The research results presented in this thesis provide tools and methods to aid in the design of developable mechanisms. This work will help engineers design compact mechanisms onto developable surfaces, making it possible for them to be used in future applications. The thesis introduces terminology and definitions to describe conical developable mechanisms. Models are developed to describe mechanism motion with respect to the apex of the conical surface, and connections are made to cylindrical developable mechanisms using projected angles. The Loop Sum Method is presented as an approach to determine the geometry of the cone to which a given spherical mechanism can be mapped. A method for position analysis is presented to determine the location of any point along the link of a mechanism with respect to the conical geometry. These methods are also applied to multiloop spherical mechanisms. This work created tools and methods to design cylindrical and conical developable mechanisms from flat, planar patterns. Equations are presented that relate the link lengths and link angles of planar and spherical mechanisms to the dimensions in a flat configuration. These flat patterns can then be formed into curved, developable mechanisms. Guidelines are established to determine if a mechanism described by a flat pattern can exhibit intramobile or extramobile behavior. A developable mechanism can only potentially exhibit intramobile or extramobile behavior if none of the links extend beyond half of the flat pattern. The behavior of a mechanism can change depending on the location of the cut of the flat pattern. Different joint designs are discussed including lamina emergent torsional (LET) joints. It is shown that developable mechanisms on regular cylindrical surfaces can be described using cyclic quadrilaterals. Mechanisms can exist in either an open or crossed configuration, and these configurations correspond to convex and crossed cyclic quadrilaterals. Using equations developed for both convex and crossed cyclic quadrilaterals, the geometry of the reference surface to which a four-bar mechanism can be mapped is found. Grashof mechanisms can be mapped to two surfaces in open or crossed configurations. The way to map a non-Grashof mechanism to a cylindrical surface is in its open configuration. Extramobile and intramobile behavior can be achieved depending on selected pairs within a cyclic quadrilateral and its position within the circumcircle. Selecting different sets of links as the ground link changes the potential behavior of the mechanism. Different cases are tabulated to represent all possibilities.

This thesis provides resources that enable the design of novel and compact mechanical devices by providing terminology, engineering models, and design methods in the fields of developable mechanisms and origami-based engineering.The first part of this work presents engineering models to aid in the design of cylindrical developable mechanisms. These models take into account the added spatial restrictions imposed by the developable surface. Equations are provided for the kinematic analysis of cylindrical developable mechanisms. A new classification for developable mechanisms is also presented (intramobile, extramobile, and transmobile) and two graphical methods are provided for determining this classification for single-DOF planar cylindrical developable mechanisms. Characteristics specific to four-bar cylindrical developable mechanisms are also discussed. The second part addresses a key challenge in origami design: how to achieve stability while maintaining the desired folding motion. The origami stability integration method (OSIM) provides an approach for graphically combining various techniques to achieve stability. This thesis presents improvements and additions to the OSIM that allow it to be applied to many different scenarios. Existing stability techniques are also categorized into four groups based on whether they are intrinsic or extrinsic to the origami pattern and whether they exhibit gradual or non-gradual energy storage behaviors. These categorizations can help designers select appropriate techniques for their application. Four case studies are presented which use the OSIM and the technique categorization to conceptualize stability in origami-based devices.