Direct Position Analysis of the 4–6 Stewart Platforms

1994 ◽  
Vol 116 (1) ◽  
pp. 61-66 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Although Stewart platforms have been applied in the design of aircraft and vehicle simulators and parallel robots for many years, the closed-form solution of direct (forward) position analysis of Stewart platforms has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral, the 3–6 and the 4–4 Stewart platforms, of which the forward position solutions were derived as an eighth or a twelfth degree polynomials with one variable in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more general case of the Stewart platform, the 4–6 Stewart platforms, in which two pairs of the upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of the 3–6 and 4–4 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

Direct Position Analysis of the 4-6 Stewart Platforms

1992 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Unknown Variable ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Abstract Although Stewart platforms have been applied in the designs of aricraft and vehicle simulators and parallel robots in many years, their closed-form solution of direct (forward) position analysis has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral Stewart and the 4-4 Stewart platforms, in which two pairs of both upper and lower joint centers are coincident. The former results in in an eighth degree polynomial and the latter results in an eighth and a twelfth degree polynomials for different cases. The single unknown variable is in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more genearl case of the Stewart platform, the 4-6 Stewart platforms, in which two pairs of upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of 4-4 and and 3-6 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

Direct Position Kinematics of the 3-1-1-1 Stewart Platforms

1992 ◽  
Author(s):  
Muqtada Husain ◽  
Kenneth J. Waldron
Keyword(s):  
Closed Form ◽  
Special Class ◽  
General Feature ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Numerical Example ◽  
Geometric Argument ◽  
Stewart Platforms ◽  
Base Platform

Abstract In this work, a closed form solution for the direct position kinematics problem of a special class of Stewart Platform is presented. This class of mechanisms has a general feature that the top platform is connected to the six limbs at four locations. Three limbs connect at one location and the remaining limbs connect to the top platform singly at three separate locations. The base platform is connected at six different locations as is the case in the general platform. This particular class of mechanism is termed as 3-1-1-1 mechanism in this paper. It has been shown that there are a maximum of sixteen real assembly configurations for the direct position kinematics problem. This has been verified using a geometric argument also. The numerical example solved in this paper demonstrates that it is possible to obtain a set of solutions which are all real.

scholarly journals A closed-form solution for the position analysis of a novel fully spherical parallel manipulator – ERRATUM

Robotica ◽  
2015 ◽  
Vol 35 (5) ◽  
pp. 1137-1137
Author(s):  
Javad Enferadi ◽  
Amir Shahi
Keyword(s):  
Closed Form ◽  
Parallel Manipulator ◽  
Mechanical Engineering ◽  
Closed Form Solution ◽  
Form Solution ◽  
Position Analysis ◽  
Spherical Parallel Manipulator

There was an error in the spelling of the author's affiliation. Where the affiliation read “Department of mechanical engineering, Mashad Branch, Islamic Azad University, Mashad, Iran” it should instead have read “Department of mechanical engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran”.The publisher regrets this error.

Nonlinear Analysis of a Class of Inversion-Based Compliant Cross-Spring Pivots

2021 ◽  
Author(s):  
Shiyao Li ◽  
Guangbo Hao ◽  
Yingyue Chen ◽  
Jiaxiang Zhu ◽  
Giovanni Berselli
Keyword(s):  
Nonlinear Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Parameters ◽  
Contributing Factors ◽  
Characteristic Analysis ◽  
Center Shift ◽  
Axial Forces ◽  
Constraint Model

Abstract This paper presents a nonlinear model of an inversion-based generalized cross-spring pivot (IG-CSP) using the beam constraint model (BCM), which can be employed for the geometric error analysis and the characteristic analysis of an inversion-based symmetric cross-spring pivot (IS-CSP). The load-dependent effects are classified in two ways, including structure load-dependent effects and beam load-dependent effects, where the loading positions, geometric parameters of elastic flexures, and axial forces are the main contributing factors. The closed-form load-rotation relations of an IS-CSP and a non-inversion-based symmetric cross-spring pivot (NIS-CSP) are derived with consideration of the three contributing factors for analyzing the load-dependent effects. The load-dependent effects of IS-CSP and NIS-CSP are compared when the loading position is fixed. The rotational stiffness of the IS-CSP or NIS-CSP can be designed to increase, decrease, or remain constant with axial forces, by regulating the balance between the loading positions and the geometric parameters. The closed-form solution of the center shift of an IS-CSP is derived. The effects of axial forces on the IS-CSP center shift are analyzed and compared with those of a NIS-CSP. Finally, based on the nonlinear analysis results of IS-CSP and NIS-CSP, two new compound symmetric cross-spring pivots are presented and analyzed via analytical and FEA models.

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.

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.

Closed-Form Solution to the Noncoplanar P4P Problem for Uncalibrated Camera

2011 ◽  
Vol 145 ◽  
pp. 6-10
Author(s):  
Yang Guo
Keyword(s):  
Closed Form ◽  
Affine Transformation ◽  
Closed Form Solution ◽  
Polynomial Equation ◽  
Form Solution ◽  
Potential Candidate ◽  
Degree Polynomial ◽  
Fast Determination ◽  
Hand Eye Coordination

This paper presents a closed-form solution to determination of the position and orientation of a perspective camera with two unknown effective focal lengths for the noncoplanar perspective four point (P4P) problem. Given four noncoplanar 3D points and their correspondences in image coordinate, we convert perspective transformation to affine transformation, and formulate the problem using invariance to 3D affine transformation and arrive to a closed-form solution. We show how the noncoplanar P4P problem is cast into the problem of solving an eighth degree polynomial equation in one unknown. This result shows the noncoplanar P4P problem with two unknown effective focal lengths has at most 8 solutions. Last, we confirm the conclusion by an example. Although developed as part of landmark-guided navigation, the solution might well be used for landmark-based tracking problem, hand-eye coordination, and for fast determination of interior and exterior camera parameters. Because our method is based on closed-form solution, its speed makes it a potential candidate for solving above problems.

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.

Nonlinear Analysis of a Class of Inversion-Based Compliant Cross-Spring Pivots

2021 ◽  
pp. 1-29
Author(s):  
Shiyao Li ◽  
Guangbo Hao ◽  
Yingyue Chen ◽  
Jiaxiang Zhu ◽  
Giovannni Berselli
Keyword(s):  
Nonlinear Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Parameters ◽  
Contributing Factors ◽  
Characteristic Analysis ◽  
Center Shift ◽  
Axial Forces ◽  
Constraint Model

Abstract This paper presents a nonlinear model of an inversion-based generalized cross-spring pivot (IG-CSP) using the beam constraint model (BCM), which can be employed for the geometric error analysis and the characteristic analysis of an inversion-based symmetric cross-spring pivot (IS-CSP). The load-dependent effects are classified in two ways, including the structure load-dependent effects and beam load-dependent effects, where the loading positions, geometric parameters of elastic flexures, and axial forces are the main contributing factors. The closed-form load-rotation relations of an IS-CSP and a non-inversion-based symmetric cross-spring pivot (NIS-CSP) are derived with consideration of the three contributing factors for analyzing the load-dependent effects. The load-dependent effects of IS-CSP and NIS-CSP are compared when the loading position is fixed. The rotational stiffness of the IS-CSP or NIS-CSP can be designed to increase, decrease, or remain constant with axial forces, by regulating the balance between the loading positions and the geometric parameters. The closed-form solution of the center shift of an IS-CSP is derived. The effects of axial forces on the IS-CSP center shift are analyzed and compared with those of a NIS-CSP. Finally, based on the nonlinear analysis results of IS-CSP and NIS-CSP, two new compound symmetric cross-spring pivots are presented and analyzed via analytical and FEA models.

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