**1. Introduction**

In mechatronic systems, operational forces and moments are essential quantities in the different stages of the development cycle and strongly impact the design, durability, diagnostic, prognostic, maintenance, and advanced control strategies [1]. However, forces and moments are also difficult, even impossible, quantities to measure. This is due to high force sensor costs and the geometrical constraints (space limitations) that would make the sensor integration impossible without influencing the overall system design and behavior.

In past decades, different test-driven and model-based inverse force methods have been presented in the literature to overcome these limitations. Initially, the challenge of inverse load identification was tackled in offline test-based strategies. One of the most commonly used technique for mechanical applications relates to the field of Transfer Path Analysis (TPA) [2].

TPA summarizes the family of test-based methodologies to study the vibration transmission in mechanical systems where the Matrix Inversion and Mount Stiffness approaches are the most commonly used to estimate inputs and transmitted forces respectively. Despite the wide variety of methods and extensive use in the industrial world, TPA still remains quite expensive from an experimental point of view in terms of preparation and execution time.

**Citation:** Adduci, R.; Vermaut, M.; Naets, F.; Croes, J.; Desmet, W. A Discrete-Time Extended Kalman Filter Approach Tailored for Multibody Models: State-Input Estimation. *Sensors* **2021**, *21*, 4495. https://doi.org/10.3390/s21134495

Academic Editor: Giuseppe Ferri

Received: 21 May 2021 Accepted: 28 June 2021 Published: 30 June 2021

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The growing computational power of modern computers opened new opportunities to exploit numerical model-based methods. These models can be exploited in virtual sensing approaches [3] which enable the exploitation of low cost, accessible and non-collocated measurements together with first-principle/physically inspired models to obtain state and input estimates.

State estimation techniques such as the Kalman Filter (KF) methods allow the joint estimation of unknown inputs and model states [4] in an efficient manner. By regularly feeding back the measurements on a physical asset, KF techniques enable the compensation of drift in the model while reducing the noise from the direct measurements.

Multibody (MB) modeling approaches are regularly used in the literature and industry [5–7] for full scale rigid and flexible mechanical systems where conventional Finite Element (FE) based methods would be unnecessarily expensive. The MB methods establish a good trade-off between model fidelity and computational cost. Moreover, MB models disclose 3D system-level information, enabling dynamic interaction phenomena among bodies due to distributed and/or localized flexibilities.

However, the link between MB models and estimation algorithms is nontrivial since most estimators require an ordinary differential state-space representation of the system dynamics. Instead, the MB model dynamics, depicted by the Equations of Motion (EOMs), are generally described by a set of Implicit-Differential Algebraic Equations (I-DAEs) that makes the state-space representation difficult to be met. On the other hand, Explicit-Ordinary Differential Equations (E-ODEs) are well suited for a state-space representation but specific MB formulations should be employed to obtain this structure (e.g., [8]), otherwise, dedicated manipulations of the EOMs are demanded to achieve an ODE form.

In [9] the Matrix-R method was proposed to eliminate the constraint equations of the MB model reducing the initial EOMs to an ODE form in independent coordinates. The aim of this work was to combine an extended KF estimator with detailed MB models to obtain an automotive real-time observer. Despite the high accuracy of the estimated quantities, the real-time target was not achieved due to the costly solver iterations.

Similarly in [10–12] the Matrix-R method was used to deal with the DAE structure of the EOMs. Here, different KF estimators are compared in terms of accuracy and performance on a rigid 4 and 5-bar linkage mechanisms.

Alternatively, in [13] a kinematic state observer is presented. It combines the constrained kinematic MB equations with nonlinear estimators. Here, the dynamic equations of the MB system are not considered therefore leading to an estimation which is less sensitive to input and mechanical (properties) uncertainties. Moreover the system accelerations are treated as random walk models and augmented to the kinematic discrete state vector that imply the use of a more extensive number of measurement sensors.

In [14,15] an interesting approach based on the combination of deep learning and MB dynamics information was proposed to achieve this transformation. It allows reducing a generic MB model to minimal coordinates allowing the description of the EOMs through E-ODEs while not requiring a specific formulation or access to the constraint equations. However, the methodology depends on a reference numerical simulation as training data which must cover the mechanism workspace; moreover only rigid MB systems can be tackled by the technique.

In [16] an Augmented Discrete Extended Kalman filter (ADE-KF) approach tailored for flexible MB models to construct a state-input estimator is presented.The methodology demonstrates the advantages of using analytical expressions to cover the necessary linearized and explicit EOMs. However, this approach relies on the use of a penalty constraint formulation to achieve E-ODE type of equations. This leads to a relatively poorly conditioned problem and introducing additional model parameters, namely the penalty factors, in comparison to a Lagrange-multiplier approach.

In this work, a generalization of the methodology described in [16] is presented which is compatible with a Lagrange multiplier approach for the constraint equations.

The proposed methodology starts from a novel linearization approach of the EOMs that includes the algebraic variables (Lagrange multipliers) to the system states. Consequently, the resulting unconstrained discrete-time state-space model is employed in a constraint KF scheme where the kinematic constraints are enforced trough the augmented measurement equations, therefore eliminating the effort of selecting effective penalty factors.

The scientific contribution is structured as follows: in Section 2 a general overview of the governing EOMs of the MB system dynamics is given; in Section 3 the implicit constrained EOMs are linearized and made explicit through a first order Taylor expansion; in Section 4 the system and measurement Kalman filter equations are introduced. Here, the system and measurement matrices are analytically assembled thanks to the use of the in-house Multibody Research Code (MBRC) [17]; finally, in Section 5 the methodology is validated on an industrial relevant application comparing the estimated quantities with the experimentally measured one.

#### **2. Multi-Body Model and Time-Disretization**

This section summarizes the derivation of the EOMs of flexible multibody systems, as they will be employed in this work.
