A sensitivity analysis of empirically derived car-following models

Trunspn Res.

Vol. 2, pp. 363-373.

A SENSITIVITY

Pergamon

Press 1968. Printed in Great Britain

ANALYSIS OF EMPIRICALLY CAR-FOLLOWING MODELS?

THOMAS H. ROCKWELL, RONALD L. ERNST

and

DERIVED

ALBERT HANKEN

The Ohio State University, Columbus, Ohio, U.S.A.

(Received

24 December

1967)

1. INTRODUCTION CONSIDERABLE interest

has been evidenced in recent years in traffic flow and car-following theory. Two classes of models are used to describe these phenomena: macroscopic models usually involve structures which relate volume, speed and density for a large number of vehicles; microscopic models concern relation of the driver-vehicle system to intervehicular dynamics. Analogies to kinetic theory of gases, diffusion processes, compressible fluids, etc., are just a few of the approaches to macroscopic modeling that have been studied. This paper concerns itself with microscopic models of the car-following situation and how the driver of the following car responds to the action of the lead car. Microscopic models have importance in (a) better understanding traffic flow concepts, (b) designing automatic systems which closely match human performance and (c) designing systems to aid or assist the driver in the car-following task, i.e. partially automated systems and sensory supplementation. At least three approaches can be taken to the study of the driver-vehicle car-following function. One is to use physical models, e.g. Newtonian-forcemass-acceleration models (Herman, 1961). A second approach, and the one taken in this paper, is curve-fitting from empirical data in order to describe the driver-vehicle system in terms of intervehicular dynamics. Lastly, this paper will conclude with an introduction to a third approach to car-following models, which is presently underway at the Systems Research Group at The Ohio State University. This combines traffic dynamic information with driver psychophysical data to predict control movements from intervehicular dynamics. The purpose of this study was to examine the sensitivity of empirically derived carfollowing models to (a) time delays on predictor variables, (b) lead-car velocity programs, that is, the forcing velocity function (open road versus traffic), (c) subjects and (d) replications within subjects. Emphasis in this work was with regression models of the driver-vehicle system as shown in Fig. 1 using acceleration of the following vehicle as the dependent variable and headway (spacing) lead-car velocity, following-car velocity and lead-car acceleration as possible predictor variables. Time was also used as a predictor variable in the original experiment to measure learning effects. Models predicting the change in following-car acceleration, A& were also investigated. In general, time did not improve data fit which suggests little t Based on a paper presented before the Professional and Technical Group on Human Factors Electronics, Institute of Electrical and Electronic Engineers, Palo Alto, California, May 1967. 363

in

364

THOMASH. ROCKWELL,RONALD L. ERNSTand ALBERTHANKEN

learning or adaptation took place during the experiment. Although acceleration change models are generally better fitted to the data, most of the analysis of this paper is directed to following-car acceleration prediction because of its relevance to previous research in car-following.

r----------7

’

I

LEAD CAR +

$t)=F[i,(t-i;),k2(t-;),%,(t-T),H(t-?;)I FCY-FOLLOWING

CAR VELOCITY

FCA-FOLLOWING

CAR ACCELERATION-

= i,

FCJ-FOLLOWING

CAR JERK’

ii,

LC V- LEAD CAR VELOCITY : i, LCA - LEAD CAR ACCELERATION

= 2,

X,

FIG. 1. Generalized car-following system. 2. THE

EXPERIMENT

The experiment consisted of lo-min car-following runs in both open-road and urban expressway traffic. The equipment consisted of two instrumented vehicles (Rockwell and Snider, 1965). The following car was equipped with a fifty-one channel oscillograph recorder which measured steering wheel position, gas and brake pedal deflection, velocity, relative velocity, spacing (headway) and acceleration. Spacing and relative velocity were measured using a version of the GMC take-up reel (Rockwell and Snider, 1965) with a tachometer generator for measuring the rate of uptake and payout of a fine wire (not visible to the driver) and a ten-turn potentiometer to measure the amount of wire played out at any point in time. The lead car was equipped with a copilot system which permitted the experimenter to dial in specific velocities for non-brake maneuvers and a brake-pressure gauge and accelerometer to provide given braking maneuvers. For the open-road carfollowing studies, a standard lead-car velocity and acceleration pattern were produced such that each subject in each trial received the same randomized pattern. For car-following in traffic, the two vehicles operated in 5 p.m. (“rush hour”) traffic on an urban expressway in Columbus, Ohio, accepting whatever velocity perturbations occurred. Thus, no experimental control was possible although mean velocity and headway performance were remarkably similar for each trial under these conditions. Subject drivers were college students with considerable experience as subjects in this kind of research. They were instructed to maintain a minimum safe headway throughout the runs. 3.

The basic data at 0.5-set intervals in a direct manner. sensitive to motion

METHOD

OF ANALYSIS

were headway, velocity and acceleration of the following car sampled from the oscillograph recording. Other data were computed from these The sample interval had been established in prior research as adequately changes without propagating error.

Empirically

derived car-following

365

models

The resulting open-road data were first tested to ascertain if derived lead-car velocity patterns were equivalent in each trial. Overall mean velocity and headway were next computed. The method employed in the analysis of the data was derived by Hanken (1965) and identified as “piecewise linear regression”. A given system is represented by a set of linear regression equations, valid over a limited range of values of variables. The method requires a partition of the data space using a method which minimizes the total unbiased estimates of the residual variance within the partition.

.5

0

.I x,-

.2

.3

.4

.5

.6

2 .7

.8

c

62

FIG. 2. Piecewise linear regression

for partitioned

.S

I.0

6,

data spaces.

Given the parameters of car-following, it is possible to determine relationships between the parameters from data collected in various car-following modes through a piecewise regression procedure. The procedure, while complex in execution, is simple in form. Basically, the space occupied by the parameters is fit, subdivided, refit, rest÷d, etc. -recursively-by linear regression, until the fit no longer improves. The resultant equation or set of equations obtained with recursive fits thus constitutes an empirically derived model; the decrease in residual mean-square error is an expression of nonlinearity. In effect, nonlinear data space can be partitioned such that linear regression functions can be fit on each of the partitioned data spaces. Using a digital computer, the method involves a search procedure for finding the optimal cutting point. The result is illustrated graphically in Fig. 2. 4. RESULTS

4.1. Data analysis For the original subject in open-road car-following, it was found that the percentage reduction in the residual due to partitioning was surprisingly small, only 8-O per cent from one cell to eight cells (see Table 1). This was true despite shifts in the time lags (to be discussed later). The multiple regression coefficient, R, improved only from 0413 to 0450 with sixteen partitions. Thus, the implications were that nonlinearities in the data were not sufficient to rule out a simple linear regression on the entire data space. In the original analysis, time lags were introduced on the predicted variable 2,. A 2-see lag produced the minimum residuals when the lag was varied from 0 to 6 set in 0.5set 27

366

THOMAS H. ROCKWELL,

RONALD L. ERNST and ALBERT HANKEN

intervals. The reduction in residual from zero lag to 2 set was 50 per cent. This 2-set finding supports earlier research on response lags by Herman (1961) and Helly (1959) and is consistent with subject detection and response time and vehicle response time (Rockwell and Snider, 1965; Algea and Ernst, 1965). TABLE I.

LINEAR AND NONLINEAR MULTIPLE CORRELATION COEFFICIENTS FROM PIECEWISE LINEAR REGRESSION ANALYSlS Model

function

(1) (2)

X,(t) n,(t)

(3) (4)

X,(t) = F[(h, it, %)1+2, iI&(f) = F[h, X,, -i-,, q,

R (1 cell)

R (16 cell)

0.640 0.813 0.816 0.976

0.694 0.850 O-865 0.978

= F[/r, _?,, 1,], = F[h, il, &It-2

tl &I,._,.,

Table 1 summarizes the analysis of the first subject in open-road car-following. Note that R (1 cell), i.e. one partition-simple linear regression-and R (16 cell)-the final division of the data space into sixteen partitions and corresponding regression fits-are not markedly different for any of the models used. A substantial improvement in R occurs with a 2-set lag, as in equation (2). Equation (3) which involves time as an independent variable shows little improvement over equation (2) and suggests a time invariant condition. Equation (4) predicts AZ, and gives the best fit using two added predictor variables, jt, and Z,, with a 0.5-set lag for this third-order differential equation. The change in the time lag for equations (2) and (4) is probably explained by the fact that the change in vehicle acceleration occurs at about O-5 set and reaches its maximum I.5 set later. TABLE 2. EFFECT OF2-SEC LEAD ON PREDICTOR VARIABLE(BASED ON 1200DA~~ POINTS) The equation is R,(t+ 2) = 0.016 + 0.058 [H(t) - 134]+ 0.498 [n, (t) - 32]- 0.546 [a,(t) - 32]+ Optimal 7

Condition SO

H

2

Hanken original R= 134ft i = 32 m.p.h.

LCV

2

FCV

2

(t)

Multiple - (PI + Pz) 0.048

a&

711

I.769

Residual

Coefficients

1285

0.0058

R 0.813

0.498

45

- 0.546

Table 2 shows the form of the results for a 2-set lead on each predictor variable. These performance measures will be discussed below. The acceleration noise, us2 = I.76 ft/se?, compares well with previous research in the Holland tunnel where lead-car variability was much less than in this experiment (see for example, Herman and Potts, 1961). In effect, the following car responds weakly to changes in headway, perhaps due to its magnitude, but strongly to changes in _i-, and _i-,. 4.2. Time sensitiaity In an effort to ascertain if the 7’s (time lags) for each of the predictor variables could be differentially adjusted to improve fit, the program was adapted to permit each of the predictor variables to lead or lag the dependent variable. Of interest here was whether memory or anticipation effects existed. Several procedures were used: (a) trial and error, (b) partial cross-correlation and (c) a selective procedure. The last determines optimal T’S on the first variable, and finds the T for the second variable such that the minimal residual and variance recursion produces no additional reduction in residual variance.

Empirically

derived car-following

367

models

Figures 3 and 4 show the effect of T slips on residuals for S-2 for headway and the following-car velocity. These curves show that the headway T’S have little effect on the residual while a 50 per cent reduction in residual occurs when T on ~7~is slipped from 15cO-

TIME

-ADVANCE

DELAY

IN SECONDS

DELAY -

HEADWAY

FIG. 3. Headway residual as a function of time delay (7). zzoo2000

-

I800

-

SUBJECT

“? g

2

1600-

5 z

1400-

9 @ (f 1200-

1000 -

Eloo-

I

-10 ---FCA

I

-8 LEAD

t

-6

I

-4

I

I

I

-2 0 2 TIME DELAY IN SECONDS

FOLLOWING

I

4

I

I

F6CA

8 LAG -

I

W

CAR VEL&TY

FIG. 4. Velocity residual as a function

of time delay (7).

0 to 2 sec. It should be emphasized that optimality of time slips is a relative factor: the difference between a 1-set and a 2-set lag may be trivial in terms of residual reduction and could easily lead to unrealistic implications. For example, an optimum 7 might well be 6 set or 0 set, both of which are inconsistent with knowledge of human response. Figure 5 shows the consistency of T slips for replications of S-3 in traffic for the variable ~7,. The practical significance of this is that it serves as a validity check. If good car-following performance exists, we would expect a good match of jt, and jt, and further expect some consistency for a given subject.

368

H. ROCKWELL, RONALD L. ERNST and ALBERT HANKEN

THOMAS

Tables 3 and 4 show the results of the overall effects of traffic vs. open road, subject and replications on derived regression models. It should be noted that although the optimal T’Slisted produced minimum residuals, these are not necessarily statistically different. This is particularly true of 7 on headway with open-road car-following.

I

I

-10

,

-8

I

I

-6

I

-4

TIME

-ADVANCE

LEAD

I

I

2

0

-2

DELAY CAR

s-2 Run 1 R= 71.8

0.5 0.5

0.5 0.5 2.5 0.0

0.001

R = 78.4 .zz = 55

H LCV FCV LCA

s-2

H

0.5

0.006

Run 3 I? = 66.3

LCV

0.0

FCV LCA

1.5 0.0

s-2

Run 2

I, = 51 s-3

Run 1 IT=49 22 = 47 ___~ s-3

Run 2 IT = 50.0 ,Gz = 50

H LCV FCV LCA ~~.~~_ .~~ H LCZ’ FCC’ LCA

0.015

aa,

0.0 0.0 0.0 0.0

OK

0.969

2.0 0.0

0.0 2.0 I.0 0.0 ~_~~._~~

-

IN TRAFFIC (1200 DATA

POINTS)

Multiple -(B1 +A)

H LCV FCV LCA

2% = 48

LO

residual as a function

3. COEFFICIENTS FOR DERIVED REGRESSION MODELS

Optimum 7

I

8 DaAY

IN SECONX

on acceleration of time delay (7).

Condition

6

ACCELERATIDN

FIG. 5. Effect of replication

TABLE

4

Residual 421.6

664.7

27.4 0.870

R

0.0097 0.2274

0.769

- 0.2423 0448 I

18.2 1.016

Coefficients

271.2

0.0049 0.1287 -0.1295 0.6630

0,645

0.0099

0.824

0.2223 - 0.2265 -0.6312

16.1 0.016

0.843

435.5

0.0045 - 0.0973 0~0811 0.1093

0.616

369.0

-0.0117 - 0.0838 - 0.0875 -04418

0.828

12.9 ~_~ -.-. 0~004

1.033

11.5

--

369

Empirically derived car-following models

In the open-road car-following conditions, S-1, S-2, S-3 have different optimal T’S and different degrees of fit (R = 0.920 for S-l and 0.876 for S-3) which demonstrates significant subject effects. This may be due to the mean headway maintained. With longer average headway, data fit improves. Shorter T’S exist for smaller average headway (S-3 replication vs. S-l) as we might expect from a psychophysical basis since the subject can detect dynamic changes earlier when H is short. Equation coefficients change substantially with subjects, lead-car effects and replications. This casts some doubt on the generality of such car-following models. TABLE 4. COEFFICIENTSFOR DERIVEDREGRESSION MODELSIN OPEN-ROADDRIVING Optimum 7

Condition

--@1+/Q

S-l Hanken original R = 134.1 zz = s-2

H LC V FCV LCA

1.0 2.0 2.5 0.0

0.029

s-2 Hanken H = 96.6 zz = s-3

H LCV FCV LCA

1.0 0.0 2.0 1.6 ~___

0026

s-3 Hanken run 1 R = 47.6 x’ = 34

H LCV FCV LCA

1.0 0.5 0.0 0.0

0.015

s-3 Hanken replicate Run 2. H = 30.3 -i-, = 20

H LCV FCV LCA

1.0 0.0 0.0 0.0

0.011

oz.

Tb!

1.769

Coefficients

507.6

0.0075 0.2387 - 0.2680 0.6120

0920

597.1

0.0036 0.1984 - 0.2242 0.2257

0.909

736.9

0.0159 - 0.2724 0.2579 0.2094

O-895

751.4

0.0114 0.2312 0.2203 0.3597

45 1.728

40 1.834

17.4 1.714

12.6

Multiple R

Residual

0.876

It appears that the greater demand on the driver in the traffic lead-car case produces poorer fit to the data and substantial differences in the coefficients for the predictor variables. In traffic acceleration noise (Q) is smaller due to the more constrained car-following conditions found there. Lead-car velocity T’S appear to decrease in traffic replications suggesting anticipation of the behavior of cars further downstream. The safety index [ - (fl, + /3,), the coefficients on the two velocity predictions] taken from the work of Komentani and Sasaki (1959) decreases in traffic. As the safety index approaches zero, collisions are more likely since the driver of the following car maintains a given headway independent of stream velocity under an equilibrium condition. In traffic with corresponding short time headways, this tends to be the case. Tables 3 and 4 also show significant differences in coefficients for different subjects. Note for the open-road case where the lead-car program is standardized, sign changes occur on the coefficients for S-3 compared to S-l and S-2. Replication effects are less significant (see S-3 open-road) and probably reflect H and & effects. 4.3. Simulation studies of derived regression models To test derived models under different lead-car patterns and to test model stability, the equations were programmed on an IBM 7094 computer for graphical output of headway, acceleration of both cars and velocity of both cars. Figures 6-9 show the output of

370

THOMASH. ROCKWELL, RONALD L. ERNST and ALBERTHANKEN

L

5

I

!

1

! I25

35 55 95

I 155

TIME WNKEN CoEFFS.-

3-2

TlMf

H =.0058

I I35

2-5

IN SECONDS

DELAY = 2.0SEC LCV=.49R

i;ANKfN

FC” =.5+8

LEAD

CAR

[So-ORIGd

FIG. 6. Output of model .S,-ORIG in time with open-road

car-following

program.

I 45 r

TIME HBNKEN COEFFS-

3-2

IN SECONDS

TIME DELAY =2 0 SEC

H=.0058

LC”=.498

SNIDER

FC”=-,546

LEAD

CAR

[S,,-ORl6j

FIG. 7. Output of model S,-ORTG in time with Snider lead-car program.

Empirically

derived car-following

models

I /HEADWAY

r

70-

NSC

5

I

L

35

HANKEN3-2

I I I I 65 95 TIME IN SECONDS

TIME DELAY=2.0 SEC HANKEN LEAD CAR

COEFFS.- H’I.0

LCV= 2.M)

FCV = 2.5 k.with

FIG. 8. Output of modeliwith

I 75

1

125

S, TAUS]

S, taus.

1

HEADWAY

65 F

TIME IN SECONDS TRAFFlC S-2 RUN 2 TIME DELAY= 2 0 SEC HANKEN LEAD CAR COEFFS.- Hs.0049 LC”;.l267 FCV: -.12% LCA z.5530 FIG.

9. Output of model with optimal time lags.

371

THOMAS H. ROCKWELL, RONALD L. ERNST

372

and

ALBERT HANKEN

these simulations. The stochastic element of the model is not included in these simulations. The lead-car program from the open-road case was used in addition to a second hypothetical one of lesser accelerations called the Snider lead-car program (Rockwell and Snider, 1965) which is shown in Fig. 7 and used to test model generality against other lead-car inputs. The simulation for the original model called S,,-ORIG (given in Table 2) is depicted in Fig. 6 which shows the acceleration of the two cars over time and their velocities over the same 330-set condition. The bottom headway plot shows what the National Safety Council (NSC) code (20 ft/each 10 m.p.h.) for car following would dictate, and the top curve shows the predicted headway with target headway the initial condition. Note the following car can be said to barely “car follow”, as such, since it has difficulty in maintaining its target headway of 134 ft. In following the Snider lead-car program, a more stable lead-car pattern as described in the acceleration plot of Fig. 7, we note this model (S,-ORIG) is considerably better in velocity and headway performance, probably because the lead-car accelerations are much less severe. Starting with a 280-ft headway, the model finally established a headway about 200 ft higher than the NSC rule. It might be noted from previous research on drivers in actual car-following that it has been found they cannot achieve the NSC rule in traffic and prefer to exceed it in open-road driving. When the original model of Table 2 (2-set lead for H, FCV and LCV) is given the optimal time lags of S-l original, namely, 1 set for H, 2 set for LCV and 2.5 set for FCV, it fails to achieve stable car-following as noted in Fig. 9. This suggests that the T’S, while not crucial for fit, are critical for stability. When LCA at a 2-set lead is added to the original case (Fig. 6), it was found that performance again deteriorates and target headway cannot be maintained. Finally, a traffic model S-2, run 2 was tested under the open-road lead-car program. Results are shown in Fig. 9. Again, we note that car-following performance rapidly deteriorates. It was found that the Snider lead-car program can be followed by a wide variety of models because it makes few severe changes in velocity. A subsequent attempt to relate driver control movements to system dynamics involved the use of three lead-car patterns, two target headways and two drivers (Keister, 1966). In this case the predicted variable was gas pedal deflection. A stepwise regression analysis was employed using H, RV, RA, LCV and FCV as independent variables. Using the Hanken piecewise method, cutting the data space reduced residuals 20 per cent suggesting Autocorrelation analysis showed high that significant non-linearities were present. correlation for gas pedal deflection at 60-set lag (1 c/min) which was precisely the squarewave velocity patterns of the lead-car in this experiment. This result was more pronounced at short headways rather than long. Is this nonlinearity a contradiction to the earlier linear fit for the driver-vehicle system? It is hypothesized that the nonlinearities of the vehicle may be linearized by gas pedal fluctuations yielding a dynamic system which can be described by linear functions. This conclusion has been suggested in previous research where a describing function approach has been taken in the analysis of velocity control (Algea and Ernst, 1965). 5. In

CONCLUSIONS

AND

IMPLICATIONS

summary, the major findings were: (a) the driver-vehicle system was surprisingly linear with the lead-car maneuvers produced or with those encountered as a result of traffic; (b) the degree of fit of any empirical car-following data is affected by (i) lead-car velocity patterns (open road vs. traffic), (ii) stream velocity and mean headway, (iii) subjects, (iv) subject replications (in traffic) and (v) the time lags on each of the independent variables (a,, ~?r, R,, H in that order); (c) reproducibility of the performance of a subject driver for his road driving is much better than for city traffic driving; this is understandable as the

Empirically

derived car-following

models

373

interaction between driver and traffic on the open road is limited so that the former is more or less free to drive according to his own habits, and (d) the optimum time lags, i.e. the time lags which give minimum residual error, are not very pronounced: similar studies indicate that certain sets of time lags are ‘forbidden’ as they give rise to instabilities; additional work is clearly needed to align theoretical and practical results. Optimal time lags for data fit can be misinterpreted since many residual changes are slight with time lag shifts. This becomes evident when derived optimal T’S are in disagreement with our knowledge of human sensing and responding. Pseudo T’S can lead to distorted results when the models are exercised in a computer simulation of car following. Here, when dealing with the deterministic component of the model, excessively long or short response lags lead to rapid car-following instability (i.e. H becomes zero or too large). Current approaches to the question of microscopic models now involve intensive data collection on trained subjects to ascertain their ability to detect changes in intervehicular dynamics FCV, FLC-LCV, FCA and H. This psychophysical data is combined with reaction time data to relate control action (usually gas pedal) to those motion parameters which are suprathreshold. It is not surprising that simple, consistent, general car-following models of the drivervehicle system derived from empirical results did not emerge from this study. This leads to the conclusion supported by others perhaps on more intuitive bases that any car-following model will require severa submodes of operation if it is to describe human performance under the varying conditions of car following experienced on the highway. Acknowledgements-This report includes data collected on several driving analysis projects separately supported by the Ohio Department of Highways, U.S. Bureau of Public Roads, the Accident Prevention Grants Division of the National Institutes of Health, and by the College of Engineering of The Ohio State University. The authors are indebted to R. S. Keister, J. Arnold and P. N. Myers, who developed computer programs for much of the analysis. REFERENCES ALGEA C. W. and ERNST R. L. (1965). Explorations in the transfer function analysis of driving. Report EES 202B-3. The Ohio State University, Columbus, Ohio. HANKEN A. F. (1965). An approach to system analysis and a proposed piecewise linear regression technique applied to car-following. Unpublished Ph.D. Dissertation, The Ohio State University, Columbus, Ohio. HELLY W. (1959). Dynamics of single-lane vehicular traffic flow. Proceedings of the Symposium on Trafic Flow. Elsevier, Amsterdam. HERMANR. (editor) (1961). Theory of Traffic Flow. Elsevier, Amsterdam. HERMANR. and Porrs R. B. (1961). Single lane traffic theory and experiment. In Theory of Traffic Flow (edited by HERMANR.). Elsevier, Amsterdam. KEISTER R. S. (1966). An investigation of the accelerator control activity in a car-following situation. Unpublished M.S. Thesis, The Ohio State University, Columbus, Ohio. KOMENTANIE. and SASAKIT. (1959). A safety index for traffic with linear spacing. Ops Res. 7,704720. ROCKWELLT. H. and SNIDER J. N. (1965). An investigation of variability in driving performance on the highway. Report RF 1450 AC 000-28-02, The Ohio State University, Columbus, Ohio.