[Ether]

Originally Posted by maths222

Spoiler for Justification:

For a given time interval, the robot turns from Q_i to Q_f. The average velocity over that time period is not in the direction of Q_i or Q_f, but somewhere in between; within the sample rate of the idler wheels, the most accurate angle is halfway in between. Therefore, half of dQ is added before, then the sin and cos of Q are used to calculate the new X and Y positions, and then the other half is added.

It is not the best proof, but it explains the general reasoning.

Excellent. A very intuitive explanation.

See attachment for additional explanation using geometry and a bit of calculus.