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 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.

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.

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.

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.

scholarly journals Closed-Form Solution to the Position Analysis of Watt–Baranov Trusses Using the Bilateration Method

2011 ◽  
Vol 3 (3) ◽  
Author(s):  
Nicolás Rojas ◽  
Federico Thomas
Keyword(s):  
Closed Form ◽  
Characteristic Polynomial ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations ◽  
Characteristic Polynomials ◽  
Position Analysis ◽  
Exact Position ◽  
Kinematic Loops ◽  
First Time

The exact position analysis of a planar mechanism reduces to compute the roots of its characteristic polynomial. Obtaining this polynomial almost invariably involves, as a first step, obtaining a system of equations derived from the independent kinematic loops of the mechanism. The use of kinematic loops to this end has seldom been questioned despite deriving the characteristic polynomial from them requires complex variable eliminations and, in most cases, trigonometric substitutions. As an alternative, the bilateration method has recently been used to obtain the characteristic polynomials of the three-loop Baranov trusses without relying on variable eliminations nor trigonometric substitutions and using no other tools than elementary algebra. This paper shows how this technique can be applied to members of a family of Baranov trusses resulting from the circular concatenation of the Watt mechanism irrespective of the resulting number of kinematic loops. To our knowledge, this is the first time that the characteristic polynomial of a Baranov truss with more that five loops has been obtained, and hence, its position analysis solved in closed form.

Development and Kinematic Position Analysis of a Novel Class 2 Tensegrity Robot

2018 ◽  
Author(s):  
Carlos G. Manríquez-Padilla ◽  
Karla A. Camarillo-Gómez ◽  
Gerardo I. Pérez-Soto ◽  
Juvenal Rodríguez-Reséndiz ◽  
Carl D. Crane
Keyword(s):  
Closed Form ◽  
Kinematic Analysis ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Novel ◽  
Universal Joint ◽  
Position Analysis ◽  
Experimental Prototype ◽  
Inverse Position Analysis ◽  
Kinematic Position

This paper presents a novel class 2 tensegrity robot which has contact between its rigid elements with a universal joint. Also, an strategy to obtain the forward and inverse position kinematic analysis using the parameters Denavit–Hartenberg in the distal convention is presented, obtaining the closed–form solution for the inverse position analysis and it was validated through simulation where a point of the robot followed the desired trajectory. Finally, the results were implemented in the experimental prototype of the novel class 2 tensegrity robot.

scholarly journals External Kinematic Calibration of Hybrid Kinematics Machine Utilizing Lower-DOF Planar Parallel Kinematics Mechanisms

2019 ◽  
Vol 21 (6) ◽  
pp. 995-1015 ◽  
Author(s):  
Abdur Rosyid ◽  
Bashar El-Khasawneh ◽  
Anas Alazzam
Keyword(s):  
Least Squares ◽  
Closed Form ◽  
Closed Form Solution ◽  
Nonlinear Least Squares ◽  
Form Solution ◽  
Geometric Parameters ◽  
Kinematic Calibration ◽  
Parallel Kinematics ◽  
Linear Least Squares ◽  
Hybrid Kinematics

AbstractThis paper presents the implementation of nonlinear least squares and iterative linear least squares algorithms for external kinematic calibration of a hybrid kinematics machine composed of two 3PRR planar parallel kinematics mechanisms by utilizing a laser tracker. First the hand-eye and robot-world transformations were obtained by a separable closed-form solution and refined by the nonlinear least squares. Subsequently, the geometric parameters of the machine’s mechanisms were estimated using the two algorithms. Due to the rank deficiency, we implemented the nonlinear least squares algorithm through a subset selection approach in which we performed the estimation in two steps. We iterated the closed-form solution of the linear least squares until the solution converges to the actual values. We have shown that the nonlinear least squares algorithm successfully refined the hand-eye and robot-world transformations and outperformed the iterative linear squares algorithm in the estimation of the geometric parameters of the mechanisms.

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.

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