Delft University of Technology

A perceptually inspired Driver Model for Speed Control in curves

Gruppelaar, Virgilio; van Paassen, Rene; Mulder, Max; Abbink, David

DOI10.1109/SMC.2018.00220Publication date2018Document VersionAccepted author manuscriptPublished inProceedings 2018 IEEE International Conference on Systems, Man, and Cybernetics

Citation (APA)Gruppelaar, V., van Paassen, R., Mulder, M., & Abbink, D. (2018). A perceptually inspired Driver Model forSpeed Control in curves. In Proceedings 2018 IEEE International Conference on Systems, Man, andCybernetics (pp. 1253–1258). Piscataway, NJ, USA: IEEE. https://doi.org/10.1109/SMC.2018.00220

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A perceptually inspired Driver Model for SpeedControl in curves

Virgılio Gruppelaar, M. M. (Rene) van Paassen, Max MulderFaculty of Aerospace Engineering, TU Delft

Delft, the Netherlands 2629 HSEmail: {M.M.vanPaassen; M.Mulder}@tudelft.nl

David A. AbbinkFaculty of Mechanical, Materials and Maritime Engineering, TU Delft

Delft, the Netherlands 2629 HSEmail: D.A.Abbink@tudelft.nl

Abstract—Understanding speed control in driving is importantfor analysis of road geometry and for the development of driversupport assistance devices. Current models for speed selectionare primarily based on the relation between road geometry andobserved speeds. This study proposes a more detailed modelthat relates individual speed control to accelerator and brakepedal, based on perception of the visual scene as captured by theExtended Tangent Point (ETP). We investigated the potential ofthe the Time to ETP (TETP) as input for accelerator and brakepedal control. Based on observations from driving studies, wepropose a model, and tuning rules to adjust the model parametersto observed behavior. A simulator experiment showed that, afterindividualization of the thresholds using a binary classificationmethod, the model is capable of accurately capturing individualspeed adaptation of 15 drivers on single lane roads with multiplecurves.

I. INTRODUCTION

When driving on a curved road, people need to adjust theirspeed to the road shape. Field studies show a correlationbetween curve radius and speed [1]–[3]. Other geometricalfactors that can influence speed choice are the road width [4],the lengths of the straight road sections between curves, thecurve deflection angle, the presence of other curves in closeproximity, and the curve superelevation [5]. In addition to this,the lateral position of the vehicle on the road can alter theeffective turn radius, impacting drivers’ speed choice [6].

In the context of road design and road safety, many studiesmodel the speed at which a driver can safely negotiate a turnbased on the geometrical characteristics of that turn. While thisis a useful limit for assistance systems to take into account asan upper boundary, it does not provide information about adriver’s desired speed and is consequently not directly usablein Advanced Driver Assistance Systems (ADAS) applications[7]. We wish to model the manner in which drivers themselveschoose their speed, depending on their preference of drivingstyle, risk, or safety margins.

The majority of models for speed adaptation when ap-proaching a curve propose a relationship between road cur-vature and speed, fitting parameters to experimental data [1],[2], [7], [8]. In order to account for the effects of the geometryof the turn, adjacent road segments, and lateral position, model

complexity will increase [5], [6], [9]. An issue with thesemodels is that human drivers have been shown to be ratherpoor judges of road curvature [10], which suggests that othercues must play a role.

During driving, humans tend to control their risk by em-ploying certain safety margins [11]. In situations where speedneeds to be adjusted, these are usually time margins such asthe Time to Line Crossing (TLC) in lane keeping [12] or theTime Headway and Time To Contact in car following [13].Contrasting with their poor performance in judging distances,velocities, and curvatures [10], humans can very accuratelyjudge visual angles and time margins to target or contact points[14]. This leads us to believe that speed adjustment in turnsis very likely dependent on a time margin to a salient point.

Land and Lee [15] introduced the idea that the Tangent Point(TP) on the inside of a curve is key to curve negotiation, anda substantial amount of research has shown that drivers focustheir gaze towards the area containing the apex of the curve,employing what is called TP orientation [16]. However, drivereye movement studies have shown that drivers do not fixateonly on this point, but also scan areas farther up the road in acontrolled pattern [17]. A particular area of interest is the farroad triangle [18], comprised of the TP, the Occlusion Point(OP), which is the farthest point of the road that is not blockedby obstacles in the field of vision, and the point where thedriver’s line of vision through the TP intersects the oppositelane edge, the Extended Tangent Point (ETP).

In this paper we present a novel model for driver speedchoice based on ETP perception. The model additionallyrelates this speed choice to driver’s longitudinal control inputs(depression of brake pedal and gas pedal), based on thresholdvalues. Section II describes the approach taken in designing themodel, the developed speed control algorithm, and predictionsfrom simulations. To validate the model an experiment was setup and performed, as shown in Section III. The results fromthis experiment are shown and discussed in Sections IV and V.

II. MODEL DESIGN

Our model for speed choice is based on the Time toExtended Tangent Point (TETP), which is defined as the time

ETPTP

Fig. 1. Location of Tangent Point (TP) and Extended Tangent Point (ETP)for two curves with different radii and deflection angles.

it would take for the vehicle to reach the ETP in a straightline, if speed would not be changed. This time margin can beinterpreted as a measure of how much the road ‘opens up’ aftera turn, see Figure 1. As can be imagined, the TETP will alsodepend on the lateral position of the vehicle with respect to theroad, making it a parameter sensitive to the driven trajectory,rather than merely depending on the road geometry.

To investigate the viability of this model approach, wecalculated TETP for data obtained from an earlier experimentperformed at the Human-Machine Interaction (HMI) Labora-tory of TU Delft [19], which studied driver’s speed choice forcurved roads. Our preliminary analysis illustrated that speeddecrease before a curve occurred first in a ’deceleration zone’,potentially following by a ’braking zone’, each correlated withtwo different time margins. Drivers first released the accelera-tor when the TETP decreased beyond a certain threshold value(deceleration zone), and would only begin to press the brakepedal when the TETP exceeded a second, lower threshold(the braking zone). Observations of mid-corner and cornerexit behavior also showed a correlation between accelerationbehaviour and TETP. More spefically, close to the end ofthe decelerating or braking zones, the TETP decreased moreslowly, eventually becoming constant, before increasing againas a consequence of the road opening up after the turn. Driversappeared not to use the same time thresholds for acceleratingas for decelerating, but became comfortable once the rate ofchange of the TETP approached zero. Then, drivers beganaccelerating, and only if the TETP started decreasing againdid they resume decelerating or braking.

The analysis also illustrated that, as could be expected,each driver behaves differently. Some drivers never adaptedtheir speed to the given road, while others felt the need tobrake in certain curves. This can be attributed to the idea thatdifferent drivers feel comfortable at different TETP values.By identifying these different thresholds, and combining themwith different gains on gas and brake pedal deflection, itcould be possible to use a TETP-based approach to modeleach driver’s individual preferences in speed choice and theassociated longitudinal control actions.

These findings led us to develop our model of driver speedcontrol through pedal actuation, based on TETP thresholds asthe primary perceptual variables.

A. Model architecture

The speed control model distinguishes five phases of speedcontrol during cornering: acceleration, deceleration, braking,brake release, and re-acceleration. Which of the five phasesof the model is active, is determined by three time thresholdsparameters: a TETP threshold for deceleration (releasing thegas pedal) and a lower one for braking, and a threshold on theTETP rate of change.

The model regulates vehicle speed using these three timethreshold parameters, along with three associated actuationgains to describe the magnitude of the corresponding pedalactuations in each phase. The six parameters are:

• Td [s]: The minimum TETP before a driver releases theaccelerator pedal to enter the deceleration phase.

• Kd [-]: The accelerator pedal release gain, determininghow quickly the driver lets go of the gas pedal during thedeceleration phase.

• Tb [s]: The minimum TETP before a driver presses thebrake pedal to enter the braking phase.

• Kb [-]: The brake pedal depression gain, determininghow strongly the driver presses the brake pedal duringthe braking phase.

• dTa [-]: The rate of change of TETP at which the driverfeels comfortable to begin accelerating out of a curve.It determines when the brake release and re-accelerationphases begin, if the model is already in the braking ordeceleration phases, respectively.

• Ka [-]: The accelerator pedal depression gain. Thisparameter etermines how strongly the driver presses theaccelerator pedal during the acceleration, brake releaseand re-acceleration phases.

A typical example of how the model works is shown inFigure 2, where a simulated vehicle negotiates a 150m radiusleft turn with a target speed of 27ms−1 on the straight sections.In this figure, we can clearly see the differences between thephases of the model.

B. Speed control algorithm

The algorithm the model uses to select one of the fivephases, based on the three thresholds Td, Tb and dTa, isillustrated in Figure 3. The calculations for the different phasesare described below:

1) Acceleration: If the current TETP is above the decelera-tion threshold Td, the model will go into the accelerationphase, where the brake pedal deflection is δb = 0 and theaccelerator pedal deflection, δa, is limited only by themaximum speed on the road segment, as described by:

δa = δa,EB +Ka

(1− V

Vmax

)(1)

2) Deceleration: If the TETP is below the deceleration limitTd, but above the braking limit Tb, the model is in thedeceleration phase. In this case, the brake pedal deflectioncontinues to be zero, while the desired accelerator pedaldeflection is a function of how far the TETP is from both

100 200 300 400 500 600 700-150

-100

-50

0ra

dius

[m

]

100 200 300 400 500 600 7000

5

10

TE

TP

[s]

Deceleration zoneBraking zone

100 200 300 400 500 600 700-20

0

20

/a,/

b [%

]

Deceleration zoneAccelerationRe-accelerationDecelerationBrakingBrake release

100 200 300 400 500 600 700

distance [m]

22

24

26

v [m

s-1]

Fig. 2. Driver model simulation of speed control for a 150m radius leftturn (negative radius in the top panel), showing TETP, pedal deflections,and speed. In the third panel, positive pedal deflections correspond to theaccelerator pedal, while negative deflections represent brake actuation. Thefive phases are shown with different line styles, and the shaded area representsthe deceleration zone: accelerator pedal position values that correspond to azero or negative acceleration.

thresholds, as described by Equation (2). The resultingvalue is bound between the accelerator pedal deflectionrequired to maintain the current speed, δa,EB , and 0.

δa = min

(δa,EB ,Kd

(TETP− TbTd − Tb

))(2)

This phase will continue until either the TETP goes underthe braking threshold Tb and the braking phase starts, orthe (filtered) rate of change of the TETP approaches zero,at which point the re-acceleration phase begins.

3) Braking: If the TETP is below the braking threshold, theaccelerator pedal deflection is set to zero, while the brakepedal deflection is dependent on the ratio between thecurrent TETP and Tb, as described in:

δb = Kb

(1− TETP

Tb

)(3)

4) Brake release: As the vehicle brakes, the TETP willgradually stop decreasing. If the rate of change of TETPbecomes larger than dTa while the braking phase isactive, the brake release phase begins. The brake pedaldeflection is the same as in the braking phase (Equa-tion (3)), with the added limitation that the deflectioncannot increase while this phase is active. While thevehicle is braking, there should be no accelerator pedalactuation (δa= 0). During this phase, once δb becomes0, the model will begin pressing the accelerator pedal.The deflection δa is then the minimum value between thepedal position from the acceleration phase, and a directfunction of the rate of change of the TETP:

Start

TETP < Td Acceleration 1

TETP < TbdTETP

dt < dTa

Re-acceleration 5

dTETPdt < dTa Brake release 4

Deceleration 2

Braking 3

no

yes

no

no

yesyes

no

yes

Fig. 3. Model phase decision flowchart

(4)δa = min

(δa,EB +Ka

(1− V

Vmax

),

Ka

(dTETP

dt− dTa

))5) Re-acceleration: This phase is entered when the TETP

is between Tb and Td, but its rate of change is abovedTa. In this case, the driver feels safe enough to beginaccelerating again. The brake pedal deflection is δb = 0,while the accelerator pedal is the minimum value betweenthe one found for the acceleration phase, and the valuederived from Eq. 5. This calculation is similar to the onein the deceleration phase, but uses the acceleration gainKa instead of Kd.

(5)δa = min

(δa,EB +Ka

(1− V

Vmax

),

Ka

(TETP− TbTd − Tb

))

Fig. 4. The fixed-base driving simulator at the Faculty of AerospaceEngineering of the Delft University of Technology.

III. EXPERIMENT

An experiment was set up at the fixed-base driving sim-ulator in the HMI Laboratory, at the Faculty of AerospaceEngineering of TU Delft, in order to satisfy three goals:

1) Show that the proposed TETP-based speed model cancapture general trends of driver speed adaptation to arange of road geometry (road width, corner radii anddeflection angles).

2) Provide evidence for the phases of speed adaptation, andshow that TETP triggers can explain these phases moreaccurately than TLC (an alternative time-based metric).

3) Provide validation data in order to individualize themodel, and measure its performance in reproducing indi-vidual drivers’ speed control strategies.

A. Apparatus

In the fixed-base driving simulator, Figure 4, an LCDscreen was used to show speedometer information, while threeprojectors displayed the driving scene, generating a field ofview over 180 degrees. Engine sound was played throughspeakers to aid in speed perception. The simulated vehicle wasa heavy sedan equipped with an automatic 4-speed gearbox.

B. Design

The experiment was performed by sixteen participants (3female, 13 male), with mixed experience (age 25 ± 2 years,licensed since 6.1±2.6 years, driving 4600±4200km per year).Data from one participant were incomplete due to motionsickness.

Six different roads were used, with alternating turns andstraight sections, with two road widths (3.6m and 2.4m), andtwo corner radii for the turns (150m and 300m), and twodeflection angles (45 and 90 degrees), an example is givenin Figure 5. In order to improve visual speed perception, theroad was lined with poles spaced at regular intervals, and treeswere placed in the scenery.

Subjects were instructed to drive as they would in a realworld driving situation, adapting their speed and steeringbehavior in order to stay within the lane boundaries. They weretold the roads allowed only for one-way traffic, and that theycould use all of the available road width. They were requested

C1C2

C3

C4C5C6

C7 C8

Fig. 5. Road 3 of the experiment (road width not to scale).

to treat the speed limit of 100kmh−1 in the same manner theywould treat a speed limit on a real road.

IV. EXPERIMENTAL RESULTS AND MODEL FIT

Using the driving experiment data, the relevant parametersfor the speed control model were calculated and plotted forinspection. Figure 6 shows a typical result. The pedal traceshows the distinct phases. As the car approaches the curve,the TETP decreases linearly. The driver releases the gaspedal once a certain TETP threshold, indicated by the dashedvertical line, has been passed, at approximately 50m beforethe corner. After releasing the gas pedal, the TETP continuesto decrease, eventually causing the driver to apply the brakes(solid vertical line). Once the TETP stops decreasing, thedriver feels comfortable enough to begin re-accelerating, andafter the TETP has increased past any thresholds, the drivereventually increases the acceleration.

The figure shows the value of using TETP triggers formodeling speed adaptation: at a TETP of approximately 3sthe driver releases the gas pedal, followed by braking at aTETP of approximately 2.3s. Comparison to the TLC, shownin the same figure, show that TETP is a better candidate forcurve speed modeling applications.

A. Individual driver parameter fit

Two-thirds of the data were used to fit the TETP thresholdsfor decelerating and for braking, Td and Tb respectively.We used the Matthews Correlation Coefficient (MCC) as ameasure for the classifier [20], since this has been shown toperform well even when the sizes of the two classifications arevery different [21], as was the case for our data. A confusionmatrix was set up (as in Table I), and the classifier:

MCC =nTP · nTN − nFP · nFN√

(nTP + nFP )(nTP + nFN )(nTN + nFP )(nTN + nFN )

(6)

was evaluated. Td and Tb were varied to find the maximumclassifier values. The remaining parameters were found by a

3550 3600 3650 3700 3750 3800 3850 3900 3950 40000

50ra

dius

[m

]Deceleration triggerBraking trigger

3550 3600 3650 3700 3750 3800 3850 3900 3950 4000

5

10

TE

TP

[s]

TLCTETP

3550 3600 3650 3700 3750 3800 3850 3900 3950 4000

-20

0

20

/a,/

b [%

]

Deceleration zone

3550 3600 3650 3700 3750 3800 3850 3900 3950 4000

distance [m]

20

25

v [m

s-1]

Fig. 6. Typical example (S15) of the variation of TETP, TLC, speed, and pedaldeflections on a 75m radius right turn. Positive pedal deflections correspondto the accelerator pedal, while negative deflections represent brake actuation.The shaded area represents the deceleration zone, that is, the accelerator pedalposition values that correspond to a zero or negative acceleration.

constrained least squares fit between the combined acceleratorand brake pedal trace of the model and recorded data. dTa wasallowed to vary from −0.5 to 0, Ka, Kd and Kb from 0.01to 10. Since differences in speed behavior between subjectswas expected, and also observed later in the data, the modelfit was performed for each subject individually. The groupaverage and standard deviation of the parameter values aregiven in Table II.

TABLE ICONFUSION MATRIX FOR ACCELERATOR PEDAL RELEASE

TETP ≤ Td TETP > Tdδa ≤ δa,EB TP FNδa > δa,EB FP TN

B. Validation

One third of the data was reserved for validation. Inliterature, speed control models are usually judged on theirperformance in a single turn [2], [4], [7]. In this case, themodel was initialized on a straight segment approximately500m before a 75m radius 90 degree turn. The global resultsof this validation are given in Figure 7, while Figure 8 showsan example of the model performance. Variance AccountedFor (VAF) for the speed (V ) shows fair results, predicting

TABLE IIAVERAGE IDENTIFIED PARAMETERS FOR ALL SUBJECT, AND SET OF

PARAMETERS FOR THE INITIAL MODEL SIMULATION

Td [s] Tb [s] Kd [-] Kb- Ka [-] dTa [-]Mean 4.6673 3.2597 0.6504 4.1488 0.9801 -0.2259

StdDev 1.4342 1.1286 0.7130 1.4776 0.0460 0.0657Simulation 3.8 2.8 1 5 1 -0.25

VAF(V) VAF( a) VAF( b) VAF( ) 0%

20%

40%

60%

80%

100%

Fig. 7. Evaluation of the TETP speed control model performance for a wideand a narrow isolated turn with 75m radius.

0 100 200 300 400 500 600 700 800 9000

20

40

60

radi

us [

m]

0 100 200 300 400 500 600 700 800 900

-20

0

20

40/

a,/b [

%]

Deceleration zone

0 100 200 300 400 500 600 700 800 900

distance [m]

15

20

25

v [m

s-1]

ModelExp.

Fig. 8. Example (S4) of the performance of the model on an isolated75m radius turn. Positive pedal deflections correspond to the acceleratorpedal, while negative deflections represent brake actuation. The shaded arearepresents the deceleration zone, that is, the accelerator pedal position valuesthat correspond to a zero or negative acceleration.

the accelerator pedal trace (δa) and in particular brake timingworks well for only a part of the participants. Combinedacceleration/brake pedal prediction ((V AR(δ)) is fair. In theexample, the model accurately tracks the speed and pedaldeflections, the only significant mismatch is the fact that thereal driver releases the gas pedal slightly earlier than themodel and coasts for approximately 50m before applying thebrakes. The braking event timing is also accurate, with a smalldifference in magnitude of brake pedal actuation caused bythe difference in speed at the onset of braking due to the extracoasting. The short release of the accelerator pedal on curveexit is also reproduced.

Validation results showed that, for most subjects, the TETP-based model can accurately track driver speed choice in anisolated turn, with an average speed VAF above 60%. Theacceleration and gas pedal deflection VAF values are lower,

0 100 200 300 400 500 600 700 8000

20

40

60ra

dius

[m

]

0 100 200 300 400 500 600 700 800

-50

0

50

/a,/

b [%

]

Deceleration zone

0 100 200 300 400 500 600 700 800

distance [m]

10

15

20

25

v [m

s-1]

ModelExp.

Fig. 9. An example of decreased model performance, subject 6, narrow road.

but the fact that they are, on average, above 30% can beseen as a corroboration for the speed results. The brake pedaldeflection VAF values are low, but this can be attributed to thefact that there are only few occasions where drivers actuallyuse the brake pedal, resulting in low variance overall. VAFresults show large variability, with a standard deviation of thespeed VAF of approximately 30%. For four of the subject/roadcombinations the speed VAF is below 25%. An exampleof poorly fitted data is shown in Figure 9, indicating thatmodeling the relationship between TETP and pedal actuationmust still be improved. Some subjects use larger values ofTd, for different reasons. Subject 6 drove at consistently highspeeds (vavg = 26.8ms−1), released the accelerator pedalrelatively early and braked as little as possible, while Subject9 drove at comparatively low speeds (vavg = 20.1ms−1)throughout the experiment and started the brake phase early,as evidenced by a Tb value of 6.29s.

V. CONCLUSIONS

Unlike many existing models, this research directly relatesvisual perception to pedal control when driving on a curvedroad. The two thresholds on a single perceptual variable,the Time to Extended Tangent Point, allow for a model thatdistinguishes acceleration, coasting and braking phases. Bymodeling individual driver’s speed adaptation including pedalcontrol behaviour, we enable future research towards individ-ualized ADAS and self-driving vehicles. More specifically,this research paves the way for haptic shared control systemsthat integrate longitudinal [13], [22] and lateral support [23],potentially allowing drivers to feel the tightness of upcomingturns through forces on the gas pedal.

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