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

[Mr. N]

Very nice work!

I think, in fact, we're in good agreement. Here are some points of clarification:

Originally Posted by Jared

...curve cannot be simplified to simple cubic curves. A trajectory can contain a large number of parametric quintic curves, which can have specified end headings, specified end heading derivatives, as well as (because they're parametric) dy/dt, which affects how sharp the curves are. If these are optimized, I believe it is faster to drive in a curve.

Yes, a trajectory can be defined by an arbitrary parametric curve. I was merely pointing out that the lowest order polynomial spline that satisfies the end conditions is a cubic. However, the analysis is generalized for any function that has either (1) no inflection points or (2) one inflection point (a curve with more inflection points has superfluous arc length and is always longer). So, within the context of the original assumptions, the conclusions hold true for any spline.

It is also not accurate to disregard robot acceleration.

Agreed. However, in my earlier post, I attempted to set out general boundaries for when the assumption is still valid. For longer paths --- on the order of many meters --- there is closer agreement to real results than for shorter paths. It also depends on what gear you're in. The assumption is closer to reality for low-speed gears.

...a robot can decelerate faster than it can accelerate...

Very true, but a faster deceleration is in fact closer to my "instantaneous" assumption.

...and jerk limits...

This I'm really curious about. In my dynamic model, the velocity/acceleration limits are set by the motor characteristics. Speed and torque are coupled. However, I don't see how third-order derivatives (jerk) enter into it.

...This path takes 1.77 seconds to drive, and travels 5 feet down and 5 feet to the left...

In your example, the total path length is a little over 7 feet or about 2.2 meters. In my previous post, I point out that, in hi-speed gear, it takes about 1.4 meters to accelerate (and likely something similar to decelerate). So, this example is what I would classify as a "short" path (where acceleration definitely matters).

Can you re-run your example for 25x25 foot run (e.g., a cross-field maneuver) with similar rates of turn (remembering that the speed constraint must be imposed on the outermost bank of wheels during a turn, not the centroid).

What if I do something like this? To stop and turn at each waypoint would be really slow.

Agreed , but is this representative of a real game strategy? I would argue that a typical game involves significantly fewer way points per scoring cycle that what you've shown.

--- --- ---

All that said, it looks like your research and programming has led to a very powerful and useful modelling tool. Congrats!