Computational Laser Forming Origami


Yue Hao and Jyh-Ming Lien


Laser forming is a fabrication method that uses a laser to fold sheets into 3D structures. To overcome the limitations in the traditional practice that relies on tedious manual design, this work advances laser forming by developing computational methods that procedurally convert a polyhedron P into laser formable 2D patterns and folding instructions or report that P is not laser formable.

Laser Origami


Due to the limitation of the low-cost laser cutter considered in this paper, we will focus our discussion on laser forming convex surfaces. A 3D surface is called convex if the entire surface lies on the boundary of its convex hull. Our theoretical analysis shows that, even for convex surfaces, the laser formability can be expensive to determine. We then present a framework that efficiently computes patterns and motion instructions for laser forming convex surfaces. An end-to-end laser forming pipeline is presented with several fabrication results to demonstrate the capability and current limitations of the software and hardware framework.

Publications


Computational Laser Forming Origami of Convex Surfaces, Yue Hao and Jyh-Ming Lien, Proceedings of the ACM Symposium on Computational Fabrication (SCF), Jun 2019
Web Site / Paper(pdf) / Slides(pdf) / Poster(pdf) / BibTeX


Results


Curved Surface


Curved Surface


Convex Surface Model



Sinuous Antenna with Various Tessellations


Laser forming sinuous antennas with various resolutions. The fabrication time ranges between 4 and 5 minutes for all models. As the number of crease lines increases, the folding angle and number of laser passes decrease for each crease line. Thus the folding time is not significantly affected by the resolution. Substrate: 75 um thick stainless steel.

Sinuous Antenna in Various Tessellations


Video




Method


Laser Forming and Visibility Constraints


The laser cutter with a single laser head can translate on a 2D plane z = 0 and can only emit laser vertically downwards and focus on the z = 0 plane.

Laser Model


The picture on the right provides an example of how the existing folded structure might block the visibility between a crease line (the dashed line) and the laser.

L and L- Formability Analysis


Nets that are L− formable. Since L can only reach the sheet and focus on z = 0 from directly above. A direct consequence is that, if a net has more than one crease line whose dihedral angle is less than π/2, then the net is not laser formable by L. This is because that one of the face incident to such a crease will occlude the line-of-sight from above if the other face is fixed on z = 0 plane. Thus, the only way to fold such a crease will require both incident faces to fold but make sure that they fold less than π /2. Since L can only focus on z = 0, such a crease must only be folded at the end of the folding process. Therefore, we can only afford to have one crease line with an acute dihedral angle.

Formability


Outside-In Folding Motion


We propose a simple folding strategy that folds the net from creases that are near the perimeter of the net gradually into the center of the net. More specifically, because polyhedra nets are tree structures, we can always assign a face as a root of the tree and assign the children-parent relationships using BFS or DFS from the root face. Then the outside-in folding strategy simply requires the folding of child face into 3D before the parent face can be folded. Consequently, the leaves of the net will be folded first and the root face will be folded last (or simply fixed on z = 0).

Folding Motion



Single-Head Laser Path with Fake Parallelization


With one laser head, in order to fold multiple creases at the same time, we need a scheduling strategy to make it act as if we have multiple laser heads, we call this strategy fake parallelization.

Simulated Folding


In our fake parallelization algorithm, we give each crease a timer, starting with the initial value 0. Every time the laser passes through the crease, the timer increases by τ > 0 based on the laser strength, material and cooling conditions etc. Every time the laser passes through other creases or boundaries, the timer decreases by 1. At any given time t of the folding motion, we schedule the single-head laser path in a sequence based on the timers, the crease with lowest timer value less or equal to 0 goes first and so on so forth. If all the timers are greater than 0, we pause the laser motion or mark some random lines outside the shape to wait for the heat to dissipate. Using the fake parallelization scheduling strategy, we can fabricate the same model much more efficiently than sequential folding.

Related Work





Acknowledgement


This work is in part funded by ARO W911NF-19-2-0121. We thank Dr. Nathan Lazarus and Mr. Gabriel Smith at the Adelphi Lab Center of ARL who helped us fabricating the antennas and the curved surface.

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Computer Science @ George Mason University