CONSIDERATIONS ON MODELING DRIVER ACTION FOR SYSTEMIC STUDY OF VEHICLE DYNAMICS

7/30/2019 CONSIDERATIONS ON MODELING DRIVER ACTION FOR SYSTEMIC STUDY OF VEHICLE DYNAMICS

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CONSIDERATIONS ON MODELING DRIVER ACTION

FOR SYSTEMIC STUDY OF VEHICLE DYNAMICS

Univ. Prof. eng. Ion COPAE PhDMilitary Technical Academy, Bucharest email: [email protected]

Abstract: The paper highlights the drivers influences onto vehicle dynamicity by the way he acts on

the gas pedal and his reaction time, using experimental data gathered throughout test runs. The paper

also presents the main mathematical models that describe the drivers actions throughout vehicle

movement.

Keywords: automotive, dynamics modeling, vehicle dynamics, systemic study, driver action.

Introduction

The driver influences vehicle dynamic behavior through driving style (the way he acts onto gas and

brake pedals) and through reaction time [3]. The paper deals with the way the gas pedal is being

operated and drivers reaction time. These issues introduce nonlinearities, called in system theory

static nonlinearities of relay type. The technical literature presents different mathematical models

(transfer function) for the driver that includes his reaction time and his driving style, which differ

depending on the established goal [3].

Drivers influence onto vehicles dynamics

1. Gas pedal operation may be performed in various ways. Its operation affects the throttles angular

position (its opening and closing). The throttles position is considered to describe the engines load,because the driver acts on it to change vehicle dynamic behavior (acceleration, maintaining a constantspeed, overpassing etc.); the technical literature recommends the engine load to be described by the

inlet manifold air pressurepa.

The upper graphs from figure 1 presents the values for the throttles position and vehicle speed in the

case of two experimental test-runs[1].

Figure 1.
mailto:[email protected]:[email protected]


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Added to that, the lower graphs from figure 1 presents the functional dependency between throttles

position and vehicle speed; these graphs highlight the fact that operating the throttle has as effect the

introduction of static nonlinearities by the driver. Figure 2 presents four such examples, which consist

of four static nonlinearities of relay type [3; 4].

Thus figure 2a presents a static characteristic of ideal relay type, for exemplification on the horizontal

axes is vehicle speed, and on the vertical the throttles position p; this shows that the driver actssuddenly on the gas pedal between the two positions from which one is minimum (of c value) and a

maximum (of +c value); the maximum movement of the gas pedal is 2c (meaning 100%), but the

representation is done according to the acknowledged procedure, symmetrically towards the axis

origin.

The mentioned characteristic shows the neglecting of inertia in operating the pedal, the lack of a

constant positioning area around the origin, the existence of a maximum value and the inexistence of a

closed loop; thus, it is a static characteristic of relay type with no inertia, no insensibility (no dead

zone), saturated and with no hysteresis, which adds to the fact that we deal with a symmetric

nonlinearity, with two active positions in this case.

Figure 2.

Figure 2b presents a relay type characteristic with a dead zone (there is a zone of constant position

around axes origin), with no inertia (pedal is pressed swiftly), with saturation and without hysteresis

(lack of closed loop); added to that it is a symmetrical characteristic that has three positions (two

active positions and an inactive position). The characteristic shows that the driver does not press the

pedal in an area of2b length around the origin, and the pedal operation is performed swiftly both on its

pressing as well as on its release.

Figure 2c presents static nonlinearity of saturated relay type, with no insensibility (dead zone) and

without hysteresis, but on which inertia is taken into consideration at the time of pedal operation.

Figure 2d presents a nonlinearity of polarized relay type, without inertia, saturated with hysteresis and

which includes an area of insensibility (dead zone).

All nonlinearities are described through nonlinear analytic expressions. For example, the polarized

relay type nonlinearity from figure 2d is described by the expressions ( represents the logicaloperator for and, V represent the speeds derivation):

0, pentru

0

0, pentru

0

V b Vc

V b Vp

V b Vc

V b V

(1)

Practically this shows that the driver, wishing to maintain constant speed within the limits (-b,b) at an

imposed value ( 80 for example, so b=5), he performs the following movements:5 km/h- Through AB area he maintains a constant position for the pedal (of c value), after which he

swiftly releases it in point B, where he acknowledges he reached the maximum allowed deviation

value b (85 km/h in the presented example);


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- through area CD maintains a constant pedal position (of c value), after which he swiftly

presses it in point D, where he acknowledges he reached the maximum b imposed allowed value

(75km/h in the presented example).

For example we aim to establish the drivers influence by the existence of a insensibility area (dead

zone) when operating the gas pedal (see fig. 2b); for 60 test runs that were available we get the results

from figure 3 in two cases [1]: 5% and 10% dead zone, both percentages from the total value of

throttles position. As we can see from figure 3a on all experimental test runs the average speed

decreases (increase in losses) once the dead zone increases, the biggest was reached in test L57; for all

test runs, we can see a drop in average value of 7,8% when the insensibility area increased with 10%.

Similarly from figure 3b we can see that for all experimental test runs 2nd

norm is decreasing (losses

increase) once the dead zone increases, the highest still present on test run L57; for all test runs we can

see a decrease in 2nd

norm with 7,3% when the insensibility area increases with 10%. Thus we can

conclude that once the insensibility area increases the vehicle dynamic behavior decreases a pattern

which was expected!

Figure 3

Figure 4 presents the drivers influence onto vehicle dynamic behavior through the existence of a ramp

(sees fig. 2c, line AB), for its three values: 15 degrees, 45 degrees and 70 degrees; it is being

concluded that in the latter case the driver pressed the pedal suddenly, getting closer to the ideal relaycharacteristic from figure 2a.

As we can see from figure 4, as the driver presses more and more violently on the gas pedal, the

vehicles dynamic behavior increases (average speed and 2nd

norm increase)

The graph from figure 4 highlights another important aspect, which has implications in the

establishment of mathematical models for drivers actions: we can see that the average speed and 2nd

norm have lower values than the experimental ones at a 45 degrees ramp, but higher at a 70 degrees

ramp.

This means that at this test ramp we can adopt an equivalent nonlinearity of ramp type in order to

model drivers actions, which in this case has a value that ranges between 4570 degrees; this aspect is

covered by the Hammerstein-Wiener mathematical model. 2. Drivers reaction time is another influence factor of vehicles dynamic behavior. If we take into

consideration system theory, reaction time is a dead time known as timedelay; thus in this situation

we have a time-delay system. In system theory, we call a pure timedelay (often called only delaying


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element) that element that has the transfer function, for example in Laplace images, with dead-time,in this case the driver's reaction time [3; 4]:

( ) es

iW s

(2)

Nonlinearity that emerges in the case a time delay exists is due to the fact that the system has a

transcendental characteristic equation, with infinity of solutions.

Figure 4.

Figure 5 presents the influence of drivers reaction time on to the vehicles speed, in the case of anexperimental test run [1]; it has been considered two values of reaction time: =1 s and =3 s

Figure 5.


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As we can see from figure 5, the main statistic characteristics of vehicle speed (the average value, 2nd

norm, standard deviation and maximum value) do not have large variations, in the sense of their

considerable reductions once the reaction time increases; in other words put, increasing the reaction

time considerably worse vehicle dynamics. What we can notice, especially in detail D, is the fact that

the increase of reaction time leads to a delayed start in speed variation. This aspect is very important

during the movement while obeying certain conditions, for example heavy traffic, or when a certaindistance from the front vehicle is to be kept; in these cases it is necessary that the reaction time to be

reduced as possible, otherwise accidents may occur in traffic. The last aspect leads to the necessity of

studying the systems stability, including for critical reaction time determination cr.

Drivers actions mathematical modeling

In the technical literature various transfer function are presented for the driver, which include both

reaction time, as well as the drivers actions, which differ from the targeted imposed goal [2; 3].

Thus on often used model is that of Pipes, a model which was later validated by Chandler. According

to this model, it is considered that each driver reacts the same as the driver in front of them; so all

drivers will react as the column driver reacts, which obviously is a simplifying hypothesis. Based on

this hypothesis Pipes proposes the following transfer function in Laplace images (ofs argument):

1,5

1,5

0,37e( )

0,37e

s

sW s

s

(3)

As we can see the reaction time of the driver was considered to be =1,5 s according to expression (2).Figure 6 presents Nyquist diagram (frequency characteristic in phase and amplitude) for

Pipes Chandler model from expression (3).

Figure 6.


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We can see from figure 6 that for the A-B area of frequencies, ( )A amplitude for the transfer

function is greater than 1; this means that for these frequencies, according to the given definition for

transfer function amplitude, there is a risk of a systemic instability. This aspect is easily highlighted

from the graph because we drew a circle for greater than 1 amplitude (R=1, 03), as well as the circle of

range equal to one that passes through critical point C (-1;0). Also the graph presents the first 6 roots

of the characteristic equation:1,5

( ) 0,37 e 0s

A s s

(4)

We have to mention that based on the first 6 roots, all of which being positioned in the LCS (left

complex semi-plane) we can appreciate that the system is stable; but having an infinity of roots due to

the existence of the exponential function, the mentioned instability may occur.

Figure 7 presents the step response of Pipes-Chandler model [2; 3]. The graph presents the systemic

performances of the model: drivers response time tr=8,87 s, rise time tc=2,43 s, stationary valueyst=1,

overshoot =7,98%. As we can see, this model has three disadvantages: consider all drivers to be

identical (same reaction time), systemic instability may occur and the index response may present

overshoot (although within limits).

Figure 7.

Another model that of Burnham, targets traffic optimization and has the following functions:

2

0,09

0,09

(0,5 1,64) e( )

(2, 3696 1, 64) e

s

s

sW s

s As s

(5)

WhereA [1/s] coefficient is dependent of vehicle speed v[m/s], and reaction time is considered to be

=0,09 s [2; 3].If frequency response is presented in the case of optimal model (figure 8 with Nyquist diagram), we

can see that for all frequencies the transfer functions amplitude (6) is under the value of one, which

means that this model satisfying the imposed conditions regarding stability. Indeed, we havent got

like previous cases frequency values on which their amplitude to be greater than the value of one.

Figure 9 presents the step response afferent to Burnham model for the driver. As we can see from the

graph, we have no overshoot, and the response time has lower value than in the previousPipes Chandler model.


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Figure 8.

Figure 9.

Out of the necessity of optimizing traffic, separate transfer functions were established for traction and

for braking. In the first case optimization leads to a transfer function that ensures the control of

throttles angular movement and in the second case ensures the operation of ABS. To this purpose,

Ioannou and Xu established transfer functions for traction under the following form (reaction time is

considered null):


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2

1 3 2

1, 2 0, 24 0,012( )

1, 4 0, 25 0,012

s sW s

s s s

(6)

And for breaking the following form:

2 20,25( )

1, 25 0, 25

sW s

s s

(7)

On both frequency responses confirm the algorithms correctness, as we can see from figure 10, where

Nyquist diagrams are presented forW1 and W2 from the latter two expressions.

Figure 10.

Just as the previous cases, it can be established step response afferent to Ioannou Xu model for the

driver, using the two data transfer functions; thus we can conclude that there is no overshoot, and

response times are reduced and approximately equal.

Conclusions

Similarly we can analyze other mathematical models that describe the drivers actions, on which we

can consider his reaction time. Considering the vehicles movement equation we can study its systemic

dynamic behavior, on which we take into consideration the drivers influence and the roads influence.

References

1. Bivol G.C. Considerations about establishing mathematical models using system identifications

procedures. The 36th

International Scientific Symposium of the Military Equipment and Technologies

Research Agency, 2005, Bucharest.

2. Copae I. Teoria reglrii automate cu aplicaii la autovehiculele militare. Performanele sistemelor

automate. Editura Academiei Tehnice Militare, 1997, Bucureti.

3. Copae I. Teoria reglrii automate cu aplicaii la autovehiculele militare. Sisteme automate

neliniare. Editura Academiei Tehnice Militare, 1998, Bucureti.

4. Copae I., Lespezeanu I., Cazacu C.Dinamica autovehiculelor. Editura ERICOM, 2006, Bucureti.

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CONSIDERATIONS ON MODELING DRIVER ACTION FOR SYSTEMIC STUDY OF VEHICLE DYNAMICS