Finite element simulation of robotic origami folding

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Abstract

In this paper, we focus on some aspects of the finite element simulations of robotic paper folding and the reconstruction of models from the origami crease patterns by the robot arms. The paper highlights the simulation problems, which should be solved in developing our recent study in mechanical and geometrical design of the origami-performing robot. The basic premise underlying the study is that folding operations with the origami crease patterns are considered as the functions of the mechanical systems such as a robot. Manipulations with the foldable objects, such as a sheet of paper (the origami crease pattern), by the robot arms in the simulation environment lead to understanding the design of the origami-performing robot without testing physical prototypes at each design stage. In this case, dynamic and kinematic behavior of the robot arms in forming the 3D origami objects is modelled by using the finite element method (FEM) in LS-DYNA solver. For simulating, two forms of origami are considered: flexible, if bending is used for paper folding, and rigid, if origami patterns are considered as the kinematic systems. Results of the simulation are presented and provided by the illustrations.

Introduction

Since origami has many advantages, its applications are now used widely in industry and everyday life. Origami starts from a two-dimensional layer and transforms to the three-dimensional structure via folding. Origami principles have broad and varied applications: from solar arrays [20] and building facades [7] to robotics [9], [17], mechanisms in stent grafts [12], and DNA-sized boxes [1]. The materials and methods, which are used for fabricating, actuating, and assembling these products, can vary greatly with a length scale. Large-scale origami structures can be constructed from the thickened panels connected by hinges and can be actuated with mechanical forces. The benefit of origami structures is their ability to support weight with enough stiffness and to pack a large surface area into a compact flat shape. With developing the origami structures, the material using for folding currently is not only an ordinary paper with a small thickness 0.1 mm. Special paper materials, such as cardboard or coated paper, which thickness is bigger (1–2 mm), are used for forming origami structures to increase the stiffness and still keep the lightweight structure. There are some approaches for folding the origami patterns with different thickness [6] and developing self-folding machines [9]. Hence, building a robot that can help people to fold the target origami patterns is a trend all over the world.

The experimental folding machine presented by [2] includes a blade press for forming the creases and a working table. Two robot hands are used for folding paper in Elbrechter et al. [8]. The authors apply a method for real-time detection, physical modelling of paper, and suggest an approach to recognize the shape of a sheet of paper. In [18] the robot design for folding the origami model, which uses a rubber ball to form creases, is described; that robot design should be redesigned for each origami model.

To our knowledge, there are no publications related to the simulation-based design for the origami-performing robotic system. In the previous publication [19], we have introduced our vision and detail investigation of the simulation-based design of the origami-performing robot, where each origami folding operation is considered as a function of the mechanical systems such as a robot and performing the folding operations by the robotic arms by numerical simulation. The preliminary simulation results show that the proposed approach for designing the origami-performing robot can be considered as a basis for the design of the realistic robot. The design of a robot that is based on experiment only is difficult, expensive, and time-consuming. We focus on the finite element (FE) simulation of paper folding and forming the 3D origami models from the 2D origami crease patterns by the robot arms with the aim to design the origami-performing robot.

The present paper highlights the calculation problems that are related to FE simulation of forming the origami models by the robot arms such as: a simulation of folding the multi-intersecting origami patterns, kinematic modelling the cardboard origami patterns, planning the folding operations, and folding conditions for the real crease patterns. The FEM in LS-DYNA software [14] allows us to study the complicated geometrical shapes and simulate the performance of the formation of the origami models by the robot arms.

The main contributions in this paper are:

  • (a) Meshing template to solve FE problems related to the strong deformed sheet of paper: multi-thickness computational model (the origami multi-intersecting crease pattern);

  • (b) The FE simulation approach for the production of the cardboard origami-based structures with straight and curve folds by the robotic arms;

  • (c) Generating unique folding sequences from origami crease pattern for robot manipulation.

The presented simulation results of forming the 3D origami objects by robotic arms show the applicability of the suggested approach for the origami-performing robot design.

The following is a brief overview of this paper: Section 1 introduces the FE simulation approach for the design of the robotic system for the origami applications. In Section 2, fundamentals of the approach are described. Description of the general computational models for the flexible origami is provided in Section 3. The simulation problems related to the formation of the multi-intersecting origami crease patterns are discussed in Sections 4 and 5. Section 6 presents virtual testing of forming origami models by the origami- performing robot. Discussion and conclusions can be found in Section 7.

Section snippets

Fundamentals of method

The engineering design process includes series of steps for creating functional product. In this section, we briefly describe the simulation-based method for mechanical and geometrical design of the origami-performing robot (Fig. 1). FE simulations and mechanical engineering problems, which were solved in the previous phase of our research, are explained here. The suggested robot design process includes three main stages: schematic design that presents a conceptual design, computational

General computational model for flexible origami

The FE model is prepared by LS-DYNA for forming single crease, which is parallel to the boundary of the sheet of paper. The basic assumption is that the sheet of paper is thin in the sense that its thickness t (prior deformation) much less than its length. Paper model is considered as an elastic shell with 2t thickness. The FEM is the technique that is applied to solve all discussed simulation problems. The implementation of the shell structure is based on the formulation of the 4-node

FE simulation of formations of the multi-intersecting creases

The most of all the origami models are based on the intersecting creases, for instance, the origami crease pattern in Fig. 5 (two mountains in red and two diagonal valleys in blue).

In this Section, we discuss the problems of the simulation of forming crease line, which intersects already formed one. Correctness of FE simulation of bending the sheet of paper depends on shell elements [3], which are used to model the structure, when two dimensions are much greater than the third one.

Simulation problems and FE analysis

For the FE model with the two-intersecting crease lines, which is described in the Section 4.1, 2 steps involve in simulating the formation of the intersecting creases:

  • (1) Fixing and flattening the deformed sheet of paper on the working table by the holders;

  • (2) Forming the upcoming crease line on the deformed paper by the grippers.

Problems with fixation and flattening the deformed sheet of paper are related to kinetic friction between the shell structure of the sheet of paper and solid

Virtual testing of the origami robot

In this Section, we demonstrate possibilities of robotic folding paper for some real origami models by using the simulation. Folding steps are defined by the robot functions: rotation, translation, or flipping the sheet of paper, which represent manipulation sequences: folding paper according to one type of creases (mountain or valley) one by one; turning over the sheet of paper; folding paper again according to one type of creases (mountain or valley). Modifications of the design of the robot

Discussion and conclusions

We have presented a new study in the development of our research: the simulation-based approach for designing the origami-performing robot. The development of the general simulation–based methodology of folding origami structures by robot manipulations is considered as a basis for designing a robotic system. We consider implementation of the methodology of virtual origami folding for developing real robotic system. The methodology of robotic origami folding in FE simulation can be used as a

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