paper: Driving a Robot — Fastest Path from A to B

[Mr. N]

Originally Posted by Jared Russell

What you have brought up is a special case of a well known theorem from optimal control that says (essentially) that you always want to be saturating your inputs to get to the final state in a time optimal way.

Aw, man... you made me dig out my old Optimal Control textbook (thanks, actually)

Indeed, if the question had been posed as an optimal control problem, Pontryagin's Minimum Principle leads to the conclusion that "bang-bang" control inputs (where the motors are always at + or - max speed) are the optimal solution, where the cost functional is simply total time (and the upper limits of the inputs are constrained). This implies that the robot must either be in straight-line motion (both motors +) or pure rotation (one + and the other -), which corresponds to a "turn-straight-turn" approach. Any other motor state generates a curvilinear path, and would be sub-optimal (for this specific cost functional).

There's a nice description here:
http://www.cds.caltech.edu/~murray/b...al_04Jan10.pdf

This property holds with an infinite acceleration limit and, indeed even with limits on acceleration, jerk, etc.

Interesting.

If you look at 254's acceleration profile generation code from this year...

I'll definitely have a look.

...if you ignore practicalities...

Certainly, this analysis was performed in an idealized context, with a highly simplified cost functional. Your points are well taken.