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.

Forward displacement analysis of a new 1CCC–5SPS parallel mechanism using Gröbner theory

2009 ◽  
Vol 223 (5) ◽  
pp. 1233-1241 ◽  
Author(s):  
D Gan ◽  
Q Liao ◽  
J S Dai ◽  
S Wei ◽  
L D Seneviratne
Keyword(s):  
Parallel Mechanism ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Geometrical Constraint ◽  
Degree Polynomial ◽  
Forward Displacement ◽  
Constraint Equations ◽  
Moving Platform ◽  
Forward Kinematic ◽  
Forward Displacement Analysis

A new parallel mechanism 1CCC–5SPS which has distance and angle constraints is introduced in this article. Degree of freedom and forward kinematic analysis of this new parallel mechanism are presented, in which four equivalent polynomial equations are obtained from the original six geometrical constraint equations. The Gröbner basis theory is used with the four equations and the problem of forward displacement is reduced to a 40th degree polynomial equation in a single unknown from a constructed 10 × 10 Sylvester's matrix which is small in size, from which 40 different locations of the moving platform can be derived. A numerical example confirms the efficiency of the procedure.

Forward Displacement Analyses of the 4-4 Stewart Platforms

1992 ◽  
Vol 114 (3) ◽  
pp. 444-450 ◽  
Author(s):  
W. Lin ◽  
M. Griffis ◽  
J. Duffy
Keyword(s):  
Closed Form ◽  
Stewart Platform ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Half Angle ◽  
Stewart Platforms ◽  
Forward Displacement Analysis

A forward displacement analysis in closed-form is performed for each case of a class of Stewart Platform mechanisms. This class of mechanisms, which are classified into three cases, are called the “4-4 Stewart Platforms,” where each of the mechanisms has the distinguishing feature of six legs meeting either singly or pair-wise at four points in the top and base platforms. (This paper only addresses those 4-4 Platforms where both the top and base platforms are planar.) For each case, a polynomial is derived in the square of a tan-half-angle that measures the angle between two planar faces of a polyhedron embedded within the mechanism. The degrees of the polynomials for the first, second, and third cases are, respectively, eight, four, and twelve. All the solutions obtained from the forward displacement analyses for the three cases are verified numerically using a reverse displacement analysis.

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.

Forward Displacement Analysis of a Parallel Manipulator

2011 ◽  
Vol 217-218 ◽  
pp. 1061-1065
Author(s):  
Xi Guang Huang
Keyword(s):  
Closed Form ◽  
Parallel Manipulator ◽  
Algebraic Method ◽  
Polynomial Equation ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Closed Form Solutions ◽  
Displacement Problem ◽  
Forward Displacement Analysis

A new algebraic method for the solution of the forward displacement analysis of a parallel manipulator is presented in this paper. Based on the algebraic method, the problem of the forward displacement problem is reduced to a polynomial equation in a single unknown from a constructed matrix which is relative small in the size. From the univariate equation, all closed-form solutions of the different locations of the mechanism can be derived.

Forward Displacement Analysis of Nine-Link Barranov Truss Based on Anti-Control of Chaos Newton Downhill Method

2011 ◽  
Vol 230-232 ◽  
pp. 749-753
Author(s):  
You Xin Luo ◽  
Ying Yang
Keyword(s):  
Body Motion ◽  
Control Of Chaos ◽  
Vector Method ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Real Solutions ◽  
Chaotic Sequences ◽  
Motion System ◽  
Constrained Equations ◽  
Forward Displacement Analysis

The anti-control of chaos Newton downhill method finding all real solutions of nonlinear equations was proposed and the forward displacement analysis on the 25th nine-link Barranov truss was completed. Four constrained equations were established by vector method with complex numbers according to four loops of the mechanism and four supplement equations were also established by increasing four variables and the relation of sine and cosine function. The established eight equations are that of forward displacement analysis of the mechanism. Combining Newton downhill method with chaotic sequences, anti-control of chaos Newton downhill method based on utilizing anti-control of chaos in body motion system to obtain locate initial points to find all real solutions of the nonlinear questions was proposed. The numerical example was given.The result shows that all real solutions have been quickly obtained, and it proves the correctness and validity of the proposed method.

Forward Displacement Analysis of Three New Classes of Analytic Spherical Parallel Manipulators

1998 ◽  
Author(s):  
Xian-Wen Kong
Keyword(s):  
Characteristic Polynomial ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Dual Solutions ◽  
Fourth Degree ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
The Third ◽  
Spherical Parallel Manipulator ◽  
Forward Displacement Analysis

Abstract The analytic manipulator is a manipulator the characteristic polynomial of which is of fourth degree or lower. Three new classes of analytic spherical parallel manipulators with prismatic actuators are proposed. The first is the spherical parallel manipulator with non-similar planar platforms, the second is the spherical parallel manipulator with similar planar platforms, and the third is the spherical parallel manipulator with orthogonal platforms. The forward displacement analysis of these new classes of spherical parallel manipulators is investigated in sequence. Polynomials of degree 4, 2 and 2 in one unknown respectively can be obtained to inscribe this problem. Due to dual solutions of other unknowns, a maximum of eight solutions might be possible for each of the new analytic spherical parallel manipulators.

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