Keywords

1 Introduction

Rear-wheel steering is available and has demonstrated improved performance in maneuverability at low speeds and driving stability at higher speeds [1, 2]. Further, even more rear-wheel steering capability has been introduced, e.g., in vehicles such as the GMC Hummer or the Hyundai Ioniq 5 with diagonal driving. These new developments in rear-wheel steering may also be used for increased safety, and the potential for this is the research question of this paper.

Dynamic optimization has been used to study critical vehicle maneuvers in at-the-limit driving to find control principles and devise control schemes [3,4,5]. To specifically study safety, recent articles have used the optimization criterion of maximum entry speed into a scenario (e.g., a constant radius 90\(^\circ \)-curve) [6, 7]. By definition, this computation gives the maximum speed that can still be handled when entering a situation. Furthermore, it has been used to analyze safety potential using data from crashes [8], and for developing control principles and schemes [7]. This paper uses this methodology to investigate the characteristics and opportunities of adding rear-wheel steering capabilities.

2 Steering Topologies, Scenarios, and Optimization

Here, critical cornering at the limit of friction is studied for different steering topologies with varying amounts of steering capability and different curve radii. The vehicle is described by a double-track dynamics model as depicted in Fig. 1. The dynamic equations and parameters are the same as in [5], with fixed distribution between rear and front braking based on the proportional weight on each axle, hence one control input for the total amount of braking. The steering topologies studied are the traditional front-wheel steering (FWS), rear-wheel steering (RWS), all-wheel steering (AWS), and individual-wheel steering (IWS). The difference between the two latter is that in AWS, wheels on the same axle have the same steering angle, whereas in IWS, the steering angle is individually controlled for each wheel. The front-wheel steering angle is, for all cases, limited to a standard \(\pm 30^\circ \). However, the study varies the rear-wheel steering capability by examining maximum rear steering angles with 0, 1, 3, 5, 7, 10, 15, 20, 30, 40, and 50\(^\circ \), respectively. Also, [9] studied different steering architectures in combination with differential braking for a lane-change maneuver. The scenarios here, all consist of a constant radius 90\(^\circ \)-curve as shown in Fig. 1, and it is studied for the curve radii 10, 20, 30, 50, and 70 m. The maximum entry speed that can be managed for these vehicles and scenarios is computed. This computation is based on dynamic optimization as shown in Fig. 1, where the optimization formulation, the modeling used, and the numerical solution principles follow the methods described in [5,6,7].

Fig. 1.
figure 1

The studied driving scenario, the optimization formulation, and the vehicle dynamics model for the scenario, [5].

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3 Results

It turns out that the most interesting results from this research are about the initial driving strategy to avoid going off-road. Therefore, for Figs. 2, 3 and 4, the focus is on the first part of the constant radius 90\(^\circ \)-curve.

3.1 FWS, RWS, AWS, and IWS

For the same case as in Fig. 1, i.e., a curve with a radius of 30 m, the first part of the optimal solutions are presented in Fig. 2 for the four steering topologies FWS, RWS, AWS, and IWS. The first part of the curve is shown in the top left plot. The lane’s width indicates the allowed area for the vehicle’s CoG, and the box’s width indicates the vehicle’s track width (tire-to-tire). The angle of the vehicle box shows the vehicle’s orientation.

Fig. 2.
figure 2

Optimal solutions for FWS, RWS, AWS, and IWS. The first row shows initial paths (left) and velocities (right). The second row shows the individual steering angles, the third row the heading, the total longitudinal and lateral forces on the vehicle, and the total moment acted on the vehicle. All variables are presented in the vehicle coordinate system, referring to the notation in Fig. 1.

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It is seen that for this scenario and these parameters (with a maximum rear steering angle of 5\(^\circ \)), the rear wheels change \(\delta _3, \delta _4\) from negative to positive, which means that the optimal strategy for both AWS and IWS is to change from out-of-phase steering to in-phase diagonal driving. This will be studied in more detail in Sect. 3.2. Another major conclusion from Fig. 2, and in other performed computations, is that the difference between AWS and IWS is minor. Thus, the following will only present detailed results for AWS.

3.2 Characteristics for Different \(\delta _{\max }\) and R

For AWS, wheels on the same axle have the same steering angle, so \(\delta _1=\delta _2\) with maximum \(\pm 30^\circ \) and \(\delta _3=\delta _4\) with maximum denoted \(\delta _{\max }\). As pointed out in Fig. 2, there is a shift from out-of-phase steering (where \(\delta _1,\delta _3\) have different signs) to in-phase diagonal driving (where \(\delta _1,\delta _3\) have the same sign). It can also be seen (top left) from the vehicle orientation that AWS prioritizes diagonal driving compared to FWS. To study these aspects more in detail, Fig. 3 presents the optimal solutions for \(R=30 \ \textrm{m}\) and for maximum rear-wheel steering capability \(\delta _\textrm{max}=0,\ 1,\ 3,\ 5,\ 7,\ 10,\ 15,\ 20,\ 30,\ 40, \textrm{and}\ 50^\circ \). As before, the case of \(\delta _{\max }=5^\circ \) leads to a shifting strategy in the rear-wheel steering, whereas for \(\delta _{\max }=30^\circ \), the strategy is diagonal driving during the whole start of the maneuver. Thus, it seems advantageous to start rotating the vehicle for less steering capability of the rear wheels. Still, for larger steering capability, it seems better to move sideways directly and then rotate in the turn when the critical part is cleared. This is manifested in the top left part of Fig. 3, where it can be seen that 10 m into the curve, the vehicle rotates less when \(\delta _{\max }\) is increased, and thus, has focused more on diagonal driving than rotation.

Fig. 3.
figure 3

Initial paths, velocities, and steering angles for AWS (for \(R=30 \, \textrm{m}\)) with different maximum rear steering angles, \(\delta _\textrm{max}\). FWS corresponds to \(\delta _\textrm{max}=0^\circ \).

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Concerning the shifting in steering strategy, Fig. 4 shows the rear-wheel steering solutions for the curve radii \(R=\ \)10, 20, 30, and 50 m. The sharper the curve, i.e., the smaller the R is, it is natural that there is more out-of-phase steering on the rear wheels to get the vehicle to rotate. The switching point from out-of-phase steering (\(\delta _3<0^{\circ }\)) to in-phase steering (\(\delta _3>0^{\circ }\)) depends on the curve radius. For \(R=10 \ \textrm{m}\), the switching point comes earlier with smaller \(\delta _{\max }\). On the other hand, for \(R\ge 20 \ \textrm{m}\), the switching point comes later with smaller \(\delta _{\max }\).

Fig. 4.
figure 4

Rear steering angle for FWS and AWS for different radii R. The color coding for \(\delta _{\max }\) is the same as in Fig. 3.

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Fig. 5.
figure 5

Left: Maximum entry speed for different maximum allowed rear-steering angles and curve radii. Right: The relative difference in kinetic energy for different maximum allowed rear-steering angles and curve radii.

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3.3 Improved Safety

Having observed interesting steering behaviors in the previous subsection, the next question is what crash avoidance or mitigation could mean regarding saved lives and reduced injury for increased traffic safety. As seen in the top right of Fig. 2, higher entry speeds can be handled when using rear-axle steering in AWS or IWS, which is an indicator of improved safety. To provide more details, the left plot in Fig. 5 shows the maximum entry speed as a function of \(\delta _{\max }\) and R. Recall that \(\delta _{\max }=0^{\circ }\) corresponds to the regular FWS, so the curves in the left plot show the improvement in maximum entry speed compared with FWS as an evaluation of safety. When it comes to saved lives and mitigated injuries, a full analysis of crash databases could be made [8]. However, a simple way to get a first estimate is to compare the kinetic energy, which is made in the right plot of Fig. 5, where the relative improvement in terms of kinetic energy, \(\varDelta v^2/v^2\), is plotted. Compared to regular FWS, being the origin, the improvement is almost linear when adding just some steering capability in terms of \(\delta _{\max }\) for the rear wheels, but then it flattens out. Using this simple way would indicate a safety improvement in the order of 10 % already for \( \delta _{\max }=7^{\circ }\), and then it starts to flatten out at \(\delta _{\max }=20^{\circ }\) to eventually reach about 20 %. Another observation that was not foreseen is that all curves in the right plot of Fig. 5 are so similar, except for \(R=10\) m.

4 Conclusions

The steering capabilities of the rear wheels are introduced mainly to improve maneuvering or stability. Still, here, they are investigated in terms of improved handling of critical situations even at the limit of friction. The main takeaways are as follows. It was shown that already small steering capabilities on the rear wheels could give significant safety improvements, e.g., for seven degrees as the maximum rear-wheel steering angle it provides in the order of approximately 10 %, and for additional steering capability, it gives approximately double. Interestingly, the initial steering strategy for avoiding going off-road depends on the steering and scenario parameters and varies from initial out-of-phase steering (to achieve vehicle rotation) to initial in-phase steering (parallel driving) to get the vehicle away from the critical road border.