**5. Conclusions**

This work presents an adaptive Kalman filter for state and input estimation based on multibody models (AerrorEKF-FE). The aim is to accurately estimate the noise covariance matrix of the estimator, which has shown to be a critical factor for accurate and robust observer performance. The method is tested on two different planar mechanisms: a fourbar linkage and a five-bar linkage. In addition, four different sensor configurations are considered, increasing the general application of the proposed solution. Several tests are presented in order to analyze the behavior of the filter in terms of accuracy and robustness.

The AerrorEKF-FE combines an error-state Kalman filter with an adaptive method based on the maximum likelihood criteria. This adaptive technique, commonly used for navigation estimation, has several assumptions which are not fulfilled in multibody dynamics. Even though adaptive algorithms have the capacity of overcoming some of the main difficulties of multibody based Kalman filtering, its application has never been explored. In this work, the selected adaptive method is adjusted to fit with multibody particularities, such as time-variant transition and observation matrices. Furthermore, the estimated matrix should be adapted to the shape of the covariance matrix (if known), increasing the accuracy of the observer.

The first test evaluates the accuracy of the filter under different initial values of the noise covariance matrix. The results are compared against the estimations obtained with a non-adaptive version of the filter (errorEKF-FE). The results show an improvement in accuracy with the use of adaptive techniques. In addition, it can be seen how the adaptive method allows to achieve similar results in terms of accuracy in spite of the initial assumption of the covariance noises.

In a second test, the robustness of the filter is studied through a simulation in which an unexpected event changes the system. The filter shows a quick reaction and corrects with accuracy the new error, without loosing the stability of the estimator.

The tests executed in this work and their results show the potential of adaptive Kalman filtering for multibody based estimation. Determining the noise covariance matrix (specially the noise of the process) in multibody based estimators is an actual limitation for the general use of this techniques in multibody applications. Through adaptive estimation, the uncertainties on the process noise covariance matrix can be solved and the robustness and accuracy of the estimators can be increased and guaranteed during different scenarios. However, this method also implies a reduction of the computational efficiency that should be considered in real-time applications.

In future works, the proposed adaptive filter should be applied in systems of more complexity, such as vehicle dynamics, where the state and input estimation is of high interest. In addition, techniques for reducing the associated increment in computational cost can be studied.

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

**Funding:** This research was partially financed by the Spanish Ministry of Science, Innovation and Universities and EU-EFRD funds under the project "Técnicas de co-simulación en tiempo real para bancos de ensayo en automoción" (TRA2017-86488-R), and by the Galician Government under grant ED431C2019/29.

**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. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**

