Aerospace and Electronic Systems - November 2018 - 11

De Vivo, Battipede, and Gili

Figure 4.

Safety bubble inflation when the receiver moves in the occluded region.

confidence domain around the estimated vector . This ellipsoid,
or safety bubble, is indicated as Φ(σ), where σ = σ x σ y σ z  is
the vector of the predicted standard deviations, obtained from the
square root of the diagonal elements of
. From the inequality
relation of (24) it is easy to figure out that the safety bubble associated to R2 is bigger than the one associated to R1.
Starting from these considerations, the SBOA algorithm can
be explained. The example shown in Figure 4 is considered. If the
platform flies in normal conditions, with all the satellites in view,
the sensor accuracy is the highest, and consequently the safety bubble has the minimum dimensions. For simplicity, we assume that
σ x = σ y = σ z, and for this reason the safety ellipsoid degenerates into
a sphere with radius rΦ. In order to take into account also the DEM
uncertainty (it can span from few centimeters to some meters), the
safety bubble is increased accordingly. By including the DEM uncertainty in the safety bubble, the DEM uncertainty becomes zero. An
obstacle avoidance algorithm has the aim to optimize the platform
trajectory, or the evasive manoeuvre, so that the distance dΦ between
the platform and the obstacle is greater than the safety bubble size
Φ

≥

(25)

This condition is shown for the time step tk = 1 in Figure 4. Considering that the safety bubble radius is a confidence interval associated to a variance σ, its value is calculated once a probability p(σ) is
chosen. For safety reasons, it is not allowed to violate the inequality
of (25), and the minimum flying distance is dΦ = rΦ, shown in Figure 4 for tk = 2. If the platform moves inside the satellite occluded
region, the conventional navigation system does not change the size
= )
of the safety bubble associated with the estimated state
because R is kept constant. In the real case, the GNSS performance
is degraded because of the occlusion effect and the increase of the
DOP caused by the reduction of visible satellites. By including this
sensor degradation information in the measurement covariance matrix R, the safety bubble increases as shown by (24) (violet ball for
NOVEMBER 2018

Figure 5.

Scheme to calculate the Safety Bubble and Digital Morphing Model.

tk = 3 in Figure 4). By neglecting this bubble inflation, the navigation system performs a manoeuvre with the constrain dΦ = rΦ, where
rΦ is the value associated to the nominal situation. By performing
this manoeuvre, the real safety bubble intersects the obstacle surface. This means that the probability of impacting with the obstacle
is higher than the probability assured by the safety bubble radius.
If the SBOA algorithm is applied, as soon as that the platform flies
in the occluded region, the DOP value in that region increases the
covariance matrix R, increasing immediately the safety bubble dimensions. As a consequence of this bubble inflation, the minimum
allowed distance dΦ becomes bigger than the previous one, pushing
the platform far from the obstacle surface (tk = 4). This "repulsive
effect" increases with increasing values of the local DOP, forcing
the body to fly at a safer distance from the obstacle.
In Figure 5, the block scheme that summarizes the procedure to
calculate the SBOA is presented. The input values are the satellite
constellation, DEM and UAV position. The algorithm to calculate
the occlusion points, and consequently the nonvisible satellites in
each point of the control volume Ω, is applied. At this point, a list
of visible satellites is associated to each voxel of Ω. By applying
(16) to the visible satellites in each voxel, the DOP map is ob-

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Aerospace and Electronic Systems - November 2018

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