Decentralized Control for Self-driving Cars That can Freely Move on Two-dimensional Plane

In the current traffic rules, cars have to move along lanes and to stop at red traffic lights. However, in the future when all cars become completely driverless, these traffic rules may vanish and cars may be allowed to move freely on two-dimensional plane by avoiding others like pedestrian flow. This innovation could greatly reduce traffic jams. In this study, we propose a decentralized control scheme for future self-driving cars that can freely move on two-dimensional plane, based on the social force model widely used as the model of pedestrian flow. The performance of the proposed scheme is validated via simulation. Although this study is still conceptual and does not consider realistic details, we believe that it paves the way to developing novel traffic systems.


Introduction
Technologies for self-driving cars have been developed within the framework of the conventional traffic rules in which cars have to move along lanes and to stop at red traffic lights. However, these traffic rules, which limit traffic flow and often cause traffic jams, may not be necessary in the future when all cars become completely driverless, because self-driving cars will be able to move more accurately than cars driven by humans. It is expected that traffic jams will be greatly reduced if self-driving cars move freely on two-dimensional plane by avoiding others like pedestrian flow.
In this study, we propose a decentralized control scheme for future self-driving cars that can freely move on two-dimensional plane. Because this study looks ahead to distant future, we employ minimal assumptions, rather than consider realistic and detailed problems, to capture the essence of the control. Social force model [1,2], a simple model of pedestrian flow, would be a suitable platform for considering this problem. We construct the control scheme by improving the social force model and demonstrate that the simulated self-driving cars can move fast, smoothly, and safely.

Model
In the social force model [1,2], each pedestrian on a two-dimensional plane is regarded as a circular particle with the radius of , and its time evolution is described as (1) where is the mass of the particle, is the position of the particle , is the unit vector that represents the direction particle wants to move, is the target speed, and is a positive constant. The second and third terms on the right-hand side denote forces generated by the interaction with the other particles and that with the environment, e.g., walls, respectively. Superscripts "phys" and "soc" for these forces denote physical and social forces, respectively. Details are provided in [1,2].
The model of self-driving cars proposed here is also described in the same form as Eq. (1). Here, whereas the social force originates from psychological effect, i.e., desire to avoid others, in the case of pedestrian flow [1,2], it is redefined as the control input to self-driving cars and is designed based on the prediction of future motion of itself and nearby cars (see Steps 1-4 below). Note that the basic idea of the design is similar to our previous work on the decentralized control of traffic signals [3].
The social force (control input) in the proposed model is calculated according to the following 4 steps (f soc can be calculated in a similar manner as f soc ): Step 1: Each car detects the relative position and velocity of cars within the distance from itself every time interval .
Step 2: Based on the information obtained in Step 1, each car predicts the future motion of cars within the distance from itself. Because the interaction terms (the second and third terms on the right-hand side) in Eq. (1) are hard to predict, the equation in which the interaction terms are omitted from Eq. (1) , , is solved numerically for duration , wherein the initial position and velocity of the cars are the same as those obtained in Step 1 (Fig. 1).
Step 3: Based on the numerical calculation performed in Step 2, the expected time until the distance between cars and takes a minimum value or until car collides car , , is derived (Fig. 1). Here, when the distance is expected to increase monotonically, and when the distance is expected to decrease monotonically without any collision. Further, the expected distance between cars and after the time interval , , is also derived.
Step 4: The social force is derived according to the following equation: where is the expected position of car after the time interval (Fig. 1). The first and second terms on the right-hand side in Eq. (3) denote the exclusive volume effect and the avoidance force based on the prediction, respectively. The term in Eq. (3) was introduced because the avoidance force should be large when the distance between cars and is expected to become small in near future. Note that the proposed model is consistent with the original social force model [1,2] when and .

Simulation
Simulation of the proposed model was performed. The deflection angle of was set to be random and did not vary during the simulation. The periodic boundary condition was employed. The performance was evaluated by the following indices 1 , 2 , and 3 : where is the time step, is the maximal time step, and is the number of particles. Index is large when cars move fast in the direction they want to move, while indices and are small when cars move smoothly and safely, respectively.
The result when parameters and are varied with the other parameters fixed is shown in Fig. 2. It is found that increases as decreases and increases ( Fig. 2(a)), while decreases as and increase (Fig. 2(c)). In contrast, is small in the area where is small and is around 3.0 [m 4 kg s -2 ]. White and yellow arrows in Fig. 2  and at the region where the yellow arrows point). Thus, the avoidance force based on the prediction, which was newly introduced, plays a crucial role for enabling self-driving cars to move fast, smoothly, and safely.

Conclusion
We proposed a decentralized control scheme for future self-driving cars that can freely move on two-dimensional plane, based on the social force model. We demonstrated via simulations that cars with the proposed control scheme can move fast, smoothly, and safely. It should be noted that this study is still conceptual and does not consider realistic issues (e.g., calibration of parameters to realistic values of current cars etc.). However, we believe that our new concept becomes a breakthrough for developing novel traffic systems. Finally, we would like to note that the model proposed here could be also used as the model of pedestrian flow because pedestrians change their path by predicting the movement of their nearby pedestrians. Hence, we believe that this study contributes to the field of pedestrian and evacuation dynamics.