Inspiring videos: SpaceX rockets landing autonomously, inspiring video of a Falcon 9 RC model hovering.
Physics simulator
We make use of a custom C++ physics simulator that makes use of numerical integration to run a n-body simulation and to predict what is going to happen. This library will be shared by the rocket model (that uses it to predict and adjust trajectory) and by a mission-control-like PC that receives live data.
Random stuff
Hardware
The hardware choice will probably be some combination of an Arduino Mega, a Raspberry 3B and/or a Teensy 3.6.