The basic form of Cassini Attitude information is quaternions. For a brief explanation, see quaternions. As with the other forms of the Attitude information, they provide a coordinate transformation between inertial space (J2000) and actual Cassini spacecraft body coordinates. For a graphic of the Cassini trajectory relative to the J2000 inertial reference frame, see J2000.
The four quaternions can be used to create a rotational transformation matrix. For the quaternion-to-matrix equations, see quaternion-to-matrix. This matrix can be used to transform a vector V, given in the J2000 inertial reference frame, to a vector V', given in the Cassini body coordinate reference frame:
V' = T * V
+- -+ +- -+ +- -+
| Vx' | | T11 T12 T13 | | Vx |
| Vy' | = | T21 T22 T23 | * | Vy |
| Vz' | | T31 T32 T33 | | Vz |
+- -+ +- -+ +- -+
The other form of Cassini Attitude Information is the corresponding Euler angles which will rotate from the inertial reference frame to the spacecraft body frame. These three angles are called by JPL the Right Ascension (which I call Phi), Declination (which I call Theta), and Twist (which I call Omega). The reason I have given new designations to these angles is because although the JPL TLM Dictionary defines these as the Right Ascension and Declination of the Spacecraft Z-axis, this is incorrect. In fact, these are Euler angles for performing three rotations to define the new coordinate system. The so-called "RA" angle (referred to as channel B-1001) represents the first turn about the z-axis. The so-called "Dec" angle (referred to as channel B-1002) represents the next turn about the x'-axis. The so-called "Twist" angle (referred to as channel B-1003) represents the final turn about the z''-axis. These three Euler angles can be used to derive the same transformation matrix T given above. For the Euler-to-matrix equations, see Euler-to-matrix.
The derived angles Phi, Theta, and Omega are easily derived from the components of the transformation matrix. For JPL's equations, see matrix-to-Euler. But in these equations, note that in fact the Spacecraft Z-axis is related to the Euler angles as follows:
Right Ascension of the Z-axis = Phi - 90
Declination of the Z-axis = 90 - Theta
The Twist angle Omega is the rotation angle about the Z-axis, as can be seen for example during the roll maneuver (roll rate 0.25 degree/sec) at SCET 1999-008T00:00.
For RPWS applications, the directions of the antenna elements are:
+X antenna element:
( sin(60), ( cos(37) * cos(60) ), ( -sin(37) * cos(60) ) )
OR
( 0.866025, 0.399318, -0.300908 )
-X antenna element:
( -sin(60), ( cos(37) * cos(60) ), ( -sin(37) * cos(60) ) )
OR
( -0.866025, 0.399318, -0.300908 )
Z antenna element:
( 0, cos(53) , cos(37) )
OR
( 0, 0.601815, 0.798636 )
Any one of these unit vectors could be multiplied by the transpose of the transformation matrix T to give the direction of the antenna element in the J2000 reference frame.