Finding the best path in a F1 circuit

[raa]

Hello folks!
How can we find the best path to drive in a circuit in order to do the best lap time? :)
I've been trying to solve this problem for 2 weeks, trying to find some functional to minimize, but I'm stuck!
Little help? :)

P.S.
This is not homework :) I just want to know how to beat my colleagues when we go out for some go-kart races xD

 

[Ranger Mike]

you can spend $ 2 million to replicate the course in software like Ford Motor Co. did in the 1960s ..to win Le Mans..it took AJ Foyt and Dan Gurney to win it though...like the weather,,too many variables to nail it down..but this is my opinion

 

[raa]

What if we try to simplify the problem a little bit?
I don't want to get iper-accurate results, only approximate ones, in order to give approximatively the best path to drive, but not the best one ever :) So we have to not consider a variety of parameters.
For example, maybe we can assume we are in standard climate conditions, with constant daylight weather and optimal tyres temperature that let the driver perform the best lap without loosing grip with the asphalt (in reality this is very irrealistic! but it could be a first approximation).

 

[Ranger Mike]

You basically have a low flying fighter jet plane going around a closed course race circuit. If one were to quantify factors that contribute to ultimate fast time to complete one lap one would have to :
Know every degree of angle ( especially corners) for every foot of the course
know traction capability of tires at certain track temperature at every point on the course
know weight transfer of vehicle at every point on the circuit
know power band of engine at the optimum transaxel gearing every point on the circuit
know down force for optimum speed on from and rear of vehicle every point on the circuit
know shock ( damper to the Euro) movement and vehicle load during transient every point on the circuit
know effect of quick changing track conditions ( clouds, wind..even rain) during the lap
just to name a few items be calculated..

 

[berkeman]

Hello folks!
How can we find the best path to drive in a circuit in order to do the best lap time? :)
I've been trying to solve this problem for 2 weeks, trying to find some functional to minimize, but I'm stuck!
Little help? :)

P.S.
This is not homework :) I just want to know how to beat my colleagues when we go out for some go-kart races xD

You just need some basic driving tips, not calculations. Get good at driving first, and then later you can analyze particular turns or combinations with more math.

http://www.google.com/search?source...z=1T4GGLL_enUS301US302&q=go+kart+driving+tips

.

 

[raa]

You just need some basic driving tips, not calculations. Get good at driving first, and then later you can analyze particular turns or combinations with more math.

http://www.google.com/search?source...z=1T4GGLL_enUS301US302&q=go+kart+driving+tips

.

Thank you for the good tip :)

But besides all of your "REAL" and "matter-of-facts" tips, I still want to compute the best racing line with math and calculations :)

In order to simplify the problem, we could assume that the kart is a particle, and is moving at maximum permitted speed by tyres' grip, so assuming that the whole centripetal force is given by the following relation:

Frictional Force = Centripetal Force;
mass * g * frictional_coefficient = mass * velocity ^ 2 / radius_of_curvature;

so we could find the maximun speed the particle must drive the circuit in order to not loose grip and go off-road, that is:

speed = square root (radius_of_curvature * frictional_coefficient * g);

Any suggestions about how to proceed from now on?

 

[xxChrisxx]

The best line is very simple. The one that allows maximum speed through corners and maximum speed on a straight.
You can't calculate it by hand, as there are so many variables that calculation becomes meaningless.

 

[raa]

I don't think this is a good idea, maybe I'm wrong but I will tell you why I think so:
in a "circle" corner, made by two radius, R0 and R1 with R0 < R1, the maximum speed allowed is on the external rim of the corner on the radius R1, that is v_R1 = sqrt( g* k * R1), but the path to be driven is l_R1 = 2*pi*R1! The ratio l_R1 / v_R1 is greater than the ratio l_R0 / v_R0, so the best racing line in a "circle" corner is not the one that allows maximum speed!

We all know that there are a lot of algorithms that find the minimum path to be driven when the track is modeled by a grid of point (Dijkstra, A* etc.) but I don't know any of them concerning tracks that are modeled by a grid made by infinite points (in the continuum).

 

[Ranger Mike]

i regret that you are thinking in 2D...any race course turn has three elements, corner entry. mid point power on and turn exit and these will change after a few laps as the tires ability to negotiate diminishes during the race...its all about TIRE contact.

 

[xxChrisxx]

I don't think this is a good idea, maybe I'm wrong but I will tell you why I think so:
in a "circle" corner, made by two radius, R0 and R1 with R0 < R1, the maximum speed allowed is on the external rim of the corner on the radius R1, that is v_R1 = sqrt( g* k * R1), but the path to be driven is l_R1 = 2*pi*R1! The ratio l_R1 / v_R1 is greater than the ratio l_R0 / v_R0, so the best racing line in a "circle" corner is not the one that allows maximum speed!

You are right in what you are saying for a particle to get round a circle between two radii, but I didn't word what I wanted to say very well.

Lets say this circle 'corner', is a hairpin before a long straight. You can calcualte the best radius to take the corner at to maximise speed per distance traveled in the corner. You'd calcualte a smooth curve, with a constant velocity.

However the best line through that corner would be to slow down more on the way in and go past the optimum radius, cut back and take a late apex, which allows the most amount of time whilst the car is 'straight' through the corner. You get a higher exit speed and consequently carry more speed down the stright.

However if it was just a hairpin followed by a slow section, the fastest line may not be to maximumse exit speed, and maintain a higher average corner speed (ie the optimal radius).

That's what I meant by 'maximumse seed through corners and down straights'.

If you can find a way to mathematically model that, then fair enough, programs can be written to do it, lap time simulators and chasis simulators will find the maximum theoretical laptime. You still aren't going to get something accurate to real life. A good drver can feel lines out anyway and will naturally tend towards the fastest line.


EDIT: I can't wait till you calcualte this, then try to drive it and get stumped when someone on a less 'correct' line just flies past you :P :P :P

 

[Ranger Mike]

good point Chris...also what happens when you add a HILL to the equation..or the course has a negative camber turn you have to commit to on the down side of the hill?
some road courses have change of elevation up to 700 feet over 2.25 miles

 

[xxChrisxx]

good point Chris...also what happens when you add a HILL to the equation..or the course has a negative camber turn you have to commit to on the down side of the hill?

You go out a drive it. The line that puts you in the armco is the wrong one.

 

[Ranger Mike]

Ha Ha..good one Chris..example of negative camber turn..instead of being banked to help the transition as typical in round track racing - Grattan race way in Michigan ..this is a Right Turn , the cars enter from the right side of the photo. If one used 2D calculations and did not take into account the banking , i suspect the calculated path would put the car in the marbles on the outside of this corner!

 

[raa]

First of all, I thank you all for the driving knowledge you are adding to this topic: those notions will be very useful when we will be able to add more variables to the issue and then we will make the problem more complex.

But for now, if I want to have some kind of result, even highly irrealistic, I MUST switch from 3D to 2D, and I must make a huge amount of irrealistic hypothesis like constant weather, perfect grip and so on, because I want to solve the simplest problem I could think about now, that is finding the best racing line for a particle moving at the maximum speed allowed by tyres' grip assuming speed = sqrt( g * friction_coefficient * radius_of_curvature), AND THEN, after I solve this, add one after another all the variables and factors, increasing the complexity a huge problem like finding the real best path to drive is, in order to improve my results by "successive approximation".

I hope this will clarify my intentions: I did not want to trivialize a complex problem, but I wanted to solve it step by step. I will be very glad if you could help me in this way :)

 

[turbo]

i regret that you are thinking in 2D...any race course turn has three elements, corner entry. mid point power on and turn exit and these will change after a few laps as the tires ability to negotiate diminishes during the race...its all about TIRE contact.

And if you are on a road-course, the traction will change as your car's tires and those of you competitors' shed rubber onto the road. The deposited rubber acts like traction compound that you see used at drag-ways. Add in heat-related changes in tire pressure and contact patch along with tire wear, and you have a LOT of variables to consider.

 

[raa]

And if you are on a road-course, the traction will change as your car's tires and those of you competitors' shed rubber onto the road. The deposited rubber acts like traction compound that you see used at drag-ways. Add in heat-related changes in tire pressure and contact patch along with tire wear, and you have a LOT of variables to consider.

We are in a Engineering Forum, so we cannot be as accurate as we want: we must do some approximations if we want to achieve some results! So i will not take in account this phenomenon at the moment.

 

[Ranger Mike]

good point Turbo..thanks

 

[xxChrisxx]

raa what you have been doing is fine. It'll tell you nothing about a racing line, but it'll show the quickest route a particle could take.

Just do what you did before for corners and connect corner exits and entrances with as straight a line as possible. As all your corners will be an arc so shift the line so it touches an apex.

 

[turbo]

good point Turbo..thanks

I don't mean to be pedantic about this, but you might have nice soft, sticky tires at the beginning of the race, but still feel "loose" in the turns because your tires are not warm enough yet. As everybody's tires get warm, and they get more and more aggressive in the turns, they will deposit more and more rubber on the asphalt, giving you a more-grabby surface on the turns. At some point, running the track with VERY worn tires might look attractive if you are living on lap-times.