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.

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

**1**- Taheri, S.,

*An Investigation and Design of Slip Control Braking Systems*, Ph.D. Thesis, Clemson

Integrated with Four Wheel Steering

University, 1990. **2**- American Honda Motor Company Inc.,

*Motor Vehicle Specifications, Passenger Car, Accord 1984*,

Technical Report, Honda, Windsor Locks, 1984.