Solving the Input/Output Problem for Planar Mechanisms

1999 ◽  
Vol 121 (2) ◽  
pp. 206-211 ◽  
Author(s):  
J. Nielsen ◽  
B. Roth
Keyword(s):  
Solution Procedure ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms

This paper presents a method for solving the input/output problem for all planar mechanisms composed of revolute and slider joints. The solution procedure is a modification of the Dixon resultant method, which was developed to solve sets of polynomial equations; in this paper the method is applied to sets of equations which are linear in the sines and cosines of unknown angles. A particular planar multicircuit mechanism is analyzed to illustrate the solution procedure, and implementation details are discussed.

Solving the Input/Output Problem for Planar Mechanisms

1998 ◽  
Author(s):  
James Nielsen ◽  
Bernard Roth
Keyword(s):  
Solution Procedure ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
Revolute Joints

Abstract This paper presents a method for solving the input/output problem for all planar mechanisms composed of revolute joints. The solution procedure is a modification of the Dixon resultant method, which was developed to solve sets of polynomial equations; in this paper the method is applied to sets of equations which are linear in the sines and cosines of unknown angles. A particular planar multi-circuit mechanism is analyzed to illustrate the solution procedure, and implementation details are discussed.

Solving the Kinematics of Planar Mechanisms

1998 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
General Method

Abstract This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a standard eigenvalue problem, or if preferred, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.

Solving the Kinematics of Planar Mechanisms

1999 ◽  
Vol 121 (3) ◽  
pp. 387-391 ◽  
Author(s):  
C. W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
Generalized Eigenvalue Problem ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
Generalized Eigenvalue ◽  
General Method

This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a generalized eigenvalue problem, or in some cases, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.

scholarly journals Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation

2000 ◽  
Vol 123 (3) ◽  
pp. 382-387 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
Numerical Solutions ◽  
Minimal Size ◽  
Input Output ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Revolute Joints ◽  
Generalized Eigenvalue ◽  
General Method

This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute joints. The method combines a complex plane formulation [1] with the Dixon determinant procedure of Nielsen and Roth [2]. The result is simple to derive and implement, so in addition to providing numerical solutions, the approach facilitates analytical explorations. The procedure leads to a generalized eigenvalue problem of minimal size. Both input/output problems and the derivation of tracing curve equations are addressed, as is the extension of the method to treat slider joints.

scholarly journals Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

2014 ◽  
Vol 6 (3) ◽  
Author(s):  
Mark M. Plecnik ◽  
J. Michael McCarthy
Keyword(s):  
Parallel Computation ◽  
Design Procedure ◽  
Polynomial Equations ◽  
Output Function ◽  
Logarithmic Function ◽  
Input Output ◽  
Function Generator ◽  
Kinematic Synthesis ◽  
Isotropic Coordinates ◽  
Polynomial Degree

This paper presents a design procedure for six-bar linkages that use eight accuracy points to approximate a specified input–output function. In the kinematic synthesis of linkages, this is known as the synthesis of a function generator to perform mechanical computation. Our formulation uses isotropic coordinates to define the loop equations of the Watt II, Stephenson II, and Stephenson III six-bar linkages. The result is 22 polynomial equations in 22 unknowns that are solved using the polynomial homotopy software Bertini. The bilinear structure of the system yields a polynomial degree of 705,432. Our first run of Bertini generated 92,736 nonsingular solutions, which were used as the basis of a parameter homotopy solution. The algorithm was tested on the design of the Watt II logarithmic function generator patented by Svoboda in 1944. Our algorithm yielded his linkage and 64 others in 129 min of parallel computation on a Mac Pro with 12 × 2.93 GHz processors. Three additional examples are provided as well.

scholarly journals Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation

2000 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
Numerical Solutions ◽  
Generalized Eigenvalue Problem ◽  
Minimal Size ◽  
Input Output ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Generalized Eigenvalue ◽  
General Method

Abstract This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute and slider joints. The method combines the complex plane formulation of Wampler (1999) with the Dixon determinant procedure of Nielsen and Roth (1999). The result is simple to derive and implement, so in addition to providing numerical solutions, the approach facilitates analytical explorations. The procedure leads to a generalized eigenvalue problem of minimal size. Both input/output problems and the derivation of tracing curve equations are addessed.

scholarly journals Computational Design of Stephenson II Six-Bar Function Generators for 11 Accuracy Points

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Mark M. Plecnik ◽  
J. Michael McCarthy
Keyword(s):  
Design Methodology ◽  
Computational Design ◽  
Polynomial Equations ◽  
Input Output ◽  
Degree Polynomial ◽  
Walking Gait ◽  
Quadratic Equations ◽  
Input And Output ◽  
Linkage Design ◽  
Function Generators

This paper presents a design methodology for Stephenson II six-bar function generators that coordinate 11 input and output angles. A complex number formulation of the loop equations yields 70 quadratic equations in 70 unknowns, which is reduced to a system of ten eighth degree polynomial equations of total degree 810=1.07×109. These equations have a monomial structure which yields a multihomogeneous degree of 264,241,152. A sequence of polynomial homotopies was used to solve these equations and obtain 1,521,037 nonsingular solutions. Contained in these solutions are linkage design candidates which are evaluated to identify cognates, and then analyzed to determine their input–output angles in each assembly. The result is a set of feasible linkage designs that reach the required accuracy points in a single assembly. As an example, three Stephenson II function generators are designed, which provide the input–output functions for the hip, knee, and ankle of a humanoid walking gait.

Automatic Generation of the Input-Output Equations of Planar Mechanisms

1994 ◽  
Author(s):  
Kunter A. Kanberoglu ◽  
Resit Soylu
Keyword(s):  
Closed Form ◽  
Functional Relation ◽  
Automatic Generation ◽  
Computational Effort ◽  
Input Output ◽  
Symbolic Manipulation ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Transmission Angle ◽  
Output Equation

Abstract In this article, a methodology, which yields (in closed-form) the functional relation between “any” two joint variables of a one degree-of-freedom planar mechanism, is developed. For instance, the transmission angle and crank-rotatibility synthesis algorithms (Soylu, 1993; Soylu and Kanberoğlu, 1993) require such a generic input-output equation. The equation is obtained in an optimal manner which minimizes the computational effort associated with it. The tools of theory of elimination and symbolic manipulation are also used in the developed method.

Solving the Inverse Position Problem of a 7R Robot

2010 ◽  
Vol 29-32 ◽  
pp. 961-965
Author(s):  
Xi Guang Huang ◽  
Duan Ling Li ◽  
Guang Pin He
Keyword(s):  
Computational Technique ◽  
Total Degree ◽  
Polynomial Equations ◽  
Input Output ◽  
Numerical Example ◽  
Whole Process ◽  
Position Problem ◽  
Output Equation

In this paper a new computational technique for the inverse position problem of a 7R robot is presented. Instead of reducing the problem to one highly complicated input-output equation, we work with a system of 10 very simple polynomial equations. We show the total degree of the system is 16, in agreement with previous works. Moreover we present a numerical example confirms the technique. The whole process is simple and easy to program.

scholarly journals Inverse kinematic analysis for triple-octahedron variable-geometry truss manipulators

2001 ◽  
Vol 215 (2) ◽  
pp. 247-251 ◽  
Author(s):  
L J Xu ◽  
G Y Tian ◽  
Y Duan ◽  
S X Yang
Keyword(s):  
Solution Procedure ◽  
Inverse Kinematic ◽  
Variable Geometry ◽  
Input Output ◽  
Output Variable ◽  
Output Displacement ◽  
Displacement Equation ◽  
Inverse Kinematic Analysis ◽  
Variable Geometry Truss Manipulator ◽  
Variable Geometry Truss

In this paper, a new triple-octahedron variable-geometry truss manipulator is presented. Its inverse kinematic solutions in closed form are studied. An input-output displacement equation in one output variable is derived. The solution procedure is given in detail. A numerical example is illustrated.

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