**3. Cable-Driven Prototype**

The prototypes presented in the last section validate the design of this proposal; however, the bistable behavior is binary. The link has two possible positions, completely collapsed and completely deployed. Therefore, the change of size does not have intermediate steps. Our proposal aims to obtain a self-scaling simple link with a continuous change of the size. For this purpose, a spring and a cable-driven mechanism have been included in the link. The final movement of each cylinder is composed by 2 *DoF*: displacement in *z*(*h*) and rotation in *θ*. Using three tendons and only one motor provides both movements if attached correctly. The internal spring is needed because of the structure of the link, that is not capable of maintaining an intermediate position and will fold without an external agent providing an extension force. Figure 9 illustrates the simple link prototype with an internal spring and a cable-drive with three tendons. Between the top of the single link and the top of spring, a bearing has been installed, which allows a free spring rotation movement when *θ* changes during collapse and deployment.

**Figure 9.** Cable-driven single link prototype.

The cable-drive is actuated by a DC motor with a 210:1 gear and no load speed of 75(*RPM*). It also features an encoder composed by two hall effect sensors displaced 90◦ between each other and a wheel with seven switching magnets. Therefore, one motor rotation corresponds to 28 quadrature pulses. As the gear ratio is 210:1, a 360◦ turn in the gear part corresponds to 5880 counts of the encoder. That makes the resolution of the encoder on the outside part 0.06◦.

The spring elastic constant has been obtained experimentally. Masses between 0(*gr*) and 400(*gr*) have been placed in the upper base of the spring and the deformation lengths corresponding to the compression have been registered in Table 1, from which the elastic constant has been approximated to a straight line *F* = *K* · *x*.

A simple linear regression has been applied to the collected data shown in Figure 10, from which the polynomial in Equation (5) is obtained, where the slope corresponds to the value of the constant *K*.


**Table 1.** Experimental data from spring compression.

**Figure 10.** Linear regression to obtain the spring elastic constant K from experimental data.

$$F(x) = -70.9522x + 9.5803\tag{5}$$

$$K = -70.9522 \, [N/m] \tag{6}$$

The prototype is expected to validate self-scaling in an autonomous and controlled manner. Given that the considered prototype physical definition is complex and consists of several subsystems, starting from the actuator used for the robot positioning and followed by the complete link, a system identification was performed in order to obtain the plant model.

The origami link is divided into two physical systems, the link itself, and the DC motor driving the mechanism. Given the link geometry described in the previous sections, we can neglect its effects in the final behavior and model the plant based on the DC motor only. Due to the lack of information from the motor provider, the DC motor model was obtained using recursive least squares (RLS) identification to the input-output captured data.
