A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis

2012 ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar ◽  
Xiangyun Li
Keyword(s):  
Least Squares ◽  
Fast Algorithm ◽  
Linear Formulation ◽  
Image Space ◽  
Kinematic Synthesis ◽  
One Dimensional ◽  
Least Squares Fitting ◽  
Novel Approach ◽  
Motion Approximation ◽  
Algebraic Manifolds

The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for least-squares fitting of these algebraic manifolds. This paper studies the problem of using algebraic distance for least-squares fitting of quadrics defining the constraint manifolds associated with Planar 4R mechanisms. It shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given one dimensional set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.

A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar ◽  
Xiangyun Li
Keyword(s):  
Fast Algorithm ◽  
Nonlinear Least Squares ◽  
Linear Formulation ◽  
Image Space ◽  
Kinematic Synthesis ◽  
Novel Approach ◽  
Motion Approximation ◽  
New Formulation ◽  
Algebraic Manifolds ◽  
Fitting Problem

The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for nonlinear least squares fitting of these algebraic manifolds. This paper presents a new formulation for the manifold fitting problem using algebraic distance and shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.

scholarly journals One-dimensional constrained inversion study of TEM and application in coal goafs’ detection

2020 ◽  
Vol 12 (1) ◽  
pp. 1533-1540
Author(s):  
Si Yuanlei ◽  
Li Maofei ◽  
Liu Yaoning ◽  
Guo Weihong
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Data Interpretation ◽  
Volume Effect ◽  
Underground Space ◽  
Attenuation Curve ◽  
One Dimensional ◽  
Transient Electromagnetic ◽  
Geological Conditions ◽  
Geological Information

AbstractTransient electromagnetic method (TEM) is often used in urban underground space exploration and field geological resource detection. Inversion is the most important step in data interpretation. Because of the volume effect of the TEM, the inversion results are usually multi-solvable. To reduce the multi-solvability of inversion, the constrained inversion of TEM has been studied using the least squares method. The inversion trials were performed using two three-layer theoretical geological models and one four-layer theoretical geological model. The results show that one-dimensional least squares constrained inversion is faster and more effective than unconstrained inversion. The induced electromotive force attenuation curves of the inversion model indicate that the same attenuation curve may be used for different geological conditions. Therefore, constrained inversion using known geological information can more accurately reflect the underground geological information.

scholarly journals An Error-Reduction Method for Temporal Phase Unwrapping Based on Double Three-Step Phase-Shifting and Least-Squares Fitting

2012 ◽  
Vol 6-7 ◽  
pp. 76-81
Author(s):  
Yong Liu ◽  
Ding Fa Huang ◽  
Yong Jiang
Keyword(s):  
Least Squares ◽  
Reduction Method ◽  
Intermediate Phase ◽  
Error Reduction ◽  
Phase Unwrapping ◽  
Phase Shifting ◽  
Surface Measurement ◽  
Acquisition Time ◽  
Least Squares Fitting ◽  
Temporal Phase

Phase-shifting interferometry on structured light projection is widely used in 3-D surface measurement. An investigation shows that least-squares fitting can significantly decrease random error by incorporating data from the intermediate phase values, but it cannot completely eliminate nonlinear error. This paper proposes an error-reduction method based on double three-step phase-shifting algorithm and least-squares fitting, and applies it on the temporal phase unwrapping algorithm using three-frequency heterodyne principle. Theoretical analyses and experiment results show that this method can greatly save data acquisition time and improve the precision.

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

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