Geometric representation of the swept volume using Jacobian rank-deficiency conditions

doi:10.1016/S0010-4485(96)00097-8

Computer-Aided Design

Volume 29, Issue 6, June 1997, Pages 457-468

https://doi.org/10.1016/S0010-4485(96)00097-8 Get rights and content

Abstract

A broadly applicable formulation for representing the boundary of swept geometric entities is presented. Geometric entities of multiple parameters are considered. A constraint function is defined as one entity is swept along another. Boundaries in terms of inequality constraints imposed on each entity are considered which gives rise to an ability of modeling complex solids. A rank-deficiency condition is imposed on the constraint Jacobian of the sweep to determine singular sets. Because of the generality of the rank-deficiency condition, the formulation is applicable to entities of any dimension. The boundary to the resulting swept volume, in terms of enveloping surfaces, is generated by substituting the resulting singularities into the constraint equation. Singular entities (hyperentities) are then intersected to determine sub-entities that may exist on the boundary of the generated swept volume. Physical behavior of singular entities is discussed. A perturbation method is used to identify the boundary envelope. Numerical examples illustrating this formulation are presented. Applications to NC part geometry verification, robotic manipulators, and computer modeling are discussed.

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The digital models are the fundamental elements used in the virtual simulation of CNC machining for process planning, result prediction, analysis, etc. The essential work to build the digital models of machined parts is the calculation of the envelope surface generated by moving a cutter surface relative to workpieces. This calculation is usually complicated, and the requirements in different cases are also varied. Hence, it is very beneficial to develop an effective approach for different requirements. In this work, a comprehensive envelope approach is established, and it adaptively choose 1) the elimination parameter among two cutter surface parameters, 2) the representation form of envelope surface, and 3) the discretization way, so it is robust to satisfy different requirements. The elimination parameter and representation form are adaptively chosen to improve the calculation efficiency by obtaining the closed-form representation of the envelope surface prior to the implicit one. The discretization way is further adaptively chosen to obtain approximately even point distribution to the applications of 3D and FEA modeling. Finally, comprehensive algorithms are proposed to effectively implement this adaptive envelope approach for different applications. The examples are given to the manufacturing of spiral bevel gears and five-axis CNC milling.
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Citation Excerpt :
Such an approach lacks bounds on the approximation error. Abdel-Malek and Yeh [26] propose a swept volume method based on rank deficiency condition of the Jacobian of the sweep map. This method readily generalizes to arbitrary dimensions.
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Nous proposons une méthode générale et analytique de description des limites de l'espace de travail plan, c'est-à-dire la frontière de la partie de l'espace du plan atteinte par l'extrémité d'un bras humain ou de robot. La méthode proposée est décomposée en trois étapes indépendantes, aucune ne faisant appel à des calculs de dérivées, de déterminants ou de valeurs propres, ou utilisant des méthodes de résolution numérique de problèmes non linéaires, tous souvent utilisés dans la littérature. La première étape, déjà présentée dans des travaux précédents, consiste à interpréter de façon géométrique la nécessaire dégénérescence de la matrice jacobienne de la fonction de position, au bord de l'espace de travail et une condition d'alignement de certaines articulations permet de déterminer une réunion d'arcs de cercle qui contiennent la frontière. Ensuite, pour chacun des arcs de cercle, l'étude de la variation infinitésimale d'un point par rapport à ce cercle permet d'éliminer tout ou partie des arcs de cercle précédemment définis, qui n'est pas sur la frontière. Enfin, un parcours global de la réunion des arcs de cercle permet de déterminer la frontière extérieure ainsi que la frontière intérieure de l'éventuel trou de l'espace de travail qui contient l'origine.
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Geometric representation of the swept volume using Jacobian rank-deficiency conditions

Abstract

Computer-Aided Design

Computer-Aided Design

Computer Aided Geometric Design

Computer-Aided Design

Computer-Aided Design

Swept volumes and their use in view-point computation in robot work-cells

Analysis of swept volume via Lie groups and differential equations

The International Journal of Robotics Research

Modeling of swept solids using sweep differential techniques

On algebraic methods for implicit swept solids with finite extent

ASME DE

Generating swept volumes with instantaneous screw axes