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Bicycle Model
The non-linear bicycle model
considers longitudinal (x), lateral (y), and yaw 
motion under the assumption of
negligible lateral weight shift, roll and compliance
steer while traveling on a smooth road. Our design of control strategy is to
control both longitudinal and lateral motions during hard braking and steering
maneuvers. Angular velocities of front and rear tires are
added to the states in order to investigate directional interactions
between longitudinal and lateral tire forces. In addition to these five states,
longitudinal and lateral positions and yaw angle with respect to the fixed
inertial coordinates are added to the dynamic equation
in order to refresh the vehicle position and orientation in the simulation
scene. Thus, the bicycle model used in our simulator has 5 Degrees Of Freedom with 8 state equations. The bicycle model developement presented here is based
on reference [1]. Figure 1 shows side and top views of the vehicle using this
bicycle model. Using free body diagram shown in top view of Figure 1, the
equations of motion are derived.
Figure 1: Free Body Diagram of a Vehicle
Summing the longitudinal forces
along the body x axis leads to
Where m is the mass of a
vehicle, 
and 
are the longitudinal and lateral
components of the vehicle velocity resolved along the body axis, r is
the yaw rate, and 
and 
are the front and rear wheel
steering angles. Summing the lateral forces along the body y
axis gives
The sum of the yaw moments about
the car CG yields
Figure 2: Free Body Diagram of a Wheel
For the front and rear wheels,
the sum of the torque about the axle, as shown in Figure 2, results in
Where, 
and 
are the angular velocities of
the front and rear wheels, 
is the inertia of the wheel
about the axle, 
is the wheel radius, 
and 
are the applied braking torques,
and 
and 
are the applied throttling
torques for the front and rear wheels. All the vehicle specifications are based on the 1984 Honda Accord [2] with reasonable
braking torques for front and rear tires. Yaw angle is
directly found by integrating the yaw rate. Since yaw angle is with
respect to the fixed coordinates, longitudinal and lateral position with
respect to the inertial fixed coordinates are also found
as follows.
Where, 
and 
denote the velocity components
with respect to the fixed inertial coordinates. Simple integration based on the
forth-order Runge-Kutta method is
used to integrate the above eight states in the simulation loop.
Tire Force Calculation
The longitudinal and lateral
forces from front and rear tires are derived from the
non-linear tire model discussed earlier. The input variables for the tire model
are front and rear normal loads ( 
and 
), slip angles ( and ), and longitudinal slip ratios
( 
and 
). The normal forces of front
and rear tires are determined according to the instantaneous longitudinal
acceleration. Summing the moments about the rear contact patch using the side
view of Figure 1, normal load of front tire is found
as
Summing the moments about the
front contact patch,
Where 
is the
instantaneous longitudinal acceleration and h is the height of the car CG
from the ground.
Figure 3: Slip Angle of Front Wheel
Figure 4: Slip Angle of Rear Wheel
From Figure 3 and Figure 4,
velocities of front and rear tires are determined by summing the velocity at CG
and the velocities effected by the yaw rate. Thus, the
slip angles of front and rear tires are found as
Also, the speed of the front and
rear tires are calculated by the following equations.
Where, 
and 
represent the magnitude of the
front and rear tire axle velocities. To calculate the longitudinal slip,
longitudinal component of the tire velocity should be derived.
The front and rear longitudinal velocity components are found
by
Then, the longitudinal slip is
determined according to the equation in tire model. Under braking conditions,
longitudinal slip of front and rear tires are calculated by
Using the normal load, slip
angle, longitudinal slip, and non-linear tire model realistic longitudinal and
lateral forces are generated for the 5 DOF bicycle model.
References
Taheri, S., An Investigation and Design of Slip
Control Braking Systems Integrated with Four Wheel Steering, Ph.D. Thesis,
American Honda Motor Company Inc., Motor Vehicle Specifications,
Passenger Car, Accord 1984, Technical Report, Honda,