Bioinspired Soft Spiral Robots for Versatile Grasping and Manipulation

Abstract

Across various species and different scales, certain organisms use their appendages to grasp objects not through clamping but through wrapping. This pattern of movement is found in octopus tentacles, elephant trunks, and chameleon prehensile tails, demonstrating a great versatility to grasp a wide range of objects of various sizes and weights as well as dynamically manipulate them in the 3D space. We observed that the structures of these appendages follow a common pattern - a logarithmic spiral - which is especially challenging for existing robot designs to reproduce. This paper reports the design, fabrication, and operation of a class of cable-driven soft robots that morphologically replicate spiral-shaped wrapping. This amounts to substantially curling in length while actively controlling the curling direction as enabled by two principles: a) the parametric design based on the logarithmic spiral makes it possible to tightly pack to grasp objects that vary in size by more than two orders of magnitude and up to 260 times self-weight and b) asymmetric cable forces allow the swift control of the curling direction for conducting object manipulation. We demonstrate the ability to dynamically operate objects at a sub-second level by exploiting passive compliance. We believe that our study constitutes a step towards engineered systems that wrap to grasp and manipulate, and further sheds some insights into understanding the efficacy of biological spiral-shaped appendages.

Full text

Title: Bioinspired Soft Spiral Robots for Versatile Grasping and Manipulation
Authors:
Zhanchi Wang 1, Nikolaos M. Freris1*
Affiliations:
1 School of Computer Science and Technology, University of Science and Technology of
China, Anhui, China.
*Corresponding author. Email: nfr@ustc.edu.cn.
Abstract: Across various species and different scales, certain organisms use their appendages to
grasp objects not through clamping but through wrapping. This pattern of movement is found in
octopus tentacles, elephant trunks, and chameleon prehensile tails, demonstrating a great versatility
to grasp a wide range of objects of various sizes and weights as well as dynamically manipulate
them in the 3D space. We observed that the structures of these appendages follow a common pattern
- a logarithmic spiral - which is especially challenging for existing robot designs to reproduce. This
paper reports the design, fabrication, and operation of a class of cable-driven soft robots that
morphologically replicate spiral-shaped wrapping. This amounts to substantially curling in length
while actively controlling the curling direction as enabled by two principles: a) the parametric
design based on the logarithmic spiral makes it possible to tightly pack to grasp objects that vary in
size by more than two orders of magnitude and up to 260 times self-weight and b) asymmetric cable
forces allow the swift control of the curling direction for conducting object manipulation. We
demonstrate the ability to dynamically operate objects at a sub-second level by exploiting passive
compliance. We believe that our study constitutes a step towards engineered systems that wrap to
grasp and manipulate, and further sheds some insights into understanding the efficacy of biological
spiral-shaped appendages.
One-Sentence Summary: Design, fabrication, and operation of spiral soft robots at variable scales
that can manipulate objects through wrapping.
Main Text:
INTRODUCTION
Wrapping as a paradigm for grasping and manipulation (1), which are two key objectives in robotics
(2, 3), is found in the prehensile tail of chameleons and seahorses with length scales as small as a
few millimeters (4), as well as in the tentacles of octopuses and the trunks of elephants as large as
a meter (Fig. 1A) (5, 6). These structures curl from their tips, tightly wrap around objects of different
sizes and shapes, and continuously control the curling direction to manipulate them with unmatched
efficiency (Fig. 1B). Roboticists have successfully achieved effective control of an object's motion
using grasping - defined as the force-closure of an object (7). For example, clamping (8), digging
(9), hooking (10), vacuum suction (11), magnetic attraction (12), and so on. Nevertheless, grasping
and manipulating objects through wrapping remains by-and-large unaddressed, noting the succinct
difference between the continuous deformation/compliant interaction (13) exhibited by biological
systems and the discrete joints/rigid materials employed in robotic systems (7) that scientists and
engineers have long worked on.
Soft robots (13–17) made of flexible materials are well tailored to produce continuous deformations
and interact with the environment (18–20). In particular, soft manipulators have shown high
compliance and adaptability, enabling the development of grippers capable of grasping in an

enveloping (21–23) or twisting (24–27) manner. Although there is a growing interest in developing
systems that can grasp/manipulate objects through wrapping (28–33), existing soft robots fail to
achieve biologically comparable versatility. Taking an elephant as an example, its trunk can wrap
a carrot with a diameter of 3 cm, while it can also grasp and stack 300 kg stumps over half a meter
in diameter. We argue that this gap can be filled by morphologically replicating the spiral patterns
that are ubiquitous in nature.
Fig. 1. Overview of the motivation and inspiration of the soft spiral robot. (A) Examples of animal parts that
follow the logarithmic spiral pattern, which is found primarily used for grasping and manipulating organs. (B)
The octopus moves its tentacle to grasp an object in four stages: i) extending toward the object, ii) making contact,
iii) climbing to wrap, and iv) grasping (original video credits: https://www.youtube.com/watch?v=GdCOYToDqfM).
We describe two principles that help enable a basic recreation of the spiral-shaped curling behavior
in an artificial system. First, we present a logarithmic-spiral-based design that allows the curling to
be achieved with single cable actuation; this can be further extended to designing multi-cable spiral
robots that are capable of more complex deformations in the 3D space. Second, we develop a
bioinspired grasping strategy that adapts to a variety of objects without requiring prior knowledge
about their shape. The combination of these two principles gives rise to a series of cable-driven soft
spiral robots (SpiRobs) that can effectively grasp and dynamically manipulate objects of various
sizes, shapes, and weights.
RESULTS
Design of SpiRobs
The first principle is based on the uncurling of a logarithmic spiral and is the key to enable a
parametric design scheme that allows easy fabrication across scales. The uncurled spiral results in
a tapered body, which can be curled and wrapped back into a spiral under the forces acting through
a cable (Fig. 2A). This design concept is a goal-directed reverse engineering process that ensures a
large continuous change in the curvature of the body which, in turn, constitutes the basis for
grasping objects of different sizes. The cable passes through the body of the robot with one end
fixed to the tip through a knot and the other connected to the motor (Fig. 2B). A thin elastic layer
serves to connect discrete units. This design allows to achieve basic curling and uncurling with just
one cable; when the cable is relaxed, the elastic layer provides the restoring force for uncurling (Fig.

2B and movie S1). Notably, the passive compliance inherent in the elastic layer enables the robot
to interact safely with the environment and further allows to generalize the design to multi-cable
robots with simple control laws.
The logarithmic spiral (represented in polar coordinates (, ) by =, where  and  are
scaling parameters), also known as growth spiral, is found in nature across species and scales (Fig.
2C). SpiRobs are designed directly from the mathematical expression for the spiral via the following
process. The rays starting from the origin that correspond to fixed angle intervals () intersect
with points on the spiral that are connected to form quadrilateral areas: these are the design blocks
for building the robot units (Fig. 2D). Note that once the spiral parameters and discretization step
() are given, the dimensions of the robot units are fully determined; in fact, adjacent units are in
a fixed ratio (see MATERIALS AND METHODS section 'Fabrication of SpiRob') which allows
for a modular fabrication. The curvature of SpiRobs changes exponentially fast with  (it can be
arbitrarily large/small when the value of  tends to /+) and this provides a justification for
the ability to grasp at different scales. Regarding the effect of design and fabrication, note that most
soft robots use homogeneous materials that are generally incompressible (such as silicone) to build
continuous bodies and are thus prone to buckling (34). In contrast, the proposed SpiRobs feature
(by design) uniform gaps among the discrete units which provide space for the body to deform;
when the gaps are fully squeezed, the robot curls into a spiral shape. Based on the same principle,
we can design and fabricate spiral-shaped robots capable of movements in the 3D space with multi-
cable actuation (Figs. 2E, S1, and S2).
Fig. 2. Design of bioinspired soft spiral robots. (A) A computer-aided design (CAD) model of a 25 cm long
SpiRob driven by a cable. The robot consists of a series of discrete units with a cable running through them. The
top-to-bottom image sequence shows curling from the tip when the cable is stretched. The blue circled region of
the robot's tip transmission is magnified in (B): Enlarged top view of the robot's motor–cable–body assembly.
The body of the robot consists of a series of units, connected by an elastic layer. The contour (dashed lines in the
first panel) forms a cone with a fixed taper angle. A motor is connected to the cable which is attached to the
tipmost unit by a knot. The cable contraction and relaxation (second and third panels, respectively) are translated
into the robot curling and uncurling motion. (C) Illustrations of the length scales and taper angles of spiral-shaped
wrapping appendages in nature. The length information was obtained from Encyclopedia Britannica, and the taper
angle information was measured based on pictures of different individuals. (D) Illustration of a SpiRob tightly
packed into a logarithmic spiral (=);  is the discretization step. (E) Images of soft spiral robots driven
by a single cable, two cables, and three cables, with corresponding cross sections shown in the upper right corner.

A salient feature of our proposed design lies in the choice of taper angle. This trait has also been
observed in nature, for example different taper angles exist across various species of the octopus.
We designed three spiral robots of variable taper angle (5°, 10°, and 15°) with the same length and
tip diameter (Fig. 3A). We found that the envelope of the workspace of a spiral robot also follows
a spiral shape (Fig. S5 and Text S5). Moreover, simulations validate that all of the interior is
reachable (Fig. 3B). The following conclusions are drawn for fixed length and tip diameter. The
smaller the taper angle, the larger the workspace (Fig. 3B). At the same time, the larger the taper
angle, the smaller the diameter of the smallest object that can be grasped and the larger the
maximum load capacity; besides, for fixed diameter, the larger the weight (with the difference more
pronounced for large-sized objects, see Fig. 3C and Text S5). Taking the robot with a taper angle
of 15° as an example, it can grab objects whose diameters vary by two orders of magnitude (from
5.6 mm to 108 mm) and weigh roughly 260 times the weight of the robot (38.4 g self-weight and
10 Kg load capacity, see Fig. 3D, and Text S5 for theoretical justification).
Fig. 3. The effect of taper angle on the
workspace and grasping ability. (A) SpiRobs
with different taper angles: 5°, 10°, and 15°;  is
the length of the robot and (0) is the width of its
tip (  = 25 cm and (0) = 5.5 mm in this
case). (B) Workspace: the smaller the taper angle,
the larger the workspace, which translates to more
flexibility. The envelope of the workspace also
follows a logarithmic spiral (plotted in different
colors for each taper angle). The green dots
illustrate the workspace for a taper angle of 10°, as
generated by randomly sampling cable forces in
simulation. (C) Theoretical predictions of the
object size and weight that can be grasped with a
maximum actuation force of 100 N: a larger taper
means larger weight for fixed diameter. (D)
Images of a SpiRob (15° taper angle) grasping a
5.6 mm diameter cable, 108 mm diameter can, and
10 kg weight. Stars in (C) and (D) are used to
match the experiments to the weight-diameter plot.
Bioinspired grasping strategy
The second principle that we leverage enables the active control of the curling direction so as to
wrap to grasp and manipulate different objects. A similar movement pattern is observed in octopus
tentacles when interacting with the environment (Fig. 1B and movie S2). In our system, we
implement this by controlling the cable forces. Take the robot actuated by two cables as an example
(Fig. 4A): when only the left or right cable is stretched, it is capable of curling in the two directions
and can be tightly packed into a spiral shape. By jointly controlling the forces exerted by the two
cables, the robot reaches out, contacts the object, and uncurls the packed body along the object
surface to wrap it (Fig. 4 B, C, S3, and movie S2). In quasi-static operations, the tip of the robot
forms a stable structure under the action of antagonistic forces from the two cables. A distinctive
attribute of the robot operation is that it always starts to curl/uncurl from the unit closest to the root
when changing the force on one side (while keeping the other one fixed). To this end, the elastic
layer on the central axis is instrumental to the uncurling of the packed body in order to reach out in
different directions (Fig. S3, movie S2, and Text S2). Based on this working principle, it is possible
to control the robot to grasp objects on a two-dimensional plane (Fig. 4C) by varying the actuation

forces through the two cables (Fig. 4D). We have verified the effectiveness of this grasping and
manipulation principle on both real robots (Fig. 4E) and in simulations (where we use a model
based on serial elastic joints, see Fig. S4 and text S4). Different from existing fingertip-based
grasping paradigms (8), the proposed strategy utilizes all the surfaces of the robot to contact and
wrap around the object. This is advantageous because a larger contact area means greater load
capacity and grasping stability. Moreover, this strategy is distinguished from the "whole-arm-
grasping" paradigm (35), since it is not only the contact between the robot surface and the object
but also the contact and extrusion with itself that are leveraged for grasping and manipulation. We
would like to highlight that when the packed body uncurled along the surface of the object, a
behavior similar to the "climbing" of a plant vine that navigates without friction from sliding against
the environment was generated: this enables the robot to adapt to objects of different geometries
regardless of their surface roughness.
Fig. 4. The principle of asymmetric cable forces enabling active curling, wrapping, and grasping. (A)
Illustrations of the design of a SpiRob driven by two cables passing through the body and fixed to the tip with
two knots. The central axis of the robot comprises an elastic layer. The design parameters of the robot can be
determined by a logarithmic spiral along with the discretization step (), 30° in our case. (B) Block diagram of
the bioinspired grasping strategy. (C) Illustration of a biologically inspired grasping and manipulation strategy
that can be implemented by controlling the cable forces (plotted in (D)). Starting from a state where the robot is
packed into a spiral shape (Packing), increase the force applied to the right cable () while keeping the left ()
unchanged. The robot starts to uncurl from the root, reaching out towards the object (Reaching). After in contact
with the object, by slowly reducing the force of the left cable while keeping the right unchanged the robot uncurls
across the surface of the object (Climbing) to wrap around it. After this, the force of the right cable is increased
to achieve a firm grasp (Grasping). (D) Illustration of the corresponding actuation pattern for the two cables used
to produce the grasping and manipulation motion in (C). (E) An overview of the robot system and an image
sequence for the real robot.
In order to implement the grasping strategy, we propose a state-aware approach based on motor
current detection for automatically switching between the different behaviors (Fig. 4B). Starting
from the Packing state, the robot unfolds and reaches out as the right cable is pulled at a constant
speed (Stage 1: Reaching) (Fig. 5). When there is no contact with the object, the current of the right
motor remains rather stable. If there is contact with an object, the robot is subjected to an external
force so that the motor current increases to maintain a constant speed of motion. By detecting such
an increase in current, the robot can perceive contact with an object. The threshold for contact
detection is set through multiple experiments, where the maximum value of the current is recorded
as the robot is running without contact. This simple mechanism is very effective and capable of
detecting even the slightest contact (Fig. 5A and movie S3). After detecting the contact, the motor
on the left switches from speed mode to torque mode and gradually decreases the torque down to

zero (Stage 2: Wrapping). That is, it slowly relaxes the left cable, letting the robot uncurl and climb
along the surface of the object. When the torque of the left motor is zero, the wrap is completed.
Then, the motor on the right pulls the cable at a constant speed to grasp the object to the root of the
robot (Stage 3: Grasping). This method does not require external sensors, e.g., cameras or force
sensors, and can realize high-precision contact sensing when grasping objects of different shapes
(Fig. 5B and movie S4).
Fig. 5. Contact detection and automatic
grasping based on onboard current sensing. (A)
By detecting changes in the current, the robot can
sense contact with a feather. We placed a feather
in three different positions and the robot retracted
to the packing state as soon as it detected contact.
The result shows that the same threshold can be
used for contact detection across the workspace
while keeping high sensitivity. The current signal
of the right motor is plotted below the screenshots.
(B) The robot automatically grasps objects of
different geometries. Stage 1: Reaching. The
cables are pulled at a constant speed to make the
robot reaching out. Contact is detected by
monitoring the current value of the right motor.
Stage 2: Wrapping. The left motor changes to the
torque control mode, and the current is gradually
reduced to zero and the wrapping of the object is
completed. Stage 3: Grasping. The motor on the
right moves at a constant speed until the current
increases beyond a set value, whence the object is
grasped to the root of the robot. Stage 4: holding.
The robot holds the object by keeping cables at a
constant length. The current signal of the motors
and the length of the cables are plotted below the
screenshots. The three black dashed lines from top
to bottom are 0, contact detection threshold and
grasping detection threshold.
Manipulating various objects
To demonstrate the ability to grasp and manipulate a variety of objects, we first designed and
fabricated a 2-cable SpiRob (45 cm in length) by 3D printing using soft filaments (Fig. S2A; another
origami-based fabrication method is described in detail with Text S1). This robot was used to
demonstrate the potential of the proposed spiral-shaped design principle and operating strategy. We
tested for numerous objects varying in size, weight, and material. Unlike existing multimodal
grasping mechanisms (31, 36, 37), the same spiral robot with the same strategy can successfully
grasp and manipulate a wide variety of objects in size and weight (Fig. 6A-D, movie S5). Even in
a confined space, the packed body can navigate through obstacles without relative sliding (Fig. 6 E,
F, S6, and movie S6). Interestingly, we observed that the robot takes advantage of its interaction
with the obstacles to reach the target instead of trying to avoid it. Since visual perception and control
were not the focus in these experiments, we manually controlled the two cables like a puppet
(implementation of a remote-control scheme with a joystick is detailed in Fig. S2, movie S7, and
text S3). Evidently, given that the base of the robot is fixed in our experiments, objects need to be
positioned in a suitable range.

Fig. 6. SpiRobs grasping and manipulating various objects. (A)- (D) Image sequences of a two-cable spiral
robot grasping and transporting a raw egg, a pen, a soft cube, and a paper cup filled with water, respectively. (E)
The robot navigates through a door to transport a pingpong ball to its target in an S-shape trajectory. (F)
Manipulation in confined space: the robot actively interacts with obstacles before reaching to grasp. These
experiments demonstrate that the spiral robot can perform complex manipulation tasks by capitalizing on
interactions with the environment. Scale bar in (F) represents 10 cm.
3-cable SpiRobs
To further explore the capabilities of the spiral robots, we designed and fabricated a robot actuated
by 3 cables evenly spaced on the circle corresponding to the cross section (Fig. 7A). The length of
the robot is 75 cm (tip/root diameter is 5 mm/72 mm), demonstrating that the spiral design principle
can be easily extended to different scales. The choice of the cross section serves to reduce inertia
and also increase the contact surface when grasping. Using a strategy similar to the 2D grasping,
the 3D robot unfolds the curled body on the surface of the object to wrap and grasp it (Fig. 7B and
movie S12). By controlling the cables complex paths can be achieved (Fig. 7C). Even very thin
objects can be gripped with high stability for a non-slip grip (Fig. 7D, E and, movie S13).
Fig. 7. 3-cable SpiRobs. (A) Photographs
of a 75 cm long 3-cable spiral robot and its
wrapping around a 500 g hammer. (B) Image
sequence of a 3-cable SpiRob grasping a ball.
(C) Illustration of a non-slip grasping
strategy for the case of a tree-fork-shaped
object. (D) Image sequence of a 3-cable
spiral robot implementing the strategy to lift
a hexagonal wrench. (E) Images showing the
robustness of the strategy under external
forces. The scale bars in (A) and (B)
represent 10 cm and in (D) and (E) represent
5 cm.

SpiRobs conducting dynamic tasks
We further illustrate that the presented class of soft robots portrays similar capabilities as in living
organisms: high-speed dynamic movement and impact resistance with robust grasping. On sub-
second or second timescale, octopuses can grab and catch a fish, while elephants can use their trunks
to spray water all over their bodies. For most existing soft manipulators, achieving dynamic
efficiency comparable to biological structures is merely impossible, despite having soft bodies. We
observed that the open-loop behavior (movie S8) of SpiRob enables grasping within 60 ms (Fig.
8A, movie S1) and throwing at a speed of 8 m/s (Fig. 8B, S7, and movie S9). Its ability to passively
deform allows grasping objects moving at high speeds (Fig. 8C, movie S10). Last but not least, the
robot shows excellent grasping stability that can resist out-of-plane torsion and dynamic impact
(Fig. 8D, movie S11). For dynamic operation in the 3D space, the shapes and moving patterns that
the cable actuation can generate are more complicated due to the intrinsic inertia and stiffness of
the body as well as the effect of gravity (movie S12). The curled body "shoots" out to reach a
specific position (Figs. 8E, S8, Text S6, and movie S14) or grasp objects (Fig. 8F and movie S15).
Fig. 8. Dynamic tasks. (A) Image sequence of dynamic grasping. The experiment shows that the robot finishes
a fast-grasping within 0.06 s. The actuation pattern is displayed in the upper right corner where 'On' means that
the cable is stretched and 'Off' means that the cable is slack. (B) Elephant throw. A robot 35 cm in length throws
a piece of rubber at a speed of 8 m/s. (C) Passive grasping. Exploiting its passive compliance, the robot is capable
of grasping a high-speed moving object within 100 ms even without control (the left cable is fixed to maintain a
constant length while the right cable is slack). (D) Robust grasping. The robot holds the racket steadily as a 60 g
tennis hits at 11.5 m/s (producing an average out-of-plane impact force of 138 N). (E) Whipping. The robot
dynamically reaches a point in the 3D space on a sub-second timescale to pounce a pingpong ball. (F) The robot
grasps and lifts a headset within 1 s. Scale bars in (A)-(F) represent 10 cm.

DISCUSSION
This paper proposed a modular and interpretable principle for designing soft robots across different
scales and presented a series of spiral robots that can grasp and manipulate objects through
wrapping. By tensioning the cables, the robot reproduces a spiral-shaped wrapping behavior that is
common in nature. A bioinspired strategy allows controlling the curling direction so as to handle
objects that vary in size by more than two orders of magnitude and up to 260 times self-weight. Our
results corroborate that the spiral-shaped principle constitutes an effective solution for the structural
design of soft-engineered systems to exploit wrapping as a paradigm for versatile grasping and
manipulation.
The design principle of the logarithmic spiral a key novelty in this paper. Different from most soft
robotic systems (13) where the hardware is designed first and the models are developed afterwards,
in our system modeling (logarithmic spiral) comes first, and design/fabrication is a direct outcome
of the model. This allows designing SpiRobs to meet the requirements of a large diversity of
application scenarios (i.e., in terms of grasping size and load capacity, see Table S1), instead of
using a trial (physically building the robot) and error (testing) approach. In addition, unlike the
Piecewise Constant Curvature (PCC) model commonly used in the field (38, 39) (Fig. S9 A and B),
the logarithmic spiral features faster curvature changes along the body (Fig. S9 C), thus resulting
in a higher degree of flexibility. Most existing soft manipulators gain increased flexibility through
the combination of multiple independently actuated segments (40–42). However, increasing the
number of segments also increases the complexity of modeling and control (43). We show that
based on a simple control strategy, SpiRob can achieve complex operations by controlling only
two/three cables.
The presented SpiRobs feature regular discrete units, which is a distinctive feature that distinguishes
them from other existing soft manipulators (19, 29, 30, 40, 44). The discretization introduces a
series of internal constraints, which physically constrained the maximum curvature of each part of
the robot body (that is, the curvature reaches the maximum when adjacent units touch each other).
We found that the internal constraints not only improve the stability of grasping but also allow the
robots to achieve wrapping with simple control laws. These constraints are key to realizing the
logarithmic spiral-shaped wrapping. In nature, this internal constraint is reflected in the limited
rotation angle between the adjacent bones of the skeletal organs (e.g., the tail of chameleons and
seahorses) and the limited distance of muscle contraction in the muscular hydrostatic structure (e.g.,
the tentacles of octopuses and the trunks of elephants). However, for most existing soft octopus-
tentacle or elephant-trunk inspired designs (Fig. S9A), whether driven by tensile actuators or fluids,
deformations of the robot always increase with increasing actuation in the absence of structural
curvature regulations. A similar design was presented in (45), where the cable-driven robot also
features discrete units. However, the length of each unit is the same making the robot deform in a
PCC manner. We emphasize that all geometric parameters of SpiRobs can be fully determined by
the equation of the logarithmic spiral (e.g., the length and shape of units, the size of gaps, see Table
S1).
The bioinspired grasping strategy shows large adaptability and high grasping stability. To climb on
the surface of an object, the SpiRob makes full use of its interaction with it. However, these
interactions exert forces on the object. If the object is not fixed, it will be moved when the robot
touch it. This strategy has a higher success rate in grasping objects that are handed over by humans.
The 3-cable SpiRob shows its ability to grasp a ball placed on the table without bumping it away.
Moreover, our current work does not yet involve the visual perception of the robot and its
environment, thus the motion in some experiments is controlled through direct manipulation of
cables like a puppet (Figs. 6, 7, and 8).

MATERIALS AND METHODS
Conceptual design of a soft spiral robot
SpiRob consists of three main components: the robot body, the cables, and the motors. The two
ends of each cable are connected to the tipmost unit and the motor, and the cable's contract/relax
actuation is translated to the curling/uncurling motion of the robot, respectively. In the following,
we describe the design process and determine the key parameters of the robot body.
We take a spiral robot with two cables that can wrap in two directions on a plane as an example.
Recall the expression of a logarithmic spiral ((,) are polar coordinates):
() = , =cot , (1)
where  is a scaling factor and  is the cotangent of the constant polar tangential angle  - defined
as the angle between the tangent of a point on the spiral and the line connecting the point to the
origin (Fig. S1A). Equation (1) can be re-written as =󰇡
󰇢, so that  can be interpreted
as translation in the angle domain and  as scaling. For design, we restrict attention to the
range 0. The regime < 0 corresponds to the extension from the tip to the point where
the outer edges of the robot meet (). We define the 'central' spiral characterizing the
central axis of the robot (Fig. S1B) by:
()=
()+=
( + 1). (2)
The rays starting from the origin and at fixed angle intervals (=30° in our case) intersect with
points on the original spiral and the central spiral which are connected to form quadrilateral areas
(Fig. S1C and D). This forms one part of the robot, to which we attach an elastic layer that provides
the restoring force that allows the robot to reach different directions (Fig. S1F). The other part is
obtained by mirroring with respect to the central axis (and thus its contour follows an 'outer' spiral).
In our design, the elastic layer's thickness is 5% 10% of the unit width (5% for the 2-cable and
10% for the 3-cable robot in our implementation, see also Fig. S1F).
The taper angle () of the robot has the following relationship with the spiral parameters:
= 2 arctan 󰇧
()
()

 󰇨= 2 arctan 󰇡
 󰇢, (3)
()=(), (4)
where () is the width of the robot at angle . The integral in the denominator in equation (3)
captures the length of the central spiral starting = (see Fig. S1E). The taper angle  is
independent of , see equation (3), indicating that the spiral is tapered when expanded (Fig. S1E).
The length () of the (central axis of the) robot has the following relationship with the spiral
parameters: =()

=
 . (5)
The lower bound of the integral here is = 0, which corresponds to the tip of the robot (Fig. S1E).
The curvature of the robot () (when packed into a logarithmic spiral) can be calculated from the
expression of the central spiral to be: ()=
. (6)
In particular, the curvature changes exponentially fast with . This renders inappropriate for our
case the Piecewise Constant Curvature (PCC) model (38) that is commonly used for modeling and
control of soft robots. Moreover, the deformation rate () of the robot’s surface when it shifts from
the packed state pointing to the left to the packed state pointing to the right (or vice versa) can be
calculated as:

=()


()

=, (7)
For example, when = 0.2199 (i.e., taper angle =15° in this case), = 3.98, which indicates
the challenges to design and fabricate spiral robots with a continuous homogeneous body to undergo
such large deformation. This is because repeated stretching/shortening at such a deformation rate
can cause the material used to construct the robot (for instance, Silica gel) to break. We believe that
this phenomenon also provides a justification for the wrinkles and folds observed on the surface of
the elephant trunk that serve to provide additional flexibility for large deformations (46).
The design of a 3-cable robot that can curl in the 3D space follows the same concept as building a
2-cable robot (Fig. S1G). The difference is that the mirroring is in 3D (i.e., it gives a cone). Each
three-dimensional unit is a fragment of the cone, so that without any alteration this would give a
small contact surface when grasping an object. For this reason, we reshaped the cross section to
increase the contact area (Fig. S1G). A similar feature of "square" cross section that improves
mechanical performance in grasping has been observed in seahorse tails (4).
Fabrication of SpiRob
The robot bodies are built using an existing 3D printing method. Specifically, we use a desktop 3D
printer (X1 Carbon, Bambu Lab) together with 1.75mm TPU (Thermoplastic polyurethanes)
filament (eTPU-95A, Esun) to fabricate our robots (Fig. S2A). We designed small holes on each
unit for the cable to pass through (we provide a printable .stl file in the supplementary material).
We use UHMWPE (ultra-high molecular weight polyethylene) cables for the actuation of the robot.
This type of cable is wear-resistant and smooth, thus reducing friction as it moves through the
robot's body. Furthermore, the cable is fastened to the tip unit with a fisherman knot to ensure that
the actuation force is transmitted without slipping. We have made several observations that are
pivotal in achieving modularity of the fabrication process. First, note that the scales of adjacent
units of the robot are in a fixed ratio , given by:
=()
()=, (8)
where () is the width as at angle  defined in equation (4) and  is the discretization step. Thus,
we only need to design one unit, and then scale up/down according to the factor  to obtain the
design of all other units. Second, longer robots can be fabricated in segments that are then connected
by a dovetail tenon connector (Fig. S2A). The 2-cable robot used in Figs. 4-6 consists of two
segments, and the 3-cable robot in Fig. 7 consists of three segments.
Control terminal
We built a motor control terminal (Fig. 4E), which consists of two motors (GM6020, DJI), an
embedded controller (Robomatser Development Board, type A, DJI), and a 24V battery. The motors
are direct-drive brushless without gearbox and can be controlled in terms of torque, speed, and
position. This provides the basis for our current-based contact detection and wrapping-based
grasping.
Data analysis
The motor current, speed, and position data are captured using STM32Cube Monitor (data files S1
and S2). The length of the cable (Fig. 5B) is converted from the recorded rotor position of the
motors in MATLAB. For dynamic tasks (Fig. 8), videos and screenshots are captured using a high-
speed camera (ACS-3, NAC Image Technology).

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