Legendre Polynomials Of Order Research Articles

Comparative analysis on pulse compression with classical orthogonal polynomials for optimized time-bandwidth product

Abstract The theme of this paper is to analyze and compare the pulse compression with classical orthogonal polynomials (Chebyshev, Laguerre, Legendre and Hermite polynomials) of different orders. Pulse compression is used in radar systems to improve the range resolution by increasing the time-bandwidth product of the transmitted pulse. It is done by modulating the instantaneous angle of the transmitted pulse. Three types of angle modulations are considered in this paper. Initially, the angle is varied in proportional to the original polynomials. Secondly, the angle is proportional to integral of the polynomial and thirdly, the angle is proportional to derivative the polynomial. The main purpose of this analysis is to obtain and use the best of all these polynomials in pulse compression. This is done by comparing the quantitative parameter of pulse compression - time-bandwidth product. Optimization to maximize the time-bandwidth product is also considered in the analysis.

Genetic evaluation of weekly body weight in Japanese quail using random regression models

ABSTRACT1. A total of 11 826 records from 2489 quails, hatched between 2012 and 2013, were used to estimate genetic parameters for BW (body weight) of Japanese quail using random regression models. Weekly BW was measured from hatch until 49 d of age. WOMBAT software (University of New England, Australia) was used for estimating genetic and phenotypic parameters.2. Nineteen models were evaluated to identify the best orders of Legendre polynomials. A model with Legendre polynomial of order 3 for additive genetic effect, order 3 for permanent environmental effects and order 1 for maternal permanent environmental effects was chosen as the best model.3. According to the best model, phenotypic and genetic variances were higher at the end of the rearing period. Although direct heritability for BW reduced from 0.18 at hatch to 0.12 at 7 d of age, it gradually increased to 0.42 at 49 d of age. It indicates that BW at older ages is more controlled by genetic components in Japanese quail.4. Phenotypic and genetic correlations between adjacent periods except hatching weight were more closely correlated than remote periods. The present results suggested that BW at earlier ages, especially at hatch, are different traits compared to BW at older ages. Therefore, BW at earlier ages could not be used as a selection criterion for improving BW at slaughter age.

Genetic parameters for growth characteristics of free-range chickens under univariate random regression models

Repeated measures from the same individual have been analyzed by using repeatability and finite dimension models under univariate or multivariate analyses. However, in the last decade, the use of random regression models for genetic studies with longitudinal data have become more common. Thus, the aim of this research was to estimate genetic parameters for body weight of four experimental chicken lines by using univariate random regression models. Body weight data from hatching to 84 days of age (n = 34,730) from four experimental free-range chicken lines (7P, Caipirão da ESALQ, Caipirinha da ESALQ and Carijó Barbado) were used. The analysis model included the fixed effects of contemporary group (gender and rearing system), fixed regression coefficients for age at measurement, and random regression coefficients for permanent environmental effects and additive genetic effects. Heterogeneous variances for residual effects were considered, and one residual variance was assigned for each of six subclasses of age at measurement. Random regression curves were modeled by using Legendre polynomials of the second and third orders, with the best model chosen based on the Akaike Information Criterion, Bayesian Information Criterion, and restricted maximum likelihood. Multivariate analyses under the same animal mixed model were also performed for the validation of the random regression models. The Legendre polynomials of second order were better for describing the growth curves of the lines studied. Moderate to high heritabilities (h(2) = 0.15 to 0.98) were estimated for body weight between one and 84 days of age, suggesting that selection for body weight at all ages can be used as a selection criteria. Genetic correlations among body weight records obtained through multivariate analyses ranged from 0.18 to 0.96, 0.12 to 0.89, 0.06 to 0.96, and 0.28 to 0.96 in 7P, Caipirão da ESALQ, Caipirinha da ESALQ, and Carijó Barbado chicken lines, respectively. Results indicate that genetic gain for body weight can be achieved by selection. Also, selection for body weight at 42 days of age can be maintained as a selection criterion.

A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn–Hilliard equation

In this paper, we develop and analyze a nodal discontinuous Galerkin method for the linearized fractional Cahn–Hilliard equation containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative. The linearized fractional Cahn–Hilliard problem has been expressed as a system of low order differential/integral equations. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization using a high-order nodal basis set of orthonormal Lagrange–Legendre polynomials of arbitrary order in space on each element of computational domain. Moreover, we prove the stability and optimal order of convergence N+1 for the linearized fractional Cahn–Hilliard problem when polynomials of degree N are used. Numerical experiments are displayed to verify the theoretical results.

Random regression model with heterogeneous residual variance reduces polynomial order fitted for permanent environmental effect but does not affect genetic parameters for milk production in Sahiwal cattle

The random regression test-day models can accelerate the genetic improvement of Sahiwal cattle as test-day milk yield models offer a faster, accurate and economical approach of genetic evaluation. First three lactation monthly test-day records of Sahiwal cows calved between 1961 and 2012 at ICAR-National Dairy Research Institute, Karnal were used to fit random regression model (RRM) with various order of legendre polynomial, and a constant (RRM-HOM) or heterogeneous residual variance (RRM-HET). For both RRM-HOM and RRM-HET third order legendre polynomial for modelling additive genetic effects were found best. There was reduction in order of fit for modelling permanent environmental effects due to assumption of heterogeneous residual variance, as legendre polynomial of sixth order for RRM-HOM and fourth or fifth order for RRM-HET was found to be best, for modelling the permanent environmental effect. First two eigenvalues of additive genetic random regression coefficient matrix explained more than 99% of the additive genetic variation, whereas four eigenvalues explained ~98% of the permanent environment variations. First eigenfunction from both the models did not show any large change along lactation, suggesting most variation can be explained by genes that act in same way during lactation. The heritability estimates were generally low but moderate for some test-day milk yields, and it ranged from 0.007 to 0.088 for first, 0.044 to 0.279 for second, and 0.089 to 0.129 for third lactation from RRM-HOM. The values of genetic correlation between test-day milk yields ranged from 0.06 to 0.99 for first, 0.77 to 0.99 for second, and 0.07 to 0.99 for third lactation, from RRM-HOM. The value of permanent environment correlation ranged from 0.30 to 0.98 for first, 0.07 to 0.99 for second, and 0.16 to 0.98 for third lactation. The genetic correlations between two adjacent test-days were high (≥0.90). RRM-HET also gave similar heritability and correlation estimates. The rank correlation between sire breeding values for first, second, and third lactation, estimated using two models were 0.98, 1.00, and 0.99, respectively, indicating there was no difference in the ranking of animals using two models. Thus the random regression model with lower order of polynomial for modelling additive genetic effect and higher order polynomial for modelling animal permanent environmental effect was found suitable for genetic evaluation and both RRM-HOM and RRM-HET can be used for modelling test-day milk yield and breeding value prediction in Sahiwal cattle.

Sensitivity of breeding values for carcass traits of meat-type quail to changes in dietary (methionine + cystine):lysine ratio using reaction norm models.

Our objective was to evaluate changes in breeding values for carcass traits of two meat-type quail (Coturnix coturnix) strains (LF1 and LF2) to changes in the dietary (methionine+cystine):lysine ([Met+Cys]:Lys) ratio due to genotype by environment (G×E) interaction via reaction norm. A total of 7000 records of carcass weight and yield were used for analyses. During the initial phase (from hatching to day 21), five diets with increasing (Met+Cys):Lys ratios (0.61, 0.66, 0.71, 0.76 and 0.81), containing 26.1% crude protein and 2900kcalME/kg, were evaluated. Analyses were performed using random regression models that included linear functions of sex (fixed effect) and breeding value (random effect) for carcass weight and yield, without and with heterogeneous residual variance adjustment. Both fixed and random effects were modelled using Legendre polynomials of second order. Genetic variance and heritability estimates were affected by both (Met+Cys):Lys ratio and strain. We observed that a G×E interaction was present, with changes in the breeding value ranking. Therefore, genetic evaluation for carcass traits should be performed under the same (Met+Cys):Lys ratio in which quails are raised.

Higher-order beam model for stress predictions in curved beams made from anisotropic materials

A higher-order beam model for analyzing the flexural response of curved multilayered beams with constant curvature and arbitrary constant thickness is developed. The new model is derived from the Hellinger–Reissner mixed variational statement and predicts inherently equilibrated 3D stresses from an equivalent single-layer model. As a starting assumption, the hoop stress is formulated as a series of higher-order stress resultants multiplied by Legendre polynomials. The governing equations are derived in a generalized manner such that the modeling order can be adjusted and is not defined a priori. Hence, the highest order Legendre polynomial determines the modeling order. The through-thickness shear and normal stresses are derived by integrating the generalized hoop stress in Cauchy’s polar equilibrium equations. As a result, all stress fields are based on the same set of variables, thereby considerably reducing the computational effort. The three stress fields, and two displacements in the radial and hoop directions are used in the Hellinger–Reissner functional to derive a new set of stress-displacement relations. The enforcement of the classical membrane and bending equilibrium equations of curved beams in the Hellinger–Reissner functional guarantees that all interlaminar and surface traction equilibrium conditions are satisfied exactly. A validation study of a composite laminate using a high-fidelity 3D finite element model shows that the stresses are captured very accurately by the present model, but with much less computational effort than the finite element model. As a result, the developed model can provide rapid and accurate insights into the expected damage onset behavior of curved laminates.

Estimation of genetic parameters for test-day milk yield in Khuzestan buffalo

Abstract: The objective of this work was to estimate covariance functions for additive genetic and permanent environmental effects, as well as to obtain genetic parameters for buffalo test-day milk yield using random regression models on Legendre polynomials (LPs). A total of 2,538 test-day milk yield (TDMY) records from 516 first lactation records of Khuzestan buffalo, calving from 1993 to 2009 and belonging to 150 herds located in the state of Khuzestan, Iran, were analyzed. The residual variances were modeled through a step function with 1, 5, 6, 9, and 19 classes. The additive genetic and permanent environmental random effects were modeled by LPs of days in milk using quadratic to septic polynomial functions. The model with additive genetic and animal permanent environmental effects adjusted by cubic and third order LP, respectively, and with the residual variance modeled through a step function with nine classes was the most adequate one to describe the covariance structure. The model with the highest significant log-likelihood ratio test (LRT) and with the lowest Akaike information criterion (AIC) and Bayesian information criterion (BIC) was considered to be the most appropriate one. Unexpected negative genetic correlation estimates were obtained between TDMY records of the twenty-fifth and thirty-seventh week (-0.03). Genetic correlation estimates were generally higher, close to unity, between adjacent weeks during the middle of lactation. Random regression models can be used for routine genetic evaluation of milk yield in Khuzestan buffalo.

Absolute hydrogen depth profiling using the resonant 1H(15N,[formula omitted])12C nuclear reaction

Resonant nuclear reactions are a powerful tool for the determination of the amount and profile of hydrogen in thin layers of material. Usually, this tool requires the use of a standard of well-known composition. The present work, by contrast, deals with standard-less hydrogen depth profiling. This approach requires precise nuclear data, e.g. on the widely used 1H(15N,αγ)12C reaction, resonant at 6.4MeV15N beam energy. Here, the strongly anisotropic angular distribution of the emitted γ-rays from this resonance has been re-measured, resolving a previous discrepancy. Coefficients of (0.38±0.04) and (0.80±0.04) have been deduced for the second and fourth order Legendre polynomials, respectively. In addition, the resonance strength has been re-evaluated to (25.0±1.5)eV, 10% higher than previously reported. A simple working formula for the hydrogen concentration is given for cases with known γ-ray detection efficiency. Finally, the absolute approach is illustrated using two examples.

Random Regression Models Using Legendre Polynomials to Estimate Genetic Parameters for Test-day Milk Protein Yields in Iranian Holstein Dairy Cattle.

The objective of this study was to estimate the genetic parameters of milk protein yields in Iranian Holstein dairy cattle. A total of 1,112,082 test-day milk protein yield records of 167,269 first lactation Holstein cows, calved from 1990 to 2010, were analyzed. Estimates of the variance components, heritability, and genetic correlations for milk protein yields were obtained using a random regression test-day model. Milking times, herd, age of recording, year, and month of recording were included as fixed effects in the model. Additive genetic and permanent environmental random effects for the lactation curve were taken into account by applying orthogonal Legendre polynomials of the fourth order in the model. The lowest and highest additive genetic variances were estimated at the beginning and end of lactation, respectively. Permanent environmental variance was higher at both extremes. Residual variance was lowest at the middle of the lactation and contrarily, heritability increased during this period. Maximum heritability was found during the 12th lactation stage (0.213±0.007). Genetic, permanent, and phenotypic correlations among test-days decreased as the interval between consecutive test-days increased. A relatively large data set was used in this study; therefore, the estimated (co)variance components for random regression coefficients could be used for national genetic evaluation of dairy cattle in Iran.

A robust nonparametric framework for reconstruction of stochastic differential equation models

In this paper, we employ a nonparametric framework to robustly estimate the functional forms of drift and diffusion terms from discrete stationary time series. The proposed method significantly improves the accuracy of the parameter estimation. In this framework, drift and diffusion coefficients are modeled through orthogonal Legendre polynomials. We employ the least squares regression approach along with the Euler–Maruyama approximation method to learn coefficients of stochastic model. Next, a numerical discrete construction of mean squared prediction error (MSPE) is established to calculate the order of Legendre polynomials in drift and diffusion terms. We show numerically that the new method is robust against the variation in sample size and sampling rate. The performance of our method in comparison with the kernel-based regression (KBR) method is demonstrated through simulation and real data. In case of real dataset, we test our method for discriminating healthy electroencephalogram (EEG) signals from epilepsy ones. We also demonstrate the efficiency of the method through prediction in the financial data. In both simulation and real data, our algorithm outperforms the KBR method.

Application of Fractional Order Legendre Polynomials: a New Procedure for Solution of Linear and Nonlinear Fractional Differential Equations under $m$-point Nonlocal Boundary Conditions

In this paper, we have proposed a new formulation for the solution of a general class of fractional differential equations (linear and nonlinear) under $\hat{m}$-point boundary conditions. We derive some new operational matrices and based on these operational matrices we develop scheme to approximate solution of the problem. The scheme convert the boundary value problem to a system of easily solvable algebraic equations. We show the applicability of the scheme by solving some test problems. The scheme is computer oriented.

Photoelectron circular dichroism observed in the above-threshold ionization signal from chiral molecules with femtosecond laser pulses

Photoelectron circular dichroism is investigated experimentally as a function of the number of absorbed circularly polarized photons. Three structurally different chiral molecules yet showing similar absorption spectra are studied. They are isotropically distributed in the gas phase and ionized with femtosecond laser pulses. We measure and analyze the photoelectron angular distribution of threshold electrons ionized with three photons and compare them to those of above-threshold (ATI) electrons ionized with four photons. Additionally to an increase in high even order Legendre polynomials the coefficients of the high odd order Legendre polynomials rise with increasing photon number. Consequently, the ATI electrons also carry the chirality signature. All investigated chiral molecules reveal an individual set of coefficients for the threshold and ATI signatures despite their similarities in chemical structure. The presented data set can serve as a guideline for theoretical modeling of the interaction of circularly polarized light with chiral molecules in the multiphoton regime.

Reaction norms in weights at 365 days old in nellore bulls in northern Brazil

<p>Genotype by environment interaction (GxE) studies are of particular interest in Brazil because of the regional diversity of environmental effects and the wide variety of management systems. The present study evaluates GxE effects on 365 d weight (365W) of Nellore cattle raised on pasture in northern Brazil. The analysis utilized random regression techniques to model the reaction norm. Fixed effects consisted of sex, contemporary group, and the covariate of age of cow at calving. The environmental gradient, defined by the concatenation of a bull and the state in which the calf was born, was modeled by second order Legendre polynomials. Direct additive genetic and residual effects were fit as random. Results showed differences in the magnitude of expression of genotype in proportion to decreasing favorability of the environment. As the environment became more unfavorable, the correlation of breeding value to different environments decreased. The correlations between the intercept and the level slope for 365W feature were of moderate magnitude, predominantly indicating the reclassification of sires in different environments. Reaction standard model was coherent from a technical and biological view point and enabled the perception of GxE in the genetic evaluation of Nellore cattle in the states of Maranhão, Pará and Tocantins.</p><p> </p>

Reaction norms in weights at 365 days old in nellore bulls in northern Brazil

Genotype by environment interaction (GxE) studies are of particular interest in Brazil because of the regional diversity of environmental effects and the wide variety of management systems. The present study evaluates GxE effects on 365 d weight (365W) of Nellore cattle raised on pasture in northern Brazil. The analysis utilized random regression techniques to model the reaction norm. Fixed effects consisted of sex, contemporary group, and the covariate of age of cow at calving. The environmental gradient, defined by the concatenation of a bull and the state in which the calf was born, was modeled by second order Legendre polynomials. Direct additive genetic and residual effects were fit as random. Results showed differences in the magnitude of expression of genotype in proportion to decreasing favorability of the environment. As the environment became more unfavorable, the correlation of breeding value to different environments decreased. The correlations between the intercept and the level slope for 365W feature were of moderate magnitude, predominantly indicating the reclassification of sires in different environments. Reaction standard model was coherent from a technical and biological view point and enabled the perception of GxE in the genetic evaluation of Nellore cattle in the states of Maranhão, Pará and Tocantins.

Genotype×dietary (methionine+cystine):Lysine ratio interaction for body weight of meat-type quails using reaction norm models

Our objective was to evaluate the effect of genotype by environment (G×E) interaction for body weight (BW) of meat-type quails (Coturnix coturnix) during the growing period using two strains (LF1 and LF2) to changes in the dietary (methionine+cystine):lysine ([Met+Cys]:Lys) ratio. A total of 4512 records, collected during the initial phase (day 14 and 21), and final phase (day 28 and 35), was used for analyses. During the initial phase, quails were fed diets with increasing (Met+Cys):Lys ratios (0.61, 0.66, 0.71, 0.76, and 0.81), containing 26.1% crude protein (CP) and 2900kcal ME/kg; and during the final phase, quails were fed a single diet containing 22% CP and 3050kcal ME/kg. A single-trait animal model, including sex as the fixed effect, (Met+Cys):Lys ratio as the control variable, and breeding value as the random effect, was utilized for analyses. The Legendre polynomial of second order was utilized for the analysis of the G×E (nutritional levels) interaction using reaction norm. Genetic variance and heritability estimates varied according to the nutritional gradient. The LF2 strain had greater sensitivity to changes in the dietary (Met+Cys):Lys ratio compared with LF1, and greater residual effect of diets provided during the initial phase of the growing period. Genetic evaluations for quails fed different dietary (Met+Cys):Lys ratio provided changes in the predicted BW breeding values. We conclude that selection population must be fed with same dietary (Met+Cys):Lys ratio that commercial populations are fed.

Random Regression Models Are Suitable to Substitute the Traditional 305-Day Lactation Model in Genetic Evaluations of Holstein Cattle in Brazil.

The aim of this study was to compare two random regression models (RRM) fitted by fourth (RRM4) and fifth-order Legendre polynomials (RRM5) with a lactation model (LM) for evaluating Holstein cattle in Brazil. Two datasets with the same animals were prepared for this study. To apply test-day RRM and LMs, 262,426 test day records and 30,228 lactation records covering 305 days were prepared, respectively. The lowest values of Akaike’s information criterion, Bayesian information criterion, and estimates of the maximum of the likelihood function (−2LogL) were for RRM4. Heritability for 305-day milk yield (305MY) was 0.23 (RRM4), 0.24 (RRM5), and 0.21 (LM). Heritability, additive genetic and permanent environmental variances of test days on days in milk was from 0.16 to 0.27, from 3.76 to 6.88 and from 11.12 to 20.21, respectively. Additive genetic correlations between test days ranged from 0.20 to 0.99. Permanent environmental correlations between test days were between 0.07 and 0.99. Standard deviations of average estimated breeding values (EBVs) for 305MY from RRM4 and RRM5 were from 11% to 30% higher for bulls and around 28% higher for cows than that in LM. Rank correlations between RRM EBVs and LM EBVs were between 0.86 to 0.96 for bulls and 0.80 to 0.87 for cows. Average percentage of gain in reliability of EBVs for 305-day yield increased from 4% to 17% for bulls and from 23% to 24% for cows when reliability of EBVs from RRM models was compared to those from LM model. Random regression model fitted by fourth order Legendre polynomials is recommended for genetic evaluations of Brazilian Holstein cattle because of the higher reliability in the estimation of breeding values.

A new SPn theory formulation with self-consistent physical assumptions on angular flux

The conventional SPn theory cannot provide the explicit angular flux solution. Chao and Yamamoto (2012) proposed the explicit angular flux representation for the SPn theory as cylindrically symmetric with respect to the net current direction at any point in space, such that the angular flux is an expansion in Legendre polynomials of the cosine of the polar angle with respect to the local net current. Such a model however cannot lead to the SPn equations without further ad hoc assumptions because the multiple directions of the spatial gradients of the flux moments cannot be all parallel to the net current. In this paper we relax the assumption of the total angular flux being locally one-dimensional and generalize it to each of the flux moments being locally one-dimensional along the direction of the spatial gradient of each individual flux moment. The angular distribution of the nth order flux moment is the nth order Legendre polynomial of the cosine of the polar angle with respect to the direction of the spatial gradient of the nth order flux moment, ∇ϕn(r). With this physical model one can rigorously derive the equations for the current and the boundary conditions. The SPn equations can also be derived with the additional assumption of the total cross-section being locally constant, which is practically always valid when the spatial variation is discretized in numerical calculations. However the boundary conditions turn out to be different from the conventional ones, containing some non-linear factors. The internal interface boundary conditions are not affected by the non-linear factors as they cancel out on the interface. But the external boundary condition does get affected by the non-linear factors. The effect of non-linear factors is of higher order and if neglected, the external boundary condition also reduces to the conventional one. The non-linear external boundary condition can be iteratively updated to estimate the correction effect. A numerical calculation problem is suggested to test the new SPn theory and to assess the non-linear effect.

Distortions in EEG interregional phase synchrony by spherical spline interpolation: causes and remedies

Measures of the coherence of electroencephalography (EEG) time-series recorded at spatially distant points on the scalp are often used by researchers to characterize the dynamic interactions of brain regions. In dense-array EEG recordings, one or more electrode signals often contain prominent artifact necessitating replacement of the recorded data with an estimated signal using interpolation from data in valid recordings from the surrounding electrodes. Typically the signal estimation is carried out using spherical spline interpolation (SSI; Perrin et al., 1989); however, it is shown that this can introduce an erroneous increase in coherence between signals because the estimated signal is derived from elements in common recordings from other electrode sites. Although SSI performance depends on three SSI parameters, including interpolation order m, the Legendre polynomial order n, and regularization parameter λ, clear guidelines on how to optimally choose parameters have yet to be established for ensuring the temporal features of interpolated signals are accurate. Using a cross validation approach we compared interpolated and original signals in 64 and 128 channel resting EEG data systematically examining the effect of data interpolation across various sets of SSI parameters, frequency bands, and locations of interpolated electrodes and electrode pairs on a standard measure of phase coherence (phase locking value: PLV). We found substantial PLV inflation with SSI, which was greater in 128 channel than in 64 channel EEG data and greater for neighboring electrodes than distant electrodes. Furthermore, the amount of PLV distortions varied across the locations of interpolated signals. Performance of SSI was highly dependent on the three SSI parameters, and the effect of each parameter was dependent on the levels of other parameters. Although SSI parameters commonly recommended in the literature led to notable and concerning PLV inflation, application of SSI with alternative and optimal parameter values led to accurate and nearly distortion-free phase coherence values (i.e., PLV) from estimated signals.

Large deformation analysis and synthesis of elastic closed-loop mechanism made of a certain spring wire described by free curves

Recently novel mechanisms with compact size and without many mechanical elements such as bearing are strongly required for medical devices such as surgical operation devices. This paper describes analysis and synthesis of elastic link mechanisms of a single spring beam which can be manufactured by NC coiling machines. These mechanisms are expected as disposable micro forceps. Smooth Curvature Model(SCM) with 3rd order Legendre polynomial curvature functions is applied to calculate large deformation of a curved cantilever beam by taking account of the balance between external and internal elastic forces and moments. SCM is then extended to analyze large deformation of a closed-loop curved elastic beam which is composed of multiple free curved beams. A closed-loop elastic link is divided into two free curved cantilever beams each of which is assumed as serially connected free curved cantilever beams described with SCM. The sets of coefficients of Legendre polynomials of SCM in all free curved cantilever beams are determined by taking account of the force and moment balance at connecting point where external input force is applied. The sets of coefficients of Legendre polynomials of a nonleaded closed-loop elastic link are optimized to design a link mechanism which can generate specified output motion due to input force applied at the assumed dividing point. For example, two planar micro grippers with a single pulling input force are analyzed and designed. The elastic deformation analyzed with proposed method agrees very well with that calculated with FEM. The designed micro gripper can generate the desired pinching motion. The proposed method can contribute to design compact and simple elastic mechanisms without high calculation costs.