DBN-MACTraj: Dynamic Bayesian Networks for Predicting Combinations of Long-Term Trajectories with Likelihood for Multiple Agents

Abstract

Accurately predicting the long-term trajectories of agents in complex traffic environments is crucial for the safety and effectiveness of autonomous driving systems. This paper introduces DBN-MACTraj, a probabilistic model that takes historical trajectories and surrounding lane information as inputs to generate a distribution of predicted trajectory combinations for all agents. DBN-MACTraj consists of two main components: a single-agent probabilistic model and a multi-agent risk-averse sampling algorithm. The single-agent model utilizes a dynamic Bayesian network, which incorporates the driver’s maneuvering decisions along with information about surrounding lanes. The multi-agent sampling algorithm simultaneously generates predictions for all agents, using a risk potential field model to filter out samples that may lead to traffic accidents. Ultimately, this results in a probability distribution of the combinations of long-term trajectories. Experimental evaluations on the nuScenes dataset demonstrate that DBN-MACTraj delivers competitive performance in trajectory prediction compared to other state-of-the-art approaches.

1. Introduction

Predicting a vehicle’s future trajectory with precision is essential for autonomous driving systems. Such accurate predictions facilitate better decision-making, help prevent accidents, contribute to a smoother traffic flow, and enhance safety, stability, and efficiency [1]. When predicting trajectories, the vehicle’s current physical state, which includes factors such as speed, acceleration, and heading angle, is crucial for determining its immediate future trajectory [2]. However, as time goes by, the number of potential trajectories increases exponentially, leading to a reduced influence of the current physical conditions. The trajectory of the distant future is mainly determined by two factors: (i) deterministic traffic environment information, such as static information like lanes and dynamic information like traffic signals; and (ii) non-deterministic traffic participants, such as other vehicles, pedestrians, etc. Consequently, trajectory prediction can be classified into two types [3]: short-term prediction (less than 1 s) and long-term prediction (more than 2 s).

Short-term predictions are primarily grounded in physics [4]. Related studies often utilize theoretical mechanical models, including those based on constant turn rates and accelerations [5], with some advanced approaches incorporating vehicle dynamics [6]. Thanks to centuries of advancements in physics and the capabilities of modern computing, the mathematical frameworks and algorithms designed to tackle short-term prediction challenges are well-developed. The current challenge lies in addressing engineering issues related to precise data collection and ensuring stable automated control.

The long-term prediction problem falls into a multi-agent system problem (where the agent refers to non-deterministic traffic participants) constrained by the deterministic traffic environment. Compared with short-term prediction, the difficulty of the long-term prediction problem comes from two main aspects: (i) Subjective maneuvers by the driver can affect the trajectory, and this effect amplifies in long-term predictions. The driver can only directly change the agent’s acceleration and yaw rate. According to the laws of physics, instantaneous changes in second-order derivative information, such as acceleration and yaw rate, cannot immediately modify the observable motion state of the intelligent agent. The speed and orientation of the agent can only gradually change over time, and cannot undergo sudden changes. Therefore, subjective maneuvers by the driver will not affect the short-term prediction of the trajectory. However, as time elapses, the influence of these subjective maneuvers will be fully reflected in the trajectory. (ii) The long-term future trajectories of all agents will influence each agent’s current subjective maneuvering decisions. In real-world scenarios, experienced drivers ensure safe driving by predicting the movement trends of other agents. In summary, the challenge of long-term prediction in multi-agent systems lies in the need to consider the overall situation when predicting future trajectories.

Numerous studies that utilize machine learning have employed temporal probabilistic models to tackle long-term prediction challenges. Examples of these models include Gaussian processes [7,8], hidden Markov models [9], and dynamic Bayesian networks [10]. These methods are usually based on maneuvers that classify trajectories into different prototypes from real data, and then complete the prediction by selecting prototype trajectories based on recognizing the driver’s intention. The advances of these methods are interpretability and efficiency, as well as the ability to provide confidence in trajectory predictions. However, since maneuvers are incorporated into the model as prior information, the precision of the predictions is constrained by how comprehensive this prior knowledge is. Additionally, these methods often struggle with accurately modeling interactions among different agents.

In recent years, various deep learning-based techniques have been developed. These methods not only incorporate physical principles and deterministic traffic environments, but also account for interactions among traffic participants, enabling the modeling of more complex scenarios. For instance, the approach proposed in [11] employs convolutional neural networks to extract map-related information, in conjunction with recurrent neural networks for trajectory prediction. VectorNet [12] leverages graph neural networks to collect data on interactions between agents and the traffic environment for predictive purposes. Additionally, some techniques utilize the Transformer architecture to analyze relationships among historical trajectories [13]. While deep learning has shown the ability to achieve satisfactory prediction outcomes, it requires substantial amounts of accurate data for the extensive training process of the model. Meanwhile, these deep learning models lack the ability to quantify the confidence of their predicted trajectories or estimate the probability distribution of multiple predicted trajectories. This limitation means they cannot offer more options for downstream decision-making systems. Most critically, autonomous driving systems are extremely demanding in terms of safety. Uninterpreted models can make mistakes in scenarios not covered by the training dataset, potentially resulting in serious consequences like traffic accidents.

In summary, long-term prediction remains a key fundamental challenge in the field of autonomous driving. The essence of long-term prediction is concatenating multiple short-term predictions. The difficulty of concatenation lies in the consideration of the traffic environment and the interaction between multiple agents. Therefore, this paper proposes a new approach, DBN-MACTraj, which is a novel probabilistic model for the long-term trajectory prediction of multiple agents under lane constraints. The input for DBN-MACTraj includes lane information and the positions of all agents over a short duration. The output consists of predicted multiple trajectory combinations and their probability distributions for all agents over the long-term future. DBN-MACTraj consists of two parts: (i) Probabilistic model for single agent. This model employs dynamic Bayesian networks to represent short-term physical principles and driver maneuvering decisions, incorporating probabilistic corrections based on lane information from the traffic environment. And (ii) Risk-averse sampling algorithm for multiple agents. According to the single-agent probabilistic model fitted on historical trajectories, the positions of all agents at the next moment are sampled simultaneously. A risk potential field model is used to eliminate high-collision-risk position combinations, and the probability distribution of the predicted trajectory is obtained through iterations. DBN-MACTraj was evaluated on the nuScenes [14] dataset and achieved competitive performance in trajectory prediction compared with other state-of-the-art methods.

2. Methods

2.1. Overview

The formal definition of the long-term prediction for multi-agent systems under lane constraints is described as follows. Define the current time as $t=0$ and the interval as $\Delta T$. For agent i, given its historical state vector sequence $o_i^{-m\Delta T},\cdots ,o_i^0$ in the past $m\Delta T$ time, the task is to predict its state vector sequence in the future $n\Delta T$ time (shown as Formula (1)).

$$o_i^{\Delta T},o_i^{2\Delta T},\cdots ,o_i^{n\Delta T}$$

(1)

The state vector $o_i^t=(x_i^t,y_i^t,v_i^t,{\beta }_i^t)$ describes the agent’s current motion state, where $x,y$ are position coordinates, v is the velocity, and $\beta$ is the heading angle. From the current environment, lane l can be represented as a directed curve $f_l$. Although there is no closed-form expression for $f_l$, it is possible to obtain any point $(x,y)\in f_l$ on the curve together with its tangent direction (indicating the lane direction).

Figure 1 describes the workflow of DBN-MACTraj. Firstly, as shown in Figure 1a, the inputs are the positions of multiple agents in the past $m\Delta T$ time and the centerline of lanes. The output is several combinations of predicted trajectories for multiple agents in the future $n\Delta T$ time, along with the probability of each combination. DBN-MACTraj first uses a dynamic Bayesian network to model each agent according to the driving motion state and obtains the distribution of short-term trajectory predictions (as shown in Figure 1b). This short-term trajectory distribution is then fitted to the lane centerlines as prior to generate the posterior trajectory distribution. After that, each agent follows its predicted short-term trajectory distribution, allowing for the simultaneous sampling of their positions after a time interval, $\Delta T$ (see Figure 1c). In some cases, the sampled results at $\Delta T$ may indicate potential collisions between agents. To address this, a risk potential field model is utilized to eliminate samples that could lead to collisions. The trajectory distribution obtained after filtering involves ensuring safe driving while also considering the distribution of the original predicted trajectories. After n iterations of multi-agent risk-averse sampling, several feasible combinations of multi-agent long-term trajectories are generated, along with the likelihood of each combination.

2.2. Single-Agent Probabilistic Model

The single-agent probabilistic model predicts the distribution of possible trajectories based only on the agent’s historical trajectory and lane information in the environment. The probabilistic model comprises two components: a dynamic Bayesian network for modeling the agent’s motion, and a correction model according to lanes. The dynamic Bayesian network is the fundamental framework for concatenating tandem short-term prediction to long-term prediction. Meanwhile, the lane correction model accounts for a crucial influence of traffic environment information on driver maneuvering decisions. In this section, the subscripts for the agents are not necessary, as the discussion focuses solely on a specific agent.

2.2.1. Dynamic Bayesian Network

When driving, the driver can directly control two key variables: acceleration and yaw rate. These variables, which are second-order derivatives, significantly influence the agent’s future trajectory. In the dynamic Bayesian network, the acceleration and yaw rate at a given moment t are represented as random variables $a^t$ and ${\psi }^t$, respectively. The agent’s current motion state, denoted as $o^t=(x^t,y^t,v^t,{\beta }^t)$, can be directly determined using the previous motion state $o^{t-\Delta T}$, along with the random variables $a^t$ and ${\psi }^t$. This calculation is based on a straightforward physical model, as illustrated in Formula (2).

$$\begin{array}{c}\hfill \left[\begin{array}{c}{x}^t=x^{t-\Delta T}+(v^{t-\Delta T}\Delta T+\frac{1}{2}{a}^t\Delta T^2)\cos ({\beta }^t)\hfill \\ y^t=y^{t-\Delta T}+(v^{t-\Delta T}\Delta T+\frac{1}{2}{a}^t\Delta T^2)\sin ({\beta }^t)\hfill \\ v^t=v^{t-\Delta T}+a^t\Delta T\hfill \\ {\beta }^t={\beta }^{t-\Delta T}+{\psi }^t\hfill \end{array}\right]\end{array}$$

(2)

In Formula (2), the velocity $v^{t-\Delta T}$ and the heading angle ${\beta }^{t-\Delta T}$ of the agent are calculated from $o^{t-\Delta T}$ and $o^{t-2\Delta T}$.

The driver’s maneuvers over the acceleration and yaw rate are defined as discrete random variables, namely $Q^t$ and $R^t$. The values of these hidden variables represent different driving maneuvering intentions, such as using braking to maneuver the acceleration $a^t$ as a negative value or using steering to maneuver the yaw rate ${\psi }^t$. To establish the relationship between $a^t,{\psi }^t$, and the hidden variables, normal distributions are employed. Accordingly, their probability densities are represented by the following equations, respectively

$$P\left(a^t\right|Q^t=q)=norm(a^t;{\mu }_q,{\sigma }_q)$$

(3)

$$P\left({\psi }^t\right|R^t=r)=norm({\psi }^t;{\mu }_r,{\sigma }_r)$$

(4)

where $norm$ is the probability density function of the normal distribution.

The current maneuvering mode is closely linked to the maneuvering mode at the previous moment $t-\Delta T$. For example, transitioning from a left steering maneuver to a right steering maneuver does not occur instantly. Therefore, intermediary states between different moments are calculated using a state transition matrix, as follows:

$$P\left(Q^t\right)=\sum _{Q^{t-\Delta T}}P\left(Q^t\right|Q^{t-\Delta T})P\left(Q^{t-\Delta T}\right)$$

(5)

$$P\left(R^t\right)=\sum _{R^{t-\Delta T}}P\left(R^t\right|R^{t-\Delta T})P\left(R^{t-\Delta T}\right)$$

(6)

$$P(Q^t=i|Q^{t-\Delta T}=j)=M_Q[i,j]$$

(7)

$$P(R^t=i|R^{t-\Delta T}=j)=M_R[i,j]$$

(8)

$M_Q$ and $M_R$ denote the state transition matrices, $M[i,j]$ is the probability when $Q^t=i$ and $Q^{t-\Delta T}=j$. Both $P\left(Q_{init}\right)$ and $P\left(R_{init}\right)$ follow a uniform distribution.

The above definitions are depicted in Figure 2 as a probabilistic graphical model. Each moment comprises five random variables, and only the motion state variable $o^t$ is observable. The relations from the acceleration $a^t$ and the yaw rate ${\psi }^t$ to the motion state variable $o^t$ are both deterministic. The parameters are estimated by maximizing the likelihood, which utilizes information gathered from the historical trajectory.

2.2.2. Lane Correction Model

Dynamic Bayesian network establishes a probabilistic framework to concatenate short-term predictions to long-term predictions by the physical model. One significant contribution of this probabilistic framework is that it represents the driver’s maneuvering decisions as discrete hidden states. This allows for the seamless incorporation of traffic environment information into the agent’s trajectory while adhering to physical laws.

The traffic environment is both diverse and complex. This study focuses specifically on the most influential lane information. As a static element of traffic, lanes play a crucial role in determining an agent’s trajectory and direction. Agents are naturally drawn to lanes, and typically move along them. Incorporating lane information into the probabilistic model involves two main steps. First, it is necessary to define the distance between the agent and the various lanes. Second, the probability that the agent will navigate toward a specific lane in the future is calculated based on this distance.

When evaluating the distance between an agent and a lane, it is definitely inappropriate to use the shortest straight-line distance from the agent’s position to the lane. This is because the movement of the agent is restricted by its kinematics, and the trajectory to reach a lane must be a smooth curve. Thus, finding the optimal trajectory for an agent to travel from the starting motion state to the target motion state leads to a variational optimization problem involving both endpoint and point-by-point constraints from physical laws, which is highly challenging and even intractable.

To make it practical, DBN-MACTraj simplified the problem based on the following three considerations: (i) The trajectory curve has a closed form and it is easy to calculate the curvilinear integration to obtain its length. (ii) The trajectory should be smooth enough to resemble a result constrained by physical laws. And (iii) at least four strict constraints should be met: the trajectory must pass through the starting position and the target position of the agent, and the heading angles at both positions must be the same. As a result, the third-order Bézier curve meets the above considerations. Given the starting motion state $o^t$ and an arbitrary position q on the lane, the third-order Bézier curve is denoted as $\mathcal{B}(o^t,q)$, and its control points are determined by Formula (9).

$$\left[\begin{array}{c}(x^t,y^t)\\ (x^t+0.5d_u\cos \left({\beta }^t\right),y^t+0.5d_u\sin \left({\beta }^t\right))\\ (x_q-0.5d_u\cos \left({\beta }_q\right),y_q-0.5d_u\sin \left({\beta }_q\right))\\ (x_q,y_q)\end{array}\right]$$

(9)

The coordinates of position q are $(x_q,y_q)$. ${\beta }_q$ indicates its tangent direction, and $d_u$ represents the Euclidean distance between the position of agent i and q. The constant 0.5 is empirically selected to make the resulting curve similar to the trajectory constrained by physical laws.

Therefore, the shortest distance from the agent’s current position $o^t$ to the lane $f_l$ can be appropriately defined as Formula (10).

$$d(o^t,f_l)=\underset{q\in f_l}{\min }C\left(\mathcal{B}(o^t,q)\right)$$

(10)

The notation $C(·)$ denotes the curve length, which can be determined through curvilinear integration. Next, the distance between the agent’s position and nearby lanes is transformed into a probability distribution by Formula (11).

$$P(lane=l|o^t)\propto \frac{1}{1+\exp \left(d(o^t,f_l)\right)}$$

(11)

The modified future trajectory distribution can be obtained by a Bayesian formula (as shown in Formula (12)),

$$P\left(o^t\right|lane=l)\propto P(lane=l|o^t)P\left(o^t\right)$$

(12)

So far, the probabilistic model for a single agent is now complete. This model employs physical principles to perform short-term predictions in a probabilistic manner. It connects these short-term predictions to long-term outcomes using a dynamic Bayesian network and incorporates lane information from the traffic environment to make probabilistic adjustments.

2.3. Multi-Agent Risk-Averse Sampling

In a single-agent probabilistic model, predicting future trajectories involves straightforward sampling from a probability distribution. However, the challenge arises because the future movements of each agent are interdependent. It is crucial to avoid the risk of collisions between agents as they move. Since it is impractical to directly describe the conditional probabilistic relationships among multiple agents in a model, these interactions must be taken into account during the sampling process. To address this issue, a multi-agent risk-averse sampling algorithm is proposed.

The sampling process uses historical data from the moment $t-m\Delta T$ up to the present, along with the optimal hidden variable state trajectory reconstructed by the Viterbi algorithm, to determine the position at the moment $\Delta T$. During each sampling step, a combination of possible positions for all agents is sampled simultaneously. Each agent’s position probability is derived from the corresponding single-agent model. At this stage, the sampled combinations may pose a risk of collision, which must be assessed. Combinations that have a high risk of collision should be rejected to ensure safety.

A risk potential field model is utilized to evaluate the risk of collisions among multiple agents. This model is widely applied in microscopic traffic modeling, considering the repulsive forces between agents to avoid collisions [15]. The repulsion is typically represented as a repulsive potential field generated by each agent. In this study, the kinematics of the agents indicate that the repulsive potential field created by an agent is small in the lateral direction and larger in the longitudinal direction. Additionally, the magnitude of the force in the longitudinal direction is influenced by the agent’s current speed. The repulsive potential field $F_i(x,y)$ at the position $(x,y)$ of agent i is defined as follows:

$$F_i(x,y)=\kappa \exp \left(-{\left(\frac{w_i(x,y)}{{\sigma }_W}\right)}^2-{\left(\frac{l_i(x,y)}{{\sigma }_L}\right)}^2\right)$$

(13)

where $\kappa$ is a constant. $w_i(x,y)$ represents the lateral projection distance between the point $(x,y)$ and agent i, while $l_i(x,y)$ indicates the longitudinal projection distance between the same point and agent i. The variables ${\sigma }_W$ and ${\sigma }_L$ are the distance correction factors for the lateral and longitudinal directions, respectively, and are defined as follows:

$${\sigma }_W=\frac{W_i}{2}$$

(14)

$${\sigma }_L=\frac{L_i}{2}+\tau v_i^2\left(2-\frac{{\theta }_i(x,y)}{\pi }\right)$$

(15)

where $\tau$ is a constant, $W_i,L_i$ are the width and length of the agent i, respectively. ${\theta }_i(x,y)$ is the angle between the direction from the agent’s position to $(x,y)$ and the direction of motion of the agent i.

If the repulsive interaction between two agents exceeds a certain threshold during the sampling of positions, the sample will be discarded. By repeating the sampling process, multiple combinations of positions are generated at the time interval $\Delta T$, which helps avoid collision risks and adheres to the single-agent probabilistic model. After clustering the nearby positions, the algorithm proceeds with n iterations, starting from various position combinations. Ultimately, it generates the trajectories of all agents over the next $n\Delta T$ time, as shown in Figure 3.

3. Results

The evaluation of DBN-MACTraj is conducted using the nuScenes dataset [14], a public resource for autonomous driving research. The nuScenes is a large-scale public dataset for autonomous driving developed by Motional, an autonomous driving company. The dataset was retrieved from the nuScenes official website (https://www.nuscenes.org/nuscenes, (accessed on 1 June 2023)). The nuScenes dataset features 1000 real driving scenarios collected in Boston and Singapore, capturing a wide range of traffic information and providing high-resolution maps. These scenarios encompass various traffic environments, such as intersections and construction areas, as well as diverse agent behaviors, including changing lanes, turning, and stopping. This comprehensive dataset covers most driving situations encountered on the road.

DBN-MACTraj uses the PyMC library for modeling and parameter learning. Since the nuScene only provides the ground truth of the 6-s trajectory starting from a 2 s history, the experiments were carried out with the parameters set as $\Delta T=0.5$ s, $m=4$, and $n=12$. The probabilities $P\left(Q^0\right|o^{[-m\Delta T:0]})$ and $P\left(R^0\right|o^{[-m\Delta T:0]})$ are calculated using the historical trajectory and the state transition matrix. The predicted trajectory is then obtained through parallel sampling using DBN-MACTraj. The parameters $\kappa$ and $\tau$ for the risk-averse model are set to 1 and 0.1, respectively.

Metrics for evaluating prediction performance include final displacement error ($FDE$) and average displacement error ($ADE$). $FDE$ means $L_2$ distance between the predicted final position and the final position in the ground truth. $ADE$ means $L_2$ distance between the ground truth and predicted trajectories. When predicting k trajectories for an agent, ${FDE}_k$ and ${ADE}_k$ indicate the minimum $FDE$ and $ADE$ value among k predicted trajectories. Another metric to consider is the missing rate ($M{R}_k$), which indicates the percentage of predicted trajectories whose last position is more than 2 m away from the actual final position.

To evaluate the performance of DBN-MACTraj, five representative methods are considered for comparison as the benchmark.

• Physics Oracle [16] is a method based on physics principles. For comparison, the chosen result is the best prediction from four different approaches: (i) constant velocity and yaw, (ii) constant velocity and yaw rate, (iii) constant acceleration and yaw, and (iv) constant acceleration and yaw rate.
• CoverNet [16] utilizes a fixed set of trajectories to reformulate the trajectory prediction problem as a classification problem.
• MTP [17] utilizes rasterized images as input, which enhances the accuracy of predictions.
• Trajectron++ [18] is a graph recurrent neural network model that incorporates the dynamics and semantics of the agents involved.
• ALAN [19] uses the lane as a reference point for predicting agent trajectories.

Table 1. The performance of different methods on nuScenes dataset.

Method	${\mathit{ADE}}_1$	${\mathit{ADE}}_5$	${\mathit{ADE}}_{10}$	${\mathit{FDE}}_5$	${\mathit{FDE}}_{10}$	${\mathit{MR}}_5$	${\mathit{MR}}_{10}$
Physics Oracle	-	3.70	3.70	-	-	0.88	0.88
CoverNet, $ϵ=2$	5.41	2.62	1.92	-	-	0.76	0.64
MTP	4.42	2.22	1.74	4.83	3.54	0.74	0.67
Trajectron++	-	1.88	1.51	-	-	0.70	0.57
ALAN	4.62	1.87	1.22	3.54	1.87	0.60	0.49
DBN-MACTraj	4.34	1.81	1.37	3.61	2.39	0.61	0.37

shows the performance of the methods for $k=5$ and $k=10$.

It has been demonstrated that DBN-MACTraj can achieve performance that is comparable to, or even better than, other methods. Both CoverNet and MTP use rasterized images of the map as inputs to their convolutional neural network models. However, the results indicate that using images directly does not effectively extract the necessary information. Instead, when lane nodes are utilized as input (as performed in Trajectron++ and ALAN), performance improves significantly.

The most notable finding is that the metric $M{R}_{10}$ indicates the method achieves only 37% of predicted trajectories within 2 m of the actual position at the final moment. This performance is significantly lower than that of other methods, the best of which reaches 49%. Additionally, an analysis of the 37% of failed scenarios shows that many of these failures occur due to the absence of lanes or because the agent is not traveling within a designated lane. In other words, these failed cases fall outside the initial assumptions.

4. Discussion

To visually illustrate the effects of DBN-MACTraj, the results will be presented in various common and representative traffic environments. In the upcoming visualizations:

• The orange trajectory indicates the five most probable pathways predicted by DBN-MACTraj.
• The orange star marks the endpoint of this predicted trajectory.
• The green trajectory represents the true future path.
• The green star denotes the endpoint of the actual trajectory.
• The blue blocks symbolize surrounding obstacles, typically other agents.
• The red block signifies the location of the main agent.

Although DBN-MACTraj predicts the trajectories of all dynamic agents in the scene simultaneously, the visualizations will focus solely on the predicted trajectories of the main target for clarity in discussion and comparison.

4.1. Understanding the Risk Potential Field

The principles and formulas of the risk potential field model are explained in the Methods section. To enhance understanding, an example analysis is provided that focuses on an agent navigating a turning scenario. Figure 4a,b shows the trajectory predictions with and without the risk potential field model. In Figure 4a, where the model is applied, the main agent is forced to stop. In contrast, Figure 4b illustrates that without the risk potential field, the main agent continues past an obstacle at high speed, which could lead to a serious collision in a real-world situation. Additionally, Figure 4c visualizes the risk potential field, demonstrating that the field of the main agent is already in contact with the obstacle in front. As a result, the prediction is adjusted to slow down and avoid a rear-end collision.

4.2. Case Studies

4.2.1. Light Traffic on a Straight Lane

Figure 5 shows the predictions made by DBN-MACTraj in a light traffic scenario on a straight lane. In this situation, the agent maintains a consistent speed throughout the lane. The predictions from DBN-MACTraj closely match the agent’s actual motion trajectory, demonstrating excellent predictive performance. The main reason for this accurate result is the use of the physically constrained Bayesian model.

4.2.2. Heavy Traffic on a Straight Lane

Figure 6 shows the predictions made by DBN-MACTraj in heavy traffic scenarios with straight lanes. In these situations, the agent moves along designated lane lines, but there are often other agents both in front of and behind it. Since DBN-MACTraj predicts the behavior of all agents in the scene simultaneously, it can effectively account for the agent in front. This approach prevents our agent from viewing that agent as an obstacle impeding its progress. Furthermore, it takes into consideration the slower speed of the leading agent, as rapid acceleration could result in collisions. Consequently, the predicted trajectory closely matches the actual trajectory, demonstrating that DBN-MACTraj achieves strong predictive performance in congested and complex following scenarios.

4.2.3. Intersection

Figure 7 illustrates the predictions made by DBN-MACTraj in an intersection scenario. In this context, lane attributes play a crucial role, as agents may either turn or continue straight ahead. DBN-MACTraj takes these lane influences into account when predicting agent movement. It can generate trajectories for different lanes during the sampling process, resulting in various types of predicted trajectories, including those for turning and going straight. As demonstrated in the figure, DBN-MACTraj is capable of accurately predicting reasonable trajectories, even in situations where agents make turns at intersections.

Here are the predictions of DBN-MACTraj in three typical traffic environments. In these settings, it is clear that the predicted trajectory generated by DBN-MACTraj closely matches the actual trajectory. Additionally, it offers a range of realistic predictions based on various acceleration, deceleration, and steering intentions of the agents. This demonstrates that DBN-MACTraj performs effectively in common traffic scenarios.

5. Conclusions

DBN-MACTraj is a long-term trajectory prediction model that utilizes dynamic Bayesian networks and multi-agent risk-averse sampling. It takes historical trajectories and surrounding lanes as inputs and generates a distribution of predicted trajectory combinations for all agents. Experiments conducted on the nuScenes dataset demonstrate that DBN-MACTraj is effective and reliable in a variety of traffic scenarios. Compared to other methods, DBN-MACTraj offers several advantages: (i) it is an interpretable probabilistic framework, ensuring both the accuracy and feasibility of predictions; (ii) it can produce predictions of arbitrary length due to its step-wise sampling scheme; and (iii) it can simultaneously predict the trajectory combinations of all agents within the scenario.

Future work involves the enhancement of the single-agent probabilistic model and the optimization of the multi-agent risk-averse sampling algorithm. In the single-agent model, current research has not taken into account special agents like drunk drivers and running animals, etc. These agents fundamentally constitute a new single-agent probabilistic model, which demands suitable new physical models and a substantial amount of data to accurately capture the agents’ behavioral intentions. For instance, in the case of animals, the physical model permits instantaneous directional alterations, rendering the yaw rate unnecessary. For drunk driving, it becomes essential to either update the state transition within the model or introduce disturbance noise to effectively model the behavior of drivers. The most significant obstacle to this subsequent research is the lack of corresponding training data.

Regarding the multi-agent risk-averse sampling algorithm, future work focuses on its improvement, aiming to expand its functionality to handle various traffic information that frequently emerges in the real-world traffic environment, such as roadblocks, accidents, and other unforeseen circumstances. Additionally, given the existence of numerous similar or repetitive sample pools in the sampling prediction process, dynamic programming methods should be employed to optimize the accept-reject sampling process, thereby improving efficiency.