Extraction of Feature Points for Non-Uniform Rational B-Splines(NURBS)-Based Modeling of Human Legs

Methods of digital human modeling have been developed and utilized to reflect human shape features.However,most of published works focused on dynamic visualization or fashion design,instead of high-accuracy modeling,which was strongly demanded by medical or rehabilitation scenarios.Prior to a high-accuracy modeling of human legs based on non-uniform rational B-splines(NURBS),the method of extracting the required quasi-grid network of feature points for human legs is presented in this work.Given the 3 D scanned human body,the leg is firstly segmented and put in standardized position.Then re-sampling of the leg is conducted via a set of equidistant cross sections.Through analysis of leg circumferences and circumferential curvature,the characteristic sections of the leg as well as the characteristic points on the sections are then identified according to the human anatomy and shape features.The obtained collection can be arranged to form a grid of data points for knots calculation and high-accuracy shape reconstruction in future work.

An Approach to Representing Heterogeneous Non-Uniform Rational B-Spline Objects

The representation method of heterogeneous material information is one of the key technologies of heterogeneous object modeling, but almost all the existing methods cannot represent non-uniform rational B-spline （NURBS） entity. According to the characteristics of NURBS, a novel data structure, named NURBS material data structure, is proposed, in which the geometrical coordinates, weights and material coordinates of NURBS heterogene- ous objects can be represented simultaneously. Based on this data structure, both direct representation method and inverse construction method of heterogeneous NURBS objects are introduced. In the direct representation method, three forms of NURBS heterogeneous objects are introduced by giving the geometry and material information of con- trol points, among which the homogeneous coordinates form is employed for its brevity and easy programming. In the inverse construction method, continuous heterogeneous curves and surfaces can he obtained by interpolating discrete points and curves with specified material information. Some examples are given to show the effectiveness of the pro- posed methods.

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

Shape Control and Modification of Rational Cubic B-Spline Curves

This paper considers the construction of a rational cubic B-spline curve that willinterpolate a sequence of data points x'+ith specified tangent directions at those points. It is emphasisedthat the constraints are purely geometrical and that the pararnetric tangent magnitudes are notassigned as in many' curl'e manipulation methods. The knot vector is fixed and the unknowns are thecontrol points and x'eightsf in this respect the technique is fundamentally different from otherswhere knot insertion is allowed.First. the theoretical result3 for the uniform rational cubic B-spline are presented. Then. in theplanar case. the effect of changes to the tangent at a single point and the acceptable bounds for thechange are established so that all the weights and tangent magnitUdes remain positive. Finally, aninteractive procedure for controlling the shape of a planar rational cubic B-spline curve is presented.

A New Isogeometric Finite Element Method for Analyzing Structures

High-performance finite element research has always been a major focus of finite element method studies.This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method.Firstly,the physical field is approximated by uniform B-spline interpolation,while geometry is represented by non-uniform rational B-spline interpolation.By introducing a transformation matrix,elements of types C^(0)and C^(1)are constructed in the isogeometric finite element method.Subsequently,the corresponding calculation formats for one-dimensional bars,beams,and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping.The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis,eliminating the need for mesh generation and maintaining flexibility in element construction.Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method.Finally,the test results of several examples show that:(1)Under the same degree and element node numbers,the constructed elements are almost consistent with the results obtained by traditional finite element method;(2)For bar problems with large local field variations and beam problems with variable cross-sections,high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method;(3)The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom,while as degrees of freedom increase,the computational efficiency between the two is similar.

An adaptive generation method for free curve trajectory based on NURBS

To realize the high precision and real-time interpolation of the NURBS （non-uniform rational B-spline） curve, a kinetic model based on the modified sigmoid function is proposed. The constraints of maximum feed rate, chord error, curvature radius and interpolator cycle are discussed. This kinetic model reduces the cubic polynomial S-shape model and the trigonometry function S-shape model from 15 sections into 3 sections under the precondition of jerk, acceleration and feedrate continuity. Then an optimized Adams algorithm using the difference quotient to replace the derivative is presented to calculate the interpolator cycle parameters. The higher-order derivation in the Taylor expansion algorithm can be avoided by this algorithm. Finally, the simplified design is analyzed by reducing the times of computing the low-degree zero-value B-spline basis function and the simplified De Boor-Cox recursive algorithm is proposed. The simulation analysis indicates that by these algorithms, the feed rate is effectively controlled according to tool path. The calculated amount is decreased and the calculated speed is increased while the machining precision is ensured. The experimental results show that the target parameter can be correctly calculated and these algorithms can be applied to actual systems.

Methods of configuration test and deformation analysis for large airship

In recent years, high-altitude aerostats have been increasingly developed in the direction of multi-functionality and large size. Due to the large size and the high flexibility, new challenges for large aerostats have appeared in the configuration test and the deformation analysis. The methods of the configuration test and the deformation analysis for large airship have been researched and discussed. A tested method of the configuration,named internal scanning, is established to quickly obtain the spatial information of all surfaces for the large airship by the three-dimensional(3D) laser scanning technology. By using the surface wrap method, the configuration parameters of the large airship are calculated. According to the test data of the configuration, the structural dimensions such as the distances between the characteristic sections are measured. The method of the deformation analysis for the airship contains the algorithm of nonuniform rational B-splines(NURBS) and the finite element(FE)method. The algorithm of NURBS is used to obtain the reconfiguration model of the large airship. The seams are considered and the seam areas are divided. The FE model of the middle part of the large airship is established. The distributions of the stress and the strain for the large airship are obtained by the FE method. The position of the larger deformation for the airship is found.

Spline-based automatic path generation of welding robot

This paper presents a flexible method for the representation of welded seam based on spline interpolation. In this method, the tool path of welding robot can be generated automatically from a 3D CAD model. This technique has been implemented and demonstrated in the FANUC Arc Welding Robot Workstation. According to the method, a software system is developed using VBA of SolidWorks 2006. It offers an interface between SolidWorks and ROBOGUIDE, the off-line programming software of FANUC robot. It combines the strong modeling function of the former and the simulating function of the latter. It also has the capability of with on-line robot. The result data have shown its high accuracy and strong reliability in experiments. This method will improve the intelligence and the flexibility of the welding robot workstation.

DESCRIPTIONS OF THE CYLINDRICAL HELIX WITH NURBS

The representation of a cylindrical helix by Non-Uniform Rational B-Spline (NURBS) curves is presented in this paper. A method is proposed to assess the influences produced by different ways to determine the control ver-texes positions of helix. The error distribution cases between the helix approximated by NURBS curves and the original theoretical one are also analyzed. Meanwhile a computational method that guarantees the precision requirements is presented.

FREE-FORM SURFACE RECON-STRUCTION BASED ON NURBS TO SERIAL CROSS-SECTIONS

A new method for recovering shape from cross-sectional contours with complexbranching structures is presented. First, each branching problem by providing an intermediatecontour using distance function and image processing technology is solved. Then, all contours aredivided into several groups of simple contours. For each group, a NURBS curve is fitted to contourpoints in each section within a given accuracy on a common knot vector. Finally, the NURBS surfaceskinning of these contours is performed for providing a smooth geometric model. The method issuitable to reproduce the object by NC machining or rapid prototyping. Some results demonstrate itsusefulness and feasibility.

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.

G^2 Continuity Algorithms between Adjacent NURBS Patches along Common Cubic Boundary Curve

Non-uniform rational B-spline (NURBS) curves and surfaces are becoming increasingly widespread. The author have explored G^1 continuity condition between adjacent NURBS surface patches along common cubic boundary curve. On the basis of the research performed, this paper presents a G^2 continuity condition between adjacent NURBS patches along common cubic boundary curve and deduces a specific algorithm for contro1 points and weights of NURBS patch. For making another NURBS patch and one given NURBS patch to attain G^2, according to algorithm condition, one can adjust another patch control points and weights. It is much more convenient for engineers to apply.

Study on Real-time Look-ahead and Adaptive Parametric Curve Interpolator

The principle of real-time look-ahead was introduced and analysed. An adaptive parametric curve interpolator with a real-time look-ahead function was developed for non-uniform rational B-spline (NURBS) curves interpolation, which considering the maximum acceleration/deceleration of the machine tool. In order to deal with the acceleration/deceleration around the feedrate sensitive corners, the look-ahead function was designed and illustrated. It can detect and adjust the feedrate adaptively. With the help of real-time look-ahead, the acceleration/deceleration can be limited to the range of the machine tool capacity. Thus, feedrate fluctuation is reduced. A NURBS curve interpolation experiment was provided to verify the feasibility and advantages of the proposed interpolator with a real-time look-ahead function.

PRACTICAL G^2 CONTINUITY ALGORITHMS BETWEEN ADJACENT NURBS PATCHES

Non-uniform rationa1 B-spline (NURBS) curves and sdrices are becomingincreasingly widespread. 'The authors have explored G1 continuity condition between adja-cent NURBS surface patches along common boundary curve. This paper presents a G2continuity condition between adjacent NURBS patches along common quadratic boundarycurve and deduces a specific algorithm for control Points and weights of NURBS patch.For making another NURBS patch and one given NURBS patch to attain G2, according toalgorithms condition, one can adjust another patch control ponts and weights. It is muchmore convenient for engineers to apply.

Robust fault diagnosis for non-Gaussian stochastic systems based on the rational square-root approximation model

The task of robust fault detection and diagnosis of stochastic distribution control （SDC） systems with uncertainties is to use the measured input and the system output PDFs to still obtain possible faults information of the system. Using the rational square-root B-spline model to represent the dynamics between the output PDF and the input, in this paper, a robust nonlinear adaptive observer-based fault diagnosis algorithm is presented to diagnose the fault in the dynamic part of such systems with model uncertainties. When certain conditions are satisfied, the weight vector of the rational square-root B-spline model proves to be bounded. Conver- gency analysis is performed for the error dynamic system raised from robust fault detection and fault diagnosis phase. Computer simulations are given to demon- strate the effectiveness of the proposed algorithm.

An efficient data structure for calculation of unstructured T-spline surfaces

To overcome the topological constraints of non-uniform rational B-splines,T-splines have been proposed to define the freeform surfaces.The introduction of T-junctions and extraordinary points makes it possible to represent arbitrarily shaped models by a single T-spline surface.Whereas,the complexity and flexibility of topology structure bring difficulty in programming,which have caused a great obstacle for the development and application of T-spline technologies.So far,research literatures concerning T-spline data structures compatible with extraordinary points are very scarce.In this paper,an efficient data structure for calculation of unstructured T-spline surfaces is developed,by which any complex T-spline surface models can be easily and efficiently computed.Several unstructured T-spline surface models are calculated and visualized in our prototype system to verify the validity of the proposed method.

RESEARCH AND REALIZATION OF MUTUAL TRIMMING OF NURBS

Since regular surface is defined by a mapping from 2-D parameter plane to 3-D space, trimming of NURBS surface is equivalent to changing a valid parameter region ofthe surface with the same mapping. In this paper, by presenting a rigid and comprehensi-ble definition of union, difference and intersection of two intersecting loops in the parame-ter region of NURBS surface, and by working out the corresponding algorithm, a rebui1d-ing algorithm of the valid parameter region of NURBS surface is obtained.

Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities

In the paper [1], the geometrical mapping techniques based on Non-Uniform Rational B-Spline (NURBS) were introduced to solve an elliptic boundary value problem containing a singularity. In the mapping techniques, the inverse function of the NURBS geometrical mapping generates singular functions as well as smooth functions by an unconventional choice of control points. It means that the push-forward of the NURBS geometrical mapping that generates singular functions, becomes a piecewise smooth function. However, the mapping method proposed is not able to catch singularities emerging at multiple locations in a domain. Thus, we design the geometrical mapping that generates singular functions for each singular zone in the physical domain. In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in C0?space. Also, we modify the B-spline functions whose supports include the interface between/among them. We put the idea in practice by solving elliptic boundary value problems containing multiple singularities.

Parametric Selection and Continual Combination of Building Surface

A method to reparametrize G retional curve to obtain a C^1 curve is given. A practical G^1 continual connective between adjacent NURUS patches along common guadratic boundary curve is presented in this paper, and a specific algorithm for control points and weights of NURBS patches is discussed.

NEW DESIGN METHOD FOR HYPOID GEARS

A digital model is presented for the purpose of design, manufacture and measurement of hypoid gear, based on the non-uniform rational B-spline surface （NURBS） method. The digital model and the function-oriented active design technique are combined to form a new design method for hypoid gears. The method is well adaptable to CNC bevel gear cutting machines and CNC-controlled gear inspection machines, and can be used to create the initial machine tool cutting location data or program measurement path. The presented example verifies the method is correct.