Rigid Body Attitude Estimation- An Overview and Comparative Stud.pdf

It should be noted that the matrix j defined as j a i

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It should be noted that the matrix J defined as J = ( A I - ˆ RA B ) A T B , (4.176) is highly dependent on the matrix A B that is given by A B = [ b m , b m × b a , b m × ( b m × b a )] , (4.177) which shows high sensitivity to the measurement noise. Both vectors of magnetic field b m and apparent acceleration b a in the body frame are contaminated with noise and the cross product between these vectors makes the use of this matrix unreliable in generating an orthogonal basis. On the other hand, because of the special structure of this observer, the estimated ro- tation matrix ˆ R may not start its trajectory inside SO (3). This makes it vulnerable to mea- surements noise since it might not get the chance to overcome the initial gap and make its way into the special group. In this case, the correction matrix J will never converge to zero and the estimated matrix ˆ R remains out of the SO (3). Figure (4.16) shows how the choice
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C hapter 4. D ynamic A ttitude F iltering and E stimation 116 Figure 4.13: (a) Convergence of the auxiliary matrix A to R SO (3), (b) Convergence of the attitude error norm to zero.
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C hapter 4. D ynamic A ttitude F iltering and E stimation 117 Figure 4.14: Performance of the velocity-aided observer with auxiliary matrix not belong- ing to SO (3) in noisy measurements condition.
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C hapter 4. D ynamic A ttitude F iltering and E stimation 118 Figure 4.15: Performance of the global observer non-evolving on S O (3) with ideal mea- surements (Accelerated mode) of the gain matrix Γ results in the estimated rotation matrix to even increase its distance from the special group rather than decreasing it. In this simulation, it was also observed that a large gain Γ led to instability in the estimation system. 4.9.6 Invariant Observer An invariant observer of the form given in (4.63) is simulated under accelerated motion assumption. The inputs of the filter are taken as the magnetic field vector measurement and rigid body velocity, both in B . The observer gains are chosen in a way that the linearized error system is stable. The chosen gains are L Q V = 0 - 1 . 6 0 1 . 6 0 0 0 0 0 , L Q b = 0 0 0 0 0 0 - 8 B (2) 8 B (1) 0 , L V V = - 8 0 0 0 - 8 0 0 0 4 , L V b = 0 0 0 0 0 0 0 0 0 , (4.178)
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C hapter 4. D ynamic A ttitude F iltering and E stimation 119 Figure 4.16: Performance of the global observer non-evolving on S O (3) with noisy mea- surements (Accelerated mode) where B (1) and B (2) are the Earth magnetic field vector values in the directions X and Y , respectively. Figures (4.17) and (4.18) show the performance of this observer in noise-free and noisy IMU measurements, respectively. In Fig. (4.18), the e ff ect of measurement noise is visible. The invariant observer is also sensitive to noise and based on the noise intensity, its various gains have to be tuned to maintain the stability of the observer. The performance of this observer in estimating the Euler angles is quite comparable to that of the extended Kalman filters.
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