A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints

2015 ◽  
Vol 7 (1) ◽  
Author(s):  
Xiangyun Li ◽  
Xin Ge ◽  
Anurag Purwar ◽  
Q. J. Ge
Keyword(s):  
Linkage Analysis ◽  
Least Squares ◽  
Efficient Algorithm ◽  
Image Space ◽  
Cartesian Space ◽  
Closure Equation ◽  
Unified Algorithm ◽  
Generalized Manifolds ◽  
Four Bar Linkage ◽  
Best Fit

This paper presents a single, unified, and efficient algorithm for animating the coupler motions of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation for each type of four-bar linkages separately. Recently, we developed a unified algorithm for synthesizing various four-bar linkages by mapping planar displacements from Cartesian space to the image space using planar quaternions. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of generalized manifolds (or G-manifolds) that best fit the image points in the least squares sense. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple's iOS platform.

A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints

2014 ◽  
Author(s):  
Xiangyun Li ◽  
Xin Ge ◽  
Anurag Purwar ◽  
Q. J. Ge
Keyword(s):  
Linkage Analysis ◽  
Least Squares ◽  
Efficient Algorithm ◽  
Image Space ◽  
Loop Closure ◽  
Closure Equation ◽  
Unified Algorithm ◽  
Four Bar Linkage ◽  
Best Fit ◽  
Linkage Synthesis

This paper presents a single, unified and efficient algorithm for animating the motions of the coupler of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation associated with each type of four-bar linkages separately. In our previous paper on four-bar linkage synthesis, we map the planar displacements from Cartesian to image space using planar quaternion. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of Generalized- or G-manifolds that best fit the image points in the least squares sense. The three planar dyads associated with Generalized G-manifolds are RR, PR and RP which could construct six types of four-bar mechanisms. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple’s iOS platform.

A Task Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

2013 ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Singular Values ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Motion Approximation ◽  
Value Decomposition ◽  
Four Bar Linkage ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of planar four-bar motion approximation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the Image Space of planar displacements, we obtain a class of quadrics, called Generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using Singular Value Decomposition. The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

scholarly journals A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Q. J. Ge ◽  
Anurag Purwar ◽  
Ping Zhao ◽  
Shrinath Deshpande
Keyword(s):  
Least Squares ◽  
Singular Values ◽  
Singular Value ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Value Decomposition ◽  
Four Bar Linkage ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of planar four-bar motion generation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the image space of planar displacements, we obtain a class of quadrics, called generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using singular value decomposition (SVD). The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

The line of best fit

2019 ◽  
pp. 26-37
Author(s):  
M. D. Edge
Keyword(s):  
Least Squares ◽  
Statistical Estimation ◽  
Best Fit

One way to visualize a set of data on two variables is to plot them on a pair of axes. A line that “best fits” the data can then be drawn as a summary. This chapter considers how to define a line of “best” fit—there is no sole best choice. The most commonly chosen line to summarize the data is the “least-squares” line—the line that minimizes the sum of the squared vertical distances between the points and the line. One reason for the least-squares line’s popularity is convenience, but, as will be seen later, it is also related to some key ideas in statistical estimation. The derivations of expressions for the intercept and slope of the least-squares line are discussed.

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

1998 ◽  
Vol 64 (6) ◽  
pp. 2044-2050 ◽  
Author(s):  
Laurence H. Smith ◽  
Perry L. McCarty ◽  
Peter K. Kitanidis
Keyword(s):  
Least Squares ◽  
Mixed Culture ◽  
Substrate Concentration ◽  
Reaction Rate ◽  
Initial Rate ◽  
Weighted Least Squares ◽  
Nonlinear Least Squares ◽  
Rate Coefficients ◽  
Monod Equation ◽  
Best Fit

ABSTRACT A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.

Kinematic Mapping Based Evaluation of Assembly Modes for Planar Four-Bar Synthesis

2005 ◽  
Author(s):  
Hans-Peter Schro¨cker ◽  
Manfred L. Husty ◽  
J. Michael McCarthy
Keyword(s):  
Image Space ◽  
Synthesis Problem ◽  
New Method ◽  
Kinematic Mapping ◽  
Position Synthesis ◽  
Four Bar Linkage

This paper presents a new method to determine if two task positions used to design a four-bar linkage lie on separate circuits of a coupler curve, known as a “branch defect.” The approach uses the image space of a kinematic mapping to provide a geometric environment for both the synthesis and analysis of four-bar linkages. In contrast to current methods of solution rectification, this approach guides the modification of the specified task positions, which means it can be used for the complete five position synthesis problem.

AN AUTOMATIC LEAST‐SQUARES MULTIMODEL METHOD FOR MAGNETIC INTERPRETATION

Geophysics ◽  
1973 ◽  
Vol 38 (2) ◽  
pp. 349-358 ◽  
Author(s):  
Peter H. McGrath ◽  
Peter J. Hood
Keyword(s):  
Least Squares ◽  
Magnetic Anomalies ◽  
Magnetic Data ◽  
Model Parameters ◽  
Hudson Bay ◽  
Northern Ontario ◽  
Magnetic Interpretation ◽  
Hudson Bay Lowlands ◽  
Sverdrup Basin ◽  
Best Fit

The magnetic anomalies caused by such diverse model shapes as the finite strike length thick dike, the vertical prism, thes loping step, the parallelepiped body, etc., may be obtained through an appropriate numerical integration of the expression for the magnetic effect produced by a finite thin plate. Using models generated in this manner, an automatic computer method has been developed at the Geological Survey of Canada for the interpretation of magnetic data. Because the magnetic anomalies produced by the various model shapes are nonlinear in parameters of shape and position, it is necessary to use an iterative procedure to obtain the values for the various model parameters which yield a least‐squares best‐fit anomaly curve to a set of discrete observed data. The interpretation method described in this paper uses the Powell algorithm for this purpose. The procedure can sometimes be made more efficient using a Marquardt modification to the Powell algorithm. Examples of the use of the method are presented for an elongated anomaly in the Moose River basin of the Hudson Bay lowlands in northern Ontario, and for an areally large elliptical anomaly in the Sverdrup basin of the Canadian Arctic Islands.

A more efficient algorithm for Convex Nonparametric Least Squares

2013 ◽  
Vol 227 (2) ◽  
pp. 391-400 ◽  
Author(s):  
Chia-Yen Lee ◽  
Andrew L. Johnson ◽  
Erick Moreno-Centeno ◽  
Timo Kuosmanen
Keyword(s):  
Least Squares ◽  
Efficient Algorithm
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