Centimeter-scaled Self-Assembly: A Preliminary Study

Martin J

ılek, Miroslav Kulich and Libor P

reu

cil

Czech Institute of Informatics, Robotics, and Cybernetics, Czech Technical University in Prague, Czech Republic

Keywords:

Passive Self-assembly Systems, Robot Swarms, Multi-robot Systems, Smart Materials, Tile Assembly.

Abstract:

Passive self-assembly represents a general kind of bottom-up assembly process for objects, where the assem-

bling particles exhibit no explicit, active ”sense and affect” featuring, but embedded properties only. Although

the majority of the previous work in the ﬁeld has been performed on a microscopic scale, in the ﬁeld of

chemistry and nanotechnology, we identify a strong relation to macroscopic cases and principles studied in

robotics. We show that passive self-assembly processes might be promising also towards the development

of new, completely passive (multi)robot systems, driven entirely by environmental perturbations. This work

sketches insight into fundamental principles of driving a centimeter-scale self-assembly system while observ-

ing the behavior of single particles. Investigations show, that not all the principles previously studied in the

microscopic scale do hold also in the macroscopic cases of centimeter-scale particles. Thus, this article pro-

poses the macroscopic problem speciﬁcation and an experimental self-assembly system design consisting of

entirely passive elements. We tackle a theoretical description of the system model and the necessary simpliﬁca-

tion towards a two-handed tile assembly model (2HAM) together with real-world experimentation design. We

evaluate experimental results, discuss the feasibility of shake-driven macroscopic self-assembly, and elaborate

their major properties together with estimated future work.

1 INTRODUCTION

Self-assembly is a natural behavior of many systems

in our universe, from ensembles of elementary parti-

cles through biological systems to astronomical ob-

jects (Whitesides and Grzybowski, 2002) – yet it is

not utilized much past the nanometer scale. This is

probably going to change in future years when we

master the potential hidden in this technology. The

main principle of self-assembly, minimization of the

potential energy of the assembling system, makes it

very general and applicable at all scales, as long as

it is physically feasible to produce individual compo-

nents of the system which will assemble into larger

parts.

Research of self-assembly was pioneered in the

area of biology at the end of 20

century. Since the

beginning, self-assembling elements are often made

of DNA (like in (Winfree, 1998), (Chen and Seeman,

1991), (Jiang et al., 2017)). This, together with a

mathematical understanding of the process, led to the

design of systems where the result of the assembly

can be precisely controlled. These shapes range from

DNA assembly of Sierpinski triangle (Rothemund

et al., 2004) to an assembly of complex 2D and 3D

shapes using a technique called DNA origami (Rothe-

mund, 2006), (Douglas et al., 2009). Nowadays, self-

assembly is not only limited to DNA. For example,

colloidal self-assembly of spherical polystyrene par-

ticles can be used on the micrometer scale (McGorty

et al., 2010), (van Dommelen et al., 2018). Mi-

croscale also permits usage of capillary forces (like

in (Hosokawa et al., 1996)), which even caused fold-

ing of macroscale objects (Wei et al., 2016) and self-

assembly of tiles in a size of millimeters (Rothemund,

2000). Unlike most common manufacturing methods,

self-assembly offers high positioning accuracy, it also

scales very well with the number of assemblies we

need to build because of its parallel nature (Boncheva

and Whitesides, 2005).

1.1 Motivation

We explore the possibility of utilization of (at least

a part of) knowledge from the area of tile-based

self-assembly, as known in nanotechnology, in the

centimeter-scaled world of robotics. Our idea is a

construction of a universal assembling machine –

a machine assembling arbitrary structures based on

stochastic excitation of individual parts, which are en-

tirely passive. The target structure is encoded into the

structure of individual parts at the beginning of the ex-

438

Jílek, M., Kulich, M. and P

reu

cil, L.

Centimeter-scaled Self-Assembly: A Preliminary Study.

DOI: 10.5220/0009830104380445

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 438-445

ISBN: 978-989-758-442-8

periment and all the parts are being excited until they

form the target pattern.

Our long term practical goal is the utilization of

tile-based self-assembly methods in the manufactur-

ing of objects made from modular metamaterials, like

in (Ne

zerka et al., 2018). Such emerging materials

posses interesting physical properties (like negative

Poisson ration) and because their manufacturing re-

quires precise placement of a large quantity of geo-

metrically similar modules, we see this as an impor-

tant application of tile-based self-assembly methods.

1.2 Related Work

We are interested in the physical implementation of

simple models of tile-based self-assembly, like the

abstract tile assembly model (aTAM) which is al-

ready underlaid by a heavy body of theory. Such im-

plementations were already shown in chemistry and

nanotechnology as successful, nevertheless, in the

centimeter-scaled world there is not much literature

about the topic and most of it reports only assem-

bly of simple shapes, like in (Masumori and Tanaka,

2013), (Daiko Tsutsumi and Satoshi Murata, 2007),

(Egri and Bihari, 2018) or (L

othman, 2018).

Such passive self-assembly approaches can be

divided into two groups (Cademartiri and Bishop,

2015): puzzle-based and strand-based.

The puzzle-based approach operates with freely

moving tiles, which can bind together and form bigger

assemblies. This approach originated from biology

((Winfree, 1998), (Chen and Seeman, 1991), (Jiang

et al., 2017), (Rothemund et al., 2004), (Rothemund,

2006), (Douglas et al., 2009)). Here, elementary

building tiles are constructed using DNA. Most im-

plementations use rectangular tiles, where each side

has some kind of binding site. The type of the site en-

sures reactivity just with the matching sites of other

tiles. This passive branch of self-assembly is backed

by mathematical models and guarantees on assembly

feasibilities and complexities. One of the most theo-

retically explored models is the abstract tile assembly

model (aTAM), which models the tile system as a set

of massless non-rotating tiles, binding themselves to

seed according to their binding site parameters and a

system-wide temperature parameter (Winfree, 1998).

Possible generalization of aTAM is a 2HAM model

(Patitz, 2014), which removes the need for seed and

enables interaction of all assemblies in parallel.

The strand-based approach is inspired by protein

folding. Tiles are the same as in the puzzle-based as-

sembly but are also connected to the 1D chain. This

can lower the number of encoded speciﬁc interactions

in the system but poses another difﬁculty during the

design of ﬂexible interconnections of tiles in the chain

(Cademartiri and Bishop, 2015). It was proven that it

is possible to ﬁll any continuous area or volumetric

shape with such a technique (Cheung et al., 2011).

Such passive systems have several plausible prop-

erties. They can be manufactured cheaply, which per-

mits large scale experiments. They are an interest-

ing addition to manufacturing technology, offering a

possibility to assemble 3D parts too small for con-

ventional manufacturing. (Boncheva and Whitesides,

2005) They also mimics fundamental processes on

which our world is built upon. Research of the behav-

ior of such a system can result in knowledge transfer-

able to other scientiﬁc areas like biology. (Boncheva

et al., 2003)

1.3 Contributions

The main contribution of this article is the devel-

opment of a prototype of a passive robotic swarm

platform exhibiting self-assembly behavior. The sys-

tem resembles the theoretical models of the tile-based

self-assembly. It consists of a set of independent (and

completely passive) tiles and a reactor, in which the

assembly takes place.

We tested the system on two assembly scenarios

resembling the sHAM model, which we derived from

the commonly used 2HAM model. The results lead us

to the conclusion that the assembly of at least simple

structures is possible in a reasonable time.

1.4 Content of the Article

The article is structured as follows. In Section 2 we

described individual steps in the design of the self-

assembly system. Firstly, we propose a simpliﬁed

model of the tile-based self-assembly system, derived

from the 2HAM model. Secondly, the mechanical de-

sign of the system is described along with a theoret-

ical model of tile-tile interaction. In the third part,

we describe the reactor used to supply kinetic energy

to the system during experiments. The last part de-

scribes the visual tracking system. Section 3 qualita-

tively evaluates the behavior of our prototype. All the

results are summarized in section 4.

2 METHODOLOGY

Our research question can be formulated as follows:

is it possible to create a centimeter-scale passive

robotic system exhibiting a self-assembly behavior, as

described by the 2HAM model?

Centimeter-scaled Self-Assembly: A Preliminary Study

439

Informally, tile-based self-assembly assumes that

the assembling system consists of a set of tiles

(Fig. 1), moving freely in a reactor (either in 2D or

3D). Sides of the tiles contain binding sites, which

can cause that two tiles stick together upon contact.

The binding sites can be of different types, which re-

stricts the bonding interactions between the tiles, and,

as a result, the assembly will result in different ob-

jects. Many mathematical models of such a process

exist, as in (Patitz, 2014). We derived our own assem-

bly model (based on assumptions closer to the physi-

cal reality) in the ﬁrst part of the work. The physical

prototype was designed in a way that resembles the

model as much as possible. We also proposed a solu-

tion to the problem of ﬁnding sets of suitable encod-

ings of tile-tile interactions. Constructed prototype

was placed into the reactor (section 2.4) equipped

with visual tracking system (section 2.5) and several

experiments were performed.

2.1 Formal Self-assembly Model

Our tile design is inspired by a group of often men-

tioned tile assembly models, namely abstract tile as-

sembly model (aTAM) (Winfree, 2006), kinetic tile

assembly model (kTAM) (Winfree, 2006) and, espe-

cially, two-handed assembly model (2HAM) (Patitz,

2014). These models have some limiting restrictions,

but are often mentioned in literature and a large cor-

pus of a theory is dedicated to them.

Informally, the 2HAM model represent tiles as

non-rotating objects with 4 sides (see Fig. 1a), placed

on integer-valued lattice, which can bind to other

(matching) sides or assemblies iff the resulting bond

will be strong enough (as in Fig. 1b). Nevertheless,

the 2HAM model forbids rotation of the tiles and as-

sumes an inﬁnite number of tiles (which is not our

case). Thus, we present a relaxed variant, the simpli-

ﬁed two-handed assembly model:

Deﬁnition 2.1 (sHAM). Simpliﬁed two-handed as-

sembly model (sHAM) is a tuple (T,S, I, τ), where T

is a set of tile types, S is an initial state, I is a glue

strength function and τ ∈ N is a system temperature.

Deﬁnition 2.2 (Tile type). Tile type t is a 5-tuple of a

label l and four glues (l, σ

,σ

), where σ ∈

Σ. Each glue is associated with one direction from

{(0,1),(−1,0),(0,−1),(1,0)}.

Deﬁnition 2.3 (Tile instance). Each tile instance is

a tuple (t, p), where p is a position of a tile in an

integer-valued unit 2D orthogonal lattice. Tile type

t of a particular tile instance can change during the

experiment because of a rotation - glue assignment in

t will shift to the left or to the right. The position p

can change because of a movement of a tile instance.

Each position in the sHAM lattice can be occupied by

at most one tile instance.

Deﬁnition 2.4 (Initial state). The initial state S is a

multiset of tile instances.

Deﬁnition 2.5 (Interaction of glues). Two tile in-

stances a, b are in neighborhood iff their distance is

equal to one. Glues g

of a, b can interact iff a,b

are in neighborhood and iff for direction vectors

and

, associated with g

, holds

= −

. Let

all such pairs of glues be called interacting glues of

a,b.

Deﬁnition 2.6 (Interaction strength). Interaction is

ruled by I : Σ ×Σ → Z, where Σ is the set of glue

types (arbitrary symbols, we restricted ourselves to

integers). Two tiles (or assemblies) can form a stable

assembly iff the sum of strengths of all their interact-

ing glues is at least equal to τ.

An example of one such tile and a whole assembly

is shown in Fig. 1.

North

South

West

East

Label

(a) Abstract model of

a tile.

N:1

S:2

W:1

E:2

N:1

S:2

W:1

E:2

N:1

S:2

W:1

E:2

N:1

S:2

W:1

E:2

(b) Example of an as-

sembly.

Figure 1: Model of a tile (1a) and example of a valid as-

sembly. Each instance of tile type carry 4 glues on its side.

The tiles form bigger assemblies on an orthogonal 2D lat-

tice whenever matching glues (1,2) meet together and their

combined strength is higher than external disturbances (1b).

2.2 Design of a Tile

Tile design signiﬁcantly contributes to the success of

the assembly. All tiles are geometrically the same and

roughly follow the 4-sided rectangular shape of the

tile of our idealized model (Figure 1a). The imple-

mentation of the bonding mechanism is purely mag-

netic. Each side of the tile has three channels for re-

placeable magnet holders. The conﬁguration of mag-

nets on each side of each tile determines the behavior

of the system.

Since our physical realization encodes the interac-

tions into 2D arrangements of magnets, it could hap-

pen that two such binding sites stick together only

partially, which leads to an invalid assembly. Thus,

the sides of our experimental tiles are curved to pre-

vent horizontal misalignment. The shape is inspired

by (Haghighat et al., 2015), but is more curved, which

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

440

makes the potential energy landscape of the system

smoother. Thus it contains less local minima which

can trap the evolution of the system in an undesired

state. Moreover, to prevent vertical shifts of magnetic

codes, the tile is also equipped with vertical locks.

Vertical

locks

AprilTag

Holes for

magnet holders

Magnet

holder

(a) (b)

Figure 2: The physical realization of a tile (left) and a com-

plete set of magnet holders that encode bond on one side of

the tile.

The width of the tile is 3 cm, height 1.5 cm.

Weight is 6.28 g. Tiles were printed from PLA plas-

tics on a Prusa MK3S printer with a 0.4 mm extruder.

No inﬁll was used to make them as light as possible

for possible future experiments in a liquid environ-

ment. The printing of one tile without magnet hold-

ers takes 54 minutes. AprilTag (Krogius et al., 2019)

ﬁducial marker is glued on top of each tile for au-

tomatic evaluation of experiments via a vision-based

system.

2.3 Encoding of Interactions between

Tiles

Each side of each tile carries code assembled from

permanent magnets. The magnets are assembled into

a 3 ×3 matrix with various orientation of south and

north poles. Formally:

Deﬁnition 2.7 (Glue plate). Glue plate A is a matrix

of size m ×n. Glue plate contains numbers from the

set {−1,0, 1}, where −1 is a magnetic south pole, 1 a

north pole and 0 no magnet.

Let us propose the (abstract) magnetic interaction

model, which assumes that magnets on glue plates in-

teract only when the tiles touch together and neglects

the effect of magnetic crosstalk:

Deﬁnition 2.8 (Magnetic interaction model). Let H

be a matrix formed as a Hadamard product of two

glue plates A,B with dimensions m ×n:

= A B (1)

Interaction strength between interacting (neighbor-

ing) glue plates A and B is then:

F(A, B) = −

∑

r=0

∑

c=0

(2)

Interaction between non-interacting (non-

neighboring) plates is 0.

Thus, for the practical implementation of a tile

assembly model, we need to determine all the glue

plates in the system. In simple cases (where only a

small number of glue plates is needed) they can be de-

signed manually. For complex cases, the task can be

formulated as a constraint satisfaction problem, where

we search for the set of glue plates satisfying con-

straints given by the Deﬁnition 2.8.

Magnetic plates were realized with the use of Nd-

FeB magnets (N50) with a size 3 ×3 ×1 mm. The

magnets are arranged into bars of size 18 ×5 ×1.6

mm. Each such bar can support 3 magnets. An exam-

ple of 3 such bars, ready to be placed into the holder

on one side of a tile, can be seen in Figure 2b. Each

magnet bar, printed from PLA with a 0.4 mm ex-

truder, weights 0.14 g. The printing time of each is

2.5 minutes.

2.4 Design of the Reactor

The purpose of the reactor (Fig. 3) is to supply me-

chanical energy to the system and to achieve a suf-

ﬁciently random movement of tiles. For our initial

experiments, we constructed a circular cardboard re-

actor (with a diameter of 30 cm), which is mounted

on a 6DoF robotic manipulator (UR5) to provide ex-

citation energy. The reactor is also equipped with a

camera, observing the process from the top (such that

the optical axis is perpendicular to the plane on which

the tiles move).

Figure 3: The reactor mounted on a robotic manipulator.

The reactor coordinate frame, used through this work, is

marked.

Centimeter-scaled Self-Assembly: A Preliminary Study

441

2.5 Visual Tracking of a Self-assembly

Process

A possibility of observation of the self-assembly on

the scale of individual tiles in real-time is one of the

reasons why macroscopic self-assembly could bring

insight into similar processes on a smaller scale. It

could be also used for ﬁne-grained feedback control

of the reactor.

Our setup assumes that the motion of each tile is

planar and that the excitation energy is provided by

the movement of the whole reactor. Because of that,

the camera was ﬁxed to the reactor above the plane

of motion of tiles such that its optical axis is per-

pendicular to the plane. This solution keeps all the

tiles in the focal plane. The ﬁrst step of the visual

Model of

a system

Assembly

detection

Filter

...

AprilTag

detector

2D, 3D poses

Frame 1

AprilTag

detector

Frame n

2D, 3D poses

Figure 4: Visual tracking pipeline.

tracking pipeline (Fig. 4) is the detection of AprilT-

ags (with the usage of apriltags3-py library) on each

frame from the camera. It outputs poses of tiles in 3D

and also in image coordinates. The ﬁltering stage then

linearly interpolates missing values (where the detec-

tor failed to identify the tag) and the kinematic model

of the system is ﬁnally used to perform assembly de-

tection. The kinematic model of a system is similar to

the abstract model (Deﬁnition 2.1). Each tile instance

is modeled as a matrix T , where s is the width of the

tile. Each column d

represents the position of one

glue w.r.t. the local coordinate frame of the tile:

T =



−s



(3)

The tracking system determines tile translation

t and

rotation R. Let positions of glues of tiles 1 and 2 in

the camera reference frame be

= RT

(4)

The bond between tiles 1, 2 is detected only be-

tween glues with position vectors (w.r.t. camera

frame)

which satisfy the following conditions:

−

| < ε

trans

(

−

) ·(

−

) < ε

rot

(5)

where ε

trans

,ε

rot

are predeﬁned thresholds. Moreover,

the detector also permits detection of bonds using the

same criteria, but in image coordinates (which can be

beneﬁcial with noisy observations).

This procedure outputs one bonding graph G

(V,E) for each frame, where V is a set of tile instances

and E is a set of edges formed between pairs of tiles

with stable bonds. All edges in G

must satisfy condi-

tions (1) and (2). Complex assemblies are then found

as connected components of G

3 RESULTS

We designed two experiments to test the system. In

both scenarios, a set of 20 tiles was placed into the

reactor, which performed back-and-forth movement

along the reactor x-axis with small swings around y-

axis. The amplitude of linear acceleration along the

x-axis was chosen as a parameter related to the system

temperature. It was experimentally set to the lowest

value where the individual tiles overcome the friction

forces and start moving (almost) independently of the

reactor (see Fig. 5). The amplitude of the position

change (Fig. 6) during movement was ﬁxed at a value

where the tiles start to collide with opposing walls of

the reactor. The reactor was slightly rotated around

its x-axis during the whole motion to keep the tiles in

one cluster to maximize the probability of collisions.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Time (s)

−15

−10

−5

Acceleration (m/s

)

Figure 5: Acceleration of the reactor measured in prevailing

(x) axis of movement.

Glues used in both experiments were of type G

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

442

Figure 6: Motion of a manipulator.

and G

. In a formalism of the Deﬁnition 2.8:





0 0 0









0 0 0

0 1 0

0 0 0









0 0 0

0 −1 0

0 0 0





(6)

Both experiments were repeated with different

random initial placement of tiles and evaluated by the

visual tracking system incorporated into the assembly

reactor.

3.1 Self-assembly of Boxes

In this experiment, we aimed for an assembly of boxes

composed of 2 ×2 tiles. Our system consisted of one

tile type (Fig. 7a), with two types of glues, which

permits four assemblies to be present during the as-

sembly process (Fig. 7b).

(a)

(b)

Figure 7: Tile model and a set of possible products (A, B,

C, D) for the box assembly experiment.

The experiment was ended when the number of

completed boxes stabilized for a sufﬁciently long

time. Equidistantly sampled images of the process

are visible in Figure 8. Results of automatic analysis

of the whole experiment are shown in Fig. 9 and Fig.

10.

Figure 8: Attempt to self-assemble 2 ×2 boxes. The images

(from left to right, top to bottom) were captured in approx.

24-second intervals.

Time (s)

100

120

140

Assembly size (-)

Number of assemblies (-)

Figure 9: Number of assemblies of all possible sizes during

the assembly of boxes in the experiment depicted in Fig. 8.

0 20 40 60 80 100 120 140

Time (s)

Size of largest assembly (-)

Figure 10: Size of the largest assembly present at each

timestamp during three runs of the assembly of boxes. The

spike at 30 s is an artifact caused by the visual tracking sys-

tem.

3.2 Self-assembly of Linear Chains

This experiment explores the behavior of a tilesystem

which is designed to form linear segments of unlim-

ited length. Tiles were designed according to the Fig-

ure 11a. Evaluation of assembly progress resulted in

ﬁgures 13 and 14.

Centimeter-scaled Self-Assembly: A Preliminary Study

443

(a)

...

(b)

Figure 11: Tile model (a) and a set of possible linear assem-

blies (A, B, C, D, ...) for the chain assembly experiment (b).

Figure 12: Attempt to self-assemble linear segments of un-

limited length. The images (from left to right, top to bot-

tom) were captured in approx. 19-second intervals.

Time (s)

105

120

Assembly size (-)

Number of assemblies (-)

Figure 13: Number of assemblies of all possible sizes dur-

ing the assembly of linear chains as in Fig. 12.

0 50 100 150 200 250

Time (s)

Size of largest assembly (-)

Figure 14: Size of the largest assembly present at each

timestamp during four runs of the assembly of linear chains.

3.3 Observations

The following facts are visible from both types of ex-

periments and captured data:

• Single tiles are highly reactive, after the start of

the experiment almost all of them are incorporated

into dimers in a few seconds.

• L-shaped trimers appear after the concentration

of dimers exceeds a certain limit. Nevertheless,

they are not stable and quickly form tetramers or

dissolve into dimers and monomers.

• Box-shaped tetramers are extremely stable.

They exhibit a kind of self-repairing behavior.

Their destruction often results in two dimers,

which again form the same tetramer in a matter

of seconds.

• Longer chains tend to form clusters, where all

assemblies are parallel. This leads to a higher sta-

bility and self-repair behavior because chains in

these clusters have a degree of structural ﬂexibil-

ity.

• Both experiments ended up in equilibrium,

which was similar for different trials of the same

experiment.

4 CONCLUSIONS

This paper presents a novel macroscopic robotic sys-

tem capable of tile-based 2D self-assembly together

with preliminary experiments. The system is com-

posed of entirely passive elements – tiles – whose in-

teractions are ﬁxed before the start of the assembly.

The whole system is continuously excited by uncon-

trolled mechanical disturbances from the environment

and the formation of patterns (pre-programmed into

interactions between tiles) is observable as the poten-

tial energy of the system decreases.

Our experimental results suggest that the pro-

grammable tile-based assembly is feasible in the

centimeter-scaled world. Unlike our expectations, the

macroscopic self-assembly of simple shapes is fast

(assembly times of structures containing 4-6 tiles are

in order of tens of seconds) and assemblies are sta-

ble on time intervals of seconds for unstable shapes to

tens of seconds for stable shapes.

Future work should be focused on multiple ar-

eas: (1) Assembly of more complex shapes within the

framework of aTAM and kTAM models, (2) real-time

control of the external disturbance to optimize assem-

bly times, (3) improvement of the tracking system,

which in the current state requires manual prepara-

tion (search for optimal detection thresholds, precise

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

444

lightning of the experiment) when used with a noisy

or low-resolution camera. (4) The evolution of the

system should be also studied over a longer time-scale

since some experiments (like in Figure 12) suggest the

presence of cyclic behavior. (5) Future work should

also focus on the quantitative analysis of the process

in the context of the proposed model.

ACKNOWLEDGEMENTS

The research leading to these results has re-

ceived funding from the Czech Science Foundation

(GA

CR) under grant agreement no. 19-26143X.

The work of Miroslav Kulich has been supported

by the European Regional Development Fund un-

der the project Robotics for Industry 4.0 (reg. no.

CZ.02.1.01/0.0/0.0/15 003/0000470).

REFERENCES

Boncheva, M., Bruzewicz, D., and Whitesides, G. (2003).

Millimeter-scale self-assembly and its applications.

Pure Appl. Chem, 75:621–630.

Boncheva, M. and Whitesides, G. (2005). Making things by

self-assembly. MRS Bulletin, 30:736–742.

Cademartiri, L. and Bishop, K. J. M. (2015). Programmable

self-assembly. Nature materials, 14 1:2–9.

Chen, J. and Seeman, N. C. (1991). Synthesis from DNA

of a molecule with the connectivity of a cube. Nature,

350:631–633.

Cheung, K. C., Demaine, E. D., Bachrach, J. R., and Grif-

ﬁth, S. (2011). Programmable assembly with univer-

sally foldable strings (moteins). IEEE Transactions on

Robotics, 27(4):718–729.

Daiko Tsutsumi and Satoshi Murata (2007). Multistate part

for mesoscale self-assembly. In SICE Annual Confer-

ence 2007, pages 890–895.

Douglas, S. M., Dietz, H., Liedl, T., Hogberg, B., Graf, F.,

and Shih, W. M. (2009). Self-assembly of DNA into

nanoscale three-dimensional shapes. In Nature.

Egri, S. and Bihari, G. (2018). Self-assembly of magnetic

spheres: a new experimental method and related the-

ory. Journal of Physics Communications, 2.

Haghighat, B., Droz, E., and Martinoli, A. (2015). Lily: A

miniature ﬂoating robotic platform for programmable

stochastic self-assembly. In 2015 IEEE International

Conference on Robotics and Automation (ICRA),

pages 1941–1948.

Hosokawa, K., Shimoyama, I., and Miura, H. (1996).

Two-dimensional micro-self-assembly using the sur-

face tension of water. Sensors and Actuators A: Phys-

ical, 57(2):117 – 125. Ninth International Workshop

on Micro Electro Mechanical System.

Jiang, S., Hong, F., Hu, H., Yan, H., and Liu, Y. (2017).

Understanding the elementary steps in DNA tile-based

self-assembly. ACS Nano, 11(9):9370–9381. PMID:

28813590.

Krogius, M., Haggenmiller, A., and Olson, E. (2019). Flex-

ible layouts for ﬁducial tags. In 2019 IEEE/RSJ In-

ternational Conference on Intelligent Robots and Sys-

tems (IROS), pages 1898–1903.

othman, P. A. (2018). Macroscopic magnetic self-

assembly. PhD thesis, University of Twente, Nether-

lands.

Masumori, A. and Tanaka, H. (2013). Morphological com-

putation on two dimensional self-assembly system.

In ACM SIGGRAPH 2013 Posters, SIGGRAPH ’13,

New York, NY, USA. Association for Computing Ma-

chinery.

McGorty, R., Fung, J., Kaz, D., and Manoharan, V. N.

(2010). Colloidal self-assembly at an interface. Mate-

rials Today, 13(6):34 – 42.

zerka, V., Somr, M., Janda, T., Vorel, J., Do

r, M.,

Anto

s, J., Zeman, J., and Nov

ak, J. (2018). A jigsaw

puzzle metamaterial concept. Composite Structures,

202:1275–1279.

Patitz, M. J. (2014). An introduction to tile-based self-

assembly and a survey of recent results. Natural Com-

puting, 13(2):195–224.

Rothemund, P. W. K. (2000). Using lateral capillary forces

to compute by self-assembly. Proceedings of the Na-

tional Academy of Sciences, 97(3):984–989.

Rothemund, P. W. K. (2006). Folding DNA to create

nanoscale shapes and patterns. Nature, 440:297–302.

Rothemund, P. W. K., Papadakis, N., and Winfree, E.

(2004). Algorithmic self-assembly of DNA Sierpin-

ski triangles. PLoS Biology, 2:1021 – 1024.

van Dommelen, R., Fanzio, P., and Sasso, L. (2018). Sur-

face self-assembly of colloidal crystals for micro- and

nano-patterning. Advances in Colloid and Interface

Science, 251:97 – 114.

Wei, S., Qin, M., and Zhang, J. (2016). Mechanism

and application of capillary-force self-assembly mi-

cro/nanofabrication. In 2016 IEEE International Con-

ference on Manipulation, Manufacturing and Mea-

surement on the Nanoscale (3M-NANO), pages 256–

260.

Whitesides, G. M. and Grzybowski, B. A. (2002). Self-

assembly at all scales. Science, 295:2418–2421. 793.

Winfree, E. (1998). Algorithmic self-assembly of DNA. PhD

thesis, Caltech.

Winfree, E. (2006). Algorithmic self-assembly of DNA. In

2006 International Conference on Microtechnologies

in Medicine and Biology, page 4.

Centimeter-scaled Self-Assembly: A Preliminary Study

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