Accuracy requirements for the road friction coefficient estimation of a friction-adaptive automatic emergency steer assist (ESA)

The number of traffic accidents resulting in personal injury and property damage is increasingly being reduced by effective advanced driver assistance systems (ADAS). Nevertheless, many traffic accidents still cannot be prevented today because they are due to wet, snow- and ice-covered roads. For this reason, the Institute of Automotive Engineering (IAE) of the Technical University of Braunschweig is investigating the road friction coefficient sensitivity and adaptation of advanced driver assistance systems (ADAS) currently in series production from 2018 to 2021 as part of the ‘Road Condition Cloud’ research project funded by the German Research Foundation (DFG) to increase driving safety, particularly on wet, snow- and ice-covered roads. In this article, the road friction coefficient sensitivity and adaptation of an automatic emergency steer assist is simulatively investigated. This assist overrides the driver to automatically execute an evasive maneuver. The driving maneuver used is a standardized obstacle-avoidance maneuver that is simulatively repeated on a dry, wet, snow- and ice-covered road. The road friction coefficient sensitivity shows that this test is already failed on a wet road because the simulated vehicle does not pass the second lane without errors. Subsequently, a road friction coefficient adaptation of the emergency steer assist is investigated. This adaptation varies the maximum lateral acceleration of the evasive trajectory depending on an estimated value of the road friction coefficient in order not to exceed the maximum adhesion coefficient of the wheels during the evasive maneuver. Ideally, the estimated value matches the true road friction coefficient so that the second lane is passed without errors even on a wet, snow- and ice-covered road. In contrast, an existing difference determines whether the second lane is reached. Finally, the necessary accuracy requirements of the road friction coefficient estimation are determined in an novel estimation error diagram. A road friction coefficient adaptation increases the driving safety of driver advanced assistance systems (ADAS) that are in series production today and future highly automated driving functions (HAF) and is necessary for automated driving because the driver is not present as a fallback level. The described results were presented before in [1].


Introduction
The number of traffic accidents resulting in personal injury and property damage is increasingly being reduced by effective advanced driver assistance systems (ADAS). Nevertheless, many traffic accidents still cannot be prevented today because they are due to wet, snow-and ice-covered roads [2]. For this reason, the road friction coefficient in particular determines driving safety in road traffic.
For legal reasons, automatic emergency brake assists (AEB) currently in series production only become active when it is no longer possible to take evasive action and the remaining relative distance falls below the last point to steer distance. Because the collision-avoiding braking distance exceeds the last point to steer distance at high driving speeds, automatic emergency brake assists cannot always avoid a collision, but only at low driving speeds.
In contrast, emergency steer assists (ESA) in series production today become active when rapid steering by the driver is detected at low relative distances. These assists superimpose an additional torque on the driver's steering wheel torque, which supports the driver in taking evasive action [3, pp. 755-758]. As a result, a possible collision can always be avoided if the driver begins to steer before the last point to steer distance is reached.
Currently in series production, the last point to brake and steer distance is calculated by the constant road friction coefficient of a dry road, so that collisions on wet, snow-and icecovered roads are not prevented. For this reason, a frictionadaptive emergency steer assist (ESA) that automatically execute an evasive maneuver has high potential to increase driving safety even on wet, snow-and ice-covered roads. This friction-adaptive assist can be implemented through technical advances in the development of sensors for reliable clearance detection, since all other sensors and actuators are already available in series production [3, pp. 553-577] [4, pp. 1021-1040] [5].
For example, active lane keeping assist (LKA) systems in series production today use one or more cameras to reliably determine the vehicle position within the lane. This position determines the additional torque that is superimposed on the driver's steering wheel torque [6][7][8].
Today, a difference is made between two groups of emergency steer assists. The assists of the first group superimpose an additional torque on the driver's steering wheel torque, which assists the driver in taking evasive action. These assists are described in [9][10][11][12][13][14][15] and use as actuator, for example, the electric motor of an electromechanical steering system (EPS). Because the assists in this group cooperate with the driver, driver models are required that represent human driving behavior, especially in emergency situations such as an evasive maneuver. These models are described, for example, in [16][17][18][19][20]. Further, in [21][22][23][24][25] different evasive trajectories are compared with the human driving behavior.
The assists of the second group superimpose an additional angle on the steering wheel angle of the driver, which cannot be overridden by the driver. These assistants are studied in [26][27][28][29][30][31], among others, and use, for example, a superposition transmission as actuator. Because the assists in this group do not interact with the driver, driver models are not required.
Critical driving situations occur when the driver fails to steer appropriately in an emergency situation such as an evasive maneuver or a sudden lane change. In [32], traffic accidents are studied and recognized that over 70 % of deadly collisions occur on the shoulder or median strip when the driver leaves his lane. In [33,34], accidents involving a sudden lane change are examined and it is described that many drivers often recognize an impending collision too late. In [35] it is explained that the average driver often notices other road users too late to avoid a collision.
Already in [36] it is recognized in the context of driving tests with a suddenly appearing obstacle that only experienced drivers stabilize the vehicle during an evasive maneuver. Inexperienced drivers are often unable to do so, controlling the vehicle response that occurs during evasive action. In [37], a user study recognizes that the average driver often reacts too late to avoid an impending collision by taking evasive action. The results of this study indicate that only an automatic emergency steer assist that overrides the driver can avoid an impending collision, especially on wet, snowand ice-covered roads.
In [11,14,28,38,39], driving simulator studies are described that superimpose an additional torque on the driver's steering wheel torque during evasive action when rapid driver steering is detected. The driving simulator study in [38] illustrates that an emergency steering assist improves driver behavior when a collision is imminent. The study in [39] examines the vehicle dynamics during an evasive maneuver and highlights that driving stability is increased because the emergency steering assist reduces the overshoot of the yaw rate. In [28], over 70 % of users favorite an automatic emergency steering assist that overrides the driver in emergency situations. The study in [14] illustrates that an emergency steering assist reduces the number of collisions and decreases the humans reaction time by at least 100 ms.
In [11], two studies with a sudden lane change are described. In the first study, the driver's steering wheel torque is superimposed by an additional torque. To increase driving stability, the steering wheel angle is superimposed in the second study by an additional angle. The first study emphasizes that an additional torque is not always sufficient to prevent the vehicle from leaving the lane during a sudden lane change. In contrast, the second lane are almost always traversed without error when the driver is supported by an additional angle in the second study.
In [40], an emergency steer assist investigated in real traffic is described. This is activated during lane changes with a high probability of leading to a critical driving situation to stabilize the driving situation or prevent a collision. This assist is able to override the driver.
More advanced approaches in [41][42][43] investigate the combination of an emergency brake assist with an emergency steer assist that is in series production today. The trade-off between braking and evasion is described in [9] as a function of relative speed and road friction coefficient. In [27,31,44] approaches are described that determine the last point to steer distance as a function of the road friction coefficient in order not to exceed the maximum adhesion coefficient of the wheels, especially on wet, snow-and icecovered roads.
In [42], a steer assist is described that additionally brakes individual wheels as desired in order not to exceed the maximum adhesion coefficient of the inside and outside wheels of the curve. This additionally increases the yaw rate and shortens the length of the calculated evasive trajectories.
In this paper, the road friction coefficient sensitivity and adaption of an automatic emergency steer assist is simulatively investigated using a validated dual track model. This assist overrides the driver to automatically execute an evasive maneuver along a calculated evasive trajectories. The driving maneuver used is the obstacle-avoidance maneuver standardized in ISO 3888-2, which is transferred to the simulation and repeated on a dry, wet, snow-and ice-covered roadway [45].

Method
The method used takes into account the true road friction coefficient R and an estimated value E of the true road friction coefficient (Fig. 1).
In this article, the road friction coefficient sensitivity and adaptation for the four classes of a dry, wet, snow-covered and ice-covered road are investigated (Fig. 2) and representative road friction coefficients are assigned to them (Table 1).

Function
The trajectory used by the emergency steer assist is calculated by an optimizer. This optimizer minimizes the evasive trajectory length x e depending on two input parameters. These two input parameter are the lateral offset y e to be achieved and the predefined maximum lateral acceleration a y,max of the evasive trajectory (Fig. 3).
The y-coordinate y(x) of the evasion trajectory are calculated by a seventh degree polynomial, the lateral offset y e to be achieved, and the trajectory length x e .
The curvature (s) of the evasive trajectory is determined by the first and second derivatives of the coordinates x and y(x) with respect to the arc length s.
For this, the differential quotient ds of the arc length s is calculated by the differential quotients dx and dy of the coordinates x and y(x).
The yaw angle (s) of the evasive trajectory is calculated by integrating the curvature (s) . This is used for the yaw angle control of the emergency steer assist and described in Sect. 5.
The yaw rate ̇(s) of the evasive trajectory is calculated by the curvature (s) and initial driving speed v 0 and a side slip angle is neglected.  The lateral acceleration a y (s) of the used evasive trajectory is calculated like the yaw rate ̇(s) by the curvature (s) and initial vehicle speed v 0 .
The maximum curvature max of the evasive trajectory is determined by the maximum lateral acceleration a y,max and is not exceeded by this optimizer.
The maximum lateral acceleration a y,max of the evasive trajectory is calculated in series production today by the acceleration of gravity g and in particular the constant road friction coefficient R of a dry road. The factor k describes the ellipse of kamm's circle and takes values between 1.0 and 0.8 for current summer and winter tires [3, pp. 899-904].

Driving maneuver
The road friction coefficient sensitivity and adaption are investigated simulatively with a validated dual track model as part of the obstacle-avoidance maneuver of ISO 3888-2.
The test parameters and dimensions of this driving maneuver are described in [45]. The dual track model used includes a validated Magic Formula tire model of version 5.2 according to [46] with the individual parameters of a current winter tire. This test is used to subjectively evaluate the vehicle dynamics during an evasive maneuver. An initial lane is driven through at a constant driving speed and then changed to a second lateral offset lane. This test is passed if the lane markings are not overrun.
The lengths of the initial and offset lanes are constant. The widths are determined depending on the individual vehicle width ( Table 2).
In this paper, a vehicle width of 2 m is used. Thus, the lane centers of the initial and offset lanes are 3.725 m apart. The described dimensions are shown in the simulation and additionally marked by pylons (Fig. 4).
The evasive trajectory from Fig. 4 is calculated by Eq. 1 to Eq. 8 from Sect. 3 and the representative road friction coefficient R of the dry road (Table 1) from Sect. 2 (Fig. 5a). Figure 5b shows the lateral acceleration a y (x) of the evasive trajectory calculated by Eq. 7. This is limited by Eq. 8 and an assumed factor k of 0.9 [47]. Figure 6a shows the yaw angle (x) of the evasive trajectory calculated by Eq. 4. This is determined by integrating the curvature (s).

Control
The architecture of the steering wheel angle control used in this paper is described in detail in [48]. This includes a feedforward control (FFC) as well as a yaw rate and lateral offset control.
In the obstacle-avoidance maneuver, the used lateral offset control described in [48] is replaced by a simple yaw angle control to increase driving stability when the simulated driving lane of the dual-track model increasingly deviates from the calculated evasive trajectory (Fig. 7). Figure 8 shows the operation of the steering wheel angle control when the yaw angle (x) and the yaw rate ̇(x) of the evasive trajectory from Fig. 6 are used in the simulation. The yaw angle and yaw rate differences between the evasive trajectory and the dual-track model (DTM) are converted into a steering wheel angle by the steering wheel angle control. Figure 9a shows the steering wheel angles resulting from the yaw angle and yaw rate differences. Additionally, the steering wheel angle of the feedforward control is shown. It can be seen that the steering wheel angle of the feedforward control relieves the yaw rate and yaw angle control. The maximum lateral offset occurring between the evasive trajectory and the dual-track model (DTM) during simulation is 0.1 m, confirming the operation of the steering wheel angle control on a dry road.
Finally, Fig. 10 complements the evasive trajectory from Fig. 4 with the simulated lane of the dual-track (DTM) model.

Road friction coefficient sensitivity analysis
The road friction coefficient sensitivity analysis determines to what extent the true road friction coefficient R influences the operation of the automatic emergency steer assist (ESA). To determine this, the true road friction coefficient R is gradually reduced and the lateral offset y end at the obstacle position is calculated.
In this chapter, a function currently in series production without a friction coefficient adaption with a constant evasive trajectory is investigated. The maximum lateral acceleration a y,max of this evasive trajectory is calculated by Eq. 8 and the representative friction coefficient of the dry road from Table 1. Figure 11 shows the road friction coefficient sensitivity determined in the simulation. The drawn lines illustrate a failure to reach the second lane and an occurring rear collision with the obstacle at the obstacle position (Table 3).
It can be seen that the lateral offset y end continuously decreases with the road friction coefficient R . The second lane is not reached when the road friction coefficient R decreases from 1.0 to 0.7. A rear collision is caused when the friction coefficient R is 0.4. Table 4 contains the lateral offset y end of the four road condition classes from Table 1 and illustrates that the obstacleavoidance maneuver is only passed on a dry road. For this reason, the operation of a emergency steer assist has to be calculated in a friction-adaptive manner to reach the second (9) y end = y(x = 100 m)    lane even on wet, snow-and ice-covered roads and to pass the obstacle-avoidance maneuver.

Road friction coefficient adaption
The road friction coefficient adaption calculates the maximum lateral acceleration a y,max of the evasive trajectory by an estimated value E of the true road friction coefficient. For this purpose, the road friction coefficient R from Eq. 8 is replaced by the estimated value E . Figure 12a shows the four evasive trajectories calculated by Eq. 8 and the representative estimated values of a dry, wet, snow and ice covered road from Table 1. All evasive trajectories achieve the lateral offset y end of the second lane of 3.725 m at the obstacle position. This friction adaption reduces the maximum lateral acceleration a y,max on wet, snow-and ice-covered roads so as not to exceed the maximum adhesion coefficient of the wheels. The maximum lateral acceleration a y,max decreases continuously with the estimated value E (Fig. 12b).
To investigate the evasive trajectory calculated with the estimated value E of the wet road, the road friction (10) a y,max = E ⋅ g ⋅ k coefficient R is gradually reduced and the lateral offset y end at the obstacle position (Fig. 13a) and the additional evasive distance x add are calculated (Fig. 13b).
The second lane is always achieved as long as the true road friction coefficient R equals or exceeds the estimated value E . The second lane is not reached if the true road friction coefficient R decreases from 1.0 to 0.4.
The additional evasive distance x add is used as a second characteristic parameter if the second lane is achieved although the estimated value E does not match with road friction coefficient R .
This method is repeated for the evasive trajectories calculated using the estimated value E of the snow-and ice-covered road, and the lateral offset y end at the obstacle position and the additional evasive distance x add are calculated. These characteristic parameters are used in Chap. 8 to determine the necessary accuracy requirements of the road friction coefficient estimation. (11) x add = x e E − x e R Fig. 12 Variation of the calculated evasion trajectory Fig. 13 Road friction coefficient adaption of the evasion trajectory with the estimated value of the wet road 1 3

Accuracy requirements of the road friction coefficient estimation
To determine the accuracy requirements of the road friction coefficient estimation, the characteristic parameters calculated in Chap. 7 are presented in an estimation error diagram. For this purpose, the estimation error Err is calculated as the difference between the estimated value E and road friction coefficient R and a tolerance band of the characteristic parameters is defined.
This diagram was developed in [49] as part of an industrial cooperation with the Research Association of Automotive Technology (FAT) and was presented for the first time in [50] using the example of a friction-adaptive automatic emergency brake (AEB). Figure 14 shows the estimation error diagram of the friction-adaptive automatic emergency steer assist with a tolerance band determined by the lateral offset of 3.225 m from Table 3 and an additional evasive distance x add of 10 m. For this reason, the accuracy requirements for reaching the second lane are approximated in this estimation error diagram.
The four lines are equal to the constant estimated value E of a dry, wet, snow and ice covered road ( Table 1). The line with the estimated value E of the dry road runs only to the right, because the road friction coefficient R can only be equal or smaller than the estimated value E of the dry road. The largest estimation error Err corresponds to the road friction coefficient E of the ice-covered road.
In contrast, the lines with the constant estimated values E of the wet and snow-covered road run both to the right and to (12) Err = E − R the left, because the road friction coefficient R can be both smaller and larger than the estimated values E of these roads.
The lines with the estimated value E of the ice-covered road run only to the left, because the road friction coefficient R can only be equal or greater than the estimated value E of the ice-covered road. The largest estimation error Err corresponds to the true road friction coefficient E of the dry road.
To determine the accuracy requirements, the maximum permissible estimation errors Err,max in the tolerance band are determined. The second lane is not reached if these are exceeded (Table 5).
Subsequently, the maximum permissible estimation errors Err,max from Table 5 are converted into relative estimation errors Err,rel to present them as a function of the estimated value E ( Table 6). The relative estimation error Err,rel must not exceed about 30 % of the estimated value E at a driving speed of 50 km/h to reach the second lane.  determined by the lateral offset of 2 m from Table 3. For this reason, the necessary accuracy requirements to avoid an approximate rear collision are determined in this diagram. The additional evasive distance x add is consistent with that in Fig. 14.
The maximum permissible estimation error Err,max increases when the used permissible lateral offset y end is increased. The relative estimation error Err,rel must not exceed about 60 % of the estimated value E at a driving speed of 50 km/h to avoid a rear collision ( Table 7).
The tolerance band used in Fig. 14 and Fig. 15 is exemplarily determined by an additional evasive distance x add of 10 m. The acceptance of this band is currently being determined in further user studies on the institute's own Dynamic Vehicle Road Simulator (DVRS).

Conclusion
The number of traffic accidents with personal injury and property damage is increasingly being reduced by effective advanced driver assistance systems (ADAS). Nevertheless, many traffic accidents still cannot be prevented today because they are due to wet, snow-and ice-covered roads.
In this paper, the road friction coefficient sensitivity and adaption of an automatic emergency steer assist are simulatively investigated. This assist overrides the driver to automatically execute an evasive maneuver. The driving maneuver used is the obstacle-avoidance maneuver standardized in ISO 3888-2, which is transferred to the simulation and repeated on a dry, wet, snow-and ice-covered roadway.
The road friction coefficient sensitivity shows that this test is already failed on a wet road because the simulated vehicle does not pass the second lane without errors. It can be seen that the lateral offset decreases continuously with the true road friction coefficient. The second lane is not reached when the road friction coefficient decreases from 1.0 to 0.7. A rear collision is caused when a road friction coefficient of 0.4 is present. For this reason, the operation of an emergency steer assist must be calculated in a friction-adaptive manner to reach the second lane even on wet, snow-and ice-covered road.
The road friction coefficient adaption calculates the maximum lateral acceleration of the evasive trajectory using an  estimated value of the true road friction coefficient. This adaption reduces the maximum lateral acceleration on wet, snow-and ice-covered roads so that the maximum adhesion coefficient of the wheels is not exceeded. Ideally, the estimated value matches the true road friction coefficient so that the second lane is passed without errors even on a wet, snow-and ice-covered road. In contrast, an existing difference determines if the second lane is reached. In order to evaluate the adaption, the additional evasive distance is calculated as a second characteristic parameter in addition to the lateral offset. This distance is used as a characteristic parameter if the second lane is reached although the estimated value does not match with road friction coefficient.
To determine the necessary accuracy requirements of the road friction coefficient estimation, the calculated characteristic parameters are presented in an estimation error diagram. For this purpose, the estimation error is calculated as the difference between the estimated value and the true road friction coefficient. The tolerance band is determined exemplarily by a lateral offset of 3.225 m and 2 m as well as exemplarily an additional evasive distance of 10 m. Subsequently, the maximum permissible estimation errors are converted into relative estimation errors to present them as a function of the estimated value. The relative estimation error must not exceed about 30 % of the estimated value at a driving speed of 50 km/h to reach the second lane. The maximum permissible estimation error increases as the tolerance band is increased. The relative estimation error must not exceed about 60 % of the estimated value at a driving speed of 50 km/h to avoid a rear collision.
Author contributions TA and JI wrote the manuscript and prepared the figures. All authors reviewed the manuscript.
Funding Open Access funding enabled and organized by Projekt DEAL.

Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.