Computational Kinematic Geometry of NURBS Motions

该项目旨在为几何驱动的运动学近似创建一个框架，该框架将为可制造性设计提供统一的处理方法，解决制造中的运动学问题，如雕刻表面的多轴加工、零件匹配和装配规划、齿轮设计和制造、任务驱动的机器设计的任务规范，以及CAD/CAM集成中的自由曲面加工特征建模。主要任务是发展NURBS（非均匀有理b样条）运动的计算运动学几何，并探索其在形状生成中刚体运动的分析，综合，微调和工程设计中几何和运动之间的相互作用起着关键作用的询问过程中的应用。NURBS运动的计算、算法和微分几何将从刚体位移的射影几何和运动学几何的角度进行充分的探索。利用NURBS运动的算法几何，将研究制造中常用刀具形状的单参数和多参数NURBS运动的轨迹和扫描体积。这将导致第一个综合理论的计算运动学几何的一个和多参数NURBS运动。此外，该结果将代表第一个真正基于运动学的全局规划方法，用于无量规的5轴刀具路径，该方法完全将全局曲面逼近问题与局部刀具位置生成的逐步过程解耦。通过生成学习空间可视化和推理的工具箱，以及探索几何和运动之间的相互作用，虚拟设计和制造的研究活动与教育和推广活动相结合。预计这些软件工具将特别吸引本科生和高中生，创造一种设计游戏方法，从根本上讲是严谨的，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.