*Model Results*

By way of example, the results of the model are presented in the present paragraph in the case of a finger consisting of three modules 40 mm long each, placed in series and having a Polytetrafluoroethylene (PTFE) central rod, with the characteristics summarized in Table 1.


**Table 1.** Actuator main data.

When the actuator reached the theoretical equilibrium condition, the model gave the results summarized in Table 2.

> **Table 2.** Model results.


Figure 7 shows the final deformed configuration of the finger, both along the main directions (dotted line) and the secondary ones (solid line), considering the actuator axis in a rest condition laying along the Z-axis. This graph represents the deformation of the actuator axis on the deformation plane; this plane contained the axis of the actuator, and the forces exerted by the SMA wires and by the antagonist wires.

**Figure 7.** Axis of central rod of the actuator in equilibrium condition.

Figure 8 shows the projection of the deformation on a plane perpendicular to the actuator axis (X-Y).

As can be seen, and with reference to Figure 2, the greater deformations occurred along the three main directions, which, starting from the X-axis, were arranged to form three 120◦ angles between them. Instead, along the three secondary directions that were also arranged at 120◦ between them but 60◦ out of phase with the previous ones, the deformations were smaller. This was due to the fact that along the secondary directions, although there were two SMA wires activated at a time and there was only one antagonist wire, the arm of action of the force exerted by the two SMA wires was very small with respect to the arm of the force exerted by the antagonist wire. On the contrary, in the main directions, only one SMA wire was operated, but the arm of the force exerted by the SMA wire was bigger.

**Figure 8.** XY projection of the workspace.

## **4. Prototypes Designs**

In Table 3, the characteristics of the materials of three different realized prototypes are presented, and Figure 9 shows the prototypes.


**Table 3.** Prototypes characteristics.

**Figure 9.** Prototypes A with nylon central rod (**A**), B with PTFE central rod (**B**), and C with LIM centralrod and PEEK1000 disks (**C**).

Due to a Young's modulus of the nylon central rod of about 800 MPa, prototype A was characterized by a quite good flexibility. The advantage of this solution is that the material is inexpensive and its machine tooling process is simple, but the disadvantage is that its melting temperature, around 120 ◦C, is close to the temperatures reached by the activated hot wires. Some results of experimental testing on this prototype can be found in Maffiodo and Raparelli [32]. Prototype B, which had a central module made of PTFE, was then built. This material is more expensive, in particular with regard to tool machining. The advantages, however, are a similar Young's modulus, excellent flexibility of the structure, and a decidedly higher melting temperature.

Prototype C was created with the idea of separating the need to have a high melting temperature of the areas in contact with the SMA hot wires and the need to have a high flexibility of the central shaft. Then, the bases and intermediate disks were made with a technopolymer, PEEK1000, with a high Young's modulus and with a high melting temperature, while the flexible elements were made of silicone rubber (LIM) with a low Young's modulus.

#### **5. Experimental Tests Bench, Procedure, and Results**

The experimental set up, sketched in Figure 10, is composed of a flat square plate with four vertical rods at its edges. The prototype is clamped in the middle of the plate and four low-friction nylon wires are fixed to the actuator upper end. The opposite end of the wire is connected to a load with the aim of tightening the wire itself and to calculate the actuator end position in space using simple algebraic and trigonometric calculations.

**Figure 10.** Sketch of the test bench.

The test aimed to evaluate the actuator displacement along the three main directions and the three secondary directions (cf. Figure 2) during heating and cooling sequences. The test was carried out with a step supply current of 1 A for the heating phase (duration 30 s). The activation sequence for each direction was the following: activation of the lower module, activation of the central module, activation of the higher module. Cooling was in still air. Each sequence was repeated at least three times. The power supply was 2.8 W.

The tests performed on prototype A highlighted its stable and repeatable behavior. The 3D workspace, visible in Figure 11, was sufficiently large. The melting temperature close to the temperatures reached by the SMA wire in the hot phase was an important limit of the prototype, which after a relatively low number of cycles had been damaged in these areas. In a first instance, a local insulation can be added, but design changes have to be implemented.

**Figure 11.** 3D workspace of prototype A.

Prototype B, despite having mechanical characteristics similar to prototype A, had a slightly lower working space. It was observed that this difference can be attributed to a different pretensioning of the antagonist wires in the two prototypes. This observation has led to the need to investigate in more depth the contribution of inactive antagonistic SMA wires, which seemed to significantly limit the deformation of the entire actuator. Figure 12 shows the comparison of the tests obtained on prototype B in the case of the presence of antagonist wires (a), and in the absence of antagonist wires (b). Obviously, this situation, which was artificially generated by disassembling the antagonist wires, is not compatible with the normal use of the device, but is used only for the purpose of investigation. In the result obtained, namely a considerable increase in the working space in the second case, encourages the realization of a new device in which an additional locking/unlocking device allows for the disabling of the antagonist wire when necessary.

**Figure 12.** (**a**) 3D workspace of prototype B, and (**b**) 3D workspace of prototype B without antagonistic wires.

Figure 13 shows the comparison of the two different workspaces on an X-Y projection.

**Figure 13.** X-Y projection of the workspaces for prototype B (cross: experimental, circle: mean values; solid line: with antagonistic wires, dotted line: without antagonistic wires).

Figure 14 shows the results obtained with prototype C, which was also tested in the case of the presence of antagonistic wires (a) and in the absence of antagonist wires (b). It can be observed that, also in this case, the 3D workspace, already quite large in case (a), significantly increased in case (b). The device, having good thermomechanical characteristics, was resistant to prolonged use over time. However, silicone rubber was shown to be insufficient to exercise the correct elastic recall for the recovery of undeformed conditions at rest.

**Figure 14.** 3D work space of prototype C: (**a**) complete; (**b**) without antagonistic wires; (**c**) X-Y projection of the workspaces, (cross: experimental, circle: mean values; solid line: with antagonistic wires, dotted line: without antagonistic wires).

#### **6. New Prototype D**

Based on the results of the experimental tests and the observation that the absence of antagonistic wires would benefit the width of the workspace, a fourth prototype was designed. In this new device, the wires could be blocked for the active phase and released during inactive phases, so as not to work against the active wire. It was, however, necessary that a bias force was present for the return to the undeformed condition of the device. This new device must be as simple and compact as possible, even if the introduction of these new features necessarily involves a greater complexity and an increase in the number of components.

The structure of each module of the new prototype was similar to that of the previous prototypes, except for the addition of the locking/unlocking device housed in a base seat. Figure 15a shows the assembly of the prototype. As can be seen, the addition of a locking/unlocking device has been added below each "traditional" module. In order to facilitate the connection of the wire ends to the cylinder, the distance between the holes for the passage of one wire was modified with respect to the previous solutions (25◦ instead of 40◦). The materials chosen for the new prototype were the following: PTFE for the module, PEEK100 for the base seat, and PES (Polysulfone) for the cylinders.

**Figure 15.** (**a**) New prototype D. (**b**) Detail of activation/deactivation device on the module base.

Figure 15b shows the detail of the locking/unlocking device. Both ends of each of the three SMA wires were connected to a cylinder (1) located below the lower base. A cylinder (1) was placed inside a gripper (2), which was driven by an additional SMA wire (3). When a current was sent to the additional SMA wire of the gripper, the cylinder was locked and could not move vertically anymore. Subsequently, current was sent to the corresponding SMA wire of the module, which would consequently flex along a main direction. During this phase, the cylinders of the two inactive SMA wires were free to slide vertically because the corresponding grippers were also inactive. Therefore, outside of the friction between the cylinder and gripper, the active wire was not impeded by antagonist wires. When the power supply was interrupted, the clamping action also stopped and the cylinder was free to move again. A similar behavior occurred for the activation along the secondary directions. The bias force for the cylinders and rod was guaranteed by using appropriately sized springs (k = 2.28 N/mm) (4).

A set of preliminary tests of the device has been carried out and the results are shown in Figure 16 (continuous black line). The projection in the XY plane of the work space is here compared with the results obtained with prototype B, which had a central rod made of the same material. Compared to the solution with antagonistic wires (green solid line), the results of the new prototype were better. Obviously, the presence of sliding friction in the contact between the gripper and cylinder meant that it was not possible to reach the deformation obtained by completely eliminating the antagonist wires from prototype B (green dotted line).

The difference in orientation of the flexion directions of the new prototype with respect to prototype B could be attributed to the sum of two problems related to the experimental procedure: the pretensioning of the SMA wires and the difficulty in orienting the two devices exactly in the same way with respect to the test bench.

**Figure 16.** X-Y projection of the workspace with prototype D in black and prototype B in green (solid line: with antagonistic wires, dotted line: without antagonistic wires).
