*Case A*

The displacement along the main directions, as mentioned, is achieved by operating the SMA wires one at a time. In this case, therefore, there will be a force given by the active SMA wire and a double antagonist action determined by the two SMA wires at rest, which is added to the elastic antagonist action determined by the central rod (Figure 4a).

To build a simplified 2D model, the following assumptions were made. With reference to the sketch of Figure 4a, the force of action of the SMA wires (shown in red) is applied to the midpoint of the segmen<sup>t</sup> that connects the centers of the two holes in which the SMA wire is running and is directed along the wire itself in the shown direction. The two antagonist forces (shown in gray) are also applied in the midpoints of the segments that connect the centers of the two holes in which the two antagonist SMA wires are running, and are directed along the direction of the wires themselves

and have the sense represented in the picture. These forces remain parallel to each other during the module deformation; therefore, instead of considering their separate e ffect, they are replaced by an overall antagonistic force (represented in blue). This antagonistic force has direction and sense equal to the direction of the two antagonistic 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 original antagonistic forces. As can be seen in Figure 4a, the actuating force exerted by the SMA wire, the overall antagonist force, and the axis of the actuator are then in the same plane, and therefore the deformed actuator will move along this plane.
