Computational Kinematic Geometry of NURBS Motions

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

This project seeks to create a framework for geometry driven kinematic approximation that will provide a unified treatment for design for manufacturability, addressing kinematic problems in manufacturing such as multi-axis machining of sculptured surfaces, part-mating and assembly planning, gear design and manufacturing, task specification for task-driven machine design, as well as freeform machining feature modeling in CAD/CAM integration. The primary task is to develop the computational kinematic geometry of NURBS (Non-Uniform Rational B-Spline) motions and explore its applications in the analysis, synthesis, fine-tuning of rigid-body motions in shape generation and interrogation process in engineering design where the interplay between geometry and motion plays a critical role. The computational, algorithmic and differential geometry of the NURBS motions will be fully explored from the perspectives of projective geometry and kinematic geometry of rigid body displacements. The trajectories and swept volumes of commonly used tool shapes in manufacturing will be studied for one- and multi-parameter NURBS motions taking advantage of the algorithmic geometry of NURBS motions. This will lead to the first comprehensive theory for the computational kinematic geometry of one- and multi-parameter NURBS motions. In addition, the results will represent the first truly kinematics based, global planning methodology for gauging-free 5-axis tool paths that completely decouples the problem of global surface approximation from the step-by-step process of local tool position generation. The research activities in virtual design and manufacturing are integrated with educational and outreach activities through the generation of a toolbox for learning spatial visualization and reasoning, and exploring the interplay between geometry and motion. It is expected these software tools will be particularly attractive to undergraduate students and high school students, creating a design game approach that is fundamentally rigorous while providing experiential learning.