provide and feature applications vary in the level of geometric operations they can support internally, a multi-layered communication architecture is proposed. The methodology is analogous to the X-Window standard for graphics. At the lowest level is a library of functions named Geo-lib, which are translated into geometric modeler specific commands. If there was to be a future dynamic interfacing standard, such as STEP-SDAI, these specific calls could be replaced by standard calls, analogous to Geo-Protocol. At the next layer is a library, called Geo-widgets, which are written entirely using Geo-lib functions. At the highest level Geo-Tools, functions used commonly by generic applications. Feature applications can choose to use the library at any level, as necessary. This multi-layered geometric toolkit creates a seamless object oriented bond between the feature application and the geometric modeling core, in such a way that either one could be replaced without requiring any changes to the other.

A Double-Faced 6R Single-Loop Overconstrained Spatial Mechanism

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Xianwen Kong ◽  
Xiuyun He ◽  
Duanling Li
Keyword(s):  
Kinematic Analysis ◽  
Isosceles Triangle ◽  
The Other ◽  
Dual Quaternion ◽  
Single Loop ◽  
Plane Symmetric ◽  
Spatial Mechanism ◽  
Revolute Joints ◽  
Full Turn ◽  
Overconstrained Mechanisms

This paper deals with a 6R single-loop overconstrained spatial mechanism that has two pairs of revolute joints with intersecting axes and one pair of revolute joints with parallel axes. The 6R mechanism is first constructed from an isosceles triangle and a pair of identical circles. The kinematic analysis of the 6R mechanism is then dealt with using a dual quaternion approach. The analysis shows that the 6R mechanism usually has two solutions to the kinematic analysis for a given input and may have two circuits (closure modes or branches) with one or two pairs of full-turn revolute joints. In two configurations in each circuit of the 6R mechanism, the axes of four revolute joints are coplanar, and the axes of the other two revolute joints are perpendicular to the plane defined by the above four revolute joints. Considering that from one configuration of the 6R mechanism, one can obtain another configuration of the mechanism by simply renumbering the joints, the concept of two-faced mechanism is introduced. The formulas for the analysis of plane symmetric spatial triangle are also presented in this paper. These formulas will be useful for the design and analysis of multiloop overconstrained mechanisms involving plane symmetric spatial RRR triads.

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.

The Conceptual Design of Differential Mechanisms

2013 ◽  
Vol 284-287 ◽  
pp. 973-978
Author(s):  
Chia Chun Chu
Keyword(s):  
Conceptual Design ◽  
Design Process ◽  
Degrees Of Freedom ◽  
Geometric Constraints ◽  
The Other ◽  
Design Approach ◽  
Process Step ◽  
Two Degrees Of Freedom ◽  
Number Synthesis

The purpose of this paper is to present a design approach based on the geometric constraints of joints for synthesizing differential mechanisms with two degrees-of-freedom, including some mechanisms with the same functions but distinct structures. The concept of virtual axes is presented. And, there are five steps in the design process. Step 1 is to decide fundamental entities by the properties of existing mechanisms and the technique of number synthesis, and 10 suitable fundamental entities of differential mechanisms are available. Step 2 is to compose geometric constraints, and 14 items are obtained. Step 3 is to compose links, and 15 items are derived. Step 4 is to assign fixed constraints for inputs or outputs, and 15 results are found. The final step is to particularize the obtained events by the properties of existing mechanisms and the structures of fundamental entities. As a result, 8 feasible results for differential mechanisms with two degrees-of-freedom and two basic loops are obtained in which 2 are existing designs and the other 6 are novel.

scholarly journals Accurate Computation of Periodic Regions' Centers in the General M-Set with Integer Index Number

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Wang Xingyuan ◽  
He Yijie ◽  
Sun Yuanyuan
Keyword(s):  
Polynomial Equation ◽  
Negative Integer ◽  
The Other ◽  
Index Number ◽  
Polynomial Equations ◽  
Decimal Point ◽  
Numerical Accuracy ◽  
Index Numbers ◽  
Accurate Computation ◽  
The Relationship

This paper presents two methods for accurately computing the periodic regions' centers. One method fits for the general M-sets with integer index number, the other fits for the general M-sets with negative integer index number. Both methods improve the precision of computation by transforming the polynomial equations which determine the periodic regions' centers. We primarily discuss the general M-sets with negative integer index, and analyze the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric interior. We can get the centers' coordinates with at least 48 significant digits after the decimal point in both real and imaginary parts by applying the Newton's method to the transformed polynomial equation which determine the periodic regions' centers. In this paper, we list some centers' coordinates of general M-sets'k-periodic regions(k=3,4,5,6)for the index numbersα=−25,−24,…,−1, all of which have highly numerical accuracy.

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.

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