Calculating the backlash in a gearbox

Hello everybody,

I am trying to find a way to calculate the backlash in a gearbox. If manufacturing is accurate there will be no backlash. However due to wear and limited manufacturing accuracy the geometry will not be optimal. Therfore I have given shapes (the physical parts of the gearbox) which can be shifted and turned without overlapping. The actual, not optimal geometry is given. Having these constraints I want to calculate the maximum angle that the output shaft can rotate, when the input shaft is fixed.

Already I managed to calculate (not 100 % accurate) the backlash, when only one part is to small and able to rotate (see picture backlash.png). However I am having difficulties to find a way how to calculate/find the backlash when more parts are smaller/bigger then expected or in another way out of shape. My collegue suggested an optimization. So I tried a nonlinear optimization with optimproblem() etc. minimizing the distance between to very simply forms. However I did not get far. The distance is calculated as something like minD: 1.0723e+09 - which does not make sense. The value is far to big. Also the visualization shows, that the shapes overlap (solution in red in the png "Optimization_Attempt.png").

Therefore I would appreciate help and ideas on the question:

In the basic case: How can I find the maximum angle the ellipse can turn until it touches the circle when it can also shift in a given area? (Optimization_Attempt.png)

In the final case: How can I find maximum angle a shape (the output shaft) can turn, when several parts are involved? (backlash.png)

I also started searching for packing problems/nesting problems, such as the container or bin packing problem, nesting of polygons and Cutting and Packing problems. However most of the time the solutions are meant for a greater number of parts, and the orientation of each part is fixed. In my case the parts have 3 degrees of freedom, and individual contraints for the boundaries of these movements (e.g. due to clearance to a bearing 1mm or down, and free to turn until contact of other part). The parts are also not allowed to overlap.

Thank you in advance for Your help :)