How Do You Convert 2-Axis Rotation to 3-Axis Lift for a Disc?

[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 vertical "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.