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Tensegrity-Blocks: Modular Shape-changing Blocks
Enable Self-assembling Robotic Structures
Luyang Zhao1∗, Yitao Jiang1, Muhao Chen2, Kostas Bekris3, Devin Balkcom1
1Department of Computer Science, Dartmouth College,
Hanover, NH 03755, USA
2Department of Mechanical and Aerospace Engineering, University of Kentucky,
Lexington, KY 40506 USA
3Department of Computer Science, Rutgers University,
New Brunswick, NJ 08901 USA
∗To whom correspondence should be addressed; E-mail: luyang.zhao.gr@dartmouth.edu
Modular robots are currently designed to perform a variety of tasks, primarily
focusing on locomotion or manipulation through the reconfiguration of rigid
modules. However, the potential to integrate multiple functions, such as mak-
ing each robot deployable and capable of building lattice structures for self-
construction and infrastructure creation, remains largely unexplored. To ad-
vance the field, we hypothesize that combining tensegrity principles with mod-
ular robotics can create lightweight, deformable units capable of integrating
three critical functions within a single design: navigating varied terrains, ma-
nipulating arbitrary shape objects, and assembling weight-sustainable, active
large infrastructures. Here, we designed untethered modular robots that are
deformable, lightweight, deployable, outdoor-scale, capable of bearing loads,
and capable of 3D attachment and detachment. With these characteristics, the
1
system can form various 3D structures using different assembly methods, such
as walking into position or being transported by rotorcraft. The deformability
and lightweight nature of each block enable the assembled structures to dy-
namically change shape, providing new capabilities such as added compliance
during locomotion and manipulation and the ability to interact with the envi-
ronment in tasks like tent and bridge assemblies. In summary, we suggest that
integrating lightweight and deformable properties into modular robot design
offers potential improvements in their adaptability and multi-functionality.
Multi-functional blocks that integrate modular design with tensegrity properties for robotic
locomotion, manipulation, and structure formation.
INTRODUCTION
Traditional robots are normally highly specialized and effective in controlled environments,
designed for specific tasks that require precision, repeatability, and reliability (1). However,
in emergency situations, there is a pressing need for robotic systems that are not only versa-
tile but also quickly deployable to address a wide range of challenges. These scenarios often
require the rapid assembly of temporary structures, such as antennas, scaffolding, and shel-
ters, as well as the deployment of robots capable of navigating and transporting supplies across
unstructured terrain. Drawing inspiration from the capabilities of biological insects, such as
army ants (Eciton genus), which link their bodies to form bridges across gaps in their foraging
paths (2), and fire ants (Solenopsis invicta), which form rafts to survive floods (3), researchers
have developed modular robots that, though still at the proof-of-concept stage, present several
potential advantages. Currently, these robots can adapt to various tasks through reconfigura-
tion and are reusable across different missions, often built with rigid modules with a focus on
2
Figure 1: Robot capabilities and examples. (A) System capabilities include locomotion over various terrains and
obstacles, manipulation, such as stretcher transportation, as well as structure formation. (B) A single block. (C)
Object carrying. (D) Time-lapse transportation of a stretcher mock-up on blacktop. (E) Whole-body wave-like
motion transferring a ball.
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one or two specific functions, such as locomotion or manipulation (4–6). Recent designs, such
as SMORES (7, 8), Sambot (9), showcase how those untethered, self-assembling, rigid modu-
lar robots can reconfigure into different configurations to achieve various locomotion patterns.
Multi-legged robot swarms (10) successfully incorporated appendages such as limbs into the
robot design and showed the capabilities of modular robots to navigate rough outdoor terrains.
While modular robotic systems have mainly focused on locomotion (6, 11), there are a few ex-
amples of systems that attack manipulation, either using grippers formed from the modules (11),
turning a screw with aerial rotorcraft (12), or transporting a table by lifting (13). Despite these
advances, a significant gap remains in creating modular robots that are not only adaptable for
locomotion and manipulation functions but also easily deployable, packable, and capable of
constructing temporary structures on a human scale.
To address this gap, tensegrity structures (14, 15), known for their lightweight design, can
sustain significant weight while also being able to deform and adapt to different shapes, mak-
ing them an ideal complement to modular robotic systems that require both versatility and
robustness. They typically consist of rigid components, such as rods, held together by flexi-
ble elements like cables or strings, allowing them to be both lightweight and compliant (14).
NASA’s Super Ball Bot exemplifies the potential of active tensegrity robots, utilizing cable-
driven systems for both landing and locomotion (16). Research studies on pre-assembled teth-
ered module-based tensegrity robots have also demonstrated their effectiveness in outdoor loco-
motion and as robotic grippers (17, 18), further highlighting the practical applications of these
structures. Moreover, the lightweight and deformable properties that are crucial to our design
enable efficient transport and deployment via rotorcraft with very limited loading capacities,
representing an early but important step toward the development of flexible, rapidly deployable
robotic solutions for various applications, including emergency response.
Untethered operation is also crucial for making each module more practical. Small-sized,
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shape-changing soft modular robots primarily use three actuation methods. The first is high-
current Shape Memory Alloys (SMAs), which allow for quick demonstrations but are difficult
to design for untethered use, making outdoor testing challenging (17, 19). Despite this limi-
tation, tethered shape-changing soft modular robots actuated by SMAs excel in manipulation
through deformation, whereas rigid robots often need extra parts to achieve similar function-
ality (6, 20). Pneumatic systems, such as air pumps, also struggle with untethered operation,
though Foambot (21) manages untethered vibration using an air pump. A more complex but
effective approach is cable-driven systems, which adjust string lengths via motor-pulley mech-
anisms, offering broader control bandwidth, lower cost, and greater environmental robustness.
Eciton Robotica (22) demonstrates untethered operation using this method, showcasing soft
modular robots capable of self-assembling. However, fully adaptable, deformable untethered
soft modular robots face ongoing challenges such as self-recognition, module communication,
and the complexities of assembly and disassembly (11), which need to be addressed for practical
deployment in unstructured environments.
Scaling up soft modular robots to human size is beneficial for making them applicable
in people-centric and outdoor applications. Some manually assembled modular flexible sys-
tems offer meter-scale solutions, such as legged locomotion across various terrains via shape-
changing capabilities (23–25). Achieving self-assembly allows for more complex, responsive
behaviors, requiring durability in diverse environments and the ability to autonomously navigate
and interact with complex terrain. This scale-up also unlocks new possibilities for constructing
human-scale infrastructure, such as shelters and bridges. Although the use of active modular
robots for human-sized infrastructure construction remains largely unexplored, promising ad-
vancements in related fields highlight its potential. For example, passive structures have been
successfully assembled using mobile robots and aerial rotorcraft, such as the construction of a
6-meter-tall tower from 1500 foam blocks by quadcopters (26–29). Our work explores the inte-
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gration of active modular blocks into rotorcraft-assisted construction, with the goal of enabling
the formation of active 3D structures, such as active scaffolding equipped with an antenna that
can dynamically adjust to point toward a satellite to increase signal reception.
In this work, we have integrated the properties of modular and tensegrity robots to create
a system that embodies five key characteristics essential for each module: (a) lightweight and
easily deployable, (b) deformable, (c) untethered operation, (d) designed for outdoor use and
capable of bearing loads, and (e) capable of 3D attachment and detachment. With these char-
acteristics, our system can form various 3D structures using different assembly methods, such
as walking into position or being transported by rotorcraft. The deformability of each block
allows assembled structures to dynamically change shape, while the lightweight nature enables
the blocks to be deployable by rotorcraft. These capabilities allow the modules to achieve three
distinct functions (Figure 1): (i) effective locomotion across different terrains by adapting their
shape, (ii) versatile object manipulation through various methods (grasping and non-prehensile
manipulation), and (iii) rotorcraft-assisted assembly into active 3D lattice structures. These
functions are not isolated; rather, they interact synergistically, allowing an assembled active
structure to potentially perform multiple tasks through whole-body deformation, with the po-
tential to interact adaptively with humans and the environment without requiring reassembly.
For example, a snake-like configuration of blocks can move through open spaces, contract to
navigate narrow openings, and transport objects along its path. A chain of blocks can form a
bridge over a gap by locomotion, with the active bridge also capable of undulating to transport
objects across. A human-scale shelter skeleton can lower to facilitate fabric placement and then
rise to its full height. These examples showcase the versatility and potential of active structures.
6
RESULTS
Robot design and characteristics
To meet diverse functional requirements, especially for block deployment and 3D structure
formation, we opted for a simple, cubic design for each block, as its symmetrical properties
facilitate tiling into larger structures using axis-aligned connectors, reducing the complexity of
assembly. Figure 1 illustrates the physical design of a single block, which differs from tradi-
tional tensegrity robots like the well-known 3-bar or 6-bar designs where rods are connected
solely by strings or cables. Each block in our design features a flexible central joint (TPU
printed), which behaves similarly to a ball joint, from which eight rigid rods extend outward
in a 3D radial pattern, classifying it as a class-8 tensegrity structure according to Tensegrity
Systems by Skelton and De Oliveira (14). Each rod terminates in an endcap, with the twelve
adjacent pairs of endcaps connected by strings that can be adjusted in length by motors housed
within the endcaps. This 3D rotational symmetry across various axes allows for the connection
of adjacent blocks in multiple orientations, enabling versatile assembly configurations.
Aiming to achieve untethered operation, all 12 actuators and 12 connectors are integrated
into the 8 endcaps, and each block is powered by an onboard Lithium-ion battery capable of
supporting locomotion on flat ground for up to 3.5 hours. A customized PCB includes onboard
sensing components comprising an Inertial Measurement Unit (IMU), a Wi-Fi module, and an
RP2040 microcontroller for twelve-channel encoder processing.
With the focus of creating a lightweight, easily deployable, and outdoor-capable module
with a relatively good thumb of load-bearing capacity, we used carbon fiber bars and high-
stiffness strings in the design. This choice of carbon fiber enables each module to weigh less
than 1.2 kg, with dimensions of 52.11 cm per side length, resulting in a density of 8.53 kg/m³.
The blocks are also designed to withstand drops of up to 3 meters onto various outdoor surfaces
7
without damage (Movie S12), ensuring robustness during rotorcraft-assisted vertical assembly.
Each module can support a load of approximately 153.53 N, demonstrating a load-bearing ca-
pacity of 13 times its own weight. Additionally, similar to the approach taken for AuxBots (30),
we also tested the actuated lifting forces, finding that our blocks can exert forces 7 to 11.5 times
their own weight (AuxBots can exert forces 23 to 76 times their weight). Failures typically
occur at the carbon-fiber rods. Depending on the specific applications, design adjustments such
as using thicker rods or strings may be required for enhanced load-bearing capacity or drop
resistance, or employing longer rods for larger-scale module designs.
Flexible central joints and 12 adjustable-length cables (actuated by 12 motors) enable the
structure to adapt to various configurations. For instance, shortening four parallel strings while
accordingly extending the other eight strings compacts the robot into a flattened shape, reduc-
ing its height to 30% (Figure 2(A), Movie S11), while shortening eight strings (extending the
other four correspondingly) on two parallel faces compresses it further into a bundle, reducing
its volume to 41% of its original size (Figure 2(B), Movie S11). Additionally, four strings on
a single face can be shortened to perform a gripping action on external objects. By actuating
strings in specific sequences, the robot can also achieve continuous motions to achieve loco-
motion. To achieve a desired configuration, the controlled string lengths can be determined by
solving the nonlinear static equation, Kn =fex −g, using the Lagrangian method described
in (31). In this equation, Krepresents the stiffness matrix, nthe nodal coordinates, fex the ex-
ternal force matrix, and gthe gravitational force matrix. Using the matrix-based form-finding
method (32) to solve the static equation allows us to determine all feasible shapes within the
robot’s workspace. The workspace for string movements from 0 to l(initial length of the string
between two endcaps) and from 0.5lto lis depicted in Figure 2(C) and (D), respectively, based
on 500 samples. These deformation capabilities allow each robot to not only locomote but also
be packed into a compact form for transport and function as a gripper for object manipulation.
8
To further assess whether the deformation is primarily due to the central joint or if the rods also
bend, we conducted experiments measuring the displacement versus force relationship for a
single carbon fiber rod, both with and without the central joint. The results show that achieving
a 5 cm displacement required approximately 0.5 N with the central joint, compared to 12.8 N
for the rod alone (Figure S5). The results indicate that the deformation is primarily due to the
central joint, as the rods exhibit significantly higher stiffness and resistance to bending.
Figure 2: Two packing strategies and the workspace of a single module subject to different minimum string
lengths. (A) Flattened shape. (B) Bundle shape. (C) Workspace of a single module when min string length equal
to 0, (D) and 0.5l.
For the purpose of enabling robust 3D structure formation, self-assembly, and self-
disassembly, we require the connectors between blocks to have four properties: secure con-
nection, error-tolerant attachment, reliable detachment, and power efficiency. Due to the lack
of inherent mobility of each individual unit, most existing shape-changing modular robotic sys-
tems either depend entirely on manual assembly (23, 33–39) or partially (40–42). In addition,
permanent magnets are commonly employed for connection (19, 20, 43), but strong magnets
are difficult to separate for detachment. Additionally, the larger the robot, the larger and more
impractical the required magnets become. Electromagnets (44) and electro-permanent mag-
nets (45) have been used in rigid modular robots but require heavy coils or substantial electrical
9
current.
In pursuit of the four desired properties, we designed the connector (Figure 3) with four spe-
cific features, each tailored to meet one of our requirements. First, the connector incorporates a
mechanically interlocking design that sustains at least 370 N of force per pair of endcaps, ensur-
ing a secure connection that can withstand loads at least 125 times the robot’s weight. Second,
permanent magnets are included to mitigate alignment errors, contributing to error-tolerant at-
tachment. Third, the connector allows for reliable detachment through the combination of the
first two designs that facilitates easy separation when necessary. Lastly, the connector operates
with low power consumption, requiring about 0.06 J of energy per connection via a latch-servo
mechanism, and once locked, it does not require additional power to maintain the connection.
To evaluate the effectiveness of the magnetic alignment, we conducted experiments by fixing
one block’s position and placing another at various angles (0, 15, 30, 45, and 60 degrees) and
distances at 1 cm intervals (Figure 4(B)). Our results, shown in Figure 4(D), indicate that at
angles of 45 degrees or less, at least one pair of endcaps attaches, enabling further movement
toward complete docking. However, at 60 degrees, no attachment is observed. The magnetic
force between a pair of endcaps (Figure 4(A(ii)), endcaps a and b) on two modules is shown
in Figure 4(C), with the model used for calculation described in the “Supplementary methods”
section.
Specific gaits are developed for both the docking and undocking processes. For docking, we
designed two specific gaits: turn left fix vertical left front and turn left fix vertical left back.
Testing these gaits demonstrated that even when only one pair of endcaps initially makes con-
tact, the appropriate gait ensures the successful attachment of the remaining endcap pairs, re-
sulting in 10 successful attachments out of 10 trials. For undocking, an unscrewing motion is
employed. This process begins with the shortening of the horizontal strings on the first module
and the vertical strings on the second module, followed by reversing the sequence: shortening
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the vertical strings on the first module and the horizontal strings on the second module (Fig-
ure 4(E), Movie S11)).
Figure 3: Endcap design and attachment. (A) Exploded-view drawing of the active connector. (B) Two states
of the active connector: unlocked, with latches rotated inward, and locked, with latches rotated outward. (C)
Magnetic alignment to attach and mechanical lock process.
Structure formation
What kinds of active structures can our module design achieve? This section demonstrates sev-
eral possibilities, including bridges that enable non-prehensile manipulation, tents that expand
or contract for use and disassembly, and scaffolding that can rotate to direct an antenna or solar
panel.
Rotorcraft plays a key role in the demonstrated deployment, vertical assembly, and provid-
ing camera perception. For deployment, the rotorcraft must have a payload capacity exceeding
the weight of the module; our modules weigh 1.2 kg, while the DJI Matrice 350 RTK rotorcraft
used in this study has a payload capacity of 2.7 kg. The size of the rotorcraft (unfolded, without
propellers) is similar to that of one module: 81 cm ×67 cm ×43 cm (L×W×H) compared
to 52.1 cm ×52.1 cm ×52.1 cm. We tested the flight duration, which was about 30 minutes,
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Figure 4: Attachment and detachment. (A) State estimation and detection of attachment. (B) Alignment experi-
ment setup for a pair of blocks at different configurations. (C) The magnetic forces between two endcaps at varied
distances and angles of 0◦, 15◦, 30◦, and 45◦.(D) Alignment robustness contour for a pair of blocks at different
configurations. (E) Detachment between two modules.
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imposing a constraint on extended assembly tasks.
Figure 5: Planning and control. (A) A rotorcraft hovers at 4.5 m. Below, three blocks are positioned: one
malfunctioning block on the left and two rescue blocks. (B) Side and top views of initial block positions and
movement. (C) Operational workflow: on-board control, off-board state estimation, and motion planning.
Terrestrial formation of structures: block connection
The structure formation on land involves attaching pairs of blocks, where one block remains
stationary while the other, the active block, selects gaits from predefined gait primitives gen-
erated by the gait generation helper to approach the target. This attachment process consists
of two critical phases: the approach phase, guided by real-time motion planning and low-level
re-planning to mitigate errors, and the connection phase, where blocks are precisely aligned and
joined. The transition between these phases is determined by the distance and angular differ-
ences between the start and goal positions of the active block. If the start and goal are within
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a threshold of 350 pixels (a unit derived from image-based sensing) and 10 degrees, the pro-
cess advances to the second phase; otherwise, it continues refining the approach phase until
proximity is achieved.
To ensure accurate real-time sensing during both phases, we employ a rotorcraft-mounted
camera to provide a broad field of view and enable rapid repositioning for large modules. Hov-
ering at 4.5 meters, the rotorcraft detects color-coded fiducials—blue for the middle joint and
pink/yellow for endcaps—on each block. To estimate the state of each block, we first identify
white endcaps via brightness thresholding, establishing boundaries for the colored fiducials.
Color detection is then performed within predefined HSV ranges, adjusted for lighting varia-
tions. We select 4nregions optimized for the desired area-to-perimeter ratio, where nis the
number of visible modules. Finally, a global distance minimization algorithm associates each
blue joint with its closest color fiducials, allowing precise calculation of each module’s position
(x,y) and orientation (θ). The system operates with a latency of approximately one second,
primarily due to image data transmission.
In the approach phase, an A* search algorithm is used to find a path to the goal using five
selected gait primitives. The modules’ symmetrical design, lacking a defined front, left, right,
or back, allows any side to act as the ‘front,’ facilitating directional changes. To reduce the
search space, a ‘front’ face is defined based on the color fiducial on each module, with active
modules consistently using the left face to connect with passive robots. This simplification,
however, comes with a cost— in tight spaces, it may be beneficial to use motions outside this
limited set of primitives. Nonetheless, these primitives have been sufficient for the demonstrated
examples. We set a maximum of 10,000 iterations for the A* algorithm. If A* fails due to overly
tight constraints on reaching the goal, the blocks continue with the gait from the last successful
search. If there is no recent path, the blocks use a greedy algorithm to choose the action that
most decreases the distance to the goal. The primitives used by A* include turn left,turn right,
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turn slow left,turn slow right, and slide left slow.
In the connect phase, the active module employs three sliding mechanisms to make
the final approach: slide left slow,slide left slow front more, and slide left slow back more.
When the absolute angular difference between the active and passive modules is large,
slide left slow back more is selected if the active module has a larger angular value; otherwise,
slide left slow front more is used. When the angular difference is small, the module typically
chooses slide left slow. However, to avoid the possibility of the slide left slow primitive be-
coming stuck on a terrain obstacle, there is a small probability that the module randomly selects
one of the other two primitives, as shown in the data collected (Figure S2).
To showcase both the system’s capabilities in structure formation and its ability to rescue
malfunctioning robots, we present an example of a rescue scenario involving a damaged block
(see Figure 5 and Movie S1). In this scenario, the far-left block is broken, and the other two
blocks act to rescue it. Figure 5(B) shows the side and top views of the blocks, highlighting
their automatically planned movement trajectories. The first rescue block attaches to the broken
block (0 s - 164 s). Since the malfunctioning block lacks power, only the active connectors on
the rescue block are actuated during attachment, resulting in only two pairs of endcaps being
connected. The second rescue block then joins (186 s - 1,387 s), and together they pull the
broken block to safety (1,446 s - 1,736 s).
Aerial formation of structures: vertical lifting and assembly
To use rotorcraft for vertical assembly, the block grasping mechanism was designed to avoid
interfering with the rotorcraft’s landing. The mechanism needed to be compliant to handle the
challenges posed by wind conditions, which make precise docking difficult. A cable and inflated
ball mechanism was developed and attached to the rotorcraft. The rotorcraft positions the ball
on top of the block, which then grasps the ball for secure flight. For terrestrial connections, the
15
Figure 6: Structure formation with rotorcraft assistance. (A) Adaptive scaffolding formation: block-rotorcraft
pair transports the block (t = 23 s), assembles the structure (t = 147 s), and deploys a solar panel (t = 238 s - 247
s), with the completed scaffolding rotating up to 360 degrees. (B) Rotorcraft-assisted bridge formation: modules
start in a box (t = 0 s), the rotorcraft forms blocks into a vertical structure (t = 63 s), tilts the structure (t = 800 s),
and completes a bridge over a 2-block width gap (t = 814 s). (C) (i) Rotorcraft aiding in tent skeleton assembly (t
= 165 s), dropping a block with covering cloth (t = 229 s), and tent completion with the stretched skeleton (t = 366
s). (ii) Top and 3D views of the block-based tent skeleton.
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sensing, planning, and execution loop automates module connections. However, the 1-second
latency in the vision system complicates dynamic aerial connections. As a result, the examples
in this section were demonstrated using human remote control of the rotorcraft.
Figure 6 shows several examples of how 3D structures, such as tents, adaptive scaffolding,
and bridges, can be assembled using a rotorcraft. Figure 6(A) and Movie S4 show a rotorcraft
constructing adaptive scaffolding that provides targetable support for solar panels. Once blocks
have been assembled into a scaffolding tower, a block then grasps the solar panel and lifts it to
position; the tower can then deform to track the sun. Figure 6(B) and Movie S5 show an example
of forming a bridge. First, blocks are assembled into a vertical stack on one island; the rotorcraft
then tilts the stack to form the bridge. Figure. 6(C) and Movie S3 show the construction of a
shelter from nine blocks on fairly level grass. The rotorcraft assembles a 2-meter tall structure
and delivers the 3 kg fabric. Then, the shelter compresses to be short enough (about 70%) that
a human can attach the fabric.
To evaluate the structural stability under varying loads, we analyzed the critical buckling
load, which is defined as the load at which a structure is susceptible to global buckling. This
critical load was determined by solving the generalized eigenvalue problem for the scalar α, as
expressed in the following equation: −(KG2+KE)dn=αKG1dn, where KG1and KG2rep-
resent the geometric stiffness matrices due to external loading and prestress, respectively, and
KEdenotes the material stiffness matrix (46). Using this formulation, we computed the critical
buckling compressive loads for various structural configurations. As illustrated in Figure S7, a
single-unit configuration exhibited a critical buckling load of 179.07 N. For horizontally com-
bined configurations of two, three, and four units, the critical loads were calculated to be 286.06
N, 286.21 N, and 286.06 N, respectively. Horizontal bridge configurations comprising three and
four units demonstrated loading capacities of 187.58 N and 295.85 N, respectively. In contrast,
vertically combined units exhibited slightly lower critical loads, with values of 176.83 N, 176.24
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N, and 175.93 N for two, three, and four units, respectively. Additionally, the critical buckling
load for a tent structure was determined to be 202.59 N. Experimental validation for a single
unit under compressive load resulted in a measured critical load of 153.53 N, closely approxi-
mating the predicted value of 179.07 N. The observed discrepancy can be primarily attributed
to imperfections in the structural bars, strings, and 3D-printed joints. Moreover, the eigenvalues
of the stiffness matrices for the ten structural configurations are provided in Figure S7.
Manipulation
A primate may grasp a tool tightly to manipulate it (47). A dolphin playing with a ball em-
ploys a whole-body non-prehensile approach to manipulation (48). Ants perform cooperative
transport by synchronizing individual actions with the group’s collective force (49). Different
arrangements of robotic blocks and tasks motivate the use of each of these strategies. In this
way, the developed system provides a platform for exploring different types of manipulation.
A single module is capable of functioning like a gripper (Figure 6). This capability is
demonstrated in rotorcraft deployment, where a block acts as a gripper to hold onto a ball
mounted on the rotorcraft. Apart from the ball, the module can grip additional robots, boards,
fabric, or balls. This also enables autonomous construction of active structures, similar to how
prior rotorcraft systems have constructed passive architectural structures (28, 29).
To demonstrate the potential of modular robots for transporting objects, similar to ware-
house robots such as those developed by Kiva Systems - Amazon Robotics (50), which transport
shelving units, we present the following examples. As illustrated in Figure 1(C), a single block
can transport two boxes. Further, Figure 1(D) and Movie S9 show a pair of blocks transporting a
manikin on a stretcher. The manikin used, along with the stretcher, weighs approximately 5 kg,
which is significantly lighter than an average human. While the current blocks are neither strong
enough to carry a human nor fast enough for emergency response, with further fine-tuning to
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improve their weight-sustaining capability, they have the potential to be used for transporting
heavier objects in the future.
Biological systems use manipulation capabilities to rescue individuals. For example, ants
have been observed to engage in complex behaviors to assist and free trapped members of their
colony (51). The previously discussed robotic rescue scenario depicted in Figure 5(B) likewise
serves as an example of carrying manipulation.
In addition, manipulation in biology is not limited to animals with opposable thumbs –
dolphins can play with a ball (48). Similarly, Figure 1(E) and Movie S10 show how blocks
can link to form a dynamic, non-prehensile conveyor system, moving a ball (diameter: 72 cm,
weight: 430 g) without the need for direct grasping. This is achieved by the synchronized
movements of the connected blocks, simulating the wave motion of a surface that propels a
ball. To further explore the adaptability and limitations of this conveyor system, with the same
control sequence, we extended our testing to three other objects: a cylinder (perimeter: 180 cm,
height: 61 cm, weight: 2,050 g), an irregularly shaped bean bag cushion (max perimeter: 280
cm, height: 1 m, weight: 745 g), and a cuboid box (51 cm ×54 cm ×47 cm, weight: 1,712 g)
as shown in Movie S10. The success rate we tested was 5/5 for both the ball and cylinder. The
cushion’s success rate was lower, 4/5, due to its tendency to fall from the middle of the conveyor
before reaching its destination, indicating challenges in maintaining stability for objects with
shifting centers of gravity. We tried different initial orientations for the box; manipulation was
successful for 2 of the 5 configurations we tried. The box’s failures were attributed to one of its
corners becoming lodged in the face with four strings, showing the limitations of the system’s
ability to handle objects with sharp edges and rigid structures.
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Locomotion
Robotic locomotion uses two primary strategies. Statically stable approaches, used by e.g.
Honda’s ASIMO (52), maintain constant balance by keeping the center of mass above a well-
defined support polygon. Dynamic gaits, utilized by e.g. Boston Dynamics robots (53), allow
faster motion, such as running or jumping, but require more sophisticated control approaches.
In the paper, we focus on statically stable gaits for their simplicity and ease of control. We note,
however, that lightweight robots are also well-suited to dynamic motion strategies (54).
Control and planning strategies in the literature vary, from fully pre-programmed sequences,
such as Sony’s dancing robots (55), to gaits generated by machine learning methods that au-
tonomously optimize motion for diverse environments (56). As this work focuses on robot
design and capabilities, we implement an approach that sits somewhere in between. A human
specifies the basic pattern as a time-dependent sequence of constraints (e.g., the left front foot
should be lifted at time 2 s and the left back foot at 6 s while maintaining a maximum width
profile of 0.37 m) and an automated gait generation helper algorithm constructs the control se-
quence, determining the string lengths needed to satisfy these constraints. In multi-robot setups,
the gait generation helper treats the connected endcaps between robots as single units to ensure
synchronized movement. More details about gait generation helper can be found in the “Sup-
plementary Methods” and Figure S3. For single robots, the gait is inspired by the quadrupedal
”amble” pattern (57), where feet on the same side are lifted and moved forward sequentially. In
multi-robot setups, the gait generation helper coordinates synchronized movement by treating
connected endcaps between robots as single units, with feet grouped into two sets based on a
zigzag pattern of diagonally adjacent feet. The gait generation helper algorithm adapts the con-
trol lengths to different environmental constraints, ensuring smooth and coordinated movement.
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Locomotion dynamic models
To understand how slopes and surface friction impact walking behavior, we developed a dy-
namic locomotion model that helps explain why these differences occur and predict their effects.
This model not only allows us to create a mechanical theory of locomotion but also provides
a framework for testing our hypotheses through experiments. The dynamic model is based on
the Lagrangian method (31) and is represented by equation M¨
n+D˙
n+Kn =fex −g,
where M,n,K,D,fex, and gare the mass, nodal coordinates, stiffness, damping, exter-
nal, and gravitational matrices. The external forces fex at the contact points can be divided
into parallel (fex,∥) and perpendicular (fex,⊥) components, satisfying: fex =fex,∥+fex,⊥. To
model how the robot interacts with the ground, we treat the ground as a spring-damper sys-
tem. Assume the ith node is contacting the ground, the fexi,⊥and fexi,∥can be written as:
fexi,⊥= (KG|nzi|+CG|˙nzi|)⊗0 0 1Tand fexi,∥=µ|fexi,⊥|sgn(˙
ni−0 0 |˙nzi |T),
where KG,CG, and µare the stiffness, damping, and friction coefficients of the ground, and
sgn(v) is an operation that takes the direction of the vector v.
To validate this model, we conducted physical experiments on three different surfaces:
wood, a coir vinyl mat, and sandpaper, each at slopes of 0, 5, 10, and 15 degrees. The sur-
faces are modeled by adjusting only the friction coefficient in the simulator. We estimated these
coefficients by tilting each surface until the robot began to slip, resulting in friction values of
µwood = 0.354,µcoir = 0.854, and µsand = 1.412. This method does not distinguish between
kinetic and static friction coefficients. Comparisons of the simulator’s output with real-world
tests are shown in Figure 7. The results show that the simulation provides a useful prediction,
with the shapes of the curves for the various surfaces for the simulation qualitatively matching
the curves from the experiments. The lowest friction surface (wood; red curve) has the shortest
walking distances for the robot per gait cycle for all non-zero slopes in both experiment and
simulation. Steeper slopes result in shorter walking distances.
21
Figure 7: The historical data for the upper node’s x-coordinate across various surfaces (wood in red, coir
vinyl in orange, and sandpaper in blue) and slope angles are presented as follows. From left to right, slope
angles are θ= 0◦,θ= 5◦,θ= 10◦, and θ= 15◦, respectively.
Locomotion efficiency and cost of transport
To evaluate the energy efficiency of our locomotion system, we calculated the Cost of Transport
(CoT), a dimensionless metric that compares energy efficiency across robotic and biological
systems (the details of the calculation can be found in the “Supplementary methods”). The
results show that our module is less efficient than some of the aquatic robots, such as octopus-
inspired and reconfigurable armed robots (58), but more efficient than a prior motor-driven soft
six-bar tensegrity robot (59) and significantly more efficient than soft modular robots actuated
by SMAs (20). Specifically, the CoT for single-, two-, and four-module locomotion is 163, 143,
and 178, respectively. Further details can be found in Figure S8.
Outdoor locomotion demonstrations
Outdoor environments present unique challenges and opportunities for modular robot locomo-
tion. Depending on how blocks are arranged, different locomotion strategies can be employed to
navigate various terrains and obstacles. Figure 8 shows several modes of locomotion: Travers-
ing a stream by lifting the front blocks while walking with the rear blocks on non-level dirt
surfaces with leaves and stones (A, Movie S6), traversing a log tunnel (B, Movie S7) and a
narrow corridor between trees (C, Movie S8) by changing shape before walking, and a “stan-
22
Figure 8: Locomotion in natural environments. (A) Four blocks cross a 0.6 m wide stream, indicated by a blue
arrow, forming a bridge, shown by a yellow arrow (t = 647 s). (B) Two blocks compress vertically to pass under
a log tunnel (0.35 m high) from their original height (0.52 m) and then traverse (t = 244 s). (C) To navigate a
narrow alley (0.37 m wide), the blocks compress horizontally. (D) Speed comparison for different block numbers
on grass, soil, blacktop, snow, and ice, with linear regression estimating speeds for each configuration.
23
dard” locomotion gait (D, Movie S2, Table S2) with speed measurements across grass, soil,
asphalt, snow, and ice. For traversing narrow alleys and tunnels, the same gait is employed but
with different directional constraints to accommodate the specific environments. In tunnels, the
primary constraint is the height (z-coordinate), which must remain below a certain threshold to
avoid collisions with the tunnel ceiling. Narrow alleys limit the robot’s maximum width dur-
ing the gait. From the experimental results, the two-block system is the fastest for all terrains
except in the snow scenario, where the four-block system is the fastest. We surmise that the
single-block system is slow because only a single foot is lifted at a time, leaving the other three
feet down to form a support triangle. For the four-block system, the problem is the opposite
– when lifting all diagonal feet in a zigzag pattern, more feet are left on the ground to permit
perfect compliance to the surface; some of the “ground” feet are in fact slightly lifted and slip.
Different gaits or lower-level control of the compliance of feet to the surface might enable faster
locomotion for multi-block systems.
DISCUSSION
Existing modular robot systems have demonstrated the ability to use simple components to
achieve a variety of tasks (6, 10). However, they have been primarily limited to locomotion
and manipulation tasks (6), lacking the deployability and the capability to build 3D temporary
infrastructures. Our study bridges this gap by integrating the principles of tensegrity — char-
acterized by inherent lightweight and deformable properties — into modular blocks, enabling
the robots to combine three critical functions within a single design: navigating challenging
terrains, performing complex manipulation tasks, and constructing temporary structures with
real-world applications.
The mechanical design incorporates eight rods linked by flexible joints and is complemented
by active connectors on the endcaps, facilitating 3D structure formation and whole-body defor-
24
mation, improving manipulation and locomotion in unstructured outdoor environments. Transi-
tioning from indoor to outdoor applications highlighted several environmental challenges, such
as fluctuated temperatures affecting battery life and low light conditions impairing state esti-
mation. These issues, coupled with the challenges of navigating diverse terrains like rugged
woodland, underscore the need for more robust and adaptive gait optimization strategies for
modular systems.
The deployment system, assisted by rotorcraft, enables the rapid assembly of large-scale
3D structures. However, our experiments indicate that further system refinements are needed to
handle a larger array of robotic modules and more dynamic tasks. To address these issues, we
propose enhancing local processing capabilities on the rotorcraft to reduce latency and improve
real-time decision-making, which is vital for dynamic and complex task environments. More-
over, introducing onboard cameras and sensors on the modules themselves could foster a more
autonomous and integrated approach to self-assembly, similar to the methods demonstrated by
Daudelin et al. (60).
Our research provides a starting point for exploring the use of modular robots in building
temporary active structures intended for human use. Inspired by the cooperative behaviors of
insect swarms, this study demonstrates the potential of simple, modular units to collaboratively
construct functional setups like emergency shelters without complex control systems. The in-
sights from our experiments suggest practical avenues for further development, particularly in
enhancing the structural integrity and deployment efficiency of these systems. For instance, a
modular-assembled tent structure successfully housed an adult, showcasing its immediate util-
ity. However, other configurations, such as bridges assembled with four modules and stretchers
transported by two modules, though effective for specific tasks, currently lack the strength to
support human weight. We anticipate that modular shape-changing lightweight blocks will en-
hance multi-functional robotics, supporting applications such as automated infrastructure con-
25
struction with integrated structural and actuation elements, compact building blocks for space
systems, and deeper insights into biological systems to drive bio-inspired robotics design.
MATERIALS AND METHODS
Block fabrication and design
Each block consists of a flexible central joint, eight magnetic endcaps, and a Printed Circuit
Board (PCB). The central joint is a flexible core of the robot, fabricated from Thermoplastic
Polyurethane (TPU) material (1.75 mm, Amazon Basics), selected for a combination of flexi-
bility and durability. The central core features an internal cavity sized to house a battery (21,700
3.7 V 4,200 mAh). Carbon-fiber rods (diameter = 3.5 mm; length = 30 cm) were chosen for
their high strength-to-weight ratio.
We created two designs for robot endcaps: 2A1P and 1A2P. The 2A1P type comprises two
active and one passive connector, whereas the 1A2P type consists of one active and two passive
connectors. These endcaps provide structural support for multi-block assemblies and house the
motors. Each 2A1P endcap holds two servo motors for locking and unlocking connections and
one primary motor for controlling string lengths. For weight balance, the 1A2P type contains
one servo motor and two primary motors. The primary motor is an N20 DC motor with a
magnetic encoder (12V/30,000 rpm with a 1:298 gear ratio); servo motors are micro 3.7g servo.
Two different sizes of magnets are used in the design: for the active connector, a magnet with the
N pole facing outward (Amazing Magnets, product number D063J-N42, thickness: 1.59 mm,
diameter: 31.75 mm), and for the passive connector, a magnet with the S pole facing outward
(Amazing Magnets, product number D125J-N42, thickness: 3.18 mm, diameter: 31.75 mm).
The PCB, detailed in Figure S1, is a six-layer design with a Wi-Fi module, one microcon-
troller for control commands and sensor data, a second microcontroller for actuation signals, an
inertial measurement unit, and temperature sensors.
26
Experimental design and data analysis
This section presents the experimental design and data analysis approaches utilized to evaluate
the performance of robotic modules in locomotion and rotorcraft-based state estimation experi-
ments across different outdoor terrains, employing tracking and computer vision techniques for
measurement and analysis.
For the drop test, we tested the block on four different surfaces (Movie S12): hard-packed
snow (tensile strength ranging from 0.1 to 1 MPa (61)), grass-covered soil (stiffness from 240
to 1,693 kN/m (62)), hard-packed gravel (stiffness modulus from 126 to 426 MPa (63)), and
soil (stiffness from 3 to 22.1 MN/m (64)). To conduct the test, we first used a rotorcraft to grasp
the robot from the ground, recording the initial height. The rotorcraft then ascended 3 meters
above this height before releasing the robot.
For experiments testing the compressive and tensile properties of a block with a half-meter
width, we developed a customized test platform according to standard testing protocols, as
shown in Figure S6. In the compression test, a wooden platform was horizontally suspended
by strings attached to each corner, which were tension-adjusted to ensure planarity, with a level
used to verify horizontality. The load was incrementally increased by filling a centrally-placed
bucket with sand (each time 500 g), while a vernier caliper attached to the left T-slots of the
support frame measured displacement. A counterweight system was employed to establish an
initial load of zero, using a second bucket pre-filled with a calculated amount of sand, connected
via a twin-pulley system to balance the weight of the empty bucket and platform. For the tensile
test, the module was suspended by a central string that passed through a freely sliding connector
within the frame’s T-slots to ensure vertical alignment, confirmed with a mounted level. Four
equal-length strings were attached to the endcaps on the right face of the module, converging
into a single strand that anchored to the right T-slots. Similarly, four strings connected to the
left endcaps were merged and routed through a pulley to a bucket, which was gradually filled
27
with sand to increase the load.
In the alignment experiment (see Figure 4(D)), a paper with grids representing various an-
gles was glued on the ground. We fixed the left block in position and manually positioned the
right block to achieve specific initial positions and angles. For each angle, we conducted at least
three tests at different positions and used the median value for the final results. Our locomotion
experiments were conducted outdoors across a variety of terrains to evaluate the performance
of different robotic modules. To quantify the speed of each module on different surfaces, we
recorded their movements using a camera system. The robots’ central joints were distinctly
marked with blue/green tape to facilitate tracking during video analysis.
For accurate distance measurement and speed calculation, we equipped the testing area with
boards featuring AprilTags (65) within the camera’s field of view. These served as reference
points, enabling us to employ computer vision techniques to track the trajectory of the central
joints of the blocks. The resulting plots illustrating the distance-time relationship are presented
in Figure 8 and Movie S2, where the speed is indicated by the slope of the linear regression.
For consistency in multi-block locomotion experiments, we track the first block relative to the
direction of movement in all cases.
Rotorcraft-camera state estimation experiments were consistently performed over soil inter-
spersed with small stones, vegetation, or snow. We utilized the DJI Matrice 300 RTK rotorcraft.
The rotorcraft was maneuvered to a fixed position at a height of 4.5 m and was equipped with
its default camera to capture images at a frequency of 30 Hz. We used a Rybozen 4K audio
video capture card to connect the laptop to the remote controller for image transmission. These
images with a resolution of 2,560×1,440 pixels were then utilized for state estimation purposes.
Outdoor structure formation with a rotorcraft includes the construction of bridges, tents, and
actuatable scaffolding. An operator manually piloted the rotorcraft using a remote controller.
Bridge constructions and actuatable scaffolding were erected over a soil environment, whereas
28
the tent formation was carried out on grass partially covered with snow.
To classify how each gait affected the displacement and orientation of a block, 25 trials
were conducted for each gait to measure the expected results. The resulting net translation and
rotation for each trial are illustrated in Figure S2, where 5 outliers for each gait are removed.
The mean displacement values among 20 trails for each gait are used for planning.
29
Supplementary Materials
This PDF file includes:
Methods
Figures S1 to S7
Tables S1 to S2
Other Supplementary Material for this manuscript includes the following:
Movies S1 to S12
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Acknowledgments: We thank R. Kramer-Bottiglio, J. Booth, X. Huang, S. Lu, E. Os-
egueda, for helpful discussion; A. Quattrini Li, and M. Jeong for equipment support. Funding:
Supported by National Science Foundation (NSF): Robust Assembly of Compliant Modular
Robots (Award 1954882). Author contributions: Conceptualization: LZ, DB, KB; Method-
ology: LZ, YJ, DB, MC; Investigation: LZ, DB, YJ, KB, MC; Visualization: LZ, YJ, MC;
Experiments Design and Implementation: LZ, YJ, DB, MC; System Design and Implementa-
tion: LZ, YJ, DB, KB, MC; Algorithm Design and Implementation: LZ, YJ, DB, MC; Funding
acquisition: DB, KB; Project administration: DB, LZ; Supervision: DB, KB, MC; Writing:
original draft: LZ; Writing: review & editing: LZ, YJ, DB, KB, MC. Competing interests:
The authors declare that they have no competing interests. Data and materials availability:
All data needed to evaluate the conclusions in the paper are present in the paper or the Supple-
mentary Materials.
38
Supplementary Methods
Gait generation helper
The gait generation helper works for both single and multi-module setups. In multi-module
configurations, the helper considers connected endcaps as one unit and synchronizes the lengths
of two strings that connect the same pairs of connected endcaps. The operational workflow
of the helper is as follows. Input: A sequence of constraints, {(PC1,PL1),(PC2,PL2),· · · ,
(PCi,PLi),· · · ,(PCn,PLn )}, where each (PCi ,PLi)represents constraints configurations and
constraints on string lengths at time step i. Some string lengths or configurations may be left
unconstrained based on human-designed requirements. The current configuration, modified by
the required constraints, is used as the starting point for the optimizer. Output: Optimized
control sequences U={U1,U2,· · · ,Ui,· · · ,Un}, where each Uigives string lengths for a
single time step.
The gait generation helper iteratively invokes the configuration optimization. The flowchart
of configuration optimization is shown in Figure S3; an outer controller optimizes the string
lengths to minimize the difference from the desired partial configurations, and the inner simu-
lator calculates the configuration that satisfies the physical constraints as well as string length
constraints defined by the outer controller. The Broyden–Fletcher–Goldfarb–Shanno (BFGS)
algorithm (66) is utilized for optimization in both the controller and simulator.
Let Ldenote string lengths, and let C(·)represent the inner simulator, where the input is the
string lengths. The output is the configuration. P(x,y)takes a full configuration xand a partial
configuration yas inputs, and extracts elements from xthat are exactly matched in y. Then,
the optimization of the outer controller can be expressed as: minLP(PCi −P(C(L),PCi ))2.
The inner simulator calculates the robot’s configuration Xthat minimizes the elastic potential
energy Vof the middle joint, which is approximated as the sum of squared differences of
39
the initial string lengths in cube configuration L0, given the inequality distance constraints L
provided by the outer controller. Let f(X)denote the calculated string lengths based on the
configuration X. The objective function for simulator optimization is C(L) = argmin
X
V(X) =
argmin
XP(f(X)−L0)2, subject to the inequality constraints: f(X)≤L.
For illustrative purposes, consider a representative example detailing a right-turn maneuver.
To describe the right turn, a human engineer selects a sequence of requirements for partial
configurations and partial string lengths, which are then sent to the gait generation helper.
Assume the distance between two near endcaps of a module is l. Here is an example input
sequence of this locomotion type, using the zheights of various connectors (used as feet) above
the ground. (i) The robot starts by elevating its left back foot, with the other three feet remaining
on the ground. This configuration is described by z1= 0.5l, while z2,z3, and z4are all zeros.
(ii) Following this initial setup, the robot propels its left back foot forward, leading to z1= 0.1l,
sleft bottom = 0.3l, with z2,z3, and z4all zeros. (iii) The robot then raises its left front foot,
ensuring the remaining three feet retain ground contact. This stage is described by z2= 0.5l,
with z1,z3, and z4all zeros. (iv) As a final move, the robot advances its left front foot forward,
described by z2= 0.1l,sleft bottom =l, while z1,z3, and z4continue to be zeros. After the
engineer generates this sequence, each requirement is used as the input for the configuration
optimization to generate the optimized control of the robot, given by the target string lengths.
The output string lengths serve as the control commands sent to the blocks.
Modeling of the magnetic forces
To accurately facilitate the connection of modules using the four pairs of magnets, we first
analytically model the forces between two magnets at a distance r(x, y)and angle αto quantify
the interaction force between them, as shown in Figure 4(B). According to the Biot-Savart Law,
the magnetic flux density dBat position rin 3D-space generated by a filamentary current Iin
the magnet is given by: dB=µ0I
4π
dl×r
r3, where dlis a vector along the electric currents flow
40
and µ0is the magnetic constant. The magnetic force between two flat cylindrical magnets is
calculated by: F=RS1RS2dI×dB, where S1and S2are the interacting surface areas of
the two magnets (67). In our robot setup, we use two flat cylindrical magnets with the same
radius but different magnetic induction values. By varying the distance and angles between two
modules, we can determine the magnetic forces between two endcaps (endcaps a and b shown
in Figure 4(A)). As seen, the contour in Figure 4(C) shows a good match with the experiment
given in Figure 4(D). The color bar denotes the magnetic forces.
Cost of Transport
Cost of Transport is calculated by dividing the energy input into the system by the work
done, which can be expressed as: CoT =Rtf
0P(t)dt+U0−Uf
Wa,U0and Ufare the total initial and
final elastic energy of the structure, and P(t)is the power consumption by the robot with respect
to time (in Watts, W), and Wais the total work done by the active cables at the end of the
actuation process: U0=1
2(lp−lp0)T(ˆ
Eˆ
Aˆ
l−1
p0)(lp−lp0), Uf=1
2(lp−lpf )T(ˆ
Eˆ
Aˆ
l−1
pf )(lp−lpf ),
vectors E∈Rne(neis the number of elements) and A∈Rneare Young’s modulus and
cross-section area of all the elements, lpand lp0are the actual length and rest length of all the
elements after prestress but before the active actuation strategy. lfis the actual final length
of all the elements. Notably, smaller efficiency ratios indicate a more efficient cable-actuation
process. The work done by the robot depends on the operation. For example, the work done
while walking, Wa, can be calculated as Wa=µmgd, where µis the coefficient of friction, m
is the mass of the robot, gis the gravitational acceleration, and dis the distance traveled by the
walking block.
41
Supplementary Figures
Main Controller
RP2040
PWM Controller
STM32F030
Wi-Fi Module
ESP8266
IMU
3.3V Power Supply
Active Connector
Power Supply
Motor Power Supply
Unused Power Supply
Main Controller PWM Controller
21700 3.7V
Battery
3.3V Power
Supply Active Connector
Power Supply
Motor Power
Supply
Motor Driver
Wi-Fi Module
IMU
Tem per atu re
N20 DC Motor
Servo
......
Sensors and inputs x12
x12
Signal
3.7V
3.3V
6V
12V
Figure S1: PCB board layout and functional framework of the power delivery and signal
transmission.
42
20 10 0 10 20
(degrees)
0
10
20
30
40
50
60
70
Distance (pixels)
turn_left
mean of turn_left
turn_right
mean of turn_right
turn_left_slow
mean of turn_left_slow
turn_right_slow
mean of turn_right_slow
slide_left_slow
mean of slide_left_slow
slide_left_slow_front_more
mean of slide_left_slow_front_more
slide_left_slow_back_more
mean of slide_left_slow_back_more
Figure S2: Collected data points for seven different gaits. The displacement and angle
differences that each gait can achieve.
43
Start
Obj(current configuration)
< tolerance?
Begin configuration
optimization with current configuration
Input: Required partial
configuration and partial
string lengths
Output: Optimized
string lengths
Yes No
Controller: Adjust string
lengths
Simulator: Compute configuration
satisfying physical constraints
with given string lengths
End
Figure S3: Flow chart of configuration optimization. The process begins with an input
generated by an engineer, which is the required partial configuration and string lengths of the
blocks. The configuration optimization step checks if the current configuration’s objective value
is within the tolerance limit. If not, the controller adjusts all string lengths. These adjusted
lengths are then provided as input into a simulator to generate a configuration that satisfies both
physical constraints and the given string length inequality constraints. The objective value is
reassessed until it falls below the tolerance threshold. Once this condition is met, the optimized
string lengths are outputted.
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Figure S4: 3D reconstruction of real-world environments, generated by Polycam soft-
ware. Stream: https://poly.cam/capture/A26AB51B-AE8D-4135-BF28-94FD225D5521; Nar-
row Alley: https://poly.cam/capture/67A7FA8A-37BF-4FAC-809E-428B47BAE1C5; Tunnel:
https://poly.cam/capture/66B5090B-433E-42BF-B46B-1BA1C57D3887.
45
Figure S5: Displacement versus force for a carbon fiber rod in two configurations: with
and without a middle joint (TPU).
46
Figure S6: Displacement versus force for a single robot under compression and tension.
(A) Compression experiment setup. (B) Tension experiment setup. (C) Results of displacement
versus force subject to compression. (D) Results of displacement versus force under tension.
During the tests, we measured 500 g of sand in each cup and then carefully poured the sand
into the barrel each time, ensuring that the wooden plate remained parallel to the ground. We
recorded the displacement from the vernier caliper each time until the modular robot failed.
47
Figure S7: Various structural configurations and stiffness plots. (A) Single unit. (B) Two
horizontally connected units. (C) Three horizontally connected units. (D) Four horizontally
connected units. (E) A bridge: three horizontal units with fewer ground supports. (F) A 4-
bridge. (G) Two vertically stacked units. (H) Three vertically stacked units. (I) Four vertically
stacked units. (J) Tent structure.
48
Figure S8: Comparison of our tensegrity modular robots with other robots based on the
Cost of Transport (CoT). All referenced data can be found in (20, 58, 59).
49
Supplementary Tables
Table S1: Comparative analysis of actuation and lifting forces in varied block configura-
tions.
Configuration Dimensions (cm) Actuation & Lifting Force
Compressed 58.5×58.5×38 ≤8.4 kg (≈7 robots)
Initial 52.1×52.1×52.1 ≤13.3 kg (≈11 robots)
Stretched 63×47×47 ≤13.8 kg (≈11.5 robots)
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Table S2: Quantitative analysis of module-based locomotion efficacy on various terrains.
Terrain Type Locomotion Speed (cm/s)
1 Module 2 Modules 4 Modules
Grass 0.27 0.54 0.40
Soil 0.26 0.54 0.36
Blacktop (3◦slope) 0.26 0.66 0.47
Snow 0.29 0.43 0.54
Ice 0.35 0.63 0.57
51
