Figure 1: Analog Circuit of Auditory Display
In order to process the auditory displays for the driver assistance tool, a simple analog circuit is constructed. The control signal is generated by the different control algorithms in order to assist a driver. This signal is sent to the data acquisition system, where it is converted into the voltage output from the DAC (Digital-to-Analog Converter) built in the data acquisition board. 1000 Hz sine wave is generated from the function generator in order to produce a simple driver-assisting tone such that the amplitude is a multiple of the sine wave and the control signal. Two diodes are used in order to distinguish between the left and the right tones according to the sign of the control signal. The analog circuit is shown in Figure 1.
Figure 1: Analog
Circuit of Auditory Display
To study the affect of visual cues to the driver two samll red lights are created in the two corners of the dashboard. This lights can vary their intensity so as to represent the control signal.
Slip angle is an important state variable in vehicle dynamics
and in our vehicle simulation. Since the slip angle is the most
important input to produce the lateral forces, it can greatly
affect the vehicle stability. The behavior of a stable vehicle
shows that the slip angle closely follows the actual steering angle.
By keeping the steering angle and slip angle nearly equal, the
lateral force of the tire can be operated in the linear range. Thus,
the desired lateral force can be generated as required by the
driver. In slip angle feedback design, slip angle of the front tire
is directly used as a desired steering angle such that
. Then, the audio and visual assistant display
is used to keep the desired steering and actual steering angle
nearly equal. The front slip angle,
, can be computed
from the non-linear vehicle model. The block diagram for this
design is shown in Figure 2.
Figure 2: Slip Angle Feedback without Driver Model
Next, the driver model is included the design and its block diagram is shown in Figure 3.
Figure 3: Slip Angle Feedback with Driver Model
The steering dynamics represents the response of a front tire steering
angle (
) with a steering input of the steering wheel (
).
The steering dynamics are considered negligible and a constant gain of
0.0625 is used as shown in Figure 3.
The driver model and vehicle model are coupled with each other. In
order to apply an optimal regulator design, complete state-space
representation of the driver-vehicle model is required. For
calculation purpose, the driver model is
broken into two first order elements shown in Figure 4, where
is the inaccessible internal driver
state variables.
Figure 4: Development of Inaccessible State Variable
The linear vehicle model
is integrated with the internal driver states by defining the rest
of the states such that
,
,
,
and
. Converting the block diagram representation
in Figure 4 into state-space form, the linear
driver-vehicle model in state-space representation become
The state-space representation of the system in Equation 1 can be written as
The LQR problem is to find the optimal gain matrix such that the state-feedback law minimizes the quadratic cost function.
The matrices F and G are referred to as the weighting matrices on the state and the input respectively. A smaller F increased the relative weighting on the input matrix. This should decrease the magnitude of the input necessary to maintain control. In order to insure that all the states go to zero as time goes to infinity, F must be chosen to be a positive-definite matrix. G is also chosen as a positive-definite matrix to insure the control is finite. The weighting matrices are chosen based on matlab simulations and driving simulator tests such that
where I is the identity matrix. The constant gain optimal control is
where, P is the steady-state solution to the matrix differential Riccati equation of the form
The boundary condition at terminal time is 0 such that
.
References
