**4. Results**

A Matlab simulation scheme has been created to validate the kinematics for both simple and nested links with two and three sections. The main parameters *a*, *α* and *β* were used to compute the origami behavior while the height *h* was the input data to fold or deploy the origami. Taking Equation (4) into account, the maximum height is when *δ* = 90 and the structure is completely deployed. This position is not achieved in the prototype because the spring force does not generate enough rotation for the links to change position at a positive *θ* angle and reach *δ* = 90.

Figure 17 shows the kinematics simulation for a single link with different heights. The change of the rotation angle *θ* can be seen while the origami is deploying. The simulation results are shown in Table 3.

**Table 3.** Data from single link kinematics simulation.


The nested link has been simulated, too. In this case the total link height is the sum of each single link. To achieve a certain height position, we can actuate each individual link or all at once. Figure 18 shows three different configurations: completely deployed link, only one section folding, and three sections folding. The data results from this simulation are shown in Table 4.

**Table 4.** Three-sections link kinematics simulation data.


**Figure 17.** Origami Kresling single link kinematics vadidation, with parameters *α* = 38◦, *β* = 30◦ and *a* = 30 mm. (**a**) Collapsed state. (**b**) Deploying state. (**c**) Deployed state.

**Figure 18.** Three-sections link kinematics validation, with parameters *α* = 38◦, *β* = 30◦ and *a* = 30 mm. (**a**) Completely deployed link. (**b**) Only one section folding. (**c**) Three sections folding.

## *Single Link Cable-Driven Prototype Experimental Results*

Position control tests have been carried out with both the integer and the fractional controllers adjusted. The tests were made according to encoder data in *rad*; however, the origami linear displacement is easy to obtain knowing that the motor coupling radius is 7.5 (mm) and the encoder resolution is 5580 counts per turn. The linear displacement corresponds to the origami folding, and height *h* and rotation angle *θ* can be indirectly calculated.

Two tests have been designed. The first consists of giving the system individual targets in position: 0.5, 1.0, 1.5, 2.0 and 2.2 (*rad*). The physical behavior of the system can be seen in Figure 19, where the initial reference of the upper part of the origami and the union of the three tendons to visualize the displacement has been marked with a red dashed line.

**Figure 19.** Prototype test results with different target position.

The data results obtained are shown in Figure 20. The saturation of the system is evident in the higher set points, showing an overshoot in the response of the system. The control signals of the internal loop in speed and the external loop in position are shown in Figure 21.

**Figure 20.** Test results with different target positions. (**a**) Integer controller. (**b**) Fractional controller.

**Figure 21.** *Cont.*

**Figure 21.** Control signals. (**a**) Position loop control signals with integer controller. (**b**) Position loop control signals with fractional controller. (**c**) Velocity loop control signals with Integer controller. (**d**) Velocity loop control signals with fractional controller.

Table 5 shows the experimental kinematic data, which have been calculated indirectly according to the experimental position changes in the first test. For this purpose, the initial position of the prototype with angle *θ* = 0.0514 (rad) and initial height *h* = 39.44 (mm) has been considered, resulting in a maximum *δ* = 85.64 (deg). The position data were measured for all targets once the system reached its permanent state, at time *t* = 4 s.

A simulation was performed with the obtained *δ* data in order to make a comparison between experimental and simulation data in terms of height *h* and angle *θ*. The resulting errors are shown in Table 5. The difference between the real measured and simulated values is clearly the value of *lBC*, since in the measured value *lBCreal* = 39.56 (mm) and in the simulated value *lBCsim* = 43.09 (mm). This difference is due to the fact that in the assembled prototype the link representing *lBC* is slightly compressed to maintain the desired position of the structure.


**Table 5.** Experimental kinematic data indirectly computed.

The second test consists in giving the system sequential targets between 0.5 and 2.5 rad with steps of 0.5 rad. The results are shown in Figure 22, where the real position data is in red. Here we can see an expected behavior for the designed controllers. The overshoot in each step is lower than in the first test because the sequential targets are lower than the individual ones.

On the other hand, when the loop is restarted and the position must change from 2.5 to 0.5 rad, the system behavior is the opposite, that is, it must change from folding to deploying. In that case, the position reaches the target with an initial overshoot but

quickly stabilizes. In this way, the efficiency and robustness of the control system have been validated.

**Figure 22.** Test results with sequential target positions. (**a**) Integer controller. (**b**) Fractional controller.

The video of the performance of this test can be visualized in the following link: https://cutt.ly/rjW0Obi, available since 14 January 2021.

To verify the robustness of both the control and the structure, the steps were tested with different payloads between 100 and 400 g, as shown in Figure 23.

**Figure 23.** Prototype test results with different payloads in 2.5 (rad) position.

The results obtained allow for the determination of the behavior of the integer and fractional controllers (Figure 24), where the most relevant behavior is shown in the return zone from 2.5 (rad) to 0.5 (rad).

In the case of the integer controller (Figure 24a), the lower peak position reaches −0.54 (rad) with a stabilization time of 0.63 (s) and a maximum overshoot of 16%, while in the case of the fractional controller (Figure 24b), the minimum position is −0.44 (rad); it has no overshoot and the stabilization time is 0.43 (s). Therefore, it can be concluded that the fractional controller is not only robust but also faster than the integer controller.

**Figure 24.** Prototype test results with different payloads. (**a**) Integer controller. (**b**) Fractional controller. The control signals for both controllers are shown in Figure 25.

**Figure 25.** *Cont.*

**Figure 25.** Control signals test results with different payloads. (**a**) Position loop control signals with integer controller. (**b**) Position loop control signals with fractional controller. (**c**) Velocity loop control signals with Integer controller. (**d**) Velocity loop control signals with fractional controller.

It is interesting to mention that in this last test the loads were placed in a distributed way so that the center of mass does not change. In case the loads were placed off-center or unbalanced, the dynamics of the system would change because an additional DoF would be generated, allowing the upper base of the platform to tilt in the direction of the weight. For these scenarios more than one actuator would be required in order to compensate this tilt movement (and other actuators could be included to generate other different DoF). This is an interesting topic to be developed in future works.
