In part 16, we introduced RARS, the Robot Auto Racing Simulator. We talked
about the clever and simple tyre-friction model in RARS and gave a terse
presentation of its details in the big table in the article. Here, we'll explain
in a little more detail why the model is cool.
First, consider RARS' idea of a tyre when there is no steering applied. In
the following diagram, we look down on a RARS tyre from above, using "X-Ray
Vision" to see the contact patch:
There are only two interesting quantities at this point: the velocity of the
car with respect to the ground, and the velocity of the contact patch with
respect to the car. If there is no power, braking, or cornering applied, then
these two vectors are equal and opposite; in other words, the velocity of the
contact patch with respect to the ground is zero. In general, if you have
a velocity of some thing, any thing, with respect to the car
and you have the velocity of the car with respect to the ground, all you
have to do to get the velocity of the thing with respect to the ground
is add the vectors, and we show how to add vectors immediately below. You
eliminate the middleman - the car - by so doing. This is relativity, not
the Einstein kind, but the Galileo kind-hence the title of this article. In the
Einstein kind of relativity, you correct the vector sum with some quantities
depending on the speed of light, a constant, and this is not relevant for auto
racing because the speeds are so low compared to the speed of light, which is
about 670 million miles per hour.
What happens when you apply a little steering input? Look at the next
diagram.
The velocity of the contact path with respect to the car,
,
gets a little angle-the slip angle or the grip angle, as I called it in part 10
of the Physics of Racing.
still points
directly back along the plane of the tyre, and the velocity of the car with
respect to the ground, ,
still heads forward, that is, up on the page, at least for
an instant. To find out the velocity of the contact patch with respect to the
ground, , we
add the vectors
and , just as
before, but now there's an intervening angle. Here's the
procedure for adding the vectors: Transport one of them, without changing its
direction, so that its tail touches the head of the other one. It turns out that
not all kinds of vectors can be transported like this, but velocity in this
context is one of the kinds that we can freely transport. We'll transport
over to the
head of , as
in the following diagram:
Draw a new vector from the tail of
to the head of
in its new location. This
new, little vector is defined as the vector sum,
, drawn from the
tail of one to the head of the other. Note it would have the same direction and
length if it were drawn from the tail of
to the head of
. Because of this
fact, we can write the following equation:
=
+
=
+
This procedure for adding vectors works even when the vectors are collinear,
in which case the triangle is flat, the opposite corners coincide, and the
vector sum is zero-the mathematically unique vector zero. It also
turns out that this procedure has a very simple equivalent in algebraic form. To
do computations, we need to represent vectors with numbers. To do so, we measure
the length of the projection of the vector against the axes of a coordinate
frame, as in the following diagram:
So, we get just what we need: numbers, also called components
because they are the component parts that make up the perpendicular, independent
projections of the vector. Numbers, in general, are also called scalars
since they can be used to scale vectors. In general, we'll get three numbers for
any vector in a 3D world, and two numbers in a 2D world. In either case, adding
vectors is trivial. If we write
= (Vx,
Vy, Vz)
and =
(Wx, Wy,
Wz), then
+
= (Vx +
Wx + Vy
+ Wy + Vz + Wz)
Just add them up componentwise. Couldn't be easier. I'll leave it to you to
show that the head-to-tail method is equivalent to the numerical
add-'em-up-componentwise method.
Let's go back to the tyres. Here's what makes RARS' model so clever: It's
undeniable that the contact patch moves with respect to the ground if we assume
that it continues to move in the circumferential direction with respect to the
wheel and the car. We can summarize all the complex motion of the contact patch
in a single velocity
and we can approximate our friction model so that it depends only
on the magnitude of .
This is a simpler model than the one presented in part 10, but
also potentially less accurate. Let's review that one briefly. Again, looking at
the contact patch from above, as if by X-Ray vision:
In the diagram above, we're looking at a wheel steered slightly to the left
of the direction of travel. Assuming that there's a little acceleration in
addition to the steering, both sides of the contact patch move backwards with
respect to the ground. The left-hand side (LHS) of the patch moves a little more
slowly than the right-hand side (RHS) because the RHS crabs around the corner.
The wheel, through the carcass of the tyre, twists the patch to the left,
generating a force couple illustrated by the grey arrows. The ground resists,
through friction, producing the right-twisting, restoring, force couple in
black. Since the patch continues to twist leftward with respect to an inertial
reference frame in a steady-state turn, the grey couple must be a little larger
than the black one.
This model is not by any means complete, and it's already MUCH more complex
than the RARS model, which does a decent job in practice. RARS computes just one
quantity, the vector ,
and accounts for all forces and torques on the tyre through that
one variable. The advantages of the approach, when it comes to computer
simulation, are
very simple math, easy to code and debug
very fast conversion from velocity to force
one table lookup and one interpolation
The limitations, of course, are that RARS cannot account for detailed tyre
physics and important effects like suspension geometry and dynamics, so the
whole scheme trades off accuracy for simplicity. However, as a quick-and-dirty
approximation, it's remarkably effective.
,
gets a little angle-the slip angle or the grip angle, as I called it in part 10
of the Physics of Racing.
,
still heads forward, that is, up on the page, at least for
an instant. To find out the velocity of the contact patch with respect to the
ground, 
, we
add the vectors 
