An evolved robot generates forward movement alone (a,e), and then attaches
with clones (b,f) to form a fractal robot (c,g) with the same behavior at a
larger size scale (d,h).

# Scale invariant robot behavior with fractals

Author Names Omitted for Anonymous Review. Paper-ID [add your ID here]  Sam
Kriegman1, Amir Mohammadi Nasab2, Douglas Blackiston3, Hannah Steele2,
Michael Levin3, Rebecca Kramer-Bottiglio2, Josh Bongard1 1University of
Vermont, 2Yale University, 3Tufts University

###### Abstract

Robots deployed at orders of magnitude different size scales, and that retain
the same desired behavior at any of those scales, would greatly expand the
environments in which the robots could operate. However it is currently not
known whether such robots exist, and, if they do, how to design them. Since
self similar structures in nature often exhibit self similar behavior at
different scales, we hypothesize that there may exist robot designs that have
the same property. Here we demonstrate that this is indeed the case for some,
but not all, modular soft robots: there are robot designs that exhibit a
desired behavior at a small size scale, and if copies of that robot are
attached together to realize the same design at higher scales, those larger
robots exhibit similar behavior. We show how to find such designs in
simulation using an evolutionary algorithm. Further, when fractal attachment
is not assumed and attachment geometries must thus be evolved along with the
design of the base robot unit, scale invariant behavior is not achieved,
demonstrating that structural self similarity, when combined with appropriate
designs, is a useful path to realizing scale invariant robot behavior. We
validate our findings by demonstrating successful transferal of self similar
structure and behavior to pneumatically-controlled soft robots. Finally, we
show that biobots can spontaneously exhibit self similar attachment
geometries, thereby suggesting that self similar behavior via self similar
structure may be realizable across a wide range of robot platforms in future.

##  I Introduction

Fractals are ubiquitous in nature [20] and increasingly prevalent in
artificial structures [23, 4]. Coastlines, rivers, and trees all exhibit self-
similar structures—accurate replicas of themselves within themselves. Fractals
are also prevalent across length scales in animals and cellular structures:
respiratory and vascular systems, brains [11], and DNA [16] all exploit space
filling fractal networks. At the largest known length scale, the universe
itself consists of a fractal clustering of spiraling galaxies [31]. However,
the ways in which fractals may be useful in robotics is mostly unexplored.

One desirable property of fractals derives from their ability to represent
infinite complexity within a compact representation: a simple rule applied to
itself, recursively. Hornby and Pollack [7] exploited this phenomenon in
robotics by evolving branching kinematic chains in simulation and then
building physical instances of the most promising designs [8]. The resulting
robots yielded an order of magnitude more physical elements than any other
simulated robots at that time: Whereas Sims [28] and Bongard and Pfeifer [1]
evolved simulated robots composed of no more than 14 and 50 parts,
respectively, Hornby and Pollack [7] utilized Lindenmayer-systems [17] to
produce branching structures of up to 350 parts.

The ubiquity of self-similar forms in living systems suggests that fractals
may confer adaptive benefits in addition to reduced descriptive complexity,
which could be useful in artificial systems such as robots. One such potential
use of fractals in robotics is that self similar structure can, in some cases,
result in self similar behavior. Branching arteries and capillaries maximize
efficient blood flow at the macro and micro scales, respectively. In the
gecko, nanoscale spatulae branch from microscale setae, which in turn branch
from toes, which in turn branch from feet, and together maximize the
probability of adhesion of spatulae, setae, toes, and ultimately the animal
itself to vertical surfaces. However, with the exception of artificial gecko
feet [32] and Moravec and Easudes’ hypothetical trillion-fingered Bush Robot
[21], this particular property of fractals remains unexplored in robotics.
Here, we explore whether self similar structure can confer self similar
behavior in modular robots.

Conventional robots contain numerous smaller components (legs, wheels, end
effectors, servomotors, sensors, battery packs) but these parts are highly
specialized, interdependent, and incapable of independent or self-similar
behavior. Modular robots differ in that they consist of repeated robotic
elements that are to some degree self-sufficient, able to behave and survive
on their own. They often autonomously detach, move about each other, and
reattach to reconfigure the robot’s overall geometry [24, 9, 22]. But robot
modules tend to be indivisible cubes or other simple geometries that do not
reprise the shape of the whole robot, unless the robot’s overall structure is
a cube of cubes. In all cases, the target behavior of a module is, by design,
different from that of the whole robot.

Figure 1: Fractalizing mass-produced robots. As an example of how basal robots
are here combined in fractals, the well-known Menger sponge is shown in place
of an evolved design (1 sponge) at three fractal scales. A single (basal)
sponge is a 3-by-3-by-3 cube of voxels with the middle voxel missing on each
face, and no center voxel (a total of 20 voxels). Replacing each voxel with a
copy of the entire sponge creates a fractal: a self-similar aggregation of 20
sponges. Repeating this operation yields larger and larger fractals.

Swarms of hundreds or thousands of individual robots [25] have loosely coupled
to form amorphous phototatic aggregates [15] or moved within a shared shell
[26], however these robots have largely been restricted to cylindrical
geometries and 2D interaction in-plane. Most robot swarms comprise rigid-
bodied electromechanical robots which, in order to be produced in large
quantities, are designed to be as simple as possible. A consequence of this is
that each robot may be more or less behaviorally static (functionless) in
isolation [15, 26].

Here, we test whether self-similar forms can facilitate the evolution of scale
invariant behavior as follows. Shapes of basal robots (e.g. Fig Scale
invariant robot behavior with fractalsa) are evolved in a voxel-based
simulation; each design can be composed with copies of itself, using the same
pattern, indefinitely to realize self-similar form at a range of size scales.
However, if the desired behavior is lost and needs to be relearned de novo at
each level of recursion, then training such a system to operate on more than
one size scale becomes computationally infeasible. Thus, an evolutionary
algorithm is here employed to find behaviorally scale-invariant fractal
robots: they must demonstrate similar behavior, normalized for scale, at three
different size scales. The results demonstrate that robots with this property
can be automatically designed and in some cases, manufactured in reality.

In the first attempts to transfer simulated scale invariant behavior to a
physical system, computer-generated mold designs were 3D printed and then used
to rapidly fabricate identical, hollow elastomeric modules that can be
de/pressurized to induce volumetric actuation [14]. Modules behaved in
isolation and together when attached to form a fractal soft robot.

Scale invariant fractal design principles could also provide novel solutions
to challenges facing in vitro bioengineering. One such challenge is the size
limit of biobots without vascular systems in which diffusion does not bring
sufficient oxygen and other key metabolites beyond approximately 1 mm. As
proof of principle, we show that living xenobots, motile biological machines
built from amphibian stem cells, can be mechanically joined forming permanent
multi-individual subunits that do not need vacular systems because they always
keep their cells close to an interface with a nutrient medium. These subunits
could then be stacked and multiplexed to form increasingly large fractal
biobots.

Figure 2: Evolved single-actuator fractal. An asymmetrical quadrupedal fractal
was the best fractal found by evolution when every basal robot (a) must
synchronize their volumetric actuation in time. The entire structure (b) can
thus be treated as a single bladder, at any size scale, pressurized via a
single air inlet. Videos: the simulation; the physical robot at the basal and
middle scales. At atmospheric pressure, the basal robot fits inside a
2$\times$2$\times$2 cm cube; its chamber walls are 2 mm thick.  Figure 3:
Evolved fractal robots. Four designs (columns; a-b) evolved for scale
invariant locomotion in fractal aggregations at three size scales (rows): one
robot (top; 3 cm wide), robots inside of a robot (middle; 9 cm), and robots
inside of robots inside of a robot (bottom; 27 cm) (video). Voxels are 1 cm in
length in simulation. Cyan and red voxels actuate in antiphase at 5 Hz. The
fitness of a scale invariant design is the least amount of net displacement it
generates across the three size scales (body lengths per minute; Eq. 2).
Figure 4: Manufacturing fractal robots. (1) 3D printed mold was partially
filled with Dragon Skin 10 Slow and then placed in a vacuum oven to degas for
5 minutes. (2) Second 3D printed mold was inserted into the first mold slowly
to avoid introducing air bubbles into the silicone. (3) Flanges on the second
mold were adjusted on the first mold to keep a 2 mm clearance between the
walls of the molds. Silicone was cured in the oven at 60∘C for 75 minutes. (4)
The module was removed from the mold and the edges were trimmed. (5) The
module was flipped over an uncured layer of silicone and its top side was
sealed. (6) The module was removed from the substrate and its edges were
trimmed. (7a) A hole was punched in one of the sides of the module and a
rubber tube was inserted and glued using Sil-poxy as the bonding agent to make
a robot (video). (7b) Basal robots join together to form a fractal robot
(video).

Figure 5: Evolving scale-invariant behavior at three fractal scales. The mean
fitness of the best design (in body lengths per minute) is plotted for ten
independent evolutionary trials (95% confidence intervals). Fitness was taken
to be the worst performance out of the three size scales: a single robot (3
cm), a fractal aggregation of robots (9 cm), and a fractal aggregation of
fractal aggregations of robots (27 cm). The null hypothesis is that of no
evolutionary improvement in fitness (scale-invariant locomotion). On the basis
of two samples (randomly-generated designs (generation 0) versus evolved
designs (generation 325))—and a distribution-free rank sum test (Wilcoxon)—we
reject the null hypothesis of no evolutionary effect ($p<0.001$).

Figure 6: Evolving scale-invariant behavior without the fractal assumption.
The mean fitness of the best design (in body lengths per minute) is plotted
for 15 independent evolutionary trials (95% CIs). Fitness is the worst
performance out of the three fractal size scales (3 cm, 9 cm, 27 cm). The null
hypothesis of interest is that there is no difference in fitness (scale-
invariant locomotion) whether or not fractals are imposed. Thus, we have two
samples: one that assumes fractals (the performance curves in Fig. 6) and
another that does not explicitly utilize fractals (the curves here in Fig. 6)
(video). Using a distribution-free rank sum test (Wilcoxon), we reject the
null hypothesis that there is no evolutionary effect of fractals ($p<0.001$).

Figure 7: Hausdorff dimension across ten evolutionary trials. Smooth fractal
geometries such as points, straight lines, flat planes, and cubes have
Hausdorff dimension 0, 1, 2 and 3, respectively. More rough fractal surfaces
have non-integer Hausdorff dimension $(H)$. Each individual color plots $H$ of
the most scale invariant design, at each generation, of a single evolutionary
trial. Half of the trials ended in a plane of voxels ($H=2$), the the other
half found rougher, more geometrically interesting designs with $2.5<H<2.7$.

##  II Methods

###  II-A Fractalizing robots.

Robots in this paper are polycubes with at most $c$ cubes (voxels) connected
face-to-face inside a bounding box with length $m$ voxels. A fractal
aggregation is made by replacing each voxel in a basal robot with a copy of
itself. This operation can be repeated (to form an aggregate of aggregates of
elements) either recursively—replacing voxels with the current aggregate—or
iteratively—replacing voxels with the basal robot. We use iteration here to
achieve additional size scales with fewer elements. However, the results of
this paper are not reliant on the distinction between recursion and iteration.

The Menger sponge provides a familiar example (Fig 1). A basal sponge is a
box, $m=3$ voxels in length, with a hole through the center of each plane.
Replacing each of the $c=20$ voxels within an basal sponge generates a fractal
sponge of 20 sponges. Repeating this operation a second time generates a
sponge of 20 sponges composed of 20 smaller sponges (400 sponges in total).
And so on.

The Menger sponge has Hausdorff dimension:

$H=\log_{m}(c)=\frac{\log c}{\log m}=\frac{\log 20}{\log 3}\approx 2.727,$ (1)

which describes the space filled by a fractal geometry in the limit. The
Hausdorff dimension is zero for a single voxel; one for a column of three
voxels $(\log_{3}(3)=1)$; two for a 3-by-3 plane of nine voxels
$(\log_{3}(9)=2)$; and three for a 3-by-3-by-3 cube of 27 voxels
$(\log_{3}(27)=3)$. The design space here spans these (smooth) structures and
many more in between. Hausdorff dimension provides an interpretable measure of
their structural complexity (or “roughness”). However since all fractals here
have $m=3$, $H\propto\log c$.

###  II-B Simulating fractal robots.

Voxels are simulated by a point mass and up to six Euler-Bernoulli beams
connecting to neighbors on each voxel face. There are two kinds of voxels (red
and blue) which actuate in antiphase. Volumetric actuation is implemented by
treating the beams as pistons and adjusting their rest length at every time
step according to a sine wave with a phase shift stored at each voxel. A given
voxelyzed robot can be fractalized by replacing each voxel with a copy of the
entire robot, just like the Menger sponge.

To simulate the high mechanical resolution of fractal robots we used voxcraft-
sim [18], a GPU-accelerated re-implementation of Voxelyze [6]. In Voxelyze,
voxels are evaluated sequentially on a single thread of a CPU, one after
another. In voxcraft-sim, voxels are evaluated concurrently on GPUs.
Collisions in Voxelyze are resolved in a double loop of all $n$ surface
voxels, with time complexity $\mathcal{O}(n^{2})$. In voxcraft-sim, collisions
are handled using a bounding volume hierarchy tree [10] with
$\mathcal{O}(n\log n)$.

The basic simulated elements of Voxelyze and voxcraft-sim are the same (beams
and masses) but computing dynamics in parallel and handling collisions with a
BVH tree makes it possible to simulate geometrically interesting robots with
complicated surface areas that self collide. Here, evolution operates in a 273
voxel workspace (up to 19,683 total voxels). Due to past computational limits,
previous evolutionary robotics work was largely restricted to voxel workspaces
with less than a thousand voxels.111The lone exception was Hiller and Lipson
[5], who evolved soft robots within a 203 voxel workspace (up to 8000 voxels).

###  II-C Evolving fractal robots.

Encoding. Each basal robot design is generated by an evolved Composition
Pattern-Producing Network (CPPN; [29, 2]). Each network “paints” a pair of
floating point values in the range [-1,1] at every (x,y,z) location in the
3-by-3-by-3 workspace. The first value is thresholded at 0 to determine the
presence or absence of material at that point. The second is thresholded at 0
to determine the type of material (if any) at that point. The largest
connected component of material output by a network (genotype) is taken to be
the morphology of the basal robot (phenotype). For more details see [13].

Fitness. Each basal robot design is evaluated three times: first alone, and
then twice more in fractal aggregations of increasing size. Performance at the
$i$-th size scale is measured by the net displacement $d_{i}$ the robot or
aggregate traveled (measured in body lengths) at the end of the 5 second
evaluation period (i.e. 25 actuation cycles at 5 Hz). A design’s fitness, $F$,
is taken to be the least performance achieved across the three size scales:

$F=\min(d_{1},d_{2},d_{3}).$ (2)

Simulation hyperparameters. Actuation here occurs at 5 Hz, with amplitude
$A=\pm 50\%$ resting volume. Material was simulated with 10 kg/m3 density,
Young’s modulus $10^{4}$ Pa, and Poisson’s ratio 0.5. Each voxel is 1 cm in
length. A Coulomb friction model is used for the surface plane. Coefficients
of static and dynamic friction were 1 and 0.5, respectively.

Algorithm hyperparameters. Populations of CPPNs were evolved with “age
protection” [27], an additional selection pressure that maintains diversity by
relaxing competition on newer genetic lineages. A randomly-generated CPPN,
with age 0, is injected into the population at every generation. Mutations
add/remove/modify vertices and edges within a selected network. The activation
functions at each vertex are randomly chosen from the following functions:
sine, absolute value, square, square root, and a step function.

###  II-D Manufacturing fractal robots.

Previous work [12, 14] used a rotational molding technique to fabricate hollow
cubic silicone modules (physical voxels) as building blocks of pneumatic
robots. Here we introduce a molding technique, which reduces the weight of
silicone required for module fabrication, and allows for the creation of
custom-shaped basal modules.

Evolved fractal robots were fabricated using Dragon Skin 10 Slow in a
multistep molding and assembly process (Fig. 4). Two rigid molds for casting
were 3D printed. The outer mold was treated with a release agent (Ease Release
200), half-filled with silicone, and then degassed in a vacuum oven at room
temperature for 5 min. Next, the inner mold was treated with release agent,
and inserted into the outer mold. Four flanges on four sides of the inner mold
ensure that it remains centered with a 2 mm clearance from the outer mold.
Silicone was then allowed to cure in an oven at 60∘C for 75 min.

Once cured, the module was removed from the mold, flipped over, and placed on
a draw-coated layer of uncured silicone with 2 mm thickness to seal its top
side. A hole was poked into one of the side walls of the module and a rubber
tube was inserted and glued with Sil-Poxy to create the basal robot.

To assemble a fractal aggregation, single modules were attached to each other
using silicone elastomer as bonding agent. Finally, holes were punched in the
side walls of the adjacent modules located in each horizontal plane such that
one tube can be used to pressurize all the modules in each plane at the same
time.

###  II-E Culturing fractal biobots.

Living biobots, called xenobots [13], were produced using two methods. In the
first, the animal cap of Xenopus embryos (Nieuwkoop and Faber stage 10) were
removed using surgical forceps and cultured in a mild saline solution (0.75x
Marc’s Modified Ringers [MMR], pH 7.8) for 24 hours at 14∘C. Individuals do
not naturally adhere to each other but can be induced to do so via wounding
the outer surface with forceps. Following injury, individuals are manually
held together for five seconds, after which they heal together permanently.

For the second method, animal cap tissue is harvested from Xenopus embryos as
above and moved to a calcium/magnesium free dissociation media. This solution
allows the pigmented outer layer of cells to be separated and discarded,
leaving the white inner layer which form spheroids when moved to a mild saline
solution (0.75x Marc’s Modified Ringers [MMR], pH 7.8). After 24 hours of
development at 14°C, these spheroids will naturally adhere to neighbors upon
touching, allowing multi-individual chains to be formed through simple
contact.

Figure 8: Culturing fractal biobots. Embryonic amphibian stem cells (Xenopus
laevis species) form spheres of motile (ciliated-driven) epidermis (a) without
additional intervention. Spheroids can be combined into various macro-
combinations (a’, a”) through the induction of a laceration, which allows
individual xenobots to fuse at the wound site (video). Using a separate
construction method which removes the outer pigmented cell layer (b),
spheroids are naturally sticky and can be daisy-chained through simple contact
(b’) enabling rapid connection of tens or hundreds of individuals (b”)
(video). Scale bars: 500µm.

##  III Results

###  III-A Simulated fractal robots.

Ten independent evolutionary trials were conducted, each starting from a
different set of random initial conditions. Basal robots were evolved inside a
3-by-3-by-3 voxel workspace ($m=3$) using up to $c=27$ voxels, which are each
1 cm in length. Basal robots were recursed twice to yield 9-by-9-by-9 (up to
729 voxels) and 27-by-27-by-27 (up to 19,683 voxels) fractal robots.222Source
code: github.com/skriegman/2021-RSS.

Each trial used 8 NVIDIA Tesla V100s. No trial took less than 40 hours or more
than 43 hours to complete 325 generations of evolution. Each generation
consisted of 48 simulations (16 designs in the current population, each
evaluated at three size scales), for a total of $325*48=15{,}600$ design
evaluations per trial.

Scale invariant behavior. Fig. 6 shows evolutionary improvement occurring on
all three size scales. Usually (but not always) the largest fractals (27 cm)
generated less forward movement than the middle (9 cm) and basal (3 cm)
scales, as evidenced by the low orange and blue curves in Fig, 6. There was
not a significant difference in performance between the middle (9 cm) and
largest (27 cm) size scales. Because fitness is driven by the size scale with
the worst performance (Eq. 2), after $\sim$50 generations of evolution, there
seems to be, on average, little to no selection pressure on the performance of
the basal and middle fractal scales. This can be seen to some extent in the
aggregate curves of Fig. 6. Whenever a one of the curves drops in performance
over successive generations, the average improvement in scale invariant
behavior was driven by another size scale. The largest fractal scale more
consistently trends upward in performance, relative to the middle and basal
fractal scales.

Evolved fractals. The confidence bands visible in Fig. 6 indicate that
evolution did not converge to a single fractal design across the ten trials.
The diversity of form can be seen in the four evolved robots drawn in Fig. 3
at the three fractal scales. These four robots have interesting roughness:
fractal geometries that fill much more than a 2D plane but less than a full 3D
cube. Their roughness is formally measured by their Hausdorff dimension (Eq.
1), which is plotted over evolutionary time in Fig. 7. Early in evolution
there were some full cubes $(H=3)$. At the end of evolution there were five
full planes $(H=2)$ and five designs with more roughness $(2.5<H<2.7)$.

Comparison to non-fractals. To determine how much (if at all) fractals aid the
evolution of scale invariant behavior, we compared against an otherwise
equivalent algorithm that freely combines 3 cm basal robots into any aggregate
structure within a 9 cm2 bounding box, without imposing self-similarity. At
the third fractal size scale, aggregates are freely combined within a 27 cm2
bounding box to form an aggregate of aggregates of basal robots. Because we
removed the assumption of self-similarity, some rule must be introduced to
orchestrate aggregation at every size scale. Because additional selection
pressures are necessary to induce modularity in separate outputs of the same
CPPN [3], we here use three independent CPPNs to dictate the shape at the
three size scales.

Note that if modularity was not artificially induced, then the aggregation
strategies produced by a single CPPN would by default be strongly correlated,
yielding fractals, and thus defeating the purpose of this control experiment.
However, fractals can emerge spontaneously under these conditions if the
independent CPPNs converge to identical aggregation functions that mirror the
basal robot’s internal voxel structure.

The behavior of a random nonfractal robot (before evolution) can be seen here.
The results after evolution indicate that searching for multiple aggregation
policies, even just a two size scales, can significantly slow the evolution of
scale invariant behavior compared to assuming fractality (Fig. 6).

Transferal to reality. Two additional evolutionary trials were conducted. In
the first, we used phase shifted actuation: Instead of using cyan and red
voxels that actuate in anti-phase, a wave of phase shifted actuation
propagates through the body, from anterior to posterior, repeating every 0.2
seconds. This experiment was conducted a few weeks before the experiments
using anti-phase materials (Fig. 3). Thus, the robot with the most scale
invariant behavior that evolved under these conditions (Fig. 9) was the first
to be manufactured in reality (Fig. Scale invariant robot behavior with
fractals).

After observing the transferal of a phase shifted actuation, a single bladder
design was evolved in which all basal robots synchronized their actuation in
time (zero phase-offsets across the body). The evolved single actuator fractal
robot with the most scale invariant behavior was then fabricated. The winning
design is shown in simulation and reality in Figs. 2 and 10, respectively.

###  III-B Physical fractal robots.

The best evolved fractal robot utilizing a phase shifted actuation (Fig. Scale
invariant robot behavior with fractals) was fabricated first (Fig. 4). At the
basal fractal scale, the robot was actuated with $\simeq$ 6 kPa at a frequency
of 3 Hz and a locomotion behavior was observed similar to the prediction from
simulation. At the middle fractal scale, the robot was first actuated using a
cam system which was powered by a DC motor (Pololu 131:1) with maximum
rotational speed of 72 RPM and designed to operate two piston-cylinder
assemblies (Clippard Inc.) for pressurizing/depressurizing the robot. This
design led to a controlled actuation of the robot with maximum $\simeq$ 4 kPa
pressure at a maximum frequency of 1.2 Hz. This combination of the pressure
and frequency did not lead to locomotion for the robot. Due to this limitation
of the cam system, a secondary testing method was also employed in which the
pressure inlet was manually switched between an airline with 20 kPa pressure
and a vacuum pump for 50 cycles resulting in an average actuation frequency of
0.55 Hz. The robot was then able to locomote at the middle fractal scale, but
at a much lower speed compared to the basal scale, due to hardware constraints
in applying similar combination of input pressure and actuation frequency.

The best evolved single actuator fractal robot was the second design to be
fabricated (Fig. 2). In the single actuator case, scale invariant behavior
successfully transferred: the physical robot behaved similarly at the basal
and middle fractal scales. However, the direction of the robot’s movement did
not match the prediction from simulation. We suspect that this is due to an
overly simplistic (Coulomb) friction model. Given that the direction of
locomotion for soft robots can be reversed by tuning the tribological
properties of the surface plane [19], future work should either modify the
simulator, or the physical surface on which the robot operates, or both.

###  III-C Living fractal biobots.

In Fig. 8 living xenobots [13] were formed from amphibian stem cells harvested
from embryos of the African clawed frog Xenopus laevis. The apical surface of
individual spheroids are not naturally adherent to neighbors, however,
following mechanical laceration with microsurgery forceps individuals can be
attached to one another during the healing process. This attachment is stable
(connected members can be pushed with forceps or moved via pipetting without
detaching) and permanent, although members can be manually decoupled through
bisection with forceps or a scalpel.

If the pigmented outer epidermal layer is removed during construction, the
resultant biobot is naturally adherent to neighbors through contact—allowing
for the rapid connection of many individuals. Long term stability is less
using this construction method as the increased adherence can result in
neighbors fusing into one large spheroid. Future work would allow these
connected systems to be stacked vertically, creating three dimensional
fractalized designs.

##  IV Discussion

In this paper, we achieved scale invariant robot behavior using fractals. The
results demonstrate that, under certain conditions, the nonrandom behavior of
a single robot can be preserved when connected with copies of itself
fractally. However, this depends on the morphology of the basal robot; most
robot morphologies do not exhibit scale invariant behavior. We found that
evolutionary algorithms333It is unlikely that evolutionary algorithms are
uniquely capable of designing fractal robots with scale invariant behavior.
Other computational search methods (e.g. reinforcement learning [33] and
simulated annealing [30]) are capable of producing self similar robot forms,
which in principle could be modified to generate scale invariant behavior. can
automatically discover morphologies that facilitate scale invariant robot
behavior in simulation.

Simulated fractal robots were transferred to reality in two systems. Using
pneumatically actuated silicone bladders, two centimeters across, self similar
structure was readily assembled, and the physical robot exhibited scale
invariant behavior. However, the amount of locomotion was limited and the
direction did not always match the simulated prediction. The micrometer length
scale biobots investigated here exhibited spontaneous attachment and scale
invariant behavior up to supermillimeter length scales. However, self similar
structure was more challenging to produce out of plane.

###  IV-A Limitations of current fractal design.

For locomotion on a flat surface plane, larger robots can move farther in
terms of raw distance but not necessarily in terms of body lengths (Fig. 6),
and they cannot fit into smaller 3D spatial constraints. If instead the task
environment is constrained spatially in one dimension (e.g. a low overhang) or
two dimensions (e.g. a narrow pipe) then the Mengerian fractalization (Figs. 1
and 3) used here may not be appropriate. Rather than uniformly repeating basal
robots in 3D space, future work will explore allometry: how fractals may
expand at different rates along different axes, as dictated by physical
constraints and selection for specific target behaviors.

In addition to self-similar geometry and movement, a system’s statics (e.g.
structural integrity) could also be fratalic. Such a robot may not look the
same at all scales but it would resist forces passively in the same way. Scale
invariant statics might be a better target than scale invariant movement for
calibrating simulations for transferal to reality.

###  IV-B Limitations of current physical fractal robots.

By increasing the number of modules in an aggregate robot, the volume of the
air required to increase the internal pressure of the robot to a certain level
increases and pressurizing the aggregate robot at a high frequency becomes a
challenge. In this paper, the module fabricated Fig. Scale invariant robot
behavior with fractalse had a chamber volume of 18 cm3. When 18 of these
modules are assembled to form the aggregate robot shown in Fig. 4h, the
chamber volume increases to $18^{2}=324\;$cm3 which makes it very challenging
to actuate at a high frequency. In order to facilitate the actuation of the
physical fractal robots, the volume of the base module should be decreased as
much as the fabrication procedure allows. The number of pistons in the cam
testing system can also be increased to allow for a higher pressure within
robot while maintaining a high frequency.

###  IV-C Limitations of current fractal biobots.

The construction of fractalized biobots faces a number of biological
challenges and constraints. First, the developmental timing permissive for
attachment is narrow and requires frequent manual manipulation. If development
is too early (less than 24 hours of culturing), individuals fuse completely
forming one large sphere, and if development is too late (48 hours or more)
the outer surface becomes covered naturally occurring mucus, making attachment
difficult. Second, individuals must be manually connected and observed for
healing for 10-20 minutes each, necessitating significant time for husbandry
from the investigator. Finally, living biological system are not static and
continue to remodel throughout their lifespan. Cell division and cell death
are natural components of all developmental systems, which makes the long-term
stability of such fractalized designs an interesting area of study.

## Acknowledgements

This work was supported by DARPA contract HR0011-18-2-0022, NSF EFRI award
1830870, and NSF NAIRI award 2020247.

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Figure 9: Instructions for building the evolved fractal with the most scale
invariant behavior using phase shifted actuation. The local volume of the
robot fluctuates according to a sine wave with static (preselected rather than
evolved) phase offset at each module (dark pink): a wave of actuation
propagates through the body, from anterior to posterior repeating every 0.2
seconds (video). The basal robot (pictured in step 19) is designed in
simulation by combining up to 27 voxels inside a 3-by-3-by-3 workspace (steps
1-18). The same steps are repeated using the basal robot instead of a single
voxel (steps 19-36) to create a self similar aggregate machine. Note that the
voxels are purely design tools; there are no physical voxels; custom mold is
made according to the basal robot’s shape (Fig. 4). The physical robot
assembled according to these instructions can be seen, turned right side up,
in Fig. Scale invariant robot behavior with fractalse-h.

Figure 10: Instructions for building the evolved single bladder fractal robot
with the most scale invariant behavior. The entire robot is a single actuator:
it expands and contracts in unison according to a central pattern generator
(video). The basal robot (pictured in step 19) is designed in simulation by
combining up to 27 voxels inside a 3-by-3-by-3 workspace (steps 1-18). The
same steps are repeated using the basal robot instead of a single voxel (steps
19-36) to create a self similar aggregate machine. The physical robot
assembled according to these instructions can be seen, flipped right side up,
in Fig. 2.