Higher Order Polynomial Models Research Articles

A novel iterative model updating for jointed structures using nonlinear FRFs

A novel iterative model updating method is developed to identify the local nonlinear joint properties using the frequency response functions (FRFs) of assembled structures. In this paper, the least-squares fitting method was used to transform the sensitivity matrix into a square iteration matrix to match the dimensions of the objective functions and nonlinear joint model parameters. Leveraging the Newton’s iteration, the adaptive successive over-relaxation (A-SOR) was used to ensure iteration convergence while the multi-scale parameter adjusting (MPA) strategy was developed to degrade the ill-condition of the iteration matrix. Two updating examples of phenomenological equivalence models were applied to demonstrate the effectiveness of the proposed method. The nonlinear FRFs of a lap-type bolted joint beam system with Iwan model were simulated as the objective functions to identify the local nonlinear joint properties, as well as experimental investigations of a metal rubber isolation system with a high-order polynomial model. The proposed method was validated by the good agreement of the comparison results, and it indicated a better model updating performance with a much smaller condition number of the iteration matrix.

Experimental Investigation of Characteristics and Development of Nonlinearity Models for FR1-Range Radio-Frequency Amplifiers

Experimental studies of nonlinear properties of low-power radio-frequency amplifiers similar to those used in 4G/5G mobile communication equipment of the FR1 frequency range are carried out. The measurements of the characteristics of the amplifiers are performed by using the double-frequency testing technique at frequencies of the n7 band (2500–2570 / 2620–2690 MHz), which is allocated in Belarus for 4G mobile communication systems, and the n78 band (3300–3800 MHz), which is planned to be used in Belarus for 5G mobile communications. Based on the results of the measurements of double-frequency characteristics of the amplifiers, their single-tone amplitude characteristics, as well as two-tone characteristics and values of the dynamic range of 3-, 5-, 7-, and 9th order intermodulation in the first harmonic zone, high-order polynomial models of transfer characteristics of the investigated amplifiers are synthesized. The synthesized models are suitable for use in a wide dynamic range of input signals in case of simultaneous modeling of nonlinear effects of all kinds that pose a danger to radio reception in a complex electromagnetic environment created in the frequency bands of mobile (cellular) communications: both “subtle” effects (intermodulation) and “rough” effects (desensitization, cross-modulation). When using the technology of discrete nonlinear analysis of the behavior of radio equipment in a complex electromagnetic environment, the obtained models provide high efficiency of simulation and quantitative analysis of nonlinear processes and radio interference occurring in 4G/5G/6G radio equipment and networks in a complex electromagnetic environment.

ADVANCED THERMOCOUPLE LINEARIZATION METHOD USING ADVANCED POLYNOMIAL FITTING

In this study, this paper presents a new method for linearizing thermocouple data using Python and compares the performance of higher-order polynomial models in achieving linearization. It involves fitting a non-linear model to the thermocouple data using the curve fit function from Python and then calculating the linearized temperature values using the optimized parameters. The paper also presents a comparative analysis of different polynomial models, ranging from 3rd to 12th order, and evaluates their performance in achieving linearization. The results show that higher-order polynomial models generally perform better than lower-order models in achieving linearization, but also have a higher risk of overfitting. The paper concludes that the presented method provides an effective way of linearizing thermocouple data using Python and that the choice of polynomial model should be carefully considered based on the data characteristics and the desired level of accuracy.

Kinks in higher-order polynomial models

We consider a family of field-theoretic models with a real scalar field in (1+1)-dimensional space–time. The field dynamics in each model is determined by a polynomial potential with two degenerate minima. We obtain exact general formulas for kink solutions with power-law asymptotic behavior. We also write out formulas for the asymptotics of all found kinks. In addition, we analyze some other properties of the obtained kinks: stability potentials, zero modes, positions of the centers of mass.

Phase-unwrapping method based on local polynomial models and a maximum a posteriori model correction.

Recently, a theory on local polynomial approximations for phase-unwrapping algorithms, considering a state space analysis, has been proposed in Appl. Opt.56, 29 (2017)APOPAI0003-693510.1364/AO.56.000029. Although this work is a suitable methodology to deal with relatively low signal to noise ratios observed in the wrapped phase, the methodology has been developed only for local-polynomial phase models of order 1. The resultant proposal is an interesting Kalman filter approach for estimating the coefficient or state vectors of these local plane models. Thus, motivated by this approach and simple Bayesian theory, and considering our previous research on local polynomial models up to the third order [Appl. Opt.58, 436 (2019)APOPAI0003-693510.1364/AO.58.000436], we propose an equivalent methodology based on a simple maximum a posteriori estimation, but considering a different state space: difference vectors of coefficients for the current high-order polynomial models. Specific estimations of the covariance matrices for difference vectors, as well as noise covariance matrices involved with the correct estimation of coefficient vectors, are proposed and reconstructions with synthetic and real data are provided.

High-precision shape approximation low-thrust trajectory optimization method satisfying bi-objective index

The shape approximation method has been proven to be rapid and practicable in resolving low-thrust trajectory; however, it still faces the challenges of large deviation from the optimal solution and inability to satisfy the specific flight time and fuel mass constraints. In this paper, a modified shape approximation low-thrust model is presented, and a novel constrained optimization algorithm is developed to solve this problem. The proposed method aims at settling the bi-objective optimization orbit involving the twin objectives of minimum flight time and low fuel consumption and enhancing the accuracy of optimized orbit. In particular, a transformed high-order polynomial model based on finite Fourier series is proposed, which can be characterized as a multi-constraint optimization problem. Then, a novel optimization algorithm is specifically developed to optimize the large-scale multi-constraint dynamical equations of shape trajectory. The key performance indicators of the index include minimum flight time, low fuel consumption and bi-objective optimization of the two. Simulation results prove that this approach possesses both the high precision achievable by numerical methods and low computational complexity offered by shape approximation techniques. Besides, the Pareto front of the fuel-time bi-objective optimization orbit is firstly introduced to analyze an intact optimal solution set. Furthermore, we have demonstrated that our proposed approach is appropriate to generate the preliminary orbit for pseudo-spectral method.

A Motion Parameter Estimation Method for Radar Maneuvering Target in Gaussian Clutter

It is known that accurate motion parameter estimation makes significant benefits to radar target tracking, imaging and recognition. However, maneuvering targets usually cause across-range-unit and across-Doppler-unit effects, which make it difficult to accurately estimate the motion parameters of the target. The generalized Radon-Fourier transform (GRFT) has been proposed to estimate the motion parameter vector of maneuvering targets, but only the noise scenario is considered. In this paper, the normalized adaptive GRFT (NAGRFT) is first proposed for motion parameter estimation, which is proven to be the maximum likelihood estimator (MLE) of the motion parameter vector in Gaussian clutter. Then, we derive the Cramer-Rao lower bound (CRLB) for the motion parameter vector as a comparison of the estimation mean square error (MSE) of the NAGRFT. In this derivation, the target motion is described by a high-order polynomial model, and a Gaussian-shaped function is adopted to characterize the power spectrum density of the homogenous clutter. Finally, to demonstrate the optimal estimation performance of the proposed NAGRFT, numerical experiments are provided, which show that the estimation MSE of the NAGRFT can reach the CRLB. Experiments are also provided to analyze the relationship between the CRLB and clutter parameters.

A Reliable Gravity Compensation Control Strategy for dVRK Robotic Arms With Nonlinear Disturbance Forces

External disturbance forces caused by nonlinear springy electrical cables in the Master Tool Manipulator (MTM) of the da Vinci Research Kit (dVRK) limits the usage of the existing gravity compensation methods. Significant motion drifts at the MTM tip are often observed when the MTM is located far from its identification trajectory, preventing the usage of these methods for the entire workspace reliably. In this paper, we propose a general and systematic framework to address the problems of the gravity compensation for the MTM of the dVRK. Particularly, high order polynomial models were used to capture the highly nonlinear disturbance forces and integrated with the Multi-step Least Square Estimation (MLSE) framework. This method allows us to identify the parameters of both the gravitational and disturbance forces for each link sequentially, preventing residual error passing among the links of the MTM with uneven mass distribution. A corresponding gravity compensation controller was developed to compensate the gravitational and disturbance forces. The method was validated with extensive experiments in the majority of the manipulator's workspace, showing significant performance enhancements over existing methods. Finally, a deliverable software package in MATLAB and C++ was integrated with dVRK and published in the dVRK community for open-source research and development.

Smooth and time-optimal S-curve trajectory planning for automated robots and machines

In this paper, a smooth and time-optimal S-curve trajectory planning method is proposed to meet the requirements of high-speed and ultra-precision operation for robotic manipulators in modern industrial applications. This method utilizes a piecewise sigmoid function to establish a jerk profile with suitably chosen phase durations such that the generated trajectories are infinitely continuously differentiable under the given constraints on velocity, acceleration and jerk. All the trajectory parameters are derived with an analytical algorithm to ensure an acceptable computational cost. This S-curve model achieves a greater efficiency than the trigonometric models, while avoiding the high complexity presented by conventional high order polynomial models. The trade-off between efficiency and smoothness can be modulated by the limit value of the snap (the derivative of jerk), this feature is advantageous for adapting to different task requirements. Furthermore, a synchronization strategy is suggested to coordinate multi-axis motions, enabling the full capabilities of the actuators to be exploited. The feasibility and practicality of the proposed approach is evaluated by the simulation and experimental studies in comparison with other benchmark techniques in the literature.

A new framework for characterizing landslide deformation: a case study of the Yu-Kai highway landslide in Guizhou, China

This paper presents a new framework for characterizing landslide deformation. Here, the deformation of a landslide is interpreted as a summation of three components: rigid deformation, within-mass deformation, and residual deformation. On the basis of the monitored data of the landslide deformation, these three components may be characterized separately: the rigid deformation is simulated by a summation of a trend term and a periodic term, the within-mass deformation is simulated by a high-order polynomial model, and the residual deformation is simulated by a conditional random field model. In particular, the characterization of the residual deformation, the third component of the landslide deformation, with the random field allows for a probabilistic assessment of the landslide deformation in the face of geological uncertainties. With the proposed framework, the evolution of landslide deformation in both geometric and time domains may be established, which allows for an assessment of the sliding mechanism of the landslide. Further, evolution in the geometric domain may allow for an assessment of the serviceability of infrastructures in the landslide area. To illustrate this new landslide deformation characterization framework, a case study of the Yu-Kai highway landslide in Guizhou, China is presented, through which the effectiveness of the proposed framework is demonstrated.

A new sequential sampling method for constructing the high-order polynomial surrogate models

PurposeThis paper aims to study the sampling methods (or design of experiments) which have a large influence on the performance of the surrogate model. To improve the adaptability of modelling, a new sequential sampling method termed as sequential Chebyshev sampling method (SCSM) is proposed in this study.Design/methodology/approachThe high-order polynomials are used to construct the global surrogated model, which retains the advantages of the traditional low-order polynomial models while overcoming their disadvantage in accuracy. First, the zeros of Chebyshev polynomials with the highest allowable order will be used as sampling candidates to improve the stability and accuracy of the high-order polynomial model. In the second step, some initial sampling points will be selected from the candidates by using a coordinate alternation algorithm, which keeps the initial sampling set uniformly distributed. Third, a fast sequential sampling scheme based on the space-filling principle is developed to collect more samples from the candidates, and the order of polynomial model is also updated in this procedure. The final surrogate model will be determined as the polynomial that has the largest adjusted R-square after the sequential sampling is terminated.FindingsThe SCSM has better performance in efficiency, accuracy and stability compared with several popular sequential sampling methods, e.g. LOLA-Voronoi algorithm and global Monte Carlo method from the SED toolbox, and the Halton sequence.Originality/valueThe SCSM has good performance in building the high-order surrogate model, including the high stability and accuracy, which may save a large amount of cost in solving complicated engineering design or optimisation problems.

A note on estimating single-class piecewise mixed-effects models with unknown change points

Piecewise mixed-effects models are useful for analyzing longitudinal educational and psychological data sets to model segmented change over time. These models offer an attractive alternative to commonly used quadratic and higher-order polynomial models because the coefficients obtained from fitting the model have meaningful substantive interpretation. The current study thus focuses on the estimation of piecewise mixed-effects model with unknown random change points using maximum likelihood (ML) as described in Du Toit and Cudeck (2009). Previous simulation work (Wang & McArdle, 2008) showed that Bayesian estimation produced reliable parameter estimates for the piecewise model in comparison to frequentist procedures (i.e., first-order Taylor expansion and the adaptive Gaussian quadrature) across all simulation conditions. In the current article a small Monte Carlo simulation study was conducted to assess the performance of the ML approach, a frequentist procedure, and the Bayesian approach for fitting linear–linear piecewise mixed-effects model. The obtained findings show that ML estimation approach produces reliable and accurate estimates under the conditions of small residual variance of the observed variables, and that the size of the residual variance had the most impact on the quality of model parameter estimates. Second, neither ML nor Bayesian estimation procedures performed well under all manipulated conditions with respect to the accuracy and precision of the estimated model parameters.

Simplification of high order polynomial calibration model for fringe projection profilometry

In fringe projection profilometry systems, high order polynomial calibration models can be employed to improve the accuracy. However, it is not stable to fit a high order polynomial model with least-squares algorithms. In this paper, a novel method is presented to analyze the significance of each polynomial term and simplify the high order polynomial calibration model. Term significance is evaluated by comparing the loading vector elements of the first few principal components which are obtained with the principal component analysis, and trivial terms are identified and neglected from the high order polynomial calibration model. As a result, the high order model is simplified with significant improvement of computation stability and little loss of reconstruction accuracy. An interesting finding is that some terms of 0 and 1st order, as well as some high order terms related to the image direction that is vertical to the phase change direction, are trivial terms for this specific problem. Experimental results are shown to validate of the proposed method.

Fundamental comparison of time-domain experimental modal analysis methods based on high- and first-order matrix models

All time-domain methods for experimental modal analysis (EMA) begin with a mathematical model. Based on either a high-order matrix polynomial model or a first-order state-space model, this paper emphasizes the comparison of numerical conditioning and stability, as well as the modal parameter estimation, among EMA methods. Numerical conditioning pertains to the perturbation behavior of a mathematical problem (model) itself and stability pertains to the perturbation behavior of an algorithm used to solve that problem on a computer. As various EMA methods are modeled differently with distinct solution algorithms, implementing these methods would have different conditioning and stability. In this paper, both deterministic and stochastic EMA methods are covered. Three different scenarios for the response signal are considered: (1) clean response from impulse loading, (2) noisy response from impulse loading, and (3) noisy response from ambient noise excitation. Comparing the numerical conditioning of various EMA methods, this paper theoretically illustrates that methods based on first-order state-space models are more likely to be well-conditioned (with a smaller conditioning number) than those based on high-order polynomial models. Furthermore, the numerical observation of a case study for a 6 degree-of-freedom system also suggests that first-order state-space model methods are more robust and accurate for the estimation of modal frequency and damping.

Simultaneous Noninvasive Clinical Measurement of Lens Autofluorescence and Rayleigh Scattering Using a Fluorescence Biomicroscope

Lens autofluorescence increases with the age of the subject, and the fluorophores responsible are associated with cataract, retinopathy, and other complications of diabetes. We built a scanning confocal lens fluorescence biomicroscope suitable for routine clinical measurement of lens autofluorescence and light scattering and report data from 127 healthy subjects. The fluorescence biomicroscope focuses a beam of light from a blue light-emitting diode on the lens and measures fluorescent green light and blue scattered light using a sensitive silicon photomultiplier. The system includes a target fixation light and a video camera for alignment and automatic pupil tracking. Under software control, a volume of measurement is scanned from behind the posterior lens capsule, through the lens to the aqueous humor, and then back again. Software computes the average ratio of lens autofluorescence to scattered light in the central portion of the lens. Self-reported healthy nondiabetic subjects were examined by an optometrist; if their eyes were healthy and without significant cataract, they were entered into the study. Valid lens autofluorescence data were collected from 127 subjects between 21 and 70 years of age. A linear model for lens autofluorescence intensity with age was highly statistically significant, and the improvement in fit for higher-order polynomial models was not statistically significant. The ratio of lens autofluorescence to light scatter was also calculated; regression analysis showed significant curvature for the relationship of the fluorescence ratio to age, so a nonlinear model was used to estimate the mean ratio of autofluorescence to scatter and its prediction intervals as a function of age. Our observation of a strongly significant linear regression of fluorescence intensity with age of the subjects agrees with the results from previous studies, as does a nonlinear model for the fluorescence ratio. The fluorescence biomicroscope enables the clinician to identify patients with fluorescence ratio significantly higher than expected for their age.

High-accuracy band to band registration method for multi-spectral images of HJ-1A/B

Band-to-band registration accuracy is an important parameter of multispectral data. A novel band-to-band registration approach with high precision is proposed for the multi-spectral images of HJ-1A/B. Firstly, the main causes resulted in misregistration are analyzed, and a high-order polynomial model is proposed. Secondly, a phase fringe filtering technique is employed to Phase Correlation Method based on Singular Value Decomposition (SVD-PCM) for reducing the noise in phase difference matrix. Then, experiments are carried out to build nonlinear registration models, and images of green band and red band are aligned to blue band with an accuracy of 0.1 pixels, while near infrared band with an accuracy of 0.2 pixels.

Modeling the growth pattern of in-season and off-season Ross 308 broiler breeder flocks

Growing the Ross broiler parent according to the target growth curve ensures that males and females achieve optimum lifetime performance and well-being. Accurate control of growth will lead to uniformity and sexual maturity, which are of crucial importance for the production of hygienic, healthy, and fertile eggs of high quality. This study examined the growth of Ross 308 broiler breeder flocks from hatch to 35 wk of age to identify which growth model would describe the growth of these animals most accurately. Growth was measured and modeled using linear and nonlinear functions, and the experimental growth curves were compared with target curves from the Parent Stock Management Manual for Ross 308 (Aviagen). Broiler breeder flock R6 (in-season from February until October) and flock R7 (off-season from August until April) were kept in an environmentally controlled breeder house from hatch until 35 wk of age. Three nonlinear growth functions (logistic, Gompertz, and Richards) and 3 polynomial functions (linear, second-order, and third-order) were applied. Parameters of the models were estimated by the least squares procedure. The fit of growth curves to experimental data was assessed using R(2). A t-test was used to identify significant differences in the goodness of fit of the model to the different data sets (breeder manual, R6, and R7). The third-order polynomial gave the best fit to the Ross 308 parent broiler BW data, with R(2) ranging from 0.992 to 0.998. Among the nonlinear growth functions, the Richards model gave the best fit to the data, with R(2) ranging from 0.992 to 0.995. The advantage of second- and higher-order polynomial models is that they can be linearized and their parameters estimated by linear regression.

Another view of dual response surface modeling and optimization in robust parameter design

Robust parameter design (RPD) based on the concept of building quality into a design has received much attention from researchers and practitioners for years, and a number of methodologies have been studied in the research community. There have been many attempts to integrate RPD principles with well-established statistical techniques, such as response surface methodology, in order to model the response directly as a function of control factors. In this paper, we reinvestigate the dual response approach based on quadratic models Vinning and Myers (J Qual Technol 22:38–45), which is often referred to in the RPD literature and demonstrate that higher-order polynomial models may be more effective in finding better RPD solutions than the commonly-used quadratic model. We also propose optimization models for each of the three classes of quality characteristics (i.e., nominal-the-best, larger-the-better, and smaller-the-better). The optimal solutions obtained using the proposed models are compared with the solutions obtained using the RPD techniques in the current literature.

Use of Genetic Algorithms for Contrast and Entropy Optimization in ISAR Autofocusing

Image contrast maximization and entropy minimization are two commonly used techniques for ISAR image autofocusing. When the signal phase history due to the target radial motion has to be approximated with high order polynomial models, classic optimization techniques fail when attempting to either maximize the image contrast or minimize the image entropy. In this paper a solution of this problem is proposed by using genetic algorithms. The performances of the new algorithms that make use of genetic algorithms overcome the problem with previous implementations based on deterministic approaches. Tests on real data of airplanes and ships confirm the insight.

How to select polynomial models with an accurate derivative

Derivatives of estimated static relations are often used for linearization in control and in extended Kalman filtering. However, the structure of selected models may only be an approximation to the true relationship, which can cause problems in taking derivatives. Polynomial models, estimated from noisy observations, may give accurate descriptions of the data while at the same time their derivatives may be poor approximations of the true derivative. The explanation of the strong degradation of the derivative of selected models is straightforward: estimating polynomial models of increasing order from a set of data gives not only a description of the true underlying process, but also of the accidental realization of the additive noise. The higher order polynomial models will crinkle around the true process; therefore, they will mostly have an irregular derivative. Models with a better derivative can be selected by using a higher penalty factor in the selection criterion.