To simulate the agent, you will need to track its position and orientation on the track, as well as its steering angle and current velocity. That means that velocity will be in speed per unit time. Noise should be a parameter to the simulation. V-old is the current velocity. Direction is in radians off of going straight ahead.
Velocity is in speed per unit.
To update position, you will need to keep track of where the vehicle is in the track grid and what the previous speed and orientation was. The above calculation assumes you have set an origin in your track and you change position and orientation based on it. Again, to simplify the calculation, just round the result to obtain the discrete position within the array. For this, we will be using a simple race track. Just a slight serpentine of length 50 and total width of 34 as shown in the racetrack figure.