Figure 1 Tire Kinematics
Normal penetration depends on the normal direction of contact coordinate.
(1)
Where, 
is the carcass radius of toroidal and
is the contact radius
from carcass center.
Figure 2 Normal Penetration
Inclination angle was defined 
(2)
The contact point position vector 
was defined as following
equation.
(3)
where,
is
the position vector of tire.
is
the position vector from the tire center to contact point.
Also, the contact point velocity vector 
is defined.
(4)
(5)
(6)
(7)
(8)
(9)
Where,
is
the angular velocity vector of tire.
spin
direction unit vector.
The circumferential velocity of the contact point p, 
(10)
(11)
(12)
(13)
The lateral slip velocity of the contact point p, 
(14)
The normal penetration velocity, 
,
(15)