Interpolation and Extrapolation of Optimally-Fitted Kinematic Error Model for Five-Axis Machine Tools

Machine tool geometric errors are frequently corrected by populating compensation tables that contain position-dependent offsets to each commanded axis position. While each offset can be determined by directly measuring the individual geometric error at that location, it is often more efficient to compute the compensation using a volumetric error model derived from measurements across the entire workspace. However, interpolation and extrapolation of measurements, once explicit in direct measurement methods, become implicit and obfuscated in the curve fitting process of volumetric error methods. The drive to maximize model accuracy while minimizing measurement sets can lead to significant model errors in workspace regions at or beyond the range of the metrology equipment. In this paper, a novel method of constructing machine tool volumetric error models is presented in which the characteristics of the interpolation and extrapolation errors are constrained. Using a typical five-axis machine tool compensation methodology, a constraint bounding the tool tip modeled error slope is added to the error model identification process. By including this constraint over the entire space, the geometric errors over the interpolation space are still well-identified. Also, the model performance over the extrapolation space is consistent with the behavior of the geometric error model over the interpolation space. The methodology is applied to an industrial five-axis machine tool. In the experimental implementation, for measurements outside of the measured region, an unconstrained model increases the mean residual by 40% while the constrained model reduces the mean residual by 40%.

Geographical Distribution of Cutaneous Leishmaniasis and Its Relationship With Climate Change in Southeastern Morocco

The current chapter deals with one of the most neglected tropical diseases in Morocco, the cutaneous leishmaniasis. It is based on 10-year research (2010-2017) on the evolution of leishmaniasis taking climate change into account. Epidemiological and climatological data were collected from different administrations. The Geographic Information System (GIS) is chosen for interpolation, space-time analysis of climate data and map creation. The SPSS software was used for statistical analysis and to establish the relationship between Leishmaniasis and climatic conditions. Results show that the maximum number of cases is recorded in 2010 with 4,407 people affected while the low number is recorded in 2014 with 18 cases. Results also show a clear link between climatic factors and the incidence of the disease. The distribution of the disease in the province is influenced by maximum temperature, aridity, and vegetation cover. Additionally, anthropogenic factors play a significant role in explaining the emergence or re-emergence of leishmaniasis in the region.

Variable Frequency Signal Carrying Nonlinear Transmission Line - Modeling using Machine Learning

in modeling of complex systems, manual creation and maintenance of the appropriate behavior is found to be the key problem. Behavior modeling using machine learning has found successful in modeling and simulation. This paper presents artificial neural network (ANN) modeling of transmission line carrying frequency varying signal using machine learning. This work uses proper orthogonal decomposition (POD) based reduced order modeling. In this proposed work, snapshot sets of complex mathematical model of nonlinear transmission line and also linear model are obtained at different time interval. These snapshot sets are arranged in matrix form separately for nonlinear and linear models. POD method is applied on both the matrices separately. This reduces the order of the matrix which is used as input and output data set for neural network training through machine learning technique. Trained neural network model has been verified using different untrained data set. The proposed algorithm determines the dimension of the interpolation space prompting a considerable decrease in the computational expense. The proposed algorithm doesn't force any imperatives on the topology of the appropriate circuit or kind of the nonlinear segments and hence relevant to general nonlinear systems.

A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves

For sampling values along spherical Lissajous curves we establish a spectral interpolation and quadrature scheme on the sphere. We provide a mathematical analysis of spherical Lissajous curves and study the characteristic properties of their intersection points. Based on a discrete orthogonality structure we are able to prove the unisolvence of the interpolation problem. As basis functions for the interpolation space we use a parity-modified double Fourier basis on the sphere that allows us to implement the interpolation scheme in an efficient way. We further show that the numerical condition number of the interpolation scheme displays a logarithmic growth. As an application, we use the developed interpolation algorithm to estimate the rotation of an object based on measurements at the spherical Lissajous nodes.

3D LITHOLOGICAL MODEL USING BY INTERPOLATION AND SIMULATION METHODS (ON AN EXAMPLE OF THE CU-AG DEPOSIT, LEGNICA-GŁOGÓW COPPER DISTRICT)

The article attempts to evaluate the effectiveness of three different procedures of 3D lithological model creation with the usage of: deterministic or geostatistical interpolation methods, standard interpolation method used in KGHM Polska Miedź S.A. and Plurigaussian geostatistical simulation. The 3D models of main lithology created using those methods for a part of Cu-Ag Legnica-Głogów Copper District (Rudna Mine) served as the basis for the evaluation. The effectiveness of those methods was quantified by means of relative absolute estimation errors of thickness of lithological series in the 3D models. The mean relative estimation errors of thickness of main lithological series in 3D models take values from 6% to 13%. This demonstrates the comparable accuracy of the 3 procedures used, despite the fact that they differ significantly in computational algorithms and in the different way of defining the 3D interpolation space. Moreover, no significant differences in the accuracy level of lithological borders estimation of carbonate series, shale series and sandstone series were stated.

Importance of Applicability Domain of QSAR Models

Quantitative Structure-Activity Relationship (QSAR) models have manifold applications in drug discovery, environmental fate modeling, risk assessment, and property prediction of chemicals and pharmaceuticals. One of the principles recommended by the Organization of Economic Co-operation and Development (OECD) for model validation requires defining the Applicability Domain (AD) for QSAR models, which allows one to estimate the uncertainty in the prediction of a compound based on how similar it is to the training compounds, which are used in the model development. The AD is a significant tool to build a reliable QSAR model, which is generally limited in use to query chemicals structurally similar to the training compounds. Thus, characterization of interpolation space is significant in defining the AD. An attempt is made in this chapter to address the important concepts and methodology of the AD as well as criteria for estimating AD through training set interpolation in the descriptor space.

On the Singularity of Multivariate Hermite Interpolation of Total Degree

We study the singularity of multivariate Hermite interpolation of type total degree onmnodes with3+d<m≤d(d+3)/2. We first check the number of the interpolation conditions and the dimension of interpolation space. And then the singularity of the interpolation schemes is decided for most cases. Also some regular interpolation schemes are derived, a few of which are proved due to theoretical argument and most of which are verified by numerical method. There are some schemes to be decided and left open.

Importance of Applicability Domain of QSAR Models

Quantitative Structure-Activity Relationship (QSAR) models have manifold applications in drug discovery, environmental fate modeling, risk assessment, and property prediction of chemicals and pharmaceuticals. One of the principles recommended by the Organization of Economic Co-operation and Development (OECD) for model validation requires defining the Applicability Domain (AD) for QSAR models, which allows one to estimate the uncertainty in the prediction of a compound based on how similar it is to the training compounds, which are used in the model development. The AD is a significant tool to build a reliable QSAR model, which is generally limited in use to query chemicals structurally similar to the training compounds. Thus, characterization of interpolation space is significant in defining the AD. An attempt is made in this chapter to address the important concepts and methodology of the AD as well as criteria for estimating AD through training set interpolation in the descriptor space.

Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds

Forμ≥−1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis functionΦin certain spaces of continuous functionsYn(n∈ℕ) depending on a weightw. The functionsΦandware connected through the distributional identityt4n(hμ′Φ)(t)=1/w(t), wherehμ′denotes the generalized Hankel transform of orderμ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian spaceℋμin order to derive explicit representations of the derivativesSμmΦand their Hankel transforms, the former ones being valid whenm∈ℤ+is restricted to a suitable interval for whichSμmΦis continuous. Here,Sμmdenotes themth iterate of the Bessel differential operatorSμifm∈ℕ, whileSμ0is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation(hμ′Φ)(t)=1/t4nw(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation spaceYn.