A bi-stable soft robotic bendable module driven by silicone dielectric elastomer actuators: design, characterization, and parameter study

The basic structure of a DE corresponds to the one of a parallel-plate capacitor with flexible electrodes and dielectric [36]. By applying high voltage to the electrodes, attractive Coulomb forces are generated between the charges of opposite sign on the two electrodes. As a result, the thickness of the DE decreases and, due to the incompressibility of the elastomer, the membrane surface area increases at the same time. Both thickness reduction and area expansion can be used to generate actuation.

Due to their high flexibility, DEAs can be designed in many different geometries. A popular design is represented by strip-in-plane DEAs (SIP-DEAs) [31]. The basic working principle of this type of actuators is shown in figure 1(a). SIP-DEAs can be used for applications in which a flat layout is preferred, and can also easily stacked on top of each other to increase the resulting work output [37].

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In soft robotics, principles from nature are often taken as a role model. Therefore, a reasonable approach when designing DEAs for soft robotic applications would consist of using biological systems as a guide. As mentioned in the introduction, DEs are often associated to artificial muscles. The structure of human muscles, made of bundles of several fibers, serves as a design template for the RDEA used in this work. Their basic working principle is the same as for SIP-DEAs, but here the dielectric membrane and the electrodes are tightly wound in a cylindrical shape, see figure 1(b). In this case, the high voltage activation produces a reduction in roll diameter and, in turn, an axial expansion. RDEAs can be designed with a very slim aspect ratio (diameter < 4 mm, length > 50 mm), can be easily integrated in soft robotic structures, and can be eventually bundled together to increase the total force [34]. While working as artificial muscle-like actuators, RDEAs can be simultaneously used as nerve-like sensor fibers. In fact, a linear relationship exists between their mechanical deformation and the resulting electrical capacitance [34]. This capacitance-based deformation estimation can be implemented during high voltage actuation, thus realizing the so-called self-sensing operating mode. This unique property is a further advantage of DEs in soft robotic applications, since one does not require additional sensors for controlling the robot in closed loop.

The RDEAs considered in this work are manufactured based on the process presented in [34]. Rectangular electrodes (53 mm × 58 mm) with an extension for electrical contact on one side, consisting of carbon black particles suspended in a mixture of PDMS, are screen printed on a 50 μm thin silicone film (Wacker Elastosil 2030). For the considered T-platform application, we restrict our investigation to this single electrode size. In this way, we were able to develop a special manufacturing rig which allows us to systematically manufacture four identical RDEAs at the same time. After screen printing, the films are cured in an oven. The electrodes are then tested with a voltage of 3900 V, to check for defects. If this test is passed, two electrodes with a mirrored layout are cut out and stacked on top of each other. In the subsequent step, the still planar silicone-electrodes sandwich is tightly rolled. To implement the electrical contacts, ferrules are crimped to the ends of the RDEAs and, to simplify the mounting, a nylon rod with a M2 thread is inserted at the free end of each ferrule. In order to supply a high voltage to the actuators, the RDEA ferrules need to be connected to a high voltage amplifier. This is done by screwing a cable shoe with a nut next to the ferrule on the threaded rod. A picture of the final RDEA prototype is shown in figure 2, while its electro-mechanical characterization curves are shown in figure 3 (obtained with the same experimental setup described in [34], in here a small viscoelastic hysteresis can be observed).

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In order to achieve a meaningful stroke, a DEA must be pre-tensioned with a suitable biasing system. In the simplest case, the biasing mechanism can be obtained via a constant load (i.e. a mass) or, more commonly, via a linear bias spring (LBS). A simple graphical method, based on a force equilibrium analysis, can be used to predict the stroke of a DEA based on the force-displacement curves of DE membrane and biasing element. An example of application of this method is shown in figure 4(a). In here, the blue and red lines correspond to the DE characteristics for low and high voltage, respectively, while the black line corresponds to the LBS (note that, even though the LBS is a standard Hookean spring, its slope appears negative in this diagram since it is plotted with respect to the DE strain). The intersections between bias and low/high voltage DE curves represent the equilibrium positions when the actuator is turned off and on, respectively. The horizontal distance between these points coincides with the achieved stroke. Given a pair of DE curves, one can freely design the biasing element to tune the stroke accordingly, based on this simple graphical method.

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However, note how the use of simple elements such as masses (constant forces) or LBS results in relatively small strokes (<10% actuation strain), compared to the deformation range achievable by the material. More complex DEA layouts make use of bi-stable biasing elements, such as pre-compressed buckled beam [31], to increase the actuation stroke. The bi-stability of such mechanisms results into a region of negative stiffness (i.e. slope) in their force-displacement characteristics, thus they are commonly referred to as negative bias spring (NBS). If this negative stiffness is properly matched with the positive stiffness of the DEA membrane, a very large stroke (>45% actuation strain) can be achieved, as shown figure 4(b) (also in this case, the negative stiffness of the NBS appears positive in this diagram, since its force is being plotted with respect to the DE strain). It is worthwhile mentioning how, up to date, NBS-based concepts for stroke magnification have only been applied to single-DoF DEA systems, where their design can be effectively performed graphically as described above. The desired stroke and force for the application can be set by adapting the stiffness of the LBS, or in the case of NBS the buckled beam parameters, in such a way that the force-displacement characteristic of the biasing system fits with the desired requirements into the characteristics of the DEs. For more complex DEA systems such as robots, in which several actuators are used at the same time to achieve a multi-DoF behavior, simple one-dimensional graphical methods can no longer be applied for designing the biasing elements. As a result, the high potential of DE material turns out to be highly underutilized in current DE soft robotic applications. A concept for transferring this bi-stability induced stroke magnification to multi-DoF DE systems, and demonstrating it through experiments, represents the main goal of this manuscript.

A conceptual sketch of the studied T-platform is shown in figure 5. It consists of three main components, i.e. a flexible vertical backbone, a rigid top plate, and the RDEAs, see figure 5(a). As for many soft robotic concepts, the nature is partly used as a role model for the T-platform operating principle. In the human upper arm, flexor and extensor work complementary as an agonist-antagonist pair [38]. In our system, the basic idea consists of using two RDEAs to replicate the role of the agonist-antagonist muscle pair. The RDEAs are mounted in a pretensioned state, with one end connected to the rigid top plate of the backbone while the other end is fixed on the base plate. Figure 5(a) shows the setup in its neutral position. Due to the pretensioning of the RDEAs, a compressive force acts on the top part of the flexible backbone. If a voltage is now applied to one of the actuators, the RDEAs stiffness is reduced and, as a result, the actuator force decreases accordingly. The resulting force imbalance generates a moment, which causes the backbone to bend in-plane towards the opposite direction. The motion obtained when activating the right and left RDEAs is shown in figures 5(b) and (c), respectively. Many of such T-platforms can be eventually stacked on top of each other, resulting into a soft tentacle arm.

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One of the advantages of the proposed T-platform is represented by the possibility of exploiting the beam as a bi-stable biasing element for stroke magnification. In fact, by properly tuning the RDEAs pretensioning, the normal force acting on the beam can be made slightly larger than the buckling load. In this state, the force change due to the electrical activation of a RDEA is sufficiently large to trigger the beam buckling instability, thus causing it to snap and, in turn, produce a large bending displacement. For a detailed theoretical analysis of this phenomenon, the reader may refer to [35]. By exploiting this effect, it is possible to generalize the bi-stability induced stroke magnification concept, extensively used in case of simple DEAs [31], to the considered class of multi-DoF DE robotic structure. In this way, the actuation performance can be dramatically increased given the same amount of active material and input voltage.

The T-platform is characterized by several free design parameters, which strongly affect its displacement performance. Those include the geometry of both beam and rigid plate, as well as the size and attachment points of the RDEAs. Not only those parameters are critical for determining whether the bi-stable actuation can be made possible, but can be also used to further fine-tune the resulting displacement range. A systematic understanding of the relationship between free parameters and resulting motion range is essential for the design of high-performance DE soft robotic systems. Since the system is characterized by a large number of design parameters, it is inconvenient to attempt optimizing all of them at the same time. In here, we will discuss a suitable subset of free design parameters for practical design optimization, whose effects on the T-platform performance will be investigated in subsequent sections.

The first and most relevant parameters are represented by the ones related to the RDEAs geometry. In fact, the shape of the RDEA force-displacement characteristics strongly depend on their geometry as well as their manufacturing process [34]. Since the relationship between RDEA geometry and their force-displacement behavior was already investigated in [34], it will not be further studied in this work. Conversely, for design convenience, we will focus on a specific actuator geometry that can be manufactured in a highly reliable way (see section 2.2).

In addition to the RDEAs, the flexible backbone also plays a fundamental role in determining the system bending angle. The most important parameter describing the backbone flexibility, namely the bending stiffness, strongly depends on the chosen material and beam geometry [35]. A rectangular shape is considered for the cross section, to allow exploiting known results from classic beam theory for quantitative performance predictions. This results in four main free parameters for beam, namely cross-sectional thickness, cross-sectional width, length, and material (i.e. Young's modulus). Since all those parameters affect the beam bending stiffness in a lumped way, it is sufficient to change only one of them for our study. For practical design reasons, we decided to only investigate the effects of the beam thickness on the overall bending stiffness, leaving its other geometric parameters (length, width) and beam material constant.

Further free parameters result from the assembly of the various components. The attachment points of the RDEAs on the top and bottom plates strongly affect the direction along which the actuation forces are transmitted to the flexible structure, as well as the RDEA pre-stretch. Therefore, it is expected that such points have a meaningful impact on the bending performance. Since the structure needs to be symmetrical, the two attachment points on the left- and right-hand sides are selected in an equal manner. The horizontal distances from the attachment points of the RDEAs with respect to the beam top and bottom points are then chosen as further main parameters to investigate. The vertical positions of the RDEA attachment points, as well as the beam length and width, are fixed in a heuristic way compatibly with the size of the actuators discussed in section 2.2, in order to ensure a RDEA strain of about 50% in the initial vertical position (since higher values might damage the RDEAs).

To conclude this section, we report in figure 6 a summary of all the free quantities considered for the parameter study.

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In order to obtain a preliminary assessment of the effects of the free design parameters (i.e. upper and lower attachment points, beam thickness) on the bending performance of the T-platform, a model-based study is conducted in this section. This model allows us to establish a framework for the basic design and understanding of the bendable module and its associated theoretical performance. In this way, suitable parameter ranges can be evaluated for the subsequent experimental testing campaign.

As a first step, we summarize a physics-based, quasi-static model of the considered soft robotic system. The result is grounded on the energy-based modeling framework we previously developed for the T-platform in [35], which is here updated with our most recent version of the RDEA model [39] as well as with a novel kinematic description which accounts for the actual structure of the novel experimental setup. Figures 7(a) and (b) depict a schematic representation, the T-platform in both undeformed and deformed configurations, respectively. In here, all the relevant system component and geometric parameters are also highlighted. The model essentially comprises three main elements of the T-platform, namely the rigid top plate, the flexible beam, and the RDEAs. For simplicity, we assume that the geometric state of the beam and RDEAs is uniquely determined once the configuration of the top plate is fixed (see [35] for details). For a planar motion, this allows us to completely describe the system with the following vector of generalized coordinates

$$\begin{equation}q = \left[ {\begin{array}{*{20}{c}} {{q_x}} \\ {{q_y}} \\ \alpha \end{array}} \right],\end{equation} \tag{ 1 }$$

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whose components correspond to the horizontal and vertical displacement of the beam tip, as well as the orientation of the top plate, respectively (cf figure 7(b)). According to Lagrangian mechanics, the system equilibrium configurations are given by the solutions of the following system of equations

$$\begin{equation}\frac{{\partial \mathcal{V}}}{{\partial q}} = 0,\end{equation} \tag{ 2 }$$

where ν is the total potential energy of the system. In our case, ν can be expressed as the sum of three sub-components, i.e.

$$\begin{equation}\mathcal{V} = {\mathcal{V}_{{\text{TP}}}} + {\mathcal{V}_{\text{B}}} + {\mathcal{V}_{{\text{DE}}}},\end{equation} \tag{ 3 }$$

representing the gravitational potential energy of the top plate, the elastic potential energy of the beam, and the electro-mechanical potential co-energy of the RDEAs, respectively. ν_TP and ν_B can be expressed as a function of q as follows [35]

$$\begin{equation}{\mathcal{V}_{{\text{TP}}}}\left( q \right) = {m_{{\text{TP}}}}g\left( {{l_{\text{B}}} + {q_y} + {b_{{\text{top}}}}\cos \alpha } \right),\end{equation} \tag{ 4 }$$

$$\begin{equation}{\mathcal{V}_{\text{B}}}\left( q \right) = \frac{{{E_{\text{B}}}{I_{\text{B}}}}}{{2{l_{\text{B}}}}}\left[ {{q^{\text{T}}}{S^{\text{T}}}GSq + \frac{{{{\left( {\frac{{{q_y}}}{{{l_{\text{B}}}}} + \frac{1}{2}{q^{\text{T}}}{S^{\text{T}}}PSq} \right)}^2}}}{{\frac{{{I_{\text{B}}}}}{{{A_{\text{B}}}l_{\text{B}}^2}} - {q^{\text{T}}}{S^{\text{T}}}QSq}}} \right],\end{equation} \tag{ 5 }$$

where

$$\begin{align*} S &= \left[ {\begin{array}{*{20}{c}} {l_{\text{B}}^{ - 1}}&0&0 \\ 0&0&1 \end{array}} \right],\;\;\;{} G = \left[ {\begin{array}{*{20}{c}} {12}&{ - 6} \\ { - 6}&4 \end{array}} \right], \\ P &= \left[ {\begin{array}{*{20}{c}} {\frac{6}{5}}&{ - \frac{1}{{10}}} \\[4pt] { - \frac{1}{{10}}}&{\frac{2}{{15}}} \end{array}} \right],\;\;\;{} Q = \left[ {\begin{array}{*{20}{c}} { - \frac{1}{{700}}}&{\frac{1}{{1400}}} \\[4pt] {\frac{1}{{1400}}}&{ - \frac{{11}}{{6300}}} \end{array}} \right], \end{align*}$$

m_TP is the rigid top platform mass, g is the gravitational constant, l_B is the undeformed beam length, E_B is the Young's modulus of the beam material, I_B is the beam second moment of area, and A_B is the beam cross-sectional area. By assuming a beam with a rectangular cross-section with width w_B and thickness t_B, we can simply compute the last two beam parameters as I_B = w_B t_B ³/12 and A_B = w_B t_B. In order to define the DE electro-mechanical energy, we first need to define two additional quantities, i.e. the vector of DE voltages v_DE

$$\begin{equation}{v_{{\text{DE}}}} = \left[ {\begin{array}{*{20}{c}} {{v_{{\text{DE}}1}}} \\ {{v_{{\text{DE}}2}}} \end{array}} \right]\end{equation} \tag{ 6 }$$

as well as the vector of DE lengths l_DE (cf figure 7(b))

$$\begin{align} {l_{{\text{DE}}}}\left( q \right) &= \left[ {\begin{array}{*{20}{c}} {{l_{{\text{DE}}1}}\left( q \right)} \\ {{l_{{\text{DE}}2}}\left( q \right)} \end{array}} \right],\nonumber \\ {l_{{\text{DE}}1}}\left( q \right) &= \left\| {\left[ {\begin{array}{*{20}{c}} {{q_x} - {b_{{\text{top}}}}\sin \alpha - c\cos \alpha + a} \\ {{l_B} + {q_y} + {b_{{\text{top}}}}\cos \alpha - c\sin \alpha - b} \end{array}} \right]} \right\| - 2{l_{{\text{cr}}}}, \nonumber \\ {l_{{\text{DE}}2}}\left( q \right) &= \left\| {\left[ {\begin{array}{*{20}{c}} {{q_x} - {b_{{\text{top}}}}\sin \alpha + c\cos \alpha - a} \\ {{l_B} + {q_y} + {b_{{\text{top}}}}\cos \alpha + c\sin \alpha - b} \end{array}} \right]} \right\| - 2{l_{{\text{cr}}}}, \end{align} \tag{ 7 }$$

where subscript i refers to RDEA i, i = 1, 2, l_cr takes into account the length of the passive crimps connecting the RDEAs to the T-platform pivot points, and all the other geometric parameters are reported in figure 7. Then, based on (6), (7), the total co-energy of the RDEAs can be expressed as follows [35, 39]

$$\begin{align} & {\mathcal{V}_{{\text{DE}}}}\left( {{l_{{\text{DE}}}}\left( q \right),{v_{{\text{DE}}}}} \right) \nonumber \\ & \quad = \sum\limits_{i = 1}^2 \sum\limits_{j = 1}^3 {\Bigg[ { {{L_1}{L_2}{L_3}{C_{j0}}{{\left( {\frac{{{l_{{\text{DE}}i}}\left( q \right)}}{{{L_1}}} + 2\sqrt {\frac{{{L_1}}}{{{l_{{\text{DE}}i}}\left( q \right)}}} - 3} \right)}^j}} } } \nonumber \\ & \qquad\qquad\qquad - \frac{1}{2}\frac{{{\alpha _{e1}}{\eta _b}{\in _0}{\in _{\text{r}}}{L_2}{l_{{\text{DE}}i}}\left( q \right)}}{{{L_3}}}v_{{\text{DE}}i}^2 \Bigg], \end{align} \tag{ 8 }$$

where L₁, L₂, L₃ are the RDEAs principal lengths in undeformed configuration, C_{j 0} are Yeoh parameters describing the DE material hyperelastic response, j = 1, 2, 3, α_{e 1} an η_b are volumetric loss factors of the RDEA, ε₀ is the vacuum permittivity, and ε_r is the DE relative permittivity. The complete set of equations describing the T-platform steady-state response can be obtained by replacing (4), (5), (7), (8) in (3) (explicit computations are straightforward but cumbersome, thus they are omitted here for conciseness). In this way, we can compute the system equilibrium configurations resulting from the application of a given pair of input DE voltages.

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The developed model is then used to perform some initial parameter studies, aimed at understanding the trends of how the bending angle α varies as a function of the free design parameters shown in figure 6, i.e. upper RDEA attachment point c, lower RDEA attachment point a, beam thickness t_B. All the other model parameters are assumed to be known in advance. In particular, the beam length and width, as well as the constant geometric parameters of the structure, are fixed in a heuristic according to the guidelines provided at the end of section 3.2. The DE material parameters are obtained based on the RDEA characterization in figure 3, while the beam Young's modulus and the top-plate mass are determined based on known properties of available 3D printing materials (i.e. PETG [40], Rigid 10 K Resin [41], see section 4 for more details). The numerical values used for all the constant parameters are summarized appendix. Simulations are then conducted by choosing different values of a, c, t_B, and the resulting performance of the T-platform is examined. In each case, only one RDEA is activated with a voltage of 3000 V, while the other one is kept at 0 V. For each combination of parameters, figure 8 shows the maximum deflection angle α and the corresponding maximal RDEA strain (whose computation is based on the roll axial stretch, according to [39]). From the results in figure 8, it can be seen that α increases for smaller distances between the actuators and the beam, independently of whether it is the top distance c or the bottom one a, and the same holds true for decreasing beam thickness t_B. The maximum bending performance, indeed, is obtained for c = 18 mm, a = 15 mm, and t_B = 1.35 mm, and is predicted to be of about 24°. Following the approach in [35], we analyzed the stability of the T-platform for this set of parameters, in both unactuated and actuated states. It is revealed that, for this specific combination of parameters, the unactuated T-platform is indeed bistable, while it is monostable when in its actuated configuration at 24°. The trends in figure 8 are consistent with the physics of the problem. In fact, thinner beams tend to buckle at lower compressive forces, and thus the resulting bending angle increases given the same RDEAs pre-tension. If the beam thickness is reduced too much, however, the normal stress becomes so large that activating the RDEA no longer causes a buckling to the opposite direction. In this case, not only the unactivated T-platform, but also the actuated one, turns out to be bi-stable. As a result, reversibility of the actuation is lost, and the design becomes no longer effective. Placing the actuators closer to the backbone, instead, increases the bending angle because the same change in RDEA length results in a larger beam displacement, due to the stronger kinematic coupling among the two. The corresponding maximal RDEA strain diagrams, depicted in the lower part of figure 8, show that the parameter combination corresponding to maximum bending angle is close to 50%, which represents the maximum allowed strain for the considered rolls. Smaller beam-RDEA distances or beam thickness values would result in higher maximal strains, which would then exceed our safety limit value. Conversely, configurations with bigger beam-RDEA distances of beam thickness would result into monostable configurations which lead to a poor bending angle performance, and thus they are not relevant in practical applications. By using the model, one can find the parameters which maximize the bending performance while resulting into a functional T-platform design, i.e. the actuated configuration always exhibits only one stable equilibrium and the RDEAs must not be overstretched. Based on the above considerations and practical design trade-offs, we choose the following parameter ranges for the subsequent experimental investigation: distance between RDEA upper part and beam c ∈ [18, 30] mm; distance between RDEA lower part and beam a ∈ [15, 18] mm; beam thickness t_B ∈ [1.35, 1.5] mm.

As a final remark, we include some considerations on the maximum electric field, which is related to electrical breakdown in the DE. The electric field of RDEA i, i = 1, 2, is given by [39]

$$\begin{equation}{E_i} = \sqrt {\frac{{{l_{{\text{DE}}i}}\left( q \right)}}{{{L_1}}}} \frac{{{v_{{\text{DE}}i}}}}{{{L_3}}},\;\;\;{} i = 1,2.\end{equation} \tag{ 9 }$$

For the maximum DE length obtained in simulation, a peak electric field of about 73.5 V μm⁻¹ is computed. Since this is always smaller than the safety value of 80 V μm⁻¹ recommended by the elastomer manufacturer, we can consider the system suitable from an electrical viewpoint.

A custom test rig is assembled to systematically characterize the T-platform, and evaluate the performance achieved with different design parameters. For each combination of parameters, an experiment is conducted in which different periodic voltage signals are alternately applied to the left and right RDEA. Two high voltage amplifiers available in our lab (Hivolt HA51U, Ultravolt 4HVA) are used for this purpose, both of them compatible with the voltage/capacitance range of the RDEAs. During this process, a camera is used to record the motion of the module. The test rig allows to automatically detect the position of the T-platform, track its motion over time, and compute the resulting bending angle.

To make the setup as independent as possible from focus settings and camera position, those parameters should be automatically recognizable by (and account for) the motion tracking algorithm. To this end, the MATLAB 'Camera Calibrator' app is used to extract those parameters through at least two reference coordinate systems located in different planes. In order to recognize the coordinate systems during the video post-processing, three tilted plates with a chessboard pattern are placed in the test rig. Note that two of the chessboard plates are out of plane with the T-platform and one plate in plane with the T-platform. By means of an image recognition algorithm, it is possible to identify the individual corners of the chessboard, and use them to generate a coherent coordinate system. The two out of plane coordinate systems are used in the calibration app to get the camera settings. And the third one as reference coordinate system for the T-platform. The complete arrangement of T-platform and three chessboard plates is portrayed in figure 11(a).

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To further assist the motion tracking algorithm, three markers are placed on both the base and the top plate of the T-platform (i.e. two black dots and a vertical line for each element), also visible in figure 10. The marked positions on the T-platform can now be determined with respect to the reference coordinate systems. Once the positions of all the T-platform relevant points are obtained, their motion can be tracked in the video. In the last step, the bending angle can be computed from the positions of the moving points by using standard rigid body kinematics. The result of this tracking algorithm for the angle of deflection is shown in figure 11(b).

The developed test rig can be used to determine the relationship between the bending angle and the various parameters discussed in section 3. Other than just analyzing the angular displacement performance, it is also meaningful to investigate for which combinations of parameters the system is bi-stable, and how the bi-stability feature affects the maximum deflection angle. The performance of different prototypes are measured, by considering all the possible combinations of the following parameters (for a total of 32 experiments):

• Top DE distance c: 18 mm, 21, mm, 24 mm, 30 mm;
• Bottom DE distance a: 15 mm, 18 mm;
• Beam thickness t_B: 1.35 mm, 1.4 mm, 1.45 mm, 1.5 mm.

Note that the configurations corresponding to c = 27 mm are omitted, since they led to almost identical performance compared to the 30 mm case.

During the measurements, a square wave voltage signal with a duration of 2 s and a constant amplitude of 3000 V is applied alternately to the two actuators, in such a way that both rolls remain inactive for 4 s between activations. An example of such a signal is shown in figure 12.

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In the first step of the experimental campaign, the bending angle is determined over time according to the procedure described in section 5.1. Based on the time behavior of the angle, statements can already be made concerning the bi-stability of the system. To better clarify the meaning of bi-stability in the context of our system, figure 13 shows an example of time behavior of the T-platform bending angle, for both a mono-stable (a) and a bi-stable (b) configuration. The mono-stable case corresponds to c= 30 mm, a= 18 mm, and t_B = 1.4 mm. With this combination of parameters, the T-platform reaches a maximum deflection around 6°, and the system returns to the neutral position when the voltage is switched off. In contrast, an example of bi-stable case corresponds to c= 18 mm, a= 15 mm, and t_B = 1.35 mm. With this combination of parameters, the T-platform reaches a maximum deflection around 25°. After the voltage is switched off, the system does not come back to the neutral position (as for the mono-stable case), but rather settles to a deflection angle of about 22°. This behavior can be observed with respect to the actuation along both directions. The system therefore snaps from one stable equilibrium position to a second one when the voltage is applied, and is therefore bi-stable. Other than proving a larger bending angle, the bi-stable actuation also provides energy saving features, since it allows to maintain a non-zero bending angle without the need of supplying any electrical power. This might represent a further advantage in energy-critical applications, e.g. mobile ones.

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After clarifying the role of bi-stability in determining the T-platform time behavior, we discuss the results of the experimental study. Figure 14 shows the relationships between the attachment points and the resulting maximum bending angle, for all the considered combinations of parameter. Each experiment is represented with a circle, which are superimposed to continuous lines that represent the simulation results from section 3.3. As expected from the theoretical investigation, we can achieve a wide spectrum of possible bending angles, which range from 5° up to 25°. The experimental results also show a remarkable agreement with the simulated curves. Some deviations exist, mostly visible for the t_B = 1.4 mm case, which are reasonably due to manufacturing tolerances as well as due to the introduction of simplifying assumptions in the derivation of the beam model. Nevertheless, the trends are reproduced in a remarkable way, and the maximum bending angle of 25° was predicted by the model with an error of 1°. The obtained results confirm then the effectiveness of our bi-stable design principle, and confirm that our model represents a valuable numerical tool to systematically optimize the T-platform design.

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By inspecting the results, it is observed that bending angles larger than a given threshold value always correspond to a bi-stable behavior, independently of the choice of the beam and other parameters. Such a threshold is small for thinner beams, and becomes higher for thicker beams. As an example, in the case of the thinnest beam (t_B = 1.35 mm), the system is always bi-stable for bending angles when the bending angle is larger than 7.8°. With thicker beams, the system only becomes bi-stable at larger angles of deflection (11.2° for a 1.45 mm thick beam). Figure 15 summarizes the experimental results from figure 14 in two plots, and groups together the different configurations which result in a mono-stable (blue region) and bi-stable (red region) behaviors. From this plot, it is evident how the bi-stable configurations generally lead to much higher bending angles than mono-stable designs. This confirms that bi-stable biasing mechanisms permit to increase the actuation displacement not only for single-DoF DEAs, but also to multi-DoF DE soft robotic structures. Such a result represents then the first successful experimental verification of the bi-stable actuation concept initially theorized in [35].

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In conclusion, we have verified experimentally that, in order to maximize the actuation performance, one should move the RDEAs as close as possible to the beam and select a sufficiently thin beam. This combination of parameters will favor buckling which, in turn, leads to a large bending performance. The buckling, however, should not be favored too much, otherwise the actuation becomes irreversible, causing the system no longer to snap when the opposite RDEA is activated. The optimal trade-off between large bending angle and reversible actuation can be tuned via an appropriate choice of the free parameters, and can be eventually supported by the proposed physics-based model. Those finding will provide useful guidelines for the design of future soft tentacle arms based on stacks of many T-platforms.

To conclude this section, we investigate the suitability of the T-platform for soft robotic applications. To this end, we perform experiments aimed at evaluating the prototype compliance when subject to an external deformation, since this property is important to ensure a safe human-robot interaction. To measure the T-platform compliance, the setup shown in figure 16 is used. A steel rope (depicted in red) is wired parallel to the left RDEAs, the rope is connected to a load cell (ME-Meßsysteme KD40s) via pulleys. This load cell is mounted on a linear drive, whose displacement can be arbitrarily prescribed.

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In order to measure parameters related to the structure mechanical compliance, tensile experiments are carried out for both mono- and bi-stable configurations (mono-stable: c = 18 mm, a = 18 mm, t_B= 1.4 mm; bi-stable: c = 24 mm, a = 18 mm, t_B = 1.4 mm). By prescribing a deformation profile to the linear actuator, the structure bends accordingly. During this process, we acquire the force of the load cells, as well as the angle of the rigid spacer mounted at the top of the flexible beam. In order to extract a meaningful compliance measure from those data, we use standard rigid body dynamics to convert the forces to the equivalent moment applied to the center of mass of the rigid spacer. In this way, we can plot the moment as a function of the bending angle, and extract the angular compliance by measuring the slope of this diagram. The corresponding plots are shown in figure 17.

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For the monostable case, there is a linear relationship between the deflection angle and the torque. The corresponding angular stiffness is C_T = 0.33 × 10⁻³ Nm/°. In contrast, the bi-stable design results in a non-linear relationship with a region of positive stiffness (C_T = 0.35 × 10⁻³ Nm/°) and a region of negative stiffness (C_T = −0.15 × 10⁻³ Nm/°). This behaviour is equivalent to that of a mono- and bi-stable biasing mechanism with an LBS or NBS in single-DoF. For comparison, the angular stiffness of the human muscle in the elbow is in a range of C_Tmin = −0.35 Nm/° up to C_Tmax = 0.35 Nm/° [42]. Therefore, our T-platform is softer by a factor of 10³ compared to human muscles, thus confirming its suitability for soft robotics and safe human-robot collaboration.