A question about rotation for the vehicle dynamics, engineering, or similar, folks

Forty years ago +/- I came across Piero Taruffi’s book, The Technique of Motor Racing. At one point he said (paraphrasing… it was 40 years ago after all “The car must rotate around it’s center of mass (COM) to navigate a corner.” Perhaps this should have been an obvious statement to me, but I had never thought about it. I drove (with considerable success) just by feel, by doing what needed doing to be fast, without really understanding what I was doing, or why it worked.

Anyway that statement was an epiphany for me; I suddenly understood the truth in that statement, that rotation management was one of, if not the, key to being fast, and I realized the myriad of techniques I had been using to control or influence rotation.

Fast forward 40 years (and several sessions of shower philosophising :thinking:), and I’m confused about something. In a normal circumstance (e.g. If I drive a car/kart toward the apex of a turn, load transfer increases, tire load increases, traction increases, yaw rotational force increases, and slip angle increases until there is no longer sufficient speed/energy to maintain the load/traction/slip angle on the outside front tire) at this point, the major portion of the ‘rotation’ occurs, right? You stop entering the turn, the car/kart rotates (changes its orientation on ‘the line’), and you begin the exit phase of the turn.

HOWEVER, does the car/kart actually rotate around it’s center of mass, or does it rotate around the front end, or the outside-front tire (the contact patch that is providing the most cornering force)? Empirically, it feels like the rotation is relative to the outside front tire… like (in a car/kart with front brakes) you can ‘pin’ the nose and fling the back end around the outside front tire.

In a properly executed turn, it does NOT feel like the car/kart rotates around the COM; like it just orients itself as tome angle relative to ‘the line’. That is, I don’t sense the front end rotating towards the inside of the line being driven as the back end rotates towards the outside of the line being driver. Instead, it feels like the front end stays relatively put (ignoring the small change in drift as the slip angle decreases on the front one once it has maxed out) and both the COM and rear contact patches rotate towards the outside of ‘the line’. Of course that would mean that the COM is changing location laterally due to the rotation, so ‘the line’ is also changing slightly.

THAT SAID, there are times when it feels like the car/kart is rotating around it’s COM, like, if you lock the brakes at turn-in, or are a Swedish Flick type of technique.

Anyway, sorry to go off into the weeds on this, but I apparently have too much time to think. :grin:


Jerry Seinfeld Reaction GIF


I don’t think it’s as simple as one or the other. While I don’t have a semblance of a definitive answer, I’ll still talk out loud on some of the key concepts I see here:

Sounds obvious, but the kart has FOUR tires. All four of those tires are touching the ground (i.e. “loaded”) to SOME degree for 90+% of the corner.

Any shifts in load are initiated mechanically via the steering wheel and end geometry. That said, the INSIDE front is the wheel doing the majority of the work in this context.

Add in a grip coefficient (not sure if this is technically the right term) and the remaining three tires are now doing some of the cornering work via lateral g-forces. To me, this points us a but towards COM being the most influential factor.

However, we must then consider slip angle, and I could see a number of ways to look at that. Will vary across tire compounds, and also vary across front and rear tires (assuming different sizes for now).

Look at any footage of a kart cornering in slow-mo, and you’ll notice that the point at which the inside rear physically lifts off the ground (IF it does at all) is let’s estimate 30-40% through the corner. So I think different things are happening in different phases of the corner. The point of max lift could be more COM driven, and the center-off portion of the corner could be more front-driven as you suggest.

Just my initial thoughts, so I’m curious if that adds any perspective whatsoever :sweat_smile:


Also, you’re 100% correct that “rotation management” is what sets apart the GREAT drivers from the good drivers.


The instant center is the term for the point it is rotating about and is a function of both the center of mass as well as the sum of all applied forces and moments. In fact, the IC is usually not even on the kart. Think of driving around a skid pad and holding steady yaw. The center of rotation is the center of the skid pad itself because you are rotating around it. Now, if the kart is yawing, the instant center will shift, but it won’t ever be the center of mass unless you are on a zero-turn lawn mower.

Maybe Pierro was referring to the kart’s local rotation? That is, if you look at the difference between the path arc and the total rotation of the kart, you get a local component. I think that component is about the CG.


I always thought that I was at the pointy end of the analytical spectrum, but reading this thread I can only think…

(j/k - interesting discussion, but yeah, still pretty nerdy, lol)

Neat! Well there you have it, Warren. Problem solved.

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I had an interesting experience in downforce car in sim wherein the car was driven aggressively into a corner in a V manner. At the bottom of the turn, the pointy end of the V, I experienced the distinct sensation of the car rotating almost in place, as if a stake had been driven through the car, holding the vehicle in place while the car rotated in a circular fashion around that stake point. It felt insane in VR, like the car loaded up on the front and then as the weight shifted it slowly, deliberately rotated on that COM point. Weight was not on front at that point, hard braking had occurred and then as brakes released and weight went neutral the slow rotation happened.

I dont feel like the kart rotates around the ffront tire that’s pushed down, exactly. I feel like the rotation is around the center of mass. At least intuitively that’s what I feel. I feel like the front tire that’s pressed in is a pivot enabler but not the point around which the rotation occurs.

Maybe there’s two rotations of sorts that can occur simultaneously or not. What about a scenario where you have planted front that is dug in and the kart does a rotation that appears to be around that tire dug in but then let’s say you have a release and go into flat slide. The kart then spins round its com and the tire is not involved.

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Hey guys, thanks very much for taking the time to edumacate me!

Agreed on all of this!
However, I have kind of a twisted way of relating to this stuff. I tend to not think in terms of just ‘line’; I think of a rectangle (described by the four contact patches and positioned on the COM) oriented at some angle relative to ‘the line’… so more of a trajectory than just a line. And that trajectory is typically pointing to the outside of the corner/line during the entry phase of the turn (i.e. understeering orientation), and then at the point where main rotation occurs, the trajectory transitions to pointing to the inside of the corner/line during the exit during the exit phase of the corner (e.g. oversteering orientation). That said, this ‘trajectory shift’, is very subtle stuff because if done well, it is only 6+/- degrees either way (depending on the optimal slip angle for the tire being used).

I know all that probably sounds overly complicated, but that’s more of a background thing. In practice, I just think “how is my trajectory in” (am I using my front tires to the limit? overloading them?, etc.), and then “how do I manage this rotation phase so that I have optimal orientation for the exit phase”. If I get the first two right, the exit phase doesn’t require much attention. (For example, in a shifter, it would seem important to avoid over rotation, so that the acceleration forces can be driven, as directly as possible, through the kart’s COM, instead of ‘spilling’ out into the outside rear tire, overloading it and causing unnecessary corner exit oversteer.)

I’m more of a car driver than a kart driver, (and I recognize that the inside front tire is very important in karts), but my method for simplifying all of the above is that most of the time I only really pay attention to the outside front tire on the way in, do my rotation management magic :grin: and then pay attention to the outside rear on the way out if needed. I feel like by doing this one tire at a time approach I can get a deeper connection (a higher level of sensitivity) to the tire loading and the resulting traction and slip angles.

Thanks Caleb, this makes sense. :+1:

I wonder if the local rotation component could change (or appear to change) if extreme forces are being applied? So, maybe by default the local rotation component acts on the CG, but extreme forces cause it to act on the ‘source’ of the force (or maybe the force shifts the CG towards the source of the force making it appear/feel like the rotation is around the contact patch?).

To add contest and a visual to what I’m trying to say
Laguna Seca 1981
This is the last turn at Laguna Seca. This was back before they neutered the track, so this turn was tighter than it is now. My way of dealing with this turn was to brake VERY late and hard, and then VERY aggressively but briefly trail brake the turn-in to pitch the car into the corner. When the car rotated, it felt like that was happening around the outside front tire, and it felt like because of that the ‘center’ of the car rotated ‘outside’ of my my ‘line’ - blue line in picture-, so an extreme oversteer orientation, but relative to the front end instead of the ‘center’ of the car). Once rotation stared, I dialed in correction (to ‘jump’ the slide and meet it at the desired trajectory, and then when the car arrived, I went full throttle (at about where the green X is) to transfer load to the rear tires, check the rotation, and drive the car out of the turn. It feels like if the car had been rotating around it’s GG, and the CG was where you would think it might be, I would have driven over the inside curb. But then again, what I think I was feeling might have been a sensory illusion, or just a misinterpretation of the sensory experience.

I found a brief example of this type of corner-entry technique in motion in the first few seconds of this video. The more I look at it, the more I’m wondering if the car is rotating around it’s CG, but certain driving techniques artificially force the CG forward momentarily.


I feel like we’re being sucked deeper and deeper down the rabbit hole, lol. Love your thoughts on trajectory. Responding here with some more raw thoughts after just having read your response. In a kart, I would personally think of the “line” instead as three separate components, all of which are going to vary a bit based on tire and class.

  1. Trajectory- my own take here, so slightly different from your definition; this is essentially angle of approach, i.e. angle of the path(s) from entry to apex, and alex to exit, just to keep it simple. Example would be the stereotypical arching “206 line” vs the v-shaped “shifter line”.

  2. Position- pretty simple, imagine the above trajectory overlayed with the track map. Now adjust the overlay to suit the nuances of each particular corner. Could vary based on patches, curbs, etc.

  3. Angle- similar to what you’re calling trajectory, this is the yaw angle of the kart relative to the centerline at a given point in a corner. A good example corner to help visualize the different techniques required for different classes is the T1 at NCMP (end of long straight). Much more angle is required much sooner in something like a KA100 vs a shifter.

Also really dig the small rectangles visualization. The tricky bit with that in karts vs cars is small movements are happening much faster in a kart. Not sure I can take that much further at this time, but essentially those front two rectangles are moving around quite a bit, even if just micro adjustments.

This would be good convo over a beer or five at the end of a test day :sunglasses:

You can transform any movement of a body (like a kart) in a plane (on a track) into translation and rotation and their derivatives.

A useful transformation to think about is to consider a kart going around a steady curve as being a pure rotation about an “instant center” - the distance from center of gravity to instant center would be the “r” in a=v^2/r.

Another one is to consider understeering attitudes as having an instant center in front of a line perpendicular to the fore-aft axis of the kart through the cg and oversteering as having an instant center behind it…

Yeah, that’s why I like the skidpad example. The IC is pretty obvious. I think this is a great example of where us engineers have to bite our tongues and allow some leeway in the technical terms for the sake of common understanding.

Another common mistake is the concept of “conserving momentum”. Momentum is directional, so you lose 100% of your forward momentum when turning 90° and gain 100% lateral momentum. Mathematically, this is very different. You are really trying to conserve kinetic energy, but that doesn’t roll of the tongue as well, haha. I know most people will get more confused if I use the correct terms, so I concede to using the common terms instead. Kinda the same has to be done with IC. Yes, you are rotating about a point outside the kart (and I like your definition of under/over-steer), but the driver feels the local reference frame and is really referring to that.

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Some thoughts

Another way to look at under/over steer: Understeer occurs when the front tires slip angle is greater than the rear and oversteer is when the rear tire slip angle is greater than the front tires. (The front slip angle gets bit ambiguous because the IF and OF are at different angles.) The car usually follows the prescribed line because the driver makes it so or tries to make it so through the steering wheel, brake, throttle, even body motion.

Taruffi may have been describing a balanced car, which I think would be rotating about it’s center of mass. If one considers that in a turn the car is always rotating about it’s center of mass when referenced to earth. Then it’s the vector sum of the grip of the front tires and the rear tires that generate the right amount of moment to keep the vehicle rotating about it’s center of mass for the prescribed radius (which is ever changing). The car maybe oversteering. understeering or neutral to keep the car on the prescribed line.

I think?

There are several ways to define under/over steer that are not mutually exclusive, especially you provide a little leeway knowing that common words and technical words can have different meanings.

Look at the term ‘grip’. In karting, we accept that the word can be used literally or it can be used specific to how a kart responds to jacking effects in the rear.

You can have a massive push and still have a higher slip angle at the front?

Well, my name is warren, so by definition I am a rabbit hole. :rofl:

About your Trajectory, Position, and Angle definitions; it feels like from an action/understanding/sensory experience we are on the same page, and we just ‘translate’ that knowing into words a little differently. I think one of my favorite sayings sums it up:

Words are used to express meaning, when you understand the meaning, you can forget about the words.
-Chuang Tzu

I know I talk about stuff (line, trajectory, load, traction, etc.) like they are separate things, but that’s just because it’s the only way my tiny logical/language brain can do it. Inside, I understand, and relate to, driving as a holistic thing based on two main ideas:

  • You control the car/kart by managing its energy. That is, you collect energy/speed via the accelerator, return it to the track via your brakes, and move it around with the steering & pedals (to send it through your tires to generate traction when cornering, or you may need to remove energy/load from a tire to maintain/regain control by reestablishing balance between the work being requested from a tire vs the energy/load being provided to tire). So, looking at things this way, operating the kart’s ‘controls’ is “the cause”, but the effect is produced by the controls’ influence on energy.

  • The more you relate to driving as relationships (cause/effect, and influencer, influenced by), the more complete a picture you will have of your driving. For example, say you change your trajectory (angle of the path) into a turn to make it more direct instead of ‘round.’ Doing so (manipulating the energy in your kart to make that happen), sets off a huge cascading change in the way energy flows through your kart, how load moves to the front tire, how IF caster transfers twisting load to the chassis, how that influences rotational momentum, how that influences IR lift, and on, and on).

Again, reading this, it all sounds so :face_with_symbols_over_mouth: complicated, but I really believe that you, TJ, and all really fast drivers own this knowledge, and act on it, even if you don’t consciously relate to that knowledge, or would necessarily express it in the same way I do. I feel confident saying this because I have see your onboard videos, and TJ’s, and Tanguy’s, and I can see the depth of understanding, and the intention, and the finesse with which you guys control your karts (the kart’s energy, not the ‘pedals & wheel’). I don’t think there is any way to drive at the level you guys display without being fully plugged into these dynamic relationships at least subconsciously.

100% this; in a car it takes so much longer for energy to travel through the chassis, through the spring/shock/suspension, tire and then for the forces produced to travel back and influence the chassis. I have always thought of kart drivers as having a kind of mental rolling average smoothing filter for tire loads, traction, and slip angles to make the immediacy of feedback more manageable. :grin:

I think the next layer of this is WHY is an oversteer or understeer dynamic being created. I could be wrong about this, but to me ‘slip’ is really ‘drift’ (the tire is drifting away from the path being described by the wheel). But, when you think of it this way, a tire could be drifting off target by the same amount (let’s say 7degrees for example), but it could be doing so for two reasons:

  1. The ‘work’ requested of the tire is in balance with the load being placed on the tire, which would result in optimum tire performance/traction, which results in a certain amount of ‘slip angle’; or what I call ‘loaded drift’.
  2. The work/load relationship for the tire is out of balance (insufficient load is being placed on the tire for the work requested). This would cause the tire to drift off course by that 7degrees, but it would be producing nowhere near optimal performance/traction. So in this case, would you still call the amount of this ‘unloaded drift’ the tire’s slip angle?