Non-uniform Rational B-spline Curves Research Articles

Optimization design of custom pattern based on microclimate heat transfer

PurposeThe aim of this paper is to achieve a reasonable microclimate between clothing and the human body and optimize the custom dress pattern.Design/methodology/approachAn interactive design method of 3D modeling, virtual try-on and heat transfer simulation are used. First, a 3D dress is designed with nonuniform rational B-splines curves and tried on virtually. After that, the heat transfer in the body-air-clothing microclimate and temperature distributions on the clothing surface are obtained. Based on the heat transfer in the body-air-clothing system, we design a method to improve the thermal comfort by optimizing the garment pattern digitally. Then, this paper utilized two heat transfer validating indexes to quantify the improvement of thermal comfort, and evaluate the modified model of dress.FindingsThe microclimate under the clothing is varied with the air gap distance, and the heat transfer on the area of the clavicle, bust point and front abandon are higher than other parts due to the narrow air gaps. In view of thermal comfort, the pattern optimization changes the distance ease and reforms the air circulating efficiency. The mean heat transfer and its standard variance are changing by about 10% and more than 20%. Thus, the heat transfer evaluation indexes are suitable to represent the heat transfer and thermal comfort in the microclimate system.Research limitations/implicationsIt can be concluded that the methodology proposed in this paper has the advantage of interactive design, 3D visualization and local heat transfer simulation. This technology meets the need of personalized customization and well-considered garment and has broad application prospects.Originality/valueThis study demonstrates that modifying the distance ease on body key girths based on heat transfer is a reliable way to improve thermal comfort. This method meets the consumers’ demand of the comfort of body-fit clothing under the condition of daily activities.

Small-sample Linear Profile Error Uncertainty Assessment Based On Grey System

Abstract The uncertainty assessment of the profile error of the cam profile, as defined in the national standard method, is difficult to carry out under conditions of small sample size and absence of probability distribution assumptions. This paper proposes a small-sample assessment model for the uncertainty of the profile error based on grey system. Firstly, the coordinate transformation is conducted using Vector Alignment Method（VAM）to reduce systematic errors, and the Non-Uniform Rational B-splines(NURBS)curve interpolation is utilized to fit the cam profile curve and perform error assessment. Subsequently, based on the error assessment results, Grey Information Measurement Model (GIMM) for the uncertainty of the profile error in small samples is established. This model employs Grey Relational Analysis (GRA) to eliminate outliers and evaluates the uncertainty of the profile error by solving grey correlation coefficients. Maximum-Minimum Information Measure Method (MIMM) is used to assess the optimal sample size. Finally, numerical experiments and experimental tests were conducted on the uncertainty of camshaft profile error in automobiles. A total of 15 sets of profile data were compared with Guide to the Representation of Uncertainty in Measurement(GUM) and Monte Carlo Method (MCM) under different sample sizes. The results showed that GIMM achieved evaluation with only 8 sets of data samples under small sample and poor information conditions, with an uncertainty of 0.6338, compared to 0.6346 for GUM and 0.6391 for MCM. The acceptance rate of GIMM reached 95.2%. This model outperforms other methods, providing a simplified and reliable assessment of cam profile error uncertainty.

Structural deformation dynamic simulation of weft knitted fabric basic stitch

To address the current problem of separating the structural and deformation models of weft knitted fabrics’ basic stitch, and to achieve synchronous simulation of the stitch’s structural form and deformation behavior, we studied the structural form and deformation behavior of the basic stitch, including weft plain stitch, rib stitch, and purl stitch. We established a three-dimensional structural model and conducted data analysis, improving the traditional mass-spring model to obtain a deformation model that fits better with the knitted fabric. We then investigated the connection between the structural and deformation models and constructed a correlation model, and used NURBS (Non-Uniform Rational B-Splines) curve application, force analysis, and dynamic solution to study the simulation of the correlation model. Finally, based on computer technology, we achieved dynamic simulation of the structural deformation of weft knitted fabrics’ basic stitch and demonstrated the simulation effect, proving the feasibility of the method and the rationality of the model.

Research on NURBS Curve Interpolation Algorithm Based on Mline-Gear

Aiming at the problem of speed fluctuation caused by the large error between the ideal interpolation step and the actual interpolation step in the traditional non-uniform rational B-spline ( NURBS ) curve interpolation, a NURBS curve interpolation algorithm based on Mline-Gear is proposed. The differential and derivative definitions are derived to transform the Mline and Gear formulas respectively, and the prediction and correction formulas of the interpolation parameters are obtained. Then, it is judged whether the interpolation parameter values obtained by the correction formula meet the speed fluctuation rate requirements, and the interpolation can be continued. Otherwise, the Steffensen descent formula is used to iteratively correct the interpolation parameter values. In order to ensure the calculation accuracy and avoid solving high-order derivatives, the fourth-order Runge-Kutta method is used to solve the initial two interpolation parameters. Finally, the algorithm is simulated. The simulation results show that the algorithm reduces the feed speed fluctuation rate while ensuring a small number of iterations.

Trajectory planning of tire laser engraving orthogonal robot based on an improved multi-objective grasshopper optimization algorithm

The traditional tire laser engraving robots have the low and stable running capability and discontinuous motion trajectory during operation, leading to long running time, high energy consumption, and uneven operation. In order to solve these problems, this paper proposes an orthogonal robot trajectory planning method based on an improved multi-objective grasshopper optimization algorithm (IMOGOA) with the orthogonal robot as the research object. Firstly, this paper used the quintic non-uniform rational b-splines (NURBS) curves to construct the trajectory. Secondly, the optimization objectives of the orthogonal robot’s running time, energy consumption, and smoothness were used to obtain the Pareto optimal solution by using the IMOGOA to establish the objective function. Then, the optimization method was simulated and analyzed, and the optimal solution was selected to compare with the rest of the non-dominated solution sets. The simulation results showed that the proposed method reduced the running time by 33.68%, improved the energy consumption and smoothness performance by 34.48% and 56.58%, and achieved multi-objective comprehensive optimization of the orthogonal robot. Thus, the effectiveness and feasibility of the proposed method were verified. Finally, a tire laser engraving orthogonal robot experimental platform was built for physical verification experiments. The experimental results showed that the experimental trajectory deviated from the simulated trajectory by only 5.63%, further proving the proposed method’s practicality.

Full-LSPIA: A Least-Squares Progressive-Iterative Approximation Method with Optimization of Weights and Knots for NURBS Curves and Surfaces

The Least-Squares Progressive-Iterative Approximation (LSPIA) method offers a powerful and intuitive approach for data fitting. Non-Uniform Rational B-splines (NURBS) are a popular choice for approximation functions in data fitting, due to their robust capabilities in shape representation. However, a restriction of the traditional LSPIA application to NURBS is that it only iteratively adjusts control points to approximate the provided data, with weights and knots remaining static. To enhance fitting precision and overcome this constraint, we present Full-LSPIA, an innovative LSPIA method that jointly optimizes weights and knots alongside control points adjustments for superior NURBS curves and surfaces creation. We achieve this by constructing an objective function that incorporates control points, weights, and knots as variables, and solving the resultant optimization problem. Specifically, control points are adjusted using LSPIA, while weights and knots are optimized through the LBFGS method based on the analytical gradients of the objective function with respect to weights and knots. Additionally, we present a knot removal strategy known as Decremental Full-LSPIA. This strategy reduces the number of knots within a specified error tolerance, and determines optimal knot locations. The proposed Full-LSPIA and Decremental Full-LSPIA maximize the strengths of LSPIA, with numerical examples further highlighting the superior performance and effectiveness of these methods. Compared to the classical LSPIA, Full-LSPIA offers greater fitting accuracy for NURBS curves and surfaces while maintaining the same number of control points, and automatically determines suitable weights and knots. Moreover, Decremental Full-LSPIA yields fitting results with fewer knots while maintaining the same error tolerance.

Research on the path extraction method of shoe upper grinding based on 3D vision

Shoe upper grinding is a crucial step in the shoe manufacturing process, impacting both the bonding quality between the sole and the upper and the overall appearance of the shoe. Presently, many shoe companies rely on manual or semi-automated grinding methods for shoe uppers, resulting in low production efficiency and subpar grinding quality. This study proposes a 3D vision-based shoe upper grinding path extraction method for guiding a robotic arm to polish the shoe upper and address this issue. Initially, the RGB-D camera was used to obtain the point cloud data of the shoe surface, which was then preprocessed using point cloud filtering, smoothing and registration methods. Second, by combining RGB images with 3D point clouds, we propose a point-cloud region of interest (ROI) extraction method based on RGB information to obtain the feature point cloud of the shoe upper grinding line. Finally, we slice the feature point cloud of the shoe upper grinding line, extract the grinding points using an intersection algorithm, and fit them with a non-uniform rational B-spline curve (NURBS) to obtain the shoe’s upper grinding path. The research demonstrates that this method can effectively extract the shoe upper grinding trajectory curve, significantly reduce the time required to obtain the shoe upper grinding trajectory, and achieve good accuracy, with promising application prospects.

Fast Reconstruction Model of the Ship Hull NURBS Surface with Uniform Continuity for Calculating the Hydrostatic Elements

The fast reconstruction of the ship hull nonuniform rational B-spline (NURBS) surface with uniform continuity is essential for calculating hydrostatic elements such as waterplane area and molded volume in real time. Thus, this study proposes a fast reconstruction model with uniform continuity to solve the problem of uniform continuity and splicing in the separate model of hull bow and stern surfaces. The proposed model includes the NURBS curve generation (UCG) algorithm with uniform continuity and the hybrid NURBS surface generation (HSG) algorithm. The UCG algorithm initially fits the feature points using the global interpolation algorithm and then precisely constructs straight-line segments in the curve using the improved flattening algorithm. In comparison, the HSG algorithm adaptively selects the surface knot vectors according to the parameters of the section curves. In this study, the profile of discontinuous compartments is uniformly expressed, effectively avoiding various articulation problems in separation modeling. The results of comparative experiments show that the NURBS surface generated using the HSG algorithm can accurately express the characteristics of various parts of the hull with uniform continuity, and the calculation speed of the proposed model can be increased by up to 8.314% compared with the existing best-performing algorithms. Thus, the proposed model is effective and can improve computational efficiency to a certain extent. The NURBS surfaces generated by the proposed model can be further applied to calculating the hydrostatic elements of hulls and compartments.

Modeling and Analysis of a Conical Bridge-Type Displacement Amplification Mechanism Using the Non-Uniform Rational B-Spline Curve.

This paper presents a new analytical model of a conical bridge-type displacement amplification mechanism (DAM) considering the effect of external loads and a piezostack actuator (PSA). With the merits of simple implementation and better fitting, the non-uniform rational B-spline (NURBS) is employed to parameterize conical connecting beams of the DAM, and an analytical model of the displacement amplification ratio and input stiffness is established based on Castigliano's second theorem. After that, considering the interactions with elastic loads and PSA, the actual displacement amplification ratio of the conical DAM is obtained, and the effect of the shape of connecting beams in the performance of the DAM is further analyzed. The proposed analytical model is verified by finite element analysis (FEA), and the results show a maximum error of 6.31% between the calculated value and FEA results, demonstrating the accuracy of the proposed model. A prototype of the conical DAM with optimized shape is fabricated and experimentally tested, which further validates the effectiveness and accuracy of the proposed analytical model. The proposed model offers a new method for analysis and shape optimization of the bridge-type DAM under specific elastic loads.

基于生物量的玉米可视化模型构建

Maize is one of the characteristic species in Shanxi Province. It has strong drought and cold tolerance. The study of maize growth and development law through the construction of the model is helpful to accurately understand the growth characteristics of maize, improve the yield, and further provide an important theoretical basis for the realization of precision agriculture. In this paper, "Xianyu 335" maize was taken as the research object. The data of leaf length, leaf width, stem length, stem number and tassel length, as well as leaf, stem, tassel, ear, cob and root organ morphology and biomass of maize during the whole growth period were obtained by field experiment. The maize geometric form structure module was constructed. The dynamic simulation of maize growing process was realized by object-oriented programming in Visual Studio software platform. With the help of 3D OpenGL graphic library Nurbs (Non-Uniform Rational B-Splines) curve and surface simulation technology, the growth visualization of maize organs was realized based on the idea of curve and surface control point selection parameter modeling.

A Feedrate Planning Method in CNC System Based on Servo Response Error Model

Reducing servo response error and further making reduction on contour error is crucial for high-precision computer numerical control (CNC) machine tools. For a permanent magnet synchronous motor (PMSM) servo system, there is always a response lag in feedrate tracking, which would introduce response error into the machining trajectory. Therefore, it is necessary to improve the performance of feedrate planning and interpolation for trajectory path. In this paper, a novel contour error compensation strategy is proposed. Compared with the mainstream methods, the proposed method offers a simplified alternative to existing contour error estimation techniques. Through a three-closed-loop control structure of a PMSM servo system, a response error model is founded. Afterwards, an improved S-model feedrate planning method is introduced according to the servo response error compensation. This predicted error is subsequently compensated in each interpolation cycle, resulting in a reduction of contour error. Finally, simulations and experiments are performed to demonstrate that the contour error can be reduced in both the ‘∞’-shaped Non-Uniform Rational B-Spline (NURBS) curve path and the butterfly-shaped NURBS curve path using the proposed method.

An isogeometric frictionless beam‐to‐beam contact formulation for hyperelastic Cosserat rods with unconstrained directors

AbstractWe present an isogeometric finite element formulation of frictionless beam‐to‐beam contact, using a Cosserat beam formulation with unconstrained directors. The beam's cross‐sectional deformation can be efficiently represented by director vectors. For the first order beam kinematics, Poisson locking due to the inability to represent linear in‐plane strain field in the cross‐section is alleviated by an enhanced assumed strain (EAS) method [1]. We employ a Gauss point‐to‐surface contact formulation combined with an active set iteration and a penalty method. The beam's cross‐sectional boundary as well as the axis is parameterized by a NURBS (non‐uniform rational B‐spline) curve, so that the lateral boundary surface has at least C2‐continuity, which yields a continuous surface metric and curvature in the closest point projection [2]. Since we do not employ zero stress conditions, a three‐dimensional nonlinear constitutive law can be straightforwardly utilized. We present several numerical examples, where a compressible Neo‐Hookean material is particularly considered.

Opposition-based learning boosted orca predation algorithm with dimension learning: a case study of multi-degree reduction for NURBS curves

AbstractThe approximate degree reduction of ball non-uniform rational B-splines (NURBS) curves is a pivotal and knotty technique in computer-aided design/computer-aided manufacture. As we all know, the multi-degree reduction of NURBS ones is a mathematical optimization problem that a swarm intelligence algorithm can deal with. This paper uses an improved orca predation algorithm (IOPA) to accomplish the optimal multi-degree reduction of NURBS curves. Firstly, by incorporating a dimension learning strategy and opposition-based learning strategy into the orca predation algorithm (OPA), an IOPA is developed to increase the population diversity and enhance its capability of jumping out of the local minima. Secondly, the superiority of the proposed IOPA is comprehensively verified by comparing it with the original OPA and numerous celebrated and newly developed algorithms on the IEEE Congress on Evolutionary Computation (CEC) 2014 test suite and IEEE CEC2017 benchmark functions, respectively. Meanwhile, the practicability of IOPA is also highlighted by solving three real-world engineering design problems. Furthermore, statistical testing of IOPA has been conducted to validate its significance. Finally, the optimization model of multi-degree reduction for NURBS curves is established by minimizing the distance between the original curve and the approximate curve. The IOPA is utilized to solve the optimization model, and the optimal approximate NURBS curves are obtained. Some representative numerical examples illustrate the ability of the proposed IOPA to effectively solve the multi-degree reduction problem of NURBS curves in terms of precision, robustness, and convergence characteristics.

A non‐uniform rational B‐splines enhanced finite element formulation based on the scaled boundary parameterization for the analysis of heterogeneous solids

AbstractThe contribution is concerned with a finite element formulation for the nonlinear analysis of heterogeneous solids in boundary representation. It results in an element with an arbitrary number of curved boundary edges. The curved edges can be parametrized by, for example, non‐uniform rational B‐splines (NURBS). The presented element formulation is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). In contrast to SBFEM, the proposed method uses a numerical approximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of interpolation in scaling direction. Thus, different basis functions can be used for each direction, for example, NURBS basis functions in circumferential and Lagrange basis functions in radial direction. It allows the construction of polygonal elements with an arbitrary number of curved sides, which are described by either whole NURBS curves or NURBS curves' segments. The advantage of the presented element formulation is the flexibility in mesh generation. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.

Automatic preform design and optimization for aeroengine disk forgings

To ensure a more uniform forging distribution microstructure and improve the service performance of aeroengine disk parts, an automated preform design method is proposed for integrated preform shape design and optimization based on the non-uniform rational B-spline (NURBS) curve, finite element method, and genetic algorithm (GA). First, the random preform shape graph is automatically constructed by the NURBS curve design criterion. The volume and shape complexity are used as the constraints of the preform. Second, the ratio of the mesh area within the set strain range to the total mesh area is used as the fitness function for the uniformity of deformation, and the GA is used for optimization. Finally, a large disk forging is an example of its optimal design. Results show that the deformation uniformity of the forgings is excellent, its fitness value is as high as 99.59%, and problems, such as folding, underfilling, and limited distribution of flash, do not exist, thereby verifying the effectiveness of the method. In addition, the method has the advantage of strong universality, that is, it can find the preform shape with good deformation uniformity for any shape forgings.

Jerk-Continuous Feedrate Optimization Method for NURBS Interpolation

Feedrate scheduling is one of the most critical technologies in CNC machining, requiring a reasonable balance between efficiency and quality. This paper proposes a jerk-continuous feedrate smoothing (JCFS) method to generate a low-vibration and smooth feedrate profile for non-uniform rational B-spline (NURBS) interpolation. Firstly, the segmentation concept is introduced to subdivide the entire trajectory into segments to accommodate curvature changes of the NURBS curve, accelerating the acceleration/deceleration process. Secondly, a length threshold-based curve segment classification method is proposed to overcome the complexity of the traditional acceleration and deceleration algorithms. The curve segments are divided into long, medium, and short types, and the length threshold calculation model is derived. Next, to avoid computational complexity for engineering applications, a model is established for the first time to calculate the actual maximum feedrate for different types of segments. Finally, the horizontal-8-shaped and butterfly-shaped NURBS curves are simulated and analyzed. The simulation results indicate that the machining quality is steadily improved while several key indicators remain within the given tolerances. Compared with the traditional method, the proposed method reduces the computational and interpolation time by 17.2% and 22.8%, respectively, demonstrating the feasibility and effectiveness of the method.

OPTIMISATION OF A SIX-DEGREE-OF-FREEDOM ROBOT TRAJECTORY BASED ON IMPROVED MULTI-OBJECTIVE PSO ALGORITHM, 218-230.

In order to improve the operation efficiency and mechanicalperformance of six-degree-of-freedom manipulators, an improvedmulti-objective particle swarm optimisation (IMPSO) algorithm isproposed to optimise the trajectory of the KUKA KR16 joint spacefitted with the quintic non-uniform rational B-splines (NURBS)curve.

A Novel Interpolation Method for Cutting Trajectories in Spatial Coherent Cutting Machine

Coherent lines are generally more complex spatial curves, and the study of coherent lines is the basis for studying cutting trajectories of spatial coherent cutting machines. In this paper, the spatial geometry problem is transformed into an algebraic problem solving by establishing a mathematical model of spatial coherent line to find its parametric equations. Interpolation refers to the process of data densification. In this paper, a speed planning method that adapts to the curvature change of the Non-Uniform Rational B-Splines ( NURBS ) curve is proposed. After inputting the starting points and endpoints to the Computer Numerical Control (CNC), the computer will calculate the coordinate data of each step according to the characteristics of the line segment, carry out the speed planning and then complete the curve trajectory interpolation. The accuracy of the algorithm is verified by comparing the theoretical coherent trajectory and interpolated trajectory.

Industrial Robot Contouring Control Based on Non-Uniform Rational B-Spline Curve

This paper presents a novel algorithm about the industrial robot contouring control based on the NURBS (non-uniform rational B-spline) curve. First, aiming at the error between the industrial robot’s actual trajectory and the desired trajectory, the contour error is proposed as the trajectory evaluation index, and the estimation algorithm of contour error based on the tangent approximation is proposed. Based on the tangent approximation algorithm, the estimation algorithm of contour error in the local task coordinate frame is proposed to realize the transformation from the Cartesian coordinate frame to the local task coordinate frame. Second, according to the configuration of the industrial robot, a modified cross-coupling control scheme based on the local task coordinate frame is designed. Finally, the Bernoulli’s lemniscate curves are constructed by NURBS curve and five-order polynomial curve, respectively, and they are symmetrical. The contrast experiment is designed using the two types of constructed Bernoulli’s lemniscate curves as the incentive trajectory. Through the analysis and comparison between the obtained uniaxial tracking error and the contour error curve of the two incentive trajectories, it is concluded that the incentive trajectory constructed by the NURBS curve has better contour control performance than that constructed by the five-order polynomial curve. The results drawn from this paper lay a certain foundation for the future high-precision contouring control of industrial robots.

S-Velocity Profile of Industrial Robot Based on NURBS Curve and Slerp Interpolation

This paper presents a novel algorithm for industrial robot trajectory planning based on the NURBS(Non-Uniform Rational B-Spline) curve and Slerp interpolation aiming at the problems that the trajectory of a six-axis industrial robot is not smooth enough in the operation process, the posture planning process is non-uniform, and the six-axis industrial robot starts and stops frequently. Firstly, aiming at the first problem, the trajectory planning algorithm based on the NURBS curve is presented to improve the smoothness of the trajectory curve. Combined with Slerp posture planning based on quaternion description, which realizes the uniform change of posture on the robot’s end-effector. Secondly, aiming at the second problem, the S-velocity planning algorithm is presented in the interpolation interval of the robot, which realizes the operation process of complex curves continuously, and improves the operation quality. Finally, this paper uses Bernoulli’s lemniscate as the incentive trajectory, and the contrast experiment of trajectory planning between two incentive profiles is designed, which are the NURBS curve and the five-order polynomial curve. The result of the experiment indicates that the planning algorithm proposed in this paper could effectively improve the smoothness of trajectory in a Cartesian workspace, decrease the impact and tremulous in a Cartesian workspace, and effectively improve the performance of the robot working process. The results drawn from this paper lay a certain foundation for the future high-precision control of industrial robots.