An adaptive real-time NURBS curve interpolation for 4-axis polishing machine tool

The non-uniform rational B-spine (NURBS) curve interpolation is a key technology of the advanced computer numerical control (CNC) system. NURBS curve interpolation can realize a high-speed and high-precision machining, and it can also avoid some inevitable deficiencies of the linear and circular interpolation functions which are generally used in traditional NC system. Before the interpolation, some calculation tasks are finished, which will decrease the amount of calculation during interpolation and increase the interpolation efficiency. Further, an adaptive NURBS curve interpolation with real-time and flexible S-shaped curve acceleration/deceleration (ACC/DEC) control method is added to the interpolation algorithms. The NC machining simulation conducted with the MATLAB software and the NURBS curve interpolation experiments performed on the 4-axis polishing machine tool demonstrate the validity and correctness of the adaptive real-time NURBS curve interpolation algorithm in the CNC system.     

The feedrate scheduling of NURBS interpolator for CNC machine tools

This paper proposes an off-line feedrate scheduling method of CNC machines constrained by chord tolerance, acceleration and jerk limitations. The off-line process for curve scanning and feedrate scheduling is realized as a pre-processor, which releases the computational burden in real-time task. The proposed method first scans a non-uniform rational B-spline (NURBS) curve and finds out the crucial points with large curvature (named as critical point) or G⁰ continuity (named as breakpoint). Then, the NURBS curve is divided into several NURBS sub-curves using curve splitting method which guarantees the convergence of predictor–corrector interpolation (PCI) algorithm. The suitable feedrate at critical point is adjusted according to the limits of chord error, centripetal acceleration and jerk, and at breakpoint is adjusted based on the formulation of velocity variation. The feedrate profile corresponding to each NURBS block is constructed according to the block length and the given limits of acceleration and jerk. In addition, feedrate compensation method for short NURBS blocks is performed to make the jerk-limited feedrate profile more continuous and precise. Because the feedrate profile is established in off-line, the calculation of NURBS interpolation is extremely efficient for CNC high-speed machining. Finally, simulations and experiments with two free-form NURBS curves are conducted to verify the feasibility and applicability of the proposed method.     

A simple technique for NURBS shape modification

Non-uniform rational B-splines (NURBS) have become a de-facto industry standard primarily because they offer a unified mathematical form for representing both freeform shapes and analytical curves or surfaces. Despite these advantages, designers need software that lets them work with NURBS in a natural way. The paper presents a unified approach to NURBS shape modification that builds on a perspective functional transformation of arbitrary origin and provides a homogeneous interface based on simple, easily understood geometric concepts     

Real-time NURBS interpolation using FPGA for high speed motion control

Modern motion control adopts NURBS (Non-Uniform Rational B-Spline) interpolation for the purpose of achieving high-speed and high-accuracy performance. However, in conventional control architectures, the computation of the basis functions of a NURBS curve is very time-consuming due to serial computing constraints. In this paper, a novel FPGA (Field Programmable Gate Array) based motion controller utilizing its high-speed parallel computing power is proposed to realize the Cox-de Boor algorithm for second and higher degrees NURBS interpolation. The motion control algorithm is also embedded in the FPGA chip to implement real-time control and NURBS interpolation simultaneously for multi-axis servo systems. The proposed FPGA-based motion controller is capable of performing the Cox-de Boor algorithm and the IIR (Infinite Impulse Response) control algorithm in about 46 clock cycles, as compared to the 1303 clock cycles by the traditional approach. Numerical simulations and experimental tests using an X-Y table verify the outstanding computation performance of the FPGA-based motion controller. The result indicates that shorter sampling time (10 μs) can be achieved for NURBS interpolation which is highly critical to the success of high-speed and high-accuracy motion control.     

One-sided offset approximation of freeform curves for interference-free NURBS machining

Generating valid tool path curves in NURBS form is important in realizing an efficient NURBS machining. In this paper, a method for computing one-sided offset approximations of freeform curves with NURBS format as tool paths is presented. The approach first uses line segments to approximate the progenitor curve with one-sided deviations. Based on the obtained line approximating curve and its offsets, a unilateral tolerance zone (UTZ) is constructed subsequently. Finally, a C¹-continuous and completely interference-free NURBS offset curve is generated within the UTZ to satisfy the required tolerance globally. Since all of the geometric computations involved are linear, the proposed method is efficient and robust. Interference-free tool path generation thus can be achieved in NURBS based NC machining.     

NURBS边界曲面直接生成法

由于非均匀有理B样条(NURBS)曲面的复杂性,传统NURBS边界曲面的生成是先构造孔斯曲面,再由孔斯曲面向NURBS曲面转换得到,其操作过程比较烦琐。针对此问题,提出了NURBS边界曲面直接生成算法,该算法根据给定的四条NURBS边界曲线,结合孔斯曲面生成方法直接插值生成NURBS曲面,从而避免了通过孔斯曲面向NURBS曲面转换所带来的计算代价,因此同传统方法相比,具有较低的计算代价。实验结果表明:该算法简化了曲面生成步骤,减少了曲面转换过程的计算量,生成的曲面边界信息明确,且连续性好。     

NURBS surface fitting using orthogonal coordinate transform for rapid product development

Constructing a CAD model from a physical model plays a key role in some rapid product development processes. Presented in the paper is a method of fitting NURBS surfaces for rotational freeform shapes: (1) cloud-of-points data (COP-data) representing a rotational freeform shape are transformed into an orthogonal coordinate system, (2) a single-valued B-spline surface is fitted to the transformed data, and (3) the resulting B-spline surface is converted to a 3D NURBS surface by applying a symbolic product operation with a quadratic NURBS base-geometry. Compared to the existing ‘direct’ fitting methods, the proposed method has some distinctive advantages: it provides a natural means to parameterization, enables to recover exact NURBS geometry when the COP-data represent a true surface-of-revolution, and allows an easy point-membership classification for NURBS-bounded solid objects.     

NURBS曲线自动光顺的一种有效算法

提出了一种局部光顺NURBS曲线的算法。算法建立在重复删除和插入节点的过程中,这个重复删除和插入的节点通过一个光顺准则自动选择。此算法自动找出NURBS曲线需要修改的那一点,局部修改控制多边形,使生成的新曲线更加光顺。     

Direct slicing of STEP based NURBS models for layered manufacturing

Direct slicing of CAD models to generate process planning instructions for solid freeform fabrication may overcome inherent disadvantages of using stereolithography format in terms of the process accuracy, ease of file management, and incorporation of multiple materials. This paper will present the results of our development of a direct slicing algorithm for layered freeform fabrication. The direct slicing algorithm was based on a neutral, international standard (ISO 10303) STEP-formatted non-uniform rational B-spline (NURBS) geometric representation and is intended to be independent of any commercial CAD software. The following aspects of the development effort will be presented: (1) determination of optimal build direction based upon STEP-based NURBS models; (2) adaptive subdivision of NURBS data for geometric refinement; and (3) ray-casting slice generation into sets of raster patterns. The development also provides for multi-material slicing and will provide an effective tool in heterogeneous slicing processes.     

A Study on Error Recovery Search Strategies of Electronic Connector Mating for Robotic Fault-Tolerant Assembly

In this paper, a CAD-based trajectory planning scheme for parallel machining robots is introduced using the parametric Non-uniform rational basis spline (NURBS) curves. First, a trajectory is designed via a NURBS curve then, a motion scheduling architecture consisting of time-dependent and constant feedrate profiles is advised to generate the position commands on the represented NURBS curve as the tool path. Using the generated commands, the inverse kinematics is elaborated to obtain the joints motions of the parallel machining robot. This paper investigates the NURBS trajectory generation for a parallel robot with 4(UPS)-PU mechanism as the case study. In order to evaluate the effectiveness of the proposed method, the inverse kinematic results for the parallel machining robot of 4(UPS)-PU is compared with the simulation results obtained from the CATIA software. The results confirmed that the proposed trajectory planning scheme along with the advised motion planning architecture is not only feasible for the parallel machining robots but also yields a smooth trajectory with a satisfactory performance for all the joints.     

Accurate GPU-accelerated surface integrals for moment computation

We present algorithms for computing accurate moments of solid models that are represented using multiple trimmed NURBS surfaces and triangles. Our algorithms make use of programmable Graphics Processing Units (GPUs) to accelerate the moment computations. For NURBS surfaces, we evaluate the surface coordinates and normals accurately, with theoretical bounds, using our GPU NURBS evaluator. We have developed a framework that makes use of this data to evaluate surface integrals of trimmed NURBS surfaces in real time. Since typical solid models also contain flat planes that are usually tessellated into triangles, our framework also includes GPU acceleration of the moment contributions of such flat faces. Using our framework, we can compute volume and moments of solid models with theoretical guarantees. Our algorithms support local geometry changes, which is useful for providing interactive feedback to the designer while the solid model is being designed. We can compute the center of mass and check for stability of the solid model interactively. Other applications of such real-time moment computation include deformation modeling, animation, and physically based simulations.     

NURBS曲线曲率连续延拓的一种有效算法

提出了一种局部延拓NURBS曲线的算法。从理论上探讨了使NURBS曲线获得曲率连续的延拓应满足的条件，同时又给出了在实际应用中使NURBS曲线获得曲率连续的延拓的基本算法。     

NURBS曲线曲面的显式矩阵表示及其算法

从 B样条的差商定义出发 ,提出差商展开系数的概念 ,通过差商展开系数显式解析表示式的导出 ,得到任意次 NU RBS曲线曲面系数矩阵的显式解析表示式 ,并给出了求差商展开系数和 NURBS曲线曲面系数矩阵的数值算法 .文中给出的方法适用于一切 NU RBS曲线曲面 ,包括有理和非有理的 Bézier、均匀和非均匀的 B样条曲线曲面 .相应的数值算法计算简单 ,易于实现 .差商展开系数解析表示式为 NU RBS曲线曲面的表示、转换和节点插入、升阶等基本运算以及与差商相关的问题的研究提供了一个统一的构造性工具和应用方法 .     

A unified architecture for the computation of B-spline curves andsurfaces

B-Splines, in general, and Non-Uniform Rational B-Splines (NURBS), in particular, have become indispensable modeling primitives in computer graphics and geometric modeling applications. In this paper, a novel high-performance architecture for the computation of uniform, nonuniform, rational, and nonrational B-Spline curves and surfaces is presented. This architecture has been derived through a sequence of steps. First, a systolic architecture for the computation of the basis function values, the basis function evaluation array (the BFEA), is developed. Using the BFEA as its core, an architecture for the computation of NURBS curves is constructed. This architecture is then extended to compute NURBS surfaces. Finally, this architecture is augmented to compute the surface normals, so that the output from this architecture can be directly used for rendering the NURBS surface     

曲面数控加工中面向NURBS刀具路径生成的刀位点分段算法

曲面数控加工中,NURBS刀具路径生成技术需要在大量有序刀位点中提取适合于NURBS刀具路径表示的刀位点段,刀位点的分段质量是决定NURBS刀具路径生成的前提.通过分析NURBS刀具路径的特点,对由刀位点表示的刀具路径之间的连接方式和边界点进行分类,提出通过层次聚类法将刀具路径进行分段的算法.在此基础上,通过判断连接点的类型来提取合适的刀位点段以进行NURBS刀具路径的生成.实例结果表明,该算法分段可靠、快捷,对不同曲线曲面轮廓刀具轨迹点进行分段的适应性强,分段结果可以满足NURBS刀具路径的生成.     

Optimized GPU evaluation of arbitrary degree NURBS curves and surfaces

This paper presents a new unified and optimized method for evaluating and displaying trimmed NURBS surfaces using the Graphics Processing Unit (GPU). Trimmed NURBS surfaces, the de facto standard in commercial mechanical CAD modeling packages, are currently being tessellated into triangles before being sent to the graphics card for display since there is no native hardware support for NURBS. Other GPU-based NURBS evaluation and display methods either approximated the NURBS patches with lower degree patches or relied on specific hard-coded programs for evaluating NURBS surfaces of different degrees. Our method uses a unified GPU fragment program to evaluate the surface point coordinates of any arbitrary degree NURBS patch directly, from the control points and knot vectors stored as textures in graphics memory. This evaluated surface is trimmed during display using a dynamically generated trim-texture calculated via alpha blending. The display also incorporates dynamic Level of Detail (LOD) for real-time interaction at different resolutions of the NURBS surfaces. Different data representations and access patterns are compared for efficiency and the optimized evaluation method is chosen. Our GPU evaluation and rendering speeds are more than 40 times faster than evaluation using the CPU.     

Neural Network Methods for NURBS Curve and Surface Interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS curve and surface interpolation.When all th knot spans are identical,the NURBS curve interpolation procedure degenerates into that of uniform rational B-spline curves.If all the weights of data points are identical,then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.     

NURBS曲面G~1光滑拼接算法

非均匀有理Ｂ样条（ＮＵＲＢＳ）曲线、曲面造型方法，是当前ＣＡＤ／ＣＡＭ领域研究热点之一，大量的基于ＮＵＲＢＳ的实用造型系统得到发展。对ＮＵＲＢＳ而言，虽然具有参数连续性，但为了实用需要，仍需构造具有一定光滑程度的合成曲面，满足局部设计和修改的目的。本文给出了实用的具有二次公共边界曲线的ＮＵＲＢＳ曲面片Ｇ１光滑拼接条件，得到了相应控制顶点、权系数的具体算法；对于一个已知ＮＵＲＢＳ曲面，构造另一个ＮＵＲＢＳ曲面，使其达到Ｇ１拼接是简单易行的。     

D-NURBS: a physics-based framework for geometric design

Presents dynamic non-uniform rational B-splines (D-NURBS), a physics-based generalization of NURBS. NURBS have become a de facto standard in commercial modeling systems. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom-control points and associated weights-to achieve the desired shapes. The conventional shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, and hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of computer-aided geometric design (CAGD) applications     

Efficient Hausdorff Distance computation for freeform geometric models in close proximity

We present an interactive-speed algorithm for computing the Hausdorff Distance (HD) between two freeform geometric models represented with NURBS surfaces. The algorithm is based on an effective technique for matching a surface patch from one model to the corresponding nearby surface patch on the other model. To facilitate the matching procedure, we employ a bounding volume hierarchy (BVH) for freeform NURBS surfaces, which provides a hierarchy of Coons patches and bilinear surfaces approximating the NURBS surfaces (Kim et al., 2011 [1]). Comparing the local HD upper bound against a global HD lower bound, we can eliminate the majority of redundant surface patches from further consideration. The resulting algorithm and the associated data structures are considerably simpler than the previous BVH-based HD algorithms. As a result, we can compute the HD of two freeform geometric models efficiently and robustly even when the two models are in close proximity. We demonstrate the effectiveness of our approach using several experimental results.