Hi John,

If the Sentinel-1's quaternion is interpreted as q0i + q1j + q2k i + q3, where q3 = 0.8444761 is the scalar part, then by choosing the YXZ sequence of rotation axes for Tait–Bryan angles, the equivalent Euler angles (degrees) are shown on the above figure.

https://www.andre-gaschler.com/rotationconverter/

The y and z angles have the 'wrong' sign, but the numbers hit spot on! Great!

The next question is why ESA would change both the order of the values in the quaternion (" the S/C Attitude Quaternions represent the S/C attitude wrt. the Orbital Reference Frame which, in NPM/OCM, is the Zero-Doppler Reference Frame. Q0 is the real component and Q1, Q2, Q3 are the vector components of the Attitude Quaternion " Page 24 of 85 of https://sentinel.esa.int/documents/247904/685163/Sentinel-1-SAR-Space-Packet-Protocol-Data-Unit.pdf) and the order of rotation ("Note that in general L(psi) is not uniquely defined by a given attitude vector. In order for L(psi) to be uniquely defined, a rotation order for the (phi(roll), theta(pitch), psi(yaw)) triplet must be known. For Sentinel-1, the same rotation order as for the ASAR was assumed, namely the 3-2-1 rotation order." on Page 59 of 159 in https://sentinels.copernicus.eu/documents/247904/1877131/Sentinel-1-Level-1-Detailed-Algorithm-Definition. )

My expectation on the definition of 3-2-1 is from "Euler angles (z-y′-x″ intrinsic) → rotation matrix" (https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions).

Can anyone shed light on why the definition of the quaternions in the Spacecraft Protocol document is different that the values in the Level-1 annotation XML?

ANOTHER EXAMPLE:

Given:

-

2018-04-26T06:38:18.749997

GM2000

2.043915e-01

3.931763e-01

3.003220e-01

8.446557e-01

-5.387243072618730e-05

-9.472090168856084e-04

-4.780825984198600e-04

-6.264703253252621e+00

-5.234385960129340e+01

4.222737787233338e+01

from:

PRODUCT:

S1B_IW_SLC__1SDV_20180426T063818_20180426T063845_010651_013707_1395.SAFE

FILE:

s1b-iw3-slc-vv-20180426t063818-20180426t063843-010651-013707-006.xml

LEVEL-1 XML DATA:

Q = [ 2.043915e-01 3.931763e-01 3.003220e-01 8.446557e-01 ]

LOOKING FOR ROLL:-6.2647(or perhaps 173.7353) PITCH:-52.3438(or perhaps 127.6562) YAW:42.22737(or perhaps -137.7726) (ALL IN DEGREES)

Based on Quaternion documentation then the following values are defined to be:

roll = atan2(2.0*(qw*qx+e*qy*qz),1.0-2.0*(qx*qx+qy*qy));

pitch = asin(2.0*(qw*qy-e*qz*qx));

yaw = atan2(2.0*(qw*qz+e*qx*qy),1.0-2.0*(qy*qy+qz*qz));

Permuting the order and selection of values one gets the following combinations to match the magnitudes of pitch, roll, and yaw:

MAGNITUDE IS qw=Q(1)

qx = Q(2); qy = Q(4); qz = Q(3); Roll=137.772622 Pitch=6.264700 Yaw=127.656146

qx = Q(3); qy = Q(4); qz = Q(2); Roll=127.656146 Pitch=6.264700 Yaw=137.772622

MAGNITUDE IS qw=Q(2)

qx = Q(1); qy = Q(3); qz = Q(4); Roll= 42.227381 Pitch=-6.264700 Yaw=127.656146

qx = Q(4); qy = Q(3); qz = Q(1); Roll=127.656146 Pitch=-6.264700 Yaw= 42.227381

MAGNITUDE IS qw=Q(3)

qx = Q(1); qy = Q(2); qz = Q(4); Roll= 52.343858 Pitch=-6.264700 Yaw=137.772622

qx = Q(4); qy = Q(2); qz = Q(1); Roll=137.772622 Pitch=-6.264700 Yaw= 52.343858

MAGNITUDE IS qw=Q(4)

qx = Q(2); qy = Q(1); qz = Q(3); Roll= 52.343858 Pitch=6.264700 Yaw= 42.227381

qx = Q(3); qy = Q(1); qz = Q(2); Roll= 42.227381 Pitch=6.264700 Yaw= 52.343858

NOTE #1 - No permutation matches all of the same sign values.

NOTE #2 - Pitch is the ~6 degrees and based on the math for the quaternion numbers CAN NOT BE ~52 in any combination/permutation.

It is hard to implement the documented algorithms (Sentinel-1-Level-1-Detailed-Algorithm-Definition.pdf) when "standard" notation and "defined" conventions are not followed...