Aerospace and Electronic Systems - November 2018 - 10

Nonvisible Satellite Estimation Algorithm
this assumption is confirmed by the analysis of the DOP distribution presented in the Numerical Results section. The schematic
situation presented in Figure 3 has only the intention to highlight
some algorithm peculiarities. The number of satellites has been
set equal to 3 to provide the reader with an explicative figure.
Once the 3D DOP map inside the control volume is calculated,
the DMM can be obtained as the isosurface enveloping the points
PDMM, where
PDMM = {P ∈ Ω | DOP (P ) = DOPlim }.

(19)

This new DEM has been called DMM in the Introduction to
this article because the surface shape is a time-dependent function
of the satellite motion. This technique is very useful to solve the
problem related to nondirect signal reception. If the platform is
flying in the rightmost region of Figure 3, the signal of the satellite
S1 is shielded by the mountain. In this situation, if the user antenna
receives a signal from this satellite, certainly it is corrupted by multiple reflections because the satellite is in NLOS. For this reason,
in order to avoid a measure degradation, the NLOS information
can be used to filter-out the wrong measurements to mitigate this
effect. When the DMM is available, it can be used by the path
planning algorithm to optimize the platform trajectory, in order to
avoid regions where the number of visible satellites is dramatically
reduced, the DOP value is very high, and the risk of receiving corrupted measurement is large.

SAFETY BUBBLE OBSTACLE AVOIDANCE (SBOA)

(20)

is a function of both the Kalman gain Kn, the innovation matrix
Pzzn|n−1 and the predicted state covariance matrix Pxxn|n−1. Substituting in the second term of (20) the relative expressions of Kn and
Pzzn|n−1, taken from Table 3, the following formulation is obtained
10

LKF Algorithm [40]
xˆ n|n −1

= Fxˆ n −1|n −1 + un

Pxxn|n−1

= FPxxn−1|n−1FT + Q

zˆ n|n −1

= Hxˆ n|n −1

Pzzn|n−1

= HPxxn|n−1HT + R

Pxzn|n−1

= Pxxn|n−1HT

Kn

= Pxzn|n−1(Pzzn|n−1)−1

xˆ n|n

= xˆ n|n −1 + Kn ( z − zˆ n|n −1 )

Pxxn|n

= Pxxn|n−1 − KnPzzn|n−1KnT

KP zz K T =  P xz (P zz ) −1  P zz  P xz (P zz ) −1 
=

T
T

 P xx HT

P xx HT
HP xx HT + R 
 .
xx T
xx T
HP H + R
HP
H
R
+



(

)

(21)

In (21), the subscript time steps n have been omitted for clarity.
Defining
xx T
C1 = P H

xx T
C2 = HP H

and substituting them in (21) the following equation is obtained

In this section, an interesting application, deriving from the previous algorithm, is described. This application refers to the intrinsic
capability of an unmanned platform of avoiding an obstacle. To
understand how the algorithm works, a platform equipped with a
navigation system that integrates measurements from a GNSS and
inertial measurement unit is considered. Sensor integration is generally performed using a Kalman Filter (KF), or more advanced
and sophisticated integration techniques such as the one described
in [40], [41]. The filter output is generally characterized by the
estimated state vector xˆ n|n and a covariance matrix Pxxn|n that quantifies the error associated to the state vector estimation. The Linear
KF algorithm is shown in Table 3.
The matrix F is the state transition matrix, un is a known control vector, Q is the noise covariance matrix that takes into account
the error associated to the inaccuracy of the dynamic model, H is
the sensor measurement matrix, R is the sensor noise covariance
matrix, zˆ n|n−1 is the predicted measurement vector, z is the vector
containing the sensor measurements, and Kn is the Kalman gain.
The updated state covariance matrix
P xx n|n = P xx n|n −1 − K n P zz n|n −1K nT

Table 3.

T

KP zz K T =

 C 
C1
( C2 + R )  1  .
C2 + R
 C2 + R 

(22)

Substituting (22) in (20), the state covariance matrix Pxxn|n is
written as function of the sensor covariance matrix R as follows
P xx n|n =  ( R ) = P xx n|n −1 −

C1C1T

( C2 + R )

T

.

(23)

As already introduced, the matrix R quantifies the sensor accuracy. This implies that the higher the sensor measurement error, the
higher the matrix values. For simplicity, a scalar problem is considered here, so the matrices of (23) reduce to scalar values. If two
sensors are considered, where the first one is more accurate than
the second one, it results that R1 < R2, where Ri (with i = {1,2}) indicates the covariance value associated to the sensor i. Substituting
these two parameters in (23) the following inequality is verified
P1xx n|n < P2xx n|n .

(24)

As the state covariance matrix Pxx, associated to the sensor #2,
is bigger than the one associated to sensor #1, the state estimation
uncertainty associated to sensor #2 is greater than the uncertainty
associated to sensor #1. Considering the diagonal elements of Pxx,
it is possible to draw an n-dimensional ellipsoid that defines the

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