Lateral control

2.1. Design of trajectory

Government transportation agencies have been evaluating several studies in the field of highway road planning in respect to horizontal curve design. Road design manuals determine a minimum curve radius for a predefined velocity, road superelevation and adhesion coefficient, see [76], [77]. The calculation is based on assuming the vehicle to move on a circular path, where the vehicle is subject to centrifugal force that acts away from the center of the curve as illustrated in Figure 94, see [78]. The slip angle is assumed to be small enough for the lateral force to point along the path radius, and the longitudinal acceleration is also small enough not to degrade the lateral friction of the vehicle considerably.

Figure 6.6. Counterbalancing side forces in cornering maneuver

The mass of the vehicle along with the road superelevation (cross slope) and the side friction between the tire and the road surface counterbalance the centrifugal force. Assuming that there is the same at each wheel of the vehicle the sum of the lateral forces is: , where is the gravitational constant. At cornering the dynamics of vehicle is described by the equilibrium of the two forces:

, where is the radius of the curve. Assuming that the road geometry is known by on-board devices such as GPS, it is possible to calculate a safe cornering velocity.

The following relationship holds for the maximum safe cornering velocity regarding the danger of skidding out of the corner:

(15) (15) is also used in crash reconstruction and is referred to as the Critical Speed Formula (CSF), see [79], [80].

In the so-called yaw mark method the critical speed of the vehicle is determined by using the calculated radius of the vehicle path from the tire skid marks left on the road.

Note that in the calculation of the safe cornering velocity the value of the side friction factor plays a major role. This factor depends on the quality and the texture of the road, the weather conditions and the velocity of the vehicle and several other factors. The estimation of has been presented by several important papers, see e.g. [81], [82], [83]. However, these estimations are based on instant measurements, thus they are not valid for look-ahead control design, where the friction of future road sections should be estimated.

In road design handbooks the value of the friction factor is given in look-up tables as a function of the design speed, and it is limited in order to determine a comfortable side friction for the passengers of the vehicle. For the calculation of safe cornering velocity these friction values give a very conservative result.

A method to evaluate side friction in horizontal curves using supply-demand concepts has been presented by [84]. Here the side friction has an exponential relation with the design speed as follows:

(16)

where and are constant values depending on the texture of the pavement. Note that is the estimated reference friction at a measurement speed of . Thus, by using (15) and (16) the maximum safe cornering speed can be determined on a given road surface along with the side friction factor, as it is illustrated inFigure 95. The intersections of the supply and demand curves give the safe cornering velocities and the corresponding maximum side frictions. Note that whereas the friction supply only depends on the velocity of the vehicle, the friction demand is a function of the velocity and the curve radius as well.

Figure 6.7. Relationship between supply and demand of side friction in a curve

The relationship between the curve radius and safe cornering velocity can be observed in Figure 5. The data points of the diagram are given by the intersections of the supply and demand curves. It shows that the safe

cornering velocity increases with the curve radius. However, the relationship is not linear, i.e by increasing an already big cornering radius results in only moderated growth of the safe cornering velocity.

Figure 6.8. Relationship between curve radius and safe cornering velocity

Rollover danger estimation and prevention control methods have already been studied by several authors, see [85], [86], [87]. A quasi-static analysis of maximal safe cornering velocity regarding the danger of rollover has been presented by [88]. Assuming a rigid vehicle and using small angle approximation for superelevation (

, ), a moment equation can be written for the outside tires of the vehicle during cornering as follows:

(17)

where is the height of the center of gravity, is the track width, is the load of the inside wheel at cornering.

The vehicle stability limit occurs at the point when the load reaches zero, which means the vehicle can no longer maintain equilibrium in the roll plane. Thus by reorganizing (17) and substituting the rollover threshold is given as follows:

(18)

Thus, to ensure safe cornering of the vehicle in a cornering maneuver without the danger of skidding or rollover, the velocity of the vehicle has to chosen to meet the two constraints defined by (15) and (16).

2.2. Road curve radius calculation

Another important task is to calculate the radius of curves ahead of the vehicle in order to define the safe cornering velocities in advance. The road ahead of the vehicle can be divided into number of sections. The goal is to calculate the curve radius at each section points ahead of the vehicle to determine the safe cornering velocities corresponding to the curve radius.

The calculation of the cornering radius , is as follows. It is assumed that the global trajectory coordinates and of the vehicle path are known. Considering a sufficiently small distance the trajectory of

the vehicle around a section point can be regarded as an arc, as it is shown in Figure 97. The arc can be divided into data points.

The length of the arc can be approximated by summing up the distances between data points:

These distances are calculated as: The length of

the chord is calculated as follows: Knowing the length and ,

it is possible to calculate a reasonable estimation of the curve radius . Note that the length of the arc must be chosen carefully. A too short section with fewer data points may give an unacceptable approximation of the radius. On the other hand a too big distance can be inappropriate as well, since then the section may not be approximated by a single arc. The number of data points selected is also important, and by increasing the number , the accuracy of the following calculation can be enhanced. The angle of the arc is as follows . The length of the chord is also expressed as a function of the radius: . Expressing the radius the following equation is derived:

(19)

Figure 6.9. The arc of the vehicle path

This expression can be transformed by introducing and using the Taylor series for the approximation of the function, i.e., . Then the following expression is gained for the radius:

(20)

The curve radius , can be calculated at each section point ahead of the vehicle path. The calculation method is validated through the CarSim simulation environment. Here, the vehicle follows the desired path while the curve radius is being measured and at the same time the calculation method is running giving a close approximation of the real value. The comparison of the real and the calculated radius is shown in Figure 98.

Figure 6.10. Validation of the calculation method

By calculating the radius of the curve using (20) the safe cornering velocity of the vehicle can be determined by using (15) and (18). This velocity can be considered as the maximum velocity that the vehicle is capable of in a corner without the danger of slipping and leaving the track or rolling over. It is important to state that in severe weather conditions this safe velocity may be smaller than that of the speed limit, thus the consideration of the maximum safety velocity in the cruise control design is essential.