Tire Modeling: Extracting Results from a Large Data Set

Hello, everybody. This is again The MATLAB and Simulink Racing Lounge. And as you can see, in today's session, we are again cooperating with a team. Today it's Monash Motorsport, and we are going to talk about tire modeling and their approach on that.

Basically, it all comes to the point, you have a huge amount of data and you want to extract the essence, the most important things to make your car faster. So now let's move to the content of today's session. But before introducing you to the content of today's session, I am pretty happy that I can introduce you to Marc Russouw. Hi, Marc, how are you doing?

Microsoft-- I'm great, and yourself?

I am fine, fine. Thanks for joining that episode. Thanks for making it possible. To briefly introduce you, so what is your role in the team and why are you working on tire modeling?

OK, so my role on the Monash Motorsport team has changed. I started on the team in my first year at university, which was six years ago. And I started off in the aerodynamics section of the Formula SAE team here. I started off manufacturing a lot of components. And I've also switched to the suspension team.

And I've also designed and done manufactured components there. From that, I chose a more sort of management role in terms of design and the aerodynamics section. And last year I changed my role a little bit. Or this year, actually, should I say. I changed my role over to more of a supervisor sort of role, a mentor sort of role for the aerodynamics team, and I've also done some work into vehicle dynamics for the team.

OK, great. Thanks. Seems that you have quite some experience on the Monash race car and what is going on in the Monash race team.

Yeah.

Great to have you. And what we are going to talk about today is, well, we will have a [INAUDIBLE] about available tire modeling. We will talk about the benefits, and also about the pitfalls of tire modeling. We

Will talk about how you should choose the right model for you, and also, Marc, you will introduce us to a non-dimensional tire model you have been working on using in MATLAB.

That's right.

So let's start with the technical content. So what is the motivation behind tire modeling? Why are you doing that?

So essentially we want to model the tire-- use tire modeling here at Monash to be able to predict tire response to various parameter changes and see how changing, say, camera angle, pressure, or normal load affects the output force of the tire, and also various other parameters. We want to use tire modeling as the basis for a vehicle model, using either software, industry software, or our own software developers.

And we also want to use this as the basis for the guys doing kinematics design at the beginning of the year to have an idea of what the tire operating range is, or what essentially keeps each tire happy. We want to also see what the impact of different setups changes are, and that hearks back to the vehicle model.

And the advantage of tire modeling is actually that it's a lot-- once you get it going, and simulation as well, is it's less resource intensive. You don't have to roll the car to go out and test it. You don't have to buy the tires to be able to get at least a starting point or an idea of how it behaves.

No, great. Makes perfect sense to me. So either you want to research really for setting up your car, learning how it behaves, and also embedding it to a bigger picture of vehicle modeling simulation or suspension simulation. No, looking really forward to the session, so let's start.

OK, so tire models, I wanted to introduce at first what the complexity is, and what you can find out there. So tire models come in all sorts of shapes and sizes. They can be anything from-- and complexity as well, most importantly. They can be anything from physical representations, so finite element models, as you can see in the top right, to pure theoretical models.

So we've actually had someone here at Monash do his PhD on developing a tire model from first principles. So this guy actually went out-- his name is Nick Trevorrow. He actually went out, did materials testing on the tires, scanned the road surface itself, and actually modeled the tire rubber draping over all the surface asperities and the interactions occurring between the rubber and the tire tread. And also the contact of the road and the contact patch.

So we also get something-- or the less complex strategy to adopt is to use empirical fits. So these are what I used most often in Formula SAE, because the data can be readily available. Teams can obtain it or they can buy it. And yeah, it's generally the least complex way of modeling tires. So they're empirical fits because the relations used to fit them don't have that much of a connection to-- they don't physically mean anything, or that much in reality. They're just fitting coefficients to constrain the curve to a certain set of data.

And what I am seeing, and totally agreeing with you, the easy model and the applicable models should be the ones chosen for a Formula student. Because imagine you have a FE model of a tire. Setting the model up is rather quite easy, but having it verified might be a little challenge. So totally agree that analytical or empirical, semi-empirical approaches are the ways to go for a Formula student.

Essentially, at least the best place to start.

Yep, I agree.

And there's also the constraints of there not being a lot of time available. So you've only got three months or so in which to design your car or--

Exactly.

--you've only got three or so years on the team. So--

Yeah, you have change in personnel, so.

Exactly. And so this leads onto the next slide. So I'll just present this. This is quite a good figure as presented by Pacejka, just showing the number of considerations that do exist when you're choosing a tire model. It also shows how things like insight into tire behavior, or effort into creating the tire model varies as you move up and down the scale from more empirical models to more theoretical models.

So we are essentially lying more towards the left of the diagram, around the similarity method. But you can see if you go through to a very complex physical model, you have to do things like testing the tire, material properties itself, making sure you have the road properties. You have to do a lot of special tests.

So as you can see, there's a huge range of complexity. But again, semi-empirical models on the most common ones used in SAE. And for that reason, they're easier-- they're the easiest ones to fit and to get results out of it as quickly as possible.

The Monash goals for this exercise were basically to gain a lot more insight into tire behavior than we currently have. We've just been using rules of thumb in terms of predicting tire response. And we need the model to be able to be simple enough to understand. It's no use, one person goes away for a year or two years and creates this exceedingly complex model.

The fact of the matter is you have to be able to hand over your work from one year to the next. And the people coming in, or the person getting the work might have less familiarity with it, so they might be your boss. So you need to be able to distill it into a form that's very easy to understand, or continue work on, or just to be able to get results out of. So also, something that's less complex, like a semi-empirical model, can be quite versatile. You can extend it to a vehicle simulation or a lab time simulation quite quickly.

Sometimes it even helps to make a concept decision. A very simple model.

Exactly. All we want-- we're not so focused on getting to within 5% of reality. But we are focused on being able to predict what the relative gain of a certain change is going to be. So if going to one tire is better than another, we want to be able to see that in the numbers or in the trends. So yeah. And also, less complexity means less computational resources are used in terms of time. So it's a lot better to be used as an iterative, in an iterative manner.

OK, so moving on, I just wanted to quickly also go over what the tire testing consortium means. So as I said, Formula SAE teams can test for the own data. It's been done. It's currently being done by a couple of teams in Europe. But it also can be quite resource intensive and money intensive if you don't have the sponsors.

So this testing is also performed by a company called Calspan, a tire testing and research facility in the US. And there's a small group of them who volunteer some of their time. I think that a lot of them are ex-Formula SAE students. They volunteer some of their time to collect data for Formula SAE tires.

So these tire tests are done in rounds every year or every two years. And for each tire test, they take a tire and they test at least five inclination angles, four normal loads, and three to four different pressures. So you could have anything from 60 to 80 test points per tire.

Great. And if I were a member of a Formula student team, what should I do in order to be able to access the data?

So in order to be able to access the data, it can be bought for a one-off fee. And the fee's around 250 to 500 US dollars, somewhere around there. And it's one-off fee, so you get all the previous tire testing data that's been done since 2000, early 2000s up to now.

And you also get access-- as long as you're on the list, you'll get access to future tire data. So--

Sounds like a

[INTERPOSING VOICES]

Yeah, it's in the tens of thousands of dollars that normal automotive companies pay for the stuff. So it's [INAUDIBLE].

No, great that volunteers really work on that, and great that Formula student teams can access the data, because it's a huge amount of resource I imagine.

Exactly, yeah.

OK, cool. Perfect. Good to know, good to know.

So just moving on, looking at the procedure, what I wanted to illustrate with the figure on the right at first was just the form in which this tire model or this tire data comes. So each of these tire data tests is done on the rig as I showed before. From the title, this is just one sweep, so this is a certain inclination angle, a certain vertical load, and a certain pressure. This is one of the 60 points. And essentially--

This is one of the test points you have been mentioning previously.

Exactly. So this is what a test point looks at in terms of force versus time. So what the machine does is it brings the tire down and it sweeps the tire through a certain slip angle range. Now, if you're not quite so familiar with slip angles, the slip angle of a tire, it's basically the angle of displacement between the tire's direction that it's heading in and the direction that it's pointed in.

And this is created by the rubber, the deformation of the rubber as it moves through the contact patch, that interface between the tread rubber and the ground. So essentially what's happening here-- it's not shown on the x-axis, but the slip angle is being varied. And you can see that the force varies with time. The slip angle's being varied with time and the force is being varied with time.

And essentially, if we want to fit the curve for certain data points, this is what the data looks like. So the slip is a partial-- this is a partial negative slip angle sweep, and then there's a full positive, and then there's a partial negative. So this is a form that-- this has to be distilled down to a non-dimensional model in. And I've used a non-dimensional model here because it's very convenient to compress a lot of test data down into a single relationship.

So As. You can see on the bottom left, what I've used is a magic formula. This is the overall formula to fit the force versus slip angle curve to the data. And you can see it's a bit complicated in terms of functions, but it only has two coefficients. And it's got another two parameters, which [INAUDIBLE].

OK, probably the only-- the main purpose of that formula is to fit the test data, right?

Exactly. So the B and the E coefficients don't necessarily represent anything-- they don't really represent anything in reality. They're just there to condition the curve to the best possible fit. The--

Quick question on that. Where does that formula come from, or has it evolved over time because it's, well, the best approach to fit your test data?

Exactly. So this is one of the earliest versions of the Pacejka magic formula that's been developed by Hans Pacejka, who's still at Delft. This formula was revised. I think this harks back to the late 1980s.

This has been revised with time, so every couple of years, a new version comes out. And these days, the full set of equations of all the Pacejka magic formula can include as many as 50 to 70 coefficients and 30 different equations. So the complexity has increased quite a lot.

But shows that this semi-empirical approach is still used.

It is still used, yes, exactly. It's still quite convenient. So also, what I wanted to illustrate with just a quick overview of the non-dimensional time model, parts of the graph that are significant, we're trying to compress a characteristic shape of the graph down to-- just to a single shape. So the parts that are significant are the friction coefficients, which is the non-dimensionalized force, essentially.

This has to be picked out-- essentially, it's just picking out the peak of the graph. And the non-dimensional force essentially is just making the force independent of the normal load. And we've also got the non-dimensional slip angle.

And this takes into account the gradient of the graph at the origin in terms of the cornering stiffness. And it also takes into account, once again, the vertical load. So the last point I've illustrated there, it's the cornering stiffness. And that's basically the gradient of the graph through the origin.

OK

I think it's time just to show quickly a demo, just to show what the output of all of this is. So I've prepared a script. I'll just bring those two up. I've got my window here as well.

So the script that I've prepared for this, this is a processing script. This takes in the raw data and separates it out. So essentially what I've done is I've taken in the files. They come in all different-- the TTC actually provides them in a lot of different forms. So you can get them in .csv, .mat extensions.

OK, basically it's a text file containing all the data.

Exactly. So I've just pulled out all the variables here. You can see there's quite a lot, so not only force, but there's also things like pressure and temperatures. So going down, what I've done is just use a whole lot of logic statements. And because there are a lot of test points, I've had to separate them all out.

So it becomes a bit tedious. But I find that if I use a good naming convention, you can use the the find and replace function to make it a lot simpler.

Exactly. And if it's done once properly, well, you can use it forever.

Exactly. And this can also be used-- once you've overcome this stage, this was mainly just, for me, it's trying to get results. If I want to hand it over to someone, essentially, the best way to treat this would be to transform it into some sort of graphical user interface. That's a lot easier to work with. And it doesn't rely on a lot of experience.

OK, so after all those blocks were separated out of the data, there are some functions that I've used to non-dimensionalize it, and then to fit the curves, and then to expand it out again. I'll just quickly show the results of all of this, so results of the fit. I'll run my data quickly.

OK.

My scripts. Comes up-- it takes about five seconds.

OK.

The process itself, the whole fitting process and gaining all the data generally takes around 15 seconds. For a full model, 15 to 20 seconds. It's just because the algorithm that's used to-- the nonlinear least squares method I've used to fit to the data, it has to iterate through a couple of guesses.

Exactly. It's curve fitting, it's iterative. But these 15 seconds don't scare me at all.

No, they're not too bad. And once you've got the data for the curves, the overall-- the package-- the overall formula that you're using, if you're going to use a vehicle model or a lab sim where you're iterating through the tire performance curves, the functions that you're using are really very simple. There's not a great deal of operations taking place.

OK, so what we see here is a typical performance curve of a tire. Could you guide us through the diagram? So what exactly are we seeing here?

Exactly. So what we're seeing here, this is the most common way in which a tire performance is represented. There are a lot of other parameters as well, but this is most common by which people understand it. So on the y-axis, we have the later force, just in this case. It could be anything from lateral force to longitudinal force as well, or combined, a combined force.

And on the x-axis, we have, once again, slip angle in degrees. So you can see here, the curves that I've-- or the point that I've done here is I've varied the normal force on the tire. And I've kept the inclination angle the same at zero degrees. And I've kept the pressure the same as well. So all I've varied here is inclination angle. And then fitted the black lines, or basically the formulation that's been fitted to the raw data.

OK, so let me quickly chime in. So what I see that that fit really nicely works. What I also see, that sometimes there is-- it seems that there are more tests made, or it's kind of a hysteresis behavior. And what I also see is some waviness, some noise in the test data, presumably. So could you comment on these points a bit?

Yep, definitely. So you'll see on the large curves, or on the curves with a large vertical load-- we'll just have a look at the light blue one. You can see this big hysteresis loop as it's passing through the origin.

OK, hysteresis.

And this is mainly due to the tire being swept in different directions. So slip angle is increasing in one direction or the other, and the tire behavior is very different under those conditions, because the rubber is being deformed differently. The waviness in the data, it can be down to a number of things.

It can be due to some sort of imbalance in the rim or the tire itself due to maybe some asymmetries or some out of balance forces acting on the rubber or the rims that haven't been balanced properly. But it can also be due to the fact that the tire, as it does go through the contact patch, or just before it reaches the contact patch, it tends to bulge in front of the tire, it tends to compress. And then it tends to-- as it leaves the contact patch, it tends to stretch or snap out.

And this basically creates a non-symmetrical tire shape. And it can create a bit of out of balance. But it's not too bad in terms of force magnitudes.

I would totally agree. But at the end, it all comes down to understanding what is happening even. So the fit is really nice. And you seem to know pretty well what is behind that experimental data.

Exactly.

That's good.

Yeah. Yeah, it's worthwhile reading up on how they do the tests. But the testing consortium does do a good lot of work in terms of making sure that the hysteresis and the waviness in the graphs are as little as possible. So does a good job that way.

OK, so going back to the-- I'll start it from here. So essentially what I have shown is the start point and the endpoint simultaneously of the modeling process. I just wanted to go, graphically just show what actually is happening. So we're starting here at the left, the figure right at the left with the lateral force versus slip angle curves.

And what we're doing-- what I've done with using those non-dimensional transforms is I've compressed the curves, all these curves, essentially into one characteristic shape. That's shown in the non-dimensional graph to its right.

What I'm asking myself here is you have one blue line. Also, it's quite noisy in the left diagram. And then you're making a point cloud out of it. So what actually is happening here?

So essentially what's happening here is this point cloud, or this waviness, also shows up as a variation in normal load. So what you essentially are doing when you're non-dimensionalizing the data is you're dividing through by the normal load. So it doesn't come up as such a coherent spray here, because it's independent of the normal load, essentially.

OK.

So yeah, it doesn't come up as that wavy pattern. So I think that's the main reason for it. And the graphs are also shifted to the origin. so it might not be evident in the picture on the left, but if you've got inclination angle variations or high inclination angles, these graphs can actually shift away from the origin when you-- it has to be simplified as much as possible, because the relationship that we're using, the magic formula, doesn't assume any shifts in the curve shape, to keep it as simple as possible.

So this procedure can also be performed for the raw data for a longitudinal force versus slip ratio. So this little capper here is the non-dimensionalized slip ratio. And what I've shown in this step is that all the relations-- I'll first go through the relations for the non-dimensional slip ratio, and dimensionless force are very much the same as for the slip angle and y force. So these can also be seen in the Milliken literature, or other literature out there as well.

But what I've done is I've fitted curves to each of these non-dimensionalized test points. You can see in the different colors. To make it even simpler, you can just fit one curve to the data. That would make it a little bit less accurate, but once again, a lot simpler.

Or you could just fit at not too-- it's not much of-- it's not a lot more effort, but you can fit a curve to each one of these raw data sets. And I chose it because you can actually see in the data that there is a bit of variation from one set to the other. So on the right is essentially-- and what I've shown in the demo is essentially once you gather the outputs, if you wrap it-- and it agrees quite well with the model behavior, and with the tire behavior at least.

What I'm asking myself here is the test data seems to be available for a quite big range of the slip angle, so minus 12.5 up to 12.5. What is the range of slip angle actually happening during, let's say, an endurance race?

So we've done-- that was exactly the same question I was asking myself at this stage in the modeling process. So we've actually done a bit of testing on one of our older cars. And we instrumented it with an optical slip angle sensor and were able to look at what sort of slip angles we were seeing in the different corners, or the different corner radii and slalom that we were going through in competition.

And we found that, or I found that the slip angles were well within the plus or minus 12.5 degrees for a car that is not-- yeah, that is not sliding around for a perfect driver. We found that the slip angles were below at least-- at the most 8 degrees to the side. So it was well within the range that's being tested.

OK.

For the slip ratio, however, it's-- yeah, I mean what's encountered out on the track, if you lock a wheel or if you spin a wheel, the slip ratio is greater than what they test at the TTC. And they test only up to plus or minus 0.3. If it's locked or so, it can be 1 or 2.

And essentially what we're doing when we are simulating the car, or we're doing a lab sim, is we're assuming that the drive is perfect. So this situation shouldn't, or won't, come up as often. We are extrapolating to that extent in the data, then it's worthwhile doing your own tests or so.

OK, good.

As I said, yeah, just a brief summary. So I mean, what I've used to fit this magic formula is essentially a built-in MATLAB function on the nonlinear least squares fit that I could understand, at least what the theory behind it was. There are different algorithms as part of the function as well that you can use as a convergence criteria. But it worked quite well. And essentially what I've done in the last figure is just inverse to what I did for the first two. So--

[INTERPOSING VOICES]

I would say that the quality of the fit is really good.

Yeah.

Mm-hmm, makes sense.

It's quite good.

The only thing that I would suggest, just from the fit, is that if you wanted to find, say, the peak slip angle-- say, the peak slip angle for the peak force for each one of these vertical loads, it would be better, looking at the model fits, to rather take it off the raw data than the model fit, because the peak in the model doesn't necessarily correspond to the peak in the raw data. So yeah, just have to keep in mind what sort of application you're looking at. But it's fairly simple to get the peak from the raw data anyway.

OK, that's interesting. Good.

Moving on from that, for each one of those non-dimensional parameters that I mentioned before, that accounts-- they're now to six. So there are two coefficients. There's a peak. There's one parameter that represents the peak in the graph. There's another that represents the gradients in the graph through the origin. And then there are two shifting parameters.

So for each of these-- and the graph on the right here illustrates this. For each of these tests, we obtained-- or my script has gone and grabbed all these parameters and has represented it as the points in the raw data, or as a point there. Because when we're trying to create a model, or when I'm trying to create this model for a vehicle model, or later for a lab time simulation, it's important to be able to estimate continuously between test points.

So if I want, say, a force value between an inclination-- or at an angle, inclination angle, of about 1 and 1/2 degrees that's not tested, I need to fit some sort of function. So I've done this here in the form of a surface for discrete pressures. And the surface encompasses inclination angle and vertical load as the independent parameters, or independent variables.

This can be fitted for discrete pressures. And essentially what we've done here, or what I've done here is condense it down to six coefficients. And the only thing that needs to be saved from this data or from this fit is the coefficients of the two-- or the quadratic surface that I've fitted to this. So you can fit any sort of surface, whatever you see fit.

But I've gone simple and just got a quadratic surface. And it's only about five coefficients that need to be fitted for this, so that can then be stored. Those can then be stored in any form you want. MATLAB is quite flexible in that regard. So I've stored it as a .mat extension, but it's up to you.

Yeah, text files, whatever. Good.

So just moving on, so with any sort of testing, with any technique, you have-- all the tire testing techniques have their inherent weaknesses. And they also have their strengths. So constraint testing has the strength that it's very easy. It can very easily keep one variable constant, like inclination angle. Out on track, if you were to try and get inclination angle, it's varying all the time.

And the measurements of the data is a lot more accurate for constraint testing. And it's a lot easier in terms of effort. Some of the weaknesses, or some of the disadvantages, or the things to watch out for-- actually, they're not really disadvantages, but you just have to be aware of them-- is that the friction coefficient-- one of them is that the friction coefficient that's tested is quite high.

Where does that come from? Why is it not a realistic value?

It's quite a-- well, I'll answer. It's mainly because the belt itself is a different surface to that that you normally encounter out on track. So what you encounter out on track is normally a parking lot sometimes, a very dusty, oily surface that sometimes has a bit of water in it as well it's been raining, or it hasn't got any rubber on it.

OK, so what you're really experiencing is by far not an ideal surface. And they rather test on an ideal surface.

Yeah, I mean, there's a whole range of surfaces that you can come up against. And the TTC, it's unreasonable to expect that they test for all of those surfaces.

Yeah, [INAUDIBLE], right.

But the surface here is more like a sort of safety walk. So what's essentially happening as well, as I showed in the picture earlier on of the tower above the rig, or the drum and belt setup, is that the belts continuously-- it's continuously circulating. So the tire itself is putting down rubber.

Ah, OK.

And this rubber that's left on the belt, it makes for super grippy conditions. And the belt itself--

Got it.

--it's not as much of a-- I mean, it warms up as well a lot more than, say, the road might over a short amount of time. So [INAUDIBLE].

And unfortunately, the area of the belt is a lot smaller than the area of a racetrack. So to put as much rubber down would be quite an effort.

That's right.

OK, got it.

The model itself, as well, it's a steady state model, so it doesn't take into account transient effect. So it only really takes into account the variables themselves, not their first order derivatives in reality, something that's very important if you're going through a transit maneuver, such as a slalom, or you're continuously various steer input to the vehicle, or you're turning in, maybe to a corner. The thing is, what becomes more very dominant is the rate at which you transfer load from one side of the vehicle to another, the rate at which load increases on the tire, the rate at which pressure changes sometimes. And also, the rate at which temperature builds up, either internally, but for, say, it's mostly on the surface that the tires get a lot of temperature from just grabbing on the surface. So--

OK, what I'm just asking myself-- I'm sorry to interrupt-- this steady state, if I would imagine to go and have a look at dynamic behavior, I would have to change so many things. I would have to adapt the testing procedure. the procedure itself should be somehow dynamic. The whole evaluation would be more effort. So could you comment a bit on what the impact, or what you could actually gain from dynamic testing, and what the efforts would be?

Yeah. So if you were trying to capture transit performance of the tire, it would mostly be worth your while-- it would probably be worth a while, actually, to just go straight to the theoretical model or a physical representation. And that's what one of the students, the PhD students, a few years ago did. You have to go back to first principles and actually extract the material quantities.

You have to be able to extract surface parameters as well. And his simulation was actually able to predict tire performance as temperature changed on turning, and as a function of [INAUDIBLE]. The Pacejka formulation that I've shown doesn't have provision for that. A lot of the semi-empirical formulae does, but it just-- if you have to include the 50 or so variables out there, and then you include their first order derivatives, the model just becomes-- it becomes huge. You really need to go back to first principles.

But yeah, I mean leading on from that, the steady state is, for an initial model, for SAE purposes, an easy model. It's the way I think it's where you normally start.

Yeah, totally agree on that. Totally agree on that. Starting with that, there's no other way of considering it, so you say.

Exactly. So another thing that you have to keep an eye on in the data is that on the tire constraint-- or on the constrained testing rig, the tire deforms a lot. So this is shown here on the graph on the below right. So on the x-axis, we have vertical load. On the y-axis, we have friction coefficient. Think of that as your friction, or your lateral force.

So looking at each of these sort of stripes, the stripes-- we'll look at the stripe right on the left for the lower vertical force, this shows very little variation from the test. So each one of these stripes represents a test normal force. And little variation from this line is what we're looking for.

But as you get to higher normal forces, the machine actually-- or the tire itself deforms a lot. All tires do, but SAE tires, when you take into account the machine tests-- the test head is about 12,000 pounds, and it normally tests truck tires. Formula SAE tires are very soft and they do deform a lot under high loads and when you're sweeping them around.

So the machine tries to compensate by changing height, but it does-- I mean, it does so by changing vertical load on the tire. And so--

OK, but the question that I'm asking myself is, what is the range of realistic normal forces during an endurance race? Are we really touching that 1,600 newtons, or is it more to the left of the diagram what is actually happening?

That's a good-- that's a very good question. With error, it can be more than that.

OK.

So I would say for a wingless car, 1,600 newtons would be a lot of weight transfer onto the front. 99% of the time, you'd be less than that, unless you did some sort of special maneuver where you're running on one wheel. But for error, it could actually reach-- error, to be honest, it could actually reach the point where it's exceeding these values a little bit.

OK.

I don't think it would exceed the value by an amount that's a lot more than the jumps or increments that they've tested here.

That's a very interesting information. And well, it's just saying that the tire deforms and the model is not capturing that per se, but this is not necessarily a bad thing.

No. No, not necessarily. I mean, it's not too bad. The TTC's done a lot of good work in trying to minimize it. So only really, it's only a variation of 100 newtons either side of the test data. So it's-- or above the desired point. So it's not really that bad.

OK, got it. [INAUDIBLE]

Another point that's very important is that, of course, one of the most rogue variables is surface temperature, road surface temperature, tire temperature. So unfortunately, these are very difficult to control or to keep constant during a test. So what I've shown here is the variation of surface temperature for all the different lateral loads or different friction coefficients that I've encountered during a test.

And you can see that there's a lot of variation. So as the tire is slipped along the belt, the temperature rises and falls quite abruptly. And yeah, the tire actually reaches an operating window. So essentially what's happening on the-- what we hope for what's happening, on the lower-- on the left side of the graph, is that the tire's being warmed up.

The TTC does a procedure for that. And more towards the right, where the peak is and beyond that, is where it's going through the proper slip angle

[INTERPOSING VOICES].

But the main point from all of this is just to-- and then the last point as well is just to-- the model is only really going to be as good as the on-track data that you have. So you need sort of validation or some sort of sanity check. You need to go out and test, really, to confirm that the data you're using is good, or does fall-- your tire or your car does fall within the range that's being tested.

No, my impression on that is actually very good. So you seem to be pretty aware of what your model can do, and where are the constraints. And as long as you know about the impact of these constraints, you are on the safe side. Then you have the right expectations.

No, actually, I'm quite impressed on that. So great, thanks.

So yeah, just moving on from that, I mean, like the slide says, it's not only the pitfalls of constraint testing, but it's any form of time modeling there are going to be errors, even pure theoretical models.

Yeah.

So just quickly before I move to the takeaway, I'd just like to acknowledge the role that Calspan, the company that does the testing for the Formula SAE cars. And that testing forms under volunteer work that is done by a couple of the guys, a couple of the engineers at Calspan. So I think some of them have been on SAE teams before, and they volunteer their time to test the tires. So thanks to them.

Yeah, they seem to be a really good job, a reasonable amount of money for Formula student teams, but a huge amount of data. That's cool.

Exactly. You get a-- it's a treasure trove of data for the-- it's a little bit of pain. I mean, the money is a one-off fee, for smaller teams can be a bit prohibitive. But it's a little bit of pain for a lot of gain in the long term, and especially if you can make the model work and you can extract the data from it. [INAUDIBLE]

It's a very nice approach, and a nice form of sponsorship. That's good.

Yeah.

Mm-hmm.

And the key takeaways from this presentation, so what I just wanted to say at the end of it is you goals determine which model you choose. And you can-- it's a notion of choice there. You can choose anything. There's a wide variety of models.

And what essentially constrains, or what essentially narrows you down to a certain model is what your constraints are. So in SAE, we don't have a lot of time. We don't have a lot of time on the team.

We also, sometimes we don't have the money to do one sort of testing. And we also need to keep the tests as simple for the model, and the information and the knowledge as simple as possible so that it can be passed on from one generation to the next. That's one of the main weaknesses in the SAE project. And where the good teams really excel is where they carry over the knowledge and work from one year to--

[INAUDIBLE].

So how we use the data. Well, in a nutshell, we've shown how we can take constrained data from tire testing and fit a very simple curve to it. And the curve can predict what a change in vertical load is going to have on the force slip performance of the tire, and what pressure, and what inclination angle can also have on the tire. So this essentially allows us to see what actually makes different tires happy.

And the last point is that just the advantage of using MATLAB to do this. The modular coding structure means that if you get one set of code right, it can easily be copied over and changed, small changes made. It just makes working with large arrays that we have from this data, it just makes it a lot easier. And yeah, it simplifies it.

And once you get the code going, it's just very, very easy to run. You can also extend this via the range of apps that-- or the range-- or the functionality that MATLAB has these days, so a graphical user interface is very handy. And this can be a very easy way of getting information over to, say, your boss, or getting the new guys on the team to maybe even just play with the data, and to get an understanding without having to go through all the hours that you did trying to get your script working or--

Totally agree on that. And what you're mentioning is just the basic form of working on with your scripts, generate reports from that, use the stuff to plot data, working in the grid. But also have a look at the bigger picture. You could use all that data for Simulink models where you use physical bodies, multi-body systems.

I think you can go far, way up in the hierarchy, for more complex modeling tasks. And this is, I think, where really the strength of MATLAB and Simulink show up.

Exactly. It's very flexible in terms of what forms it can output the data in as well. So whatever sort of a program you might use. You might use one that you've developed in-house or one that you've got through a sponsorship. The output, or the output formats that MATLAB has covers pretty much all of them out there. So.

Yeah, fine. So impressive, Marc. Really impressive. Totally agree with the key takeaways. Thank you very much for giving us such a nice insight to your work.

You're welcome.

Thanks a lot really. So really cool. So that brings us to the end of today's session. As usual, I will point you to our resources. So first of all, the Matlab and Simulink Racing Lounge, the Racing Lounge webpage where you'll find all the videos in the context of the Racing Lounge.

I also would like to point you to our Formula Student webpage, where you will find all information, including the software offer. Of course, and we are really appreciating that, send us your feedback to formulastudent@mathworks.com. We are constantly working on the Racing Lounge. If you have an interesting topic to contribute, or if you have some feedback, just let us know.

And last but not least, if you use our software, or if you use our support, we really would appreciate if you put our logo on your car and your reports. Thanks for watching. And thanks again, Marc. Hope to see you next time.

You're welcome.

Bye-bye.

Bye. See you later.