direct conversion from az/el to ecliptic coordinates

Background:

I’m trying to build a lightweight antenna tracker with two servos. For mechanical reasons, I’m first mounting servo 1 on a base so that it tilts forward/backwards, then mount servo 2 on it twisted 90˚so it can tilt left/right.

I can basically use the first servo to select a from the great circles that go through $(az, el) = (0,0)$ and $(pi, 0)$, and use the 2nd servo to move on the chosen great circle. This way, I should be able to point at the entire upper hemisphere even though I might need to reposition the antenna when crossing the midline (the servos only have 180 degrees of movement.)

I’m trying to find the function that maps az/alt to the tilt/tilt servo angles, which I think would be $(epsilon, lambda)$ in ecliptic coords (with $beta =0$)

Looking at https://aas.aanda.org/articles/aas/full/1998/01/ds1449/node3.html i do find transforms for az/el to equatorial, and transforms from equatorial to ecliptic.

In my case, I think I want to use an ecliptic transform where I force the angle $beta$ to be $0 $ and treat the ecliptic angle $epsilon$ as a variable instead of constant.

Does that sound reasonable ? If so, is there any self-contained code around to implement these transforms, or even better is there a single transform that I can use ?

Thanks in advance!

EDIT:

I think I am a dolt. The equatorial – az/el transformation is a red herring. It’s used in astronomy to map between the conceptual coordinate system centered in the middle of the earth to the one supporting a human sitting someone on top the planet with a telescope. With the antenna tracker, we don’t have this problem, we just map between equatorial and ecliptic coordinates.

So the equations should look like

$sin beta = sin delta cdot cos epsilon – cos delta cdot sin epsilon cdot sin alpha$

$cos lambda = (cos alpha cdot cos delta) / cosbeta$

$sin lambda = [sindelta cdot sinepsilon + cosdeltacdot cosepsiloncdotsinalpha] / cosbeta$

Then I want $beta=0$, so that the $cosbeta$ terms become 1 and the $sinbeta$ terms become 0, this simplifies to

$epsilon=arctan(tandeltacdotsinalpha)$

$lambda = arccos(cosalphacdotcosdelta)$

It would be great to get a confirmation that this is indeed correct (for some part of the sphere at least)