Variables associated with the tire kinematics as well as the contact patch forces/moments are shown in the figure below.

Tire kinematics and force/moment calculations are performed with respect to ISO coordinates. The origin of the ISO coordinate system is the tire-surface contact point, C. The unit vector for the z-axis, ${\stackrel{\^}{e}}_z^{\mathrm{ISO}}$, aligns with the surface normal vector which in turn is a result of defined surface geometry as explained in Surface. The unit vectors for the x and y axes, ${\stackrel{\^}{e}}_x^{\mathrm{ISO}}$ and ${\stackrel{\^}{e}}_y^{\mathrm{ISO}}$, are calculated using the following vector algebra equations:

${\stackrel{\^}{e}}_x^{\mathrm{ISO}}\={\stackrel{\^}{e}}_{\mathrm{spin}}\times {\stackrel{\^}{e}}_z^{\mathrm{ISO}}$

${\stackrel{\^}{e}}_y^{\mathrm{ISO}}\={\stackrel{\^}{e}}_z^{\mathrm{ISO}}\times {\stackrel{\^}{e}}_x^{\mathrm{ISO}}$

where ${\stackrel{\^}{e}}_{\mathrm{spin}}$ is the unit vector for the tire spin axis.

The inclination angle or camber, $\mathrm{\gamma }$ is

$\mathrm{\gamma }\={\sin }^{\mathrm{-1}}\left(\left({\stackrel{\^}{e}}_y^{\mathrm{ISO}}\times {\stackrel{\^}{e}}_{\mathrm{spin}}\right)·{\stackrel{\^}{e}}_x^{\mathrm{ISO}}\right)$

Tire revolution speed or spin rate $\mathrm{\Omega }$, which is needed for slip calculations, is derived using the following equation

$\mathrm{\Omega }\=\stackrel{\&conjugate0;}{\mathrm{\omega }}·{\stackrel{\^}{e}}_{\mathrm{spin}}-\stackrel{\&conjugate0;}{\mathrm{\omega }}·{\stackrel{\^}{e}}_z^{\mathrm{ISO}}\sin \left(\mathrm{\gamma }\right)$

where $\stackrel{\&conjugate0;}{\mathrm{\omega }}$ is the angular velocity vector of the tire.