Cones in theory

If you’ve read my book (linked at the right) you know I love quotes. Here’s one of my favorites that I find applies equally well to my day job (bioinformatics if you didn’t know) and my weekend hobby (racing of course).

In theory, there’s no difference between theory and practice. In practice, there is.

I can’t tell you how many times I’ve come up with an optimal solution to some problem only to find out that it isn’t really optimal. Real life is unfortunately always more complex than a mathematical/computational model. We make models to simplify, but modeling isn’t simple. So let me tell you about the theoretical optimal path around a set of cones, fully aware that the model isn’t completely accurate. Here are the key points to the model.

  1. Corners are semi-circular and driven at constant speed
  2. Straights feature a throttle zone to get to maximum speed and a braking zone that reduces speed to exactly the corner speed
  3. It takes no time to switch from throttle to brake and vice-versa
  4. Tire grip is the same for accelerating, braking, and cornering
  5. Gearing and RPM are ignored

As models go, this one is pretty simple. The only complex part is figuring out exactly where to transition from throttle to brake so that the car enters the corner at the correct speed (which is the hard part of cornering in real life too). There’s probably an elegant solution to this problem, but I cheated and wrote a program to do find the switch point.

So let’s get to the theoretically optimal solutions. Question 1 from last week featured a Spec Miata going around a set of cones that are pretty close to each other. Should the path be oblong (A), circular (D), or some hybrid? First, we have to know a little about the performance of a Spec Miata. How much grip does it have and how quickly does it accelerate. Let’s say it has 1.1G of grip and 0.25G of acceleration. It will be accelerating on the straights at 0.25G but braking and cornering at 1.1G. Well, except path D which is cornering only.

Turns out the answer is A, the shortest path.

circles1

After investigating a variety of scenarios, the car and surface hardly ever matter, the answer is almost always A. Corvette? A. Rain? A. Snow? A. WRX? A. In almost every common situation, the answer is the shortest path. One situation where A is not optimal is when tire grip is really high, acceleration is really low, and the cones are farther apart. So if you find yourself competing in some weird cone-circling competition where you’re only allowed to use 4th gear, take path F or G and you’ll save a few hundredths of a second.

circles2

So is there any situation where circular paths (D and H) are optimal? Yes! If you don’t have brakes, your top speed is limited by the corner speed. So if you ever find yourself in a no-brakes competition, take a circular line.

 

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3 thoughts on “Cones in theory

    1. In theory, the distances and radii don’t matter. The ratio of the radius to the length of the straight does matter though. This model should apply to toy cars as much as real cars. Thinking back to my days of slot cars, the inside car always wins. But the cones are slightly farther apart for the outside car, so it’s not exactly a fair fight. I think I did the math correctly, but I make mistakes. So I’m fully prepared for the model to be a pile of crap. On the other hand, it might be spot on. But even if it is, theory and practice are sometimes far apart. What I need to do next is to test this in real life (or at least a racing sim).

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