Keywords

1 Introduction

After the steering system, the use of braking systems offer a high potential for the lateral control of a vehicle [1]. In the event of a failure in the steering system (e.g. Steer-by-Wire system) official standards describe potential redundancies [2]. With the help of a functional redundancy in form of the brake system already present in the vehicle, a fallback level can be created. This allows actuators already installed in the vehicle to be used and saves costs in the manufacturing process. Wheel-specific braking interventions generate longitudinal forces that influence the yaw moment of the vehicle and thus influence lateral control of an automated vehicle along a given trajectory (Steer-by-Brake, SbB). In order to continue to ensure driving safety in road traffic despite a potential steering failure, the functionality of such a SbB system can be used. In critical situations the fallback system could just function like an automated emergency brake. But if the collision-avoiding braking distance exceeds the last possible avoidance distance, there is only the possibility to steer the vehicle to take evasive action along a calculated trajectory. Under these conditions, a collision could potentially be avoided. The potential of braking interventions for lateral control of automated vehicles has already been discussed in several publications. The majority of the investigations have taken place in simulative environments studies [3], but some publications also validate their results experimentally using real driving tests [4, 5]. The investigations in [5] have shown, that the road friction coefficient is a relevant influencing variable. In order to continue to ensure driving safety in road traffic despite a potential steering failure, the functionality of such a SbB system can be adapted to the road friction coefficient. The focus of the investigations in this paper is not on the possibilities of measuring or estimating the road friction coefficient, but rather on the effects of the SbB function being able to calculate and adapt to the road friction coefficient.

2 Method

The used methodology takes into account the true road friction coefficient μR and an estimated value μE of the road friction coefficient which are integrated into the SbB function. This method is already used in [6] to investigate the road friction coefficient sensitivity and adaption of automatic emergency braking (AEB) [7] and lane keeping assist (LKA). In this paper, the road friction sensitivity and the friction coefficient adaption for the four classes of a dry, wet, snow- and ice-covered road are simulatively investigated. The four classes of road friction are assigned to four representative characteristic parameters (Table 1).

Table 1. Representative road friction coefficients.
Full size table

3 Function

The trajectory used by the SbB function is calculated depending on two input parameters: the lateral offset \({y}_{e}\) which should be achieved and the predefined maximum yaw rate \({\dot{\psi }}_{max}\). With these two parameters an optimizer is minimizing the best fitting trajectory, so that it achieves the lateral offset and is not exceeding the maximum yaw rate during the maneuver. The y-coordinates \(y(x)\) of the trajectory are calculated by a polynomial of the seventh degree [8], the lateral offset \({y}_{e}\) which should be achieved and the evasive trajectory length \({x}_{e}\).

$$y(x)={y}_{e}\left[-{20}{\left(\frac{x}{{x}_{e}}\right)}^{7}+{70}{\left(\frac{x}{{x}_{e}}\right)}^{6}-{84}{\left(\frac{x}{{x}_{e}}\right)}^{5}+{35}{\left(\frac{x}{{x}_{e}}\right)}^{4}\right]$$
(1)

The curvature of the trajectory is determined by the first and second derivatives of the coordinates \(x\) and \(y(x)\) in relation to the arc length \(s\).

$$\kappa (s)=\frac{\dot{x}(s)\cdot \ddot{y}(s)-\ddot{x}(s)\cdot \dot{y}(s)}{{\left({\dot{x}(s)}^{2}+{\dot{y}(s)}^{2}\right)}^{3/2}}$$
(2)

The yaw rate \(\dot{\psi }(s)\) of the trajectory is calculated by the curvature \(\kappa (s)\) and the initial driving speed \({v}_{0}\).

$$\dot{\psi }(s)=\kappa (s)\cdot {v}_{0}$$
(3)

The maximum yaw rate \({\dot{\psi }}_{max}\) of the evasive trajectory is derived by the context from the kamm’s circle, which in this paper is based on the assumption that at the friction limit the same amount of force can be applied in the lateral direction as in the longitudinal direction. As a result, the maximum possible yaw rate \({\dot{\psi }}_{max}\) of the trajectory is calculated depending on the acceleration of gravity \(g\), the constant road friction coefficient \({\mu }_{R}\), a scaling factor \(k\), which describes the ellipse of the kamm’s circle and takes the value 0.9 in this investigation and the initial driving speed \({v}_{0}\) [9].

$${\dot{\psi }}_{max}=\sqrt{{\left({\mu }_{R}\cdot g\right)}^{2}-\frac{{\left({\mu }_{R}\cdot g\right)}^{2}}{2\cdot {k}^{2}}}/{v}_{0}$$
(4)

4 Driving Maneuver

The driving maneuver under investigation (Fig. 1) corresponds to an evasive maneuver with a safety distance of 2.5 m between the vehicles center of gravity (CoG). In this case, a stationary vehicle is approached at initial driving speed \({v}_{0}\) of 80 km/h. Compared to the evasive test according to the standard [10], the lateral safety distance is chosen to be smaller in order to investigate an emergency avoidance maneuver. The test is performed with a non-functioning steering system and the driver does not apply any steering torque during the test procedure. In the beginning the initial lane is driven through a constant driving speed and then changed to a second lane with an offset which is achieved exclusively via differential braking.

Two characteristic parameters are used to investigate the road friction coefficient sensitivity (Fig. 1). The first parameter describes the lateral offset \({d}_{Y}\) that is reached at the end of the evasive trajectory, while the second parameter describes the additional distance \(\Delta {d}_{E}\) if the vehicle moves out too early due to an incorrectly estimated road friction coefficient \({\mu }_{E}\).

Fig. 1.
figure 1

Driving maneuver of an emergency evasion with differential braking and associated characteristic parameters.

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The road friction sensitivity and adaption are simulated using a validated dual-track model. In the dual-track model there is a validated steering-model and a Magic Formula tire model of Version 5.2 with the measured parameters of a current winter tire.

5 Road Friction Coefficient Sensitivity

To find out the sensitivity of differential braking, the road friction coefficient between the selected classes (Table 1) is reduced step by step and the selected characteristic parameter of the lateral offset \({d}_{Y}\) is calculated. At the beginning, an SbB function without friction coefficient adaption is examined. This is calculated by the friction coefficient of a dry road surface. It can be seen that the evasive trajectory on the dry road with a road friction coefficient of 1.0 can be driven on with the SbB function, but if the road friction coefficient is now reduced, the evasive action is taken too late on the wet, snow- and ice-covered road and the required lateral offset \({d}_{Y}\) can no longer be maintained. Table 2 contains the characteristic parameters of the road friction coefficient sensitivity. It can be seen that the SbB function can only generate the required lateral offset on a dry road, in all other cases it leads to a collision with the target vehicle due to non-compliance with the lateral offset \(d_Y = 2.5\,{\text{m}}\). The analyses show that the SbB function is very sensitively influenced by the road friction coefficient. In order to nevertheless ensure vehicle safety, the SbB function is to be implemented in a friction-adaptive manner in the further course.

Table 2. Characteristic parameters for road friction coefficient sensitivity of a SbB function.
Full size table

6 Road Friction Coefficient Adaption

The road friction coefficient adaption replaces the previously constant parameter of the road friction coefficient \({\mu }_{R}\) in the calculation of the maximum possible yaw rate \({\dot{\psi }}_{max}\) of the evasive trajectory with the estimated value \({\mu }_{E}\). This estimated value can take the road friction coefficient of a dry, wet, snow- or ice-covered road. The maximum yaw rates to be maintained are now calculated for the evasive trajectories on the various road friction coefficients according to the following correlation.

$${\dot{\psi }}_{max}=\sqrt{{\left({\mu }_{E}\cdot g\right)}^{2}-\frac{{\left({\mu }_{E}\cdot g\right)}^{2}}{2\cdot {k}^{2}}}/{v}_{0}$$
(5)

Figure 2 shows four different evasive trajectories that are calculated with the estimated value of a dry, wet, snow-covered or ice-covered road. It can be seen that all calculated trajectories achieve the desired lateral offset \(d_Y = 2.5\,{\text{m}}\). To guarantee this, the evasive trajectories are lengthened as the estimated value decreases, so that the SbB function evades at comparatively large relative distances to the target vehicle.

Fig. 2.
figure 2

Evasive trajectories of the SbB function for the road friction coefficients of a dry, wet, snow- and ice-covered road for a vehicle with winter tires.

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Due to the correlations, the evasive trajectory of the dry road has the largest and the trajectory of the ice-covered road has the smallest maximum yaw rate \({\dot{\psi }}_{max}\) that the vehicle must withstand in order to negotiate this path. In the following, the described friction coefficient adaption is examined for an initial driving speed \({v}_{0}\) of 80 km/h. For this purpose, the selected driving maneuver is repeated for the friction coefficients of a dry, wet, snow- or ice-covered road as well as for the estimated values of these classes. The characteristic parameters of the friction coefficient adaption of each individual combination are then calculated and shown in Table 3. It can be seen that the characteristic parameters of the first row correspond to the characteristic parameters of the road friction coefficient sensitivity. For the characteristic parameters of the other rows, the distances of the evasive maneuver were adapted to the estimated values of a wet, snow- or ice-covered road. If the estimated value \({\mu }_{E}\) corresponds to the true road friction coefficient \({\mu }_{R}\), the required lateral offset \({d}_{Y}\) can always be maintained. The same applies if the estimated value \({\mu }_{E}\) is smaller than the true road friction coefficient \({\mu }_{R}\). Although the lateral offset can be achieved in this case, the SbB function deviates too early by the distance \(\Delta {d}_{E}\). If the estimated value \({\mu }_{E}\) is greater than the true road friction coefficient, the required lateral offset \(d_Y = 2.5\,{\text{m}}\) is not reached.

Table 3. Characteristic parameters for road friction coefficient adaption of a SbB function.
Full size table

7 Conclusion

The simulation results show, that the evasion test can be passed even on snow- or ice-covered roads. The prerequisite for this is an adaption of the SbB system with regard to an estimated road friction coefficient μE. With this method, the required accuracy of the friction value estimation can be determined and also information regarding the driving strategy (e.g. limitation of the maximum speed) can be generated in the event of a steering failure. It can also be seen that a mistake from wet to dry road does not lead to the required lateral offset \({d}_{Y}=\text{2.5 m}\), but to a lateral offset of 2 m. This means that the safety distance of 0.5 m between the target and the ego vehicle can no longer be maintained, but a collision with the target vehicle is basically avoided. If the SbB function is compared with that of an AEB, it can be seen that in the event of a steering error on a dry road, the triggering of an AEB is sufficient. However, if the road friction coefficient is reduced, the strengths of the SbB function come into play. In this case, the length of the avoidance distance is shorter than the required braking distance of an AEB, so that potential collisions due to faults in the steering system can be avoided.