EDRF: Enhanced Driving Risk Field Based on Multimodal Trajectory Prediction and Its Applications

The model is trained with Argoverse 2 dataset. The prediction precision of models used in this paper and the state-of-the-art (SOTA) are shown in Table I, where Final Displacement Error (FDE) and Average Displacement Error (ADE) are chosen as the metrics.

It is worth noting that we have not made efforts to improve the accuracy of trajectory prediction for several reasons. Firstly, the gap between results of model used in this paper and the SOTA is already minimal. Secondly, trajectory prediction inherently possesses a high degree of uncertainty, and our focus is on modeling this uncertainty rather than enhancing the precision of deterministic outcomes. Thirdly and most importantly, trajectory prediction serves merely as an independent input module within our method. Should more precise results be required in the future, a simple model replacement could be readily implemented. Examples of the trajectory prediction results are illustrated in Fig. 2, where the orange line represents the vehicle’s historical trajectory, the red line denotes the ground truth, and the green lines depict the multimodal trajectory prediction results.

In [11], the authors proposed the driver’s risk field and modelled it as a torus with a Gaussian cross-section along a straight line or an arc, which is determined by the current vehicle’s steering angle and velocity. We expand the model by extending its future trajectory from a kinematic-based straight line or arc to the predicted trajectory. With the current vehicle position as the origin, a Frenet coordinate system $sod$ is established along the predicted trajectory (Fig. 3). The height and width of the Gaussian cross-section are functions of the predicted trajectory length, denoted as $s$ (Fig. 4).

Here, $(s,d)$ represents the corresponding coordinates of point $(x,y)$ when transferred from the Cartesian coordinate system to the Frenet coordinate system. The parameters $a$ and $\sigma$ are the height and width of the Gaussian distribution respectively, and they are defined as follows:

$s_{pt}$ denotes the total length of the predicted trajectory. The height $a(s)$ is modelled as a parabola, based on the assumption that the risk decreases relatively rapidly around the vehicle [22], and approaches zero at the end of the trajectory. Parameter $q$ represents the steepness of the parabola. The width parameter $\sigma$ is modeled as a linear function of the path length $s$, with the slope determined by the parameters $b$ and $k$, as well as the average curvature $\overline{\kappa_{pt}}$ of points along the trajectory. The parameter $c$ represents the initial Gaussian width. This modeling approach implies that the width of the Gaussian distribution increases along the trajectory, reflecting increasing uncertainty in the predictions, which aligns with intuitive understanding. Additionally, incorporating the $k\cdot\overline{\kappa_{pt}}$ term into the slope of the linear model suggests that higher average curvatures lead to more pronounced uncertainty dispersion, consistent with the greater uncertainty observed in turning trajectories in reality.

Fig. 3 and Fig. 4 illustrate the establishment of the Frenet coordinate system and the evolution of the torus cross-section when the predicted trajectory follows a sinusoidal curve. This example clearly demonstrates the advantages of prediction-based trajectory methods over kinematic-based methods (Fig. 5). Furthermore, to provide a visual understanding of the modeling results, we take the scenario corresponding to Fig. 2(b) as an example, and the $DRP$ distributions for each predicted trajectory (with the probability) are depicted in Fig 6. The parameters to model $EDRF$ are shown in Table II (according to [11, 26]).

In multimodal trajectory prediction, the complete $DRP$ is the weighted sum of the distributions corresponding to each individual trajectory, that is:

where $p_{i}$ represents the probability associated with the $i$-th trajectory, while $s_{i}$ and $d_{i}$ denote the coordinates of point $(x,y)$ in the Frenet coordinate system corresponding to the $i$-th trajectory.

$DRP$ can be seen as a model representing the probability of an event occurring. Modeling the driving risk additionally requires accounting for the consequences of that event. Generally, the consequences of an event are directly related to parameters such as the vehicle’s mass, type, and velocity. According to [22], the consequence can be modelled by virtual mass of the vehicle, which is define as:

Therefore, the complete $EDRF$ model is the product of the $DRP$ and the virtual mass $M$:

In summary, the $EDRF$ is parameterized by $q,b,k,c,\alpha,\beta,\gamma$ (with their respective numerical values detailed in Table II), and is determined by the results of multimodal trajectory prediction as well as ego-vehicle state parameters such as mass, type and velocity. To evaluate the effectiveness of the model, we first defined two scenarios: potential lane changing and potential turning. The $EDRF$ for these scenarios are shown in Fig 7. Subsequently, scenes from the dataset are used. We calculate the $EDRF$ in scenarios (a) and (b) of Fig 2 and depict them in Fig 8. The results demonstrate that the $EDRF$ is capable of representing the uncertainty in driving intentions, as well as the propagation of this uncertainty along the predicted trajectory.

In the traffic environment, risk is composed of interactions among various entities. The vast majority of traffic accidents can be decomposed into conflicts between two traffic entities. Therefore, we primarily focus on the interaction risk between pairs of entities. For any two traffic participants $i$ and $j$, we first establish the $EDRF$ for each, then the interaction risk between them, denoted as $IR_{ij}$, is defined as follows:

This implies that interaction risk exists only where the $EDRF$ of both entities is non-zero. Additionally, for location $(x,y)$ where interaction risk is present, its calculation can be decomposed into $IR_{ij}(x,y)=DPR_{i}^{c}(x,y)\cdot DPR_{j}^{c}(x,y)\cdot M_{i}\cdot M_{j}$. Here, $DPR_{i}^{c}(x,y)\cdot DPR_{j}^{c}(x,y)$ quantifies the probability of both entities being simultaneously present at $(x,y)$, and thus colliding, while $M_{i}\cdot M_{j}$ quantifies the severity of the collision. Finally, we determine the level of risk $F_{ij}$ between entities $i$ and $j$ by taking the maximum of the interaction risk at all positions:

The scenario of two vehicles traveling head-on is utilized as an example. Fig 9 illustrates the respective $EDRF$ of the two as well as the distribution of their interaction risk ($IR$). In practical applications, a threshold $F_{thld}$ can be set for the risk $F$. if the risk intensity between two entities in the traffic environment exceeds $F_{thld}$, it may lead to a traffic accident and should call attention to the driver of HDV or intervene in the motion of AV.

A significant difference between ego-vehicle risk analysis and traffic risk monitoring is that the uncertainty associated with the ego-vehicle is substantially lower than that of predicted trajectories of other vehicles. Moreover, ego-vehicle trajectory prediction does not require multimodal approaches. Therefore, a kinematic-based can be employed for ego-vehicle trajectory prediction: assuming that the vehicle will continue to move according to its current state (position, orientation, steering angle and velocity) over a look-ahead time period $t_{la}$ (6 seconds in this paper).

According to the simplified bicycle model, the turning radius of the vehicle can be calculated as:

where $L$ represents the wheelbase and $\delta$ is the steering angle. Then, combined with other state parameters of ego-vehicle, the future trajectory can be determined (Fig 10(a)).

Furthermore, due to the lower uncertainty of ego-vehicle, using a Gaussian distribution to quantify is not appropriate. Hence, we replace the Gaussian distribution with a Laplace-like one (Fig 10(b)) to represent the cross-section of the torus. The height and width of the Laplace-like distribution are also functions of the predicted trajectory s.

The height $a_{ego}$ also decreases along $s$, but is modeled as a linear function for the lower uncertainty. The trend in width $\lambda$ variation follows the same pattern as equation (3), that is:

The parameters are set as Table III. The calculation of the interaction risk also adheres to equation (7) and (8). Here we take the potential cut-in of adjacent vehicle as an example scenario. Fig 11 shows the $EDRF$ for both the ego-vehicle and the cut-in vehicle , as well as the $IR$ distribution between them. A threshold $F_{thld}$ can similarly be set as a criterion for analyzing the risk situation of ego-vehicle.

For motion and trajectory planning of AVs, the $EDRF$ and $IR$ can be used in various manners, depending on the specific framework of the planning module. The most straightforward method to integrate the $EDRF$ and $IR$ is to use them as safety assessment metrics for trajectories, thereby guiding the planning module to select safer trajectories.

For instance, the $EDRF$ and $IR$ can be combined with a trajectory sampling framework, calculating the $EDRF_{ego}$ for each sampled trajectory. Here, as trajectories have already been sampled, there is no need to compute future trajectories based on the current state. Additionally, due to the low uncertainty associated with the ego vehicle, a Laplace-like distribution is still employed for the cross-section of the torus. Subsequently, the $IR$ and $F$ for each trajectory, based on the $EDRF$ of surrounding vehicles, can be determined. These metrics serve as safety indicators for feasible trajectories, which, along with other criteria such as comfort, efficiency, and energy conservation, allow for the selection of the optimal trajectory according to customized standards.

We design a scenario where a leading vehicle decelerates as an example, in which different planning methods may yield trajectories for either following with deceleration or overtaking by lane changing. We sample nine feasible trajectories in front of the vehicle, representing options for left lane change / following / right lane change, and acceleration / maintaining velocity / deceleration. Figure 12 shows the $IR$ distribution for each of these trajectories. From the safety perspective, deceleration is the best choice; however, when combined with other criteria and their respective weights, different outcomes, such as overtaking by lane changing, may also be recommended.