Hi guys, I was having problems computing slip ratio (SR) and slip angle at
time = 0, when my simulated vehicle starts from rest, with the throttling
torque applied at that instant � i.e., the problematic zero terms in the
denominator of the slip ratio and slip angle equations, due to a vehicle
at rest at time = 0 (when the initial acceleration torque is applied)
I did a Google search and found lots of postings related to this problem...
I listed the known solutions below, for your review:
1. Artificially limit the long. speed that you use in the slip calculations
to some minimum value in order to prevent the infinities popping up.
2. More advanced are DE-based approaches like that described in this paper:
"SAE 950311 - 'Tire Modeling for Low-Speed and High-Speed Calculations'"
which is available from the SAE website (but not for free).
3. The Barnard way of indirecting SR. (so SR gets updated using an
indirect tan(SR) based on the wheel spin velocity). This gets the feel
of 'rubber building up'.
4. Just ignore the (quick) rubber force buildup and accelerate the
wheel. With SR=0 at the first timestep, no counterforce will be
present. The NEXT step though, SR will be >0 (probably even infinite)
and a force will push from the road back onto the tire. Eventually,
you'll reach a balance. If your timestep is small enough, it will
bounce back & forth but look good onscreen.
5. SAE paper (98-something) looks at the tire as a 3D spring.
So longitudinally, you let the wheel spin and this creates a force
because the tire ofcourse also works as a spring in longitudinal
direction. When that is found, it is verified against the friction
circle, and if the force is really not possible (too much amplitude),
then the force is reduced to the edge of the circle and the spring
offset is adjusted (shifted) accordingly (it's a predictor-corrector
method).
I think I'll try approach #3 above first, since it seems to be reasonably
elegant, with the minimal amount of work (LOL)...
Then I'll try the proposed DE-based approach (#2 above, SAE 950311)
Does anyone know of other fix-ups for starting simulations when SR and slip
angle are undefined, using the classical formulation given here:
http://www.bartellmotorsports.com/part21.pdf
I assume that my tire force algorithm is similar to the Pacejka approach,
in that both models use a polynomial fit to empirical data, and take slip,
slip angle, normal load, and stiffness as inputs.
Have you seen this force model before?
http://code.eng.buffalo.edu/dat/sites/tire/tire.html
The algorithm requires one iteration to recompute the forces, since the contact
patch length requires Fx, which is zero on the first pass... then it converges
to the correct value on the second iteration.
This approach is based on Szostak's paper "Analytical Modeling of Driver
Response in Crash Avoidance Maneuvering Volume II: An Interactive Model for
Driver/Vehicle Simulation".
I bought this publication from NIST - it's an excellent concise source for
tire force theory with derivations - published in 1988.
I wonder how this model compares with Pacejka's Magic Formula, in terms of
accuracy?
FYI, I'm working on a real-time, virtual reality human-powered vehicle simulation,
using Open Inventor for the 3d graphics interface.
Here's the preliminary web site for the project:
http://cubex.temp.powweb.com/
We have the Open Inventor scene graph fully defined (using Planet 9 Studio's
Austin, Texas Open Inventor model)...
I can move the bicycle around in 3-d space with the fixed cameras propagating
correctly as the bike moves... the last step is getting the bike to propagate
realistically, using the Szostak force model, and the 5-DOF bicycle EOMs
given on this web site:
http://code.eng.buffalo.edu/dat/sit...el/bicycle.html
As seen in the above paper, Equation #5 for angular acceleration of the back
wheel requires Fx.
So I'll use the tan(SR) approach (#3 above) when the bike is just starting
from rest, with zero speed and the pedal torque applied at time = 0, to compute
the initial Fx, with Fy = 0 in the beginning (i.e., assume Vy = 0 at time
zero)...
What do you think?
~Matt Verona
Houston, Texas
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