Welcome back. In the last video,
we discussed the basics of kinematic modeling and
constraints and introduced the notion of the instantaneous center of rotation.
In this lesson, we will develop the kinematic bicycle model,
a classic model that does surprisingly well at
capturing vehicle motion in normal driving conditions.
Let's get started. The well-known kinematic bicycle model has long been used as
a suitable control-oriented model for representing vehicles because of
its simplicity and adherence to the nonholonomic constraints of a car.
Before we derive the model,
let's define some additional variables on
top of the ones we used for the two-wheeled robot.
The bicycle model we'll develop is called the front wheel steering model,
as the front wheel orientation can be controlled relative to the heading of the vehicle.
Once again, we assume the vehicle operates on
a 2D plane denoted by the inertial frame FI.
In the proposed bicycle model,
the front wheel represents the front right and left wheels of the car,
and the rear wheel represents the rear right and left wheels of the car.
To analyze the kinematics of the bicycle model,
we must select a reference point X,
Y on the vehicle which can be placed at the center of the rear axle,
the center of the front axle,
or at the center of gravity or cg.
The selection of the reference point changes the kinematic equations that result,
which in turn change the controller designs that we'll use.
As needed, we'll switch between reference points throughout this course.
Let's start with the rear axle reference point model.
We'll denote the location of the rear axle reference point as xr,
yr and the heading of the bicycle as Theta.
We'll use L for the length of the bicycle,
measured between the two wheel axes.
As with the two-wheeled robot,
these are our main model states.
The inputs for the bicycle model are
slightly different than those for the two-wheeled robot,
as we now need to define a steering angle for the front wheel.
Let this steering angle be denoted by Delta,
and is measured relative to the forward direction of the bicycle.
The velocity is denoted v and points in the same direction as each wheel.
This is an assumption referred to as the no slip condition,
which requires that our wheel cannot move laterally or slip longitudinally either.
It is the same assumption that allows us to compute
the forward speed of the two-wheeled robot based on the rotation rates of its wheels.
Because of the no slip condition,
we once again have that Omega,
the rotation rate of the bicycle,
is equal to the velocity over the instantaneous center of rotation,
radius R. From the similar triangles formed by L and R,
and v and Delta,
we see that the tan of Delta is equal to the wheelbase L
over the instantaneous turn radius R. By combining both equations,
we can find the relation between the rotation rate of the vehicle Omega,
and the steering angle Delta,
as Omega equals v tan Delta over
L. We can now form
the complete kinematic bicycle model for the rear axle reference point.
Based on this model configuration,
the velocity components of the reference point in the x and y direction are
equal to the forward velocity v times cos Theta and sine Theta respectively.
These two equations are combined with the equation for rotation rate
derived previously to form the rear axle bicycle model.
The bicycle kinematic model can be reformulated when the center
of the front axle is taken as the reference point x, y.
This is a good exercise to try yourself to practice applying the principles of
instantaneous center of rotation and follow the rear axle derivation quite closely.
The velocity points in the direction of the front wheel this time,
which is defined by the summation of Delta and Theta.
Working through the derivation leads to the following kinematic model for the vehicle.
The last scenario is when the desired point is placed at
the center of gravity or center of mass as shown in the right-hand figure.
Because of the no slip constraints we enforce on the front and rear wheels,
the direction of motion at the cg is slightly different from
the forward velocity direction in either wheel and from the heading of the bicycle.
This difference is called the slip angle or side slip angle,
which we'll refer to as Beta,
and is measured as the angular difference between
the velocity at the cg and the heading of the bicycle.
This definition of side slip angle will also
apply when we move to dynamic modeling of vehicles,
where it can become more pronounced.
The kinematic model with the reference point at the cg can be derived
similarly to both the rear and forward axle reference point models.
We end up with the following formulation,
which we'll use as the basis for our modeling of the dynamics of vehicles as well.
Lastly, because of the no slip condition,
we can compute the slip angle from the geometry of our bicycle model.
Given LR, the distance from the rear wheel to the cg,
the slip angle Beta is equal to the ratio of LR over L times tan Delta.
Finally, it is not usually possible to instantaneously
change the steering angle of a vehicle from one extreme of its range to another,
as is currently possible with our kinematic model.
Since Delta is an input that would be selected by a controller,
there is no restriction on how quickly it can change which is somewhat unrealistic.
Instead, our kinematic models can be formulated with four states: x,
y, Theta, and the steering angle Delta.
If we assume we can only control the rate of change of the steering angle Phi,
we can simply extend our model to include Delta as
a state and use the steering rate Phi as our modified input.
Our kinematic bicycle model is now complete.
Once again, we'll use a state-based representation of the model for
control purposes later in this course
and throughout the second course on state estimation as well.
Our kinematic bicycle model takes as inputs the velocity and the steering rate Phi.
The state of the system,
including the positions XC,
YC, the orientation Theta,
and the steering angle Delta,
evolve according to our kinematic equations from the model,
which satisfy the no slip condition.
We can now use this model to design
kinematic steering controllers as we'll see in a later module in this course.
To summarize this video,
we formulated the kinematic model of a bicycle for
three different reference points on
that vehicle and Introduced the concept of slip angle.
We'll use this kinematic bicycle model throughout
the next two modules for designing of controllers for self-driving cars.
In the next video,
we'll learn about how to develop
dynamic vehicle models for any moving system. See you next time.
Steven WaslanderAssociate Professor
Jonathan KellyAssistant Professor
