A Hermite‐Lobatto Pseudospectral Method for Optimal Control

A pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre‐Gauss‐Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over‐constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush‐Kuhn‐Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.     

A type insensitive code for delay differential equations basing on adaptive and explicit Runge-Kutta interpolation methods

For the numerical solution of initial value problems for delay differential equations with constant delay a partitioned Runge-Kutta interpolation method is studied which integrates the whole system either as a stiff or as a nonstiff one in subintervals. This algorithm is based on an adaptive Runge-Kutta interpolation method for stiff delay equations and on an explicit Runge-Kutta interpolation method for nonstiff delay equations. The retarded argument is approximated by appropriate Lagrange or Hermite interpolation. The algorithm takes advantage of the knowledge of the first points of jump discontinuities. An automatic stiffness detection and a stepsize control are presented. Finally, numerical tests and comparisons with other methods are made on a great number of problems including real-life problems.     

基于Hermite的SAR卫星轨道插值干涉测量应用研究

合成孔径雷达(Synthetic aperture radar, SAR)卫星轨道参数是干涉测量技术中影像配准、基线估算、平地相位去除等环节的重要参数,但部分SAR卫星轨道参数采样间隔较大,导致干涉测量过程中产生残余相位,发生较大的系统误差;利用MTALAB编程工具,对卫星原始轨道状态矢量进行了埃尔米特插值法拟合,等距插值计算后,发现可以缩小轨道参数采样间隔,提高干涉测量精度;以覆盖巴姆地区的Envisat卫星为例,分别获取了基于粗轨、埃尔米特插值轨道参数和代尔夫特精密轨道参数得到的干涉测量图,定性判断出埃尔米特插值法可有效提高SAR卫星轨道精度;再以覆盖陕西地区的AOLS卫星为例,插值轨道矢量采样间隔分别为10秒、5秒、2秒,发现间隔为5秒时相干性最优;结果表明：采用埃尔米特插值法可有效增加SAR卫星轨道状态矢量数量,消除系统误差,提高干涉测量精度。     

Smooth Stream Surfaces of Fourth Order Precision

We introduce a novel technique for the construction of smooth stream surfaces of 4th order precision. While common stream surface techniques use linear interpolation for generating seed points for new streamlines in the refinement phase, we use Hermite interpolation. The derivatives needed for Hermite interpolation are obtained by integration along the streamlines. This yields stream surfaces of4th order precision. Additionally, we analyse the accuracy ofthe well known Hultquist approach and our new algorithm and proof that Hultquist's method is exact for linear vector fields. We compare both methods using the well known distance based and a novel error based refinement strategy. Our resulting surface is C¹ -continuous, enabling improved rendering among other benefits.     

Extensions and generalisations of rational and high order transformation methods

A review of previously published work extending the rational basis to include Hermite interpolation is given. New high order transformation methods are described for Hermite interpolation. The intimate connection between the two basic methods both in the construction of transformations and the construction of basis functions is highlighted and the basic high order transformation method is generalised to handle elements with an arbitrary number of sides.     

三次Hermite参数曲线与曲面的扩展

在给定插值条件时,标准三次Hermite参数曲线与曲面的形状无法调整。为克服标准三次Hermite参数曲线与曲面的不足,首先通过提高基函数次数的方法给出了一种带形状参数的四次Hermite基函数,然后生成了相应的带形状参数的四次Hermite参数曲线与曲面。所生成的曲线与曲面是标准三次Hermite参数曲线与曲面的扩展,不仅与标准三次Hermite曲线与曲面具有完全相同的性质,而且当插值条件给定时,其形状可通过修改形状参数的取值进行局部或整体调节,为插值曲线与曲面的构造提供了一种新方法。     

三次Hermite插值曲线的细化优化

在给定端点及其切矢方向的条件下,通过在相邻两节点之间插入一个中间节点,研究三次Hermite插值曲线的优化问题．如果以与曲率有关的二阶导数为目标,证明插入节点与不插入节点的情形是一样的,体现三次Hermite插值曲线的一种特性．如果以与挠率有关的三阶导数为目标,给出优化三次Hermite曲线的计算公式,从而提出一种新的曲线构造方法．实例表明了方法的有效性．     

Interpolants for Runge-Kutta-Nyström methods

Hermite interpolation polynomials are developed for use with low order Runge-Kutta-Nyström methods in producing continuous approximate solutions to second order initial value problems. Polynomials of degrees five and four, which interpolate te solution and derivative components, respectively, are easily computable and require no extra stages. Interpolants of the next higher orders are obtained at the cost of two additional stages per integration step. The interpolation procedure is demonstrated using two recently developed Runge-Kutta-Nyström methods. Numerical experimentation reveals some aspects of the performance of these interpolants.     

重节点埃尔米特插值算法与程序实现

研究了具有任意阶导数信息Hermite插值问题,使用广义差商的一种新的表示方法和构造广义差商表的一种新方法,给出具有任意阶导数信息Hermite插值算法和程序实现,拓展了牛顿差商插值公式和余项公式。     

基于Hermite插值的视频图像分辨率匹配电路设计

目前常用的图像分辨率匹配算法,存在着缩放效果模糊或硬件消耗过大等问题。综合缩放效果和电路消耗考虑,提出了一种基于三次Hermite插值算法的图像分辨率匹配电路。以基于数字差分分析算法的模块控制插值源像素的选取,以Hermite算法为内核完成插值系数的计算,相对双三次插值算法,保证了较小的硬件消耗。通过均方差、峰值信噪比、灰度梯度模值和傅里叶频谱对Hermite插值算法缩放效果与其余算法缩放效果进行定量分析,表明三次Hermite插值算法比双线性插值算法具有更好的缩放效果。同时通过电路综合结果,表明该电路相对基于双三次插值的电路,硬件消耗更小。     

Hermite interpolation and a new iterative method¶for the computation of the roots¶of non-linear equations

We present a new iterative method, derived from Hermite interpolation, with order of convergence p = 1+ , which requires, at each step, only two function evaluations. The efficiency index of the method is better than those of classical methods, such as the secant method or Newton's method, and those of the recent methods introduced by Costabile et al. [1,2] as well. This method has the best efficiency index in a family of methods derived from Hermite interpolation. Received: September 2002 / Accepted: January 2003 Work supported by the Italian agency MURST.     

基于Hermite曲线的机器人足球射门算法

机器人足球比赛是关于人工智能的新兴研究领域,它集数学算法、多智能体、机械设计、控制理论等多个学科于一体,是先进科学技术的发展代表。在机器人足球比赛中,射门和传球是两个最基本的动作。提高动作的速度以及准确性、连贯性是提高动作效率的关键。本文提出一种基于Hermite插值曲线的机器人足球射门算法。利用Hermite插值曲线的数学特性,可以使机器人在满足一定速度的基础上,连续、准确地完成射门或传球动作,并且能大幅度地提高射门或传球的效率。本文以FIRA5：5仿真平台为背景,结合试验证明,利用此方法可以较好地提高射门的成功率。     

六次PH 曲线G2 Hermite 插值

以其在弧长计算与等距线表示上的优势,PH 曲线成为近年来计算机辅助几何设计 研究的焦点问题之一。为此讨论了六次PH 曲线的G2 Hermite 插值问题。在指定自由参数下,对 两类六次PH 曲线分别进行复分析曲线求解,得到满足G2 插值条件的六次PH 曲线和控制顶点。 通过弧长、能量积分、绝对旋转数的衡量,选取较好的插值曲线。进一步,讨论了用六次PH 曲 线G2 Hermite 插值逼近90°和67°圆弧的问题。在同一个自由参数下,选择插值最好的曲线,可 实现六次C1 Hermite 插值逼近圆弧的效果,且逼近90°圆弧时,优于五次G2 Hermite 插值逼近的 PH 曲线,而逼近67°圆弧时,与最好的五次PH 曲线达到的效果几乎相同。     

基于Hermite 插值的多体系统动力学离散变分方法

针对多体系统动力学数值仿真问题,研究基于Hermite插值的离散变分方法.首先对广义坐标和广义速度进行Hermite插值,结合Gauss数值积分方法,利用Hamilton原理和离散力学变分原理,建立了含已知导数信息和含未知导数信息的Hermite插值离散变分数学模型,求解得到精确度较高的动力学仿真结果.该方法可以在步长较大时精确保持约束方程,并保持系统总能量在一定范围内有界变化,适用于长时间仿真情况.     

Collocation and galerkin finite element methods for viscoelastic fluid flow—I. : Description of method and problems with fixed geometry

Collocation and Galerkin finite element methods are developed for viscoelastic fluid flow in a fixed geometry. The collocation methods use Hermite cubic polynomials with a global coordinate transformation to permit irregular geometry. The Galerkin method uses isoparametric elements (transformed element by element) with bilinear polynomials for pressure and quadratic polynomials for velocity. Both methods are applied to two-dimensional flow in planar geometry and the Galerkin method is applied to axisymmetric cylindrical geometries as well. The fluid model is a nonlinear Maxwell model but is limited to small elastic components.

The two methods are applied to several test problems. Entry-length problems test the ability to model pressure singularities are velocity discontinuities. Stick-slip problems test the ability to model pressure singularities and stress discontinuities. Both test problems have analytic or accurate numerical solutions for Newtonian fluids so that the accuracy of the two methods is compared.     

四次Hermite曲线的构造及其特性

在Hermite曲线插值理论基础上,针对工程应用中的特殊要求,提出了四次Hermite捅值曲线的概念．所构造的四次Hermite曲线满足给定点位置矢量和一阶导数矢量的条件,保证构造曲线处于给定锥面上,达到准双曲面齿轮轮廓曲线的插值精度要求．     

三次Hermite插值曲线的能量优化

在给定插值点的位置矢量及切矢量的情况下,通过在两相邻节点引入两个新的节点,提出了一类保持[C1]连续的三次Hermite插值曲线的构造方法,分别通过基于曲率、挠率的能量函数对其进行优化,给出了能量最小化的参数取值公式。讨论了参数对曲线形状的影响,实例表明了方法的有效性。     

On the approximation order of a space data-dependent PH quintic Hermite interpolation scheme

Dealing with Pythagorean Hodograph quintic Hermite interpolation in the space, we deepen the analysis of the so-called CC criterion proposed in Farouki et al. (2008) for fixing the two free angular parameters characterizing the set of possible solutions, which remarkably influence the shape of the chosen interpolant. Such criterion is easy to implement, guarantees the reproduction of the standard cubic Hermite interpolant when it is a PH curve and usually allows the selection of interpolants with good shape. Here we first rigorously prove that the PH interpolant it selects doesn?t depend on the unit pure vector chosen for representing its hodograph in quaternion form. Then we evaluate the corresponding interpolation scheme from a theoretical point of view, proving with the help of symbolic computation that it has fourth approximation order. A selection of experiments related to the spline implementation of the method confirms our analysis.     

Möbius变换下四次有理抛物-PH曲线的C2 Hermite插值

目的 曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用，而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法，本文讨论了一种偶数次有理等距曲线，即四次抛物-PH曲线的C² Hermite插值问题。方法 基于M bius变换引入参数，利用复分析的方法构造了四次有理抛物-PH曲线的C² Hermite插值，给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果 通过给出"合理"的端点插值数据，以数值实例表明了该算法的有效性，所得12条插值曲线中，结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法，并以具体实例说明了与其他插值方法的对比分析结果。结论 本文构造了M bius变换下的四次有理抛物-PH曲线的C² Hermite插值，在保证曲线次数较低的情况下，达到了连续性更高的插值条件，计算更为简单，插值效果明显，较之传统奇数次PH曲线具有更加自然的几何形状，对偶数次PH曲线的相关研究具有一定意义。     

分布式光纤拉曼测温系统信噪比优化研究

针对分布式光纤拉曼测温系统(ROTDR)信噪比较低的问题,在传统累加平均算法的基础上,提出利用小波包去噪以及埃尔米特插值算法对后向拉曼散射信号进行处理来提升系统信噪比。该方法首先在数据采集端对散射信号进行累加平均去噪,然后使用埃尔米特插值算法进行色散补偿,最后选取sym6小波基对散射信号进行小波包去噪处理。实验表明,该方法将系统信噪比由27.6159 dB提升至32.6988 dB,而且有效补偿了系统的色散效应。选取8 km光纤进行实验,光纤全程温度波动范围从0.65℃~1.15℃降低至0.36℃~0.75℃,不同温度的测量误差中位数小于0.5℃。综上,本文提出的信号处理方案可有效提升系统信噪比及降低系统测温误差。