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 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.

Explicit Solution for the Forward Displacement Analysis of the Stewart Platform Manipulator

1996 ◽  
Author(s):  
Yao Jin ◽  
Fang Hai-Rong
Keyword(s):  
Explicit Solution ◽  
Computation Time ◽  
Stewart Platform ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Numerical Example ◽  
Forward Displacement Analysis ◽  
Explicit Expressions

Abstract A method is presented to directly solve the problem of the forward displacement analysis of the Stewart platform manipulator. With algebraic elimination and the use of only one extra sensor, the explicit expressions on the forward displacement solution is obtained so that the method may enable designers to get the best compromise between the supplementary cost and complexity of the manipulator and the computation time for solving the forward displacement analysis. A numerical example is given to illustrate the method presented.

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.

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 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.

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