Aerospace and Electronic Systems - November 2018 - 7

De Vivo, Battipede, and Gili

Figure 2.

Mesh geometry based on camera footprint and image formation by means of homogeneous transformation f. In the figure there are also the image place
xc - yc, the DEM matrix with the coordinate in the IRF (XI DEM) and CRF (XC DEM).

x

y
z





=




  
 =
  

=

−
x

x

y

z









(2)

and
are, respectively, the rotational matrix and the
where
translational vector (resolved in the camera reference frame) which
take into account the different orientation and origins of the two
reference frames. P is the vector containing the coordinates of the
generic point Pij of the DEM in the Image Reference Frame (IRF).
The IRF is the reference frame used to represent the DEM. It could
be either a Geodetic or a North-East-Down (NED) or ECEF reference frame. The homogeneous transformation is given by

=


=


−
x





−
x





−
x





(3)

projects the points from the inertial to satellite (body)
where
reference frame,
from body to gimbal, and
from gimbal to
camera. The rotation matrix
is given by the sequence of Euler
rotation 313
 φ ψ− φ θ ψ

= − φ θ ψ − φ ψ

θ ψ

the matrix
NOVEMBER 2018

is

φ θ ψ
φ θ ψ

−

+
−
θ ψ

φ ψ
φ ψ



φ θ

θ 

φ θ

(4)

 θ

= −

 − θ

ψ

θ

z

θ

ψ
ψ

ψ

z

−

θ

θ

z

z

ψ

θ

z







(5)

where θel is the gimbal elevation angle, defined in the xb - zb plane,
and ψaz is the gimbal azimuth angle defined in the xb - yb plane.
Considering that the satellites are generally off-nadir with respect
to the UAV platform, their Field of View (FOV) needs to be centered around P. This is achieved by introducing a gimbal rotation
that aligns the camera principal axis with the vector between the
camera and the point P. This rotation justifies the use of
in (3).
The angles ψaz and θel used in (5) are calculated by simple geometric relations as follows

ψz=
θ =

2

y −y
x −x
z −z
−

−

π

(6)

2

where the arctan 2 indicates the 4-quadrant inverse tangent. Considering that (6) is applied only to the satellites in the visibility
cone, the maximum value of ψaz is bounded by the visibility cone
aperture shown in Figure 1. This is true if the main lobe of the
satellite antenna points toward the Earth center. The matrix
is
constant if the relative motion between the antenna and gimbals is
neglected

IEEE A&E SYSTEMS MAGAZINE

7



Aerospace and Electronic Systems - November 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems - November 2018

Contents
Aerospace and Electronic Systems - November 2018 - Cover1
Aerospace and Electronic Systems - November 2018 - Cover2
Aerospace and Electronic Systems - November 2018 - Contents
Aerospace and Electronic Systems - November 2018 - 2
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