Robotics and the Mathematics of Origami
This week MIT announced further development towards self-assembling robots. Their advance is based on a trail of remarkable results in the mathematics of origami combined with breakthroughs in directionally controlling the folding behaviour of special materials under heat.
(For the impatient, bets are announced at the end...)
Unfolding a Journey in Applied Geometry
So how did this come about? And where is it going? Our Story unfolds over the past 15 years. Below are a few highlights, tracing backwards from yesterday's announcement to the early days of origami viewed as a branch of applied geometry.
2014: Bake Your Own Robot (MIT)
[LINK 1] >>> http://newsoffice.mit.edu/2014/bake-your-own-robot-0530
Here we see demonstrated a prototype of an end-to-end pipeline for creating self-assembling robots (May 2014)
Though the demonstrated pipeline is new, the concept for producing the robots is not, appearing out of MIT in 2010.
2010: Programmable Matter (MIT)
[LINK 2] >>> http://newsoffice.mit.edu/2010/programmable-matter-0805
Using special materials & folding algorithms, a new concept & mechanism for producing self-assembling robots is demonstrated. (Aug 2010)
What enabled all this?
2009: A Universal Folding Theorem (MIT CSAIL)
[LINK 3] >>> http://arxiv.org/abs/0909.5388
A universal folding theorem by Nadia Benbernou, a fifth year PhD student in applied mathematics at MIT, set up this branch of applied geometry to be launched into the world of self-assembling robotics.
A Universal Crease Pattern for Folding by Nadia Benbernou, Erik D. Demaine, et al, Sep 2009, (ArXiv, PDF Available for download)
http://people.csail.mit.edu/nbenbern/
Abstract:
We present a universal crease pattern--known in geometry as the tetrakis tiling and in origami as box pleating--that can fold into any object made up of unit cubes joined face-to-face (polycubes). More precisely, there is one universal finite crease pattern for each number n of unit cubes that need to be folded. This result contrasts previous universality results for origami, which require a different crease pattern for each target object, and confirms intuition in the origami community that box pleating is a powerful design technique.
2003-2008: The Gold Rush
Investigations into the mathematics of origami had been active well before this, in the fields of computational geometry, tilings, and computer aided design (CAD).
A surge of interest and productive activity took off in the early part of the millenium. Two examples illustrate this new energy & its applications.
2005: Automatic Cut-outs for Paper Models (Hartmut Prautzsch)
[LINK 4a] >>> geom.ibds.kit.edu/papers/cut-out-sheets.pdf
In 2005, Raphael Straub & Hartmut Prautzsch presented an Automatic Algorithm for Creating Cut-out Sheets for Paper Models at the SIAM Conference on Geometric Design & Computing.
A beautiful display of results (pictures) is here:
[LINK 4b] >>> http://i33www.ira.uka.de/pages/Research/Projects/PaperModels/index.html
2003 Mathematical Origami goes mainstream (Robert Lang)
[LINK 5] >>> http://www.amazon.co.uk/Origami-Design-Secrets-Mathematical-Methods/dp/1568811942
In 2003, Robert Lang, one of the world's foremost origami designers printed his book: Origami Design Secrets – Mathematical Methods for an Ancient Art, A K Peters, Ltd, 2003.
Review:
If you have ever wondered how complex origami models are created then look no further. Lang describes the theory behind mapping a real life object or animal to the flat square of a piece of origami paper. (RD Turner)
2001: The Tipping Point (Barry Cipra)
[LINK 6] >>> http://www.siam.org/pdf/news/579.pdf
Was there a tipping point to this surge?
One of the earliest widely read expository articles appeared in SIAM News by Barry Cipra, the famous writer on mathematics:
In the Fold, Origami Meets Mathematics, Barry Cipra, SIAM News, Vol 34, Number 8, Oct 2001 (PDF available for download)
<2001: The Pre-Years
Before this investigations were diffuse and appeared in the work of discrete mathematicians (networks, planar graphs), graph theorists (cycles, tours), operations researchers (optimal packing algorithms), topologists (polyhedra), applied geometers, computational geometers, geometrical design engineers, and 3D graphics engineers.
Where the Frontier now lies
[LINK 7] >>> http://i33www.ira.uka.de/pages/Home/index.html
There are now many centres where paper models and subdivision algorithms are a standard part of the applied geometry landscape.
In particular, the AG/CADG resarch group at the Karlsruher Institut fur Technologie in Germany, and in particular the work of Hartmut Prautzsch (IBM, RPI, KIT), integrates this newer branch of mathematics with the full breadth of classical & modern applied geometry, including paper models (mathematical origami) alongside NURBS (splines), subdivision algorithms (grids & meshes), point clouds, and more.
Epilogue
Notice how, over the space of just 15 years, we went from the diffuse proto-material in diverse investigations, through a tipping point triggered by expository writing, leading to widespread popularisation activity, and discoveries. This period then led to focused programmes driven by research centres in computational algorithms and applied geometry. But it was only a particular confluence of breakthroughs in both the technology side (materials science) and the computational side (algorithms), along with a killer-app' (self-assembling robots) that has led to these developments appearing in the public eye.
Where from here?
Whether the excitement grows or fades depends now on three factors:
(1) how readily these developments can be converted into operational piplines for producing industrial / commercial / home robots,
(2) how costly these pipelines are,
and, perhaps most importantly,
(3) what is the need that can accelerate the birth of this sci-fi inspired, mathematically enabled reality
Stay tuned as this field matures and engineering expertise is applied to designing, manufacturing, and operationalising product lines for the two sectors that drive such technologies: industrial-extreme and home-commercial.
My bet?
With the NASA objective still in place to see Mars colonised by 2035, let's just say that flat-packed, self-raising robots, IKEA-style, are a lot easier to stow than the pre-built variety.
Leave your bets in the comments...
#RoboticsTechnology #Mathematics #Robotics