In the previous video,

we covered longitudinal dynamics for the vehicle.

In this lesson, we will cover the dynamic modeling of

a four-wheel car based on the bicycle modeling approach.

By the end of this video,

you will be able to;

build a dynamic model of a car using the kinematic bicycle model as a starting point,

and represent it in a standard state-space form suitable for lateral control design.

Let's get started. We would like to extend our kinematic bicycle model

to a dynamic model by relaxing the no slip condition and force for the kinematic model.

Recall that in the full dynamic bicycle model,

we maintain two components of the motion: the first

in the longitudinal direction in the direction of the heading,

and the second in the lateral direction perpendicular to the heading.

Specifically, for the lateral vehicle model,

we are interested in modeling the rotation rate of

the vehicle based on the moments that affect the vehicle while moving.

To start modeling the lateral dynamics of the bicycle model,

the following assumptions will be made: first,

the forward longitudinal velocity is assumed constant.

This is done to decouple our lateral and longitudinal dynamic models,

which simplifies our modeling task greatly,

but does lead to modeling inaccuracies when accelerating or decelerating out of curves.

Second, as with the kinematic bicycle model,

the left and right wheels for

both front and rear axles are lumped into a single wheel each.

So, this assumption converts the four wheels to two wheels bicycle model.

Finally, other nonlinear effects such as suspension movement,

road inclination, and aerodynamic forces are assumed to be negligible.

In practice, these effects can have a significant impact on the tire forces that occur.

So, this is again,

a limiting assumption in some cases,

but it's sufficient for our purposes.

We will use the vehicle center of gravity as the reference point for

the dynamic model as it simplifies the application of Newton's second law.

We can define the total acceleration in the inertial frame as a_y,

and this includes the lateral acceleration in the body frame y double dot,

and the centripetal acceleration from rotation of the vehicle,

omega squared R. These expressions can be

rewritten in terms of the slip angle rate of change, beta dot,

and the heading rate of change,

psi dot using the definition of the slip angle and the fact that V equals omega R,

and omega is equal to psi dot respectively.

The model formulation for lateral dynamics can now be formed with the only two

forces affecting the dynamics being the lateral forces on the front and rear tires.

The vehicle longitudinal velocity is defined by V as before and

the mass is m. For the angular accelerations, psi double dot,

the moments produced by the tire forces act in

opposite directions and combine with the inertial term,

I_z times psi double dot to form this second-order equation.

The parameters, L_f and L_r define the distances to each tire from the cg.

One of the most important components in vehicle dynamics modeling or the tires.

Tire forces are generally hard to predict exactly,

and tire models tend to be nonlinear and empirically identified.

We'll explore some of

the more common tire models in detail in the final video in this module.

Fortunately however, for normal driving conditions,

a simple linear approximation can be used to model tire force generation.

This approximation is really only valid for small slip angles,

and the tire forces are modeled as varying linearly with slip angles specifically.

As we use the linear tire model in our control design,

we must make sure to not exceed

this small angle assumption by sticking to non-aggressive driving maneuvers.

In order to use these linear tire models,

we need to define the front and rear side slip angles,

alpha f and alpha r. They are defined

in exactly the same way as the vehicle slip angle beta,

but are defined relative to the direction of

the wheel and the vehicle velocity at the wheel center.