On swept volume formulations: implicit surfaces

doi:10.1016/S0010-4485(00)00065-8

Computer-Aided Design

Volume 33, Issue 1, January 2001, Pages 113-121

https://doi.org/10.1016/S0010-4485(00)00065-8 Get rights and content

Abstract

Recent developments in the formulations for generating swept volumes have made a significant impact on the efficiency of employing such algorithms and on the extent to which formulations can be used in representing complex shapes. In this paper, we outline a method for employing the representation of implicit surfaces using the Jacobian rank deficiency condition presented earlier for the sweep of parametric surfaces. A numerical and broadly applicable analytic formulation is developed that yields the exact swept volume.

Introduction

Swept volumes have become increasingly important in modern computer-aided design, because of the need to represent the trace of objects that have experienced motion. Such applications are numerous and encompass solid modeling, manufacturing automation, robot analysis, collision detection, and computer graphics. In this report we present new results pertaining to the sweep of implicit surfaces using the Jacobian rank deficiency conditions [1], [2], [3], [4], [5]. Recent work published in the special edition of computer-aided design [6], dedicated to the subject of swept volumes, has outlined a method for trimming swept volumes.

For a comprehensive review of prior work in this field, the reader is referred to Abdel-Malek and Yeh [1]. The work presented in Ref. [6] is the continuation of a robust method introduced in many reports [7], [8], [9], [10] that employs the concept of a differential sweep equation. While the method is aimed at computing trimming curves in order to remove points that are in the interior of the boundary of the swept volume, the technique by which the authors have formulated the problem is of interest. Surfaces and solids, implicitly represented, were considered in the sweep equation. We shall adapt this same representation, but we will apply the Jacobian rank deficiency method (never before used for implicitly defined equations) to obtain singular sets. These sets will describe (implicitly) surfaces that exist on the exterior and in the interior of the swept volume. Combining these surfaces (referred to as singular surfaces) yields an exact representation of the swept volume.

Although parametric representations are more widely used and are generally much simpler to treat than implicit representations, we focus our attention in this report on the latter. Swept volumes are extensively used in many fields (Solid Modeling in CAD is perhaps the most common) and implicit representations are commonly utilized to represent surfaces because of their simplicity to use and store. In such cases, a CAD program must have the ability to perform sweep operations on the implicitly represented solid without converting the equation to parametric form. Such an example is given below, characterized by the modeling and representation of a mechanical spring element, generated by the sweep of a solid sphere along a helical curve.

The Jacobian rank deficiency method was developed for parametric surfaces [1], [2], [3], [4] and was shown to handle self-intersections, multiple parameter sweeping, and solid property computations. The method was never applied to implicitly defined surfaces, because of the lack of formulations for proper representation of such surfaces in a consistent numerical algorithm. This report will outline a systematic method for computing the swept volume of implicit surfaces. The uniqueness of the method is in determining the swept volume generated by the sweep of a solid (represented in implicit form) along a curve (characterized as a function of one parameter). Prior to this work, there has never been any published results addressing the computation of implicit swept volumes with more than three parameters. In fact, the only paper that has dealt with a related subject to the sweep of implicit solids is that of Hu and Ling [11], where they have addressed the sweeping of the natural quadric surfaces (i.e. the result is still limited to three parameters and only deals with quadric surfaces).

In order to compare with the method recently presented by Blackmore et al. [6], we shall treat one of the same self-intersecting examples addressed therein. We will first address the general problem using the Jacobian rank deficiency method. It will be shown that self-intersections are inherently considered and do not require additional trimming computations.

Section snippets

Formulation for implicit sweeps

In order to remain consistent with the notation used in Blackmore et al. [6], where a smooth rigid sweep σ may be represented in the form $σ (w)=ξ(t)+ A (t) x$ with x=[x₁,…,x_M]^T and $w =[x^{T} t]^{T},$ and where ξ and A are, respectively, smooth (=C^∞) vector and matrix valued functions with ξ(0)=0, A(0)=I the identity matrix, and A(t) is a real (3×3) orthonormal matrix (when M=3) for every t∈[0,1]. For the purpose of developing the formulation, it is required only that the object being swept be defined as the

Illustrative example

Consider the solid treated in Ref. [6] using the SEDE trimming approach. Let W be the solid cylinder representing a flat-end machine tool in space defined by $W={x : − 14 ≤x_{1}^{2} +x_{2}^{2} − 14 ≤0, −1≤x_{3} ≤1}$

The goal is to compute the swept volume of W generated by the sweep $ξ(x,t)= 0 4t 0 + R (t) x_{1} x_{2} x_{3}$ where R is a basic rotation matrix about the x-axis defined by $R (t)= 1000 cos (π t) − sin (π t) 0 sin (π t) cos (π t)$ and where the sweep parameter t is bounded as 0≤t≤1. The sweep can be written as $ξ(z)= x_{1} 4t+x_{2} cos (π t)−x_{3} sin (π t) x_{2} sin (π t)+x_{3}$

Conclusions

The method based on a Jacobian rank deficiency condition developed for computing the sweep of parametric surfaces was expanded in this report to a broadly applicable formulation for implicit surfaces. This was accomplished by reformulating the parametrization method in implicit form.

Note that in the foregoing analysis, neither the dimension of x nor the number of inequality expressions has been limited. As a result, the formulation is valid for n-dimensional manifolds. Also note that the

Karim Abdel-Malek is Assistant Professor of Mechanical Engineering at the University of Iowa. He received his BS degree in ME from the University of Jordan, his MS and PhD from the University of Pennsylvania, both in Mechanical Engineering. He was a Fulbright scholar for two years at the University of Pennsylvania. Dr Abdel-Malek was a consultant for the US manufacturing industry for a number of years before joining the faculty at Iowa. He is a recipient of the SME T.J. Parsons Outstanding

References (11)

K. Abdel-Malek et al.
Geometric representation of the swept volume using Jacobian rank deficiency conditions
Computer-Aided Design
(1997)
K. Abdel-Malek et al.
Swept volumes: void and boundary identification
Computer-Aided Design
(1998)
K. Abdel-Malek et al.
Multiple sweeping using the Denavit–Hartenberg representation method
Computer Aided Design
(1999)
D. Blackmore et al.
Analysis and modeling of deformed swept volumes
Computer-Aided Design
(1994)
Z.J. Hu et al.
Swept volumes generated by the natural quadric surfaces
Computers and Graphics
(1996)

There are more references available in the full text version of this article.

Cited by (24)

A method for integrating ergonomics analysis into maintainability design in a virtual environment
2016, International Journal of Industrial Ergonomics
Citation Excerpt :
Other research models, such as the incomplete repair model and effective visualization model, are found in recent literature (Chen and Cai, 2003; Chabal et al., 2005; Kahle, 2007; Hu et al., 2011). Over the years, SVs have been extensively studied and used in a variety of applications, including ergonomics design, (Abdel-Malek et al., 2004), robot workspace analysis (Abrams and Allen, 2000), collision avoidance (Redon et al., 2004), NC tool path verification (Blackmore et al., 1997), and solid modeling in CAD (Abdel-Malek et al., 2001; Erdim and Horea, 2008). Researchers have proposed solutions to the mathematical formulation of SV by using the Jacobian rank deficiency method (Abdel-Malek and Othman, 1999; Abdel-Malek and Yeh, 1997; Blackmore et al., 2000; Shapiro, 1997), sweep differential equation (Blackmore and Leu, 1990), Minkowski sums (Elber and Kim, 1999), envelope theory (Martin and Stephenson, 1990; Rossignac and Kim, 2000), implicit modeling (Schroeder et al., 1994), and kinematics (Jüttler and Wagner, 1996).
Designers must consider human factors/ergonomics when making decisions from the perspective of maintainability. As an important aspect of maintainability, maintenance space should be made adequate at the design stage to achieve a convenient maintenance process. A maintenance space evaluation method that considers ergonomics is proposed in this study. By comparing free swept volumes and constrained swept volumes in a virtual environment, maintenance space could be evaluated quantitatively and objectively. The results of the evaluation are obtained by combining the principles of ergonomics and maintainability. These results can help designers improve product design such that it fits ergonomics and maintainability requirements. A case study is introduced at the end of this paper to demonstrate the feasibility of the proposed method in efficiently evaluating the maintenance space based on the layout design of the product components in the design stage.
For a large number of disasters caused by human errors in current industry, the result of this study contributes a guide to fully consider human factors in maintainability design through virtual environment and is beneficial to designers and engineers of industrial application fields.
Computational Topology
2007, Open Problems in Topology II
This chapter provides an overview of computational topology. The first usage of the term “computational topology” appears to have occurred in the dissertation of M. Mantyla. The focus there was upon the connective topology joining vertices, edges, and faces in geometric models, frequently informally described as the symbolic information of a solid model. These vertices, edges, and faces are discussed as the operands for the classical Euler operations. One of the basic goals in computational topology is to create computer generated procedures for obtaining representations of objects having the same shape—at least in some acceptable approximate sense—as a given geometric object. Although computational topology is a relatively new discipline, it has grown and matured rapidly partially because of its increasing importance to many vital contemporary applications areas such as computer-aided design and manufacturing, (CAD/CAM), the life sciences, image processing, and virtual reality. It is leading to new techniques in algorithm and representation theory. These applications are evoking new connections between mathematical subdisciplines such as algebraic geometry, algebraic topology, differential geometry, differential topology, dynamical systems theory, general topology, and singularity, and stratification theory.
Sweeping of an object held by a robotic end-effector
2005, Robotics and Computer-Integrated Manufacturing
A broadly applicable formulation for calculating the swept volume generated by an object held by a manipulator's end-effector is presented. While the problem of determining the workspace of a robot arm has been extensively addressed in the literature, this rarely addressed problem is of significance to path planning, collision detection, plant layout, and robot design. The totality of points touched by a geometric entity moved in space using a number of joints is defined as the swept volume. The formulation and accompanying experimental code are presented and are aimed at providing the reader with a replicable computer algorithm. Calculating the swept volume is based on The Implicit Function theorem and is shown to be any number of degrees of freedom yielding the exact representation of the swept volume. By considering the sweep equation as a vector function defined on a manifold (possibly with boundaries), it is shown that stratification of the various sub-manifolds yields varieties that can be depicted in $R^{3}$ . A measure of the computational complexity is presented to give the reader a sense of the robustness of this method as well as its difficulties. An experimental computer code is developed using a symbolic manipulator that performs the automated calculations necessary to calculate the varieties and to visualize the manifold.
Approximate swept volumes of NURBS surfaces or solids
2005, Computer Aided Geometric Design
This paper presents a method of determining the approximate swept volume of Non-Uniform Rational B-Spline (NURBS) surfaces or solids. The method consists of (1) slicing the NURBS surfaces or solids by finding the intersection of plane/surface; (2) forming the sliced curves; (3) setting up the local moving coordinate system; (4) determining the characteristic (also called singular) points or curves by obtaining local maxima points at discrete frames during motion and with respect to a local moving coordinate system; (5) fitting each NURBS singular surface; (6) trimming the singular surfaces to obtain the boundary of the final approximate swept volumes by the surface/surface intersection and perturbation method. The local moving coordinate system is set up in reference to the motion direction of the rigid body as determined from its composite velocity vector. This work aims to develop a rigorous method for identifying and visualizing the approximate swept volume generated as a result of sweeping a NURBS surface or solid. The method and numerical algorithm are illustrated through examples.
Boolean operations for 3D simulation of CNC machining of drilling tools
2004, CAD Computer Aided Design
Citation Excerpt :
The main difficulty of this option is the computation of swept volumes. There are several references [1,2,21] on this subject, that contain methods generally applied in CAD for extrusions, collision detection, and other problems but none of them can be applied to the non-trivial case of simultaneous motion of the two solids in play. The strategy proposed herein overcomes the disadvantages of these methods.
This paper addresses the simulation of drilling tools CNC machining. It describes a novel approach for the computation of the boundary representation of the machined tools. Machining consists of a sequence of boolean operations of difference between the tool and the grinding wheels through time. The proposed method performs the dynamic boolean operations on cross sections of the tool and it reconstructs the 3Dmodel by tiling between the cross sections. The method is based on classical computational geometry algorithms such as intersection tests, hull computations, 2D boolean operations and surface tiling. This approach is efficient and it provides user control on the resolution of the operations.
Multiple sweeping using quaternion operations
2002, CAD Computer Aided Design

View all citing articles on Scopus

Jingzhou Yang is a PhD student in the Department of Mechanical Engineering, University of Iowa, Iowa City and a research member of the Center for Computer Aided Design (CCAD). He received his BS and MS degrees in Automobile Engineering from Jilin University of Technology, Changchun, Jilin, P.R. China, in 1989 and 1992, respectively. He was a lecturer at the Department of Automobile Engineering, Tsinghua University, Beijing, People's Republic of China, from 1992–1997. His research interests include geometric modeling, computer aided design, robotics, ergonomics, and automobile engineering.

Denis Blackmore has been a Professor of Mathematical Sciences at the New Jersey Institute of Technology (NJIT) since 1982. Previously, he taught at the Polytechnic University of New York. He is engaged in teaching and research in computer-aided geometric design, dynamical systems, manufacturing science, fluid mechanics, granular flow dynamics, differential topology, control theory and fractal geometry, and for his work he received the Harlan J. Perlis Award from NJIT in 1993. A founding member of the Center for Applied Mathematics and Statistics at NJIT, he has a PhD in Mathematics from the Polytechnic University of New York, 1971.

View full text

Technical NoteOn swept volume formulations: implicit surfaces

Abstract

Introduction

Section snippets

Formulation for implicit sweeps

Illustrative example

Conclusions

Computer-Aided Design

Computer-Aided Design

Computer Aided Design

Computer-Aided Design

Computers and Graphics

Technical Note
On swept volume formulations: implicit surfaces