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AVEC ‘04
Models of Driver Speed Choice in Curves
Andrew MC Odhams and David J Cole Cambridge University Engineering Department
Trumpington Street, Cambridge, CB2 1PZ, UK
Phone +44 (0)1223 332600 Fax +44 (0)1223 332662
Email: [email protected]
Present understanding of driver speed choice in curves is reviewed, and published mathematical models are compared and evaluated. Preliminary data gathered from a fixed-base driving simulator at CUED are used to evaluate the models. It is found that road width and curve radius are significant influences on speed choice, although several of the models account for only one or other of these factors. It is concluded that a more sophisticated model is required to account for other important factors such as vehicle dynamics, driver skill, and driver preference for speed, risk and discomfort.
Topics / A14 1 INTRODUCTION
This paper describes part of a research programme aimed at understanding how a driver learns to steer a vehicle. An important part of the learning process is believed to be speed choice, since speed strongly influences the dynamic response of the vehicle.
Road vehicles are usually operated in their region of linear dynamic behaviour, making the vehicle easy to control by the driver. However, negotiating a curve at a speed that is too high can result in highly nonlinear behaviour. Drivers are generally not experienced in controlling vehicles in the nonlinear regime, and the result can be deviation from the intended course and direction, departure from the roadway, or collision.
An understanding and mathematical model of driver speed choice could enable the design of vehicles that are less likely to be involved in accidents, or the design of racing cars that are easier for the driver to achieve fast lap times.
The published literature on speed choice is reviewed and it is found that, whilst there is some understanding, there is insufficient knowledge for the development of detailed driver-vehicle models. In the field of experimental psychology there have been several empirical studies of driver speed choice, using on-road measurements or driving simulators. In the field of driver-vehicle dynamics and simulation, speed choice has received relatively little attention. The linkage between speed choice and steering control appears not to have been explored extensively.
This paper reviews existing published models of speed choice, and compares the models by fitting them to preliminary experimental data gathered in a driving simulator experiment. Shortcomings of the models are identified and the possibilities for improvement are
discussed. In the next section of the paper existing models are reviewed. A driving simulator experiment is described in section 3 and the results are discussed in section 4. Conclusions are given at the end of the paper. 2 REVIEW OF PREVIOUS WORK 2.1 Lateral Acceleration
Attempts to characterise speed choice first
concentrated on lateral acceleration as the major cue, with speed a simple function of curvature [1-3].
Ritchie [3] recorded lateral acceleration and speed for fifty subjects during normal road driving. He found that lateral acceleration was constant below 9 m/s (32 km/h), so that speed V was related to curve radius R by
R
VLatacc2
= (1)
At speeds above 9m/s, drivers were found to choose a lower speed than that which would yield constant lateral acceleration, such that there was a linear relationship between lateral acceleration and speed.
Herrin [2] recorded similar results to Ritchie's in on-road tests, and fitted an empirical model of the form: ( )'' Ve β=Γ (2)
'Γ is the fraction of the driver’s maximum tolerable lateral acceleration unused, and 'V is the speed reduction from the driver’s maximum comfortable straightline speed. β is an expedience parameter, reflecting the driver’s trade-off between lateral acceleration and speed. High β implies that the driver travels at his or her maximum comfortable speed unless the maximum tolerable lateral acceleration is reached. Lower β implies a more gradual trade-off.
Drivers were found to display higher β when more familiar with the road. Drivers told they were ‘late for a
AVEC ‘04
meeting’ displayed higher maximum comfortable speed, and higher lateral acceleration tolerance.
Mclean [1] fitted several models to speed vs road curvature data from previous studies. The most successful fits were: a linear variation of speed with curvature (from Taragin [4]) LCV ∆+−= .45.2.363.04.75 (3) where: =V Mean speed (km/h) =C Curvature (deg/100 m) =∆L Pavement width – 7.3 (m); and an exponential model proposed by Emmerson [5]: ( )ReV .017.01.74 −−= . (4)
Both models predict zero speed for small radii, and asymptotically approach the ‘free speed’ on straights for large radii. Though mechanisms are suggested, these models are essentially empirical fits to observations.
2.2 Available Tolerance
Research on the speed of self-paced tracking tasks
as a function of available tolerance (allowable tracking error) has considered the influence of path width [6-8].
As an analogy to the driving task Drury [8] timed subjects tracing lines and circles by hand, with various available tolerances. A linear relationship between speed and tolerance was found.
Low speed manoeuvres with tractors during negotiation of competition ‘gates’ was studied by Bottoms [7]. A linear relationship of mean speed ( V ) vs tolerance gave good agreement, but a logarithmic model performed better :
TdbaV
.1+= (5)
where: difficulty Tracking.2ln =
−
=v
v
WWWTd
widthRoad widthVehicle
==
WWv
For driving tests on a circular track, De Fazio et al [6] found speed to be proportional to road width. However, as with Drury [8] and Bottoms [7], the effect of path curvature is not included in the models. 2.3 Anticipated Error
Several studies have attempted to characterise
speed choice as a function of the driver’s anticipated path following error [9, 10].
In simulator tests by Van Winsum and Gothelp [9] the ‘time to lane crossing’ (TLC) was found to be kept above a constant minimum level by drivers. Also, steering error was found to be proportional to steer angle (hence bend radius). It was claimed that a driver model with these two features could explain speed vs. radius behaviour of drivers, and in turn the reduction of lateral acceleration with speed.
However, Reymond [10] found that TLC was not an invariant parameter in driver speed choice. Reymond derived an expression for TLC in terms of the driver’s anticipated steering (and hence radius) error:
( )V
RRTLC ∂−=
.α (6)
where: ( )( )
∂−∂−
−= −
RRRdRd
..2.21cos 1α
( )1.01
.2
hTrack widt
=+
=∂
=
kk
RkR
d
The vehicle is assumed to be travelling at speed V on the centreline of a curve, radius R , with radius error
R∂ . α is the angle traversed before the road edge is reached if this error is not corrected. However, Reymond found that assuming constant TLC gave incorrect speed choice behaviour, with little reduction of lateral acceleration at high speed.
Reymond [10] instead proposed a lateral acceleration safety margin to explain reduction of lateral acceleration with speed: ∆Γ−Γ≤Γ max (7) 2
max .VC∆=∆Γ (8) The driver chooses a lateral acceleration Γ that is
less than his or her prediction of the maximum available maxΓ by a lateral acceleration safety margin ∆Γ . The
margin is proportional to the driver’s uncertainty of the future curvature of the track maxC∆ , which is independent of speed. In on-road tests it was found that
maxC∆ decreased as drivers tried to drive more urgently. maxC∆ was found to be lower when the same road was driven on a moving base simulator, and lower still when a fixed base simulator was used. 3 EXPERIMENT
Results from driving simulator experiments at Cambridge University Engineering Department (CUED) were fitted to some of the above models to evaluate and compare their performance.
Tests were carried out on a fixed-base driving simulator running on two computers with Matlabtm XPC software, linked by ethernet. The ‘target’ computer runs a vehicle model, and receives inputs from pedals and steering wheel. It outputs steering torque, a velocity dependent tone to a speaker, and a buzzer to indicate when the road edge has been reached. The ‘host’ computer receives position information from the target computer, and renders the graphics to a wide-angle projection screen.
The lateral/yaw vehicle model was a linear bicycle model, with no tyre saturations. Parameters were similar to those of a saloon car, with slight understeer behaviour. A model of longitudinal dynamics enabled the vehicle speed to be varied, but not coupled to the lateral/yaw model. Road curves consisted of five parts: inward and outward straights, inward and outward transition sections, and a constant curvature section. Transitions had a linear increase in curvature, giving the curvature profile shown in Figure 1. Curves were 180°, with 150° constant curvature sections. The visual
AVEC ‘04
display showed the lane width and a buzzer sounded when this was exceeded.
Each experiment consisted of the test subject guiding the vehicle around twenty-five curves, comprising all permutations of five lane widths (2.4m to 4.2m), and five radii (66m to 262m). These curves were presented in random order to avoid the driver anticipating the properties of the next curve. Drivers were instructed to ‘drive as fast as possible without leaving the track’. The figures and results presented in the following sections are those of one experiment on one test subject. A training period, also consisting of twenty-five similar curves, preceded the experiment, during which the driver became familiar with the task. Data from the training period is examined in section 4.4.
4 DATA ANALYSIS
A typical measured speed profile is shown in Figure 1. On first entering the curve, an ‘open loop’ speed is achieved, and is the driver’s choice of speed based on preview of the approaching curve. The driver then modifies this speed once in the constant curvature section to a (sometimes lower) ‘closed loop’ speed. To be consistent with previous studies, the speed and lateral acceleration data points recorded in the constant curvature section ( ccV and ccΓ ) are used in sections 4.1 and 4.4. In sections 4.2 and 4.3 the mean value of the data points in the constant curvature section is used. 4.1 Effect of Lateral Acceleration on Speed Choice
Reymond’s lateral acceleration margin model
(eqns 6 and 7) was fitted to the measured data. Results for the narrowest (2.4m) and widest (4.2m) curves are shown in Figure 2 and Figure 3. The data points show measured ccΓ vs. ccV , and for comparison, radius contours for 66m and 262m are also shown. An envelope of maximum lateral acceleration at each speed is shown as a solid line, and a least squares fit of the Reymond model is shown as a dashed line. The fitted values of the model parameters and the correlation coefficient are shown in Table 1 for each curve width.
The correlation coefficients are low except for the widest road. However, the widest three roads do show a clear drop-off in lateral acceleration with speed. maxΓ is much larger than the 10ms-2 recorded by Reymond, presumably because there is no tyre saturation, and
maxΓ is seen to increase appreciably for the wider roads. maxC∆ is of similar magnitude to that of Reymond, demonstrating that lateral acceleration reduced with speed even though there was no tyre saturation.
Width /m 2.4 2.8 3.2 3.6 4.2
maxΓ /ms-2 14.6 9.61 23.9 24.2 51.5 maxC∆ /10-3 m-1 2.88 -1.83 3.09 2.48 11.5
r2 0.09 0.05 0.13 0.08 0.65
Table 1: Reymond lat. accn. model
4.2 Effect of Curve Radius on Speed Choice
The TLC criterion [9] suggests a direct relationship between speed and curve radius [10]. Reymond’s lateral acceleration margin model can also be rearranged to give such a relationship (eqn 9). In this section, these two models are fitted to the ( )ccVmean and curve radius data. An envelope function such as that shown Figures 2 and 3 is not used.
4.2.1 Reymond lat. accn. margin model
Recorded mean speed and radius for each of the twenty-five curves is shown in Figure 4, along with a least square fit of equation (9) for each curve width.
Γ+
Γ∆
=R
CV
111
maxmax
max2
(9)
The correlation coefficients (Table 2) are much better than those in Table 1, largely because the mean speed in the curve has been used, rather than the maximum speed envelope. Also the effect of radius on speed is clearer on these axes. maxC∆ and maxΓ are similar in magnitude to those of section 4.1, and again both increase with increasing curve width. It is clear that curve width affects speed significantly, although curve width is not included in this lateral acceleration based model.
Width / m 2.4 2.8 3.2 3.6 4.2 maxC∆ /10-3.m-1 -2.57 2.50 6.33 3.45 7.24
maxΓ /ms-2 4.47 16.0 28.9 25.1 38.6 r2 0.81 0.64 0.49 0.91 0.95
Table 2: Reymond speed vs radius model
4.2.2 Time to Lane Crossing model
Reymond’s model [10] of Van Winsum’s constant TLC hypothesis [9] (eqn 6) leads to the following:
( ) [ ]RkTLC
aV ..11
α+
+= (10)
The velocity offset value a has been included to allow a regression coefficient to be calculated, and should be zero if the model is correct. However, its value ranges from 0 to 25 ms-1, nearly half of the total speed change (see Table 3). The theoretical TLC, calculated from the fitted model, is also listed. Values are of the correct magnitude and similar to those found by Van Winsum [9], but increase with track width. So although curve fit is good (see Figure 5), the coefficients do not fit the assumptions in the TLC hypothesis [9].
Width /m 2.4 2.8 3.2 3.6 4.2 a /ms-1 -.79 8.24 17.7 14.1 25.2 TLC /s 1.79 1.92 2.36 2.11 3.04 r2 0.85 0.79 0.74 0.93 0.93
Table 3: Reymond TLC model
AVEC ‘04
4.3 Effect of Curve Width on Speed Choice 4.3.1 De Fazio et al linear model [6]
De Fazio et al’s linear model of speed vs. width (eqn. 11) is fitted to each radius separately in Figure 6. WbaV .+= (11) Fitted coefficients a and b are shown in Table 4. Fit is not as good as that observed in [6], with r2 from 0.66 to 0.89. Close examination suggests a curve would fit the points better, such that less increase in speed is seen at higher widths. Radius /m 66 94 129 190 262 a /ms-1 -3.12 1.58 17.0 17.7 24.9 b /s-1 11.4 10.7 8.54 9.91 8.75 r2 0.66 0.89 0.69 0.65 0.87
Table 4: De Fazio et al linear model
4.3.2 Tracking difficulty model [7]
Bottoms’ tracking difficulty model (eqns 5), is shown in Figure 7. Unlike De Fazio et al’s linear model, the curves show the appropriate reduction of gradient at high speed, and a corresponding improved fit (Table 4). Td was much lower than that recorded by Bottoms since the road is much wider relative to the vehicle. However, the model’s intended mechanism based on driver limited cognitive capacity still applies since the instructions are to go as fast as the driver is capable. Radius /m 66 94 129 190 262 a /10-2m-1s 0.75 1.90 1.47 1.25 1.41 b /10-2m-1s 1.90 0.71 0.62 0.62 0.38 r2 0.80 0.72 0.86 0.80 0.97
Table 5: Tracking difficulty model
4.3.3 Time to Lane Crossing model
The constant TLC model from section 4.2.2 was fitted to the width data, as shown in Figure 8. The resulting curves are nearly linear, having fit only slightly better than De Fazio et al’s. Again TLC is not constant, but increases with increasing radius (Table 6).
Radius /m
66 94 129 190 262
a /ms-1 -38.5 -30.1 -10.8 -15.0 -3.54 TLC /s 0.65 0.84 1.17 1.21 1.62 r2 0.70 0.87 0.72 0.68 0.90
Table 6: Reymond TLC model based on width 4.4 Effect of Combined Width and Radius on
Speed Choice
A simple model taking into account both curve width and radius is shown in equation (12). Taragin’s model (eqn. 3) is similar but with no interaction term.
RWdWc
RbaVcc ..1. +++= (12)
Least square regression of this surface was performed on the data from the training period (track 1) and from the main experiment (track 2), see Table 7, Figure 9, and Figure 10. Fit is good (r2~0.8) for both surfaces. The surfaces are qualitatively similar, showing increasing speed with increasing width, and decreasing curvature, but the fit to the training period shows significant interaction (surface warping). This suggests that speed is less dependent on width in the high curvature bends. This is less obvious in the fit to the data from track 2, where little interaction seems evident. This may be due to the driver learning that the vehicle has no tyre saturation and thus no lateral acceleration limit.
Track a /10ms-1
b /103m2s-1
c /s-1
d
/102 ms-1 r2
1 -1.12 1.40 19.7 -9.42 0.79 2 3.23 -2.75 8.45 2.55 0.81
Table 7: Least square surface fit
5 CONCLUSIONS 1. In a driving simulator experiment, in which the test
subject was instructed to drive as fast as possible around curves without leaving the lane, the achieved lateral acceleration was observed to reduce at high speed. This was despite the use of a fixed-base simulator and the vehicle having no tyre saturations.
2. Lane width and curve radius were both found to be significant determinants of speed choice, but several existing models of speed choice account for only one or other of these factors.
3. A model based on constant ‘time to lane crossing’ accounts for the effect of curve radius and width, but TLC was not found to be independent of either of these factors. This finding supported that of Reymond [10].
4. A linear surface model with interaction gives good fit ( 8.02 ≈r ) to the effects of radius and width, but is purely empirical and gives little insight into the mechanism of speed choice.
5. An improved driver speed choice model, suitable for understanding driver-vehicle interaction behaviour, should account for: i) the distinction between open-loop and closed-
loop speed choice; ii) the vehicle dynamic behaviour and the driver’s
familiarity with it; iii) the driver’s preferences for minimising travel
time, risk and discomfort. 6. A research programme to address these issues is
underway at Cambridge University Engineering Department. The programme comprises simulator experiments and model development.
AVEC ‘04
REFERENCES 1. McLean, J.R., Driver Behaviour on Curves - A
Review. ARRB Proceedings, 1974. 7(5): p. 129-147.
2. Herrin, G.D. and J.B. Neuhardt, An Empirical Model for Automobile Driver Horizontal Curve Negotiation. Human Factors, 1974. 16(2): p. 129-133.
3. Ritchie, M.L., W.K. McCoy, and W.L. Welde, A Study of the Relation Between Forward Velocity and Lateral Acceleration in Curves During Normal Driving. Human Factors, 1968. 10(3): p. 255-258.
4. Taragin, A., Driver performance on horizontal curves. Proc. Highway Research Board., 1954. 33.
5. Emmerson, J., Speeds of cars on sharp horizontal curves. Traffic Engineering and Control, 1969(July): p. 135-137.
6. DeFazio, K., D. Wittman, and C.G. Drury, Effective Vehicle Width in Self-Paced Tracking. Applied Ergonomics, 1992. 23(6): p. 382-386.
7. Bottoms, D.J., The interaction of driving speed, steering difficulty and lateral tolerance with particular reference to agriculture. Ergonomics, 1983. 26(2): p. 123- 139.
8. Drury, C.G., Movements with Lateral Constraint. Ergonomics, 1971. 14(2): p. 293-305.
9. Winsum, W.V. and H. Godthelp, Speed Choice and Steering Behaviour in Curve Driving. Human Factors, 1996. 38(3): p. 434-441.
10. Reymond, G., et al., Role of Lateral Acceleration in Curve Driving: Driver Model and Experiments on a Real Vehicle and a Driving Simulator. Human Factors, 2001. 43(3): p. 483-495.
FIGURES
2600 2800 3000 3200 3400 3600 3800 400020
40
60
Spe
ed /m
s-1
Dist /m2600 2800 3000 3200 3400 3600 3800 4000
0
0.005
0.01
Speed
Curvature
'Open loop' Speed
'Closed loop' Speed
Cur
vatu
re /m
-1
Figure 1: Speed and curvature profile for a typical curve
0 10 20 30 40 50 600
5
10
15
20
25
30
35
Speed /ms-1
Lata
cc /m
s-2
latacc66m rad262m radEnvelopeLSq fit
262 m radius
66m radius
Figure 2: Reymond model, Latacc vs speed, 2.4m width
0 10 20 30 40 50 600
5
10
15
20
25
30
35
Speed /ms-1
Lata
cc /m
s-2
latacc66m rad262m radEnvelopeLSq fit
66m radius
262 m radius
Figure 3: Reymond model, Latacc vs speed, 4.2m width
0 50 100 150 200 250 3000
10
20
30
40
50
60
Radius /m
Spe
ed /m
s-1
4.23.63.22.82.4
4.2 3.6
3.2
2.8
2.4
Figure 4: Reymond model, speed vs rad, widths 2.4 to 4.2m.
AVEC ‘04
0 50 100 150 200 250 3000
10
20
30
40
50
60
Radius /m
Spe
ed /m
s-14.23.63.22.82.4
4.2 3.6
3.2
2.8
2.4
Figure 5: TLC model, speed vs rad, widths 2.4 to 4.2m
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.40
10
20
30
40
50
60
Width /m
Spe
ed /m
s-1
2621901299466
262
190
129
94 66
Figure 6: Linear model speed vs width, rads 66 to 262m
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.40
10
20
30
40
50
60
Width /m
Spe
ed /m
s-1
2621901299466
262 190
129
94
66
Figure 7: Bottoms’ model, speed vs width, rads 66 to 262m
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.40
10
20
30
40
50
60
Width /m
Spe
ed /m
s-1
2621901299466
262
190
129
94 66
Figure 8: TLC model, speed vs width, rads 66 to 262m
00.005
0.010.015
0.02
2
3
4
510
20
30
40
50
60
70
Curvature /m-1Width /m
Spe
ed /m
s-1
Figure 9: Speed vs curvature vs width, track 1
00.005
0.010.015
0.02
2
3
4
510
20
30
40
50
60
70
Curvature /m-1Width /m
Spe
ed /m
s-1
Figure 10: Speed vs curvature vs width, track 2
