Recent Researches in Power Systems and Systems Science
COMPARISON BETWEEN TWO METHODS TO CALCULATE
THE TRANSITION MATRIX OF ORBITAL MOTION
ANA PAULA MARINS CHIARADIA1
HÉLIO KOITI KUGA2
ANTONIO FERNANDO BERTACHINI DE ALMEIDA PRADO3
1
Grupo de Dinâmica Orbital &Planetologia – FEG- UNESP – Guaratinguetá – SP- Brazil- CEP: 12 516-410
2,3
INPE, Av. dos Astronautas, 1758 - São José dos Campos - SP - Brazil - CEP: 12 227-010
1
[email protected]
2
[email protected]
3
[email protected]
Abstract: - Two methods to evaluate the state transition matrix are implemented and analyzed to verify the
computational cost and the accuracy of both methods. This evaluation represents one of the highest
computational costs on the artificial satellite orbit determination task. The first method is an approximation of
the Keplerian motion, providing an analytical solution which is then calculated numerically by solving the
Kepler´s equation. The second one is a local numerical approximation that includes the effect of J2. The
analysis is performed comparing these two methods with a reference generated by a numerical integrator. For
small intervals of time (1 to 10 s) and when one needs more accuracy, it is recommended to use the second
method, since the CPU time does not overload excessively the computer during the orbit determination
procedure. For larger intervals of time and when one expects more stability on the calculation, it is
recommended to use the first method.
Key-Words: - Kalman Filter, GPS, Transition Matrix, Astrodynamics, Celestial Mechanics
that allows the inclusion of more perturbations in a
simple way.
1 Introduction
The function of the transition matrix is to relate the
rectangular coordinate variations for the times tk and tk+1.
The evaluation of the state transition matrix presents one
of the highest computational costs on the artificial
satellite orbit determination procedure, because it
requires the evaluation of the Jacobian matrix and the
integration of the current variational equations. This
matrix can pose cumbersome analytical expressions
when using a complex force model. One method to
avoid these problems consists in propagating the state
vector using the complete force model and, then, to
compute the transition matrix using a simplified force
model. For sure, this simplification provides some
impact in the accuracy of the orbit determination. The
analytical calculation of the transition matrix of the
Keplerian motion is a reasonable approximation when
only short time intervals of the observations and
reference instant are involved. On the other hand, the
inclusion of the J2 (Earth flattening) effect in the
transition matrix can be performed adopting the
Markley’s method.
In the present work, the spherical harmonic
coefficients of degree and order up to 50 and the drag
effect are included in the reference orbit provided by the
RK78 numerical integrator (Runge-Kutta with Fehlberg
coefficients of order 7-8). For this study, the atmospheric
density is considered constant. For the simulations made
here, circular and elliptical low (up to 300 km) orbits
were used, as well as a medium altitude elliptical orbit
(like the one used for the CBERS satellite that is 780 km
high). To analyze the results, the reference orbit is
compared with the two methods developed in this
previous work.
2 State Matrix Transition
The differential equation for the Keplerian motion is
expressed by:
&r& = −
In Chiaradia [1] and Chiaradia et. al [2], two methods
have been compared for calculating the transition matrix,
considering circular (1 000 km of altitude) and elliptical
(Molnyia) orbits. Those methods considered the pure
Keplerian motion with or without the flattening effect of
the Earth. To include the flattening of the Earth in the
transition matrix, the Markley's is used. It is a method
ISBN: 978-1-61804-041-1
μr
r3
,
(1)
with r = (x, y, z),
r = x2 + y2 + z2 .
(2)
where r is the magnitude of the satellite position vector
and μ is the Earth gravitational constant. The equation
that relates the state deviations in different times is given
by:
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Recent Researches in Power Systems and Systems Science
⎛ δr ⎞
⎛ δr ⎞
⎜⎜ ⎟⎟ = Φ (t ,t0 ) ⎜⎜ 0 ⎟⎟ ,
⎝ δv ⎠
⎝ δv 0 ⎠
3.1 Calculating the functions f, g, f& , g&
(3)
Given r0 = ( x0 , y0 , z0 ) , v 0 = ( x&0 , y& 0 , z&0 ) , and the
where r and v are the position and velocity vectors at the
time t, respectively; r0 and v0 are the initial position and
velocity vectors at the time t0, respectively; and Φ is the
transition matrix given by:
⎛Φ
Φ = ⎜⎜ 11
⎝Φ 21
⎛ ∂r
⎜
Φ 12 ⎞ ⎜ ∂r0
⎟=
Φ 22 ⎟⎠ ⎜ ∂v
⎜ ∂r
⎝ 0
∂r
∂v 0
∂v
∂v 0
⎞
⎟
⎟.
⎟
⎟
⎠
(4)
⎛M
M 22 ⎞
⎟ (r0
M 32 ⎟⎠
v0 ) ,
Φ 12 = gI + (r v ) ⎜⎜ 22
⎝ M 32
⎛M
M 23 ⎞
⎟ (r0
M 33 ⎟⎠
v0 ) ,
⎛M
Φ 21 = f&I − (r v ) ⎜⎜ 11
⎝ M 21
M 12 ⎞
⎟ (r0
M 22 ⎟⎠
v0 ) ,
M 13 ⎞
⎟ (r0
M 23 ⎟⎠
v 0 ) , (5)
(7)
v0 = x&02 + y& 02 + z&02 ,
(8)
2μ
,
r0
(9)
(10)
e sin E0 =
h0
,
(11)
r0
,
a
(12)
μa
e cos E0 = 1 −
with E0 reduced to the interval 0 to 2π. The mean
anomalies for the initial and propagated orbits, M0 and
M, are given by:
M 0 = E0 − e sin E0 ,
(13)
n=
μ
,
a3
M = n Δ t + M0 ,
(14)
(15)
with M0 and M reduced to the interval 0 to 2π. The
eccentric anomaly for the propagated orbit is calculated
by the Kepler’s equation. The variation of the eccentric
anomaly is calculated and reduced to the interval 0 to 2π
by:
ΔE = E − E0 ,
(16)
T
The transcendental functions s0, s1, s2 for the
elliptical orbit, according to Goodyear [2, 4], are
calculated by:
T
T
s0 = cos ΔE ,
T
where I is the identity matrix 3×3; the Mi,j, i, j = 1,2,3
are the components of a 3×3 matrix M, which will be
shown later, and f, g, f& , g& are calculated in the next
topic.
ISBN: 978-1-61804-041-1
h0 = x0 x&0 + y0 y& 0 + z0 z&0 ,
where a is the semi-major axis. The eccentric anomaly
E0 and the eccentricity e for the initial orbit are
calculated by:
Goodyear [3] published a method for the analytical
calculation of a transition matrix for the two-body
problem. This method is valid for any kind of orbit.
Kuga [4] implemented this method using the same
elegant and adequate formulation optimized for the
Keplerian elliptical orbit problem. He performed some
simplifications in this method to increase its numerical
efficiency (processing time, memory, and accuracy).
However, this method is used to propagate the position
and velocity of the satellite and can be used in any kind
of two-body orbits. Such equations are simple and easy
to be developed.
According to Goodyear [3, 5] and Shepperd [6], the
four submatrices 3×3 are written as:
⎛M
(6)
α
1
=− ,
μ
a
3 First Method: The Analytical
Transition
Matrix
Solution
Considering the pure keplerian
motion
Φ 22 = g&I − (r v ) ⎜⎜ 12
⎝ M 22
r0 = xo2 + y02 + z02 ,
α = v02 −
The submatrices Φ11, Φ12, Φ21, Φ22 are calculated in
accordance with the two methods shown in the next
topics.
Φ 11 = fI + (r v ) ⎜⎜ 21
⎝ M 31
Δt = t − t0 :
propagation interval
49
(17)
Recent Researches in Power Systems and Systems Science
a
s1 =
s2 =
μ
a
μ
sin ΔE ,
the case where the orbit propagation time, Δt, is larger
than one orbital period:
where INT(x) provides the truncated integer number of
'
'
the real argument x, and the variables s 4 and s5 are
(18)
(1 − s0 ) .
(19)
related with the transcendental functions s4 and s5
(Goodyear [3, 5]).
Therefore, the components of the matrix M are given
by:
Therefore, the functions f, g, f& , g& are calculated by:
r = r0 s0 + h0 s1 + μs2 ,
μs
f = 1− 2 ,
r0
g = r0 s1 + h0 s2 ,
μs
f& = − 1 ,
rr0
μs
g& = 1 − 2 .
r
(20)
M 31 =
(21)
( f − 1)s1 − μ U
r0
M 23 = −(g& − 1)s 2 ,
M = − f&s
(22)
2,
22
(23)
r03 ,
⎛s
1
1 ⎞
U
M 11 = ⎜⎜ 0 + 2 + 2 ⎟⎟ f& − μ 2 3 3 ,
r r0
⎝ rr0 r0 r ⎠
⎛ f&s
(g& − 1) ⎞⎟ ,
M 12 = ⎜⎜ 1 +
r 2 ⎟⎠
⎝ r
( g& − 1 )s1
U
M 13 =
−μ 3 ,
r
r
⎛ f&s
( f − 1) ⎞⎟ ,
M 21 = −⎜⎜ 1 +
r02 ⎟⎠
⎝ r0
(24)
There is no singularity problem and the Kepler’s
equation is solved through the Newton Raphson’s
Method in double precision. The classical parameters a,
e, i, Ω, ω, are constant in the Keplerian motion and,
therefore, there is no different subscript for them.
3.2 The evaluation of the matrix M3×3
First of all, it is necessary to calculate the secular
component U including the effect of multi-revolutions in
⎛
⎝
ΔE = ΔE + INT ⎜ Δt
n ⎞
⎟2π ,
2π ⎠
(25)
s 4' = cos ΔE − 1 ,
(26)
s '5 = sin ΔE − ΔE ,
(27)
5
⎛a⎞
U = s 2 Δt + ⎜⎜ ⎟⎟ ΔEs '4 − 3 s 5' ,
⎝μ⎠
(
)
M 32 = ( f − 1)s 2 ,
M 33 = gs 2 − U .
4 Second Method: The Markley’s
Method
The Markley's method uses two states, one at the tk-1 time
and the other at the tk time, and calculates the transition
matrix between them by using μ, J2, Δt, the radius of the
Earth, and the two states. In this case, the effect of the
Earth’s flattening is the most influent factor in the
process. The Markley's method consists of making one
approximation for the transition matrix of the state vector
based on Taylor series expansion for short intervals of
propagation, Δt. This method can be used by any kind of
orbit and the equations are simple and easily
implemented, as shown next. The state transition's
differential equation is defined by:
(28)
(29)
3.3 The property of the transition matrix
I⎤
dΦ (t ,t 0 )
⎡ 0
= A 1 ( t ) Φ ( t ,t 0 ) = ⎢
⎥ Φ (t ,t 0 )
dt
⎣G(t) 0⎦
Sometimes the inverse matrix is required, such as in
backward filters. It is also easily accomplished as
−1
follows. The inverse matrix Φ , that propagates
deviations backward from t to t0, is given by:
Φ −1
⎛ Φ T22
= ⎜⎜
T
⎝ − Φ 21
−Φ ⎞
⎟,
Φ T11 ⎟⎠
T
12
where
r = (x
(31)
is the
initial
condition,
z ) and v = (x& y& z& ) are the Cartesian
state at the instant t; r0 and v0 are the Cartesian state at
(30)
y
the instant t0; 0 ≡ the matrix 3×3 of zeros; I ≡ the identity
that results from the canonic nature of the original
equations (Danby [7]; Battin [8]). This also applies to the
second method (Markley’s) if the perturbations are
derived from a potential (e.g. J2 effect).
ISBN: 978-1-61804-041-1
Φ (t0 ,t0 ) ≡ I
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Recent Researches in Power Systems and Systems Science
matrix 3×3; G(t) ≡
∂a( r ,t )
≡ the gradient matrix; and
∂r
a( r ,t ) = the accelerations of the satellite.
Performing successive derivatives of the Eq. (31),
followed by substitutions, gives the derivative of the
transition matrix:
dΦ
= A i ( t ) Φ ( t ,t 0 ) ,
dt i
(32)
& ( t ) + A ( t )A ( t ) ,
Ai ( t ) = A
i −1
i −1
1
Φ rv ⎤
Φ vv ⎥⎦ 6 x6
∂a y
∂x
,
∂a y
∂y
∂a y
∂z
∂y
∂a z
∂y
∂a x ⎤
∂z ⎥
⎥
∂a y ⎥
.
∂z ⎥
∂a z ⎥
⎥
∂z ⎥⎦
=
∂a x
,
∂y
=
y ∂a x a x
+ ,
x ∂y
x
=
y ∂a x
,
x ∂z
∂a z ∂a y
=
,
∂y
∂z
∂a z
μ ⎡
3
⎛
= 5 ⎢− r 2 + 3⎜ z 2 − J 2 re2 +
∂z
2
r ⎣
⎝
derives from a potential. The G gradient matrix including
only the central force and the J2 is given by:
∂a x
∂y
∂a y
)
∂a z ∂a x
,
=
∂x
∂z
(35)
The calculations of these matrices pose no problems,
since the gradient matrix G in the end of the propagating
interval is a function of the final Cartesian state which
should be calculated during the data processing and is
available at no extra cost. The G and, therefore,
Φrr ,Φrv ,Φvr ,Φvv are symmetric if the perturbation
15
J 2 re2 2 35 J 2 re2 4 ⎞⎤
z ⎟⎟⎥
z −
2 r4
r2
⎠⎥⎦
(39)
(36)
5 Analyses Of The Methods
First of all, the reference orbit is integrated using the
numerical integrator Runge-Kutta of eight-th order with
automatic step size control (RK78). This integrator is
implemented to integrate simultaneously the stated
vectors considering the spherical harmonic coefficients
up to 50th order and degree and the transition matrix
The accelerations due to Earth’s flattening are given
ISBN: 978-1-61804-041-1
(37)
∂a x 3 μxz ⎡ 15 J 2 re2 35 J 2 re2 2 ⎤
= 5 ⎢1 +
−
z ⎥,
2 r2
2 r4
∂z
r ⎣
⎦
Δt ≡ t – t0 and G0 ≡ G(t0).
by:
⎞⎤
⎟⎟⎥ .
⎠⎦
∂a x 3 μxy ⎡ 5 J 2 re2 35 J 2 re2 2 ⎤
= 5 ⎢1 +
−
z ⎥,
2 r2
2 r4
∂y
r ⎣
⎦
(34)
6
(Δt )3 ,
Φ rv ≡ IΔt + ( G 0 +G )
12
(
Δt )
,
Φ vr ≡ (G 0 + G )
2
(
Δt )2
Φ vv ≡ I + (G 0 + 2G )
,
6
⎡ ∂a x
⎢ ∂x
⎢
∂a(r ,t ) ⎢ ∂a y
G( t ) =
=
⎢ ∂x
∂r
⎢ ∂a
⎢ z
⎢⎣ ∂x
⎛
5z 2
⎜⎜ 3 − 2
r
⎝
(
(33)
velocity obtained after some simplifications is given by
(Markley [9]):
Φ rr ≡ Ι + (2G 0
y
ax ,
x
3 J 2 re2
− μz ⎡
a z = 3 ⎢1 +
2 r2
r ⎣
∂a x
3
15 J 2 re2 2
μ ⎡
= 5 ⎢3 x 2 − r 2 − J 2 re2 +
x + z2 −
2
∂x
2 r2
r ⎣
105 J 2 re2 2 2 ⎤
x z ⎥,
2 r4
⎦
the dot represents the derivative with respect to the time.
Developing Φ(t,t0) in Taylor's series at t = t0, using the
matrices Ai(t0) for i = 1,..,4 and the initial condition
Φ(t 0 ,t 0 ) ≡ I , the transition matrix of the position and
(Δt )2
+ G)
⎞⎤
⎟⎟⎥ ,
⎠⎦
The partial derivatives are (e.g. Kuga [10]):
where
where
⎛
5z 2
⎜⎜ 1 − 2
r
⎝
ay =
i
⎡Φ
Φ (t ,t0 ) ≈ ⎢ rr
⎣Φ vr
− μx ⎡ 3 J 2 re2
⎢1 +
r3 ⎣ 2 r2
ax =
51
Recent Researches in Power Systems and Systems Science
considering the Earth's flattening and the atmospheric
drag effects.
The transition matrix generated by the numerical
integrator is used as the reference to compare the
transition matrix generated by the two methods. To
compare the accuracy of them, let us define the global
relative error as (Kuga [4]):
ε Global =
1 6 6 (Φ ij − Φ ij )
,
∑∑ Φ
36 i =1 j =1
ij
Besides, the RK4 (Runge-Kutta of fixed 4-th order)
numerical method for computing the transition matrix is
also implemented and the transition matrix generated by
the RK4 is also included in the comparison. The results
of those three comparisons for the circular orbit are
shown in Table 3 and the results for the elliptical orbit are
shown in Table 4. The errors are shown in terms of per
step mean and standard deviations considering the whole
day sample. The comparison is done for only these two
kinds of orbits, where the Keplerian approximation for
the transition matrix provides already reasonable
accuracy. For larger times interval, the second method is
always disadvantageous because it is a local numerical
approximation, as depicted in Tables 3 and 4.
(40)
where Φij is the component i,j of the transition matrix
calculated by the RK78 considered as reference and Φij
is the component i,j of the transition matrix calculated by
one of the methods. It gives a rough idea of the number
of the common significant figures retained after the
computations.
TABLE 3 - COMPARISON FOR A CIRCULAR
ORBIT
(TOPEX)
The first method considers the pure Keplerian motion
and the second one considers the Keplerian motion and
the J2 effect. A whole day of integration is divided in
intervals of 1 to 60 seconds, which produces from 86,400
to 1,440 steps of integration, respectively, as shown in
Table 1. The total processing time of each method is also
shown in Table 1, although it depends on the code, the
computer, and the programmer skills. However, it
illustrates
roughly
the
comparative
expected
performance. One can note that the CPU time difference
at any step of integration and for any method is very
small. The first method is slightly faster than the second
one for small steps of time, like 1 and 10 seconds.
TABLE 4 - COMPARISON
ELLIPTICAL ORBIT (MOLNIYA)
To analyze the accuracy of the methods, six
comparisons are done, using two kinds of orbits: one
circular and one elliptical. The transition matrix
generated by each of the methods is compared with the
reference generated by the RK78. The circular orbit is
from the Topex/Poseidon satellite (Shapiro [11]) and the
elliptical one is from the Molniya satellite (Markley [9]).
All those tests are performed for a period of 24 hours.
The data of these orbits are shown in Table 2.
A
6 Conclusions
The goal of this research was to compare and choose the
most suited method to calculate the transition matrix used
to propagate the covariance matrix of the position and
velocity of the state estimator, (e.g. Kalman filtering or
least squares), within the procedure of the artificial
satellite orbit determination. The methods were evaluated
according to accuracy, processing time, and handling
complexity of the equations for two kinds of orbits:
circular and elliptical. The processing time of the second
method is around 30% larger than the first one; however,
this difference is not considered enough to harm the
computer load in the orbit determination tasks. The
second method is more accurate for short intervals, as Δt
= 1 and 10 seconds, for the orbits considered. For other
intervals of propagation, the first method shows to be
more stable in the sense of keeping the same accuracy
regardless of the step size. The equations of the second
method are easier to be handled, which means that it is
possible to include more perturbations easily. However,
the first method is a closed analytical solution for the
two-body problem only. The second method has no
singularity and no restriction about the type of orbit; the
first one is optimized for elliptical orbits (not optimized
TABLE 1 – PROCESSING TIME FOR ANALYTICAL
CALCULATION OF THE TRANSITION MATRIX
TABLE 2 - KEPLERIAN PARAMETERS OF THE
TEST ORBITS
ISBN: 978-1-61804-041-1
FOR
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Recent Researches in Power Systems and Systems Science
[5] Goodyear W. H.; A general method for the
computation of Cartesian coordinates and partial
derivatives of the two-body problem. Greenbelt,
MD, Goddard Space Flight Center (NASA CR522), Sept, 1966.
for parabolic and hyperbolic orbits). The second method
performs an approximation in Taylor's series; whereas in
the first there is an analytical solution for the Keplerian
motion. Therefore, for small intervals of time (1 to 10
seconds) and when one expects more accuracy, it is
recommended to use the second method, since the CPU
time does not overload excessively the computer in the
orbit determination procedure. For larger intervals of time
and when one expects more stability on the calculation, it
is recommended to use the first method. The importance
of choosing the best method can be exemplified when
there is the goal of using the orbits determined by one of
these methods to perform orbital maneuvers. In this case,
to be able to determine the orbits as fast as possible is a
key parameter of the problem. References [12] to [17]
describe in more details the orbital maneuvers problem.
[6] Shepperd S. W.; Universal Keplerian state
transition matrix. Celestial Mechanics, 35: 129-144,
1985.
[7] Danby J. M. A.; The matrizant of Keplerian
motion. AIAA Journal, 2(1): 16-19, Jan.,1964.
[8] Battin, R. H.; Atronautical Guidance. New
York, Academic, 1964.
[9] Markley F. L.; Approximate Cartesian State
Transition Matrix. The Journal of the Astronautical
Sciences. Vol. 34, n. 2, p161-169, apr-jun, 1986.
[10] Kuga H. K.; Adaptive orbit estimation applied
to low altitude satellites. Master Thesis (INPE2316-TDL/079),
INPE,
Brazil,1982.
(in
Portuguese).
Acknowlegements
The authors wish to express their appreciation for the
support provided by UNESP (Universidade Estadual
Paulista “Júlio de Mesquita Filho”) of Brazil and INPE
(Brazilian Institute for Space Research).
[11] Shapiro B. R.; Topex/ Poseidon Navigation
Team. [On-line] http://topexnav.jpl.nasa.gov., Last
access April, 1998.
[12] Prado, A.F.B.A., Optimal Transfer and SwingBy Orbits in the Two- and Three- Body Problems,
Austin. 240p. PhD Thesis, Department of Aerospace
Engineering and Engineering Mechanics –
University of Texas, Dec., 1993.
References:
[1] Chiaradia, A. P. M. Autonomous artificial
satellite orbit determination and maneuver in realtime using single frequency GPS measurements. São
José dos Campos. 202p. (INPE_875-TDI/798)
Thesis (Ph.D. in Portuguese)- Instituto Nacional de
Pesquisas Espaciais (INPE), 2000.
[13] Sukhanov AA, Prado AFBA, Constant tangential
low-thrust trajectories near an oblate planet, Journal of
Guidance Control and Dynamics, v. 24, No. 4, 2001, p.
723-731.
[2] A. P. M. Kuga, H. K; Prado, A. F. B. A. Single
Frequency GPS measurements in real-time artificial
satellite
orbit
determination.
ACTA
ASTRONAUTICA. Vol. 53 Issue: 2 Pages: 123-133
Published: JUL, 2003.
[14] Prado AFBA, Numerical and analytical study of the
gravitational capture in the bicircular problem, Advances
in Space Research, v. 36, No. 3, 2005, p. 578-584.
[15] Prado AFBA, Numerical study and analytic
estimation of forces acting in ballistic gravitational
capture, Journal of Guidance Control and Dynamics, v.
25, No. 2, 2002, p. 368-375.
[3] Goodyear W. H.; Completely general closedform solution for coordinates and partial derivatives
of the two-body problem. The Astronomical
Journal, 70(3): 189-192, Apr., 1965.
[16] Prado AFBA, Broucke R, Transfer Orbits in
Restricted Problem, Journal of Guidance Control and
Dynamics, v. 18, No. 3, 1995, p. 593-598.
[4] Kuga H. K.; Transition matrix of the elliptical
Keplerian motion,. (INPE-3779-NTE/250), INPE,
Brazil, 1986. (in Portuguese).
ISBN: 978-1-61804-041-1
[17] Prado AFBA, Third-body perturbation in orbits
around natural satellites Journal of Guidance Control and
Dynamics, v. 26, No. 1, 2003, p. 33-40.
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