Fine Tuning of Rational B-Spline Motions

1998 ◽  
Vol 120 (1) ◽  
pp. 46-51 ◽  
Author(s):  
L. N. Srinivasan ◽  
Q. Jeffrey Ge
Keyword(s):  
Computer Aided Design ◽  
Rate Of Change ◽  
Fine Tuning ◽  
Smoothing Algorithm ◽  
Third Order ◽  
B Spline ◽  
Key Points ◽  
Geometric Discontinuities ◽  
Aided Design ◽  
Three Space

This paper presents two algorithms for fine-tuning rational B-spline motions suitable for Computer Aided Design. The problem of fine-tuning of rational motions is studied as that of fine-tuning rational curves in a projective dual three-space, called the image curves. The path-smoothing algorithm automatically detects and smoothes out the third order geometric discontinuities in the path of a cubic rational B-spline image curve. The speed-smoothing algorithm uses a quintic rational spline image curve to obtain a second-order geometric approximation of the path of a cubic rational B-spline image curve while allowing specification of the speed and the rate of change of speed at the key points to obtain a near constant kinetic energy parameterization. The results have applications in Cartesian trajectory planning in robotics, spatial navigation in visualization and virtual reality systems, as well as mechanical system simulation.

Fine Tuning of Rational B-Splines Motions

1997 ◽  
Author(s):  
Lakshmi N. Srinivasan ◽  
Q. Jeffrey Ge
Keyword(s):  
Kinetic Energy ◽  
Computer Aided Design ◽  
Rate Of Change ◽  
Image Space ◽  
Fine Tuning ◽  
Smoothing Algorithm ◽  
B Spline ◽  
Spline Curves ◽  
Aided Design ◽  
Spatial Kinematics

Abstract This paper presents two algorithms for fine-tuning rational spatial motions suitable for Computer Aided Design. The rational motions are represented by rational B-spline curves in a projective dual three-space known as the Image Space of Spatial Kinematics. The problem of fine-tuning of rational motions is studied as that of fine-tuning the corresponding rational curves in the Image Space called the image curves. The path-smoothing algorithm automatically detects and smoothes out the third order geometric discontinuities in the path of a cubic rational Bspline image curve. The speed-smoothing algorithm uses a quintic rational spline image curve to obtain a second-order geometric approximation of the path of a cubic rational B-spline image curve while allowing specification of the speed and the rate of change of speed at the key points to obtain a near constant kinetic energy parametrization. The notion of kinetic energy is used in the paper as a natural way of combining the rotational and translational speed of a spatial motion. The results have applications in trajectory generation in robotics, planing of camera movement, spatial navigation in visualization and virtual reality systems, as well as mechanical system simulation.

Filling N-Sided Holes With Trimmed B-Spline Surfaces Based on Energy-Minimization Method

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Computer Aided Design ◽  
Control Points ◽  
Minimization Method ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints ◽  
Aided Design

Using a trimmed rectangular B-Spline surface to fill an n-sided hole is a much desired operation in computer aided design (CAD), but few papers have addressed this issue. Based on an energy-minimization or variational B-Spline technique, the paper presents the technique of using one single trimmed rectangular B-Spline surface to fill an n-sided hole. The method is efficient and robust, and takes a fraction of a second to fill n-sided holes with high-quality waterproof B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, the paper also presents the framework and addresses the key issues on variational B-Spline technique. Without any precalculation, the variational B-Spline technique discussed in this paper can solve virtually any B-Spline surface with up to 20,000 control points in real time, which is much more efficient and powerful than previous work in the variational B-Spline field. Moreover, the result is accurate and satisfies CAD systems' high-precision requirements.

Curve and Surface Representation by Iterative B-Spline Fit to a Data Point Set

1987 ◽  
Vol 16 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Marilyn Lord
Keyword(s):  
Computer Aided Design ◽  
Control Point ◽  
The Body ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Data Points ◽  
Point Set ◽  
Support Surfaces ◽  
Aided Design

The method of B-splines provides a very powerful way of representing curves and curved surfaces. The definition is ideally suited to applications in Computer Aided Design (CAD) where the designer is required to remodel the surface by reference to interactive graphics. This particular facility can be advantageous in CAD of body support surfaces, such as design of sockets of limb prostheses, shoe insoles, and custom seating. The B-spline surface is defined by a polygon of control points which in general do not lie on the surface, but which form a convex hull enclosing the surface. Each control point can be adjusted to remodel the surface locally. The resultant curves are well behaved. However, in these biomedical applications the original surface prior to modification is usually defined by a limited set of point measurements from the body segment in question. Thus there is a need initially to define a B-spline surface which interpolates this set of data points. In this paper, a computer-iterative method of fitting a B-spline surface to a given set of data points is outlined, and the technique is demonstrated for a curve. Extension to a surface is conceptually straightforward.

Review on Non Uniform Rational B-Spline (NURBS): Concept and Optimization

2014 ◽  
Vol 903 ◽  
pp. 338-343
Author(s):  
Ali Munira ◽  
Nur Najmiyah Jaafar ◽  
Abdul Aziz Fazilah ◽  
Z. Nooraizedfiza
Keyword(s):  
Image Processing ◽  
Literature Review ◽  
Computer Aided Design ◽  
B Spline ◽  
B Splines ◽  
Curves And Surfaces ◽  
Geometry Design ◽  
Nurbs Curves ◽  
Computer Aided ◽  
Aided Design

This paper is to provide literature review of the Non Uniform Rational B-Splines (NURBS) formulation in the curve and surface constructions. NURBS curves and surfaces have a wide application in Computer Aided Geometry Design (CAGD), Computer Aided Design (CAD), image processing and etc. The formulation of NURBS showing that NURBS curves and surfaces requires three important parameters in controlling the curve and also modifying the shape of the curves and surfaces. Yet, curves and surfaces fitting are still the major problems in the geometrical modeling. With this, the researches that have been conducted in optimizing the parameters in order to construct the intended curves and surfaces are highlighted in this paper.

Ideal Compliant Joints and Integration of Computer Aided Design and Analysis

2014 ◽  
Author(s):  
Ashraf M. Hamed ◽  
Paramsothy Jayakumar ◽  
Michael D. Letherwood ◽  
David J. Gorsich ◽  
Antonio M. Recuero ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Large Scale ◽  
Algebraic Equations ◽  
B Spline ◽  
Constraint Equations ◽  
Compliant Joints ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Computer Aided ◽  
Aided Design

This paper discusses fundamental issues related to the integration of computer aided design and analysis (I-CAD-A) by introducing a new class of ideal compliant joints that account for the distributed inertia and elasticity. The absolute nodal coordinate formulation (ANCF) degrees of freedom are used in order to capture modes of deformation that cannot be captured using existing formulations. The ideal compliant joints developed can be formulated, for the most part, using linear algebraic equations, allowing for the elimination of the dependent variables at a preprocessing stage, thereby significantly reducing the problem dimension and array storage needed. Furthermore, the constraint equations are automatically satisfied at the position, velocity, and acceleration levels. When using the proposed approach to model large scale chain systems, differences in computational efficiency between the augmented formulation and the recursive methods are eliminated, and the CPU times resulting from the use of the two formulations become similar regardless of the complexity of the system. The elimination of the joint constraint equations and the associated dependent variables also contribute to the solution of a fundamental singularity problem encountered in the analysis of closed loop chains and mechanisms by eliminating the need to repeatedly change the chain or mechanism independent coordinates. It is shown that the concept of the knot multiplicity used in computational geometry methods, such as B-spline and NURBS (Non-Uniform Rational B-Spline), to control the degree of continuity at the breakpoints is not suited for the formulation of many ideal compliant joints. As explained in this paper, this issue is closely related to the inability of B-spline and NURBS to model structural discontinuities. Another contribution of this paper is demonstrating that large deformation ANCF finite elements can be effective, in some MBS application, in solving small deformation problems. This is demonstrated using a heavily constrained tracked vehicle with flexible link chains. Without using the proposed approach, modeling such a complex system with flexible links can be very challenging. The analysis presented in this paper also demonstrates that adding significant model details does not necessarily imply increasing the complexity of the MBS algorithm.

Computer-Aided Design With Spatial Rational B-Spline Motions

1996 ◽  
Vol 118 (2) ◽  
pp. 193-201 ◽  
Author(s):  
B. Ju¨ttler ◽  
M. G. Wagner
Keyword(s):  
Computer Animation ◽  
Computer Aided Design ◽  
Control Structure ◽  
Basic Theory ◽  
Linear Control ◽  
Geometry Processing ◽  
B Spline ◽  
Computer Aided ◽  
Interpolation Techniques ◽  
Aided Design

Using rational motions it is possible to apply many fundamental B-spline techniques to the design of motions. The present paper summarizes the basic theory of rational motions and introduces a linear control structure for piecewise rational motions suitable for geometry processing. Moreover it provides algorithms for the calculation of the surface which is swept out by a moving polyhedron and examines interpolation techniques. The methods presented in this paper can be applied to various problems in computer animation as well as in robotics.

Computer-Aided Design of Curved Surfaces With Automatic Model Generation

1982 ◽  
Vol 104 (4) ◽  
pp. 817-824 ◽  
Author(s):  
S. M. Staley ◽  
R. B. Jerard ◽  
P. R. White
Keyword(s):  
Computer System ◽  
Computer Aided Design ◽  
Three Dimensional ◽  
Model Generation ◽  
Curved Surfaces ◽  
B Spline ◽  
Three Dimensional Objects ◽  
Computer Aided ◽  
Automatic Model Generation ◽  
Aided Design

The design and visualization of three-dimensional objects with curved surfaces have always been difficult. This paper describes a computer system that facilitates both the design and visualization of such surfaces. The system enhances the design of these surfaces by virtue of various interactive techniques coupled with the application of B-Spline theory. Visualization is facilitated by including a specially built model-making machine that produces three-dimensional foam models. Thus the system permits the designer to define an object and, with little additional effort, produce an inexpensive model of the object which is suitable for evaluation and presentation.

scholarly journals Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations

2018 ◽  
Vol 28 (10) ◽  
pp. 1899-1928 ◽  
Author(s):  
Stefan Takacs
Keyword(s):  
Error Estimates ◽  
Computer Aided Design ◽  
Convergence Rates ◽  
Approximation Error ◽  
Computational Domain ◽  
B Spline ◽  
B Splines ◽  
Multigrid Solvers ◽  
Aided Design ◽  
Real World Problems

In recent publications, the author and his coworkers have shown robust approximation error estimates for B-splines of maximum smoothness and have proposed multigrid methods based on them. These methods allow to solve the linear system arising from the discretization of a partial differential equation in Isogeometric Analysis in a single-patch setting with convergence rates that are provably robust both in the grid size and the spline degree. In real-world problems, the computational domain cannot be nicely represented by just one B-spline patch. In computer aided design, such domains are typically represented as a union of multiple patches. In this paper, we extend — for two-dimensional domains — the approximation error estimates and the multigrid solver to this multi-patch case.

Ideal Compliant Joints and Integration of Computer Aided Design and Analysis

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Ashraf M. Hamed ◽  
Paramsothy Jayakumar ◽  
Michael D. Letherwood ◽  
David J. Gorsich ◽  
Antonio M. Recuero ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Processing Unit ◽  
B Spline ◽  
Constraint Equations ◽  
Central Processing ◽  
Compliant Joints ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Computer Aided ◽  
Aided Design

This paper discusses fundamental issues related to the integration of computer aided design and analysis (I-CAD-A) by introducing a new class of ideal compliant joints that account for the distributed inertia and elasticity. The absolute nodal coordinate formulation (ANCF) degrees of freedom are used in order to capture modes of deformation that cannot be captured using existing formulations. The ideal compliant joints developed can be formulated, for the most part, using linear algebraic equations, allowing for the elimination of the dependent variables at a preprocessing stage, thereby significantly reducing the problem dimension and array storage needed. Furthermore, the constraint equations are automatically satisfied at the position, velocity, and acceleration levels. When using the proposed approach to model large scale chain systems, differences in computational efficiency between the augmented formulation and the recursive methods are eliminated, and the central processing unit (CPU) times resulting from the use of the two formulations become similar regardless of the complexity of the system. The elimination of the joint constraint equations and the associated dependent variables also contribute to the solution of a fundamental singularity problem encountered in the analysis of closed loop chains and mechanisms by eliminating the need to repeatedly change the chain or mechanism independent coordinates. It is shown that the concept of the knot multiplicity used in computational geometry methods, such as B-spline and NURBS (nonuniform rational B-spline), to control the degree of continuity at the breakpoints is not suited for the formulation of many ideal compliant joints. As explained in this paper, this issue is closely related to the inability of B-spline and NURBS to model structural discontinuities. Another contribution of this paper is demonstrating that large deformation ANCF finite elements can be effective, in some multibody systems (MBS) applications, in solving small deformation problems. This is demonstrated using a heavily constrained tracked vehicle with flexible-link chains. Without using the proposed approach, modeling such a complex system with flexible links can be very challenging. The analysis presented in this paper also demonstrates that adding significant model details does not necessarily imply increasing the complexity of the MBS algorithm.

Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Brian Tinsley ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Elements ◽  
Computer Aided Design ◽  
Eigenvalue Analysis ◽  
Reference Configuration ◽  
Position Vector ◽  
B Spline ◽  
Split Method ◽  
Stiffness Matrices ◽  
General Continuum ◽  
Aided Design

Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.

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