Well it's difficult to find any published info on the topic, probably because the best knowledge is kept as a professional secret. So in the end if you want something better the formula for calculating tire response forces will need to be altered. For games we can fake it however we like as long as the result is "realistic enough".
Since the definitions of slip contain denominators that can tend towards zero, near-zero means these equations are near-singularity, and numerically will not be very stable. In general dividing by very small floating point numbers is bad. Even if you have an if-statement checking for zero.
As a result near-zero speeds will probably not behave well in many simulations using slip velocities in a particular way. There's a popular SAE paper that describes taking the derivative of the slip equations to get rid of the singularity along with some artificial damping (i.e. non-physical tuning coefficient)... But this solution is completely non-realistic and can be a pain to implement and tune. Since it's not a realistic solution, maybe a simpler non-realistic solution exists:
So, following Brian Beckman's advice in his "Physics for Racing" book he proposes a solution that may be simpler and easier to implement:
( 1 )
s = longitudinal_slip / |max_longitudinal|
a = lateral_slip / |max_lateral|
rho = sqrt( s*s + a*a )
( 2 )
define
Phi0( s ) = Pacejka0( longitudinal_slip )
and
Phi1( a ) = Pacejka1( lateral_slip )
( 3 )
Force_x = s / rho * Phi0( rho )
Force_y = a / rho * Phi1( rho )
The above pseudo code in step ( 1 ) is calculating a normalized representation of slip for longitudinal/lateral. To do this a function that returns the input which generates the max force on the curve needs to be constructed. I wrote this myself with a variant of the bisection method. Then the max's can be used to scale the slip ratios between -1 and 1. Since the scale is uniform each ratio can be conceptually concatenated into a 2D vector with 2 components. The magnitude of this vector can be fed to Phi functions. Phi returns the force equivalent, or nearly equal, to the Pacejka curve. Normalized slip is fed to Phi functions, whereas regular slip values used to be fed to Pacejka curves (shown in step 2).
Step 3 denormalizes the vector by scaling the forces inversely with rho.
So to summarize: normalize slip ratio scalar values between -1 and 1, place them in a vector, feed this vector to a vector function Phi, denormalize the vector and use the resulting forces. Since step 3 uses rho, and rho cannot be negative, no singularity exists. If either slip (lat or long) is undefined using a 0 in the s or a numerator makes sense, especially when thinking about what this physically represents.
Beckman more or less leaves it up to the reader to determine what Phi is. Phi can just be the Pacejka magic function with properly scaled coefficients, or it can be a different approximation (like the thermo-dynamics equation Beckman proposes).
Resource: clicky.
