GRACE CG OFFSET DETERMINATION BY MAGNETIC TORQUERS DURING THE IN-FLIGHT PHASE

GRACE CG OFFSET DETERMINATION BY MAGNETIC TORQUERS DURING THE IN-FLIGHT PHASE by Furun Wang January 2 Center for Space Research The University of Texas at Austin CSR-TM--1

This work was supported by NASA Contract NAS5-97213. Center for Space Research The University of Texas at Austin Austin, Texas 78712 Principal Investigator: Dr. Byron D. Tapley

ABSTRACT The Gravity Recovery And Climate Experiment (GRACE mission) is scheduled for launch in June 21. 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 22 kilometers apart in a polar orbit 45 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.2mm for x axis (along track),.1mm for y axis (cross track), and.2 mm for z axis (radial). ii

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 ii iii vii viii 1 1 5 7 7 8 9 9 12 12 13 14 16 17 18 19 iii

2.4.1.3 Spacecraft Magnetic Torque 2.5 COM and CG for Small Spacecraft 3 MEASUREMENTS OF ACCELOMETER AND STAR CAMERAS 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 3.3.1 Instrumentation Design Features 3.3.2 Star Camera Data Simulation 3.3.3 Star Catalog 3.3.4 QUEST Algorithm 3.4 Magnetometer Instrument and Simulation Data 3.4.1 Instrumentation Design Features 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 2 22 24 24 25 25 28 32 32 33 34 36 38 38 39 4 4 42 42 45 47 5 v

4.6.1 ASSEST Algorithm 4.6.2 ASCFEST Algorithm 4.6.3 ASCREST Algorithm 4.7 Closing Remarks 5. SIMULATION PROCEDURE AND ASSUMPTIONS VERIFICATION 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 54 6 62 65 67 67 71 75 84 84 88 95 96 98 98 99 1 vi

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 1 19 3 35 53 7 77 78 79 8 81 82 83 86 92 93 94 95 97 vii

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 3 4 68 91 91 viii

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 21, 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 22 kilometers apart in a polar orbit 45 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 5 km and 3 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 1

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 nongravitational 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.1 m (shown in section 2.5), so small, compared to other error sources shown below, that can be neglected. 2

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.2.2.2 Remaining unbalanced.67.42.21 Accuracy of Tank Mounting.76.76.76 Difference in Tank volume.1 --- --- 1g/g effects---gravity.15.42.49 1g/g effect---temperature.83.4.27 Moisture release CFRP / shrink.53.21.11 Moisture release CFRP / mass --- ---.38 decrease Moisture release foam / mass --- ---.78 Decrease Shrink due to moisture release of --- ---.8 foam Uncertainty of boom position.7 --- --- Impact of buoyancy.53.53.53 RMS---Value of Error.27.23.24 Requirement.5.5.5 3

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.3.1 Difference in mass of tanks.462 --- --- due to gas consumption Total mass decrease.7.7.7 RMS-Value of Error.462.7.7 Requirement.1mm/ 6months.1mm/ 6months.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. 4

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 5

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. 6

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. 7

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 nongravitational 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 O I XYZ adopted in this report for the orbit dynamics model is the J2 geocentric inertial coordinate system, which is defined by the mean equator and vernal equinox at Julian epoch 2.. The Earth s body-fixed coordinate system O E 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 O b x 1 y 1 z 1, O b x 2 y 2 z 2 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 8

as: for the front GRACE satellite, - x ˆ 1 is along track by pitching up ~ 1 o, y ˆ 1 is 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 ~ 1 o, y ˆ 2 is out of orbital plane, and z ˆ 2 is radial downward. Furthermore, another set of spacecraft body-fixed coordinate systems O p x 1 y 1 z 1, O p x 2 y 2 z 2 is defined with axes parallel to O b x 1 y 1 z 1, O b x 2 y 2 z 2 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 J2. geocentric non-rotating reference system as follows r = f r g + f r ng (2-1) where r is the position vector of the COM of the satellite, f r g is the sum of the gravitational perturbations acting upon the satellite and f r ng is the sum of the nongravitational 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 9

y1 y1 BACK GRACE COM r d 2 PMC x2 x2 x1 x1 PMC r d 1 COM FRONT GRACE y2 y2 z2 z1 z1 z2 r 1 r 2 Z(z ) EARTH y OI(OE) Y α G X x Figure 2.1 Coordinate Definition 1

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) a e U = r l P r lm (sin( ))[C lm cos(m ) + S lm sin(m )] (2-2) l= l m= where a e 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 O E x y z, C lm,s lm are the geopotential harmonic coefficients of degree l and order m, P lm 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, f r g can be obtained as r f g = M XYZ x y x y z M r z U (2-3) where M XYZ x y z is the rotation matrix from Earth's body-fixed coordinate system O E x y z to inertial system O I XYZ, M r x y z is the rotation matrix from spherical coordinate ( u ˆ r, u ˆ, u ˆ ) to O E 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 XYZ x y z XYZ M x y z cos G sin G = sin G cos G 1 (2-4) 11

where x y M r α G is the right ascension of the Greenwich meridian. Besides, the rotation matrix z can be obtained as follows x y z M r cos cos sin cos sin = cos sin sin sin cos sin cos (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, 198) r f drag = 1 2 (C d A d r )v m r v r (2-6) S where is the atmospheric density, v r r is the satellite velocity with respect to the atmosphere, v r is the magnitude of v r r, m S is the mass of the satellite, C d is the drag coefficient for the satellite and A d 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 A d can be obtained as 12

6 A d = S i n ˆ i v r r / v r H( n ˆ i v r r /v r ) (2-7) i =1 where H(x) = 1, if x, otherwise H(x) =, ˆ n 3 = (,sin a, cosa) T, ˆ n 4 = (, sin a, cos a) T, S 1 = S 2 = (width _bot + width _ top) height /2, n ˆ 1 = (1,,) T, n ˆ 2 = ( 1,,) T, n ˆ 5 = (,,1) T, n ˆ 6 = (,, 1) T, S 3 = S 4 = length height /sina, S 5 = width_ bot length, S 6 = width _top length, sin a = sin(height / c 2 + height 2 ), cos a = cos(c / c 2 + height 2 ) 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, JAC7M(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., 199) r f solar = P(1+ )( A s m S )v ˆ u sun (2-8) where P is the momentum flux due to Sun, is the reflectivity coefficient of the satellite, A s is the cross-sectional area of the satellite normal to the Sun, v is the eclipse 13

factor ( v= if the satellite is in full shadow, v=1 if the satellite is in full Sun, and <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 A s can be obtained 6 A s = S i n ˆ i u ˆ sun H( n ˆ i u ˆ sun ) (2-9) i = 1 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 ( A N e m S c ) [( ae s cos s + em B )ˆ r ] j (2-1) j =1 where e is the satellite reflectivity for the Earth radiation pressure, A is the projected, attenuated area of a surface element of the Earth, A e is the cross sectional area of the satellite, c is the speed of light, is 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, E s is the solar momentum flux density at 1 A.U., is the solar zenith angle, M s B 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. 14

The nominal albedo and emissivity models can be represented as a = a + a 1 P 1 (sin ) + a 2 P 2 (sin ) (2-11) e = e + e 1 P 1 (sin ) + e 2 P 2 (sin ) (2-12) where a 1 = c + c 1 cos (t t ) + c 2 sin (t t ) (2-13) e 1 = k + k 1 cos (t t ) + k 2 sin (t t ) (2-14) where P 1, P 2 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 t is time from the epoch of the period term. The cross-sectional area A e can be obtained 6 A e = S i n ˆ i r ˆ H ( n ˆ i r ˆ ) (2-15) i =1 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. 15

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 J d dt = T (J ) (2-16) where J is the moment of inertia tensor of the spacecraft, ω ω, ω, ω ) is the ( x y z spacecraft's instantaneous angular velocity with respect to the inertial system, T (T x,t y,t z ) is the total external torque acting upon the spacecraft. Note that all these are defined in the spacecraft body-fixed coordinate system O b xyz. The moment of inertia tensor J is defined by I xx I xy I xz J = I yx I yy I yz = I zx I zy I zz (y 2 + z 2 )dm xydm xzdm xydm (x 2 + z 2 )dm yzdm xzdm yzdm (x 2 + y 2 )dm (2-17) where x, y, z are the coordinates of particles in the spacecraft body-fixed system O b xyz, and the integrals are carried out through the whole spacecraft. During the GRACE mission, the moment of inertial tensor J p with respect to O p xyz can be relatively accurately known given the knowledge of cold gas consumption. Futhermore, J can be obtained from J p according to the Huygens-Steiner parallel axes theorem. d 2 y + d z d x d y d x d z J = J p m S d x d y d 2 2 x + d z d y d z d x d z d y d z d 2 2 x + d y (2-18) 16

where d(d x,d y,d z ) 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 q (t) is used. It is defined based on the Euler axis a(a x, a y,a z ) and Euler angle φ as follows q(t) = [ q 1 q 2 q 3 q 4 ] T = asin( / 2) cos( /2) (2-19) and the attitude kinematic equations of motion is governed by q (t) = 1 Ω( )q(t) (2-2) 2 where z y x z x y Ω( ) = y x z x y z (2-21) Integrating the attitude dynamics equation (2-16) and kinematic equation (2-2) 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 ~45km, 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. 17

2.4.1.1 Spacecraft Gravitational Torque The gravitational torque on the entire spacecraft, expressed in the satellite bodyfixed system, can be obtained by T g = 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 bodyfixed system to be derived (Peter C.Hughes, 1986). The result is given by T g = 3 (I zz I yy )c 2 c 3 + I yz (c 2 2 c 2 3 ) + I zx c 1 c 3 I xy c 3 c 1 3 (I Rc xx I zz )c 3 c 1 + I zx (c 2 3 c 2 1 ) + I xy c 2 c 3 I yz c 2 c 1 (I yy I xx )c 2 c 1 + I xy (c 2 1 c 2 2 ) + I yz c 1 c 3 I zx c 2 c 3 (2-23) where R c is the magnitude of R r c which is a vector from the Earth center towards the COM of spacecraft (shown in Figure 2.2), and [ c 1,c 2,c 3 ] T is the unit vector along R r c expressed in the spacecraft body-fixed system. Thus, c i = cos( i), i =1,2,3. 18

2 α 3 ˆ u α 1 COM x y r R c z Earth Figure 2.2 Radial Unit Vector Definition 2.4.1.2 Spacecraft Aerodynamic and Radiation Torque by The solar radiation and aerodynamic torque acting upon the spacecraft is given T SA = R (df solar + df aero ) (2-24) where R is measured from the COM to surface element da of the spacecraft, and df solar df aero 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. 19

2.4.1.3 Spacecraft Magnetic Torque The instantaneous magnetic torque T M due to the spacecraft effective magnetic moment m ( in A m 2 ) is given by T M = 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 f 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 r, B θ, B φ ) at any point in space can be calculated by the following equations (Wertz, 1978) k n+ 2 n a (n +1) (g B r r n,m cos(m ) + h n,m sin(m ))P n, m ( ) n=1 m = k n +2 n B a = B r (g n, m cos(m ) + h n, m sin(m )) Pn,m ( ) n =1 m = 1 k n +2 n a sin r m( g n,m sin(m ) + h n,m cos(m ))P n,m ( ) n= 1 m = (2-26) where B, B, B r are the Earth's magnetic flux density component along vertically θ φ 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 r,θ, φ are the geocentric distance, coelevation, and east longitude from Greenwich which define any point in space, n,m n,m g,h are coefficients combined Gaussian coefficients with certain fixed factors, P n, m ( θ ) is the Gauss functions combined Schmidt functions with some fixed factor. Compared to the spherical coordinate defined in section 2

o 2.3.1, θ = 9 ϕ, and =. Equation (2-26) can be carried out if r,θ, φ and the IGRF Gaussian coefficients are given. The geocentric inertial components (B X, B Y, B Z ) can be obtained from ( B, B, B r θ φ ) by the rotation matrix XYZ M r θφ, which is given by cos cos sin cos sin M XYZ r = cos sin sin sin cos sin cos (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 q (t) defined in 2.4, can be obtained by integrating (2-16) and (2-2). Knowing q (t), one can obtain the rotation matrix xyz M XYZ rotating from the geocentric inertial system to satellite body-fixed system by the following equation, which, meanwhile, introduces a new operator R() xyz M XYZ 1 2(q 2 2 + q 2 3 ) 2(q 1 q 2 + q 3 q 4 ) 2(q 1 q 3 q 2 q 4 ) =R(q)= 2(q 1 q 2 q 3 q 4 ) 1 2(q 2 1 + q 2 3 ) 2(q 3 q 2 + q 1 q 4 ) 2(q 1 q 3 + q 2 q 4 ) 2(q 2 q 3 q 1 q 4 ) 1 2(q 2 2 + q 2 1 ) (2-29) obtained by Thus, the magnetic flux density B expressed in satellite body-fixed system can be 21

B = B x B y B z = M XYZ xyz M r XYZ B r B B (2-3) 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 v q a 13 dimension vector. As a summary, the dynamics equation is rewritten as [ ] T which is r v r f X g + f r ng =.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(t k ) = r r v q [ ] T for any time t k, thus rotation matrix xyz M XYZ can be derived, also angular acceleration expressed in satellite body-fixed system, non-gravitational acceleration f r 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 linking the origin of an arbitrary coordinate system and the c COM of a spacecraft is defined by COM = 1 m S cdm (2-32) The CG of a spacecraft is that point at which the concentrated mass m S of the spacecraft would have to be located in order to be attracted by the same gravitational 22

force as the distributed mass of the spacecraft. For the general case, it is expected that a spacecraft 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 CG expressed in the satellite body-fixed system (referred to Figure 2.1 and Figure 2.2) x CG y CG z CG (3I xx I yy I zz I c )cos 1 3 = (3I 4m s R yy I zz I xx I c )cos 2 c (3I zz I xx I yy I c )cos 3 (2-33) where I c = I xx cos 2 1 + I yy cos 2 2 + I zz cos 2 3 (2-34) In fact, equation (2-33) is valid only for satellite body-fixed system O b xyz being the coordinates representing the principal axes. For the GRACE satellite, the inertial products are every small, O b xyz can be approximately regarded as principal axes coordinate system. Taking the following nominal value, I xx = 7Kg m 2, I yy = 34Kg m 2, I zz = 39Kg m 2, 1 = 2 = /2, 3 =, m s = 42Kg, R c = 6828Km, the equation (2-33) gives x CG y CG z CG = (m) (2-35) 9.67 1 8 The very small difference between COM and CG permits neglecting of this difference. 23

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 1 1 ms 2 over the frequency bandwidth of 2 1 4 Hz to.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 24

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(t k ) = r r xyz v q M XYZ rotating from the [ ] T for any time t k, thus rotation matrix 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 bodyfixed 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.1 sec.,.5 sec. and 1 sec., respectively. 3.2 Accelerometer Instrument and Simulation Data 3.2.1 Instrumentation Design Features 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 25

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 1 9 ms 2 integrated over the frequency bandwidth of 2 1 4 Hz to.1hz. Its full-scale range is 1 3 ms 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 1 5 ms 2. This, combined with.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 2V to 1V. This results in a smaller accelerations bias by a factor of 2, and more importantly, bias fluctuations are also reduced by a factor of 2. The combined effect of these change is a resolution on the order of 1 1 ms 2 over the frequency bandwidth of 2 1 4 Hz to.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 2 A out = scale1 a out + scale2 a out 3 + scale3 a out + bias + noise (3-1) where a out 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: scale 1 = 1. ± 1% for x, y and z axes 26

and the time stability of scale1 scale 1 = 1. ±.1% / year for x axis scale 1 = 1. ± 1% / year for y axis scale 1 = 1. ±.2% / year for z axis The non linear quadratic term scale2 shall be better than: scale2 s 1 2 < 2m. for x and z axes scale2 s 1 2 < 5m. for y axis The non linear cubic term scale 3 shall be better than: scale3 s 4 2 4 < 1 m. for x and z axes scale3 s 5 2 4 < 1 m. for y axis The bias bias shall be better than: 6 2 bias < 2.1 m. s for x and z axes 5 2 bias < 5.1 m. s 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: Along y axis: PSD( f ) < (1+.5Hz ) 1 2 m 2 s 4 Hz 1 f PSD( f ) < (1+.1Hz ) 1 18 m 2 s 4 Hz 1 (3-2) f 27

3.2.2 Accelerometer Data Simulation The true non-gravitational acceleration of the accelerometer is given by a out = d + d + 2 d + ( d) + gg + a ng (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, a ng is the non-gravitational accelerations acting upon the satellite. During the nominal mission phase, a ng is the dominating term of the accelerometer outputs, all other terms are disturbance accelerations. The acceleration due to gravity gradient gg is given by xyz gg = M XYZ r f g r d (3-4) and the non-gravitational acceleration a ng is related to r f ng by a ng = M r xyz XYZ 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. 28

The Earth will be considered as a spherically symmetrical mass when considered, this permits a simple expression for gg to be easily derived. gg is Based on the second assumption made above, the disturbance acceleration due to the gravity gradient is obtained as (NASA conference publication 388) gg = r 3 d 3 r 3 ˆ u d ˆ u (3-6) where û is the unit vector along the local vertical, it has exactly the same meaning as [,c, ] T c defined in section 2.4.1.1, r is the geocentric distance of the center of mass of 1 2 c3 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 1 9 ms 2 if CG offset is about 1mm, while the disturbing acceleration due to sum of others is in order of 1 11 ms 2. The conclusion is illustrated is Figure 3.1. 29

Figure 3.1 Disturbing acceleration due to / r term and due to sum of other terms, in the RTN frame 3

Thus, the true acceleration can be reduced as a out = d + ( d) + r 3 d 3 r 3 ˆ u d ˆ u + a ng (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 z y a acc = z x y x d x d y d z (3-8) The angular velocity induced disturbance acceleration is defined as 2 2 y z x y x z a vel = y x 2 2 x z y z z x z y 2 2 x y d x d y d z (3-9) The gravity gradient induced disturbance acceleration is defined as 2 1 3c 1 3c 1 c 2 3c 1 c 3 a gg = 2 r 3 3c 2 c 1 1 3c 2 3c 2 c 3 2 3c 3 c 1 3c 3 c 2 1 3c 3 d x d y d z (3-1) The simulated accelerometer data is obtained from (3-1). 31

3.3 Star Camera Instrument and Simulation Data 3.3.1 Instrumentation Design Features The star camera instrument consists of two separate sensor heads and an electronics control and data processing unit (DPU). A sensor head, with a 22 o 16 o 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.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 7 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 32

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.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 M xyz XYZ, 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 /2M xyz XYZ. In this report, star identification procedure is bypassed by assuming that the observed stars in FOV are matched with catalog stars. Step2: choose 2 stars and add noise. Due to the large FOV of GRACE star cameras, 22 by 16, every frame can contain roughly 7 stars, at least 2 stars will be observed. An optimal choice of 2 stars from every observed frame for attitude determined is preferred to increase the determination accuracy, although, for simplicity, first 2 stars in every frame are chosen in this report. After that, gauss white noises are added to the observed 2 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. 33

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 2 Master Catalog (J.R. Myers, at al, 1997), which was developed at the Goddard Space Flight Center. In stead of using the SKY 2 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. and 6.. (3) Each star has no companion stars within.1 degree that are less than 3.. 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.15 magnitude. 34