Seems an unspoken part of your analysis approach is to account for the fact that the gear sizing will certainly be driven by the impact loading forces they will see, versus average loading over the service life.
The robot (& gears) have a service life of only few hours (maybe 4 to 20 hours).
You probably already looked at the Boston Gear specifiying material. They use the Service Factor approach & a cookbook, no calculation method. [
http://bostongear.com/ ] Not good for determining this sort of condition, hi impact with short running life.
A few things I would put in the mix for any calculations:
1. The best way to determine the impact forces is probably directly, hook up some instrumentation to a robot and crash it into a wall a few times.
If you want to do it from theory you need to pick some numbers & see how they work out.
Example - pick a number for a max. traction (lbs) your wheels might develop, and multiply it by.. 5? 10? for impact and use that to determing max allowable stress based on yield of the teeth surface (hertzian stress).
Get material specs from the supplier, Boston or whoever. For a first pass I'd use 30 ksi for steel or iron. When you get some actual values you can just scale them to the new yield.