PSI - Issue 38

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

346

5

Finally, the state vector of the multi-body system is q = ( q constraint accelerations ¨ q of the multi-body system are given by : 1 · · · q k S

) with k S the bodies number. The

   M 1

   and F =    F 1 .. . F k S

   (13)

   M G B

  

   o q λ B

   =

F L B + W B  

  

G t

B

. .

 With M =

.

0

M k S

n c × n c

Where o q

= ( ( ¨ q t 1

t · · · ( ¨ q t

t ) and where λ

B are the Lagrangian multipliers for the constraints

λ 1 )

λ k S )

k S

between bodies, G B according to the state vector q , L B corresponds to the term independent on acceleration in the second order of derivation of constraints equation according to the time domain and W B is the stabilization term of constraint g B . W B is defined : W B = − 2 ξω ˙ g B ( q ) − ω 2 g B ( q ) (14) We can find a pseudo-analytic solution of the equation (13) thanks to the knowledge of M − 1 k and G B :    o q λ B    = Ξ − 1    F L B + W B    with Ξ =    M G t B G B 0 n c × n c    (15) And Ξ − 1 =    M − 1 0 0 0    −    − M − 1 G t B I    ( G B M − 1 G t B ) − 1 − G B M − 1 I 3. Kalman filtering of the multi-body model 3.1. The Augmented and Constrained Extended Kalman Filter (ACEKF) In this section, we introduce the following notations for s a given symbol, s denote the prediction of states and ˆ s denote the Kalman estimation. The Kalman filter is a commonly used method to make a estimation of a linear dynamic system with two steps: • A prediction, denoted X is computed by the known state variable at time step n using equation (24), i.e. : X n + 1 = f ˆ X n . Where f is the function of prediction model. • A correction of the estimation is done based on the measurement equation (27) at the time n + 1, i.e. : Y n + 1 = h X n + 1 . Where h is the function of measurement model and Y is the measurement vector. ˆ X is the state estimation of the Kalman filter. ˆ X is assumed Gaussian. The covariance matrix of ˆ X noted Pˆ is estimated by the Kalman filter. f and h are nonlinear functions of the states. The Extended Kalman filter (EKF), is an approach to study the non-linear formulation. The prediction X n is computed using the system of equations (24) and (25).   X n + 1 = f ( ˆ X n ) P n + 1 = F Pˆ n F + Q (16) The correction of the prediction is computed using the equations (26) and (27):   ˆ X n + 1 = X n − K h ( X n ) − Y n Pˆ n + 1 = I ns − K H P n (17) is the jacobian matrix of the constraint function g B