Revealing bending and force in a soft body through a plant root inspired approach.

An emerging challenge in soft robotics research is to reveal mechanical solicitations in a soft body. Nature provides amazing clues to develop unconventional components that are capable of compliant interactions with the environment and living beings, avoiding mechanical and algorithmic complexity of robotic design. We inspire from plant-root mechanoperception and develop a strategy able to reveal bending and applied force in a soft body with only two sensing elements of the same kind, and a null computational effort. The stretching processes that lead to opposite tissue deformations on the two sides of the root wall are emulated with two tactile sensing elements, made of soft and stretchable materials, which conform to reversible changes in the shape of the body they are built in and follow its deformations. Comparing the two sensory responses, we can discriminate the concave and the convex side of the bent body. Hence, we propose a new strategy to reveal in a soft body the maximum bending angle (or the maximum deflection) and the externally applied force according to the body's mechanical configuration.


Supplementary Methods 1
The polydimethylsiloxane (PDMS) cylindrical body is not deformed by its own gravity. Indeed, the maximum height of a cylindrical column with density ρ and Young's modulus E standing under its own weight is 29 √ , where is the gravity acceleration, the second moment of area, and is the least positive root of the Bessel function of order -1/3. In the case of PDMS, = 965 kg/m 3 and = 1.8MPa. Moreover, the cylindrical body has mm, and it results cm. Therefore, given that the length of the soft body is 12 cm, gravity effects are not considered in our analysis.

Supplementary Methods 2
From an electrical point of view, the nominal capacitance of the single sensing element can be considered as the capacitance of two parallel electrode plates, as explained in where , and are the permittivity, the sensing area and the dielectric thickness, respectively. A natural logarithm is applied to equation (M1) to transform the product into a sum and the quotient into a subtraction, respectively, as follows The term represents the fractional change in the area due to the strain . The area is the product of the width and the length of the top electrode, then since ⁄ and ⁄ with the Poisson's ratio coefficient. Moreover, Then, the resulting fractional change in capacitance is represented by (M7) since we can consider that the permittivity does not vary when the strain is applied.
On the other hand, when an external force is applied, as discussed in the main text, the predominant effect is due to the change in the distance between the two electrodes, resulting If , then (M9)

Supplementary Methods 3
A theoretical model is presented in order to describe the mechanical and electrical behaviour of the soft sensing body when subjected to bending and/or force stimulations. In particular, some configurations of the body, representing typical mechanical stimulations are analysed; these being: (a) a cantilever and (b) an eccentrically loaded beam subjected to bending and buckling, respectively; (c) beam clamped at both extremities subjected to both force and bending.

3a. Bending of a cantilever beam
In this configuration, the body is subjected to a bending solicitation, and the aim is to correlate the capacitance variation of the sensing elements to the maximum bending angle.
From a mechanical point of view, a cantilever beam (as presented in Fig. 2a) with length and radius is clamped at one extremity and free to move at the other one; a force is applied at the beam free extremity, while the sensing elements S1 and S2 are positioned at the centre of the beam. The resulting deflection of the cantilever beam in the y-direction is given by where is the Young's modulus and the moment of inertia. The slope is determined from the first derivative of the deflection and is given by Therefore, the maximum deflection is represented by (M12) and the maximum slope is given by It is possible to assume that the predominant effect is represented by the uniaxial strain , given by where is the vertical coordinate of the considered surface with respect to the neutral axis. We can note that below the neutral axis the beam is stretched (and we have a positive strain), while above it is compressed (with a resulting negative strain). The yaxis, defining the deflection, is taken positive upward. Then, if the sensing element is positioned at the centre of the beam and on the lower surface, we have to consider and . The resulting strain  is positive and it is given by equation As previously explained in equation (M7), the nominal capacitance variation corresponds to the strain  to which the system is subjected. Therefore, the correlation between the capacitance variation and the bending angle results

(M16)
If angles are measured in degree, we have to introduce the factor /180 and the slope of equation (M16) is given by (M17) As we can note from equation (M14), in the case of a sensing element positioned on the upper surface of the beam ( ), the strain should be negative, and so the relative capacitance variation, with the same absolute value of the sensing element considered above (positioned on the opposite surface). However, the assumption that the sensing element follows the mechanical deformations of the beam surface is true only for positive strains, since the sensing element constituent materials can stretch conformably to the substrate. On the other hand, in the case of negative strain, the 6 materials are subjected to compression. By considering the thickness of the sensing element comparable (or even larger) to the beam curvature, abovementioned compression phenomenon results in mechanical deformations (i.e., wrinkles of the different constituent layers, as depicted in Fig.2a) which do not conform to the body surface. Therefore, from these considerations, we can understand that equation (M16) cannot be applied to a compressed sensing element.

3b. Buckling of an eccentrically loaded beam
In this configuration, we want to describe the behaviour of the soft sensing body when subjected to buckling due to a compressive load, and to correlate the nominal capacitance variation to the maximum deflection of the beam. In the general case of axial compressive load, the buckling of a beam occurs when the critical load ⁄ is reached. However, in many situations, the load is not perfectly axial, and an eccentricity between the load application point and the beam vertical axis is present. In this latter case, the buckling occurs even for very small compressive loads.
Consider a beam with length and radius clamped at both extremities, with an eccentricity between the beam vertical axis and the force application point, and the sensing elements S1 and S2 positioned at the beam centre (as depicted in Fig. 3a): the deflection of the beam in the y-direction is given by where √ . The maximum deflection occurs at as explained in The strain  on the stretched surface of the beam ( ) is given by and for , which relates the maximum strain to the compressive load (since √ ). By combining equations (M19) and (M21), the relation between the maximum strain and the maximum deflection can be obtained From an electrical point of view, the normalized capacitance variation can be correlated to the maximum strain, as explained in the following equation ) or, alternatively, by considering the maximum deflection , we can correlate the normalized capacitance variation to the compressive load, as described in equation Equation (M24) can also be written as follows where ⁄ . Then, expanding the function at the first order around as a Taylor's series, it is possible to obtain since, using de l'Hôpital's rule, we can find The result obtained in equation (M27) is valid for , which means that Finally, it is possible to obtain (M28) In the last case, we can observe that the strain, and consequently the capacitance variation, does not depend anymore on the eccentricity , but only on the geometrical dimensions of the beam. Also in this case, the above relations are verified only for a stretched sensing element.

3c. Bending of a beam clamped at both extremities
In this case, we want to describe the behaviour of a body clamped at both ends when subjected to both bending and force solicitation. The combination of S1 and S2 signals will let us establish which side is subjected to force, giving also its value, and the maximum deflection of the beam.
Consider a beam with length and radius clamped at both extremities (see Fig. 4a): the deflection of the beam, under a concentrated force in the middle of the beam (where the sensing elements S1 and S2 are positioned) and in the y-direction, for ⁄ , is given by equation (M29) (M29) The maximum deflection is represented by As previously, the predominant contribute is due to the uniaxial strain  given by In this case, the strain is positive above the neutral axis. Therefore, when the sensing element is positioned at the centre of the beam ( ) and on the upper surface In the case of a beam with a rigid but elastic core (i.e., a metallic spring) and a soft coating (i.e., rubber), the deviation to equation (M33) is very small, and we have 1, as shown in the section below. Otherwise, for body entirely made of a deformable material, such as PDMS, the corrective factor can differ significantly to 1 (in our experiments we found a value around 0.5).
Finally, for a sensing element positioned on the lower surface, a small negative capacitance variation should be observed in the case of pure beam buckling. However, in this case, a force is applied directly on it. As discussed in the main text, the force mainly varies the thickness of the sensing element therefore the resulting normalized capacitance variation corresponds to equation (M8).

Supplementary Data D1
In the following, we show the performance of our soft sensing body in terms of hysteresis.
In the case of a cantilever beam, we demonstrate that the response (normalized capacitance variation vs. angle) of the sensing element S1 is linear in the range 0°-60° with no hysteresis (as shown in Fig. D1). Figure D2 depicts the transfer curve (normalized capacitance variation vs. deflection) of the sensing convex side S1 in the configuration of an eccentrically loaded beam: the sensor response is linear in the deflection range of 0-8.3 mm, and it does not show hysteresis.

11
Finally, in the case of a beam clamped at both extremities, the S1 response (normalized capacitance variation vs. deflection) is linear up to a deflection of 1.32 mm, and the hysteresis is negligible, as shown in Fig. D3.
Supplementary Figure D1. Characteristics (normalized capacitance variation vs. angle) of the sensing convex side S1 in the cantilever configuration for a bending/unbending cycle in the range 0°-60°.
Supplementary Figure D2. Characteristics (normalized capacitance variation vs. deflection) of the sensing convex side S1 under a load/unload cycle in the eccentrically loaded beam configuration.
Supplementary Figure D3. Characteristics (normalized capacitance variation vs. deflection) of the sensing convex side S1 for a beam clamped at both ends when the system is subjected to both bending and force solicitations.

Supplementary Data D2
As shown above, in the case of a beam clamped at both ends, we need to introduce a corrective factor to equation (M33) for soft material bodies. To investigate the correctness of equation (M33), we made a module (with length of 120 mm and radius of 6 mm) composed of a metallic spring with a rubber coating. The length and the radius are the same of the PDMS body presented in the main text. In the same way, two stretchable capacitive sensors are placed at centre of the beam at 180° each other.
The characteristic of the sensing element S1 on the convex side of the body is shown in Fig. D4. We can note that the experimental data are consistent with the theoretical curve (dashed black line) of equation (M33). In particular, in this case we obtain 1, much closer to the unit with respect to the PDMS module. 14

Supplementary Data D3
Three experimental setups were built to investigate the three typical mechanical configurations (i.e., (a) cantilever and (b) eccentrically loaded beam subjected to bending and buckling, respectively, and (c) beam clamped at both extremities subjected to both bending and force) of the soft sensing body. They are depicted in the following figures.
Supplementary Figure D6. The schematic depicts the experimental setup (not in scale) used for applying a bending stimulation to the soft sensing body in the cantilever beam configuration and for acquiring data during the characterization (the soft sensing body subjected to such solicitation is shown in Fig. D9a). S1 and S2  subjected to buckling (the sensing body in this configuration is depicted in Fig. D9b). S1 and S2 are the sensing elements positioned at the centre of the soft body, at the convex and concave sides, respectively. A laser displacement sensor is used to measure the beam maximum deflection. The results of this experiment are shown in Fig. 4b-c of the main text.
Supplementary Figure D8. The schematic shows the experimental setup (not in scale) for applying a bending solicitation by means of an external force to the soft sensing body clamped at both extremities, together with the acquisition system employed during the characterization (the sensing body in this configuration is shown in Fig. D9c). S1 and S2 are the sensing 16 elements positioned at the centre of the soft body, at the convex and concave sides, respectively. The results of this experiment are shown in Fig. 5b-e of this work.
Supplementary Figure D9. Solicitations applied to the soft sensing body. (a) Illustration of the sensing body in the cantilever beam configuration with a concentrated force at the free end and the sensing elements S1 and S2 located at the beam centre. (b) Picture depicting the buckling of the soft sensing body with eccentricity between the beam vertical axis and the load application point, with the sensing elements S1 and S2 positioned at the centre of the beam. (c) Illustration of the sensing body clamped at both extremities with an external force applied at the middle and the sensing elements S1 and S2 positioned at the beam centre.