**1. Introduction**

Shape memory alloys (SMAs) are a particular class of metal alloys characterized by two properties: (1) the shape memory effect (SME), the ability to recover a preset geometric shape when subjected to an appropriate temperature change; and (2) superelasticity (SE), the ability to withstand large deformations (up to 10–15% compared to the initial configuration) without producing permanent effects within a certain range of temperatures. The first discovery of these phenomena dates back to 1932 thanks to the studies of Chang and Read on the AuCd alloy, and in 1938 the transformation was studied in brass (CuZn). However, it was only in 1962 that Buehler discovered the SME in NiTi alloy, and it was from then that actual research on its metallurgy and on its practical applications began. Many other alloys that presented these properties were analyzed, but from the point of view of applications, the most interesting and useful proved to be those of the NiTi group and the Cu alloys [1].

Since then, SMAs have been used in many fields of engineering: robotics [2–7], biomedical engineering [8–10], and structural engineering [11,12]. The field of innovative robotics seems to be particularly suited to exploiting the advantages that these materials can provide. These advantages are a high power/weight ratio [13], sensing ability, remotability, low driving voltage, simplicity, cleanliness, and silent actuation. Some of these advantages are emphasized when the device decreases in size. On the other hand, they also present disadvantages, such as low energy efficiency, fatigue problems, history-dependent characteristics, and low actuation frequency [14]. Much research attempted to

overcome these disadvantages, in particular, by implementing different controls in order to obtain a stable and repeatable behavior of their devices [15–17].

Today, industrial automation is entering a new phase, that of the Internet of things (IoT) and the smart factory. Robotics, which is one of the ingredients of this revolution, has the increasing need to find non-conventional solutions. Since the aim is to allow humans and machines to work together without protective barriers, manipulators must also be designed for cooperation.

The end-effector mounted on a manipulator wrist can be a working tool or a gripper. The traditional parallel or angular grippers, with two elements facing each other and operating a parallel or angular movement, are widely used in robotics since they are simple and generally economical [18,19].

Dexterous hands usually have an anthropomorphic shape and the object is grasped by closing mechanical fingers. They are more versatile and are preferred for gripping non-symmetrical objects [20–23]. The actuation system of the grippers can be traditional (electrical, pneumatic) or innovative, like piezo-actuation [24] or SMA actuator-based [16,17,25].

Moreover, flexible actuators are an interesting solution to use as fingers in the gripping hands, in particular for cooperative robots. Different solutions, operated using electric current, hydraulic fluid pressure, or pneumatic pressure [26–28], have been developed, but SMA-operated flexible fingers can also be found in the literature. In general, the solutions implemented with electric motors are less light, smooth, and silent compared to solutions with shape memory materials. Regarding the solutions with hydraulic and pneumatic actuation, they can be used only in environments where the corresponding generation plant is present. Devices with SMA actuation therefore appear to be promising where lightness both for energy system demand and the device itself is required.

As an example, Yang et al. [29] created a flexible device made of three SMA springs embedded off-axially and movably in a silicone rubber rod. A large deformation of the bending actuators are obtained. Drawbacks of their solution are the cooling time, increased by the embedded solution, and friction between the silicone and spring during the motion. Torres-Jara et al. [30] developed a compliant modular actuator based on different arrays of the same simple unit based on a folded sheet of SMA. Linear, rotational and surface actuators are obtained, depending on a different assembly. This is a very interesting solution, both for the design of the object and its performance and for the idea of amplifying the work space by means of the series or parallel composition of a certain number of modules.

The idea of a flexible actuator and that of a modular actuator have been merged together to create the new family of flexible modular actuators presented in this paper. Each module is small, light, and can be variously assembled to allow for the creation of fingers having different shapes and characteristics to satisfy a wide range of needs. Depending on the task, it will be possible to assemble a different number of fingers to constitute various grippers, from the simplest with two fingers for the grasping of simple objects to the anthropomorphic five-finger hand.

This study aims to contribute to this growing area of research by proposing a mathematical model for the design of a flexible actuator, the design and experimental tests performed on four actuators, and the comparison of their different behaviors.

## **2. SMA-Actuated Module**

A modular actuator based on shape memory wires is sketched in Figure 1a. The module has a length of 40 mm and is composed of a central rod (1) with a lower base (2), an upper base (3), and two intermediate disks (4). Three Nitinol SMA wires (5) (diameter 250 μm) are longitudinally placed. One end of each wire is fixed to the lower base and passes through holes in the intermediate disks running parallel to the central rod. This end is now looped through a hole on the upper base and then returns to the lower base where it is fixed. Suitable screws (not shown) placed on the upper base allow the proper tensioning of the SMA wires. Each of the three wires is positioned at 120◦ from the others in order to allow the module to bend in any direction when one or more wires are actuated. The module bends when the wire is heated, e.g., by means of the Joule effect, which causes the shortening of the wire. When cooled and applying a bias force, the wire is stretched to the original shape and the module

comes back to the undeformed shape. This bias force is exerted in this case both by the central rod and by the inactive wires. The wires are not embedded in the structure so as to obtain faster cooling.

**Figure 1.** (**a**) Sketch of one module: 1) central rod, 2) lower base, 3) upper base, 4) intermediate disks, and 5) three SMA wires @120◦; (**b**) Example of a finger composed by three modules in series.

The modules can be joined in order to assemble fingers. This requires a rigid support that constitutes the base on which to arrange the electric power cables, switches, and drives for the control of the structure. As an example, Figure 1b shows a finger made of three modules. The heating of one wire will bend the module along one of three *main directions* (directions 1, 2, and 3 in Figure 2). The heating of two SMA wires will bend the module along three *secondary directions* (directions 1-2, 2-3, and 1-3 in Figure 2).

**Figure 2.** View from above of a module with actuator displacement directions.

The main advantage of this arrangemen<sup>t</sup> is that it allows the actuator to move within a volume whose projection on the plane perpendicular to the actuator is 360◦; that is, it is possible to carry out all the desired positioning on this plane within the work space of the device.

However, the presence of the other inactive SMA wires, which by their function will be called *antagonists*, constitutes a physical limit to deformation. Solutions to this limitation are investigated in Section 6.

Figure 3 shows a rendered image of a three-finger gripper and its possible operation.

**Figure 3.** Three-finger gripper.

## **3. Mathematical Model**

To evaluate the behavior of a three-module finger, and in particular, the contribution of the antagonistic wires, a numerical analysis was performed. This analysis involves the construction and development of a mathematical model of the system and its implementation and solution. A simplified mathematical model, in which the contribution of the antagonistic wires was neglected, is presented in Maffiodo and Raparelli [31].

This model consists of two parts to distinguish the different behavior in the case of a single SMA wire actuation (motion along the main directions, see Figure 2) and the actuation of two wires (movements along the secondary directions, see Figure 2) where the forces involved and the position of the latter with respect to the center of the device axis varies. Figure 4 shows the sketch of the forces acting on a module in these two different situations.

**Figure 4.** Sketch of the forces acting on a module: (**a**) lateral view and plant view in the case of one-wire activation (A), (**b**) lateral view and plant view in the case of two-wire activation (B).
