PSI - Issue 38

A. Débarbouillé et al. / Procedia Structural Integrity 38 (2022) 342–351 A. De´barbouille´ / Fatigue Design (2022) 1– ??

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system taking into account only few solids was used to estimate loads at wheel center in the 3 directions. Vertical forces were well estimated but owing to the hypothesis of linear behavior of the wheel suspension, longitudinal and lateral loads could not be approximated. In our study a complete multi-body vehicle dy namic system is developed with fully nonlinear geometric behavior. We present here the methodology to estimate the longitudinal and vertical wheel center forces of the 2D multi-body model with a few sensors and an Extended Kalman Filter M. I. Ribeiro (2004) to correct the predictions. The Kalman filter is used for the estimation of attitude in many domains. In aerospace domain E. J. Le ff ens, F. L. Markley, M. D. Shuster (1982) the author presents an approach to implement Kalman filter with di ff erential equation for the state model based on the relation between gyrometer and quaternion for the attitude representation. Some works in heavy vehicle domain use the Kalman filter to predict vehicle attitude to improve technics of fuel savings C. Joly, D. Bre´taille, F. Peyret (2008). In this paper, the state model presents the evolution of position and orientation of the vehicle with the knowledge of wheels orien tation. A new branch of study with Kalman filter makes investigation on health monitoring L. Xie, Z. Zhou, L. Zhao, C. Wan, H.Tang and S.Xue (2018). In general, for the attitude estimation, the measurement model uses measures of one accelerometer and a magnetometer and the state model uses measures of a gyrometer A. Makni (2016). In the automobile domain, we find an odometer or a GPS P. Martin (2010) to improve these estimations. When quaternion is used for the estimation of attitude, the constraints equation, the nor malization of the quaternion, must be integrated to the process of the Kalman filter. In A. Makni (2016) the quaternion is reevaluated after each iteration of the Kalman filter. Other approach add constraints in the measurement model Y. Yang (2010) and D. Simon (2010), then called the perfect measurement. According D. Simon (2010) this approach has a numerical issue: the con straints uncertainty is equal to zero and the covariance matrix of the measurement is singular. Another approach changes the Kalman gain B. O. S. Teixeira, J. Chandrasekar, H. J. Palanthandalam-Madapusi (2008) in order to respect the constraints equation in the estimation state. The last approach to take account the constraints in a Kalman filter consist to implement an unconstrained estimation with the Kalman filter, then to project the unconstrained estimation into the surface of constraints. We implement our Kalman filter with this approach, called Estimate Projection D. Simon (2010), in the article. Here the accelerations of the multi-body system are computed according to the dynamic equation in order to estimate the position, speed and attitude. The vehicle model is considered as a multi-body system with rigid bodies, every body is char acterized with a mass and an inertia matrix and bodies are connected by mechanical links. The multi-body system is excited by an unknown road profile. This road profile is considered as a variable to be identified by the Kalman filter. We implement an Augmented and Constrained Extended Kalman filter with position, speed, quaternion, the quaternion first derivation and the road profile in the state vector. The constraints are implemented with the hypothesis of Estimate Projection. This work is still in progress and we aim at dealing with full 3D systems quite soon. Dealing with 2D is the first step towards the 3D, moreover 2D models are more convenient for future real time application. We are actually trying to apply our method to rolling measures simulated with Simpack on digitalised roadtracks and also to real rolling measures on our proving grounds (roadtracks). In section 2, we introduce the multi-body model with quaternion. Then in section 3 we present the Kalman approach to estimate vehicle dynamics: the implementation of a nonlinear Kalman filter with constraints in the measurement vector. In section 4, we expose an example of the methodology to verify the ability of our algorithm to estimate the vehicle wheel forces and road profile.

2. Numerical processing

2.1. Rotation representation The attitude of a body can be defined by an axis of rotation given by the unit vector u and an angle of rotation θ in a tree-dimensional system. This representation let us to avoid issues of singularity encountered