Non-uniform Rational B-spline Curves Research Articles

Application Framework for Aero-based Design Optimization of Passenger Cars using NURBS

This paper presents a new application framework for aerodynamics-based shape optimization of passenger cars. The rear geometry of a passenger car is the focus of this study due to its significant influence on the aerodynamic characteristics of vehicle. The rear body of a generic car model (the Ahmed body) was represented by Non-Uniform Rational B-Spline (NURBS) curve and NURBS parameters were employed for geometric parameterization. These geometric parameters were systematically modified to alter the geometry using the model developed through a design of experiments process. Computational Fluid Dynamics (CFD) simulations were performed on these geometries to obtain drag coefficients. A polynomial response surface model of drag coefficient was then constructed using linear regression to relate design parameters to the drag coefficient. This response surface model was then used as a starting point for the optimization process. The proposed framework was implemented on a generic notch back car model and the optimized geometric parameters for minimum drag were obtained.

Improved Empirical Mode Decomposition Algorithm of Processing Complex Signal for IoT Application

Hilbert-Huang transform is widely used in signal analysis. However, due to its inadequacy in estimating both the maximum and the minimum values of the signals at both ends of the border, traditional HHT is easy to produce boundary error in empirical mode decomposition (EMD) process. To overcome this deficiency, this paper proposes an enhanced empirical mode decomposition algorithm for processing complex signal. Our work mainly focuses on two aspects. On one hand, we develop a technique to obtain the extreme points of observation interval boundary by introducing the linear extrapolation into EMD. This technique is simple but effective in suppressing the error-prone effects of decomposition. On the other hand, a novel envelope fitting method is proposed for processing complex signal, which employs a technique of nonuniform rational B-splines curve. This method can accurately measure the average value of instantaneous signal, which helps to achieve the accurate signal decomposition. Simulation experiments show that our proposed methods outperform their rivals in processing complex signals for time frequency analysis.

Mine 3D Geological Interface Modeling by NURBS Technology

It is imperative for 3D modeling of mining or underground engineering to precisely and efficiently describe various interfaces, which is also the precondition for 3D visualization of underground engineering. NURBS (Non-Uniform Rational B-Splines) curve and surface technology are introduced in the paper. With help of analyzing 3D geologic structure of Laohutai mining area and comprehensive treatment of surfaces of tunnels and geologic structures. 3D geological surfaces of mining area are built which is the base of building complex 3D geologic modeling.

A novel scheme for flexible NURBS-based C2 PH Spline Curve contour following task using Neural Network

In this paper, a novel scheme for approximation of complicated CNC tool paths, designed by non-uniform rational B-spline (NURBS) curves, is proposed. The idea of this approximation is based on introducing flexible NURBS-based C2 PH Spline Curves. Using the specifications of nodal points on a NURBS curve and introducing the coefficients for the velocity and acceleration vectors on these points, as well as defining flexible connection nodes, 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 flexible connection nodes on the original NURBS curve are computed by genetic algorithm and Neural Network. To this end, the normal distance between the tool path and its corresponding original NURBS curve is considered as the objective function. Using combination of the time-dependent feedrate and constant feedrate interpolations, the position commands of the designed curve are generated. Several contour following tasks were implemented with pseudo-derivative feedback feed forward (PDFF) controller. Experimental and simulation results confirm that the devised interpolator 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 feedrate.

Optimal Forebody Shape for Minimum Drag in Supersonic Flow

In this work, a simple and efficient numerical approach to determine the shape of the minimum-drag axisymmetric forebody in inviscid supersonic flow with an attached shock constraint has been described. Taylor–Maccoll equation in conjunction with the tangent cone method is employed to estimate the pressure drag coefficient which is also chosen as the cost function. The forebody geometry is parameterized using a Non-Uniform Rational B-Splines (NURBS) curve whose control points are the design variables for optimisation using the steepest descent algorithm. Numerical studies demonstrate that the optimal forebody geometry for a given length and base radius has as much as 15 % lesser drag, depending on the Mach number than a cone of the same fineness ratio and that the convergence to the optimal solution exhibits a relatively weak Mach-number dependence.

Study on Five-Axis NURBS Path Generation Method and Direct-Interpolation for CNC System

In this article, a new method with Non-Uniform Rational B-Spline (NURBS) technology is presented and realized in a location and federate controlling system. In the method, the tool paths and the cutter locations (CL) were represented into NURBS curves based on the same knot vector. The above CL data was firstly calculated and then transformed to five motion commands of five axes of machine through the real-time post-procession algorithm. The acceleration/deceleration controlling method is also presented to avoid the impact of machine. The proposed 5-axis spline interpolation method is realized and the experimental result of machining shows that the method is valid.

Development and implementation of a NURBS interpolator with smooth feedrate scheduling for CNC machine tools

Parametric interpolation for Non-Uniform Rational B-Spline (NURBS) curve has become more important than ever before in the control of CNC machine tools. An effective NURBS interpolator not only can obtain accurate contour trajectories, but also have smooth dynamics performance. This paper proposes a numerically efficient NURBS interpolation scheme which consists of two stages namely preprocessing and interpolating. In the stage of pre-processing, the parameter interval is split into several blocks at breakpoints and an iterative numerical quadrature method is applied for each block. By means of the iterative quadrature method, the initial parameter intervals of each block are divided into several subintervals according to the arc length approximation error. Meanwhile, the curvature of each knot and the cubic polynomial coefficients of each subinterval are obtained. Then the critical points with large curvature of each block are found from the candidate points and the tolerated speed of each critical point is calculated according to the constraints of chord error and centripetal acceleration. Hence, the feedrate scheduling based on the S-shaped acceleration profile for each block can be preplanned via the approximate arc length of each subinterval, the tolerated speed of each critical point and kinematics characteristics such as acceleration/deceleration and jerk limits of the machine tools. In the stage of interpolating, the parameter of the next interpolation point can be calculated directly using the cumulative arc length and the cubic polynomial coefficients of each subinterval. Finally, a series of numerical simulations and real machining experiments are conducted, and the simulation and experimental results have showed the good performance of the proposed NURBS interpolator both in efficiency and accuracy.

Study on Tool Path Generation and Direct Interpolation Method

At present, Non-Uniform Rational B-Spline (NURBS) curves are mostly used to generate arbitrary contour of a complex shape in the CAD field. In this article, a new method with NURBS technology is presented and realized in a location and federate controlling system. In the method, the tool paths and the cutter locations (CL) were represented into NURBS curves based on the same knot vector. The above CL data was firstly calculated and then transformed to five motion commands of five axes of machine through the real-time post-procession algorithm. The acceleration/deceleration controlling method is also presented to avoid the impact of machine. The proposed 5-axis spline interpolation method is realized and the experimental result of machining shows that the method is valid.

NURBS Curve Extraction from 2D Laser Sensor Data and 3D Simulated Data

This paper proposes a non-uniform rational B-spline (NURBS) curve extraction algorithm from 2D laser sensor data and 3D simulated data. In robot localization and mapping application, the raw sensor data cannot be stored due to its need of large storage space. However, only a small number of control points of NURBS curve are needed to be stored to recover the geometrical feature of raw data. To comprise the number of control points and accuracy of the extracted curve, global approximation method is adopted to minimize the error between the raw data and the extracted curve. In extraction process, all the weights are set as one firstly. After find the control points, a weight calculation method is developed to update the weight values. The NURBS curve with new weight has smaller error than with original weights. Finally, NURBS curve extraction results from real 2D laser sensor data and 3D simulated data are shown to check the feasibility of proposed algorithms.

Fast Discretization Algorithm of Densely Subdivided NURBS Curves

This paper develops a discretization algorithm for non-uniform rational B-spline (NURBS) curves. The algorithm is used for CAM software which is developed for HNC-3000 five-axis woodworking NC milling machine. Functions from the CAM software divide the NURBS into dense segments with equal arc length by vertex set p1, p2pn. The algorithm simultaneously considers chord error δ and curvature of the NURBS curve to maximize the arc length of the segments to get vertex set p'1, p'2 pm'. The algorithm can be described as f (p1, p2pn, δ) = (p1', p2'pm'). The complexity of the algorithm is O (n).

Review on Non Uniform Rational B-Spline (NURBS): Concept and Optimization

This paper is to provide literature review of the Non Uniform Rational B-Splines (NURBS) formulation in the curve and surface constructions. NURBS curves and surfaces have a wide application in Computer Aided Geometry Design (CAGD), Computer Aided Design (CAD), image processing and etc. The formulation of NURBS showing that NURBS curves and surfaces requires three important parameters in controlling the curve and also modifying the shape of the curves and surfaces. Yet, curves and surfaces fitting are still the major problems in the geometrical modeling. With this, the researches that have been conducted in optimizing the parameters in order to construct the intended curves and surfaces are highlighted in this paper.

An approach for chest tube detection in chest radiographs

It is known that overlapping tissues cause highly complex projections in chest radiographs. In addition, artificial objects, such as catheters, chest tubes and pacemakers can appear on these radiographs. It is important that the anomaly detection algorithms are not confused by these objects. To achieve this goal, the authors propose an approach to train a convolutional neural network (CNN) to detect chest tubes present on radiographs. To detect the chest tube skeleton as the final output in a better manner, non-uniform rational B-spline curves are used to automatically fit with the CNN output. This is the first study conducted to automatically detect artificial objects in the lung region of chest radiographs. Other automatic detection schemes work on the mediastinum. The authors evaluated the performance of the model using a pixel-based receiver operating characteristic (ROC) analysis. Each true positive, true negative, false positive and false negative pixel is counted and used for calculating average accuracy, sensitivity and specificity percentages. The results were 99.99% accuracy, 59% sensitivity and 99.99% specificity. Therefore they obtained promising results on the detection of artificial objects.

An automatic measuring method and system using laser triangulation scanning for the parameters of a screw thread

An automatic non-contact measuring system for extracting the parameters of a screw thread was developed. It was integrated with a laser-based measuring unit for the longitudinal section of the thread profile and the dual-axis moving stage position system for the movement of the undetected object. Firstly, data points from the thread profile were collected by the measuring system, the non-uniform rational B-spline curve (NURBS) method with the minimal error constraints was used to carry out the fitting of the measured data and data partitioning was performed. Secondly, according to the processed data, the parameters, such as thread pitch, pitch diameter and half of thread angle, can be calculated. Thirdly, the errors were analyzed for the proposed system. Finally, the proposed system and method were verified for their good performance with acceptable accuracy and reliability. Compared with existing measurement methods, the designed non-contact optical measuring system was capable of measuring the parameters of the internal thread which is no less than the M4 and external thread with required accuracy.

EFFECTIVENESS OF INPUT SHAPING AND REAL-TIME NURBS INTERPOLATION FOR REDUCING FEEDRATE FLUCTUATION

In modern CAD/CAM system, NURBS (Non-Uniform Rational B-Spline) interpolation is adopted for the purpose of achieving high speed and high accuracy performance. The main problem with using NURBS curve is the relationship approximation between arc length and spline parameter. In case the toolpath planning can be assumed with no feed fluctuations, implementing of driving mechanism still can lead CNC system to vibrate. Input shaping is a method of preshaping control input to reduce residual vibration in the flexible systems. Input shaping alters the reference command by convolving it with an impulse sequence. The convolution product is then used to drive the system. In this paper, NURBS curve representation was employed to generate the planning trajectory for a two axis gantry crane which is considered as a CNC system. The effectiveness of input shaping based on genetic algorithm was investigated by following the desired NURBS curve.

A NURBS-Based Triangulation Method for 3D Ship Hull Simulation

To design and implement a system of simulating the process of charging and discharging awkward length cargo with computer, key task is to construct high precision three-dimensional model of the hull. In view of problems exiting in distance subdivision method, we proposed one adaptive subdivision method for NURBS (Non-Uniform Rational B-Splines) curve based on energy function. In view of problems from repeated storage of coordinate points and non-linear search of adjacent field in tree structure, namely, when applying binary-tree and quad-tree to recursively divide surface, the intermediate result from each division needs to be saved, each vertex coordinate is repeatedly saved 3 to 4 times, we proposed quadrangle coding method produced automatically by triangle grid of ship hull based on NURBS. Simulation results show that this algorithm has obviously saved time and space, improved the algorithm efficiency, avoided crack between the grids and proved good application in shipping industries.

The improved NURBS-based C2 PH spline curve contour following task with PDFF controller

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.

FPGA-based hardware CNC interpolator of Bezier, splines, B-splines and NURBS curves for industrial applications

Tool path interpolation is an important part of Computerized Numerical Control (CNC) systems because it is related to the machining accuracy, tool-motion smoothness and overall efficiency. The use of parametric curves to generate tool-motion trajectories on a workpiece for high accuracy machining has become a standard data format that is used for CAD/CAM (Computer Aided Design/Computer Aided Manufacturing) and CNC systems. Splines, Bezier, B-splines, and NURBS (Non-Uniform Rational B-splines) curves are the common parametric technique used for tool path design. However, the reported works bring out the high computational load required for this type of interpolation, and then at best only one interpolation algorithm is implemented. The contribution of this paper is the development of a hardware processing unit based on Field Programmable Gate Arrays (FPGA) for industrial CNC machines, which is capable of implementing the four main interpolation techniques. It allows the selection of the required interpolation technique according the application. Two CAD models are designed for test the CNC interpolations; experimental results show the efficiency of the proposed methodology.

Real-Time Coplanar NURBS Curves Fitting in Motion Controller

This paper proposes a real-time coplanar Non-Uniform Rational B-Spline (NURBS) curves fitting in CAD/CAM system to improve the fitting quality and reduce the calculation effort. The real-time NURBS curve fitting process consists of (1) the reading discrete points, (2) curve segmentation and data reshaping, (3) curve fitting, and (4) local and global modulation. According to the simulation and experimental results, the proposed approach certainly raises the fitting quality and the efficiency of calculation.

Study on Curved Wedge Diffraction in NURBS‐UTD Method

ABSTRACTThe curved wedge diffraction in nonuniform rational b‐spline (NURBS)‐uniform geometrical theory of diffraction method is studied in this article. The curved wedge is denoted as a NURBS curve while it is also treated as the intersection of two NURBS surfaces. Basic aspects are managed through numerical measures, and parameters used in the calculation of ray field are achieved by using the mathematic definition of NURBS and basic knowledge of differential geometry. Good results are achieved in the examples given and with the contribution of the wedge diffracted ray field to the total ray field, the computational accuracy can be slightly improved. Numerical results show the validation and the improvement. © 2013 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:2317–2321, 2013

Weak visibility polygons of NURBS curves inside simple polygons

Visibility computation plays an important role in applications such as architectural design, art gallery patrolling and virtual worlds. In this paper, we present an algorithm to compute the weak visibility polygons (WVP) of Non Uniform Rational B-spline (NURBS) curves inside simple polygons. The NURBS curve is first subdivided into triangular curves. We then compute the WVP of each triangular curve by shearing that of its triangle hull. Finally, all triangular curves’ WVPs are merged together to obtain the WVP of the NURBS curve. Analysis and examples are given to show the performance of our algorithm.