^{1}

^{1,2}

^{1}

^{2}

In city traffic, it is important to improve transportation efficiency and the spacing of platoon should be shortened when crossing the street. The best method to deal with this problem is automatic control of vehicles. In this paper, a mathematical model is established for the platoon’s longitudinal movement. A systematic analysis of longitudinal control law is presented for the platoon of vehicles. However, the parameter calibration for the platoon model is relatively difficult because the platoon model is complex and the parameters are coupled with each other. In this paper, the particle swarm optimization method is introduced to effectively optimize the parameters of platoon. The proposed method effectively finds the optimal parameters based on simulations and makes the spacing of platoon shorter.

With the rapid development of economy and growth of population, the number of vehicles becomes more and more in China and the increase in vehicle number is unprecedented. As a result, the traffic congestion has been becoming a serious problem. Therefore, it attracts many researchers’ attention to improve the platoon’s control which can shorten the spacing of platoon and increase traffic flow.

The best method to deal with this problem is automatic control of vehicles. One of the transportation automatic control systems is the Automated Highway System (AHS) which includes the longitudinal control, the lateral control, and the comprehensive control. An AHS is a proposed intelligent transportation system technology designed to provide driverless cars with specific rights-of-way. It is most often touted as a means of traffic congestion relief, as it would drastically reduce following distances and headway, thus allowing more cars to occupy a given stretch of road [

The idea of longitudinal vehicle control has developed very quickly and become very attractive with the increasing issues of traffic congestion and road safety. The researchers showed many technical considerations in the design of longitudinal control systems, such as external forces, process and measurement noise, and sampling and quantization of measurements [

The particle swarm optimization (PSO) method is a population based stochastic optimization method proposed by Kennedy and Eberhart in 1995 and is inspired by social behavior such as flocks of birds or schools of fish [

The remainder of this paper is organized as follows. Section

Various models for vehicles dynamics have been used in the study of longitudinal control of platoon. For a platoon travelling at a constant speed in a fixed direction, we adopt the following third-order model [

Platoon of five vehicles.

The parameter

Given the direction of platoon from right to left, the platoon variables are the velocity

From the input/output point of view of the

The objective of longitudinal control is to maintain the spacing error below a predetermined level or, if possible, at zero. As to the first vehicle, the controller [

When

Differentiating both sides of (

Taking Laplace transforms, we obtain

For the

Taking Laplace transforms, we obtain

For the second vehicle, the transfer function is

Therefore, the main design objective for the longitudinal control law is shown in Figure

Block diagram for the platoon.

We use the block diagram in Figure

Therefore, these parameters

The particle swarm optimization method uses the concept called particle and swarm [

In this paper, the PSO method is used to optimize the parameters ^{2}. Finally it reaches the value of 20 m/s. Taking inverse Laplace transforms for (

The size of the swarm is 50 and the number of iterations is 300. Parameters

The optimization procedure of the PSO method is given by the following steps.

Initialize a population of particles with random positions and velocities.

Evaluate each particle’s fitness value.

Compare each particle’s fitness with the particle’s

Compare the fitness with the population’s overall previous best

Update each particle’s velocity and position according to (

Return to Step

Through the particle swarm optimization method, we obtain the values of parameters which are

Curve of

Then transfer functions are described by

The Bode Diagrams of two transfer functions are shown in Figure

Bode diagrams.

According to the longitudinal control law, the diagram of platoon is shown in Figure

Diagrams of the platoon.

Spacing of the platoon

Position of vehicles

Velocity of vehicles

Acceleration of vehicles

The simulation results show that the deviations of vehicles from their preassigned positions do not exceed 0.06 m. The accelerations of vehicles in the platoon are within the range of acceptable comfort limits.

In this paper, a mathematical model is built for the platoon’s longitudinal movement and a longitudinal control law is analyzed in detail. It is well known that the parameter calibration of the platoon is a difficult problem. However, the PSO method effectively finds the parameters by storing the previous knowledge of particles and estimating the best positions and achieves the computation time of 2.7 s. The Automated Highway System is seen as a better way to deal with the problem of the traffic congestion. In general, the road conditions are complex. A number of studies should be done to analyze the effects of disturbances and modeling errors which may need further investigation.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work was supported by the National Natural Science Foundation of China under Grants 61040056 and 61070127 and Shanghai Key Project of Foundation funding under Grant 09JC1414600, China.