7.4.2. Sine Wave

Unlike the last results, in Section 7.4.1, as expected, they are not as clean as desired, considering that the FSS lacks these behaviors and therefore, the predictor does not emulate the given dynamics. As can be seen in Figures 22 and 23, the SM stays closer to the measured values. However, there are important gain errors and the model dynamics do not resemble the expected one, even if the results are better than those obtained by the NN model. The final fit values in these cases were: for pitch, NN = 13.5% and SM = 60%; for roll, NN = 60% and SM = 37%. These results confirm that in order to model static and non-coupled behaviours, additional dynamic signals should be considered in the FSS so that the identification data provide reliable information to increase the performance to the one shown in the experiment of Section 7.4.1.

**Figure 22.** Prediction configuration results for pitch in Test 2.

**Figure 23.** Prediction configuration results for roll in Test 2.

#### **8. Conclusions**

In this work, an improved mathematical model for a robotic soft neck has been presented. The whole soft-neck actuation range was modeled, resulting in a multi-input multi-output (MIMO) system showing a total of three inputs and two outputs. In particular, a nonlinear data-driven identification model using Set Membership, a linear model using Recursive least-squares, and a Neural Network model have been developed and discussed in this paper.

The outstanding results show that the proposed methods are suitable for estimation and control purposes when measures from the output are available to align the models. As shown, given the high level of correlation that the identification data set has over the NN training and the FSS for the set membership, additional identification data are required to use the methods as predictors over long prediction horizons, although results show that the proposed models are viable in soft nonlinear dynamics with multiple inputs and outputs.

A shown advantage of the SM identification stands in the possibility of incorporating additional signal dependency, delays, and unknown dynamics through a richer identification data set which derives from better and more complex modeling without explicit knowledge of the system. Even though the computational time might be a future consideration, there already exist approximation methods to overcome this drawback.

The accuracy difference found between the linear and nonlinear models suggests an important plant non-linearity, as expected. This issue can lead to problems at the time of defining a control strategy, although there are several options which will be explored in upcoming studies.

From the control point of view, the self-aligning characteristic of the given methods provide further knowledge on forecasting in short horizons, which is interesting for predictive and robust control techniques. Besides, the linear model accuracy is good enough to propose solutions like adaptive or robust control, which can provide excellent results. The predictive models' performance shown allows the use of the system for some applications. However, it is limited to continuous mode operation, which yet limits its utility. To overcome this issue, a more informative data set should be constructed that contains additional system behaviors to the continuous operation mode.

**Author Contributions:** Conceptualization, F.Q., J.M., J.A.C.P. and C.A.M.; methodology, F.Q., J.M., J.A.C.P. and C.A.M.; software, F.Q., J.M. and J.A.C.P.; validation, F.Q., J.M. and J.A.C.P.; formal analysis, F.Q., J.M. and J.A.C.P.; investigation, F.Q., J.M., J.A.C.P. and C.A.M.; resources, C.A.M.; data curation, F.Q., J.M. and J.A.C.P.; writing—original draft preparation, F.Q., J.M. and J.A.C.P.; writing—review and editing, F.Q., J.M., J.A.C.P. and C.A.M.; visualization, F.Q., J.M. and J.A.C.P.; supervision, C.A.M.; project administration, C.A.M.; funding acquisition, C.A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research leading to these results has received funding from the project Desarrollo de articulaciones blandas para aplicaciones robóticas, with reference IND2020/IND-1739, funded by the Comunidad Autónoma de Madrid (CAM) (Department of Education and Research), and from RoboCity2030-DIH-CM, Madrid Robotics Digital Innovation Hub (Robótica aplicada a la mejora de la calidad de vida de los ciudadanos, FaseIV; S2018/NMT-4331), funded by "Programas de Actividades I+D en la Comunidad de Madrid" and cofunded by Structural Funds of the EU.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
