Non-uniform Rational B-spline Curves Research Articles

Weld seam fitting and welding torch trajectory planning based on NURBS in intersecting curve welding

In the industrial applications of intersecting curve welding, in order to improve the accuracy and quality of welding, the automation program of weld seam positioning, data processing, and the actual welding is widely used. In this paper, a lot of research has been carried out, and a method based on non-uniform rational B-spline (NURBS) curve fitting is presented, while the trajectory of the welding torch is planned. The method can use the position data of the discrete weld seam points obtained by manual teaching or weld seam location. By the improved least squares method and the White rule, the outliers are eliminated and the weld seam points for intersecting curve fitting are obtained. These points are fitted based on NURBS, and then the parameter expressions of the fitted intersecting curve are given. For the purpose of obtaining the orientation of the welding torch, the coordinate system which can describe the feature of the fitted intersecting curve is established. The coordinate system is combined with the theoretical intersecting curve model and the fitted intersecting curve model and gives a novel approach to compute the orientation of the welding torch according to the position of the fitted intersecting curve. In the process of planning the welding trajectory, a fitted intersecting curve discrete algorithm based on the equal arc length principle is proposed to reduce the fluctuation of the feed rate according to the restriction of chord error. Finally, this method gives the position and orientation of the welding torch in the form of a homogeneous matrix and verifies the correctness and flexibility of the algorithm by MATLAB simulation and intersecting curve welding experiment.

Velocity control algorithm in glass polishing based on the cubic NURBS curve

To limit velocity fluctuations and to achieve a controllable jerk value in a glass polishing process, a new velocity control algorithm is proposed based on nonuniform rational B-splines (NURBS). The key of this algorithm is replacing the traditional linear acceleration–deceleration with flexible NURBS acceleration–deceleration. Based on the linear acceleration–deceleration algorithm, the control points of the NURBS curve are confirmed, and the final velocity of the polishing wheel center is solved using the Preston equation. With jerk continuity and limitations of the servo system, nonlinear equations are constructed, and the weighting factors corresponding to the control points are obtained. Cubic velocity control equations can be derived from the obtained feature parameters, which include the final velocity, control points, weighting factors and knot vectors. Based on the proposed NURBS acceleration–deceleration algorithm, a fourth-order Runge–Kutta formula was used to obtain the initial points, and the Milne–Hamming equation was used to predict and correct the next point. The predictor-corrector interpolation algorithm for parametric trajectory was implemented during the polishing process. The experimental results indicate that the proposed approach guarantees limited fluctuations of the relative velocity at contact points and ensure smoother velocity changes at dangerous points.

Analyses of fold profiles by changing weight parameters of NURB curves

Analyses of Non-Uniform Rational B-spline (NURB) curve by varying weights at its nodal points and projection ratio produce several kinetically plausible symmetric and asymmetric fold morphologies in 2D promptly and efficiently with varied overall geometries, curvature of limbs, sharpness/bluntness of hinges, extent of hinge zone, tightness/interlimb angles, etc. Some of these folds are new geometries what other approaches, such as those with Bézier curve, did not produce so far. Natural fold profiles can be matched with NURB curves from photographs.

Analysis on multifocal contact lens design based on optical power distribution with NURBS.

This paper aims to develop and analyze the design method of multifocal contact lenses to obtain curvature continuity in the optical surfaces with the high addition (Add) powers by adjusting non-uniform rational B-spline (NURBS) curves. The paper has developed mathematical formulae to generate the optical power distributions in which the powers continuously change from either near or distant center to the opposite focal length in the periphery of the optical region with different change rates and Add power values. This developed method can efficiently adjust and optimize three parameters, including control points, weight, and knots of the NURBS, to be anterior optical lens surface profiles to adapt for these given power profiles. The result shows that the proposed contact lenses not only achieve smooth and continuous anterior optical surfaces, but also satisfy various optical power distributions with high Add power values for different pupil diameters. Then, these designs of contact lenses can be feasibly converted to the computer-aided design format for analysis and manufacture for molding or single-point diamond turning. Experimental results of this method have been tested and proven when both the power distributions of simulation of lenses and the actual machined samples match the original specified powers provided by clinical demands of a multifocal contact lens. Future integration with variant clinical demands and optimization rules of lens design can be explored for a progressive contact lens.

Planar Pythagorean-Hodograph B-Spline curves

We introduce a new class of planar Pythagorean-Hodograph (PH) B-Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) Bézier curves, presented by R. Farouki and T. Sakkalis in 1990, including the latter ones as special cases. Pythagorean-Hodograph B-Spline curves are non-uniform parametric B-Spline curves whose arc length is a B-Spline function as well. An important consequence of this special property is that the offsets of Pythagorean-Hodograph B-Spline curves are non-uniform rational B-Spline (NURBS) curves. Thus, although Pythagorean-Hodograph B-Spline curves have fewer degrees of freedom than general B-Spline curves of the same degree, they offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. After providing a general definition for this new class of planar parametric curves, we present useful formulae for their construction and discuss their remarkable attractive properties. Then we solve the reverse engineering problem consisting of determining the complex pre-image spline of a given PH B-Spline, and we also provide a method to determine within the set of all PH B-Splines the one that is closest to a given reference spline having the same degree and knot partition.

Curve Fitting of NURBS Curve for CNC Motion Controller

This paper proposes a real-time coplanar Non-Uniform Rational B-Spline (NURBS) curve fitting in a CAD/CAM system to improve fitting quality and reduce calculation effort. Curve segmentation is applied to filter the line segments and reshape the different NURBS curve segments. Both position errors and the first derivative are considered to avoid over fitting of the NURBS curve. After the fitted NURBS curve is obtained, the corresponding interpolation for each interpolated time is further processed to obtain the motion command for each axis. The real-time NURBS curve fitting process consists of (1) discrete points import, (2) linear segment filter, (3) curve reshaping, (4) curve fitting considering both the position errors and the first derivative. Simulation results show the proposed approach effectively improves the fitting quality and calculation efficiency.

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.

Real-Time Visual Tracking of Dynamic Surgical Suture Threads.

In order to realize many of the potential benefits associated with robotically assisted minimally invasive surgery, the robot must be more than a remote controlled device. Currently, using a surgical robot can be challenging, fatiguing, and time consuming. Teaching the robot to actively assist surgical tasks, such as suturing, has the potential to vastly improve both patient outlook and the surgeon's efficiency. One obstacle to completing surgical sutures autonomously is the difficulty in tracking surgical suture threads. This paper presents novel stereo image processing algorithms for the detection, initialization, and tracking of a surgical suture thread. A Non Uniform Rational B-Spline (NURBS) curve is used to model a thin, deformable, and dynamic length thread. The NURBS model is initialized and grown from a single selected point located on the thread. The NURBS curve is optimized by minimizing the image matching energy between the projected stereo NURBS image and the segmented thread image. The algorithms are evaluated using suture threads, a calibrated test pattern, and a simulated thread image. Additionally, the accuracy of the algorithms presented are validated as they track a suture thread undergoing translation, deformation, and apparent length changes. All of the tracking is in real-time. Note to Practioners: Abstract-The problem of tracking a surgical suture thread was addressed in this work. Since the suture thread is highly deformable, any tracking algorithm must be robust to intersections, occlusions, knot tying, and length changes. The detection algorithm introduced in this paper is capable of distinguishing different threads when they intersect. The tracking algorithm presented here demonstrate that it is possible, using polynomial curves, to track a suture thread as it deforms, becomes occluded, changes length, and even ties a knot in real time. The detection algorithm can enhance directional thin features while the polynomial curve modeling can track any string like structure. Further integration of the polynomial curve with a feed-forward thread model could improve the stability and robustness of the thread tracking.

Collaborative optimization of NURBS curve cross-section in a telescopic boom

To improve the carrying capacity and reduce the weight of telescopic boom structure in a truck crane, a Collaborative optimization (CO) approach was applied to solve the problems of strength, stiffness and local stability in the telescopic boom structure. First, the complex optimization problem of the telescopic boom structure was decomposed into two-level optimizations: the system level and two subsystem levels for strength and local stability. Second, the underside curve of the boom’s cross-section was constructed by the Non-uniform rational B-Splines (NURBS) curve. 3D parametric solid model and the parametric finite element analysis model for the strength and the local stability were then established. Third, the mathematical models of the strength and local stability for the subsystem levels, and the system level were optimized, respectively. The adaptive relaxation factor algorithm and the penalty function approach were applied to improve the efficiency of CO. Next, the CO process which integrates the ANSYS package with ISIGHT platform was implemented. The optimal results show that the carrying capacity of the telescopic boom structure can be significantly improved and its weight efficiently is reduced. Finally, with the comparison of the stress values obtained from both the experimental test and the theoretical computation, highly coincident results could be obtained to verify the reliability of CO of a telescopic boom.

Servo-controlling structure of five-axis CNC system for real-time NURBS interpolating

NURBS (Non-Uniform Rational B-Spline) 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.

Topology optimization based on spline-based meshfree method using topological derivatives

Spline-based meshfree method (SBMFM) is originated from the Isogeometric analysis (IGA) which integrates design and analysis through Non-uniform rational B-spline (NURBS) basis functions. SBMFM utilizes trimming technique of CAD system by representing the domain using NURBS curves. In this work, an explicit boundary topology optimization using SBMFM is presented with an effective boundary update scheme. There have been similar works in this subject. However unlike the previous works where semi-analytic method for calculating design sensitivities is employed, the design update is done by using topological derivatives. In this research, the topological derivative is used to derive the sensitivity of boundary curves and for the creation of new holes. Based on the values of topological derivatives, the shape of boundary curves is updated. Also, the topological change is achieved by insertion and removal of the inner holes. The presented approach is validated through several compliance minimization problems.

An Efficient Approach of Time-Optimal Trajectory Generation for the Fully Autonomous Navigation of the Quadrotor

In this work, a time-optimal trajectory generation approach is developed for the multiple way-point navigation of the quadrotor based on the nonuniform rational B-spline (NURBS) curve and linear programming. To facilitate this development, the dynamic model of the quadrotor is formulated first. Then, the geometric trajectory regarding multiple way-point navigation is constructed based on the NURBS curve. With the constructed geometric trajectory, a time-optimal interpolation problem is imposed considering the velocity, acceleration, and jerk constraints. This optimization problem is solved in two steps. In the first step, a preliminary result is obtained by solving a linear programming problem without jerk constraints. Then by introducing properly relaxed jerk constraints, a second linear programming problem is formulated based on the preliminarily obtained result, and the time-optimal problem can be fully solved in this way. Subsequently, a nonlinear trajectory tracking controller is developed to track the generated trajectory. The feasibilities of the proposed trajectory generation approach as well as the tracking controller are verified through both simulations and real-time experiments. With enhanced computational efficiency, the proposed approach can generate trajectory for an indoor environment with the smooth acceleration profile and moderate velocity V≈1 m/s in real-time, while guaranteeing velocity, acceleration, and jerk constraints: Vmax=1 m/s, Amax=2 m/s2, and Jmax=5 m/s3. In such a case, the trajectory tracking controller can closely track the reference trajectory with cross-tracking error less than 0.05 m.

Transition from adjoint level set topology to shape optimization for 2D fluid mechanics

Abstract Most optimization problems in the field of fluid mechanics can be classified as either topology or shape optimization. Although topology and shape have been considered mutually exclusive optimization methods since their inception, it is conceivable that they will find choicest solutions in tandem, with shape optimization refining a solution found by topology. However, linking the topology optimization problem to that of shape is not trivial and, to the authors’ knowledge, has yet to be formally attempted. This paper proposes a novel transitional procedure that post-processes 2D adjoint topology solutions, fitting the interface between the solid and fluid topological domains to create a parameterized solution which can be used as either a CAD-compatible representation of the interface or a source for grid generation from which a shape optimization loop can be initialized. The interface to be fit can be extracted from any topological field with distinct fluid and solid domains, meaning that the proposed transition process is independent of the topology approach utilized. To conveniently describe the interface between the solid and fluid topological domains, the topology optimization process employed in this paper is filtered using the level set method. The interface is fit with non-uniform rational B-spline (NURBS) curves through application of sensitivities garnered from the solution of an auxiliary inverse design problem which aims at reducing the difference between signed-distance fields generated about both the NURBS curve being optimized and the section of interface being fit. The geometry defined by the fit NURBS curves is then (optionally) used to build a boundary-fitted grid on which a shape optimization loop is performed. The parameterized result of the topology to shape transition process is compared to that of shape optimization in 2D cases with internal, incompressible fluid flows.

Immunological Approach for Full NURBS Reconstruction of Outline Curves from Noisy Data Points in Medical Imaging.

Curve reconstruction from data points is an important issue for advanced medical imaging techniques, such as computer tomography (CT) and magnetic resonance imaging (MRI). The most powerful fitting functions for this purpose are the NURBS (non-uniform rational B-splines). Solving the general reconstruction problem with NURBS requires computing all free variables of the problem (data parameters, breakpoints, control points, and their weights). This leads to a very difficult non-convex, nonlinear, high-dimensional, multimodal, and continuous optimization problem. Previous methods simplify the problem by guessing the values for some variables and computing only the remaining ones. As a result, unavoidable approximations errors are introduced. In this paper, we describe the first method in the literature to solve the full NURBS curve reconstruction problem in all its generality. Our method is based on a combination of two techniques: an immunological approach to perform data parameterization, breakpoint placement, and weight calculation, and least squares minimization to compute the control points. This procedure is repeated iteratively (until no further improvement is achieved) for higher accuracy. The method has been applied to reconstruct some outline curves from MRI brain images with satisfactory results. Comparative work shows that our method outperforms the previous related approaches in the literature for all instances in our benchmark.

A new doubly-curved shell element for the free vibrations of arbitrarily shaped laminated structures based on Weak Formulation IsoGeometric Analysis

The main aim of the paper is to present a new numerical method to solve the weak formulation of the governing equations for the free vibrations of laminated composite shell structures with variable radii of curvature. For this purpose, the integral form of the stiffness matrix is computed numerically by means of the Generalized Integral Quadrature (GIQ) method. A two-dimensional structural model is introduced to analyze the mechanical behavior of doubly-curved shells. The displacement field is described according to the basic aspects of the general Higher-order Shear Deformation Theories (HSDTs), which allow to define several kinematic models as a function of the free parameter that stands for the order of expansion. Since an Equivalent Single Layer (ESL) approach is considered, the generalized displacements evaluated on the shell middle surface represent the unknown variables of the problem, which are approximated by using the Lagrange interpolating polynomials. The mechanical behavior of the structures is modeled through only one element that includes the double curvature in its formulation, which is transformed into a distorted domain by means of a mapping procedure based on the use of NURBS (Non-Uniform Rational B-Splines) curves, following the fundamentals of the well-known Isogeometric Analysis (IGA). For these reasons, the presented methodology is named Weak Formulation Isogeometric Analysis (WFIGA) in order to distinguish it from the corresponding approach based on the strong form of the governing equations (Strong Formulation Isogeometric Analysis or SFIGA), previously introduced by the authors. Several numerical applications are performed to test the current method. The results are validated for different boundary conditions and various lamination schemes through the comparison with the solutions available in the literature or obtained by a finite element commercial software.

Optimization of the target profile for asymmetrical rail grinding in sharp-radius curves for high-speed railways

Asymmetrical rail grinding in sharp-radius curves could reduce the side wear of railheads and enhance curve capacity of rail vehicles. The wheel/rail contact performance and curve capacity could be further improved by the optimization of the asymmetrical rail grinding target profile. In order to modify the target profile smoothly, the nonuniform rational B-spline curve with adjustable weight factors is used to establish a parameterized model of railhead curves in the asymmetrical grinding region. The indices of contact performance and curve capacity for different weight factors are obtained using experiment design and service performance simulation. Two Kriging surrogate models are proposed, in which the design variables are the adjustable weight factors, and the response parameters are the indices of contact performance and curve capacity, respectively. The multi-objective optimization model of the target profile is established, in which the objective functions are the two Kriging surrogate models of contact performance and curve capacity. The optimized weight factors are sought using a nondominated sorting genetic algorithm II, and the corresponding optimal target profile is obtained. The wheel/rail service performance simulation before and after optimization indicates that the contact performance and curve capacity are improved significantly.

A bisection method for the milling of NURBS mapping projection curves by CNC machines

The concept of non-uniform rational B-spline (NURBS) mapping projection curves (NURBS-MPCs) is proposed in this work. A NURBS-MPC is a projection curve on a NURBS surface of a NURBS curve. The bisection method is used to interpolate NURBS-MPCs. Using an assigned chord, the interpolation of a NURBS-MPC can be obtained easily, and the milling precision of the NURBS-MPC can be controlled effectively. Based on the NURBS theory, the bisection method, and the parametric programming method, an online NURBS software package (NURBS-SP) for FANUC 0i-MB/MC/MD CNC system and an offline NURBS toolbox (NURBS-T) for Matlab have been developed. Using an example of a planar NURBS curve, a NURBS-MPC is created on a NURBS surface. The simulation and milling of the NURBS-MPC show that the bisection method is feasible and effective. The online NURBS-SP endows the NURBS interpolation function for those CNC systems only equipped with linear interpolation (G01) and circular interpolation (G02/G03) and extends the interpolation functions and machining capability of low-middle level three-axis milling machines. The interactive application of the NURBS-T and NURBS-SP can accomplish the design, simulation, and milling for NURBS-MPCs. This feature makes them to have broad application prospects in CNC machining industry.

Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates

Abstract By means of Non-Uniform Rational B-Splines (NURBS) curves, it is possible to describe arbitrary shapes with holes and discontinuities. These peculiar shapes can be taken into account to describe the reference domain of several nanoplates, where a nanoplate refers to a flat structure reinforced with Carbon Nanotubes (CNTs). In the present paper, a micromechanical model based on the agglomeration of these nanoparticles is considered. Indeed, when this kind of reinforcing phase is inserted into a polymeric matrix, CNTs tend to increase their density in some regions. Nevertheless, some nanoparticles can be still scattered within the matrix. The proposed model allows to control the agglomeration by means of two parameters. In this way, several parametric studies are presented to show the influence of this agglomeration on the free vibrations. The considered structures are characterized also by a gradual variation of CNTs along the plate thickness. Thus, the term Functionally Graded Carbon Nanotubes (FG-CNTs) is introduced to specify these plates. Some additional parametric studies are also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. The validity of the current methodology is proven through a comparative assessment of our results with those available from the literature or obtained with different numerical approaches, such as the Finite Element Method (FEM). The strong form of the equations governing a plate is solved by means of the Generalized Differential Quadrature (GDQ) method.

Three-dimensional modeling and simulation of deformation behavior of fancy weft knitted stitch fabric

In order to obtain deformation behavior and volumetric characteristic of fancy weft knitted fabric, loop models are built on improved particle systems in this article. The problem of the non-uniform rational B-splines (NURBS) curves, which cannot pass through all control points, is solved by using an interpolation algorithm which can generate new auxiliary points. To simulate the twist of folded yarns, the NURBS curves are regarded as the geometric center, which is rotated with cylinders whose three relative Euler angles are calculated by the spatial coordinates of adjacent points. By analyzing the relationship between the deformation of the loop and the displacement of the particles, the deformation behavior of fancy weft knitted stitches is simulated. Velocity-Verlet, a numerical integration, is introduced to simulate fancy weft knitted stitches, and stable results are obtained. The results show that these models and algorithm accurately display the deformation behavior of fancy weft knitted stitches, as demonstrated by qualitative comparisons to measure the deformations of actual samples, and the simulator can scale up to animations with complex dynamic motion.

Isogeometric Analysis of the Nonlinear Deformation of Planar Flexible Beams with Snap-back

Based on the continuity of the derivatives of the Non-Uniform Rational B-Splines (NURBS) curve and the Jaumann strain measure, the present paper adopted the position coordinates of the control points as the degrees of freedom and developed a planar rotation-free Euler-Bernoulli beam element for isogeometric analysis, where the derivatives of the field variables with respect to the arc-length were expressed as the sum of the weighted sum of the position coordinates of the control points, and the NURBS basis functions were used as the weight functions. Furthermore, the concept of bending strip was used to involve the rigid connection between multiple patches. Several typical examples with geometric nonlinearities were used to demonstrate the accuracy and effectiveness of the proposed algorithm. The presented formulation fully accounts for the geometric nonlinearities and can be used to study the snap-through and snap-back phenomena of flexible beams.