Introduction. The latest achievements in the field of artificial intelligence (AI) are being actively implemented in various areas of activity. One of such achievements is autonomous vehicles (AV). The current research was aimed at creating algorithms that allow AV to move safely on public roads. This will significantly reduce the number of road accidents [1].

The scientific community has already identified the basic modules of an autonomous vehicle. One of them is a system for predicting the future behavior of road users (agents) [2]. A clear understanding of how the environment will develop and in which direction dynamic objects (pedestrians, cars, cyclists) will move is urgently needed for AV to search and use a safe and effective trajectory of movement.

Numerous scientific papers are devoted to the problem of predicting such trajectories [3–12]. However, there is currently no active research on the application of existing methods outside of urban conditions. And this is extremely important, since autonomous cars will be used on country roads, too [13]. Urban conditions are highly structured: cars mostly follow traffic lanes, and pedestrians move through special zones. In this sense, the area of country roads is the complete opposite, which means that it will have additional difficulties during development. In the given paper, attention is focused on these difficulties: the existing predicting methods and their applicability to new, less structured conditions are considered.

The objective of the study involved:

Materials and Methods. At first glance, it would seem that the domain presented is a simpler version of urban conditions due to the fact that country roads are characterized by less traffic. However, the absence of complex multi-level junctions, special traffic-free zones, a large number of signs, markings, etc., makes the domain of country roads less structured, i.e., fewer rules and specific traffic patterns increases the randomness and reduces the predictability of the behavior of cars and pedestrians.

The following features of the country road domain influence strongly on the selection of the architecture of the prediction module:

The prediction module implies the presence of AV recognition, tracking and localization systems and their accurate operation. The authors of the article use the Argoverse dataset, which stores the required records of the operation of all systems in a convenient form [14].

The dataset consists of recordings of road scenes observed on the streets of Miami and Pittsburgh, USA. Each of the entries contains a local part of the terrain map (lane boundaries, roads, pedestrian crossings) and a list of all recognized agents, including the current position and movement history of each of them. Each of the records is divided into two parts: two seconds of the observation history and three subsequent seconds for which prediction is made (prediction horizon). Data on the future movement of objects is also available and used to calculate the accuracy of prediction methods and model training.

Information about agents is presented in a discrete format. The time interval between measurements is fixed, in this work it is equal to 0.1 seconds (10 Hz).

For each moment of time t, the module receives the observation history for each detected agent i. The observation history consists of the agent's current and past states, where each of the states is a 2D position in the global coordinate system. The authors make the assumption that the height information is redundant.

The dataset also provides access to an HD map that contains information about road borders and roadway, pedestrian crossings. To simulate the domain of country roads, the dataset was modified in such a way as to exclude all information from road maps, except for the boundaries of the roadway D. This reduces the amount of information about the road context and complicates the task of prediction.

Hence, the context of the scene is represented as

(1)

where k — the total number of tracked agents on the scene.

This approach implies predicting the trajectory for only one agent per execution, therefore further is treated as for simplification. To generalize the model for all recognized agents, it is required to repeat the proposed approach for all k agents on the scene. The agent for which the prediction is currently being made is considered a target agent.

To assess the accuracy of prediction methods, the dataset contains recorded future trajectories for each target agent:

(2)

where H indicates the number of next time steps. In this case, parameter H will be equal to 30, since the planning horizon is three seconds with a sampling frequency of 10 Hz.

The domain of the prediction module is multimodal, i.e., the future behavior of agents may differ significantly in absolutely identical traffic situations. Let us say, a car approaching a crossroad may continue straight ahead or make a turn. To take this into account, it is required to generate M possible future trajectories and M probabilities of each of them at the model output.

Therefore, the purpose of the prediction module is to create function , that takes the context of the scene c as input and generates M pairs of possible future trajectories and their probabilities:

. (3)

Here, at least one generated trajectory should be as close as possible to the real trajectory , and the probability of its execution p should be close to unity.

Model architecture. The proposed approach involves the use of a neural network consisting of submodules of scene encoding, decoding and filtering trajectories. The architecture of the system is shown in Figure 1.

A neural network adapted to the new conditions, based on a vector representation, is responsible for encoding the scene. This selection is due to the fact that on country roads, the HD map will contain a limited amount of information (only the roadway boundaries and the history of observations of dynamic objects). Popular methods represent the context of a road scene с in an image format and process it using convolutional neural networks. However, vector coding avoids the overhead associated with image generation [4–5].

The presented encoder is based on the VectorNet model, but its input data format has been modified to receive information only about the boundaries of the roadway and the state of the agents. [3]. This encoder represents the boundaries of the road and the state of agents using polylines, which are further processed by a graph neural network. This provides encoding the interaction between polylines. Details of the implementation are described in paper [3].

A trajectory decoder is a task of regressing several possible trajectories and generating a set of probabilities. To solve this problem, a multilayer perceptron model is used. The decoder implementation is inspired by the MTP model [4], however, the authors of the article propose a different formula for calculating the best trajectory m* from the set of M trajectories. It is also proposed to use an additional mechanism that penalizes the model for predictions that go beyond the area of movement.

The authors of the original MTP model propose to train a multilayer perceptron using the loss function that represents the sum and , where:

(5)

In this case,  — mean-square error between the real trajectory and the best trajectory m* of M generated.

(6)

where si — the agent's actual future position at time i, and  — the predicted future state of the best trajectory m*.

 — loss function based on cross entropy, which increases the probability of executing the best of the predicted trajectories m* to 1 and reduces the probability of other trajectories to 0.

Ic is a binary indicator equal to 1, if condition c is true, and 0 — otherwise.

In the original article, the best of the predicted trajectories m* is defined as the one that has the minimum value of the root-mean-square error in comparison to the real trajectory:

(7)

The authors of the article suggest using the following modification:

(8)

where ∆ — subset of generated trajectories that has a similar final direction to the real trajectory .

The idea is to remove from consideration trajectories in which the final direction of the agent differs significantly from the direction in the real trajectory when calculating the best trajectory m*. If the difference in directions features less than certain threshold γ, then the generated trajectory is considered correct, i.e., m⸦∆. In the case under consideration γ=30o. Therefore, the best trajectory m* should have a similar final direction and the lowest value of the loss function.

This work also involves prior knowledge of the domain to achieve greater convergence of the model [15]. Since only information about the roadway boundaries is available from the HD map when driving in the domain of country roads, an additional variable is introduced — into the loss function. Thanks to it, the model will penalize the predicted trajectories that go beyond the road in cases where at least one state is . The model penalizes only the best trajectory, since only in this case, it is possible to determine the direction of error reduction by approximating the best of the generated trajectories m* to the real trajectory .

Thus, is defined as:

(9)

where is equal to 1, if , and 0 — otherwise.

The final loss function is defined as

(10)

where α and β — neural network hyperparameters used for training. In this case, both of these parameters are equal to 0.5.

To filter similar and duplicate trajectories, the proposed approach uses the filtering of a finite set of trajectories M at the final stage. This module is required because in some cases, the number of possible agent trajectories may be less than M, e.g., when a car is moving along a straight road at a constant speed, the model can generate only one trajectory: the car continues to move straight. However, the need to generate exactly M trajectories will result in the situation when all predictions are similar to each other.

The proposed filtering is based on the final direction and positions of states : if the direction and the sum of the deviations between states of the real and generated trajectories are less than the threshold value σ, then the trajectories are considered similar. The authors average each state of the trajectories and sum up the probabilities of the trajectories .

This approach was implemented in the Python programming language on the PyTorch deep learning framework. The model was trained on GeForce RTX 2080 Ti graphics card for 40 epochs, the training took four hours.

Research Results. To assess the accuracy of prediction models, this section applies widely used metrics for the trajectory prediction problem: average displacement error, ADE, final displacement error, FDE [6], MissRate (MR), and Offroad rate (OR).

For multimodal cases with the generation of several trajectories, ADE and FDE are taken as the minimum ADE and FDE among M trajectories (the trajectory with the lowest metric value) [5].

The prediction is considered “missed” if ADE metric of the generated trajectory is more than two meters. OR metric is calculated as the percentage of trajectories in which at least one state goes beyond the range of motion D.

To visualize the context of scene с, as well as the real and predicted future trajectories a script in the Python programming language was implemented using the Matplotlib library.

This section compared the operation of several different methods in the case of an unstructured domain. The following methods were used in comparison:

Table 1 presents comparison of the accuracy of the methods when working under unstructured conditions. Several methods are compared, including the proposed approach.

Kalman filter. The simplest way to predict behavior is to obtain the current state of the object (current lane, speed, direction, etc.) and extend this state to future steps based on some assumptions, e.g., that the car will continue to follow its lane or will have a constant speed and/or acceleration. Another popular method for such tasks is to use the Kalman filter [12].

According to Table 1, the Kalman filter works worse than all the presented methods based on neural networks.

Figure 2 shows two cases. In the first case, the Kalman filter successfully performs prediction because the vehicle is moving straight, without any turns or speed variation. In the second case, the Kalman filter mispredicts due to lack of knowledge about the context of the traffic situation.

Single trajectory output. This method involves the use of a graph scene encoder, which is identical to the one used in the proposed approach. The output of the network implies the generation of only one trajectory. This model is trained using the root-mean-square loss function.

As shown in Table 1, the neural network, even with the generation of a single trajectory, demonstrates better results in comparison to the Kalman filter.

Figure 3 shows the visualization of this prediction method operation. The image on the left shows that the model can successfully predict the agent's turn. The image on the right shows that generating one trajectory is not enough. The neural network tries to imagine both possible outcomes: going straight and turning right. As a result, the model outputs the average of the two outcomes.

Fixed set classification. The implementation was inspired by the CoverNet prediction method [5]. This model consists of a proposed vector scene encoder, followed by a different trajectory decoder. The decoder is a classification task based on a predefined set of trajectories consisting of physically realizable vehicle trajectories with sufficient coverage. Two sets were created for experiments: of 415 and 64 possible trajectories. The second set has the same coverage as the first, but provides a lower density of trajectories. Detailed information about the sets of trajectories is contained in paper [5].

The visualization of the work is shown in Figure 4. The classification model successfully copes with multimodality at crossroads, but in some cases, the lack of sufficient coverage by a set of trajectories negatively affects the results.

As shown in Table 1, this method works more accurately than generating a single trajectory, but worse than the proposed approach. In addition, increasing the density of the set of trajectories by using a set of 415 trajectories did not improve the results. The authors attribute this to the presence of noise in the dataset, which comes from the tracking system used in the data collection.

Proposed approach. The proposed approach eliminates the disadvantages of all the methods described above. This is a multimodal forecasting method that does not suffer from the limitations of a predefined set of trajectories.

Moreover, according to Table 1, the proposed approach surpasses all other methods in all indicators. As shown in Figure 5, the method successfully captures two possible outcomes at the crossroad: driving straight or making a turn.

Figure 6 shows an example of filtering similar trajectories in the case of a single possible outcome. The probability that the agent will complete the initiated turn is close to 1, since he is already in the process of turning. Therefore, in this case, the probability of other outcomes is close to 0. The proposed module successfully filters similar trajectories.

Limitations. Although the authors of the original article on the MTP model [4] indicate that their method solves the problem of mode collapse, the experiments conducted by the authors of this article do not confirm this. The problem still occurs in some cases. It is assumed to be due to the following features: the loss function does not penalize the neural network for generating all possible trajectories that the target agent can execute, as long as the best of them is as close as possible to the real trajectory. But also, the model does not encourage the network in any way to predict a variety of possible trajectories. Therefore, it is advantageous for the network to make several similar predictions in one direction, in which it is more confident than to make one prediction for each possible trajectory.

One of the possible solutions to this problem may be the use of a trajectory decoder presented in the TnT, DenseTnT models [10–11], which imply the generation of final goals at the first stages of work. In these models, all possible final goals for the agent are generated first, and then trajectories that describe the movement from the starting position to each of the goals, are generated. This provides filtering out similar final goals in the early stages, and preventing the mode collapse.

Discussion and Conclusion. In the work performed, modern methods of solving the trajectory prediction problem are investigated. The adaptability of the methods to unstructured road conditions — country roads, is considered. Insufficient accuracy of the methods is established, and a new approach to predicting is proposed.

The proposed approach is based on the VectorNet and MTP models, but has been adapted for the country road domain. In addition, a trajectory filtering module and an additional mechanism for the loss function, which penalizes trajectories for going beyond the movement zone, are proposed.

The presented comparison shows that the proposed approach is superior to other popular methods.

Limitations of the MTP approach have been identified: the output data still tends to mode collapse. The suggestion for further modifications is to use methods that generate the final goal at the early stages of prediction and thus are less susceptible to regime collapse.