Probabilistic Prediction of Collisions between Cyclists and Vehicles Based on Uncertainty of Cyclists’ Movements

Abstract

The uncertainty of cyclists’ movements has a significant impact on predicting the risk of collisions between cyclists and vehicles. The purpose of this study was to provide a method for assessing collision risk using probability, taking into account the uncertainty of cyclists’ movements. A cyclist model was first developed using a first-order Markov model. Then, based on Monte Carlo sampling, the distribution characteristics of the minimum distance and the time-to-collision (TTC) between the vehicle and the cyclist were extracted. By fitting these features, the probability density functions of the collision distance and TTC were estimated to derive the collision probabilities. The effectiveness of the collision probability prediction model was benchmarked against a deterministic crash risk prediction model (autonomous emergency braking [AEB] system) applied to three real-world cases previously reconstructed in an in-depth crash database. The results show that the collision probability prediction model can effectively predict the risk of collisions between cyclists and vehicles with better accuracy than AEB systems using a fixed trigger threshold. This study is a valuable reference for the development of advanced vehicle collision avoidance systems to protect cyclists and other vulnerable road users.

Keywords

bicyclist collision risk probability distribution uncertainty Monte Carlo

According to the World Health Organization, vulnerable road users (VRUs), that is, pedestrians, cyclists, and motorcyclists, account for more than half of global road traffic deaths, with cyclists accounting for 3% of global deaths ( 1 ), although in some countries, such as Germany (12%) and Japan (11.8%) ( 2 ), the percentage is higher. In China, the number of cyclists killed or injured in 2019 was 2,774 and 8,174, respectively, with fatalities accounting for 4.42% of all traffic fatalities ( 3 ).

Active safety systems, such as collision mitigation and collision avoidance systems, are now on the market and can effectively prevent accidents or reduce the severity of injuries ( 4 , 5 ). These systems consist of four main basic components: environmental perception; collision risk assessment; decision-making; and avoidance measures. To reduce false warnings and decision errors, accurate assessment of crash risk is an important function in addition to accurate sensing of the vehicle’s environmental. Quantitative threshold measures such as TTC ( 6 , 7 ), time-to-braking, and time-to-reaction (TTR) ( 6 – 10 ) have been widely adopted to assess crash risk. However, these metrics are typically established assuming that the vehicle and VRUs do not change travel speed before a collision occurs. It is clear that these deterministic time metrics do not take into account the effects of uncertainty in traffic accidents, which may lead to misjudgment of crash risk by the system and result in suboptimal decisions.

In the field of active safety, one of the important uncertainties is the measurement errors of sensors that detect VRUs on the road ( 11 , 12 ). If the objects are not detected clearly enough to obtain accurate measurements, the accuracy and robustness of the prediction algorithms decrease, thus reducing the performance of subsequent decisions ( 8 , 12 ). With the development of computer vision, Kooij et al. combined images with a dynamic Bayesian network. This system could efficiently calculate probabilistic path predictions considering the uncertainty in measurements and target behavior ( 13 ). Zhou et al. ( 14 ) utilized the extended Kalman filter to detect the motion status of objects, thus reducing the errors from sensor measurement noise and blind zones.

The second type of uncertainty is the change in the behavior of traffic participants over short time periods. For example, VRUs can quickly change their speed or direction according to their habits and travel intentions. To address the effect of VRUs’ uncertain movements, for example, those of pedestrians, the uncertainty can be described by establishing a stochastic model of VRU kinematics and calculating the probability distribution of the corresponding evaluation metric. Monte Carlo sampling is one of the most straightforward methods for assessing the probability of collision ( 4 ). By combining a simple but representative stochastic model for pedestrians and an offline risk assessment relying on Monte Carlo simulation, De Nicolao et al. ( 15 ) proposed a crash risk assessment method based on scenarios and pedestrian behavior. Further, Tamke et al. ( 6 ) proposed the binary search algorithm for searching for time-to-maneuver and calculating TTR. Based on Gaussian distribution ( 16 ), Berthelot et al. ( 11 , 17 ) addressed the uncertain movements of vehicles. Huang et al. applied the unscented transformation ( 18 ) as an alternative to Monte Carlo simulation to estimate collision probabilities effectively and efficiently ( 12 ). In addition, visual inspection and machine learning algorithms have been used to predict VRU intention and short-term movements, and these uncertainty estimations can be incorporated into the mathematical description of collision possibility ( 19 – 21 ). Lambert et al. ( 22 ) integrated the Gaussian set into the vehicle velocity to calculate collision probability. Eidehall and Petersson ( 4 ) used the repeated Monte Carlo sampling method to sample uncertainty in the positions of surrounding objects to calculate collision probabilities.

Previous research has focused on how path uncertainty affects pedestrian detection by vehicle sensor systems; however, the uncertainty associated with cyclists’ movements has not been systematically investigated. Kooij et al. ( 13 ) considered the uncertainty of cyclists’ movements based on diverse and relevant cues that may also have some uncertainty. However, a large amount of computation is required in model prediction inference and mathematical model construction, and this consumes a lot of computing devices’ computational power. In addition, it is not known whether the prediction models developed for pedestrians are equally applicable to cyclists, because of the higher speed and larger range of cyclists’ movements. Furthermore, as studied by Gao et al. ( 23 ), it is important to consider the uncertainty associated with the trajectory of cyclists, especially in urban environments.

The objective of this study was to develop a predictive model of collision probability associated with a dynamic model of cyclists at unsignalized intersections, taking into account the uncertainty of cyclists’ movements. The model would also be validated against three real-world cyclist accident reconstructions and benchmarked against results for three autonomous emergency braking (AEB) systems with different braking preferences. These results provide an important reference for predicting the collision risk of autonomous vehicle systems in relation to nondeterministic kinematics pertaining to VRUs such as cyclists and pedestrians.

Method

Vehicle Model

To improve the stability and accuracy of collision prediction, vehicle movement is mostly described by deterministic motion models, for example, the constant velocity model, the constant acceleration model, the constant turn rate and acceleration model, and the constant curvature and acceleration (CCA) model ( 24 , 25 ). In this study, the CCA model was applied to describe the transient movement of the vehicle to predict the collision probability. The state of the vehicle $Stat e_{veh}$ can be expressed as follows:

\begin{matrix} Stat e_{veh} = (x_{veh}, y_{veh}, v_{veh}, θ_{veh}) \end{matrix}

(1)

where $x_{veh}$ and $y_{veh}$ denote the longitudinal and lateral coordinates, respectively, of the vehicle in the real-world coordinate system as shown in Figure 1, $v_{veh}$ denotes the vehicle velocity, and $θ_{veh}$ indicates the heading angle of the vehicle. All states are obtained based on the fixed coordinate system XOY. The variable $u (r, a)$ was used as the input described in the CCA model and $r$ is the vehicle steering radius. If $r > 0$ the vehicle turns to the right, if $r < 0$ the vehicle turns to the left, and if $r = + \infty$ the vehicle drives straight on. The acceleration of the vehicle is given by $a$ . Thus, the vehicle status at any time $t$ can be further denoted as follows:

\begin{matrix} S_{veh} (t) = f (state (t_{0}), u (r, a), t) \end{matrix}

(2)

Figure 1.

Road coordinate system: (a) real-world coordinate system; and (b) parametric analysis diagram.

Cyclist Model

In this study, cyclists’ movements were considered as a first-order Markov process ( 12 , 21 ), and Gaussian white noise was used to describe the uncertainty of cyclists’ speed variation ( 12 , 15 , 26 ). The state of cyclists’ movements can be expressed as follows:

\begin{matrix} Stat e_{b} = (x_{b}, y_{b}, v_{x_{b}}, v_{y_{b}}) \end{matrix}

(3)

where $x_{b}$ and $y_{b}$ are the cyclist’s coordinates and $v_{xb}$ and $v_{yb}$ are the cyclist’s velocities in the X and Y directions. As a result, the cyclist’s movement in the X direction can be further expressed as follows:

\begin{matrix} x_{b} (t + Δ t) = x_{b} (t) + Δ t \cdot v_{x_{b}} \end{matrix}

(4)

\begin{matrix} v_{x_{b}} (t + Δ t) = v_{x_{b}} (t) + Δ v_{x_{b}} \end{matrix}

(5)

\begin{matrix} Δ v_{x_{b}} = - κ_{x} [v_{x_{b}} (t) - v_{x_mean}] + w_{x} (t) \end{matrix}

(6)

\begin{matrix} v_{x_mean} ~ N (μ_{x}, σ_{xm}^{2}) \end{matrix}

(7)

\begin{matrix} w_{x} (t) ~ WGN (0, σ_{x}^{2}) \end{matrix}

(8)

where $v_{x_mean}$ refers to the average velocity of cyclists crossing the road, following the normal distribution $N (μ_{x}, σ_{xm}^{2})$ . The velocity noise $w_{x} (t)$ follows a Gaussian distribution $WGN (0, σ_{x}^{2})$ with mean 0 and variance $σ_{x}^{2}$ . $κ_{x}$ is described as the adjustment factor of the time constant in the first-order Markov model. Correspondingly, the movement along the $y$ -axis can be represented in the same form, with the subscript $x$ replaced by $y$ .

To obtain the cyclist’s uncertain movement parameters $v_{x_mean}$ , $v_{y_mean}$ , $ω_{x} (t)$ , $ω_{y} (t)$ , $κ_{x}$ , and $κ_{y}$ , several road tests were conducted with an unmanned aerial vehicle that was stationary at high altitude at a signal-free four-lane intersection in Xiamen, China (Figure 1). Video analysis of cyclists’ movements at $Δ t$ time intervals extracted their pixel coordinates, which were converted into real-world coordinates by direct linear transformation ( 20 ). Video was recorded at 25 frames per second and data were extracted every 5 frames, yielding a $Δ t$ of 0.2 s.

Risk Assessment Metric and Distance Model

The time metric TTC is very useful for assessing the risk of traffic accidents. In this study, TTC was used to represent the risk of a collision between a vehicle and a cyclist during the future period $T = [t_{o}, t_{h}]$ , which is defined as follows:

\begin{matrix} TTC = \min {t \in T | d (t) \leq 0} \end{matrix}

(9)

where $d (t)$ represents the distance between the vehicle and the cyclist. The time $T$ is the prediction horizon, usually two to three seconds.

\begin{matrix} d (t) = d (s (S_{veh} (t)), s (S_{b} (t))) \end{matrix}

(10)

where $s (S_{veh} (t))$ and $s (S_{b} (t))$ denote the space occupied by the vehicle and the cyclist, respectively, at moment $t$ , as shown in Figure 2. Because the cyclist mainly collides with the front end of the vehicle, this study takes the position of the on-board sensor as the center of the circle representing the shape of the vehicle with a radius of about 1.7 m. Similarly, the bicycle and the cyclist are simplified to a circle with a radius of 0.8 m. Therefore, $d (t)$ is defined as follows:

\begin{matrix} d (t) = \sqrt{{(x_{veh} (t) - x_{b} (t))}^{2} + {(y_{veh} (t) - y_{b} (t))}^{2}} - 2.5 \end{matrix}

(11)

According to Equation 11, the mapping space of the distance function $d (t)$ is [ $- 2.5, + \infty)$ . If $d < 0$ , this means that a collision will occur. If $d (t) > 0$ , $\forall t \in T$ , no collision is predicted, at which point TTC is +∞.

\begin{matrix} T T C = {\begin{matrix} m i n {t \in T | d (t) \leq 0} Crash \\ + \infty No Crash \end{matrix} \end{matrix}

(12)

Figure 2.

Vehicle-to-cyclist collision distance model.

Collision Probability Distribution

Considering the uncertainty of cyclists’ movements, $P (TTC \in R)$ was used to assess the likelihood of a collision between the vehicle and the cyclist. In addition, to determine whether a collision would occur at some future time, $P (TTC \leq t)$ was used to predict the probability distribution of the collision time. When TTC is $+ \infty$ , no collision is predicted and is a singularity in the value of $P (TTC \leq t)$ . Therefore, the joint collision probability $P (TTC \leq t, TTC \in R)$ was introduced to replace $P (TTC < t)$ ( 11 ). Also, according to Bayes’ theorem, the collision probability can be derived as follows:

\begin{matrix} P (TTC \leq t, TTC \in R) = P (TTC \leq t | TTC \in R) (TTC \in R) \end{matrix}

(13)

By calculating the shortest distance ( $d_{\min}$ ) between the vehicle and the cyclist, the probability density function of the distance function $d (t)$ is obtained, and the collision probability can be further calculated; thus, the collision probability can be expressed as follows:

\begin{matrix} P (TTC \in R) = P (d_{\min} \leq 0) = \int_{- \infty}^{0} p_{d} (x) dx \end{matrix}

(14)

where $p_{d} (x)$ is the probability density function of the shortest distance $d_{\min}$ .

According to Equation 12, the TTC may not exist when no collision occurs; therefore, the direct calculation of $P (TTC \leq t | TTC \in R)$ is complicated. To simplify the calculation, a pseudo TTC ( $\bar{\underline{TTC}}$ ) was introduced into this study instead of TTC to consider all possible collision situations ( 11 , 17 ):

\begin{matrix} \bar{\underline{T T C}} = {\begin{matrix} T T C if collision is predicted \\ t_{m i n} if no collision is predicted \end{matrix} \end{matrix}

(15)

where $t_{\min}$ denotes the moment at which the distance between the cyclist and the vehicle $d (t)$ reaches its minimum value. According to Berthelot et al. ( 17 ), $P (TTC < t_{threshold})$ can be well fitted to the conditional probability $P (TTC \leq t | TTC \in R)$ . Therefore, the joint collision probability can be converted to the following equation:

\begin{matrix} P (TTC \leq t, TTC \in R) = P (TTC \in R) (TTC \leq t | TTC \in R) \\ \approx P (TTC \in R) P (\underline{\bar{TTC}} \leq t) \\ = \int_{- \infty}^{0} p_{d} (x) dx \cdot \int_{0}^{t} p_{t} (τ) d τ \end{matrix}

(16)

where $p_{t} (τ)$ is a pseudo probability density function.

Calculation of the $d_{\min}$ and TTC

To calculate $P (TTC \leq t, TTC \in R)$ , the collision probability $P (TTC \in R)$ and the probability distribution of the collision time $P (TTC \leq t | TTC \in R)$ should be determined first. Predicting these two probabilities requires finding the exact $d_{\min}$ and $\underline{\bar{TTC}}$ , which can be done by using the dichotomy method. Considering the time step $Δ t$ = 0.2 s for the cyclist model, the dichotomous search flow chart shown in Figure 3 was established to determine $d_{\min}$ . For the collision time, the earliest moment when the distance $d (t)$ crosses the zero can be found in a similar way.

Figure 3.

Flow chart of shortest distance dichotomous search method.

Collision Probability Estimation Using Monte Carlo Sampling

Monte Carlo sampling ( 27 – 29 ) with a sample of 10,000 was used to estimate the density histogram of $d_{\min}$ and $\bar{\underline{TTC}}$ and to calculate the collision probability. As shown in the results section “Distribution Characteristics of $d_{\min}$ and TTC,” the distribution of $d_{\min}$ and $\bar{\underline{TTC}}$ follows the approximate Gaussian distributions $N (μ_{MC_d}, σ_{MC_d}^{2})$ and $N (μ_{MC_t}, σ_{MC_t}^{2})$ , respectively. Further, because the mapping space of the distance function is [ $- 2.5; + \infty)$ , a Gaussian derivative function was proposed to describe and fit the probability density function, which was denoted as the left saturated Gaussian distribution $N_{d} (α, β, γ^{2})$ ( 11 ). The probability density function of the left saturated normal distribution $N_{d} (α, β, γ^{2})$ can be defined as follows:

\begin{matrix} p_{d} (x) = \frac{1}{\sqrt{2 π} \cdot μ_{M C_{d}}} \cdot \exp [- \frac{{(x - μ_{M C_{d}})}^{2}}{2 σ_{M C_{d}}^{2}}] \end{matrix}

(17)

The moments of $N_{d} (α, β, γ^{2})$ are

\begin{matrix} \binom{E [X_{α, β, γ}] = \int_{- \infty}^{+ \infty} x \cdot p_{d} (x) dx = \frac{γ}{\sqrt{2 π}} \exp [- \frac{{(α - β)}^{2}}{2 \cdot γ^{2}}]}{- \frac{α - β}{2} erfc (\frac{α - β}{\sqrt{2} γ}) + α} \end{matrix}

(18)

\begin{matrix} \binom{E [{X^{2}}_{α, β, γ}] = \int_{- \infty}^{+ \infty} x^{2} \cdot p_{d} (x) dx}{= \frac{γ (α + β)}{\sqrt{2 π}} \exp [- \frac{{(α - β)}^{2}}{2 \cdot γ^{2}}] + \frac{γ^{2} + β^{2} - α^{2}}{2} erfc (\frac{α - β}{\sqrt{2} γ}) + α^{2}} \end{matrix}

(19)

respectively, where $α$ represents the left saturation and $α = - 2.5$ . $β$ and $γ$ are determined by solving Equation 20

\begin{matrix} {\begin{matrix} μ_{MC_d} = E [X_{α, β, γ}] \\ μ_{MC_d}^{2} + σ_{MC_d}^{2} = E [X_{α, β, γ}^{2}] \end{matrix} \end{matrix}

(20)

Thus, the $p_{d} (x)$ and the calculation for integrating the probability density function can be described as follows:

\begin{matrix} P (TTC \in R) = \int_{- \infty}^{0} p_{d} (x) dx = \int_{- \infty}^{0} \frac{1}{\sqrt{2 π} γ} \cdot \exp [- \frac{{(x - β)}^{2}}{2 γ^{2}}] dx \end{matrix}

(21)

The probability density function $p_{t} (t)$ for $\underline{\bar{TTC}}$ is similar to the collision probability density function $p_{d} (x)$ . Thus, another Gaussian derivative function was proposed to describe and fit $p_{t} (t)$ , which was denoted as left saturated Gaussian with Dirac $δ$ function. The selection range of collision time is $[t_{0}, t_{h}]$ , the saturation point is at $t_{0} = 0 s$ , and $t_{h} = 3 s$ corresponds to its probability density function as $N_{t} (t_{0}, t_{h}, η, λ^{2})$ ( 17 ).

\begin{matrix} p_{t} (x) = φ (x | η, λ^{2}) Π (\frac{x - t_{0}}{t_{h} - t_{0}} - \frac{1}{2}) \\ + Φ (t_{0} | η, λ^{2}) δ (x - t_{0}) \\ + Φ^{*} (t_{h} | η, λ^{2}) δ (x - t_{h}) \end{matrix}

(22)

where $Π$ is a rectangular function, $t_{0}$ is at the initial prediction moment, and $η$ is the mean of the probability density function. The moments of $N_{t} (t_{0}, t_{h}, η, λ^{2})$ are given as follows:

\begin{matrix} E [X_{t_{0}, t_{h}, η, λ^{2}}] = \int_{- \infty}^{+ \infty} t \cdot p_{t} (t) dt \\ = t_{0} + \frac{λ}{\sqrt{2 π}} {\exp [- \frac{{(t_{0} - η)}^{2}}{2 \cdot λ^{2}}] - \exp [- \frac{{(t_{h} - η)}^{2}}{2 \cdot λ^{2}}]} \\ - \frac{t_{0} - η}{2} erfc (\frac{t_{0} - η}{\sqrt{2} λ}) + \frac{t_{h} - η}{2} erfc (\frac{t_{h} - η}{\sqrt{2} λ}) \end{matrix}

(23)

\begin{matrix} E [X_{t_{0}, t_{h}, η, λ^{2}}^{2}] = \int_{- \infty}^{+ \infty} t^{2} \cdot p_{t} (t) d t \\ = t_{0}^{2} + \frac{η^{2} + γ^{2} - t_{0}^{2}}{2} e r f c (\frac{t_{0} - η}{\sqrt{2} λ}) - \frac{η^{2} + λ^{2} - t_{h}^{2}}{2} e r f c (\frac{t_{h} - η}{\sqrt{2} λ}) \\ + \frac{λ (t_{0} + η)}{\sqrt{2 π}} e x p [- \frac{{(t_{0} - η)}^{2}}{2 λ^{2}}] - \frac{λ (t_{h} + η)}{\sqrt{2 π}} e x p [- \frac{{(t_{h} - η)}^{2}}{2 λ^{2}}] \end{matrix}

(24)

Similarly, $η$ and $λ$ are determined by solving Equation 25

\begin{matrix} {\begin{matrix} μ_{MC_t} = E [X_{t_{0}, t_{h}, η, λ^{2}}] \\ μ_{MC_t}^{2} + σ_{MC_t}^{2} = E [X_{t_{0}, t_{h}, η, λ^{2}}^{2}] \end{matrix} \end{matrix}

(25)

As a consequence, the conditional collision probability $P (TTC \leq t | TTC \in R)$ can be obtained using the following equation:

\begin{matrix} P (TTC \leq t | TTC \in R) \approx P (\underline{\bar{TTC}} \leq t) = \int_{0}^{t} p_{t} (x) dx \end{matrix}

(26)

Simulation Analysis

To validate the collision probability model, three car-to-cyclist accident cases were selected from the VRU Traffic Accident Database with Video ( 30 , 31 ) to investigate the collision probability. The database aims to obtain more detailed and objective precrash, crash, and postcrash accident information by collecting in-depth accident data with videos as well as public internet videos. All accident videos are recorded by either a drive recorder or an intersection monitor. In accordance with the high-precision reconstruction method used for the accident ( 32 , 2 ), the precrash scenarios of the three cases were first reconstructed in PC-Crash Ver. 11.0 Based on the reconstruction results, the boundary conditions, for example, relative position and velocity, were extracted for the vehicle and the cyclist at $Δ t$ intervals. These trajectories were imported into the collision prediction model as initial inputs for estimating the collision probability. In addition, the cases were also resimulated in PC-Crash by equipping accident vehicles with different AEB systems to compare the differences between collision probability prediction models and deterministic prediction models. Specifically, three AEB systems with different braking characteristics were configured: a regulation-based AEB system; an aggressive AEB system; and a conservative AEB system.

Appendix A shows the process of AEB intervention and the braking deceleration profile ( 33 ). AEB is triggered when the time-to-collision $T_{TC}$ (ratio of relative distance between vehicle and cyclist to relative speed) trigger threshold is reached. According to the references, in this study, the braking precharge time ( $t_{BPT}$ ) and the braking boost time ( $t_{BBT}$ ) were 0.1 s and 0.15 s, respectively ( 34 , 35 ). Based on the relationship between braking velocity, deceleration, displacement, and field of view (FOV) described by Equations A1–A5 in Appendix A, and the test protocol of the European New Car Assessment Program ( 36 ), in which the coefficient of friction is $μ$ =0.9, the speed of the vehicle is $v_{vehicle}$ =60 km/h to 20 km/h, and the speed of the cyclist is $v_{cyclist}$ =15 km/h, the parameters of the regulation-based AEB system were determined to be $T_{TC}$ for 1.1 s with a FOV of 75°. With reference to the regulation-based AEB system, the parameters of the other two AEB systems were then determined. Because the conservative AEB system represents a system that is triggered by a lower crash risk, the $T_{TC}$ trigger threshold is higher and the braking deceleration is lower. However, an aggressive AEB system is only triggered by a higher crash risk for which the $T_{TC}$ is lower and the braking deceleration is higher. The parameters of all three AEB systems used in this study are provided in Table 1.

Table 1.

Parameters of the Autonomous Emergency Braking (AEB)System for the Three Different Braking Preferences

AEB parameters	Braking preference			References
AEB parameters	Conservative	Regulation-based	Aggressive	References
Deceleration (g)	0.5	0.9	0.9	Zhao et al. ( 2 ) and, Rosén ( 5 )
$T_{TC}$ (s)	1.4	1.1	0.8	Zhao et al. ( 2 ) and Wu et al. ( 32 )
FOV (°)	90	75	75	Zhao et al. ( 2 ), Rosén ( 5 ), and Lenard et al. ( 37 )

Note: (g) = gravity; (s) = seconds; (°) = degrees; FOV = field of view.

Results

Uncertainty Characteristics of Cyclists’ Movements

Video data was collected for 200 cyclists crossing the road presented in Figure 1 A time series record of each cyclist’s position state was created, and the statistics describing the cyclists’ movements are as follows.

v_{x_mean} ~ N (0.051, 0.315)

v_{y_mean} ~ N (2.263, 0.182)

\begin{matrix} ω_{x} (t) ~ WGN (0, 0.302) \\ ω_{y} (t) ~ WGN (0, 0.184) \end{matrix}

\begin{matrix} k_{x} = 0.45 \\ k_{y} = 0.45 \end{matrix}

Figure 4 shows a comparison between cyclist crossing velocity ( $v_{mean}$ ) characteristics from the experiment and the simulated cyclist velocities generated during the simulations. It is clear that the mean and variance of $v_{mean}$ based on the cyclist model simulations match the data collected in the experiments, indicating the validity of the cyclist model developed in this study. A comparison of the simulated and experimental cyclist trajectories and $v_{mean}$ distributions is detailed further in Appendix B.

Figure 4.

Comparison of velocity characteristics of cyclists crossing the road during simulation and experiment: (a) velocity in X direction; and (b) velocity in Y direction.

Distribution Characteristics of $d_{\min}$ and TTC

The entire prediction model was applied to three cases. Figure 5 shows the results of Monte Carlo simulation and the fitted Gaussian distribution of $d_{\min}$ and $\bar{\underline{TTC}}$ at t = 0 for the vehicle and bicycle in Case 1. Clearly, the Gaussian distribution functions constructed with mean $μ_{MC_d}$ , $μ_{MC_t}$ and variance $σ_{MC_d}^{2}$ , $σ_{MC_t}^{2}$ in the Monte Carlo simulations (solid line in Figure 5) fit the distributions of $d_{\min}$ and $\bar{\underline{TTC}}$ well, which indicates they follow the approximate Gaussian distributions $N (μ_{MC_d}, σ_{MC_d}^{2})$ and $N (μ_{MC_t}, σ_{MC_t}^{2})$ , respectively.

Figure 5.

Results of Monte Carlo simulation and fitting of Gaussian distribution at the moment t = 0 in Case 1: (a) distribution of $d_{\min}$ ; and (b) distribution of TTC.

Simulation and Case Analysis

Figure C1 (in the appendix) shows the relative positions of the cyclist and vehicle in the simulated in-depth reconstructions of the three accidents compared with the screenshots of the accident videos. The relative positions of the car and cyclist are basically the same, which indicates that the reconstruction of the precrash phase for these three cases is reasonable. The boundary conditions were derived from the in-depth reconstruction results shown in Table 2, including the coordinates $(x_{veh}, y_{veh})$ and velocity $(v_{veh})$ of the vehicle, and the coordinates $(x_{b}, y_{b})$ and velocity $(v_{xb}, v_{yb})$ of the cyclist. These initial coordinates were limited to the FOV of the cameras recording the accidents and a defined value of time = 0 for the simulations.

Table 2.

The Initial Boundary Parameters of Three Reconstructed Cyclist Cases ( $t_{0} = 0 ms$ )

	$x_{veh} (t_{0})$ /(m)	$v_{veh_x} (t_{0})$ /(m/s)	$x_{b} (t_{0})$ /(m)	$y_{b} (t_{0})$ /(m)	$v_{x_{b}} (t_{0})$ /(m/s)	$v_{y_{b}} (t_{0})$ /(m/s)	$r$ /(m)
Case 1	−1.7	14.17	29.5	−2.5	3.2	1.32	$+ \infty$
Case 2	−1.7	10	10	−2.4	1.83	1.89	$+ \infty$
Case 3	−1.7	9.8	16.5	0	1.65	0	$+ \infty$

Note: rad = radian.

Figures 6 to 8 show the simulation results and the predicted crash probability distribution for the collision between the cyclist and the vehicle in Case 1. As shown in Figure 6, the original accident occurred at 2.66 s and the collision was avoided when the vehicle was equipped with three different AEB systems. The results of the crash probability model show that at a simulation time of 1.8 s, the crash probability would reach 1.0 for the first time in the next 0.95 s (Figure 7), which indicates a predicted crash time of 2.75 s, with an error of 3.38% from the actual accident time. Therefore, the collision probability model predicted the collision risk well.

Figure 6.

Results of the collision simulation for Case 1. In the case of using autonomous emergency braking (AEB), the collision was avoided.

Figure 7.

Prediction of the collision probability at t = 1.8 s in Case 1.

Figure 8.

Collision probability predictions of different autonomous emergency braking (AEB) preferences for Case 1.

Figure 8 shows the collision probability distribution of the whole simulated collision process for Case 1. In the original accident, the collision probability already exceeded 50% at the start of the simulation, after which the probability increased as the distance between the two parties decreased, reaching a peak of 1.0 at 1.80 s. Because all collisions were avoided when using the three different AEB systems, the collision probabilities were reduced to 0. However, when the AEB systems were triggered, the collision probabilities varied significantly. The conservative AEB system was first triggered at 1.29 s after reaching the $T_{TC}$ trigger threshold of 1.4 s with a collision probability of 0.91; the regulation-based AEB system was triggered at 1.54 s after reaching the $T_{TC}$ trigger threshold of 1.1 s with a collision probability of 0.97; and the aggressive AEB system was triggered last, with a collision probability of 1.0 at 1.84 s.

In Case 2, the original accident without the AEB system occurred at 1.17 s (Figure 9), whereas at simulation time t = 0, the crash probability model predicted that the collision probability would increase to a peak of 1.0 after 1.25 s (Figure 10), which is comparable with the actual accident situation. Because of the closeness of the collision partners when first visible in the video, the probability that a collision would occur between the cyclist and the vehicle had reached 1.0 at time = 0 for the simulation (see Figure 11). This indicated that the accident was unavoidable if no action was taken to avoid the collision. When the accident was resimulated by equipping the vehicle with three different AEB systems, as in Case 1, the accident was avoided in all cases. Despite the collision probability being 1.0 at the start of simulation, the trigger times for the three AEB systems still differed significantly. The AEB system with a conservative braking preference came into play at t = 0; the collision probability also decreased from 1.0, followed by a slower decrease in collision probability at 0.4 s. In the regulation-based AEB system, the $T_{TC}$ reached a threshold value (1.1 s) when the simulation time was 0.03 s, and the collision probability after intervention of the AEB system rapidly decreased to 0.1 within 0.1 s. In the aggressive braking preference, the AEB system intervened in braking when the simulation time was 0.33 s, causing the collision probability to decrease rapidly to below 0.1 in 0.1 s.

Figure 9.

Results of the collision simulation for Case 2. In the case of using autonomous emergency braking (AEB), the collision was avoided.

Figure 10.

Prediction of the collision probability at t = 0 s for Case 2.

Figure 11.

Collision probability predictions of different autonomous emergency braking (AEB) preferences for Case 2.

Figures 12 to 14 show the simulation results for Case 3. According to Figure 13, the collision probability prediction at the simulation time of 1.3 s shows that the collision probability increased to the maximum value after 0.85 s. Therefore, the predicted collision time of 2.13 s is comparable with the actual accident time of 2.02 s (see Figure 12), which demonstrates that the collision probability model is valid. For the whole simulation process (see Figure 14), the collision probability without AEB increased from an initial value of 0.04 to 0.99 at 1.2 s and then maintained this value to the end of the simulation. When the accident vehicle was equipped with the three different AEB systems, the collision probability was significantly reduced until it became 0, indicating that the accident was avoided in all cases. Compared with Case 1 and Case 2, the collision probability when AEB was triggered was lower in Case 3. With the conservative settings, the AEB system started to intervene at a simulated time of 0.52 s, making the collision probability instantly reduce from 0.36 to 0 within 0.1 s. In the regulation-based system, the $T_{TC}$ reached a threshold of 1.1 s, and the AEB system performed braking control at a simulation time of 0.82 s, resulting in a reduction of collision probability from 0.71 to 0 within 0.9 s. For the AEB system with aggressive braking preference, the reduction in the distance between the cyclist and the vehicle caused a $T_{TC}$ of 0.8 s. At this point in time, the intervention of the AEB system at a simulated time of 1.12 s resulted in a rapid decrease in crash risk, which dropped to less than 0.1 after 1.2 s.

Figure 12.

Results of the collision simulation for Case 3. In the case of using autonomous emergency braking (AEB), the collision was avoided.

Figure 13.

Prediction of the collision probability at t = 1.3 s for Case 3.

Figure 14.

Collision probability predictions of different autonomous emergency braking (AEB) preferences for Case 3.

Discussion

In this study, a prediction algorithm for collision probabilities associated with cyclists’ uncertain movements was proposed and validated by using PC-Crash to simulate the results of three real-world accidents involving cyclists. The model could predict the time of collision witnessed in the video when the actual behavior of the cyclist was replaced with the modeled behavior. The results show that the prediction model can be applied to real-world cases to effectively mitigate and avoid accidents.

The AEB system is one of the leading active safety systems available today and has been proved to be effective in avoiding accidents between VRUs and vehicles. It assesses the crash risk and performs avoidance measures by assuming that the VRU and vehicle maintain their current speed and direction of travel using $T_{TC}$ as an indicator ( 2 , 34 , 35 ). However, as shown in Figure B1 in the appendix, cyclists are more mobile and less constrained by the road than vehicles; their real trajectories exhibit a significant randomness that obviously cannot be represented in existing AEB systems if these assume the approaching cyclist is traveling at a constant speed. Therefore, by developing a cyclist model that considers the uncertainty of cyclists’ movements, the kinematics of cyclists can be more accurately modeled and used to predict their future trajectories.

Compared with deterministic risk assessment models, collision probability models that consider the uncertainty of cyclists’ movements can predict crash risk more accurately if applied in real time. As shown in Figures 8, 11, and 14, although AEB systems with the same braking characteristics were used, the probability of an AEB-triggered collision varied considerably from one accident case to another because of the differences between the cases, indicating that a fixed trigger threshold cannot be used to assess the risk of a collision between a cyclist and a vehicle accurately. In addition, even for the conservative AEB systems, the collision probability during AEB braking can still be very high (Case 1 in Figure 8 and Case 2 in Figure 11), which may cause driver discomfort and may also lead to driver misjudgment and more serious crash consequences.

In the current study, the CCA model was used as the vehicle model because of its accuracy and computational efficiency. However, this may not accurately describe the future movement of the vehicle because the influence of the driver was not considered. Predicting the driver’s possible operation of the vehicle based on his/her behavior will influence the future movement of the vehicle, particularly for longer simulation times. This may not have been a factor in this study because of the limited simulation time provided by the case reconstruction times (restricted by the camera’s FOV). Another shortcoming of this study is that the cyclist model did not consider the influence of the environment ( 38 ). In addition, although it is feasible to predict the trajectory of cyclists over a short period of time using the model, its accuracy and adaptability were limited.

Conclusions

In this study, a collision probability prediction algorithm considering the uncertainty of cyclists’ movements was proposed and validated based on simulations in PC-Crash of three real-world accidents involving cyclists. The results show that the prediction model can be effectively used for collision risk assessment of real-world conflicts. In addition, compared with the AEB system in the deterministic prediction model, the collision probability prediction model can assess the collision risk more accurately. This study provides information for improving collision avoidance strategies.

Supplemental Material

sj-docx-1-trr-10.1177_03611981221121270 – Supplemental material for Probabilistic Prediction of Collisions between Cyclists and Vehicles Based on Uncertainty of Cyclists’ Movements

Supplemental material, sj-docx-1-trr-10.1177_03611981221121270 for Probabilistic Prediction of Collisions between Cyclists and Vehicles Based on Uncertainty of Cyclists’ Movements by Di Pan, Yong Han, Qianqian Jin, Jin Kan, Hongwu Huang, Koji Mizuno and Robert Thomson in Transportation Research Record

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: Y. Han, H. W. Huang; data collection: Q. Q. Jin, J. Kan; analysis and interpretation of results: D. Pan, Q. Q. Jin, J. Kan; draft manuscript preparation: D. Pan, Q. Q. Jin, Y. Han, R. Thomson, K. Mizuno. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the support of the National Natural Science Foundation of China (Project No: 51775466), the Fujian Provincial Science Foundation for distinguished young scholars (Project No: 2019J06022), the State Key Laboratory of Automotive Safety and Energy under Project No. KF2005, and the Education and Research Project conducted by the Fujian Provincial Finance Department (File No. (2021)-103). The research was supported by Toyota China.

ORCID iDs

Di Pan

Yong Han

Qianqian Jin

Koji Mizuno

Robert Thomson

Supplemental Material

Supplemental material for this article is available online.

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Supplementary Material

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Probabilistic Prediction of Collisions between Cyclists and Vehicles Based on Uncertainty of Cyclists’ Movements

Abstract

Keywords

Method

Vehicle Model

Cyclist Model

Risk Assessment Metric and Distance Model

Collision Probability Distribution

Calculation of the d min and TTC

Collision Probability Estimation Using Monte Carlo Sampling

Simulation Analysis

Results

Uncertainty Characteristics of Cyclists’ Movements

Distribution Characteristics of d min and TTC

Simulation and Case Analysis

Discussion

Conclusions

Supplemental Material

sj-docx-1-trr-10.1177_03611981221121270 – Supplemental material for Probabilistic Prediction of Collisions between Cyclists and Vehicles Based on Uncertainty of Cyclists’ Movements

Footnotes

Author Contributions

Declaration of Conflicting Interests

Funding

ORCID iDs

Supplemental Material

References

Supplementary Material

Calculation of the $d_{\min}$ and TTC

Distribution Characteristics of $d_{\min}$ and TTC