I'm working with some satellite data that defines the spacecraft reference frame from two star cameras and two different measurements of orientation configuration of the cameras, and I'm trying to quantify which measurement is "better".

Here's the math behind the two options; for the first source of camera configuration:

$R_{I}^{SF}(S_1,H_1) = R_{CF}^{SF}(S_1,H_1) \times R_{I}^{CF}(H_1)$

$R_{I}^{SF}(S_1,H_2) = R_{CF}^{SF}(S_1,H_2) \times R_{I}^{CF}(H_2)$

$L(S_1)=R_{I}^{SF}(S_1,H_1) \times R_{I}^{SF}(S_1,H_2)^T$

And the same set of equations for the second source of camera configuration:

$R_{I}^{SF}(S_2,H_1) = R_{CF}^{SF}(S_2,H_1) \times R_{I}^{CF}(H_1)$

$R_{I}^{SF}(S_2,H_2) = R_{CF}^{SF}(S_2,H_2) \times R_{I}^{CF}(H_2)$

$L(S_2)=R_{I}^{SF}(S_2,H_1) \times R_{I}^{SF}(S_2,H_2)^T$

where $SF$ is the spacecraft body frame, $I$ is the J2000 inertial frame, and $CF$ is the camera frame; $H_1$ is camera head 1 and $H_2$ is head 2; $R_{CF}^{SF}(S_1)$ and $R_{CF}^{SF}(S_2)$ are the two measurements of the same cameras-to-spacecraft orientation.

My objective is to figure out which source $S_1$ or $S_2$ is better, using the value of the 3x3 rotation matrices $L(S_1)$ and $L(S_2)$, whichever is "closer" to identity; that is, which measurement $S_1$ or $S_2$ is a better approximation of the orientation of the two cameras relative to the spacecraft body.

Both matrices are small (close to identity) so currently I'm just getting a roll, pitch, and yaw from $L(2,3)$, $-L(1,3)$, and $L(1,2)$ using the small angle approximation; and then using $\sqrt{r^2+p^2+y^2}$ to get a single number for the "closeness" of the $L$ matrices to identity (which would have roll, pitch, yaw = 0).

Is there a better quantity I can use to determine which $L$ is closer to identity? Like:

  • converting the matrix to Euler axis/angle and using the magnitude of the angle? or
  • using some norm of $L-I$ to measure this closeness; $\sqrt{r^2+p^2+y^2}$ is not far from the 2-norm of that quantity

TL;DR: How can I quantify the closeness of a very-slightly-offset-from-identity 3x3 rotation matrix to the identity matrix with a single number?


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