CAREER: The Foundations of Ellipsoid Synthesis Theory

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.