# ElectroVoxel: Electromagnetically Actuated Pivoting for Scalable Modular
Self-Reconfigurable Robots

Martin Nisser  MIT CSAIL
Cambridge, MA
<EMAIL_ADDRESS>Leon Cheng  MIT CSAIL
Cambridge, MA
<EMAIL_ADDRESS>Yashaswini Makaram
MIT CSAIL
Cambridge, MA
<EMAIL_ADDRESS>Ryo Suzuki  University of Calgary
Calgary, AB
<EMAIL_ADDRESS>Stefanie Mueller  MIT CSAIL
Cambridge, MA
<EMAIL_ADDRESS>
###### Abstract

This paper introduces a cube-based reconfigurable robot that utilizes an
electromagnet-based actuation framework to reconfigure in three dimensions via
pivoting. While a variety of actuation mechanisms for self-reconfigurable
robots have been explored, they often suffer from cost, complexity, assembly
and sizing requirements that prevent scaled production of such robots. To
address this challenge, we use an actuation mechanism based on electromagnets
embedded into the edges of each cube to interchangeably create identically or
oppositely polarized electromagnet pairs, resulting in repulsive or attractive
forces, respectively. By leveraging attraction for hinge formation, and
repulsion to drive pivoting maneuvers, we can reconfigure the robot by
voxelizing it and actuating its constituent modules—termed Electrovoxels—via
electromagnetically actuated pivoting. To demonstrate this, we develop fully
untethered, three-dimensional self-reconfigurable robots and demonstrate 2D
and 3D self-reconfiguration using pivot and traversal maneuvers on an air-
table and in microgravity on a parabolic flight. This paper describes the
hardware design of our robots, its pivoting framework, our reconfiguration
planning software, and an evaluation of the dynamical and electrical
characteristics of our system to inform the design of scalable self-
reconfigurable robots.

###### Index Terms:

modular robots; self-reconfigurable robots; space robotics

##  I Introduction

Roboticists have pursued a vision of modular self-reconfigurable robots for
over 30 years [1, 2, 3]. Exhibiting unique benefits in adaptability,
scalability, and robustness, modular self-reconfigurable robots (MSRR) promise
application domains that include space exploration [4, 5], reconfigurable
environments [6, 7], search and rescue [8], and shape-changing user interfaces
[9]. Roboticists have typically built MSRR via individually actuated modules
connected by temporary joints [10, 6, 11]. MSRR based on cubic modules have
moreover achieved self-reconfigurability in two dimensions via sliding [12]
and disassembly [13], as well as in three dimensions via pivoting [14, 15].
However, a grand challenge facing self-reconfigurable robots is their
scalability [16]—existing designs often require separate mechanisms for
actuation and attachment [6, 14], and in addition, require mechanical
components such as motors, gears, and transmissions. However, these components
are often bulky, complex, and expensive, hindering their miniaturization and
scalability.

Figure 1: Elements of an electromagnetically actuated reconfigurable robot.
(Above) A single module. (Middle) An array of four modules. (Bottom) 2D and 3D
reconfiguration experiments on an air table and a microgravity environment.
Figure 2: Reconfiguration maneuvers for (above) Pivoting and (below)
Traversal. Electromagnets are shaded in red and blue to indicate polarization
w.r.t. a global co-ordinate system; like polarizations repel, unlike
polarizations attract.

One promising approach to addressing these challenges is to leverage
electromagnetic forces to connect and actuate modules at the same time with a
single component. Being solid-state—that is, having no moving parts itself—it
is also easy to maintain and manufacture for a large-scale system. In
particular, Nisser et al. [17] proposed and simulated the use of inexpensive
electromagnets embedded into cube edges to actuate pivots between adjacent
modules via repulsion while creating temporary hinges via attraction. Unlike
traditional hinges that require mechanical attachments between two elements,
this approach requires no dedicated physical mechanism and can be formed
between any electromagnet pair dynamically. Kubits [18] built the first
physical prototype leveraging this electromagnetically actuated pivoting
framework and replaced electromagnets with programmable magnets to conserve
power. They also demonstrated how the electromagnetically actuated pivoting
framework can generate forces significant enough to pivot modules against
gravity moments if large currents are applied. However, the modules in their
system were tethered to off-board electronics, and because cube edges were
only partially embedded with electromagnets, interactions between all
electromagnets in the system could not be observed, such as using
electromagnets to maintain bonds between stationary cubes during traversals.

In this paper, we contribute the first demonstration of reconfigurable robots
leveraging the electromagnetically actuated pivoting framework that are fully
untethered, supported by reconfiguration planning software and electromagnet
force predictions verified experimentally. A key goal is to validate these
robots’ use for microgravity environments to enable near-term space industry
applications [4, 16, 17, 18], where propellant-free actuation and
reconfigurability address many challenges associated with today’s limitations
on launch mass and volume, as well as facilitating stowage during launch.
Reconfigurable modules can enable the augmentation and replacement of
structures over multiple launches, form temporary structures to aid in
spacecraft inspection and astronaut assistance, function as self-sorting
storage containers, and allow spacecraft to actively change their inertia
properties. Microgravity alleviates demands placed on actuation forces,
facilitating untethering of the modules by moving electronics onboard, and we
chose electromagnet parameters such as winding number, core radius and
material to limit current. We also parameterized Ampère’s Force law in terms
of these parameters to expand the design space of electromagnetic actuators
for future modules’ force and mass requirements. We simulate a microgravity
environment using an air table, and deploy our modules on a parabolic flight
to demonstrate untethered three-dimensional reconfigurability in space.

Relative to existing self-reconfigurable robots [15, 14], our robots are light
(103g), inexpensive ($68), and easy to fabricate (80 min/cube), promising
scalability. In addition, using full assemblies, we demonstrate Sung et al.’s
[19] two reconfiguration primitives, the pivot and traversal, demonstrating
the electromagnetically actuated pivoting framework’s compatibility with
algorithms that allow reconfiguring large numbers of cube-based robots between
arbitrary 3D shapes. To show how the framework complies with [19] to
reconfigure more complex shapes, we constructed a web interface that simulates
reconfiguration between user-defined shapes.

##  II Actuation Mechanism & Simulation

Reconfiguration algorithms for pivoting cubes [19, 20] demonstrate
transitioning between arbitrarily configured 3D lattices that are provably
correct in under $\mathcal{O}(n^{2})$ moves, barring three inadmissible sub-
configurations. These reconfiguration maneuvers rely on two reconfiguration
primitives; a simple pivot between two cubes along a shared edge (Fig. 2, 1st
row) and a traversal between the face of one cube and another (Fig. 2, 2nd
row). In our work, we introduce an algorithm that describes how the
electromagnetically actuated framework for individual cubes is compatible with
these reconfiguration planning algorithms in three dimensions.

###  II-A Actuation Mechanism

There are three steps to the polarization sequence for both the pivot and
traversal maneuvers, which we call the Launch, the Travel, and Catch phases
(Fig. 2). In each of these phases, three cubes are involved: a traveling cube
(the cube selected for moving), an origin cube (from which the traveling cube
launches), and a destination cube (which catches the traveling cube). For the
pivot (Fig. 2, 1st row), the origin and destination cubes correspond to the
same physical cubes; for the Traversal (Fig. 2, 2nd row), they correspond to
different cubes. The algorithm inputs are a cube ID, its desired rotation axis
and rotation direction. Given these inputs, all electromagnet assignments
(repulse, attract, or OFF) are uniquely defined and identified by our
software.

During the Launch phase (Fig. 2 A and D), we polarize one electromagnet pair
identically to launch the maneuver, while oppositely polarizing a second pair
to form an attractive hinge. For the Traversal (Fig. 2, bottom), we energize
two additional pairs of electromagnets to keep the non-traveling cubes
attached to one another; for this, we choose electromagnets oriented
orthogonally to the launching electromagnets in order to avoid unwanted
interactions between these pairs. During the Travel phase (Fig. 2 B and E),
after a short pulse we switch the launching electromagnets off, while the
remaining electromagnet pair remain attractive to maintain the hinge. During
the Catch phase (Fig. 2 C and F), we energize a new pair of attractive
electromagnets to form a stable bond in the newly acquired configuration.

###  II-B Simulation and Control Interface

Manually planning pivoting maneuvers and their associated electromagnet
assignments becomes intractable for more than a few cubes. To let users
visualize and plan reconfiguration maneuvers, we developed a simulation (Fig.
3) that computes all electromagnet assignments based on desired
reconfiguration maneuvers specified by the user. The simulation is browser-
based and built using React, TypeScript, and Three.JS. It consists of three
parts: (A) different ways to interact with the cubes (via buttons, direct
manipulation, or code), (B) a viewport that simulates the cubes and affords
their direct manipulation, and (C) a settings panel to toggle simulation
features (e.g., different render modes for the cubes).

We provide three ways to define maneuvers. Users can initiate maneuvers
directly by clicking cubes and arrow directions, each resulting in a single
pivot. Alternatively, they can launch pre-defined scripts via the buttons that
encode multiple consecutive rotations. And finally, new buttons encoding
different reconfiguration maneuvers can be added at any time in the underlying
Typescript file. To do so, users define the number and locations of starting
cubes addressed by (x,y,z) coordinates, and define each subsequent maneuver by
specifying the cube number and pivot direction.

Cubes in the viewport are affixed to a grid of cubic cells of unit length,
with each cube occupying an integer address on coordinates (x,y,z). Rotations
are permitted along orthogonal axes X,Y,Z, in clockwise or counter-clockwise
directions. Because physical rotations require cubes to form both repulsive
and attractive edges, cubes must share a face with another cube to execute a
valid pivot. Each cube is assumed to have access to the local occupancy
information of its neighboring cells. Prior to executing a pivot, this
occupancy is checked in order to determine whether a selected rotation results
in a pivot or a traversal. If the maneuver path is obstructed, the viewport
returns an error message. Given a pivot direction and unobstructed path, there
exists for each cube a unique valid edge about which to form a hinge, but up
to two valid edges to repel and actuate the maneuver; in this case, we choose
the edge corresponding to the cube with which the hinge is formed. We
represent all rotations using quaternions in order to facilitate unordered
rotations about multiple axes.

Finally, the settings panel allows users to set rendering features. These
include displaying IDs for cubes, their electromagnets and their polarization
values, displaying occupancy requirements of neighboring cells to prevent
collisions for desired maneuvers, toggling animation speed, and setting
rendering fidelity. Once a sequence of maneuvers is defined, the associated
electromagnet assignments can be ported to a transmitting microcontroller for
deployment on the hardware.

Figure 3: Web simulation for planning reconfigurations and calculating the
associated electromagnet commands. (A) Pre-scripted maneuvers. (B) Viewport.
(C) Settings.

##  III Electromagnets

In this section, we first describe the selection of the electromagnet
parameters for our system. To support further exploration of
electromagnetically driven pivoting cubes, we then provide a force model to
allow generating candidate electromagnets by using Ampère’s Force law to
compute magnetomotive force for a given electromagnet pair parametrically. We
finally apply this force to a preliminary dynamical model of a 2-cube system.

###  III-A Electromagnet Parameters

The idealized case of Ampère’s law gives that an electromagnet’s force is
proportional to $(NI)^{2}\mu$, where $\mu$ is the permeability of its core,
$N$ the number of turns and $I$ the current applied. To limit each cube’s mass
and size, we first selected the narrowest COTS ferrite cores available, at
1.625mm radius R and initial permeability $\mu_{0}$ of 2000. In addition,
larger cores proved difficult to be supported by the rated pressure of our air
table on a 60mm-side cube footprint. We next chose the COTS SMD drivers we
found capable of delivering the highest current, at 1.2A continuous. Using
exploratory experiments, we determined that an $N$ of 800 provided sufficient
force to pivot 100g prototype cubes within interactive times of 2s. Finally,
choosing a wire gauge of AWG 34 at this $NI$ yielded a coil resistance of
$10.5\Omega$, allowing driving the electromagnets, microcontroller and
auxiliary electronics from a single untethered power source of 11.1V-12.6V.

###  III-B Electromagnet Force Model

The dipole approximation commonly used to find electromagnets’ magnetic field
strengths is invalid over short separation distances, such as when neighboring
cubes are in contact. As such, we use Ampère’s force law (1), which expresses
the force $F_{1,2}$ exerted on coil 1 due to coil 2 as a double line integral
over each coil’s geometry where infinitesmal wire elements $\textup{d}l_{1}$
and $\textup{d}l_{2}$ in wires 1 and 2 are energized with currents $I_{1}$ and
$I_{2}$, respectively, $\hat{r}_{12}$ is a unit vector from each element on
wire 1 to those on wire 2 separated by distance $r$, and $\mu(I)$ is the
permeability of the electromagnet core as a function of current.

$F_{1,2}=\frac{\mu(I)}{4\pi}\int_{1}\int_{2}\frac{I_{1}\textup{d}l_{1}\times(I_{2}\textup{d}l_{2}\times\hat{r}_{12})}{\left|\left|r\right|\right|^{2}}$
(1)

Equation (1) has no known analytical solution and is discretized to give (2),
where $D_{1}$ and $D_{2}$ represent the number of discretized elements in
wires 1 and 2.

$F_{1,2}=\frac{\mu(I)}{4\pi}\displaystyle\sum_{p\,=\,1}^{D_{1}}\displaystyle\sum_{q\,=\,1}^{D_{2}}\frac{I_{p}\textup{d}l_{p}\times(I_{q}\textup{d}l_{q}\times\hat{r}_{pq})}{\left|\left|r\right|\right|^{2}}$
(2)

We parameterize (2) in terms of radius, length, turns, and pitch, and solve
this numerically for our electromagnet parameters, using 8000 elements
($D_{1},D_{2}=8000$) per coil to compute $F_{1,2}$ from the sum of 64,000,000
force vectors over separation distances of 0.5mm to 20mm in 0.5mm increments
(85 minutes/increment on a Razer Blade Intel(R) Core(TM) i7-8750H). Fig. 4
illustrates this computation for single-turn electromagnet coils with 10
elements each (100 force vectors).

###  III-C Dynamic Model

We apply these forces to a preliminary model (Fig. 5). We model two 60-mm side
length cubes as a 2-link pendulum consisting of two point masses $m$ placed at
the distal ends of two massless links (length $L$) connecting the hinge to
each cube’s center of mass. Electromagnet forces are applied at locations that
correspond to the positions of the other two electromagnet pairs associated
with the maneuver along vectors that connect them. We derive the equations of
motion and solve with Kane’s Method using Python’s SymPy package.

Figure 4: Computing electromagnet forces. Shown here for $D_{1},D_{2}$=10
between 1-turn coils (force vectors in magenta). Figure 5: Dynamical model.
Massless links of length L (solid lines) connect point masses m. Force F
actuates the pivot via repulsion; G attracts electromagnets to form a new
stable bond.

##  IV Hardware

###  IV-A Electronics

Each electromagnet is comprised of 800 turns of 34 AWG magnet wire wound
around a ferromagnetic core (fair-rite 77) of 3.25mm diameter, 55.5mm length
and initial permeability ($\mu_{i}$) of 2000, with average electrical
properties characterized by a capacitance of $118.1\mu F$, an inductance of
$21.44mH$, a resistance of $10.65\Omega$ and a Q factor of $1.265$. Each
actuator (core + winding) costs just $0.66. The circuitry for an untethered
cube with 12 electromagnets consists of a microcontroller (Arduino Nano)
integrated with a wireless transceiver (nRF24L01), two 16-channel GPIO
expanders (Semtech SX1509) and 6 full dual H-bridges (Toshiba TB6612FNG).
These are distributed evenly between two double-sided 0.78mm PCBs of square
cross-section (side length 42mm) which sandwich three serially connected 3.7V
batteries (ENGPOW 3.7v 150mAh Lipo, 4.2V at full charge). Combined, this
allows controlling each electromagnet as to enable bidirectional pivoting in
three orthogonal axes.

We use an NRF-equipped Arduino Nano as a centralized controller to transmit
commands from a laptop to modules via radio. We utilize a simple open-loop
bang-bang control scheme. To accommodate addressing N cubes, each with 12
electromagnets, where each electromagnet can be polarized in two directions or
turned off, each command consists of a 16-bit signed integer that encodes the
cube ID $[1..N]$, electromagnet ID $[1..12]$ and its polarity $[-1,0,1]$.
Individual messages are transmitted in 20 milliseconds, and separate commands
can be transmitted to configure individual cubes to drive selected
electromagnets using PWM at a chosen duty cycle $[0..255]$.

###  IV-B Mechanical Design

Each module (Figures 1 and 6) is a 60mm side length cube and can be described
as a unit cell of a primitive cubic Bravais lattice, with edges representing
electromagnets that connect to vertices representing corner connectors. In the
middle of the cube, two PCBs sandwich three batteries, centering the system’s
mass to limit moment of inertia, and are fixated by a scaffold that interfaces
via struts to all 8 corner connectors. The corner connectors are 3D printed on
a Formlabs 2 using Tough 2000 resin and the scaffold from PLA using an
Ultimaker 3. Table I details the cost and mass breakdown of a cube; mass and
cost are for total in each row, and structure costs are based on raw material
pricing.

TABLE I: Module Cost/Mass Breakdown Section | Part | Number | Mass (g) | Cost ($)
---|---|---|---|---
Electromagnet | Ferrite core | 12 | 22.7 | 6.2
Electromagnet | Coil winding | 12 | 24.1 | 1.8
PCB | MCU | 1 | 5.6 | 8.9
PCB | Cabling | 36 | 9.2 | 11.4
PCB | Boards & ICs | 2 | 20.4 | 31.6
PCB | Batteries | 3 | 15.6 | 8.2
Structure | Corners | 8 | 2.7 | 0.4
Structure | Scaffold | 2 | 2.8 | 0.2
Total | / | / | 103.1 | 68.7

Two Molex cables connect the upper and lower PCBs, and four cables of 6 wires
each (2 leads/electromagnet) harness triads of three electromagnets to the
PCBs. These four triads consolidate all electromagnet wirings to two pairs of
diametrically opposed corners to simplify wire routing; each triad connects
three orthogonally positioned electromagnets to the PCBs in order of axis to
ensure symmetry between all cubes such that they respond identically for a
given command.

Each cube took 80 minutes to assemble, discounting time to reflow-solder PCBs
and manually wind electromagnets.

Figure 6: CAD model of a cube module.

##  V Experiments & Results

###  V-A Simulation

The simulation renders pivoting maneuvers and outputs correct electromagnet
IDs and polarity assignments for any valid reconfiguration; assignments were
verified on hardware by reconfigurations across all dimensions and
electromagnets. The simulation supports real-time interaction with up to 200
modules while rendering the associated CAD files (1.1Mb .STL), and replacing
these with low resolution proxy cubes permits interaction with 1000 modules.
Fig. 7 (see supplementary video) illustrates the viewport simulating and
computing electromagnet assignments for reconfiguring 19 cubes from a chair to
a table (via 22 maneuvers) to a couch (40 maneuvers).

Figure 7: Reconfiguring between a chair, table and couch

###  V-B 2D Experiments: Air Table

The modules were deployed on an air table (ScienceFirst #12000) and programmed
to perform the two reconfiguration primitives; a pivot and a traversal (see
supplemental video). Each electromagnet drew 11.7W (nominally 11.1V x 1.06A)
to 15.1W (fully charged, 12.6V x 1.2A) for the duration of the maneuver, which
was 1.53s for pivots and 1.03s for traversals.

52 pivots and 30 traversal maneuvers were performed, with a success rate of
100% and 94%, respectively. To yield this success rate, the electromagnets of
the manually assembled cubes required careful positioning; small misalignments
resulted in failures of the cubes to generate sufficient attractive forces to
catch stably, and traversal maneuvers showed a higher likelihood of failure
due to involving more electromagnets.

###  V-C 3D Experiments: Parabolic Flight

The modules were deployed in microgravity on a parabolic flight to observe
pivoting maneuvers unimpeded by kinematic constraints from the ground plane or
sliding friction. The modules were deployed in a clear 460mm side length cubic
polycarbonate box with two 120mm diameter arm holes in one side for the
experimenter. Ten 15-second parabolas were flown. The first 7 parabolas were
utilized for calibrating the steps and timing of the experimental protocol
under microgravity conditions. Most significantly, the microgravity quality
was found to vary significantly between and during individual parabolas,
resulting in re-programming the modules for a shortened experiment window of
approximately 4 seconds of stable microgravity available before free-falling
modules would impact the polycarbonate enclosure. The last 3 parabolas were
used to demonstrate the pivoting maneuver between two modules. The procedure
for each 15-second parabola involved the experimenter rising from a lying to a
strapped-in kneeling position on the aircraft floor, inserting hands through
arm holes, and positioning the cubes while waiting for stable microgravity to
settle. A trigger button was pushed to wirelessly command execution of the
pivoting maneuver, and a second button was pushed to power down all cubes.

The pivoting maneuver was executed successfully each of the three times in
1.13 seconds, while electromagnets for launch, travel and catch phases
performed correctly in all tests. However, small oscillations orthogonal to
the axis of rotation were observed which led to the hinge disengaging early
once completing the maneuver. This result was unique to microgravity
experiments and unobserved on air tables as these kinematically constrain
cubes in the ground plane, and arose due to imperfect alignments including
protrusions of up to 0.6mm in our manually wound coils.

(a) Air table

(b) Parabolic flight

Figure 8: Two elements undergoing a pivoting maneuver (a) on an air table, a
low friction surface used to simulate microgravity environments for 2D
maneuvers, and (b) in microgravity aboard a parabolic flight for 3D maneuvers.

###  V-D Model Accuracy

We measured the force generated between two electromagnets using a Mettler
Toledo Precision Balance ME203T/00 (10$\mu$N precision) while varying their
separation distance and their currents, shown in Fig. 9. We held two
electromagnets at a fixed separation distance of 0.5mm, varying the current
applied to each electromagnet from 0A to 1.2A in 0.05A increments. We then
held the electromagnets at a fixed current of 1.2A, varying the separation
distance from 0.5mm to 20mm in 0.5mm increments. Each of these experiments
were conducted 5 times. Measured data are plotted in raw form (scatter plots)
and as mean$\pm$1 standard deviation (shaded plots) for both experiments.
Using the force-current mean at I=1.2A to extract a characterised $\mu(1.2)$
of 874, we generated predicted values for force v distance (line graph) with
our force model (2), correlating well with measured data.

Finally, tentative results of our simple dynamical model including a
simulation (supplemental video) agree qualitatively with the experiments,
however as physical experiments only utilize a single force value (for
$I$=$1.2A$), further work is required to capture more data to validate the
model.

Figure 9: Force v current and electromagnet separation distance, showing raw
data and mean $\pm$1 St.D for N=5. Predicted force-distance relationship from
(2) shown for comparison.

##  VI Discussion and Future Work

This section discusses limitations and future work for our current
implementation. First, our cubes have to date been manually assembled and
soldered, and each electromagnet hand-wound, resulting in imperfect alignments
of electromagnets. While sufficient to successfully showcase the
electromagnetic actuation framework on an airtable and in microgravity, Fig. 9
reveals the degree to which these misalignments reduce attraction forces, at
times disengaging the hinge. This resulted in a 6% failure rate during air
table traversals and minor instabilities in microgravity without the kinematic
constraint of a ground plane; in the future, coils will be machine-wound and
the assembly rigidized before deployment.

Our parameterized force model accurately predicts forces between
electromagnets, supporting tailoring these actuators for different module
designs. The concave downwards relationship in the force v current data (Fig.
9) indicates the onset of saturation of the ferrite cores in addition to
potential heating effects at higher currents. Although current can be raised
with higher voltage sources, this diminishes the returns on force that could
potentially be achieved using greater currents; by exploiting larger cores
with smaller currents, more favorable force-current relationships could be
achieved with future prototypes. Nonetheless, the lack of opposing gravity
moments in microgravity obviates the need for large forces, and our parabolic
flight deployment required preparatory testing on an air table whose air
pressure limited the total core mass.

Several avenues for future work present themselves. The success of our open-
loop bang bang control suggests self-correction and robustness of the
electromagnetic actuation method to small disturbances, however future work
could support braking the traveling cube on impact via closed-loop control.
Promising bases for control could include model-based strategies that leverage
the pivoting cube model’s tractable dynamics, combined with IMUs and
electromagnet-based inductive sensing. To conserve power in a large scale
system, future modules should embed passive permanent magnets in cube faces to
form stable face-to-face bonds, or replace electromagnets with
electropermanent [12] or programmable [18] magnets. Further applications of
our microgravity-adapted architecture, such as tangible or swarm user
interfaces [21], could be explored using air tables, caster wheels or low
friction surfaces. Of equal interest would be to incorporate power electronics
such as boost circuits for reconfiguring untethered cubes against gravity
moments for terrestrial applications.

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