Origami, the art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example of the many difficulties and general manipulation problems faced in robotics. This paper describes our initial exploration, and highlights key problems in the manipulation, modeling, and design of foldable structures. Results include the design of the first origami-folding robot, a complete fold-sequence planner for a simple class of origami, and analysis of the kinematics of more complicated folds, including the common paper shopping bag.

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Robotic origami folding

An Approach to Designing Origami-Adapted Aerospace Mechanisms

Transformable polyhedral surfaces with rigid facets, i.e., rigid origami, are useful for designing kinetic and deployable structures. In order to apply rigid origami to various architectural and other engineering design purposes, it is essential to consider the geometry of origami in kinetic motion and provide sufficiently generalized methods to produce controlled variations of shapes that suit the given design conditions. In this paper, we introduce the author’s recent studies and their extensions on the geometry of rigid origami for designing transformable and deployable structures.

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Geometric Considerations for the Design of Rigid Origami Structures

From stadium covers to solar sails, we rely on deployability for the design of large-scale structures that can quickly compress to a fraction of their size1–4. Historically, two main strategies have been used to design deployable systems. The first and most frequently used approach involves mechanisms comprising interconnected bar elements, which can synchronously expand and retract5–7, occasionally locking in place through bistable elements8,9. The second strategy makes use of inflatable membranes that morph into target shapes by means of a single pressure input10–12. Neither strategy, however, can be readily used to provide an enclosed domain that is able to lock in place after deployment: the integration of a protective covering in linkage-based constructions is challenging and pneumatic systems require a constant applied pressure to keep their expanded shape13–15. Here we draw inspiration from origami—the Japanese art of paper folding—to design rigid-walled deployable structures that are multistable and inflatable. Guided by geometric analyses and experiments, we create a library of bistable origami shapes that can be deployed through a single fluidic pressure input. We then combine these units to build functional structures at the metre scale, such as arches and emergency shelters, providing a direct route for building large-scale inflatable systems that lock in place after deployment and offer a robust enclosure through their stiff faces. Origami-inspired multistable structures that can be inflated from flat to three dimensions have been designed; a library of foldable shapes is created and then combined to build metre-scale functional structures.

Multistable inflatable origami structures at the metre scale.

It is proved that in R3 the volume of any polyhedron is a root of some polynomial with coefficients depending only on the combinatorial structure and the metric of the polyhedron. As a consequence, we have a proof of the ``bellows conjecture'' affirming the invariance of the volume of a flexible polyhedron in the process of its flexion.

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The Volume as a Metric Invariant of Polyhedra

Cauchy’s rigidity theorem states: If P and P’ are combinatorially equivalent convex polyhedra such that the corresponding facets of P and P’ are congruent, then P and P’ are congruent polyhedra. For many years it was unknown whether the same theorem was true in general for non-convex polyhedron. In 1977, more than 160 years after the work of Cauchy, Robert Connelly discovered a polyhedron P (without selfintersections) that allowed for a continuous deformation keeping the facets of P flat and congruent. It was soon noticed that the volume of Connelly’s polyhedron remained constant under the flexing motion, and the same fact was found to be true of all later-discovered flexible polyhedra. It was conjectured that this fact would hold in general, and it came to be known as the bellows conjecture. In 1995, Idjad Sabitov proved the bellows conjecture by showing that for any (oriented) polyhedron P, the volume of P is a root of a polynomial depending only on the combinatorial structure and edge lengths of P. Moreover, the coefficients of the polynomial are themselves polynomials of the squares of the lengths of the edges of P with rational coefficients, with the coefficient polynomials depending only on the combinatorial structure of P. Hence, the volume of a polyhedron P with combinatorial structure K is a finitely valued function of the edge lengths of P, and in this way the theorem may be viewed as a generalization of Heron’s formula for the area of a triangle. 1 Formulas from Antiquity Often attributed to Heron of Alexandria (though perhaps known already to Archimedes), Heron’s formula expresses the area of a triangle as a function of the side lengths. Heron’s Formula. Let T be a triangle with side lengths (a, b, c). Area(T ) = 1 4 √ (a+ b+ c)(a+ b− c)(a− b+ c)(−a+ b+ c) We all know from elementary geometry that a triangle is determined by its side lengths. With this knowledge, Heron’s formula is, perhaps, unsurprising. For quadrilaterals, we quickly observe that the side lengths do not uniquely determine the area, so there is no hope of finding a similar formula that holds in the same generality. However, if we restrict to the case of cylic quadrilaterals, i.e., quadrilaterals whose vertices all lie on a common circle, then an analogous formula does exist. Brahmagupta’s Formula. Let Q be a cyclic quadrilateral with side lengths (a, b, c, d). Area(Q) = 1 4 √ (a+ b+ c− d)(a+ b− c+ d)(a− b+ c+ d)(−a+ b+ c+ d) If we let an edge of Q have length 0, we see that Heron’s Formula appears as a special case of Brahmahgupta. It is natural to ask whether or not a similar formula exists for cyclic pentagons, hexagons, etc. Now Heron’s formula may be easily derived using nothing more than the Pythagorean theorem, and in [3], we see that Brahmagupta’s formula follows easily from Heron. Despite the relative ease of these first two cases, the formula for the cylic pentagon went unknown for over 1300 years after Brahmagupta recorded his formula for the quadrilateral. In 1994 D. P. Robbins discovered the formulas for both the cyclic pentagon and the cyclic hexagon and proved the following more general result. Theorem (Robbins [6]). For any natural number n, there exists a unique (up to sign) irreducible homogenous polynomial f (where we regard the first argument as degree 4, the rest as degree 2) with integer coefficients such that for any cyclic n-gon P , the edge lengths (a1, a2, . . . , an) and area K of P satisfy f(16K, a1, . . . , a 2 n) = 0.

A. Walz

Papers

The Bellows conjecture.