Analytic equations to make algorithm of 3D printer I have a project in the 3D printing. the aim of the project is to realize the control of the 3D printing. 

the type of the printer is a printer with cables.

the problem seems to have some development both in mechanical and robotics engineering. 

first of all, as our tutor said, we should find the dynamical and the geometrical equations and from those equations, we can find the algorithm to control the system of the 3 D printing. 

Well, my question is particularly about the dynamical equations in 2D as you can see in the picture above.

the idea  is to find equations of : 
- L1 and L2 in terms of xa and ya : which I already found as you can see in the picture
- xa and ya and teta in terms of L1 and L2: which I cannot find ! I would lilke you to help in this point.

thank you in advance



And here, I add the dynamic equations and I would like you to check if it is ok
As shown, the mechanics are under-constrained. You can't solve for theta because you have three degrees of freedom (X, Y, theta) and only two constraints (L1, L2). Gravity will tend to bias theta in a particular orientation, but the geometrical stiffness of this arrangement will be so low that it will not be possible to do 3D printing. 

To calculate the free-hanging orientation of theta, you will need to know the center of gravity of the end-effector, and solve a system of equations to find the angles and tensions for each cable that produce force vectors which sum to equilibrium with the gravity force vector through the COG. Unfortunately, the tensions will be a function of the angles, so it's not trivial to solve. As a hint, the virtual intersection of the two cables will be coincident with or directly above the COG in all equilibrium positions, and the horizontal components of the tensions in the two cables will be equal.