Geometry and Kinematics of Cylindrical Waterbomb Tessellation

Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, the theoretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a kinematic model of waterbomb tube by parameterizing the geometry of a module of waterbomb tessellation and derive a recurrence relation between the modules. Through the visualization of the configurations of waterbomb tubes under the proposed kinematic model, we classify solutions into three classes: cylinder solution, wave-like solution, and finite solution. Furthermore, we give proof of the existence of a wave-like solution around one of the cylinder solutions by applying the knowledge of the discrete dynamical system to the recurrence relation.

Folded surface elements coupled with planar scissor linkages: A novel hybrid type of deployable structures

Origami folding structures can find significant applications in the general area of building design, as they can be lightweight and deployable. An inherent property of folded surfaces, which is related to the degrees of freedom of each origami crease pattern, is form flexibility. Therefore, when building-scale applications are considered, in many instances, the folded surfaces, in order to become stiff and load-bearing, need to be constrained. A study of different types of deployable structures has led to the observation that in planar scissor linkages, hinges and pivots follow the same deployment path in space, as sets of vertices in certain origami structures. Due to the similarities in their kinematic behavior, selected origami patterns and scissor linkages can function as effective kinematic pairs, leading to structures able to transform in a controlled manner through a wide range of possible spatial configurations. A few examples of combining these two types of structures already exist [1–3]. In this paper a systematic approach for coupling origami crease patterns characterized by biaxial and rotational symmetry, with translational, polar and angulated scissor linkages, towards the development of novel forms of deployable structures, has been attempted. For the design and evaluation of the kinematic performance of the developed new structures, existing geometric modeling and calculation methods, parametric and simulation processes, as well as testing with physical models have been used. It is anticipated that research in this direction will lead to promising novel hybrid types of deployable structures.

Joining Unfoldings of 3-D Surfaces

Origami-based design methods enable complex devices to be fabricated quickly in plane and then folded into their final 3-D shapes. So far, these folded structures have been designed manually. This paper presents a geometric approach to automatic composition of folded surfaces, which will allow existing designs to be combined and complex functionality to be produced with minimal human input. We show that given two surfaces in 3-D and their 2-D unfoldings, a surface consisting of the two originals joined along an arbitrary edge can always be achieved by connecting the two original unfoldings with some additional linking material, and we provide an algorithm to generate this composite unfolding. The algorithm is verified using various surfaces, as well as a walking and gripping robot design.