This section concludes by introducing a car model that considers it as a skidding rigid body in the plane. This model was suggested by Jim Bernard. The C-space is , in which . Suppose that as the car moves at high speeds, the tires are able to skid laterally in a direction perpendicular to the main axis of the car (i.e., parallel to the rear axle). Let denote the angular velocity of the car. Let denote the lateral skidding velocity, which is another state variable. This results in a five-dimensional state space in which each state is a vector of the form .
The position of the rear axle center can be expressed as
Reasonable values for the parameters for an automotive application are: kg, , , m, m, kg/m, and m/sec. This state transition equation involves a linear tire skidding model, which is a poor approximation in many applications. Nonlinear tire models provide better approximations to the actual behavior of cars . For a thorough introduction to the dynamics of cars, see .
Steven M LaValle 2020-08-14