Non-uniform Rational B-spline Curves Research Articles

Comprehensive design method of a soft multifocal contact lens with NURBS.

This study presents a comprehensive method for designing a multifocal contact lens (MCL) with Snell's law and non-uniform rational B-spline (NURBS) curves. Instead of using thin lens approximation, general mathematical formulas have been developed to achieve the accurate coordinates of points on the anterior lens surface profile of the MCL to meet various given optical power distributions. Then the NURBS curve is adjusted to fit these data points to obtain the smooth front lens surface profile. This method not only improves the accuracy of the optical power profiles of MCLs but also reduces the spherical aberration in near/distance optical zones. The experimental results show that the power profiles of soft MCLs agree with those of the simulation results and original design requirements. The proposed method has been proven for the MCL design, and it can be feasibly applied in complex optical lens designs.

Investigation on the Bending Behavior of Tea Stalks Based on Non-Prismatic Beam with Virtual Internodes

The study aims to fully explicate the bending behavior of tea stalks under the condition of large deflection, which is crucial to improve the working performance of mechanized harvesting equipment. The mechanical model of the stalk was assumed to be a non-prismatic beam with virtual internodes that could differ from actual internodes. With the model, the stalk can be freely divided into multiple virtual internodes, whose flexural rigidities can be determined by solving an optimization problem, and deflection curves can be predicted after determining the positions of virtual nodes under given loads. Moreover, a novel method was proposed to obtain the deflection curve of the stalk based on the techniques of binocular vision and non-uniform rational B-spline (NURBS) curve fitting. The results show that R-squared values of fitted 2nd-degree NURBS curves of bending shape of tea stalks range from 0.9576 to 0.9964, with an average of 0.9797. The results indicate that flexural rigidity decreases from the bottom to the top of the tea stalk, and the deflection curve could be predicted more precisely with the model of piecewise flexural rigidities than that of average flexural rigidity. The study could be applied to the optimization design of the cutter and adaptive adjustment techniques of operational parameters for tea picking machines.

NURBS-Diff: A Differentiable Programming Module for NURBS

Boundary representations (B-reps) using Non-Uniform Rational B-splines (NURBS) are the de facto standard used in CAD, but their utility in deep learning-based approaches is not well researched. We propose a differentiable NURBS module to integrate NURBS representations of CAD models with deep learning methods. We mathematically define the derivatives of the NURBS curves or surfaces with respect to the input parameters (control points, weights, and the knot vector). These derivatives are used to define an approximate Jacobian used for performing the “backward” evaluation to train the deep learning models. We have implemented our NURBS module using GPU-accelerated algorithms and integrated it with PyTorch, a popular deep learning framework. We demonstrate the efficacy of our NURBS module in performing CAD operations such as curve or surface fitting and surface offsetting. Further, we show its utility in deep learning for unsupervised point cloud reconstruction and enforce analysis constraints. These examples show that our module performs better for certain deep learning frameworks and can be directly integrated with any deep-learning framework requiring NURBS.

Scanning Path Planning of the Robot for Breast Ultrasound Examination Based on Binocular Vision and NURBS

Ultrasonic scanning is the most commonly used method for detecting breast cancer. Traditional breast ultrasound mainly relies on manual examination, which requires doctors with rich clinical experience. The scanning path planning of the robot for breast ultrasound examination based on binocular vision and non-uniform rational B-Splines (NURBS) is proposed in this paper to solve the limitations of traditional examination methods. First, breast images are acquired by the binocular camera. The depth information of the images is obtained through stereo matching, and the 3D point cloud reconstruction of the breast model is completed. Second, the spatial index algorithm based on the K-dimension tree (Kd-tree) and uniform sampling method are selected to denoise and down-sample the point cloud model to complete the preliminary optimization of the model. The point cloud information for the scanning path is extracted according to the method of point cloud extraction and the principle of normal vector calculation. Then, the interpolation optimization of the scanning path based on the NURBS curve is completed. The optimized scanning path is transformed into the robot motion path. Finally, the robot motion is simulated and analyzed, and the experimental results show that the rationality of robot operation and the smoothness of robot motion meet the requirements of path planning defined in this paper. The error analysis experiment shows that the path planned by this method can ensure the efficiency and accuracy of ultrasound examination.

FC-NURBS curves: fullness control non-uniform rational B-spline curves

FC-NURBS curves: fullness control non-uniform rational B-spline curves

Automatic interpolation algorithm for NURBS trajectory of shoe sole spraying based on 7-DOF robot

PurposeThe purpose of this paper is to propose an automatic interpolation algorithm for robot spraying trajectories based on cubic Non-Uniform Rational B-Splines (NURBS) curves, to solve the problem of sparse and incomplete trajectory points of the head and heel of the shoe sole when extracting robot motion trajectories using structured-light 3D cameras and to ensure the robot joints move smoothly, so as to achieve a good effect of automatic spraying of the shoe sole with a 7-degree-of-freedom (DOF) robot.Design/methodology/approachFirstly, the original shoe sole edge trajectory position points acquired by the 3D camera are fitted with NURBS curves. Then, the velocity constraint at the local maximum of the trajectory curvature is used as the reference for curve segmentation and S-shaped acceleration and deceleration planning. Immediately, real-time interpolation is performed in the time domain to obtain the position and orientation of each point of the robot motion trajectory. Finally, the inverse kinematics of the anthropomorphic motion of the 7-DOF robot arm is used to obtain the joint motion trajectory.FindingsThe simulation and experiment prove that the shoe sole spraying trajectory is complete, the spraying effect is good and the robot joint movement is smooth, which show that the algorithm is feasible.Originality/valueThis study is of good practical value for improving the quality of automated shoe sole spraying, and it has wide applicability for different shoe sole shapes.

An interpolation method based on adaptive smooth feedrate scheduling and parameter increment compensation for NURBS curve

Compared with the traditional line/circle interpolation, non-uniform rational B-splines (NURBS) interpolation has more efficient and high-quality advantages in free-form curve/surface modeling. In order to obtain high efficiency and stable interpolation feed motion, a NURBS interpolation method with adaptive smoothing federate scheduling and parameter increment compensation is studied. Firstly, according to the curvature constraint, the adaptive feedrate is obtained and the segmentation of the NURBS curve is completed. Then, in order to avoid the problem of feeding speed overrun, the improved S-shape acceleration/deceleration control is adopted to adjust the jerk. The adjacent segments meeting the combined planning conditions are treated comprehensively. Secondly, aiming at the problem of velocity error in interpolation point parameter calculation, an interpolation parameter calculation method based on parameter increment compensation is proposed to reduce the velocity fluctuation and improve the interpolation point parameter calculation accuracy. Finally, the effectiveness of the proposed method is verified by simulation and experiment. The results show that the algorithm is effective.

EM Analysis of Geometrical Uncertainty for Metallic/Dielectric Bodies of Revolution Targets

An efficient EM scattering solution of geometrical uncertainty is fabricated to analyze the scattering from electrically large metallic/dielectric bodies of revolution (BoRs). The algorithm started by describing the geometrical shape of the BoR with the two-dimensional non-uniform rational B-spline curves. Then the relationship can be constructed between the local random variables and the spatial BoR basis functions. Thus, the surface integral equations were represented in terms of independent random variables. Radar cross section can be quickly obtained for perturbed geometrical shapes and the proposed algorithm can save computational resources when compared with theMonte Carlo (MC) method. The electrical field integral equation (EFIE) is applied to calculate the metallic BoR targets, whereas the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation is used to analyze the dielectric BoR targets. Numerical results are given to demonstrate the accuracy and efficiency of the proposed method.

Thermoelastic Mechanics Analysis Based on Isogeometric Boundary Element Method

Solid thermal stress caused by temperature has always been one of the important factors affecting the life of solid materials.The boundary element method, using boundary integrals instead of domain integrals, reduces the dimensionality of complex issue and the computational complexity. The non-uniform rational B-splines (NURBS) curve is used to construct the boundary of the calculation model, and the isogeometric idea is used to divide the unit, so as to overcome the geometric error caused by the unit discretization. Because of the existence of thermal stress, the constructed isogeometric boundary integral equation contains the domain integration term. The radial basis function method is used to convert the domain integration to the boundary integration. The advantage of the dimensionality reduction can be fully presented by using isogeometric boundary element method. The model calculation shows that the proposed calculation method is more reliable than the traditional boundary element method.

Parametric Design and Optimization of the Profile of Autonomous Underwater Helicopter Based on NURBS

Autonomous Underwater Helicopter (AUH) is a disk-shaped Autonomous Underwater Vehicle (AUV), and it has comparative advantage of near-bottom hovering and whole-direction turn-around ability over the traditional slender AUV. An optimization design of its irregular geometric profile is essential to improve its hydrodynamic performance. A parametric representation of its profile is proposed in this paper using Non-Uniform Rational B-spline (NURBS) curve. The parametric representation of AUH profile is described with two decision variables and several data points. Based on this parametric curve, Computational Fluid Dynamics (CFD) simulation is carried out to evaluate its hydrodynamic performance with various parameters. A predication model is established over variables’ design space using Kriging surrogate model with CFD simulation results and a Genetic Algorithm (GA) procedure is conducted to find optimal design variables, which can produce an optimum lift-drag ratio. CFD verification results confirm that AUH profile with optimized design variables can increase its lift-drag ratio by 2.11 times compared with that of non-optimized ones. It demonstrates that the parametric representation and optimization procedure of AUH profile proposed in this paper is feasible, and it has a great potential in improving AUH’s performance.

Two view NURBS reconstruction based on GACO model

The key job here in the presented work is to investigate the performance of Generalized Ant Colony Optimizer (GACO) model in order to evolve the shape of three dimensional free-form Non Uniform Rational B-Spline (NURBS) curve using stereo (two) views. GACO model is a blend of two well known meta-heuristic optimization algorithms known as Simple Ant Colony and Global Ant Colony Optimization algorithms. Basically, the work talks about the solution of NURBS-fitting based reconstruction process. Therefore, GACO model is used to optimize the NURBS parameters (control points and weights) by minimizing the weighted least-square errors between the data points and the fitted NURBS curve. The algorithm is applied by first assuming some pre-fixed values of NURBS parameters. The experiments clearly show that the optimization procedure is a better option in a case where good initial locations of parameters are selected. A detailed experimental analysis is given in support of our algorithm. The implemented error analysis shows that the proposed methodology perform better as compared to the conventional methods.

Velocity planning in multi-axis EDM based on a coder-player architecture

With a higher requirement on machining accuracy in electrical discharging machining (EDM), there is a drastically increased number of multi-axis line segments for machining complex workpieces. High speed jumping is needed for a conducive discharge gap status. The increased number of line segments, however, makes the implementation of high-speed jumping difficult. To alleviate the online computational burden for velocity planning, this paper proposes a multi-axis velocity planning based on a coder-player architecture. Under this architecture, a multi-axis feeding trajectory is represented by a series of double non-uniform rational B-spline (NURBS) curves, which are then interpolated by the unit arc length increment interpolation (UALII) method. Based on the interpolation points with a resolution of a basic length unit (BLU), a velocity limit curve is generated by considering simultaneously the constraints on chord error, acceleration and jerk. The velocity planning is implemented under a coder-player architecture. As a trajectory represented by a double-NURBS curve has C2 continuity, velocity losses can be avoided at junctions between line segments, thus smoothing of line segments at junctions is unnecessary in jumping. Machining tests showed that by the use of the velocity planning based on double-NURBS curves, the machining time for one channel of a shrouded blisk can be decreased by 29.38%.

Well-testing based turbidite lobes modeling using the ensemble smoother with multiple data assimilation

The representation of geological bodies is a difficult task, which involves a large number of parameters and assumptions that are commonly simplified in object-based modeling. A famous method that has been extensively applied for modeling geological bodies is the use of non-uniform rational B-Spline curves (NURBS) to delimit the boundaries of an object. Although NURBS provides highly detailed models, it only considers geological observations and assumptions. Moreover, the use of NURBS neglects information obtained from well-testing. This method also requires the complex and time-consuming process of determining the interior of the object in the parametric space. This is a classic problem in computational geometry, known as point location. To address these problems, this study proposes a well-testing-based object-based model of turbidite lobes using the ensemble smoother with multiple data assimilation. To escape the point location problem, we use single-valued B-Spline curves (SVBS) to build the turbidite system model. These curves are planar type B-Spline curves but they are defined as functions. The use of SVBS avoids the use of all complex algorithms and structures for solving the point location problem in the parametric space, if it is straightforward to decide if a point is in or out of the object. Consequently, it is possible to use the ensemble smoother with multiple data assimilation method to estimate the geometric parameters of the object, where a large number of realizations is required.

Boundary Element Modeling for Simulation and Optimization of Three-Temperature Anisotropic Micropolar Magneto-thermoviscoelastic Problems in Porous Smart Structures Using NURBS and Genetic Algorithm

The main objective of the present paper is to propose a new boundary element modeling technique for simulation and optimization of three-temperature micropolar magneto-thermoviscoelastic problems in anisotropic porous smart structures, where we implemented the genetic algorithm (GA), as a method of optimization based on the free form deformation (FFD) methodology to improve the performance of our proposed technique. Two numerical examples are presented herein, in order to prove that the proposed technique is able to optimize the shape of the domains with minimum computational effort. A nonuniform rational B-spline curve (NURBS) has been introduced to define the optimum boundary where it decreases the number of control points and offers a new degree of versatility in the design of various different shapes. The profiles of the items considered shall be represented by the FFD methodology. The location vectors of the FFD control points are known to be the genes, and then the chromosomes for the profiles are determined by the gene sequence. The population is made up of several chromosomes individuals, where the fitness functions of individuals are assessed using BEM. The numerical results are depicted graphical forms to show the effects of viscosity and magnetic fields on the three temperatures, displacement components, microrotation components, pore pressure, electric potential, and thermal stress components. The validity, accuracy, and computational efficiency of the proposed BEM technique were demonstrated by comparing our BEM-obtained results with the corresponding results of normal mode analysis method (NMAM), finite difference method (FDM), and finite element method (FEM).

Optimal Design of Elliptical Beam Cassegrain Antenna

An optimal design method of elliptical beam Cassegrain antenna based on differential evolution with global and local neighborhoods algorithm is proposed to achieve high efficiency, low sidelobe, and low profile at Ku/Ka dual band. A rapid shaping method of elliptical beam antenna is given by applying the algorithm of steepest descent. Constraining of the first sidelobe level, antenna efficiency is optimized with only 17 variables, where aperture field distribution functions in the major and minor axis planes are parameterized by Non-uniform rational B-spline curves, and the transition function is expressed with a sinusoidal function. A prototype of optimized antenna with an aperture of 630mm $\times1150$ mm is fabricated and measured. The measured results show that the total efficiency is greater than 54.5%, and the first sidelobe levels are lower than −14.2dB across the Ku/Ka dual-band, which agree well with simulated ones. It is verified that the optimal design method is very effective.

Real-time Simulation Methods for Robots with a Flexible Arm Based on Computer Graphics Technology

A robot with a flexible arm that is controlled with a remotely operated water-pressure mechanism has been developed for dismantling objects that are heavily contaminated by radioactive materials during decommissioning of nuclear power plants. The objective of this research is to develop a process planning support system and method that can improve the accuracy of estimating the time required by the robot to complete its dismantling activity, support recovery from delays, and determine the feasibility of conducting a dismantling process in the reactor building. Since the flexible arm has a more complex mechanism and shapes those that of multi-axis robots, unique movable arm structures were modeled with a tool for three-dimensional (3D) computer graphics (CG) technology. The 3D CG model was used to make valid operation sequences for planning the motion of the robot. With the help of a prototype system, motion planning can perform to calculate the duration needed for the robot to complete its operation. The calculated duration is then used for updating the planned duration of a specific activity in a dismantling schedule. To plan the robot behaviors for complex dismantling processes, a simplified planning method with few virtual controllers based on a spline inverse kinematics (IK) with a non-uniform rational B-spline (NURBS) curve for an arm comprised of pistons and cylinders pressurized by water was studied. A prototype system for planning the behaviors of the robot was evaluated, and it was confirmed that the movement trajectory of the robot and the three-dimensional isometric display could be visualized using the mesh model generated from point-cloud data used to make the environment model of the robot. It was also confirmed that the operations involved in a specific activity of the robot could be completed within the duration determined in the simulation.

NURBS Modeling and Curve Interpolation Optimization of 3D Graphics

In order to solve the problem of complicated Non-Uniform Rational B-Splines (NURBS) modeling and improve the real-time performance of the high-order derivative of the curve interpolation process, the method of NURBS modeling based on the slicing and layering of triangular mesh is introduced. The research and design of NURBS curve interpolation are carried out from the two aspects of software algorithm and hardware structure. Based on the analysis of the characteristics of traditional computing methods with Taylor series expansion, the Adams formula and the Runge-Kutta formula are used in the NURBS curve interpolation process, and the process is then optimized according to the characteristics of NURBS interpolation. This can ensure accuracy, and avoid the calculation of higher-order derivatives. Furthermore, the hardware modules for the Adams and Runge-Kutta formulas are designed by using the parallel hardware construction technology of Field Programmable Gate Array (FPGA) chips. The parallel computing process using FPGA is compared with the traditional serial computing process using CPUs. Simulation and experimental results show that this scheme can improve the computational speed of the system and that the algorithm is feasible.

Chord error constraint based integrated control strategy for contour error compensation

As the traditional cross-coupling control method cannot meet the requirements for tracking accuracy and contour control accuracy in large curvature positions, an integrated control strategy of cross-coupling contour error compensation based on chord error constraint, which consists of a cross-coupling controller and an improved position error compensator, is proposed. To reduce the contour error, a PI-type cross-coupling controller is designed, with its stability being analyzed by using the contour error transfer function. Moreover, a feed rate regulator based on the chord error constraint is proposed, which performs speed planning with the maximum feed rate allowed by the large curvature position as the constraint condition, so as to meet the requirements of large curvature positions for the chord error. Besides, an improved position error compensation method is further presented by combining the feed rate regulator with the position error compensator, which improves the tracking accuracy via the advance compensation of tracking error. The biaxial experimental results of non-uniform rational Bsplines curves indicate that the proposed integrated control strategy can significantly improve the tracking and contour control accuracy in biaxial contour following tasks.

A NURBS curve interpolator with small feedrate fluctuation based on arc length prediction and correction

Non-uniform rational B-spline (NURBS) curve has been widely used in manufacturing systems. A good interpolator can help the system to improve the contour accuracy and get smooth dynamics performance, but it is hard to get a balance between the interpolation performance and computational load. As the derivative and curvature of NURBS curves used in manufacturing systems are high-order continuous, it is possible to predict a desired interpolation arc length based on the relationship of historical feed chord length and its corresponding arc length in one interpolation cycle. Therefore, this paper proposes a novel interpolation method, which consists three stages. Firstly, a NURBS curve is split into several high-order continuous segments based on its degrees and control points. Secondly, a prediction model based on Newton’s divided differences interpolation equation is derived from the relationship of interpolated chord length and its corresponding arc length, so that the target arc length of the next interpolation cycle can be predicted. Finally, target parameter u of every interpolation cycle is calculated with Taylor’s expansion, whose values are corrected by iteration, and then the position of the target interpolation point can be achieved. Performance of the proposed algorithm is tested and compared with other methods, and the simulation results show the proposed method can achieve smaller velocity fluctuation with low computational load.

Dynamic multi-patch isogeometric analysis of planar Euler–Bernoulli beams

This study presents a novel isogeometric Euler–Bernoulli beam formulation for in-plane dynamic analysis of multi-patch beam structures. The kinematic descriptions involve only displacements of the beam axis, which are approximated by non-uniform rational B-spline (NURBS) curves. Translational displacements of the control points are here considered as control variables. The motivation of this work is to propose a penalty-free method to handle in-plane dynamic analysis of multi-patch beam structures. A simple relation between cross-sectional rotations at the ends of the beams and control variables is derived, allowing the incorporation of the end rotations as degrees of freedom. This improved setting can straightforwardly tackle beam structures with many rigid multi-patch connections, a major challenging issue when using existing isogeometric Euler–Bernoulli beam formulations. Additionally, rotational boundary conditions are conveniently prescribed. Numerical examples with complicated beam structures such as circular arches and frames with kinks are considered to show the accuracy and performance of the developed formulation. The computed results are verified with those derived from the conventional finite element method, and the superior convergence properties of the proposed formulation are illustrated. A possible extension of the present approach to spatial beam structures is discussed.