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Nowadays, research on Vehicular Technology aims at automating every single mechanical element of vehicles, in order to increase passengers’ safety, reduce human driving intervention and provide entertainment services on board. Automatic trajectory tracing for vehicles under especially risky circumstances is a field of research that is currently gaining enormous attention. In this paper, we show some results on how to develop useful policies to execute maneuvers by a vehicle at high speeds with the mathematical optimization of some already established mobility conditions of the car. We also study how the presence of Gaussian noise on measurement sensors while maneuvering can disturb motion and affect the final trajectories. Different performance criteria for the optimization of such maneuvers are presented, and an analysis is shown on how path deviations can be minimized by using trajectory smoothing techniques like the Kalman Filter. We finalize the paper with a discussion on how communications can be used to implement these schemes.
According to statistics of the DGT (
In this regard, the automotive industry is currently moving towards the automation of every single aspect of the driving experience [
Communications will also play an important role when disseminating information monitored by sensors installed on board to enhance the safety conditions of passengers while driving. The standard WAVE 1609/802.11p [
Within this topic, a very concerning problem is how to perform the best emergency maneuvers. Interest in this issue has increased in the last decades [
In the present study, we focus on the development of useful procedures to dynamically optimize the trajectory of a vehicle under timing constraints,
A discussion on the different ways to compute optimized realtime maneuvers for a highspeed moving vehicle subject to timing constraints (the maneuver must be performed in a maximum time interval of
The evaluation of functionals including the minimization of the final lateral speed. By keeping the final lateral speed (at
A preliminary discussion on the accuracy of the computed trajectories by an evaluation of the discretization factor
An analysis on how trajectories could be affected by random Gaussian noise, and the application of Kalman Filter theory to minimize the impact of unwanted deviations from the optimum path.
The rest of this paper is organized as follows. In
Several previous studies on active maneuvering not directly related to emergency maneuvers can be found in the open literature. In the case of [
In this section we start giving a description of the specific problem we tackle in this evaluation, formulating the problem, discussing alternative functionals and describing the tools used to solve it. We will also analyze the performance of the trajectory tracing procedure and how we can address the inconvenience of path deviations because of additive noise processes affecting sensor measurements while maneuvering. We conclude this section by providing a qualitative analysis of the connection between our proposal and how it might be integrated within some sort of vehicular networking protocols.
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
Maximization of lateral distance after
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
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:
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:
For the optimization of the aforementioned functional, we will rely on the
In this first subsection, we compare the performance of functionals
Configuration parameters and values for all evaluations.
Evaluation parameter  Meaning  Value 


Discretization factor  20 

Initial lateral position  1 m 

Initial lateral speed  0 m/s 

Initial lateral acceleration  0 m/s_{2 } 

Road width  20 m 

Longitudinal speed  120 km/h 

Maximum absolute lateral acceleration  3 m/s_{2 } 
Position, speed and acceleration
Position, speed and acceleration
Trajectory evolution of functional
Trajectory evolution of functional
In this part, we focus on analyzing the properties of the trajectories when we use the instantaneous lateral distance maximization as the optimization target (
Position, speed and acceleration
Position, speed and acceleration
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 (
Evaluating the results for the same scenario (see
Trajectory evolution of functional
Trajectory evolution of functional
Considering that using
Having a look at the
Influence of discretization factor N.
Observing this graph, it is also straightforward to infer that setting a value of
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
The shape of the traced path due to possible deviations from the optimum course, see Subsection 3.5.1.
The sensors’ measurement on the position and speed at a fixed time t, see Subsection 3.5.2.
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 [
The Kalman Filter model assumes the true state at time
At time k an observation (or measurement)
We can observe that in comparison with the initial proposal of System (1), we now model our problem as a linear system of equations whose state variables are corrupted by additive Gaussian noise, represented by variable
The position and speed in the extended System (9) is described by:
Let us now express the
Focusing on the covariance matrices regarding process and measurement noise, we get:
From this set of Expressions we are now capable of evaluating how different values of the measurement variance
We can deduce from the previous two comments that with Smoothing, the estimated trajectory will reduce the dispersion with respect to Filtering, since we count on more updated information to estimate the current position and speed of the vehicle.
Now it is time to graphically visualize some application examples of the previous concepts. In the first set of graphs, we will show how the trajectory of a vehicle (departing from a lateral position
Now we turn our attention to the evaluation of the Mean Square Error (MSE) between the optimum trajectory, and the measured, filtered and smoothed trajectories for the interval
On the other hand, we will also represent the averaged evolution (with a Degree2 polynomial like with MSE) of the lateral distance with respect to the optimum lateral position (which we call LDP) at the last time step
If we have now a look at
Degree2 polynomials for regression of MSE under measurement noise (
Percentage of distance with respect to the lateral optimum position under measurement noise (
For the MSE, we can notice that for lower values of the measurement variance
Degree2 polynomials for regression of MSE under measurement noise (
Percentage of distance with respect to the lateral optimum position under measurement noise (
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 [
In this paper, we have discussed how to trace the optimum trajectory of a high speed vehicle that wants to relocate its lateral position before a time interval of
On the other hand, the problem solved here corresponds to a scenario with no obstacles or other vehicles. It is our aim to study the problem for a higher number of vehicles (more differential equations) and obstacles. This results in functionals whose gradients are semiconvex. For this reason, depending on the initial configuration values, the
Last but not least, we have also carried out a firststep evaluation of how additive Gaussian noise can affect the shape of a single vehicle’s trajectory when measurement sensors are affected by such phenomena. We could realize the necessity to use path reconstruction techniques such as the Kalman Filter [
This work was supported by project grants MICINN/FEDER TEC201021405C0202/TCM (CALM) and 00002/CS/08 FORMA (