- Email: [email protected]
A simulation tool for the new gravity field satellite missions
0 2002 COSPAR.
Pergamon www.elsevier.com/locate/asr A
Adv. Space Res. Vol. 30, No. 2, pp. 227-232, 2002 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273-1177/02 $22.00 + 0.00
PII: SO273-1177(02)00289-2
SIMULATION
TOOL FOR THE NEW GRAVITY SATELLITE MISSIONS
H. Oberndorfer’,
J. Miiller’, R. Rummel’,
FIELD
and N. Sneeuw’
‘Institute for Astronomical
and Physical Geodesy, Technische Universittit Miinchen, 80290 Munich, Germany e-mail: helmut@,alpha.fesn.ttu-muenchen.de resp. [email protected] ‘Dept. of Geomatics Engineering, University of Calgary, 2500 University Drive, Calgary, Alberta T2N IN4, Canada ABSTRACT
In the new decade three important gravity missions will measure the Earth’s gravity field: CHAMP, GRACE and GOCE. As main sensor of these three missions accelerometers are used for measuring the non-gravitational disturbance acceleration of the satellites. Moreover, in case of GOCE, the observables are acceleration differences between six accelerometers. The accelerometer is modeled by a simulation tool in which orbit dynamics, satellite control systems, accelerometer characteristics and errors are taken into account. This tool allows to verify the projected accelerometer and gradiometer accuracies. The results of the simulator are time series of ‘measured’ accelerations and their corresponding error power spectral densities. 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
INTRODUCTION The satellite mission CHAMP was launched in July 2000. One of its goals is to measure the Earth’s gravity field. CHAMP is followed by the gravity field satellite missions GRACE and GOCE. GRACE will focus on the time variable gravity field and the medium wavelengths. GOCE will provide a model of the static Earth’s gravity field with high spatial resolution and high accuracy. The three missions are based on different measurement principles. In the case of CHAMP high-low satellite-to-satellite tracking is used to measure the satellite position from which the gravity field is derived. For GRACE the distance between two low-flying satellites is the main measurement quantity. The gravity measurements of GOCE are derived from a combination of different accelerometer readouts. To extract the gravitational signal in all three missions, it is necessary to observe the non-gravitational forces disturbing the satellite dynamics, and to control their effect by thrusters or correct for it in postprocessing. Therefore GPS, star trackers and accelerometers are employed for orbit control, attitude control, precise positioning and gravity field measurements. The accuracy of accelerometers measurements and the interaction with the other satellite sensors determine the measurement performance. The complete measurements were simulated in the time domain using the software SIMULINK. The errors to be considered are caused by the imperfection of any sensor or actuator and their measurement noise. Moreover, the geometry (position and orientation) of the several instruments with respect to each other have been taken into account with a certain accuracy. All these errors and their couplings affect the measurements and consequently the end-products. Differences between input signals and simulated accelerometer measurements served as input for the derivation of error Power Spectral Densities (PSD’s). The latter are used for further error analyses of the scientific end-products (e.g. spherical harmonic coefficients, geoid heights, gravity anomalies). The goal of the simulation is to check if the requirements for the three missions are fulfilled. In terms of Power Spectral Densities, the mission requirement for CHAMP is the measurement of the disturbing non-gravitational accelerations with an accuracy of better than 3x100~ m/(s2&) white noise with a l/f-behaviour below 5 mHz. The mission requirement for GRACE is the measurement of the disturbing non-gravitational accelerations with an accuracy of better than 3~10-‘~ m/(s2a) white noise 227
H. Oberndorfer
228
ef al
with a l/f-behaviour below 5 mHz. For the diagonal elements of the gravity gradient tensor measured by GOCE the requirement is an accuracy of 4 mE/& white noise with a l/f-behaviour below 5 mHz . SIMULATION
CONCEPT
signals _____~_____________~~~~~~~~~~~~~~~~~~
Input
Derived signals
and Sensors
On Board SignalProcessing and Regulators Down Link
Data Processing
Data Products i.4
Fig. 1. The CHAMP
A
A
satellite: sensors and signal flow.
The strategy of the simulations is explained by the example of the CHAMP mission. In the flow chart (Figure 1) the interaction between forces, sensors, satellite dynamics and control loops is shown. The flow chart represents both the mission concept and the simulator modules. It is the basis for the closed-loop simulation (Miiller and Oberndorfer, 1999). At the top of the flow chart are the external forces which act on the satellite. The forces are separated in gravitational (G) and non-gravitational (F) forces. For the simulations, they were calculated off-line and were introduced in the simulator as time series. They are called “Input Signals”.
Simulation Tool for New Gravity Field Missions
229
The central block which is called “Satellite systems and sensors” represents the satellite itself, its platform and the sensors mounted on it: in particular the accelerometer, star sensor and the GPS antenna. All linear forces working on the platform/accelerometer frame, act on the center of mass (CoM). Additionally the forces produce torques. In the case of non-gravitational forces they act on the center of pressure of the satellite. In this way torques which force the satellite to rotate were derived from input forces. The simulator solves the dynamic Euler-equations and calculates satellite pointing, angular velocity and angular acceleration of the satellite platform. These quantities are called derived signals, because they are calculated in the simulator based on the input signals. Satellite pointing is observed by’ the star sensor which is modeled by adding a characteristic colored noise to the pointing angles. This is the input for the Attitude Controller, which fires the thrusters when the satellite pointing angles are above a value of 2”. In the flow chart, the Attitude Controller can be seen in the field of “On Board Signal Processing and Regulators”. One line goes to the thrusters which are mounted on the satellite. In the simulator the thrusters are modeled with noise and misalignment. The torque produced by the thrusters is fed back to the Attitude Dynamics Block. The accelerometers are mounted on the satellite platform. An accelerometer observes the difference between testmass- and platform-acceleration. Gravity forces act on the testmass and the platform, nongravitational forces on the platform only, because the satellite walls shield the testmass. The acceleration acting on an accelerometer depends on its position 61’ in the satellite platform frame. Since the observations are performed on a rotating platform, inertial forces (centrifugal and Euler) are sensed as given by:
(1) &
is the satellite angular acceleration
nz -nv
0
f-ijk=
fiz
0 anti-symm.
Rijfljk
0
(2)
7
)
is the satellite angular acceleration
-4-g Rijfijk
due to Euler forces:
nq
R&,
QJl,
4-g - 52;
=
symm.
due to centrifugal forces:
q-2, -nq - “p
(3)
and Gjk the gravity gradient tensor. With equation (l), the input acceleration for the accelerometer module is calculated. The detailed modeling of the accelerometer is explained in the next section. The accelerometer output is filtered and converted from analog to digital, sampled and linked down to the Earth. Additional data processing can be done in post-processing. The difference of time series between input signals and simulated output will be transformed in the spectral domain and presented as error PSD. In the case of CHAMP simulations, the interesting quantity is the accuracy of the non-gravitational acceleration measurements.
THE ACCELEROMETER
MODEL
The accelerometers for all three missions are developed and produced by ONERA (Touboul et al., 1998; Willemenot, 1996). The basic working principle of these instruments is electrostatic suspension of a test mass in the center of a cage, which is controlled by a capacitive position detector. The accelerometers (see Figure 2) can be described by three blocks and one feedback loop, the test mass dynamics block for the interactions between the test mass and the electrodes, the position detector and the PID-controller. In reality, the situation is more difIicult, but the main dynamics of the accelerometer are described very well by this simple scheme. The test mass dynamics describes the sensitivity of the test mass to input and control accelerations where the limiting factor is the negative stiffness of the loop: the lower the absolute value of
H. Oberndorfer
et al.
Fig. 2. Simulation model and elements of the accelerometer.
the stiffness the higher the sensitivity of the instrument. The position of the test mass is measured by the position detector which is driven by a carrier frequency of 100 kHz. The position sensor is simulated with a transfer function of one, but with additional noise. The output of the position detector is used to drive the feedback voltage by the PID-controller. The PID-controller is operating in a bandwidth from DC to 150 Hz which limits the calculation of the derivative of the signal. As a result of the feedback, the position sensor noise, amplified by the controller, couples with the test mass dynamics and produces a large noise level which increases proportional to f 4. The accelerometer transfer function models the reaction of the accelerometer to input accelerations. In the ideal case, it should be one which is approximately true at frequencies much lower than the resonance frequency of the accelerometer. At higher frequencies the transfer function of the accelerometer decreases. All internal effects and error sources within one accelerometer are summarized by two kinds of noise, the acceleration noise and the position sensor noise. We used this simplified accelerometer model in our simulations, to avoid the implementation of all the individual effects indicated. This simplification, combining all errors into two kinds of noise sources, speeds up the simulation. The real accelerometer controls six degrees of freedom of the test mass, but we simulated just control loops for the linear motion of the test mass and implemented the coupling between linear and rotation circuits as a additional noise source. The structure of the accelerometer error PSD is a l/f increase at low frequencies with a corner frequency of 10m2 Hz and a f4 increase at frequencies higher than 0.1 Hz. When one samples the real accelerations or gradiometer measurements at 1 Hz, one obtains big aliasing errors because of the high frequency noise. TO avoid it, one has to use an anti-aliasing filter (i.e. a kind of a low-pass filter) before sampling. The sensitive axis has an accuracy of 3~10~~ m/(s2a) z in the measurement bandwidth from 5 mHz to 0.1 Hz. Simulations show good agreement with the theoretical accelerometer noise model. The error PSD of the less sensitive axis is worse than the other two. Depending on the maximum acceleration which acts on the testmass the accelerometer parameters can be tuned. A drag-free controller reduces the drag in the case of GOCE by a factor of 100 and the mod&xl accelerometer has an accuracy of lo-l2 m/(s2&) in the measurement bandwidth. GOCE SIMULATIONS For the GOCE gradiometer, six accelerometers were arranged on the satellite platform in a SO called This became necessary, because not all axes of the ONERA acnon-symmetric diamond cox@uration. celerometers achieve the same precision. The accelerometers have only two sensitive axes, the third one is less sensitive by a factor of 1000. In Figure 3 the selected gradiometer configuration is shown, where the dots indicate the less sensitive axis. As a consequence, not all tensor components can be observed with the same precision. Now the observables are difference accelerations over a baseline of 0.5 m.
Simulation Tool for New Gravity Field Missions
Fig. 3. Non-symmetric
diamond
gradiometer
231
configuration.
The acceleration differences measured by the gradiometer depend on gravity gradients, angular accelera tions and angular velocities. This is described by equation (1). The accuracy of the gradiometer measurements depends mainly on residual drag, angular velocities and angular accelerations of the satellite and of accelerometer errors. For separating the gravity gradients, the knowledge of the angular velocity of the satellite is necessary. The input for attitude and drag-free control is obtained from the high-low SST measurements with GPS and gradiometer observations. The best gradiometer configuration for this purpose is a non-symmetric diamond configuration (Figure 3). This gradiometer has the feature to measure the diagonal components Gii directly with high accuracy as well as the two off-diagonals which are needed to determine Qi from the integrated anti-symmetric measurement tensor. The result for GzyrGzz and G,, of GOCE simulations, with a system configuration described above, is presented in Figure 4. The error PSD of the diagonal gravity gradient tensor component G,, is less than 2 mE/& in the measurement bandwidth from 5 mHz to 0.1 Hz. The other two diagonal gravity gradient tensor components G,, and G,, shows similar error PSDs. The G,, component is better than 4 mE/fi However only in a frequency band from 10 mHz to 0.1 Hz. G,, and G,, reach an accuracy of 2 E/G. only a strong requirement for the diagonal components, which are the baseline observables of GOCE, has been deilned. The additional good quality of G,, is a by-product of good system engineering. SUMMARY With the simulation tool, it is possible to build up the complete system configuration of the gravity field missions CHAMP, GRACE and GOCE. The simulator makes it possible to estimate the mission performance under different configurations. In this way the simulator is an important tool for mission design studies and error investigations. ACKNOWLEDGEMENTS This work has been funded in part under the ESA/ESTEC contract No. 12735/98/NK/GD. We are mostly indebted to our colleagues from the SID-Consortium, existing of SRON (Space Research Organisation Netherlands), IAPG (Institut fiir Astronomische und Physikalische Geodkie) and DEOS (Delft institute for Earth-Oriented Space research). The help of Alenia Aerospazio, Italy, is also greatly acknowledged.
H. Oberndorfer
232
$10 &o-2
. 118 ii!!!!!!:!!!i!!i!l!.!illi!!!! : s: :I :I::I:::._..:::I :r 811 ~: I: :I :i! !:! . :III .:~ : ::::::::.::::::;:;:.:I:::::::: . . . .. .. .. ... .. .. .> ... . i. ....l..l.... ... .. ... I : :.: ... .?. .,I ,,,._..,.,..I.. ..,s .. L, , , ,,IIII1L,I,. I.,.1.1.*,,IIII !I/j!~I!/i)!i::j:/i: !! “““,“,I!,!,!,, ::::::.:::,:, :,>.:.:::;::::: ‘Ii’l!!!! .._....__......, I L .:.. .;. .:. .:. .:. (,......,... ,, , ..,......, ,. .. , ~,I , / ., . . I I ! ., !I! ! ! ,i I !:! I!!f ,. !:!,. I:! :, III ,, .!!!!!!:!IIi!fiijl:ll,l . . .._.. ::. , .I ,,, lo3 . . . . .-,..
I...
I...
10-b ,!!!!!:!!!i!!:!!:!!:!iiij!!!!!!:!!!i!!i!i!:!ii lOA
IO3
et al
P . . . ,.,
..
.
,%,
L
.
. .
i I
_....
,,.,..
II.III.I1.1.1.II.I..,I
! !.j!! ?! ::.::::::::::.::;:.;.::
._:
!!!I
:.,:..;:
!.!,
_.
?!.!.!!
1
“’ . .._.
..
fi.iifrI((fli.]Ir~biI
IO1 frequency [Hz:
-
10-l
I.
, ,.l$!!li IO2 frequency[Hz]
10-l
IO0
IO2 frequency[Hz]
10-l
IO0
Fig. 4. Error PSD of GOCE gravity gradient measurement.
REFERENCES Miiller, J., and H. Oberndorfer, Simulation of the Gradiometer Mission GOCE, Artificial SateZlites, Vol. 34, No. 1, 41-55, 1999. Touboul, P., B. Foulon, and G.-M. Le Clerc, STAR, the accelerometer of geodetic mission CHAMP, 49th International Astronautical Congress, Melbourne, Melbourne, 1998. Willemenot E., Electrostatic Space Accelerometer for Present and Future Missions, Federal Astronautical Federation, 1996.
Pergamon www.elsevier.com/locate/asr A
Adv. Space Res. Vol. 30, No. 2, pp. 227-232, 2002 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273-1177/02 $22.00 + 0.00
PII: SO273-1177(02)00289-2
SIMULATION
TOOL FOR THE NEW GRAVITY SATELLITE MISSIONS
H. Oberndorfer’,
J. Miiller’, R. Rummel’,
FIELD
and N. Sneeuw’
‘Institute for Astronomical
and Physical Geodesy, Technische Universittit Miinchen, 80290 Munich, Germany e-mail: helmut@,alpha.fesn.ttu-muenchen.de resp. [email protected] ‘Dept. of Geomatics Engineering, University of Calgary, 2500 University Drive, Calgary, Alberta T2N IN4, Canada ABSTRACT
In the new decade three important gravity missions will measure the Earth’s gravity field: CHAMP, GRACE and GOCE. As main sensor of these three missions accelerometers are used for measuring the non-gravitational disturbance acceleration of the satellites. Moreover, in case of GOCE, the observables are acceleration differences between six accelerometers. The accelerometer is modeled by a simulation tool in which orbit dynamics, satellite control systems, accelerometer characteristics and errors are taken into account. This tool allows to verify the projected accelerometer and gradiometer accuracies. The results of the simulator are time series of ‘measured’ accelerations and their corresponding error power spectral densities. 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
INTRODUCTION The satellite mission CHAMP was launched in July 2000. One of its goals is to measure the Earth’s gravity field. CHAMP is followed by the gravity field satellite missions GRACE and GOCE. GRACE will focus on the time variable gravity field and the medium wavelengths. GOCE will provide a model of the static Earth’s gravity field with high spatial resolution and high accuracy. The three missions are based on different measurement principles. In the case of CHAMP high-low satellite-to-satellite tracking is used to measure the satellite position from which the gravity field is derived. For GRACE the distance between two low-flying satellites is the main measurement quantity. The gravity measurements of GOCE are derived from a combination of different accelerometer readouts. To extract the gravitational signal in all three missions, it is necessary to observe the non-gravitational forces disturbing the satellite dynamics, and to control their effect by thrusters or correct for it in postprocessing. Therefore GPS, star trackers and accelerometers are employed for orbit control, attitude control, precise positioning and gravity field measurements. The accuracy of accelerometers measurements and the interaction with the other satellite sensors determine the measurement performance. The complete measurements were simulated in the time domain using the software SIMULINK. The errors to be considered are caused by the imperfection of any sensor or actuator and their measurement noise. Moreover, the geometry (position and orientation) of the several instruments with respect to each other have been taken into account with a certain accuracy. All these errors and their couplings affect the measurements and consequently the end-products. Differences between input signals and simulated accelerometer measurements served as input for the derivation of error Power Spectral Densities (PSD’s). The latter are used for further error analyses of the scientific end-products (e.g. spherical harmonic coefficients, geoid heights, gravity anomalies). The goal of the simulation is to check if the requirements for the three missions are fulfilled. In terms of Power Spectral Densities, the mission requirement for CHAMP is the measurement of the disturbing non-gravitational accelerations with an accuracy of better than 3x100~ m/(s2&) white noise with a l/f-behaviour below 5 mHz. The mission requirement for GRACE is the measurement of the disturbing non-gravitational accelerations with an accuracy of better than 3~10-‘~ m/(s2a) white noise 227
H. Oberndorfer
228
ef al
with a l/f-behaviour below 5 mHz. For the diagonal elements of the gravity gradient tensor measured by GOCE the requirement is an accuracy of 4 mE/& white noise with a l/f-behaviour below 5 mHz . SIMULATION
CONCEPT
signals _____~_____________~~~~~~~~~~~~~~~~~~
Input
Derived signals
and Sensors
On Board SignalProcessing and Regulators Down Link
Data Processing
Data Products i.4
Fig. 1. The CHAMP
A
A
satellite: sensors and signal flow.
The strategy of the simulations is explained by the example of the CHAMP mission. In the flow chart (Figure 1) the interaction between forces, sensors, satellite dynamics and control loops is shown. The flow chart represents both the mission concept and the simulator modules. It is the basis for the closed-loop simulation (Miiller and Oberndorfer, 1999). At the top of the flow chart are the external forces which act on the satellite. The forces are separated in gravitational (G) and non-gravitational (F) forces. For the simulations, they were calculated off-line and were introduced in the simulator as time series. They are called “Input Signals”.
Simulation Tool for New Gravity Field Missions
229
The central block which is called “Satellite systems and sensors” represents the satellite itself, its platform and the sensors mounted on it: in particular the accelerometer, star sensor and the GPS antenna. All linear forces working on the platform/accelerometer frame, act on the center of mass (CoM). Additionally the forces produce torques. In the case of non-gravitational forces they act on the center of pressure of the satellite. In this way torques which force the satellite to rotate were derived from input forces. The simulator solves the dynamic Euler-equations and calculates satellite pointing, angular velocity and angular acceleration of the satellite platform. These quantities are called derived signals, because they are calculated in the simulator based on the input signals. Satellite pointing is observed by’ the star sensor which is modeled by adding a characteristic colored noise to the pointing angles. This is the input for the Attitude Controller, which fires the thrusters when the satellite pointing angles are above a value of 2”. In the flow chart, the Attitude Controller can be seen in the field of “On Board Signal Processing and Regulators”. One line goes to the thrusters which are mounted on the satellite. In the simulator the thrusters are modeled with noise and misalignment. The torque produced by the thrusters is fed back to the Attitude Dynamics Block. The accelerometers are mounted on the satellite platform. An accelerometer observes the difference between testmass- and platform-acceleration. Gravity forces act on the testmass and the platform, nongravitational forces on the platform only, because the satellite walls shield the testmass. The acceleration acting on an accelerometer depends on its position 61’ in the satellite platform frame. Since the observations are performed on a rotating platform, inertial forces (centrifugal and Euler) are sensed as given by:
(1) &
is the satellite angular acceleration
nz -nv
0
f-ijk=
fiz
0 anti-symm.
Rijfljk
0
(2)
7
)
is the satellite angular acceleration
-4-g Rijfijk
due to Euler forces:
nq
R&,
QJl,
4-g - 52;
=
symm.
due to centrifugal forces:
q-2, -nq - “p
(3)
and Gjk the gravity gradient tensor. With equation (l), the input acceleration for the accelerometer module is calculated. The detailed modeling of the accelerometer is explained in the next section. The accelerometer output is filtered and converted from analog to digital, sampled and linked down to the Earth. Additional data processing can be done in post-processing. The difference of time series between input signals and simulated output will be transformed in the spectral domain and presented as error PSD. In the case of CHAMP simulations, the interesting quantity is the accuracy of the non-gravitational acceleration measurements.
THE ACCELEROMETER
MODEL
The accelerometers for all three missions are developed and produced by ONERA (Touboul et al., 1998; Willemenot, 1996). The basic working principle of these instruments is electrostatic suspension of a test mass in the center of a cage, which is controlled by a capacitive position detector. The accelerometers (see Figure 2) can be described by three blocks and one feedback loop, the test mass dynamics block for the interactions between the test mass and the electrodes, the position detector and the PID-controller. In reality, the situation is more difIicult, but the main dynamics of the accelerometer are described very well by this simple scheme. The test mass dynamics describes the sensitivity of the test mass to input and control accelerations where the limiting factor is the negative stiffness of the loop: the lower the absolute value of
H. Oberndorfer
et al.
Fig. 2. Simulation model and elements of the accelerometer.
the stiffness the higher the sensitivity of the instrument. The position of the test mass is measured by the position detector which is driven by a carrier frequency of 100 kHz. The position sensor is simulated with a transfer function of one, but with additional noise. The output of the position detector is used to drive the feedback voltage by the PID-controller. The PID-controller is operating in a bandwidth from DC to 150 Hz which limits the calculation of the derivative of the signal. As a result of the feedback, the position sensor noise, amplified by the controller, couples with the test mass dynamics and produces a large noise level which increases proportional to f 4. The accelerometer transfer function models the reaction of the accelerometer to input accelerations. In the ideal case, it should be one which is approximately true at frequencies much lower than the resonance frequency of the accelerometer. At higher frequencies the transfer function of the accelerometer decreases. All internal effects and error sources within one accelerometer are summarized by two kinds of noise, the acceleration noise and the position sensor noise. We used this simplified accelerometer model in our simulations, to avoid the implementation of all the individual effects indicated. This simplification, combining all errors into two kinds of noise sources, speeds up the simulation. The real accelerometer controls six degrees of freedom of the test mass, but we simulated just control loops for the linear motion of the test mass and implemented the coupling between linear and rotation circuits as a additional noise source. The structure of the accelerometer error PSD is a l/f increase at low frequencies with a corner frequency of 10m2 Hz and a f4 increase at frequencies higher than 0.1 Hz. When one samples the real accelerations or gradiometer measurements at 1 Hz, one obtains big aliasing errors because of the high frequency noise. TO avoid it, one has to use an anti-aliasing filter (i.e. a kind of a low-pass filter) before sampling. The sensitive axis has an accuracy of 3~10~~ m/(s2a) z in the measurement bandwidth from 5 mHz to 0.1 Hz. Simulations show good agreement with the theoretical accelerometer noise model. The error PSD of the less sensitive axis is worse than the other two. Depending on the maximum acceleration which acts on the testmass the accelerometer parameters can be tuned. A drag-free controller reduces the drag in the case of GOCE by a factor of 100 and the mod&xl accelerometer has an accuracy of lo-l2 m/(s2&) in the measurement bandwidth. GOCE SIMULATIONS For the GOCE gradiometer, six accelerometers were arranged on the satellite platform in a SO called This became necessary, because not all axes of the ONERA acnon-symmetric diamond cox@uration. celerometers achieve the same precision. The accelerometers have only two sensitive axes, the third one is less sensitive by a factor of 1000. In Figure 3 the selected gradiometer configuration is shown, where the dots indicate the less sensitive axis. As a consequence, not all tensor components can be observed with the same precision. Now the observables are difference accelerations over a baseline of 0.5 m.
Simulation Tool for New Gravity Field Missions
Fig. 3. Non-symmetric
diamond
gradiometer
231
configuration.
The acceleration differences measured by the gradiometer depend on gravity gradients, angular accelera tions and angular velocities. This is described by equation (1). The accuracy of the gradiometer measurements depends mainly on residual drag, angular velocities and angular accelerations of the satellite and of accelerometer errors. For separating the gravity gradients, the knowledge of the angular velocity of the satellite is necessary. The input for attitude and drag-free control is obtained from the high-low SST measurements with GPS and gradiometer observations. The best gradiometer configuration for this purpose is a non-symmetric diamond configuration (Figure 3). This gradiometer has the feature to measure the diagonal components Gii directly with high accuracy as well as the two off-diagonals which are needed to determine Qi from the integrated anti-symmetric measurement tensor. The result for GzyrGzz and G,, of GOCE simulations, with a system configuration described above, is presented in Figure 4. The error PSD of the diagonal gravity gradient tensor component G,, is less than 2 mE/& in the measurement bandwidth from 5 mHz to 0.1 Hz. The other two diagonal gravity gradient tensor components G,, and G,, shows similar error PSDs. The G,, component is better than 4 mE/fi However only in a frequency band from 10 mHz to 0.1 Hz. G,, and G,, reach an accuracy of 2 E/G. only a strong requirement for the diagonal components, which are the baseline observables of GOCE, has been deilned. The additional good quality of G,, is a by-product of good system engineering. SUMMARY With the simulation tool, it is possible to build up the complete system configuration of the gravity field missions CHAMP, GRACE and GOCE. The simulator makes it possible to estimate the mission performance under different configurations. In this way the simulator is an important tool for mission design studies and error investigations. ACKNOWLEDGEMENTS This work has been funded in part under the ESA/ESTEC contract No. 12735/98/NK/GD. We are mostly indebted to our colleagues from the SID-Consortium, existing of SRON (Space Research Organisation Netherlands), IAPG (Institut fiir Astronomische und Physikalische Geodkie) and DEOS (Delft institute for Earth-Oriented Space research). The help of Alenia Aerospazio, Italy, is also greatly acknowledged.
H. Oberndorfer
232
$10 &o-2
. 118 ii!!!!!!:!!!i!!i!l!.!illi!!!! : s: :I :I::I:::._..:::I :r 811 ~: I: :I :i! !:! . :III .:~ : ::::::::.::::::;:;:.:I:::::::: . . . .. .. .. ... .. .. .> ... . i. ....l..l.... ... .. ... I : :.: ... .?. .,I ,,,._..,.,..I.. ..,s .. L, , , ,,IIII1L,I,. I.,.1.1.*,,IIII !I/j!~I!/i)!i::j:/i: !! “““,“,I!,!,!,, ::::::.:::,:, :,>.:.:::;::::: ‘Ii’l!!!! .._....__......, I L .:.. .;. .:. .:. .:. (,......,... ,, , ..,......, ,. .. , ~,I , / ., . . I I ! ., !I! ! ! ,i I !:! I!!f ,. !:!,. I:! :, III ,, .!!!!!!:!IIi!fiijl:ll,l . . .._.. ::. , .I ,,, lo3 . . . . .-,..
I...
I...
10-b ,!!!!!:!!!i!!:!!:!!:!iiij!!!!!!:!!!i!!i!i!:!ii lOA
IO3
et al
P . . . ,.,
..
.
,%,
L
.
. .
i I
_....
,,.,..
II.III.I1.1.1.II.I..,I
! !.j!! ?! ::.::::::::::.::;:.;.::
._:
!!!I
:.,:..;:
!.!,
_.
?!.!.!!
1
“’ . .._.
..
fi.iifrI((fli.]Ir~biI
IO1 frequency [Hz:
-
10-l
I.
, ,.l$!!li IO2 frequency[Hz]
10-l
IO0
IO2 frequency[Hz]
10-l
IO0
Fig. 4. Error PSD of GOCE gravity gradient measurement.
REFERENCES Miiller, J., and H. Oberndorfer, Simulation of the Gradiometer Mission GOCE, Artificial SateZlites, Vol. 34, No. 1, 41-55, 1999. Touboul, P., B. Foulon, and G.-M. Le Clerc, STAR, the accelerometer of geodetic mission CHAMP, 49th International Astronautical Congress, Melbourne, Melbourne, 1998. Willemenot E., Electrostatic Space Accelerometer for Present and Future Missions, Federal Astronautical Federation, 1996.