[Jared Russell]

This is the right question to ask!

In the presentation, we mostly talked about the case where major disturbances are not expected, because this is simpler (still, I think the presentation was a bit too technically deep for most of the audience and we'll tweak our next one).

There are lots of strategies for how to deal with this, but generally they all boil down to some variation of the following:

1. Estimate where you believe your robot to be (from some combination of sensors as you describe)

2. Determine where you would like your robot to be right now (based on elapsed time in a trajectory, or the nearest point on a path, or something else).

3. Figure out the components of the error between these two poses (with respect to your current pose). There are naturally three parts:

a. Along-track error. Your robot is lagging behind (or leading) the desired pose.
b. Cross-track error. Your robot is deviating to the side of the desired pose.
c. Angular error. Your robot orientation is deviating from the desired pose.

4. Construct a controller to drive these sources of error to zero. PID controllers work just fine. For a holonomic robot, this is easy: put a PID controller on the x velocity (fwd/rev) based on along-track error; a PID controller on the y velocity (strafe) based on cross-track error; and a PID controller on the angular velocity based on the angular error.

For a non-holonomic robot, you can't strafe, but you can steer back towards the path...you actually want a PID controller that maps from cross-track error to a steering adjustment (if you are to the right, steer a bit left, etc.).

5. Execute your PID controllers, add in feedforward gains (from your original trajectory, for example), and now you have new desired velocities for your motors.

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Note that, as with most things, the devil is in the details and you will need to make sure you choose your desired pose carefully (using the time-indexed trajectory is simplest), and ensure your controllers don't end up fighting each other (since in the non-holonomic case, you will have two controllers that both want to affect steering...the easiest way to do this is make the output from the cross-track error controller be the input to the angular-error controller).