Swept volumes: Theory and implementations

Swept volumes arise in computer aided design, manufacturing automation, robotics and human modeling and simulation, where a geometric entity is swept in space and where the totality of points touched by the entity is called the swept volume. The goal of this work is to find systematic mathematical formulations to determine and visualize swept volumes of parametric, implicit, and free form surfaces or solids.; A rank deficiency method of the Jacobian matrix is introduced and used in parametric and implicit sweeping. By considering the sweep equation as a vector function defined on a manifold (possibly with boundaries), this work shows that further stratification of the various submanifolds yields varieties that can be depicted in R3. The code is developed using a symbolic manipulator and is presented with a number of examples. It is also shown that a rigorous mathematical formulation using manifold stratification is capable of treating consecutive sweep problems.; This method is then implemented to better understand human motion in ergonomics design. A rigorous “closed form” kinematic formulation is proposed to model human limbs, to understand their workspace, and to delineate barriers where a path becomes difficult or impossible to follow due to physical constraints.; Further implementation of this method has yielded a systematic method for characterizing and visualizing the volume generated as a result of sweeping a Non-Uniform Rational B-Splines (NURBS) surface (or solid) in space. A rigorous method has been established calculating a local moving coordinate system from the velocity vector along the path. Local maxima points (also called singular or grazing points) are determined and used to formulate the envelope singular surfaces.; Research on sweeping NURBS has led to a method for representing free form surfaces under dynamic loading that are affected and respond to external deformations. The presented method provides a unique approach to calculating the deformations with forces of a NURBS surface at any point. This dynamic surface model incorporates mass distributions, internal deformation energies, forces and other physical quantities into the NURBS geometric description. The dynamic behavior results from the discrete integration of a set of nonlinear ordinary differential equations using explicit Euler integration.; Conclusions are drawn concerning swept volumes as a viable method that can be used in human reach studies, solid modeling, and robotic analysis. Future considerations for continued research are also indicated.