Closed-Form Displacement Relations of a Five-Link R-R-C-C-R Spatial Mechanism

Using (3 x 3) matrices with dual-number elements, closed-form displacement relationships are derived for a spatial five-link R-C-R-C-P mechanism. The input-output closed form displacement relationship is obtained as a second order polynomial in the output displacement. For each set of the input and output displacements obtained from the equation, all other variable parameters of the mechanism are uniquely determined. A numerical illustrative example is presented. The derived input-output relationship can be used to synthesize an R-C-R-C-P function generating mechanism for a maximum of 15 precision conditions.

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Kinematic Analysis of Spatial Six-Link 3R-2P-C Mechanisms

Transactions of the Canadian Society for Mechanical Engineering ◽

10.1139/tcsme-1976-0012 ◽

1976 ◽

Vol 4 (2) ◽

pp. 71-76

Author(s):

M.O.M. Osman ◽

R. V. Dukkipati

Keyword(s):

Closed Form ◽

Kinematic Analysis ◽

Input Output ◽

Variable Parameters ◽

Output Displacement ◽

Loop Equation ◽

Dual Number ◽

Input And Output ◽

Order Polynomial ◽

Output Equation

Using (3 x 3) matrices with dual-number elements, closed-form displacement relationships are derived for a spatial six-link R-C-P-R-P-R mechanism. The input-output closed form displacement relationship is obtained as a second order polynomial in the output displacement. For each set of the input and output displacements obtained from the equation, all other variable parameters of the mechanism are uniquely determined. A numerical illustrative example is presented. Using the dual-matrix loop equation, with proper arrangement of terms and following a procedure similar to that presented, the closed-form displacement relationships for other types of six-link 3R + 2P + 1C mechanisms can be obtained. The input-output equation derived may also be used to generate the input-output functions for five-link 2R + 2C + 1P mechanisms and four-link mechanisms with one revolute and three cylinder pairs.

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Displacement Analysis of Spatial Five-Link Mechanisms Using (3×3) Matrices With Dual-Number Elements

Journal of Engineering for Industry ◽

10.1115/1.3591499 ◽

1969 ◽

Vol 91 (1) ◽

pp. 152-156 ◽

Cited By ~ 60

Author(s):

A. T. Yang

Keyword(s):

Closed Form ◽

Algebraic Equation ◽

Input Output ◽

Fourth Degree ◽

Displacement Analysis ◽

Dual Number ◽

Closure Equation

A closure equation, in terms of matrices with dual-number elements, for spatial five-link mechanisms, is presented in this paper. From the equation, a set of displacement equations for a RCRCR mechanism with general proportions is obtained; the input-output relationship is expressed as a fourth-degree algebraic equation and formulas to determine other linkage variables are expressed in closed form.

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Dual Polynomials and Complex Dual Numbers for Analysis of Spatial Mechanisms

Volume 2B: 24th Biennial Mechanisms Conference ◽

10.1115/96-detc/mech-1221 ◽

1996 ◽

Cited By ~ 1

Author(s):

Harry H. Cheng ◽

Sean Thompson

Keyword(s):

Numerical Solutions ◽

Polynomial Equation ◽

Mathematical Tool ◽

Polynomial Equations ◽

Dual Numbers ◽

Output Displacement ◽

Spatial Mechanism ◽

Dual Number ◽

Spatial Mechanisms ◽

Language Environment

Abstract Complex dual numbers w̌1=x1+iy1+εu1+iεv1 which form a commutative ring are for the first time introduced in this paper. Arithmetic operations and functions of complex dual numbers are defined. Complex dual numbers are used to solve dual polynomial equations. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism. Like the dual number, the complex dual number is a useful mathematical tool for analytical and numerical treatment of spatial mechanisms.

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Closed-Form Direct Position Analysis of a 5–5 Parallel Mechanism

Journal of Mechanical Design ◽

10.1115/1.2919220 ◽

1993 ◽

Vol 115 (3) ◽

pp. 515-521 ◽

Cited By ~ 31

Author(s):

C. Innocenti ◽

V. Parenti-Castelli

Keyword(s):

Closed Form ◽

Parallel Mechanism ◽

General Feature ◽

Polynomial Equation ◽

Stewart Platform ◽

Parallel Mechanisms ◽

Complex Field ◽

Output Link ◽

Degree Polynomial ◽

Displacement Analysis

The paper presents the closed form direct displacement analysis for a class of Stewart platform-type parallel mechanisms whose general feature consists of six legs which meet five distinct points both in the base and in the movable output link. Out of the two possible arrangements, only one is here analyzed in detail. Given a set of actuator displacements the analysis provides all the possible locations of the platform relative to the base. The analysis results in a 40th degree polynomial equation in one unknown. The roots of the equation provide in the complex field forty closures of the mechanism. This new result has been numerically verified by the inverse displacement analysis.

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Finding Zeros of Quartic Polynomials by Using Radicals

10.32920/14639649.v1 ◽

2021 ◽

Author(s):

Sureyya Sahin

Keyword(s):

Polynomial Equation ◽

Lower Degree ◽

Polynomial Equations ◽

Single Variable ◽

Degree Polynomial ◽

Numerical Example ◽

General Polynomial ◽

Quartic Polynomial ◽

Quartic Polynomials

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

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Displacement Analysis of the RPRCRR Six-Link Spatial Mechanism

Journal of Applied Mechanics ◽

10.1115/1.3408906 ◽

1971 ◽

Vol 38 (4) ◽

pp. 1029-1035 ◽

Cited By ~ 2

Author(s):

M. S. C. Yuan

Keyword(s):

Algebraic Equation ◽

Input Output ◽

Variable Parameters ◽

Displacement Analysis ◽

Output Displacement ◽

Spatial Mechanism ◽

Numerical Example ◽

Input And Output ◽

Displacement Equation

Using the method of line coordinates, the input-output displacement equation of the RPRCRR six-link spatial mechanism is obtained as an algebraic equation of 16th order. For each set of the input and output angles obtained from the equation, all other variable parameters of the mechanism are also determined. A numerical example is presented.

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Finding Zeros of Quartic Polynomials by Using Radicals

10.32920/14639649 ◽

2021 ◽

Author(s):

Sureyya Sahin

Keyword(s):

Polynomial Equation ◽

Lower Degree ◽

Polynomial Equations ◽

Single Variable ◽

Degree Polynomial ◽

Numerical Example ◽

General Polynomial ◽

Quartic Polynomial ◽

Quartic Polynomials

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

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

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.145.6 ◽

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.

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A new closed-form kinematics of the generalized 3-DOF spherical parallel manipulator

Robotica ◽

10.1017/s0263574799001630 ◽

1999 ◽

Vol 17 (5) ◽

pp. 475-485 ◽

Cited By ~ 23

Author(s):

Zhen Huang ◽

Y. Lawrence Yao

Keyword(s):

Closed Form ◽

Analytical Method ◽

Parallel Manipulator ◽

Degree Of Freedom ◽

New Method ◽

Numerical Example ◽

Three Degree Of Freedom ◽

Spherical Parallel Manipulator

This paper presents a new method to analyze the closed-form kinematics of a generalized three-degree-of-a-freedom spherical parallel manipulator. Using this analytical method, concise and uniform solutions are achieved. Two special forms of the three-degree-of-freedom spherical parallel manipulator, i.e. right-angle type and a decoupled type, are also studied and their unique and interesting properties are investigated, followed by a numerical example.

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