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

2016 ◽  
pp. 911-924
Author(s):  
Ying Zhang ◽  
Qizheng Liao ◽  
Shimin Wei ◽  
Feng Wei ◽  
Duanling Li
Keyword(s):  
Geometric Modeling ◽  
Solution Method ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Stewart Platforms ◽  
Forward Displacement Analysis

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 Kind of 2-Coupled–degree Nine-Link Barranov Truss Based on Hyper-Chaotic Least Square Method

2011 ◽  
Vol 230-232 ◽  
pp. 754-758
Author(s):  
You Xin Luo ◽  
Qi Yuan Liu ◽  
Xiao Yi Che
Keyword(s):  
Solution Method ◽  
Least Square Method ◽  
Least Square ◽  
Vector Method ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Real Solutions ◽  
Constrained Equations ◽  
Parameter Coupling ◽  
Forward Displacement Analysis

The hyper-chaotic least square method finding all real solutions of nonlinear equations was proposed and the forward displacement analysis on the 31th 2-coupled–degree 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 least square method with hyper-chaotic sequences, hyper-chaotic least square method based on utilizing parameter coupling hyper-chaotic discrete system to obtain locate initial points to find all real solutions of the nonlinear questions was proposed. The numerical example was given. Comparison was also done with other finding solution method. The result shows that all real solutions have been quickly obtained, and it proves the correctness and validity of the proposed method.

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.

Displacement Analysis of Nine-Link Barranov Truss Based on Hyper-Chaotic Least Square Method

2012 ◽  
Vol 507 ◽  
pp. 260-264
Author(s):  
You Xin Luo ◽  
Qi Yuan Liu
Keyword(s):  
Solution Method ◽  
Least Square Method ◽  
Least Square ◽  
Vector Method ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Real Solutions ◽  
Chaotic Sequences ◽  
Constrained Equations ◽  
Forward Displacement Analysis

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 30th 2-coupled–degree nine-link Barranov truss mechanism. Combining least square method with hyper-chaotic sequences, hyper-chaotic least square method based on utilizing two-dimensional discrete hyper-chaotic system to obtain locate initial points to find all real solutions of the nonlinear questions was proposed. The numerical example of forward displacement analysis was given. Comparison was also done with other finding solution method. 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 Non-Plane Two Coupled Degree Nine-Link Barranov Truss Based on Hyper-Chaotic Newton Downhill Method

2010 ◽  
Vol 20-23 ◽  
pp. 659-664 ◽  
Author(s):  
You Xin Luo ◽  
Qi Yuan Liu
Keyword(s):  
Discrete System ◽  
Solution Method ◽  
Complex Numbers ◽  
Vector Method ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Real Solutions ◽  
Chaotic Sequences ◽  
Constrained Equations ◽  
Forward Displacement Analysis

The hyper-chaotic Newton downhill method finding all real solutions of nonlinear equations was proposed and the forward displacement analysis on the 33th non-plane 2-coupled–degree 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 hyper-chaotic sequences, hyper-chaotic Newton-downhill method based on utilizing hyper-chaotic discrete system to obtain locate initial points to find all real solutions of the nonlinear questions was proposed. The numerical example was given. Comparison was also done with other finding solution method. 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 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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