Consider the system in the picture, where the green structure is subjected to two body rotations around the origin, ending in the position showed in black lines. The first rotation (alpha) is performed around the original 'Z' axis, and the second rotation (beta) is performed around the rotated Y axis (the body's y axis follows the first rotation). Both rotations are limited from -90 to 90 degrees. If the goal is to calculate distances L1 and L2 (from P1 and P2 to Py and Pz, respectively) from known alpha and beta, the problem is solved by calculating the positions of points P1 and P2, which can be done by multiplying the original vector positions of P1 (-L0,k1,0) and P2(-L0,0,k1) by the rotation matrix of the combined Z and Y rotations, and then using the trivial distance formula between the points desired.
I am now trying to invert the problem, i.e., to generate an analytical solution to calculate angles alpha and beta, knowing the distances L1 and L2. However, my different approaches always result in a numerous system of non-linear equations of herculean proportions. Writing down the equations of the rotation matrix and inverting them results in a relatively small system of equations, but with various cos and sin multiplications, while my other approaches (purely pythagorean) result in a very large system of quadratic equations.
Could some of you perhaps offer a solution or propose an approach that will simplify the problem ?
Thanks in advance,
Cheers
