B-spline backstepping control with derivative matrix estimation and its applications

A B-spline backstepping controller is proposed for a class of multiple-input multiple-output (MIMO) nonlinear systems. The control scheme incorporates the backstepping design technique with a B-spline neural network which is utilized to estimate the system dynamics. The B-spline neural network has the advantage of locally controlling its output behavior compared with other neural networks; therefore, it is very suitable to online estimate the system dynamics by tuning its interior parameters, including control points and knot points. Based on the mean-value theorem, the derivative of B-spline basis functions in relation to parameters can be estimated to online adjust these parameters. In addition, the validity of the proposed scheme is verified through an experiment on a servo motor system which is controlled by the output voltage of the Buck DC-DC converter.     

Reducing control points in lofted B-spline surface interpolation using common knot vector determination

A new algorithm for reducing control points in lofted surface interpolation to rows of data points is presented in this paper. The key step of surface lofting is to obtain a set of compatible B-spline curves interpolating each row. Given a set of points and their parameterization, a necessary and sufficient condition is proposed to determine the existence of interpolating B-spline curves defined on a given knot vector. Based on this condition, we first properly construct a common knot vector that guarantees the existence of interpolating B-spline curves to each row of points. Then we calculate a set of interpolating B-spline curves defined on the common knot vector by energy minimization. Using this method, fewer control points are employed while maintaining a visually pleasing shape of the lofted surface. Several experimental results demonstrate the usability and quality of the proposed method.     

The conversion of a dynamic B-spline curve into piecewise polynomials in power form

The evaluation of points and the computations of inflection points or cusps on a curve are often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed into a set of piecewise polynomial curves in power form. The usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in power form is done either by a knot refinement followed by basis conversions, or by applying a Taylor expansion on each knot span of a B-spline curve.Presented in this paper is a new algorithm to convert a B-spline curve into a set of piecewise polynomial curves in power form. Experiment shows that the proposed algorithm significantly outperforms the conventional approach when one or more control points of a B-spline curve are continuously moving.     

Target shape design optimization by evolving B-splines with cooperative coevolution

With high reputation in handling non-linear and multi-model problems with little prior knowledge, evolutionary algorithms (EAs) have successfully been applied to design optimization problems as robust optimizers. Since real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) have been frequently used as efficient miniature model to check algorithmic performance for general shape design. There are at least three important issues in developing EAs for TSDOPs, i.e., design representation, fitness evaluation and evolution paradigm. Existing work has mainly focused on the first two issues, in which (1) an adaptive encoding scheme with B-spline has been proposed as a representation, and (2) a symmetric Hausdorff distance based metric has been used as a fitness function. But for the third issue, off-the-shelf EAs were used directly to evolve B-spline control points and/or knot vector. In this paper, we first demonstrate why it is unreasonable to evolve the control points and knot vector simultaneously. And then a new coevolutionary paradigm is proposed to evolve the control points and knot vector of B-spline separately in a cooperative manner. In the new paradigm, an initial population is generated for both the control points, and the knot vector. The two populations are evolved mostly separately in a round-robin fashion, with only cooperation at the fitness evaluation phase. The new paradigm has at least two significant advantages over conventional EAs. Firstly, it provides a platform to evolve both the control points and knot vector reasonably. Secondly, it reduces the difficulty of TSDOPs by decomposing the objective vector into two smaller subcomponents (i.e., control points and knot vector). To evaluate the efficacy of the proposed coevolutionary paradigm, an algorithm named CMA-ES-CC was formulated. Experimental studies were conducted based on two target shapes. The comparison with six other EAs suggests that the proposed cooperative coevolution paradigm is very effective for TSDOPs.     

基于二进制GA的B样条重构曲线节点优化

带法向约束的自由曲线曲面重构在光学反射面设计中起至关重要的作用.本文为解决法向约束下的曲线重构问题提出了一种优化方案,使得重构出的曲线在逼近数据点的同时,亦能满足相应法向约束.首先,利用惩罚函数的方法将带法向约束的优化问题转化为无约束的优化问题.然后,引入二进制编码的遗传算法（GA）,建立合适的适应度函数,自适应产生优化节点向量,如此迭代进化,直到产生令人满意的重构曲线为止.考虑到节点向量非递减的特性,而遗传算法在寻找最优节点向量的过程中有可能打乱节点向量的顺序,所以在建立适应度函数的时候将变量调整为无序有界变量.通过与传统最小二乘方法和粒子群智能优化方法的比较,本文方案在解决带法向条件约束的曲线重构问题上优势明显,且对于任意形状的曲线重构都行之有效.     

Smooth surface approximation to serial cross-sections

The reconstruction of the surface model of an object from 2D cross-sections plays an important role in many applications. In this paper, we present a method for surface approximation to a given set of 2D contours. The resulting surface is represented by a bicubic closed B-spline surface with C² continuity. The method performs the skinning of intermediate contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The proposed method provides a smooth and accurate surface model, yet realizes efficient data reduction. Some experimental results are given using synthetic and MRI data.     

Modelling and control of Hammerstein system using B-spline approximation and the inverse of De Boor algorithm

In this article a simple and effective controller design is introduced for the Hammerstein systems that are identified based on observational input/output data. The nonlinear static function in the Hammerstein system is modelled using a B-spline neural network. The controller is composed by computing the inverse of the B-spline approximated nonlinear static function, and a linear pole assignment controller. The contribution of this article is the inverse of De Boor algorithm that computes the inverse efficiently. Mathematical analysis is provided to prove the convergence of the proposed algorithm. Numerical examples are utilised to demonstrate the efficacy of the proposed approach.     

基于差分进化算法的B 样条曲线曲面拟合

应用B 样条曲线曲面拟合内在形状带有间断或者尖点的数据时,最小二乘法得到的 拟合结果往往在间断和尖点处误差较大,原因在于最小二乘法将拟合函数B 样条的节点固定。本 文在利用3 次B 样条曲线和曲面拟合数据时,应用差分进化算法设计出一种能够自适应地设置B 样条节点的方法,同时对节点的数量和位置进行优化,使得B 样条拟合曲线曲面在间断和尖点处 产生拟多重节点,实现高精度地拟合采样于带有间断或尖点的曲线和曲面数据。     

B-spline surface fitting based on adaptive knot placement using dominant columns

By expanding the idea of B-spline curve fitting using dominant points (Park and Lee 2007 [13]), we propose a new approach to B-spline surface fitting to rectangular grid points, which is based on adaptive knot placement using dominant columns along u- and v-directions. The approach basically takes approximate B-spline surface lofting which performs adaptive multiple B-spline curve fitting along and across rows of the grid points to construct a resulting B-spline surface. In multiple B-spline curve fitting, rows of points are fitted by compatible B-spline curves with a common knot vector whose knots are computed by averaging the parameter values of dominant columns selected from the points. We address how to select dominant columns which play a key role in realizing adaptive knot placement and thereby yielding better surface fitting. Some examples demonstrate the usefulness and quality of the proposed approach.     

基本样条循环神经网络及其非线性建模

提出一种新的基于基本样条逼近的循环神经网络,该网络易于训练且收敛速度快。此外为克服定长学习步长训练速度慢的问题,提出一种用于该网络训练的自适应权值更新算法,给出了学习步长的最优估计。该最优学习步长的选择可用于基本样条循环神经网络的训练以及对非线性系统的建模。     

System identification of Wiener systems with B-spline functions using De Boor recursion

In this article a simple and effective algorithm is introduced for the system identification of the Wiener system using observational input/output data. The nonlinear static function in the Wiener system is modelled using a B-spline neural network. The Gauss–Newton algorithm is combined with De Boor algorithm (both curve and the first order derivatives) for the parameter estimation of the Wiener model, together with the use of a parameter initialisation scheme. Numerical examples are utilised to demonstrate the efficacy of the proposed approach.     

Adaptive knot placement in B-spline curve approximation

An adaptive knot placement algorithm for B-spline curve approximation to dense and noisy data points is presented in this paper. In this algorithm, the discrete curvature of the points is smoothed to expose the characteristics of the underlying curve. With respect to the integration of the smoothed curvature, knots are placed adaptively to satisfy a heuristic rule. Several experimental results are included to demonstrate the validity of this algorithm.     

基于B样条曲线的植物叶片几何建模

针对基于图像的植物叶片真实感建模进行了研究, 提出对植物叶片图像的轮廓特征点进行三次均匀B样条曲线插值的算法。该方法提取出标本叶片图像的角点作为特征点, 采用三次均匀B样条曲线进行初步插值拟合, 并根据拟合误差采用改进的节点插入算法进行反复迭代, 直到达到要求的拟合精度。与传统方法相比, 该方法能够更快地达到逼近精度, 并且具有良好的模拟效果, 有一定的实际应用价值。     

Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation

One of the key problems in using B-splines successfully to approximate an object contour is to determine good knots. In this paper, the knots of a parametric B-spline curve were treated as variables, and the initial location of every knot was generated using the Monte Carlo method in its solution domain. The best km knot vectors among the initial candidates were searched according to the fitness. Based on the initial parameters estimated by an improved k-means algorithm, the Gaussian Mixture Model (GMM) for every knot was built according to the best km knot vectors. Then, the new generation of the population was generated according to the Gaussian mixture probabilistic models. An iterative procedure repeating these steps was carried out until a termination criterion was met. The GMM-based continuous optimization algorithm could determine the appropriate location of knots automatically. A set of experiments was then implemented to evaluate the performance of the new algorithm. The results show that the proposed method achieves better approximation accuracy than methods based on artificial immune system, genetic algorithm or squared distance minimization (SDM).     

A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method

In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions. This work extends the idea of Tamsir et al. [An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation, Appl. Math. Comput. 290 (2016), pp. 111–124] for 3D nonlinear wave type problems. Expo-MCB-DQM transforms the 3D nonlinear wave equation into a system of ordinary differential equations (ODEs). To solve the resulting system of ODEs, an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) scheme is used. Stability analysis of the proposed method is also discussed and found that the method is conditionally stable. Four test problems are considered in order to demonstrate the accuracy and efficiency of the algorithm.     

N-sided patches with B-spline boundaries

We present a method for constructing an n-sided patch of parametric surface, with n greater than 2. The main property of the resulting patch is that its boundary coincides with a B-spline. Thus, it can easily be connected to given B-spline surfaces with fixed continuity conditions.

The patch is built from a star-shaped input mesh that outlines a generic n-hole and a surface in a vicinity of the hole. The main advantages of the method are the following: continuity conditions of arbitrary order k can be imposed; the mesh involved can have an arbitrary number of sides and an arbitrary shape (convex or not); the simplicity of the construction process makes it an easy and flexible method; and finally, the surface near the boundary is a B-spline with piecewise uniform knot sequences and whose control points are vertices of the mesh (both knot sequences and control points are easily computed). We give implementation details for evaluating a surface point and show that the de Boor algorithm can be exploited for efficiency.     

Lofted B-spline surface interpolation by linearly constrained energy minimization

This paper proposes a new approach for lofted B-spline surface interpolation to serial contours, where the number of points varies from contour to contour. The approach first finds a common knot vector consisting of fewer knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Then, it computes from the contours a set of compatible B-spline curves defined on the knot vector by adopting B-spline curve interpolation based on linearly constrained energy minimization. Finally, it generates a B-spline surface interpolating the curves via B-spline surface lofting. As the energy functional is quadratic, the energy minimization problem leads to that of solving a linear system. The proposed approach is efficient in computation and can realize more efficient data reduction than previous approaches while providing visually pleasing B-spline surfaces. Moreover, the approach works well on measured data with noise. Some experimental results demonstrate its usefulness and quality.     

B样条乘积计算的区间跳跃算法及应用

利用B样条基函数节点区间的对应关系,首先给出了B样条基函数间的转换矩阵的计算方法,进而给出了计算B样条乘积的区间跳跃算法。该算法仅需计算部分节点区间上的转换矩阵,因此称其为区间跳跃算法。这一方法解决了分段多项式与B样条曲线乘积的计算问题,可应用到B样条曲线的升阶、曲面间光滑拼接等问题中。通过算例验证了该方法计算简捷、易于实现。     

An Extension of Chaiken′s Algorithm to B-Spline Curves with Knots in Geometric Progression

Chaiken′s algorithm is a procedure for inserting new knots into uniform quadratic B-spline curves by doubling the control points and taking two successive averages. Lane and Riesenfeld showed that Chaiken′s algorithm extends to uniform B-spline curves of arbitrary degree. By generalizing the notion of successive averaging, we further extend Chaiken′s algorithm to B-spline curves of arbitrary degree for knot sequences in geometric and affine progression.     

B样条曲面的双向插值造型算法

基于Coons-Gordon造型原理,研究了插值两族相交截面线采样点的B样条曲面双向插值造型算法。参数化各采样点并计算每条截面线的节点矢量,估算每条截面线对应的曲面参数,根据每条截面线的节点分布以及另一族截面线对应的曲面参数统一节点矢量。分别插值两族截面线采样点及其公共点得到三张B样条曲面,其布尔和即为插值两族截面线采样点的B样条插值曲面。实例表明,得到的双向插值曲面控制顶点数少,光顺性好。