Producing Follower Motions Through Their Digitized Cam Contours

2002 ◽  
Vol 2 (2) ◽  
pp. 98-105 ◽  
Author(s):  
Der-Min Tsay ◽  
Meng-Hung Huang ◽  
Hui-Chun Ho
Keyword(s):  
Least Squares ◽  
Contact Point ◽  
Spline Approximation ◽  
B Spline ◽  
Theoretical Design ◽  
Analytical Results ◽  
Surface Data ◽  
Cam Follower ◽  
Data Points ◽  
Computing Procedure

A computing procedure that can be used to generate follower motions with a higher degree of accuracy by using their digitized 2-D cam surfaces is presented. Based on the geometric relationships at the contact point between a planar cam and its follower, the reverse functions that can be solved to find the follower motion of the cam-follower mechanism are established. To approximate the digitized cam surfaces and to create their derivatives required in determining the follower motions, a least squares periodic B-spline approximation is also introduced. In comparison with analytical results obtained from a theoretical design, the procedure is verified for its accuracy and reliability. Furthermore, by using the cam surface data points taken from a CMM, a practical example is given to show the effectiveness and usefulness of the proposed techniques.

Determination of Follower Motions Using Measured Cam Profiles

1998 ◽  
Author(s):  
Der Min Tsay ◽  
Meng Hung Huang ◽  
Hui Chun Ho
Keyword(s):  
Least Squares ◽  
Contact Point ◽  
Spline Approximation ◽  
Measured Data ◽  
Profile Data ◽  
B Spline ◽  
Systematic Procedure ◽  
Cam Profile ◽  
Geometric Relationship

Abstract A versatile, systematic procedure that can be used to determine follower motions from measured discrete planar cam profile data is presented. Based on the geometric relationship at the contact point between a cam and its follower system, the reverse functions for planar cams with various types of followers are established. Then, a periodic B-spline approximation in a form of least squares is introduced to approximate the measured data and to generate the data derivatives required in finding their corresponding follower motions. The accuracy and reliability of this method is verified, and furthermore, a practical example is given to show the application of this procedure.

Development of a Measuring System for Planar Cam Profiles and Their Follower Motions

2004 ◽  
Author(s):  
Der-Min Tsay ◽  
Kuo-Shu Tseng ◽  
Hsin-Pao Chen
Keyword(s):  
Contact Point ◽  
Coordinate Measuring Machine ◽  
Measuring System ◽  
Displacement Curve ◽  
Practical Application ◽  
Test Bed ◽  
Theoretical Design ◽  
Cam Profile ◽  
Computing Procedure ◽  
Measuring Machine

A measuring system that can be used to inspect planar cam contours and to evaluate their follower displacement, velocity, and acceleration curves with a higher degree of accuracy without the aid of approximating follower displacements in traditional methods is constructed. Based on the geometric relationships at the contact point between a planar cam profile and its follower, analytical descriptions that can be utilized to determine the follower displacement curve and its derivatives are first identified. To verify the feasibility and accuracy of the algorithms proposed for the measuring system, analytical results generated from a theoretical design are compared to those obtained by the application of the computing procedure. To demonstrate the effectiveness and usefulness of the developed procedure, a measuring test bed has been constructed for a practical application example. Furthermore, the results obtained by using the built measuring system are also compared to those obtained by using a coordinate measuring machine (CMM) with the proposed algorithms.

scholarly journals Combination of regional and global geoid models at continental scale: application to Iranian geoid

2021 ◽  
Vol 64 (4) ◽  
pp. GD434
Author(s):  
Mahin Hosseini-Asl ◽  
Alireza Amiri-Simkooei ◽  
Abdolreza Safari
Keyword(s):  
Least Squares ◽  
Spline Approximation ◽  
Functional Model ◽  
Geoid Model ◽  
Continental Scale ◽  
Spline Surface ◽  
Check Points ◽  
Data Points ◽  
High Degree ◽  
Geometric Surface

High precision geoid determination is a challenging task at the national scale. Many efforts have been conducted to determine precise geoid, locally or globally. Geoid models have different precision depending on the type of information and the strategy employed when calculating the models. This contribution addresses the challenging problem of combining different regional and global geoid models, possibly combined with the geometric geoid derived from GNSS/leveling observations. The ultimate goal of this combination is to improve the precision of the combined model. We employ fitting an appropriate geometric surface to the geoid heights and estimating its (co)variance components. The proposed functional model uses the least squares 2D bi-cubic spline approximation (LS-BICSA) theory, which approximates the geoid model using a 2D spline surface fitted to an arbitrary set of data points in the region. The spline surface consists of third- order polynomial pieces that are smoothly connected together, imposing some continuity conditions at their boundaries. In addition, the least-squares variance component estimation (LS- VCE) is used to estimate precise weights and correlation among different models. We apply this strategy to the combined adjustment of the high-degree global gravitational model EIGEN-6C4, the regional geoid model IRG2016, and the Iranian geometric geoid derived from GNSS/leveling data. The accuracy of the constructed surface is investigated with five randomly selected subsamples of check points. The optimal combination of the two geoid models along with the GNSS/leveling data shows a reduction of 21 mm (~20%) in the RMSE values of discrepancies at the check points.

A B-Spline Least-Squares Surface-Fitting Method for Articular Surfaces of Diarthrodial Joints

1993 ◽  
Vol 115 (4A) ◽  
pp. 366-373 ◽  
Author(s):  
G. A. Ateshian
Keyword(s):  
Least Squares ◽  
Articular Surface ◽  
Three Dimensional ◽  
Surface Fitting ◽  
B Spline ◽  
Spline Surface ◽  
Articular Surfaces ◽  
Diarthrodial Joint ◽  
Surface Data ◽  
Fitting Method

The B-spline least-squares surface-fitting method is employed to create geometric models of diarthrodial joint articular surfaces. This method provides a smooth higher-order surface approximation from experimental three-dimensional surface data that have been obtained with any suitable measurement technique. Akima’s method for surface interpolation is used to provide complete support to the B-spline surface. The surface-fitting method is successfully tested on a known analytical surface, and is applied to the human distal femur. Applications to other articular surfaces are also shown. Results show that this method is precise, highly flexible, and can be successfully applied to a large variety of articular surface shapes.

scholarly journals Complex Uncertainty of Surface Data Modeling via the Type-2 Fuzzy B-Spline Model

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1054
Author(s):  
Rozaimi Zakaria ◽  
Abd. Fatah Wahab ◽  
Isfarita Ismail ◽  
Mohammad Izat Emir Zulkifly
Keyword(s):  
Complex Surface ◽  
Complex Data ◽  
Fuzzy Data ◽  
Control Points ◽  
B Spline ◽  
Spline Surface ◽  
Data Control ◽  
Surface Data ◽  
Model Complex

This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex uncertainty of surface data in type-2 fuzzy data/control points. These type-2 fuzzy data/control points are blended with the B-spline surface function to produce the proposed model, which can be visualized and analyzed further. Various processes, namely fuzzification, type-reduction and defuzzification are defined to achieve a crisp, type-2 fuzzy B-spline surface, representing uncertainty complex surface data. This paper ends with a numerical example of terrain modeling, which shows the effectiveness of handling the uncertainty complex data.

scholarly journals On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations

2020 ◽  
Vol 10 (1) ◽  
pp. 110-123
Author(s):  
Gaël Kermarrec ◽  
Hamza Alkhatib
Keyword(s):  
Shape Control ◽  
Simple Procedure ◽  
Spline Approximation ◽  
Autoregressive Process ◽  
Optimal Number ◽  
Information Criteria ◽  
General Effect ◽  
Control Points ◽  
B Spline ◽  
Correlated Observations

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.

EXPRESS: Baseline Correction Based on a Search Algorithm from Artificial Intelligence

2020 ◽  
pp. 000370282097751
Author(s):  
Xin Wang ◽  
Xia Chen
Keyword(s):  
Artificial Intelligence ◽  
Objective Function ◽  
Least Squares ◽  
Least Squares Method ◽  
Search Algorithm ◽  
Absolute Error ◽  
Search Problem ◽  
Baseline Correction ◽  
Current Spectrum ◽  
Data Points

Many spectra have a polynomial-like baseline. Iterative polynomial fitting (IPF) is one of the most popular methods for baseline correction of these spectra. However, the baseline estimated by IPF may have substantially error when the spectrum contains significantly strong peaks or have strong peaks located at the endpoints. First, IPF uses temporary baseline estimated from the current spectrum to identify peak data points. If the current spectrum contains strong peaks, then the temporary baseline substantially deviates from the true baseline. Some good baseline data points of the spectrum might be mistakenly identified as peak data points and are artificially re-assigned with a low value. Second, if a strong peak is located at the endpoint of the spectrum, then the endpoint region of the estimated baseline might have significant error due to overfitting. This study proposes a search algorithm-based baseline correction method (SA) that aims to compress sample the raw spectrum to a dataset with small number of data points and then convert the peak removal process into solving a search problem in artificial intelligence (AI) to minimize an objective function by deleting peak data points. First, the raw spectrum is smoothened out by the moving average method to reduce noise and then divided into dozens of unequally spaced sections on the basis of Chebyshev nodes. Finally, the minimal points of each section are collected to form a dataset for peak removal through search algorithm. SA selects the mean absolute error (MAE) as the objective function because of its sensitivity to overfitting and rapid calculation. The baseline correction performance of SA is compared with those of three baseline correction methods: Lieber and Mahadevan–Jansen method, adaptive iteratively reweighted penalized least squares method, and improved asymmetric least squares method. Simulated and real FTIR and Raman spectra with polynomial-like baselines are employed in the experiments. Results show that for these spectra, the baseline estimated by SA has fewer error than those by the three other methods.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

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