DIRECTIONAL CONTROL OF A PLATOON OF VEHICLES FOR COMFORT SPECIFICATION BY CONSIDERING PARASITIC TIME DELAYS AND LAGS

The problem of controlling a string of vehicles moving in one dimension is considered so that they all follow a lead vehicle with constant spacing between successive vehicles. The goal of this paper is to consider the effects of parasitic time delays, lags and comfort specification. The contribution of this paper is two-fold. First, negative effects of the parasitic time delays and lags on the string stability have been investigated since most physical systems usually involve parasitic time delays and lags. Second, restrictions on control parameters, to be well-suited with bounds on the dynamic variables as well as the requirement for string stability have been presented. Finally, a simulation example of multiple vehicle platoon control is used to demonstrate the effectiveness of the proposed method.


INTRODUCTION
Highway congestion is imposing an intolerable burden on urban residents.Increasing traffic density has been causing traffic congestion on the roads, which will reduce travel time, traffic safety, while increasing air pollution, and energy consumption.Congestion occurs when vehicle velocity variation propagates to the following vehicles.It is difficult for drivers to recognize small changes in the preceding vehicle velocity.There are various approaches to reduce congestion such as vehicle platoon control.One possible solution to this problem is to use the Adaptive Cruise Control (ACC) concept.The ACC system has been proposed as an enhancement over the existing cruise controllers on ground vehicles.Autonomous cruise control devices use the distances and velocity differences between vehicles by radar to calculate the required action and actuate cars accordingly [1].
The control system for a string of vehicles consists of the guidance model depicting communication between vehicles and individual vehicle control.There are many guidance models for communication structure between vehicles in a platoon.Interaction structures may be restricted by sensing, communication, or computational limitations.For example, vehicles only interact with their nearest neighbours rather than with all vehicles in the system in many implementations.The effects of either the interaction structures or information flow on the system stability were investigated using graph theory [2][3][4].
For multi agent systems, many properties may become significant in addition to the usual stability and robustness analysis such as string stability, and must be taken into account.The term string stability was initially introduced in 1974 [5].String stability explains how errors propagate through the group of vehicles as result of disturbances or the reference trajectory of the formation lead.A string-stable control signifies that spacing errors between adjacent vehicles do not amplify along the vehicle string.String stability has been widely used in automated highway systems [6,7].The relationship between string stability and spacing policies was an important issue [9,10].There are two major strategies for controlling of a platoon of vehicles; these are constant time headway control and spacing distance control.In spacing distance control, the distance between two adjacent vehicles in the platoon is controlled to be the same as a specified constant that is independent of the velocity in the platoon travelling.Constant spacing strategies are usually used due to high traffic capacities.Swaroop and Hedrick investigated constant-spacing control strategy, in that the control purpose is to preserve a constant distance between adjacent vehicles in a string [6].
In this paper, the desired trajectory of the platoon is considered to be of constant-velocity type.String stability cannot be guaranteed when constant spacing policy is used [11,12].Also, extra information is needed.The focus of this paper is to analyze the centralized directional adaptive cruise.Directional ACC monitors the behaviour of the preceding vehicle.The centralized control receives relative distance and relative velocity information with respect to the preceding vehicle as well as velocity information of the preceding vehicle.
Time delays appear in those systems due to sensing computation and actuation.The parasitic delays and lags are the unavoidable natural characteristics of actuators and sensors in the mechanical and control systems, which may considerably affect the stability of systems in some situations.Xiao and Gao investigated string stability of a platoon of adaptive cruise control vehicles by considering the parasitic time delay and lags of the actuators and sensors [13,14].
The passenger comfort in public ground transportation is affected by the changes in movement felt in all directions.Typically, the value of acceleration is taken as a comfort criterion [15].However, comfort based on the movement changes in vehicle longitudinal direction (the "jerk") was used in [16].So, the jerk is significant when assessing the discomfort caused to the passengers in a vehicle.Cook [17] investigated control of vehicle convoy with a selection of control parameters to ensure string stability while satisfying passenger comfort constraints.
Another purpose of this paper is to take a fresh look at the string stability with these considerations in mind, with the aim of obtaining control parameters to guarantee string stability of a vehicle convoy system, whilst fulfilling restrictions compelled by considerations comfort criterion by taking into account the effect of the time delays and lags.Note that the previous analyses of homogenous string stability are conducted without using the parasitic time delays and lags to keep the comfort of passengers and take account of restrictions on the achievable performance of the driving force system.This is inspired by the work [13,17] that inspected the string stability analysis of 1-D vehicular platoons in a similar framework.Compared to [13], this paper makes some novel contributions.Due to practical design and implementation, the constraints imposed by the driving force system are considered.Since the comfort criterion is taken as the result of these constraints, it is important to consider it during the design of vehicles.Also, [17] did not consider parasitic time delay and lags that are the inevitable natural characteristics of the mechanical and control systems.
Vehicles require a complex high-order nonlinear model to describe the complete dynamic behaviours.The analyses are usually based on linear system theory which does not take into account the limitations of the performance of the propulsion system, such as bounds on the forces.This subject was also not considered in [13,17].In this paper, a hierarchical platoon controller design framework is established comprising a feedback linearization controller at the first layer and a linear controller at the second layer.By reducing the non-linear system to a linear model using the top layer feedback linearization controller, a controller is designed.Also, a platoon consisting of identical vehicles is considered that are modelled by non-linear differential equations.
The remainder of the paper is structured as follows.Section 2, briefly introduces the longitudinal vehicle model that considers the parasitic time delays and lags.Section 3 handles the development of a control law when such parasitic delays and lags are taken into account.Section 4, presents an analysis of string stability.Section 5 deals with the selection of control parameters to satisfy passenger comfort constraints.In Section 6, the simulations are presented to show the efficiency of the proposed method.

VEHICLE MODEL
Consider a group of vehicles in dense traffic with no overtaking and assume each vehicle uses only the information of the preceding vehicle.The formation control of N 1 + homogeneous string of vehicles is considered so that they all follow a lead vehicle (Figure 1).
The position, velocity and acceleration of the lead vehicle are denoted by x t 0 ^h, v t 0 ^h, a t 0 ^h, respectively.Also, x t i ^h, v t i ^h, a t i ^h denote the position, velocity and acceleration of the i-th vehicle, respectively.
A model for the motion of a vehicle must take into account the power-train, tyre forces, aerodynamic drag forces, rolling resistance forces and gravitational forces.The power-train consists of the internal combustion engine, the torque converter, the transmission and the wheels [16].The dynamics of the i-th following vehicle can be modelled by the following non-linear differential equations ( [18][19]) where ci is the engine input and , f v a where v is the specific mass of the air, Ai , cdi , dmi , mi and i x are the cross-sectional area, drag coefficient, mechanical drag, mass, and engine time constant of the i-th vehicle, respectively, / A c m 2 i di i v stands for air resistance.Note that the vehicles considered here are not necessarily identical.
The following control law has been adopted where ui is additional input signal to be designed.Obviously, this control law can be achieved by feedback linearization, since after introducing (2), equation (1) becomes a a u The feedback linearization controller in (2) plays the role of the first layer controller in our architecture.It helps to simplify the system model by excluding some characteristic parameters of the vehicle from its dynamics.
Due to the pure information delay or sensing delay, the controller can access just the delayed information but not the instant information to achieve the desired acceleration or deceleration.It means that the expression of the control law is u t i ^h replaced by u t i D -^h [20][21].Therefore, equation (3) has to be rewritten in the following form a a u t One may treat i D and i x as "lumped" parasitic time delay and lag, respectively, which are usually caused by several sources such as the pure time delay and lag in the engine response, of the throttle actuator, brake actuator and bandwidth of low-pass filters used for other sensors such as engine manifold pressure sensor, wheel speed sensor [22][23].Because of the parasitic time delays and lags, the commanded acceleration may not be accessible instantly.
For simplicity, all the vehicles are supposed to be identical, although this assumption may be neither entirely necessary for the discussion nor satisfied in practice.Therefore, Assuming that a model of single vehicle is described by equation ( 4), the resultant model for a platoon is shown in Figure 2.
In this paper, only linear models will be used for the dynamics, but constraints will be forced on the magnitudes of the variables, representing bounds on what can be achieved in practice.Distinctively, the acceleration will be assumed to satisfy a boundary a t i # a ^h (5) This is for the worst case scenario in an urgent situation.While the jerk is required to be restricted by Because the imposition of constraints does not contradict the assumption of linear dynamics, it is supposed that the accelerations are always maintained within the boundary (5), and the investigation is concerned with the conditions which should be imposed on the control parameters in order for the bounds such as (6) then to remain satisfied.

LOOK-AHEAD CONTROL LAW
There are many feasible control strategies for vehicle following in a platoon, depending on the information about the motion of other vehicles.In this paper, the controller on each vehicle utilizes only the information about itself and the one directly ahead.For the development of a control law, it will be supposed that there is a lead vehicle, indexed by 0, that executes manoeuvres and the following vehicles are controlled to keep a desired following distance.Constant spacing strategy is used for the desired following distance of any following vehicle.
The spacing error can be defined as: It is to be noted that the desired distance in between is not necessarily identical for vehicles.However, it is assumed to be constant in this study.
s s e -Ds 1 1

Figure 2 -Transfer Function representation of a platoon of vehicles (open loop)
It is clear then that . The objective of the controller is to asymptotically drive the spacing error to zero.With the aim of extracting the control law, the following equation is considered: where Kc and Kv are constant proportional and derivative gains, respectively.If gains are positive, the spacing error will converge to zero.Then the control input can be generated from Eq. ( 8) using the simplified model of a vehicle.
Because of the parasitic time delays of the actuators and sensors, the control law ( 9) must be replaced with The goal of the controller as defined above is to follow the lead vehicle with a constant spacing between successive vehicles.

STRING STABILITY ANALYSIS
The question of string stability/error amplification is usually answered by looking at the transfer functions that relate the spacing errors between two successive vehicle pairs.The string-stability ensures that errors range decreases as they propagate along the vehicle stream.The string stability performance is mainly determined by the structure of the controller and headway policy.In other words, the string stability performance is mostly dependent on the information received and used by the longitudinal controllers.
The velocity and spacing error dynamics models can be derived based on the vehicle dynamics model (4).
By differentiating both sides of (11), we obtain The Laplace transforms can be used for analyzing the conventional notation and the regular assumption of zero initial conditions for the derivation of transfer functions.Therefore, by taking the Laplace transformation on both sides of (12), we obtain This describes the velocity dynamics model of two successive vehicles in the string of vehicles.
It is clear that the range error output must be smaller than or equal to the range error input to avoid range errors from propagating indefinitely along the string.For this uniform vehicle string, a string-stability definition is widely used [24][25] and described as follows: where G i~h is derived from the spacing error transform function (13) by substituting s i= .
will be obtained.
The spacing errors have most of their energy in the region of low frequencies, which is also called the key region of the string stability [26]. Knowing Here, the proof of the theorem is completed.
By comparing G i x ~~D + -+ = --- At low frequencies, when the lumped parasitic time delay and the lumped parasitic time lag are the same, Eq. ( 14) can be rewritten as follows Then the G i Here, the proof of the theorem is completed.Also, theorem 2 is true for a string of vehicles by considering constant time headway [13].

CONSTRAINT FORCE
For the sake of practical restrictions on dynamic variables, it is essential to consider the restrictions in order to be compatible with bounds on the dynamic variables and requirement of the string stability.
In order to assess the implications of bounds on acceleration and jerk, it is necessary to examine the relation between the acceleration of the (i 1 -)-th vehicle and the jerk of the i-th vehicle, given by the transfer function [17] which becomes Then, if the jerk is to remain within acceptable limits, for any allowable acceleration of the preceding vehicle, it is required that [17] Theorem 3: Comfort criterion is satisfied if the following condition is met: The nominator iC^h by setting Kv ~= will be: Also, the denominator iC^h by setting Kv ~= is Then, equation ( 20) can be written as the following inequality By taking the upper boundary of 21) can be written in the following form With some simplifications, inequality (22) can be written as follows: By considering inequalities ( 18),( 19) and ( 23) (24) The jerk is significant when evaluating the discomfort brought about to passengers in a vehicle.Then, an acceptable criterion is that the restricted longitudinal accelerations and jerks can satisfy a definite degree of comfort in longitudinal control.Because of comfort criteria, the bounded jerk is maintained at less than / m s 3 3 i.e. ( 3 b = ) and a is the deceleration of the i-th vehicle under the maximum brake action which can be obtained from the vehicle manufacturers or by experiments.In this paper the value of a has been considered as equal to / m s 7 2 .Inequality (24) should be maintained in addition to string stability requirement.Then, for obtaining comfort criteria, by substituting variable as x Kv = and y K K Here, the proof of the theorem is completed.

SIMULATION
In order to validate the performance of the proposed control algorithm, computer simulations have been carried out for the platoon system containing nine followers, i.e., N = 9.As stated before, the ability of maintaining the distance between vehicles is important for the safety of the vehicle platoon system.The most important disturbances in a platoon control system include the lead vehicle acceleration/deceleration. The disturbance is defined as any source that causes the vehicle string to lose maintaining constant velocity.In the simulations, the desired vehicle spacing was set as D m 8 ; other parameters used in the simulations are the same as parameters mentioned in [27], that is, / kg m Figure 3 shows tracking of velocity and it also shows the spacing errors of the vehicles smoothly reduce upstream in the string.also show the performances of the vehicle platoon control system so that the bounds on acceleration and jerk are preserved.The ability of maintaining the distance between vehicles regardless of the measurement signal noises is important for the vehicle platoon.In this paper, apart from the measurement signal noises the robustness is considered.Even though the information is measurable via sensors, the extraneous noises were unavoidably included in the measurement process, and the overall performance of the string of vehicle control system can be worse as result of the noises.In this simulation, in order to take into account the robustness against the noise in the measurement signals, it is assumed that the measurement signals of the position and the velocity are added into the random number with a mean of 0 and a variance of 0.01 for the position and a mean of 0 and a variance of 0.1 for the velocity.The performance of the vehicle platoon control system under the noise in the measurement signals is shown in Figure 8 which represents the string stability state.

CONCLUSION
The paper addresses the coordination problem of a multi-vehicle system with a leader.The analysis of error accumulation was also performed.The paper has extended the results related to the string stability analysis of a platoon of vehicles based on the more practical scenarios such as the parasitic time delays and lags.Also, the restriction which needs to be imposed on the control parameters is considered, in order to be well-suited with bounds on the dynamic variables.The results have shown that when the condition K K i i i ^h and g v i i ^h are A. Ghasemi et al.: Directional Control of a Platoon of Vehicles for Comfort Specification by Considering Parasitic Time Delays and Lags

p 1 -
^h indicate spacing (the following distance) between the i-th vehicle and the (i 1 -)-th vehicle, li indicates the length of the (i 1 -)-th vehicle, di ref indicates the spacing distance reference, t i d ^h indicates the spacing error of the i-th vehicle which is deviation between the spacing and the desired spacing.

Theorem 2 :
G i~h with 0 D = , and value of G i 0 ~x = ^h denotes value of G i~h with 0 x = , the most negative effect of the parasitic time delays and lags can be recognized.Greater values denote higher negative effect.The parasitic time delay has more negative effect on the string stability if the lumped parasitic time delay and the lumped parasitic time lag are the same.independent of parasitic time delays and lags.Therefore each one has a smaller denominator which has the most negative effect on the string stability.

A
. Ghasemi et al.: Directional Control of a Platoon of Vehicles for Comfort Specification by Considering Parasitic Time Delays and Lags

2
We can obtain that inequality (23) is true for .

Figure 5 illustrates
Figure3shows tracking of velocity and it also shows the spacing errors of the vehicles smoothly reduce upstream in the string.Figure5illustrates Case2 in Table1which represents the string unstable condition of K K

Figure 5
Figure 5  illustrates worse instability in tracking velocity, as well as in keeping the spacing.

Figures 6 and 7 9 Figure 3 - 10 Figure 4 -
Figures 6 and 7 illustrate acceleration and jerk of the vehicles along the string, respectively.The figures

9 Figure 5 -
Figure 5 -Performance of string instability of ten vehicles the string stability is obtained.At the end, the robustness of the proposed algorithm is validated through computer simulations under noise in the sensor measurement signal.

Table 1 -
Parameters for the platoon