Keywords

1 Introduction

Controllability in vehicle dynamics is generally considered as the driver’s ability to follow a desired course by having a direct feedthrough of control inputs via the steering wheel [1]. It is the task of functional safety to ensure this controllability even in the event of failures by limiting their effectiveness [2]. For this purpose, driving situation dependent limits for maximum permissible dynamic inputs are determined with the help of subject studies where system faults are systematically injected [3]. With the use of new actuators such as steer-by-wire systems, which offer additional degrees of freedom for lateral control, the limits of driving dynamics are of increasing interest for such controllability studies. Thus, this paper presents the design and the results of a subject study investigating the effects of faults of a steer-by-wire superposition function.

Previously published studies have largely defined the driving situation by variations of vehicle speed and lateral acceleration and are therefore based on common standards [4]. Neukum et al. [5] have previously investigated superposition steering system faults, but restricted their analysis to a maximum steering fault of 3° at tire level and only considered straight-ahead driving. Schneider [6] also examines higher yaw moment disturbances, triggered by faults in the ESC actuators, but also refers to straight-ahead driving. In both cases, the authors argue that straight-ahead driving represents a worst-case scenario when assessing lateral dynamic malfunctions [7]. However, it remains open to what extent the limits shift during transient maneuvers. To investigate the impact of steering torque recommendation, Mehrjerdian et al. [8] investigated the driver's reaction in an oversteer situation, reproducing side slip angles of approx. 3° using a kickplate.

In contrast to the research mentioned above, the study presented in this paper is intended to take the increased potentials of steer-by-wire based superposition systems into account by assessing larger fault amplitudes in more critical driving situations characterized by higher side slip angles. It focusses on faults which decrease the yaw reaction since this is the effective operating area of superposition steering systems in oversteering driving situations. The following research questions are mainly addressed: (1) Do steering faults of a superposition steer-by-wire system which decrease the yaw reaction lead to a reduction in controllability at the limits of driving dynamics? (2) How do the effects of such steering angle faults differ depending on the criticality of the initial driving situation?

To answer the research questions a subject study on a semi-dynamic driving simulator was conducted. The study design uses a new methodical approach to conduct subject tests at the limits of driving dynamics in a reproducible manner. Thus, the evaluation of the applied methods and evaluation criteria is also part of this paper’s contribution. Methods and design of the subject study are explained in more detail hereinafter.

2 Study Design

In total, 52 (age range = 20–65 years) participants took part in the study. The participants had a valid driving license for an average of 18 years and reported a yearly driving amount from 5.000 to more than 50.000 km. Two thirds of the participants reported having prior experience in simulated driving environments. 33% of the participants attended safe-driving trainings.

2.1 Driving Maneuver and Fault Injection

To investigate the research questions described above, the participants experienced the following driving situation: Driving on a rural road with a reference trajectory defined by a pylon lane and a center line. The road consisted of 32 consecutive curves with a length of 240 m and a radius of 75 m. To simplify the driving task and to standardize the driving situation, a constant speed of 86 km/h was set using cruise control. This speed results in stationary circular driving close to the maximum adhesion utilization of the tires.

The cornering was divided into three phases (Fig. 1b). In phase I, the vehicle was set to a defined side slip angle \({\beta }_{0}\) by a controlled traction stimulus at the rear axle in order to create a reproducible unstable driving situation. A reversible, initially yaw-decreasing steering angle fault \({\delta }_{\text{F}}\) was then injected near the curve apex over a period of 1 s. It is known from literature that compensatory actions of a driver generally start 250 ms after fault injection and are completed after 1.5 s [5]. Thus, the chosen fault duration is within the most critical time frame. Figure 1a shows an example of stimulus \(\overline{M }\) relative to the maximum drive torque and steering fault \({\delta }_{F}\).

Fig. 1.
figure 1

Side slip angle \(\beta \) and lateral deviation \(\Delta y\) based on the lateral stimuli \(\overline{M }\) and \({\delta }_{F}\) (a) during an example cornering with separated phases for destabilization (Phase I), fault injection (Phase II) and evaluation (Phase III) (b).

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2.2 Study Design and Procedure

A 4 × 4 within-subject design including additional baseline conditions was used to examine the independent variables initial side slip angle \({\beta }_{0,\text{m}}\) at the end of phase I and amplitude of steering angle fault \({\widehat{\delta }}_{\text{F},\text{n}}\). Each combination of \({\beta }_{0,\text{m}}\) and \({\widehat{\delta }}_{\text{F},\text{n}}\) defines a specific test condition (TCmn). During baseline conditions (BLm0), the participants experienced solely an unstable driving situation without any steering angle fault. The baseline conditions serve as controllability references of the nonlinear driving situation to objectively evaluate the criticality of injected steering angle faults. Each participant experiences conditions shown in Table 1 two times, once for left and once for right hand curves.

Table 1. Test conditions of the 4 × 4 within-subject design including baseline conditions (\({\widehat{\delta }}_{F,0}=\) 0°)
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Two dependent variables were chosen to evaluate the criticality of the test conditions defined above. First, the integral of the absolute side slip angle \(\left|\beta \right|\) during phase III generally indicates the instability of the driving situation [9]. Second, the maximum lateral deviation \({\left|\Delta y\right|}_{\text{max}}\) from the reference line is used to evaluate the ability to follow the track. In the following, both values are used as indicators for the criticality of a steering angle fault. In contrast to former controllability studies, no absolute criterion was used to assess pass or fail of a test condition. Instead, we use baseline-related values for the dependent variables:

$$ \overline{\smallint \left| \beta \right|}_{{{\text{TCmn}}}} = \frac{{\smallint \left| \beta \right|_{{{\text{TCmn}}}} }}{{{\text{max}}\left( {\smallint \left| \beta \right|_{{{\text{BLm}},{\text{R}}}} ;\smallint \left| \beta \right|_{{{\text{BLm}},{\text{L}}}} } \right)}} $$
(1)
$$ \overline{{\left| {\Delta y} \right|_{{{\text{max}}}} }}_{{{\text{TCmn}}}} = \frac{{{\Delta }\left| y \right|_{{{\text{max}},{\text{TCmn}}}} }}{{{\text{max}}\left( {{\Delta }\left| y \right|_{{{\text{max}},{\text{BLm}},{\text{R}}}} ;{\Delta }\left| y \right|_{{{\text{max}},{\text{BLm}},{\text{L}}}} } \right)}} $$
(2)

For each \({\beta }_{0,\text{m}}\) there is a baseline value for a left-hand (L) and a right-hand (R) curve. In Eqs. (1) and (2), the maximum of these two values is used to consider the most critical cornering without steering angle fault. According to these definitions, a baseline-related value greater than 1 corresponds to increased criticality as a consequence of a steering angle fault.

3 Results

For the analysis a data set of 48 valid test runs was used. A significance level of α = 0.05 was defined for all inferential statistical tests. The curve direction had no significant effect on the dependent variables (p∫β = 0.26, pΔy = 0.21). Therefore, the measurements for left and right curves were combined into a repeated measures model in the following analyses. The Mauchly test revealed that the requirement for sphericity of the measurement data is not fulfilled. The differences in variance are plausible due to the changed criticality of the driving situations caused by the independent variables. However, in the following evaluation Huynh-Feldt (HF) corrected p-values are used to compensate the violation of sphericity.

3.1 General Inferential Statistics

The results of a two-way ANOVA carried out for the repeated measures model are listed in Table 2. The results of the significance test (\({p}_{HF}\)) reveal significant effects of \({\beta }_{0}\) and \({\delta }_{\text{F}}\) on both dependent variables (\({p}_{HF}<.001\)). Based on the explained variation (\({\eta }^{2}p\)), the effect size \(f\) can be determined according to Cohen [10]. Following Cohen’s classification of the effect size \(f\), the measurement data shows an almost entirely strong effect (\(f > 0.4\)) of the within-subject factors on \(\overline{\smallint \left| \beta \right|}\) and \(\overline{{\Delta \left| y \right|_{max} }}\). A medium effect can be observed for the interaction \(\delta_{{\text{F}}} :\beta_{0}\) on the dependent variable \(\overline{\smallint \left| \beta \right|}\) (\(f > 0.25\)). These results show that the study design effectively captures the relationship between independent and dependent variables in a nonlinear driving situation.

Table 2. Metrics for the assessment of the chosen study design
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3.2 Descriptive Analysis

To determine maximum permissible yaw-decreasing faults, the medians of the baseline-related maximum lateral deviation \(\overline{{\Delta \left| y \right|_{max} }}\) and integral of side slip angle \(\overline{\smallint \left| \beta \right|}\) are examined. Figure 2 summarizes the results for all test conditions. Values greater than 1 correspond to an increase in the lateral deviation or integral of side slip angle compared to the baseline condition and thus indicate reduced controllability resp. Increased instability of the driving situation. The comparison shows that the impact of the steering angle fault \({\delta }_{F}\) on both dependent variables tend to diminish as the initial side slip angle \({\beta }_{0}\) increases. For lower initial side slip angles (\({\beta }_{0}=\left\{0^\circ , 5^\circ \right\})\) the median values of both dependent variables increase with increasing steering fault. Whereas for \({\beta }_{0}=\left\{10^\circ , 15^\circ \right\}\) even improvements in stability and controllability are observable.

Fig. 2.
figure 2

Median values for baseline-related values of the maximum lateral deviation and the integral of side slip angle

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This descriptive observation is supported by one-tailed t-tests. The results show a significant increase in the mean values of \(\overline{\smallint \left| \beta \right|}\) for all test conditions compared to the baseline conditions (\(p<.01\)) at lower initial side slip angles (\({\beta }_{0}=\left\{0^\circ , 5^\circ \right\})\). For higher initial side slip angles \({\beta }_{0}=\left\{10^\circ , 15^\circ \right\}\), no significant difference in mean values could be detected (\(p>.99\)).

No significant differences between mean values of baseline and test conditions were also found for \(\overline{{\Delta \left| y \right|_{max} }}\) at \({\beta }_{0}=15^\circ \) (\(p=.28\)). For \({\beta }_{0}=10^\circ \), there is a significant increase for test conditions with \({\delta }_{F}=\{50^\circ , 75^\circ \}\) (\(p<.01\)). For a steering angle fault of 25°, there is no significant difference in the means (\(p=.78\)).

4 Discussion

The subject study identified significant effects caused by steering angle faults which decrease the yaw reaction in the nonlinear region of driving dynamics. Due to the high effect sizes, the proposed study design is suitable for investigating controllability in oversteering driving situations. Based on the descriptive analysis, the initially formulated research questions can be answered as follows: Yaw decreasing steering angle faults can reduce controllability at the limits of driving dynamics. The impact depends on the amplitude of the steering angle fault and the criticality of the initial situation. The steering angle faults investigated in this study were even found to improve controllability as the initial side slip angle increased. In summary, only the test conditions TC31, TC41, TC42 and TC43 (see Table 1) show an improvement in stability and controllability according to the analysis in Sect. 3.2 and can therefore be assessed as permissible. In further investigations, additional grid points of the independent variables, especially at smaller side slip angles, should be examined to enable correlation analysis.