calculating position using follower wheels

[Ether]

Question 4:

In the code block highlighted in blue at the bottom of post 11 the following code appears for calculating position and heading:

Q+=dR/2.0;
X+=dF*sin(Q)+dS*cos(Q);
Y+=dF*cos(Q)-dS*sin(Q);
Q+=dR/2.0;

Give the mathematical justification for updating the heading "Q" twice in half-steps, instead of doing it like this:

Q+=dR;
X+=dF*sin(Q)+dS*cos(Q);
Y+=dF*cos(Q)-dS*sin(Q);

[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.

[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.

[Ether]

Just for fun.

[RyanCahoon]

Originally Posted by Ether

Ryan: what CAS did you use for that? The syntax you used was rejected by Maxima, Octave, and SciLab.

It's Mathematica's syntax, which is also accepted by Wolfram Alpha, making it useful for quick one-line integral problems.