CAREER: The Foundations of Ellipsoid Synthesis Theory

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。这个教师早期职业发展（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造一种新的方法。工程师和数学家先前已经注意到，输入处施加的力（或运动）与输出处产生的力（或运动）之间的关系可以用椭圆和相关的形状来概念化。该项目将这些形状转变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表述为构成机构尺寸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来探索这些受限的设计空间。由于该项目推进了基础设计科学研究，因此它在具有多输入和多输出的机制的任何地方都具有广泛的适用性。这包括在机器人（康复机器人、工业机器人、定位器）、外骨骼、被动辅助装置、驱动假肢和用于腹腔镜手术中力传感的机制中的许多应用。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。该项目的精神是通过提供核心方法及其相关计算技术来提高我国机械设计工程师的能力。计划的项目活动包括学生设计项目的发起，新课程的课程设置，以及STEM机器人设计比赛。这些努力旨在提高学生的设计教育和研究经验，扩大弱势群体的参与。椭圆（或超椭球，一般情况下）用于可视化多自由度机械臂的方向力和速度特性。这个项目将这样的椭球体转化为几何约束，形成一个机构的设计空间。新理论将展示如何将各种设计规范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范，还包括反驾驶性、刚度、灵敏度和执行器功率要求。然而，目前尚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）中挖掘椭球体映射信息，可以抑制使用符号操作来形成实现关键计算搜索技术所需的综合方程。相反，可以将球体上随机选择的点映射到指定的椭球上，以便以符号形式获得具有设计参数的代数方程。一系列的计算搜索策略将被调查，从代数几何，优化理论，并包括图形界面。比较指标将通过应用新的设计方法来评估案例研究。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.