Miura and Miura-derivative rigid origami patterns are increasingly used for engineering and architectural applications. However, geometric modelling approaches used in existing studies are generally haphazard, with pattern identifications and parameterizations varying widely. Consequently, relationships between Miura-derivative patterns are poorly understood, and widespread application of rigid patterns to the design of folded plate structures is hindered. This paper explores the relationship between the Miura pattern, selected because it is a commonly used rigid origami pattern, and first-level derivative patterns, generated by altering a single characteristic of the Miura pattern. Five alterable characteristics are identified in this paper: crease orientation, crease alignment, developability, flat-foldability, and rectilinearity. A consistent parameterization is presented for five derivative patterns created by modifying each characteristic, with physical prototypes constructed for geometry validation. It is also shown how the consistent parameterization allows first-level derivative geometries to be combined into complex piecewise geometries. All parameterizations presented in this paper have been compiled into a matlab Toolbox freely available for research purposes.

This paper develops an origami based mechanical metamaterial with programmable deformationdependent stiffness and shape modulation, leading to the realization of a distant actuation feature.Through computational and experimental analyses, we have uncovered that a waterbomb based tubular metamaterial can undergo mixed mode of deformations involving both rigid origami motion and structural deformation. Besides the capability of achieving a near-zero stiffness, a contact phase is identified that initiates a substantial increase in the stiffness with programmable features during deformation of the metamaterial. Initiation of the contact phase as a function of the applied global load can be designed based on the microstructural geometry of the waterbomb bases and their assembly. The tubular metamaterial can exhibit a unique deformation dependent spatially varying mixed mode Poisson's ratio, which is achievable from a uniform initial configuration of the metamaterial. The spatial profile of the metamaterial can be modulated as a function of the applied far-field global force, and the configuration and assembly of the waterbomb bases. This creates a new possibility of developing a distant actuation feature in the metamaterial enabling us to achieve controlled local actuation through the application of a single far-field force. The distant actuation feature eliminates the need of installing embedded complex network of sensors, actuators and controllers in the material. The fundamental programmable features of the origami metamaterial unravelled in this paper can find wide range of applications in soft robotics, aerospace, biomedical devices and various other advanced physical systems.

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A modified planar foldcore geometry was developed by introducing sub-folds into a standard foldcore pattern. It was demonstrated using numerical simulation that the new geometry, deemed the indented foldcore, successfully triggered a high-order travelling hinge line failure mode. This was found to have a much higher and more uniform energy absorption than the plate buckling failure mode seen in a standard foldcore structure. A numerical analysis also established optimum standard and indented geometries with maximum energy absorption. Prototypes were constructed to experimentally validate numerical findings. Prototypes with no visible geometric imperfections displayed the travelling hinge line behaviour as predicted. Prototypes with visibly-buckled plates showed no change in failure mode compared to a standard foldcore, confirming numerical findings that the travelling hinge line failure mode is highly sensitive to geometric imperfections.