Home

Simultaneous longitudinal and lateral tire forces (or slip).

Introduction

Each simulator using wheels (whether car or flying or motorbike) will have to deal with combined slip. This happens when there is both a slip angle (wheel turned with respect to its forward velocity) and a slip ratio.

Internally, Racer contains multiple tire models:

• Pacejka ~1987-1996; coefficients are named a0/a1/a2/b0/b1/b2 etc. This method only calculates pure Fx and Fy (longitudinal and lateral) forces.
• Magic Formula 5.2; a well-known version of the Magic Formula. The changes coefficient naming to Pcx1 etc with lambda values to scale effects based on a more physical background.
• Noronha; a 3D spring-based variant. Only the lateral and longitudinal forces are used; the vertical tire forces is already dealt with in other parts of Racer.

As for combined slip, only Magic Formula 5.2 has built-in properties to handle mixed forces (=combined slip). Since these coefficients are hard to measure and are highly suspect (even when measured), I would advise against using them really.

The combined slip methods (that are more or less independent of the chosen pure longitudinal/lateral tire models) that are present in Racer are described here. The are quite a few of them, and as much as tire modeling is a big part of car modeling, combined slip is a big part of tire modeling (for generic use, which in simulator terms means: large and enduring slip).

Magnus Gäfvert's thesis from 2003 - Construction of Novel Semi-Empirical Tire Models for Combined Braking and Cornering

This thesis is very interesting; it presents a mix between Pacejka and the brush model. It divides tire forces up into adhesive (sticky/springy) and sliding forces, based on results obtained from the brush model. It is implemented in Racer (v0.9.0 RC6+) as friction_circle_method 8 in racer.ini.

You can find the paper in the 2003 edition here. I've found another mix of papers (3 papers combined) where this thesis is also more or less included, but it is unclear if either of them is more prevalent, since they're both from 2003.

It calculates a 'psi' value for X and Y directions; this is the amount of slip. The psi value for both X and Y is the slip amount, normalized.

I found a few things that I needed to tweak (perhaps correct?) from the equations in the paper:

• The algorithms describing implementations of the properties seem to go bad when using as-is. For example, when using Algorithm 4 to implement Property 9, it states 'if(psi_sigma_x_sigma_y<1) eq 144'. Even if psi for both directions combined is >=1, it can mean that either direction is still less. The paper though goes on to set theta for X and Y to 1. This means that theta_y for example can jitter between ~0 and 1 if you're slipping (slipping = slip angle > optimal slip angle and slip ratio > optimal slip ratio).
  When fixing this and trying to combine more situations (only psi_sigma_x or psi_sigma_y < 1), be careful for the adhesion vs sliding formula to explode on you. Using Wolframalpha for example to plot the function x*(3-2*x)/(3*(1-x)*(1-x)+x*(3-2*x)), which is used to calculate which part of the tire is sliding, and which is not (adhesion). Clamp your inputs to be between 0..1.

(last updated March 11, 2013 )