[KenWittlief]

Most commercial controllers I have seen that use PID control loops put the system that is being controlled through a brief test, where the open loop characteristics of the system are measured.

One way they do this is to hit the system with a squarewave: full forward, full reverse, full forward, full reverse - and the response of the system is measured in real time

this information is then used to tune the PID loop to the system

The problem you are facing is, you want this information for a 'typical' robot (drive train)

what is typical? Every drive train will be different, and the friction it will encounter will be different.

Back to your equations, I dont know what type of friction you are describing with those equations. There are two types of fricition normally encountered:

Static friction: This is the 'over the bump' friction that you must overcome to get an object (or mechanism) to move from a standstill. It is normally a function of the weight that is being applied perpendicular to the force that is trying to move the object.

Kinetic friction: This is the force that is encountered opposite to the force that is moving the object, once it starts to move. This friction is constant, it is not a function of velocity, or acceleration.

FIRST robots do not move fast enough to have to consider friction from the air, and we have not yet had a water competition - so I dont think you need to worry about friction that is a function of velocity

but if you did, you would have to delve into the advanced mathematics known as differential equations. The friction is not simple a function of velocity, its an exponential equation.