Non-uniform Rational B-spline Curves Research Articles

Motion Instruction Generation along Two NURBS Curves in Each Cycle Period

Abstract Arbitrary types of curve on a free-form surface can be accurately represented by the non-uniform rational b-spline (NURBS) curves with use of the huge advantages of NURBS. Accordingly, an interpolation method using NURBS technology in the CAD field is proposed and realized in a five-axis computer numerical controlled (CNC) system. In this method, the tool path was represented to two NURBS curves, tool-tip and tool-orientation spline curve respectively, based on the same knot vector. The cutter location (CL) data was firstly calculated and then transformed to five motion instructions of five axes of machine through the real-time post-procession algorithm. The acceleration/deceleration controlling method is also presented to avoid the mechanical impacts on machine tools. The proposed five-axis spline interpolation method is realized in our developing five-axis machine tool and the result of machining shows that the proposed method is reasonable and valuable.

Automatic unstructured all-hexahedral mesh generation from B-Reps for non-manifold CAD assemblies

This paper describes an automatic and robust approach to convert non-manifold CAD assemblies into unstructured all-hexahedral meshes conformal to the given B-Reps (boundary-representations) and with sharp feature preservation. In previous works, we developed an octree-based isocontouring method to construct unstructured hexahedral meshes for arbitrary non-manifold and manifold domains. However, sharp feature preservation still remains a challenge, especially for non-manifold CAD assemblies. In this paper, boundary features such as NURBS (non-uniform rational B-Splines) curves and surface patches are first extracted from the given B-Reps. Features shared by multiple components are identified and distinguished. To preserve these non-manifold features, one given surface patch may need to be split into several small ones. An octree-based algorithm is then carried out to create an unstructured all-hexahedral base mesh, detecting and preserving all the sharp features via a curve and surface parametrization. Two sets of local refinement templates are provided for adaptive mesh generation, along with a novel 2-refinement implementation. Vertices in the base mesh are categorized into four groups based on the given non-manifold topology, and each group is relocated using various methods with all sharp features preserved. After this stage, a novel two-step pillowing technique is developed for such complicated non-manifold domains to eliminate triangle-shaped quadrilateral elements along the curves and “doublets”, handling non-manifold and manifold features in different ways. Finally, a combination of smoothing and optimization is used to further improve the mesh quality. Our algorithm is automatic and robust for non-manifold and manifold domains. We have applied our algorithm to several complicated CAD assemblies.

Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM

This paper focuses on the numerical integration of polynomial functions along non-uniform rational B-splines (NURBS) curves and over 2D NURBS-shaped domains, i.e. domains with NURBS boundaries. The integration of the constant function f=1 is of special interest in computer aided design software and the integration of very high-order polynomials is a key aspect in the recently proposed NURBS-enhanced finite element method (NEFEM). Several well-known numerical quadratures are compared for the integration of polynomials along NURBS curves, and two transformations for the definition of numerical quadratures in triangles with one edge defined by a trimmed NURBS are proposed, analyzed and compared. When exact integration is feasible, explicit formulas for the selection of the number of integration points are deduced. Numerical examples show the influence of the number of integration points in NEFEM computations.

5-axis Interpolation and Synchronous Servo Process with NURBS Method

Abstract Nowadays, NURBS (Non-Uniform Rational B-Spline) method is widely used in CAD/CAM (Computer-Aided Design/Computer-Aided Manufacturing) to represent sculptured curves or surfaces. In this paper, we develop a 5-axis NURBS real-time interpolator and realize it in our developing CNC(Computer Numerical Control) system. At first, we use two NURBS curves to represent tool-tip and tool-axis path respectively. According to feedrate and Taylor series extension, servo-controlling signals of 5 axes are obtained for each interpolating cycle. Then, generation procedure of NC(Numerical Control) code with the presented method is introduced and the method how to integrate the interpolator into our developing CNC system is given. And also, the servo-controlling structure of the CNC system is introduced. Through the illustration, it has been indicated that the proposed method can enhance the machining accuracy and the spline interpolator is feasible for 5-axis 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 G 0 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 novel hole patching algorithm for discrete geometry using non-uniform rational B-spline

A hole patching algorithm for discrete geometry using a parametric approach based on Non-Uniform Rational B-Spline (NURBS) curves and surfaces is being presented in this paper. This algorithm is focused on recovering the missing geometric information based on the neighboring points surrounding the topologically simple but geometrically complex holes. The neighboring points are utilized to reconstruct a set of three-dimensional (3D) NURBS surface patches covering the hole region which in turn are used to produce a smooth surface patch covering the hole. This algorithm can automatically identify the hole, obtain surrounding points, create the NURBS surfaces, and perform the projection of points onto the surface to complete the process. The algorithm also provides a way for the user to control the size and density of triangulation in the patches matching that of the surroundings. This paper describes the algorithm step by step in detail with a test case for description purposes. The validation of the algorithm is performed on several analytical geometries to assess the accuracy of the algorithm. Several complex geometries are used to demonstrate the success and robustness of the algorithm. Finally, a conclusion provides the overall assessment of this research and identifies its weaknesses for future improvements. Copyright © 2011 John Wiley & Sons, Ltd.

Real-time PH-based interpolation algorithm for high speed CNC machining

Various methods for parametric interpolation of non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, the errors caused by the approximate nature of the NURBS interpolator were rarely taken into account. This paper proposes an integrated look-ahead algorithm for parametric interpolation along NURBS curves. The algorithm interpolates the sharp corners on the curve with the Pythagorean-hodograph (PH) interpolation. This will minimize the geometric and interpolator approximation errors simultaneously. The algorithm consists of four different modules: a sharp corner detection module, a PH construction module, a feedrate planning module, and a dynamics module. Simulations are performed to show correctness of the proposed algorithm. Experiments on an X–Y table confirm that the developed method improves tracking and contour accuracies significantly compared to previously proposed algorithms.

Development of integrated acceleration/deceleration look-ahead interpolation technique for multi-blocks NURBS curves

Modern motion control adopts acceleration/deceleration before interpolation (ADBI) motion planning to eliminate path command errors. However, the individual velocity profiles might not be continuous at the junction of the blocks. Acceleration/deceleration after interpolation (ADAI) method may provide an alternative for solving the discontinuous problems, but it causes path command errors. In this paper, an integrated acceleration/deceleration interpolation (IAD) scheme which integrates the ADBI and ADAI modules is proposed. The ADBI provides a look-ahead function which plans the feedrate profiles based on chord errors, command errors, curvatures, and acceleration limits. Within the look-ahead function, the command error equation is utilized to determine the feedrate at the junction of adjacent blocks. Then the ADBI performs non-uniform rational B-spline (NURBS) interpolation using the planned feedrate profile and outputs the position points to the ADAI module. The ADAI module processes the points by a digital convolution technique such that the continuity of the block junction velocity is ensured. Finally, the IAD is applied to the multi-block NURBS interpolation to validate its effectiveness. Simulations and experiments are conducted to demonstrate the IAD scheme. It is shown that the IAD scheme can reduce the acceleration significantly at the junctions of the blocks under the given tolerance of the command error. Furthermore, the proposed algorithm can improve tracking and contour accuracies as compared to the hybrid multi-blocks look-ahead approach.

The application of muscle wrapping to voxel-based finite element models of skeletal structures

Finite elements analysis (FEA) is now used routinely to interpret skeletal form in terms of function in both medical and biological applications. To produce accurate predictions from FEA models, it is essential that the loading due to muscle action is applied in a physiologically reasonable manner. However, it is common for muscle forces to be represented as simple force vectors applied at a few nodes on the model's surface. It is certainly rare for any wrapping of the muscles to be considered, and yet wrapping not only alters the directions of muscle forces but also applies an additional compressive load from the muscle belly directly to the underlying bone surface. This paper presents a method of applying muscle wrapping to high-resolution voxel-based finite element (FE) models. Such voxel-based models have a number of advantages over standard (geometry-based) FE models, but the increased resolution with which the load can be distributed over a model's surface is particularly advantageous, reflecting more closely how muscle fibre attachments are distributed. In this paper, the development, application and validation of a muscle wrapping method is illustrated using a simple cylinder. The algorithm: (1) calculates the shortest path over the surface of a bone given the points of origin and ultimate attachment of the muscle fibres; (2) fits a Non-Uniform Rational B-Spline (NURBS) curve from the shortest path and calculates its tangent, normal vectors and curvatures so that normal and tangential components of the muscle force can be calculated and applied along the fibre; and (3) automatically distributes the loads between adjacent fibres to cover the bone surface with a fully distributed muscle force, as is observed in vivo. Finally, we present a practical application of this approach to the wrapping of the temporalis muscle around the cranium of a macaque skull.

Research on CNC Key Technology for High Speed Milling

At present, high speed machining (HSM) features in high efficiency, high precision, high flexibility and high quality. HSM technology involves many factors, including cutting mechanisms, machining performance, tool path and other aspects. The key techniques on HSM such as linear interpolation, circular interpolation, cubic B-spline curve interpolation, non-uniform rational B-spline (NURBS) curve interpolation and their respective characteristics are paid more attention and expatiated.

Interpolation Algorithm on Numerical Control Technology

High speed machining (HSM) features in high efficiency, high precision, high flexibility and high quality. The key techniques on HSM such as linear interpolation, circular interpolation, cubic B-spline curve interpolation, non-uniform rational B-spline (NURBS) curve interpolation and their respective characteristics are paid more attention and expatiated. Circle arcs are approximated by using intersecting polygons instead of general inscribed polygons. NURBS interpolation algorithms can reduce feedrate fluctuation and improve machining accuracy.

Robust Multi-Objective Wing Design Optimization Via CFD Approximation Model

:A robust multi-objective wing design optimization procedure using CFD is presented in this work. Instead of directly applying CFD to wing design optimization, a Kriging aerodynamic model that approximates CFD result is proposed to save computational cost. By introducing 6σ robust approach as a sub-objective function, multi-objective wing design problems can be solved with aerodynamic robustness. Non-Uniform Rational B-Spline curves (NURBS) are introduced to depict the wing geometry in design optimization process. The most important parameters of the wing geometry can then be screened by design of experiment and treated as design variables due to the local variation property of NURBS. Under this formulation, a multi-objective genetic algorithm based on non-dominated sorting is employed to handle multiple flight conditions in wing design. The optimum wing geometry is selected according to trade-off among design objects on the non-dominated front. To validate the proposed approach, two robust design optimization cases are studied for ONERA-M6-Wing. It turns out that the drag coefficient is insensitive to Mach number between Ma0.8–Ma0.9 after robust optimization. The result also verifies the effectiveness of our method in boosting performance and robustness of multiple flight conditions: the lift curve slope (Ma0.3) of optimum wing and the 6σ objective function of transonic drag (Ma0.8–Ma0.9) increase by 11.9% and 25.4%, respectively. Observations in the optimization cases are concluded as follows: 1) For wing aerodynamic design problems, the multi-objective robust optimization can both improve the performance at different flight conditions and provide robustness. 2) The Kriging aerodynamic model derived from CFD result can satisfy the precision requirement of wing design. 3) Even though the optimization result is subject to the weights of the 6σ function in sub-object, the influence is not comparable to the trade-off among design objects. 4) The optimum solution obtained by proposed approach is superior to gradient based optimization method, while the computational cost is acceptable.

Real-Time Optimum Feedrate Planning for Coplanar NURBS Curves Considering Motion Dynamics

AbstractThis paper proposes a real-time optimum feedrate planning for NURBS (Non-Uniform Rational B-Spline) motion trajectory considering motion dynamics including maximum feedrate and the limitations of acceleration and jerk. First, the estimated feedrate, which satisfies the chord accuracy and maximum feedrate limitation, can be evaluated by adaptive feedrate. Then, to achieve optimum feedrate planning, the estimated feedrate is fed into our proposed lowest feedrate first priority planning (LFFPP) algorithm, which can adapt any kind of acceleration/deceleration (ACC/DEC) profile. The curvature of an NURBS curve is calculated, and the estimated feedrate corresponding to different curvature and chord error is evaluated by adaptive feedrate. In addition, the break points, which have the local maximum curvatures and where the corresponding estimated feedrate is smaller than the maximum feedrate limitation, are picked up. The NURBS curve is separated into many segments by these break points and the estimated...

Development of a hybrid iterative learning control for contouring NURBS curves

AbstractThis paper develops a hybrid iterative learning control (ILC) algorithm which uses both tracking and contour errors as correction signals during the learning process. The tracking, contour and combined ILCs are first introduced and analyzed. A hybrid ILC is then proposed to predict contour errors based on the identified models and then used to determine whether the tracking or contour errors should be utilized for the correction signals. The criteria for selecting different signals are based on the predicted root mean square (RMS) value of the contour errors for non‐uniform rational B‐spline (NURBS) curves. Simulations performed on a butterfly trajectory represented by NURBS curves show that the hybrid ILC adopts tracking errors as correction signals for the first five iterations, even if the contour error RMS values are used as the objective function. As the iterations evolve, the hybrid ILC switches from using tracking errors to using contour errors as correction signals. Geometric interpretations are given to illustrate the switching behavior. It is shown that the hybrid ILC can significantly reduce contour errors as compared to the process without learning. Finally, validation experiments using the butterfly curve show that the hybrid ILC outperforms the tracking, contour and combined ILCs algorithms.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Aerodynamic shape optimization of airfoils in unsteady flow

Aerodynamic shape optimization of airfoils is carried out for two values of Reynolds numbers: 103 and 104, for an angle of attack of 5° . The objective functions used are (a) maximization of lift (b) minimization of drag and (c) minimization of drag to lift ratio. The surface of the airfoil is parametrized by a 4th order non-uniform rational B-Spline (NURBS) curve with 61 control points. Unlike the efforts in the past, the relatively large number of control points used in this study offer a rich design shape with the possibility of local bumps and valleys on the airfoil surface. The airfoils obtained via the optimization exhibit significantly improved aerodynamic performance compared to the NACA 0012 airfoil. Maximization of lift results in airfoils with corrugations on the upper surface. These corrugations are accentuated in the airfoils that are optimal at Re=104. The improved performance is due to the formation of small vortices for the optimal airfoil. These vortices lead to an increase in suction on the airfoil surface. The time-averaged flow field shows that these vortices occupy the valleys in the corrugated upper surface of the optimal airfoil. Compared to a NACA 0012 airfoil, the time-averaged lift coefficient increases by 36.58% at Re=103 and 220.72% at Re=104, over the NACA 0012 airfoil. Minimization of drag results in airfoils with flatter upper surface. Compared to the NACA 0012 airfoil, the time-averaged drag coefficient for the optimal airfoil is 2.1% smaller at Re=103. The reduction at Re=104 is 15.36% compared to the NACA 0012 airfoil. The effect of the time period for time averaging of the flow, on the convergence of the optimization cycle, is investigated. It is found that time period for which the unsteady data is sampled and time-averaged should be sufficiently long time to achieve good convergence in the optimization cycle. A continuous adjoint based method is used for shape optimization. The flow as well as the adjoint equations are solved via a stabilized finite element method. The limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm is used to minimize the objective function.

Design of Feedforward Compensator for the Travelling Wave Ultrasonic Motor Based on Non-uniform Rational B-splines Curve Neural Network

Design of Feedforward Compensator for the Travelling Wave Ultrasonic Motor Based on Non-uniform Rational B-splines Curve Neural Network

Fast and accurate NURBS fitting for reverse engineering

Fast and accurate fitting of non-uniform rational B-spline (NURBS) curves and surfaces through large sets of measured data is an important problem in applications such as reverse engineering and geometric modelling. This paper presents a method for realising significant improvements in the computational efficiency of this task. The basic idea is that the sparsity structures of the relevant matrices that are specific to the problem of NURBS fitting can be precisely defined and that full exploitation of these structures leads to significant savings in both computational and storage requirements. These savings allow for a large number of control points to be used in order to define the surface and consequently to improve the accuracy of shape representation. The achieved computational complexity is linear in both the number of measured points and the number of control points while the storage requirements of the algorithm are linear with the number of control points only. The complexity analysis, as well as the analysis of actual running times is presented. The results demonstrate that, using this approach, highly complex shapes may be modelled accurately with a single NURBS surface.

Mathematical modeling for the design of a generic custom-engineered form mill

This paper presents a mathematical model for the geometric design of a custom-engineered form milling (CEFM) cutter and develops an accurate 3D surface-based generic definition of the form mill. The proposed geometric definition of the CEFM cutter is developed in terms of biparametric surface patches using newly defined 3D rotational angles rather than the conventional 2D angles. The non-uniform rational B-spline curve fairing and sweep surface approaches are used to design the cutting edge and flank surface, respectively, of the cutter. To validate the methodology, an interface is developed that converts the proposed 3D parametric definition of the cutter into an intermediate neutral CAD format and then renders the cutter model in any CAD modeling environment. The accuracy of the mathematical model is verified by evaluating the deviation plot between the surfaces of the proposed cutter model and the surfaces developed using digitized data of an existing actual CEFM cutter. The satisfactory comparison verifies the shape design methodology for custom cutters presented in this paper. The method described here offers a simple and intuitive way of generating shape design of custom cutters for possible use in machining process simulations, finite element analysis, and other applications.

Real-time NURBS curve interpolator based on section

Parameter interpolation is more capable of modern computer numerical control (CNC) than traditional linear/circular interpolation with higher speed and higher precision. Most of non-uniform rational B-spline (NURBS) interpolation algorithms were developed based on the chord error and machines capability, where interpolation points are calculated beforehand to overcome acceleration/deceleration (acc/dec) and jerk problem, which needs large memory. In this paper, a NURBS interpolator based on feedrate section is proposed. Instead of a single interpolation point, this interpolator aims to feedrate section, which makes it possible to run on a digital signal processor or field programmable gate array (FPGA) whose memory is limited. Experiment on FPGA showed the performance of interpolation. A mould experiment verifies the feasibility of application.

Five-axis NURBS Path Real-time Generation Method in CNC System

Abstract A real-time path generation method with Non Uniform Rational B Spline(NURBS) technology in the Computer Aided Design(CAD) field is presented and realized in a five-axis Computer Numerically Controlled(CNC) system. In this method, the tool path was represented into two NURBS curves, tooltip and tool-orientation spline curve, based on the same knot vector. The Cutter Location(CL) data was firstly calculated and then transformed to five motion commands of five axes of machine tool through the real-time post-procession algorithm. The acceleration/deceleration controlling method was also presented to avoid the impact of machine tool. The proposed five-axis spline interpolation method is realized in our developing five-axis CNC machine tool and the result of machining shows that the method is feasible.