A Soft Continuum Robotic Arm with a Climbing Plant‐Inspired Adaptive Behavior for Minimal Sensing, Actuation, and Control Effort

A key challenge in designing soft continuum robotic arms is the realization of intelligent behavior while minimizing sensing, actuation, and control effort. This work investigates how soft continuum arms can benefit from mimicking the distribution of flexural rigidity of searcher stems in climbing plants to accomplish this goal. A modeling approach is presented to tune both the structural design and the tactile sensor design of a soft continuum arm inspired by the flexural rigidity distribution of Mandevilla cf. splendens’ searcher stems. The resulting soft continuum arm, named Mandy, can detect suitable supports along its length and twining around them using a single sensor and actuator. Through simulations and experiments, it is shown such behavior cannot be achieved with a soft continuum arm possessing uniform structural stiffness and a standard tactile sensor design. Thus, the significance of investing greater effort in structural design, leveraging biological data, to improve the design of soft continuum arms with more compact actuation and sensing hardware, is highlighted.


Introduction
Slender and continuously deformable soft continuum arms hold significant potential for addressing major challenges in robotic manipulation within extreme and unstructured environments.These can include outer space, [1,2] in vivo scenarios, [3] underwater applications, [4] and confined spaces. [5]Constructing such possess a flexural rigidity that decreases from the base toward the tip of the structure.This distribution enables them to have the necessary flexibility at the tip to manipulate objects while maintaining the required stiffness at the base to support the rest of the structure.[19] However, the implementation and effectiveness of incorporating bioinspired distributions of flexural rigidity in soft continuum robotic arms have not been extensively studied before.While some previous works have presented bioinspired soft continuum arms with tapered bodies (e.g., other studies [4,[20][21][22] ), such as those mimicking material density and constant-volume property, [23] or focusing on taper angle, [16] the distribution of flexural rigidity has not been the primary focus.This study is the first to investigate the potential benefits of mimicking the flexural rigidity distribution found in soft continuum structures in nature for reducing the required actuation and sensing effort in soft continuum robotic arms.
Climbing plants provide intriguing biological models for investigating the distribution of flexural rigidity because they rely on external supporting structures, thereby avoiding the need to allocate resources for developing their own stiff and mechanically stable bodies.This nonself-supporting growth allows climbing plants to reach greater heights and advantageous positions for crucial biological processes such as photosynthesis, pollination, and fruit or seed dispersal, in some cases at the expenses of other plants (see, e.g., other studies [24][25][26] ).Climbing plants employ their most apical organs, known as searcher stems, to explore their surroundings and achieve successful attachment.The searcher stems typically exhibit a characteristic rotating and periodic growth movement known as circumnutation, which helps them locate a suitable support.Once a suitable support is detected, the plants alter their growth pattern and establish attachment to the support.This behavior is also referred to as thigmomorphogenesis.[28][29] A common attachment principle is that the searcher twines around the support.These plants are also referred to as twiners.As plants do not possess a central control unit, that is, a brain, to regulate and switch between these movements, the body of the plant plays a crucial role in the realization of its behavior, which can be interpreted as physical intelligence embodied in the plants' materials system. [17,30]To bridge distances between neighboring support structures, the searcher stems must be sufficiently stiff and lightweight to prevent drooping under their own weight. [26,31]On the other hand, they must be sufficiently flexible, especially in their most apical parts, to allow for circumnutation movements covering a large volume in space, which is essential for finding new supports.
In nature, different searchers are found depending on the plant species.For instance, the climbing cactus Selenicereus setaceus is characterized by an apical star-shaped stem: such shape provides the necessary flexural rigidity to cross a void of one meter without relying on any support. [32]Other portions of the stem exhibit a different shape of the cross section (triangular or circular), depending on the distance from the apex.Recently, this feature has inspired researchers to build shapechanging soft robots, [33] whose rigidity can be conveniently tuned to implement the desired mechanical behavior.Other climbing plants have instead thin stems with circular cross section that can grow vertically, or protrude from the plant, until they collapse.Consistently, robots inspired by such plants are characterized by slender, thin bodies.An example can be found in an existing work, [34] in which a tendon-driven continuum robot performs circumnutation movements to find a support and engage it by means of artificial prickles, consisting in bent metal nails.Other authors [35] have proposed a plant-inspired robot that can perform shape-locking, stiffening the basal part of the robot while leaving the apical part relatively soft, as observed in plants.In all these works, a particular behavior or element of the plant has been taken as the starting point to develop a technology, such as the class of performed movements, or the presence of prickles, or macroscopically evident geometrical features (e.g., slenderness).However, no work has yet focused on the structural properties of climbing plants, even though they play a crucial role and can provide useful insights for building soft continuum robotic arms.
In this work, we demonstrate how the structural properties found in the searcher stem of a climbing plant species, Mandevilla cf.splendens (syn.: Dipladenia cf.splendens), can aid in the design of soft continuum robotic arms capable of performing twining movements with minimal actuation and a single distributed sensor.To our knowledge, this is the first soft continuum arm with a body that mimics, in a quantitative way, structural properties observed in a living organism.Our study started with the biomechanical characterization of Mandevilla stems, to investigate how the flexural rigidity, a key structural property for continuum arms, varies along the stem.These results have guided the design of a robot, named Mandy, with a fully soft body and driven by a single tendon, enabling it to twine around a support.Through our research, we demonstrate that investing more effort in the structural design, especially when informed by biological data, yields significant advantages in building soft continuum robotic arms.These benefits include more compact actuation and sensing hardware, simplified control, and easier integration with other robotic systems.

Biomechanical Characterization of Searcher Stems
The samples of Mandevilla cf.splendens used in this study were cultivated at the Botanic Garden at the University of Freiburg, Germany.As shown in Figure 1, searcher stems of this plant can either explore the surroundings individually or form braided structures with other stems of the same plant, probably to increase their stiffness and cover larger distances. [18] searcher braid consisting of three intertwined stems (from now called S1, S2 and S3) was collected by cutting the individual stems basal to the intertwined domain.Their relative position within the "braid" was preserved by fixing with cotton stings.Three-point bending tests were carried out on individual stem sections and segments of 2 or 3 intertwined stems using a material testing machine (Instron 4466-10 kN, with a retrofit kit to inspect-DC standard, Hegewald & Peschke Mess-und Prüftechnik GmbH, Nossen, Germany) fit with a 10 N load cell.After measuring the segments of intertwined stems, they were taken apart and the individual constituent stem segments were measured if they were reasonably straight.The flexural rigidity (EI) was calculated as where L is the distance between the supports of the three-point bending setup and b is the slope of the force versus deflection curve.The cross-sectional shape of the stems was assumed to be circular and the axial second moment I ax was calculated as where D is the mean diameter of the stem segment measured at three positions in two orientations perpendicular to each other.The elastic modulus can be calculated as For the intertwined segments (i.e., the braids), the elastic modulus could not be determined.The results for each segment are summarized in Figure 2. It was found that the flexural rigidity in S1 decreased from the basal section of the stem to the apex, of more than two orders of magnitude.The same trend was shown by the younger stems S2 and S3.To design our robot, we initially referred to the properties of S1 and later discussed the properties of the braid in the Discussion.
As we were interested in how the flexural rigidity varies along S1, rather than on its absolute value, we normalized the obtained quantities with respect to the value of the first segment.The trend can be fit with a fourth-degree polynomial (see Figure 3) with p 4 ¼ 1.254, p 3 ¼ À3.426, p 2 ¼ 4.065, p 1 ¼ À2.881, and p 0 ¼ 0.995.

Bioinspired Robot: Design and Fabrication
This section describes the integrated design and fabrication approach that was taken to incorporate the distribution of flexural rigidity that was observed in the plant stem (shown in Figure 3) in a sensorized soft continuum arm capable of twining upon detecting a suitable contact using only a single actuator and sensor.The body of the arm was hollow, as this brings two advantages.First, all the components needed to enable actuation and sensing can be hosted inside the robot's body, allowing for a compact design.Also, a hollow structure exhibits a greater flexural rigidity compared to a full solid body with the same mass.
The gradient in flexural rigidity from the plant stem was approximated by eight segments with a common design parameterization but a unique flexural rigidity and tactile sensor sensitivity (see Figure 4).The rationale behind the design parameterization allowing for the tuning of the flexural rigidity and sensor sensitivity is discussed in Section 2.2.1 and 2.2.2 respectively.Finally, Section 2.2.3 discusses the fabrication of the segments and the assembly of the arm.

Structural Design of the Segments
Mandy is a fully soft continuum robotic arm resulting from the assembly of multiple rings resembling the wave spring washers commonly used by mechanical engineers.Our design took inspiration from an existing work, [36] in which it was assessed that series of soft wavy rings were suitable to build bendable robotic segments.The thickness of the rings changed along segments, resulting in a gradient of flexural rigidity from the base to the apex.The variation from segment to segment was chosen such that the distribution shown in Figure 3 was implemented.In principle, such distribution can be used to design a robot with an arbitrary number of segments; the greater the number, the more accurately the distribution is reproduced.In this work, Mandy was composed of eight segments of equal length L S .However, the design method that we report in the following has a general validity.
For the scope of this work, we developed a simplified structural model to adjust the flexural rigidity of the segments along the robot's length.
Here we estimated the deformation generated by pulling a tendon passing through points A and G (see Figure 5) and exerting a force 2F.Let us consider the path depicted by the broken line in figure.The path consisted of six arcs along the segment.Each piece of the line was here modeled as a beam, with length l and with square cross section having edge b.The length l depends on the number of waves of the springs and on the radius of the segment.
Under the force F, the beam underwent a deflection that is here computed as  The same holds for the wall thickness of the air chambers.All the chambers are connected to the same silicone tube and hence constitute a single touch sensor.
Here E is the Young's modulus of the material of the segment and I is the moment of inertia of the cross section.
Since the path consisted of six beams, the total deflection obtained with a force F would be f ¼ 6 f .Since the robot had no backbone, here we worked under the simplification that each segment bent because the distance between the points A and G shortened, while the diametrically opposite distance remained unchanged (for more rigorous considerations, see Discussion).
Adopting the notation in Figure 5, Here L M is the length of the undeformed module.After some manipulation, and being If we approximate the moment generated by the pulled tendon by M ¼ 2F⋅2R 0 , the bending stiffness k θ is as in the following.
Recalling Equation ( 5), and considering that the square cross section of the ring has moment of inertia I ¼ b 4 =12, the bending stiffness that appears in Equation ( 9) can be rewritten as a function of all the fundamental design parameters for the j-th module.
Here, we imposed that all the segments share the same Young's modulus, radius R 0 , and number of waves, although this is not mandatory.Equation ( 10) was used to find the dimension b for each segment of the robot.As we aimed at reproducing the distribution of k observed in the biological sample, we chose arbitrarily the flexural rigidity of the first segment of Mandy, and we computed the remaining accordingly.This value was chosen such that the ratio stiffness/weight of the robot's body was sufficiently high to allow neglecting the static deflection and make the robot self-standing.
Table 1 reports the dimensions of the wavy rings in the eight segments that constitute Mandy's body.Segments were numbered from 1 to 8 from the base to the apex.The value for the first segment was chosen arbitrarily.The total length of the robot was %535 mm, each segment being %67 mm long.The average radius R 0 was 12 mm for all the segments.All the rings had 5 waves.

Sensor Design
To investigate thigmomorphogenesis, the soft continuum robotic arm should be capable of detecting mechanical stimuli along its length.This was achieved through integrating small air chambers around the tip of each segment (Figure 4).When a mechanical stimulus was applied to one of the air chambers, it resulted in indentation and an increase in air pressure inside the chamber.These tactile sensors were omnidirectional and located on every segment except for the tip, as applying a mechanical stimulus at the tip should not trigger the twining.
Twining should be initiated whenever one of the air chambers detects a mechanical stimulus that suggests the presence of a support that is sufficiently stiff for the arm to twine around.It should be noted that the required stiffness of the support varies greatly along the length of the arm; a support near the base should be much stiffer to prevent it from being pushed away by the arm compared to a support near the apex.Instead of measuring the air pressure inside each air chamber individually and calibrating each sensor to determine the air pressure indicating a sufficiently stiff support, we embody this intelligence into the arm.All air chambers are connected to a single air pressure sensor, and the sensitivity of each chamber is controlled by adjusting the wall thickness.This ensures that a sufficiently stiff support at any air chamber results in the same air pressure.This air pressure is then used as a threshold value for initiating the twining behavior.As a result, the number of sensors is reduced to one, and a single-voltage threshold can be set to initiate the twining.
An important question that remains open is what the relationship between the stiffness of the plant stem at the point of contact and the contact force is at which plants initiate twining.As literature is inconclusive on this question, we initially adopted an intuitively sensible relationship, where a deflection y of any point j along the arm will always result in the same sensor signal, regardless of the location of the force causing this deflection.To implement this, the thickness of the air chambers decreases from the base to the apex of the robot.The wall of the j-th Table 1.Edge b of the square cross section of waves and thickness t of the wall of the air chamber, for each segment.As the biomimetic flexural distribution is implemented, waves and air chambers become thinner from the base to the apex of the robot.chamber, located at the abscissa x j (see Figure 6), was designed such that a force F j that needs to be applied on it in order to produce the air pressure threshold ΔP threshold should generate the same deflection y j as observed at that section when the force F TIP produces ΔP threshold in the air chamber located at x TIP , as clarified by Figure 6.
To tune the thickness of the chambers along Mandy's length as explained above, we recurred to linear beam theory again.If a force F TIP is applied at the tip of the robot, the following relation holds.
with m being the number of segments.As we wanted to integrate this expression, here we approximated the distribution of flexural rigidity implemented in Mandy as the continuous parabolic arc.
, the subscripts referring to the segments.The deflection caused by the force F TIP can be calculated by integrating Equation ( 11) twice.Such deflection is a function of x; we considered that the deflections of the sections at x j ¼ j⋅L S , with j ¼ 1, 2, : : : , 7, that is, at the top of each segment (where the chambers are located).As we could treat the force as a multiplying factor in Equation ( 11), we rewrote the latter as Since we imposed the condition Being then we computed how the forces F j rescaled for our purpose as All the integrations were performed numerically in MATLAB.From these results, the thickness t j of the j-th chamber was obtained by the following expression.
as we assumed that the indentation of the chamber was inversely proportional to the cube of the thickness (as it happens in bending plates).The constant Q was here taken based on empirical observation on the seventh segment of the robot and on the constraint introduced by the fabrication process.In fact, we had to impose that t 7 ¼ 0.8 mm, as it was the minimum thickness printable by the SLA printer used for this work.All the other thicknesses are shown in Table 1.As expected, the thickness of the chambers increased from the apex to the base of the robot.Thicknesses t 5 and t 6 were the same due to their difference resulting to be smaller than the 3D printing resolution.

Fabrication
Mandy's segments were fabricated through additive manufacturing.In the current work, we utilized stereolithography (SLA) and used a commercial material (Flexible 80 A resin by Formlabs) with a shore hardness of 80 A. The outlets of the air chambers were located on the inner side of the segments to facilitate the routing of the silicon tube through the arm's segments.The segments were connected by 3D-printed rigid rings with longitudinal holes, which served as guides for the tendon, and a radial hole that interlocked around the inlet of the air chamber.After connecting two segments, a pneumatic connector was plugged into the air chamber through the radial hole in the rigid ring.Finally, all pneumatic connectors were connected to a silicone tube (2 mm OD, 1 mm ID).The total mass of the robot was %0.2 kg.As the motor to pull the tendon as well as the air pressure sensor were located outside of the body of the robot, the arm was free of electronics and can therefore be deployed in wet and dirty environments as well as in the presence of a magnetic field.

Kinematics and Control
As inspired by the climbing plant, Mandy was designed to perform two types of movements: circumnutations and twining.Circumnutations allow the plant to explore its surroundings in search of support, while twining is used by climbing plants to sustain their body while growing around a support.In the robotic artefact, circumnutations can be achieved either by means of three (or more) tendons at the base segments, as usually demonstrated in tendon-driven continuum robotic arms (e.g., as shown in ref. [36] or in ref. [37]), or using a single tendon to make the body of the robot perform planar bending, while the base of the robot rotates thanks to an additional mechanism.However, in this work, we specifically focused on the twining behavior enabled by a biomimetic distribution of structural properties; hence, circumnutations were not discussed.Twining was here achieved using a single tendon, deployed from the base to The deflecting lines represent the longitudinal axis of the robot, which bends under F TIP , applied at the apex, producing a deflection y x j À Á at x j , where the j-th chamber is located.We find the force F j applied at x j that produces the same deflection, and we tune the thickness of corresponding air chamber to obtain the same pressure variation that would be detected by an air chamber located at the tip under the action of F TIP .The same is performed for all the chambers.
the second to last segment, following helical routing: it ran spanning an angle ϕ i between adjacent segments.By pulling this tendon, if ϕ i was sufficiently small, each segment deformed performing approximatively planar bending on a different plane, resulting in twining, as shown in the following.

Kinematic Model
Let us assume that in its initial, undeformed state, the robot was in the straight configuration along the z-axis of a global coordinate system x, y, z ð Þ.In addition to such coordinates, we defined a local system of coordinates x i , y i , z i ð Þfor the i-th segment, having origin at the center of the base of the segment and z along its longitudinal axis.Here, to simplify the model, we approximated the helicoidal path as piecewise: within each segment, the tendon ran along a straight path, that is, it was parallel to the axis of the segment.In the circumferential direction, the tendon was shifted of a phase angle ϕ i at the interface between segments i À 1 ð Þ-th and i-th, as shown in Figure 7.
In the following paragraphs, we denote by θ bi the bending angle of the i-th segment and by p i the coordinates of the generic point of the axis of the segment in its local system of reference.Under the assumption that each segment bends with constant curvature, the bending radius and the length of the central line of the bent segment are computed respectively as Hence, the local coordinates p i ¼ x i , y i , z i f g T are found as being 0 ≤ θ i ≤ θ b i , as denoted in Figure 7.
The same coordinates in the global system of reference will be denoted by q i ; the top point or the arc will be denoted by q ⋅ i .If we define the rotation matrices the coordinates in the global reference can be obtained as follows.
in which the rotation matrix is We implemented the problem in MATLAB R2019b, to verify whether the desired twining behavior can be obtained.The moment applied on each segment was here calculated as the product between the pulling force of the tendon and the average diameter of the segment.

Control Strategy
The structural design of the arm in combination with the tactile sensor design allows for a greatly simplified control.A single Honeywell SSCMRRD010MDAA5 air-pressure sensor with a range of þÀ1 kPa was connected to the seven air chambers along the length of the arm by a single silicone tube.We experimentally defined the change in air pressure ΔP threshold when F TIP was applied at x TIP .When the measured change in air pressure exceeded this threshold, a suitable support was present somewhere along the length of the arm (i.e., a force F j at x j ).A PID-controlled 100:1 metal gearmotor with a 64 counts per revolution encoder was used to pull the tendon by a fixed length.The tendon followed a helical routing throughout the arm.For our tests, we assembled Mandy such that the tendon spun 30°a long a single segment.The last (eighth) segment was not actuated.This configuration allowed the arm to twine while adapting its shape to the external object.

Mandy: Simulation Study
In this subsection, we investigate the convenience of Mandy's biomimetic flexural rigidity in performing twining, as well as the adopted tendon angle through simulations.Mandy's deformation is compared with the one achieved by other continuum arms of the same length, and with the same number of segments, but not provided with the biomimetic flexural rigidity.In fact, we consider 1) Mandy, with the biomimetic distribution found through mechanical characterization; 2) a robot with linearly decreasing rigidity, from the base to the apex; 3) a robot with constant rigidity along its length, equal to k max (unitary, due to the aforementioned normalization); and 4) a robot with constant distribution equal to the average value of the biomimetic distribution, k avg .
Figure 8a shows the four simulated distributions.In all the robots, the tendon follows the same helical routing, spanning 180°from the base to the apex of the robot; however, due to the extremely low rigidity at Mandy's apex, the last segment is not actuated.The results, shown in Figure 8b, are normalized with respect to the length of the robot; hence, the applied force is merely a multiplying factor, equal for all the simulations.It can be noticed that the biomimetic flexural rigidity seems to provide an advantage for twining: continuum arms with constant rigidity are not able to achieve the desired configuration.The same holds if the rigidity decreases linearly, as better discussed in the next Section.Figure 8c shows how Mandy's body deforms under a progressive action of the tendon, starting from the initially straight pose to the configuration in which its distal portion creates a sort of loop.In this simulation, the angle ϕ is 30°; for this value, we will show Mandy's experimental behavior.This angle, as expected, plays a major role in the achievable deformation of the robot.For comparison, Figure 8d-g shows Mandy's progressive deformation for different values.

Mandy: Experimental Study
We have tested Mandy to assess whether it is able to perform twining as expected, by relying on only one tendon and as a response to a touch stimulus.In this section, we first provide the sensor readings obtained when the contact occurs on the different air chambers, to further discuss the validity of our design approach.Then, we show the result of the implemented actuation scheme.

Touch Detection
The validation of the design of the chambers consisted of assessing whether the voltage variations read by the sensor were constant across chambers, that is, that the sensor signal for a deflection y at any point j along the arm did not depend on the position of the force causing this deflection.For this scope, we made Mandy perform planar bending by displacing its tip and marking on a board the points reached by each of the chambers (see Figure 9a).In this way, a point P i was associated with the i-th chamber.Then, in the subsequent tests, the i-th chamber was touched and displaced to make it reach P i (Figure 9b,c).The contact was performed manually, using a metal rod with hexagonal cross-section.Each time, the sensor reading was recorded before the touch occurred, to obtain a reference value V 0 i , and after the displacement was complete, to obtain V f i .The difference ΔV i ¼ V f i À V 0 i is here reported.For each chamber, the test was repeated five times.The results are shown in Figure 9d, in terms of mean value and standard deviation.Although the ΔV still varies across the sensor chambers, the difference has been significantly reduced with respect to an approach where sensor chambers with uniform wall thickness were adopted.The difference in ΔV can be explained by the assumptions in the model presented in Section 2.2.2, the measurement and fabrication inaccuracies, and the unaccounted-for influence of the bending deformation on the pressure inside the air chambers that are not being touched.These limitations, and how they can be overcome, will be discussed later.Finally, a simple control system was implemented where twining is initiated at a ΔV of 0.08 V.

Twining Behavior
Here, we show that Mandy performs twining as predicted by our kinematic model.For this test, the tendon has been deployed to span an angle of 30°within each segment (the reason for this choice will be motivated in the next section).Figure 10 shows how Mandy progressively deforms as the tendon is pulled.The configuration in Figure 10e resembles the one plotted in Figure 8c, as Mandy initially leans toward the support (Figure 10a) and then its distal segments gradually embrace the support (Figure 10b-d), eventually bringing its tip behind it.The diameter of the wooden support is %60 mm.

Discussion and Conclusions
While continuum arms are a topic of interest in robotics, related works rarely focus on their structural design in a quantitative way.Most of the effort, so far, has been dedicated to the development of efficient control strategies, the investigation of novel actuation patterns, and the modeling of their kinematics and dynamics.However, as previously discussed, one of the drawbacks of continuum robotic arms is that they often require bulky, heavy actuation hardware, along with all the electronic components needed for sensing and control.For example, tendon-driven continuum arms that can perform twining are often actuated by 3⋅M tendon, where M is the number of modules of the robot.Here, we suggest including an initial structural design stage to build continuum robotic arms showing complex behavior with reduced cost in terms of actuation and control.
As an example, we have investigated the advantage of providing the continuum robotic arm with a biomimetic distribution of flexural rigidity.Our study is based on the consideration that twining plants demonstrate remarkable capabilities such as exploring their surroundings, finding a suitable support and twine around it, despite having no brain.Such capabilities are related to the structural properties of their stems.Despite the different actuation mechanisms adopted by plants and by our robot (plants are not tendon driven), the properties of Mandevilla cf.splendens stem have turned out to be conveniently reproduced in a twining robot with only one actuator and one pressure sensor.Moreover, in our robot, the motor that pulls the tendon can be encapsulated inside the body of the robot (inside the first segment, if large enough), and the sensing does not require a control board.The wavy rings that compose the segments result in the presence of voids that allow for a lighter structure, while ensuring the arm's standability and sufficient friction between the robot and the external object, also due to the employed material (a resin).As a result, the robot is very compact and can be easily integrated into another robotic system (e.g., a mobile platform or as an end effector of another manipulator).Although other authors have highlighted the advantages of implementing helical routing in addition to straight tendons, [38] in Mandy's case, a class of deformed configurations is obtained using only one helical tendon and no straight tendons.The dexterity and the possibility to perform other movements (such as planar bending or assuming a particular nonplanar configuration) are sacrificed here to explore the capabilities that can be achieved with minimal hardware.Some readers will see in Mandy an example of physical intelligence [30] or of morphological computation. [39]Indeed, the reduction of complexity of the control has been achieved here at the cost of a greater effort in designing the robot's body.
Rather than having seven uniform sensors along the length of the soft continuum arm and a central computer that solves Equation ( 10)-( 16) to determine the twining activation voltage threshold for each of the individual sensors, this computation is embedded in its physical attributes (the tuned thickness of the chambers) and the interaction with the environment.Hereby, the need for a microcontroller and six air pressure sensors (and their corresponding tubes and cables) was circumvented.
Moreover, our investigation shows that an arm with a different structure would not be able to perform twining as Mandy demonstrates.Figure 8b reveals that a single helically deployed tendon would generate nonplanar bending in any case, but not twining, and not at the same energy cost (as we mentioned, in all the simulations, the tendon is pulled by the same force).Energy consumption is indeed another frequently neglected aspect of continuum arms.As living beings, plants tend to spend their resources very carefully, as their primary purpose is survival.This consideration about energy-related issues represents another significant motivation for roboticists to draw inspiration from nature to design the next generation of robots.The twining behavior is certainly dependent on the helical path followed by the tendon.To twine around cylindrical objects with a diameter in the range of 2 ÷ 3 times Mandy's diameter, we recommend ϕ ¼ 20°÷ 60°.Too small (<20°) or too large (>60°) values produce deformations that could be hardly used in common robotic applications (see Figure 8d-g), such as grasping.Interesting behaviors are also obtained by varying the angle among segments.For example, the robot shown in the Supplementary Video performs twining having a routing angle of 30°in its first three segments and of 60°in the remaining ones.The interaction with the support when the tendon is pulled does not compromise the integrity of the robot's body; on the contrary, twining is achieved also thanks to this interaction, as the underactuation allows the robot to adapt to the shape of the support.

How to Improve Mandy's Performance
As the first of its kind, Mandy has significant potential for improvement in multiple aspects.First, Mandy's body is segmented, meaning its rigidity varies discretely along its length.To effectively implement the distribution of flexural rigidity, each wavy ring should have a different thickness (as far as the 3D printing technique allows for the necessary resolution).By varying the flexural rigidity continuously, Mandy's body would perform better in twining.In the current design, each segment is joined to the adjacent ones by a relatively rigid interface (stiffer than the wavy rings), which limits the deformation.Without this limitation, Mandy would achieve smaller curvature radii along all its body, enabling it to twine around supports with smaller diameters.With the current design, Mandy can only partially wrap the supports used for our tests (see Figure 10).Although the number of loops could be increased by increasing the length of the robot, an excessive length could compromise the standability of the robot, hence complicating its maneuvers and operability.
Mandy's segments have been designed under strong assumptions: linear beam theory has been used, with an approximate bending moment.Additionally, it was assumed that, during bending, the segment maintains a hollow circular cross section.However, a finite-element analysis would reveal that the deformation mode is more complex: as the tendon is pulled, the cross section tends to warp, and the assumption of constant curvature becomes less reliable.It is worth noting that this assumption of constant curvature was adopted here because of Mandy's standability, a concept found in botany as well. [40,41]This means that the gravitational deflection of Mandy is sufficiently smaller than the bending generated by the actuation force, preventing the robot from buckling under its own weight.Hence, the stiffness-to-weight ratio is a crucial factor to consider in the design process of a Mandy-like robot.This ratio can be increased, for example, using a different material or a different additive manufacturing technique.For instance, filament deposition modeling (FDM) can produce relatively stiff segments with reduced density compared to SLA, if a low infill density is set.
If FDM is chosen for manufacturing, the air chambers may need to be produced separately using a technique that ensures their integrity and continuity to prevent air leakages.This is why, for this work, we opted for SLA.However, an important aspect to improve Mandy is the design of the chambers, which requires more accurate modeling than that presented in this article.While our approach yielded good results for the chambers in the distal segments of the robot (see Figure 9d), the thickness of the proximal chambers should be better tuned to enable positionindependent sensing, as described earlier.In future works, the chamber's wall should be modeled as a nonlinear membrane, and any imperfections introduced during the fabrication process should be taken into account.Additionally, the results in Figure 9 were obtained for a large deflection of Mandy, which may introduce some errors.Another crucial consideration in the design of the air chambers is that the softer the chambers, the smaller the contact force that the robot can detect.
In summary, although it is commonly believed that soft robots are challenging to model and design, Mandy demonstrates that modeling techniques commonly used in linear structural analysis (such as linear beam theory) can still be successfully applied in soft robotics, serving as a valuable tool for designers.Despite the strong modelling assumptions mentioned above, Mandy is able to perform twining as predicted by our model.A key lesson from this work is to design soft robots in a way that allows them to undergo large displacements but experience small strains, thus avoiding material nonlinearities (if the used material can be treated as linear for small strains).This is precisely what happens in Mandy: each of its wavy rings deflects significantly under the action of the pulled tendon, while strains remain low, locally, throughout Mandy's body.

Biomimetic Robots and Robots for Biology
Regardless of the adopted biological model, our goal is to permeate artificial systems with the remarkable capabilities and versatility observed in their biological counterparts.However, replicating these features is not a trivial matter.While the relation between a bioinspired robot and its related living organism is always evident intuitively, it might be challenging to assess the convenience of a bioapproach compared to traditional engineering methods.This difficulty may stem from two main factors: biological systems are inherently complex, making it not always easy to isolate and accurately extract the key features and principles we wish to translate into our technology. [42,43]dditionally, roboticists often design robots for specific tasks or applications, while biological systems must demonstrate great adaptability to a variety of conditions, making them less effective for a low number of goals as typically done in engineering.As shown, in the case of climbing plants such as Mandevilla, understanding the key features is further complicated by the interaction among stems of the same plant, which form braids.As mentioned earlier, Mandy is designed solely based on the properties of the main stem (S1) of the braided structure considered in this study.This choice is justified by the biomechanical characterization results (Figure 3).When considering segment n. 6 of the main stem (S1), it intertwines with the apical portion of S2; however, S2's contribution to the overall flexural rigidity of the braid is minimal.It is reasonable to assume that S2 benefits from the braid, while representing an additional load carried by S1.
Segments n. 4 and 5 of S1 are braided with both S2 and S3, but even in this case, the presence of S2 and S3 does not lead to a significant increase in the total flexural rigidity compared to S1 taken individually.Interestingly, in Mandevilla, different stems may come from different directions to converge to the braid.For example, in the considered sample, S2 and S3 form a two-braided structure before joining S1.Along this intertwining, S2 and S3 exhibit flexural rigidities of similar magnitude (despite S2 appearing as the youngest stem of the braid), and therefore, they mutually benefit from their interaction.By joining with S1 and braiding with it, S2 and S3 seem to find a reliable support from which they can continue growing.
Overall, S1 does not gain a significant direct mechanical advantage from the formation of the braid, as its distribution of flexural rigidity is almost equal to that of the braid: segment n. 6 of S1 has a rigidity equivalent to 8.84% of the value found for segment n. 1.This percentage increases to 9.30% for the braided portion, which is a negligible difference.The increase in flexural rigidity that S1 pays for sustaining the mass of S2 and S3 on its own stem is minimal.If Mandy was designed based on the properties of the braid instead of just S1, we would observe the same mechanical behavior described above, with small differences in the deformation being neglected.Since stems come from different directions to converge and form a braid, there is another important mechanical aspect to consider: while S1 may not directly benefit from the braid in terms of flexural rigidity, it could still gain an advantage owing to the tensile action exerted by all the individuals of the braid.In other words, the point at which braiding occurs may maintain its reached height, thanks to such action rather than to an increase of flexural rigidity.This may be a way by which the plant generates a sort of self-support to reach further objects.Another significant factor to consider is friction: stems may or may not slide on each other even after the braid formation is established, and this influences the overall mechanical behavior.As this condition can also change over time, it is not easy to assess how (and if ) it affects the plant.Certainly, such tensile behavior induces a parallelism with tensegrity robots (see, e.g., other studies [44] ), besides with continuum arms.Possibly, robots like Mandy can be used to better understand twining plant's behavior in terms of tropisms and twining capabilities.Similarly, tensegrity robots can help biologists investigate plant mechanics.This two-way connection between biology and robotics is exemplified in Figure 11.A biomimetic artificial system certainly brings the advantages observed in its natural counterpart; at the same time, it can be purposely provided with, or deprived of, a particular feature that is found in the living being, allowing for experiments aimed at validating or discarding biological hypotheses in an approach known as reverse biomimetics. [45,46]In conclusion, the benefits of biomimetics can be twofold: for robotics, it helps build better robots, and for biology, it provides deeper insight into natural organisms.

Figure 1 .
Figure 1.Braided stems of Mandevilla cf.splendens in the greenhouse, a) three days before it was cut, b) just before the cut, and c) after the cut.

Figure 2 .
Figure 2. Values of flexural rigidity obtained by three-point bending tests.Each of the three lines represents a stem in a braided structure composed of three.The main stem is denoted by n.1.The dashed rectangles enclose braided stems, with the flexural rigidity resulting from the braiding reported below. k

Figure 3 .
Figure 3. Normalized flexural rigidity observed in Mandevilla stem S1.The trend is described by means of a fourth-degree polynomial.

Figure 4 .
Figure 4. Drawing of Mandy's body and of the air chambers for tactile sensing.The size of the wavy rings decreases from the base to the apex of the robot.The same holds for the wall thickness of the air chambers.All the chambers are connected to the same silicone tube and hence constitute a single touch sensor.

Figure 5 .
Figure 5.The bending stiffness of the segment (left) is estimated considering the path depicted by the dashed line as a series of beam subject to a vertical load (center), that results in the bending of the segment (right).

Figure 6 .
Figure 6.Schematic representation of the design principle of the air chambers in Mandy.The deflecting lines represent the longitudinal axis of the robot, which bends under F TIP , applied at the apex, producing a deflection y x j À Á at x j , where the j-th chamber is located.We find the force F j applied at x j that produces the same deflection, and we tune the thickness of corresponding air chamber to obtain the same pressure variation that would be detected by an air chamber located at the tip under the action of F TIP .The same is performed for all the chambers.

Figure 7 .
Figure 7. Robot in undeformed straight configuration along the vertical axis of the global system of coordinates (left).The local system of coordinates for the generic segment is shown together with the notation used to describe the planar bending of the segment (bottom right).The phase between the tendon routes in adjacent segments is denoted by ϕ i (top right).

Figure 8 .
Figure 8.For comparison, a) four continuum arms with different distributions of flexural rigidity are considered to simulate the mechanical response under b) the actuation of a helically deployed tendon with ϕ ¼ 22.5°.c) Helical routing in Mandy with ϕ ¼ 30°produces a progressive deformation, in which each segment is depicted with a different color; d-g) the results for ϕ ¼ 10°, 22.5°, 45°, and 60°.

Figure 9 .
Figure 9. a) When Mandy is bent by displacing its tip, its air chambers reach the points marked in red.In the tests, each chamber is brought to its corresponding point, as visible in b) for chamber n.7 and in c) for chamber n.3.Each test is carried out five times, to investigate repeatability: the mean value and the standard deviation of the sensor readings for each chamber are in d).

Figure 10 .
Figure10.From a-e), Mandy's progressive deformation as the helical tendon is pulled.

Figure 11 .
Figure 11.Biology provides inspiration to robotics; similarly, robotic artefacts can be used to validate biological hypotheses.