IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020

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Rolling Soft Membrane-Driven Tensegrity Robots
Robert L. Baines , Joran W. Booth , and Rebecca Kramer-Bottiglio

Abstract—We present a methodology for designing, fabricating,and controlling rolling membrane-driven tensegrity robots. Thismethodology is enabled by pneumatic membrane actuators and ageneralized path planning algorithm for rolling polyhedra. Membrane actuators are planar, assembled in a scalable fashion, andamenable to arbitrary geometries. Their deformation trajectories
can be tuned by varying the stacking sequence and orientationof layers of unidirectional lamina placed on their surfaces. We
demonstrate the application of the same set of membrane actuators
consisting of polygonal faces of Platonic Solids to create polyhedral
tensegrity variants. Three specific tensegrities in the forms of cube,
dodecahedron, and rhombicuboctahedron are chosen to demonstrate the path planning algorithm, though the algorithm is generalizable to any uniform or non-uniform n-sided polyhedra. The
membrane-driven tensegrities are able to roll in unique trajectories
and circumvent obstacles contingent on the distribution and typesof polygons which constitute their faces.
Index Terms—Soft robot applications, tensegrity, motion andpath planning.

I. INTRODUCTION

T

Fig. 1. With square, triangle, and pentagon -shaped membrane actuators asbuilding blocks, we can construct a variety of polyhedra tensegrities. Based on
their face arrangement and shapes, tensegrities may be created to more accuratelyroll on specified paths.

Manuscript received February 24, 2020; accepted July 18, 2020. Date of
publication August 7, 2020; date of current version August 20, 2020. Thisletter was recommended for publication by Associate Editor R. Niiyama and
Editor K.-J. Cho upon evaluation of the Reviewers’ comments. The workof R.L. Baines was supported by the National Science Foundation Graduate
Research Fellowship under Grant DGE-1333468. The work of J. W. Booth was
supported by NASA through the Early Career Faculty Program under Grant
80NSSC17K0553. (Corresponding author: Rebecca Kramer-Bottiglio.)
The authors are with the Department of Mechanical Engineering andMaterials Science, School of Engineering and Applied Science, Yale University, New Haven, CT 06520 USA (e-mail: robert.baines@yale.edu;
joran.booth@yale.edu; rebecca.kramer@yale.edu).
This article has supplementary downloadable material available at http://ieeexplore.ieee.org, provided by the authors.
Digital Object Identifier 10.1109/LRA.2020.3015185

To impart motion onto a tensegrity, actuation schemes usually
incorporate rigid motors that spool and contract the tensile
elements [5]–[9]. Contracting tensile elements shifts the centerof mass of a tensegrity, making it statically unstable, and as aconsequence, it rolls to an adjacent face. One group inverted
convention, making a tensegrity with linear actuators as struts
to displace the cables [10]. A marked departure from controlling
tensions in cables to destabilize the center of mass used vibration
as a means for small tensegrities to move [11].
Considering previous work, it may be advantageous to rely
on soft material actuators for tensegrity locomotion, rather thanrigid motors. For example, replacing motors with soft actuators could enable new locomotion modes. A tensegrity driven
by soft contracting McKibben actuators introduced the firstplatform that rolled without motors [12]. More recently, ourgroup showed a tensegrity driven by contraction of robotic
skins consisting of McKibben actuators and soft sensors embedded into a single substrate [13]. In addition, groups havedemonstrated tensegrities that actuated using shape memory
alloy [14], [15] and shape memory polymer struts with elastomer tensile elements [16]. The aforementioned demonstrations relied on several actuators per face to impart deformation, making control highly complex. In addition, the designs

HE word tensegrity was coined by Buckminster Fuller, asa blend word to describe a type of structure composedof rigid parts under compression and compliant elements intension [1]. The tensegrity paradigm is widespread, arising in
biological structures, interpretations in art and architecture, as
well as mechanisms with high strength-to-weight ratios [1]–[4].
More recently, tensegrities have gained traction in robotics asa means to navigate in unstructured or dynamically changing environments, places where traditional rigid robots often
under-perform because external disturbances may damage orentirely destroy them. Tensegrities distribute external forces
or disturbances internally through their network of tensile andcompression elements, mitigating stress concentrations thatmight otherwise lead to mechanical failure in a traditional
robot [5].

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Fig. 2. Fabrication of membrane actuators is simple and scalable. Membrane actuators operate via volumetric expansion, which can be either constrained or
unconstrained. After steps 1-4, step 5 shows unconstrained membrane actuators that expand out-of-plane. With the addition of unidirectional laminae patches (step5a), the actuator expansion can be constrained and directed into areal extension or contraction of the membrane. In step 6a, the depicted triangular membrane isdivided into three cavities, each with its own inlet. All three tubes are connected to the same air supply; the cavities inflate together.

relied on contractile motion between the nodes, overlooking thewealth of possible locomotion strategies that could be achievedby other motion primitives, such as extension or out-of-plane
expansion.
This letter’s first contribution is a design space for a newclass of tensegrity, which we coin the membrane tensegrity. In
contrast to a typical tensegrity where every member experiences
exclusively axial forces concentrated at nodes, a membranetensegrity’s struts are held in compression by entirely soft
membranes that sustain forces in a plane, rather than an axis,
and simultaneously serve as actuators. Membrane actuators canimpart motion onto tensegrities in new and unexplored ways, including out-of-plane expansion to change center of mass without
displacing the nodes, and areal extension and contraction thatchange center of mass by displacing the nodes. The same set ofmembranes and struts can be used to create myriad tensegrities
in the space of uniform and non-uniform polyhedra (Fig. 1). An
outcome of using membrane actuators is that the arrangement ofpolygons which constitute a tensegrity’s faces can be modulated
to tailor performance in a given set of path planning problems.
Incidentally, the problem of path planning for tensegrities is arelatively unexplored field. Thus, the second contribution of thisletter is a generalized path planning algorithm for rolling polyhedra that goes hand-in-hand with the modular nature of membrane
actuators.

II. MEMBRANE ACTUATORS
A. Fabrication of Membrane Actuators
Membrane actuators form the basis of the tensegrities constructed herein. Membrane actuators are simple to fabricate,easily formed to unique geometries, lightweight, and, due to theirthinness, can be compacted into a small volume. Fig. 2 illustrates
the fabrication process for membrane actuators. Membrane actuators operate via volumetric expansion, which can be either constrained or unconstrained. Unconstrained membranes result in
out-of-plane expansion (top), while constrained membranes (via
the addition of unidirectional laminae) result in motions such as

in-plane areal extension and contraction (top plus bottom). Bothfabrication procedures follow the same first four steps.
First, a 2 mm layer of elastomer resin (Dragon Skin 10 Fast,
SmoothOn Inc.) was rod-coated onto polyethelene terephthalate(PET) film with a precision threaded rod. Then, we flipped therod-coated sheet over, PET-side up, and laser-cut away the PET
to selectively mask the area of a desired internal bladder, or multiple segmented bladders (VSL 2.30, Universal Laser Systems).
Next, another 2 mm thick layer of elastomer was applied via
rod coating. To make out-of-plane expanding membranes, welaser cut out the desired shape contour, removed the internal
PET sheet, and inserted silicone tubing to enable inflation. For
directionally constrained membranes (Fig. 2, bottom), we additionally cut out elastomer laminae with embedded unidirectional
fibers [17], and adhered those to the both sides of the membrane
to govern its inflation trajectory.
B. Motion Primitives by Varying Fiber Orientation
We are able to elicit a number of motion primitives frommembrane actuators by adhering directionally-constraining
laminae onto membrane surfaces (Fig. 3). In addition to expansion, extension, contraction, twist, and bending motion primitives, it is possible to elicit highly complex deformations bysuperposition and localization of lamina stacking sequences.
We created oblong rectangular membrane actuator specimens(125 mm x 25 mm) to better visualize the motion primitives. The
far left of Fig. 3 presents a side view of a deflated membrane
actuator for reference. To the right are side views of actuators
inflated with 150 mL of air, numbered as specimens 1–5. Fiberorientation of the laminae influenced axial strain a , transverse
strain (i.e. bulge) t , and if applicable, twisting γr and bending
γb . Specimen 1, with no laminae, exhibits t /a = 69. For theextension actuator, specimen 2, t /a = 11.32. The ratio is small
relative to that of specimen 1 and testifies to the fact that muchof the deformation has been forced in the desired direction.
Essentially a 2D McKibben muscle, specimen 3 maintains a
relatively small t /a = −22 as well, yet undergoes contraction
comparable to traditional McKibben muscles [18]. Of course, as

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BAINES et al.: ROLLING SOFT MEMBRANE-DRIVEN TENSEGRITY ROBOTS

Fig. 3. Oblong 2D rectangular membrane actuators to better visualize motion
primitives. The schametic above each specimen shows the angle, relative tovertical, at which fibers are embedded in any attached laminae. Left to right: at
rest, inflated without any lamina, with fibers at 90◦ to force extension, ±30◦ on
both sides to create a McKibben-style contraction, 45◦ mirrored on each side
for twist, and 90/0◦ on one side for bending. All were inflated with 150 mL ofair. Scale bar: 30 mm.

with traditional McKibben muscles, the exact strains achievable
are a function of the fiber braid angle. Owing to the simplicity of the fabrication process, the braid angle can easily betuned. Moving on to the next motion primitives, specimen 4
twists one complete revolution while maintaining a smallera = −0.11. However, the lack of axial deformation manifested
itself transversely; t was comparable to that of specimen 1,
making t /a = 102. Lastly, specimen 5 bends at γb = 350◦ ,nearly completing a full curl, with t /a = −25.
The exact strain-to-pressure or -volume relationship for amembrane actuator depends on geometry, pre-strains, and forcesinduced when they are initially placed between tensegrity nodes.
Due to the extent of the membrane actuator design space,data presented in Fig. 3 is intended to indicate general relationships between fiber placement and motion primitives, notdefinitive specifications for implementation. In this letter, we
utilize square, pentagon, and triangle out-of-plane expanding
membrane actuators for hardware implementation of path planning.

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Fig. 4. Rolling mechanisms for tensegrities with out-of-plane expanding
membrane actuators, which minimally shift the struts (top dodecahedron),and extension membrane actuators, which significantly shift the struts (bottom
icosahedron). Scale bars: 25 mm (top), 40 mm (bottom).

In the present letter, we explore the utility of out-of-plane
expanding membrane actuators on a rolling tensegrity because:
1) among the motion primitives in Fig. 3, we believe expansion
to dislodge tensegrity center of mass is most conducive tomaking tensegrities roll, 2) the expansion membrane lends itself
to a simplistic actuation strategy compared to the constrained
actuators, and 3) tensegrities utilizing out-of-plane expansion tomove have not yet been reported. Though we proceed with thislimited scope, we believe the motion primitives present in Fig. 3,
and combinations thereof, should provide engineers an ample
tool set to devise novel locomoting membrane tensegrities.
Polyhedra tensegrities were constructed using the same set ofmembrane actuators (out-of-plane expanding square, pentagon,
and triangle with side lengths 50 mm) and 1/16 in diameter carbon fiber composite struts (McMaster Carr) of 125 mm length.
In our specific hardware implementation, the cube, dodecahedron, and rhombicuboctahedron tensegrities consisted of four,ten, and twelve struts, respectively. At each end of each strut,
3D-printed nodes provided an interface site for the membranes.
To preserve modularity and interchangeability, we did not bondthe membranes to the nodes. Rather, compression supplied by
the membrane network held the structure intact.
III. PATH PLANNING FOR MEMBRANE TENSEGRITIESC. Assembly of Membrane-Driven Tensegrity Robots
The mechanism by which a tensegrity outfitted with out-ofplane expanding membrane actuators rolls is depicted in the
top of Fig. 4. Inflating a downward membrane creates a bubble
of curvature k, an unstable equilibrium which, perturbed byinflation of neighboring faces, biases the center of mass to favorrolling to a particular face. Note the bubble is highlighted green.
On the other hand, the bottom of Fig. 4 shows how a tensegrity
with constrained extension membrane actuators rolls. Inflationof a face displaces nodes with a strain , shifting center of gravity
and toppling the robot. Both rolling mechanisms are shown inreal-time in Supplementary Video Part 1.

A sizable body of literature employs machine learning techniques for control of tensegrity robots. These works focus on
how to generate actuator commands required to travel the furthest by some heuristic within a given time frame (usually euclidean distance between two points over 60 s) [19]–[21]. Other
research developed actuator policies for rolling icosahedron andrhombicuboctahedron tensegrities, but in consideration of thelatter shape, did not allow for rolling to and from triangularfaces that are a source of great complexity in movement [22].
In contrast to the large amount of literature developing actuator control policies for tensegrities, scant work pertains topath planning in the traditional sense of solving for a route

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IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020

from point A to B in the presence of obstacles. Prior researchmost related to the tensegrity path planning domain presented
algorithms for a specific geometric configuration: the popular
six-strut icosahedron tensegrity. One group introduced a steerable locomotion controller for an icosahedron tensegrity which
mapped user joystick inputs to a rolling sequence [23]. Anothergroup presented a sampling-based kinodynamic planning approach for an icosahedron tensegrity approximated as a sphericalentity [24].
The use of membrane actuators—modular units adaptable
to arbitrary shapes and easily assembled on-the-fly to create
unique polyhedra tensegrities—demands a path planning algorithm agnostic to shape and configuration to truly realize theirpotential. Pioneering work furnished mathematical context for
generalized polyhedra rolling on a plane yet conceded they
were not able to test their algorithm on a system, simulated or
physical [25]. We build on prior work and create a generalized
path planning algorithm for rolling n-sided polyhedra, uniform
or non-uniform. The algorithm combines the A* graph searchwith geometric constraints of a given polyhedra. Instead of
relying on hard-coded coordinate frame assignments and a seriesof affine transformations, variables that change based on shape,
our algorithm maintains generality by evoking an optimization
routine to keep track of the orientation of a polyhedra as it rolls.
A. Geometric Reasoning for Planning Algorithm
As a polyhedra tensegrity rolls, its downward face polygon P will occupy a unique orientation on the ground. The
orientation-invariant possible set of moves from any downward
face contacting the ground to its neighboring faces which sharean edge can be considered as its geometric footprints F. Fig. 5
gives examples of different polyhedra downward faces and theirneighbors flattened into footprints. Notice that the cube has
one such footprint, and it forms a perfectly tessellating squaregrid. In contrast, the dodecahedron and rhombicuboctahedron
do not have perfectly tessellating grids. Furthermore, F for therhombicuboctahedron contains three items.
The subset of F that is a grid of perfectly tessellating polygons, like the triangle, square and hexagon, is easy to visualize
and a simple space in which to plan movement from point A topoint B. Consider the cube. Evoking the famous A* algorithm,
an evaluation function is used to find the optimal path across thegrid in the presence of obstacles:f (n) = g(n) + h(n)(1)

Here, g(n) is the cost from the initial node to the present noden, and h(n) estimates the cost of an optimal path from node n to
the goal node; a value that is defined by the heuristic (commonly
euclidean distance). At each step, the algorithm picks the nextn to move to with the lowest f [26].
We cannot constrain our tensegrity design space to uniform
grids of perfectly tessellating shapes because membrane actuators can achieve any shape and tensegrities can be composedof multiple differently shaped polygonal membrane actuators.
Imperfectly tessellating polyhedra footprints and those which
are non-uniform (constituted of disparate polygonal faces) may

Fig. 5. Each polyhedra has a unique set of footprints that determine thepossible set of paths it can achieve via rolling. Imperfectly tessellating polyhedra
pose the challenge of a large, complex grid for planning motions.

exhibit non-holonomy. That is, rolling from point A to point
B, each step of the tensegrity depends on the previous step(s)taken. The sheer magnitude of configurations of F on the planeprohibits explicitly developing a grid of possible moves in F apriori for larger search spaces. For visualization, the bottom rowof Fig. 5 shows the stacked contours F for two moves. It is clearthat planning optimal movements in large grids—where rolls
may be on the order of hundreds—becomes computationallyexpensive. Thus we take an alternative approach. We sacrificeguarantees of admissibility for practicality and fast convergenceon a viable path.
B. General Path Planning for Polyhedra Tensegrity
In our path planning pipeline, we first impose a dense squaregrid on the environment and perform a traditional A* search defined by Equation 1 to compute a path oblivious to the geometricrolling constraints of the tensegrity. Next, Algorithm 1 begins.
The path generated by A* is further discretized into a series ofcheckpoints via a createCheckP oints method. The number ofcheckpoints is arbitrarily set to lie between the A* path length
divided by the shortest edge of the polyhedra and the A* path
length divided by the longest edge. This technique ensures that
there is a sufficient number of checkpoints to prevent straying
from the A* path or, conversely, excessive meandering. Next, fora given polyhedra comes a set of adjacency relationships A. We
can use A to find centroid-to-centroid distances lj for each of
the footprints, where j = 1, . . ., k and k is the number of uniquecentroid-to-centroid distances for a polyhedra. We can also use

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Algorithm 1: Rolling n-sided Tensegrity Following A*Generated Path.
Input: A∗ , O, FA∗ Path, Obstacles, Footprints
Output: storedF aces  Vector of Faces Traversed
1: cpoints = createCheckP oints(A∗ )2: d = [x, y]F ace , [x, y]cpoint L2  Initialize Distanceto Checkpoint
3: for length(cpoints) do
4: while d > tol do  Roll Until Within tol Tolerance
5: [l, φ] = f indCentroids(F, A)
6: [ξ, f aces] = generalOrient([x, y]prevF ace )7: ctroid = [x, y]F ace ∗ l ∗ cos(φ + ξ)
8: for n ∈ P do n-sided Polygon on Ground
9: if ctroid(n) ∈ O then
10: ctroid(n) = ∞  Penalize Collisions
11: [[x, y]nextF ace , i] = min ctroid, [x, y]cpoint L2
12: f aceN umselected = f aces(i)13: rollT oN extF ace([x, y]nextF ace )14: [x, y]prevF ace = [x, y]F ace
15: [x, y]F ace = [x, y]nextF ace
16: d = [x, y]F ace , [x, y]cpoint L2  Update
17: storedF aces.append(f aceN umselected )A to find the angles φ of neighboring centroids relative to P.
Both tasks are accomplished via the f indCentroids method.
Note that A depends on a user-defined numbering scheme forfaces. For example, on the cube, we could have labeled each of
the sides 1–6. If P = 1 then A contains 2, 3, 4, and 5—faces
sharing an edge with 1. l is a centroid-to-centroid distance
between neighboring faces and is constant for uniform polyhedra
like the cube but can change for non-uniform polyhedra like therhombicuboctahedron.
After delineating neighboring faces’ centroid coordinates relative to P, the generalOrient method is invoked. We keep trackof the orientation of the polyhedra as well as user-numberedfaces across which a tensegrity rolls while retaining generality
for any n-sided polyhedra, uniform or non-uniform. To do so,in generalOrient we employ a 1D optimization that finds theangle ξ which minimizes the distance between [x, y]prevF ace
and [x, y]F ace :min [x, y]prevF ace − ([x, y]F aceξ+ [lcos(ξ + φ), lsin(ξ + φ)])

(2)ξ is propagated throughout the rolling sequence as an additive
term to compensate for generic changes in orientation to F. The
generalOrient method also outputs the face numbers f aces
within the footprint that can be used later to convert a computed
path to a sequence of actuator commands on the physical system.
After the optimization routine, the footprint of possible single
moves is searched for the face which lies closest to the current
checkpoint in line 11. If there are no obstacles O contained
within the same face as a potential roll, the tensegrity rolls
across the corresponding edge. The chosen face number is stored
in storedF aces. Rolling reveals a new footprint of potential

rolls. The process of expanding the footprint and evaluating forthe closest face that does not contain obstacles loops until thetensegrity rolls within a threshold tol of the checkpoint. Whenthe tensegrity is within tol, the algorithm increments to the next
checkpoint on the A* path. The geometric constraints imposed
during rolling are that the tensegrity can only roll over edges,
not corners, and it must chose a next move bounded by F.
Using this planning pipeline, we exploit a fast initial search
for an optimal path on a grid that is perfectly tessellating,followed by subsequent ordered searches constrained to thenumber of elements in F that strive to best follow the optimal
path. Convergence on a viable path can occur faster than it
does compared to performing an A* with polyhedra geometricconstraints defined at each step, especially for polyhedra with
F which do not tessellate the plane.
C. Implementing Membrane Tensegrity Path Planning in a
Simulated Maze
To validate the generalized algorithm, we used it in path planning of simulated cube, dodecahedron, and rhombicuboctahedron -shaped membrane tensegrities. In MATLAB, we simulated
the same maze for each type of membrane tensegrity which
had equivalent edge lengths, matching those of actual hardware
used later (50 mm). A* was broken into 70 checkpoints basedon the side length of the tensegrity and the size of the grid.
Fig. 6 presents simulation results. Gray sections denote walls
and the A* generated path is the dashed red line. Polyhedra
footprints are shown by alternating colors and the black line
connects downward faces step-wise over time. Starting at thebottom left, each case was able to reach the goal at the top left.
The percentage error between the distance of the A*generated path on the imposed fine square grid and the distancetraveled by our simulated tensegrity indicates extent of deviationfrom an optimal path. For the cube, dodecahedron, and rhombacuboctehedron, the percentage error between the optimal pathand the actual route taken via Algorithm 1 was 11.5%, 21.6%,
and 28.9%, respectively. The increasing error with complexityof shape is evidenced visually by the straighter routes the cube
takes relative to the meandering routes of the other shapes.
Another finding from Fig. 6 is that the centroid-to-centroid
distance l between each downward face in a footprint governs
the number of total rolls required to reach the goal. For instance,
the dodecahedron only took 74 rolls to reach the goal comparedto the cube’s 91 rolls. Interestingly, the rhombicuboctahedron
took 116 rolls to reach the goal. Consisting of both triangles andsquares, it has greater mobility than the cube. While greater mobility is advantageous for more closely following checkpoints,
it can encourage meandering in between those checkpoints,
causing polyhedra to take indirect paths toward the goal.
In a similar vein, we note that as a polyhedra tensegrity
decreases in θ it approximates a sphere more, and is able to
make both sharper turns and smoother diagonal movements.
This fact is underscored when comparing the dodecahedronand cube’s diagonal movement portions in the maze. The cube
makes two movements to approximate a diagonal, owing tothe θ = 90◦ edge spacing of its footprint. In contrast, theAuthorized licensed use limited to: Yale University. Downloaded on September 11,2020 at 15:20:12 UTC from IEEE Xplore. Restrictions apply.

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Fig. 6. Solving a simulated maze: rolling polyhedra can traverse a generated A* path broken into checkpoints, irrespective of shape. The geometry of the shapedetermines the ability to move in straight lines, take sharp or smooth corners, and also influences the number of rolls needed to reach the goal.

dodecahedron has much more movement flexibility due to its
smaller θ = 36◦ (Fig. 5), though it cannot maintain a purely
straight path like the cube. In practical applications, there exists
a balancing act between size and shape of a membrane tensegrity
to minimize energy of traversal across a path. Larger or fewer
faces allow a tensegrity to reach a point with fewer rolls, andtherefore fewer actuation cycles. Yet on physical systems larger
size implies greater mass; and fewer faces means more shiftmust occur to destabilize center of mass. On the other hand,smaller or more faces may require more rolls but have lessmass to move and may more accurately approximate a path.
Here the advantage of membrane actuators comes to the fore:ease adapting them to custom geometries and sizes should help
the engineer faced with the unique challenges of tensegrity
navigation.
D. Translating Simulated Path Planning to Reality
To further evaluate the generalized algorithm pipeline, we
simulated a simple path planning problem on the three tensegrity variants presented in Fig. 5 and translated the solution to
hardware (Supplementary Video Part 2). We conducted three
trials for each tensegrity to understand the variance associatedwith implementing a fixed generated path on actual hardware. To
track differences between simulation and hardware, we recorded
the tensegrity executing the path with an HD camera facing
downward. The obstacle was chosen to be a flat wooden blockthat would indicate whether a collision occurred if it were
occluded from above.
Having stored the faces that the tensegrity traversed as the output of our path planning algorithm, we translated the calculated
path to a series of actuator commands. Of course, the exact natureof the path-to-actuator command mapping is dependent on thetype of actuator. For out-of-plane expanding membrane actuators, we simply inflated the membrane on the downward face ofthe tensegrity, leveraging the rolling mechanism depicted in the
top of Fig. 4, one similar to the rolling mechanism proposed forother membrane-driven robot systems [27]. The system becomes

unstable and the direction it will roll is unpredictable. However,
we can direct the roll by also inflating all the adjacent faces
except the one in the direction of the desired roll. We utilized
an Arduino Uno in communication with miniature pressure
regulators [28] to command rolling sequences.
Fig. 7 juxtaposes experimental and simulated path planning
results for one of the three conducted trials. Each tensegrity
experienced propagation of error in rolling, evidenced by thedifference between the simulated trajectory (black line with solid
dots pinpointing centroid location) and experimental trajectory(dotted line with white dots).
Fig. 8 plots the cumulative absolute angle error as well as the
cumulative distance error step-wise throughout the experiment
for each tensegrity shape. Plotted points represent an averageof the three trials. Confidence bars give one standard deviationfrom the mean. Step-wise error accrual is due to a number offactors. First, the pneumatic tether and wires emanating froma tensegrity bias direction. To achieve longer and unimpeded
rolling sequences, tether-less operation is necessary. Second, ouralgorithm assumes perfect rolling. As can be seen from Supplementary Video Part 2, sliding and bounce stray the tensegrities
from their theoretical perfect rolling trajectories. By switching
the membrane type to extension or contraction, it is likely we
could reduce bouncing and sliding because in those types ofactuation, the tensegrity “relaxes” to its next face rather than“falling” as with inflation. Additionally, the 3D-printed nodes
on the tensegrities were made from thermoplastic. Choosing amaterial for the nodes with a higher friction coefficient could alleviate sliding issues. Third, face edge lengths of the tensegrities
were not entirely uniform and therefore did not match those of
the perfect polyhedra in simulation. Disparate tensions causedby the arrangement of internal struts skewed edge lengths. The
results of Fig. 7 and 8 and observation of variance in hardware
reinforces that a noise-free estimate of the path-length heuristic
cannot be guaranteed admissible.
While conducting the experiment, we found that slow, quasistatic inflation of the actuators lead to more stable and repeatablerolls. Fast inflation imparted significant angular momentum toAuthorized licensed use limited to: Yale University. Downloaded on September 11,2020 at 15:20:12 UTC from IEEE Xplore. Restrictions apply.

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Fig. 7. Path planning algorithm implemented in simulation and translated to actual tensegrities. The same simple problem is shown for each polyhedral variant
to home in on idiosyncrasies of operation. The dashed line with white dots at the end of the frames indicates the experimental trajectory of the centroids while thesolid line with black dots, the theoretical trajectory. Scale bar in top left of each experiment sequence: 50 mm.

Fig. 8. Cumulative absolute error between simulated and experimental values of the three tensegrities as they roll. Plotted points are a mean of three experimental
trials. Confidence bars indicate one standard deviation above and below the mean.

the tensegrities and caused them to roll multiple times, deviatingfrom steps in the generated path. Although multiple rolls for asingle inflation may not be desirable when carefully following
a generated path, the behavior could be leveraged in the interestof dynamic or unusual modes of locomotion.
As a last note on practical implementation, though it exhibitedthe least average angle and length error step-to-step on its
trajectory due to its sharp contours, (cumulative angle and length
errors of 32.4° and 53.3 mm, respectively) tipping the cube took
significantly more inflation than the other polyhedra (which approximate spheres to a greater degree). The initial high-pressure
hurdle to tip sometimes prevented the downward membrane’s

inflation from tipping the cube over the appropriate edge. The
opposite problem occurred with the rhombicuboctahedron: it
rolled too easily and freely. In step 7 of the trial visualized inFig. 7, the rhombicuboctahedron deviated from the final step of
its path substantially, even though the correct actuators inflated.
Consequently, the angle error for the rhombicuboctahedron instep 7 of Fig. 8 has a large standard deviation. Since a rhombicuboctahedron is more spherical than a cube, rolling to acertain point requires less pressure but is associated with higher
variability. Especially with neighboring actuators inflated, therhombicuboctahedron assumes a very rounded surface that tips
with little bias.

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IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020

Due to the confounds mentioned above, particularly
nonequivalent edge lengths induced by disparate tensions,collisions occurred. In the trial pictured in Fig. 7, the dodecahedron collided with the right portion of the obstacle in step 2of its trajectory. The rhombicuboctahedron also collided withthe obstacle in step 3. In spite of the collisions and deviations
from simulation, the results of Fig. 7 serve as a testament to thecompatibility of membrane actuators and the generalized polyhedra path planning algorithm. Knowledge of how membranetensegrities replicate paths in real life can help inform the designof robots for specific path planning problems that require, say,the location certainty of the cube, the maneuverability of thedodecahedron, or the spontaneity of the rhombicuboctahedron.
IV. CONCLUSION
We presented a new class of tensegrity robot composed ofmembrane actuators. Membrane-driven tensegrity robots areeasy to fabricate, able to be tailored to arbitrary geometries,
and deform in numerous ways. The membrane actuators workin tandem with a generalized path planning algorithm, providing
a tool set for rapidly designing and implementing tensegrities forrolling. In future work, we intend to utilize an IMU to accountfor the disparity between theoretical and actual orientation of adownward face to correct the path planner in real-time. We will
also explore path planning with different membrane actuatortypes introduced herein, such as extension, contraction, andcombinations of these types on the same system. Lastly, we
should remark that the A* to Algorithm 1 pipeline is by no stretch
optimal. The objective of the pipeline was to circumvent thescaling problem associated with large search spaces of potential
moves for non-tessellating polyhedra footprints. For smallerscale problems, we found that A* with step-wise geometricconstraints can be used to generate guaranteed admissible paths.
Since the intention of the present work was to introduce theconcept of complementary hardware and generalized path planner for customized polyhedra membrane-driven tensegrities, weleave heuristic tuning to future work.
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