THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

An adaptive generation method for free curve trajectory based on NURBS

To realize the high precision and real-time interpolation of the NURBS （non-uniform rational B-spline） curve, a kinetic model based on the modified sigmoid function is proposed. The constraints of maximum feed rate, chord error, curvature radius and interpolator cycle are discussed. This kinetic model reduces the cubic polynomial S-shape model and the trigonometry function S-shape model from 15 sections into 3 sections under the precondition of jerk, acceleration and feedrate continuity. Then an optimized Adams algorithm using the difference quotient to replace the derivative is presented to calculate the interpolator cycle parameters. The higher-order derivation in the Taylor expansion algorithm can be avoided by this algorithm. Finally, the simplified design is analyzed by reducing the times of computing the low-degree zero-value B-spline basis function and the simplified De Boor-Cox recursive algorithm is proposed. The simulation analysis indicates that by these algorithms, the feed rate is effectively controlled according to tool path. The calculated amount is decreased and the calculated speed is increased while the machining precision is ensured. The experimental results show that the target parameter can be correctly calculated and these algorithms can be applied to actual systems.

眼镜镜片尖边轮廓的五轴数控加工

为了改善和优化眼镜镜片的加工流程,避免镜片尖边加工过程中刀轴矢量变化剧烈、影响工件表面的加工质量,提出了一种基于UG NX的五轴数控加工路径。利用智能镜框扫描仪对镜框边缘进行扫描,获得镜框凹槽点云数据文件,编写相应程序对符合眼镜片加工标准的OMA文件中的球坐标数据进行坐标及格式转换等处理。基于NURBS曲线拟合算法,实现刀路轨迹的连续性并保证拟合曲线的精度。设计和建立一种控制刀轴矢量平滑变化的刀轴驱动约束方法,依靠曲线驱动刀轴及镜片曲面法向量约束刀轴使刀头平滑进给,并将刀轨文件经过后处理转换为NC文件。最后,对加工后的镜片进行误差测量,结果表明,通过点云数据拟合刀路曲线的方法提高了加工面的精确度和光滑性,有效缩减了加工流程,提高了加工效率。

Study on Real-time Look-ahead and Adaptive Parametric Curve Interpolator

The principle of real-time look-ahead was introduced and analysed. An adaptive parametric curve interpolator with a real-time look-ahead function was developed for non-uniform rational B-spline (NURBS) curves interpolation, which considering the maximum acceleration/deceleration of the machine tool. In order to deal with the acceleration/deceleration around the feedrate sensitive corners, the look-ahead function was designed and illustrated. It can detect and adjust the feedrate adaptively. With the help of real-time look-ahead, the acceleration/deceleration can be limited to the range of the machine tool capacity. Thus, feedrate fluctuation is reduced. A NURBS curve interpolation experiment was provided to verify the feasibility and advantages of the proposed interpolator with a real-time look-ahead function.

A New Isogeometric Finite Element Method for Analyzing Structures

High-performance finite element research has always been a major focus of finite element method studies.This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method.Firstly,the physical field is approximated by uniform B-spline interpolation,while geometry is represented by non-uniform rational B-spline interpolation.By introducing a transformation matrix,elements of types C^(0)and C^(1)are constructed in the isogeometric finite element method.Subsequently,the corresponding calculation formats for one-dimensional bars,beams,and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping.The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis,eliminating the need for mesh generation and maintaining flexibility in element construction.Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method.Finally,the test results of several examples show that:(1)Under the same degree and element node numbers,the constructed elements are almost consistent with the results obtained by traditional finite element method;(2)For bar problems with large local field variations and beam problems with variable cross-sections,high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method;(3)The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom,while as degrees of freedom increase,the computational efficiency between the two is similar.

复杂型腔类零件电火花加工电极运动轨迹搜索算法研究

针对复杂型腔类零件电火花加工中电极无干涉运动路径难规划的问题,提出一种电极运动轨迹搜索算法。使用NURBS参数化曲线曲面建模方法拟合叶片型面,建立叶片精确数学模型并计算流道中心线,基于流道中心线表达式提出轨迹搜索算法,开发了复杂型腔类零件多轴联动电火花加工专用CAM系统,成功实现4轴、5轴、6轴复杂型腔类零件的电极运动路径规划和无干涉加工过程仿真。进行整体带叶冠涡轮盘的加工试验,结果表明,叶形加工精度≤0.03 mm,总耗时249.3 h,满足闭式整体叶盘类零件指标要求。试验证明该算法可以快速搜索电极无干涉运动轨迹,有效避免电极干涉,从而提高轨迹搜索效率,工况适应能力强,可用于4~6轴各种工况下叶轮、叶盘等复杂型腔类零件的电极运动轨迹搜索。

Spline-based automatic path generation of welding robot

This paper presents a flexible method for the representation of welded seam based on spline interpolation. In this method, the tool path of welding robot can be generated automatically from a 3D CAD model. This technique has been implemented and demonstrated in the FANUC Arc Welding Robot Workstation. According to the method, a software system is developed using VBA of SolidWorks 2006. It offers an interface between SolidWorks and ROBOGUIDE, the off-line programming software of FANUC robot. It combines the strong modeling function of the former and the simulating function of the latter. It also has the capability of with on-line robot. The result data have shown its high accuracy and strong reliability in experiments. This method will improve the intelligence and the flexibility of the welding robot workstation.

A Methodology for Curve and Surface Fitting with Adaptive Fairing for Digitized Points

A new approach for NURBS(Non-uniform rational B-spline) curve and surface fitting for measured points was presented which employs a fairing method applied to digitized point data with discrete curvature. If measured points are used as control points to construct an NURBS curve, then the curvature of each data point corresponding to control point of the constructed curve can be computed. According to the convex hull and local properties of NURBS, based on the curvatures obtained, the measured points can be faired. If faired measured points are used as target points to modify, the constructed curve passing through these faired points can produce a smooth NURBS curve. This paper also presented the justification for utilizing the curvatures of constructed NURBS curve instead of the conventional interpolated curve to fair the measured points. Based on the presented algorithms, some qualities of the constructed curves can be improved.

基于原子轨道搜索算法的6自由度装配机器人轨迹规划

针对6自由度装配机器人多目标优化和轨迹规划问题,首先,将原子轨道搜索(Atomic Orbit Search,AOS)算法融入到多目标优化中,构建了时间、能耗和冲击的多目标优化函数;其次,轨迹规划采用路径点插值非均匀有理B样条曲线的方法,使用AOS算法测定原子的结合态和结合能,并确定原子中能级最低的候选解,输出最优的电子位置,优化了Pareto取值集合空间,获得多目标优化后的轨迹曲线;最后,通过Matlab仿真验证了所提方法的有效性。结果表明,结合AOS算法优化后的轨迹规划方法能保证装配机器人按最优路线运行,具有一定的工程应用价值。

基于NURBS曲线的超高速电梯导流罩多目标气动优化设计

随着电梯梯速的提高,轿厢受到的气动阻力和升力越来越大,这不仅制约着梯速的进一步提高,也成为影响电梯舒适性和安全性的关键因素。为优化电梯的气动性能,利用NURBS曲线构建了包括导流罩、轿厢、对重和井道的电梯系统参数化模型。以气动阻力系数C_(d)和气动升力系数C_(l)为优化目标,利用敏感性分析筛选出对优化目标影响最大的4个关键设计参数。通过网格独立性检验选取合适的仿真计算网格,为缩短计算时间,结合最优空间填充设计的样本点建立Kriging模型对导流罩进行多目标优化,得到一系列Pareto解,并对优化前后导流罩的气动性能进行了对比分析。结果表明,降低导流罩侧面和后部凸凹度,增大导流罩总高和后部高,可以改善电梯系统的气动性能,优化后导流罩的C_(d)减小26.67%,C_(l)减小34.27%,并且在不同梯速下具有较好的适用性。

基于最少控制点的非均匀有理B样条曲线拟合

针对叶片型线的优化设计,提出采用自适应方法提取合适的节点来插值非均匀有理B样条(NURBS)曲线的算法,实现了满足一定精度要求的数据点云拟合以及控制点的计算.该方法首先通过点云外形特征提取主特征点,把主特征点作为节点插值NURBS曲线,通过德布尔递推公式求解控制点,然后根据误差及曲率信息自适应地增加节点反复迭代,直到达到要求的拟合误差精度,从而简洁有效地实现了大量数据点云的拟合.相比传统方法,该方法能够更快地达到要求的逼近精度,同时将误差与曲率信息结合起来调整节点,不仅适合于有局部大曲率及有噪声点的数据点云的曲率计算,而且可用于估计插值节点的数量和工业逆向设计中空间曲面控制点的提取,为优化设计奠定了良好的基础.

非均匀有理B样条曲线直接插补进给速度规划

在非均匀有理B样条直接插补算法中,为了提高插补实时性、克服前瞻规划算法的缺点,提出了基于进给速度预处理曲线的进给速度规划方法。该方法在插补前,通过建立进给速度预处理曲线来取代非均匀有理B样条曲线实时插补进给速度规划的前瞻算法。在速度规划过程中,为了防止进给加速度和加加速度超过机床性能要求,在曲线曲率变化较大的区域,采用基于逐点比较计算的方法规划每个参数点处的进给速度。通过MATLAB仿真表明所提方法具有较高的加工精度、良好的进给速度平滑效果和实时插补性能。

改进的非均匀有理B样条曲面片拼接算法

研究了已网格化的曲面片之间的拼接技术并对现有的拼接算法进行了改进。首先,根据哈特利-贾德的弦长参数化算法求取非均匀有理B样条(NURBS)曲线上的节点向量,利用节点向量和曲线反求出控制顶点,并采用投影变换的方法求出权因子。其次,根据要求调整曲面片上的控制顶点,使曲面片之间达到G1连续。最后,再对已拼接好的曲面片上的控制顶点进行整体修改,重新排列整张曲面片上的控制顶点,从而实现曲面片的无缝拼接。用专业的三维造型软件对拼接后合并的曲面片进行了测试,结果表明,该方法在保证曲面片之间G1连续的情况下可使多块曲面片实现无缝拼接,无缝拼接试验中的平均误差为0.0049°。

NURBS曲面上的曲线精确表达

给出一种用显式方程精确表示NURBS曲面上曲线的方法,利用该方法NURBS曲面上曲线仍然用NURBS形式来表示,此外,可用显式公式来计算该曲线的控制顶点、权因子和节点矢量。基本思想是:运用代数基本定理避免通常B样条基函数复合所采用的极形式(Polar form)以及非常烦琐复杂的开花(Blossom)运算,把B样条基函数的复合、乘积运算转化为求解线性方程组。通过实例验证了该方法的有效性,可用于CAD/CAM领域中的曲面求交、剪裁以及数控加工刀轨生成等。

圆弧曲线的三次NURBS表示

本文首次提出三次ＮＵＲＢＳ曲线精确地表示圆弧的充要条件，解决了两方面的问题：一是已知三次ＮＵＲＢＳ曲线，如何判断它是否是圆弧；二是已知一圆弧曲线，怎样用三次ＮＵＲＢＳ曲线精确地表示．给出了圆弧曲线的三次ＮＵＲＢＳ表示的几何构造算法．均匀有理Ｂ样条曲线和有理Ｂｅｚｉｅｒ曲线精确地表示圆弧曲线的充要条件可作为ＮＵＲＢＳ曲线的特殊情形得到．这些研究结果为ＮＵＲＢＳ应用于ＣＡＧＤ，ＣＡＤ／ＣＡＭ提供了一个有用的工具．

弯曲激波压缩型面的设计及数值分析

分别采用等压缩角和递增压缩角的小折线构成压缩型面。研究了二维均匀超声来流流过压缩面时的波后超声流场,结果表明,该方法能够形成弯曲激波且波后气流沿流向的壁面静压近似为等压力梯度,其等压力梯度程度取决于各小折线压缩角的配置。采用该方法生成的曲面压缩型面进气道附面层稳定性好,优于常规的平面压缩进气道。与二维常规平面压缩进气道相比,设计工况下,性能相当;非设计工况下,性能优于二维常规平面压缩进气道。

纬编针织物线圈建模与变形三维模拟

为使纬编针织物具有更真实的力学效果和体积感,在改进的弹簧-质点模型基础上对线圈进行结构建模。同时采用插值算法,在相邻型值点间插入2个新的型值点,用非均匀有理B样条曲线(NURBS)连接线圈型值点,解决了NURBS不能穿过所有型值点的问题。为模拟股线的捻度,以此样条曲线为几何中心,用4根圆柱围绕,通过对相邻型值点的空间坐标计算3个相对欧拉角,对圆柱进行相应的旋转变换。测量了真实线圈的形变量,分析出线圈形变量与质点位移量的关系来模拟线圈变形。用Velocity-Verlet数值积分方法求解力学方程,计算各质点位移。借助于Microsoft Visual Studio 2010集成开发工具和Open GL三维绘图函数库,在计算机上实现了纬编织物的三维建模和变形仿真。

一种实时快速NURBS插补算法研究与实现

非均匀有理B样条刀具轨迹参数u与弧长S之间没有精确解析表达式,且是非线性映射现象,为了减少速度波动,减轻在线插补计算量,提出了基于双圆弧引导曲线的快速非均匀有理B样条插补算法。利用步长参数和高斯积分对刀具轨迹进行采样,采样点是基于参数u与弧长S的坐标点,利用双圆弧拟合方法对采样点拟合成插补引导曲线,建立参数u和弧长S解析表达式。在插补时依靠建立的双圆弧引导曲线进行快速的非均匀有理B样条插补。仿真实验结果表明了所提方法的可靠性和有效性。

基于NURBS曲线拟合的刀具路径优化方法

为解决复杂曲面高速、高精度加工中由于微小直线段链逼近自由曲线产生的刀具路径不连续问题,提出一种基于非均匀有理B样条曲线拟合的刀具路径优化方法。在该方法中,通过分析刀具路径的几何特征鉴别需要平滑的连续直线段链,对其分段进行基于最小二乘法的非均匀有理B样条曲线拟合优化,并设置相邻段连接点处的边界平滑条件,还原刀具路径应有的连续性。拟合算法采用特征点搜索法设定曲线参数,通过误差控制的曲线节点向量和控制顶点调整保证拟合曲线精度。实验表明,经过该算法拟合优化后的刀具路径具有良好的平滑性,显著提高了曲面加工的精度和效率。

三次NURBS空间曲线的插值研究

基于三次NURBS曲线的矩阵表示给出了1个三次NURBS空间曲线的插值算法,即反求三次NURBS曲线权因子和控制顶点的完整算法。最后,通过给出2个三次NURBS空间曲线的插值算例验证了该算法的正确性与可行性。