System to Assist the Driver During a Single Lane Change Maneuver, in the Conditions of Danger Arising from a Change in the Condition of the Road Surface

Abstract

The article describes the application of a vehicle simulation model to assess the driving hazards that arise due to changes in the tyre–road adhesion during a lane change. An experimentally verified vehicle dynamics model was used. The typical single lane change maneuver, performed on dry concrete, wet concrete, and ice-covered roads, was simulated for three vehicle speeds. The disturbance was introduced as a changed tyre–road adhesion on the target lane. Typical sinusoidal (one-period) input was applied to the steering wheel. Adhesion change may lead to an accident when considering the driver’s reaction time. An original control algorithm has been built. It is the driver assistance system with a double PID controller of the steering wheel angle. The system parameters were determined based on the principles recommended in the automatic control engineering monographs. The controller fulfils its tasks for motion at very high speeds on a homogeneous road surface and for the case where the tyre–road adhesion is changed during a maneuver. Thanks to this, the threat can be avoided.

1. Introduction

Modern motor vehicles are being equipped with various driver assistance systems: Anti-lock Brake Systems (ABS), Acceleration Slip Regulation or Traction Control Systems (ASR-TCS), Electronic Stability Programs (ESP), Electronic Brake Distribution (EBD), Brake Assist Systems (BAS to shorten the vehicle response time), Adaptive Cruise Control (ACC to keep a safe distance to the frontward vehicle moving on the same road lane), Lane Departure Warning systems (LDW), etc. The vehicles with a high autonomization level (Level 4 and Level 5, according to the SAE classification [1]) can be driven almost without driver’s participation.

The road accident risk situations must be defined for the driver training process. Also, in the development process of driver assistance systems, it is important to include the scenarios in which the hazards are identified.

In publications [2,3,4,5], the authors highlight the importance of the lane change maneuver as the one that poses a great hazard. The collisions resulting from it are accountable for one-tenth of all the traffic delays caused by accidents and often lead to traffic jams [5,6]. The course of such maneuvers was examined in real conditions [3], by simulation methods [5], using car driving simulators [2,4].

In publications [6,7,8,9,10,11,12,13] (and many others), the introduction of automatic control systems modeling driver’s actions or assisting the driver during the lane change maneuver has been proposed.

In publication [14], the author unequivocally points to the “evident impact of the road surface and shoulders’ condition on the traffic safety”, including “the tyre–road adhesion conditions” and the surface irregularities (ruts, folds, humps). The problem of modeling the straight-line braking with a possible lane change when the driver spots a sudden change in the tyre–road adhesion has been discussed and the process has been modeled in [11]. The simulation represented the pre-accident situation rather than the change in the vehicle behaviour during a normal lane change. A similar problem (but without an analysis of the impact of the tyre–road adhesion) was addressed in [15]. Publication [16] covered the examination of a PID controller within the scope of per-forming a double lane change on a homogeneous road surface. In [8], a collision avoidance assistance system was proposed, including lane changing as a defensive action. The work [17] reviews a two-axle autonomous vehicle’s (AV) control methods (PID, MPC, SMC) to drive according to a given trajectory. An essential aspect of this work is to pay attention to the vehicle’s motion on a low-friction surface. The examples focused on controlling the steering wheel angle will be discussed hereafter.

This publication is to show the real threat that may arise in the road traffic from changes in the road surface condition, if the changes are encountered when the vehicle is driven without any influence of other vehicles or other road users. The single lane change on a straight road section has been described. Many simulations of the motion of a medium-class passenger car were carried out for a few typical speed values and different road surface conditions; the quantities that warn of the potential collision situation in question were pointed out when any intervention of the driver or the assistance systems is necessary.

An algorithm has been proposed for the operation of the assistance system that is to perform the maneuvers of this kind in a safe way when the driver cannot avoid the danger of lane or road departure.

The structure of the paper is organized as follows: Section 2 (Materials and Methods) presents a synthetic description of the vehicle simulation model and the reference trajectory model for the assistance system, while Section 3 (Scope and Parameters of the Tests) presents assumptions for the scope of calculations. The next section, Section 4 (Results and Discussion), presents the calculation results for the vehicle without the assistance system, followed by the model of the proposed assistance system and the results after its application, along with the analysis. Section 5 (Conclusions) summarises the authors’ most important observations on the results.

2. Materials and Methods

2.1. The Vehicle Under Test and Its Representative Model

The vehicle used for the tests was a KIA Ceed SW motor car (Kia Corporation, Žilina, Slovakia). The simulation model representing this vehicle (Figure 1), also shown in [18,19,20], consists of nine mass elements, i.e., car body solid, four material points O₁, O₂, O₃, and O₄ (assumed as the points of concentration of car’s “unsprung masses”), and four rotating solids (wheels). The following coordinate systems have been adopted (see also works [21,22,23,24] with a similar approach to the subject):

• Oxyz—a system attached to the road (horizontal axes Ox and Oy and vertical axis Oz pointing upwards);
• O_Cx_Cy_Cz_C—a system with axes parallel to the Ox, Oy, and Oz axes (respectively), with its origin situated at the car body centre of mass Oc;
• coordinate systems fixed to model’s rigid bodies, i.e., body solid (O_Cξ_Cη_Cζ_C) and four road wheels (O₁ξ₁η₁ζ₁, O₂ξ₂η₂ζ₂, O₃ξ₃η₃ζ₃, O₄ξ₄η₄ζ₄);
• extra systems that define the transformation matrices.

The translational motion is described using the positions of the centres of mass (O_C, O₁, O₂, O₃, O₄) of the said solids. The axes O_iξ_i, O_iη_i, O_iζ_i (i = C, 1, 2, 3, 4) are treated as the principal central axes of inertia of the corresponding rigid bodies.

To describe the spherical motion of the car body solid around the O_C centre, the “aircraft angles”, also referred to as “quasi-Euler angles”, have been used (see also works [21,22,23,24] with a similar approach to the subject):

• yaw angle ψ_C (rotation around the O_Cζ_C axis);
• pitch angle φ_C (rotation around the O_Cη_C axis);
• roll angle ϑ_C (rotation around the O_Cξ_C axis).

The purpose of introducing the O_Cx_Cy_Cz_C system was to show body’s c.g. and the reference system that enables the definition of body angular motion coordinates (“quasi-Euler angles”): ψ_C, φ_C, ϑ_C.

The axes of rotations are treated as the principal central axes of inertia. The Lagrange equations of the second kind have been applied to derive equations of motion. Fourteen generalized coordinates were adopted (q_i; i = 1, 2, …14):

• q₁ = x_OC, q₂ = y_OC, q₃ = z_OC—the position of the centre of mass of the car body solid (O_C) in the inertial reference system Oxyz;
• q₄ = ψ_C, q₅ = φ_C, q₆ = ϑ_C—the spherical motion of the car body solid around its centre of mass O_C (yaw, pitch, and roll angle, respectively);
• q₇ = ζ_CO1, q₈ = ζ_CO2, q₉ = ζ_CO3, q₁₀ = ζ_CO4—the motion of points O₁, O₂, O₃, O₄ relative to the car body solid in the direction of axis O_Cζ_C of the O_Cξ_Cη_Cζ_C coordinate system;
• q₁₁ = φ₁, q₁₂ = φ₂, q₁₃ = φ₃, q₁₄ = φ₄—wheel angles of rotation.

The model describes steering system flexibility and the steered wheels parameters: camber, caster, kingpin, and toe-in. The HSRI-UMTRI tyre–road contact model [25,26] has been adopted together with the IPG-Tire model of transient states of tyres [23,27]. The presented model has been successfully experimentally verified for typical vehicle maneuvers: steady-state cornering, step input on the steering wheel, double lane change, single lane change, straight-line braking [28,29,30,31]. So, it also considered the single lane change maneuvers.

2.2. The Road on Which the Lane Change Maneuver Is Carried Out

Figure 2 shows a straight section of a two-lane road, where the right and left lanes, separated from each other by a broken line, are labelled R and L, respectively. Oxyz is an inertial reference system, attached to the road. At the beginning of the single lane change maneuver under analysis, the vehicle under test is moving straightforward in the centre of the right lane.

2.3. Model of the Desired Vehicle Trajectory During the Single Lane Change Maneuver

In publications [32,33,34,35], the following models to describe the vehicle trajectory (represented by the trajectory of vehicle’s centre of mass) during the single lane change maneuver (1LCM) have been proposed: model 1 by Papić et al. [33] and Bascunan [32], model 2 by Nelson [35], model 3 by Zellner [34], model 4 by Fett, Frick and Daily [33], and model 5 by Chee [33]. They have trigonometrical forms based on sin/cos functions or circular/polynomial functions. As a result, they show very similar shapes. Authors of this paper decided to apply model 3 by Zellner [34] presented in the paper by Papić et al. [33]—see Equation (1). The reason was the closest (in the mean square sense) distance of this analytical description to the motion trajectory of the simulated vehicle during the lane change maneuver (1LCM).

$$\mathrm{y}\left(\mathrm{x}\right)=\mathrm{H}·[{\displaystyle \frac{\mathrm{x}}{{\mathrm{L}}_{\mathrm{m}}}}-{\displaystyle \frac{1}{2\mathsf{\pi }}}\sin \left({\displaystyle \frac{2\mathsf{\pi }}{{\mathrm{L}}_{\mathrm{m}}}}·\mathrm{x}\right)],$$

(1)

where x—longitudinal vehicle displacement, [m]; y—lateral vehicle displacement, [m]; H—planned maximum lateral vehicle displacement, [m]; L_m—longitudinal vehicle displacement during the performing of the single lane change maneuver, [m].

In this case, the vehicle yaw angle during the maneuver can be expressed as below:

$$\mathsf{\psi }\left(\mathrm{x}\right)=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\{{\displaystyle \frac{\mathrm{H}}{{\mathrm{L}}_{\mathrm{m}}}}·\left[1-\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{2\mathsf{\pi }}{{\mathrm{L}}_{\mathrm{m}}}}·\mathrm{x}\right)\right]\}.$$

(2)

If the vehicle speed (v [m/s] or V [km/h]) is constant during the 1LCM (single lane change maneuver), then Equations (1) and (2) may be transformed into the time domain t [s] thanks to the relation:

$$\mathrm{t}={\displaystyle \frac{\mathrm{x}}{\mathrm{v}}}$$

(3)

The parameter of the 1LCM is the maneuver duration time T [s]:

$$\mathrm{T}={\displaystyle \frac{{\mathrm{L}}_{\mathrm{m}}}{\mathrm{v}}}.$$

(4)

The 1LCM duration time T is estimated based on experimental tests. In publication [3], this time is 2–7 s. In [36], its full range, i.e., 1–13.3 s, is considered, but particular attention is paid (based on a statistical analysis) to the range of 2.54–6.28 s. In [37], the range of 1.0–2.6 s is covered by the analysis. In the normative act withdrawn by ISO [28], the value of 2 s was adopted. In the simulation tests reported herein, the authors adopted a value of T = 3 s, thus emphasizing the low dynamics of the maneuver under analysis, which corresponds to the normal smooth driving on a straight road section. This is usually connected with an identical steering wheel input duration time T_w = 3 s (a single period of the sinusoidal input). In [36], attention is directed to the fact that the 1LCM duration time T does not depend on the vehicle speed.

The final lateral vehicle displacement H during the 1LCM depends on the lane width. In the normative act [38], values from the range of 2.5–3.75 m were specified; actually, the value of H = 3.5 m often occurs on higher-class roads and this value was adopted in the simulation tests reported herein.

In [39], the range of the maximums of the lateral acceleration values recorded during the overtaking maneuvers in the real traffic has been specified as 0.3–3.2 m/s². The averaged values of these maximums were in the range of 0.99–2.01 m/s².

2.4. Model of the Driver Assistance System

The model of the driver assistance system proposed, together with a review of similar models found in the literature, will be presented in Section 4.4, where the results of the tests carried out within this scope will be discussed.

3. Scope and Parameters of the Tests

The vehicle data adopted for the tests corresponded to a fully loaded KIA Ceed car (1890 kg) with “symmetrical” specifications, i.e., a car where its centre of mass is situated in the car symmetry plane and the parameters of setting the left and right steerable wheels are identical to each other.

The scope of the tests included the following, in succession:

• simulation tests of the single lane change maneuver (1LCM) for the vehicle with no driver assistance system, on homogeneous and heterogeneous road surfaces;
• verification of the vehicle trajectory model for the driver assistance system;
• simulation tests of the single lane change maneuver (1LCM) for the vehicle with a driver assistance system, on homogeneous and heterogeneous road surface.

The simulation tests were carried out for three initial vehicle speeds, i.e., 50, 90, and 140 km/h (the last is the speed limit on motorways in Poland). These values were adopted as the typical speed limits that occur in the road traffic. Initially, the vehicle was moving in the centre of the right lane (R) and then it changed the lane to the left one (L) (see Figure 2).

Three types of the road surface condition were adopted for the tests: dry concrete, wet concrete, and ice. In the case of homogeneous road surface, the surface condition was identical for both the left (L) and right (R) lanes (see Figure 2). For the case with the heterogeneous road surface (also referred to as being of the μ split type), six combinations were considered, as shown in

Table 1. The road lane surface combinations adopted.

Road Surface	R (Right Lane)	L (Left Lane)	Notation
homogeneous	Dry concrete	Dry concrete	Dry
	Wet concrete	Wet concrete	Wet
	Ice-covered surface	Ice-covered surface	Ice
heterogeneous (μ-split)	Dry concrete	Wet concrete	Dry→Wet
	Wet concrete	Dry concrete	Wet→Dry
	Dry concrete	Ice-covered surface	Dry→Ice
	Ice-covered surface	Dry concrete	Ice→Dry
	Wet concrete	Ice-covered surface	Wet→Ice
	Ice-covered surface	Wet concrete	Ice→Wet

.

The maneuver was forced by changing the steering wheel angle. For the vehicle with no driver assistance system, the input to be applied was adopted as a single period of a sinusoid with a duration time of T_w = 3 s. The amplitude of changes in the steering wheel angle for the homogeneous road surface was selected by iteration so that the lateral vehicle displacement (in the y direction, see Figure 2) was about 3.5 m. For the heterogeneous road surface, it was assumed that the driver performed the maneuver by applying an input identical (as regards its period and amplitude) to that used for the surface of the right lane (R) in the case of the homogeneous road surface (dry concrete, wet concrete, or ice), keeping this input unchanged in spite of entering the other lane, i.e., the left one (L) with a different surface.

For the vehicle with a driver assistance system, the input applied to the steering wheel was as determined by the system.

As mentioned, the HSRI-UMTRI model [25,26], together with the IPG-Tire model of transient states of tyres [23,27], was used for the description of the forces and moments of forces.

Table 2. The adopted values of the main coefficients of the Dugoff–Fencher–Segel model of the forces and moment in the tyre–road contact zone.

Surface	μ0 [25,26]	kv [s/m]	cx [N/slip unit]	cy [N/rad]	kp [N/m]
Dry concrete	0.95	0.007	125,000	68,000	123,000
Wet concrete	0.75	0.040	62,500	34,000	123,000
Ice-covered surface	0.25	0.015	42,000	23,000	123,000

shows the values of the main coefficients of the model, which is also referred to as the Dugoff–Fancher–Segel model [25,26], for the three road surface types adopted. The meaning of the symbols used in the table are as follows: μ₀ is the maximum value of the tyre–road adhesion coefficient (for v being close to zero), k_v defines the dependence of the tyre–road adhesion coefficient on the slip velocity, cx is the kinematic circumferential stiffness of the tyre, c_y is the cornering stiffness of the tyre, and k_p is the lateral tyre stiffness.

The characteristics of the longitudinal and lateral tangential unit forces and of the aligning moment in the tyre–road contact zone for the vehicle speeds under consideration have been illustrated in Figure 3.

4. Results and Discussion

4.1. The 1LCM for the Vehicle with No Driver Assistance System, for the Homogeneous Road Surface

Figure 4 shows the results of the simulation of the lane change maneuver for a vehicle with no driver assistance system on a homogeneous road surface for three surface types (dry, wet, ice) and three vehicle speeds (50, 90, and 140 km/h) in the form of time histories of the input applied, i.e., steering wheel angle α_k, and of the response obtained, i.e., lateral displacement of vehicle’s centre of mass (y_Oc) and time derivative of the displacement (y_OcPrim ≡ dy_Oc/dt).

Satisfactory results were obtained for all three road surface types and the three vehicle speeds under consideration. Even for ice and 140 km/h, the assumed effect was obtained, and this is an extreme case (it is hard to imagine that there is someone who would drive on ice at such a speed).

As mentioned before, the assumed lateral displacement of 3.5 m was obtained by iterating the value of steering wheel angle amplitude. The results for the dry and wet concrete pavement were similar to each other, which may indicate a lack of interesting results for the changing of lanes from dry to wet concrete and vice versa. However, they were achieved for different amplitudes of the sinusoidal input applied to the steering wheel.

The results for the ice-covered surface differed from the other two for each of the vehicle speeds considered, especially for 140 km/h.

The maximum values of lateral acceleration (the time histories of this quantity have not been provided here) were in the range of 1.47–2.55 m/s². This means that they corresponded to their variability ranges in real conditions (0.3–3.2 m/s²) given in [39].

4.2. The 1LCM for the Vehicle with No Driver Assistance System, for the Heterogeneous Road Surface

Lane change (1LCM) simulation results without a driver assistance system on a heterogeneous road surface have been presented in an analogous form in Figure 5, Figure 6 and Figure 7. They are time histories of steering wheel angle α_k, lateral displacement of vehicle’s centre of mass (y_Oc), and time derivative of the displacement (y_OcPrim ≡ dy_Oc/dt), obtained for three vehicle speeds (50, 90, and 140 km/h). In this case, as previously mentioned, the steering wheel input applied to carry out the maneuver was adopted as identical to that used for the homogeneous road surface and kept unchanged as chosen for the surface of the lane where the maneuver was started.

The change in the road surface condition encountered when the vehicle changed the right lane to the left one significantly affected the time histories of the quantities represented in the graphs, i.e., lateral displacement and lateral velocity of vehicle’s centre of mass (y_Oc and dy_Oc/dt, respectively, the latter being denoted in the drawings as y_OcPrim). Due to the use of the same form of the steering wheel input as it was in the case of the homogeneous road surface (based only on the right lane), the intended lateral displacement of the vehicle of 3.5 m could not be achieved. For the differences to be better illustrated, the trajectories of vehicle’s centre of mass plotted for 6 variants of the maneuver carried out on heterogeneous road surfaces have been compared in Figure 8 with the trajectories obtained for the homogeneous road surfaces.

The differences between the lateral vehicle’s position relative to the line y = 3.5 m (i.e., the centreline of the left lane) reached (at the end of the maneuver) c.a. +2 m and −3 m, which indicates a great hazard, including even the risk of lane departure. Having noticed such a situation (excessive deviation from the desired trajectory), the driver should correct the input (after the necessary reaction time) in order to implement the lane change as assumed. Due to the typical reaction times of average drivers (1 s or even more [10,40,41,42,43]), it may be too late for any driver’s defensive action to be effective. However, we can imagine the use of an assistance system that could support or replace the driver in such situations. This will be the subject of the subsequent part of this article.

4.3. Verification of the Model of the Vehicle Trajectory to Be Adopted in the Assistance System

Figure 9 shows exemplary results for two trajectories, i.e., the one calculated for the theoretical model 3 (M3, but shifted by +0.22 s, to ensure conformity with the vehicle dynamics model) and the one obtained from the simulation of y_Oc(t) on the wet concrete road surface for the vehicle speed of V = 90 km/h. Here, the conformity can be seen very clearly.

Attention should also be paid to the fact that for the maneuver carried out on an ice-covered road surface at high speeds (140 km/h in this case), the vehicle trajectory significantly differs from that observed in the other cases under analysis (see Figure 4c, Figure 6b and Figure 7b). The maneuver duration time (T) increases although the steering wheel input remains unchanged (T_w = 3 s). This will have an impact on the assistance system’s parameters adopted (see Section 4.4).

4.4. The 1LCM for the Vehicle with a Driver Assistance System

4.4.1. Examples of the Steering Wheel Angle Control Systems for the Lane Change Maneuver

The systems for the automatic controlling of the steering wheel angle during the lane change have been described in many publications, e.g., [6,7,8,9,11,12,13,17].

In the control strategy of the system described in [9], the lane change process is decomposed into time-related phases, which is in line with the driving practice of experienced drivers. In the first phase, the controlling is performed in a partly open system (“blindly”, “just fast”) by generating an appropriate turn of the steering wheel. The precision of this phase of the maneuver is ensured by the previously identified reference model of the vehicle dynamics. During this phase of the control process, corrective controlling also takes place. The course of changes in the steering wheel angle is corrected on the principle of regulation based on comparing the current should-be values of the variable that describes the lateral vehicle displacement according to the reference model with the current values of this variable, which are actually measured. In [6], the lane-changing maneuver is also analyzed as a multi-stage process carried out by controlling the steering angle of the steered wheels. This work highlights the benefits of using MPC (model predictive control) and SMC (Sliding Mode Control) algorithms and the use of neural networks to adjust the parameters of the control algorithm.

In publication [11], a model for controlling the steering system in a closed loop has been presented, where controllers of two types, i.e., proportional-integral (PI) and proportional-derivative (PD) ones, and a low-pass filter with the delay time at a level of 0.1 s, were used. In that work, a straight-line braking maneuver, with a possibility of changing it into a lane change when the driver notices a sudden change in the road surface grip, was successfully simulated.

In the work reported in [12], the object of control and regulation was a vehicle model with 9 degrees of freedom and a driver model was added to it. Preliminarily, the vehicle trajectory for the maneuver of keeping the lane at a constant speed was defined. The model of the controller was developed to control the steering wheel angle. The proposed structure of the controlling in the major loop of the controller model is a combination of the proportional gain control (P) with the adaptive control, using the fuzzy logic to control the yaw angle. In [7], a dual PID controller is considered to regulate the deviation of lateral displacement and yaw angle. The PID controller is also used in [8], where an assistant system called ACAS (adaptive collision avoidance strategy) was proposed, enriched with additional adaptive fuzzy-logic systems. The system is intended to ensure operation in conditions of variable speed and wheel adhesion to the road surface.

Other publications, e.g., [13,44,45] deal with V2V (vehicle-to-vehicle) and connected and automated vehicles (CAV) technology.

As it can be seen, various control methods, including very advanced and complex ones, are employed to address the issues of controlling the vehicle movement during the lane change maneuver. For this purpose, models of proportional-integral-differential (PID) controllers are used because of their relative simplicity, tried, proven, and documented operational reliability, and known methods of identification of their parameters [46,47,48]. In most cases, the predefined and planned vehicle trajectory is used as the reference.

4.4.2. The Steering Wheel Angle Control System Adopted for the Lane Change Maneuver

It was assumed that the car under test would be provided with a lane change assistance system as proposed below. Its operation has been illustrated in Figure 10. If it is ON, then the reference trajectory y_p(x) is determined at the instant when a lane change is signalled. The trajectory parameters (H and L_m) depend on: (1) the lane width and the initial vehicle position on the first lane at the instant of signalling the intention to carry out the maneuver (it was assumed that these are determined by means of optical systems similar to those used for the parking and lane keeping sensors); (2) readings of the ambient temperature and rain presence sensors (based on these readings, the road surface condition is assessed, i.e., the question whether the road is dry, wet, or ice-covered is answered); and (3) the current vehicle speed V. Then, the reference trajectory (y_p(x)) is transformed to the time domain (y_p(t)) and the system starts the execution of the lane change maneuver. Two proportional-integral-differential (PID) controllers operating in parallel have been provided: for the lateral position y_Oc and the yaw angle ψ_C. The current values of y_Oc and ψ_C are calculated based on the readings of optical sensors (similar to those used in the parking and lane keeping sensors). The input signals are the deviations Δy and Δψ calculated for the lateral position y_Oc and the yaw angle ψ_C. The resultant value of the steering wheel angle is the weighted average of the components obtained from the two control systems operating in parallel, where the weighting factors are w_y and w_ψ, as appropriate, according to Equation (5):

$${\mathsf{\alpha }}_{\mathrm{k}}\left(\mathrm{t}\right)={\mathsf{\alpha }}_{\mathrm{k}\mathrm{y}}\left(\mathrm{t}\right)·{\mathrm{w}}_{\mathrm{y}}+{\mathsf{\alpha }}_{\mathrm{k}\mathsf{\psi }}\left(\mathrm{t}\right)·{\mathrm{w}}_{\mathsf{\psi }}$$

(5)

The settings of the PID controllers were determined using a step input, where a controlled trajectory y_P = 0 was applied for the instant t = 1 s, at the initial value of y_P(0) = −H. The proportional control gains K_py and K_pψ were selected to come close to the stability limit, i.e., to the appearance of non-decaying oscillation. Then, the equivalent delays of the object (vehicle) controlled and the equivalent time-constants of the vehicle under test were determined. Based on them, the controllers’ parameter values (K_py, K_iy, K_dy and K_pψ, K_iψ, K_dψ) were determined by the Ziegler–Nichols method [46,47,48]. Apart from the controllers’ parameter values, the weighting factors for the summands α_ky(t) and α_kψ(t) in Equation (5) had also to be determined. The raising of w_y at the expense of w_ψ led to reductions of the phase shift of the trajectory (after the transformation to the time domain) relative to the reference one at the expense of the time histories of α_k(t) becoming more irregular. When the opposite happened, the said phase shift increased but the courses of the steering wheel angle became smoother. Using the trial and error method, the controller parameters were slightly modified to values K_py = 0.1 rad/m, K_iy = 38 s, K_dy = 6 s, K_pψ = 100 rad/rad, K_iψ = 80 s, and K_dψ = 12 s and the weighting factors equal to w_y = w_ψ = 0.5 were used. In result of that, the upward trend in the phase shift was reduced and the α_k(t) curves obtained were only slightly disturbed.

The real exemplification of the system proposed will require the use of a few protective elements to limit the values of y_Oc(t) and ψ_C(t) and, in consequence, of α_k(t) as well as of their first derivatives in respect to time.

4.4.3. The 1LCM for the Vehicle with an Active Driver Assistance System, for the Homogeneous Road Surface

Figure 11 shows results for the single lane change maneuver, where an assistance system was used in comparison with the results with a sinusoidal input (T_w = 3 s), on a homogeneous road surface. The final lateral displacement and the vehicle speed were pre-set as H = 3.5 m and V = 90 km/h = 25 m/s, respectively. The maneuver began at the instant of t = 1 s. The time histories of the input, i.e., steering wheel angle α_k(t), and of the response, i.e., lateral displacement of vehicle’s centre of mass y_Oc(t) and yaw angle ψ_C(t), have been shown. The curves represent the reference course (Reference), the result obtained in the maneuver controlled by the open system with a sinusoidal input (Sine), and the result of operation of the assistance system described herein (Assist. S.). After performing the maneuver, the car should move parallel to the lane centre line and a sketch of the car should not leave the lane. Knowing vehicle width gives limits for the vehicle’s c.g. lateral displacement.

These results indicate that the assistance system correctly executes the vehicle motion along the preset trajectory treated as a reference. However, the curve y_Oc(t) is shifted along the time axis with decreasing tyre–road adhesion coefficient (from dry concrete through wet concrete to ice). This indicates that on the road surfaces with less grip, the vehicle responds with some delay and a bit slower, which results from an increase in the tyre sideslip angles. This is reflected in the graphs that show the time histories of the yaw angle ψ_C(t) and of the controlled quantity, i.e., the steering wheel angle α_k(t). Such a phenomenon could also be observed when comparing the curves in Figure 4 (Section 4.1), which represented the results obtained in the maneuver controlled by the open system with a sinusoidal input applied to the steering wheel, carried out on roads with less and less grip.

Moreover, a small overshoot in the y_Oc(t) curve can be seen for the ice-covered road surface (Figure 11c). Nevertheless, the effect of operation of the assistance system may be considered satisfactory. The system can substitute for the driver in executing the maneuver planned.

4.4.4. The 1LCM for the Vehicle with an Active Driver Assistance System, for the Heterogeneous Road Surface

For the car provided with a lane change assistance system described herein, simulations were carried out that corresponded to those presented in Section 4.2 (Figure 5, Figure 6, Figure 7 and Figure 8), i.e., for the μ split road surfaces in combinations as specified in Table 1, and the effect of the operation of such a system was assessed. For illustration, the most spectacular cases were selected, including those that highlight the possible imperfections of the control type employed (the Dry→Ice and Ice→Dry cases).

Figure 12 shows the results obtained for the maneuver with dry concrete surface of the right lane and ice-covered surface of the left lane (the Dry→Ice case) and for the case with ice-covered right lane and dry concrete left lane (the Ice→Dry case). All the other test parameters were as before, i.e., maneuver duration time T_w = 3 s, anticipated final lateral vehicle displacement H = 3.5 m, the maneuver began at the instant of t = 1 s, and the vehicle speed was V = 90 km/h = 25 m/s. The curves in the graphs represent steering wheel angle α_k(t), lateral displacement of vehicle’s centre of mass y_Oc(t), and yaw angle ψ_C(t) and show the reference course (Reference), the result obtained in the maneuver controlled by the open system with a sinusoidal input and carried out on a homogeneous dry concrete road surface (Sine Dry), and the result of operation of the assistance system under consideration on the heterogeneous road surface (Assist. S.).

It can be seen that the vehicle response is significantly delayed when the vehicle is on the ice-covered surface at the start of the maneuver (similar vehicle behaviour can be observed when the vehicle is moving without the assistance system, especially at high speeds—see Figure 4c, Figure 6b and Figure 7b).

The results presented indicate that the assistance system definitely improves vehicles’ behaviour compared to that observed during the maneuver carried out on a μ-split road surface and controlled by the open system with a sinusoidal input applied to the steering wheel. The vehicle remains within the target lane, although the actual final lateral displacement values differ from that having been preset (the difference is about 0.4 m, both positive and negative), and the vehicle moves parallel to the road direction (Ox).

The error of the lateral vehicle’s position can be reduced by lowering the proportional control gain K_pψ (for the yaw angle). However, this would raise the variability of the steering wheel angle and oscillations of the yaw angle ψ_C(t). The oscillations, in turn, can be reduced by extending the lane change time. For vehicle speeds of V = 90 km/h and higher, the lane change time (the T_w period) was extended to a value of 4 s (without changing the values of K_py and K_pψ). Such an action improves the conditions of operation of the assistance system and lowers the intensity of turning the steering wheel and the values of lateral acceleration and yaw velocity. The effects of these actions have been presented in Figure 13, which corresponds to Figure 12. Thus, the functioning of the assistance system is markedly improved, and the error of the lateral vehicle’s position is decreased.

It was decided to check whether the assistant system proposed would work correctly for the most difficult case of those considered (the lane change at a speed of V = 140 km on the “μ-split” road surface, Dry→Ice and Ice→Dry lane change combinations). Figure 14 shows the results obtained for a maneuver carried out at such a speed, where an assistance system was used, for a maneuver duration time of T_w = 4 s. As in the previous cases, the curves represent the input applied, i.e., the steering wheel angle α_k(t) in the sinusoidal form controlled by the open system (Sine) and in the form controlled by the assistant system (Assist. S.), and the responses, i.e., lateral displacement y_Oc(t) and yaw angle. The curves include those adopted as the reference (Reference), those obtained as a result in the maneuver controlled by the open system with a sinusoidal input and carried out on a homogeneous road surface (Sine Dry, i.e., on dry concrete, see Figure 14a, and Sine Ice, i.e., ice, see Figure 14b) and carried out on a heterogeneous road surface (Sine) and in the maneuver controlled by the assistance system under consideration (Assist. S.). The maneuver began at the instant of t = 1 s.

As in the case with the vehicle speed of V = 90 km/h, the extending of the lane change time from 3 s to 4 s and the reducing of the proportional control gain K_pψ, without changing K_py, led to very satisfactory results, i.e., the vehicle trajectories were implemented correctly. A disadvantage of the second of the mentioned changes to the PID controller of the assistance system is an increase in the intensity of changes in the yaw angle ψ_C and in the steering wheel angle α_k (which can be seen in Figure 14). This slightly reduces the desirable effects of extending the lane change time T_w. Such functioning of the assistance system may cause some anxiety for the driver. Nevertheless, the system fulfills its task.

5. Conclusions

A change in the condition of the road surface on which the vehicle is moving during the lane change maneuver (from the right lane to the left one) significantly affects the time histories of the lateral displacement of vehicle’s centre of mass and of the yaw angle as well as of their time derivatives, i.e., velocities. If the input applied to the steering wheel is in the same form as it would be in the case of the homogeneous road surface (i.e., where the left lane surface is identical to that of the right one), then it will be impossible to achieve the intended lateral displacement and the yaw angle of the vehicle. Having noticed such a situation, the driver should correct the input (after the necessary reaction time) in order to implement the lane change as assumed. Unfortunately, in many cases of the vehicle motion under analysis, it is too late (due to the driver reaction time of 1 s or more) to eliminate the threat that has emerged in the form of large differences in vehicle’s position and yaw angle from those expected (intended). This leads to a very high risk of an accident resulting from vehicle’s departure from the occupied lane or even the road.

The proposed assistance system based on a PID controller correctly implements the motion along the assumed trajectory, treated as a reference (a pre-set one) on a homogeneous surface. It has been observed, however, that the course of the vehicle trajectory (transformed to the time domain) is shifted along the time axis with decreasing tyre-road adhesion coefficient (from dry concrete through wet concrete to ice). This indicates that on the road surfaces with less grip, the vehicle responds with some delay and a bit slower, which results from an increase in the tyre sideslip angles. This is reflected in the graphs showing the yaw angle and the time history of the controlled quantity, i.e., the steering wheel angle. Moreover, a slight inaccuracy in the course of vehicle’s lateral position can be seen for the ice-covered road surface. Nevertheless, the effect of operation of the assistance system may be considered satisfactory. The system can substitute for the driver in executing the maneuver planned. The results presented indicate that the assistance system definitely improves vehicle’s behaviour in comparison with that observed during the maneuver carried out on a heterogeneous road surface and controlled by the open system with a sinusoidal input applied to the steering wheel. The vehicle remains within the second lane. In this work, the components of the control system have been indicated that have an impact on the error of lateral vehicle’s displacement. The said error can be reduced, but the reduction may be accompanied by greater variability of the course of the steering wheel angle and oscillations of vehicle’s yaw angle. Such functioning of the assistance system may cause some anxiety for the driver. This possible negative aspect should be tested when checking prototype devices or during driving simulator tests.

It is worth emphasizing that the assistance system proposed in this article fulfils its tasks even for the motion with a very high speed (140 km/h) on a homogeneous road surface (even if ice-covered) as well as for the case where the tyre–road adhesion is changing during a lane change maneuver. Thanks to this, it is possible to avoid the risk of an accident, which (as shown) is real when the vehicle is driven in a routine way. The presented results show that traditional, well-known (and tested in many real-world applications) PID controllers are enough effective compared with newer methods (e.g., MPS—Model Predictive Control or SMC—Sliding Mode Control).

The authors claim that for reproducibility of the presented work, the most important are the described method, simulation model, block diagram of the proposed lane change assistant system (Figure 10), and description of the way how to determine the controller parameters (see Section 4.4.2). This gives a big chance to reproduce the described control system for other vehicles, for another author’s research.

The proposed system has (of course) some limitations. These are the minimum and maximum values of the steering wheel angle, steering wheel angular velocity, and torque on the steering wheel. In the presented research, the mentioned variables did not reach the limits, but it is important to remember such limitations. In the paper, the authors wanted to recognize the main problems dealing with the application of the presented idea. The simulation method is a very good tool for it. In further works, it is planned to consider uncertainty problems (due to a number of passengers or amount of load in the vehicle, etc.), e.g., adaptive controllers.