GRACE CG OFFSET DETERMINATION BY MAGNETIC TORQUERS DURING ... · GRACE CG OFFSET DETERMINATION BY...

GRACE CG OFFSET DETERMINATION BY MAGNETICTORQUERS DURING THE IN-FLIGHT PHASE

by

Furun Wang

January 2000

Center for Space ResearchThe University of Texas at Austin

CSR-TM-00-01

This work was supported by NASA Contract NAS5-97213.

Center for Space ResearchThe University of Texas at Austin

Austin, Texas 78712

Principal Investigator:

Dr. Byron D. Tapley

ii

ABSTRACT

The Gravity Recovery And Climate Experiment (GRACE mission) is scheduled

for launch in June 2001. Within the 5-year lifetime, the GRACE mission will map

variations in the Earth’s gravity field with unprecedented accuracy. The mission will

have two identical spacecraft flying about 220 kilometers apart in a polar orbit 450

kilometers above the Earth.

The accelerometer, one of key instruments on board GRACE, serves to measure

all non-gravitational accelerations. In combination with the position measurements of the

GPS receiver assembly, purely gravitational orbit perturbations can be derived for use in

gravity field modelling. However, The Proof-Mass Center (PMC) of the accelerometer

needs to be positioned precisely at the Center of Gravity (CG) of the GRACE satellites in

order to avoid measurement disturbances due to rotational accelerations and gravity

gradients. Affected by a lot of unfavorable factors, the CG offset, defined by the

difference between PMC and CG, cannot be zero, in fact, even large enough to affect the

mission target. Therefore, CG offset needs to be measured, and then be reset to zero by

mass balancing during the mission lifetime.

Based on CG calibration approach which uses a magnetic moment with harmonic

time dependence at some fixed frequency for a time interval without the thruster torque,

an efficient method to activate the magnetic moment along two axes is put forward, and

three different estimation algorithms , ASSEST, ASCFEST and ASCREST, are brought

out to estimate the CG offset. The optimal timing for estimating CG offset along each

axis has been found. By operating the CG calibration at the optimal time , the estimation

accuracy of CG offset could be better than 0.02mm for x axis (along track), 0.01mm for

y axis (cross track) , and 0.02 mm for z axis (radial).

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Table of Contents

Abstract

Table of Contents

List of Figures

List of Tables

1 INTRODUCTION

1.1 Background and Motivation

1.2 Outline of Research

2 SPACECRAFT DYNAMICS

2.1 Introduction

2.2 Coordinate System

2.3 Spacecraft Orbit Dynamics Model

2.3.1 Geopotential Gravitational Perturbation

2.3.2 Non-gravitational Perturbation

2.3.2.1 Atmospheric Drag

2.3.2.2 Solar Radiation Pressure

2.3.2.3 Earth Radiation Pressure

2.4 Spacecraft Attitude Dynamics Model

2.4.1 Spacecraft Torques

2.4.1.1 Spacecraft Gravitational Torque

2.4.1.2 Spacecraft Aerodynamics and Radiation Torque

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2.4.1.3 Spacecraft Magnetic Torque

2.5 COM and CG for Small Spacecraft

3 MEASUREMENTS OF ACCELOMETER AND STARCAMERAS AND MAGNETOMETER

3.1 Introduction

3.2 Accelerometer Instrument and Simulation Data

3.2.1 Instrumentation Design Features

3.2.2 Accelerometer Data Simulation

3.3 Star Camera Instrument and Simulation Data


3.3.2 Star Camera Data Simulation

3.3.3 Star Catalog

3.3.4 QUEST Algorithm

3.4 Magnetometer Instrument and Simulation Data


3.4.2 Magnetometer Data Simulation

4 OPTIMAL ESTIMATION OF GRACE CG OFFSET

4.1 Introduction

4.2 Magnetic Moment Activating

4.3 Dynamics Fitting Model and Partial Derivative

4.4 Observation Fitting Model and Partial Derivative

4.5 Data Preprocessing and Interpolation

4.6 Batch Estimation of CG Offset

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4.6.1 ASSEST Algorithm

4.6.2 ASCFEST Algorithm

4.6.3 ASCREST Algorithm

4.7 Closing Remarks

5. SIMULATION PROCEDURE AND ASSUMPTIONSVERIFICATION

5.1 Simulation Procedure

5.2 Parameters and Initial Values Used in Verification and Simulation

5.3 Assumption Verification

6 SIMULATION RESULTS AND ANALYSIS

6.1 Parameters and Initial Values in Simulation

6.2 Simulation Results and Analysis

6.3 Main Error Sources of CG Calibration

6.4 Loss of Magnetometer Data Impact

7. CONCLUSIONS

7.1 Summary and Conclusions

7.2 Recommendations for Future Work

REFERENCES

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List of Figures

2.1 Coordinate Definition

2.2 Radial Unit Vector

3.1 Disturbing acceleration due to / r term and due to sum of other terms , in the RTN frame

3.2 Flowchart of the Star Cameras Data Generation

4.1 Batch Processor Algorithm Flow Chart

5.1 The flowchart for the CG calibration simulation procedure

5.2 Non-gravitational Acceleration of Front GRACE

5.3 Non-gravitational Acceleration of Back GRACE

5.4 Disturbance Acceleration of the Front GRACE Due to CG offset

5.5 Disturbance Acceleration of the Back GRACE Due to CG offset

5.6 External Torque of Front GRACE

5.7 External Torque of Back GRACE

5.8 Angular Velocity and Acceleration of Front and Back GRACE

6.1 The location and corresponding magnetic flux density in the three cases for front GRACE and Back GRACE

6.2 Simulation Results of Front GRACE for three cases

6.3 Simulations Results of Back GRACE for three cases

6.4 The Angular Acceleration if the Magnetic Moment Activated During the Nominal Phase in one Orbit period

6.5 Estimation Accuracy with respect to Various Error Sources

6.6 Loss of Magnetometer Data Simulation

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List of Tables

1.1 Ground of Error of GRACE COM

1.2 In Flight Stability of GRACE COM

5.1 Perturbations and Torques Applied in Simulation

6.1 The CG Offset RMS of x, y and z axes for Front GRACE

6.2 The CG Offset RMS of x, y and z axes for Back GRACE

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1

Chapter 1

INTRODUCTION

1.1 Background and Motivation

The Gravity Recovery And Climate Experiment (GRACE) mission was selected

as the second mission under the NASA Earth System Science Pathfinder (ESSP) Program

in May 1997. Launching in June of 2001, the GRACE mission will accurately map

variations in the Earth's gravity field over its 5-year lifetime. The GRACE mission will

have two identical spacecraft flying about 220 kilometers apart in a polar orbit 450

kilometers above the Earth.

From scientific point of view, GRACE will succeed the German CHAMP in the

field of Earth gravimetric measurements with unprecedented accuracy. Besides using an

advanced accelerometer, the required dramatic step in accuracy will be achieved by using

two satellites, following each other on the same orbital track. These satellites are

interconnected by a microwave RF link to measure both the exact separation distance and

it’s rate of change to an accuracy of better than 1 m/ s . Therefore, the satellites

themselves become the experiment, allowing a precise ‘snapshot’ of the gravity field to

be measured about every two weeks for a mission life of 5 years over a decreasing orbit

altitude between approximately 500 km and 300 km. The results from this mission will

yield crucial information about the distribution and flow of mass within the Earth and it's

surroundings.

The precise accuracy of GRACE measurements allows scientists to use the

GRACE mission to weigh various parts of the Earth system. The gravity variations that

GRACE will study include: changes due to surface and deep currents in the ocean; runoff

and ground water storage on land masses; exchanges between ice sheets or glaciers and

the oceans; and variations of mass within the Earth. Another goal of the mission is to

2

create a better profile of the Earth's atmosphere. The results from GRACE mission will

make a huge contribution to the goals of NASA's Earth Science Enterprise, Earth

Observation System (EOS) and global climate change studies.

The accelerometer instrument on board GRACE serves to measure all non-

gravitational accelerations. These forces include air drag, solar radiation pressure, Earth

radiation pressure, attitude control activator operation, etc. In combination with the

position measurements of the GPS receiver assembly, purely gravitational orbit

perturbations can be derived for use in gravity field modelling. A by-product of the

accelerometer measurements is the determination of upper atmospheric densities.

The accelerometer uses the basic principle of any electrostatic

microaccelerometer: a proof-mass is free floating inside a cage supported by an

electrostatic suspension. The cavity walls are equipped with electrodes thus controlling

the motion (both translational and rotation) of the proof-mass by electrostatic forces.

Electric signals proportional to the accelerations acting onto the proof-mass are picked up

by these electrodes and fed to the experiment electronics. By applying a closed loop-back

inside the sensor unit it is intended to keep the test body motionless in the center of the

cage.

The Proof-Mass Center (PMC) of the accelerometer needs to be positioned

precisely at the Center of Gravity (CG) of the GRACE satellites in order to avoid

measurement disturbances due to rotational accelerations and gravity gradients.

Unfortunately, before satellite launching, the location of the Center Of Mass (COM) and

the CG of the satellite cannot be precisely fixed to where they are supposed to be, even

so, they still keep moving during the in flight mission due to satellite distortion, gas

consumption, even related to attitude, and so on. Therefore, in reality, the CG offset,

defined by the difference between the CG and PMC, inevitably exits. It has been shown

that the difference of COM and CG of the GRACE satellites during the nominal phase

shall be less than 0.1 m (shown in section 2.5), so small, compared to other error

sources shown below, that can be neglected.

3

From the satellite distortion analysis ( Riede, Tenhaeff, Settelmeyer, 1999), the

ground error of GRACE COM resulting from various sources is shown in Table 1.1, and

the in flight stability of GRACE COM is shown in Table 1.2.

Table 1.1 Ground Error of GRACE COM

Effect Ground Error

Dx in mm Dy in mm Dz in mm

COM measurement uncertainties 0.2 0.2 0.2

Remaining unbalanced 0.067 0.042 0.021

Accuracy of Tank Mounting 0.076 0.076 0.076

Difference in Tank volume 0.01 --- ---

1g/0g effects---gravity 0.105 0.042 0.049

1g/0g effect---temperature 0.083 0.004 0.027

Moisture release CFRP / shrink 0.053 0.021 0.011

Moisture release CFRP / mass

decrease

--- --- 0.038

Moisture release foam / mass

Decrease

--- --- 0.078

Shrink due to moisture release of

foam

--- --- 0.008

Uncertainty of boom position 0.07 --- ---

Impact of buoyancy 0.053 0.053 0.053

RMS---Value of Error 0.27 0.23 0.24

Requirement 0.5 0.5 0.5

4

Table 1.2 In Flight Stability of GRACE COM

Effect In-Flight Stability

Dx in mm Dy in mm Dz in mm

Impact of cold gas piping 0.0003 0.0001 0

Difference in mass of tanks

due to gas consumption

0.0462 --- ---

Total mass decrease 0.0007 0.0007 0.0007

RMS-Value of Error 0.0462 0.0007 0.0007

Requirement 0.1mm/ 6months 0.1mm/ 6months 0.1mm/ 6months

From the above two tables, it can be seen that the CG offset may be large enough

to negatively affect the accelerometer measurements by including some disturbance

accelerations. Consequently, it will decrease the accuracy of mapping the Earth's gravity

fields. An approach adopted in practice, called CG calibration, to avoid large disturbance

accelerations due to the CG offset is to measurement the CG offset and then move the

CG of the GRACE to the PMC of the accelerometer by mass balancing. During the whole

mission, the CG calibration should be done several times, say, roughly once every 6

months.

JPL scientist L.Romans (1997) brought out an approach for GRACE CG offset

determination with the magnetic torques, in which a magnetic moment with harmonic

time dependence at some fixed frequency for a time interval , realized by magnetic torque

rods onboard the GRACE, is employed during the CG calibration. Based on this

illuminating CG calibration approach, an efficient method to activate the magnetic

moment along two axes is put forward, and three estimation algorithms are developed in

this report to estimate the CG offset during the in flight phase. The accelerometer data ,

star camera data and the magnetometer data are used in these three algorithms as

observation data.

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1.2 Outline of Research

The considerations discussed in the previous section are the motivation for this

work. The goal of the research is to simulate the observation data and establish estimation

algorithms to optimally determine the CG offset , and meanwhile to find efficient way to

activate the magnetic moment and to seek the optimal calibration timing.

The spacecraft orbit and attitude dynamics are described in chapter 2, in which the

perturbations and external torques acting upon the spacecraft are presented in detail. A

dynamics model, which includes gravitational perturbation, atmospheric drag, solar

radiation pressure and Earth radiation pressure for orbit dynamics, and magnetic torque,

aerodynamics torque, solar radiation torque and gravitational torque for attitude

dynamics, is built up. The orbit and attitude dynamics model described in this chapter is

used to generate the real orbit trajectory and attitude orientation of GRACE satellites.

Besides, the difference between CG and COM of the GRACE during the nominal phase

is calculated. The extremely small difference allows it to be neglected. In fact, the CG

offset estimated in this report actually turns out to be the difference between PMC of

accelerometer and COM of the satellite.

The performance characteristic of measurement instrument system, including the

accelerometer, star cameras and magnetometer, is outlined, and furthermore the

measurement models for these three instruments are established in Chapter 3. The

accelerometer data and magnetometer data could be easily generated given the real orbit

and attitude information of GRACE satellites, while the star cameras data are much more

complicated. The star cameras data generating process, which gives quaternion data

given the attitude of the satellite, is presented in detail in this chapter.

Setting the magnetic moment efficiently and developing data processing

algorithm to estimate the CG offset, which are most of the research, are presented in

chapter 4. The observed data coming from the measurement models, equivalent to the so

called level 1 data in most of other documents, are further preprocessed to be used in

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estimation program. Three different estimation algorithms, ASSEST, ASCFEST and

ASCREST, for determining the CG offset are put forward in detail.

In chapter 5, the whole simulation procedure is summarized, parameters and

initial values are specified for the simulation program. Some assumptions used in the

report are further verified.

Simulations have been done for three different cases in chapter 6. Estimation

results are presented and comparisons to the real values of CG offset are made, and then

are analyzed to find the optimal timing for CG offset estimation along each axis.

Furthermore, the effects of some main error sources of CG calibration on the estimation

accuracy are simulated, and loss of magnetometer data simulation is also made.

Finally, summary and conclusions are made from the research and the simulation

results in Chapter 7, and also some recommendation work to be investigated are

presented.

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Chapter 2

SPACECRAFT DYNAMICS

2.1 Introduction

In reality, orbit dynamics and attitude dynamics of near Earth spacecraft are

mutually coupled. Different orbit has different gravitational torque, aerodynamic torque,

radiation torque, and magnetic torque for the same spacecraft orientation, and different

attitude orientation induces different gravitational force, atmospheric drag, radiation

pressure for the same spacecraft orbit. Thus orbit dynamics affects attitude dynamics, and

vice versa.

GRACE CG calibration involves both orbit dynamics and attitude dynamics.

Simply speaking, because they are coupled. To be much more precise, from the point of

view of generating the observation data, the accelerometer measurements include the

non-gravitational acceleration related to the orbit dynamics, the star cameras output the

quaternion data involved the attitude dynamics, furthermore, while integrating the

attitude dynamics equations, the magnetic torque exerting upon the spacecraft has to be

known , simply implying that knowledge of the Earth magnetic field experienced by the

satellites, involved both orbit and attitude information, is needed. On the other hand, from

the point of view of data processing of the CG offset, optimal estimation methods put

forward in this report need spacecraft’s both orbit and attitude information. Due to these

reasons, in this report, the spacecraft orbit dynamics and attitude dynamics , modelled in

the following sections, are integrated together to simulate the real orbit trajectory and

attitude orientation, from which the accelerometer, star cameras and magnetometer

measurement data are derived. Eventually, determination of CG offset, when applied to

the GRACE real mission, also needs orbit and attitude information of the satellite.

8

Many formulations of dynamics exist. Most of the orbit dynamics models used in

this report are extracted from MSODP (Multi-Satellite Orbit Determination Program) , a

sophisticated orbit determination software developed in CSR. Furthermore, an elegant

subprogram AMA/LaRC , which can output the normalized torque and perturbation

acceleration due to atmosphere and solar radiation, are included to increase the non-

gravitational perturbations models, and what is more, to obtain the torque experienced by

spacecraft. During the CG Calibration, the dominating torque, magnetic torque, is

produced by activating the magnetic torque rods, as will be discussed later in this chapter.

As usual, the spacecraft orbit and attitude equations of motion in this report are

written with respect to the COM of spacecraft. However, the proof mass of the GRACE

accelerometer is supposed to keep in the CG , rather than COM, to avoid disturbance

accelerations induced by CG offset. Fortunately, the difference between COM and CG is

small enough to be neglected, typically, for the GRACE satellite, it is less than 2 m ,

mainly along the radial direction, during the nominal phase. The derivation of CG of

spacecraft is referred to ( F.P.J.Rimrott , 1989), and the main result is quoted and

applied to GRACE satellite.

2.2 Coordinate System

The reference system OI − XYZ adopted in this report for the orbit dynamics

model is the J2000 geocentric inertial coordinate system, which is defined by the mean

equator and vernal equinox at Julian epoch 2000.0. The Earth’s body-fixed coordinate

system OE − ′ x ′ y ′ z is defined by a simple rotation with respect to the reference system,

which implies that the effect of Earth's precession, nutation, polar motion, and the true

sideal time correction , small enough indeed, are neglected in this report. The spacecraft

body-fixed coordinate systems Ob − x1y1z1 , Ob − x2y2z2 are defined differently for the two

GRACE satellites, although the origins of both systems are located in COM of

corresponding spacecraft. The axes directions of GRACE body-fixed system are defined

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as: for the front GRACE satellite, - ˆ x 1 is along track by pitching up ~ o1 , ˆ y 1is out of

orbital plane, and ˆ z 1 is radial downward; for the back GRACE satellite, ˆ x 2 is along track

by pitching down ~ o1 , ˆ y 2 is out of orbital plane, and ˆ z 2 is radial downward. Furthermore,

another set of spacecraft body-fixed coordinate systems Op − x1y1z1 , Op − x2y2z2 is

defined with axes parallel to Ob − x1y1z1 , Ob − x2y2z2 and origins at PMC of

accelerometer of front and back GRACE satellite , respectively. These coordinate

systems are shown in Figure 2.1. For the later parts of this report, the subscript 1 and 2,

which indicate the front GRACE and back GRACE, respectively, will be omitted if no

confusion occurs.

2.3 Spacecraft Orbit Dynamics Model

The spacecraft orbit equations of motion can be described in J2000.0 geocentric

non-rotating reference system as follows

r ˙ r =

r f g +

r f ng (2-1)

where rr

is the position vector of the COM of the satellite, gfr

is the sum of the

gravitational perturbations acting upon the satellite and r f ng is the sum of the non-

gravitational perturbations acting upon the surfaces of the spacecraft.

2.3.1 Geopotential Gravitational Perturbation

In this report, the gravitational perturbation is considered only due to the

geopotential of the Earth. Perturbations due to the solid Earth tides, the ocean tides,

rotational deformations, the planets including Sun and Moon and general relativity are

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Figure 2.1 Coordinate Definition

r r 1

r r 2

y2

x2

z2

z2

x2

y2

COM

PMC

y1y1

x1

x1

z1 z1

COMPMC

r d 1

r d 2

X

Z(z’)

x’

Y

y’

OI(OE)

FRONT GRACE BACK GRACE

αG

EARTH

11

neglected. This neglecting would not affect the calibration accuracy too much simply

because they are gravitational perturbations.

The spherical harmonic representation of the Earth gravitational field is referred

as ( Kaula,1966; Heiskanen and Moritz,1967)

U =r

ae

r

l= 0

∞

∑l

Plm(sin( ))[Clm cos(m ) + Slm sin(m )]m= 0

l

∑ (2-2)

where ae is the semi-major axis of the Earth's reference ellipsoid, µ is the Earth

gravitational constant , r, , is the radius, latitude, and longitude of the satellite in

Earth's body-fixed coordinate system OE − ′ x ′ y ′ z , Clm ,Slm are the geopotential harmonic

coefficients of degree l and order m , Plm is the Legendre associate functions.

The gravitational perturbation of the satellite due to the attraction of the Earth can

be expressed as certain transformations of gradient of the potential U . In fact, gfr

can be

obtained as

r f g = M ′ x ′ y ′ z

XYZ Mr′ x ′ y ′ z ∇U (2-3)

where M ′ x ′ y ′ z XYZ is the rotation matrix from Earth's body-fixed coordinate system OE − ′ x ′ y ′ z

to inertial system OI − XYZ , Mr′ x ′ y ′ z is the rotation matrix from spherical coordinate

( ˆ u r , ˆ u , ˆ u ) to OE − ′ x ′ y ′ z , ∇U is the gradient of the geopotential.

Neglecting the effect of Earth's precession, nutation, polar motion, and the true

sideal time correction yields the following expressions for the rotation matrix M ′ x ′ y ′ z XYZ

M ′ x ′ y ′ z XYZ =

cos G − sin G 0

sin G cos G 0

0 0 1

(2-4)

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where Gα is the right ascension of the Greenwich meridian. Besides, the rotation matrix

Mr′ x ′ y ′ z can be obtained as follows

Mr′ x ′ y ′ z =

cos cos − sin cos −sin

cos sin −sin sin cos

sin cos 0

(2-5)

2.3.2 Non-gravitational Perturbation

In this report, the non-gravitational perturbations acting on the satellite include

perturbations due to atmospheric drag, solar radiation pressure, the Earth radiation

pressure.

2.3.2.1 Atmospheric Drag

A near-Earth satellite of arbitrary shape moving with some velocity r v in an

atmosphere of density will experience both lift and drag forces. The lift forces are

small compared to the drag forces, which can be modeled as (Schutz and Tapley, 1980)

r f drag = −

1

2(Cd Ad

mS

)v r

r v r (2-6)

where is the atmospheric density, r v r is the satellite velocity with respect to the

atmosphere, vr is the magnitude of r v r , mS is the mass of the satellite, Cd is the drag

coefficient for the satellite and Ad is the cross-sectional area of the main body

perpendicular to r v r .

For the trapezoid-shaped GRACE with size length,height,width _bot and

width _top , the cross-sectional area Ad can be obtained as

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Ad = Si

i =1

6

∑ ˆ n i ⋅r v r / v rH( ˆ n i ⋅

r v r /vr ) (2-7)

where H(x) = −1, if x ≤ 0 , otherwise H(x) = 0 , ˆ n 1 = (1,0,0)T , ˆ n 2 = (−1,0,0) T ,

ˆ n 3 = (0,sin a,− cosa)T , ˆ n 4 = (0, −sin a,− cos a)T , ˆ n 5 = (0,0,1)T , ˆ n 6 = (0,0, −1)T ,

S1 = S2 = (width _bot + width _ top) ⋅ height / 2, S3 = S4 = length ⋅height / s i na ,

S5 = width_ bot ⋅length , S6 = width _top ⋅ length, sin a = sin(height / c2 + height2 ),

cos a = cos(c / c2 + height2 ) and c = (width _bot − width_ top) /2 .

Another way to compute the atmospheric drag is from AMA/LaRC. The air drag

model program GETAFT outputs the normalized atmospheric drag vector in satellite

body-fixed frame for a given wind velocity vector in the body-fixed frame, and then by

unnormalizing the unit atmospheric drag vector can obtain the drag acceleration .

There are a number of empirical density models used for computing the

atmospheric density. There include the Jacchia77 (Jacchia,1977), the Drag Temperature

Model(DTM)(Barlier et al., 1977), Exponential Density Model, JAC70M(Mike

P.Hickey) and AMSIS Model. The wind model from AMSIS and short period

atmospheric density perturbations are included.

2.3.2.2 Solar Radiation Pressure

The direct solar radiation pressure from the Sun on a satellite is modeled as

(Tapley et al., 1990)

r f solar = −P(1+ )(

As

mS

)v ˆ u sun (2-8)

where P is the momentum flux due to Sun, is the reflectivity coefficient of the

satellite, As is the cross-sectional area of the satellite normal to the Sun, v is the eclipse

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factor ( v=0 if the satellite is in full shadow, v=1 if the satellite is in full Sun, and 0<v<1

if the satellite is in partial shadow) and ˆ u sun is the unit vector pointing from the satellite to

the Sun.

The cross-sectional area As can be obtained

As = Sii = 1

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∑ ˆ n i ⋅ ˆ u sunH( ˆ n i ⋅ ˆ u sun) (2-9)

Another way to compute the solar radiation pressure is from AMA/LaRC. The

solar radiation pressure model program GETAFT outputs the normalized solar radiation

pressure vector in satellite body-fixed frame for a given Sun-satellite unit vector in the

body-fixed frame, and then by unnormalizing the unit solar radiation pressure vector can

obtain the solar radiation pressure.

2.3.2.3 Earth Radiation Pressure

The Earth radiation pressure model used can be summarized as follows (Knocke

and Ries, 1987; Knocke,1989)

r f erp = (1+ e) ′ A (

Ae

mSc) [( aEs cos s + eMB)ˆ r ]j

j =1

N

∑ (2-10)

where e is the satellite reflectivity for the Earth radiation pressure, ′ A is the projected,

attenuated area of a surface element of the Earth, Ae is the cross sectional area of the

satellite, c is the speed of light, is 0 if the center of the element j is in darkness and 1 if

the center of the element j is in daylight, a,e are the albedo and emissivity of the element

j, Es is the solar momentum flux density at 1 A.U., s is the solar zenith angle, MB is the

exitance of the Earth, ˆ r is the unit vector from the center of the elements j to the satellite

and N is the total number of segments.

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The nominal albedo and emissivity models can be represented as

a = a0 + a1P10 (sin ) + a2P20(sin ) (2-11)

e = e0 + e1P10 (sin ) + e2 P20(sin ) (2-12)

where

a1 = c0 + c1 cos (t − t0 ) + c2 sin (t − t0) (2-13)

e1 = k0 + k1cos (t − t0) + k2 sin (t − t0) (2-14)

where P10 , P20 are the first and second degree Legendre polynomial, is the latitude of

the center of the element on the Earth’s surface, is the frequency of the periodic terms

(period=365.25 days) and t − t0 is time from the epoch of the period term.

The cross-sectional area Ae can be obtained

Ae = Sii =1

6

∑ ˆ n i ⋅ ˆ r H ( ˆ n i ⋅ ˆ r ) (2-15)

This Earth radiation pressure model, based on analyses of Earth radiation budgets

by Stephens et al. (1981), characterizes both the latitudinal variation in Earth radiation

and the seasonally dependent latitudinal asymmetry.

There is no AMA/LaRC program for computing the Earth radiation pressure.

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2.4 Spacecraft Attitude Dynamics Model

The equation of the spacecraft attitude dynamics can be written in the spacecraft

body-fixed coordinate system as follows

Jd

dt= T − ×(J ) (2-16)

where J is the moment of inertia tensor of the spacecraft, ω ),,( zyx ωωω is the

spacecraft's instantaneous angular velocity with respect to the inertial system, T

(Tx ,Ty,Tz) is the total external torque acting upon the spacecraft. Note that all these are

defined in the spacecraft body-fixed coordinate system Ob − xyz .

The moment of inertia tensor J is defined by

J =I xx I xy Ixz

I yx I yy Iyz

Izx I zy Izz

=

(y2 + z2 )dm∫ − xydm∫ − xzdm∫− xydm∫ (x2 + z2 )dm∫ − yzdm∫− xzdm∫ − yzdm∫ (x2 + y2)dm∫

(2-17)

where z,y,x are the coordinates of particles in the spacecraft body-fixed system

Ob − xyz , and the integrals are carried out through the whole spacecraft. During the

GRACE mission, the moment of inertial tensor Jpwith respect to Op − xyz can be

relatively accurately known given the knowledge of cold gas consumption. Futhermore,

J can be obtained from Jp according to the Huygens-Steiner parallel axes theorem.

J = J p − mS

dy2 + dz

2 −d xdy −d xdz

−d xdy dx2 + dz

2 −d ydz

−dxdz −dydz d x2 + dy

2

(2-18)

17

where d(d x ,d y ,dz ) is the CG offset between PMC of accelerometer and the COM of

satellite.

To describe the relationship between the spacecraft body-fixed system and the

inertial system, here the attitude quaternion )t(q is used. It is defined based on the Euler

axis a(ax , ay ,az) and Euler angle φ as follows

q(t) = q1 q2 q3 q4[ ]T=

asin( / 2)

cos( /2)

(2-19)

and the attitude kinematic equations of motion is governed by

˙ q (t) =1

2Ω( )q(t) (2-20)

where

Ω( ) =

0 z − y x

− z 0 x y

y − x 0 z

− x − y − z 0

(2-21)

Integrating the attitude dynamics equation (2-16) and kinematic equation (2-20)

yields the compete information about the satellite angular motion and attitude orientation.

2.4.1 Spacecraft Torques

The external torque can be produced by various sources. For the near-Earth

satellite, say, the GRACE satellites altitude ~450km, the main sources of disturbance

torques include the Earth's gravitational torque, solar radiation torque, Earth radiation

torque, aerodynamic torque, magnetic torque. There will be no thruster torque during the

GRACE CG calibration.

18

2.4.1.1 Spacecraft Gravitational Torque

The gravitational torque on the entire spacecraft, expressed in the satellite body-

fixed system , can be obtained by

Tg = r∫ × ∇Udm (2-22)

where r is measured from COM to the mass element dm of the spacecraft, the gradient

of potential ∇U represents the gravitational perturbations acting upon the element.

To get a simplified result for the gravitational torque, the following four

assumptions are made :

(a) Only one celestial primary (Earth) needs be considered.

(b) This primary (Earth) possesses a spherically symmetrical mass distribution.

(c) The spacecraft is small compared to its distance from the mass center of the primary

(Earth).

(d) The spacecraft consists of a single body.

These assumptions permit simple gravitational torque expressed in satellite body-

fixed system to be derived (Peter C.Hughes, 1986). The result is given by

Tg = 3Rc

3

(Izz − I yy)c2c3 + I yz(c22 − c3

2 ) + Izxc1c3 − Ixyc3c1

(Ixx − Izz)c3c1 + Izx(c32 − c1

2 ) + I xyc2c3 − I yzc2c1

(I yy − I xx)c2c1 + Ixy (c12 − c2

2 ) + Iyzc1c3 − Izxc2c3

(2-23)

where Rc is the magnitude of cRr

which is a vector from the Earth center towards the

COM of spacecraft (shown in Figure 2.2) , and c1,c2 ,c3[ ]Tis the unit vector along cR

r

expressed in the spacecraft body-fixed system. Thus, ci = cos( i) , i =1,2,3 .

19

Figure 2.2 Radial Unit Vector Definition

2.4.1.2 Spacecraft Aerodynamic and Radiation Torque

The solar radiation and aerodynamic torque acting upon the spacecraft is given

by

TSA = R ×(dfsolar + dfaero )∫ (2-24)

where R is measured from the COM to surface element dA of the spacecraft, solardf

and aerodf represent the solar pressure perturbation and aerodynamic perturbation,

respectively, acting upon this element. The solar radiation and aerodynamic torque

expressed in satellite body-fixed system are to be obtained from AMA/LaRC model.

However, there is no AMA/LaRC model for Earth radiation torque, it is just neglected in

this report.

α1

2 α3

Earth

r R c

x

y

z

COM

ˆ u

20

2.4.1.3 Spacecraft Magnetic Torque

The instantaneous magnetic torque TM due to the spacecraft effective magnetic

moment m ( in A ⋅ m2) is given by

TM = m × B (2-25)

where B is the magnetic flux density expressed in satellite body-fixed system, and m is

the magnetic dipole moment. During the CG calibration, the magnetic torque rods is

used to generate a harmonic time dependence magnetic dipole moment at some fixed

frequency f0 to produce the magnetic torque, which will dominate over other torques.

In this report, for the dynamics integration, the Earth’s magnetic field is obtained

from the spherical harmonic model by taking the IGRF 95 Gaussian coefficients. The

predominant portion of the Earth's magnetic flux density )B,B,B(B r φθ at any point in

space can be calculated by the following equations (Wertz, 1978)

Br

B

B

=

a

r

n+ 2

(n +1) (gn ,m cos(m ) + hn ,m sin(m ))Pn, m ( )m =0

n

∑n=1

k

∑

−a

r

n +2

(gn, m cos(m ) + hn, m sin(m ))∂Pn ,m( )

∂m = 0

n

∑n =1

k

∑

−1

sin

a

r

n +2

m(−gn ,m sin(m ) + hn,m cos(m ))Pn ,m( )m = 0

n

∑n= 1

k

∑

(2-26)

where φθ B,B,B r are the Earth's magnetic flux density component along vertically

upward, local south direction, and local east direction, respectively; a is the equatorial

radius of the Earth (6371.2km adopted for the International Geomagnetic Field, IGRF);

and φθ,,r are the geocentric distance, coelevation, and east longitude from Greenwich

which define any point in space, m,nm,n h,g are coefficients combined Gaussian

coefficients with certain fixed factors, )(P m,n θ is the Gauss functions combined Schmidt

functions with some fixed factor. Compared to the spherical coordinate defined in section

21

2.3.1, ϕ−=θ o90 , and = . Equation (2-26) can be carried out if φθ,,r and the IGRF

Gaussian coefficients are given.

The geocentric inertial components (BX, BY , BZ ) can be obtained from

( φθ B,B,B r ) by the rotation matrix XYZrM θφ , which is given by

MrXYZ =

cos cos sin cos − sin

cos sin sin sin cos

sin −cos 0

(2-27)

where is the right ascension and is the declination, which is equal to latitude .

is related to longitude by

= + G (2-28)

The attitude quaternion )t(q defined in 2.4 , can be obtained by integrating (2-

16) and (2-20). Knowing )t(q , one can obtain the rotation matrix xyzXYZM rotating from the

geocentric inertial system to satellite body-fixed system by the following equation,

which, meanwhile, introduces a new operator ℜ()

MXYZxyz = ℜ(q) =

1− 2(q22 + q3

2) 2(q1q2 + q3q4 ) 2(q1q3 − q2q4 )

2(q1q2 − q3q4) 1 − 2(q12 + q3

2) 2(q3q2 + q1q4 )

2(q1q3 + q2q4) 2(q2q3 − q1q4 ) 1 − 2(q22 + q1

2)

(2-29)

Thus, the magnetic flux density B expressed in satellite body-fixed system can be

obtained by

22

B =Bx

By

Bz

= MXYZxyz Mr

XYZ

Br

B

B

(2-30)

Given all perturbations and external torques acting upon the satellite, the

dynamics equations can be integrated to generate the true orbit and attitude for the

simulation. In this report, the integration state is chosen as X =r r

r v q[ ]T which is

a 13 dimension vector. As a summary, the dynamics equation is rewritten as

˙ X =

r v

r f g +

r f ng

0.5Ω( )q(t)

J −1(T − × (J ))

(2-31)

The integrator adopted is Runge-Kutta (7) 8 (Fehlberg, E., 1968). Integrating the

above equation can give X(tk) =r r

r v q[ ]T for any time t k , thus rotation matrix

xyzXYZM can be derived, also angular acceleration ˙ expressed in satellite body-fixed

system, non-gravitational acceleration r f ng expressed in inertial system and the Earth’s

magnetic flux density B expressed in the satellite body-fixed system can be obtained for

any time t k as byproduct information.

2.5 COM and CG for Small Spacecraft

The position vector c linking the origin of an arbitrary coordinate system and theCOM of a spacecraft is defined by

COM =1

mScdm∫ (2-32)

The CG of a spacecraft is that point at which the concentrated mass mS of thespacecraft would have to be located in order to be attracted by the same gravitational

23

force as the distributed mass of the spacecraft. For the general case, it is expected that aspacecraft changes its attitude continually and thus the location of its center of gravity.

A complete derivation ( F.P.J.Rimrott, 1989) yields the final result for the CGexpressed in the satellite body-fixed system (referred to Figure 2.1 and Figure 2.2)

xCG

yCG

zCG

=3

4ms Rc

(3Ixx − I yy − I zz − Ic )cos 1

(3I yy − I zz − Ixx − Ic) c o s 2

(3Izz − Ixx − Iyy − Ic )cos 3

(2-33)

where

Ic = I xx cos21 + Iyy cos2

2 + Izz cos23 (2-34)

In fact, equation (2-33) is valid only for satellite body-fixed system Ob − xyzbeing the coordinates representing the principal axes. For the GRACE satellite, theinertial products are every small, Ob − xyz can be approximately regarded as principal

axes coordinate system. Taking the following nominal value, Ixx = 70Kg ⋅ m2,

Iyy = 340Kg ⋅ m2, Izz = 390Kg ⋅ m2 , 1 = 2 = / 2, 3 = , ms = 420Kg, Rc = 6828Km ,

the equation (2-33) gives

xCG

yCG

zCG

=0

0

9.67 ×10−8

(m) (2-35)

The very small difference between COM and CG permits neglecting of thisdifference.

24

Chapter 3

MEASUREMENTS OF ACCELEROMETER

AND STAR CAMERAS AND MAGNETOMETER

3.1 Introduction

The GRACE instrument subsystem, including accelerometer (ACC) , star camera

(SCA) and magnetometer (MAG) , provides all the observables necessary for GRACE

CG calibration. The Super STAR accelerometer (ACC) measures non-gravitational

accelerations of the spacecraft, the star camera (SCA) determines the spacecraft attitude

from the observed images , and the magnetometer (MAG) senses the Earth’s magnetic

field.

The GRACE accelerometer is derived from the ASTRE and STAR

accelerometers. The accelerometer works by electrostatically controlling the position of a

proof mass between capacitor plates that are fixed to the spacecraft. It is intended to

measure all non-gravitational accelerations with a resolution on the order of

10−10ms−2over the frequency bandwidth of 2 ×10− 4 Hz to 0.1Hz .

The orientation of the satellite is sensed using two DTU Star Camera Assemblies

(SCA), with a field of view of 22°by 16°. These are rigidly attached to the accelerometer,

and view the sky at 45° angle with respect to the zenith, on the port and starboard sides.

The star camera is vital to GRACE. It provides the information to allow accelerometer

measurements to be transferred from the body-fixed system into the inertial frame of the

reference and to allow the satellites to be pointed to each other.

During the nominal mission phase, the prime purpose of the magnetometer

(MAG) is to allow the satellite ‘s Attitude and Orbit Control System (AOCS) to adjust

25

the three magnetic torque rod currents according to the attitude control needs. However,

during the CG calibration, the MAG provides the information of the Earth’s magnetic

field.

Integrating orbit dynamics and attitude dynamics equations (2.31) can yield the

real orbit trajectory, attitude orientation and angular velocity vector

X(tk) =r r

r v q[ ]T for any time t k , thus rotation matrix xyz

XYZM rotating from the

geocentric inertial system to satellite body-fixed system can be calculated by (2-29), also

a piece of byproduct information , angular acceleration ˙ expressed in satellite body-

fixed system, non-gravitational acceleration r f ng expressed in inertial system and the

Earth’s magnetic flux density B expressed in the satellite body-fixed system, can be

obtained for any time t k . Based on the above knowledge, the observed accelerometer

data, star cameras data and magnetometer data are generated with the frequency of 0.1

sec., 0.5 sec. and 1 sec. , respectively.

3.2 Accelerometer Instrument and Simulation Data


The GRACE accelerometer is derived from the ASTRE and STAR

accelerometers that have been developed by the Office National d’Etudes et de

Recherches Aerospatiales (ONERA) for the European Space Agency (ESA) and for the

French Space Agency CNES. While the configuration of the sensor head has been

adapted to the GRACE environment, the operation and technology are identical.

The accelerometer works by electrostatically controlling the position of a proof

mass between capacitor plates that are fixed to the spacecraft. While gravitation affects

both the proof mass and the spacecraft, non-gravitational forces affect only the

spacecraft. In order to keep the proof mass centered, the voltages suspending must be

26

adjusted using a control loop. Thus the suspension control voltage is a measure of the

non-gravitational forces on the spacecraft.

The STAR accelerometer, which is the French contribution to the German

CHAMP mission, has a planned resolution of 10−9 ms−2 integrated over the frequency

bandwidth of 2 ×10− 4 Hz to 0.1Hz . Its full-scale range is 10−3ms−2 . The expected

resolution is based on accepted error source analysis, and the sensor head geometry is

based on results from the ASTRE model.

The GRACE accelerometer model (Super STAR) benefits from this development.

Because of the GRACE orbit and the low-vibration design of the spacecraft, the full-scale

range has been reduced to 5 ×10− 5ms−2 . This, combined with 0.1-K thermal control,

allows the sensor core capacitive gaps to be increased from 75 m to 175 m and the

proof mass offset voltage to be reduced from 20V to 10V. This results in a smaller

accelerations bias by a factor of 20, and more importantly, bias fluctuations are also

reduced by a factor of 20. The combined effect of these change is a resolution on the

order of 10−10ms−2over the frequency bandwidth of 2 ×10− 4 Hz to 0.1Hz .

The accelerometer is intended to measure the non-gravitational acceleration.

However, the accelerometer output is the true acceleration corrupted by scale, bias, and

noise as follows

Aout = scale1⋅ aout + scale2 ⋅ aout2 + scale3 ⋅ aout

3 + bias + noise (3-1)

where aout represents the true non-gravitational acceleration; and

scale1,scale2,scale3,bias can be determined with the following precision:

The scale factor scale1 shall be better than:

%10.11 ±=scale for x, y and z axes

27

and the time stability of scale1

yearscale /%01.00.11 ±= for x axis

yearscale /%10.11 ±= for y axis

yearscale /%2.00.11 ±= for z axis

The non linear quadratic term 2scale shall be better than:

21.202 smscale −< for x and z axes

21.502 smscale −< for y axis

The non linear cubic term 3scale shall be better than:

424 .103 smscale −< for x and z axes

425 .103 smscale −< for y axis

The bias bias shall be better than:

26 .10.2 −−< smbias for x and z axes

25 .10.5 −−< smbias for y axis

noise is the measurement noise. It is assumed that the noise power spectrum

density shall be better than:

Along x and z axes: PSD( f ) < (1+0.005Hz

f) ×10− 20 m2s−4Hz−1

Along y axis: PSD( f ) < (1+0.1Hz

f) ×10− 18m2s −4Hz−1 (3-2)

28

3.2.2 Accelerometer Data Simulation

The true non-gravitational acceleration of the accelerometer is given by

aout = ˙ d + ˙ × d + 2 × ˙ d + × ( × d) + gg + ang (3-3)

where d is CG offset between the PMC of accelerometer and the COM of the satellite,

d& and d&& are time derivative carried out with respect to the satellite body-fixed system,

ω is the spacecraft's instantaneous angular velocity with respect to the inertial system, ω&

is the spacecraft's instantaneous angular acceleration with respect to the inertial system,

gg is the acceleration due to gravity gradients, ang is the non-gravitational accelerations

acting upon the satellite. During the nominal mission phase, ang is the dominating term of

the accelerometer outputs, all other terms are disturbance accelerations.

The acceleration due to gravity gradient gg is given by

gg = MXYZ

xyz

r f gr r

d (3-4)

and the non-gravitational acceleration ang is related to r f ng by

ang = MXYZxyz

r f ng (3-5)

During the calibration, the following assumptions are made:

• The CG offset is constant, thus the terms including center of mass variations will be

vanished.

29

• The Earth will be considered as a spherically symmetrical mass when gg is

considered, this permits a simple expression for gg to be easily derived.

Based on the second assumption made above, the disturbance acceleration due to the

gravity gradient is obtained as (NASA conference publication 3088)

gg =r3 d − 3

r 3ˆ u ⋅ d ˆ u (3-6)

where u is the unit vector along the local vertical, it has exactly the same meaning as

[ ]T321 c,c,c defined in section 2.4.1.1, r is the geocentric distance of the center of mass of

satellite. If all higher order and degree of geo-potential is carried according to (3-4), the

disturbance acceleration will not be different from (3-6) too much. In fact, it has been

shown that the disturbing acceleration due to / r term is in order of 10−9 ms−2 if CG

offset is about 1mm , while the disturbing acceleration due to sum of others is in order of

10−11ms−2 . The conclusion is illustrated is Figure 3.1.

30

Figure 3.1 Disturbing acceleration due to / r term anddue to sum of other terms , in the RTN frame

31

Thus, the true acceleration can be reduced as

aout = ˙ × d + × ( × d) +r3 d − 3

r3ˆ u ⋅ d ˆ u + ang (3-7)

To better understand the nature of every terms of measured acceleration, several

new variables are introduced as below.

The angular acceleration induced disturbance acceleration is defined as

aacc =0 − ˙

z˙

y

˙ z 0 − ˙

x

− ˙ y

˙ x 0

dx

dy

dz

(3-8)

The angular velocity induced disturbance acceleration is defined as

avel =− y

2 − z2

x y x z

y x − x2 − z

2y z

z x z y − x2 − y

2

d x

dy

dz

(3-9)

The gravity gradient induced disturbance acceleration is defined as

agg =r 3

1 − 3c12 −3c1c2 −3c1c3

−3c2c1 1 − 3c22 −3c2c3

−3c3c1 −3c3c2 1 − 3c32

dx

dy

dz

(3-10)

The simulated accelerometer data is obtained from (3-1).

32

3.3 Star Camera Instrument and Simulation Data


The star camera instrument consists of two separate sensor heads and an

electronics control and data processing unit (DPU). A sensor head , with a 22o × 16o

Filed Of View (FOV), consists of a lens optics which images a sky portion to a CCD

chip. The image of a sensor head is integrated for 0.5 second, read out, pre-amplified at

the sensor head electronics board, and the signals are routed to the DPU. The DPU

processes the data. The on-chip location and, thus, the on-sky projection, of all objects

found in the image are determined, the constellations of up to 70 stars are compared with

a reference star catalogue stored in the DPU and tried to match with a catalogue

constellation. If the match process has been successful, a 3-axis attitude solution can be

derived.

The attitude determination is done by the autonomous star camera, which outputs

the quaternion building up the coordinate system relationship between the inertial

reference system and star camera fixed system. This task will be performed by the

software in the DPU. Every frame of observed stars will be processed onboard to

determine attitude. In essence, the software attempts to identify measured stars and match

the camera picture with a simulated picture using stars from a catalog in computer

memory. By matching the two views, one simulated, the other measured, the camera

attitude can be derived.

Attitude determination proceeds through several steps. First, using an estimate of

the satellite attitude, a group of stars is retrieved from an onboard star catalog contained

in the flight computer memory. Second, measured stars are matched with catalog stars by

comparing the angle between each pair of measured stars with the angle between pairs of

catalog stars. Third, when a match is found between a measured and catalog pair, the

initial attitude estimate is adjusted so that the catalog pair of stars, when mathematically

projected on the focal planes, lies over the measured pair. Forth, a search is made for

33

other catalog stars that lie close to other measured stars when projected. If other matches

are found, the probability of an incorrect attitude is essentially zero and the attitude is

considered to be uniquely determined. Finally, an adjustment of the attitude is made

using the QUEST algorithm (Shuster and Oh , 1981) or others with all matched stars to

improve the attitude accuracy.

3.3.2 Star Camera Data Simulation

Step1: generate the observed stars position data for every frame.

Every 0.5 seconds, the real attitude orientation quaternion q is available from

integrating (2-31). By Mutiplying given fixed rotation matrices ROT1,ROT 2 , rotating

from satellite body-fixed system to star camera 1 and 2, with xyzXYZM , the star cameras

attitude orientations are obtained in the inertial system, thus the Boresight Direction

(BD) and Field Of View (FOV) for both star cameras can be determined. Selecting stars

from star catalog inside the FOV can give the observed stars position in inertial system ,

then they can be transferred in star camera frame by mutiplying ROT1 / 2MXYZxyz . In this

report, star identification procedure is bypassed by assuming that the observed stars in

FOV are matched with catalog stars.

Step2: choose 20 stars and add noise.

Due to the large FOV of GRACE star cameras, 22°by 16°, every frame can

contain roughly 70 stars, at least 20 stars will be observed. An optimal choice of 20 stars

from every observed frame for attitude determined is preferred to increase the

determination accuracy, although , for simplicity, first 20 stars in every frame are chosen

in this report. After that, gauss white noises are added to the observed 20 stars positions

in very frame, more specifically, 1 arcsecond to account for the star catalog position

error, and 3 arcsecond for the observation noise. Aberration , proper motion and parallax

have not been added for corrections of star measurements.

34

Step3: use QUEST to determine the attitude.

The QUEST algorithm is used to get the maximum-likehood estimate of

quaternion . It has been shown that the minimization of the loss function can be

transformed into an eigenvalue problem of a 4 by 4 matrix where the components of the

eigenvector corresponding to the largest eigenvalue are the attitude quaternion

(Davenport, 1978).

The flowchart showing the procedure to get the simulated star cameras data is

illustrated in Figure 3.2.

3.3.3 Star Catalog

The star catalog is a fundamental part of the attitude determination process that

uses measurement data obtained from any star sensor. The most famous star catalog is the

SKY 2000 Master Catalog (J.R. Myers, at al, 1997), which was developed at the Goddard

Space Flight Center. In stead of using the SKY 2000 Master Catalog, Stauffer Catalog is

used in this report. The special features of the Stauffer Catalog are summarized as

follows (J. Stauffer, 1994):

(1) It contains 4853 stars.

(2) The instrument magnitude of stars are between 1.0 and 6.0.

(3) Each star has no companion stars within 0.1 degree that are less than 3.0. In order to

allow the search for companions to be made, the Position and Proper (PPM) Catalog

was used as an auxiliary star catalog.

(4) They have positions in the sky known to better than one arcsecond, and magnitude

accuracy is 0.15 magnitude.

35

Yes

Yes

Figure 3.2 Flowchart of the Star Cameras Data Generation

START

Integrate the orbit and attitude dynamicsEquation (2-31), get the satellite attitude

Form the BD of star cameras ,and select stars from theStar catalog seen in FOV

Choose 20 stars from observed stars

Add noise of the observed starsposition. And get the positions in

Inertial frame of these stars

Get star cameras attitude orientation

CalibrationFinished STOP

Needs Star CamerasObs

Use QUEST algorithm tocompute the quaternion as OBS.

36

3.3.4 QUEST Algorithm

Basically, an orthogonal matrix A rotating from inertial system to the star camera

fixed system is sought which satisfies,

k,ik,i WVA = )n...2,1i( k= (3-11)

where k,nk,1 kV,...V are a set of reference unit vectors, which are kn known direction in

the reference coordinate system , and k,nk,1 kW,...W are the observation unit vectors, which

are the same kn directions as measured in the star camera fixed coordinate system, k is

the time index.

To take advantage of multiple unit vector simultaneously obtained by a CCD star

camera, a least square attitude problem was suggested in the early 1960’s by

Wahba(G.Wahba, 1986) in stead of solving (3-11). That is, to find an orthogonal matrix

optA that minimizes the loss function

L(A) =1

2ai

ˆ W i ,k − A ˆ V i,ki=1

nk

∑2

(3-12)

where

aii =1

nk

∑ = 1 (3-13)

An efficient attitude determination algorithm, QUEST (QUaternion ESTimator)

developed by Davenport (G. M. Lerner, 1978) , is used to get the maximum-likehood

estimate of (3-12). He has shown that the minimization of the loss function can be

transformed into an eigenvalue problem of a 4 by 4 matrix where the components of the

eigenvector corresponding to the largest eigenvalue are the attitude quaternion.

37

The best estimate q which minimizes the loss function (3-12) is given by

Kˆ q = maxˆ q (3-14)

where maxλ is the maximum eigenvalue of K, and the 4 by 4 matrix K is given by

K =S − I Z

ZT

(3-15)

where

= ˆ W i,k ⋅ ˆ V i ,ki =1

nk

∑ (3-16)

S = ( ˆ W i, kˆ V i , k

T

i =1

n k

∑ + ˆ V i ,kˆ W i ,k

T ) (3-17)

Z = ai(ˆ W i,k × ˆ V i ,k )

i = 1

nk

∑ (3-18)

The covariance for the quaternion is defined as follows: Let qδ be the quaternion

of the small rotation that takes the true quaternion into the optimal quaternion calculated

by (3-14). qδ is assumed to be unbiased, thus

E( q) = EQ

=

03×1

1

(3-19)

By this definition, the covariance of quaternion is defined as

PQQ = E( Q QT ) (3-20)

Furthermore

38

PQQ =1

4 tot2 [I − ˆ W i , k

ˆ W i ,kT ]−1

i =1

n k

∑ (3-21)

where

( tot2 )−1 = ( i, k

2 )− 1

i=1

nk

∑ (3-22)

where 2k,iσ is the sum of covariance of measurement and reference unit vector at time kt .

3.4 Magnetometer Instrument and Simulation Data


The prime purpose of the magnetometer is to allow the satellite’s AOCS to adjust

the three magnetic torque rod current according to the attitude control needs. For the CG

calibration, it is intended to measure the Earth’s magnetic field for processing the CG

offset determination.

The magnetometer hardware consists of a sensor head containing a three axes

sensing assembly including pertinent coils, and pertinent electronics. The magnetic field

sensing characteristics are listed below:

• Measurement axes: 3 orthogonal axes

• Measuring Range, each axis: -50 micro Tesla to +50 micro Tesla

• Resolution: 25 nano Tesla

• Measurement Bandwidth: 4.5Hz ±1Hz

• Maximum measurement disturbances (each independent from the others):

1. Bias error 100 nano Tesla

2. Linearity error 25 nano Tesla

3. Noise 3 nano Tesla

39

3.4.2 Magnetometer Data Simulation

As mentioned before, integrating the orbit and attitude dynamics equation (2-31)

will give the Earth’s magnetic flux density B expressed in the satellite body-fixed system.

The observed magnetic flux density B can be modeled as follows

B = B + bias _ B + noise _ B (3-23)

where bias _ B is the measurement bias, and noise _ B is gaussian measurement noise.

40

Chapter 4

OPTIMAL ESTIMATION OF GRACE CG OFFSET

4.1 Introduction

During a time interval T0 (60 seconds adopted in this report ) of calibration, the

magnetic torque rods are activated to produce the magnetic torque to vibrate the satellite.

The magnetic torquer system employs the induction of a reaction torque vector T on a

magnetic dipole moment m when exposed to the Earth magnetic field vector

B( zyx B,B,B ), m and B are vectors, according to the vector product formula T = m × B ,

as described before. The magnetic dipole moment m is given by vectorial superposition

established of the three individual dipole moment. Any direction of m thus can be

established using by at least three magnetic rods such that the rods’ longtitudinal axes

mark an orthogonal coordinate frame. The minimum linear momentum produced by the

magnetic torque rods ranges from −30Am2 to +30Am2 .

The key point for the GRACE CG calibration is that by applying suitable

magnetic torque to the satellite without using the thruster torque, the magnetic torque will

be the dominating torque over the sum of gravitational torque, solar radiation torque,

Earth radiation troque and aerodynamic torque, which cannot be accurately known for

the GRACE mission. Thus a very accurate model could be built up to fit the attitude

dynamics. Also the magnetic dipole moment m will be activated with harmonic time

dependence at some fixed frequency f0 to create a dominating fixed frequency over the

noise wide band frequency of the accelerometer data, which allows to estimate the CG

offset with high accuracy.

An optimal way to activate the magnetic dipole moment m is to activate along

two axes with harmonic time dependence at some fixed frequency f0 such that the

41

magnetic torque is maximum for the Earth's magnetic filed B( zyx B,B,B ) experienced by

the magnetic rods. However, an alternative way , described in section 4.2, is used in

this report.

It has been demonstrated that activating the magnetic dipole moment along only

one axis cannot yield good CG calibration accuracy since the magnetic torque is no

longer the dominating torque, the fitting attitude model is not good enough.

During the CG calibration, there is no thruster torque, thus the magnetic torque

will be the dominating torque if the magnetic moment is activated as section 4.2. Also,

acca and vela are the dominating disturbance acceleration of the accelerometer data for

the periodic terms, especially acca , and all the other acceleration terms can be

approximately fitted by linear terms. In fact, during the GRACE nominal phase, this is

valid only for a limited time span, that is why the CG calibration cannot take too long.

Theoretically, more time which means more observation data can improve the CG

calibration accuracy, but the non-gravitational acceleration cannot be approximately

fitted by linear terms any more, consequently, the CG offset estimation method put

forward in this report will not yield good result. So it is suggested that there should be a

tradeoff between more time, more data and the validity of linearly fitting of the non-

gravitational acceleration. For this reason, in this report, the time interval of calibration is

adopted to be 60 seconds, also recommended by L. Romans. In fact, 60 seconds is long

enough to get good estimation accuracy of the CG offset, which will be demonstrated

later in the simulation.

In terms of the star cameras, it is required that as the star cameras perform a short

integration of the sky image, the satellite movement (rotation about any axis ) must not

exceed a value of s/1.0 o in order to obtain the optimum resolution and accuracy. In

fact, it has been demonstrated that the angular velocity of the satellite will satisfy the

required range for star cameras during calibration. Thus, the star cameras data can be

used for calibration of CG offset. It is essential to combine the observation data of star

42

cameras to estimate the CG offset, because it provides the attitude information, which is

required for estimating CG offset by the algorithms put forward in this report.

4.2 Magnetic Moment Activating

As mentioned in introduction 4.1, the magnetic moment is activated along two

axis in order to make the magnetic torque dominating over other disturbance torques. The

problem and the solution can be stated as follows:

Question: Find the magnetic moment magnitude (mx ,m y,m z) to make the

magnitude of magnetic torque T = m × B large, which is subject to

(mx ,m y,m z) ∈(0, ±1,±1) , where m = [mx my m z]T30sin(2 f0t), and

B = [Bx(t0), By(t0 ), Bz (t0 )]T , which is known from the magnetometer measurement or from

the Earth’s magnetic field model given the position and attitude of the satellites.

Solution: It is assumed that kji BBB >> , where i, j,k ∈ x ,y ,z . The magnetic

moment magnitude is activated to be: 0m i = ,mk = ±1, m j = m1sign(Bj Bk ), where

sign(s) is 1 if s > 0 , is 0 if s = 0, is –1 if s < 0 . In fact, it has been found that magnetic

moment activated only along y and z axes can yield good estimation. Thus, in this report,

the magnetic moment magnitude is activated to be: mx = 0 ,my = ±1, mz = m1sign(ByBz).

4.3 Dynamics Fitting Model and Partial Derivative

Since the magnetic torque is the dominating torque, the attitude dynamics fitting

model can be simplified as

d

dt= J −1 (m × B − × (J )) (4-1)

43

where

J = J p + ∆J (4-2)

∆J = −mS

dy2 + dz

2 −d xdy −d xdz

−d xdy d x2 + dz

2 −d yd z

−dxdz − dydz d x2 + dy

2

(4-3)

J −1 = (Jp + ∆J)−1

= (Jp + ∆J)−1 (Jp−1)− 1J p

−1

= (I + Jp−1∆J)−1Jp

− 1 (4-4)

The inverse of moment of inertial tensor J could be simplified, because Jp−1 is

very small, in order of 10−3 Kg ⋅m2 , also the 2-norm J p−1∆J <1 , which satisfy the

condition for the Neuman series (Greenberg).

Neuman Series: If M is linear and M 2 < 1, then (I + M)− 1 = I − M + M2 − ...

Thus

J −1 = Jp−1 − J p

−1∆JJp−1 + ... (4-5)

≈ Jp−1 (4-6)

Furthermore, since Jp >> ∆J , the attitude dynamics fitting model can be

simplified as

d

dt= F1 = Jp

−1 (m × B− × (J p )) (4-7)

The attitude kinematic equations of motion is used as the same formula as (2-20),

rewritten as follows

44

˙ q (t) = F2 =1

2Ω( )q(t) (4-8)

where

Ω( ) =

0 z − y x

− z 0 x y

y − x 0 z

− x − y − z 0

(4-9)

Since equations (4-7) and (4-8) will be used to fit the dynamics model for the

estimation, the partial derivative is needed to linearize the equations. Carrying on the

partial derivative of (4-7) and (4-8) about q(t), (t) yields

F1

q= 03 × 4 (4-10)

F1 = Λ3 ×3 = −Jp−1Q (4-11)

F2

q=

1

2Ω( )4×4 (4-12)

F2 = Γ4 ×3 (4-13)

where

Q = Π( ) ⋅ Jp − Π(Jp ) (4-14)

Γ =1

2

q4 − q3 q2

q3 q4 −q1

−q2 q1 q4

−q1 −q2 −q3

(4-15)

and the anti-symmetric matrix operator Π(v) operating on v = [vx ,vy , vz]T is defined by

45

Π(v) =0 −vz v x

vz 0 −vy

−v x vy 0

(4-16)

The subscript i × j in equations (4-10) through (4-13) stands for the dimension of the

corresponding matrix , i rows and j columns.

4.4 Observation Fitting Model and Partial Derivative

Since the non-gravitational acceleration is fitted by the liner terms, the

acceleration observation model from the accelerometer can be simplified as

Aout =− y

2 − z2

x y − ˙ z x z + ˙

y

x y + ˙ z − x

2 − z2

z y − ˙ x

x z − ˙ y z y + ˙

x − y2 − x

2

dx

d y

d z

+ t + + noise (4-17)

or

noisetdA~

A out +β+α+= (4-18)

And the attitude quaternion observation model from the star camera can be

rewritten as

qobs = q (4-19)

Equations (4-18) and (4-19) will be used to fit the data observation model for the

estimation, the partial derivative is needed to linearize the equations. Carrying on the

partial derivative of (4-18) and (4-19) about q(t), (t), d, , yields

46

Aout

q= 03 ×4 (4-20)

Aout =( ˜ A 1 + ˜ A 2)

x

d( ˜ A 1 + ˜ A 2 )

y

d( ˜ A 1 + ˜ A 2 )

z

d

(4-21)

Aout

d= ˜ A 3× 3 (4-22)

Aout = tI3 × 3 (4-23)

Aout = I3 ×3 (4-24)

qobs

q= I4 × 4 (4-25)

qobs =qobs

d=

qobs =qobs = 04 × 3 (4-26)

where

˜ A 1

x

=0 y z

y −2 x 0

z 0 −2 x

(4-27)

˜ A 1

y

=−2 y x 0

x 0 z

0 z −2 y

(4-28)

˜ A 1

z

=−2 z 0 x

0 −2 z y

x y 0

(4-29)

˜ A 2

x

= Π(Λ1) (4-30)

˜ A 2

y

= Π(Λ2) (4-31)

47

˜ A 2

z

= Π(Λ3) (4-32)

The subscript in equations (4-30) through (4-32) stands for the column , and the subscript

i × j in equations (4-20) through (4-26) stands for the dimension of the corresponding

matrix , i rows and j columns.

4.5 Data Preprocessing and Interpolation

The star cameras output the attitude quaternion of the star camera fixed system

relative to the inertial system, which can uniquely determines the attitude orientation of

the star camera. However, for a given attitude orientation, the attitude quaternion cannot

be uniquely determined, simply the sign difference, although. Even the quaternion data is

forced to give the unique form, say it is required that q4 ≥ 0 , the quaternion data still can

have problem if q4 ≈ 0 when the star camera observation noise can make a sign flip of

quaternion. Furthermore, the attitude quaternion should be transformed back to describe

the satellite body-fixed system relative to the inertial system. Finally, in case that the star-

camera observation quaternion data are not unit vectors, they need to be normalized.

All those work should be done during data preprocessing. In this report, the quaternion

data is chosen from either one of the star cameras.

For a set of quaternion data from one of the star camera, q(t1),q(t2),...q(tm) ,

the following steps need to be done before being used to the CG offset estimation.

Step1: Transform the quaternion data into q (t1),q (t2),...q (tm) to describe satellite

body-fixed system relative to inertial system.

48

For data at time t k , by applying the rotation matrix ROT , formed is the new

rotation matrix ROTIB , which rotates from the inertial system to the satellite body-fixed

system.

ROTIB = ROT −1ℜ(q(t k)) (4-33)

The quaternion , subject to q4 ≥ 0 , can be obtained from rotation matrix ROTIB

by the following way(Peter C. Hughes, 1986)

q 4 =1

21 + Trace(ROTIB) (4-34)

q 1q 2q 3

=1

4q 4

ROTIB(2,3) − ROTIB(3,2)



, q 4 ≠ 0 (4-35)

or

q 1q 2q 3

=1+ ROTIB(1,1) / 2

1+ ROTIB(2,2) / 2

1 + ROTIB(3,3) / 2

, q 4 = 0 (4-36)

Step2: check sign flip of the quaternion data q (t1),q (t2),...q (tm) and normalized if

necessary.

If q 4 ≈ 0 , it is necessary to check the quaternion data to see if quaternion sign flip

happens because of the observation noise, if happens, just change the sign of all

quaternion at this observation time; on the other hand, if q 4 is large, there is no need

checking sing flip for the observed quaternion because it could not happen.

After flip check, the quaternion data for any time t k can be normalized by

49

q (t k ) = q (t k )/ q 12 (t k) + q 2

2 (t k ) + q 32(t k ) + q 4

2(tk ) (4-37)

As mentioned before, the star cameras output observation attitude quaternion data

every 0.5 second, and the magnetometer outputs the Earth’s magnetic field data every 1

second, while the accelerometer outputs observed acceleration data every 0.1 second.

Therefore, the star camera data and magnetometer data needs to be interpolating to

construct data every 0.1 second, same data frequency as the accelerometer data.

Applied to the quaternion data and magnetometer data is a simple linear

interpolator as follows

• Loop t from 0 to T0 ( 60sec.) by step 0.1 sec..

• Construct index k as k = int(t / ∆t) + 1, where int() indicates the smallest integer less

then ( or equal to ) the number inside, for star camera data ∆t = 0.5, and for the

magnetometer data ∆t = 1.0 .

• If index k = 1, quaternion data interpolate as

q (t ) = q (t1) − 2 ⋅(q (t2) − q (t1)) ⋅ (0.5 − t) (4-38)

and the magnetometer data interpolate as

B (t) = B (t1 ) − (B (t2 ) − B (t1)) ⋅(1− t) (4-39)

Note that for the quaternion data, (t1,t2 ,...t120 ) = (0.5,1,...60) seconds, while for

the magnetometer data, (t1,t2 ,...t60 ) = (1,2,...60) seconds.

• For the quaternion data , if index 1 < k < 121, then the quaternion data interpolate as

q (t ) = q (t k −1) + 2 ⋅ (q (t k ) − q (t k −1) ⋅(t − 0.5 ⋅ (k −1)) (4-40)

50

For the magnetometer data , if index 1 < k < 61, then the magnetometer data

interpolate as

B (t) = B (tk −1 ) + (B (tk) − B (tk −1 )) ⋅ (t − k +1) (4-41)

• For the quaternion data , if index k = 121, then the quaternion data interpolate as

q (t ) = q (t120) (4-42)

For the magnetometer data , if index k = 61, then the magnetometer data interpolate

as

B (t) = B (t60) (4-43)

• If t < 60sec . go back to first step; otherwise, stop.

Then, again normalize the quaternion data for any time t k to construct the final

quaternion observation data

qobs(t k ) = q (t k )/ q 12 (tk ) + q 2

2 (t k) + q 32 (t k ) + q 4

2(t k ) (4-44)

4.6 Batch Estimation of GRACE CG Offset

After preprocessing the observed star camera data and the Earth magnetic field

data, we can move on to the estimation step. In this report, three different batch

algorithms of CG offset estimation are put forward, Accelerometer and Star camera data

Separate ESTimation (ASSEST), Accelerometer and Star camera data Combined Full-

dimensioned ESTimation (ASCFEST) and Accelerometer and Star camera data

51

Combined Reduced-dimensioned ESTimation (ASCREST). All of these three

estimators use accelerometer data, star camera data and magnetometer data.

ASSEST uses the accelerometer data and star camera data separately, star camera

data is used for estimating the initial angular velocity and accelerometer data is used for

fitting the observation model to estimate the CG offset. The advantage is that the later

estimator for CG offset estimation does not involve the star camera data, it means that the

star camera data could not be used if the angular velocity is known from other means,

but it may yield accuracy less than the other two algorithms.

Both ASCFEST and ASCREST process the accelerometer data and star camera

data simultaneously . The difference is that the former estimator tries to estimate the full-

dimensioned quaternion, while the later estimates the small angles difference between the

real attitude orientation and the nominal attitude orientation. The disadvantage of

ASCFEST is that it may cause singular problem because only 3 variables are independent

in the 4 quaternion vector. ASCREST overcomes such problem by reducing one

dimension , however, it may not yield result , as good as ASCFEST, because the

quaternion has to be normalized in every iteration, thus increasing the errors.

Theoretically, both estimators should yield better estimation if suitable weights are

applied to the observation data.

To describe the batch estimation algorithm ASSEST, ASCFEST and ASCREST

easier, the computational algorithm for the batch processor for the general non-linear

system case is reviewed below from class notes( Bettadpur, 1998).

For no loss of generality, let’s start with a nonlinear dynamics model as

˙ X (t) = F(X ,t) & initial guess X*(t0) (4-45)

and a nonlinear observation model as

52

Y(t) = G(X,t) + (t) , & observation noise (t) with R)(E;0)(E T =εε=ε (4-46)

Simply define state residual, x(t) = X(t) − X*(t) , the difference of the real

dynamics state and the nominal trajectory state obtained by integrating (4-45), and

observation residual , y(t) = Y(t) − G( X*(t),t) , then the state transition matrix for x(t) is

obtained by integrating totally or partially ( if possible) the elements of the state transition

matrix

˙ Φ (t,tk ) = A(t)Φ(t,t k) , subject to Φ(tk ,t k ) = I (4-47)

The matrix A(t) is evaluated on the nominal trajectory, i.e.,

A(t) =F(X* (t),t)

X(t) (4-48)

where F(X*(t),t) is the time derivative of the state vector in the differential equations

governing the time evolution of the system. The observation-state mapping matrix is

given by

˜ H i =G(X*(t i),ti )

X(t) (4-49)

where G(X* (ti),t i) are the observation-state relationships evaluated on the nominal

trajectory. The flow chart for the batch processor computational algorithm is given in

Figure 4.1.

53

yes

Figure 4.1 Batch Processor Algorithm Flow Chart

START & Initialize at t0

Read next observation at ti

If all obs havebeen read A

Integrate nominal trajectory (4-45) and statetransition matrix (4-47) from ti −1 to ti to get

X*(ti) and Φ(ti ,t0 )

Form Hi = ˜ H iΦ(t i ,t0 ) where ˜ H i is from (4-49)

Accumulate HiT

i∑ WiHi , Hi

T

i∑ Wiyi where Wi is the

observation weight, and yi is observation residual

If ti < t f go to B, otherwise go to A

B

A

Solve Norm Equations for ˆ x (to ) = [ HiTWiHi ]

−1

i =1

m

∑ [ HiTWiy i]

i =1

m

∑

Update ˆ X (t0) = X*(t0 ) + ˆ x (t0 )

ConvergedInitialization& Iteration BSTOP

54

4.6.1 ASSEST Algorithm

This Estimator needs the following input information:

• accelerometer data Aout(t1), Aout(t2),... Aout (tm ) • star camera data qobs(t1), qobs(t2 ),...qobs(tm) • magnetometer data B (t1), B (t2 ),...B (tm) • moment of inertial Jp

• magnetic dipole moment m

This Estimator outputs the following estimation vector:

• [q(t0), (t0)]T

• [d, , ]T

ASSEST can be separated into two estimation processing parts, one for

estimating the angular velocity from the star camera data, and the other for estimating

the CG offset.

Let’s start with the angular velocity estimation using a estimator called Full

Dimension Estimation (FDE).

The dynamics model :

˙ q (t) =1

2Ω( )q(t) (4-50)

d

dt= Jp

−1(m × B − × (Jp )) (4-51)

The observation model:

55

qobs = q (4-52)

Let estimated parameter vector be X(t) = q(t), (t)[ ]T , and given a initial guess

of the estimated vector X*(t0) , then integrate equations (4-50) , (4-51) and the

following state transition matrix (4-53) through (4-55) from t k −1 to t k , where t1 ≤ tk ≤ tm ,

and the dimension of Φ11 ,Φ12 and Φ22 are 4 by 4, 4 by 3 and 3 by 3, respectively.

˙ Φ 11 =1

2Ω( )Φ11 , with Φ11(t0,t0) = I (4-53)

˙ Φ 12 =1

2Ω( )Φ12 + ΓΦ22 , with Φ12 (t0 ,t0) = 0 (4-54)

˙ Φ 22 = ΛΦ22 , with Φ22(t0,t0) = I (4-55)

Then, the state transition matrix Φ(tk ,t0 ) can be constructed as

Φ(tk ,t0 ) =Φ11 Φ12

0 Φ22

7× 7

(4-56)

and the observation-state mapping matrix is given by

˜ H (t k) = I 0[ ]4 × 7 (4-57)

then by following the batch processor computational algorithm outlined in Figure

4.1 can give the estimate of the initial angular velocity of the satellite.

This state vector , however, may cause a singular covariance matrix since only

three of the four quaternion components are independent. Thus this may create a serious

problem because the covariance matrix itself maybe singular. However, several

simulations were done by this method, there is no singular problem. To overcome the

potential singularity problem, a alternative method , Reduced Dimension Estimation

(RDE), is put forward.

56

The dynamics model is rewritten as

˙ q (t) =1

2Ω( )q(t) (4-58)

d

dt= Jp

−1(m × B − × (Jp )) (4-59)

and the observation model

qobs = q (4-60)

Let state vector be X(t) = q(t), (t)[ ]T , but the state residual vector is defined as

x(t) = ∆ (t),∆ (t)[ ]T , where ∆ (t) = (t) − *(t) and T321 ],,[)t( ς∆ς∆ς∆=ς∆ is

defined as a set of small rotation angles, by the 3-2-1 sequence of the Euler angles,

carrying the nominal attitude matrix into the true attitude matrix. It is assumed that ∆ (t)

is unbiased and infinitesimal. Thus

ℜ(q(t)) = C(∆ (t)) ⋅ ℜ(q*(t)) (4-61)

where

C(∆ (t)) =c2c3 c2s3 −s2

s1s2c3 − c1s3 s1s2s3 + c1c3 s1c2

c1s2c3 + s1s3 c1s2s3 − s1c3 c1c2

(4-62)

where the shorthand ci = cos(∆ i(t)) and si = sin(∆ i(t)). A new operator ⊗ , which has

the exactly the same meaning as equation (4-61), is introduced such that equation (4-61)

can be rewritten in the following form

q(t) =1

1 + ∆ (t)2

/ 4

∆ (t) / 2

1

⊗ q*(t) (4-63)

57

The differential of x(t) = ∆ (t),∆ (t)[ ]T is carried out as follows

˙ x (t) =Π( *(t)) I

Λ 0

x (4-64)

Thus, given a initial guess of the state vector X*(t0) , then one can integrate

equations (4-58) , (4-59) and the following state transition matrix (4-65) through (4-67)

from t k −1 to t k , where t1 ≤ tk ≤ tm , and the dimension of Φ11 ,Φ12 and Φ22 are 3 by 3.

˙ Φ 11 = Π( (t* ))Φ11 , with Φ11(t0,t0) = I (4-65)

˙ Φ 12 = Π( *(t))Φ12 + Φ22 , with Φ12 (t0 ,t0) = 0 (4-66)

˙ Φ 22 = ΛΦ22 , with Φ22(t0,t0) = I (4-67)

The measurement model given by the equation (4-60) furnishes an maximum-

likehood estimate (MLE) quaternion ˆ q obs(t k )and an observation error covariance, which

may be written as

qobs(t) =1

1+ ∆v(t) 2 / 4

∆v(t) / 2

1

⊗ q(t) (4-68)

The new derived measurement )t(u∆ , equivalent to the observation residual, is

defined such that

qobs(t) =1

1+ ∆u(t) 2 / 4

∆u(t) / 2

1

⊗ q* (t) (4-69)

or explicitly given by

∆u2 = −sin−1 c13

58

∆u1 = tan−1(c23

cos(∆u2),

c33

cos(∆u2 )) (4-70)

∆u3 = tan−1(c12

cos(∆u2),

c11

cos(∆u2 ))

where the matrix elements in the right hand of above equation is from matrix C ,

C = ℜ(qobs) ⋅ ℜ(q*)−1 (4-71)

Thus

y(tk ) = ∆u(tk ) = ∆ (tk ) + ∆v(tk ) (4-72)


Φ(tk ,t0 ) =Φ11 Φ12

0 Φ22

6× 6

(4-73)


˜ H (t k) = I 0[ ]3 ×6 (4-74)


4.1 can give the estimate of the ˆ x (t0 ) = ∆ ˆ (t0), ∆ ˆ (t0)[ ]T by given the initial guess

X*(t0) = q*(t0), * (t0 )[ ]T. The final estimate of ˆ X (t0) = ˆ q (t0), ˆ (t0)[ ]T

can be obtained as

follows

ˆ q (t0)

ˆ (t0)

= X*(t0) +

∆q

∆ ˆ (t0)

(4-75)

59

where

∆q4 =1

21+ Trace(C) (4-76)

∆q1

∆q2

∆q3

=1

4∆q4

c23 − c32

c31 − c13

c12 − c21

, ∆q4 ≠ 0 (4-77)

or

∆q1

∆q2

∆q3

=1+ c11 / 2

1 + c22 / 2

1 + c33 / 2

, ∆q4 = 0 (4-78)

where the matrix elements in the right hand of above equations is from matrix C ,

C = C(∆ ˆ (t0 )) ⋅ ℜ(q* ) (4-79)

Then, if necessary, the initial satellite angular velocity can be improved by

iteration, but what is important is that before iteration , the nominal quaternion vector

has to be normalized.

After the initial angular velocity is obtained, the CG offset can be estimated by

ASSEST.


d

dt= Jp

−1(m × B − × (Jp )) (4-80)


60

noisetdA~

A out +β+α+= (4-81)

Let estimated parameter vector be X(t) = d , ,[ ]T , and given a initial guess of

the estimated vector X*(t0) and the initial angular velocity ˆ (t0 ) obtained from the

above estimation, then integrate equations (4-80) to get the angular velocity (tk ) at

any time t k .


H(t k) = ˜ A tk I I[ ]3× 9

(4-82)


4.1 can give the estimate of the CG offset. Note there is no need of iteration for this

approach because of the linearity between observation and constant estimated vector.

4.6.2 ASCFEST Algorithm





• [q(t0), (t0),d, , ]T

61


˙ q (t) =1

2Ω( )q(t) (4-83)

d

dt= Jp

−1(m × B − × (Jp )) (4-84)


noisetdA~

A out +β+α+= (4-85)

qobs = q (4-86)

Let estimated parameter vector be X(t) = q(t), (t), d, ,[ ]T , and given a initial

guess of the estimated vector X*(t0) , then integrate equations (4-83) , (4-84) and the

following state transition matrix (4-87) through (4-89) from t k −1 to t k , where t1 ≤ tk ≤ tm ,

and the dimension of Φ11 ,Φ12 and Φ22 are 4 by 4, 4 by 3 and 3 by 3, respectively.

˙ Φ 11 =1

2Ω( )Φ11 , with Φ11(t0,t0) = I (4-87)

˙ Φ 12 =1

2Ω( )Φ12 + ΓΦ22 , with Φ12 (t0 ,t0) = 0 (4-88)

˙ Φ 22 = ΛΦ22 , with Φ22(t0,t0) = I (4-89)


Φ(tk ,t0 ) =

Φ11 Φ12 0 0 0

0 Φ22 0 0 0

0 0 I3×3 0 0

0 0 0 I3×3 0

0 0 0 0 I3× 3

16 ×16

(4-90)


62

˜ H (t k) = 0Aout ˜ A tkI3 × 3 I3 ×3

I4× 4 0 0 0 0

7 ×16

(4-91)


4.1 can give the estimate of the CG offset. Furthermore, the CG offset estimate can be

improved by iteration.

4.6.3 ASCREST Algorithm





• [q(t0), (t0),d, , ]T


˙ q (t) =1

2Ω( )q(t) (4-92)

d

dt= Jp

−1(m × B − × (Jp )) (4-93)


63

noisetdA~

A out +β+α+= (4-94)

qobs = q (4-95)

Let state vector be X(t) = q(t), (t), d, ,[ ]T , but the state residual vector is

defined as x(t) = ∆ (t),∆ (t),∆d,∆ , ∆[ ]T , where ∆ (t) = (t) − *(t) , ∆d = d − d* ,

∆ = − * , ∆ = − * and T321 ],,[)t( ς∆ς∆ς∆=ς∆ is defined exactly the same as (4-

61) in section 4.6.1.

The differential of x(t) = ∆ (t),∆ (t),∆d,∆ , ∆[ ]T is carried out as follows

˙ x (t) =

Π( *(t)) I 0 0 0

Λ 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

x(t) (4-96)

Thus, given a initial guess of the state vector X*(t0) , then one can integrate

equations (4-92) , (4-93) and the following state transition matrix (4-97) through (4-99)

from t k −1 to t k , where t1 ≤ tk ≤ tm , and the dimension of Φ11 ,Φ12 and Φ22 are 3 by 3.

˙ Φ 11 = Π( (t* ))Φ11 , with Φ11(t0,t0) = I (4-97)

˙ Φ 12 = Π( *(t))Φ12 + Φ22 , with Φ12 (t0 ,t0) = 0 (4-98)

˙ Φ 22 = ΛΦ22 , with Φ22(t0,t0) = I (4-99)

The measurement model given by the equation (4-95) furnishes an maximum-

likehood estimate (MLE) quaternion ˆ q obs(t k )and an observation error covariance, which

is same as (4-68). The new derived measurement )t(u∆ , equivalent to the observation

residual, is defined the same as (4-69).

64

Thus

y(tk ) = ∆u(tk ) = ∆ (tk ) + ∆v(tk ) (4-100)


Φ(tk ,t0 ) =

Φ11 Φ12 0 0 0

0 Φ22 0 0 0

0 0 I3×3 0 0

0 0 0 I3×3 0

0 0 0 0 I3× 3

15×15

(4-101)


˜ H (t k) = 0Aout ˜ A tkI3 × 3 I3 ×3

I3× 3 0 0 0 0

6 ×15

(4-102)


4.1 can give the estimate of the ˆ x (t0 ) = ∆ ˆ (t0), ∆ ˆ (t0),∆ ˆ d ,∆ ˆ ,∆ ˆ [ ]T

by given the initial

guess X*(t0) = q*(t0), * (t0 ),d*, *, *[ ]T. The final estimate of

ˆ X (t0) = ˆ q (t0), ˆ (t0), ˆ d , ˆ , ˆ [ ]T

can be obtained as follows

ˆ q (t0)ˆ (t0)

ˆ d ˆ ˆ

= X*(t0) +

∆q

∆ ˆ (t0)

∆ ˆ d

∆ ˆ

∆ ˆ

(4-103)

where ∆q is defined the same as (4-76) through(4-79).

65

Then, if necessary, the CG offset estimate can be improved by iteration, but what

is important is before iteration , the nominal quaternion vector has to be normalized.

4.7 Closing Remarks

By applying the CG calibration manoeuvre and estimation algorithm to GRACE

described in above sections, the CG offset can be determined with a good accuracy

which will be demonstrated in later chapter. However, several important points should

be brought out.

First, CG calibration while Magnetometer Failure

The magnetometer data is used for the Earth’s magnetic field in the estimation

algorithms in this report. However, the magnetometer could be failure during the GRACE

mission life. In this case, the spherical harmonic model for the Earth ‘s magnetic field can

be alternatively used to create the magnetic field data. Given the satellite position,

determined highly accurately from GPS observation, the satellite attitude orientation,

determined from the star cameras, and a good IGRF gaussian coefficients, the Earth

magnetic field model can be used to produce the Earth magnetic field data.

Second, Adjustment of CG offset and Iterating Calibration Manoeuver

After the CG offset is determined, the adjustment of CG offset can be done by

moving the balanced mass to counteract it. Then, another follow-up CG calibration can

be done to test the CG offset if zero or not. If the CG offset is still large, due to inaccurate

CG offset determination and/or inaccurate mass balancing, the CG calibration should be

done again, until the CG offset is small enough, within the requirement range.

66

Third, Some restrictions of Initial Guess

There are some restrictions about the initial guess for the estimation algorithms

put forward in this report. The initial guess for CG offset and the fourth component of the

quaternion cannot be zero, otherwise, the norm equation will be singular. However, this

can be easily overcome by using very small numbers.

67

Chapter 5

SIMULATION PROCEDURE AND

ASSUMPTIONS VERIFICATION

5.1 Simulation Procedure

Two polar, near circular orbits with altitude of ~450km , typical orbits for

GRACE satellites, are used for the simulation. Since the GRACE CG calibration can be

carried out to each satellite individually, most of the description is about one satellite.

However, everything described in previous chapters and here below can be applied to

each satellites. Simulations are made for both satellites in this report.

The applied perturbed accelerations and external torques to generate the real orbit

and attitude are summarized in Table 5.1.

The simulation procedure is summarized as:

• STEP1 Generate real Orbit and Attitude

Integrate the 13-dimension dynamics equation (2-31) with the initial values given

in section (5.2) to generate state X(tk) =r r

r v q[ ]T , rotation matrix xyz

XYZM , angular

acceleration ˙ , non-gravitational acceleration r f ng and the Earth’s magnetic flux density

B for time kt , where t k = N∆t , N is a integer, and ∆t is step size(0.1second adopted).

• STEP2 Generate Observed Accelerometer Data, Star Camera Data and

Magnetometer Data

68

The simulated accelerometer data is obtained every 0.1 second from equation

(3-1) by taking parameters described in section 5.2 ; and the simulated star camera

data is carried out every 0.5 second through procedures described in section 3.3.2;

and the magnetometer data is obtained every 1 second from (3-23) by taking

parameters defined in section 5.2.

The observed data include:

• accelerometer data Aout(t1), Aout(t2),... Aout (tm ) , where (t1,t2 ,...tm) = (0.1,0.2,...60)sec. .

• star camera data q(t1),q(t2),...q(tp ) , where (t1,t2 ,...tp ) = (0.5,1,...60)sec..

• magnetometer data B (t1), B (t2 ),...B (tq ) , where (t1,t2 ,...tq ) = (1,2,...60)sec..

Table 5.1 Perturbations and Torques Applied in Simulation

Sources Orbit Dynamics

Perturbation Accelerations

Attitude Dynamics

External Torques

Earth Geo-

potential

20 by 20 Earth gravitational

perturbation, .EGM96.GEO adopted

Gravitational Troque

Earth as spherical mass

Atmosphere

Atmosphere Drag

Conventional Drag Formula

(Atmosphere Model: DTM)

Wind (From AMSIS)

and Short period Density

Perturbations are included

Atmospheric Torque

Obtained form AMA/LaRC

(Atmosphere Model: DTM)

Wind (From AMSIS)

and Short period Density

Perturbations are included

Solar Radiation Solar Radiation Pressure(SRP)

Conventional SRP Formula

Solar Radiation Torque

Obtained from AMA/LaRC

Earth Radiation Earth Radiation Pressure(ERP)

Conventional ERP Formula

None

Earth Magnetic

Field

None Magnetic Torque

(IGRF 95 adopted)

69

• STEP3 Observed Data Preprocessing and Interpolation

Applying the approach described in section 4.5 to star camera data and

magnetometer data yields the following induced data with a time interval of 0.1 sec,

• accelerometer data Aout(t1), Aout(t2),... Aout (tm ) ,where (t1,t2 ,...tm) = (0.1,0.2,...60)sec.

• star camera data qobs(t1), qobs(t2 ),...qobs(tm) , where (t1,t2 ,...tm) = (0.1,0.2,...60)sec.

• magnetometer data B (t1), B (t2 ),...B (tm) , where (t1,t2 ,...tm) = (0.1,0.2,...60)sec.

• STEP4 Applying ASSEST , ASCFEST and ASRCEST to estimate the CG offset

Given the above observation data and some parameters about the satellite and

calibration maneuver , such as moment of inertial Jp and magnetic dipole moment m,

by using the algorithms defined in 4.6.1, 4.6.2, and 4.6.3, the GRACE CG offset can be

determine.

• STEP5 Adjustment of CG offset and Iterating Calibration Manoeuver if needed

This step may be necessary in practice, simply because that the CG offset

determination and the error of the mass balancing cannot be perfect. The simulation of

this step was not done in this report.

As a summary, the flow chart for the CG calibration simulation procedure is

illustrated in Figure 5.1.

70

No Yes(0.5sec) No Yes(0.1 sec) No Yes(1sec)

No, k=k+1

Figure 5.1 The flow chart for the CG calibration simulation procedure

START

Integrate orbit and attitude dynamicsequation (2-31) to kt

SCA OBS? ACC OBS?

Aout(tk)

MAG OBS?

q(tk ) B (tk )

END OBS (60s)

Data Preprocessing and Interpolation , to get Aout(t1), Aout(t2),... Aout (tm ) ,

qobs(t1), qobs(t2 ),...qobs(tm) , B (t1), B (t2 ),...B (tm) , where (t1,t2 ,...tm) = (0.1,0.2,...60)sec.

Applying ASSEST , ASFCEST andASCREST to estimate the CG offset

STOP

71

5.2 Parameters and Initial Values Used in Verification and Simulation

The satellite mass, mass properties and trapeziod cross section size are used as

below

ms = 420Kg

Ixx = 70.23Kg ⋅m 2 Ixy = −3Kg ⋅ m2

Iyy = 345.14Kg ⋅ m2 Iyz = −0.348Kg ⋅ m2

Izz = 388.84Kg ⋅ m2 Izx = −2.883Kg ⋅ m2

length = 3122mm

height = 720mm

Width (bottom) width _bot = 1942mm

Width (top) width _top = 693mm

The performance characteristic parameters for the GRACE accelerometer is

adopted as below

scale1 = [1.0101, 0.9841,0.9881]T

scale2 = [7.251,33.1796,-7.8153]T

scale3 = [-6.1314e+03,-6.9825e+04,3.6445e+03]T

bias =[-1.2047E- 6 , - 2 . 2 7 8 1 E - 5 , 0 . 9 8 7 1 E - 6 ]T

PSD( f ) < (1+0.005Hz

f) ×10− 20 m2s−4Hz−1 for x, z axes

PSD( f ) < (1+0.1Hz

f) ×10− 18m2s −4Hz−1 for y axis

Satellite Mass

Satellite

Moment of

Inertial

Satellite

Trapezoid

Cross

Section

Accelerometer

Observation

Scale, Bias

And

Noise

Parameters

72

The performance characteristic parameters for the GRACE star cameras is

adopted as below

Star position error due to Star Catalog: 1 arcsec. (1 )

Star position error due to measurements: 3 arcsec. (1 )

FOV: 22 o × 16o

BD of Star Camera 1: elevation from xy-plane 45o

BD of Star Camera 2: elevation from xy-plane 135o

Number of stars processed each frame: 20

The performance characteristic parameters for the GRACE magnetometer is

adopted as below

Bias error : bias _ B = [50, −30,25]TnanoTesla

Noise noise _ B : 3 nano Tesla (1 )

The magnetic torque rods’ parameters during the CG calibration is specified as

Calibration Period: 60sec.

Frequency 0f for magnetic moment: 0.1 Hz.

Magnitude of the magnetic moment: 30.

Magnetic moments activated for front GRACE:

m = [0 1 1]T 30sin(2 f0t )A ⋅ m2

Magnetic moments activated for front GRACE:

m = [0 −1 1]T30sin(2 f0t)A ⋅ m2

Star Cameras

Observation

Noise

Parameters

And

Boresight

Direction

Magnetometer

Observation

Noise

Parameters

Magnetic

Torque

Rods

And

Magnetic

Moment

Parameters

73

The initial values of the front GRACE for the orbit and attitude dynamics

equation (2-31) are

r r (t0 ) = [ 6837865.32398033

0.

0. ]m

r v (t0) = [0.

-0.066613973791

7638.807717665099]m/s

q(t0) = [-0.7008547308853075

-3.115068773005844E-6

0.7133040348803767

3.059634305881649E-6]

r (t0) = [0.

1.114901594E-3

0. ]rad/s

The initial values of the back GRACE for the orbit and attitude dynamics

equation (2-31) are

r r (t0 ) = [6833646.00660327

2.09260535

-239624.73079019]m

r v (t0) = [267.425734562641

-0.066573212320

7634.133605517418]m/s

q(t0) = [3.114586367226337E-6

-0.7008547308853075

-3.060227453483457E- 6

0.7133040348803732 ]

r (t0) = [0.

-1.114907355E-3

0. ]rad/s

Initial

Values

Of

the

Orbit

and

attitude

for

the

Front

GRACE

Initial

Values

Of

the

Orbit

and

attitude

for

the

Back

GRACE

74

The parameters for non-gravitational perturbations of front and back GRACE are

as follows, note that they are defined by the MSODP input cards (referred to MSODP

manual)

Drag model:

DRAG 5 2.0

Solar Radiation Pressure model:

FLUX 0

ORIENT 1

RADPR 1 0.45 0.5

RADPR1 1 1.07 1.9 4.4

Earth Radiation Pressure model:

ERADP 1 7 4.4 0.5

ERADP1 2 2444960.5

AL 0 0 .34

AL 1 0 .00

AL 2 0 .29

EM 0 0 .68

EM 1 0 .00

EM 2 0 -0.18

AL 1 0 365.25 .10 .00

EM 1 0 365.25 -0.17 .00

The

Parameters

For

Non-

gravitational

Perturbations

Of

Front

GRACE

And

Back

GRACE

75

Finally, some other parameters are specified as follows

Initial Greenwich sideal time G : o60748558.101

Earth's gravitational constant µ : 23 s/km4415.398600

5.3 Assumption Verification

As pointed out in previous chapters, several assumptions are made to estimate the

CG offset by using the estimation methods put forward in this report. Various

simulations are done to verify the validity of the assumptions made. Below are Some

simulation results coming from one typical simulation case, from which the assumptions

can be found out to be valid. Note that the parameters and initial values for verification

simulation are specified in section 5.2.

Some principle assumptions made include:

• The non-gravitational acceleration can be fitted by linear function during CGcalibration period;

• The acca and vela are the dominating disturbance accelerations of the accelerometer

observation for the periodic terms, especially acca , and all the other disturbance

acceleration terms can be approximately fitted by linear term;

• Magnetic torque is the dominating torque over other torques during CG calibrationperiod when the magnetic torque rods are activated, since thruster torque is absent;

• The satellite angular rotation about any axis satisfy the requirement for star camerasto obtain the optimum resolution and accuracy .

The figure 5.2 through 5.8 are intended to verify the four above assumptions.

Some

Other

Parameters

76

Figure 5.2 and 5.3 show that the non-gravitational acceleration ang which include

atmosphere drag, solar radiation pressure and Earth radiation pressure. It can be seen that

ang could be fitted by linear functions, although not perfectly, especially along x and y

axes. The main source for being not perfect linear functions is from the short period

atmospheric density perturbations. However, it will be demonstrated that this non-linear

non-gravitational acceleration has only a small impact upon the estimation accuracy

when it is fitted by the linear function.

Figure 5.4 and 5.5 show that the angular acceleration induced disturbance

acceleration acca , angular velocity induced disturbance acceleration vela and the gravity

gradient induced disturbance acceleration gg, from which it can be seen that acca and

vela , especially acca , are the dominant term of the disturbance acceleration while gg can

be linearly approximated.

Figure 5.6 and 5.7 show that the magnetic torque, the aerodynamic torque, solar

radiation torque and gravitational torque, it can be seen that the magnetic torque is the

dominant torque of the total external torques.

Figure 5.8 shows that the real angular velocity )t(ω and acceleration )t(ω& of

GRACE satellites during the calibration, it can be seen that the angular rotation about

any axis does not exceed a value of s/1.0 o , required for star cameras in order to obtain

the optimum resolution and accuracy, thus the star cameras data are valid to be used

during CG calibration.

77

Figure 5.2 Non-gravitational Acceleration of Front GRACE

0 20 40 606.4

6.5

6.6

6.7x 10-7 DRAG ACCEL.

DR

AGx(M

/S2 )

0 20 40 60-4.65

-4.6

-4.55x 10-8

DR

AGy(M

/S2 )

0 20 40 60-1.162

-1.16

-1.158

-1.156

-1.154x 10-8

DR

AGz(M

/S2 )

TIME(SEC.)

0 20 40 60-2.204

-2.203

-2.202

-2.201

-2.2x 10-8

SRP x(M

/S2 )

SRP ACCEL.

0 20 40 60-5.15

-5.1

-5.05

-5

-4.95x 10-8

SRP

y(M/S

2 )0 20 40 60

1

1.05

1.1

1.15

1.2x 10-8

SRP z(M

/S2 )

TIME(SEC.)

0 20 40 60-8.1

-8

-7.9

-7.8

-7.7x 10-10 ERP ACCEL.

ER

Px(M

/S2 )

0 20 40 60-1.53

-1.525

-1.52

-1.515

-1.51x 10-9

ER

Py(M

/S2 )

0 20 40 60-1.62

-1.6

-1.58

-1.56x 10-8

ER

Pz(M

/S2 )

TIME(SEC.)

0 20 40 606.1

6.2

6.3

6.4x 10-7

NG

x(M/S

2 )

NON GRAV. ACCEL.

0 20 40 60-9.9

-9.8

-9.7

-9.6x 10-8

NG

y(M/S

2 )

0 20 40 60-1.8

-1.7

-1.6

-1.5x 10-8

NG

z(M/S

2 )

TIME(SEC.)

78

Figure 5.3 Non-gravitational Acceleration of Back GRACE

0 20 40 60-6.5

-6.4

-6.3

-6.2x 10-7 DRAG ACCEL.

DR

AG

x(M/S

2 )

0 20 40 604.48

4.49

4.5

4.51

4.52x 10-8

DR

AG

y(M/S

2 )

0 20 40 601.08

1.1

1.12

1.14x 10-8

DR

AG

z(M/S

2 )

TIME(SEC.)

0 20 40 602.2

2.201

2.202

2.203

2.204x 10-8

SR

Px(M

/S2 )

SRP ACCEL.

0 20 40 604.95

5

5.05

5.1

5.15x 10-8

SR

Py(M

/S2 )

0 20 40 601

1.05

1.1

1.15

1.2x 10-8

SR

Pz(M

/S2 )

TIME(SEC.)

0 20 40 602.2

2.4

2.6

2.8x 10-10 ERP ACCEL.

ER

Px(M

/S2 )

0 20 40 601.5

1.51

1.52

1.53x 10-9

ER

Py(M

/S2 )

0 20 40 60-1.64

-1.62

-1.6

-1.58x 10-8

ER

Pz(M

/S2 )

TIME(SEC.)

0 20 40 60-6.25

-6.2

-6.15

-6.1x 10-7

NG

x(M/S

2 )

NON GRAV. ACCEL.

0 20 40 609.5

9.6

9.7

9.8x 10-8

NG

y(M/S

2 )

0 20 40 605

5.5

6

6.5

7x 10-9

NG

z(M/S

2 )

TIME(SEC.)

79

Figure 5.4 Disturbance Accelerations of Front GRACE Due to CG offset

0 20 40 60-3.75

-3.7

-3.65

-3.6

-3.55x 10-10 AV INDUCED ACC.

CG

-VE

L x(M/S

2 )

0 20 40 600

2

4

6x 10-12

CG

-VE

L y(M/S

2 )

0 20 40 606.1

6.2

6.3x 10-10

CG

-VE

L z(M/S

2 )

TIME(SEC.)

0 20 40 604.065

4.07

4.075x 10-10

CG

-GG x(M

/S2 )

CG INDUCED ACC.

0 20 40 604.985

4.99

4.995

5

5.005x 10-10

CG

-GG y(M

/S2 )

0 20 40 601.2245

1.225

1.2255

1.226

1.2265x 10-9

CG

-GG z(M

/S2 )

TIME(SEC.)

0 20 40 60-4

-2

0

2

4x 10-10 AA INDUCED ACC.

CG

-AC

C x(M/S

2 )

0 20 40 60-4

-2

0

2

4x 10-9

CG

-AC

C y(M/S

2 )

0 20 40 60-4

-2

0

2

4x 10-9

CG

-AC

C z(M/S

2 )

TIME(SEC.)

0 20 40 60-2

0

2

4x 10-10

AC

Cx(M

/S2 )

ALL CGOFFSET ACC.

0 20 40 60-5

0

5x 10-9

AC

Cy(M

/S2 )

0 20 40 60-2

0

2

4

6x 10-9

AC

Cz(M

/S2 )

TIME(SEC.)

80

Figure 5.5 Disturbance Accelerations of Back GRACE Due to CG offset

0 20 40 60-5.04

-5.02

-5

-4.98

-4.96x 10-10 AV INDUCED ACC.

CG

-VEL

x(M/S

2 )

0 20 40 600

2

4

6x 10-12

CG

-VEL

y(M/S

2 )

0 20 40 60-5.1

-5

-4.9

-4.8x 10-10

CG

-VEL

z(M/S

2 )

TIME(SEC.)

0 20 40 604.72

4.722

4.724

4.726x 10-10

CG

-GG x(M

/S2 )

CG INDUCED ACC.

0 20 40 60-4.99

-4.985

-4.98x 10-10

CG

-GG y(M

/S2 )

0 20 40 60-1.0236

-1.0234

-1.0232

-1.023x 10-9

CG

-GG z(M

/S2 )

TIME(SEC.)

0 20 40 60-2

-1

0

1

2x 10-9 AA INDUCED ACC.

CG

-AC

C x(M/S

2 )

0 20 40 60-4

-2

0

2

4x 10-9

CG

-AC

C y(M/S

2 )

0 20 40 60-2

-1

0

1

2x 10-9

CG

-AC

C z(M/S

2 )

TIME(SEC.)

0 20 40 60-2

-1

0

1

2x 10-9

AC

Cx(M

/S2 )

ALL CGOFFSET ACC.

0 20 40 60-4

-2

0

2

4x 10-9

AC

Cy(M

/S2 )

0 20 40 60-3

-2

-1

0x 10-9

AC

Cz(M

/S2 )

TIME(SEC.)

81

Figure 5.6 External Torques of Front GRACE

0 20 40 603.45

3.5

3.55x 10-6 AERO TOR

T x(N*M

)

0 20 40 605.7

5.75

5.8

5.85

5.9x 10-6

T y(N*M

)

0 20 40 60-1.05

-1.04

-1.03

-1.02

-1.01x 10-5

T z(N*M

)

TIME(SEC.)

0 20 40 601.65

1.7

1.75

1.8

1.85x 10-6 SOL. TOR

0 20 40 601.25

1.3

1.35

1.4x 10-7

0 20 40 607

7.2

7.4

7.6x 10-7

TIME(SEC.)

0 20 40 601.45

1.5

1.55

1.6

1.65x 10-6 GRAV. TOR

0 20 40 60-3.24

-3.22

-3.2

-3.18

-3.16x 10-5

0 20 40 60-4

-3

-2

-1

0x 10-8

TIME(SEC.)

0 20 40 60-5

0

5x 10-4 MAGNETIC TOR

T x(N*M

)

0 20 40 60-1

-0.5

0

0.5

1x 10-3

T y(N*M

)

0 20 40 60-1

-0.5

0

0.5

1x 10-3

T z(N*M

)

TIME(SEC.)

0 20 40 60-5

0

5x 10-4 TOTAL TORQUE

0 20 40 60-1

-0.5

0

0.5

1x 10-3

0 20 40 60-1

-0.5

0

0.5

1x 10-3

TIME(SEC.)

82

Figure 5.7 External Torques of Back GRACE

0 20 40 60-3.89

-3.88

-3.87

-3.86

-3.85x 10-6 AERO TOR

T x(N*M

)

0 20 40 601.6

1.65

1.7x 10-6

T y(N*M

)

0 20 40 60-1.505

-1.5

-1.495

-1.49

-1.485x 10-5

T z(N*M

)

TIME(SEC.)

0 20 40 60-1.85

-1.8

-1.75

-1.7

-1.65x 10-6 SOL. TOR

0 20 40 60-10.5

-10

-9.5

-9

-8.5x 10-8

0 20 40 606.6

6.8

7x 10-7

TIME(SEC.)

0 20 40 601.4

1.45

1.5x 10-6 GRAV. TOR

0 20 40 60-3.16

-3.14

-3.12x 10-5

0 20 40 60-4

-3.5

-3

-2.5

-2x 10-8

TIME(SEC.)

0 20 40 60-4

-2

0

2

4x 10-4 MAGNETIC TOR

T x(N*M

)

0 20 40 60-1

-0.5

0

0.5

1x 10-3

T y(N*M

)

0 20 40 60-1

-0.5

0

0.5

1x 10-3

T z(N*M

)

TIME(SEC.)

0 20 40 60-4

-2

0

2

4x 10-4 TOTAL TORQUE

0 20 40 60-1

-0.5

0

0.5

1x 10-3

0 20 40 60-1

-0.5

0

0.5

1x 10-3

TIME(SEC.)

83

Figure 5.8 Angular velocity and Acceleration ofFront GRACE (Top) and Back GRACE(Bottom)

0 20 40 600

2

4x 10-5 ANGULAR VELOCITY

AVx(R

AD/S

)

0 20 40 601.1

1.11

1.12x 10-3

AVy(R

AD/S

)

0 20 40 60-5

0

5

10x 10-6

AVz(R

AD/S

)

TIME(SEC.)

0 20 40 60-1

-0.5

0

0.5

1x 10-5 ANGULAR ACCEL.

AAx(R

AD/S

2 )

0 20 40 60-4

-2

0

2

4x 10-6

AAy(R

AD/S

2 )

0 20 40 60-2

-1

0

1

2x 10-6

AAz(R

AD/S

2 )

TIME(SEC.)

0 20 40 60-2

-1.5

-1

-0.5

0x 10-5 ANGULAR VELOCITY

AV

x(RA

D/S

)

0 20 40 60-1.13

-1.12

-1.11

-1.1x 10-3

AV

y(RA

D/S

)

0 20 40 60-5

0

5

10x 10-6

AV

z(RA

D/S

)

TIME(SEC.)

0 20 40 60-5

0

5x 10-6 ANGULAR ACCEL.

AA

x(RA

D/S

2 )

0 20 40 60-4

-2

0

2

4x 10-6

AA

y(RA

D/S

2 )

0 20 40 60-2

-1

0

1

2x 10-6

AA

z(RA

D/S

2 )

TIME(SEC.)

84

Chapter 6

SIMULATION RESULTS AND ANALYSIS

6.1 Parameters and Initial Values in Simulation

There are 3 simulation cases have been investigated. The first case is simulated

near equator, in fact, the verification simulation in chapter 5 is this case. The second one

is simulated near north pole, and the third is simulated between the north pole and

equator. The location and corresponding magnetic flux density in the three cases for front

and back GRACE is shown in Figure 6.1. Note that the magnetic flux density is

expressed in the satellite’s body fixed system.

Most of the parameters used in the three simulation cases are same as specified in

chapter 5, except that the orbit and attitude initial values and the way magnetic moments

activated. The initial orbit and attitude values for the case1 can be referred to the section

5.2, and for the case2 and case3 are specified as:

In the case2, the initial values of the front GRACE for the orbit and attitude

dynamics equation (2-31), and magnetic moments activated are

r r (t0 ) = [53166.9012866642, 50.3762086975, 6838048.9781353170]

r v (t0) = [-7627.3588408146, -0.0096034177, 63.3233882824]

q(t0) = [-0.9999051930, -0.0000000609, 0.0137697130, 0.0000043659]

r (t0) = [0.000000000E+00, 0.111490159E-02, 0.000000000E+00]

m = [0 1 −1]T30sin(2 f0t)A ⋅ m2

In the case2, the initial values of the back GRACE for the orbit and attitude


85

r r (t0 ) = [291680.8661387600, -48.5674705798, 6831570.5338286750]

r v (t0) = [-7621.0475706862, -0.0144524709, 329.1764019231]

q(t0) = [0.0000000601, -0.9999052485, -0.0000043660, 0.0137656809]

r (t0) = [0.000000000E+00, -0.111490736E-02, 0.000000000E+00]

m = [0 1 1]T 30sin(2 f0t )A ⋅ m2

In the case3, the initial values of the front GRACE for the orbit and attitude


r r (t0 ) = [ -6739303.8283807930, 48.9237966735, 1198310.5772177500]

r v (t0) = [ -1337.7622725168, 0.1452268716, -7512.2443788538]

q(t0) = [-0.7728330709, 0.0000027703, -0.6346093637, 0.0000033749]

r (t0) = [0.000000000E+00, 0.111490159E-02, 0.000000000E+00]

m = [0 1 1]T 30sin(2 f0t )A ⋅ m2

In the case3, the initial values of the back GRACE for the orbit and attitude

dynamics equation (2-31) , and magnetic moments activated are

r r (t0 ) = [ -6693453.2487186490, 45.0419246185, 1431899.2109399360]

r v (t0) = [ -1598.8879354469, 0.1417855019, -7461.0147865065]

q(t0) = [ -0.0000027710, -0.7728282124, -0.0000033745, -0.6346152804]

r (t0) = [0.000000000E+00, -0.111490736E-02, 0.000000000E+00]

m = [0 −1 1]T30sin(2 f0t)A ⋅ m2

86

Figure 6.1 The location and corresponding magnetic flux density in the three

cases for front GRACE (Top) and back GRACE (Bottom)

0 1000 2000 3000 4000 5000-5

-4

-3

-2

-1

0

1

2

3

4

5x 104 THE MAGNETIC FIELD ALONG SBF DURING NOMINAL PHASE IN ONE REV.

TIME(SEC.)

MAG

NET

IC F

LUX

DEN

SITY

(nT)

101 oW,0oN107 oW,89oN67oE,10 oNBxByBz

0 1000 2000 3000 4000 5000-5

-4

-3

-2

-1

0

1

2

3

4

5x 104 THE MAGNETIC FIELD ALONG SBF DURING NOMINAL PHASE IN ONE REV.

TIME(SEC.)

MAG

NETI

C FL

UX D

ENSI

TY(n

T)

101 oW,0oN107 oW,89oN67oE,10 oNBxByBz

87

The initial guess values of the front GRACE for the fitting attitude dynamics and

estimated parameters are

Case1 q*(t0 ) = [-0.7, 0.000001, 0.713, 0.000001]

Case2 q*(t0 ) = [-0.99, 0.000001, 0.013, 0.000001]

Case3 q*(t0 ) = [-0.75, 0.000001, -0..63, 0.000001]

Case1, Case2, Case3:

*(t0) = [0

0.00122

0 ]rad/s

d* = [0.00001

0.00001

0.00001]m

* (m / s3 ) = *(m/ s2 ) = [0,0,0]T

The initial guess values of the back GRACE for the fitting attitude dynamics and

estimated parameters are

Case1 q*(t0 ) = [0.000001, -0.7, 0.000001, 0.713]

Case2 q*(t0 ) = [0.000001, -0.99, 0.000001, 0.013]

Case3 q*(t0 ) = [0.000001, -0.77, 0.000001, -0.63]

Case1, Case2, Csae3:

*(t0) = [0

-0.00122

0 ]rad/s

d* = [0.00001

0.00001

0.00001]m

* (m / s3 ) = *(m/ s2 ) = [0,0,0]T

Initial

Guess

Values

Of the

Attitude

& Estimated

parameters

for

the Front

GRACE

Initial

Guess

Values

Of the

Attitude

& Estimated

parameters

for

the Back

GRACE

88

The weight matrices of the observation adopted in estimation are as follows

For ASSEST algorithm:

SCA weight I4 × 4 for FDE , or I3 ×3 for RDE

ACC weight

10 0 0

0 1 0

0 0 145

For ASCFEST algorithm:

ACC & SCA weight

10 0 0 0

0 1 0 0

0 0 145 0

0 0 0 90I4 × 4

For ASCREST algorithm:

ACC & SCA weight

10 0 0 0

0 1 0 0

0 0 145 0

0 0 0 90I3 ×3

6.2 Simulation Results And Analysis

Since the CG calibration of the two GRACE satellites does not affect each other, it

can be operated individually at same or different time. In this report, for simulation

simplicity, CG calibrations are operated simultaneously. Using the parameter values

described in section 5.2 and 6.1, a lot of CG offset determination simulations are done

for the GRACE twin to testify the accuracy of the estimation by approach put forward in

this report.

From the Earth’s magnetic field model, the magnetic field can be obtained for all

three cases, which are shown in Figure 6.1, then the magnetic moments can be activated

according to rules described in 4.2.

The

Weight

Matrix

Of

Observation

Data

For

ASSEST

ASCEST1

And

ASCEST2

Estimation

Algorithm

89

The CG offset values for front GRACE is assumed as d = [0.3 0.4 −0.5]T mm ,

while a different CG offset for back GRACE is set to d = [0.4 −0.4 0.4]T mm . By

running the whole program using the same noise model but different values, the CG

offset estimation gives different result. The result can be shown in Figure 6.2 and 6.3.

The rms value of the estimate of CG with respect to the real value is summarized in Table

6.1 and Table 6.2, for front GRACE and back GRACE, respectively.

From the simulation results shown in Figure 6.2, Figure 6.3, Table 6.1 and Table

6.2, it can be seen that the estimation accuracy is different for different estimation

algorithms and different spatial position where CG calibration is done. In fact, the

estimation accuracy is highly related to the angular acceleration when the magnetic

moments are activated. The larger the angular acceleration along one axis , the poorer the

CG offset estimation along that corresponding axis. This is true because the highly

correlation between the angular acceleration and CG offset.

The angular acceleration magnitude along three axes when the magnetic moments

are activated along y and z axes during the nominal phase are shown in Figure 6.4. From

this figure, it can be seen that , in the first case, angular acceleration along x is larger than

along y and z axes, it turns out be that the accuracy along x axis is poorer than along y

and z axes; in the second case, angular acceleration along x is much larger than along y

and z axes, in fact, in this case, the angular acceleration along x is almost largest during

one revolution, thus the accuracy along x axis is much poorer, worst indeed , than along

y and z axes; in the third case, angular acceleration along x is smaller than along y and z

axes of front GRACE, in fact, in this case, the angular acceleration along y and z axes are

almost largest while the angular acceleration along x axis is smaller than along y and z

axes during one revolution, thus the accuracy along x axis is better, almost best indeed ,

than along y and z axes, but for back GRACE, the angular acceleration along x is still

larger than along y and z axes, it turns out be that the accuracy along x axis is poorer than

along y and z axes for back GRACE in this case.

90

Thus, in one orbit period, the optimal timing for estimating CG offset along x axis

can be sought such that the angular acceleration along x is smaller than along y and z

axes if the magnetic moments are activated, and the optimal timing for estimating CG

offset along y and z axes can be sought such that the angular acceleration along x is much

larger than along y and z axes if the magnetic moments are activated. The optimal timing

can be found from Figure 6.4.

From Figure 6.4 , it can be seen that in one revolution , most of time the angular

acceleration along x is larger than along y and z axes if the magnetic moments are

activated, while in a very small period (~100 sec.) , the angular acceleration along x is

smaller than along y and z axes if the magnetic moments are activated.

Therefore, the CG calibration can be done in two step, in one step, the CG offset

along y and z axes can be estimated highly accurate, while in the other step, the CG offset

along x axis can be estimated very good. From the Table 6.1 and 6.2, it can be seen that

the accuracy along x axis can be better than 0.02mm, accuracy along y axis can be better

than 0.01mm, and accuracy along z axis can be better than 0.02 mm. Note that these

accuracy values are obtained from front GRACE in the best chance to estimate the CG

offset along corresponding axis.

91

Table 6.1 The CG Offset RMS of x,y and z axes for Front GRACE

The CG

Offset RMS of

x,y,z, axes

ASSEST with

Angular Velocity

from FDE

ASSET with

Angular Velocity

from RDE

ASCFEST ASCREST

x (mm) 0.14294 0.10983 0.12011 0.20461

y (mm) 0.05456 0.04251 0.04450 0.08070CASE1

z(mm) 0.07768 0.07761 0.08064 0.01112

x (mm) 1.5 1.2 1.4 1.3

y (mm) 0.00552 0.00586 0.00644 0.00677CASE2

z(mm) 0.01773 0.01714 0.01549 0.01365

x (mm) 0.01485 0.01382 0.01251 0.01483

y (mm) 0.03873 0.05168 0.04435 0.04213CASE3

z(mm) 0.04831 0.05794 0.03914 0.04150

Table 6.2 The CG Offset RMS of x,y and z axes for Back GRACE

The CG

Offset RMS of

x,y,z, axes

ASSEST with

Angular Velocity

from FDE

ASSET with

Angular Velocity

from RDE

ASCFEST ASCREST

x (mm) 0.10802 0.13614 0.11878 0.10739

y (mm) 0.05473 0.06991 0.06111 0.05414CASE1

z(mm) 0.07773 0.08159 0.06680 0.06432

x (mm) 1.5 1.3 1.6 1.5

y (mm) 0.00988 0.00999 0.00849 0.01118CASE2

z(mm) 0.02109 0.02180 0.02311 0.02167

x (mm) 0.03657 0.03989 0.03849 0.02976

y (mm) 0.03204 0.04447 0.04409 0.03888CASE3

z(mm) 0.04025 0.04242 0.04041 0.03079

92

Figure 6.2 Simulation Results of Front GRACE for three CasesCase1 ( star), Case 2( diamond), Case 3 (square)

0 5 10 15-4

-2

0

2

4x 10-3

CG

-OFF

x(M)

ASSEST WITH FDE

0 5 10 152

3

4

5x 10-4

CG

-OFF

y(M)

0 5 10 15-6

-5

-4

-3

-2x 10-4

CG

-OFF

z(M)

0 5 10 15-4

-2

0

2

4x 10-3

CG

-OFF

x(M)

ASSEST WITH RDE

0 5 10 153

4

5

6x 10-4

CG

-OFF

y(M)

0 5 10 15-7

-6

-5

-4

-3x 10-4

CG

-OFF

z(M)

0 5 10 15-4

-2

0

2

4x 10-3

CG

-OFF

x(M)

ASCFEST

0 5 10 153

3.5

4

4.5

5x 10-4

CG

-OFF

y(M)

0 5 10 15-6

-5

-4

-3x 10-4

CG

-OFF

z(M)

0 5 10 15-4

-2

0

2x 10-3

CG

-OFF

x(M)

ASCREST

0 5 10 152

3

4

5x 10-4

CG

-OFF

y(M)

0 5 10 15-6

-5

-4

-3x 10-4

CG

-OFF

z(M)

93

Figure 6.3 Simulation Results of Back GRACE for three CasesCase1 ( star), Case 2( diamond), Case 3 (square)

0 5 10 15-4

-2

0

2

4x 10-3

CG

-OFF

x(M)

ASSEST WITH FDE

0 5 10 15-5

-4

-3

-2x 10-4

CG

-OFF

y(M)

0 5 10 153

4

5

6x 10-4

CG

-OFF

z(M)

0 5 10 15-2

0

2

4x 10-3

CG

-OFF

x(M)

ASSEST WITH RDE

0 5 10 15-6

-5

-4

-3

-2x 10-4

CG

-OFF

y(M)

0 5 10 153

4

5

6x 10-4

CG

-OFF

z(M)

0 5 10 15-4

-2

0

2x 10-3

CG

-OFF

x(M)

ASCFEST

0 5 10 15-6

-5

-4

-3

-2x 10-4

CG

-OFF

y(M)

0 5 10 153

4

5

6x 10-4

CG

-OFF

z(M)

0 5 10 15-2

0

2

4x 10-3

CG

-OFF

x(M)

ASCREST

0 5 10 15-6

-5

-4

-3x 10-4

CG

-OFF

y(M)

0 5 10 153

4

5

6x 10-4

CG

-OFF

z(M)

94

Figure 6.4 The Angular Acceleration if the Magnetic MomentActivated During the Nominal Phase in one Orbit Period

Front GRACE (Top) and Back GRACE (Bottom)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5x 10-5 AA. IF MAGNETIC MOMENT ACTIVATED DURING NOMINAL PHASE IN ONE REV.

TIME(SEC.)

ANG

ULAR

ACC

.(RAD

/S2 )

101 oW,0oN107 oW,89oN67oE,10 oNAAxAAyAAz

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5x 10-5 AA. IF MAGNETIC MOMENT ACTIVATED DURING NOMINAL PHASE IN ONE REV.

TIME(SEC.)

AN

GU

LAR

AC

C.(R

AD

/S2 )

101 oW,0oN107 oW,89oN67oE,10 oNAAxAAyAAz

95

6.3 Main Error Sources of CG Calibration

There are a lot of error sources sneaking into the estimation results, such as theaccelerometer observation noise, star camera observation noise, magnetometermeasurement noise, and the non-linearity of the non-gravitational acceleration whentreated as linear function.

Simulation from case1 using ASCFEST for front GRACE has been done fordeactivating one error source in each simulation, the estimation accuracy is shown inFigure 6.5, from which it can be seen that the ACC noise is the largest error source .

Figure 6.5 Estimation Accuracy with respect to Various Error Sources

1 2 3 4 50

0.5

1

1.5x 10-4

RM

Sx(m

m)

ESTIMATION COMPARISION WRT NOISES

1 2 3 4 50

2

4

6x 10-5

RM

Sy(m

m)

1 2 3 4 50

0.5

1x 10-4

RM

Sz(m

m)

ALL NOISELINER NGNO ACC NOISENO SCA NOISENO MAG NOISE

96

6.4 Loss of Magnetometer Data Impact

As mentioned before, the magnetometer could be failure during the GRACEmission life. In this case, the spherical harmonic model for the Earth ‘s magnetic fieldwill be alternatively used to create the magnetic field data. Given the satellite position,determined highly accurately form GPS observation, the satellite attitude orientation,determined from the star cameras, and a good IGRF gaussian coefficients, the Earthmagnetic field model can be used to produce the Earth magnetic field data.

The spherical harmonics model can only represent the main field. In fact, thereare a lot of perturbations. The primary source of geomagnetic field perturbations is theSun. The Sun constantly emits a neutral plasma called solar wind, which compresses thefield ahead of it until the plasma energy density equals the magnetic field energy densityat a distance of about 10 Earth radii. Although the solar wind is fairly constant, it isfrequently augmented by energetic bursts of plasma emitted by solar flares. When thisplasma encounters the geomagnetic field, it compresses the field further giving a rise infield intensity on the surface of the Earth. This rise a magnetic storm. The geomagneticfield is monitored continuously at a series of magnetic observatories, which report theobserved magnetic activity as an index K , then can be averaged to be planetary indexKp. The magnetic field deviation can be roughly obtained as

∆B(nT ) = 4exp(( Kp + 1.6)/1.75) (6.1)

Plugging 2.5 as Kp into the above equation , and considering other perturbationssuch as polar electrojet and the equatorial electrojet, the magnetic flux density deviationalong each axis is assumed to be 150 nT as the spherical harmonics model error, whichwill be used to simulate the impact on the estimation if magnetometer is failure.

ASCFEST and ASCREST algorithms are used in the second and third cases forthe loss of magnetometer data simulation. The simulation result is shown in Figure 6.6,from which it can be seen that the impact on the estimation accuracy is every small.

97

Figure 6.6 Loss of Magnetometer Data Simulation

0 5 10 15-4

-2

0

2

4x 10-3

CG

-OF

F x(M)

ASCFEST

0 5 10 152

3

4

5x 10-4

CG

-OF

F y(M)

0 5 10 15-6

-5.5

-5

-4.5

-4x 10-4

CG

-OF

F z(M)

0 5 10 15-4

-2

0

2

4x 10-3

CG

-OF

F x(M)

ASCREST

0 5 10 152

3

4

5x 10-4

CG

-OF

F y(M)

0 5 10 15-7

-6

-5

-4x 10-4

CG

-OF

F z(M)

REALCASE2CASE2-LMCASE3CASE3-LM

98

Chapter 7

CONCLUSIONS

7.1 Summary and Conclusions

As a summary, this report deals with generating the real trajectory and attitude

orientation, obtaining the observation data from the accelerometer, star cameras and

magnetometer, and estimating the CG offset by three different estimation algorithms.

Programs coded in Fortran 77 to simulate every step have been done.

Based on the CG calibration approach by using magnetic torque rods, a method to

activate the magnetic moment along two axes is put forward( seen in section 4.2), and

three different estimation algorithms (seen in section 4.6) , ASSEST, ASCFEST and

ASCREST, are brought out to estimate the CG offset.

Although a lot of assumptions are made in this report, most of them are verified

by simulations ( shown in section 5.3) , while others are reasonably stated in the report

where they are made. The assumptions make the simulation easier, estimation simpler,

but does not affect the conclusions too much.

There are three important conclusions made from this report:

(1) The magnetic moment should be activated along two axes in order to get good

estimation accuracy from the algorithms, ASSEST,ASCFEST and ASCREST.

(2) The estimation accuracy is highly related to the magnetic field sensed by the satellite.

The optimal timing for estimating CG offset along x axis can be sought such that the

angular acceleration along x is smaller than along y and z axes if the magnetic

moments are activated, and the optimal timing for estimating CG offset along y and z

99

axes can be sought such that the angular acceleration along x is much larger than

along y and z axes if the magnetic moments are activated.

(3) The estimation accuracy of CG offset could be less than 0.02mm for x axis, 0.01mm

for y axis, and 0.02 mm for z axis.

7.2 Recommendations for Future Work

Although this report has been undergoing a complete simulation procedure, there

are a lot of research work open to investigate in the future, basically, including:

(1) The optimal weight for the accelerometer data and star cameras data needs to be

sought. It has been found that the estimation accuracy is relatively highly

dependent on the weight for the accelerometer data and star cameras data.

Although a relatively good weight has been adopted in this report, a optimal

weight yielding better accuracy is under future investigation.

(2) All these estimation algorithms put forward in this report are batch algorithm. The

sequential algorithm is needed to develop to see if it can improve the CG

estimation accuracy.

(3) Of course, most of important, the data from future GRACE mission should be

processed by the algorithms put forward in this report. Hopefully, it can yield good

result to make GRACE mission to fulfill the target held now.

100

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101

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