Computational Kinematic Geometry of NURBS Motions

该项目旨在创建一个框架，几何驱动的运动近似，将提供一个统一的处理设计的可制造性，解决制造中的运动问题，如多轴加工的雕塑表面，零件配合和装配规划，齿轮设计和制造，任务规范的任务驱动的机器设计，以及自由形式的加工特征建模的CAD/CAM集成。本文的主要任务是发展NURBS（Non-UniformRationalB-Spline）运动的计算运动几何学，并探索其在工程设计中形状生成和查询过程中刚体运动的分析、综合、微调等方面的应用。将从刚体位移的射影几何和运动几何的角度充分探讨NURBS运动的计算、算法和微分几何。在制造中常用的工具形状的轨迹和扫掠体积将被研究为一个和多参数的NURBS运动，利用NURBS运动的算法几何。这将导致第一个全面的理论计算运动几何的一个和多参数NURBS运动。此外，结果将代表第一个真正的运动学为基础的，全球规划方法的测量免费5轴刀具路径，完全从一步一步的过程中，局部刀具位置生成的全局表面近似的问题。在虚拟设计和制造的研究活动与教育和推广活动相结合，通过生成一个工具箱，用于学习空间可视化和推理，并探索几何和运动之间的相互作用。预计这些软件工具将特别吸引本科生和高中生，创造一个设计游戏的方法，是从根本上严格，同时提供体验式学习。

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.