2-axis rotation to 3-axis lift

[dav_i]

Homework Statement

Hi there,
In my experiment I have a disc which needs to be pitched and rolled about it's centre. The disk is mounted by three verticle "pistons" one every 120degrees. I need to convert a tilt in the x-direction $$\theta_x$$ and a tilt in the y-direction $$\theta_y$$ to how much each piston should be raised and lowered by. An additional constraint is that the centre of the disc must stay at the same height.
I've been bashing my head against a wall with this for what seems like forever and any help will be appreciated...

Homework Equations

We name the positioners A, B and C at height $$z_{a,b,c}$$ and the angle from origin O to A is $$\alpha$$ O to B as $$\beta$$ and O to C $$\gamma$$.
The direction O to A is at an angle of $$\Omega$$ to the x-axis.
The length of O to A is r.

The Attempt at a Solution

I figured out that
$$\alpha = \theta_x\cos\Omega + \frac{\sqrt{3}}{3}\sin\Omega+ \theta_y\frac{\sqrt{3}}{3}\cos\Omega-\theta_y\sin\Omega$$
and
$$\beta = \theta_x 2 \frac{\sqrt{3}}{3} \sin \Omega + \theta_y 2 \frac{\sqrt{3}}{3} \cos\Omega$$
From here I get stuck: I've tried many things but I know they're wrong (as in my experiment it doesn't work properly!)
Any help please, thanks in advance

 

[dav_i]

Solution found!
Measure the average height of the three points.
Use trig to move z_a and z_b.
Remeasure average height.
Move z_a, z_b and z_c all the difference between the two.