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.

Lamina Emergent Mechanisms (LEMs) are a class of compliant mechanisms that can be manufactured from sheet goods and possess motion out of the plane of fabrication. LEMs can be designed to perform sophisticated motions. This thesis expands LEM understanding and increases the ability to utilize them in applications by introducing the fundamentals of spherical LEMs, creating joints suitable for LEMs, and providing an example of a LEM application. In this thesis, the fundamentals of spherical LEMs are developed. This includes classification of all possible spherical 4R LEMs and a discussion of the motion characteristics of the various mechanisms. The motion characteristics associated with spherical 4R LEMs are then used to predict the motion of spherical 6R LEMs and arrays of spherical LEMs. Multiple spherical LEM prototypes are shown and discussed. A common difficulty of working with compliant mechanisms, especially LEMs, is creating suitable joints. There is often a trade off between flexibility in the desired direction of deflection, and stiffness in directions of undesired deflection. For this thesis, LEM joints that possess higher off-axis stiffness, especially in tension and compression, than previous designs were developed: the I-LET, the T-LET, and the IT-LET. Joint geometries were optimized and then modeled in commercial finite element analysis (FEA) software capable of nonlinear analysis. These models were used to predict the bending of tensile/compressive stiffnesses of the joints. As a benchmark, lamina emergent torsional (LET) joints were modeled and optimized for maximum tension and compression loading while maintaining the same bending stiffness as the joint being compared. Mechanisms that utilized the new joints were created and are briefly discussed. The use of these joints allows for minimized parasitic motion under tension and compression loads and expands the capability of LEM joints. The Lens Lift™ was developed to demonstrate an application of LEMs. The Lens Lift™ is a LEM device that allows for easier and more sterile use of disposable contact lenses. It possesses a monolithic structure and can be fabricated using simple manufacturing processes. As the contact lens user opens the blister pack used to store the lens, the lens is lifted out of the pack and presented to the user. The user can then lift the lens with one touch and place it in the eye. A provisional patent has been filed for the device and the device currently being evaluated by a major contact lens manufacturer for further development.

Origami, the Japanese art of paper folding, has been used previously to inspire engineering solutions for compact, deployable designs. Action origami, the subset of origami dealing with models designed to move, is a previously unexplored area for engineering design solutions that are deployable and have additional motion in the deployed state. A literature review of origami in engineering is performed, resulting in seven key areas of technical origami literature from a wide variety of disciplines. Spherical mechanisms are identified as the method by which most action origami models achieve complicated motion while remaining flat-foldable. The subset of action origami whose motion originates from spherical mechanisms is termed "kinematic origami''. Action origami is found to contain large coupled systems of spherical mechanisms. All possible action origami models are classified by their spherical mechanism structure, resulting in eight possible categories. Viewing action origami as spherical mechanisms allows the use of established equations for kinematic analysis. Several kinematic origami categories are used to demonstrate a method for the position analysis of coupled systems of spherical mechanisms. Input-output angle relationships and coupler link motions are obtained for a single spherical mechanism, two spherical mechanisms coupled together, and four spherical mechanisms coupled in a loop arrangement. This lays a groundwork from which it is possible to create compact, deployable mechanisms with motion in the deployed state.

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