Suboptimal Robot Joint Interpolation Within User-Specified Knot Tolerances

Approximation of a desired robot path can be accomplished by interpolating a curve through a sequence of joint-space knots. A smooth interpolated trajectory can be realized by using trigonometric splines. But, sometimes the joint trajectory is not required to exactly pass through the given knots. The knots may rather be centers of tolerances near which the trajectory is required to pass. In this article, we optimize trigonometric splines through a given set of knots subject to user-specified knot tolerances. The contribution of this article is the straightforward way in which intermediate constraints (i.e., knot angles) are incorporated into the parameter optimization problem. Another contribution is the exploitation of the decoupled nature of trigonometric splines to reduce the computational expense of the problem. The additional freedom of varying the knot angles results in a lower objective function and a higher computational expense compared to the case in which the knot angles are constrained to exact values. The specific objective functions considered are minimum jerk and minimum torque. In the minimum jerk case, the optimization problem reduces to a quadratic programming problem. Simulation results for a two-link manipulator are presented to support the results of this article.     

Splines over regular triangulations in numerical simulation

We investigate the use of smooth spline spaces over regular triangulations as a tool in (isogeometric) Galerkin methods. In particular, we focus on box splines over three-directional meshes. Box splines are multivariate generalizations of univariate cardinal B-splines sharing the same properties. Tensor-product B-splines with uniform knots are a special case of box splines. The use of box splines over three-directional meshes has several advantages compared with tensor-product B-splines, including enhanced flexibility in the treatment of the geometry and stiffness matrices with stronger sparsity. Boundary conditions are imposed in a weak form to avoid the construction of special boundary functions. We illustrate the effectiveness of the approach by means of a selection of numerical examples.     

Knot calculation for spline fitting via sparse optimization

Curve fitting with splines is a fundamental problem in computer-aided design and engineering. However, how to choose the number of knots and how to place the knots in spline fitting remain a difficult issue. This paper presents a framework for computing knots (including the number and positions) in curve fitting based on a sparse optimization model. The framework consists of two steps: first, from a dense initial knot vector, a set of active knots is selected at which certain order derivative of the spline is discontinuous by solving a sparse optimization problem; second, we further remove redundant knots and adjust the positions of active knots to obtain the final knot vector. Our experiments show that the approximation spline curve obtained by our approach has less number of knots compared to existing methods. Particularly, when the data points are sampled dense enough from a spline, our algorithm can recover the ground truth knot vector and reproduce the spline.     

Subdivision Surfaces with Creases and Truncated Multiple Knot Lines

We deal with subdivision schemes based on arbitrary degree B‐splines. We focus on extraordinary knots which exhibit various levels of complexity in terms of both valency and multiplicity of knot lines emanating from such knots. The purpose of truncated multiple knot lines is to model creases which fair out. Our construction supports any degree and any knot line multiplicity and provides a modelling framework familiar to users used to B‐splines and NURBS systems.     

Algorithm by hybrid splines

Hybrid splines whose bases are constructed by multi-order B-splines are presented. B-splines of order 2 are induced for representing corners, and those of orders 3 or 4 for representing curved parts. The hybrid splines can represent not only smooth parts (class C^m?22:m≥3) but also corners (class C°): Hybrid splines of order 2 and 4 cannot be represented by conventional splines with multiple knots. Satisfactory approximation results are obtained by hybrid splines of order 2 and 4 determined by a least-squares method.     

L1 approximations of strictly convex functions by means of first degree splines

The L₁ approximation of strictly convex functions by means of first degree splines with a fixed number of knots is studied. The main theoretical results are a system of equations for the knots, which solves the problem, and an estimate of the approximation error. The error estimation allows the determination of bounds for the number of knots needed so that the L₁ approximation error does not exceed a given number. Finally, an algorithm is used, by means of which a solution to the system can be obtained.     

Approximation and estimation bounds for free knot splines

The fitting to data by splines has long been known to improve dramatically if the knots can be adjusted adaptively. To demonstrate the quality of the obtained free knot spline, it is essential to characterize its generalization ability. By utilizing the powerful techniques of the empirical process and approximation theory to address the estimation and approximation error bounds, respectively, the generalization ability of the free knot spline learning strategy is successfully characterized. We show that the Pseudo-dimension of free knot splines is essentially a linear function of the number of knots. A class of rather general loss functions is considered here and the squared loss is specially treated for its excellent property. We also provide some numerical results to demonstrate the utility of these theoretical results in guiding the process of choosing the appropriate knot numbers through the training data to avoid the overfitting/underfitting problem.     

Allgemeine interpolierende Splines vom Grade 3

Choosing a special case of a general Hermitian interpolating operator, an interpolating spline is constructed with respect to the usual transient-conditions within the knots of the spline. The resulting spline in general is not a polynomial spline. The polynomial spline is contained as a special case as well as e. g. rational, trigonometrical, and exponential splines. A sufficient criterion for existence and uniqueness is given for general interpolating splines of third degree. A statement concerning convergence is added.     

Surface reconstruction using bivariate simplex splines on Delaunay configurations

Recently, a new bivariate simplex spline scheme based on Delaunay configuration has been introduced into the geometric computing community, and it defines a complete spline space that retains many attractive theoretic and computational properties. In this paper, we develop a novel shape modeling framework to reconstruct a closed surface of arbitrary topology based on this new spline scheme. Our framework takes a triangulated set of points, and by solving a linear least-square problem and iteratively refining parameter domains with newly added knots, we can finally obtain a continuous spline surface satisfying the requirement of a user-specified error tolerance. Unlike existing surface reconstruction methods based on triangular B-splines (or DMS splines), in which auxiliary knots must be explicitly added in advance to form a knot sequence for construction of each basis function, our new algorithm completely avoids this less-intuitive and labor-intensive knot generating procedure. We demonstrate the efficacy and effectiveness of our algorithm on real-world, scattered datasets for shape representation and computing.     

Splines in Higher Order TV Regularization

Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m−1 solve also a minimization problem with quadratic data term and m-th order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving m-th order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter λ. More precisely, the spline knots are determined by the contact points of the m–th discrete antiderivative of the solution with the tube of width 2λ around the m-th discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W _2,0 ^m . From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.     

Shape-preserving knot removal

Starting with a shape-preserving C¹ quadratic spline, we show how knots can be removed to produce a new spline which is within a specified tolerance of the original one, and which has the same shape properties. We give specific algorithms and some numerical examples, and also show how the method can be used to compute approximate best free-knot splines. Finally, we discuss how to handle noisy data, and develop an analogous knot removal algorithm for a monotonicity preserving surface method.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

Spherical Triangular B-splines with Application to Data Fitting

Triangular B-splines surfaces are a tool for representing arbitrary piecewise polynomial surfaces over planar triangulations, while automatically maintaining continuity properties across patch boundaries. Recently, Alfeld et al. [1] introduced the concept of spherical barycentric coordinates which allowed them to formulate Bernstein-Bézier polynomials over the sphere. In this paper we use the concept of spherical barycentric coordinates to develop a similar formulation for triangular B-splines, which we call spherical triangular B-splines. These splines defined over spherical triangulations share the same continuity properties and similar evaluation algorithms with their planar counterparts, but possess none of the annoying degeneracies found when trying to represent closed surfaces using planar parametric surfaces. We also present an example showing the use of these splines for approximating spherical scattered data.     

Image approximation by variable knot bicubic splines

This paper presents a degree of freedom or information content analysis of images in the context of digital image processing. As such it represents an attempt to quantify the number of truly independent samples one gathers with imaging devices. The degrees of freedom of a sampled image itself are developed as an approximation problem. Here, bicubic splines with variable knots are employed in an attempt to answer the question as to what extent images are finitely representable in the context of digital sensors and computers. Relatively simple algorithms for good knot placement are given and result in spline approximations that achieve significant parameter reductions at acceptable error levels. The knots themselves are shown to be useful as an indicator of image activity and have potential as an image segmentation device, as well as easy implementation in CCD signal processing and focal plane smart sensor arrays. Both mathematical and experimental results are presented.     

Shape-preserving interpolation of irregular data by bivariate curvature-based cubic splines in spherical coordinates

We investigate C¹-smooth bivariate curvature-based cubic L₁ interpolating splines in spherical coordinates. The coefficients of these splines are calculated by minimizing an integral involving the L₁ norm of univariate curvature in four directions at each point on the unit sphere. We compare these curvature-based cubic L₁ splines with analogous cubic L₂ interpolating splines calculated by minimizing an integral involving the square of the L₂ norm of univariate curvature in the same four directions at each point. For two sets of irregular data on an equilateral tetrahedron with protuberances on the faces, we compare these two types of curvature-based splines with each other and with cubic L₁ and L₂ splines calculated by minimizing the L₁ norm and the square of the L₂ norm, respectively, of second derivatives. Curvature-based cubic L₁ splines preserve the shape of irregular data well, better than curvature-based cubic L₂ splines and than second-derivative-based cubic L₁ and L₂ splines. Second-derivative-based cubic L₂ splines preserve shape poorly. Variants of curvature-based L₁ and L₂ splines in spherical and general curvilinear coordinate systems are outlined.     

Degree elevation of B-spline curves

An algorithm is presented which expresses any given linear combination of B-splines of order k as a linear combination of B-splines of order k + 1. This algorithm consists of two parts: (1) A simple representation of the curve by a sum of B-splines of order k + 1, and (2) a representation of these splines on a common knot vector by inserting new knots.     

Modification of the shape of piecewise curves

A simple mechanism is presented that simulates qualitatively the more sophisticated and mathematically complicated methods of Nielson, Cline and Schweikert, who have discussed more rigorously the problem of splines under tension. it is not, however, intended to copy their methods or to yield the same results. Its advantages are its simplicity and its ability to modify the shapes of any piecewise sequence of curve segments, regardless of their nature. It is designed to maintain the order of continuity where the separate curve segments join (at the knots).     

Practical box splines for reconstruction on the body centered cubic lattice

We introduce a family of box splines for efficient, accurate and smooth reconstruction of volumetric data sampled on the Body Centered Cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C(0) reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representation of the C(0) and C(2) box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space---generated by BCC-lattice shifts of these box splines---is {twice} as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order, and with the same sampling density). Practical evidence is provided demonstrating that not only the BCC lattice is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.     

Algorithm/algorithmus 42 an algorithm for cubic spline fitting with convexity constraints

In this paper an algorithm is presented for fitting a cubic spline satisfying certain local concavity and convexity constraints, to a given set of data points. When using theL ₂ norm, this problem results in a quadratic programming problem which is solved by means of the Theil-Van de Panne procedure. The algorithm makes use of the well-conditioned B-splines to represent the cubic splines. The knots are located automatically, as a function of a given upper limit for the sum of squared residuals. A Fortran IV implementation is given.     

对流-扩散方程数值解的四次 B 样条方法

基于四次 B 样条函数，提出一种求解一类对流-扩散方程的四次 B 样条方法。首 先利用光滑余因子协调法，得到有界闭区间上具有均匀节点的一元四次 B 样条基函数表达式。 接着计算在有界闭区间两端点处具有重节点的几种不同情况下的 B 样条基函数表达式，这些样 条基函数具有非负性、单位分解性等良好的性质。然后将一元四次 B 样条函数应用于求解一类 一维对流-扩散方程，其中对于对流-扩散方程的离散过程，对于时间变量的离散采用向前有限 差分，而对于空间变量的离散，引入参数 δ，建立四次样条逼近格式。之后利用四次 B 样条函 数去求解该对流-扩散方程。最后通过具体算例，将四次样条逼近方法与有限差分方法进行比较， 且给出直观的数值误差对比，由此说明样条逼近方法更加简便实用。