Altair Driver Mathematical Methods

Input driver parameters specify the control priorities of the driver. Vehicle parameters are used to estimate the vehicle’s future states, based on a simple linear bicycle model. The following paragraphs describe vehicle and driver parameters and their effects on the trajectory planning.

Usage

Altair Driver is implemented as a Control State Equation (CSE). The outputs from the CSE (Throttle, Brake, Steer, Gear and Clutch) are used in the Powertrain, Brake and Steering system. Its value is then used in the steering system motion statement. When a vehicle event is selected from the Analysis Wizard in MotionView, the driver is automatically added to the multi-body model as are the required entities (arrays, solver variables, and so on).

Theory

The well-known bicycle model is used as a first approximation to represent a multi-wheeled road vehicle.

Kinematic Bicycle Model

In Kinematic bicycle model, the equations of motion are based purely on geometric relationships governing the system. The major assumption used in the development of the kinematic model is that the velocity vectors at points A and B are in the direction of the orientation of the front and rear wheels respectively. This is equivalent to assuming that the "slip angles" at both wheels are zero. This is a reasonable assumption for low speed motion of the vehicle (for speeds less than 5 m/s). This model does not consider the vehicle’s dynamics such as slipping and skidding.

The kinematic equations of the bicycle model are written in terms of coordinates of the vehicle which gives our three degrees of freedom non-linear model .

The equations on motion written as a set of three differential equations where the state vector is:

Figure 1.

Equations of motion can be written as:

where,

- Velocity of the vehicle

- Rear axle to CG distance

- Slip angle at the center of gravity

Dynamic Bicycle Model

At higher vehicle speeds, the assumption that the velocity at each wheel is in the direction of the wheel can no longer be made. In this case, instead of a kinematic model, a dynamic model for lateral vehicle motion is needed.

The force equations of the bicycle model are written in terms of coordinates of the vehicle which gives our five degrees of freedom non-linear model .

Figure 2.

The equations on motion are written as a set of five differential equations where the state of vector is:

Using Newton's second law of motion, the equations can be written as:

where,

In case of multi-axle vehicles (>2), and are calculated according to the vehicle’s axle configuration and are used as a and b in the above equations.

Once the equations of motion have been written, the problem becomes that of classical control theory. The goal is to calculate the steering wheel angle that would cause the vehicle to follow a prescribed path as closely as possible.

This can be achieved in many ways. One strategy could be to use optimal control theory and minimize a cost function measuring a series of weighted factors (including lateral position error, hand-wheel steering angle, and heading error) evaluated at different time steps.

With Altair Driver, MotionSolve computes the error between the predicted path and the desired path at some point in the future. The controller has full knowledge of the vehicle dynamics equations as described above and can compute the predicted position based on those equations at some future time.

To compute the predicted path, the equations of motion are integrated for look-ahead times using the Runge-Kutta integration scheme. The path is obtained by spline interpolation using the spline utility in MotionSolve.

The difference between the predicted path and desired path at look-ahead is the error.

Thus:

The algorithm attempts to estimate the error by evaluating its sensitivity to the steering wheel angle. Convergence is achieved when the error is less than a certain tolerance.

The final steering wheel angle is essentially used as an increment to the current steering wheel angle as shown below. When computing the new steering wheel angle value (at time t+1), the previous value of the steering wheel angle (at time t) is updated by the quantity:

Limitations

The Kinematic model is valid only for low speed motion of the vehicle. At higher velocities, considerable slip angles are generated on the wheels which is not accounted in this model.

For the Dynamic bicycle model, we have assumed that the lateral tire force is proportional to slip angle. This relationship is valid for smaller slip angles. Since the controller uses this simplified linear representation of the vehicle, its applicability is limited to the linear range.

User Inputs

You can enter vehicle parameters and driver parameters in the Altair Driver panel. The vehicle parameter array contains quantities that are used in the definition of the bicycle model.

You can also click the Calculate button on the Altair Driver panel to update these (except and ) as well as brake and drivetrain parameters from the vehicle.

In addition to these, the mathematical model also uses Steering ratio (ratio between steering wheel angle input and the road wheel steer angle) which is calculated by the driver before running transient maneuver.

The maximum steering wheel angle is ultimately used as cut-off value for the computed steering wheel angle.