Robotics: Science and Systems 2021
Held Virtually, July 12–16, 2021

Scale invariant robot behavior with fractals
Sam Kriegman1 , Amir Mohammadi Nasab2 , Douglas Blackiston3 , Hannah Steele2 ,Michael Levin3 , Rebecca Kramer-Bottiglio2 , Josh Bongard1
1

University of Vermont, 2 Yale University, 3 Tufts University

Behavior alone

Mass produce

Fractal aggregation

Behavior when combined

b

c

d

e

f

g

h

realitysimulationaFig. 1. 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). More information, videos and code can be found at fractalrobots.github.io.

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 demonstratethat this is indeed the case for some, but not all, modular soft
robots: there are robot designs that exhibit a desired behavior ata 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 insimulation 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 ofself similar structure and behavior to pneumatically-controlled
soft robots. Finally, we show that biobots can spontaneously
exhibit self similar attachment geometries, thereby suggestingthat self similar behavior via self similar structure may berealizable across a wide range of robot platforms in future.

I. I NTRODUCTION
Fractals are ubiquitous in nature and increasingly prevalent
in artificial structures [6, 27]. Coastlines, rivers, and trees all
exhibit self-similar structures—accurate replicas of themselves
within themselves. Fractals are also prevalent across lengthscales in animals and cellular structures: respiratory and vascular systems, brains [14], and DNA [20] all exploit spacefilling fractal networks. At the largest known length scale,the universe itself consists of a fractal clustering of spiraling
galaxies [36]. 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 andPollack [9] exploited this phenomenon in robotics by evolving
branching kinematic chains in simulation and then building
physical instances of the most promising designs [10]. The
resulting robots yielded an order of magnitude more physicalelements than any other simulated robots at that time: Whereas
Sims [32] and Bongard and Pfeifer [3] evolved simulated
robots composed of no more than 14 and 50 parts, respectively,
Hornby and Pollack [9] utilized Lindenmayer-systems [21] toproduce 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 somecases, result in self similar behavior [13]. Branching arteries
and capillaries maximize efficient blood flow at the macroand micro scales, respectively [24]. In the gecko, nanoscale
spatulae branch from microscale setae, which in turn branchfrom toes, which in turn branch from feet, and together maximize the probability of adhesion of spatulae, setae, toes, andultimately the animal itself to vertical surfaces [1]. However,with the exception of artificial gecko feet [37] and Moravec

aFig. 2. Fractalizing mass-produced robots. As an example of how basal
robots are here combined in fractals, the well-known Menger sponge is shownin 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 eachface, and no center voxel (a total of 20 voxels). Replacing each voxel witha copy of the entire sponge creates a fractal: a self-similar aggregation of 20
sponges. Repeating this operation yields larger and larger fractals.

and Easudes’ hypothetical trillion-fingered Bush Robot [25],this particular property of fractals remains unexplored inrobotics. Here, we explore whether self similar structure canconfer 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 elementsthat are to some degree self-sufficient, able to behave andsurvive on their own. They often autonomously detach, move
about each other, and reattach to reconfigure the robot’s overall
geometry [11, 26, 28]. But robot modules tend to be cubesor other simple geometries that do not reprise the shape of
the whole robot, unless the robot’s overall structure is also a
simple geometry (e.g. a cube of cubes). In all cases, the target
behavior of a module is, by design, different from that of thewhole robot.
Swarms of hundreds or thousands of individual robots haveloosely coupled to form amorphous phototatic aggregates [18]or moved within a shared shell [30], however these robots
have largely been restricted to cylindrical geometries and
2D interaction in-plane [29]. Most robot swarms compriserigid-bodied electromechanical robots which, in order to be
produced in large quantities, are designed to be as simple aspossible. A consequence of this is that each robot may be moreor less behaviorally static (functionless) in isolation [18, 30].
Here, we test whether self-similar forms can facilitate
the evolution of scale invariant behavior as follows. Shapesof basal robots (e.g. Fig 1a) 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 ofrecursion, then training such a system to operate on more than
one size scale becomes computationally infeasible. Thus, anevolutionary 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 inreality.
In the first attempts to transfer simulated scale invariant behavior to a physical system, computer-generated mold designsbFig. 3.
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 and pressurized by a single air inlet,
at any size scale. Videos: the simulation; the physical robot at the basal andmiddle scales. At atmospheric pressure, the basal robot fits inside a 2×2×2
cm cube; its chamber walls are 2 mm thick.

were 3D printed and then used to rapidly fabricate identical,
hollow elastomeric modules that can be de/pressurized toinduce volumetric actuation [17]. Modules behaved in isolationand 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 vascularsystems in which diffusion does not bring sufficient oxygenand other key metabolites beyond approximately 1 mm. As
proof of principle, we show that “xenobots” (motile biological machines built from amphibian stem cells [2, 16]) canbe 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 can then be stacked and multiplexed to form
increasingly large fractal biobots.
II. M ETHODS
A. Fractalizing robots.
Robots in this paper are modeled as polycubes with at
most c cubes (voxels) connected face-to-face inside a boundingbox with length m voxels. A fractal aggregation is made byreplacing each voxel in a basal robot with a copy of itself. This
operation can be repeated (to form an aggregate of aggregatesof 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 scaleswith 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 2). Abasal sponge is a box, m = 3 voxels in length, with a holethrough 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 (400sponges in total). And so on.
The Menger sponge has Hausdorff dimension:
H = logm (c) =log 20log c=≈ 2.727,
log m
log 3

(1)

Fig. 4. 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).

which describes the space filled by a fractal geometry in thelimit. The Hausdorff dimension is zero for a single voxel; onefor a column of three voxels (log3 (3) = 1); two for a 3-by-3
plane of nine voxels (log3 (9) = 2); and three for a 3-by-3-by3 cube of 27 voxels (log3 (27) = 3). The design space herespans these (smooth) structures and many more in between.
Since all fractals here have m = 3, H ∝ log c.
Hausdorff dimension provides an interpretable measureof the structural complexity (or “roughness”) of the robots
evolved in this paper. It will allow us to analyze how fractals
with scale invariant behavior can emerge from initially random
fractals over evolutionary time.
B. Simulating fractal robots.
Voxels are simulated by a point mass and up to six EulerBernoulli 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 byreplacing each voxel with a copy of the entire robot, just likethe Menger sponge.
To simulate the high mechanical resolution of fractal
robots we used voxcraft-sim [22], a GPU-accelerated reimplementation of Voxelyze [8]. In Voxelyze, voxels areevaluated 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 O(n2 ). In voxcraftsim, collisions are handled using a bounding volume hierarchy
tree [12] with 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 topast computational limits, previous evolutionary robotics work
was largely restricted to voxel workspaces with less than aFig. 5. 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 wassealed. (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).

thousand voxels. The lone exception was Hiller and Lipson[7], who evolved soft robots within a 203 voxel workspace (upto 8000 voxels). However, strong morphological convergenceled to a scooting mass that filled less than a quarter of theworkspace (less than 2000 voxels). Here, we demonstrate how
fractals can provide a more scalable encoding.
C. Evolving fractal robots.
Encoding. Each basal robot design is generated by an
evolved Composition Pattern-Producing Network (CPPN; [4,33]). 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) atthat point. The largest connected component of material output
by a network (genotype) is taken to be the morphology of thebasal robot (phenotype). For more details see [16].
Fitness. Each basal robot design is evaluated three times:
first alone, and then twice more in fractal aggregations ofincreasing size. Performance at the i-th size scale is measured
by the net displacement di 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 acrossthe three size scales:
F = min(d1 , d2 , d3 ).

(2)

Simulation hyperparameters. Actuation here occurs at 5 Hz,
with amplitude A = ±50% resting volume. Material was
simulated with 10 kg/m3 density, Young’s modulus 104 Pa, andPoisson’s ratio 0.5. Each voxel is 1 cm in length. A Coulomb
friction model is used for the surface plane. Coefficients ofstatic and dynamic friction were 1 and 0.5, respectively. Beams
and collisions are damped with ζ = 1 and 0.8, respectively.
Algorithm hyperparameters. Populations of CPPNs were
evolved with “age protection” [31], an additional selection
pressure that maintains diversity by relaxing competition on
newer genetic lineages. A randomly-generated CPPN, withage 0, is injected into the population at every generation.
Mutations add/remove/modify vertices and edges within aselected network. The activation functions at each vertex are
randomly chosen from the following functions: sine, absolutevalue, square, square root, and a step function. For more details
see [16]. The radial distance from the center of the workspace
and a bias of 1 were also input to each network, in addition
to cartesian coordinates x,y,z.
Source code can be found at fractalrobots.github.io/code.
D. Manufacturing fractal robots.
Previous work [15, 17] used a rotational molding techniqueto 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 ofcustom-shaped basal modules.

10

10
3 cm
9 cm
27 cm

8

Body lengths

Body lengths

8

3 cm
9 cm
27 cm

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4

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6

4

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Generation

Fig. 6. 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).

Hausdorff dimension (H)

3.0

2.5

2.0

1.5

1.0
0

50

100

150

300

400

500

600

Generation

200

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300

Generation
Fig. 8. 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 noninteger Hausdorff dimension (H). Each individual color plots H of the mostscale invariant design, at each generation, of a single evolutionary trial. Halfof the trials ended in a plane of voxels (H = 2), the the other half foundrougher, more geometrically interesting designs with 2.5 < H < 2.7.

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

Fig. 7. Evolving scale-invariant behavior without the fractal assumption.
The mean fitness of the best design (in body lengths per minute) is plottedfor 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. 7) (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).

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 wasinserted and glued with Sil-Poxy to create the basal robot.
To assemble a fractal aggregation, single modules wereattached to each other using silicone elastomer as bonding
agent. Finally, holes were punched in the side walls of theadjacent modules located in each horizontal plane. This allows
groups of modules within the same plane to be pressurized bya single air inlet. For example, the robot in Fig. 1g,h has twogroups of modules in the lowest horizontal plane (two “feet”),
one group of modules in the middle plane, and one group on
the highest plane, requiring just four independent air inlets,
instead of 18. Once assembled, robots were placed on topof a sheet of ABS Plastic (MyStudio MS20CYC Background
Cyclorama), which was covered in cornstarch (Argo®, ACH
Food Companies, Inc.) to reduce the friction between thesilicone body of the robot and the plastic substrate [17].
E. Culturing fractal biobots.
Living biobots, called xenobots [16], were produced using
two methods. In the first, the animal cap of Xenopus embryos(Nieuwkoop and Faber stage 10) were removed using surgicalforceps 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.
III. R ESULTS
A. Simulated fractal robots.
Ten independent evolutionary trials were conducted, eachstarting 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-9by-9 (up to 729 voxels) and 27-by-27-by-27 (up to 19,683
voxels) fractal robots.
Each trial used 8 NVIDIA Tesla V100s. No trial took lessthan 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 threesize 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 notalways) 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 ∼50 generations of evolution, there seems to be,
on average, little to no selection pressure on the performanceof the basal and middle fractal scales. This can be seen to someextent in the aggregate curves of Fig. 6. Whenever a one of thecurves drops in performance over successive generations, theaverage improvement in scale invariant behavior was driven byanother size scale. The largest fractal scale more consistently
trends upward in performance, relative to the middle and basal
fractal scales. This is likely due to the increased mass, but fixedelastic modulus and actuation amplitude, of the robot at larger
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 beseen in the four evolved robots drawn in Fig. 4 at the threefractal 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 overevolutionary time in Fig. 8. Early in evolution there were somefull 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). As the basal shape changes in each columnof Fig. 4, so too does the distribution of performance at eachsize scale. For some shapes, performance was highest at thesmallest fractal scale (Fig. 4a,c). For others, performance washighest at the middle fractal scale (Fig. 4b,d). Some have more

uniform performance across size scales (Fig. 4b,c). Others
have one size scale in which performance is two or three times
as high as their next best (Fig. 4a,d).
Comparison to non-fractals. To determine how much (if atall) 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 withina 9 cm2 bounding box, without imposing self-similarity. At thethird fractal size scale, aggregates are freely combined within
a 27 cm2 bounding box to form an aggregate of aggregatesof basal robots. Because we removed the assumption ofself-similarity, some rule must be introduced to orchestrateaggregation at every size scale. Because additional selection
pressures are necessary to induce modularity in separate
outputs of the same CPPN [5], we here use three independent
CPPNs to dictate the shape at the three size scales.
Note that if modularity was not artificially induced, thenthe 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 indicatethat 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. 7).
Transferal to reality. Two additional evolutionary trials wereconducted. In the first, we used phase shifted actuation: Insteadof using cyan and red voxels that actuate in anti-phase, awave 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. 4). Thus, the robot withthe most scale invariant behavior that evolved under these
conditions (Fig. 10) was the first to be manufactured in reality(Fig. 1).
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 acrossthe body). The evolved single actuator fractal robot with themost scale invariant behavior was then fabricated. The winning
design is shown in simulation and reality in Figs. 3 and 11,
respectively.
B. Physical fractal robots.
The best evolved fractal robot utilizing a phase shifted actuation (Fig. 1) was fabricated first (Fig. 5). At the basal fractal
scale, the robot was actuated with ' 6 kPa at a frequency of
3 Hz and a locomotion behavior was observed similar to theprediction from simulation. At the middle fractal scale, therobot 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-cylinderassemblies (Clippard Inc.) for pressurizing/depressurizing theFig. 9. 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.

robot. This design led to a controlled actuation of the robot
with maximum ' 4 kPa pressure at a maximum frequency of
1.2 Hz. This combination of the pressure and frequency did notlead to locomotion for the robot. Due to this limitation of the
cam system, a secondary testing method was also employedin 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 tohardware 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. 3). In the single actuator case,scale invariant behavior successfully transferred: the physicalrobot 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 thatthe direction of locomotion for soft robots can be reversed
by tuning the tribological properties of the surface plane [23],future work should either modify the simulator, or the physicalsurface on which the robot operates, or both.

C. Living fractal biobots.
In Fig. 9 living xenobots [16] were formed from amphibianstem 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 pushedwith forceps or moved via pipetting without detaching) andpermanent, although members can be manually decoupledthrough bisection with forceps or a scalpel.
If the pigmented outer epidermal layer is removed during
construction, the resultant biobot is naturally adherent toneighbors through contact—allowing for the rapid connectionof many individuals. Long term stability is less using thisconstruction method as the increased adherence can result inneighbors fusing into one large spheroid. Future work would
allow these connected systems to be stacked vertically, creating
three dimensional fractalized designs.
IV. D ISCUSSION
In this paper, scale invariant robot behavior was achieved
using fractals. The results demonstrate that, under certain
conditions, the nonrandom behavior of a single robot canbe preserved when connected with copies of itself fractally.

However, this depends on the morphology of the basal robot. A
majority of the fractals tested did not exhibit the same behavior
at multiple size scales. An evolutionary algorithm was used to
weed out robots that behaved incorrectly at any of the tested
size scales.
Fractal robots were fabricated in reality using two distinct
hardware platforms: pneumatic soft robots and ciliated biobots.
Using pneumatically actuated silicone bladders, two centimeters across, self similar structure was readily assembled, andthe 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.
The following discussion highlights the key limitations of
the current methods and important opportunities for future
work.
A. Limitations of current fractal design.
For locomotion on a flat surface plane, larger robots canmove 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 a robot’s task environment isconstrained spatially in one dimension (e.g. a low overhang)or two dimensions (e.g. a narrow pipe) then the Mengerianfractalization (Figs. 2 and 4) used here may not be appropriate.
Thus, instead of 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 physicalconstraints and selection for specific target behaviors.
It is also possible that the coordinate system itself was apoor choice. A cubic grid was assumed because the simulator
we used was voxel-based. However, fractals formed using
using spherical modules [19, 34] and spherical coordinates(Mandelbulbs) may be better suited for forms and functions
that depend on radial symmetries.
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 itwould resist forces passively in the same way. Scale invariant
statics could also provide an easier target than scale invariant
movement for system identification and calibrating simulations
prior to optimization and mass manufacture.
Finally, it should be noted that the choice of a synchronous
evolutionary algorithm [31] was somewhat arbitrary. It is likelythat other computational search methods that are capable ofproducing self similar robot forms [35, 38] could be modified
to generate scale invariant behavior in a similar fashion.
B. Limitations of current physical fractal robots.
As the number of modules within a fractal robot increases
at higher size scales, so does the volume of the air required toattain a given pressure level. This makes pressurizing a larger
fractal robot at high frequency more challenging. The modules

fabricated here (Fig. 1e) had a chamber volume of 18 cm3 .
When 18 of these modules were assembled to form a fractal(Fig. 5h) the chamber volume increased to 182 = 324 cm3 ,which was more difficult to actuate at higher frequencies andthus slowed the robot’s locomotion speed. To facilitate theactuation of the physical fractal robots, the volume of thebase module should be miniaturized as much as the fabrication
procedure allows. The number of pistons in the cam testing
system could also be increased to allow for a higher pressure
within robot while maintaining a high frequency.
The designs transferred from simulation to reality in thispaper served as a proof of principle that fractal robots withscale invariant behavior could in fact be realized. However, it
may be impossible in practice to retain a useful behavior acrossa meaningful range of length scales using current soft robot
technologies, such as those that rely on pneumatic actuation.
C. Limitations of current fractal biobots.
The construction of fractalized biobots faces a number ofbiological 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 forhealing for 10-20 minutes each, necessitating significant time
for husbandry from the investigator. Finally, living biologicalsystem are not static and continue to remodel throughout theirlifespan. Cell division and cell death are natural componentsof 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 HR0011-182-0022, NSF EFRI award 1830870, and NSF NAIRI award
2020247.
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Fig. 10. Instructions for building the evolved fractal with the most scale invariant behavior using phase shifted actuation. The local volume of therobot fluctuates according to a sine wave with a preselected phase offset at each module (dark pink): each wave of actuation propagates through the body,from anterior to posterior, repeating every 0.2 seconds (video). The basal robot (pictured in step 19) was designed in simulation by combining up to 27 voxels
inside a 3-by-3-by-3 workspace (steps 1-18). The same steps were repeated using the basal robot instead of a single voxel (steps 19-36) to create a self similar
aggregate machine. Note that the voxels only exist in simulation as design tools; there are no physical voxels; a custom mold is made according to the basal
robot’s shape (Fig. 5). The physical robot assembled according to these instructions can be seen, turned right side up, in Fig. 1e-h.

Fig. 11. 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) tocreate a fractal assemblage. The physical robot fabricated according to these instructions can be seen, flipped right side up, in Fig. 3.