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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With:

K = P X n + 1 Where Y is the vector of real measurements, K is the Kalman gain matrix, Q is the covariance matrix of prediction model and R the covariance matrix of measurement model. It is an indicator of the trust in the measurements Y compared to the first state estimation X given by the state space model. H and F are respectively the Jacobian of h and f at the considered respective states. The CEKF is a EKF where the estimated state ˆ X is conditioned to follow exact equation represented by the function g K : g K X = 0 N c × 1 (18) Where N c is the number of equations of constraints in the Kalman filter. The equations of constraints are non-linear. We use the methode of Estimate Projection D. Simon (2010) to implement the constraints in the CEKF. The EKF return an estimation ˆ X of the real state vector X , then ˆ X is projected onto the surface of constraints. The constrained estimation ˆ X c is then written as : ˆ X c = argmin s ∈ R n s s − ˆ X t W s − ˆ X (19) Where W is a positive-definite weighting matrix. Here we choose W = Pˆ − 1 in order to obtain an estimation which minimize the covariance. Finally the constrained estimation and its covariance are then given by ;   ˆ X c = ˆ X − Γ g c ˆ X Pˆ c = I − Γ G Pˆ with Γ = Pˆ G t G Pˆ G t − 1 and G = ∂ g ∂ X ˆ X (20) The constraints equations implemented in the CEKF are the quaternion norm constraints g p defined in equation (9), the constraints between bodies g B and the first derivation. g B is a constraint of coincidence of two points P k 1 j 1 and P k 2 j 2 from di ff erent bodies S k 1 and S k 2 , it is written as : g B ( X ) = P k 1 , j 1 P k 2 , j 2 B 0 = x k 2 − x k 1 + R 0 , 2 G k 2 P k 2 , j 2 B 2 − R 0 , 1 G k 1 P k 1 , j 1 B 1 (21) Here, the multi-body model, a car, is excited by external environment, the road. We choose to charac terize the road by the vertical position z r and the variation speed of the road height δ z r for each wheel. The equation describing the road evolution in the prediction model is given by : z n + 1 r δ z n + 1 r = A z n r δ z n r with A = 1 dt 0 1 (22) 3.2. Kalman prediction We use the constant time step dt to describe the time evolution equations of the state space model. They are given by the following discrete-time equations:   q n + 1 k = q n k + dt ˙ q n k + ( dt 2 / 2) ¨ q n k ˙ q n + 1 k = ˙ q n k + dt ¨ q n k (23) Let be X = ( q t ˙ q t ζ ) t the state of the muti-body system where n s the size of state vector and ζ the vector of unknown parameters describing some characteristics of the multi-body system. For example, the height of the road. Considering f the function describing the relation between consecutive time steps based on the equation (23) for q and ˙ q . In addition, we introduce a noise denoted X on the model estimation and its covariance matrix Q , assumed independent of time. Then : n H t ( H P n H t + R ) − 1 , F = ∂ f ∂ X X n and H = ∂ h ∂ X