Efficient Stochastic Orbit Modeling Techniques using Least Squares ...

Efficient Stochastic Orbit Modeling Techniques using Least Squares Estimators A. J¨aggi, G. Beutler, U. Hugentobler Astronomical Institute, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland email: [email protected]

Abstract. Reduced-dynamic orbit determination for spaceborne GPS receivers of low Earth orbiting satellites is a successful method promising highest precision. We review the principles of (reduced) dynamic orbit determination and develop the mathematical background for different efficient stochastic orbit parametrizations (e.g., piecewise constant accelerations which provide not only continuous but also differentiable orbits) using least squares methods. Simulated as well as real data from the CHAMP GPS receiver show, to some extent, the equivalence of the different parametrizations and reveal the impressive performance of stochastic orbit modeling techniques. Independent comparisons with orbits determined by other groups and validations with SLR measurements show that our orbits are of high quality. Key words. Low Earth orbiter, reduced-dynamic orbit determination, stochastic orbit parametrization

1 Introduction The idea of using spaceborne GPS receivers for precise orbit determination (POD) of low Earth orbiters (LEOs) was demonstrated for the first time almost a decade ago (Bertiger, 1994) for dynamic and reduced-dynamic orbits of the TOPEX/POSEIDON satellite. This proof of concept stimulated an increasing number of follow-up missions of Earth observing satellites to carry on-board GPS receivers, not only for POD, but also for purposes like atmosphere sounding (Kursinski, 1997) and gravity field recovery. For the latter purpose the analysis of CHAMP GPS receiver and accelerometer data proved that it is possible to separate gravitational from nongravitational perturbations (Reigber, 2002). This article focuses on POD, namely on the promising approach of reduced-dynamic orbit determination and in particular on the stochastic orbit modeling part, which defines the degree of “strength” reduction of the dynamic laws by allowing for a stochastic component in the equations of motion. The results for simulated and real GPS data of the CHAMP GPS receiver underline the applicability of the theoretical developments.

2 Dynamic Orbit Determination The equation of motion of an Earth orbiting satellite including all perturbations reads as


 
 

   ! "$#   

(1)


&% ')( +* # 
% '( -,./021354627289* :2+*$#;=<  > 5? ,.=/@134A728 * are the six osculating elements where +*    !  " are the unknown pertaining to and where in the inertial frame with initial conditions

dynamical parameters describing the perturbing acceleration acting on the satellite (e.g., gravity field coefficients, air-drag or radiation pressure parameters, ...). is available Let us assume that an a priori orbit (e.g., from a GPS code solution). As must be a solution of the equation of motion (1) a priori values for the unknown orbit parameters are available, as well. The actual orbit may therefore be linearized by a truncated Taylor series


CB  2#
B -2# DFE *HGJI , * / * 1 * 4 * A7 * 28 *=* $ *  ! " *LK DE
-2#

N -2#
- 2# M 
CB - 2# O RQ
B D ETS  DFE  DFE * #U (2) !E P Q X  W where V

ZY denotes   D.[total  !  " K . I   !the  I ,.of/0unknown \ " K number orbit parameters D

Equation (2) allows it to improve the orbit if the partial derivatives of the a priori orbit w.r.t. the unknown orbit parameters are known. 2.1 Variational Equations

D

Let us assume that is one of the parameters defining the initial values or the dynamics in Eq. (1) and that the partial derivative of the orbit w.r.t. the parameter is designated by the function

D

2# ]0^  2#   Q
B D  Q


CB 2#

(3)

The initial value problem associated with the partial derivative (3) is referred to as the system of variational equations in this article and may be obtained

2

‡ E f"

by taking the partial derivative of Eq. (1). The result may be written as

of zero and an a priori weight variance

is inhomogeneous with zero initial values. Note that in the latter case the homogeneous part of Eq. (4) is the same as for parameters defining the initial values.

ror (RMS) of unit weight. The a priori weights in Eq. (9) constrain the estimates of the piecewise constant accelerations , not allowing them to deviate too much from zero. Note that Eq. (9) allows for an estimation of the parameter in Eq. (7). As indicated in section 2.1, every orbit parameter additionally set up requires in principle the simultaneous integration of one additional differential equation system (4). Facing the possibly large number of stochastic accelerations to be set up, it is mandatory to develop efficient methods to calculate the partial derivatives w.r.t. these parameters in order to avoid inefficiency due to a computational overload.

 ]0^`_ B S ]0^ _  S ]0 ^ Q D  (4) Q  are defined by where the 3 a 3 matrices _ B and _ b *c E '=d  *5Ef  b  c E '=d  *Ef  (5) Q& e ' Q9 e ' Q QZg where E denotes the i-th component of the total ace  in Eq. (1). For D GhI ,./02134A789* K Eq. celeration (4) is a linear, homogeneous, second orderdiffereni k ] ^ "  * #jl tial equation system with initialGnvalues   5  !   !  =      +   * # i I K 0 ] ^ m  k D and , whereas for Eq. (4) D

, E f"

,0*5f "

3.1 Pseudo-Stochastic Accelerations

3 Pseudo-Stochastic Orbit Modeling Apart from the six initial conditions, accelerations are set up as the only orbit parameters. Without taking into account any a priori models for the nonconservative forces, piecewise constant stochastic accelerations partially absorb these accelerations (e.g., air-drag, radiation pressure, ...) as well as accelerations due to a mismodeling in the gravitational attractions (e.g., due to an imperfect gravity field model). The parametrization subsequently used for the residin the inertial system reads as ual acceleration the sum of three accelerations in (r)adial, (a)longtrack and (c)ross-track directions in the satellite corotating system

o 2#

o 2#  oqp  2# osr - 2# so t 2#U 2# , ` G I ,.=v K where the acceleration osu Y

given by an a priori

&ˆ Š)‰ ‹Œ Ž  , E f " # X> : ‡ E f "  ˆ *‰  (9) ˆ Š‰ ‹Œ Ž * where ˆ denotes the a priori root mean square er-

(6) is repre-

sented by

y , f "5z  E{   E ## u -2#U (7) osu  2#  -,0*5f " x S|

E!Pw  E where | u denotes the unit vector in one of the directions specified above and

z -2 E-{ L2 E # ~} > ? :0:  E { € ‚ ƒ E  (8) else † ?$# orbit paramAltogether a total number of W …„

eters, are set up. As opposed to the constant accelera† *f " acting over the whole arc in Eq. (7), tion the " , f E piecewise constant accelerations are referred to as pseudo-stochastic accelerations in this article, because they are characterized by an expectation value

†

, E f"  E-{   ƒ  ‘ E 1 

Let us now develop the mathematical background for constant accelerations in predeestimating termined directions for . For the sake of simplicity we drop the inin the following and focus on one accelerdices in ation . The contribution of this parameter in Eq. (1) is of the form for . Neglecting any possible velocity-dependent forces, the corresponding variational equation thus reads as

?’5!  2†



13 , Y

2# |u

,

S|

2#

E  ,  E-{ € “” E

]r •  _–B S ] r } |k - 2#—:0:  E{   “” E  (10) else -2#š™  ?’  ! W Let us assume that the functions ]@˜

are the partial derivatives of the a priori orbit w.r.t. the six parameters defining the six initial conditions at . As these six functions form one complete system of homogeneous solutions of Eq. (10) the solution of the inhomogeneous system (10) and its first time derivative can be obtained by the method of variation of constants and may therefore be written as a linear combination of the homogeneous solutions

+*

]’˜ 2#

[ ] r % '( -2#  › y P  œ ›  2# @] ˜%'( 2#‚:ž< `> 5?R (11) -2# are functions of time where coefficients œ ›  to be the  ˜ š™ matrix    W and Ÿ   ¡£¢£¢!¢£¡¥¤$determined. ¦ ˜§ ¨] ˜  Ÿ   Introducing ©L¡£¢£¢!¢£¡¥ª$¦ ˜§  ] the  ?’5!notation  k ¬  2# ¬ # the solution may vector notation «­¬  | be obtained by definite integrals ® 2# °± ¯ Ÿ {  A³# « A³# Y A³  ±¯´ Ÿ {  -A³ # « A³# Y A³9 ¯² ¯ ‹µ0¶ (12)

3

  œ  5!   œ [ # and where  :@“” E-{  :@ E-{   “” E  (13) :@“»” E -2# for the parameter This implies that the solution ] r , reads as :0“ƒ E-{  k[ ¸¼ y -2# 2#½:0 E-{ € ‚ƒ E  (14) ] r 2#  ¹ ¼º › P [  œ › ]’˜ ¼¼ y P  œ › - E # ] ˜ 2#—:0“»ƒ E › “»ƒ E the coefficients œ › - E # are Note that in the case ® ¬  E-{ · J¹ º¸  E

where

constant in time.

3.1.1 Efficient Solution for

 E-{   ‚¾ E

Let us introduce an auxiliary problem and write the function (14) as a function of the solution of this auxiliary problem. The parameter underlying the auxiliary problem is a constant acceleration over the enin Eq. tire arc. Note that can be identified with (7). The corresponding variational equation (see Eq. (10)) reads as

,¿

,¿

]&rÀ •_–B S 9] rÀ | - 2#ÂÁ.Ã As the difference Ä Å] r À Æ] r

, *f "

(15)

 _ B SÄ Ä ` ] ˜ 2# [ ] r % ')(  2# `  ] rÀ % '( - 2#  › y P CÇ › ]’˜% '( -2#È:Ã< M> 5 ?R

solves the homogeneous differential equation system in the designated time interval, its solution can be written as , therefore a linear combination of the functions

(16)

 E-{  Ǜ

and taking into acEvaluating Eq. (16) at time count Eq. (14), the coefficients may be obtained as a solution of the following linear system of algebraic equations:

y[  › P[ y ›P 

Ç › ’] ˜   E-{  # ½] rÀ -  E{  # Ç › ]  ˜   -E { 5 #  ]& rÀ -  E {  #U

(17)

Ǜ

,¿

 E{ 

be obtained as a solution of the following linear system of algebraic equations:

y[  œ › P[ y › P œ

 ] r  E # ›  E # ] ˜   E # ` ›  E #q]’ ˜  E #  ]  r   E #È

(18)

›  E # ] r œ  2# ]’˜ 2#
2#

As the above equations again form a linear system of six scalar equations for the six unknowns it is possible to write the partial derivative as a linear combination of the six partial derivatives only. In conclusion not only the position vector but also the velocity vector of the improved orbit remains continuous over the whole arc.


 2#

3.2 Pseudo-Stochastic Pulses

ÉE

E

Let us briefly outline the special case of instantaneous velocity changes at times in predetermined directions (Beutler, 1994). Focusing on one pulse at time , the contribution of in in Eq. (1) may formally be written as where denotes Dirac’s delta function. The corresponding variational equation thus reads as

É

2#

Ê

 # | EE

 E  É   É SÊ -   E # S |  2#

]CËÌ`_ B S ]CË Ê -   E # | -2#È

(19)

Using the same notation the solution of (19) may be written again in the form of Eq. (11). The coefficients may be obtained in analogy to Eq. (12) by defby inite integrals which are given for

œ › 2#

‚»¾ E

® 2#  ± ¯ Ê -A³   E # Ÿ {  -A³ # « A³# Y A³  Ÿ {   E # « - E # ¯² (20) 2# are constant in time for »Å E . Obviously the œ › -2# may be written Therefore the partial derivatives ] Ë

as a linear combination of the six partial derivatives w.r.t. the initial conditions only. The drawback however is that and therefore are no longer continuous. The discontinuities lie at the epochs .

]  Ë 2#

As the above equations form a linear system of six scalar equations for the six unknowns it is possible to write the partial derivative (16) w.r.t. the parameter as a linear combination of the partial derivative w.r.t. the parameter and the six partial derivatives . Note that Eq. (11) together with Eq. (12) implies not only continuity but also differentiability of at time .

, ’] ˜  2# ] r 2#

‚»¾ E 2# is continuEquations (11) and (14) imply that ] r  ous and differentiable at time E , as well, which means    # ] % ) ' ( E that r - # evaluating Eq.  may be computed by (16) at time E . The coefficients œ › E may therefore 3.1.2 Efficient Solution for


 2#

E

4 Simulation Studies We discuss two experiments using simulated GPS phase zero-difference observations for the CHAMP satellite. In both simulations the same physical and mathematical models were used as in the real data

4

15

simulated signal estimated signal along-track deviation

5

accelerations pulses

4 10

50

5

0

0

-50

-5

-100

-10

-150

-15 200

3

Í

radial deviation (mm)

100

along-track deviation (mm)

acceleration (nm/s^2)

150

Ï

2 1 0 -1 -2 -3 -4

0

50

100 time (min)

150

-5 0

20

40

60

80

100

time (min)

Fig. 1. Piecewise constant accelerations every 15 minutes compensate for an unmodeled along-track signal. The dotted curve denotes deviations w.r.t. the “true” orbit (simulated data).

Fig. 2. Radial deviations w.r.t. the “true” orbit for pseudostochastic accelerations resp. pulses set up every 6 minutes. Epochs of velocity changes are easily recognized on the dashed curve (simulated data).

processing. In addition the error-free GPS data were affected by an artificial once per rev. along-track acm/s for the celeration with an amplitude of first simulation. An orbit with identical initial values without being affected by the artificial signal served as a priori orbit. To illustrate the feasibility of our approach for this scenario, piecewise constant accelerations in along-track direction were estimated every fifteen minutes. For a selected time interval of about two revolution periods, Fig. 1 shows that the estimated piecewise constant accelerations follow the artificial (“true”) signal and that the deviations from the “true” orbit in along-track direction are mostly below the one millimeter level. High correlations with the time intervals of the accelerations can be recognized as well as a once per rev. periodicity. Both effects are caused by the unavoidable deficiencies of the parametrization. The second simulation illustrates the applicability and, to some extent, the equivalence of both, pseudostochastic accelerations and pulses in a more realistic environment. Again GPS observations were simulated but as opposed to the first experiment the phase observations were affected by a white noise random error of 1 mm RMS. A dynamical orbit using the gravity field model EIGEN-1S (Reigber, 2002) up to degree/order 120 served as “true” orbit, whereas the same gravity field model, but truncated at degree/order 80, was used in the orbit improvement procedure. To illustrate the properties of both parametrizations two different orbits were generated, one with the parametrization defined in Eq. (6) (where stochastic accelerations were set up every six minutes) and another with pulses instead of accelerations with the same spacing in time. By optimizing the constraints in Eq. (9), both orbits could be fitted equally well (postfit RMS of about 1.02 mm).

The differences w.r.t. the “true” orbit yield almost the same RMS values over one day, smaller than 2 mm in all three directions. The largest differences occur in the radial component (caused by the large number of rather loosely constrained stochastic parameters, which reduce the dynamic to the extent that the resulting orbit tends to be more sensitive to the observation geometry (like a kinematic orbit) which penalizes the radial direction). Fig. 2 shows the radial deviations w.r.t. the “true” orbit over a selected time interval of about one revolution period where the differences for both parametrizations can be observed well. The orbit generated with piecewise constant accelerations shows a smooth behaviour, whereas epochs with velocity changes can be identified quite easily on the other orbit. Nevertheless, the differences are very small (below a few millimeters in the vicinity of the epochs of the stochastic parameters) and do not significantly affect the overall orbit quality. Consequently, no significant gain or loss has to be expected using one or the other orbit parametrization with a reasonably large number of stochastic parameters, when processing real data. The real errors are much higher due to the non-gravitational forces, which were neglected in this simulation, and therefore exceed the subtleties reported above.

? > {.Î

‰

5 CHAMP Orbit Test Campaign Using GPS final orbits and high-rate clocks (Bock, 2002) from the CODE analysis center (Center for Orbit Determination in Europe) real CHAMP GPS zero-difference phase observations were processed to study the impact of constrained stochastic parameters on the orbit. The orbit quality was assessed by internal (e.g., formal accuracies along the orbit) and external indicators (e.g., comparisons with orbits from

5 Table 1. RMS of plain differences to an orbit from CSR resp. RMS of SLR residuals (day 141).

тÒ$Ó3ԒРÕCÖ ÓÃÒ$Ó3ԒÕCÖ Ñ‚Ò$Ó3Ô ÕC× ÓÃÒ$Ó3ԒÕC×

RMS w.r.t. CSR (cm) 6.72 5.09 4.89 5.32

RMS of SLR residuals (cm) 3.07 1.77 1.99 3.38

other groups, SLR residuals). For this purpose we used the time span of the CHAMP orbit test campaign from May 20 to 30, 2001 (days 140-150), which was selected as the test period within the IGS LEO Pilot Project. As the orbits from the Center of Space Research (CSR) at Austin were found to be the best of the twelve contributing centers (Boomkamp, 2002) they serve here as benchmark orbits. 5.1 Internal Quality Assessment In order to assess the orbit quality, the full parametrization (defined in Eq. (6)) with stochastic parameters set up every six minutes was used to generate CHAMP orbits for both, accelerations and pulses with different (but in each direction equal) constraints. To obtain best results, the gravity field model EIGEN-1S was used and the attitude measurements from the star sensors were introduced. The dependency of the postfit RMS on the constraints was found to be equal for both parametrizations, obviously allowing a better fit for looser constraints. However this does not necessarily imply that such orbits are really better. On the contrary, it turns out that setting up a large number of loosely constrained stochastic parameters may result in worse orbits, which may be verified by an external comparison. Table 1 shows for day 141 the RMS of plain 1-d differences (no Helmert transformation applied) to an orbit from CSR as well as the RMS of SLR residuals for four different types of constraints (in each direction equal) when setting up stochastic accelerations every six minutes. According to this comparison an optimal solution exists in the range considered and m/s are most probably the constraints of quite close to the optimum. Note that almost identical results can be obtained if pulses are set up. It is instructive to analyze the formal errors along the orbit as an additional indicator of quality. Applying the general law of error propagation together with the variance-covariance information allows the computation of formal errors in the system co-rotating with the satellite (i.e., in radial, along-track and cross-track direction). Fig. 3 shows as an example the errors in radial direction for the four different constraints from Table 1. Apart from the common characteristics (e.g., the increased error level around 3h and 10h caused by

{.Ù Ø S ?>

‰

5e-8 m/s^2 1e-8 m/s^2 5e-9 m/s^2 1e-9 m/s^2

18 16 radial accuracy (mm)

Constraint (m/s )

20

Ú

14 12 10 8 6 4 2 0

5

10

15

20

time (h)

Fig. 3. Formal accuracies of orbit positions in radial direction for differently constrained pseudo-stochastic accelerations set up every 6 minutes using the gravity field model EIGEN-1S (day 141).

a reduced number of successfully tracked satellites), Fig. 3 implies that good solutions in Table 1 turn out to have low formal accuracies along the orbit as well. The accuracies in the other directions show a similar behaviour, whereas the highest error level can be found in the along-track direction indicating that the orbits (as opposed to the simulations in section 4) are all governed by the dynamical laws due to the relatively tight constraints. Nevertheless the “most” m/s kinematic solution with constraints of already suffers considerably from the limit of the CHAMP BlackJack receiver at that time to track only up to 8 satellites simultaneously.

{.Û Ø S?>

‰

5.2 External Quality Assessment Table 2 shows the daily RMS of plain 1-d differences w.r.t. CSR orbits (middle column) for the time span of the CHAMP orbit comparison test campaign. Over the eleven days an RMS of 5.29 cm w.r.t. the orbits from CSR (corresponding to a 3-d RMS of 9.16 cm) is achieved. Solving for a seven parameter Helmert transformation shows that our orbits agree with the CSR orbits on a RMS level better than 5 cm. However both orbit types show a mean shift in the earth-fixed z-direction of 3.32 cm which may be due to slightly different reference frames the used GPS orbits and clocks refer to. Table 2 shows the daily RMS of SLR residuals (right column) for the time span of the CHAMP orbit comparison test campaign. A total of 1837 SLR measurements (normal points) of 16 stations were used for this validation. Due to the sparse SLR tracking the differences of SLR measurements (corrected for the tropospheric delay) and our orbit positions give only snapshots of the 1-d orbit accuracy in selected directions during the short tracking passes. For the eleven

6 Table 2. RMS of plain differences to orbits from CSR resp. RMS of SLR residuals (days 140-150). An overall RMS of 5.29 cm resp. 3.37 cm results.

140 141 142 143 144 145 146 147 148 149 150

RMS w.r.t. CSR (cm) 4.87 4.89 5.85 5.19 4.91 6.07 4.85 5.12 4.73 5.71 5.74

RMS of SLR residuals (cm) 3.19 1.99 2.77 2.51 4.99 4.61 4.15 2.47 2.81 3.08 3.30

days an RMS of 3.37 cm over all SLR residuals is obtained without showing any significant SLR bias. 5.3 Comparison with Accelerometer Data If a perfect gravity field could be used the estimated accelerations should directly represent the non-conservative forces acting on the satellite. Fig. 4 shows for a selected time interval of about three revolution periods how the piecewise constant accelerations in along-track direction agree with the measured accelerations (bias and scale applied) from the STAR accelerometer, when the gravity field model EIGEN1S is used. Despite the crude approximation with constant accelerations over six minutes the correlation (77.8%) is quite remarkable whereas the agreement in the other directions is less significant (correlation of 36.1% resp. 34.7% in radial resp. crosstrack direction). At least in along-track direction reliable bias and drift parameters may be derived by such and 0.0013 day comparisons (-3.518 for the time span of the orbit comparison campaign) for internal validation purposes.

{ Ü †ÞÝ ‰

{ Ü †ßÝ ‰$à

6 Summary We developed the mathematical background for a stochastic orbit modeling in the environment of least squares estimators and presented two possible pseudo-stochastic orbit parametrizations (piecewise constant accelerations resp. instantaneous velocity changes) which may be set up in an efficient way. Due to the density and high accuracy of GPS phase tracking data both parametrizations are well suited for reduced-dynamic LEO POD and may even be considered to some extent as equivalent as shown with simulations. The processing of CHAMP zero-difference phase data (days 140-150 in 2001) showed that the estimated reduced-dynamic orbits may be validated with SLR residuals on a level of 3.37 cm RMS without any significant SLR bias and that they agree on

measured estimated

-140

á

along-track acceleration (nm/s^2)

Day of year

-120

-160 -180 -200 -220 -240 -260 -280 -300 -320 0

50

100

150 time (min)

200

250

Fig. 4. Piecewise constant accelerations every six minutes in along-track direction compared with measurements from the STAR accelerometer (bias and scale applied) using the gravity field model EIGEN-1S (day 141).

an RMS level of better than 5 cm with orbits from CSR. The additional comparison with accelerometer data showed a high correlation (77.8%) between the measured and the estimated piecewise constant accelerations in along-track direction.

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