Flexible Polyhedral Surfaces with Two Flat Poses

Abstract

We present three types of polyhedral surfaces, which are continuously flexible and have not only an initial pose, where all faces are coplanar, but pass during their self-motion through another pose with coplanar faces ("flat pose"). These surfaces are examples of so-called rigid origami, since we only admit exact flexions, i.e., each face remains rigid during the motion; only the dihedral angles vary. We analyze the geometry behind Miura-ori and address Kokotsakis' example of a flexible tessellation with the particular case of a cyclic quadrangle. Finally, we recall Bricard's octahedra of Type 3 and their relation to strophoids.

Jigsaw puzzle design of pluripotent origami

“…Such crease patterns are also inherently pluripotent, and the multiplicity of their branches is exactly countable (see Table 1 for expressions and the Supplementary Information for exact counting arguments). For example, the Huffman pattern features two folding branches regardless of its m × n size 28 , whereas the number of branches in the Mars pattern grows as 2 m+1 + 2 n+1 − 2. In Supplementary Videos 1 I I II II II II II II II II II II II II II II II I I I I I Ba, anticlockwise-ordered base vertex with sector angles α j , fold angles ρ j,j+1 and folding operators P j ; Bc, clockwise-ordered base vertex, with inverse fold operators; Sa, supplemented vertex with anticlockwise-ordered sector angles α 0 j :¼ π � α j I and negated fold operators; Sc, supplementedclockwise vertex, with inverse negated operators.…”

Hexapods with Plane-Symmetric Self-Motions

“…Note that a further prominent example of a hexapod, which possesses flat poses during its self-motion, is Bricard's flexible octahedron of type 3 (cf. [41]).…”

Kinematics and Robot Design I, KaRD2018

“…In particular, to obtain the command torques, instead of using proper filtering as is usually done, we formulate an optimal control problem with the aim of minimizing the positioning error of the tip manipulator (see, e.g., [40]). This variational approach has been adopted for trajectory planning both for flexible [41,42] and for rigid arms [6]. Herein, instead of obtaining a trajectory with the desired mechanical characteristics, the best possible input command is found to follow a given trajectory.…”