Transient Response 6.1

Here’s the proof for the statement I made in the earlier post: The fastest way through any infinite slalom is by crossing over mid-way between the cones. Any other strategy, i.e. being early or late, at any point, will be slower overall.

Let’s assume we are driving a car with infinite transient response, which means we drive through an infinite slalom on perfectly circular arcs. (It doesn’t really matter, but I drew the figure this way.) We can make our crossover either at the midpoint between cones, or before the midpoint (being early) or after the midpoint (being late.)  Figure 1, below, shows the path crossing over at the midpoint.

6point1 fig 1

Figure 1- Infinite Slalom with Midpoint Crossover

For purposes of illustration I’ve drawn the cone supersized. The point is the same even if the cone is very small as compared to the track width of the car.

If we try to be early on the cone but use the same path then we get the situation shown below in Figure 2.

6point1 fig 2

Figure 2- Crossing Early Before the Midpoint Between Cones in an Infinite Slalom

Do you see the problem? To be early we can no longer drive a path with the minimum lateral movement of the car. The car has to move farther laterally to miss the cone. This means a longer path with a smaller radius. That’s a slower path.

Being late causes exactly the same problem. Most autocrossers think of getting late in a slalom as the kiss of death. We watch novices do it over and over again and shake our heads at how slow it makes them as the car slews sideways and slows terribly in the attempt to not hit the (usually) last cone in the slalom. But, are we really so smart being early?

Any deviation from crossing at the midpoint will cost some time. So, when we think we have a good reason for not crossing at the midpoint it needs to Save More Time than we lose.

Transient Response 6.0

I will try something a little different with this final part 6 of my transient response series. I’ll be making a series of slightly raw and unfinished posts that build on each other rather than work it all out, edit it all down and then post it all at once.

I’ll begin by freaking out some people with the following general statement which has arisen out of this work on transient response:

*** The fastest way through any infinite slalom is by crossing over mid-way between the cones. Any other strategy, i.e. being early or late, at any point, will be slower overall. ***

I think this statement is easy to prove and I plan to. But notice I said an “infinite” slalom.

Here’s another statement:

*** The fastest way through any generic, non-infinite slalom is probably achieved by starting early and ending late. ***

Notice the word “probably.” I say this because I’m not sure I can prove it. Notice the word “generic.”  What I mean by this is not considering the course design immediately before and after the slalom.

What happens before (the entry) and after (the exit) of a slalom, like all other features in autocross, must always be considered because autocross is nothing if not an exercise in optimizing multiple variables. But we need to understand the two baseline, generic statements above first in order to have a clear way to optimize all three parts, i.e. the entry, the slalom itself and the exit. Common, relatively simple rules such as “enter early and stay early until the next to last cone, then accelerate out” is a rule that may actually be correct for 2 out of 3 slaloms in 2 out of 3 cars but if the 1 out of 3 times it’s wrong means the difference between a Nationals trophy and no trophy then maybe it would be good to know something more so you can know when to do something different.


Pinchy Entry 2

At the end of the last post I said I didn’t know how wide to go on entry. That’s not exactly true.

I was trying to explain this to a friend last night and I realized that maybe I owed it to the readers to get down off the theoretical high horse and explain the process I use during the course walk to figure it out, i.e. when is a chicane necessary and how wide to go. It’s a little bit complicated, but it really is what I do. You can, as always, decide for yourself whether to take it or leave it. If you think any of this helps, fine. If not, forget about it.  Most people do it basically right anyway, at least after a some experience. Only a poor few of us have to really think it through before we can do it at all correctly.

Let’s go back to the beginning and get a few things straight. Maybe this will be all too obvious to you. If so, I apologize.

fig 1

Figure 1

First thing: in Figure 1 above, we can rest assured that cone B is the position of our proper apex.

Oh, I better make this clear: the definition of apex. The apex is the location and angle at which the trail-braking entry spiral, during which the car is slowing and the path radius is tightening, ends and the exit spiral, during which the car is accelerating forward and the exit spiral is opening, begins. It is therefore, by definition, also the location of minimum corner speed. Furthermore, for each car, corner and set of conditions, there is only one correct apex in the sense of the fastest way through the corner.

This is actually a peculiar situation and one of the key differences between autocross and track driving. In the track corner, say the dotted lines shown, we will find the correct apex somewhere along the inside curve, maybe at B, but it takes some thought to figure out exactly where. This is because the apex is as much the angle of the car as a position and with the paved border of a track the angle changes with position, thus complicating things. Not so in autocross.

In autocross the apex position is the cone in the situation shown. I’m going to assume you understand this. If you aren’t completely clear why this is the case, go read Brouillard. If you’re a beginner, do me a favor, give me the benefit of the doubt and assume that cone B is the location of the freakin’ apex.

But, what about the car’s angle at cone B, the second part of the apex definition? Ah, there’s the rub. That’s what we have to figure out in autocross. 

Second thing: regarding track limits, we know the almost-always-correct rule for a race track is to use all the available space. That is, go as wide as the track limits allow, at least 99% of the time if you aren’t limited by, say, another car in an actual race or some weird traction situation. There’s almost never enough track so the basic rule is simple: use it all.

In modern autocross we often find the opposite. There can be too much track. So the road racer’s rule to use all the available track is of dubious utility. In autocross we actually have to decide whether to use all the available space or not. This is one of the cool things about the sport. (Unless you’re at one of those retrograde locations where they line the whole course with cones.)

Third thing: to be slightly ridiculous, we need to be clear that the path shown in Figure 2, below, might be possible in an autocross, but is probably not optimal. We could brake at entry cone A, turn left, do a loop and accelerate to the apex cone. Perhaps we pass the apex cone going 100mph and we’re doing a buck-thirty-five at the exit. Now that would be maximizing your exit speed! 

fig 2

Figure 2

I’m going to assume that we all know it wouldn’t Save Time, however, even if the following straight was of infinite length. You never catch up to the guy that did it right yet exited the corner at 45mph. If you don’t understand this then, once again, go read Brouillard. I sometimes think of it like this: it is rarely correct to go in the wrong direction.

Didn’t Yogi Berra say that?

So, the path we need to drive may be something like what’s shown in Figure 3, below. The question is whether and if so how far “outside the track limit” should we go. I’m going to give you  my process for figuring that out. This will necessarily be a qualitative treatment, however, not quantitative. (Thank God for that, I heard you mumbling.)

fig 3

Figure 3

So, here’s what I do during the course walk, in two parts. (If you find this backwards, it’s perfectly reasonable to reverse the steps as it’s a back and forth, iterative process by nature. Brouillard, in fact, starts the other way around, which may be easier for some.)

First part: optimize the exit, that is, the path from B to C

Second part: figure out how to drive the entry, A to B, to ensure the optimal exit

I start by standing at cone B and looking toward the exit like in Figure 4.

fig 4

Figure 4

Now I ask myself what angle the car (my car, not your car) needs to be at cone B so that when I add throttle at B and start opening the wheel I’m able to use as much acceleration (both forward and lateral) in the new ideal direction as the car is capable of and yet not hit cone C.

Here in Figure 5 is our angle and an estimated (Euler spiral) acceleration path in red to the exit at C.

fig 5

Figure 5

But, in order to draw this exit path I necessarily have to imagine one more thing: how fast I’ll be going at B when I hit the accelerator and start opening the wheel.

You may ask, how can I know how fast I’ll be at B, when I haven’t planned the path from A to B yet?

That’s a good question. The answer is that you have to sneak a peek back to A, estimate it and then iterate. (I know that some people say to never look back during your course walk. Sorry, no can do. You gotta study the course from all angles.)

Now that we have an estimated, optimized exit path, we think about how fast we will be going at A and what path and how much braking we need to get to B at the speed we estimated in step 1. Simple, right?

But, that’s what we do. And we draw the green path as shown in Figure 6. In our head. During the course walk.

fig 6

Figure 6

Did the green path exceed the track limits? Hell if I know. This is autocross. There is no track. All I know is that the green path gets me to the apex in the least amount of time and at the right speed and the right angle to produce the fastest possible exit.

Okay, I played a little fast and loose with you there. Let me make it up to you. In the next figure, Figure 7, I’ve moved cone A over to make it clear that the original green path won’t work. What do we do now?

fig 7

Figure 7

Now we clearly have to create a chicane by turning left at cone A in order to get any kind of reasonable angle at cone B. Otherwise we will be forced into a very small radius corner which will be really, really slow. In theory we’d like to get to the same apex angle and same speed at cone B as before, but we probably can’t. The problem is that having to turn left at A and then back to the right (unless there’s a whole lot of distance) means that our speed when we get to B will be a little bit slower. All because the location of cone A is pinching our entry.

If our speed at B is a little bit slower it means that we may be able to reduce the apex angle a little bit and still get full power on the exit. So, in Figure 8 I rotated the dashed angle line a little bit counter-clockwise from the original angle and modified the red exit path accordingly. The original exit line was for a different speed at the apex, so it must change to use a different part of the Euler spiral.

fig 8

Figure 8

The exit is not quite as fast as before thanks to the pinchy entry. There’s no way it can be.

The answer to the original question is, therefore: Chicane it out only as far as necessary to produce an optimized exit for your particular car, namely one where you can create as much speed as possible at the apex at an angle that allows you to use all the car’s lateral and forward acceleration ability after the apex and still make it past cone C. We must play off the entry path and the exit path against each other, taking into account the braking ability, cornering ability, acceleration ability and transient response ability of our particular car. We have to mutually and concurrently optimize a chicane entry into a corner for minimum total time through the combination.

Going out any farther in order to increase the speed at the apex will not Save Time. Going out any farther will be a variation of using the crazy path shown in Figure 2, except so small that only the stop-watch will notice.

Or, you can just wing it. You get three attempts.

Pinchy Entry

I’ll get back to Transient Response real soon, but in the meantime let’s discuss that all too common and commonly disdained autocross feature called the Pinchy Entry. Turns out, pinched entries to corners are one of those places where fast transient response and low yaw inertia play a key role in Saving Time.

Let’s start by assuming a standard 90 degree corner on a road course, as shown in Figure 1, where our entry speed is much faster than the corner can be taken. We must find our braking and turn in points and most of us know that we should trail-brake to the apex to produce an Euler spiral entry path. We spiral in and then beyond the apex we accelerate and naturally spiral out.  The track edges are the “ideal” directions in Adam Brouillard terms and we are maximizing the car’s acceleration backwards along the entry edge up to the apex and maximizing the car’s acceleration forward along the track edge beyond the apex. No issues, right? We all got this? (If not, go read Brouilliard.) The approximate path is shown in thin solid black.

figure 1

Figure 1- A Standard Road Course Corner

Now let’s replace the road course with 3 cones like we might find in an autocross, as shown in Figure 2. Has anything changed? I think it has. Or, at least, for some corners something has changed. We might take the corner exactly the same way, but most of the time it wouldn’t be correct. This is probably a key difference between road-racing and autocrossing and it is caused by the freedom (and complications) autocrossers deal with due to the lack of track edges.

figure 2

Figure 2- Autocross 3-Cone Corner

To better illustrate my future point as to why something has changed let’s move the entry cone inward to create a “pinched” entry as shown in Figure 3. We see this often in autocross courses where the entry is pinched as compared to the exit. Most of us know not to just charge down to the corner by taking the blue dotted path. If we do, we have to severely slow the car to negotiate the resulting small radius section just before the apex. Not only are we then slow to the apex, but we are then slow after the apex as well because our minimum cornering speed is reduced. I made this mistake for way too long!

figure 3

Figure 3- Pinchy Entry

Instead of taking the blue dotted line, experience leads most of us to give up just a little distance and turn left at the entry cone to get a better angle on the apex cone as shown in the red dashed line in Figure 4. We can then get out to the “original” line and therefore be able to hit the apex at the same speed and angle as before. We might have to slow a little bit as well for this “hidden” maneuver (not evident on the course map) but at least corner exit has not been compromised.

But do you see what we did? We created a chicane where there wasn’t one before. Per Brouillard’s definition a chicane is two corners of opposite direction that are so close together they cannot be optimized separately. They must be optimized together. Autocrossers are nothing if not chicane creators. Also, this “hidden” maneuver is aided and abetted mightily by a car with fast transient response and low yaw inertia.

figure 4

Figure 4- Creating A Chicane

But, wait a minute: this is autocross and there are no track edges, or at least there may not be much of a track edge-like limitation. Experience (and data) tells me that in many cases I should go even further out and take something like the green dashed path in Figure 5 to Save Time to the apex and increase the speed at the apex, even though I’m adding distance and having to slow earlier.

figure 5

Figure 5- Exceeding Track Edges?


How far should we go to the left? Beats me. Only experience can tell us how much is best for the particular car and particular corner geometry.

Transient Response 5

Note: A mistake in my calculations was pointed out to me by the observant S. J. Fehr. I pulled this post down for a while and fixed it. I apologize to those that read it earlier.


Now I’d like to assume a different, very simple, path model for a car transiting a slalom. Not because it’s more accurate than the sine curve but because it will be useful.

Let’s assume a car has infinite Transient Response (TR). Such a car could switch from turning left at max lateral-G to turning right at max lateral-G in no time at all. The path for such a car in a slalom would be a series of back to back circular arcs. Overlaid on the previous picture of a sine curve it would look like this where the two curves are almost, but not quite, the same. The red is the sine curve and the black is a circular arc.

car on path

Figure 1- 80′ Slalom Paths

I’ve done the calculations and can tell you that, assuming circular arcs, my 5th gen Corvette, given the same width and same 2″ distance off the cones as before, cornering at a constant 1.2Gs in a 80′ slalom would travel at 62.3mph and take 0.88 seconds between cones.

Here are the calculated times for different slalom cone spacing for a car the width of the C5:

new slalom time

Figure 2- Circular Arc Slalom Times Per Cone

Interesting that the distance between cones makes almost no difference, just as in the sine wave model discussed previously. Between 90′ and 45′ the difference is only +/- .001s.

Recall from Transient Response 4 that for a sine curve with a peak of 1.2Gs it should take 0.93 seconds for each cone, not far off what cars regularly achieve and only slightly longer than the times in Figure 2.

Remember as well that the fastest data for my car at one particular event was 1.05 seconds between cones. I think I’m most interested in the difference between the perfect arc/no transient zone path time of 0.881s (for the slalom spacing in that data) and my actual time of 1.05s. What’s the source of this difference?

Let’s back up and make things clear. The 0.881s time is for a car that has perfect, instantaneous TR, driven perfectly, with perfect balance, etc.  The 1.05s time was as measured in a real, Street-class car (with a real driver of limited skill) that takes time to switch from turning left to turning right and probably never reaches its maximum steady-state lateral G, so it goes slower through the slalom. The relative ability to switch directions is what I think of as Transient Response. So, I think we now have an easy-to-get and real-world-accurate measure of TR for any car.

The measure of TR is then the ratio of how much time a car would take to negotiate each slalom cone if it had infinite TR, given it’s actual maximum lateral cornering capability, to how much time it actually takes. I’ll call this the Slalom Efficiency. In the case of my Corvette it would be

Slalom Efficiency = .881s / 1.05s = 0.84 or 84%

If my car was actually capable of 1.3Gs, as my V-Box said it was reaching, then the Slalom Efficiency is less.  At 1.3Gs I calculate that my car should pass each cone in only 0.846s, given perfectly circular arcs, so the new efficiency is:

Slalom Efficiency (for 1.3Gs) = .846s / 1.05s = .81 or 81%

For the Street Prepared car whose data showed it reaching only 1.2G steady-state, but consistently reaching this level in the slalom and taking just less than 1s, let’s say it was 0.98s, its efficiency on that day on that course with that driver was:

Slalom Efficiency (SP car) = 0.881s / 0.98s = .90 or 90%

The more time a car can spend in a slalom carving a circular arc (at its maximum lateral capability) the higher its Slalom Efficiency will be. Only fast TR allows reaching and/or spending any significant time at max Lateral G around each cone. That’s why Slalom Efficiency is a good measure of transient response.

We can easily rate any car for transient response by finding its Slalom Efficiency.  Many autocrossers already have the data they need for their own car. For more perfect data we need only install a data-device (even a 10hz GPS-only device should work fine) and run it through two exercises: 1) a 200′ skidpad and 2) a longish slalom. Run the car on the skid pad to find its steady-state lateral G capability and then run it in the slalom and find the average time between cones. Plug in the numbers and produce a Slalom Efficiency percentage. A car with infinite transient response, perfectly driven and with perfect balance, would score 100%. All real cars (with real drivers) will score less.