SOFT ROBOTICS
Volume 00, Number 00, 2020ª Mary Ann Liebert, Inc.
DOI: 10.1089/soro.2019.0142

Surface Actuation and Sensing of a Tensegrity
Structure Using Robotic Skins

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Joran W. Booth,* Olivier Cyr-Choinière,* Jennifer C. Case,* Dylan Shah,*
Michelle C. Yuen, and Rebecca Kramer-Bottiglio

Abstract

Tensegrity robots comprising solid rods connected by tensile cables are of interest due to their flexible androbust nature, which potentially makes them suitable for uneven and unpredictable environments where traditional robots often struggle. Much progress has been made toward attaining locomotion with tensegrity robots.
However, measuring the shape of a dynamic tensegrity without the use of external hardware remains a challenge. Here we show how robotic skins may be attached around the exterior of a tensegrity structure, to bothcontrol and measure its shape from its surface. The robotic skins are planar, skin-like membranes with integrated actuators and sensors, which we use to transform a passive tensegrity structure into an active tensegrity
robot that performs tasks such as locomotion. In addition, sensors placed on the ends of the tensegrity rods areused to directly measure orientation relative to the ground. The hardware and algorithms presented herein thus
provide a platform for surface-driven actuation and intrinsic state estimation of tensegrity structures, which wehope will enable future tensegrity robots to execute precise closed-loop motions in real-world environments.
Keywords: tensegrity robot, state reconstruction, robotic skins

signing new arrangements of rods and tensile elements fortensegrity robots is not straightforward, leading some researchers to leverage machine-learning techniques for robot
design.8,9 In this work, we focus on the six-bar tensegrity
structure, which has a rest configuration of an irregular
icosahedron,10–15 whose faces consist of 8 equilateral triangles and 12 isosceles triangles (as shown in Fig. 1).
Common locomotion strategies generally fall into twoclasses: rolling and vibration. In the rolling approach, adjustable rods or tension elements are used to shift the center ofmass past the polygon of stability, which is also the downward
face, and tip the robot from face to face.16–25 Most of these
proposed designs use cables24 or pneumatic actuators.10,26,27
The vibration-driven locomotion strategy exploits the tensegrities’ dynamic asymmetries to produce translational motion.16,21,28–32. In one unique approach that does not fit neatly
into these two common categories, a tensegrity was placed as acage around a two-propeller drone, creating a robot that could
hop over obstacles, roll upon landing, and right itself before
hopping again.23 Finally, we note that most of the gaits used inthese proposed designs were selected by a human designer;

ObjectiveS

oft robots offer functional advantages such as thepotential for safe human/robot interactions and robustness to falls, impacts, and vibrations. One avenue of exploration in soft robotics is to leverage the advantages of tensegrity
structures, a concept originally used to create lightweight architectural elements.1,2 A tensegrity structure consists of rigid
compression elements (which we call rods) that are linked bytensile elements (e.g., cables, strings). These networks form
lightweight,3 compliant, and load-bearing structures4 due tothe equilibrium between compression and tensile forces. The
concept of tensegrities has also been applied to explain biological phenomena5,6 and to create novel rolling robots.7
Several locomoting tensegrity robot designs have been
proposed, spanning a range of rod connectivity patterns, actuation mechanisms, and locomotion strategies. Tensegrity
structures are typically described by the number of rods usedin the arrangement, and common arrangements include 3-, 6-,10-, and 12-bar tensegrities. While most proposed tensegrity
designs borrow from well-established arrangements, de-School of Engineering and Applied Science, Yale University, New Haven, Connecticut, USA.

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BOOTH ET AL.

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FIG. 1. (A) Six-bar tensegrity covered by robotic skins. (B) Interior view of the robot, showing the onboard pneumatic
valves. (C) Node mapping for use in the state reconstruction algorithm. Sensors (thin black lines) measure the distances
between nodes (extremities of bars). Color images are available online.
however, several studies demonstrated the possibility oflearning locomotion strategies using machine-learning techniques, often paired with a robot simulator.33–36
While prior work has led to many successful designs andlocomotion strategies, real-time state reconstruction of tensegrity robots has remained difficult to achieve without use of
external sensors. For example, NASA’s ‘‘Super Ball Bot’’
tensegrity robot is actuated by adjusting the length of its
tensile elements with a motor and winch.13,37 To accurately
estimate the spatial distance between two nodes in that design, the changing length of the tensile elements between rods
must be measured, and there are few sensors that can easily
measure such extreme deformations. Measuring the amountthe motor has turned is perhaps the most direct measurement,
but does not account for stretch in the cables or deformationof the bars. ‘‘Time of flight of light’’ schemes have been
proposed as another alternative; however, these approaches
require external sensors, markers, or other points that can beused as base stations to improve the resolution of the system,
limiting their applicability in unstructured environments.38
Another proposed design used a single-strain sensor woven
through a modular, spine-like tensegrity robot for detecting
the bending states of the entire system, but no attempt wasmade at shape or orientation reconstruction.39
In this article, we present a six-bar tensegrity robot capableof rolling locomotion that is able to achieve both shape andorientation reconstruction without the use of external sensors.
Our tensegrity uses robotic skins27 to provide actuation andstrain sensing capabilities between its nodes, and uses contact
sensors placed at the ends of each rod to detect direct contact
with the environment. In contrast to prior pneumatic tensegrity
structures, we also placed the pneumatic control hardware
onboard the robot (Fig. 1b). Our state reconstruction algorithms
solve an optimization problem given the constraints imposedby the connectivity of our 6-bar tensegrity, shown in Figure 1c.
In the following sections, we describe the materials andmethods used (Materials and Methods section), including thegeneral principle of the robotic skins (Robotic Skins section).
Next, we describe the state reconstruction model (State Reconstruction Model section) using the embedded strain sensors and node contact sensors to estimate node position andface orientation, and show validation of the model (Model
Validation section). Finally, we present teleoperated rollingof the tensegrity robot on a declined plane (Rolling section).

Materials and Methods

Our six-bar icosahedron tensegrity robot is primarily madeof six wooden bars (9.5 mm diameter, 350 mm long,
McMaster-Carr) and 20 triangular fabric-based robotic skins.
The robotic skins include McKibben-type pneumatic actuators and capacitive strain sensors (Fig. 2a), which are positioned along the edges (Fig. 2b). While all 20 faces on thetensegrity include 3 actuators along each edge, only the 8equilateral faces are equipped with strain sensors. We madethis choice because the edges of the isosceles faces are either
shared with an equilateral face or always slack and unable toyield a meaningful sensor value. In the fully assembled system, the diameter is 410 mm (node to node, e.g., between N5, 1
and N6, 1 ) and the entire system has a mass of 2.55 kg.
In this section, we describe each of the elements comprising the robotic skins and the rest of the robot.
Robotic skins

Our previous work introduced the concept of robotic skins,
which are planar substrates with embedded actuation andsensing that can be attached to the surface of a deformablebody to impart controlled motion onto that body.27 This capability enables surface-driven motion and state estimationof tensegrity robots. The robotic skins demonstrated in thisarticle are built on a substrate of two layers of spandex fabric
cut into a truncated equilateral triangle (long sides, 120 mm;
short sides, 50 mm), and sewn together along the long edges
to form a pocket that houses the other components of therobotic skin (Fig. 2c, d). Muslin fabric pieces are sewn into
the vertices to provide a location to anchor the components
and provide a place to attach the skin externally. Several
anchor points are provided on the muslin fabric using snap
fasteners, allowing the modular actuators and strain sensors
to be exchanged. In addition, the fabric snaps allow each skin
to independently attach to the wooden support bars and toeach other, as discussed in the Supplementary Data. The
modular actuators and sensors are attached in a triangularpattern inside the fabric skin. In this triangular arrangement,
the strain sensor readings are closely coupled to the true edge
length, simplifying the system modeling.
Pneumatic actuators. The pneumatic actuators in the
robotic skins are McKibben-type pneumatic artificial

SURFACE ACTUATION AND SENSING OF A TENSEGRITY

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FIG. 2. (A) Main components ofa robotic skin: fabric, modular
sensors, and modular actuators. (B)Robotic skin with modular pieces
attached to the fabric using fabricsnaps. In this photograph, the fabricenvelope is turned inside-out toexpose the connection points. (C)Assembled robotic skin. (D) Hand
grasping a robotic skin to demonstrate the skin’s compliant andflexible nature, even with themodular components inside. (E)Sensing layers of the node contact
sensor are placed inside the moldbefore casting. (F) Completed nodecontact sensor. (G, H) Node contact sensor (arrow) making contact
with the ground exhibiting a lower
resistance (multimeter). Color images are available online.

muscles,40 built using similar methods as in our prior work.27
The active length of each actuator is 150 mm. The actuators
are attached to the robotic skins with a 3D-printed polylactic
acid (PLA) bracket that has an integrated snap fastener. An
external tether supplies air at 172 kPa (25 psi), which results
in a contraction of 40 mm. Complete inflation is achieved in
0.5 s, while deflation takes 0.8 s. The longer deflation time isexpected, since the flow rate during inflation is driven by aconstant pressure supply, whereas the deflation is driven by achanging pressure gradient dependent on the elastic energy
stored in the actuator and the corresponding compression of
the gas inside the actuator.

Although the tensegrity robot used 20 robotic skins, not alledges need to be active. In its equilibrium irregularicosahedron shape, the robot’s 12 isosceles faces are thosethat share two nodes with the endpoints of each pair of parallel bars, with the short side of the triangle positioned between the ends of the parallel bars (e.g., the face connecting
N3, 1 with N6, 1 and N5, 1 , later named FQ in Table 2). In oursystem, the 20 robotic skins are all equilateral at rest, which
creates slack between the ends of parallel bars and reduces
the performance of those sensors and actuators. Therefore,out of 60 potential edges for actuator and sensor placement,
12 actuators and 12 sensors are located where they cannot be4Downloaded by YALE UNIVERSITY from www.liebertpub.com at 10/03/20. For personal use only.

used and only 48 can be controlled and queried. In addition,
we simplify the pneumatic hardware by plumbing adjacentactuator pairs to the same pneumatic regulator. This reduces
the number of required regulators from 48 to 24, without
sacrificing output force. Each actuator pair was independently controlled by a pneumatic regulator circuit board,41
which was mounted on the rods (Fig. 1b). The advantage ofmounting the pressure regulators inside the robot is that we
can operate the robot with a single pneumatic tether, rather
than several tethers as shown in prior pneumatic tensegrity
robots,10,26 and is an important step toward creating an untethered pneumatic tensegrity robot.
Strain sensors. The robotic skins are equipped with
conductive composite-based capacitive strain sensors andappropriate signal conditioning boards.42,43 The ends of thesensors are sewn to polystyrene tabs with integrated snap
buttons to facilitate attachment to the tensegrity’s nodes. The
sensor outputs are transmitted to a microcontroller (Arduino
Uno) using the I2C protocol and passed through a digital
exponential filter. Since the sensor data are only meaningful
if the sensors are taut, the sensors are prestrained so that they
are in tension when their associated actuator is fully contracted. The unstretched active length of the sensors is
52 mm, while the at-rest length of a sensor while on thetensegrity is roughly 90 mm. As mentioned in the Materials
and Methods section, the system does not use all the pneumatic actuators to operate. The same is true for strain sensors.
Sensors are only installed on the equilateral triangle robotic
skins, which results in 24 active strain sensors.
Node contact sensors

Twelve node contact sensors are installed on the ends ofthe six bars in the tensegrity robot. Each node contact sensor
is made from Smooth-Sil 950 silicone (Smooth-On, Inc.) in a
3D-printed hemisphere mold with a radius of 29 mm. The
radius of the hemisphere is chosen to ensure that the tensegrity will only contact the ground with the node contact
sensors and not with other hardware. The elastomer provides
compliance and impact absorption. The sensing element
consists of a pressure-sensitive conductive polyolefin film
(Velostat, Adafruit 1361) sandwiched between adhesivebacked copper foil electrodes. This sensing element is cut
into the shape of a truncated octagram, sealed between twolayers of double-sided VHB tape (0.5 mm thick, 3M 4905)
and curled to fit the contour of the hemisphere (Fig. 2e).
Connecting wires were secured to the copper electrodes. The
sensing element was then embedded in the silicone during
the casting process, ensuring that the wires protruded fromthe molded material to allow for connection (Fig. 2f).
The node contact sensors decrease electrical resistancewhen pressure is applied (Fig. 2g, h). A threshold resistancevalue is chosen after calibration to determine whether thenode is in contact with something or not. Each finished nodecontact sensor is connected to one of two onboard microcontrollers (Arduino Pro Mini; Arduino AG) via a voltage
divider.
Electronics

The electronics system architecture consists of a microcontroller, strain sensor signal conditioning boards, and

BOOTH ET AL.

pneumatic regulator boards. The microcontroller communicates with the strain sensors and pneumatic regulators via theI2C protocol, and relays data to a PC via serial. Two control
methods are implemented: open-loop control with a preprogrammed actuator sequence and teleoperated control ofeach individual actuator.
Results

The primary result of this work is the demonstration thatrobotic skins can be used to actuate, sense, and control atensegrity from its surface. The strain sensors in the robotic
skins can be used to reconstruct the state of the robot, including both the spatial location of each node and the orientation of the tensegrity with respect to the ground. In thissection, we introduce a state reconstruction model that is
enabled by our sensor platform. We describe the validation of
our state reconstruction model and demonstrate the use of thepneumatic actuators on the robotic skins to roll the entire
system.
State reconstruction model

Our sensor-driven state reconstruction approach is unique from prior approaches for six-bar tensegrity robots,
which rely on physics models or machine vision or similar
external sensors.11,12,15,44 These physics-based models are
excellent for designing a new tensegrity system and balancing the required tension forces within the system.
However, these proposed physics-based models are notwell suited for state reconstruction since they rely on
known spring forces for system components, tension on thecables, and torques on the bars, which are difficult tomeasure. In contrast, our strain sensor approach takes direct measurements that can be correlated with distances
between nodes, and enables state reconstruction of thetensegrity robot without any knowledge of the forces in thesystem.
The state of the six-bar tensegrity can be described using
two mathematical objects: (1) a matrix of node positions in a
locally defined coordinate system, which gives the shape ofthe system, and (2) a single number representing the face incontact with the ground, which gives a global orientation of
the system. While this orientation system does not fully define the system in Cartesian coordinates, it does provide thesystem with the essential components to allow future tensegrity robots to plan their motion policies and track past
motions to continue locomotion. The state reconstruction canrun in real time on a computer, using the algorithm summarized in Algorithm 1. In brief, the sensor readings (dk) are fed
to a neural net, which estimates the face pointing down to theground (Fd). The sensor readings are also fed to an algorithm that estimates the node positions (N) by minimizing
the error between estimated positions and the sensor
readings, while satisfying the constraints imposed by therods. In addition to a neural net, we also demonstrate direct
detection of the downward face using node contact sensors,which identify the downward face using a lookup table.
Finally, the node positions are rotated according to the faceestimation, to yield the tensegrity’s state relative to thesurface it is resting upon.

SURFACE ACTUATION AND SENSING OF A TENSEGRITY

sensor failure. Regardless of the method used, the downward
face Fd can be described using three nodes (Fig. 1) as

Algorithm 1: Real-time State Reconstruction
while true do
fdk g ¼ MeasureStrainSensors();
fdl g ¼ MeasureNodeContactSensors();
Fd ¼ FaceEstimation (dl);
N ¼ NodePositions (dk,Ni  1 )Nrotated, i ¼ RotationMatrix (Fd , N);

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5

Fd ¼ fNl1 , l2 , Nm1 , m2 , Nn1 , n2 g(2)where Fd describes the face according to Table 2.

Node position estimation. Node positions are estimatedby feeding strain sensor readings to a least-squares error
minimization algorithm. Let Ni, j ¼ fxi, j , yi, j , zi, j gT represent
the node position for the ith bar and jth node on the bar (wherei ¼ 1, 2, ::, 6 and j ¼ 1, 2, see Fig. 1c) and L represent thelength of the rigid bars. Without loss of generality, we initially define the position of node N1, 1 as the local origin (i.e.,
N1, 1 ¼ f0, 0, 0gT ). The location of the other 11 nodes is foundwith respect to this node using an optimization algorithm.
The node positions in subsequent iterations are calculated
from the prior iteration. Our optimization seeks to minimize
the error between the simulated edge lengths and the measured sensor lengths (dk) and is constrained by the rigid bars,
which cannot change in length.
24

minimizeNi, j+k ¼ 1 ( k Nm1 , m2  Nn1 , n2 k  dk )2s:t:

k Ni, 1  Ni, 2 k  L ¼ 0, i ¼ 1, ::, 6(1)where the nodes Nm1 , m2 and Nn1 , n2 are found for each sensor
identifier (sk) according to Table 1. Upon initialization, thenonlinear optimization routine (sequential least-squares
quadratic programming) is seeded with node positions, whichapproximates an equivalently sized regular icosahedron.
During optimization, the prior reconstructed state acts as aseed for the next iteration.
Orientation estimation. Determination of the face in contact with the environment (ground, obstacles, etc.) is criticalfor motion planning of tensegrity structures. In this work, we
use the strain sensor readings in a neural network to determine
the downward face and we use node contact sensors to directly
detect the downward face. While an inertial measurement unit
can detect the direction of gravity, it cannot be used to also
determine shape, and thus, to determine shape. Here we show
how the strain sensors can be used to detect both shape andorientation with respect to ground. In our demonstration of theneural network, we pause for the tensegrity to stop moving todetect the face, which significantly slows the potential speed of
the system. Therefore, we also demonstrate node contact
sensors that can quickly and directly detect contact with theenvironment. However, direct contact sensing is susceptible to(a) Orientation Estimation Using the Strain Sensors: Our
first approach to detecting the orientation of the tensegrity
robot uses the robotic skin-embedded strain sensors andrelies on a model-free simulation.45,46 While this approach has the disadvantage of needing to be trained on
every individual robot, it offers flexibility if the robot isdamaged, since the model-free approach can be retrained
during a mission.47
When all actuators are at rest, the tensegrity structure is
stretched more toward the top than the bottom of thestructure due to gravity. The strain sensors embedded in
the robotic skins can detect the stretching and we can feed
this information to supervised machine-learning algorithms, including regression and neural networks, tomodel the direction of gravity or even directly predict the
downward face. It is beyond the scope of this work tocompare different machine-learning algorithms, and we
instead focus on this simple implementation to demonstrate that it is possible to use surface-based sensing fororientation estimation.
We created a simple neural network to estimate thedownward face using strain sensor data only. To train the
network, we collected 60s of strain sensor data for each of
the 20 tensegrity faces. We then trained a neural network in
MATLAB using the Low-Memory Broyden-FletcherGoldfarb-Shanno algorithm. The neural network is builtwith an input layer consisting of 24 sensor input nodes, 1
for each sensor, a single hidden layer consisting of 10
nodes, and an output layer consisting of 20 nodes, 1 foreach face. The algorithm returns the face with the highest
probability of being downward. One limitation of thisneural network is that it is trained when the system is at
rest, meaning that the tensegrity robot must come to acomplete stop and relax all actuators before it can correctly
estimate the downward face. With a more sophisticatednetwork, such as a recurrent neural network, it would bepossible to train the system to estimate the downward faceduring dynamic motions. Despite these limitations, thiscurrent pairing of a simple neural network with real-timestrain sensors allows the system to estimate its orientation
without any dedicated orientation sensors.
In future work, the neural network could also be improvedby including the actuator pressures in the input layer. By
including the actuator pressures and expanding the number of conditions for which the neural network data wereTable 1. Mapping from Sensors sk to Nodes Ni, j
sk
s1
s2
s3
s4
s5
s6

Nm1 , m2
N1, 1
N1, 1
N1, 1
N1, 1
N1, 2
N1, 2

N n1 , n2
N3, 1
N4, 1
N5, 1
N5, 2
N3, 1
N4, 1

sk
s7
s8
s9
s10
s11
s12

Nm1 , m2
N1, 2
N1, 2
N2, 1
N2, 1
N2, 1
N2, 1

N n1 , n2
N6, 1
N6, 2
N3, 2
N4, 2
N5, 1
N5, 2

sk
s13
s14
s15
s16
s17
s18

Nm1 , m2
N2, 2
N2, 2
N2, 2
N2, 2
N3, 1
N3, 1

N n1 , n2
N3, 2
N4, 2
N6, 1
N6, 2
N5, 1
N6, 1

sk
s19
s20
s21
s22
s23
s24

Nm1 , m2
N3, 2
N3, 2
N4, 1
N4, 1
N4, 2
N4, 2

N n1 , n2
N5, 1
N6, 1
N5, 2
N6, 2
N5, 2
N6, 2

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BOOTH ET AL.

Table 2. Mapping from Faces Fa to Nodes Ni, j

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Fa
FA
FB
FC
FD
FE
FF
FG
FH
FI
FJ

Nl1 , l2
N1, 1
N1, 1
N1, 1
N1, 1
N1, 1
N2, 1
N2, 1
N2, 1
N1, 2
N1, 2

Nm1 , m2
N3, 1
N3, 1
N2, 1
N4, 1
N4, 1
N5, 1
N3, 2
N4, 2
N3, 1
N3, 1

N n1 , n2
N4, 1
N5, 1
N5, 1
N5, 2
N5, 2
N3, 2
N4, 2
N5, 2
N4, 1
N6, 1

trained, the future system would likely be capable ofdetecting the downward face even when actuators areactive. If the downward face can be detected when actuators are active, that would reduce the sensors’ settling
time for the face detection neural network, since nowaiting time would be needed for the actuators to deflate
before reading reliable sensor values. However, we would
expect that measurements would only be accurate after
the system stopped oscillating after a roll.
(b) Orientation Estimation Using the Node Contact Sensors: The node contact sensors detect which tensegrity
nodes are in contact with the ground, allowing for direct
detection of contact without the need to pause between
each locomotion step, as was the case with the current
neural net that used the strain sensors. When three nodes
are in contact, we infer that the corresponding face isdown, and by tracking history, there is the potential for
determining whether a node is touching the ground or awall. The algorithm for detecting the downward face inthis work is to return the downward face from a lookup
table using the last known valid triad of nodes. In otherwords, if more or fewer than three nodes are in contact,
the previously predicted face is retained until a new triadof nodes is found.
The node contact sensors are not able to detect the orientation of gravity directly, and so, determination of thedownward face can only be inferred. On the contrary, thedirect sensing of contact allows for continuous predictionof not only which face is contacting the ground but also
the edge over which the tensegrity is tipping and contacts
with walls. In the future, the node contact sensors andstrain sensors could be used jointly to detect anomalies interrain, such as coming into contact with a wall.
Model validation
Strain sensor calibration. In the physical tensegrity robot,
the distance between the nodes (lk) is not the same as theactive length of the sensors (dk), due to the way that thesensors are attached to the structure. There are passive connections between the strain sensors and the node end caps(dg ), thus requiring calibration. To calibrate the system, weaffixed one LED motion capture marker per node (Ni, j ) on
both extremities of a given edge (lk), centered the tensegrity
robot in the field of view of the motion capture system(Improv, PhaseSpace Inc.), and measured the distance andsensor values at five different actuated lengths. The above
procedure was repeated for each of the 24 sensors to extract a

Fa
FK
FL
FM
FN
FO
FP
FQ
FR
FS
FT

Nl1 , l2
N1, 2
N1, 2
N2, 2
N2, 2
N1, 2
N2, 2
N3, 1
N3, 2
N4, 1
N4, 2

Nm1 , m2
N2, 2
N2, 2
N3, 2
N4, 2
N4, 1
N3, 2
N5, 1
N5, 1
N5, 2
N5, 2

Nn1 , n2
N6, 1
N6, 2
N6, 1
N6, 2
N6, 2
N4, 2
N6, 1
N6, 1
N6, 2
N6, 2

linear relationship to map from the sensor value to the edge
length between nodes (lk ¼ Ck  dk þ dg ), where the sensor
length is multiplied by a calibration constant, Ck. This calibration approach does not fully model the physical considerations in the system, since an active actuator may slightly
displace a sensor leading to some nonlinearity, but we determined it was sufficiently accurate for our needs.

Validation of state reconstruction. To validate the model,
we simultaneously measured the position of the nodes asestimated by the state reconstruction model and the positionof the nodes as measured in a motion capture system(Fig. 3). We treated the motion capture data as ground truth.
To collect the motion capture data, we affixed an LED
motion capture marker on each node (the same way as in theStrain Sensor Calibration section), and ran a preprogrammed actuation sequence (Fig. 3a). Since not all themarkers are simultaneously visible to the motion capturesystem, the analysis was conducted only for the nodes that
could be directly verified. Figure 3b presents the reconstructed state using data from the sensors, which arepartially represented in Figure 3d, and Figure 3c presents thespatial distribution of the motion capture markers superimposed onto the position of the nodes extracted by the state
reconstruction model. The Supplementary Video S1 showsin real time the deformation and state reconstruction of thetensegrity as well as the strain sensor and motion capture
data (four panels of Fig. 3). The LED markers are not shownin the photographs (Fig. 3a) to reduce the number of distracting elements in the image (see Supplementary Video S2
presenting the footage of the actuated sequence with themarkers on the nodes).
To determine the accuracy of the model, we compared the
motion capture data and the points estimated using the strain
sensors and the node position estimation algorithm. Before
that could occur, however, we needed to transform the coordinate systems of the motion capture data into the coordinate system of the state reconstruction. To calculate thespatial transformation, we solved the constrained Procrustes
problem for aligning the positions of known reference points
on the top face of the initial data frame, while minimizing theleast-squares error.48 The result of the least-squares algorithm is a transformation matrix (T) that can convert the coordinates of the motion capture data (PMC ) into thecoordinate system of the tensegrity state reconstruction (Pt),by left-multiplying the three markers of the top face in motion
capture coordinates as PSR ¼ T  PMC .

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SURFACE ACTUATION AND SENSING OF A TENSEGRITY

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FIG. 3. (A) Tensegrity robot at different stages of a preprogrammed open-loop actuation sequence showing different
segments actuated. (B) State reconstruction from the above sequence extracted from the strain sensors and node position
estimation algorithm. (C) Comparison of the above state reconstruction (darker dots connected by lines) with data recorded
simultaneously from a motion capture system (lighter dots) to validate the strain sensors’ data and reconstruction model.
Similar colors indicate corresponding nodes. Motion capture system was unable to simultaneously detect all the nodes, andthus, only the nodes that are visible are plotted. (D) Strain sensor data measured during the actuation sequence readilypresent the deformations of the system. For clarity, 7 of the 24 strain sensors corresponding to activated actuators areplotted. The full video sequence is available in Supplementary Video S1. Color images are available online.
We trained the rotation matrix on a single frame when thetensegrity was static instead of the entire data set to avoid
overfitting, which is representative of initializing the tensegrity from an initial known configuration. Once the transformation matrix was calculated, the transformation wasapplied to the entire motion-capture data set and the rootmean-squared-error (RMSE) was calculated between all corresponding data points. The RMSE for the entire system is45.8 mm, or 13.1% of the rod length. Note that a part of thiserror could come from the fact that not all markers were placed
exactly at the center of their respective nodes in an effort to
optimize an overall detection by the motion capture system.

strain sensors of the robot in its stable state at each of the 20
faces down. The trained network has an accuracy of 99.8%,
conducted on data collected in the same experimental session
as the training data. As described above, this approach islimited to static unactuated states. In practice, we observe thatthe prediction stabilizes within a few seconds, after the system stops oscillating. To account for the 0.2% of incorrect
predictions during operation, the algorithm returns the statistical mode of a moving window of the last five measurements in the data.

Validation of the orientation estimation using the neuralnetwork. To validate the neural network, we measured thewas manually rotated while collection of video and sensor
data (Fig. 4a, b). The test lasted *35 s and traversed 7 face

Validation of the face detection using the node contact
sensors. To test the node contact sensors, the tensegrityDownloaded by YALE UNIVERSITY from www.liebertpub.com at 10/03/20. For personal use only.

8

BOOTH ET AL.

FIG. 4. (A) Sequence of manual rotations of the tensegrity robot, tipping from face to face, to validate the downward facedetection algorithm. (B) State reconstruction from the above sequence extracted from the strain sensors, node position
estimation algorithm, and node contact sensors. (C) Node contact sensor data (thin lines) measured during the rotation
sequence, plotted as normalized resistance (R  Rmin )=(Rmax  Rmin ). Resistance decreases when the node is in contact withthe ground. Rows of colored bars on top show the timing interval when different faces are in contact with the ground, byvideo analysis and from node contact sensors. The full video sequence is available in the Supplementary Video S3. Color
images are available online.

transitions. The video data were manually coded and translated
into truth data using video editing software (Adobe PremierPro) to determine the time stamps associated with individual
nodes contacting or leaving the ground (Fig. 4c). The individual
node contact sensors had an average accuracy of 95.0% over alltime stamps. Most of the error in the node contact sensors camefrom either timing ambiguities between the truth data and thesensor data or from sensor malfunctions.
To efficiently remove many of the errors, we output theface with the last known valid triad of nodes and only update
the face prediction when a new valid triad is sensed. In our
trials, the face detection algorithm correctly identified the
downward face at each transition. The overall moment-tomoment accuracy of the face detection algorithm describedabove—retaining a predicted face until a new valid triad ofnodes is detected—is *89.8%. Without using our algorithm,
the moment-to-moment accuracy of each prediction against
the truth data at each time stamp without retaining the previous valid triad yields an accuracy of 62.5%. This level ofaccuracy was primarily due to a single sensor behaving inconsistently, resulting in two correct sensor values, but not avalid triad of nodes and therefore a reduced accuracy.
The overall accuracy of our approach is highly dependent
on the independent reliability of each node contact sensor. As
a result, directly measuring the downward face using contact

is very fast and reliable, but it is also sensitive to sensor
failures. In future work, we will improve the manufacturing
process to increase the reliability of sensors.
Rolling

Locomotion of a tensegrity robot using robotic skins wasdemonstrated in our prior work, although without sensors orpressure control.27 In the present work, all the control electronics and hardware were mounted in the interior of therobot to reduce the number of tethers and off-board hardware
required. Furthermore, this work demonstrates a teleoperated
tensegrity robot, whereas the prior work used manually operated pneumatic valves.
Our hardware implementation did, however, limit thetensegrity’s locomotion capability. The inclusion of allpressure regulators onboard increased the system mass andcaused the robot to sag, therefore lowering the center ofgravity by a few centimeters. In addition, the actuators we
implemented were only capable of 12% strain in either direction, and therefore not able to displace the center ofgravity over the polygon of stability when starting from anequilateral face and rolling to any other face while on flatground.10–12 To enable rolling from any face, the robot waslaid on a declined slope (-8.7). Figure 5 presents snapshots,

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SURFACE ACTUATION AND SENSING OF A TENSEGRITY

FIG. 5. Teleoperated rolling locomotion of the tensegrity
robot (from right to left) on a downward inclined plane(-8.7 slope) shown as superposition of screenshots of thecorresponding video in the Supplementary Video S3. Due tothe weight of the onboard pressure regulators, the robot was
only able to independently roll across declined surfaces.
Color images are available online.
and Supplementary Video S4 shows the tensegrity robot actuating its segment, destabilizing its structure, and rolling
down the plane in a sequence of six rolls. While the full
locomotion capability of this tensegrity robot was only possible with a downward slope, in future work this could be
addressed by using actuators capable of higher strains anddecreasing the system mass.
The actuation policy uses two pneumatic actuators, whichis the minimum required to achieve rolling in a pneumatic
tensegrity.10,26 It is noted that actuators located between twoparallel rods in the six-bar tensegrity structure are neverutilized. To reduce the size of the polygon of stability, one of
the two pneumatic actuators must be located on the facetouching the ground. Using lower friction nodes may help in
future work. The second pneumatic actuator is used to shift
the center of mass toward the desired locomotion direction,and should be located on a side of the tensegrity, close to thebase face (Supplementary Fig. S1e). Since the structure hascompliance inherently built into the system, other actuators
do not need to be engaged.
Conclusion

In this article, we demonstrate a robotic six-bar tensegrity
structure, actualized by covering its surface with robotic
skins, which are two-dimensional, modular, robotic membranes with embedded actuators and sensors. The robotic
skin-clad tensegrity robot is able to individually activate its
pneumatic actuators and reconstruct its state—position andorientation—using strain sensors, and it achieves teleoperated rolling locomotion on a declined plane. This uniqueapproach is the first demonstration of real-time state reconstruction of a tensegrity robot using onboard proprioceptive
sensors.
We assess that the high stability of the robot when its baseface is an equilateral triangle is primarily caused by the limit
of force and displacement that the actuators can delivercompared with the system’s mass and size. Such high stability makes rolling from these orientations difficult. The next

9

logical research target would be to improve systematic rolling from any face on any surface (flat, downhill, uphill,rugged). Potential ways to achieve this include reducing themass of components (number and size) and also replacing thesurface material of the nodes with a lower friction material.
In addition to the mechanical limitations, the current
sensor-processing pipeline has limitations that make dynamic
control challenging. Although the shape and face reconstruction algorithms operate quickly on our off-board PC, thecommunication between the tensegrity and the PC introduces
lag in the state estimation, as seen in Supplementary Videos
S2 and S3. However, preliminary results suggest that there
are several areas for improvement. For example, in the current network topology, the complete set of 24 sensors can be
sampled sequentially at *30 Hz when actuation is disabled.
However, even this is not a lower bound on the achievablesystem update frequency—a single capacitive strain sensor
can be sampled at a rate of 119 Hz.42 Faster communication
rates could be attained, for example, by using a dedicatedmicrocontroller for actuation, which would only receive datafrom the PC, and a separate dedicated microcontroller thatwould only send data from the sensors to the PC. This improvement could potentially operate quickly enough to use
the sensor feedback for dynamic control, for instance, using
insights gained from other high-dimensional, soft robotic
systems such as continuum manipulators.49 Further improvements in state estimation could be gained by taking into
account sequence information, using algorithms such as longshort-term memory or vanilla recurrent neural networks.
Previous application of similar algorithms to sensor networks
has been shown to be beneficial in other settings, including
human-body motion capture garments50 and soft robotic
segments.51,52
Future work would also include the implementation ofclosed-loop control and dynamic orientation estimation.
Since this tensegrity robot is a soft and compliant nonlinear
system, machine learning tools would likely be the easiest
path foward to realize closed-loop control. In this optic, cooperative coevolutionary algorithms are a promising approach. The accuracy of the neural network used with strain
sensors to estimate orientation could also be enhanced bycontinuously training the network with ground truth datafrom contact sensors attached to the node ends. Finally, toeliminate the robot’s dependence on an external tether, it
could be equipped with an onboard source of compressed air,batteries, and Wi-Fi connection. An untethered robot would
be particularly convenient for extensive training sessions
under various conditions.
By introducing the concept of proprioceptive, surfaceactuated tensegrities, we hope to enable future advances toward controlled, autonomous locomotion of compliant
structures. In remote locations such as caves, mountains, androcky extraterrestrial terrains, there is often a lack of certainty
about the environment and lack of external sensors for localization and orientation estimation. Thus, with this combination of inherent compliance and the ability to sense theirown state, tensegrity robots such as the one introduced in thisarticle could allow exploration of previously inaccessible
locations. Furthermore, this works serves as an additional
step toward passive tensegrity structures that can be easilyclad in removable and reconfigurable robotic skins to control
the motions of these structures from their surface.

10

BOOTH ET AL.

Acknowledgments

We thank Jonathan Bruce and Massimo Vespignani at theNASA Ames Research Center for their assistance with tensegrity systems, and Nguyen Pham for his assistance inmaintaining and improving the robot between experiments.
Author Disclosure Statement

No competing financial interests exist.
Funding Information

Downloaded by YALE UNIVERSITY from www.liebertpub.com at 10/03/20. For personal use only.

This work was supported by a NASA Early Career Faculty
grant (grant numbers NNX14AO52G and 80NSSC17K0553).
D.S.S. and J.C.C. were supported by NASA Space Technology
Research Fellowships (grant numbers 80NSSC17K0164 andNNX15AQ75H). M.C.Y. was supported by an NSF Graduate
Research Fellowship (grant number DGE-1333468).
Supplementary Material

Supplementary Data S1
Supplementary Figure S1
Supplementary Video S1
Supplementary Video S2
Supplementary Video S3
Supplementary Video S4References

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Address correspondence to:Rebecca Kramer-Bottiglio
School of Engineering and Applied Science
Yale University
10 Hillhouse Avenue
New Haven, CT 06520
USA
E-mail: rebecca.kramer@yale.edu