I have a robot with a 4 wheel drive using mecanum wheels to allow for more mobility. For the engineering documentation I am looking to find both the translational velocity and rotational velocity of the robot. To begin I started with the (incorrect) assumption that the tangential velocity of the wheels was the overall linear velocity, but that yielded unreasonably high values. What is the correct way to mathematically evaluate the translational and rotational velocity of the robot?
The problem you describe, calculating the cartesian velocity form the wheel velocities is known as the direct kinematic problem of the mobile robots. You can find a detailed explanation on who to do this for 4 wheeled robots with mecanum wheels here.