**Controller Description**

The vehicle controller consists of four sections: speed proler, lateral controller, longitudinal

controller, and a traction control, shown in figure below.

The speed proler calculates desired velocities based on the path characteristics a certain distance ahead.

The lateral controller uses a full state feedback scheme to reduce the vehicle’s heading error and lateral

error. The concept of Centre of Percussion (COP) is used as reference point to introduce

some stability when tires are slipping. The longitudinal controller monitors the lateral

force required, and uses the spare forces to maximize longitudinal performance. On most

vehicles, a traction controller is a standard equipment; it monitors differences in the wheels’

spins to predict slip, and distribute longitudinal power to counter/minimize the slippage.

Many methods can be used to generate a speed prole for the vehicle. For racing

purposes, the desires paths are generated oine to optimize vehicle performance. Similar

characteristics can be achieved with a sophisticated path planning algorithm. For realtime

path planners, a spline method can be used to calculate the desired velocity. The

paths ahead of the vehicle is tted through a high order piece wise spline; the curvatures

of each piece is compared; and a velocity prole can be generated for each splines using

equation below.

For the purpose of controller design, the simplest case is used for verications

only. A constant curvature is t of three points ahead of the vehicle. Using the vehicle

coordinates, the lateral distances between the path and three specied points ahead (0m,

5m, and 30m)of the vehicle are found. A circle is t through the three points; depending

on the lateral errors, the radius of the circle changes. This radius is used to calculate a

reference velocity based on the centrifugal force and the maximum frictional force supported

by the tires where Ft is the maximum available tire force, is the coecient of friction, Fc is the

centrifugal force generated around a circle with radius R at the desired velocity Vd.

The lateral controller consists of a feedforward controller and a feedback controller.

The feedforward controller uses the centre of percussion of the rear tires, shown below to simplify calculations

The centre of percussion with respect to the reartires is a special point

where the sum of lateral acceleration from the rear tire, ay, is zero.

The rear tires produce two eects on the vehicle system. They create a lateral acceleration

on the CG, and they produce an angular acceleration about CG. At the COP, these two

eects cancel out, and the ay from the rear tires is zero.

The feedforward controller takes in current path curvature, path distance and velocity,

and calculate the forces needed to go through a curve.The feedforward

force can be derived below

where ep is the lateral error a xed distance, xp, away from the CG. is the heading

error. Vx is the vehicle longitudinal velocity, and s is the distance travelled by the vehicle.

Assuming that Vx 0, using a small angle assumption, the following can be obtained:

where y is the lateral acceleration of the vehicle in vehicle coordinate frame. r is the

vehicle yaw rate, K is the instantaneous curvature of the path, and _K is the change in

curvature.

Substitute in the Single Track Vehicle Model the following

error dynamic can be written:

Using the concept of centre of percussion, the error dynamics can

be simplied.

At the COP, the eect of rear tire forces cancel each other out:

Thus, at the centre of percussion, the error dynamics becomes:

The feedforward command is intended to reduce the error dynamics to zero; therefore, the

feedforward law is obtained by equating ecop in Eq. (5.7) to zero. The lateral feedforward

force, F_fw is dened as:

The first four feedback states are purely for lane keeping purposes. When regulated

to zero, the vehicle will be tracking the path in the correct heading. The last two terms

act as a future feedback term. These terms aim to start steering early when a curve is

coming up. Much like the human driving behaviour, when a curve is coming up, one would

steer slightly in the direction of the curve, thus reducing the amount of steering needed

otherwise. The two future terms act like a damper on the original feedback system.

The overall feedback law is described below:

It should be noted that in this setup, the feedforward command takes priority, feedback

force ensures that the vehicle is always operating near the limit of friction. With the limit

set, the actual feedback amount is described below:

The last task of the lateral controller is that it needs to inform the longitudinal controller

how much tire friction is left, Fm, a similar calculation to the F is used:

The longitudinal controller is a small feedback controller that calculates the desired

longitudinal force, based on the lateral force required and the desired speed prole.

From the speed proler described by Equation (5.1), a desired speed, Vd is obtained.

This speed is fed back via a proportional controller with linear gain kspd:

where Fspd is the force generated from the desired speed. This force is then compared to

the Fm value calculated in Eq. (5.15), the minimum of the two is selected as the longitudinal

output.

When the gains are tuned correctly, simulation has shown that this simple structure

is able to maintain the vehicle near its tire frictions on most paths. Fm ensures that the

vehicle is always going as fast as it can; however, in situations where a sharp curve appears

right after a straight long path, the vehicle would not slow down until it reaches the end of

the path. Fspd ensures that the vehicle slows down to an acceptable speed that can travel

through the turn, before reaching the turn itself.

The last piece in the longitudinal control aspect is a traction controller. The traction

controller monitors the longitudinal slip ratio, and ensures that the desired torque at the

wheel is reached through a proportional controller.

Furthermore, it ensures that the longitudinal slip ratio does not exceed a dangerous

level, in this case, 0.7. The controller would ramp down the acceleration needed if this

threshold is exceeded.

The full thesis may be found here