Kinematic and static characterization of everting Möbius kaleidocycles with slightly incongruent links

Abstract

A Möbius kaleidocycle is a closed kinematic chain of $n\ge 7$ identical links connected by revolute joints, forming a linkage with the nonorientable topology of a Möbius band. If its joints are set at a critical, $n$-dependent twist angle — the smallest that allows closure without forcing — then, despite formally having $n-6$ internal degrees of freedom, the linkage admits only a single one: a reversible, periodic everting motion. Focusing on the case $n=7$, we determine the kinematic matrix via the Denavit–Hartenberg construction, under closure and congruence constraints. A geometric mechanism arises alongside the topological one due to a matrix-rank deficiency, accompanied by a corresponding state of self-stress. The geometric mechanism is infinitesimal and stiffened by self-stress, while eversion is enabled by the finite mechanism. Using a variational argument, we confirm that the sum of squared joint rotations remains constant throughout eversion. We further categorize the states of self-stress, identifying conserved quantities — including the sum of twisting moments raised to any positive integer power $\lambda \ge 1$ — which enable estimates of self-stresses in moderately incongruent linkages requiring elastic forcing to close.