GRAVITY RECOVERY AND INTERIOR LABORATORY (GRAIL) 

Title: LIB-10: ACS Hardware Coordinate Frame Definitions and Transformations 

GRA-AC-09-0013 
6/15/2009 
-Rev 1 
-7/08/2009 

Author: R. Olds Phone: 1-9694 Team: ACS 


REFERENCES 


1.) SIRTF (Spitzer) Reaction Wheel Control Algorithm Description Document 
(ADD). PCS-100 Rev. A. 


CONTENTS 


1. INTRODUCTION ............................................................2 
2. REFERENCE FRAMES ........................................................2 
2.1 EME-J2000 COORDINATE FRAME {ECI}........................................2 
3. ACS FRAMES ..............................................................4 
3.1 MECHANICAL FRAME {M}....................................................4 
4. ACS COMPONENT FRAMES AND TRANSFORMATIONS ................................5 
4.1 REACTION WHEEL COORDINATE SYSTEM {RWA} .................................6 
4.2 INERTIAL MEASUREMENT UNIT COORDINATE SYSTEM {IMU}.......................9 
4.3 STAR TRACKER COORDINATE SYSTEM {STA} ..................................10 
4.4 SUN SENSOR COORDINATE FRAME {SSA}......................................12 
5. ACS COMPONENT POSITIONS.................................................14 
6. THRUSTER ORIENTATIONS...................................................15 
6.1 ACS THRUSTERS .........................................................15 
6.1.1 Throat Positions.....................................................16 
6.1.2 Nozzle Pointing Unit Vectors ........................................17 
6.1.3 Thrust Vectors.......................................................17 
6.2 MAIN ENGINE ...........................................................18 
6.2.1 Throat Position......................................................18 
6.2.2 Thruster Nozzle Pointing Unit Vector ................................18 
6.2.3 Thrust Vector .......................................................18 
7. FUEL TANK ..............................................................19 
8. KA-BAND PAYLOAD.........................................................19 
9. LGA POINTING............................................................21 
10. REVISION LOG...........................................................22 



1. 
INTRODUCTION 
This document details all the coordinate frames used for the GRAIL mission. 
The coordinate frames include one inertial reference coordinate frame, one 
spacecraft body-fixed (mechanical) coordinate frame, and ACS hardware 
component frames. The transformations between hardware component frames and 
spacecraft frames are defined in this document. Also included are the 
mounting locations of the ACS hardware components with respect to the origin 
of the body-fixed mechanical frame. The thruster positions, thrust vectors, 
and fuel tank parameters are defined for general reference. Fuel tank data 
is included to aid fuel slosh analysis. Ka-band payload and LGA pointing 
unit vectors are also included for reference. All data in this document with 
the exception of component level coordinate frames and drawings references 
revision 8 of the GRAIL IDEAS model. 

Coordinate Frame Transformation Nomenclature 
This document will use the following nomenclature for describing coordinate 
frame transformations: 

See PDF version of this document for equations. 


2. 
REFERENCE FRAMES 
The GRAIL spacecraft will be operated entirely within the Sun/Earth/Moon 
system from launch to end of mission. For this reason the GRAIL mission uses 
the EME-J2000 frame as its sole inertial reference frame. There are three 
flight ephemerides that utilize this reference frame: the Sun ephemeris, 
Earth ephemeris, and spacecraft/Moon orbital ephemeris. The Sun ephemeris 
represents the inertial position vector from the Sun to the spacecraft and 
is relative to the center of the Sun. The Earth ephemeris is also Sun 
relative and represents the inertial position of the Earth. The orbital 
ephemeris represents the inertial position of the spacecraft as it orbits 
the Moon and is taken relative to the center of the Moon (see Figure 2.2). 

2.1 
EME-J2000 COORDINATE FRAME {ECI} 
The Earth-Centered Inertial Coordinate System used by GRAIL is the 
Earth-Centered Earth Mean Equator and Vernal Equinox of Epoch J2000, also 
known as the EME, IAU Reference of Epoch J2000, denoted {ECI}. The Epoch 
J2000 corresponds to the Julian Ephemeris Date (JED) 2451545.0. The axes are 
defined as follows: 

+XECI = Parallel to Vernal Equinox of Earth Mean Heliocentric Orbit of Epoch 
J2000 
+ZECI = Mean Earth Equator normal of Epoch J2000 
+YECI = +ZECI x +XECI 


Figure 2.1 -EME-J2000 Inertial Frame (ECI) 
See PDF version of document for figure.


During the cruise phase, the spacecraft state is represented by the Sun and 
Earth ephemerides relative to the center of the Sun, expressed in the 
EME-J2000 frame (see Figure 2.2, pane A). After LOI, the spacecraft will be 
captured into lunar orbit and the spacecraft state will be nominally 
represented by the orbital ephemeris relative to the center of the Moon as 
depicted in Figure 2.2, pane B. 


Figure 2.2 -EME-J2000 Reference Bodies 
See PDF version of document for figure.


3. ACS FRAMES 
There is only one body fixed frame used for the GRAIL spacecraft: Mechanical 
Frame {M}. It is important to note that the mechanical frame is identical for 
both GRAIL-A and GRAIL-B so only a single definition is needed. The 
mechanical frame acts as a general purpose body fixed frame used both for 
controlling each of the spacecraft and defining the mounting locations and 
orientation of the ACS hardware. 

3.1 
MECHANICAL FRAME {M} 
The origin of the mechanical frame is located on the surface of the -X bus 
plate and centered in the Y-Z plane (see Figure 3.1 -GRAIL Mechanical {M} 
Frame). The coordinate axes are defined as follows: 

+XM = Parallel to and in opposite direction of solar array normal 
+ZM = Normal to bus plate where star tracker is mounted 
+YM = +ZM x +XM 


Figure 3.1 -GRAIL Mechanical {M} Frame 
See PDF version of document for figure.


4. 
ACS COMPONENT FRAMES AND TRANSFORMATIONS 
There are 4 ACS components per spacecraft, each with its individual local 
coordinate frame: 

  1. Reaction Wheels (4 per spacecraft) 
  2. Inertial Measurement Unit (1 per spacecraft) 
  3. Star Tracker Assembly (1 per spacecraft) 
  4. Sun Sensor Assembly (1 per spacecraft) 

All spacecraft models illustrated herein are taken from revision 8 of the 
GRAIL IDEAS model. 


Figure 4.1 -Star Tracker (STA), Inertial Measurement Unit (IMU), and Sun 
Sensor (SSA) View 
See PDF version of document for figure.


Figure 4.2 -Reaction Wheel (RWA), Star Tracker (STA), and Sun Sensor (SSA) 
View 
See PDF version of document for figure.


Each component reports its associated sensor data in its corresponding local 
frame. The reaction wheels each apply torque on the spacecraft in their own 
associated local wheel frame. Each local component coordinate frame is 
mapped to the mechanical frame for use in the ACS control system. The 
following subsections provide detailed descriptions of each of the ACS 
component coordinate frames and list transformations for each to the 
mechanical frame. 

4.1 REACTION WHEEL COORDINATE SYSTEM {RWA} 
Each spacecraft houses 4 reaction wheels oriented in a pyramid configuration. 
The pyramid configuration provides an extra degree of freedom allowing the 
wheels to be spun up or down in tandem without applying any torque to the 
spacecraft. The reaction wheel pyramid configuration is described in Figure 
4.3. The geometry of this configuration was chosen to provide an equal gain 
for momentum storage in the y and z axes of the mechanical frame, while also 
boosting momentum capacity in the x-axis of the mechanical frame for storing 
momentum due to gravity gradient torques. More momentum storage was devoted 
to the y and z body-axes because the dominant external torque due to solar 
radiation acts in these axes. Gravity gradient torques act primarily in the 
x-body axis, but are a factor of 3 to 4 less than those of solar torques. 


Figure 4.3-4-Wheel Pyramid Configuration 
See PDF version of document for figure.


In the diagram above, the a angle defines the elevation of each wheel spin 
axis (how steep the pyramid is), while the Beta angle defines the azimuth of 
each wheel spin axis. The pyramid configuration with a = 60 deg, and Beta = 
45 deg, increases the capability possible with only a single wheel in each 
axis by as much as 245% in the y and z axes, and 200% in the x-axis. The 
pyramid also causes a set of minimum torque axes in the mechanical frame that 
can only provide 98% the capability of a single wheel. 

Figure 4.4 below shows the phasing of each wheel. When viewed from above on 
the anti-connector side, positive spin is defined as counter-clockwise. When 
viewed from the connector side, positive spin is defined as clockwise. 
See PDF version of document for figure.


Figure 4.4 -Wheel Local Coordinates and Phasing (schematics taken from 
Goodrich drawing 45159 rev.2) 
See PDF version of document for figure.


The unit vectors for the wheel spin-axes (defined in the body frame) are 
listed below in Table 4. 1 for a = 60 deg, Beta = 45 deg. These unit vectors 
are used to define the columns of the transformation matrix that maps the 
4-wheel coordinate system to the spacecraft body axes. This transformation is 
identical for GRAIL-A and GRAIL-B. 


See PDF version of document for equations.


Table 4. 1 -Reaction Wheel Spin Axis Unit Vectors (Map from 4-wheel 
coordinates to body axis) 
See PDF version of document for figure.


The inverse transformation, which maps the spacecraft body axes onto the 
4-wheel coordinate system, is defined using the pseudo-inverse of the 
transformation above. 


See PDF version of document for equations.


Table 4.2 -Map from body frame to 4-wheel coordinates 
See PDF version of document for figure.


In this orientation, the mapping from the 4-wheel system to the body frame 
creates an underdetermined system since there are 4 wheels controlling 3 axes 
in the body frame. This creates a non-zero null space in the transformation 
defined in Table 4.1 so that the wheels can be commanded in tandem without 
imparting any net torque on the spacecraft. The null space is spanned by the 
4-vector defined below. 


See PDF version of document for equations.


This shows that if the wheels are commanded to provide equal magnitude 
torques, with RW-1 and RW-3 being opposite in sign to RW-2 and RW-4, no net 
torque will be imparted on the spacecraft. This extra degree of freedom is 
used to avoid zero crossings and increase the momentum capacity of the 
reaction wheel system. 

4.2 INERTIAL MEASUREMENT UNIT COORDINATE SYSTEM {IMU} 
The local IMU coordinate frame, which is coincident with the IMU sensing 
frame, is associated with the physical design of the IMU and defined as 
follows: 

+YIMU = Collinear with the outward normal to Y axis alignment mirror 
(primary mirror), Anti-Normal to J1 and J2 Connectors 
+ZIMU = Perpendicular to the Y axis and in the plane formed by the outward 
normal of the Y axis mirror and the Z axis mirror (secondary mirror), 
positive in the outward normal direction. 
+XIMU = +YIMU x +ZIMU 


Figure 4.5- IMU Local Coordinate Frame and Component Views
(schematics taken from Honeywell ICD rev. C - June 14, 2004) 
See PDF version of document for figure.


The IMU local coordinate frame {IMU} is pictured in Figure 4.5, panes A, B, 
and C. The {IMU} frame orientation with respect to the {M} frame will be 
identical in the case of both GRAIL-A and GRAIL-B. The nominal transformation 
from the {IMU} frame to the {M} frame is below. 


See PDF version of document for transformation.


The nominal transformation from the {M} frame to the {IMU} frame will be: 


See PDF version of document for transformation.


4.3 STAR TRACKER COORDINATE SYSTEM {STA} 
The star tracker alignment is spacecraft dependent (GRAIL-A or GRAIL-B) since 
each spacecraft uses a different cant angle for the unit. The unique cant 
angles are designed so that the tracker for each spacecraft will be pointed 
away from the Moon while in the science taking attitude (orbiter-to-orbiter 
pointing) when GRAIL-B is leading the formation and GRAIL-A is following 
GRAIL-B. Figure 4.6 shows the cant angle for each spacecraft. 


Figure 4.6 -Star Tracker Cant Angle for GRAIL-A and GRAIL-B 
See PDF version of document for figure.


The local mechanical star tracker coordinate frame is associated with the 
physical design of the star tracker and defined as follows (matches the 
reference frame indicated in Figure 4.7, with the "STA" subscript): 

+ZSTA = Parallel to tracker bore sight, pointing towards observed sky 
+XSTA = Orthogonal to +ZSTA, and oriented such that the reference cube is in 
the +X, -Y corner of the unit (towards unit electronics) 
+YSTA = +ZSTA x +XSTA 

Figure 4.7, pane A illustrates the Star Tracker Local Mechanical Coordinate 
Frame, viewed from the star's perspective, with the tracker bore sight out 
of the page: 


Figure 4.7 -Star Tracker Local Coordinate Frame and Component Views (drawings 
taken from Phoenix A-STR ICD rev 0, dated 3/31/05) 
See PDF version of document for figure.


The nominal coordinate frame transformations from the {STA} frame to the {M} 
frame for each spacecraft are below: 


See PDF version of document for transformations.


The nominal transformations from the {M} frame to the {STA} frame for each 
spacecraft will be:


See PDF version of document for transformations.


4.4 SUN SENSOR COORDINATE FRAME {SSA} 
The local sun sensor assembly coordinate frame is associated with physical 
design of the sun sensor assembly and is defined as follows. The sun sensor 
assembly mounting location and coordinate frame transformations are identical 
for each spacecraft. 

+YSSA = Anti-Normal to Connector 
+ZSSA = Parallel to Bore sight 
+XSSA = +YSSA x +ZSSA 

Figure 4.8 illustrates the local sun sensor assembly coordinate frame and the 
mounting of the sun sensors with respect to the spacecraft mechanical axes: 


Figure 4.8 -Sun Sensor Assembly Local Coordinate Frame and Orientation 
(drawings taken from Phoenix Adcole SSA ICD rev E, dated 4/24/97) 
See PDF version of document for figure.


There is a single sun sensor mounted on the +Z-side of the -X panel extension 
of each spacecraft. Each Sun sensor houses 4 detector arrays and each 
detector array normal is 45 deg from the mounting plane. The detector normal 
unit vectors in the mechanical {M} frame are listed below. 

Detector 0 = (- 0.7071 0.0000   -0.7071 ) 
Detector 1 = (- 0.7071 - 0.7071 0.0000  ) 
Detector 2 = (- 0.7071 0.0000   0.7071  ) 
Detector 3 = (- 0.7071 0.7071   0.0000  ) 

The baffles on the sun sensor faces allow the entire unit to have a field of 
view half angle of 80 deg. The sun sensor is mounted on the deck such that 
the axes of the sun sensor are rotated with respect to the spacecraft 
mechanical frame. The sun sensor orientation with respect to the mechanical 
frame has been defined as follows: 

+XSSA = -YM 
+YSSA = +ZM 
+ZSSA = -XM 

The transformation from the {SSA} frame to the {M} is as follows: 


See PDF version of document for transformation.


The transformation from the {M} frame to the {SSA} frame is the transpose of 
the transformation above: 


5. ACS COMPONENT POSITIONS 
ACS component locations in the mechanical frame {M} are listed below in Table 
5.1 for GRAIL-A and Table 5.2 for GRAIL-B. Locations are defined based on the 
GRAIL IDEAS model revision 8. All component locations are measured from the 
origin of the {M} frame to the center of each component mounting plate. 


Table 5.1 -GRAIL-A ACS Component Locations 
See PDF version of document for table.


Table 5.2 -GRAIL-B ACS Component Locations 
See PDF version of document for table.


The values in Table 5.1 and are expected to change once the true locations 
are measured during ATLO. Component locations should be updated with future 
revisions of this document. 


6. THRUSTER ORIENTATIONS 
GRAIL-A and GRAIL-B are each equipped with 8 ACS thrusters and one main 
engine. The main engine is used for large delta-v maneuvers such as TCMs, 
LOI, etc. The ACS thrusters are used for thrust vector control during burns, 
small orbit phasing correction maneuvers, and reaction wheel desaturation 
events. They are also used for thruster based control during rate damping, 
Sun search and Sun point slews, etc. The ACS thrusters are coupled so that 
they can apply a pure torque on the spacecraft while imparting only 
negligible delta-v. Specific sets of 4 thrusters ideally create perfect 
couples about each axis in the {M} frame. 

6.1 ACS THRUSTERS 
The ACS thrusters are 1 N hydrazine engines. The baseline thruster 
configuration is shown in Table 6.1 and defined by the unit vectors and 
thruster nozzle locations listed in the tables that follow. Table 6.1 
provides the thruster throat locations and tags each thruster with an 
identifier to be referenced during the GRAIL design. Table 6.2 lists the 
thruster pointing vectors and Table 6.3 lists the thrust unit vectors. The 
coordinates in Tables 6.1 through 6.3 are measured in the spacecraft 
mechanical {M} frame. 


Figure 6.1 -Thruster IDs and Mounting Locations 
See PDF version of document for figure.


6.1.1 Throat Positions 


Table 6.1 -Thruster IDs and Throat Locations in Mechanical Frame 
See PDF version of document for table.


Figure 6.2 -Thruster IDs and Pointing Directions 
See PDF version of document for figure.


6.1.2 Nozzle Pointing Unit Vectors 
The thruster pointing directions were chosen such that when viewed in the 
x-z plane, the pointing vectors make approximately a 1 deg. angle with the 
x-axis. When viewed from the x-y plane, the pointing vectors make 
approximately a 35 deg. angle with the x-axis (as shown in Table 6.2). 


Table 6.2 -Thruster Nozzle Pointing Unit Vectors [Mechanical Frame] 


6.1.3 Thrust Vectors 

Table 6.3 -Thrust Unit Vectors [Mechanical Frame] 
See PDF version of document for table.


6.2 MAIN ENGINE 
The main engine consists of one 22 N thruster mounted on the -X bus panel as 
shown in Figure 6.1. 

6.2.1 Throat Position 

6.2.2 Thruster Nozzle Pointing Unit Vector 

6.2.3 Thrust Vector 
See PDF version of document for tables.


7. FUEL TANK 
The GRAIL fuel tank is a titanium diaphragm tank used to store liquid 
hydrazine propellant. The tank is pictured below. Tank dimensions and 
location in the {M} frame is listed in the following tables. 


Figure 7.1 -GRAIL Fuel Tank 
See PDF version of document for figure.


Table 7.1 -GRAIL Fuel Tank Dimensions and Location 
See PDF version of document for table.


8. KA-BAND PAYLOAD 
The Ka-band payload bore sight unit vectors and mounting orientations are 
referenced below due to the impact the payload has on spacecraft pointing 
during the science phase of the mission. During science activities, each 
spacecraft will be required to point its associated Ka-band payload bore 
sight towards the other spacecraft so that an RF link between payloads can 
be established. To optimize pointing when pointing the payload bore sights 
toward the other spacecraft, a 2.1 deg cant is built into the payload 
mounting bracket. The GRAIL-A payload is canted 2.1 deg towards the -YM axis 
while the GRAIL-B payload is canted 2.1 deg towards the +YM axis as shown in 
Figure 8.1 below. 


Figure 8.1 -Ka-Band Payload Bore sight Orientation 


The unit vectors of the Ka-band payload bore sight for each spacecraft is 
shown below in the mechanical frame. 


Table 8.1 -Ka-Band Payload Bore Sight Unit Vectors [Mechanical Frame] 
See PDF version of document for table.


9. LGA POINTING 
Each spacecraft is equipped with two Low-Gain Antennas (LGA). One antenna is 
mounted on the -X face of each spacecraft and one antenna is mounted on the 
+X face of each spacecraft. 


Figure 9. 1 -LGA Orientation 


The bore sight unit vectors for the LGAs are identical for each spacecraft. 
The unit vectors are shown below. 


Table 9. 1-LGA Bore Sight Unit Vectors [Mechanical Frame] 
See PDF version of document for table.


10. REVISION LOG 
See PDF version of document for table.