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
p.palmer@eim.surrey.ac.uk

Abstract
Low cost, fast access and multi-functional small satellites are being 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 dierent application areas needs to be powered on and contact to
specialised satellites to exchange data at speci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
location 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 operation planning. The same algorithm is
also needed onboard satellites for autonomous on-board data management. The principal diculty of predicting
satellite passes over such long timescales is to take account of the eects of atmospheric drag.
In this paper, we present a fast algorithm for the prediction of passes of a LEO satellite over any given location
which provides high accuracy over a long period. The method exploits sophisticated analytic models 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 aects the accuracy for long-term estimation for satellite
in LEO, this model not only includes secular perturbation and periodic perturbations, on the other hand a drag
model based on the well acknowledged NASA atmosphere statistics is incorporated. Dierent from those in other
orbit prediction methods, for example, the most widely used SGP4, the drag model here has a variable parameter
which is subject to modify as time being on according to periodical atmosphere properties changing. Simulation
result shows it can provide quite accurate estimation for long look-ahead period.

1 Introduction cost, fast access and multi-functional small satellites to

provide and exchange information for a wide range of
Small satellites are becoming more and more exible applications, which includes communication in a very
and powerful to enable military and civil applications remote area for changing conditions, disaster warning
such as low cost store-and-forward communication, re- for approaching ships in the sea and so on. Dier-
mote facility metering, disaster warning for global ship- ent from ordinary communication stations which has
ping service and some Earth Observation missions. The sucient power supply, the communication module in
replacement of traditional spacecraft in these applica- these applications, say, a rescue team trekking in a
tions is motivated by the reality of shrinking govern- south American forest, has only very limited power
mental budgets and commercial interest in deploying capacity, therefore requires the receiver/transmitter to
low-cost small satellites for a wide variety of profes- be powered on and contact to the spacecraft at spe-
sions. cic time rather than consuming valuable power at all
the time. This, therefore, requires accurate estimation
In particular, there has been 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

a given location on the Earth and similarly, the time computers on-board spacecraft. Furthermore, in addi-
when the satellite disappears from the horizon (set), tion to taking account of secular perturbation and peri-
over a timescale of months in some cases. Meanwhile, odic perturbations, this algorithm includes a straight
long term and highly accurate orbit estimation, espe- forward atmospheric drag model derived from the well
cially rise-and-set time computation, also plays a key acknowledged NASA atmosphere statistics in order to
part in the pre-request information for mission analysis overcome the diculties involving long-term prediction
and on-board resources management in more general without incurring complex computational overhead. Dif-
communication, Earth observation and scientic space- ferent from those in other orbit prediction methods, for
craft. example, the most widely used SGP4, the drag model
One conventional way to solve this problem is to here has variable parameter which is subject to modify
let the satellite run through its ephemeris, and check- as time being on according to periodical atmosphere
ing at each instant to see whether it just becomes vis- properties changing. Simulation result shows this drag
ible/invisible to a specic ground location. An orbital model works satisfactorily in prediction over long times-
propagation is advanced in time by some small time cale.
increment,
t, and a possibility check is performed at The paper will be organised as follows: in section 2,
each step, this kind of scheme is called trajectory check- we describe the rst phase of the new method, which
ing. This method, however, is fairly computational ex- is called \Coarse Search", it works in two-body, secular
pensive and therefore not suitable in the circumstances perturbations arising from the Earth's oblateness and
where powerful processing resources are absent. Esco- atmospheric drag perturbations. In section 3, we intro-
bal [1] proposed a faster method to solve this problem duce the second phase of the method, which is called
by developing a closed-form solution for the visibility \Renement" which improves the accuracy of the new
periods. He introduced a single transcendental equa- method. A method to update the atmospheric drag
tion as a function of the eccentric anomaly of the satel- parameter consistently for long-term prediction is ad-
lite orbit which he called the controlling equation. Nu- dressed in section 4. Simulation results are presented
merical methods were then used to nd the rise and in section 5, as well as the comparison of CPU pro-
set times. The advantage of this equation is that it is cessing time between the conventional method and this
solved only once per orbital period, in contrast with new method. Finally, in section 6, we set out our con-
the hundreds of times the Keplerian equation must be clusions.
solved with the standard step-by-step technique of hill
climbing. The controlling equation, however, is only
valid for two-body motion. 2 Coarse Search
Besides the controlling equation method, Lawton [7]
has developed another method to solve for satellite-
satellite and satellite-ground station visibility periods
2.1 Fundamental Algorithm - Two-Body
for vehicles in circular or near circular orbits by ap- Analysis
proximating the visibility function (t) , by a Fourier
series. More recently, Alfano [8] further developed the
(t) function to suit all orbital types. A signicant
diculty, however, of predicting LEO satellite passes
over long period is to take account of the eects of at-
mospheric drag.
In this paper, a fast algorithm for the rise-and-
set time prediction for LEO satellite is proposed. It
provides high accuracy over a long period. By some
further extension, this algorithm also has the poten-
tial to provide maximum elevation angle time predic-
tion (or nadir tracking problem solving), which is very Figure 1: Satellite orbiting around the Earth showing crossings
useful for imaging planning using small satellites. The of the Target Latitude Line (TLL).
new method exploits sophisticated analytic models 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 by checking the satellite ascending and descend-
suitable for low-end processors in hand-held sets and ing passage once respectively per day. Set T (= 2=n)

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to be the orbital period of the satellite and t0 the time
when the satellite rst crosses over a given latitude line
on the ascending pass (see gure(1)). We call the circle
of constant latitude that runs through the target loc-
ation the Target Latitude Line (TLL). The key point
of our approach is to use the fact that for two-body
motion, a satellite will revisit exactly the same point in
an inertial co-ordinate system after each orbital period
T (see gure(1) ). This means that the satellite will
make another ascending-pass over the TLL at time
(t0 + T ). To simplify the discussion we shall ignore the
descending passages over the TLL and include them
again only at the end. Note, in this method, satellite
position is expressed by the redundant epicycle coordin- Figure 2: This gure shows the basic idea of our new method
ates: (r; ; I;
) [2][3]. for satellite rise-and-set times. When satellite longitude is within
T ,  and T +  , the passes are visible.
If the location of a target on the Earth is (T ; T ),
where T and T are the geodetic longitude and lat-
itude respectively, then the satellite will pass over the
TLL every t0 + NT (or t0 + N 2=n), where N is an T , v  S  T + v (3)
integer representing the number of satellite passages, n
is satellite's orbital mean motion. In order to test whether the passes are visible, we
start from S ' T , v , if this is a visible pass we add
At time t0 the satellite is over the TLL and the it to our coarse search list. When S < T ,  we add
initial longitude dierence between the satellite foot- 2 to S . When S > T +  we can compute the
print S and target T is
 = S , T . After each dierence in longitude
v = S , (T + v ) that will
orbital period 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 position. The satellite will see ible estimation:
the target approaching by an amount ! T or ! 2=n,
where ! is the Earth's rotation rate. The Target-  
Closest-Satellite-Passage (TCSP) occurs when the lon- N =
2v !n + 1 (4)
gitude dierence d is smaller than !2=n. Therefore 
we obtain the following fundamental equation:

 = N! 2=n + d (1) 2.1.1 Finding Initialisation Argument of Lat-
itude 0
where d is the longitudinal dierence between the sub-
satellite point and the target at TCSP. z

So:  
N =
2 !n (2)

where square brackets implies the integer part. TLL

In other words, the closest approach to the target

will occur when 0  d < ! 2=n. Therefore as long λ0
ϕc

as we know the initial passage time t0 of the TLL and

y

the satellite's orbital period T, we can derive the pos- i

sible closest approach time over long intervals of time.

We name the procedure of TCSP estimation as coarse x

search. γ

To determine the rise-and-set times of the satellite

over a given ground station, we need to set an angle Figure 3: Geometry of 0 in ECI coordinate. i is the orbit
margin, v (described in section 2.1.2). See gure(2), inclination and 'c is the target latitude.
when satellite is visible, its longitude S must satisfy
the following condition:

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In the previous section, we pointed out that we need drift which signicantly changes the long term predic-
to know the initial passage time t0 of the TLL. In our tion of maximum elevation angle. We can adopt the
approach, we only need to calculate the corresponding method we have outlined in section 2.1 to take proper
initial S0 . Therefore we need 0 (the initial argument account of all these secular variations. In the following
of latitude) for TLL. This is found from the spherical description we will introduce the formulae for satellite
triangle shown in gure(3). rise-and-set times.
Firstly we can easily add secular perturbations to
sin0 = sin'C (5) the coarse search procedure for the eect on argument
sinI of latitude  which changes the nodal period of satellite
comes back to the same TLL:
2.1.2 The Longitudinal Oset Angle Margin
 = (1 + ) (7)
where  is the coecient of secular drifts in the epicycle
T S
equations [2] and  = nt. So there is a change in  for
h each TLL crossing of
 = 2=(1 + ).
The second eect is the precession of the orbital
R

θυ
_ This moves the target away from the orbital
R

plane (
).
plane (
_ > 0). We can incorporate this eect into the
O

rotation rate of the Earth.

!eff = ! ,
_ (8)
Figure 4: This gure shows within longitude angle v satellite In the epicycle description of the orbit[2], the vari-
is visible to the ground target. ation in
is expressed as:
The rise time of a satellite should occur when the satel-
=
0 +  (9)
lite, at a given orbital height, crosses the horizon plane. where  is the secular coecient of plane precession[2][3].
In this case we set up another angle margin v as shown Hence
_ = n
in gure(4) and simplied the calculation for it.
If the orbital radius of the satellite S is a(= R + h) We can incorporate these results into equations (1)
then: and (2) for the coarse search to get:
cosv = Ra (6)
 = (! , n)N
n (10)
We therefore wish to estimate the times when the
satellite reaches the target longitude within v . However, Therefore:
because this a simplied calculation for satellite longit- 
 n(1 + ) 
udal angle margin, to avoid missing some low passes we N = 2 (! , n) + 1 (11)

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 important other eect comes from the Earth's
itational law has truly constant orbital elements. In atmosphere, which still has a signicant eect 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,
eect of this is to introduce a short period oscillation we need to consider the eects of atmospheric drag.
of the orbital elements, which we can ignore in most Drag is very dicult to model because of the many
cases. The argument of perigee, !, and longitude of factors aecting the Earth's upper atmosphere and the
the ascending node,
, however, experience a secular satellite's attitude which aects the cross sectional area.
In this paper, we only consider the eect 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 gure(5), we show the geometry of a satellite
and include the eect on r in the renement. In order pass. The target ground station, T , is located on the
to test our result, the SGP4 model[10] has been used surface of an oblate Earth, and the vector ~zT is the
for drag modelling. local normal to the ground target surface. The po-
The eect of drag on the argument of latitude can sition of the satellite is S . We have the position of
be incorporated into the epicycle equations as: both the target and the satellite in Earth centred, Earth
xed (ECEF) coordinates [5] expressed in r, I ,
, ,
 from the epicycle equations, from which we compute
 = (1 + ) + 1:5B2 (12) the slant vector P~ :
where B is the drag coecient. P~ = X~ S , X~ T (15)
We start by nding the change in the epicycle phase
 over one nodal period. By setting  to be 2 we nd This gives the position of the satellite as seen from
the solution for (=
) from equation (12): the target. The elevation angle is the angle measured
from the horizon up to the satellite. If this angle is h,

 = 1 4+ 
p 1 (13) then:
1 + 1 + 12B
P~
Z~ T = P sin h (16)
Using this in equation (10) we obtain:

  n  Therefore, we name a new controlling equation:
N =
 ! , n + 1 (14) ~ ~

F () = sin h = P
PZT (17)
This completes our discussion of the coarse search
where we have included the secular perturbations and F is a function of  only through X~ s ; while Z~ T
atmospheric drag. and X~ T are constant vectors in the ECEF coordinate
system. X~ s varies with  both because the satellite
moves along its orbits and through the Earth's rotation
3 Renement in the transformation from ECI to ECEF coordinates.
It is obvious that the zero points of the elevation angle
Having estimated the approach time to the target at h represents the zero of function F () . Therefore, to
TLL, we now need a procedure that will rene this nd the rise-and-set time we just need to nd 0 such
estimate to an application set tolerance. For this we that F () = 0.
extend Escobal's [1] approach to determine the rise-
and-set time, by introducing a new controlling equation
based on the epicycle equations. 3.1 Computation of Satellite location X~ S
ZE The epicycle equations which express (r, I ,
, ) as
functions of time can be written as:
ZT r = a(1 + ) , A cos( , p ) + a sin  (18)
T + a
r cos 2 , 2B
I = I0 +
I (1 , cos 2)
P
XT 00
11
11
00
00S
11 (19)
XS
=
0 +  +
 sin 2 (20)
 =  + 22A [sin( , p ) + sin p ]
O

(21)
YE

, 2(1 , cos  ) +
 sin 2 + 23 B 2

XE

Figure 5: This gure describes the geometry of ground target where we have included the eects of atmospheric
(T ) and satellite (S ) in ECEF coordinate. drag [paper in preparation], and
 = (1 + ) (22)

P.L.Palmer 5 14th Annual AIAA/USU Conference on Small Satellite

,  and  are the coecients for secular perturbation;
 is long periodic perturbation coecients;
repres-
ents the short periodic terms; B is the epicycle drag
coecient.
We dene the satellite position ( , ) on the orbital
plane using Cartesian co-ordinates with the  axis along
the ascending node of the orbit. Hence:
 = r cos 
 = r sin 
X~ S can be expressed in ECEF coordinates as X~ S =
(XE ; YE ; ZE ), where: Figure 6: This gure shows the probability density distribu-
8 tion of real drag data for NOAA-10 satellite from 1/94 to 12/97
< XE =  cos( ,
) +  cos I sin( ,
) exclusive solar maximum.
: YE = , sin(Z,E
) +  cos I cos( ,
) (23)
=  sin I
and  is the local ephemeris time (the angle between
the rst point of Aries and the XE axis in the ECEF
frame). These equations together describe the depend-
ence of F on .

4 Drag Variability Modelling

The predictions for satellite passes over long timescales
are sensitive to the eects of atmospheric drag. Al-
though we have taken account of the drag modelling Figure 7: This gure shows the probability density distribution
the variability of the drag causes signicant timing er- of one-month smoothed drag data for NOAA-10 satellite over a
rors for satellite passes over long look ahead times. In look ahead time of 1 year.
order to reduce the variability of atmospheric drag we
have averaged the historic drag data over time windows
to smooth it. The smoothed drag data then represents We now show how the epicycle model can cope with
real variability in the drag eect on the satellites, and a distribution of drag rates. Let the length of the
we have assumed that this historic record is a reason- time window over which we average be dT and sup-
ably accurate model for future variability. The drag pose that the historic data record starts at a time t0
model we have employed so far uses a constant drag (see gure(8)). At some later epoch tN we read the
coecient. This drag coecient represents the average NORAD 2-line element. We can dene a dimensionless
drag rate over the look ahead time. In reality, however, time from the start epoch t0 :  = n(t , t0 ). When
the drag rate has some probability density distribution t = tN ,  = N . If we dene d
= ndT then we can
and this is represented by the tabulated drag rates from determine which element of the table B [k] is appro-
smoothing the drag data in time windows. As long as priate for a given epoch t from: k = d
. We also
the look ahead time is much longer than the window
size over which we have averaged, the distribution of dene a dimensionless time from the Norad epoch tN
drag rate for the prediction should accurately re ect as:  = n(t , tN ). This is the time we have used in the
the true drag distribution. Figure (6) and (7) show the epicycle equations described in the previous section.
probability density distribution for both the historic We start by considering the drag as the average drag
drag data and the smoothed data. We can see that for rate over the look ahead time. Let us denote this by
the one year look ahead time, the probability distribu- B . We replace the symbol  by  for this constant
tion of smoothed drag data is very similar as the real drag rate model. Replacing this into equation (21) and
data. This conrms that for a single pass prediction ignoring periodic terms gives:
over the look ahead time, smoothed drag data we will
use has a similar distribution as the real drag data.

P.L.Palmer 6 14th Annual AIAA/USU Conference on Small Satellite

 t0 tN
dT
+ B [n]( , n,k )]
nX,1
, 32 [ B [r](2r,k+1 , 2r,k )
δ=0 d∆
δN

α 0 α1
=0
α2 α3 α4 α5
r=k
+ B [n](2 , 2n,k )]
Figure 8: This gure shows the time blocks for long term pre- We can write equation (26) as:
diction.
(1 + ) + 32 B
 2 = 3 B [n]2 + Yn  + Xn
2 (27)
 = (1 + ) + 32 B
 2 (24) Let:
where =0 when  = 0 = 0. To relate  and  we
wish  to be xed -ie satellite position should be kept
Xn = 23 B [n]2n,k
consistent: nX,1
, 32 B [r](2r,k+1 , 2r,k )
(1 + ) + 23 B
 2= (25) r=k
Z Z Yn = 1 +  , 3B [n]n,k
(1 + ) + 3 B (0 )d0 , 3 B (0 )0 d0 nX ,1
0 0 + 3 B [r](r,k+1 , r,k )
r=k
So if B is constant, then  = . To check for  we
replace the integral by a sum, with intervals dT or d
. Then we will have
R R
Let I = 3 0 B (0 )d0 , 3 0 B (0 )0 d0 . If 0 <
 < 1 , then B (0 ) = B [k], where 0 = 0: Xn+1 , Xn = 23 n,k+1 (B [n + 1] , B [n]) (28)
Yn+1 , Yn = ,3n,k+1 (B [n + 1] , B [n]) (29)
I = 32 B [k]2
So we keep track of the current value of n and when
If m <  < m+1 , then we have: n > n0 , we increment X and Y using the above relation.
mX,1 n,k = 0 + (n , k)d

I = 3[ B [k + r](r+1 , r ) (26) n+1 = n + d

r=0
+ B [k + m]( , m )] From equation (27), we can compute  from . Be-
mX,1
, 32 [ B [k + r](2r+1 , 2r ) cause:
r=0  2 = 3 B2 + Y  + X
(1 + ) + 23 B 2
+ B [k + m](2 , 2m )]
The range of  can be determined as follows: Therefore:
Given an input value of  we compute  =  + N , = (30)
2
then compute:
 +   p 2 2X , 2(1 + ) , 3B 2
Y + 3B (2X , 2(1 + ) , 3B ) + Y
n = d
N ; n  k

If n = k:
5 Test and Result
I = 32 B [n]2 5.1 Results for Two-body and Secular
Perturbation Expansion
If n > k:
,1
nX For many practical problems, the approximation of two-
I = 3[ B [r](r,k+1 , r,k ) body motion is sucient, especially if two closely neigh-
r=k bouring points on a trajectory are under investigation.

P.L.Palmer 7 14th Annual AIAA/USU Conference on Small Satellite

 drag can be ignored the dierence between our predic-
tion and SGP4 arises from the fact that the accuracy
of SGP4 is only 10,6 and there is a small drift of 
between the epicycle equations and SGP4 which builds
up to a signicant error. This demonstrates that over a
look-ahead time of a few days, when drag eects can be
ignored, we have achieved one-second timing accuracy.

Figure 9: The black curve shows the timing error of the two-
body prediction when compared with the SGP4 model, while the
grey curve shows the error when J2 is incorporated.

However, in our case for the long term prediction of

satellite passes for communication, we could not ig-
nore the cumulative eect of the gradual variation of
elements from their two-body values. In gure(9) we
show the prediction of our method compared with the
SGP4 model [10], the black curve clearly indicates that Figure 10: This gure shows timing errors when short and long
only after a few hours the timing error of our pre- variations are included, when comparing with SGP4.
diction based on two-body theory is already up to 8
seconds, and within one day the timing error is around
one minute!
To reduce the timing errors, we have included the 5.2 Results for Including Atmospheric
secular eects into our coarse search. Unlike Escobal's Drag
original controlling equation our function F () not only
includes secular drift but also has short-and-long peri- We next consider the drag compensation that we in-
odic perturbations taken into account. We present in troduced in section 2 and 3. In gure(11), we show
table(1) a comparison of the epicycle prediction with the timing errors compared with SGP4, now with drag
an accurate propagator [15] to look at the timing errors included in the model. Both predictions are based on
from the prediction when atmospheric drag is ignored. the same set of initial conditions taken for the same
This table shows that the timing errors are as small as NORAD le and the prediction extend over 100 days.
0.15 seconds for a look-ahead time of almost 300 days. With a look-ahead time of 100 days, the timing error
has now been reduced to about 2 seconds, while without
drag compensation, for the same accuracy level, the
Look-ahead Time [day] Timing Error [sec] look-ahead time is only one week. So for communica-
0.96 8.7e-3 tion applications, we can predict rise-and-set times for
4.53 2.2e-2 up to one month with sucient accuracy.
9.26 3.8e-2 To remove the drift errors in SGP4 we performed






one another experiment where we compared the pre-
298.99 1.5e-1 dictions of our algorithm with itself, using two dierent
NORAD les. The separation in time between the two
Table 1: Table of timing errors as a function of look-ahead time, NORAD les was anything up to 40 days, and the tim-
comparing the predictions with an accurate orbit propagator. ing errors for the SAME pass are shown in gure(12).
One of these predictions was based upon a NORAD
data set from just before the pass. The dates used for
In gure(10) we show a comparison of our predic- this experiment were from May to July of 1997. The
tion with SGP4, using an exhaustive search approach variability in prediction time is due to the variability
we see that the timing dierence between our method of atmospheric drag.
and SGP4 is less than one second for two months look-
ahead time. As pointed out in [16], when atmospheric

P.L.Palmer 8 14th Annual AIAA/USU Conference on Small Satellite

 particular pass on a particular day and trying to predict
the time of this pass using dierent NORAD 2 line ele-
ments from the historical archive ranging from 1 day
before hand to 550 days before hand. The results of
each of these experiments are presented in the form of
two graphs, gure (13), (14), (15). The rst one shows
a histogram of the timing errors. The second one dis-
plays the timing error for each prediction as a function
of look ahead time. All the timing errors are expressed
in minutes and the look ahead times are in days.
A large number of experiments were performed, but
here we only present a selection of the results while
other results show consistent trend as these presented
ones:
Figure 11: This gure shows timing errors when atmospheric
drag is included, when comparing with SGP4.

Figure 12: This gure shows the prediction errors of a single

pass using our method, for look-ahead times of up to 40 days
using NORAD data at dierent epochs.

5.3 Test and Result for Drag Variability

Modelling Figure 13: Histogram of timing errors and timing errors as a
function of look ahead time for a pass in September 1994.
Although this method is targeted for small satellite ap-
plication, we mainly use NOAA satellites data to test September 1994 This experiment was chosen be-
the performance for long term prediction. This is be- cause the 18 months prior to September 1994 all lie
cause: 1) this satellite family has very typical 800- in the period of solar minimum. In this case the drag
900km Sun Synchronous Orbit, which is common for rate on the satellites remains fairly stable and so good
small satellites. 2) NOAA satellites have fairly com- estimates of the pass times are expected. The results
plete back data log over long period which is suitable in gures (13) show that prediction times for this pass
for this test. In particular, we performed the major are accurate to within 4 minutes over the 18 month
tests on NOAA-10 as this satellite has remained in or- interval.
bit for longer than 11 years and therefore the historic
record of NORAD data covers a complete solar cycle. March 1995 This experiment also covers the period
This enables us to test the performance of the method of solar minimum, and so represents a fairly stable drag
at dierent phases of the solar cycle. regime. During this period, however, the NORAD ele-
The experiments performed consisted of xing a ments showed some drag variation which has caused

P.L.Palmer 9 14th Annual AIAA/USU Conference on Small Satellite

Figure 14: Histogram of timing errors and timing errors as a Figure 15: Histogram of timing errors and timing errors as a
function of look ahead time for a pass in March 1995. function of look ahead time for a pass in July 1997.

the prediction times, for long term predictions, to in- long look ahead time, fairly rapid changes in the drag
crease to 8 minutes in 18 months. The results are still rate were experienced by the satellite. We can see the
accurate to within 2 minutes for over 1 year look ahead eect of this on the prediction data in upper graph
time. The histogram shown in gure (14) shows a very of gure (17), in which drag parameter is a constant.
strong peak at small times. The graph shows that the timing error increases rap-
idly after 80 days. It means from the begining we have
over estimated the drag parameter because the previ-
July 1997 Another example of a prediction during ous period was in solar maximum. However, after we
solar minimum was the July 1997 pass. In gure (15) modelled drag using the method introduced in section
we show once again similar results to the previous ex- 4, the lower graph of gure (17) clearly shows the tim-
periment. These results were the best case found and ing accuracy has been greatly improved. By adjusting
show timing errors less than 1 minute over 18 months. the drag parameter according to the smoothed drag
The plot as a function of look ahead time shows a series statistic data, we avoided over-estimating drag para-
of sudden jumps. This is an artifact of the simple me- meter too much - the prediction timing error is reduced
dian ltering that has been done on the historic drag to less than 4 minutes after 300 days even when solar
data. maximum period is included. Therefore above exper-
We then continued this experiment to show the pre- iment conrms that this drag modelling method has
diction errors for look ahead times of up to 3 years. The substaintially decreased the eect of the variability of
results of this extended experiment are presented in g- atmospheric drag.
ure (16) as a function of look ahead time. We see from Above experiments show that the long term pre-
this experiment that predictions of order 4 minutes diction of satellite passes over a given ground target
can be maintained over this time period. These ex- can be made to high accuracy even for long look ahead
periments provide a fair representation of performance times up to 18 months or more. This kind of perform-
during solar minimum. ance, indeed, is satisfactory to low-cost communication
applications as addressed in the introduction.
June 1993 After couple of experiments concerning
solar minimum period, the next experiment extends
back into the previous solar maximum. Therefore, for

P.L.Palmer 10 14th Annual AIAA/USU Conference on Small Satellite

Figure 16: The timing errors as a function of look ahead time
for a 3 year prediction.

5.4 Result for Computation Time

The algorithm is several order of magnitude faster to
run than the exhaustive search using SGP4 we have
employed. In table(2) we present some timings for the
estimation on a Pentium II. These timings are su-
ciently short for this algorithm to be used on hand
held receivers and are suciently accurate to control
imaging devices on satellites.
Current Programme Proposed Method Figure 17: Timing errors for a single pass in June 1993: The
SGP4 two-body Secular upper graph shows the result when drag parameter B is a con-
786sec 2.86sec 3.71sec stant; The lower graph shows by using drag modelling, the timing
error has been signicantly reduced.
Table 2: Processing time on a Pentium II, averaged over 10,000
experiments. in one year period.
We have shown elsewhere [16] how to translate NORAD
elements, which are freely available for all traded satel-
6 Conclusions lites over the Internet, to epicycle elements. Hence this
method can be used by any system that has access to
these NORAD les.
In this paper, we have introduced a new method to
predict the passes of satellite to a specic target on the
ground which is useful for solving the satellite visibil-
ity problem. We have rstly described a coarse search References
phase of this method including two-body motion, secu-
lar perturbation and atmospheric drag. We have then [1] Pedro Ramon Escobal, \Methods of Orbit De-
described the second phase { renement, which uses a termination", pub. Wiley, New York, 1976.
further developed controlling equation F () = 0 based [2] Y. Hashida & P. Palmer \Epicyclic Motion of
on the epicycle equations. We have shown that ignor- Satellites about an Oblate Earth", submitted to
ing drag eects, we can achieve timing accuracies of AIAA J. Guidance, Control & Dynamics, 2000.
1 second for look-ahead times of 60days. When drag
compensation is included, we provide suciently accur- [3] Y. Hashida & P. Palmer \Post Epicyclic Motion of
ate timing estimates, on the order of a few seconds, for Satellite under J2 Potential" submitted to AIAA
over one month ahead. For most low-cost communic- J. Guidance, Control & Dynamics, 2000.
ation applications using small satellites, the tolerable
error is only required on the order of a few minutes, [4] Fouquet M., Sweeting M.N.,University of Surrey,
which lead this method valid to predict satellite passes UK. \Earth Observation Using Low Cost SSTL

P.L.Palmer 11 14th Annual AIAA/USU Conference on Small Satellite

 Microsatellites", Symposium on Eearth Observa- [17] Chen Fung-Yun, et al, \Composite Satellite Sys-
tion Sensors & Technology, 47th IAF Congress, 7- tem for Observation of Earth Environment and
11 October 1996, Beijing China. International Cooperation", IAA-96-IAA.3.3.02,
47th IAF Congress, Oct 1996, Beijng, China.
[5] Vladimir A. Chobotv, \Orbital Mechanics (Second
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[6] Shea JJ, \On-orbit Determination of Focal 47th IAF Congress, Oct 1996, Beijing, China.
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[7] Lawton, J.A., \Numerical Method for Rapidly De-
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[8] Salvatore Alfano , David Negron, Jr., \Rapid De-
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[9] Unwin M.J., University of Surrey. \The PoSAT
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[14] Fouquet M., Sweeting M.N., \UoSAT-12 Min-
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[15] P.L. Palmer, S.J. Aarseth, S. Mikkola, and Yoshi
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[16] Yan Mai, P.L. Palmer, \On the Conversion of
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2000.

P.L.Palmer 12 14th Annual AIAA/USU Conference on Small Satellite