moorepants
commented
7 months ago Comment from the code:
# We need to solve the dynamic equations and the auxiliary equations
# simultaneously to avoid having to solve the dynamic equations first and then
# substitute in the derivatives of the speeds. So reconstruct the equations of
# motion to this form:
# [Mp 0, 0, -Mf]*[up ] = [F ]
# [0, Cf, 0, Cz] [fyp] [-Fy ]
# [0, 0, Df, Dz] [mzp] [-Mz ]
# [Ap, 0, 0, Af] [fz ] [-B_aux]The dynamical equations end up depending on fz in this formulation and are not decoupled from the noncontributing force equations like in https://moorepants.github.io/learn-multibody-dynamics/noncontributing.html#augmented-dynamical-differential-equations
I forget why that is the case.
A general nonlinear tire model takes the lateral slip angle, camber angle, and normal force as inputs and outputs the lateral force.
$$ F_y = f(\alpha, \gamma, F_z) $$
If the tires are modeled as rigid, then the normal force between ground and wheel is a noncontributing force. This means you get equations that look like
$$ \begin{bmatrix} M & c \ c & 1 \end{bmatrix} \begin{bmatrix} \dot{\bar{u}} \ F_z \end{bmatrix} = \begin{bmatrix} F(F_y) \ f \end{bmatrix} $$
I currently solve the above system numerically at each time step. So that creates a circular dependency for F_y(F_z) but F_z depends on known F_y if you want to implement an arbitrary nonlinear function for F_y.
We could solve the system symbolically and it would give $F_z(\bar{x})$ which you can replace in the first equation and that gives F_y as a pure function of state. This will make for some nasty long equations with chances of divide by zero.
A second option is to introduce a vertical tire compliance and a new state to track this spring deformation. Then F_z = k*x and F_z is no longer a noncontributing force.
I guess a third option would be to iteratively solve the above to equations to home in on the value of F_z and F_y that satisfy both equations.