Among the four factors that affect vehicle trajectory, the legal driving direction of the lane and geometric characteristics of the road are almost constant. This relevant information can be obtained through maps, high-precision maps, networks, or vehicle perception. The remaining three factors are uncertain with time and environment, which is the key and difficult point of other vehicles' trajectory prediction. It is also the key and difficult point of collision risk assessment. In this article, these three factors are summarized as three kinds of uncertainties that affect the trajectory of other vehicles and are analyzed, modeled, and calculated.

At present, the intelligent vehicle cannot accurately sense the driving intention of its driver, so the driving intention of its own vehicle is uncertain. As shown in Figure 1A, when turning around, turning left, going straight, and turning right are allowed in the lane in which other cars travel, the driving intention of the other cars is strongly uncertain when compared with that of the own car. Generally speaking, this uncertainty will decrease as the vehicle travels. For example, when a vehicle is at an intersection, there are possibilities of turning left, going straight, and turning around. When the vehicle travels halfway along the left-turn route, it can be judged that the vehicle is more likely to turn left. When the vehicle turns left and enters the other direction lane, the driving uncertainty disappears completely. Generally, an ego-vehicle’s driving intention will not change in a driving process, but in a few special cases, the driving intention may also change. As shown in Figure 1B, because the ego-vehicle is unfamiliar with the road environment or misreads the road signs, it may suddenly switch to the left turn after driving on the straight line for a period of time.

Because people’s perception, attention, operation ability, and other abilities cannot be as accurate as computers, and the vehicle’s power system and control system cannot control the vehicle absolutely accurately, there is always some uncertainty in the vehicle trajectory during the driving process. This kind of uncertainty is interfered by the external environment to a certain extent, for example, talking with the ego-vehicle will distract the driver’s attention, and fatigue driving will reduce the ego-vehicle’s driving ability. Another characteristic of this kind of uncertainty is that it only appears when the vehicle is moving and disappears naturally when the vehicle is stationary. This kind of uncertainty is called the uncertainty of driving process in this article. There are two types of uncertainties in driving process, each with different characteristics: the first type is mainly caused by the lack of ability of the drivers or vehicles, which is objective and inevitable. However, this kind of uncertainty is often small and occurs randomly near the true value of the driving intention; the second type is mainly caused by the ego-vehicle’s inattention. This kind of uncertainty may accumulate and enlarge and even eventually force drivers to change their driving intentions. For example, the vehicle originally wanted to go straight but because of the ego-vehicle’s mistake, the vehicle continued to turn left and finally entered the left-turn lane. At this time, the ego-vehicle had to temporarily change its driving intention to turn left.

Due to the widespread existence of obstacles, vehicles often cannot drive according to the scheduled route, causing them to change the driving route constantly. However, obstacles may appear in any motion at any time and place, so the uncertainty caused by them is the most complicated and difficult to deal with. This kind of uncertainty is caused by external objects. In this article, it is called the uncertainty of other vehicles’ track invasion. Compared with the uncertainty of driving intention, obstacles appear to more frequently lead to a higher frequency of vehicle route change. Compared with the uncertainty of driving process, the change of driving trajectory caused by obstacle intrusion is bigger and more significant.

In addition to the above factors and uncertainties, there are many factors that affect the driving of vehicles in reality. The most common of which are the right-of-way rules. The right-of-way rules stipulate the priority right-of-way of different vehicles on the same road in a specific scene. The right-of-way division affects the running of vehicles and thus the assessment of collision risk. Based on the actual driving situation, this article focuses on the influence of vehicle arrival time, vehicle type, and vehicle driving intention on the right of way. In this article, it is determined that when a vehicle arrives at a certain position at different times, the vehicle that arrives first enjoys the right of way, and the vehicle that arrives later should give way; when the vehicle arrives at a certain position at the same time, consider the driving intention of the vehicle, such as turning left and right should be straight; and if two cars have exactly the same right of way, they are in a constant competitive state, and the vehicles stop at random for a period of time and then compete for the right of way again.

Besides the right of way, when vehicles avoid each other, it also affects each of their driving. In order to make the calculation easy to realize, when predicting the driving track of other vehicles V _i, only the influence of the remaining vehicles on V _i is considered while the influence of V _i on the other vehicles is not considered. For example, if the driving track of V _i is invaded by V _j, V _i may change the driving route to avoid V _j. At this time, the influence of the V _i route change on V _j is no longer considered. If V _j is affected by V _i and the driving route is adjusted, the influence of V _j on V _i after changing the route should be reconsidered. So repeated recursion will fall into an infinite loop.

The calculation process of the risk assessment framework proposed in this article can be divided into three steps as a whole, as shown in Figure 2. The framework needs the injection of the sensing module and the driving track of the vehicle generated by the decision control algorithm in the future. The calculation results of the risk assessment can be returned to the decision control module for correcting or reestablishing the new driving track. Other safety algorithms can also be triggered when necessary such as directly starting the collision avoidance system when the collision probability is high. The whole calculation process needs to be iterated in real time while driving according to the changed environments.

The first step of calculation is to realize the perception of the surrounding environment. In order to facilitate the calculation, this article uses a two-dimensional coordinate system to realize the gridding description of the whole environment space. As shown in Figure 3, Figure 3A is a realistic intersection simulation diagram, while Figure 3B is a gridded coordinate space. The origin of the coordinate system can be chosen arbitrarily when gridding. It only affects the calculation process and does not affect the calculation result. In Figure 3, the lower left corner is taken as the origin of coordinates.

The second step is the focus of the whole calculation. It needs to deal with the three kinds of uncertainties mentioned above. Let the current time be t and the interval of iterative calculation be ∆t. Then, the evaluation of the risk of collision between the own vehicle and other vehicles in the next n cycles has to be done. Let us assume that there are n other cars at the current intersection and take any other car V as an example to explain the prediction and calculation process of its driving track in the intersection.

The essence of distribution probability lies in quantifying the uncertainty of driving intentions of various vehicles. This method holds that the probability distribution of the expected route is mainly affected by two factors: first, the statistical results of historical driving records, and second, the vehicle’s trajectory in the past period of time. As shown in Figure 4, let the vehicle drive from s to S₀ and enter the intersection at S₀. When the vehicle is located at S₀, because the driving track from S to S₀ is not helpful in distinguishing and identifying the driving intention of the vehicles, the probability of the vehicles traveling on various paths can be quantified more accurately by using historical statistical data. When the vehicle is located in S₁, it can be considered that the possibility of turning around decreases while the probability of turning left along P₂ and going straight along P₃ increases. When the vehicle is located at S₂, it can be considered that the probability of turning around and going straight continues to decrease and the probability of turning left along P₂ increases. When the vehicle is located at S₃, it can almost be considered that the vehicle will definitely turn left. When the vehicle finally arrives at S₄, it turns left. After the turn is completed, the driving intention of the vehicle is determined to turn left, and the uncertainty is completely eliminated.

K possible driving paths are planned when the vehicle V enters the intersection. Assuming that a total of N vehicles have passed from the S₀ position in the past period of time, and the number of vehicles traveling along route I is N_i, the driving probability P i = N i / N can be assigned to route P_i, and when the vehicle travels along the expected path, the uncertainty of driving intention can be estimated and quantified by using the driving trajectory in the past period of time. As shown in Figure 5, the vehicle enters the intersection from S₀, and arrives at S₁ after driving for a period of time, with a driving distance of L. As P₂ and F₃ completely overlap in the L part, the process from S₀ to S₁ can be considered as follows: the vehicle has traveled a distance of L length along P₂, traveled a distance of L along P₃, and a distance of 0 along P₁. Theoretically, the probability of different expected paths of vehicles can be calculated by using the following formula: N = ∑ i = 1 K N i ,    1 = ∑ i = 1 K P i { P around = 0 L + L + 0 P left = L L + L + 0 P straight = L L + L + 0 (1)

However, the above formula is prone to the extreme case of zero probability, which is inconsistent with the reality. Because even if the vehicle is located in S₁, there is still a certain probability of turning around, but this probability is smaller than turning left and going straight. Therefore, the above formula is adjusted based on Laplace’s smoothing idea to avoid the situation that the probability is 0 or 1. The adjusted formula is as follows: { P around = 0 + 1 L + 1 + L + 1 + 0 + 1 P left = L + 1 L + 1 + L + 1 + 0 + 1 P straight = L + 1 L + 1 + L + 1 + 0 + 1 (2)

The formula can be extended to the general situation, if the vehicle has k possible driving intentions at a certain position at the intersection, and the driving distance under the guidance of the i-th driving intention in the past period is L _i, the driving probability of the corresponding path under each intention in the future period is P i = L i + 1 ∑ i = 1 K L + K ,    1 = ∑ i = 1 K P i (3)

The above method can deal with the general case of path probability allocation, but there are two special cases to be explained: 1) when a vehicle enters an unplanned location for some reason, the distance from the previous location to the current location should be specially treated when calculating; 2) some planned routes may disappear and some new planned routes may come into being during the driving process of the vehicles.

As shown in Figure 6, when the vehicle is located at S₁, it should arrive at S₂ at the next moment according to the planned expected route. However, due to some reasons, such as the ego-vehicle’s inattention or obstacles at S₂, the vehicle actually enters S₃ at the next moment. However, the distance from S₁ to S₃ does not belong to any path among P₁, P₂, and P₃, so there will be ambiguity when using the above formula to calculate the probability. In order to ensure the smoothness of calculation, this article stipulates that d should be added to all L_i in this case, and the expected path of the vehicle should be replanned at the next moment.

In the second case shown in Figure 7, when the vehicle is in S₀, three expected paths are planned, namely P₁, P₂ and P₃. When the vehicle travels to S₁, because the direction of the front of the vehicle changes, the target area corresponding to P₁ is no longer visible (not within the 180-degree visual field), so P₁ disappears from the expected path and P₄ becomes the new expected path. At S₀, because the traffic rules stipulate that the inner lane is not allowed to turn right, there is therefore no driving intention to plan a right turn; at S₁, with the gradual exposure of the ego-vehicle’s driving intention, it is more and more possible for the vehicle to turn right, so a right-turn path is added. Since there is no P₄ path in the process from S₀ to S₁, the right turn probability cannot be calculated by the above formula. It is stipulated in this article that the probability of the disappearing path is no longer calculated, the initial probability of a new route is calculated.

When calculating the expected path probability, it is not necessary to calculate the entire driving distance of the vehicle at the intersection. A time window can be set, and only the driving conditions within the time window can be considered, for example, only the driving conditions of the vehicle in the past 3 s can be considered.

Driving intention describes the uncertainty of vehicle trajectory from a macro perspective, while driving process describes the uncertainty of vehicle trajectory from a micro perspective. For the gridded intersection, the driving process of vehicles is equivalent to entering from one grid to another. Since the vehicle can travel to any feasible position in theory, it is also possible to enter a cell from any direction. As shown in Figure 8A, the vehicle may enter s from any direction of d1–d8. Similarly, as shown in Figure 8B, the vehicle may leave a cell from any angle. The essence of driving process uncertainty modeling is to measure the probability of leaving S from S along D₁–D₈. The traveling direction of the vehicle is controlled by the steering angle of the front wheel, and the direction of entering S affects the steering angle when the vehicle leaves. Therefore, the uncertainty modeling of driving process should consider both the direction when the vehicle enters and the direction when the vehicle leaves.

Uncertainty can be considered as random error based on driving intention. As shown in Figure 9, the actual trajectory of straight vehicles is often not an absolute straight line but an up-and-down disturbance. In this article, Gaussian distribution is used to describe the uncertainty of driving process. The sample of distribution is the steering angle of the vehicle, that is, the deflection angle of the front wheel during steering. In this article, negative values are used to indicate left turn, while a positive value indicates a right turn. For example, −3 means 3° to the left, +3 means 3° to the right, and 0° means straight without deflection. For Figure 9, the corresponding steering angle distribution during driving is shown in Figure 10.

There are two main reasons for describing the uncertainty of driving process with Gaussian distribution: 1) Gaussian distribution is a continuous distribution, and the steering angle which determines driving intention is taken as the distribution mean value, which can not only show the characteristics that the steering angle mainly changes around the mean value during driving but also allow the vehicle to deflect to any other direction, and the greater the deviation from the mean value, the smaller the probability. 2) Gaussian distribution is symmetrical about the mean, and its sampling probability on both sides of the mean is equal. This shows the randomness of driving deviation. As shown in Figure 9, the vehicle may shift above or below the centerline.

When the vehicle has only one expected path direction at the next moment, the steering angle distribution during driving from the current position to the next position can be described by Gaussian distribution, but there is often more than one expected path direction. This article uses Gaussian mixture distribution to deal with this situation. As shown in Figure 11A, when the vehicle is at S₁, there are three possible potential paths. Therefore, it is feasible to enter the next position from S₁ in three directions: D₂, D₁, and D₈. Let the probabilities in all directions be P₂, P₁, and P₈, and their specific values can be calculated by the method in the path probability allocation, in the previous section. Firstly, the steering angle distributions in the D₂, D₁, and D₈ directions are modeled, and the corresponding deflection angles are, respectively, μ ₂, μ ₁, and μ ₈. The variance of the corresponding Gaussian distribution is σ ₂, σ ₁, and σ ₈, so the steering angle Gaussian mixture model in the next period from S₁ is shown as follows, with its corresponding distribution graph shown in Figure 12: { P ( X ) = P 2 G ( μ 2 , σ 2 ) + P 1 G ( μ 1 , σ 1 ) + P 8 G ( μ 8 , σ 8 ) 1 = P 2 + P 1 + P 8 (4)

Although the probability is assigned to each expected path (the sum of the probabilities of each path is 1), this result cannot be directly used to measure the probability of vehicles traveling in all directions. Because under this result, the probability of the vehicle traveling beyond the expected direction is 0. As shown in the following figure, theoretically, there is only one expected S₂ direction in Figure 13, but in practice, the vehicle may still drive up to the light blue position. Combined with the uncertainty of driving process, this article uses the steering angular distribution function to calculate the driving probability of vehicles in all directions.

As shown in Figure 14A, when the vehicle is located at S₁, there are three possible paths, namely D₂, D_I, and D₈. However, due to the uncertainty of the driving process, there is a certain probability that the vehicle will drive in the D₃ and D₇ directions. As shown in Figure 14B, taking the 180-degree visual field in the front direction as all possible driving directions, the corresponding angles of D₁, D₂, D₈, D₇, and D₃ are α, β, and γ. If the side length of the cell square is 2, then sin ⁡ α = 1 / 10 ,    α = arcsin ( 1 / 10 ) The (radian) angle is about 18.4° (−90° to −72°, 72°–90°). According to the cosine formula of a triangle, β = arccos ( a 2 + b 2 − c 2 / 2 a b ) , then β ≈ 0.93 ; it is about 53.1° (−72° to −19°, 19°–72°).

According to α, β, there is γ ≈ 0.66, which is about 38° (−19°, 19°). After calculating the values of α, β, and γ, combined with the probability density function of Gaussian mixture distribution, the corresponding probability areas at different angles can be obtained, which can be used as the probability of the vehicle traveling in this direction.

According to the above calculation results, the value of β is larger than α and γ, which is because the direction angles of D₂ and D₈ are larger than those of D₁, D₃, and D₇ in this scheme. The values of α, β, and γ can be adjusted according to the actual situation, or the variance value can be adjusted in the corresponding Gaussian distribution, to correct the probability of vehicles traveling in all directions. Generally speaking, the values of α, β, and γ and the variance of Gaussian components corresponding to each direction can be set as required or can be obtained by simulation and statistics according to the real situation. Let the probabilities of the preset directions D₂, D₁, and D₈ in Figure 14A be 0.2, 0.5, and 0.3, respectively, in which the variance of Gaussian distribution in D₁, D₃, and D₇ is 1.5, and the variance of Gaussian distribution in D₂ and D₈ is 2. The values of the corresponding steering distribution and the corresponding distribution of α, β, and γ are shown in Figure 15. In this example, the probability areas in the D₃ and D₇ directions corresponding to angle A are very small, so they are almost invisible in the figure.

Since the vehicle may enter a certain cell (position) from any direction, the probability of the vehicle arriving at a certain cell at time t is the sum of the probabilities of entering the cell in all directions, which can be used for the evaluation of the later collision risk. When the vehicle arrives at the t+∆t time position from the t time position, its probability calculation includes two steps: 1) based on the direction of the vehicle entering the current cell at time t, the probability of entering the next cell from this direction is calculated. In this calculation, firstly, we were required to establish the steering distribution based on the driving direction, and then use the steering distribution to calculate the probability of going to the next cell and multiplying it with the probability of entering the cell at t time. 2) Iterative calculation, always entering the current cell in all directions and driving into the next cell at t+∆t, and accumulating all the probabilities of reaching a certain cell at t+∆t to obtain the final probability of entering the cell at t+∆t.

As shown in Figure 16, if the vehicle could enter s from d₄, d₅, and d₆ directions at time t and their respective probabilities be P _d4, P _d5, and P _d6, the final probability of the vehicle reaching s at the moment of anti-engraving is P _s = P _d4 + P _d5 + P _d6, which in calculating the position and probability of time t+∆t is according to the calculation steps introduced above:

1) Firstly, based on the situation that the vehicle enters s based on d₅ direction, the expected driving direction of the vehicle at t+∆t time is only D₁ and the expected deflection angle of the vehicle when entering s from d₅ and leaving along D₁ is 0°. Therefore, the steering distribution of the vehicle in this case is a single Gaussian distribution with a mean value of 0. Let the standard deviation of the Gaussian distribution be assumed to be 16 (which can be set according to actual statistical results or simulation results in specific application), then the steering distribution of t time entering s along d₅ and t+∆t time leaving s along D₁ is shown in Figure 17. Under this distribution, the probability distribution of vehicles leaving S along D₁, D₂, D₈, D₃, and D₇ is the probability area of the steering distribution at the corresponding angles (the angles corresponding to each direction have been calculated and explained in the previous section), which is calculated as P _d1 ≈ 0.765, P _D2 = P _D8 ≈ 0.118, and P _D3 = P _D7 ≈ 0. Here, the sum of P _D1, P _D2, and P _D8 is greater than 1 due to rounding, not calculation error. The value of P _D3, P _D7 is 0 because of the limited calculation accuracy. Theoretically, their values are all greater than 0. Based on the above results, it can be obtained that the probability of the vehicle entering S from d₅ at t time and entering S₁ at t+∆t time is P _S1 = P _d5 × P _d1; the calculation of S₂, S₈, S₃, and S₇ is entered and then analogized to P _si = P _di × P _Di.

2) Because it is possible for a vehicle to enter S from d₄, d₅, and d₆, the probabilities of entering S₁ at t+∆t are calculated in these cases, the probability of the vehicle at S₁ at t+∆t is the cumulative sum of the probabilities in all cases, P S 1 t + Δ t = ∑ P ( S , d i ) t P D j d i , where P S 1 t + Δ t means the probability of reaching S₁ at time t+∆t, P ( S , d i ) t indicates the probability of reaching S along di direction at time t, which corresponds to P_d5, P_d4, and P_d6 in Figure 16. In the above example, P D j d i indicates the probability of entering S along the di direction.

Combining the above two calculation steps, based on the uncertainty of path planning and driving process, the probability formula of the vehicle arriving at S at t time is as follows P S t = ∑ i = 1 8 P d i t . The result can be used to evaluate the risk of collision at time t and position S. At t+∆t, the probability of the vehicle arriving at Sj is as formula, P^di _Dj is the path probability according to the planned path, the steering distribution of driving vehicles leaving S at time t is obtained, and then the probability area of the corresponding direction is obtained by using the distribution function.

In theory, vehicles can travel to any position in any direction, which not only makes trajectory prediction difficult but also increases the amount of calculation greatly. However, through analysis, it can be found that most of the calculations are at a very low level of probability. Therefore, this part of the calculations is almost meaningless to the final risk assessment and can be optimized by probability pruning and truncation (to reduce the calculation amount and improve calculation efficiency). As shown in Figure 18, dark green is the expected driving direction of the vehicle, that is, the planned path direction, while the other directions are unexpected driving paths caused by the uncertainty of driving process. If the probability of driving from the current cell to the next cell on the expected path decreases by 0.6 times, the probability of reaching a specific cell from t ₁ to t ₄ becomes 0.6, 0.36, 0.216 and 0.130, respectively. These probability values correspond to the most likely driving path of the vehicle, which is very valuable for risk assessment. However, under the other two paths, the probability value decreases exponentially and quickly drops to a very low level, such as 0.0001 > 0.00001, etc., and the probability value will only be lower after further calculation. In theory, the probability value of reaching any position at any time will be very low but not zero, but the probability value below a certain level is almost meaningless for practical application. If a collision probability of 0.00000000001 is evaluated, it can be almost considered that the collision will not occur. In this article, the threshold R _low is set as the minimum value of probability pruning truncation. When R _low is less than or equal to, the calculation will be cut off to reduce the amount of calculation. As shown in Figure 18, it can be set that when the probability of the vehicle arriving at s = 0.0001, the probability of leaving from S and arriving at the new cell at t5 will no longer be calculated.

Probability truncation can not only reduce the amount of calculation but also avoid many meaningless calculations. As shown in Figure 19, the expected path of the vehicle is marked by dark green, but it is possible for the vehicle to enter the upper right cell at t ₂ under the driving process uncertainty. Then, under the influence of driving process uncertainty, the vehicle may form a circle along the diamond; as shown in the figure, the gray arrow indicates the direction. This situation can continue indefinitely with the expansion of the time window, but under normal circumstances, almost no driver will drive the vehicle around the intersection.

A direct manifestation of probability truncation is that the sum of probabilities of vehicles arriving at all positions at time t is less than 1 (without considering the loss of calculation accuracy and rounding), and the value of probability sum will gradually decrease with time. As shown in Figures 20A–C, the vehicle enters S at time t along the straight direction, theoretically departs from S, and reaches S_i, S₂, S₃, S₈, and S₇ at t ₁. Let the probability be P₁–P₈. If only S_L, S₂, and S₈ are left to reach after probabilistic pruning, then the sum of probabilities of vehicles arriving at each feasible position at t ₁ is the sum of P₁, P₂, and P₈, which is obviously less than the sum of P₁–P₈ and less than 1, which is the probability loss caused by pruning and truncation. Furthermore, the theoretically accessible positions at t ₂ and S₁, S₂, and S₈ are shown in Figure 20E. After pruning, only the cells as shown in Figure 20F are left. The probability values of all accessible positions of the vehicle at t ₂ are also determined by P₁′–P₈′. The sum becomes P₁′, P₂′, and P₉′. In addition, there is a probability loss. The loss of probability can be understood as the cost of precision due to reducing the amount of calculation.

The uncertainty of driving intention and driving process mainly aims at the uncertainty of vehicle interior and driver, while the uncertainty of trajectory invasion focuses on the impact of real-time change of the external environment on vehicle trajectory. When obstacles (possibly, motor vehicles, bicycles, motorcycles, pedestrians, suddenly dropped goods, etc.) intrude into the expected driving position of the vehicle, the vehicle will be forced to respond, such as make detours and stops, all of which will change the vehicle’s trajectory and thus change the time, location, and probability of collision with the vehicle.

In this article, we define trajectory intrusion from two aspects of time and space. In time, it includes the current time or future time in the time window. In space, it refers to the position where the current or future time of the obstacle overlaps with the possible entry position of other vehicles. As shown in Figure 21, if the other vehicle is located at s at the current moment, it may drive along D₁, D₂, D₃, D₄, and D₅ directions. The barrier is located at S₄, and it could be heading for S₃ at the next moment. When other vehicles drive into S₄ from S along the direction of D₄ because the position of obstacles at the current moment overlaps with the possible driving position of other vehicles at the next moment, the track of the other vehicles at S₄ is invaded at the next moment. When other vehicles enter S₃ from S along the direction of D₃, it is possible to drive into S₃ at the next moment due to obstacles. Therefore, the track of its car in S₃ is also invaded at the next moment. In short, if the obstacle is located at A at the moment, and if other vehicles drive to A at the next moment, then A is considered as a trajectory intrusion; if the obstacle and other vehicles are likely to drive to point B at the next moment, then point B is also a trajectory intrusion. In some special cases, such as different rights of way classes between vehicles, it can be considered that the tracks of vehicles with a higher right of way will not be invaded because they enjoy the right of way. At this time, only the tracks of vehicles with a lower right of way are invaded.

When encountering the track invasion, the actions taken by the vehicle is to stop and wait, that is, the vehicle stays still at the current position until the obstacles leave and then resumes driving, or detour, that is, the vehicle does not stop but travels in a nonintrusive direction to bypass the obstacle. Trajectory intrusion changes the route of vehicles, and the route and its probability are determined by path planning and steering distribution. Therefore, the trajectory intrusion will change the original steering distribution of the vehicle and the probability of driving along each direction. When a vehicle enters the next cell with a probability greater than the cutoff probability, it may be necessary to delete or add new paths and reallocate the probability of each path. So, trajectory intrusion uncertainty modeling should be combined with path planning and steering distribution to quantify different situations.

Here, the probability of obstacles reaching the intrusion position is taken as the rejection rate, and the steering distribution is regenerated by the rejection sample. As shown in Figure 22A, at time t, other vehicles are located at S, obstacles are located at S₄, and obstacles may be located at S₃ at the next time. S₄ and S₃ are both the positions where the tracks of other vehicles are invaded at the next moment. Let it have three expected driving directions, say D₂, D₃, and D₄ and the distribution probabilities of the three expected paths are P_D4 = 0.3, P_D3 = 0.5, and P_D2 = 0.5, respectively, and their steering distribution is shown in Figure 22B. It can be seen that the probability areas in the D₃ and D₄ directions are large and the driving process in the D₄ direction is uncertain, so the Gaussian distribution variance in the steering D₄ area is large, and it is “stout” on the graph. D₅ and D₁ are not preset directions, therefore, the corresponding probability area is small, and the eye flesh is almost invisible.

Let the probabilities of obstacles reaching S₄ and S₃ at time t be P₀₁ = 0.7 and P₀₂ = 0.6, respectively, and P₀₁ and P₀₂ are D₄, sampling steering with rejection rate in the D₃ direction, and relearning steering distribution with sampled samples as shown in Figure 23. In the figure, the red dotted line indicates the steering distribution before rejecting sampling, and the blue solid line indicates the new steering distribution learned after rejecting sampling. It can be seen from the figure that the probability areas in the D₄ and D₃ directions decrease to a certain extent, while the probability areas in the D₂ direction increase greatly. The probability areas in D₁ and D₅ directions increase slightly.

Although the new steering distribution can be obtained by taking the probability of obstacles reaching a certain position as the rejection rate, the simulation test shows that the discrimination between many new distributions and the old distributions is not obvious enough. As shown in Figure 24, let the arrival probabilities of obstacles in Figure 24A be P₀₁ = 0.3 and P₀₂ = 0.2, and the distribution probabilities of vehicles along the planned paths D₄, D₃, and D₂ before sampling rejection be P_D4 = 0.3, P_D3 = 0.5, and P_D2 = 0.2. With P₀₁ and P₀₂ as rejection rates, the obtained steering distribution after sampling is shown in Figure 24B, and the curve difference between before-sampling and after-sampling is not obvious. Especially, along the D₃ direction, the probability of the vehicles entering is hardly affected by obstacles, but in fact, P₀₁ = 0.3 and P₀₂ = 0.2 are already relatively high probability values. This shows that it is not enough to show the influence of obstacles on the driving track by directly using the probability of arrival of obstacles as the rejection rate. In this article, it is considered that squeeze operation can be performed according to a certain mapping rule P_oi, to realize the influence of obstacles on the vehicle trajectory.

The squeeze mode can be flexibly selected according to the situation. For example, the analytic equation of a circle can be the squeezed function, and the value curve of P_oi can be extruded into the circular arc shape. As shown in Figure 25A, the red straight line is the original P_oi, according to the equation of circle, (x − 1)² + y² = 1 y = √(1 − (x − 1)²). The rejection rate after Squeeze is shown as the blue curve. Figure 25B shows the steering distribution generated by the rejection rate sampling after squeeze, which shows that the distribution difference is more obvious. Except for the equation of the circle, other functions that map P_oi to the range of 0–1 can be used as squeeze functions, such as deformed log function, logistic function, sigmoid function, etc., and the rejection rate is amplified in different ways by different functions. To influence the new steering distribution.

It is also a common in driving to choose to stop when facing obstacles, so it is necessary to allocate probabilities between stopping and detouring. Let ∆t be the time of iterative calculation of risk assessment, P_wait, P_bypass, then there is 1 = P_wait + P_bypass, indicating that it is possible for other vehicles to stop moving in the next time. In reality, the probability of stopping is related to the distribution of obstacles. If the possibility of obstacles invading the track is high or more positions in the track are invaded by obstacles, the possibility of stopping and so on will increase accordingly. As shown in Supplementary Figure S1, there are five possible driving directions within the visual field of other vehicles 180, and the probability of each direction can be obtained from the steering distribution before it refuses to sample, which is set as P_D1–P_D5. In extreme cases, these five directions may be invaded by obstacles. Let the invasion probability be P₀₁–P₀₅. If the values of P₀₁–P₀₅ are all 1, it is determined that the vehicle will be surrounded by obstacles at the next moment, then the probability of the vehicle stopping at this time is 1 to ensure driving safety. If the values of P₀₁–P₀₅ are all 0, it is determined that no obstacle will invade the trajectory of the other vehicles at the next moment. Then, the probability of the vehicle stopping should be 0, which accords with the driving habits of people. Because P_D1–P_D5 are calculated from the same steering distribution, there are 1 = ∑ i = 1 5 P D i . However, P₀₁–P₀₅ may be invaded by five different obstacles, so there are 5 = max ∑ i = 1 5 P o i because there must be 0 ≤ P_wait ≤ 1, where P_wait can be calculated from the weighted sum of P₀₁–P₀₅, and the calculation formula is P w a i t = ∑ i = 1 5 P D i P o i , corresponding to P_bypass = 1 − P_wait.

The overall flow chart of calculation framework is shown as Supplementary Figure S2.