The general problem we are interested in deals with computing a trajectory tracing procedure to handle critical situations under which a vehicle circulating at high speeds has to avoid the collision against a suddenly appearing obstacle (another vehicle, a pedestrian or an animal) in the middle of the road.

However, we think it is instructive to examine in detail alternative formulations, thus in this paper we will focus on the simplest version of the problem in order to get an understanding on how trajectories for a single vehicle evolve according to the specific optimization requirements we set, and how the behavioral restrictions to which the car’s mobility is constrained affect the maneuver. As a straightforward version of the problem, we assume there is only one vehicle circulating on a W-meter-wide road (regardless of the number of lanes). This vehicle has to trace a trajectory in an interval of t_f seconds after the maneuver has begun (see Figure 1). During this interval, the vehicle will try to adjust its mobility evolution according to a set of constraints and optimizing a given functional. In this particular case, our goal would be to achieve a given degree of safety while keeping the maneuver as comfortable as possible. To combine these two features, we look for the maximization of the lateral distance with respect to the lateral boundaries of the road, while minimizing the final lateral speed at the end of the path, all within a given time interval of t_f seconds. Maximizing the lateral distance implies in this case leaving the maximum possible road width on both sides of the vehicle, thus minimizing the probability of colliding with the lateral protections. On the other hand, by simultaneously minimizing the lateral speed at t_f , we can change the general appearance of the trajectory in such a way that the inertial impact on the mobility of the passengers is reduced when reaching the optimum lateral position.

For the analysis of the problem, we first need to assume a model for the vehicle’s motion, which is represented by the system of differential equations that describes the lateral movement of the car (only x axis, see Figure 1): (1) where x₁(t) refers to the position of vehicle 1 in axis x at time t; v₁(t) denotes the current speed of vehicle 1 in axis x as a function of t; and a₁(t) defines the lateral acceleration of vehicle 1 along time. We take a₁(t) as the control variable that we have to manipulate in real time in order to modify the trajectory traced by the vehicle under consideration. As was specified before, we assume that these mobility equations only govern the lateral displacement of the vehicle (we consider the longitudinal speed to be constant and fixed to 120 km/h during the whole trajectory).

Furthermore, the car’s mobility has to respect the physical limits imposed by the inertial laws of kinematics. This means that the vehicle can only turn at a maximum established acceleration, which depends on the longitudinal speed (the higher the speed, the harder the maneuver). Considering this, we can establish the mechanical restrictions of the problem at hand:

Lateral acceleration restrictions. The absolute value of the lateral acceleration cannot take a value higher than the limit c(vi) m/s², where vi is the constant longitudinal speed of the vehicle and c(·) is a function of the longitudinal speed.(2)

The third and most important aspect of the statement of the problem is the functional we want to optimize. Although it is possible to formulate different functionals according to the specific target we want to optimize, we will focus on just the previously mentioned main objectives: minimization of the variance of the lateral distance (as we will see, by minimizing the lateral variance of the distances we simultaneously maximize lateral distance), and minimization of the final lateral speed. As we said, this can be justified due to the fact that the lower the variance of the distances of the lateral gaps (between two vehicles, two obstacles or vehicle to obstacle), the higher the lateral distance to other elements on the road will be. On the other hand, minimizing the final lateral speed will turn into a null lateral inertia when approaching the end of the path as we will see later.

Even though we restrict ourselves to these parameters, alternative functionals can be constructed. The four proposed functionals are described next:

Final lateral distance maximization and final lateral speed minimization. In this case we want to minimize the final variance of the lateral distances left by the vehicle after t_f , while minimizing the lateral speed at the end of the trajectory. The equation corresponding to this functional takes the form:(4)

For the optimization of the aforementioned functional, we will rely on the Gradient Projection Algorithm, a class of Gradient Descent procedure that includes functionals that can only be optimized inside a region determined by the constraints of the problem [11]. This method requires to discretize the trajectory into N steps. Apart from the evaluation of trajectories, in the next subsections we will explore how the discretization factor N influences on the determination of trajectories and the resolution we need according to the specific functional in order to reach a reasonable trade-off between curve error and computational load.

In this first subsection, we compare the performance of functionals J_D1 and J_D2 for the configuration of parameters in Table 1, where the instantaneous lateral distance of the vehicle along the trajectory is not maximized, but the final (at t_f ) lateral distance to the lateral protections is maximized. If we obtain the mobility evolution (acceleration, speed and position evolution along the trajectory) for two different values of t_f (10 and 2 seconds), we can notice from Figure 2 and Figure 3 that for longer time intervals to execute the maneuver, the vehicle does not need to reach the maximum acceleration stated by the model’s restrictions. On the other hand, for lower values of t_f , the steering maneuver needs to use the maximum allowable values of the lateral acceleration to maximize the functional at the end of the trajectory. The most illustrative differences between both functionals can be read from Figure 2 and Figure 3, where we see that J_D1 (blue) reaches a higher lateral peak speed than J_D2 just at the middle of the time period. The explanation for this fact is that by flipping between two opposite values of the acceleration (see Figure 2 and Figure 3) until the final position is reached, the vehicle can get there with null lateral speed. However, for J_D2, the speed will increase more smoothly from the first instant, but at the cost of not having null lateral speed at the end of the trajectory, which could lead to a further potential risk because of the inherent inertial dynamics.

Figure 4 and Figure 5 show the trajectories traced by the two functionals for different values of the final time (t_f ). Analyzing the results in these figures, we can conclude that functional J_D1 can reach the optimum lateral position at a later time but with a null lateral speed, whereas through functional J_D2 it is possible to reach the optimum lateral position earlier, but at the expense of having a non-zero speed. The first one will not have problems to follow the trajectory, but for the second one, safety relies on how will the inertial mobility be from the instant t_f onwards.

In this part, we focus on analyzing the properties of the trajectories when we use the instantaneous lateral distance maximization as the optimization target (J_D3 and J_D4). As we can infer from Figure 6 and Figure 7, in this case the main objective consists of updating the lateral position as soon as possible in order to meet the requirements set by the functionals.

This implies using alternatively the two highest (absolute) values of acceleration during the trajectory, which naturally makes the vehicle build a more sensitive path that can be obviously more affected by additional disturbances (i.e., bad weather conditions). This is explained by assuming that the vehicle will try to adapt its lateral position instantaneously in order to maximize the lateral distance. Thus, it is possible that if there is a sudden change in the width of the road, or another car gets too close, the trajectory is modified very harshly, so it could undoubtedly end in a lateral collision. With J_D1 and J_D2 there could effectively be lateral collisions, but they are less prone to disturbances, because lateral distance is maximized regarding only the final lateral positions.

Evaluating the results for the same scenario (see Table 1) for J_D3 and J_D4, we can see that for longer time intervals (Figure 8) until reaching the final position, a vehicle focus on achieving the final lateral position very quickly (thus the rapid changes in acceleration, which oscillate between the two extreme values although there is enough time to make a smoother maneuver). Besides, it is surprising that for both functionals the final lateral speed reached at the end of the path is null, and both functionals provide very similar behaviors. On the other hand, if we analyze Figure 9, we see that for very short time intervals, the evolution of mobility is very similar to what we saw for the functionals J_D1 and J_D2.

Considering that using Gradient Projection [11] to solve the optimization problems at hand implies the discretization of the trajectory in N steps (of constant acceleration), here we focus on the influence of the discretization factor N. We will compare here the trajectories traced by the four proposed functionals for a discretization factor N ∈ {1, ..., 20} and evaluate the error (as the sum of the square of the difference between one curve and the reference one, set to N = 100, see Equation 8). We consider the curve N = 100 as accurate enough for the calculation of the error evolution. The parameter configuration for this case is the same as for previous subsections (Table 1). (8) where ε(N) refers to the square error for an N-discretized curve x_N(t), in comparison with the N_ref = 100 reference trajectory x_Nref(t).

Having a look at the Figure 10, it is easy to realize the correctness of the trajectories calculated by the optimization algorithm (Gradient Projection), i.e., the highest errors are located for low values of the discretization factor, whereas the error value decreases as N gets higher.

Observing this graph, it is also straightforward to infer that setting a value of N = 20 should be enough to achieve a proper resolution in the curve for the four functionals. We can see that the functional J_D1 needs a higher value for the discretization factor than that of J_D2 to achieve the same level of accuracy. Due to the need of higher accelerations for the trajectories obtained by J_D1 in comparison with those calculated by J_D2, a higher value for the discretization factor N is thus required to keep an acceptable accuracy. On the other hand, both functionals regarding instantaneous lateral distance maximization (J_D3 and J_D4) show that they need a higher value of the discretization factor to reach the same level of accuracy. This is explained by the same reasons as regards the comparison between J_D1 and J_D2, that is, the greater acceleration values of the trajectories, the higher N must be to maintain accuracy.

Until now we have been dealing with the optimization of trajectories for a single vehicle according to two possible performance criteria. Needless to say, once the optimum trajectory is determined (here for the sake of simplicity, we disregard the time to compute the trajectories), the vehicle obviously start running the calculated path. In real cases, the vehicle will have to face undesirable phenomena during the maneuver’s execution, which could perturb the previously calculated optimum trajectory and make it divert from the calculated path. This is usually called the Random Noise Disturbances [12] in the literature. In this subsection we will extend the initial System of Equations describing the mobility of our one-dimensional vehicle to accomplish for this variability in the execution of the maneuvers. We will model such phenomena by assuming that an additive Gaussian noise process produces this variability by influencing:

The shape of the traced path due to possible deviations from the optimum course, see Subsection 3.5.1.

Kalman Filter theory has been proven to be a very reasonable option for state estimation and path reconstruction under the aforementioned circumstances. Moreover, it is the core technology that implements basic car localization and motion on autonomous vehicles, like the famous Google Car [13]. Before discussing in detail the characteristics of the Kalman Filter, we will first express our extended model for the System (1) by accounting for the two influencing factors presented earlier:

Analyzing the procedure of automatic trajectory tracing for one vehicle is the first step to completely evaluate a traffic scenario where vehicles interchange information to update their trajectories in common while simultaneously adapting to the unpredictable phenomena that could alter the normal transit. Even though we have not considered a real cooperative system, the previous discussion provides us with a valuable insight into the performance of alternative functionals for optimum trajectories. In the context of CCA, the proposed method involves the exchange of computed trajectories at discretization steps, or at least updated information that would require a high frequency of beacons’ interchange between vehicles in order to keep an updated history of the evolution of cars along the path. The IEEE 802.11p [3] is based on a CSMA (Carrier Sense Multiple Access), a medium access technique whose performance degrades as the number of transmitting nodes increases [14]. Thus, it is important to determine which are the minimum requirements that this system should have in order to guarantee appropriate performance (necessary in situations in which having this kind of information is critical). It might also be interesting to study the 3GPP/LTE (next generation, 4G) cellular infrastructure, which is already under deployment, in order to complement such technologies to properly relieve WAVE-based communications from the channel overload that V2I and V2V might entail [15]. This is something we will treat in a future work.