*2.2. Model of EDR Records*

The EDR design solutions that can be met in reality differ from each other in the number of the recorded components of the vectors that define the vehicle position. In these terms, two characteristic EDR types have been distinguished (Figure 3). The devices of the first type (denoted herein by EDR1) are similar (within the scope as considered here) to flight recorders: they record three acceleration components, i.e., longitudinal one aw, lateral one ap, and "vertical" one aζ, and three angles, i.e., yaw angle ψC, pitch angle ϕC, and roll angle ϑC, or the corresponding angular velocities. The devices of the second type (denoted herein by EDR2) can be frequently met in motor vehicles and are a simplified version in comparison with the aircraft solution. The vehicle motion is treated as a two-dimensional one, with the parameters recorded being two acceleration components, i.e., longitudinal one aw and lateral one ap, and one angle, i.e., yaw angle ψC, or its time derivative, i.e., yaw velocity.

**Figure 3.** Two EDR types: (**a**) EDR1 (6 parameters); (**b**) EDR2 (3 parameters).

The assumptions adopted in the mathematical model of acceleration sensors' readings and the angular quantities have been described in dissertation [45] (and in [27] as well). An updated description can be found in [29,30]. Here, the main features of the description and fragments of its formal mathematical approach have been presented. The kinematic relations have been formulated by taking, as a basis, the model of kinematics shown in

Figure 4. The vehicle motion is treated as a composition of translational motion of the centre OC of vehicle mass and of the rotation of vehicle body solid around point OC. Thus, 6 degrees of freedom of the vehicle body solid (3 translations and 3 rotations) are taken into account. One more reference system has also been added to the coordinate systems mentioned above (see Section 2.1):

**Figure 4.** Model if the kinematics of motion of a vehicle provided with an EDR, where the set of sensors is placed at point P; notation: r—position; V—velocity; a—acceleration in translational motion; ω—angular velocity; ε—angular acceleration.

PξPηPζP—a moving one, fixed to the EDR; the PξP, PηP, Pζ<sup>P</sup> axes define the directions of operation of the sensors of longitudinal and lateral acceleration and the yaw angle.

The vector symbols have the following meanings: *rC* ≡ *x<sup>C</sup>* = [*xC*, *yC*, *yC*] *<sup>T</sup>*—Position of OC in the inertial system Oxyz; *rP* ≡ *x<sup>P</sup>* = [*xP*, *yP*, *yP*] *<sup>T</sup>*—Position of point P in the inertial system Oxyz; *ρ* ≡ *ρ* = [*ξCP*, *ηCP*, *ζCP*] *<sup>T</sup>*—Position of point P in the OCξCηCζ<sup>C</sup> system; *ω* ≡ *ω* =  . *ψC*, . *φC*, . *ϑC T* —Angular velocity; *ε* ≡ *ε* =  .. *ψC*, .. *φC*, .. *ϑC T* —Angular acceleration; *VC* <sup>≡</sup> . *x<sup>C</sup>* = . *xC*, . *yC*, . *zC T* —Velocity of point OC; *aC* <sup>≡</sup> .. *x<sup>C</sup>* = .. *xC*, .. *yC*, .. *zC T* —Acceleration of point OC; *aP* <sup>≡</sup> .. *x<sup>P</sup>* = .. *xP*, .. *yP*, .. *zP T* —Acceleration of point P.

The kinematics of point P relative to the inertial system Oxyz is described as follows in matrix notation (point P is motionless in relation to the vehicle):

$$\mathbf{x} \text{ position}: \ x\_P = \mathbf{x}\_{\mathbb{C}} + A \cdot \boldsymbol{\rho} \tag{1}$$

$$\mathbf{i}\text{ velocity}:\ \dot{\mathbf{x}}\_P = \dot{\mathbf{x}}\_\mathbf{C} + \dot{\mathbf{A}} \cdot \boldsymbol{\rho}\tag{2}$$

$$\text{acceleration}: \ \ddot{\mathbf{x}}\_P = \ddot{\mathbf{x}}\_\mathbf{C} + \ddot{\mathbf{A}} \cdot \boldsymbol{\rho} \tag{3}$$

where the matrix of rotation *A* (from the OCξCηCζ<sup>C</sup> system to the Oxyz system) has the form (4):

$$A = \begin{bmatrix} \cos\psi\_{\mathbb{C}} \cdot \cos\phi\_{\mathbb{C}} & \cos\psi\_{\mathbb{C}} \cdot \sin\phi\_{\mathbb{C}} \cdot \sin\theta\_{\mathbb{C}} \cdots \sin\psi\_{\mathbb{C}} \cdot \cos\theta\_{\mathbb{C}} & \cos\psi\_{\mathbb{C}} \cdot \sin\phi\_{\mathbb{C}} \cdot \cos\theta\_{\mathbb{C}} + \sin\psi\_{\mathbb{C}} \cdot \sin\theta\_{\mathbb{C}} \\\sin\psi\_{\mathbb{C}} \cdot \cos\phi\_{\mathbb{C}} & \sin\psi\_{\mathbb{C}} \cdot \sin\phi\_{\mathbb{C}} \cdot \sin\theta\_{\mathbb{C}} + \cos\psi\_{\mathbb{C}} \cdot \cos\theta\_{\mathbb{C}} & \sin\psi\_{\mathbb{C}} \cdot \sin\phi\_{\mathbb{C}} \cdot \cos\theta\_{\mathbb{C}} - \cos\psi\_{\mathbb{C}} \cdot \sin\theta\_{\mathbb{C}} \\\ - \sin\phi\_{\mathbb{C}} & \cos\phi\_{\mathbb{C}} \cdot \sin\theta\_{\mathbb{C}} & \cos\phi\_{\mathbb{C}} \cdot \cos\theta\_{\mathbb{C}} \end{bmatrix} \tag{4}$$

The positions of the sensors and their axes are defined by the positions of point P and coordinate axes of the PξPηPζ<sup>P</sup> system. The position of this system relative to the OCξCηCζ<sup>C</sup> one is defined by its translation by vector *ρ* and by its rotation described by matrix *C* (see (5)). The rotations have been adopted in a way similar to that adopted previously, but in reverse order: ϑ<sup>E</sup> (rotation about the longitudinal axis ξP), φ<sup>E</sup> (rotation about the lateral axis ηP), and ψ<sup>E</sup> (rotation about the "vertical" axis ζP)—see Figure 5b. Such a sequence of rotations is convenient because of the easy levelling of the sensor axes (oriented in relation to the vehicle).

**Figure 5.** Angular situation of the coordinate systems: (**a**) *OCξCηCζ<sup>C</sup>* in relation to *Oxyz*; (**b**) *PξPηPζ<sup>P</sup>* in relation to *OCξCηCζC*.

The basic component of the EDR model is the model of acceleration records. In this model, a system of transducer axes perpendicular to each other has been adopted, with the transducers being situated, as previously mentioned, freely in relation to the vehicle body. The fact has been taken into account that the acceleration transducers measure not only the real component of the acceleration of the EDR fixing point but also the pertinent components of the gravitational acceleration. Being inertia sensors, they measure a value proportional to the sum of the inertial and gravity force components acting along the sensor axis. The component resulting from the gravity force may be treated as the indication error.

Assuming a general formula for the sensor readings in the form (6):

$$a^{\mathcal{L}} = \left[a\_{w^{\prime}}^{\mathcal{L}}, a\_{p^{\prime}}^{\mathcal{L}} a\_{z}^{\mathcal{L}}\right]^T \tag{6}$$

We may write that vector *a<sup>c</sup>* is equal to:

$$\mathfrak{a}^{\mathfrak{c}} = \mathfrak{a}\_{P\mathfrak{c}} - \mathfrak{g}\_{\mathfrak{c}} \tag{7}$$

where:

$$\boldsymbol{a}\_{P\boldsymbol{c}} = \left[a\_{P\mathbb{E}\_{\mathbb{P}}\prime}, a\_{P\eta\_{P}\prime}a\_{P\mathbb{E}\_{\mathbb{P}}\prime}\right]^T = \mathbf{C}^{-1} \cdot \boldsymbol{a}\_P = \mathbf{C}^{-1} \cdot \mathbf{A}^{-1} \cdot \ddot{\boldsymbol{x}}\_P \tag{8}$$

$$\mathbf{g}\_c = \left[ \mathbf{g}\_{\mathbb{S}^p}, \mathbf{g}\_{\mathbb{J}^p}, \mathbf{g}\_{\mathbb{J}^p} \right]^T = \mathbf{C}^{-1} \cdot \mathbf{g}\_{\mathbb{S}} = \mathbf{C}^{-1} \cdot \mathbf{A}^{-1} \cdot \mathbf{g} \tag{9}$$

and *g* = [0, 0, −*g*] T—gravitational acceleration vector (*g* = 9.81 m/s2).

Vectors *aPc* and *g<sup>c</sup>* represent the acceleration of point *P* and gravitational acceleration, expressed in the reference system PξPηPζ<sup>P</sup> (in the OCξCηCζ<sup>C</sup> system attached to the vehicle, the same acceleration vectors are denoted by *a<sup>P</sup>* and *g<sup>ξ</sup>* , respectively). The acceleration sensors' readings have been graphically illustrated in Figure 6 (where the EDR2 unit has been used as an example for simplification).

**Figure 6.** Graphic interpretation of the longitudinal and lateral acceleration readings (aw<sup>c</sup> and apc, respectively).

In the description presented, the sensors have been assumed to be free of intrinsic errors. A model applicable to the readings describing angular position (angles or angular velocities) has been developed as well. Its formal description can be found in [27,45].

#### *2.3. Reconstruction of Motion (DPM Model)*

The reconstruction of motion consists of appropriate (for the specific recorder configuration, e.g., EDR1 or EDR2) processing of the recorder's output signals. A schematic diagram of the data processing procedure has been shown in Figure 7. The procedure is based on the integration of acceleration records transformed to the inertial system attached to the road. The numerical integration ([46,47]) is carried out from a definite final instant corresponding to a known vehicle position and velocity (e.g., post-accident vehicle position specified by witnesses). If possible, transducer readings are adjusted by adding the values of gravitational acceleration components (Δ**a**<sup>c</sup> <sup>=</sup> −**g**c, see Equation (7)). The results of successive quadrature operations define, as appropriate, the velocities and positions of point P of fixing the EDR, expressed in the inertial system Oxyz. Afterwards, they can be transformed in order to obtain time histories of the velocities and trajectories of any point in the vehicle body.

**Figure 7.** Schematic diagram of the processing of data recorded by the EDR: **a**x, **V**x, **r**x,—vectors of acceleration, velocity, and position, respectively, in the inertial reference system Oxyz; **V**k, **r**k velocity and position at the final instant; "P" in the subscript indicates the values for point P of fixing the EDR; ψCc, ϕCc, ϑCc—values of angles ψC, ϕC, ϑC, determined from EDR records.

Figure 7 shows a schematic diagram of the processing of data produced by a recorder of angle values. In such a case, the angular velocities are obtained by numerical differentiation of the angles. If, on the contrary, angular velocities are recorded then the angle values are obtained by differentiation of time histories of the velocities.

#### **3. Results**

The calculations were carried out for data of a medium-class compact passenger car KIA Ceed SW, i.e., mass 1870 kg, wheelbase 2.65 m, and distance between the centre of vehicle mass and the front axle 1134 m, with simulating vehicle motion on dry, level, and ideally even road surface with good tyre-road adhesion (equivalent to asphalt concrete). A specific vehicle motion (manoeuvre) was forced by applying control signals representing time histories of the steering wheel angle and brake pedal effort (in the case of braking). For "constant sped" curvilinear motion tests, a driver model (driving torque controller) was activated to maintain constant vehicle speed.

Four typical vehicle motion cases were examined:


In each of the cases, various intensities of the manoeuvres (measured by the level of longitudinal or lateral acceleration) were applied.

Both EDR types mentioned in Section 2.2 were taken into consideration: EDR1 (6 parameters of the motion) and EDR2 (3 parameters of the motion). For the analysis, an option was adopted where angles (instead of angular velocities) were used to define the angular position of the vehicle body solid. Moreover, to avoid the introduction of additional disturbing factors that might affect the calculation results, an assumption was made that the EDR sensors were placed at the centre of vehicle mass (see Figure 4,P=OC) and the motion of the centre of the vehicle mass was analysed (see Figure 1,R=P=OC). In addition to the above, an assumption was also made that the sensors had been calibrated for the vehicle standing on a horizontal road and loaded as in the tests. The frequency of recording the parameters under analysis was assumed as 20 Hz.

#### *3.1. Straight-Line Braking*

Figures 8 and 9 show the analysis results obtained for the case of braking the vehicle from an initial speed of 90 km/h with a deceleration of about 6 m/s2 (this is approximately equal to the absolute value of the longitudinal acceleration of the centre of the vehicle mass). The vehicle braking was caused by applying a force to the brake pedal, where the driver reaction time and the reaction of braking system components were reflected in the time history of the brake pedal force. Figure 8a shows the actual (simulated, considered as "accurate") value of longitudinal acceleration aP<sup>ξ</sup> and the value recorded by the EDR (awc). The difference between them (denoted by ϕawc) resulted from variations in the pitch angle *φ*<sup>1</sup> (see Figure 8b).

Figure 9 shows the reconstruction results obtained by using recorders of both types (EDR1 and EDR2). The time histories of vehicle velocity and position were reconstructed "backwards", i.e., from the known final vehicle position and velocity to the initial instant of the simulation of the vehicle motion. The data presented in Figure 9a,b concern vehicle velocity and position, respectively. The differences that can be seen at the beginning of the velocity and position curves illustrate the reconstruction errors, i.e., errors of determining the initial velocity (ΔV0) and distance travelled (ΔS).

**Figure 8.** Straight-line braking. Initial vehicle velocity 90 km/h, deceleration level 6 m/s2. Time histories of: (**a**) longitudinal acceleration (accurate values and EDR sensor indications); (**b**) car body pitch and roll angles (accurate values).

**Figure 9.** Straight-line braking. Initial vehicle velocity 90 km/h, deceleration level 6 m/s2. Reconstruction of time histories of: (**a**) vehicle velocity; (**b**) distance travelled by the vehicle. Reconstruction based on EDR1 and EDR2 records.

Very good conformity can be seen between the reconstructed and accurate values for the EDR1 device. This is thanks to the fact that in this case, the accelerometer indication error Δaw<sup>c</sup> is automatically corrected in the EDR1 calculation algorithm, because the vehicle pitch angle, which causes this error, is known and, in consequence, the value of this error can be determined. In the case of the EDR2 device, the pitch angle is not known and the acceleration signals being integrated are burdened with the error arising from that. In the case under consideration, this translates into an overestimation of the values of initial velocity and distance travelled by the vehicle.

Analysis results were obtained for more cases, differing from each other in the pre-set initial vehicle velocity (50, 90, and 140 km/h) and braking intensity (deceleration level of about 2, 4, 6, and 8 m/s2) have been summarised in Table 1. Additionally, the errors in estimating the initial velocity and stopping distance of the vehicle have been directly shown in the absolute and relative form in Figures 10 and 11.


**Table 1.** Straight-line braking. Summary of the accurate and reconstructed values of initial velocity and stopping distance of the vehicle for different preset levels of the initial velocity (50, 90, and 140 km/h) and braking intensity (deceleration level of about 8, 6, 4, and 2 m/s2).

**Figure 10.** Straight-line braking. Errors of the reconstruction of the initial velocity of the vehicle motion: (**a**) in the absolute form; (**b**) in the relative form.

**Figure 11.** Straight-line braking. Errors of the reconstruction of the vehicle stopping distance: (**a**) in the absolute form; (**b**) in the relative form.

In the case of EDR1, the errors are practically negligible (they arise from different integration frequencies in the vehicle motion simulation process and during the reconstruction of vehicle motion parameters). For the initial velocity and EDR2 records, growth

in the relative and absolute errors with increasing braking intensity can be seen. Such an effect may be explained by the fact that an increase in braking deceleration values results in a growth in the pitch angle and, in consequence, in the longitudinal acceleration estimation error, which translates into bigger errors of the integration output signals. A somewhat different situation takes place for stopping distance errors. In this case, the absolute errors do not depend so visibly on deceleration values and a relatively more significant role is played by the initial velocity value. However, the situation becomes similar to that observed for velocity errors already when relative errors of estimating the distance travelled are concerned: higher braking intensity results in bigger, in percentage terms, relative errors of estimating the braking distance.

Nevertheless, it should also be emphasised that, from the practical point of view, the reconstruction errors in the case of EDR2, although being bigger, are still not very serious (they did not exceed 3 % in the tests under consideration). This is significantly influenced by the general inertial characteristics of the vehicle body and by the design and stiffness of the vehicle suspension system, which translate into relatively small changes in the pitch angle of the vehicle during the braking manoeuvre. In similar research works described in [27,28], the errors in estimating the initial vehicle velocity and stopping distance were bigger (even twice). Those results were obtained for another passenger car, which was also more susceptible to vehicle body pitch and roll.

#### *3.2. Vehicle Lane-Change Maneuvre*

In this test, the vehicle is to perform a manoeuvre similar to the lane-change one. The test manoeuvre was forced by applying a control signal representing the steering wheel angle in the form of one period of sinusoidal input (with reference to the Sinusoidal Input test according to ISO 7401) with a reaction time of 1 s. The sinusoid parameters (period and amplitude) were selected individually for a specific vehicle test velocity (50, 90, 140 km/h) so that the vehicle virtually changed a lane about 3.5 m wide (i.e., the centre of vehicle mass, often referred to as centre of gravity—C.G., changed its lateral position by about 3.5 m) and for a chosen intensity of the manoeuvre (low, medium, or high, with the manoeuvre intensity being defined by the level of the maximum lateral C.G. acceleration). The total manoeuvre observation time was 5 s. Figures 12 and 13 show the results obtained for the vehicle moving with an initial velocity of 90 km/h and performing the manoeuvre with an intensity defined as "medium" (with the maximum lateral acceleration level of about 5 m/s2). Similarly, as in the case of braking, Figure 12a shows the actual (accurate) value of the lateral acceleration aP<sup>η</sup> and the value recorded by the EDR ap c. The difference between these two values (denoted by Δap c) mainly results from changes in the roll angle ϑ<sup>C</sup> and, to a lower degree, from changes in the pitch angle ϕC, shown in Figure 12b.

**Figure 12.** Lane-change manoeuvre, "medium" intensity level. Initial vehicle velocity 90 km/h. Time histories of: (**a**) lateral acceleration (accurate value and EDR sensor indication); (**b**) car body pitch and roll angles (accurate).

**Figure 13.** Lane-change manoeuvre, "medium" intensity level. Initial vehicle velocity 90 km/h. Reconstruction of: (**a**) vehicle velocity; (**b**) vehicle body C.G. trajectory. Reconstruction based on EDR1 and EDR2 records.

Figure 13 shows the results of reconstruction of a 5 s time period of the vehicle motion, obtained by using recorders of both types (EDR1 and EDR2). As previously, the time histories of vehicle velocity and position were reconstructed "backwards", i.e., from the known final vehicle position and velocity to the initial instant of the simulation of the vehicle motion. The data presented in Figure 13a,b concern vehicle velocity and C.G. trajectory, respectively. The differences that can be seen at the beginning of the velocity and position curves illustrate the reconstruction errors, i.e., errors of determining the initial velocity (ΔV0) and distance travelled (ΔS).

Similarly, as in the case of braking, very good conformity can be seen between the reconstructed and accurate values for the EDR1 device. The reason for this fact is identical. The accelerometer indication errors are automatically corrected because the vehicle pitch and roll angles are known. In the case of the EDR2 device, the said angles are not known, and the acceleration signals being integrated are burdened with the errors arising from that. In the case under consideration, this translates into an overestimation of the values of the lateral vehicle displacement and a velocity estimation error.

Analysis results obtained for more cases, differing from each other in the pre-set initial vehicle velocity (50, 90, and 140 km/h) and manoeuvre intensity (three lateral acceleration levels: about 2.4–2.5, 5.0–5.4, and 7.1–8.2 m/s2), only for the EDR2 device, in this case, have been summarised in Table 2. Additionally, the errors of estimating the initial velocity and distance travelled by the vehicle during the manoeuvre have been directly shown in the absolute and relative form in Figure 14. In quantitative terms, these errors may be considered quite small (tenths of percentage units), the reasons for which may be sought, as previously, in small changes in the roll angle (in this case) and in the resulting small errors of estimating the lateral acceleration. However, the trend of these errors rising with the intensity of this manoeuvre can be noticed here as well.


**Table 2.** Lane-change manoeuvre. Summary of the simulated (accurate) and reconstructed values of initial vehicle velocity, initial position estimation error, and distance travelled by the vehicle for different preset levels of the initial velocity (50, 90, and 140 km/h) and manoeuvre intensity (low, medium, high). Manoeuvre observation time: 5 s.

**Figure 14.** Lane-change manoeuvre. Errors of the reconstruction of the initial vehicle velocity and of the distance travelled by the vehicle: (**a**) in the absolute form; (**b**) in the relative form. Reconstruction based on EDR2 records.

23 23

#### *3.3. Entering a Turn*

The test consists in dynamic making the vehicle under test follow a curve. Such a manoeuvre was forced by applying a control signal representing a constant steering wheel angle preceded by a linear angle raising period (with reference to the Step Input test according to ISO 7401), where the steering wheel angle values were selected individually for a specific vehicle test velocity (50, 90, 140 km/h) so that the vehicle performed the manoeuvre with various intensities defined by the maximum lateral C.G. acceleration level in the steady phase of the motion (2, 4, 6, 8 m/s2). The angle-raising time was so adopted that the steering wheel rising-rate fell within the limits stipulated by the said ISO standard. The total manoeuvre observation time was 5 s. In this case, only the results obtained by using the EDR2 recorder have been presented (for the EDR1 device, the reconstruction results were practically identical with the accurate data). The reconstruction results have been presented in Figures 15–18, in the form as before, for two options of the manoeuvre, i.e., for initial velocities of 50 km/h and 140 km/h, and for the maximum lateral acceleration level of about 8 m/s2.

**Figure 15.** Entering a turn, intensity level 8 m/s2. Initial vehicle velocity 50 km/h. Time histories of: (**a**) lateral acceleration (accurate values and EDR2 sensor indications); (**b**) car body pitch and roll angles (accurate values).

**Figure 16.** Entering a turn, intensity level 8 m/s2. Initial vehicle velocity 50 km/h. Reconstruction of: (**a**) vehicle velocity; (**b**) vehicle C.G. trajectory; reconstruction based on EDR2 records.

**Figure 17.** Entering a turn, intensity level 8 m/s2. Initial vehicle velocity 140 km/h. Time histories of: (**a**) lateral acceleration (accurate values and EDR2 sensor indications); (**b**) Car body pitch and roll angle values (accurate).

**Figure 18.** Entering a turn, intensity level 8 m/s2. Initial vehicle velocity 140 km/h. Reconstruction of: (**a**) vehicle velocity; (**b**) vehicle C.G. trajectory; reconstruction based on EDR2 records.

Results of all the tests carried out with entering a turn have been presented in Table 3. For better illustration of the differences obtained, the errors of estimating the initial velocity and distance travelled by the vehicle during the manoeuvre have been directly shown in the absolute and relative form in Figure 19; Figure 20 presents the initial positions of the centre of vehicle mass (C.G.), reconstructed with EDR2 records being used.

**Table 3.** Entering a turn. Entering a turn. Summary of the simulated (accurate) and reconstructed values of initial vehicle velocity, errors of estimating the initial vehicle position, and distance travelled by the vehicle for different pre-set levels of the initial velocity (50, 90, and 140 km/h) and manoeuvre intensity (2, 4, 6, 8 m/s2). Manoeuvre observation time: 5 s.


**Figure 19.** Entering a turn. Errors of the reconstruction of the initial vehicle velocity and of the distance travelled by the vehicle: (**a**) in the absolute form; (**b**) in the relative form. Reconstruction based on EDR2 records.

**Figure 20.** Entering a turn. Initial positions of the centre of vehicle mass (C.G.), reconstructed with EDR2 records being used.

When analysing the results obtained for this manoeuvre, it should be noted that the reconstruction errors are here markedly bigger than those occurring in the case of the lane-change manoeuvre. For the motion, parameters range under consideration, the errors of estimating the initial velocity or distance travelled by the vehicle reach from a few to even more than ten percent (Figure 19). The initial vehicle position obtained from the reconstruction may differ from the actual (accurate) one by a few meters (Figure 20). Noteworthy is the fact that the manoeuvre observation time was identical for both manoeuvre types (5 s). The reasons for such an effect may be sought in the fact that, similarly as it is in the case of braking and longitudinal deceleration, no reversal of the sign of vehicle acceleration and its error takes place (Figures 15 and 17), contrary to the situation that takes place during the lane-change manoeuvre, where the errors of the reconstruction (acceleration quadrature operations) generated in one phase of the manoeuvre are partly compensated by the reconstruction errors generated in the other manoeuvre phase.

Another factor worth consideration is the impact of manoeuvre intensity. Here, similarly as in other manoeuvres, the errors increase with rising manoeuvre intensity; this may be directly associated with an increase in the errors of the EDR acceleration records due to growing vehicle body pitch and roll angles.

The last regularity that can be noticed is a decrease in error values with increasing vehicle velocity, which distinguishes this manoeuvre from a similar case (in terms of absence of reversal of the acceleration sign), i.e., from the straight-line braking. To explain this finding, it is worth noticing that, apart from the initial phase of the manoeuvre, the vehicle motion under analysis is similar to the steady-state motion along a circle. Therefore, it is worth paying attention to another variable that characterises the vehicle motion, i.e., the yaw angle. Figure 21 presents the data collected that show the "angular distance" travelled by the vehicle in every case. When comparing the data plotted in this graph with the reconstruction errors presented previously in Figure 19, their qualitative convergence can be noticed: the higher the velocity, the smaller the change in the yaw angle in the manoeuvre under consideration. Based on the said qualitative convergence, a hypothesis may be formulated that the value of errors, especially the errors of the vehicle trajectory, are affected by the "angular distance" travelled by the vehicle. This aspect will be presented in broader terms in the next example, where the steady-state circular motion will be analysed.

**Figure 21.** Entering a turn. Range of changes in the vehicle yaw angle during the manoeuvre.

#### *3.4. Driving in a Roundabout (Circular Motion)*

In such a test, the vehicle is driven with a constant speed along a curvilinear path close to a circle (to some extent, the test conditions are close to those of the ISO 4148 test [42]). The manoeuvre is forced by applying a constant non-zero steering wheel angle. The test parameters (vehicle speed, steering wheel angle) were so selected that, on the one hand, the vehicle moved with a prescribed lateral acceleration (2, 4, or 6 m/s2) and, on the other hand, the vehicle path was in accordance with the roundabouts that can be met in real roads [48] (and simultaneously the actual radius of the vehicle path was in conformity with the recommendations of the said ISO 4138 standard). Actually, the vehicle path radius was set at about 30 m, and the values of the vehicle speed on the virtual traffic circle were about 30, 40, and 50 km/h. The manoeuvre was considered as completed when the vehicle travelled a "full circle", i.e., the vehicle yaw angle changed by 2π in the simulation.

This time, the trajectories reconstructed ("backwards" from the final vehicle position) have been presented as the first thing (see Figure 22). Figure 23 shows a reconstruction of the time history of the vehicle speed ("backwards" from the final instant). The direct causes of the visible differences between the accurate and reconstructed vehicle trajectories and speed vs. time curves mainly lie in errors of the EDR2 acceleration records (differences between the accurate and recorded acceleration values) shown in Figure 24.

At first, the differences between the accurate and reconstructed curves can be found to be similar to each other in qualitative terms: the differences increased in the first phase of the manoeuvre and decreased afterwards. This resulted in very good conformity between the accurate and reconstructed value of the vehicle speed and in quite good conformity between the accurate and reconstructed initial vehicle position (the vehicle C.G. coincided with the accurate trajectory). Here, reference may be made to the hypothesis proposed in the previous subsection about the dependence of errors on the "angular distance" travelled by the vehicle and, in consequence, about the "periodicity" of the errors. Therefore, the vehicle speed and position errors have been presented in Figure 25 not in the time domain but as functions of changes in the angular position (yaw angle) within the range from the final position (for which the change is equal to zero) to the initial position (where the change is equal to −2π). Additionally, the estimated values of the vehicle body C.G. path radius and the vehicle yaw velocity have been presented in Figure 26.

**Figure 22.** Driving in a roundabout, R = 30 m. Reconstruction of vehicle body C.G. trajectory based on EDR2 records. Lateral acceleration level: (**a**) 2 m/s2; (**b**) 4 m/s2; (**c**) 6 m/s2. "•"—start of the reconstruction (end of the motion under analysis); ""—end of the reconstruction (reconstructed initial position); grey dotted lines represent edges of the roundabout carriageway 6 m wide.

**Figure 23.** Driving in a roundabout, R = 30 m. Reconstruction of vehicle C.G. speed based on EDR2 records for three lateral acceleration levels: 2 m/s2; 4 m/s2; 6 m/s2.

**Figure 24.** Driving in a roundabout, R = 30 m. Comparison between time histories of accurate values and EDR2 sensor readings of: (**a**) lateral acceleration; (**b**) longitudinal acceleration.

**Figure 25.** Driving in a roundabout, R = 30 m. Errors of: (**a**) vehicle body C.G. speed; (**b**) vehicle body C.G. distance travelled as a function of changes in the vehicle yaw angle (zero—vehicle final position, −2π—vehicle initial position).

**Figure 26.** Driving in a roundabout, R = 30 m. (**a**) Estimated radius of the vehicle body C.G. trajectory; (**b**) Yaw velocity as a function of changes in the vehicle yaw angle (zero—vehicle final position, −2π—vehicle initial position).

It can be seen that on the interval from zero to −π, the speed reconstruction error is increasing; in consequence, the rate of growth in the error of estimating the distance travelled is increasing as well. On the interval where the vehicle has already "turned back", i.e., from −π to −2π, the speed reconstruction error is decreasing to values close to zero; the distance error is still increasing, but the growth rate is decreasing. Since the vehicle yaw velocity is approximately constant (Figure 26b), the changes in the reconstructed vehicle speed value translate into changes in the reconstructed trajectory radius value (Figure 26a), i.e., growth in the radius on the yaw angle range from 0 to −π and decline on the range from −π to −2π.

Noteworthy is also the fact that the reconstruction errors are also affected by the error in estimating the longitudinal acceleration (Figure 24b). In the example presented, this can be seen, e.g., in Figures 25b and 26a, when the cases with 4 m/s2 and 6 m/s<sup>2</sup> are compared with each other. For the manoeuvre with 6 m/s2, the longitudinal acceleration level, in terms of its absolute value, is lower than that for the manoeuvre with 4 m/s<sup>2</sup> (see Figure 24b). This has contributed to the fact that in spite of a higher level of lateral acceleration, the errors in estimating the distance travelled are here somewhat lower (see Figure 25b). Similarly, the radius of the reconstructed trajectory is also somewhat smaller (see Figure 26a).

It may be assumed that during the next circling, this situation may repeat itself again and again. To illustrate this effect, Figure 27 presents the vehicle trajectory and speed curves for the simulation of driving in a runabout with a speed of about 50 km/h (lateral acceleration level of 6 m/s2) for an "angular distance" exceeding the value of 2π as shown previously (the yaw angle was now 3.02 rad at the initial instant and 14.03 rad at the final instant, i.e., it changed by 11.01 rad ∼= 3.5π). The vehicle speed error "oscillates" with a period of 2π, the distance error is increasing cycle after cycle, and the reconstructed trajectory "shifts" by a certain distance with every cycle.

**Figure 27.** Driving in a roundabout, R = 30 m. Lateral acceleration level 6 m/s2 (vehicle speed ca. 50 km/h). Reconstruction based on EDR2 records of: (**a**) vehicle body C.G. trajectory; (**b**) vehicle body C.G. speed and error of the distance travelled. "•"—start of the reconstruction (end of the motion under analysis); ""—end of the reconstruction (reconstructed initial position); grey dotted lines represent edges of the roundabout carriageway 6 m wide.

In the context of the above findings, therefore, attention should be paid to the fact that at the reconstruction of manoeuvres like the one discussed here (or the entering a turn, as discussed previously), a factor of considerable importance for the magnitude of errors in the event reconstruction based on records obtained by using EDR2-like devices is a change in the yaw angle during the manoeuvre. Depending on the scope of the change, the errors may be small or very big. For the vehicle trajectory, the positions determined by reconstruction may so differ from the actual ones that correct interpretation of the situation under analysis would become impossible (errors would be of the order of several meters, such that the vehicle position thus determined would be situated e.g., outside of the road area or of any realistic vehicle trajectory).

#### **4. Discussion**

The results obtained have been discussed in detail immediately after their presentation. To synthesise the findings in the context of the accuracy of the reconstruction of vehicle motion, the following may be stated.



To some extent, especially as regards the quantitative findings, the results presented confirm the conclusions expressed in [27–29] as well as in [26]. The directions of further research on the accuracy of reconstruction of vehicle motion based on records obtained from EDR-type devices, concerning such issues as the impacts of current load and design features of the vehicle, accuracy of the measuring and recording apparatus, angular positioning of EDR sensors, or EDR configuration (e.g., recording of angular velocities instead of angles, recording frequency), can also be formulated.

#### **5. Conclusions**

The large-scale introduction of automotive "black boxes" into service will potentially bring many benefits. It will increase the resource of information about the course of an accident and will provide knowledge of the parameters of vehicle motion. Other advantages may consist in preventive influence on the driver, who will be less prone to risky behaviours in road traffic. The EDR solutions offered now in the automotive market are often a simplified version of the solutions having been used in aviation for many years. In the EDRs offered in the market, apart from those intended for research applications, the vehicle motion is generally treated as two-dimensional (as it is in the EDR2).

In the research carried out, the interest was focused on the concept of the EDR, without addressing the problems related to the measuring and recording apparatus (such as intrinsic sensor errors or errors arising in the recording process). The calculations carried out have shown that the said simplifications may result in significant errors in the reconstruction of vehicle motion, sometimes totally disqualifying the reconstruction results. This mainly applies to the reconstruction of vehicle trajectory; nevertheless, a result significantly differing from the actual value is also possible when the vehicle velocity is reconstructed. The basic reason for that lies in the fact that the devices of the EDR2 type do not provide information about some angular movements of the vehicle body solid, i.e., the pitch and roll angles. In the case of the EDR1 devices, no considerable reconstruction errors have been revealed.

**Author Contributions:** Conceptualization, M.G. and Z.L.; methodology, M.G. and Z.L.; software, M.G. and Z.L.; validation, M.G. and Z.L.; formal analysis, M.G. and Z.L.; investigation, M.G. and

Z.L.; writing—original draft preparation, M.G. and Z.L.; writing—review and editing, M.G. and Z.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. APC was funded by Warsaw University of Technology.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The simulation model of motor vehicle motion and dynamics, used in this work, was built during the execution of a project carried out by the Faculty of Transport, Warsaw University of Technology (WUT) to commission of ETC-PZL Aerospace Industries Ltd. (being now owned by a US company Environmental Tectonics Corporation and AIRBUS Poland S.A.). The superior goal of the project was to build a simulator of driving emergency service vehicles within project No O ROB 0011 01/ID/11/1 "Simulator of driving emergency service vehicles during standard and extreme actions" (in the years 2012–2013). The experimental laboratory and road tests were carried out by teams of the Cracow University of Technology (managed by Wiesław Pieni ˛azek), Automotive ˙ Industry Institute, now ŁUKASIEWICZ-PIMOT (managed by Dariusz Wi ˛eckowski), and Military University of Technology (managed by Jerzy Jackowski), with active participation of the authors of this paper and Witold Luty (WUT, Faculty of Transport).

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

#### **References**

