The Atmega2560 should be able to handle anything robot related thrown at it short of video processing. We used one this past summer for the sparkfun AVC.
video here.
This involved:
- bringing in the info from a GPS and a gyro.
- Doing all control loops for autonomous navigation.
- Tunable parameters over a wireless link.
- Send telemetry information over the wireless link for debugging
However, beware that the Arduino does not support double precision math. Everything is done single precision floating point.