IEEE - Aerospace and Electronic Systems - December 2021 - 8

Estimation of the CubeSat's Available Energy for Free-Orientation Scenario
In fact, the authors numerically integrate the functions
S not in a spherical, but in a quasi-cylindrical coordinate
system
Seff ¼ 0
Rp=2 Rp=2
Rp=2 Rp=2
Sx þ Sy þ Sz d' du
d' du
¼ 1:45
and when integrating in a spherical one, we should take
into account the size of the differential element of the area
dS ¼ R2sinðuÞd' du, then the unbiased estimate gives
Seff ¼ 0
Rp=2 Rp=2
Rp=2 Rp=2
Sx þ Sy þ Sz sinðuÞ d' du
sinðuÞ d' du
¼ 1:5
(1)
where R is the radius of the unit sphere. In the denominator,
we will get the area of the sphere, bounded by one
octant.
In addition, if the distribution ofthe area coefficients S
differs from the normal, then the most probable value
(mode) will not be equal to the mean. Moreover, the free
satellite is still spinning while in orbit, not frozen.
As we can see, the power estimations differ, although
Figure 1.
Average power estimation for 1U CubeSat, from [13].
the value for Seff of 1.38. It is clear that such an estimate
will be biased relative to the most probable power
value-it is unlikely that the positions of the CubeSats in
Figure 1 are equally probable.
In the technical report [22], the 1U CubeSat is
assumed to be randomly rotated relative to the sunlight
direction and retaining this attitude (given by the angles '
and u, see calculation scheme in Figure 2). The authors
estimate the mean power statistically-for each combination
of angles ð'; uÞ from the range 0... þ 90, the matrix
of effective area coefficient Sð'; uÞ is calculated (in fact,
it is the sum of the projections faces on the plane perpendicular
to the sunlight) and, then, find the arithmetic mean
of the effective area. Face projections are calculated using
the formulas
Sxð'; uÞ¼ cos ð'Þ sin ðuÞ
Syð'; uÞ¼ sin ð'Þ sin ðuÞ
SzðuÞ¼ cos ðuÞ
which represents spherical coordinates (hereinafter they
will be written without arguments). The authors give an
estimate for Seff of 1.45, but it is also biased.
The point is that when using evenly spaced angles
' and u [22, App. A], the top area of the sphere will
be more densely filled with calculated dots (as shown
in the color in Figure 2) and this will shift the average
value.
8
Figure 2.
Average power estimation scheme for 1U CubeSat, from [22].
IEEE A&E SYSTEMS MAGAZINE
DECEMBER 2021
not much. But the small CubeSat's format limits the size
of the SA, and the low earth orbits (usual for CubeSats)
have the longest relative shadow duration among other
types of orbits-as a result, there is always not enough
energy on low-orbit small satellite. The accuracy and adequacy
of the energy budget estimate becomes very important
here.
Using a similar statistical approach, we will estimate
the most probable generated power value of the free-oriented
CubeSat considering previously unknown attitude
parameters.
IEEE - Aerospace and Electronic Systems - December 2021

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