Numerical determination of freedom spaces from any set of twists

Abstract

We introduce a complete numerical algorithm to find the principal twists of all freedom spaces given an arbitrary set of twists without constructing the constraint space. Additionally, we propose principal geometry as a visual representation of all possible principal twists for a freedom space. Principal twists are unique to each space and can be used to identify freedom spaces by comparing their relative forms. Identifying freedom spaces has been a central problem in screw theory since its conception. Until now, there was no autonomous tool that can identify all freedom spaces. This tool can be integrated into solid modeling software, such as SolidWorks, to help design mechanisms using the freedom and constraint topologies (FACT) approach by linking desired motions to freedom space type and orientation. We accomplished this by converting existing analytical methods to numerical solutions, improving existing numerical tools, deriving new methods to find principal twists, and strengthening prior definitions. Finally, principal geometries give an orthogonal set of degrees of freedom for mechanism designers and formalize the idea of principal twists in the FACT design approach.