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Fig. 8.4 Instability of critical points. Plain line: the true homography velocity A. Dashed
line the observed homography velocity A
H k +1 = H k exp( A
Δ
t + Q k Δ
t )
where A
sl(3) is a constant velocity, Q k sl(3) is a random matrix with Gaussian
distribution, and
Δ
t is the sampling time (in the simulation we set the variance to
σ
= 0
.
1). By building the homographies in this way, we guarantee that the measured
H k
k .
We implemented a discretized observer in order to process the data. In all exam-
ples the gains of the observer were set to k H = 2and k A = 1.
Figure 8.1 shows the elements of the measured and estimated homography matri-
ces. Figure 8.2 shows the elements of the associated homography velocities. In this
simulation the initial “error” for the homography is chosen at random and it is very
large. The initial velocity estimate A 0 is set to zero. Figure 8.1 shows that after a fast
transient the estimated homography converges towards the measured homography.
Figure 8.2 shows that the estimated velocity also converges towards the true one.
SL (3),
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