SSC00-VI-5
A Fast Prediction Algorithm of Satellite Passes
P. L. Palmer & Yan Mai
Surrey Space Centre
University of Surrey,Guildford,GU2 7XH, UK
Tel +44 1483 259278, Fax +44 1483 259503
Abstract
Low cost, fast access and multi-functional small satellites are b eing increasingly used to provide and exchange
information for a wide variety of professions. They are particularly useful, for example, as a resource in very
remote areas where they can provide useful information such as to rescue teams for changing conditions in a
disaster zone and monitoring the sea state to warn approaching shipping. Unlike terrestrial communication
systems, the receiver/transmitter in these di erent application areas needs to be powered on and contact to
sp ecialised satellites to exchange data at sp eci c time rather than consuming valuable power at all the time.
This, therefore, requires accurate knowledge of when these satellites will pass over the horizon of the given
lo cation over a timescale of months in some cases. On the other hand, long term orbit estimation with high
accuracy is also a key part for mission analysis and Earth observation op eration planning. The same algorithm is
also needed onb oard satellites for autonomous on-b oard data management. The principal diculty of predicting
satellite passes over such long timescales is to take account of the e ects of atmospheric drag.
In this pap er, we present a fast algorithm for the prediction of passes of a LEO satellite over any given lo cation
which provides high accuracy over a long p erio d. The metho d exploits sophisticated analytic mo dels of the orbit
and provides direct computation of rise-set times and nadir tracking without the need of orbit propagation for
hill climbing. This provides for a very small fast algorithm so more suitable for low-end computers and hand-held
sets. Since the atmospheric drag is the key factor that a ects the accuracy for long-term estimation for satellite
in LEO, this mo del not only includes secular p erturbation and p erio dic p erturbations, on the other hand a drag
mo del based on the well acknowledged NASA atmosphere statistics is incorp orated. Di erent from those in other
orbit prediction metho ds, for example, the most widely used SGP4, the drag mo del here has a variable parameter
which is sub ject to mo dify as time b eing on according to p erio dical atmosphere prop erties changing. Simulation
result shows it can provide quite accurate estimation for long lo ok-ahead p erio d.
1 Intro duction cost, fast access and multi-functional small satellites to
provide and exchange information for a wide range of
applications, which includes communication in a very
Small satellites are b ecoming more and more exible
remote area for changing conditions, disaster warning
and p owerful to enable military and civil applications
for approaching ships in the sea and so on. Di er-
suchaslow cost store-and-forward communication, re-
ent from ordinary communication stations which has
mote facility metering, disaster warning for global ship-
sucientpower supply, the communication mo dule in
ping service and some Earth Observation missions. The
these applications, say, a rescue team trekking in a
replacement of traditional spacecraft in these applica-
south American forest, has only very limited power
tions is motivated by the reality of shrinking govern-
capacity, therefore requires the receiver/transmitter to
mental budgets and commercial interest in deploying
be powered on and contact to the spacecraft at sp e-
low-cost small satellites for a wide variety of profes-
ci c time rather than consuming valuable p ower at all
sions.
the time. This, therefore, requires accurate estimation
In particular, there has b een big trend to use low
of when the satellites will start to be visible rise to
P.L.Palmer 1 14th Annual AIAA/USU Conference on Small Satellite
computers on-b oard spacecraft. Furthermore, in addi- a given lo cation on the Earth and similarly, the time
tion to taking account of secular p erturbation and p eri- when the satellite disapp ears from the horizon set,
o dic p erturbations, this algorithm includes a straight over a timescale of months in some cases. Meanwhile,
forward atmospheric drag mo del derived from the well long term and highly accurate orbit estimation, esp e-
acknowledged NASA atmosphere statistics in order to cially rise-and-set time computation, also plays a key
overcome the diculties involving long-term prediction part in the pre-request information for mission analysis
without incurring complex computational overhead. Dif- and on-b oard resources management in more general
ferent from those in other orbit prediction metho ds, for communication, Earth observation and scienti c space-
example, the most widely used SGP4, the drag mo del craft.
here has variable parameter which is sub ject to mo dify
One conventional way to solve this problem is to
as time b eing on according to p erio dical atmosphere
let the satellite run through its ephemeris, and check-
prop erties changing. Simulation result shows this drag
ing at each instant to see whether it just b ecomes vis-
mo del works satisfactorily in prediction over long times-
ible/invisible to a sp eci c ground lo cation. An orbital
cale.
propagation is advanced in time by some small time
The pap er will b e organised as follows: in section 2, increment, t, and a p ossibilitycheck is p erformed at
we describ e the rst phase of the new metho d, which each step, this kind of scheme is called tra jectory check-
is called \Coarse Search", it works in two-body, secular ing. This metho d, however, is fairly computational ex-
p erturbations arising from the Earth's oblateness and p ensive and therefore not suitable in the circumstances
atmospheric drag p erturbations. In section 3, weintro- where p owerful pro cessing resources are absent. Esco-
duce the second phase of the metho d, which is called bal [1] prop osed a faster metho d to solve this problem
\Re nement" which improves the accuracy of the new by developing a closed-form solution for the visibility
metho d. A metho d to up date the atmospheric drag p erio ds. He intro duced a single transcendental equa-
parameter consistently for long-term prediction is ad- tion as a function of the eccentric anomaly of the satel-
dressed in section 4. Simulation results are presented lite orbit which he called the control ling equation. Nu-
in section 5, as well as the comparison of CPU pro- merical metho ds were then used to nd the rise and
cessing time b etween the conventional metho d and this set times. The advantage of this equation is that it is
new metho d. Finally, in section 6, we set out our con- solved only once per orbital p erio d, in contrast with
clusions. the hundreds of times the Keplerian equation must b e
solved with the standard step-by-step technique of hill
climbing. The controlling equation, however, is only
valid for two-b o dy motion.
2 Coarse Search
Besides the control ling equation metho d, Lawton [7]
has develop ed another metho d to solve for satellite-
2.1 Fundamental Algorithm - Two-Bo dy
satellite and satellite-ground station visibility p erio ds
Analysis
for vehicles in circular or near circular orbits by ap-
proximating the visibility function t,byaFourier
series. More recently, Alfano [8] further develop ed the
t function to suit all orbital typ es. A signi cant
diculty, however, of predicting LEO satellite passes
over long p erio d is to take account of the e ects of at-
mospheric drag.
In this pap er, a fast algorithm for the rise-and-
set time prediction for LEO satellite is prop osed. It
provides high accuracy over a long p erio d. By some
further extension, this algorithm also has the p oten-
tial to provide maximum elevation angle time predic-
tion or nadir tracking problem solving, whichisvery
Figure 1: Satellite orbiting around the Earth showing crossings
useful for imaging planning using small satellites. The
of the Target Latitude Line TLL.
new metho d exploits sophisticated analytic mo dels of
the orbit and therefore provides direct computation of
We can easily estimate the satellite closest approach
rise-set times and nadir tracking. This makes it very
time bychecking the satellite ascending and descend-
suitable for low-end pro cessors in hand-held sets and
ing passage once resp ectively per day. Set T = 2=n
P.L.Palmer 2 14th Annual AIAA/USU Conference on Small Satellite
to b e the orbital period of the satellite and t the time
0
when the satellite rst crosses over a given latitude line
on the ascending pass see gure1. We call the circle
of constant latitude that runs through the target lo c-
ation the Target Latitude Line TLL. The key p oint
of our approach is to use the fact that for two-body
motion, a satellite will revisit exactly the same p ointin
an inertial co-ordinate system after each orbital p erio d
T see gure1 . This means that the satellite will
make another ascending-pass over the TLL at time
t + T . To simplify the discussion we shall ignore the
0
descending passages over the TLL and include them
again only at the end. Note, in this metho d, satellite
Figure 2: This gure shows the basic idea of our new metho d
p osition is expressed by the redundant epicycle co ordin-
for satellite rise-and-set times. When satellite longitude is within
ates: r; ; I; [2][3].
and + , the passes are visible.
T T
If the lo cation of a target on the Earth is ; ,
T T
where and are the geo detic longitude and lat-
T T
itude resp ectively, then the satellite will pass over the
+ 3
T v S T v
TLL every t + NT or t + N 2=n, where N is an
0 0
integer representing the numb er of satellite passages, n
In order to test whether the passes are visible, we
is satellite's orbital mean motion.
start from ' , if this is a visible pass we add
S T v
At time t the satellite is over the TLL and the
0
it to our coarse search list. When < we add
S T
initial longitude di erence between the satellite fo ot-
2 to . When > + we can compute the
S S T
print and target is = . After each
S T S T
di erence in longitude = + that will
v S T v
orbital p erio d the satellite revisits the TLL and the
bring it to within the visibility of the ground station.
Earth rotates under it bringing the target closer to the
Therefore we get the following formula for satellite vis-
satellite's longitudinal p osition. The satellite will see
ible estimation:
the target approaching by an amount ! Tor! 2=n,
where ! is the Earth's rotation rate. The Target-
n
v
Closest-Satel lite-Passage TCSP o ccurs when the lon-
N = +1 4
2 !
gitude di erence d is smaller than ! 2=n. Therefore
we obtain the following fundamental equation:
2.1.1 Finding Initialisation Argument of Lat-
= N! 2=n + d 1
itude
0
where d is the longitudinal di erence b etween the sub-
t and the target at TCSP.
satellite p oin z
So:
n
2 N =
2 !
kets implies the integer part.
where square brac TLL
In other words, the closest approachto the target ϕ
λ c
d < ! 2=n. Therefore as long
will o ccur when 0 0
y
as we know the initial passage time t of the TLL and
0
T,we can derive the p os-
the satellite's orbital p erio d i
sible closest approach time over long intervals of time.
e name the pro cedure of TCSP estimation as coarse W x
γ
search.
To determine the rise-and-set times of the satellite
Figure 3: Geometry of in ECI co ordinate. i is the orbit
over a given ground station, we need to set an angle 0
inclination and ' is the target latitude.
c
margin, describ ed in section 2.1.2. See gure2,
v
when satellite is visible, its longitude must satisfy
S
the following condition:
P.L.Palmer 3 14th Annual AIAA/USU Conference on Small Satellite
In the previous section, we p ointed out that we need drift which signi cantly changes the long term predic-
to know the initial passage time t of the TLL. In our tion of maximum elevation angle. We can adopt the
0
approach, we only need to calculate the corresp onding metho d wehave outlined in section 2.1 to take prop er
initial . Therefore we need the initial argument account of all these secular variations. In the following
S 0 0
of latitude for TLL. This is found from the spherical description we will intro duce the formulae for satellite
triangle shown in gure3. rise-and-set times.
Firstly we can easily add secular p erturbations to
sin'
C
the coarse search pro cedure for the e ect on argument
5 sin =
0
sinI
of latitude whichchanges the nodal period of satellite
comes back to the same TLL:
2.1.2 The Longitudinal O set Angle Margin
= 1 + 7
where is the co ecient of secular drifts in the epicycle
TS
equations [2] and = nt. So there is a change in for
h
each TLL crossing of =2=1 + .
R
second e ect is the precession of the orbital R The
θυ
_
. This moves the target away from the orbital
O plane
_
plane > 0. We can incorp orate this e ect into the
rotation rate of the Earth.
_
! = ! 8
ef f
In the epicycle description of the orbit[2], the vari-
Figure 4: This gure shows within longitude angle satellite
v
ation in is expressed as:
is visible to the ground target.
= + 9
0
The rise time of a satellite should o ccur when the satel-
lite, at a given orbital height, crosses the horizon plane.
where is the secular co ecient of plane precession[2][3].
In this case we set up another angle margin as shown
v
_
Hence = n
in gure4 and simpli ed the calculation for it.
We can incorp orate these results into equations 1
If the orbital radius of the satellite S is a= R + h
and 2 for the coarse search to get:
then:
R
cos = 6
v
10 =! nN
a
n
We therefore wish to estimate the times when the
Therefore:
satellite reaches the target longitude within . However,
v
b ecause this a simpli ed calculation for satellite longit-
n1 +
+1 11 N =
udal angle margin, to avoid missing some low passes we
2 ! n
reduce R by a xed fraction.
2.3 Accounting for Drag
2.2 Adding Secular Perturbations
Gravity is not the only force acting on the satellite.
A satellite under the in uence of an inverse square grav-
The most imp ortant other e ect comes from the Earth's
itational law has truly constant orbital elements. In
atmosphere, which still has a signi cant e ect on orbits
reality,however, there is a gradual change in the orbital
up to altitudes as high as 1000km. Because most of
elements due to the Earth's oblateness. The principal
our small satellites orbit at altitudes lower than this,
e ect of this is to intro duce a short p erio d oscillation
we need to consider the e ects of atmospheric drag.
of the orbital elements, which we can ignore in most
Drag is very dicult to mo del b ecause of the many
cases. The argument of p erigee, ! , and longitude of
factors a ecting the Earth's upp er atmosphere and the
the ascending no de, , however, exp erience a secular
satellite's attitude which a ects the cross sectional area.
In this pap er, we only consider the e ect of drag on the
P.L.Palmer 4 14th Annual AIAA/USU Conference on Small Satellite
satellite's argument of lattitude for the coarse search In gure5, we show the geometry of a satellite
and include the e ect on r in the re nement. In order pass. The target ground station, T , is lo cated on the
to test our result, the SGP4 mo del[10] has b een used surface of an oblate Earth, and the vector ~z is the
T
for drag mo delling. lo cal normal to the ground target surface. The po-
sition of the satellite is S . We have the p osition of
The e ect of drag on the argument of latitude can
b oth the target and the satellite in Earth centred, Earth
b e incorp orated into the epicycle equations as:
xed ECEF co ordinates [5] expressed in r , I , , ,
from the epicycle equations, from whichwe compute
2
~
= 1 + +1:5B 12 the slantvector P :
~ ~ ~
where B is the drag co ecient.
P = X X 15
S T
We start by nding the change in the epicycle phase
This gives the p osition of the satellite as seen from
over one no dal p erio d. By setting to b e 2 we nd
the target. The elevation angle is the angle measured
the solution for = from equation 12:
from the horizon up to the satellite. If this angle is h,
1 4
then:
p
13 =
1+
1+ 1+12B
~ ~
P Z = P sin h 16
T
Using this in equation 10 we obtain:
Therefore, we name a new control ling equation:
n
+1 14 N =
~ ~
! n
P Z
T
17 F = sin h =
P
This completes our discussion of the coarse search
~ ~
where wehave included the secular p erturbations and
F is a function of only through X ; while Z
s T
~
atmospheric drag.
and X are constantvectors in the ECEF co ordinate
T
~
system. X varies with b oth b ecause the satellite
s
moves along its orbits and through the Earth's rotation
in the transformation from ECI to ECEF co ordinates.
3 Re nement
It is obvious that the zero p oints of the elevation angle
h represents the zero of function F . Therefore, to
Having estimated the approach time to the target at
nd the rise-and-set time we just need to nd such
0
TLL, we now need a pro cedure that will re ne this
that F =0.
estimate to an application set tolerance. For this we
extend Escobal's [1] approach to determine the rise-
and-set time, byintro ducing a new control ling equation
~
3.1 Computation of Satellite lo cation X
S
based on the epicycle equations.
epicycle equations which express r , I , , as
ZE The
functions of time can b e written as:
r = a1 + A cos +a sin 18
ZT p
+ a cos 2 2B
T r P
XT
I = I + 1 cos 2 19 I
S 0
= + + sin 2 20 XS 0 O
YE
2A
= + [sin + sin ] 21
p p
2
3
2