*Case B*

The movement along the secondary directions is obtained by operating the SMA wires two at a time. In this case, therefore, there will be a force due to the two active SMA wires and a single antagonistic action determined by the inactive SMA wire, which is added to the elastic antagonist action determined by the central rod (Figure 4b). To build the 2D model, with reference to the sketch of Figure 4b, each one of the two SMA forces (shown in gray) is applied at the midpoint of the segmen<sup>t</sup> that joins the centers of the two holes along which the SMA wire runs. Similarly to the previous case, these forces remain parallel to each other throughout the deformation; therefore, instead of considering their separate e ffect, they are replaced by a total SMA force (represented in red). This force has direction and sense equal to the direction of the two SMA forces, it has a magnitude that is the sum of the two, and its point of application is located at the midpoint of the segmen<sup>t</sup> that connects the two points of application of the two SMA forces. The antagonist force (represented in blue) is generated by a single wire (the wire at rest); it is applied at the midpoint of the segmen<sup>t</sup> that connects the center of the two holes in which the antagonist SMA wire runs, as shown in Figure 4b.

Also in this case, the total SMA force, the antagonistic force, and the actuator axis are in the same plane; therefore the deformed actuator will move along this plane.

With these assumptions, in both cases the actuator is subjected to forces lying on a plane that will be called the "deformation plane." Therefore, the model is a two-dimensional (2D) model, based on the analysis of the actions during the actuation in the two cases, and it is subsequently extended to the other directions thanks to the symmetry that the actuator presents.

Moreover, in the model, other simplifying hypotheses have been made. The first one concerns the upper head of the actuator, which even after deformation, is supposed to remain perpendicular to the central rod; the second one consists in neglecting the inertia of the system.

Figure 5 is a sketch of the 2D model.

**Figure 5.** Sketch of the 2D model.

The axis of the actuator is represented by the segmen<sup>t</sup> OE, having length L. A generic deformed condition brings the point E (end-e ffector) to the position E'.

To study this deformation, the actuator is modelled as a set of N rigid bodies having a length equal to L/N each. These elements are connected to each other via torsional springs [31]. For simplicity, in the sketch, the deformed configuration starts from point O, the starting point of the first element, and arrives at point E', the final point of the Nth element.

During the actuation, each of these N elements undergoes a rotation of an angle ϕ with respect to the element that precedes. Therefore, the last element of the rod will be rotated by an angle that is Nϕ with respect to the horizontal; the upper head, in the final position (segment A'B'), will form the same angle of value Nϕ with respect to initial position (segment AB).

The CA segmen<sup>t</sup> (drawn in blue) represents the antagonist wire in the rest condition; once the deformation has occurred, this wire changes its configuration, moving to the CA segmen<sup>t</sup> and exerting the antagonistic force Fant.

The DB segmen<sup>t</sup> (drawn in red) represents the SMA wire in the rest condition; once the deformation has taken place, this wire also changes its configuration, leading to it coinciding with the DB segmen<sup>t</sup> and exerting the FSMA.

The length *a* represents the distance between the point of application of the force exerted by the antagonist wire and the axis of the actuator at rest. The length *r* represents the distance between the point of application of the force exerted by the SMA wire and the axis of the actuator at rest.

It should be noted that the configuration shown in Figure 5 applies both in *Case A* (main directions) and in *Case B*(secondary directions). In the latter, parameters *a* and *r* and the value of the multiplicative coefficient to be given to the forces involved are different, obtained simply from the different geometry of the system. In *Case A*, the value of the single antagonistic force must be multiplied by two, and in *Case B*, it is necessary to multiply the value of the single SMA force by two.

The device reaches the equilibrium configuration when the sum of the moment of the antagonistic force and of the moment of the elastic reaction of the central rod balances the moment of the SMA force, with respect to the module base (Figure 6). This will occur for a particular angle ϕ; in fact, the three moments all depend, obviously in a different way, on this angle. Once the ϕ equilibrium value is obtained, it is possible to determine the elastic deformation of the system, and therefore, its effective working space.

**Figure 6.** Moments acting on the actuator in the equilibrium condition.

The three considered moments have a different dependence on the deformation angle; therefore, in the following, the procedure that led to the relationships between the forces and the deformation is presented.

#### *Moment of the SMA Force (One Wire)*

In our model, the value of the force exerted by the SMA wire is considered to be constant; in fact, it is assumed that, after a short transient, the wire completes the phase transition and is able to exert the nominal force (value given by the manufacturer). The value of the arm of the force applied by the

SMA wire depends by the deformation angle ϕ. The moment exerted by a SMA wire with respect to the actuator end point is described by the following formula:

$$M\_{SMA} = F\_{SMA} \cdot r \cdot \cos\left[a \tan\left(\frac{L \cdot \sum\_{i=1}^{N} \sin(iq\rho)}{\left(L \cdot \sum\_{i=1}^{N} \cos(iq\rho)\right) - r \cdot \sin(N\rho)}\right)\right] \tag{1}$$

*Moment of the Antagonistic Force (One Wire)*

The moment of an antagonist wire with respect to the actuator base is described by the following formula:

$$M\_{\rm ANT} = F\_{\rm ANT} \cdot L\_{\rm ANT} = \varepsilon \cdot EA \cdot L\_{\rm ANT} \tag{2}$$

where *A* is wire section area, *E* is wire's Young modulus and

$$\kappa = \frac{\mathcal{L}\_{final} - \mathcal{L}\_{start}}{\mathcal{L}\_{start}} \tag{3}$$

is the longitudinal deformation of the wire.

While the section of the wire and the linear elastic modulus of the material can be considered constant parameters, the value of the longitudinal deformation varies with the angle of deformation because the final length of the wire is a parameter that depends on the angle ϕ.

As can be seen, in this case, the deformation angle does not only influence the arm of the force with respect to the final point of the rod but also affects the value of the antagonist force applied by the wire, so the final equations are the following:

$$F\_{ANT} = \frac{EA}{L} \cdot \left[ \sqrt{\left(L \sum\_{i=1}^{N} \cos(iq\rho) + a \cdot \sin(N\rho)\right)^2 + \left(a - a \cdot \cos(N\rho) + L \sum\_{i=1}^{N} \sin(iq\rho)\right)^2} - L} \right] \tag{4}$$

$$L\_{ANT} = a \cdot \cos\left[a \tan\left(\frac{L \cdot \sum\_{i=1}^{N} \sin(iq\rho)}{L \cdot \sum\_{i=1}^{N} \cos(iq\rho) - r \cdot \sin(Nq\rho)}\right)\right] \tag{5}$$

#### *Moment of the Elastic Reaction of the Central Rod*

The stiffness of the spring that represents the flexibility of the central rod depends on the material (Young modulus), on the section area, and on the length of the rod itself according to the following relation, so the final moment of the central rod can be described using the following equation:

$$M\_{\rm rad} = \frac{E\_{\rm r}}{L} N \wp \tag{6}$$

where

> *Er* is the Young modulus of the material constituting the rod;

*I* is the moment of inertia of the rod section;

*L* is the length of the rod.
