Non-uniform Rational B-spline Curves Research Articles

SU-GG-I-117: NURBS-Based Deformable Image Registration

Purpose: The current trend of incorporating multi-modality and multi-instance imaging in treatment planning for external beam radiotherapy has highlighted the need for fast and accurate deformable image registration techniques. One current implementation of non-rigid registration uses uniform B-Splines to parametrically represent the displacement vector field (DVF). The piecewise B-spline segments are joined at “knots” that define the local region of support and influence for each segment. This approach has limited flexibility and can require a fairly large number of control points to describe local complexity in the DVF. The authors present the implementation of a more general functional representation of B-Splines, Non-Uniform Rational B-Splines (NURBS) as an alternative DVF model. Method and Materials: To demonstrate the improved accuracy that NURBS can provide, we have made numerical fits to a one-dimensional scalar DVF in the shape of a Gaussian. We first fit the DVF with a uniform B-Spline to establish baseline accuracy. We then fit the Gaussian with a 1D NURBS model with the same number of control points but now with non-uniform knots and weighted control points. The metric of comparison of the two fitting routines is the sum of the squared differences between the curve fit and the target Gaussian. Results: For the same number of control points, the NURBS fit produced a sum of squared differences of .0019 compared to 1.625 for the uniform B-Spline fit. Conclusion: NURBS offer an attractive alternative to uniform B-Splines in modeling the DVF. They carry forward the mathematical compactness of B-Splines while simultaneously introducing new degrees of freedom. The additional free parameters gained from the generalization to NURBS curves offers increased local control as well as the ability to explicitly represent topological discontinuities.

Optimal aerodynamic design of airfoils in unsteady viscous flows

A continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at Re = 1 × 10 4. The Reynolds number is based on the free-stream speed and the chord length of the airfoil. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure-stabilized Petrov/Galerkin (PSPG) stabilizations is used to solve both the flow and adjoint equations. The airfoil is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve. Three different objective functions are used to obtain optimal shapes: maximize lift, minimize drag and minimize ratio of drag to lift. The objective functions are formulated on the basis of time-averaged aerodynamic coefficients. The three objective functions result in diverse airfoil geometries. The resulting airfoils are thin, with the largest thickness to chord ratio being only 5.4%. The shapes obtained are further investigated for their aerodynamic performance. Maximization of time-averaged lift leads to an airfoil that produces more than six times more lift compared to the NACA 0012 airfoil. The excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface. Minimization of drag results in an airfoil with a sharp leading edge. The flow remains attached for close to 70% of the chord length. Minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface. It leads to a fairly large value of the time-averaged ratio of lift to drag (∼ 17.8). The high value is mostly achieved by a 447% increase in lift and 16% reduction in drag, compared to a NACA 0012 airfoil. Imposition of volume constraint, for the cases studied, is found to result in airfoils that have lower aerodynamic performance.

NURBS Curves Interpolation Based on Dynamic Characteristics of Machine Tools

This paper presents an interpolation algorithm for non-uniform rational B-spline (NURBS) curves. A real-time look-ahead function is introduced to deal with the abrupt change of feedrate. In the proposed algorithm, feedrate can be adaptively adjusted according to the radius of curvature of curves. Dynamic characteristic of machine tools is considered while the feedrate is adjusted. The experimental results show that this algorithm significantly reduced the machining error brought about by the instantaneous acceleration fluctuations of machine tools.

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.

An adjoint method for shape optimization in unsteady viscous flows

A new method for shape optimization for unsteady viscous flows is presented. It is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions. The objective function is minimized or maximized subject to the satisfaction of flow equations. The shape of the body is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve and is updated by using the gradients obtained from solving the flow and adjoint equations. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure stabilized Petrov/Galerkin (PSPG) stabilization techniques is used to solve both the flow and adjoint equations. The method has been implemented and tested for the design of airfoils, based on enhancing its time-averaged aerodynamic coefficients. Interesting shapes are obtained, especially when the objective is to produce high performance airfoils. The effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied. It is found that when the window of time integration is insufficient, the gradients are most likely to be erroneous.

A Minimalist Path Detection Approach for Wireless Sensor Networks

Most object tracking techniques find the exact locations of an object, while many other applications are only interested in the object's path rather than its locations. This provides an opportunity to reduce the existing tracking methods' resource usage by only focusing on the path detection. Taking advantage of this opportunity, this article presents two new minimalist approaches for accurately detecting unknown available passages in a sensor field without requiring the exact locations of the objects. In the first approach, each sensor sends its own location to a base station when it senses an object of interest. The base station uses b-spline curves to build the object's path online. Since each sensor sends its location data just once per new path, the first approach is a minimalist approach. The second approach is offline and uses non-uniform rational b-spline (NURBS) curves. Since NURBS needs weighted locations, each sensor sends its own location in addition to the number of times it has sensed an object based on the object's weight. Using the same simulation models, both approaches greatly reduce power consumption and improve the accuracy of the computed paths. The NURBS approach has proved to be robust on false alarms and improves the accuracy of path detection up to 95 percent, which is very close to detecting the object's actual path.

Optimal cutter size determination for 2½-axis finish machining of NURBS profile parts

A curve model of non-uniform rational B-spline (NURBS) has been widely adopted in mainstream CAD/CAM software systems to design complicated geometries of mechanical parts, for example, the curved profiles of pockets, sides, and islands. NURBS profile parts (the profiles include NURBS curves for pockets and islands) are produced in 2½-axis rough and finish machining. In rough machining of the parts, several end-mills with different sizes are employed for high cutting efficiency, and in finish machining, a single end-mill is usually used to cut along the profiles for high surface quality. To accurately produce the geometries with NURBS curves in finish machining, the cutter size should be optimised in order to eliminate gouging and save machining time. Although this topic has been a research focus for a decade, optimal cutter size determination still remains as a technical challenge. To rise to this challenge, our work proposes a new approach to determining the largest allowable size for the cutter to move along all the profiles (including NURBS curves) in 2½-axis finish machining without global and local gouging. The salient feature of this approach is that an original model of the cutter size is formulated and an effective solver–the particle swarm optimisation method–is employed to compute the largest allowable cutter size. This intelligent approach is more efficient and accurate than the conventional computational method based on the test examples in this work. It can also be applied to global and local gouging detection for NURBS profile machining. Our research work has great potential to advance CNC machining techniques.

Induced charge electro osmotic mixer: Obstacle shape optimization

Efficient mixing is difficult to achieve in miniaturized devices due to the nature of low Reynolds number flow. Mixing can be intentionally induced, however, if conducting or nonconducting obstacles are embedded within the microchannel. In the case of conducting obstacles, vortices can be generated in the vicinity of the obstacle due to induced charge electro-osmosis (ICEO) which enhances mixing of different streams: the obstacle shape affects the induced zeta potential on the conducting surface, which in turn influences the flow profile near the obstacle. This study deals with optimization of the geometric shape of a conducting obstacle for the purpose of micromixing. The obstacle boundary is parametrically represented by nonuniform rational B-spline curves. The optimal obstacle shape, which maximizes the mixing for given operating conditions, is found using genetic algorithms. Various case studies at different operating conditions demonstrated that the near right triangle shape provides optimal mixing in the ICEO flow dominant regime, whereas rectangular shape is the optimal shape in diffusion dominant regime. The tradeoff between mixing and transport is examined for symmetric and nonsymmetric obstacle shapes.

Concise Airfoil Representation via Case-Based Knowledge Capture

The cost of exploring a k-dimensional design space at a sampling density n is of the order nk. In view of this “curse of dimensionality,” the paper explores the possibility of reducing the dimensionality of parametric airfoil definitions to as low as one, to minimize their cost impact on full-airframe conceptual design studies. A formulation based on nonuniform rational B-splines is introduced in which the parametric airfoil is constructed in a reduceddimensionality space defined in terms of a small number of basis geometries. These bases are airfoils themselves, selected to represent key features of their class; we use them as learning cases or case-based representations of our knowledge of what makes a nonuniform rational B-splines curve an airfoil: moreover, an airfoil belonging to a certain class. This study focuses on supercritical sections, but the template presented here can also be applied to other types of airfoils and, indeed, to other classes of curves and surfaces.

A new computer geometric modelling approach of yarn structures for the conventional ring spinning process

A new computer modelling approach for three-dimensional yarn structures produced by the conventional ring spinning system is presented. The concept of virtual locations for modelling yarn cross section is extended. Virtual locations are ring configuration with ellipsoid shapes defined by yarn-twist level in each ring layer. A mathematical model based on migrating helix is developed. Fibre migration is governed by geometric and tension mechanisms. The migration parameters include twisting tension, migration period and amplitude. The initial phase and frequency of migration pattern are used as the parameters of fibre migration. Based on the results obtained from the virtual locations concept and mathematical model of twist, a computer-aided design modelling approach for fibre assemblies is presented for geometric and visual simulation of yarn structures. The fibre paths in each interval between two successive cross sections are approximated by non-uniform rational B-spline curves. The geometric model of each fibre is created by sweeping a closed curve along a centreline path. The new model is validated by using process parameters of spun rayon yarn.

Wavelets-based NURBS simplification and fairing

This paper describes a novel simplification and fairing method for NURBS (Non-Uniform Rational B-Spline) curves and surfaces using orthogonal non-uniform spline wavelets. The compression nature of wavelets is utilized for NURBS simplification. Sufficient conditions for C 0 - and G 1 -continuity at the shared boundary of multiple NURBS patches are proved and utilized to preserve the boundary continuity. Our fairing method filters out unwanted imperfections and noise on the NURBS curves and surfaces while preserving geometric details via wavelet decomposition and selective reconstruction. The continuity between neighboring patches is preserved automatically during fairing. Our algorithm is accurate, automatic, robust, efficient, and can be extensively applied to any NURBS models in various research fields such as Computer Aided Design (CAD), Reverse Engineering (RE), finite element analysis and isogeometric analysis. Several examples are presented in this paper, including a propeller CAD model, a Navy boat model and a G 1 -continuous NURBS model of the thoracic aorta constructed from medical CT (Computed Tomography) data.

NURBS curve blending using extension

Curve and surface blending is an important operation in CAD systems, in which a non-uniform rational B-spline (NURBS) has been used as the de facto standard. In local corner blending, two curves intersecting at that corner are first made disjoint, and then the third blending curve is added-in to smoothly join the two curves with G1-or G2-continuity. In this paper we present a study to solve the joint problem based on curve extension. The following nice properties of this extension algorithm are exploited in depth: (1) The parameterization of the original shapes does not change; (2) No additional fragments are created. Various examples are presented to demonstrate that our solution is simple and efficient.

Surface forming and virtual model of simulation based on IGES

Modelling, sharing and transmission of three-dimensional (3D) graphics data are essential for extensive application of 3D virtual simulation technology. To achieve data transmission between general modelling software and a 3D virtual simulation system, and real-time interaction with the simulation model, the topology of the Initial Graphics Exchange Specification (IGES) has been studied in depth, and the storage structure of all geometry element information has been given. Non-uniform rational B-spline (NURBS) surfaces are used to construct complicated dimensional surfaces. The arithmetic of extracting the equation parameters of NURBS surfaces from the general rotation surfaces’ parameters is derived in detail. The respective data structures for the NURBS curve entity, line entity and composite curve entity and their program frameworks consistent with IGES are defined and designed. All data structures are stored as nodes in the respective chain lists and, in turn, all nodes are finally encapsulated in different classes. Description of the base surfaces of the IGES data and trimming of the outer and inner boundaries are realised with an Open Graphics Library, which is based on the reading data stored in the data chain list. Regeneration of complicated dimensional surfaces and interaction with the model are accordingly realised. This program reads an IGES file and the model is displayed successfully.

Robust Control of an LUSM-Based $X\hbox{--}Y\hbox{--}\theta$ Motion Control Stage Using an Adaptive Interval Type-2 Fuzzy Neural Network

The robust control of a linear ultrasonic motor based X-Y-thetas motion control stage to track various contours is achieved by using an adaptive interval type-2 fuzzy neural network (AIT2FNN) control system in this study. In the proposed AIT2FNN control system, an IT2FNN, which combines the merits of an interval type-2 fuzzy logic system and a neural network, is developed to approximate an unknown dynamic function. Moreover, adaptive learning algorithms are derived using the Lyapunov stability theorem to train the parameters of the IT2FNN online. Furthermore, a robust compensator is proposed to confront the uncertainties including the approximation error, optimal parameter vectors, and higher order terms in Taylor series. To relax the requirement for the value of lumped uncertainty in the robust compensator, an adaptive lumped uncertainty estimation law is also investigated. In addition, the circle and butterfly contours are planned using a nonuniform rational B-spline curve interpolator. The experimental results show that the contour tracking performance of the proposed AIT2FNN is significantly improved compared with the adaptive type-1 FNN. Additionally, the robustness to parameter variations, external disturbances, cross-coupled interference, and frictional force can also be obtained using the proposed AIT2FNN.

Implementation of a CNC NURBS curve interpolator based on control of speed and precision

An efficient real-time Non-Uniform Rational B-Spline (NURBS) curve interpolator for high-speed and high-accuracy curve machining is developed and implemented, which considers the confined contour error, acceleration/deceleration (ACC/DEC) planning and the machine dynamic response simultaneously. A practical method is proposed to adjust the feed rate to the curvature radius of NURBS curves to achieve higher machining precision. However, at the sharp corner of the curves, the feed rate fluctuation might result in large impacts to machine tools. The offline pre-interpolation for sharp corners is presented. The feed rate is reduced in advance according with the machine's acceleration capability as well as satisfying the confined contour error. The velocity profile can be chosen dynamically to meet the requirements of the machining process by means of flexible ACC/DEC planning. In addition, a new strategy based on the geometrical properties of NURBS curves is suggested to predict the deceleration point. Therefore, the process of acceleration and deceleration is so smooth that it can satisfy the need of the machining process. This NURBS interpolator has been validated on a CNC system, which greatly improves machining accuracy for both position and velocity control.

Research of whole fairing and approximation algorithm of NURBS curve

With regard to the accuracy and smoothness of a set of discrete data points approximation, this paper gave a whole fairing and approximation algorithm of cubic Non-Uniform Rational B-Spline (NURBS) curve. The algorithm constructed an objective function composed of least square, sum of curvatures on discrete points and sum of curvature varieties on discrete points, obtained a series of the best control points, adjusted weights using nonlinear optimization method, set up an approximation approach to verify the fitting errors and constructed a calculation procedure. At last, this paper displayed and analyzed the obtained curve in UG NX 4.0.

Real-time NURBS interpolator: application to short linear segments

This study proposes the use of a real-time non-uniform rational B-spline (NURBS) interpolator with a look-ahead function to handle numerous short linear segments. The short linear segments conforming to the continuous short block (CSB) criterion can be fitted into NURBS curves in real time. A modified maximum feedrate equation based on the geometric characteristics of the fitting curves and the dynamics of the servo control system has been derived in this paper. Taking advantage of the multi-thread design and the look-ahead function, the real-time NURBS interpolator can process enough G01 block information and complete feedrate planning before interpolation. In addition, the S-shaped jerk-limited acceleration method is adopted for smoother feedrate profiles. Two part shapes, which possess more than 1,000 short linear segments, are tested on our PC-based real-time control system. Both simulation and experimental results verify the feasibility and precision of the proposed interpolation algorithm.

Placement Optimization of Distributed-Sensing Fiber Optic Sensors Using Genetic Algorithms

In this paper, we present methodology for the placement optimization of both discretely and continuously distributed fiber-optic sensors in which many point or quasi-point sensors can be multiplexed along a single optical fiber. A unified sensor performance metric is defined for vibration monitoring and fault detection as the integration/ summation of a weighted functional of some strain measure over the optical fiber length for both discretely and continuously distributed fiber-optic sensors. The optical fiber is represented by a nonuniform rational B-splines curve. The design variables include the control point coordinates of the nonuniform rational B-splines curve and the arclength coordinates of the point sensing element positions along the fiber. The constraints common to all kinds of optical sensors include the maximum fiber length and maximum allowed initial curvature. The constraints are treated as the exact penalty functions in the fitness function for any genetic algorithm. For discretely distributed fiber-optic sensors, we briefly show by an example how to use a genetic algorithm-based traveling salesman problem solver to optimally connect those optimally distributed sensors using one single optical fiber by minimizing the total fiber length. For continuously distributed fiber-optic sensors, two sets of examples are given as applications to fluttering or vibrating panels in aerospace engineering and a clinical smart bed for patient vital signal monitoring in biomedical engineering. In both cases, the integrating bend-sensing fiber-optic sensors are used.

Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery

Methodologies for planning motion trajectory of parametric interpolation such as non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, most of the algorithms were developed based on the constraints of feedrate, acceleration/deceleration (acc/dec), jerk, and chord errors. The errors caused by servo dynamics were rarely included in the design process. This paper proposes an integrated look-ahead dynamics-based (ILD) algorithm which considers geometric and servo errors simultaneously. The ILD consists of three different modules: a sharp corner detection module, a jerk-limited module, and a dynamics module. The sharp corner detection module identifies sharp corners of a curve and then divides the curve into small segments. The jerk-limited module plans the feedrate profile of each segment according to the constraints of feedrate, acc/dec, jerk, and chord errors. To ensure that the contour errors are bounded within the specified value, the dynamics module further modifies the feedrate profile based on the derived contour error equation. Simulations and experiments are performed to validate the ILD algorithm. It is shown that the ILD approach improves tracking and contour accuracies significantly compared to adaptive-feedrate and curvature-feedrate algorithms.

Implementation of online NURBS curve fitting process on CNC machines

The study aims to develop a method for implementing an online non-uniform rational B-spline (NURBS) curve fitting process on CNC machines for improving the quality and efficiency of machining. Conventional CAD/CAM/CNC systems usually induce some machining difficulties and limit the machining results in real applications. Therefore, some researchers have proposed various methods for improving the machining results. An online NURBS curve fitting process consists of the fetching and fitting stages for CNC machines is developed in this paper. The fetching stage fetches a block of NC code and generates motion commands for obtaining data points. The fitting stage executes the NURBS curve fitting method along with a given set of data points. Moreover, the optimal search method is designed for obtaining good results in the NURBS curve fitting process. Simulations and machining tests carried out on a vertical machining center show that the proposed approach reduces the machining time to about 23% while maintaining the machining quality for the sample test.