NURBS CURVE INTERPOLATOR WITH ADAPTIVE ACCELERATION-DECELERATION CONTROL

The feedrate profile of non-uniform rational B-spline （NURBS） interpolation due to the contour errors is analyzed. A NURBS curve interpolator with adaptive acceleration-deceleration control is presented. In interpo- lation preprocessing, the sensitive zones of feedrate variations are processed with acceleration-deceleration control. By using the proposed algorithm, the machining accuracy is guaranteed and the feedrate is adaptively adjusted to he smoothed. The mechanical shock imposed in the servo system is avoided by the first and the second time derivatives of feedrates. A simulation of NURBS interpolation is given to demonstrate the validity and the effectiveness of the algorithm. The proposed interpolator can also be applied to the trajectory planning of the other parametric curves.

A Geometric Proof of the Convexity Preserving Property of NURBS Curves

NURBS curves are convexity preserving, i.e. once the control polygon is convex, the associated NURBS curve will also be convex. In this paper this property is proved geometrically.

The Relationship Between Weights and Control Vertices of Two Rational NURBS Curves Representing the Same Curve Parametrically and Geometrically

The paper discusses the relationship between weights and control vertices of two rational NURBS curves of degree two or three with all weights larger than zero when they represent the same curve parametrically and geometrically, and gives sufficient and necessary conditions for coincidence of two rational NURBS curves in non-degeneracy case.

The new method for generating pen drawing with NURBS Curve

With the improving of people＇s artistic aesthetic level, the art of pen drawing in all walks of life has been spread and ap- plied widely. However, the artistic effect of pen drawing is still far from perfect in the current methods for generating image-based pen drawing automatically by using computer technology. Therefore, the new method for generating pen drawing with NURBS Curve defines the characteristics of pen drawing in detail and redesigns the algorithm flow chart of pen drawing. It solves the prob- lem of the description of details of pen drawing and improves curve effect of pen drawing with NURBS Curve. It＇s able to better ex- press the artistic conception of pen drawing and solve the problem of simulating pen drawing of any image by using computer art simulation technology.

Calculation of Space NURBS Curve Interpolation Error

Aiming at the problem of low accuracy of interpolation error calculation of existing NURBS curves, an approximate method for the distance between a point and a NURBS interpolation curve is proposed while satisfying the accuracy of the solution. Firstly, the minimum parameter interval of the node vector corresponding to the data point under test in the original data point sequence is determined, and the parameter interval is subdivided according to the corresponding step size, and the corresponding parameter value is obtained. Secondly, the distance from the measured point to the NURBS curve is calculated, and the nearest distance is found out. The node interval is subdivided again on one side of the nearest distance. Finally, the distance between the data point to be measured and each subdivision point is calculated again, and the minimum distance is taken as the interpolation error between the point and the NURBS curve. The simulation results of actual tool position data show that this method can more accurately obtain the error of spatial NURBS interpolation curve.

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.

Approximation of NURBS Curves and Surfaces Using Adaptive Equidistant Parameterizations

Non-uniform rational B-spline (NURBS) curves and surfaces are very important tools for model- ling curves and surfaces. Several important details, such as the choice of the sample points, of the parame- terization, and of the termination condition, are however not well described. These details have a great in- fluence on the performance of the approximation algorithm, both in terms of quality as well as time and space usage. This paper described how to sample points, examining two standard parameterizations: equi- distant and chordal. A new and local parameterization, namely an adaptive equidistant model, was pro- posed, which enhances the equidistant model. Localization can also be used to enhance the chordal parameterization. For NURBS surfaces, one must choose which direction will be approximated first and must pay special attention to surfaces of degree 1 which have to be handled as a special case.

Intelligent NURBS Interpolator Based on the Adaptive Feedrate Control

To satisfy the need for high-speed and high-accuracy machining of NURBS curve. Firstly the form of NURBS curve is analyzed and Talor＇s expansion of the parameter u with respect to time t is used to obtain the algorithm of the first order approximation interpolation. Secondly, based on the algorithm of the controlled chord error interpolator, an intelligent interpolation algorithm of the adaptive feedrate control is proposed. According to the actual machining capacity of machine tools, this algorithm uses look-ahead method, which dispenses with the complicated computation of the end point estimation of NURBS curve, to analyze the curve segment required by the maximum deceleration distance. Thus, the feedrate could decrease in advance and vary with the curvature and the variation ratio of curvature, which makes machining motion quite smooth. Not only could high accuracy and fine surface quality be achieved during high-speed machining, but also the overload of cutter tools is avoided on comers. Finally, in order to facilitate the calculation of interpolation, the dynamic matrix representation and efficient algorithm of curvature computation of the NURBS curve are presented,

Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations

In the present paper, the isogeometric analysis（IGA） of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables（displacement and rotation） in a finite element analysis are considered to be the same as the non-uniform rational basis spline（NURBS） basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.

A NURBS Fitting Optimization Method for High⁃Speed Five⁃Axis NC Machining Path Based on Curvature Smoothing Preset Point Constraint

Existing curve fitting algorithms of NC machining path mainly focus on the control of fitting error,but ignore the problem that the original discrete cutter position points are not enough in the high curvature area of the tool path.It may cause a sudden change in the drive force of the feed axis,resulting in a large fluctuation in the feed speed.This paper proposes a new non-uniform rational B-spline(NURBS)curve fitting optimization method based on curvature smoothing preset point constraints.First,the short line segments generated by the CAM software are optimally divided into different segment regions,and then the curvature of the short line segments in each region is adjusted to make it smoother.Secondly,a set of characteristic points reflecting the change of the curvature of the fitted curve is constructed as the control apex of the fitted curve,and the curve is fitted using the NURBS curve fitting optimization method based on the curvature smoothing preset point constraint.Finally,the curve fitting error and curve volatility are analyzed with an example,which verifies that the method can significantly improve the curvature smoothness of the high-curvature tool path,reduce the fitting error,and improve the feed speed.

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.

Representing Conics Using NURBS Curves

The necessary and sufficient conditions are presented for NURBS currves of an arbitrary degree to precisely represent circular arcs. NURBS curves of degree 2 or degree 3 representing circular arcs can be regarded as special cases of the conditions. It is studied whether two NURBS curves of degree three are equivalent. Classifications of conic section curves represented by cubic or quadratic NURBS curves are proposed.

A Geometric Representation Scheme Suitable for Shape Optimization

A physical approach is used to provide shape modification of NURBS curve. The movement of the control polygon of NURBS is simulated by a physical frame structure. The shape of the design boundaries is described by a linear combination of n mode shape vectors, which are introduced by modal analysis. The control points of NURBS curve are then mapped into mode coordinate system, and the curve deformation is then carried out through the changes of mode coordinates. Numerical examples are used to demonstrate the applicability of this approach to structural shape optimization.