# All models are wrong, but....

I swear to the FSAE gods, if one more person tells me that all models are wrong, but some are useful, I'll drive E95 into the Duke Pond. That said, today I want to talk about lap simulation!

### What are lap simulators?

Lap time simulators are computational tools that attempt to model the vehicle completing dynamic events. Like any model, fidelity of the simulation comes at the cost of complexity and computational time. Generally, the most core parameters of the vehicle like mass, some function of tire grip, and a power curve are required to characterize the car. This information would be sufficient to generate a point mass model of the vehicle. A point mass model, however, neglects load transfer, kinematics, and the tire sensitivity that accompanies these effects. A model that captures these parameters is great, but what about modeling unsprung mass? Or rotational inertia of wheels and drivetrain? Aero yaw moments? Tire degradation? What about modeling the driver who's strapped to this death-trap? The math can get pretty scary pretty quickly.

To model the tracks, the circuit is broken down into a series of straights and arcs. From there, the vehicle is simulated along each segment of the track based on its longitudinal acceleration capacity (for straights) and lateral capacity (for corners). But what about optimizing a racing line between gates? Or simulating a rubbered in track? Or the bump on the exit of T18? Isn't that important? Yes. But the math can get pretty scary quickly. Like any simulation tool, the engineering lies in deciding where to draw the line.

### Drawing the line

The motivation behind using a lap sim for this season stems from a need to quantify the impacts of design changes this summer. Because we have a team goal of 625 points at competition, the unit of quantification is points. Therefore, we want to be able to talk about design impacts in the language of competition points whenever possible.

You'll notice that a motivation for lap simulation was not lap time prediction. This is crucial. While a certain lap time correlates to a certain point value at a given competition, we aren't too concerned about what the stopwatch says. This is because the autocross and endurance tracks change every year, so simulating based on previous tracks isn't much help. That's okay, because if we simulate old tracks and measure the relative changes in points as we make parameter changes, we can extrapolate these changes to expected points and apply that to our 625 points goal. Of course, points are relative to your competition and the competition changes each year as well, but we work with what we have. We need to draw the line somewhere, and relative points changes (or parameter sensitivities, a phrase I will abuse here on out) does the trick.

So with this intended use case of the sim established, we can target a desired fidelity. We don't need something to accurately predict lap times to the hundredth, but we do need something powerful enough to such that we can put faith in parameter sweeps and compare sensitivities. Enter, the Clemson Lap Sim.

### Thank You, Clemson

A quick google search of "lap simulation fsae formula sae formula student fsg fsa fs east" yielded a MATLAB lap sim put together by Clemson's chief engineer in 2019. I'd like to extend my utmost gratitude towards Clemson for putting in all the leg work of this simulation tool and sharing it. The simulation is a quasi steady state approximation. This means that the math doesn't capture transient conditions of the lap during corner entry and exit. These stages of the lap of course are extremely critical to vehicle performance, but are also extremely complex to model. Instead, this steady state discretization of the track compromises on the transient effects, many of which are controlled by VD parameters, but provides respectable representation of powertrain, aero, and overall vehicle architecture (wheelbase, track width, weight) parameters. The documentation for the simulation is very thorough. Again, thank you, Clemson.

### GGV Diagram: The heart of the lap sim

The core concept of this steady state lap sim is the GGV diagram. A GGV diagram is a graph that defines the performance envelope of a vehicle via a surface constrained by lateral g, longitudinal g, and velocity. Much like a friction ellipse of a tire, a GG diagram illustrates the lateral and longitudinal capabilities of the vehicle. For example, if the car is purely braking, all of the available grip will be used longitudinally. Say, 1g. Conversely, in a corner, the vehicle may use all of the available grip to corner at 1g. There also exists the possibility of cornering and braking simultaneously, say at 0.7g longitudinal and 0.7g laterally. All the combinations of lateral and longitudinal accelerations form a closed curve that is the GG diagram. However, the vehicle's capability to accelerate depends on speed. 1g of forward acceleration may be possible from a launch at low speed, but as the vehicle approaches top speed, acceleration decreases. Likewise, as speed increases, aerodynamic downforces increases the vehicles capability to accelerate. This is where the "V" part of the GGV diagram comes in. At any given speed, a GG diagram is calculated and these level sets are stacked along the velocity axis to form the GGV surface. Here's why the GGV is so key. If we can define the vehicle's maximum capacity to accelerate at any given speed, we have the information needed to simulate a lap around a given path (neglecting transient behavior and all other phenomena covered above).

So, putting this all together, the GGV diagram defines the performance envelope of the vehicle. And the GGV surface holds all the necessary information to simulate a lap. To increase performance, the surface needs to extend to higher G's or V's.

### Let's get simulating

Alright, we know what the sim is and kind of how it works. Let's get cracking. To perform the sweeps, the vehicle parameter inputs were based on the 2024 car. Then, one parameter at a time is varied in increments I deemed reasonable. The times in each of the 4 dynamic events are recorded. Points for Autocross and Endurance are based on the 2019 field, since the sim runs the 2019 track. Points for Acceleration and Skidpad are based on the 2024 field, since these tracks do not vary year to year and the 2024 field probably is most representative of 2025. Here's what the data collection looks like for aero parameters as an example.

The sweeps are centered around the 2024 spec of the car. We observe that for these parameters, the effects on lap time are linear across the whole range of simulation. This is extremely important for how we intend to use the lap sim. If the effects are linear, we can calculate a single time or points sensitivity for each parameter and treat their effects as independent and additive to each other. This is not strictly accurate! For example, let's say we have just two parameters of interest: power and drag. If we increase power, our sensitivity for drag is no longer valid. More power means less power limited sections of the track, which means drag sensitivity decreases in magnitude. But in general, these effects are small. And ignoring them really makes this whole process very seamless. We're building a big boy go kart, not trying to land on the moon.

In total, 10 parameters where swept. The sensitivities are are compiled below. No VD kinematic attributes like caster, camber, KPI, etc were swept because the effects of these parameters are too complex to be analyzed in isolation or with a steady state simulation.

It's tempting to sort these values by total point sensitivities by value to evaluate impact. But note that none of these have the same unit. We can't compare points per inch of track width with points per percent of center of pressure. Instead, I picked units which were relatively intuitive. For example, thinking about shift time in terms of tenths of seconds is more manageable than a full second, as we would never shave a full second off our existing shift time of 0.2 s. So what's the takeaway from all these numbers? Power. Power is the takeaway. Drivetrain efficiency is just a scaling factor on the power curve, and 2.6 competition points for each percent of scale on the existing power curve is monumental. Notable runner's up include weight at a respectable -0.5 points per pound as well as the aero parameters. Track width also came through as a bit of a dark horse, with decreasing track width providing significant maneuverability benefits. But it's easy to go in the weeds and compare every parameter on every event. This analysis is coarse. And a coarse evaluation yields power as a parameter of focus for this year.

### Does any of this matter?

In addition to comparing the parameter sensitivities relative to each other, itâ€™s valuable to step back and consider how sensitive is sensitive enough to be worth analysis at all. In other words, Parameter A is not worthy of months of redesign if a modest improvement in Parameter A is worth less than a point, even if Parameter A is 5 times more valuable than Parameter B. As an example, track width at -4.5 looks awfully tasty, but track width isn't really a parameter that can be varied by much more than an inch without extreme architecture changes. In contrast, for some teams, that -0.5 per pound might have much more room for growth than -4.5 on track width (like if they've been rocking Keizer wheels for example). To determine how many points is a worthwhile amount of points, we can look at the points deltas between teams at the 2024 Michigan competition. The following histogram bins the Top 50 teams by how many points they finished behind the team in front of them, excluding outliers. While our season goal is in the units of points, not finishing place, this histogram gives a temperature check of how big of a points improvement is needed to shuffle the leaderboard. Looking at both points and finishing place helps define how useful these parameters will be. Again, the bottom line is that we want to use this lap sim tool to discover whatâ€™s the most bang for our buck. And to find that, we need to know what is a big enough bang.

Looking at the histogram, the average points gap among the Top 50 was just under 8 points. However, over half of the deltas were within 5 points. The takeaway here is that for a change to really matter, we certainly need something on the scale of points, and not tenths of points. Perhaps a more useful metric is the delta between every 5 finishers. Placing 15thÂ over 16thÂ is a data point. Placing 15thÂ over 20thÂ is a story.

Unsurprisingly, the average 5 place delta is 40 (5 times the 1 place delta of 8). But more important to note is that almost no deltas dip below 10 points. Letâ€™s synthesize from these histograms. We learned that:

1. We should focus on parameters that make points.

2. We should focus on a sum of parameters that make tens of points.

### Let's hunt for some points

We're getting close. We have all the sensitivities of interest and know how important they are to the leaderboard. Let's tie it all back to those points goals. As mentioned above, we can take a page out of the linear algebra textbook and add these parameters together assuming independence. Again, not true, but convenient. Traffic cones and parking lots, not moon landing. Now we have a quick calculator to display the effects of any design changes without having to re-run the lap simulator, which takes up to 10 minutes. Here's where the 2024 spec car lands.

And here's what a few bold design changes could do.

This very simple calculator accomplishes our goal. We can quantitatively see which parameters matter and how they effect our event goals.

### Is this helpful?

Absolutely. Let's take an example. This summer VD considered changing track width. This decision would be rooted in a host of kinematic and architectural effects. Additionally, changing track width changes the aero boxes. Before VD can make a decision, it's reasonable to say aero should see how a potential change could affect their performance. But full car aero sims are computationally expensive. An aero sweep across a few track widths and maybe a few design alterations would take up to a week. Not to mention the opportunity cost of chewing up that many computers and not simulating other things of aero interest, like maybe new wing geometry. So let's put this coarse analysis to use. If track width increased by an inch, aero would gain an inch worth of planform area. This scales their CL and CD by about 2%, assuming they use the extra 2% of width as efficiently as the other 98%, a moderately safe assumption. The total effect of that change would be 0.3 points. 0.3 points! Compared to 4.5 points from track width. Even if aero really nails that extra 2% they still have an ocean of points between them. So VD doesn't really need to worry about what aero thinks about their track width decision, and aero doesn't need to sweep these changes.. They'll deal with the +/- 0.3 points just fine. This is where this lap sim really shines. Quantifying what matters, and just as importantly, quantifying what doesn't.

### Going forward

The next steps are to set performance goals at the subsystem level to close the 6 point gap the sim says we have to our 625 goal. Fortunately, the good folks in powertrain are confident that we will be pumping out a fair bit more power than the sim expects despite the decision to go NA. This would more than erase the 6 point deficit and give other subsystems some breathing room. As mentioned above, weight and aero are next in line to scavenge performance. Weight will also largely be affected by the NA decision, but also some substantial chassis changes. Stay tuned til next blog where we will put numbers to these effects.

Finally, there are two factors the sim doesn't account for at the moment that it really should. DRS and threshold braking. While DRS will likely make a return for the 2025 challenger, the number of actuated elements is TBD. Additionally, the lap sim currently assumes the vehicle can brake at a grip-limited rate based on the GGV. In other words, the vehicle is capable of locking up at any speed. This isn't true either, so some brake math is in order to correct for this in the sim.

We've made a good start and have lots of work ahead.

Signing off,

Cody

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