**4. Results**

As the primary motivation of this study is to present lane-change manoeuvre duration calculation, a comparison was made between actual manoeuvre times (durations), documented during all the test series, and durations calculated based on the values of maximal lateral acceleration and lateral distance by using the original Kovaˇrík Equation (3), the Weiss Equation (4), and the adjusted Kovaˇrík Equation (6). An example of the calculation of manoeuvre durations using individual equations is provided in Table 3.

**Table 3.** A sample of actual manoeuvre times and durations, calculated using the formulas presented and data from test series 3.


The manoeuvre durations calculated using the methods presented are compared by the box diagrams shown in Figure 4.

**Figure 4.** Equation comparison.

Using SPSS Statistics software, the Kolmogorov–Smirnov test was carried out, verifying that the data set did not have a normal distribution. The results of the test are listed in Table 4.


As the data did not have a normal distribution, the Mann–Whitney nonparametric test was used for comparing the equations. Tables 5 and 6 compare the results of actual manoeuvre durations, calculated by using Equation (6).

**Table 5.** Mann–Whitney test results.


**Table 6.** Mann–Whitney test results.


The result of this analysis indicated no statistically significant difference between manoeuvre duration (*p*-value 0.846), calculated by the adjusted Kovaˇrík Equation (6) and actual documented times.

By using the Mann–Whitney test to compare other calculation methods with actual manoeuvre durations, it was found there was a difference between the manoeuvre durations, calculated by the original Kovaˇrík Equation (3) and Weiss Equation (4) compared to actual documented times, with *p*-values 0.000 and 0.000, respectively.

#### **5. Discussion**

Several mathematical methods of lane-change manoeuvre duration calculation were explored. These methods assume vehicle movement in a curve consisting of two circular arcs or that the vehicle's lateral acceleration is of sinusoidal shape and one-period duration. Driver's reaction time needed to perform the steering manoeuvre is not taken into consideration, nor is the response time of the steering mechanism to turning of the steering wheel.

The calculation of the manoeuvre duration presented was based on the results of experimental research carried out with a wide range of vehicles (from supermini vehicles to crossover SUVs) and in various road conditions (from dry to wet road surface). The calculation base includes only the manoeuvres that were successfully performed without vehicle stability loss or exceeding the test track borders.

Exploration of the lane-change duration calculation was introduced in [15]. Based on experimental testing with modern vehicles on a dry road surface (also described in [16]), the constant value was recalculated to 2.6.

Research determining lane-change distance during obstacle avoidance based on experimental measurements was introduced in [13]. This research has similar characteristics to the present study; however, the main goal was to present the calculation of longitudinal lane-change distance (i.e., not the overall distance travelled by the vehicle).

Table 7 lists the ranges of vehicle approach speeds and maximal documented lateral acceleration in the individual test series.


**Table 7.** Documented lateral acceleration in the individual test series.

The first step in the manoeuvre time calculation was to define the beginning and the end of the manoeuvre. In this paper, the driver's first steering intervention is regarded as the beginning of the lane-change manoeuvre and reaching the maximal lateral distance is considered to be the end of the manoeuvre, similar to [14]. An argument can be made that the end of the manoeuvre should include complete stabilization at the end of the manoeuvre—this, however, seems difficult to apply in actual cases, as keeping control over the vehicle depends largely on the driver's skills, vehicle speed, and severity of the manoeuvre.

The impact of tyre pressure on achievable lateral acceleration was observed during 2016 test series 1. From what was observed while testing both vehicles, when tyre pressure was set to 30% over its nominal value, the maximal achievable lateral acceleration values increased on average by 0.5 m/s2. When the tyre pressure was set 30% below its nominal value, the maximal achievable lateral acceleration was decreased on average by 0.8 m/s2.

A positive effect of electronic stability systems on crash occurrence was explored in numerous studies, such as [17–19]. After all, the generally accepted positive effects of these driving assistants are seen through mandatory equipment for this feature in all newly manufactured vehicles, given by the European Commission since 2014.

Compared to the older calculation methods, the method presented is based on tests of modern vehicles currently used on European roads. The proposed method also clearly defines the manoeuvre's beginning and end.

The statistical analysis found a new mathematical constant value C, adjusting the Kovaˇrík Equation (3). This way, it is possible to apply lane-change manoeuvre time (duration) calculation with only adjustment to its mathematical constant to 2.93, giving it the form of the Equation (6).

The calculation presented is based on several simplifications; mainly, the complex interaction of the vehicle and vehicle tyres with the road surface is represented only by lateral acceleration. This simplification should not be seen as a disadvantage since, in many crash reconstruction cases, many of the variables are unknown (such as slip angle, yaw motion, and others) and cannot be precisely determined, e.g., due to the analyzed vehicle's destruction. Determination of the maximal lateral acceleration thus depends on crash the reconstructionist, who can base this value on road surface condition, vehicle technical state, and vigour of the manoeuvre. The only other unknown variable in the equation is lateral distance, based on either on-road marks or estimated vehicle trajectory.

It is important to note that all the measurements were done in a testing (safe) environment. Thus, it was possible to reach the vehicles' driving limits. To use this calculation for the driver's avoidance option calculation (as a viable option, e.g., to prevent collision with an object in the driver's lane), it is recommended to use lower values of lateral acceleration, as an "average driver" is not necessarily able to execute this manoeuvre while driving at the vehicle's stability limit (either due to fear of slipping or due to lack of driving experience). The lane-change manoeuvre is not used as often as braking for crash avoidance, as seen in [20,21] as the manoeuvring mechanism is more complex than braking.

Thus, a guideline is proposed, similar to the one used for braking on the wet road surface, using only half of the maximal achievable acceleration on the given surface.

While modern vehicles can reach a lateral acceleration of up to 10.0 m/s2 during the lane-change manoeuvre even on the wet surface, it is the limit of the vehicle and not necessarily of the vehicle's driver.

Furthermore, it is important to note that requiring an evasive manoeuvre from the driver as possible collision prevention is complicated by manoeuvre complexity and the necessity to check not only the road ahead of the driver, but also whichever direction the driver chooses to evade, and also the immediate vicinity of the vehicle, to ensure the safety of the manoeuvre. This naturally increases the driver's reaction time (compared to "simple" braking).

#### **6. Conclusions**

Based on the analysis of the older calculation methods, it was determined that these methods were based on theory and, in some cases, outdated data.

For this reason, the applicability of these calculation methods was explored. Four series of driving tests were carried out using a wide variety of modern vehicles on the test track in dry and wet conditions. To ensure repeatability of the experiments, the research was performed according to the procedure prescribed by ISO 3888-2 standard, which defines the conditions and procedure of the obstacle avoidance manoeuvre. This procedure was later modified in the sense of variation of lateral distances.

Considering the calculation methods presented, it was concluded that the most prudent approach is to explore the value of the mathematical constant C. Based on a statistical analysis of the research results on a sample of 108 test drives, a new mathematical constant was presented, providing the most reliable results (there was no statistically discernible difference between the measured manoeuvre duration and the manoeuvre duration calculated by the modified formula).

The manoeuvre duration calculated by this formula represents the period between the driver's first steering input and reaching the vehicle's maximal lateral distance during the manoeuvre (without considering possible slight adjustments to steering in case of vigorous manoeuvring at the end of the manoeuvre).

One of the main components of the calculation is the value of maximal lateral acceleration, which encompasses both the road conditions (i.e., adhesion) and intensity of the manoeuvre, and its value should be carefully evaluated.

Future research should focus on adjusting the mathematical constant based on new data obtained by testing modern vehicles (regarding vehicle development) or providing a range of mathematical constants best suited either for different types of vehicles or for various road surfaces. Future research could also focus on developing more detailed mathematical models that would include additional parameters (such as tyre–road friction coefficient, vehicle mass, vehicle lateral stiffness) as data in this paper were insufficient for such detailed analysis.

**Author Contributions:** Conceptualization, R.M.; methodology, R.M., M.S., A.B. and R.K.; data analysis, R.M., M.K. and A.H.; formal analysis, M.B. (Martin Bilík) and M.B. (Michal Belák); writing original draft preparation, R.M.; writing—review and editing, R.M., M.S., A.B., S.T. and V.R.; supervision, M.S. and R.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

