Geometric representation of the swept volume using Jacobian rank deficiency conditions

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. (C) 1997 Elsevier Science Ltd.