Aerospace and Electronic Systems - November 2018 - 12

Nonvisible Satellite Estimation Algorithm

Figure 6.

Figure 7.

tained. From this map the DMM can be calculated using (15). The
measurement covariance matrix R is dynamically update according to the local value of the DOP as follows

chosen in order to highlight the potentiality of the algorithm when
a very complex scene is processed. The mountainous scenario is
preferred due to the variety of missions that can be performed, involving vehicles, pedestrians, or different kinds of aircraft. In order to project the DEM in the CRF, the homogeneous transformation of (2) has been used. For this simulation, the control volume
dimensions equal exactly the map size, imposing the same mesh
discretization. The control volume resolution along the three Cartesian axes in meters is δΩ = [48.5 49 12]. The DEM dimensions are
respectively 10 km × 13 km, whereas the maximum mountain altitude above the World Geodetic System WGS84 is ≈ 2,000 m. The
number of pixels along xc and yc are respectively n px = n p y = 300,
−6
whereas the pixel dimensions are δ px = δ p y = 3.6 × 10 m.
The value of θlim that defines the visibility cone aperture is θlim
= 85 deg. The results proposed in this article have been obtained
using the software Matlab on an Intel Core i7-4710HQ 64-bit, 2.50
GHz with 16 GB RAM.

2D section of the control volume showing the number of visible satellites.

 μ1

0
R = DOP 


 0

0 ...

0 

0 
   

0 ... μ N s 

μ2 0 ...

0

(26)

where μ is a proportionality coefficient. Using this matrix in (23),
the state covariance matrix Pxxn|n can be calculated. The standard
deviations σ = σ xˆ σ yˆ σ zˆ , required for the calculation of the
safety bubble Φ(σ), is obtained by taking the square root of the
diagonal components of Pxxn|n. A more conservative approach, the
one we adopt here, is to calculate a spherical safety bubble, for
which the radius is
rΦ = 

 diag ( P

n|n

(27)

where ε is an integer multiplicative coefficient, which is used to
regulate the confidence interval.

NUMERICAL RESULTS
This paragraph shows how the algorithm behaves when a more realistic scenario is presented. The 3D Digital Elevation Map is part
of Mount Mitchell in Yancey County (North Carolina), digitalized
with a map resolution of ≈30 m. For real applications the use of a
more accurate DEM is required. This particular scenario has been
12

2D section of the control volume showing the value of the DOP.

DMM RESULTS
In Figure 6 the DEM with a two-dimensional (2D) section of the
control volume is shown. The simulated Global Positioning System (GPS) almanac data refer to 02:46 pm Central European Time
(CET) of February 22, 2017. The theoretical results presented
graphically in Figure 3 are confirmed by the simulation. The color
map indicates the number of visible satellites at any point of the
control volume. As expected, the minimum number of visible satellites is registered in the valley with only 6 visible satellites over a
maximum of 12 in the upper part of the control volume. In Figure 7
the control volume color map indicates the values of the DOP relative to the visible satellites. As expected, the highest value of the

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NOVEMBER 2018

Aerospace and Electronic Systems - November 2018

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