CAREER: The Foundations of Ellipsoid Synthesis Theory

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。该教师早期职业发展（CAREER）项目的目标是为具有多个输入和输出的机械机构的几何设计创建一种新的方法。 工程师和数学家以前已经注意到，在输入处施加的力（或运动）和在输出处产生的力（或运动）之间的关系可以通过椭圆和相关形状来概念化。该项目将这些形状转化为一种新的设计方法的中心对象。这是通过在数学上将椭圆公式化为形成机构尺寸的几何设计空间的约束来实现的。这种方法成功的关键是创建和基准新的计算技术，探索这些受约束的设计空间。由于该项目推进了基础设计科学研究，因此在多输入多输出的机构中具有广泛的适用性，包括在机器人（康复机器人、工业机器人、定位器）、外骨骼、被动辅助设备、驱动假肢以及腹腔镜手术中用于力传感的机构中的许多应用。正是通过这种适用性，这个项目促进了我们国家的健康，繁荣和福利。本计画的精神是借由贡献核心方法论及其相关的计算技术，来提升我国机械设计工程师的能力。计划中的项目活动包括学生设计项目的发起，新的课程设置和STEM机器人设计竞赛。 这些努力致力于提高设计教育和研究经验的学生，并扩大参与的代表性不足的group.Ellipses（或超椭球，为一般情况下）的方向力和速度特性的可视化多自由度机械手。这个项目将这些椭圆体转换成几何约束，形成一个机制的设计空间。 新的理论将演示如何将各种设计规格汇集到雅可比椭球映射的几何合成中，不仅包括力和速度规格，还包括反向驱动能力，刚度，灵敏度和致动器功率要求。然而，目前还不清楚如何制定椭球约束，以促进下游设计空间搜索。利用经典的矩阵分解从雅可比矩阵（根据未知的设计参数定义）中挖掘椭球映射信息，抑制了符号操作的使用，以形成实现关键计算搜索技术所需的合成方程。相反，从球体中随机选择的点可以映射到指定的椭球体，以获得具有符号形式的设计参数的代数方程。一系列的计算搜索策略将被调查，拉从代数几何，优化理论，并包括图形界面。通过将新的设计方法应用于案例研究，将对比较指标进行评估。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).The objective of this Faculty Early Career Development (CAREER) project is to create a new methodology for the geometric design of mechanical mechanisms that have multiple inputs and outputs. Engineers and mathematicians have previously noted that the relationship between forces (or motion) applied at the inputs and forces (or motion) generated at the outputs can be conceptualized by ellipses and related shapes. This project transforms these shapes into the central objects of a new design methodology. This is accomplished by mathematically formulating ellipses as constraints that form the geometric design space of mechanism dimensions. Critical to the success of this approach is to create and benchmark new computational techniques for exploring these constrained design spaces. Because this project advances foundational design science research, it has broad applicability wherever mechanisms with multiple inputs and output are found. This includes many applications in robotics (rehabilitation robots, industrial robots, positioners), exoskeletons, passive assistive devices, actuated prosthetics, and mechanisms used for force sensing during laparoscopic surgery. It is through such applicability that this project promotes the health, prosperity, and welfare of our nation. The spirit of this project is to enhance the capabilities of our nation’s mechanical design engineers by contributing a core methodology and its related computational techniques. Planned project activities include the origination of student design projects, new course curricula, and a STEM robot design competition. Such efforts are bent on enhancing design education and research experiences for students, and broadening participation of underrepresented groups.Ellipses (or hyper-ellipsoids, for the general case) are used to visualize the directional force and velocity characteristics of multi-degree-of-freedom manipulators. This project converts such ellipsoids into geometric constraints that form the design space of a mechanism. The new theory will demonstrate how a variety of design specifications can be funneled into the geometric synthesis of Jacobian ellipsoid mappings, including not only specifications on force and velocity, but also on backdrivability, stiffness, sensitivity, and actuator power requirements. However, it is unclear how to formulate ellipsoid constraints to facilitate downstream design space searches. Leveraging classical matrix factorizations to unearth ellipsoid mapping information from Jacobians (which are defined in terms of unknown design parameters) inhibits the usage of symbolic manipulation to form the synthesis equations needed to implement key computational search techniques. Instead, randomly selected points from a sphere can be mapped to specified ellipsoids in order to obtain algebraic equations with design parameters in symbolic form. A range of computational search strategies will be investigated, pulling from algebraic geometry, optimization theory, and including graphical interfaces. Comparative metrics will be evaluated by applying the new design methodology to case studies.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.