*1.1. Current Calculation Methods*

Many current calculation methods use simplified calculation models based on Newton's laws of motion or empirical models based on experimental measurements (driving tests). For lane-change manoeuvre, simple equations analyzing the motion of the material point can be used. These methods are described, e.g., in [1,2].

The input variables for these methods are the lateral distance, the maximum or utilized lateral adhesion (or lateral acceleration of the vehicle), and the vehicle speed. Lateral distance limits the vehicle trajectory, and lateral acceleration (or adhesion) represent and simplify vehicle (vehicle tyres) interaction with the road surface. Some variables have the same meaning across the presented formulas but differ slightly (mostly in indexes used). For unification, the variables in different formulas are defined as follows:


Runkel [5] derived a model for a lane-change time based upon an assumption of constant acceleration inputs, resulting in a curve consisting of two circular arcs. A similar analysis was published by Jennings [6]. This model, however, does not seem to apply in noncritical situations.

$$t = \mathbf{C} \cdot \sqrt{\frac{\underline{Y}}{a\_y}} \tag{1}$$

Sporrer et al. in their study [7] mention that attempts were made to reflect the steering motion of a driver, which gave values of K varying from 2.51 to 2.83. Sporrer's paper then investigated the steering manoeuvre by way of driving trials. The trials showed that the manoeuvre was only symmetrical for an abrupt lane change. The results indicated that the average acceleration would not give a realistic time for the manoeuvre for a routine lane change.

In [8–10], almost identical Equation (2) is shown to be used for the theoretical calculation of the duration of the vehicle avoidance manoeuvre (*tsin*), assuming the lateral acceleration of the vehicle is of sinusoidal shape and one-period duration.

$$t\_{\rm sin} = 2.51 \cdot \sqrt{\frac{\mathcal{Y}}{a\_y}} \tag{2}$$

In the case of the actual manoeuvre, the curve of the lateral acceleration does not correspond fully with one period of the sine function. Therefore, the theoretical assumption for the deduction of the Equation (2) is not met.

Practical calculation of the manoeuvre duration (*trpac*) was done using the so-called Kovaˇrík formula, described in [1] and given as Equation (3). In this formula, the mathematical constant of 2.51 was recalculated using data from experimental testing, accommodating other influences (such as delay in the steering mechanism, tyre elasticity, and others) in the overall manoeuvre duration calculation.

$$t\_{\rm proc} = 3.13 \cdot \sqrt{\frac{\overline{Y}}{a\_{\rm y}}} \tag{3}$$

A more recent form of practical calculation is the Weiss formula (Equation (4)), where K coefficient is expressed as function *K* (*ay*, *v*, *y*) using Equation (5), with an approximate value of 2.67, as shown in [9,11].

$$t \ge \mathbb{K} \cdot \sqrt{\frac{y}{a\_y}} \tag{4}$$

$$K = 2.2 \cdot 10^{-4} \cdot a\_Y^2 + 2.6 \cdot 10^{-3} \cdot a\_Y - 2.1 \cdot 10^{-2} \cdot y + 2.1 \cdot 10^{-4} \cdot v + 2.72\tag{5}$$

The manoeuvre duration expressed by the Equations (3) and (4) is always considered to last longer than the calculated theoretical value, which is (among other things) the result of delays in the steering caused by steering wheel plays and the rigidity of the steering mechanism and elasticity of tyres.

### *1.2. Motivation*

Most of the investigated calculation methods of lane-change duration are based on experiments carried out more than ten years ago. In contrast, the newer experiments were carried out mainly on a dry road surface. Therefore, it is necessary to explore if (and how) modern vehicle construction, vehicle stability systems, and lowered adhesion conditions affect these methods and their applicability, i.e., whether it is necessary to change the approach in the manoeuvre duration calculation or whether the empirical models used currently are still viable.

#### *1.3. Hypothesis*

The duration of the lane-change manoeuvre is conditioned by its intensity, that is, the intensity of lateral acceleration and the lateral distance. Modern electronic devices allow precise measuring of parameters relevant for the analyses of lane-change manoeuvres, that is, speed, time, lateral distance, lateral acceleration, and others. In this way, with an adequate concept of experimental test track, it is possible to analyze the influence of the intensity of lateral acceleration and the lateral distance to the time necessary for obstacle avoidance by lane-change manoeuvre.

The goal of this research is also to obtain the latest and more precise experimentally gained results of lane-change manoeuvre duration convenient for theoretical clarification and description of manoeuvre, social relevance, and legal certainty impact of participants in terms of road accident analysis

#### **2. Methodology**

The main idea of this paper is to explore the applicability of current empirical Equations (1) and (2) used for the manoeuvre duration calculation, and it was necessary to obtain all the relevant variables used in the calculations (Table 1).

**Table 1.** Relevant variables of selected empirical formulae.


For this purpose, four series of driving tests were carried out between the years 2016 and 2020 in various conditions (both dry and wet road surfaces). To repeat the experiment under the same conditions, the test track for vehicle lane-change manoeuvre was based on the procedure described in standard ISO 3888-2, Passenger cars—Test track for a severe lane-change manoeuvre, Part 2 Obstacle avoidance (or its modifications) [12].

In all test series, the drivers were always experienced males, between 30 and 35 years old, with 10 to 15 years of driving experience. The drivers always had several practice runs before testing to become better acquainted with both the test track and the tested vehicles (thus, the influence of the driver on the driving manoeuvre was eliminated).

All relevant data of the lane-change manoeuvre were documented using GPS Data Logger V-BOX Video HD 2 and Racelogic PERFORMANCEBOX with receiver antennas placed on the roof, roughly at the center of gravity of each vehicle tested. These instruments record data at a frequency of 10 Hz, which was proven sufficient in [13] for trajectory documentation using a similar device (Performance Box Sport).

The driver's input was monitored using a VBOX Video HD2 camera located inside the vehicle. The vehicle position was documented using the same equipment and was observed by cameras, placed around the test track.

A wide variety of vehicles was used throughout the individual test series, ranging from the supermini vehicle class to crossover SUV (see Table 2).


**Table 2.** Tested vehicles.

The first two test series (2016 and 2017) were carried out following standard ISO 3888-2, Passenger cars—Test track for a severe lane-change manoeuvre, Part 2 Obstacle avoidance.

The 2016 test series was carried out on a landing strip with a dry asphalt surface, with temperatures ranging from 14 to 18 ◦C. The 2017 test series was carried out on an area with an asphalt surface used for truck testing. During the 2017 testing, permanent rain was present, with temperatures ranging from 7 to 15 ◦C.

The track was modified for the next two test series (2019 and 2020) but still based on standard ISO 3888-2. The track widths (a and b) remained following the standard; the length of the track segment used for lane change (S2) was variable and shortened compared to the ISO standard to lengths of 13.5, 11.0, and 8.5 m, and the offset (p) of the exit gate to entry gate was variable as well—offsets of 1.5, 2.5, and 3.5 m were used (the diagram of a modified test track is presented in Figure 1).

**Figure 1.** Adjusted test track schematics.

In all the test series, the goal was to achieve maximal vehicle speed without stability loss or exceeding the test track boundaries.

In this study, the driver's first steering intervention is regarded as the beginning of the lane-change manoeuvre and reaching maximal lateral distance (at this point, the vehicle has avoided the obstacle and changed lane at a certain width, taking a position similar to the initial one), regarded as the end of the manoeuvre, similar to [14] and as seen in Figure 2.

**Figure 2.** Example of determining the beginning and end of the manoeuvre during test series 3: (**a**) first steering input; (**b**) first discernible change of lateral acceleration and vehicle direction; (**c**) overview of the whole manoeuvre with traffic cones outlining the track and green dots representing the vehicle path.

#### **3. Lane-Change Manoeuvre Duration Calculation**

Based on the methodology chosen for the manoeuvre duration calculation, the analysis of the Kovaˇrík and Weiss Equations (3) and (4) was carried out. Since both of these formulas were almost identical (or could be simplified into the same form), analysis of the mathematical constant C was done by using the data from 108 successfully performed manoeuvres (out of 193 manoeuvres performed in total).

A linear regression analysis was carried out. The square root value of the ratio of the maximal lateral distance y max and the maximum lateral acceleration ay max achieved in the individual driving manoeuvres is plotted directly on the *x*-axis. The duration of the manoeuvres was again plotted on the *y*-axis, see Figure 3.

Based on 108 measurements from all test series and the linear regression (with correlation coefficient of 0.9899), the mathematical constant of the equation was adjusted with a value of 2.93.

$$t \ge 2.93 \cdot \sqrt{\frac{y\_{\text{max}}}{a\_{y\text{-max}}}} \tag{6}$$

**Figure 3.** Data regression analysis.
