High-speed contouring control with NURBS-based C2 PH spline curves

This paper presents a C² Pythagorean hodograph (PH) spline curve constructed by the non-uniform rational B-spline (NURBS) curve for high-speed contouring control. With the knot vector, weights, and control points given, the C² PH spline curve is defined to be a “good” interpolant for Hermit data obtained from a NURBS curve of degree 3 specified by the same control polygon, weights, and the knot vector. To this end, the first- and second-order derivatives are evaluated at the nodal points on the NURBS curve. These boundary conditions are imposed on the PH segments of degree 9 to preserve continuity between the connecting segments. The S-curve motion planning architecture with variable feed rate for a planar NURBS-based C² PH spline curve is also developed in this paper. In particular, C¹ cubic feed acceleration/deceleration is imposed on the first and last PH segments. Several NURBS-based C² PH spline curve-following tasks were conducted to verify the effectiveness of the proposed interpolation algorithm. Experimental results show that the proposed interpolator is not only feasible for machining the complicated parametric curves represented in the NURBS-based C² PH spline form but also yields satisfactory contouring performance under variable feed rate control.     

数控刀位点的B样条精确拟合

为了满足现代NURBS曲线加工的要求,提出了把数控刀位点进行分段并按照用户设定精度进行快速拟合的一种算法。在拟合的过程中,通过不断地分析逼近误差,调整节点矢量,进行最小二乘拟合,直至使生成曲线达到规定的精度。利用上述方法进行最小二乘拟合时,建立的矩阵方程存在大量的零元素,采用了新的数据结构来存储矩阵方程,节省了内存空间,简化了求解步骤,获得了较快的计算速度。     

High accurate interpolation of NURBS tool path for CNC machine tools

Feedrate fluctuation caused by approximation errors of interpolation methods has great effects on machining quality in NURBS interpolation, but few methods can efficiently eliminate or reduce it to a satisfying level without sacrificing the computing efficiency at present. In order to solve this problem, a high accurate interpolation method for NURBS tool path is proposed. The proposed method can efficiently reduce the feedrate fluctuation by forming a quartic equation with respect to the curve parameter increment, which can be efficiently solved by analytic methods in real-time. Theoretically, the proposed method can totally eliminate the feedrate fluctuation for any 2nd degree NURBS curves and can interpolate 3rd degree NURBS curves with minimal feedrate fluctuation. Moreover, a smooth feedrate planning algorithm is also proposed to generate smooth tool motion with considering multiple constraints and scheduling errors by an efficient planning strategy. Experiments are conducted to verify the feasibility and applicability of the proposed method. This research presents a novel NURBS interpolation method with not only high accuracy but also satisfying computing efficiency.     

Double-spiral tool path in configuration space

Using a spiral tool path is a common strategy in many NC machining applications. It can be used for high-speed machining of pockets or as a space-filling curve for finish applications. When using spirals for finishing application, a double spiral is desired to avoid having a starting or end-point at the center of the workpiece. In this paper, we present an algorithm to create a double spiral from offset curves in a precomputed configuration space (c-space). CAD/CAM systems that operate on NURBS surfaces or on triangular meshes have to deal with the issues of patch-boundary oscillations or long, stretched triangles, respectively. This can be avoided when operating in c-space. The c-space is given in the form of a regular quadrilateral heightfield mesh, which may be adaptively subdivided, where the slope is large. This simple data structure is memory efficient and has proven to be beneficial in CAD/CAM frameworks. Our algorithm creates a double spiral by blending adjacent offset curves. The center of the spiral is filled by a b-spline curve. When given offset curves split into multiple components, the algorithm creates multiple smaller spirals and connects them appropriately. The resulting tool path is one large intersection-free curve with starting and end-point on the boundary of the workpiece.     

节点矢量影响NURBS曲线的规律研究与应用

以非均匀有理B样条（Non uniform rational B-spline, NURBS）曲线理论为基础,证明修改节点矢量对NURBS曲线的影响具有局部性,推导修改节点矢量引起的NURBS曲线变化量的计算式,提出修改节点矢量影响NURBS曲线的一般规律。对计算机辅助图形设计（Computer aided geometric design, CAGD）中曲线局部修改的实例进行试验研究,证明了修改节点矢量可以实现对NURBS曲线的局部修改,实现曲线交互式设计。对NURBS插值曲线节点因子调节法中航空压气机叶片边缘测量的实例进行了试验研究,证明可以获得拟合误差最小的叶片型面测量点NURBS插值曲线。试验研究证明所提出的“节点矢量影响NURBS曲线的一般规律”是正确的和可行的。     

CNC codes conversion from linear and circular paths to NURBS curves

A novel approach is proposed for converting NC files of G01/G02/G03 codes into NURBS curves. The NC points are first acquired from an NC file composed of lines and arcs. A segmentation algorithm based on the angles and lengths evaluated from adjacent points is then used to subdivide the entire group of points into segments. An error-bounded constrained curve fitting algorithm is implemented to fit each segment of points into a curve, maintaining the accuracy, smoothness and boundary continuity between the curves. The chord errors between the NC points and the fitted curves are controlled, which ensures the accurate recovery of NC paths. Also, a knot insertion algorithm is proposed to determine the appropriate number of control points automatically, allowing effective fitting of all segments. Several examples are presented to demonstrate the feasibility of the proposed approach.     

三次NURBS曲线的插值与应用

N URBS方法已经被广泛地使用在 CA D中。本文将通过三次 NU RBS曲线的矩阵表示给出三次NU RBS曲线插值方法 ,即反求三次 N URBS曲线控制顶点的算法 ,并给出数字算例和三次 NURBS曲线插值方法在船舶型线设计中的一个应用     

平行分度凸轮轮廓曲线的三次NURBS表示

为了获得供具有非均匀有理B样条(Non-Uniform Rational B-Sphne,NURBS)插值功能的CAD/CAM绘图和数控机床加工使用的设计数据,文中讨论了NURBS技术和平行分度凸轮轮廓曲线的三次NURBS表示方法,提出了用三次NURBS表示平行分度凸轮轮廓曲线后产生的设计误差的估计方法.     

NURBS曲线曲面重构的方法

CAD/CAM中用有理多项式函数表示曲面越来越广泛。由于非均匀有理B样条(NURBS)可以精确表示解析形状和自由曲线曲面,国际标准组织(ISO)与1991年把NURBS作为表示工业产品几何形状的工业标准。这里主要讨论了NURBS曲线曲面重构的方法。     

基于Foley参数法反算三次NURBS曲线的算法研究

非均匀有理B样条(NURBS)方法已成为定义工业产品几何形状的唯一数学描述方法.工程实践中,常需要通过曲线已知的一批型值点反算曲线.针对三次NURBS曲线反算问题,采用矩阵形式来表示三次NURBS曲线,由已知的型值点及其权因子,利用Foley参数法确定节点矢量,并推导出以抛物条件即贝塞尔条件作为边界条件构造附加方程,反算出三次NURBS曲线的控制点及其权因子.曲线拟合结果表明,该方法生成的插值曲线具有良好的光滑性,很好地适用于工程实践.     

Constant cusp toolpath generation in configuration space based on offset curves

Constant cusp is a common strategy for generating tool paths in many NC machining applications. Cusps need to be regulated to ensure high precision without wasting machining efforts. Constant cusp strategies frequently operate on NURBS surfaces or triangular meshes and, thus, have to deal with the issues of patch-boundary oscillations or long, stretched triangles. To avoid these issues, one can operate in a pre computed configuration space (c-space). The c-space is given in form of a regular quadrilateral heightfield mesh, which may be adaptively subdivided, where the slope is large. This simple data structure is memory efficient and is widely used in CAD/CAM frameworks. In this paper we introduce an algorithm for creating a constant cusp tool path with the help of a given c-space. The constant cusp algorithm iteratively produces curves in the c-space by fitting a tube around the current curve and intersecting the tube with the c-space mesh to detect the subsequent curve. As tool paths are handed to the machine controller in form of point sequences, it suffices to operate on piecewise linear curves. The tube becomes a concatenation of cylinders, which we derive using geometric considerations. In each iteration of the constant cusp algorithm, intersection points of the cylinders with the not yet traversed part of the mesh are detected and checked for their validity. The validity check can efficiently remove global or local self-intersections of the new curve by just deleting the respective points. In a final step, the detected intersection points are connected to form constant cusp tool paths. Dealing with piecewise linear curves, we achieve low computation times for real-world data sets.     

A novel approach for computing C2-continuous offset of NURBS curves

Computing offset curves is an important geometric operation in areas of CAD/CAM, robotics, cam design and many industrial applications. In this paper, an algorithm for computing offsets of NURBS curves using C²-continuous B-spline curves is presented. The progenitor curve in database is initially approximated by a line-fitting curve, and then the exact offset of this line-fitting curve is introduced as an initial offset. Based on the initial offset and a set of selected knots, an intended C²-continuous B-spline curve is subsequently constructed. The method uses a new error-measuring scheme, which is based on the convex hull property of Bézier curves and the idea of cumulative errors, to calculate the global error bound of offset approximation. The method obtains offset curves with C² continuity and guarantees that the actual error bound is precisely within the prescribed tolerance. In addition, it also allows one to selectively parametrize the offset curve.     

Cutting temperature,tool wear,and tool life in heat-pipe-assisted end-milling operations

In this paper, using the specifications of nodal points on a nonuniform rational B-spline (NURBS) curve of three degrees with respect to NURBS curve parameter and defining the coefficients for the velocity and acceleration vectors on these points, a new method is presented to design a tool path via C² PH spline curves. Values of the velocity/acceleration vector coefficients corresponding to the nodal points on the original NURBS curve are computed by pattern search algorithm. To this end, the normal distance between the constructed C² PH spline curve and its corresponding original NURBS curve is considered as the objective function. Using combination of the time-dependent feed rate interpolation in the acceleration/deceleration phase of the motion and the constant feed rate interpolation in the middle region of the motion, the position commands of the designed NURBS-based C² PH spline curve are generated. Several improved NURBS-based C² PH spline curve following tasks were implemented with pseudo-derivative feedback feed forward (PDFF) controller. The experimental and simulation results confirm that the devised interpolator with designed PDFF controller is not only feasible for machining the complicated tool path represented in the improved NURBS-based C² PH spline form but also yields satisfactory contouring performance under variable feed rate.     

NURBS曲面造型中曲线曲面分割问题研究

提出一种基于ＮＵＲＢＳ齐次坐标表示来计算曲线曲面分割问题的方法，把ＮＵＲＢＳ问题转化为四维空间里非有理Ｂ样条问题，就可利用Ｂ样条分割算法来解决，其几何意义明确，求解过程简便易行，便于在ＣＡＤ／ＣＡＭ系统中实现。     

面向NURBS曲面在线检测的IGES文件解释器的开发

在线检测系统选取测量点集的常用途径是从NURBS曲面获取,从CAD模型中解析NURBS曲面信息是在线检测系统的一项关键技术。通过对由CAD系统生成IGES文件的格式进行研究,提出一种从IGES文件析取NURBS曲面属性信息和定义信息的方法,在Visual C＋＋6.0平台上开发IGES文件解释器,经过多次测试,开发的IGES文件解释器能够高效稳定地从IGES V5.2/5.3文件中析取NURBS曲面数据。     

Rapid Prototyping Applications in Medicine. Part 1: NURBS-Based Volume Modelling

Rapid prototyping (RP) technology has extended traditional manufacturing applications in areas other than product engineering. Using RP to fabricate custom implants and prostheses for surgical planning and education is now an important area of research. Although, in theory, RP is capable of producing objects of any complexity, designing freeform shapes is difficult using current CAD systems. These CAD systems are geared toward the design of parts manufactured by traditional methods; they do not help designers exploit the extended opportunities offered by RP technology. Medical data cannot be input into these CAD systems directly for further modification and manipulation. The purpose of this project is to explore a new approach for modelling and prototyping biomedical objects. The work extends from volume modelling to RP and medicine. In this paper, Part 1 of two papers, a new approach to modelling complex objects, NURBS-based volume modelling, is proposed. A NURBS representation of volumes is developed to represent not only the surface boundary but also the interior of a 3D object. NURBS-based volume modelling inherits advantages from both NURBS modelling and voxel-based modelling. The key idea of the NURBS-based volume modelling is to exploit the flexibility of NURBS modelling and use the voxelised NURBS volumes as components for constructing complex objects. Part 2 deals mainly with issues of interfacing volume models to RP systems. A new approach to generate STL files through volume modelling and iso-surface extraction is proposed. This approach guarantees the validity of the final STL file inherently. Software development and case studies are also given.     

离散数据点的B样条曲线精确拟合

给出了用Ｂ样条曲线拟合离散数据点的一种算法,首先利用所给的离散数据点,确 定出拟合Ｂ样条曲线的分段连接点,即尖点和局部曲率最大点,然后假设拟合点残余误 差为白高斯噪声,采用贝叶斯判定律确定每一段拟合曲线的最佳阶次与控制顶点数目, 顺利实现Ｂ样条曲线拟合。实验证明采用该算法可以获得更佳的Ｂ样条曲线拟合。     

Rapid Prototyping Applications in Medicine. Part 2: STL File Generation and Case Studies

Rapid prototyping (RP) technology has extended traditional manufacturing applications in areas other than product engineering. Using RP to fabricate custom implants and prosthesis for surgical planning and education is now an important area of research. Although, in theory, RP is capable of producing objects of any complexity, designing freeform shapes is difficult using current CAD systems. These CAD systems are geared toward the design of parts manufactured by traditional methods; they do not help designers exploit the expanded opportunities offered by RP technology. Medical data cannot be input into these CAD systems directly for further modification and manipulation. The purpose of this project is to explore a new approach for modelling and prototyping biomedical objects. The work extends from volume modelling to RP and medicine. In Part 1 of two papers, a new approach to modelling complex objects, NURBS-based volume modelling, is proposed. A NURBS representation of volumes is developed to represent not only the surface boundary but also the interior of a 3D object. NURBS-based volume modelling inherits advantages from both NURBS modelling and voxel-based modelling. The key idea of the NURBS-based volume modelling is to exploit the flexibility of NURBS modelling and use the voxelised NURBS volumes as components for constructing complex objects. This paper, Part 2, deals mainly with issues of interfacing volume models to RP systems. A new approach to generate STL files through volume modelling and iso-surface extraction is proposed. This approach guarantees the validity of the final STL file inherently. Software development and case studies are also given.     

基于矢量分析的数控加工轨迹设计方法研究

描述了基于矢量分析和NURBS的数控加工轨迹设计方法。基于给定的被加工曲面在其参数域上的优化走刀方向集合,并依据标量场与梯度场的转化关系,建立了精确逼近离散方向矢量的走刀矢量场拟合模型,由此给出了数控加工轨迹的矢量表达形式。以NURBS作为发生矢量场的流函数并借助其较强的局部调控能力,通过调整控制点列,可望实现数控加工轨迹拓扑形状的整体优化调控。验证实例表明该方法能够进行复杂形状数控加工轨迹的精细设计,有利于保证精度指标下曲面加工效率的最大化。     

Real-time variable feed rate NURBS curve interpolator for CNC machining

This paper presents a real-time control algorithm based on Taylor’s expansion for implementing variable feed rate non-uniform rational B-spline (NURBS) curve interpolators using a digital signal processor for precision CNC machining. To efficiently compute the NURBS curve and its derivatives in real-time, an effective method is proposed. The variable feed rate NURBS curve interpolator can be used to realise the ACC/DEC before feed rate interpolation in which the ACC/DEC (acceleration/deceleration) planning on the feed rate command executes before the interpolation takes place, so that the path command errors caused by conventional ACC/DEC planning using the post feed rate interpolation can be effectively eliminated. To demonstrate the performance of the proposed algorithm, an X-Y table driven by two servomotors is controlled to track command paths represented by multiple blocks of NURBS curves. Experimental results verify the effectiveness of the proposed method.