rec.autos.simulators

I've been studying The Physics of Racing series (great stuff!!!),
specifically Part 25 (http://www.racesimcentral.net/) and the later
part 29 and I really don't understand exactly what is being done.  I
realize that the goal is to get the total force on the tire, which can
be compared to the total force the tire can handle (to determine if
there is a skid), but I don't understand the final three formulas that
are given in part 25 as the solution.

Can anyone explain them in slightly plainer style (I can handle some
physics, I just need help relating it to code) or even better with
pseudocode?

I appreciate the help!

Aaron Liebling

--
"You're the worst thing to happen to musicals since Andrew Lloyd Webber!"
  - Stewie from The Family Guy

...

It looks like a friction circle cutoff. You take the original Pacejka
results, but as (when combined) they can exceed the maximum friction
force (mu*F_load) they are 'contained'.

Didn't read the complete article, but s and a are probably normalized
(otherwise the sqrt() wouldn't work as you'd have an ellipse). Then
you look at the contribution of s versus total slip and a versus total
slip. Say, if s=0.5, a=0.25, then 's' contributes more to the force
than 'a' does.
This is expressed in the formulae of the new Fx and Fy, where the
contributions of s and a are done mathematically.

Note the vector (s,a) that moves within a circle with radius 1 and
note that s/rho and a/rho look like cosines/sines.

It may be a bit more deep, as the way I put it above looks like you
could do something like:
v2d = (Fx,Fy)         // Fx, Fy original Pacejka outcome
maxForce=mu*F_load;
if(v2d.length()>maxForce)
{
  v2d=maxForce*Normalize(v2d);
Fx'=v2d.x;
Fy'=v2d.y;

(capping the force vector at the friction circle with radius
mu*F_load).

Ruud van Gaal
Free car sim: http://www.racer.nl/
Pencil art  : http://www.marketgraph.nl/gallery/

Thanks for taking the time to respond.  Your Racer is a great
inspiration!  I'm fine with programming and vector/3d math, but am
fairly new tocomplex physics (I'm really enjoying it though!).

I'm afraid your answer went right over my head, as I need an even
simpler explanation than what you gave.  If you think I'm biting off
more than I can chew, is there a simpler (but still effective) tire
model?  Or is there a better (more conventional) wayx to combine the
effects of the two Pacejka's algorithims?

Assuming you're willing to answer my dummy questions, please read on!

Unfortunately, I don't even understand the signifigance of the 3
equations derived at the bottom of part 25
(http://phors.locost7.info/phors25.htm).  In rho = +sqrt( squ(s) +
squ(a) ), what does the rho actually mean?  I'm assuming s is the output
from the longitudinal pacejka algorithm and a that from the lateral.  Is
this correct?

Given these three algorthims, how do I go about applying them to a
programmatical tire model?  I.e., given the algorithims and the
whatr is known about the tires (dimensions, load, speed, etc...), how do
I go about figuring if a given tire is locked up skidding and what rate
it is rotating at?

I apologize if this question seems silly, but as I said, I'm pretty new
to physics.  I've gotten load shifting and things like that down just
fine (lots of good general physics tutorials), but I'm a bit lost when
it comes to the tire model and trying to apply Pacejka algorithims to my
program.  Any helpyou can give is very much appreciated!

Aaron

: It looks like a friction circle cutoff. You take the original Pacejka
: results, but as (when combined) they can exceed the maximum friction
: force (mu*F_load) they are 'contained'.

: Didn't read the complete article, but s and a are probably normalized
: (otherwise the sqrt() wouldn't work as you'd have an ellipse). Then
: you look at the contribution of s versus total slip and a versus total
: slip. Say, if s=0.5, a=0.25, then 's' contributes more to the force
: than 'a' does.
: This is expressed in the formulae of the new Fx and Fy, where the
: contributions of s and a are done mathematically.

: Note the vector (s,a) that moves within a circle with radius 1 and
: note that s/rho and a/rho look like cosines/sines.

: It may be a bit more deep, as the way I put it above looks like you
: could do something like:
: v2d = (Fx,Fy)         // Fx, Fy original Pacejka outcome
: maxForce=mu*F_load;
: if(v2d.length()>maxForce)
: {
:   v2d=maxForce*Normalize(v2d);
: }
: Fx'=v2d.x;
: Fy'=v2d.y;

: (capping the force vector at the friction circle with radius
: mu*F_load).

: Ruud van Gaal
: Free car sim: http://www.racer.nl/
: Pencil art  : http://www.marketgraph.nl/gallery/

--
"You're the worst thing to happen to musicals since Andrew Lloyd Webber!"
  - Stewie from The Family Guy

Thanks so much for your response.  your Racer is a great inspiration!
While I have no problem with programming or vector/3d math, I'm a bit of
a newbie to *** physics.  Unfortunately, your reply went right over
my head. :-(

I've had no problem with getting my physics working up to this point
(lots of good tutorials), but the tire model and how to apply Pacejka's
algorithims has baffled me.

I have no idea how to apply the three algorithims found at the bottom of
part 25 (http://www.racesimcentral.net/) to a programmatical tire
model.

For example, in the equation

rho = + sqrt( squ(s) + squ(a) ), what is the rho that is being
calculated?  I'm assuming s is the output from the longitudinal Pacejka
algorithim and a = the lateral, but I have no idea what the output means
or how to use the second twoalgorithims.

Using these queries, how do I figure out (given the information I have
about a tire..dimensions, load, torque, etc...) if a given tire is
locked, skidding, or what rate it is rotating at?

If these questions are too dumb, is there somewhere simpler to start for
a tire model, or a good example of how Pacejka's algorithims can be used
in a real program?  I'm just not able to connect the material I've found
on it to what I need to be doing. :-(  While there are lots of other
good physics tutorials for toher parts of car related motion, I'm lost
on the Pacejka stuff.

Any help you can lend is deeply appreciated!

Aaron

: It looks like a friction circle cutoff. You take the original Pacejka
: results, but as (when combined) they can exceed the maximum friction
: force (mu*F_load) they are 'contained'.

: Didn't read the complete article, but s and a are probably normalized
: (otherwise the sqrt() wouldn't work as you'd have an ellipse). Then
: you look at the contribution of s versus total slip and a versus total
: slip. Say, if s=0.5, a=0.25, then 's' contributes more to the force
: than 'a' does.
: This is expressed in the formulae of the new Fx and Fy, where the
: contributions of s and a are done mathematically.

: Note the vector (s,a) that moves within a circle with radius 1 and
: note that s/rho and a/rho look like cosines/sines.

: It may be a bit more deep, as the way I put it above looks like you
: could do something like:
: v2d = (Fx,Fy)         // Fx, Fy original Pacejka outcome
: maxForce=mu*F_load;
: if(v2d.length()>maxForce)
: {
:   v2d=maxForce*Normalize(v2d);
: }
: Fx'=v2d.x;
: Fy'=v2d.y;

: (capping the force vector at the friction circle with radius
: mu*F_load).

: Ruud van Gaal
: Free car sim: http://www.racesimcentral.net/
: Pencil art  : http://www.racesimcentral.net/

--
"You're the worst thing to happen to musicals since Andrew Lloyd Webber!"
  - Stewie from The Family Guy

Thanks so much for your response.  your Racer is a great inspiration!  
While I have no problem with programming or vector/3d math, I'm a bit of
a newbie to *** physics.  Unfortunately, your reply went right over
my head. :-(

I've had no problem with getting my physics working up to this point
(lots of good tutorials), but the tire model and how to apply Pacejka's
algorithims has baffled me.

I have no idea how to apply the three algorithims found at the bottom of
part 25 (http://www.racesimcentral.net/) to a programmatical tire
model.

For example, in the equation

rho = + sqrt( squ(s) + squ(a) ), what is the rho that is being
calculated?  I'm assuming s is the output from the longitudinal Pacejka
algorithim and a = the lateral, but I have no idea what the output means
or how to use the second twoalgorithims.

Using these queries, how do I figure out (given the information I have
about a tire..dimensions, load, torque, etc...) if a given tire is
locked, skidding, or what rate it is rotating at?

If these questions are too dumb, is there somewhere simpler to start for
a tire model, or a good example of how Pacejka's algorithims can be used
in a real program?  I'm just not able to connect the material I've found
on it to what I need to be doing. :-(  While there are lots of other
good physics tutorials for toher parts of car related motion, I'm lost
on the Pacejka stuff.

Any help you can lend is deeply appreciated!

Aaron

: It looks like a friction circle cutoff. You take the original Pacejka
: results, but as (when combined) they can exceed the maximum friction
: force (mu*F_load) they are 'contained'.

: Didn't read the complete article, but s and a are probably normalized
: (otherwise the sqrt() wouldn't work as you'd have an ellipse). Then
: you look at the contribution of s versus total slip and a versus total
: slip. Say, if s=0.5, a=0.25, then 's' contributes more to the force
: than 'a' does.
: This is expressed in the formulae of the new Fx and Fy, where the
: contributions of s and a are done mathematically.

: Note the vector (s,a) that moves within a circle with radius 1 and
: note that s/rho and a/rho look like cosines/sines.

: It may be a bit more deep, as the way I put it above looks like you
: could do something like:
: v2d = (Fx,Fy)         // Fx, Fy original Pacejka outcome
: maxForce=mu*F_load;
: if(v2d.length()>maxForce)
: {
:   v2d=maxForce*Normalize(v2d);
: }
: Fx'=v2d.x;
: Fy'=v2d.y;

: (capping the force vector at the friction circle with radius
: mu*F_load).

: Ruud van Gaal
: Free car sim: http://www.racesimcentral.net/
: Pencil art  : http://www.racesimcentral.net/

--
"You're the worst thing to happen to musicals since Andrew Lloyd Webber!"
  - Stewie from The Family Guy

Ruud,

I hadn't realized the source of Racer was available.  It is a great help
in understanding how the tire model goes together!  Your code is very
readable and well commented (despite the plethora of #ifdefs) :-)

I think I'll probably be able to get the explanations I need from
following it (I can read code much more easily than physics texts).

If you have any further advice or thoughts, I'd love to hear them.

thanks,

Aaron

So you do have the original Pacejka formula's in your source?

A simpler system would be to just cut off the force. You know about
the friction circle?
Basically the tire can't generate more than some amount of force. From
the Pacejka formula, you get a D (the one that multiplies your sin(),
so D is also the maximum force you're ever going to get). D is then
the maximum longitudinal force you'll ever get.

To make things simple, you can take the 2D vector (Fx,Fy), where
Fx=PacejkaLongitudinal() and Fy is lateral. If both Fx and Fy are
large, the combined force (the length of vector (Fx,Fy)) is bigger
than the tire can actually handle.
So then you can also just apply a friction circle and do:
Vector2D v(Fx,Fy);
float len=v.Length();
if(len>maxForce)
  v.Scale(maxForce/len);

That is the length of the vector (s,a), used to determine whether the
combined forces go outside the friction circle (in which case the
vector needs to be capped).

I'd say s=slipRatio and a=slipAngle, so not really the forces
themselves, but the article might explain that better.

Locked up is more a question of slip ratio. Nothing to do with
combined slip. See the formulae of slip ratio: once the velocity of
the wheel becomes 0, the slip ratio becomes -1, which is locking. But
you don't deal with that directly, you just keep the wheel's speed by
rotational acceleration: once the forces are worked out you accelerate
the wheel by using F_engine-Fx (the road reaction force, Fx from
Pacejka, fights the will of the engine, F_engine as calculated from
engine_torque*gearboxRatio*differentialRatio/wheelRadius).
That's like:
wheelAcc=(F_engine-Fx)/spinInertiaOfWheel

In the end most of it is F=m*a or the torque variant (T=I*w,
torque=inertia*angularAcceleration).
Pacejka is just a way to describe how the forces vary when the wheel
doesn't roll exactly the same as the road below it.

A Google search on 'Car physics' in this group will also dig up a lot
of previous discussions!

Ruud van Gaal
Free car sim: http://www.racer.nl/
Pencil art  : http://www.marketgraph.nl/gallery/

Thansk for the clear explanation.  Between that and some hints from
the racer source I think I understand enough to muddle my way forward.

I'm sure I'll be back with more dumb questions soon.  I'll also check
out the book that was recommended as it sounds perfect!

Aaron

One more time now...  Best place to buy our books is SAE -- they
have it in stock and ship right away (unlike Amazon who don't stock).
We keep a link updated to the SAE bookstore from our site (since SAE
often re-organize their site):
  http://www.millikenresearch.com/rcvd.html

Thanks for the compliment, first time I've heard RCVD called a "shopping
mall"<grin>.  We use an advanced version of the non-dimensional (or
normalized) tire formulation discussed in Chapter 14 in several of our
models.  It has a lot of advantages for simulation, for example it is very
easy to change the peak mu in an organized manner.

-- Doug Milliken
   Milliken Research Associates Inc.

Ruud,

A followup Racer question (the source code has been a great help in
filling in some of the blank spots in my code (like the frixction
circle).

In RBody::ApplyRotations, you do the following:

for(i=0;i<car->GetWheels();i++)  {    
        w=car->GetWheel(i);
        forceCar=*w->GetForceBodyCC();

        torque.y+=forceCar.x*w->GetSuspension()->GetZ();
        torque.y+=-forceCar.z*w->GetSuspension()->GetX();
        ...

I must be missing something, but this means seems to imply that you're
going to apply a torque on the y axis based on forward or lateral
movement.

Looking through your code, though, I could find no where that the
suspension positions are set (used here via the GetZ() and GetX()).  I
assume this suspension position is supposed to reflect rotation of the
attached wheel, or is that accounted in to the body's torque elsewhere?

Where in Racer are the suspension positions modified? I assumed it
should have something to do with wheel rotation, but could find nothing.

Thanks again for making your source available.  It pointed out a number
of fundamental errors I had made and serves as a great guide when I get
stuck!

Aaron

Sorry, no papers yet on our internal tire stuff, but maybe someday.

Since we primarily write "engineering simulations", we have only a little
need for low speed solutions...different set of problems than writing
"driving simulators".  But I agree that if there is ever a second edition
of RCVD this would be a good topic to add.

So far we have been just trying to patch the current edition, every
printing has more bug fixes, mostly pointed out by professors that use it
for college courses in vehicle dynamics.  The publishers allow us to make
changes, as long as the pagination doesn't change.

-- Doug Milliken
   Milliken Research Associates Inc.

(emailed separately)

Hi Aaron,

Glad to see the source is actually of help. Might be cluttering at
times. :)

Not movement, but force. As a lateral force builds up at a tire, it is
applied to the body as a rotational torque. As T=F*r, the above code
converts the lateral force (forceCar.x) as a torque around the body's
CG. The arm ('r') is taken as the suspension Z location.
Same for longitudinal forces (Fx in Pacejka/SAE, forceCar.z in Racer,
which uses the OpenGL axis system). This time the arm is the
horizontal position:

  forceCar.z
  |
  V
  O-----O
  |     |
  |     |
  |  x  |   x=CG
  |     |
  |     |
  O-----O
  <-->
  susp.x

It probably is in rsusp.cpp: RSuspension::Load().

It is just the wheel position; no wheel turning itself. Wheel turning
is only used in determining slip angle. If the wheel is turned because
of the steering wheel, lateral (Fy) forces will be generated, which
are used in the above code in the same way. After all, the forces are
transmitted to the body through the suspension; doesn't matter if the
wheel is being turned or not.

Cheers,

Ruud van Gaal
Free car sim: http://www.racer.nl/
Pencil art  : http://www.marketgraph.nl/gallery/