Degree Polynomial Interpolation Research Articles

Comparative Analysis of Structure and Motion of Vision Logistics Robotic Arms

This paper combines visual recognition and robotic arm, optimizes the design of robotic arm structure, and researches the logistics robotic arm with six degrees of freedom and visual perception ability to enrich the logistics robotic use scene. Firstly, the visual recognition system is installed on the existing six-degree-of-freedom robotic arm, and the coordinate system is established by the light bar recognition method combined with the D-H parameter method. Secondly, a kinematic simulation model was established by using MATLAB, and the state of the joints was analyzed by using forward and inverse kinematic simulation and fifth degree polynomial interpolation. Finally, the actual logistics fixed-point transportation task is designed for trajectory planning. The results show that the robotic arm is able to be able to perform different grasping tasks in non-structural environments, around the visual analysis of the robotic arm as well as the optimization of the motion track, to enhance the efficiency of the comprehensive use of visual and tactile information, to improve the adaptability of the robotic hand grasping, and to be able to meet the precise control of the pick-up process and the use of the realization of the payload grasping and dexterous movement. The research in this paper provides a certain reference value for the propulsion robot arm to perform intelligent grasping tasks.

Неравномерность микротвердости и микроструктуры малоуглеродистой стали, прокатанной на двухклетьевом прокатно-дрессировочном стане

The relevance of the study is proved by two factors. One of them is the necessity to study the nature of the change in the microstructure of the cross-section of extra-thin tinplate made of TS 435 steel (analogue of 08ps steel) skin-passed in a new DSR-1250 mill of the Miory Metal Rolling Plant. The second factor is the need to develop an effective method for obtaining accurate geometry characteristics of an oblique cut of tinplate 0.19 mm or less thick. The purpose of the work is to determine the changes in microhardness and microstructure over the thickness of tinplate samples and identify the existence of a layer microstructure necessary for this type of flat-rolled products. The study was carried out on the selected samples of tinplate of TS 435 steel 0.19 mm thick. This tinplate was rolled from an annealed strip, 0.224 mm thick. The total magnitude of reduction in the mill was 15 %, and the reduction in the skin-pass stand was at least 3 %. The authors carried out measurements of microhardness at different points over the thickness of the selected tinplate samples. The microhardness values over the strip thickness were averaged using 6th degree polynomial interpolation. To study the grain dimension, a number of microstructure images were taken in various areas over the sheet thickness with ×500 magnification. The microstructure studies showed a pronounced strain microstructure with grains elongated in the direction of rolling. At the very boundary of metal contacting the rolls, the grains received the greatest deformation. The highest values of microhardness were identified in two zones adjacent to both strip surfaces and in the central layers along the strip thickness. The change in the microhardness values along the sheet thickness has a wave-like character with three pronounced zones of increase in hardness and two zones of a decrease in its values. The zones with the lowest microhardness values are located between the zones with the maximum values.

Padua points and fake nodes for polynomial approximation: old, new and open problems

Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square [−1,1]2[−1,1]2 that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log2(n)log2⁡(n) of the degree. One of the key features of the Padua points is that they lie on a particular Lissajous curve. Other important properties of Padua points arein two dimensions, Padua points are a WAM for interpolation and for extracting approximate Fekete points and discrete Leja sequences.in three dimensions, Padua points can be used for constructing tensor product WAMs on different compacts.Unfortunately, their extension to higher dimensions is still the biggest open problem. The concept of mapped bases has been widely studied (cf. e.g. [35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images. All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [37]. In this paper, some interesting connection of the FNA with Padua points and families of relatives nodes, that can be used as fake nodes for multivariate approximation, are presented and we conclude with some open problems.

Generalizations of the constrained mock-Chebyshev least squares in two variables: Tensor product vs total degree polynomial interpolation

The constrained mock-Chebyshev least squares interpolation is a univariate polynomial interpolation technique exploited to cut-down the Runge phenomenon. It takes advantage of the optimality of the interpolation on the mock-Chebyshev nodes, i.e. the subset of the uniform grid formed by nodes that mimic the behavior of Chebyshev–Lobatto nodes. The other nodes of the grid are not discarded, rather they are used in a simultaneous regression to improve the accuracy of the approximation of the mock-Chebyshev subset interpolant. In this paper we extend the univariate constrained mock-Chebyshev least squares interpolation to the bivariate case in two different ways, relying on the tensor product interpolation and on the interpolation at the mock-Padua nodes. Numerical experiments demonstrate the effectiveness of such extensions.

Automatic approximation using asymptotically optimal adaptive interpolation

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where 1le ple +infty . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε.

RETRACTED: Reducing energy consumption via trajectory planning method for delta high-speed parallel manipulator

This paper selects delta high-speed parallel robot with three degrees of freedom as the research object. The trajectory planning strategies of Cartesian space and angular displacement, angular velocity and angular acceleration of three joints in high-speed handling are studied. Firstly, the critical trajectory points starting point and end point, and points for obstacle avoidance height are set up, and then according to the inverse kinematics model of the robot, a point-to-point “door” type moving trajectory is established, and the mapping relationship between the motion characteristics of the operating space and the motion characteristics of the joint space is established by using the 4-3-4 degree polynomial motion law in the operating space. However, aiming at the higher energy consumption of 4-3-4 degree polynomial interpolation caused by longer the trajectory, and difficult control of obstacle avoidance height, one key point is added. Thereby, the motion rules are interpolated by 4-3-3-4 degree polynomial interpolation, and the mapping relationship between the motion characteristics of the operating space and the motion characteristics of the joint space is established. Two trajectory planning methods are simultaneously simulated under the same keys points and the same trajectory time range. The motion characteristics of the joint rotation angle of the parallel robot between polynomial interpolations are respectively compared. The results show that the trajectory planning method based on 4-3-3-4 degree polynomial interpolation in joint space has obvious advantages in improving the running state of the delta parallel robot and reducing the energy consumption of the system.

An improved PSO algorithm for time-optimal trajectory planning of Delta robot in intelligent packaging

With the advancement of the times, robotics technology is also developing rapidly. Some large enterprises in China use robots to work in manufacturing and handling positions, and also make robots more and more widely used. This paper focuses on the trajectory planning strategy for three-degree-of-freedom high-speed parallel manipulator of Delta robot in Cartesian space under high-speed operation handling, the point-to-point “door” type handling operation trajectory under the condition of ensuring control accuracy and increasing productivity in intelligent packaging is established based on the inverse kinematics model of the manipulator. The 4–3–3-4 degree polynomial interpolation is presented to control height of obstacle avoidance and trajectory length, on this basis, the mapping relationship between the motion features in operation space and those in joint space is established. Taking into account the complexity of trajectory optimization due to multiple constraints, in order to reduce the difficulty of trajectory optimization, it is necessary to ensure smoothness and constrains of angular displacement, angular velocity and angular acceleration of each joint in space, an improved particle swarm optimization algorithm is proposed to optimize trajectory running time of the 4–3–3-4 degree polynomial interpolation. The simulation results by using Matlab indicate that the accurate and stable time-optimal trajectory planning of 4–3–3-4 degree polynomial interpolation can be achieved by means of the improved particle swarm optimization algorithm. Compared with other trajectory planning algorithms, the proposed algorithm is easier to implement, which not only improves the local convergence of particle swarm optimization algorithm, achieves the time optimal trajectory planning of Delta robot, but also realizes the controllability of obstacle avoidance height, therefore which realizes the fast, accurate and safe operation in intelligent packaging.

Phase field approach for stress- and temperature-induced phase transformations that satisfies lattice instability conditions. Part I. General theory

Recently, results of molecular dynamics (MD) simulations were obtained for the crystal lattice instability conditions for the phase transformations (PTs) between semiconducting Si I and metallic Si II under action of all six components of the stress tensor (Levitas et al. (2017a, b)). These conditions are linear in terms of stresses normal to the cubic faces of Si I and are independent of the shear stresses. In the current paper, we (a) formulated the requirements for the thermodynamic potential and transformation deformation gradient tensors and (b) developed a phase field approach (PFA) for the stress-induced martensitic PTs for large strains while allowing for interfacial stresses, which are consistent with the obtained instability conditions. The general system of equations for coupled PFA and nonlinear elasticity is presented. Crystal lattice instability criteria are derived within a PFA, and it is proven that they are independent of the prescribed stress measure. In order to reproduce the lattice instability conditions obtained with MD: (a) one has to use the fifth degree polynomial interpolation functions of the order parameter for all material parameters; (b) each component of the transformation strain tensor should have a different interpolation functions; and (c) the interpolation functions for tensors of the elastic moduli of all ranks should have zero second derivatives for the parent and product phases, so that terms with elastic moduli, which are nonlinear in stresses, do not contribute to the lattice instability conditions. Specific interpolation and double-well functions have been derived for all parts of the Helmholtz free energy and for two models for the transformation deformation gradient. For these models, explicit expressions for the Ginzburg-Landau equations and lattice instability conditions are derived. Material parameters have been calibrated using results of MD simulations. In Part II of this paper, the developed model is further refined and studied, and applied for the finite element simulations of the nanostructure evolution in Si under triaxial loading.

Jimena: efficient computing and system state identification for genetic regulatory networks

BackgroundBoolean networks capture switching behavior of many naturally occurring regulatory networks. For semi-quantitative modeling, interpolation between ON and OFF states is necessary. The high degree polynomial interpolation of Boolean genetic regulatory networks (GRNs) in cellular processes such as apoptosis or proliferation allows for the modeling of a wider range of node interactions than continuous activator-inhibitor models, but suffers from scaling problems for networks which contain nodes with more than ~10 inputs. Many GRNs from literature or new gene expression experiments exceed those limitations and a new approach was developed.Results(i) As a part of our new GRN simulation framework Jimena we introduce and setup Boolean-tree-based data structures; (ii) corresponding algorithms greatly expedite the calculation of the polynomial interpolation in almost all cases, thereby expanding the range of networks which can be simulated by this model in reasonable time. (iii) Stable states for discrete models are efficiently counted and identified using binary decision diagrams. As application example, we show how system states can now be sampled efficiently in small up to large scale hormone disease networks (Arabidopsis thaliana development and immunity, pathogen Pseudomonas syringae and modulation by cytokinins and plant hormones).ConclusionsJimena simulates currently available GRNs about 10-100 times faster than the previous implementation of the polynomial interpolation model and even greater gains are achieved for large scale-free networks. This speed-up also facilitates a much more thorough sampling of continuous state spaces which may lead to the identification of new stable states. Mutants of large networks can be constructed and analyzed very quickly enabling new insights into network robustness and behavior.

A Babuška-Aziz type proof of the circumradius condition

In this paper the error of polynomial interpolation of degree 1 on triangles is considered. The circumradius condition, which is more general than the maximum angle condition, is explained and proved by the technique given by Babu\v{s}ka-Aziz.

A simplex-based numerical framework for simple and efficient robust design optimization

The Simplex Stochastic Collocation (SSC) method is an efficient algorithm for uncertainty quantification (UQ) in computational problems with random inputs. In this work, we show how its formulation based on simplex tessellation, high degree polynomial interpolation and adaptive refinements can be employed in problems involving optimization under uncertainty. The optimization approach used is the Nelder-Mead algorithm (NM), also known as Downhill Simplex Method. The resulting SSC/NM method, called Simplex2, is based on (i) a coupled stopping criterion and (ii) the use of an high-degree polynomial interpolation in the optimization space for accelerating some NM operators. Numerical results show that this method is very efficient for mono-objective optimization and minimizes the global number of deterministic evaluations to determine a robust design. This method is applied to some analytical test cases and a realistic problem of robust optimization of a multi-component airfoil.

Increasing of surface quality in friction free-form surfaces of knee prosthesis

This paper deals with increasing of the quality of knee prosthesis friction surfaces, in order to reduce wear debris production. Before manufacturing, data file format exchange often generates geometrical discontinuities which lead to imperfections on machined surfaces. A new five degree polynomial interpolation method has therefore been developed to limit such defect. A Design of Experiment has then been carried out to optimize the discretisation intervals of the CAD/CAM model and to test the capacity of this new strategy to smooth undesired typical geometrical discontinuities. The method is also compared to standard techniques. The results show a significant improvement of the surface quality provided by the new polynomial interpolation.

Analysis of stationary subdivision schemes for curve design based on radial basis function interpolation

This paper provides a large family of interpolatory stationary subdivision schemes based on radial basis functions (RBFs) which are positive definite or conditionally positive definite. A radial basis function considered in this study has a tension parameter λ > 0 such that it provides design flexibility. We prove that for a sufficiently large λ ⩾ λ 0 , the proposed 2 L -point ( L ∈ N ) scheme has the same smoothness as the well-known 2 L -point Deslauriers–Dubuc scheme, which is based on 2 L - 1 degree polynomial interpolation. Some numerical examples are presented to illustrate the performance of the new schemes, adapting subdivision rules on bounded intervals in a way of keeping the same smoothness and accuracy of the pre-existing schemes on R . We observe that, with proper tension parameters, the new scheme can alleviate undesirable artifacts near boundaries, which usually appear to interpolatory schemes with irregularly distributed control points.

The Pseudospectral Legendre Method for a Class of Singular Boundary Value Problems Arising in Physiology

This paper presents a computational technique for the solution of a class of nonlinear singular boundary value problems arising in physiology. The approach is based on a pseudospectral Legendre method in which Legendre—Gauss—Lobatto nodes are used to construct the N th degree polynomial interpolation to approximate the solution. The method is easy to implement and yields accurate results. Illustrative examples are included to demonstrate the capability of the proposed method, and a comparison is made with existing methods in the literature.

An SVD analysis of equispaced polynomial interpolation

Runge showed more than a century ago that polynomial interpolation of degree N on a uniform, evenly-spaced grid may diverge as N → ∞ . Here, we present a new angle on polynomial interpolation and the Runge Phenomenon by performing an SVD analysis of the interpolation matrix. We prove that the observed exponential-with- N error growth of the Runge Phenomenon requires that at least the smallest singular value of the interpolation matrix must decay to zero exponentially fast with N, and the matrix condition number κ must grow exponentially with N. Experimentally, four-fifths of the singular values are O ( 1 ) ; the remaining one-fifth of singular values decay exponentially fast with increasing mode number j. The corresponding U modes of the SVD factorization are well approximated by Hermite functions. SVD filtering, which is a common strategy for coping with ill-conditioned matrices by omitting modes of tiny singular value, gave errors no smaller than 7% for Runge's exemplary f ( x ) . Any similar anti-Runge strategy that is based on treating the “good” modes (of O ( 1 ) singular values) differently from the “bad” modes of small singular value is likely to fail because these two groups of modes are strongly coupled. A better strategy for exploiting the SVD factorization remains elusive.

Assessing the number of mean square derivatives of a Gaussian process

We consider a real Gaussian process X with unknown smoothness r 0 ∈ N 0 where the mean square derivative X ( r 0 ) is supposed to be Hölder continuous in quadratic mean. First, from selected sampled observations, we study the reconstruction of X ( t ) , t ∈ [ 0 , 1 ] , with X ˜ r ( t ) a piecewise polynomial interpolation of degree r ≥ 1 . We show that the mean square error of the interpolation is a decreasing function of r but becomes stable as soon as r ≥ r 0 . Next, from an interpolation-based empirical criterion and n sampled observations of X , we derive an estimator r ̂ n of r 0 and prove its strong consistency by giving an exponential inequality for P ( r ̂ n ≠ r 0 ) . Finally, we establish the strong consistency of X ˜ max ( r ̂ n , 1 ) ( t ) with an almost optimal rate.

Estimation du nombre de dérivées d'un processus Gaussien

We consider a real Gaussian process X with unknown smoothness r 0 ∈ N where the mean-square derivative X ( r 0 ) is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations X ( t 1 ) , … , X ( t n ) , we study reconstruction of X ( t ) , t ∈ [ 0 , 1 ] , with X ˜ r ( t ) , a piecewise polynomial interpolation of degree r ⩾ 1 . We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as r ⩾ r 0 . Next, from an interpolation-based empirical criterion, we derive an estimator r ˆ of r 0 and prove its strong consistency by giving an exponential inequality for P ( r ˆ ≠ r 0 ) . Finally, we prove the strong convergence of X ˜ r ˆ ( t ) toward X ( t ) with a similar rate as in the case ‘ r 0 known’. To cite this article: D. Blanke, C. Vial, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Smolyak's algorithm for weighted L1-approximation of multivariate functions with bounded rth mixed derivatives over ℝd

We are interested in numerical algorithms for weighted L1 approximation of functions defined on \(\mathcal{D}=\mathbb{R}^{d}\) . We consider the space ℱr,d which consists of multivariate functions \(f:\mathcal{D}\to \mathbb{R}\) whose all mixed derivatives of order r are bounded in L1-norm. We approximate f∈ℱr,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L1-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as \(\mathcal{O}((\ln n)^{(r+1)(d-1)}n^{-r})\) , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.

Polynomial interpolation of minimal degree

Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing. This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view.

An Alternative Method for a Classical Problem in the Calculus of Variations

A numerical technique for determining the solution of the brachistochrone problem is presented. The brachistochrone problem is first formulated as a non-linear optimal control problem. Using Chebyshev nodes, we construct the Mth degree polynomial interpolation to approximate the state and the control variables. Application of this method results in the transformation of differential and integral expressions into some non-linear algebraic equations to which Newton-type methods can be applied. Simulation studies demonstrate computational advantages relative to existing methods in the literature.