Second-order divided difference filter for vision-based relative navigation

A second-order divided difference filter (SDDF) is derived for integrating line of sight measurement from vision sensor with acceleration and angular rate measurements of the follower to estimate the precise relative position,velocity and attitude of two unmanned aerial vehicles (UAVs).The second-order divided difference filter which makes use of multidimensional interpolation formulations to approximate the nonlinear transformations could achieve more accurate estimation and faster convergence from inaccurate initial conditions than standard extended Kalman filter.The filter formulation is based on relative motion equations.The global attitude parameterization is given by quarternion,while a generalized three-dimensional attitude representation is used to define the local attitude error.Simulation results are shown to compare the performance of the second-order divided difference filter with a standard extended Kalman filter approach.

Square-root divided difference Rauch-Tung-Striebel smoother

A square-root version of the divided difference Rauch-Tung-Striebel (RTS) smoother is proposed in this paper. The square-root variant essentially propagates the square roots of the covariance matrices and can consistently improve the numerical stability because all the resulting covariance matrices are guaranteed to stay positive semi-definite. Furthermore, the square-root form ensures reliable implementation in an embedded system with fixed or limited precision although it is algebraically equivalent to the standard form. The new smoothing algorithm is tested in a challenging two-dimensional maneuvering target tracking problem with unknown and time-varying turn rate, and its performance is compared with that of other de-facto standard filters and smoothers. The simulation results indicate that the proposed RTS smoother markedly outperforms the associated filters and gives slightly smaller error than an unscented-based RTS smoother.

Hierarchical robot control structure and Newton's divided difference approach to robot path planning

A hierarchical robot control is proposed for robot soccer system. The Newton’s divided difference is utilized in robot path planning. This paper describes the problems encoutered, software design considerations, vision algorithm and controls of individual robots. The solutions to the problems implemented are simple and direct. It is observed that many of the ideas and solutions can be evolved based on simple theories and concepts. This paper focuses on software structure of multi agent controls, vision algorithm and simple path planning method.

Application of Adaptive Divided Difference Filter on GPS/IMU Integrated Navigation System

The efficient and accurate approximate nonlinear filters have been widely used in the estimation of states and parameters of dynamical systems. In this paper, an adaptive divided difference filter is designed for precise estimation of states and parameters of micromechanical gyro navigation system. Based on the investigation of nonlinear divided difference filter the adaptive divided difference filter(ADDF) was designed, which takes account of the incorrect time-varying noise statistics of dynamical systems and compensation of the nonlinearity effects neglected by linearization. And its performance is superior to that of DDF and extended Kalman filter(EKF). Simulation results indicate that the advantages of the proposed nonlinear filters make them attractive alternatives to the extended Kalman filter.

Quadrature formulas for Fourier-Chebyshev coefficients

The aim of this work is to construct a new quadrature formula for Fourier Chebyshev coef ficients based on the divided differences of the integrand at points 1, 1 and the zeros of the n th Chebyshev polynomial of the second kind. The interesting thing is that this quadrature rule is closely related to the well known Gauss Turn quadrature formula and similar to a recent result of Micchelli and Sharma, extending a particular case due to Micchelli and Rivlin.

New sigma point filtering algorithms for nonlinear stochastic systems with correlated noises

New sigma point filtering algorithms, including the unscented Kalman filter （UKF） and the divided difference filter （DDF）, are designed to solve the nonlinear filtering problem under the condition of correlated noises. Based on the minimum mean square error estimation theory, the nonlinear optimal predictive and correction recursive formulas under the hypothesis that the input noise is correlated with the measurement noise are derived and can be described in a unified framework. Then, UKF and DDF with correlated noises are proposed on the basis of approximation of the posterior mean and covariance in the unified framework by using unscented transformation and second order Stirling＇s interpolation. The proposed UKF and DDF with correlated noises break through the limitation that input noise and measurement noise must be assumed to be uneorrelated in standard UKF and DDF. Two simulation examples show the effectiveness and feasibility of new algorithms for dealing with nonlinear filtering issue with correlated noises.

Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.

A meshless scheme for partial differential equations based on multiquadric trigonometric B-spline quasi-interpolation

Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into ac- count the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant （multiquadric trigonometric B-spline quasi-interpolant） to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation （requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain）. Moreover, the scheme also overcomes the dif- ficulty of the meshless collocation methods （i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points）. The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.

CURVE AND SURFACE INTERPOLATIONBY SUBDIVISION ALGORITHMS

Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference analysis, a systematic method to construct Cn (n≥ 0) interpolatorycurves by subdivision from given data is described and the mask (filter) of the algorithm is presentedexplicitly. This algorithm generates a Cn smooth curve which interpolates the initial control points.Control parameters are also provided so that the shape of the final curve can be adjusted according torequirements. An immediate generalisation of the method is the construction of smooth interpolatorysubdivision algorithms over uniform triangular networks (tensor product type data) in Rm. The mainresults of this algorithm for smooth interpolatory surface subdivision algorrthm are also included.AMS(MOS) : 65D05 , 65D15 , 65D17.

ON THE DIVIDED DIFFERENCE FORM OF FAA DI BRUNO＇S FORMULA

The n-divided difference of the composite function h ：= f o g of functions f, g at a group of nodes t0,t1,…,tn is shown by the combinations of divided differences of f at the group of nodes g（t0）,g（t1）,…,g（tm） and divided differences of g at several partial group of nodes t0,t1,…,tn, where m = 1,2,…,n. Especially, when the given group of nodes are equal to each other completely, it will lead to Faà di Bruno＇s formula of higher derivatives of function h.

Some results on numerical divided difference formulas

The remainder estimates of numerical divided difference formula are given for the functions of lower and higher smoothness, respectively. Then several divided difference formulas with super-convergence are derived with their remainder expressions.

Meromorphic Functions of Finiteφ-order and Linear Askey-Wilson Divided Difference Equations

The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated.Theφ-order is used as a general growth indicator,which covers the growth spectrum between the logarithmic orderρlog(f)and the classical orderρ(f)of a meromorphic function f.

ON THE DIVIDED DIFFERENCE FORM OF FAA DI BRUNO＇S FORMULA Ⅱ

In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multivariate Newton interpolation, we generalize the divided difference form of Faà di Bruno＇s formula with a scalar argument. Moreover, a generalized Faà di Bruno＇s formula with a vector argument is derived.

Complete monotonicity of a function involving the divided diference of digamma functions

In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double inequalities for bounding the ratio of two gamma functions are discovered.

BLOCK BASED NEWTON-LIKE BLENDING INTERPOLATION

Newton＇s polynomial interpolation may be the favourite linear interpolation in the sense that it is built up by means of the divided differences which can be calculated recursively and produce useful intermediate results. However Newton interpolation is in fact point based interpolation since a new interpolating polynomial with one more degree is obtained by adding a new support point into the current set of support points once at a time. In this paper we extend the point based interpolation to the block based interpolation. Inspired by the idea of the modern architectural design, we first divide the original set of support points into some subsets （blocks）, then construct each block by using whatever interpolation means, linear or rational and finally assemble these blocks by Newton＇s method to shape the whole interpolation scheme. Clearly our method offers many flexible interpolation schemes for choices which include the classical Newton＇s polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method.

Interpolation theory of anisotropic finite elements and applications

Interpolation theory is the foundation of finite element methods.In this paper,after reviewing some existed interpolation theorems of anisotropic finite element methods,we present a new way to analyse the interpolation error of anisotropic elements based on Newton's formula of polynomial interpolation as well as its applications.

ON MATRIX AND DETERMINANT IDENTITIES FOR COMPOSITE FUNCTIONS

We present some matrix and determinant identities for the divided differences of the composite functions, which generalize the divided difference form of Faa di Bruno＇s formula. Some recent published identities can be regarded as special cases of our results.

On Formulas and Some Combinatorial Properties of Schubert Polynomials

By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.