PSI - Issue 38

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

7

348

   X n +

n ) =   

dt 2 2

dt ⊗

  

  

I dt I 0 0 I 0 0 0 A

¨ q n

X n + 1 = f ( X n ) + X

with f ( X

(24)

0

Where ⊗ is the Kronecker product and A is the matrix describing the parameters evolution. The jacobian of f , denoted F is given by : F =    I dt I 0 0 I 0 0 0 A    +      dt 2 2 dt   ⊗ ∂ ¨ q n ∂ X n 0    (25)

∂ ¨ q n ∂ X n is given by :   ∂ o q n ∂ X n ∂λ n B ∂ X n

  = − Ξ − 1

∂ Ξ ∂ X n    o q n λ B

   + Ξ −

F L + W  

∂ X n     

   with

=   ¨ q n 1 λ 1

t   t

t

¨ q n k S λ k S

n

1 ∂

o q

· · ·

After having defined the equations of the Kalman prediction step for the car body dynamics, we intro duce the measurement model in the Kalman filter.

3.3. Kalman correction In our issue, let the Solid S 1 be the car body and O the center of the terrestrial reference frame. The observations of the state space model is given by sensors, such as biaxle accelerometer, biaxle gyrometer, GPS and tachometer. The gyrometer measure the rotation’s rate of the car body, denoted y gyro , defined in the marker B 1 = e 1 x , e 1 y , e 1 z . The biaxle accelerometers measure accelerations y acc on several non-coplanar points P k 1 , j 1 on the car body, given in the marker B 1 . the accelerometers measure the acceleration of the car body plus the constant gravity vector, noted a grav . The position y GPS of the car body is mesured by the GPS. The tachometer measure the longitudinal speed of the car body denoted y tacho . In the case of a car rolling without sliding, the speed on the other directions are considered as noise without bias. We assume that measurement noises acc , gyro , GPS and tacho are zero-mean white Gaussian and uncorrelated. Then:

    y acc

+

+

+

+ p

1 j + a grav + acc

= R

¨ x 1 + 2

p¨

p˙

p˙

1 GP

+

0 , 1

1

1

1

p 1 ˙ p

y gyro

+ gyro

= Φ

1

(26)

y GPS y tacho

= x 1 + GPS

= R

˙ x 1 + tacho

0 , 1

We introduce the function h based in the equation (26): Y = y t acc y gyro y t GPS y t tacho t

= h ( X ) + Y

(27)

Where Y = ( t

t GPS

t tacho ) t denotes the noise measurement vector. Its covariance matrix R is

gyro

acc

assumed independent of time.

Finally we obtain the formulation of the Extended Kalman filter used to estimate the state of the non linear problem. However, in our study, the multi-body model is defined with constraints such as the quater nion norm and the kinematic contraints between bodies.