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SUBMITTED TO IEEE TRANS. ON CYBERNETICS 1

An Empirical Human Controller Model

for Preview Tracking Tasks

K. van der El, D. M. Pool, Member, IEEE, H. J. Damveld,

M. M. van Paassen, Senior Member, IEEE, and M. Mulder, Member, IEEE

Abstract—Real-life tracking tasks often show preview informa-

tion to the human controller about the future track to follow. The

effect of preview on manual control behavior is still relatively

unknown. This paper proposes a generic operator model for

preview tracking, empirically derived from experimental mea-

surements. Conditions included pursuit tracking, i.e., without

preview information, and tracking with one second of preview.

Controlled element dynamics varied between gain, single- and

double integrator. The model is derived in the frequency domain,

after application of a black-box system identiﬁcation method

based on Fourier coefﬁcients. Parameter estimates are obtained to

assess the validity of the model in both the time- and frequency-

domain. Measured behavior in all evaluated conditions can be

captured with the commonly used quasi-linear operator model

for compensatory tracking, extended with two viewpoints of the

previewed target. The derived model provides new insights into

how human operators use preview information in tracking tasks.

Keywords—Manual control, man-machine systems, preview con-

trol, human control models, parameter estimation

I. INTRODUCTION

PREVIEW on the future track to follow is a dominant

piece of information in many everyday manual control

tasks. Examples include car driving along a winding road and

landing an aircraft. The effect of this preview information on

the behavior of the Human Controller (HC) is still relatively

unknown. To study its contribution to HC behavior in isolation,

other visual and motion cues that are simultaneously available

are commonly removed, as well as the three-dimensional ‘real-

world’ visual perspective. In the two-dimensional preview

tracking task that results, it is shown that tracking performance

improves when the amount of preview increases [1]–[4]. The

question what control mechanisms underlie this accomplish-

ment has not yet been answered.

Manual control behavior in simple tracking tasks without

preview, such as the compensatory tracking task, is much better

understood. The quasi-linear ‘crossover model’, as proposed by

McRuer et al. [5], [6], plays a profound role in this. Derived

from measurements in the frequency domain, after application

of black-box system identiﬁcation techniques, the crossover

model reveals how HCs systematically adapt their control

behaviour and has become widely used in human-machine

systems analysis and simulation. It would be extremely useful

The authors are with the section Control and Simulation, Faculty of

Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629

HS Delft, The Netherlands. Corresponding author: d.m.pool@tudelft.nl

to also have a generic, widely applicable model for the more

relevant tracking tasks with preview.

Unfortunately, research aimed at ﬁnding such a HC model

for preview tracking has not been equally successful. Straight-

forward system identiﬁcation as applied by McRuer et al.

is impossible, due to the increased complexity of the task.

Conceptual HC models for pure preview tracking tasks [1]–[4],

[7]–[9], and for real-life ﬂying or driving tasks incorporating

preview [10]–[19], have been abundantly developed though;

all using either one, two, or all points of the previewed target

as inputs to the HC model. In general, multi-point models

can describe HC behavior better than single point models.

Which exact parts of the previewed target are used, and most

importantly, how these parts are used by the operator for

control, has remained inconclusive.

This paper aims to derive a generic, empirical HC model for

preview tracking tasks, without making a priori assumptions

on the operator’s control mechanisms. To do so, it is derived

from measurements taken from twelve subjects and similar

black-box system identiﬁcation techniques are applied as used

by McRuer et al. [5]. The measurements are collected in a

combined target-tracking and disturbance-rejection task with

no preview, the pursuit tracking task, and one second of full

preview on the displayed target, preview tracking. In order for

the model to be valid in a wide range of tasks, three basic

types of controlled element (CE) dynamics are evaluated for

each display: a gain, a single integrator and a double integrator.

The expected HC adaptation mechanisms on the effects of

preview and CE dynamics are analyzed through an instru-

mental variable, multiloop identiﬁcation method using Fourier

coefﬁcients [20]. Based on this, the HC model structure is

formulated, and the model parameters estimated. To assess

the model validity, the coherence and Variance Accounted For

(VAF) are calculated. For the ﬁrst time, rigorous system iden-

tiﬁcation is applied to allow the derivation of a mathematical

description of the HC’s response to preview, and to see whether

there is any evidence for systematic adaptation mechanisms as

found by McRuer et al. for compensatory tracking task.

The paper is structured as follows. Section II provides some

background regarding the identiﬁcation of HC behavior and

Section III explains how we acquired measurement data of

HC behavior in preview tracking tasks. Section IV describes

the system identiﬁcation technique and results. The develop-

ment of our empirical model will be extensively discussed in

Section V, followed by its validation in Section VI. The ﬁnal

two sections contain the discussion and overall conclusions.

K. van der El, D. M. Pool, H. J. Damveld, M. M. van Paassen, and M. Mulder,

“An empirical human controller model for preview tracking tasks,”

IEEE Trans. on Cybernetics, vol. 46, no. 11, pp. 2609–2621, Nov. 2016.

DOI: 10.1109/TCYB.2015.2482984

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 2

II. BACKGROUND

A. The Control Task

The control task considered here is a combined target-

tracking and disturbance-rejection task, illustrated in Fig. 1.

The HC is instructed to track the target, ft(t), as accurately

as possible, by generating control inputs, u(t), while the

CE (dynamics Hce) is perturbed by disturbance fd(t). The

HC minimizes tracking error, e(t), deﬁned as the difference

between the target and CE output, x(t):

e(t) = ft(t)−x(t)(1)

Additionally, preview information may be visible: a stretch

of the future target, ft([t, t +τp]), up to time τpahead. An

example of a preview display is shown in Fig. 2a. The CE

output (white marker) moves laterally over the screen, driven

by the operator’s control inputs and the disturbance. The

previewed target (black curve) moves down over the screen

with time, thereby laterally moving the current target (black

marker). Note that, when only the current target is available,

i.e., τp= 0 s, a pursuit tracking task results.

B. Quasi-Linear HC Model for Compensatory Tracking

The goal of this paper is to empirically derive a generic

operator model for preview tracking tasks; an approach similar

to the derivation of the model for compensatory tracking tasks

by McRuer et al. [5]. Therefore, ﬁrst some of their main

methods and ﬁndings are discussed.

In compensatory tracking tasks, only the current error e(t)

is presented to the HC, see Fig. 2b. The HC model is

characterized by a single-channel control diagram, Fig. 3,

hence only a target signal sufﬁces to allow its identiﬁcation.

McRuer et al. proposed a quasi-linear model: a linear response

to the error, Hoe(jω), to which a remnant signal n(t)is

added to account for all nonlinearities in the HC’s response.

e(t)u(t)x(t)

fd(t)

ft([t, t +τp])

x(t)

ft(t)

+

−

+

+

Human

Operator

(Dynamics Ho)

Stick

Controll ed

Element

(Dynamics Hce)

Display

Fig. 1. A HC in a previewed-target tracking and disturbance rejection task.

ref erence

controlled

target

element

output

τp

ft(t)

e(t)

x(t)

(a)

ref erence f ollower

e(t)

(b)

Fig. 2. Layouts of pursuit/preview (a) and compensatory (b) displays.

+

e(t)ft(t)u(t)x(t)

n(t)

Operator

Hce

Hoe

+

+

−

Fig. 3. Control diagram of the crossover model for compensatory tracking [5].

TABLE I. OPERATOR DESCRIBING FUNCTION Hoe(jω)[5].

CE dynamics Operator describing function

Kce Kee−τvjω Hnms (jω)/(1 + Tl,e jω)

Kce/j ω Kee−τvjω Hnms(j ω)

Kce/(j ω)2Kee−τvjω Hnms (jω)(1 + TL,e jω)

A multisine target signal was used with a limited number of

components, considered uncorrelated with the remnant at the

input frequencies ωt. The instrumental variable method then

yields the estimate of the HC describing function (DF) [5]:

ˆ

Hoe(jωt) = Sftu(jωt)

Sfte(jωt),(2)

with Sthe spectral-density function of the respective signals.

For three important types of CE dynamics, a gain, a single

integrator and a double integrator, McRuer et al. modeled the

DF as given in Table I. Here, Ke,τvand Hnms(jω)represent

the HC’s response gain, visual time delay and neuromuscular

system (NMS) dynamics, respectively. HCs additionally gen-

erate lag for gain CE dynamics and lead for double integrator

CE dynamics, characterized by lag and lead time constants,

Tl,e and TL,e. HCs thus systematically adapt their control

dynamics, Hoe(jω), to the CE dynamics in such a way that the

open-loop DF approximates single integrator dynamics around

the crossover frequency ωc[5]:

Hol(jω) = Hoe(jω)Hce (jω) = ωc

jω e−j ωτv(3)

This model is known as the crossover model. Additional

inclusion of the NMS, as in Table I, extends the validity of

the model to higher frequencies. In similar tracking tasks the

NMS dynamics are typically modeled as [21], [22]:

Hnms(jω) = ω2

nms

(jω)2+ 2ζnmsωnms jω +ω2

nms

,(4)

with ωnms and ζnms the natural frequency and damping ratio.

C. Identiﬁcation Considerations

The instrumental variable identiﬁcation method applied

here, discussed in detail in Section IV-A, allows for the

identiﬁcation of a number of DFs equal or smaller than the

number of uncorrelated inputs to the system [3], [20], [24].

To identify the HC’s response in the single-axis compensatory

tracking task, this, obviously, yields no restriction.

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 3

1) Pursuit tracking: In these tasks, the HC may respond to

the CE output, the target and the error, yielding the three DFs

HC model shown in Fig. 4 (top) [23]. A maximum of two

inputs can be inserted to the system in this particular task,

the target and the disturbance, therefore, the dynamics of only

two operator DFs can be identiﬁed. Because of this constraint,

no widely accepted, generic model has been identiﬁed to

date [25].

A workaround is possible, by introducing a two-channel

model as a ‘tool’ to identify the HC dynamics, having either

ft(t)and x(t)(‘T X ’), e(t)and ft(t)(‘ET ’) or e(t)and

x(t)(‘EX ’) as inputs to the operator model. The ‘E T ’

model is commonly thought to be the most sensible choice,

with Hoe(jω)assumed to be similar as in the crossover

model and Hot(jω)accounting for any additional feedforward

control [21], [23], [26]. Wasicko et al. [23] showed that all

three options are equally able to describe HC behavior.

In this paper we will use the ‘T X ’ model, Fig. 4 (bottom), as

it proved to be the structure in which key terms characterizing

HC behavior were most easily recognized during our analysis.

The two-channel ‘T X’ model can be expressed in terms of

the general, three-channel model:

HT X

ot(jω) = Hot(jω) + Hoe(jω),(5)

HT X

ox(jω) = Hox(jω) + Hoe(jω),(6)

which can be derived from the control diagrams in Fig. 4, for

details see [25]. When there is no chance of confusion the

‘T X ’ superscript will be omitted in the following. Equations

(5) and (6) show that the two DFs in the ‘T X’ model are not

the HC’s true responses to the target and CE output, but rather

lumped combinations of the true responses as deﬁned in the

general model, Fig. 4 (top).

2) Preview tracking: Extending the pursuit tracking task

with preview of the target signal ft(t)allows the HC to also

utilize the ‘inﬁnite’ number of points within the preview span

for control. In theory, a unique DF Hot(jω|τ)can be deﬁned

with respect to each point a certain time τahead, 0≤τ≤τp.

The control diagram is equal to that for pursuit tracking, Fig. 4,

e(t)ft(t)u(t)x(t)

n(t)

Operator

HoeHce

Hot

Hox

fd(t)

ft(t)

x(t)

Hce

fd(t)

HTX

ot(jω) = Hot(j ω) + Hoe(jω)

HTX

ox(jω) = Hox(j ω) + Hoe(jω)

+

−

+

+

−

+

+ +

+

u(t)

n(t)

Operator

+

−+

+

+

+

HTX

ot

HTX

ox

m

Fig. 4. Control diagram of the general HC model for pursuit [23] and preview

tracking tasks (top) and the equivalent ‘T X’ two-channel model (bottom).

as all points of the previewed target are related, only differing

by negative ‘time delay’ τ. The total response to the target can

then be expressed as:

Hot(jω) =

τp

X

τ=0

Hot(jω|τ)eτj ω (7)

Similar as for pursuit tracking, using equations (5) and (6),

the ‘T X ’ model can be obtained, to capture HC dynamics in

preview tracking tasks. The identiﬁed dynamics can become

rather complicated, however, as (7) shows that Hot(jω)may

contain a combination of many DFs.

D. Modeling Considerations

Most proposed HC models for preview tracking are based

on one of the three fundamental types of models proposed by

Sheridan [7]. As the aim of this paper is to learn the actual

HC model structure by recognition of individual responses in

the lumped, identiﬁed DFs, such a restriction to an existing

model is not made. It is interesting to see though, what the

lumped responses in the ‘T X ’ model structure will be if the

HC’s control mechanisms are indeed as in one of Sheridan’s

proposals.

1) The Extended Convolution: The ‘ET ’ two-channel

model, used in [1], [3], [10], [13], [14], was ﬁrst introduced

as the extended convolution model [7]. In addition to ‘com-

pensatory’ error control ue(t), feedforward control ut(t)is

exerted:

ut(t) = Zτp

Tm

ft(t+τ)wp(τ) dτ, (8)

with u(t) = ue(t) + ut(t), time Tmthe HC’s memory limit

and wp(τ)the target weighting function. The structure of (8) is

similar to (7), however, contrary to our approach, the responses

to the individual target points are explicitly related by wp(τ). If

HC behavior in preview tracking tasks is indeed characterized

by the extended convolution model, Hot(jω|τ)in (7) is a series

of gains that are related by wp(τ).

2) CE Output Predictor Models: In this type of models, the

HC is hypothesized to predict the future CE output ˆx(t+τ)

at some ﬁxed time τahead [7], [8], [11], [18]. Together with

the previewed target at the same time τahead, the HC can

internally calculate a predicted error, ˆe(t+τ):

ˆe(t+τ) = ft(t+τ)−ˆx(t+τ),(9)

which is corrected for with compensatory control. If HCs apply

linear prediction to calculate the error some time τpr ahead, for

example, Hox(jω)in our ‘T X’ model will become 1 + τpr jω

and Hot(jω|τ)in (7) will equal 1 for τ=τpr and 0 for

τ6=τpr. In practice HC’s may utilize other prediction methods

or predict the error at multiple future points.

A notable model related to the ﬁrst two types of models

is proposed in [4]. The HC is hypothesized to respond to an

internally calculated, current error e⋆(t), based on weighting

the previewed target into a single current target to steer to,

f⋆

t(t):

e⋆(t) = f⋆

t(t)−x(t) = Zτp

0

ft(t+τ)wp(τ) dτ−x(t)(10)

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 4

3) Optimal Control Models: These models [2], [15], [17],

[19] assume the HC to steer in an ‘optimal’ way. The optimum

is the minimum of a cost function which generally weights

tracking error and control effort, but can include anything.

Despite its intuitive structure its value is limited for this study,

as explicit identiﬁcation of the cost function is impossible [27].

III. DATA ACQUISITION

In this section, it is explained how the measurements,

suitable for model-based analysis using system identiﬁcation,

were collected.

A. Measurement Setup

A combined target-tracking and disturbance-rejection task

was performed to allow identiﬁcation of two operator DFs with

the instrumental variable method. Three different CE dynamics

and two different displays were evaluated, in order for the

derived operator model to be valid for a wide range of tasks.

The three CE dynamics were chosen equal to the elemen-

tary dynamics evaluated by McRuer et al. in compensatory

tracking [5]: a gain, a single integrator and a double integrator;

representing position, velocity and acceleration control, respec-

tively. Their gains, Kce, were set to 0.8, 1.5 and 5, respectively.

They were tuned such that the HC would never reach the stick

deﬂection limits, yet could give small, accurate inputs.

The preview display that was used is shown in Fig. 2a,

with the preview time τpequal to either 0 (pursuit) or 1 s.

The latter was chosen sufﬁciently higher than the reported

‘critical’ preview time for each CE dynamics: the preview time

after which additional preview yields no further performance

beneﬁt [1]–[4], suggesting constant ’preview’ control behavior.

B. Apparatus

Measurements were collected in the Human-Machine Lab-

oratory at TU Delft, in a ﬁxed-base part-task simulator. The

display resolution was 1280 by 1024 pixels, the size 36 by 29.5

cm and the update rate 100 Hz. The display was positioned

directly in front of the subjects, at approximately 75 cm, it had

bright green lines and indicators on a black background and

was projected with a time delay of approximately 20-25 ms.

An electro-hydraulic servo-controlled side-stick with a moment

arm of 9 cm, which could only rotate around its roll axis, was

used to generate the control inputs. The torsional stiffness of

the stick was 3.58 Nm/rad, the torsional damping 0.20 Nm

s/rad, the inertia 0.01 kg m2and the gain 0.175 inch/deg.

C. Forcing Functions

To analyze HC behavior with the instrumental variable iden-

tiﬁcation method, see Section IV-A, the target and disturbance

signals were quasi-random sums of Nfsinusoids:

f(t) =

Nf

X

i=1

Aisin(ωit+φi),(11)

with amplitude Ai, frequency ωiand phase φiof the ith

sinusoid. ωiwere integer multiples kiof the base frequency of

10−1 100101

10−20

10−10

100

ω, rad/s

Sf f , inch2s/rad

Sftft

Sfdfd

Sftft(ωt)

Sfdfd(ωd)

Fig. 5. Single-sided power spectra of the target and disturbance inputs.

0.0524 rad/s, corresponding to a measurement time of 120 s.

Nf= 20 sines were used for each forcing function; the

resulting signals were considered to be unpredictable [28].

The power distribution of the target signal, as well as its

total power, was chosen to be as close as possible to the signal

used by McRuer et al., with a bandwidth of approximately

1.5 rad/s [5]; the standard deviation σftwas 0.5 inch. The

standard deviation of the disturbance signal, σfd, was 0.2 inch;

the spectra are plotted in Fig. 5.

Double input frequency bands were used to allow calculation

of the coherence. Five realizations of ftwere used, differing

only by the initial phases φtof the individual sine-components,

to prevent subjects from recognizing parts of the signals

because of repeated exposure. It was unlikely that subjects

could memorize the disturbance signal, as it was not explicitly

visible on the display, therefore a single realization was used.

All forcing functions parameters are given in Table II.

D. Subjects, Instructions and Procedure

Twelve motivated, male volunteers were instructed to min-

imize tracking error e(t). Each subject performed the six

conditions in a single session, with breaks every 45 minutes.

The total experiment lasted about 2.5 to 3.5 hours per subject,

depending on the training required.

First, each condition was practiced to get the subject accus-

tomed to the task. After that, the six conditions were performed

consecutively in random order. When stable performance was

achieved in a condition, generally after three to eight runs,

the ﬁve actual measurement runs were recorded, after which

subjects moved on to the next condition. After each run the

Root Mean Square (RMS) of the error was reported to the

subjects as a measure of their performance to motivate them.

Each run lasted 128 s, of which the ﬁrst 8 s were used as run-

in time; these data were not used for analysis. The remaining

120 s from the time-traces of the error e(t), the CE output

x(t), the operator’s control actions u(t), and the target and

disturbance forcing functions were sampled at 100 Hz.

IV. OPERATOR DF IDENTIFICATION

In this section it is explained how the operator’s DFs were

obtained from the experimental data.

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 5

TABLE II. DEFINITION OF THE TARGET AND DISTURBANCE SIGNALS.

Target signals ftDisturbance signal fd

kt, - At, inch ωt, rad/s φt,1, rad φt,2, rad φt,3, rad φt,4, rad φt,5, rad kd, - Ad, inch ωd, rad/s φd, rad

2 0.240 0.105 3.646 0.174 4.878 2.917 2.709 5 0.093 0.262 2.546

3 0.240 0.157 0.030 5.953 2.868 2.040 0.508 6 0.093 0.314 6.264

8 0.240 0.419 1.277 0.655 4.205 5.857 3.369 11 0.093 0.576 6.283

9 0.240 0.471 2.367 3.526 1.921 4.188 0.477 12 0.093 0.628 1.865

14 0.240 0.733 3.901 4.809 3.786 2.602 5.165 18 0.093 0.942 3.196

15 0.240 0.785 4.287 3.391 4.201 4.110 3.286 19 0.093 0.995 5.309

26 0.240 1.361 5.554 2.965 6.014 3.640 0.619 31 0.093 1.623 3.626

27 0.240 1.414 2.411 4.446 5.538 1.626 3.397 32 0.093 1.676 3.229

40 0.057 2.094 2.551 1.730 0.183 3.703 3.131 58 0.029 3.037 0.165

41 0.057 2.147 0.490 2.952 4.354 6.108 3.926 59 0.029 3.089 0.082

78 0.057 4.084 5.431 5.432 1.588 3.406 3.812 93 0.029 4.869 4.233

79 0.057 4.136 4.238 3.697 0.270 3.801 5.548 94 0.029 4.922 5.366

110 0.057 5.760 5.014 6.200 1.858 5.698 4.196 128 0.029 6.702 5.386

111 0.057 5.812 1.768 5.910 0.585 2.903 5.239 129 0.029 6.754 4.756

148 0.057 7.749 0.163 5.952 4.385 3.622 1.333 158 0.029 8.273 1.453

149 0.057 7.802 0.156 3.563 1.825 4.269 5.889 159 0.029 8.325 3.291

177 0.057 9.268 0.214 0.077 0.325 5.841 4.290 193 0.029 10.105 3.243

178 0.057 9.320 3.687 1.961 0.269 3.608 0.164 194 0.029 10.158 3.924

220 0.057 11.519 5.176 5.484 6.179 4.261 3.196 301 0.029 15.760 3.171

221 0.057 11.572 1.266 1.050 0.304 2.325 4.165 302 0.029 15.813 1.976

A. Method

The applied black-box, instrumental variable, multiloop

identiﬁcation method is based on Fourier coefﬁcients [20]. It is

equivalent to the method based on spectral-density functions as

used by McRuer et al. [5], see (2). For the ‘T X’ two-channel

HC model in Fig. 4, the Fourier transform of the operator’s

control actions at an arbitrary target input frequency ωtis given

by:

U(jωt) = Hot(jωt)Ft(jωt)−Hox(j ωt)X(jωt).(12)

The capitals, U,Ftand X, denote the Fourier transforms of

the respective signals; remnant is neglected as its contribution

can be assumed to be small at the input frequencies. The two

unknown operator DFs can be solved for by constructing a

second equation, obtained by interpolating the same signals

from the neighboring disturbance signal input frequencies ωd

to the considered ωt. The obtained set of equations is:

U(jωt)

˜

U(jωt)=Ft(jωt)−X(jωt)

˜

Ft(jωt)−˜

X(jωt)Hot(jωt)

Hox(jωt),(13)

with the interpolated values denoted by a tilde. Solving (13)

for the estimates of the operator DFs yields:

ˆ

Hot(jωt) = ˜

U(jωt)X(jωt)−U(jωt)˜

X(jωt)

˜

Ft(jωt)X(jωt)−Ft(jωt)˜

X(jωt),(14)

ˆ

Hox(jωt) = ˜

U(jωt)Ft(jωt)−U(jωt)˜

Ft(jωt)

˜

Ft(jωt)X(jωt)−Ft(jωt)˜

X(jωt).(15)

Mutatis mutandis, replacing ωtfor ωdyields the operator DFs

at the latter frequencies.

As the method poses no a priori assumptions on the dy-

namics in the DFs, the estimates can be regarded as the

actual operator’s control actions in the frequency domain for

the chosen model inputs and outputs. The method has been

successfully applied before to identify operator DFs in similar

tasks involving multiloop HC behavior [3], [24], [25], [29].

B. Results

A selection of typical identiﬁed operator DFs, averaged over

the ﬁve runs in the frequency domain, is given in Fig. 6, 7

and 8. Each ﬁgure shows four graphs: the magnitude (top) and

the phase (bottom) of the HC’s response to target (left) and

to the CE output (right). The results for the preview (black

triangles) and the pursuit (gray dots) conditions are drawn

together, to illustrate the effects of preview on HC behavior.

In the next section, we will explain our approach to derive

the model based on the data of the shown subjects/conditions

combinations only. It equally applies to all measurements of all

twelve tested subjects, which will be shown in Section VI. Two

distinct different control strategies are found between subjects

in single integrator CE dynamics tasks with preview, hence a

representative example of each is given. For gain and double

integrator CE dynamics conditions only a single representative

subject is shown, as no structural between-subject differences

are found.

V. HC MODEL DERIVATION

Here, the HC model that captures all identiﬁed DFs in

Fig. 6, 7 and 8 is derived, by ﬁrst modeling the DF in

each individual condition separately. These are combined into

a single, comprehensive HC model towards the end of this

section. For easier understanding by the reader, the order of

individual model derivations is such that the conditions in

which the simplest dynamics are obtained are explained ﬁrst,

advancing to more complex HC dynamics throughout this

section.

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 6

10−1 100101

10−1

100

101

ω, rad/s

|Hot|, -

(a)

10−1 100101

10−1

100

101

ω, rad/s

|Hox|, -

(b)

10−1 100101

−360

−270

−180

−90

0

90

180

270

360

450

ω, rad/s

6Hot, deg

(e)

10−1 100101

−360

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

pursuit

preview

(f)

10−1 100101

10−1

100

101

ω, rad/s

|Hot|, -

(c)

10−1 100101

10−1

100

101

ω, rad/s

|Hox|, -

(d)

10−1 100101

−360

−270

−180

−90

0

90

180

270

360

450

ω, rad/s

6Hot, deg

(g)

10−1 100101

−360

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

pursuit

preview

(h)

Fig. 6. Operator DFs for single integrator CE: Subject 1 (a,b,e,f) and Subject 2 (c,d,g,h).

10−1 100101

10−1

100

101

ω, rad/s

|Hot|, -

(a)

10−1 100101

10−1

100

101

ω, rad/s

|Hox|, -

(b)

10−1 100101

−270

−180

−90

0

90

ω, rad/s

6Hot, deg

(c)

10−1 100101

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

pursuit

preview

(d)

Fig. 7. Operator DFs for gain CE: Subject 3.

10−1 100101

10−2

10−1

100

ω, rad/s

|Hot|, -

(a)

10−1 100101

10−2

10−1

100

ω, rad/s

|Hox|, -

(b)

10−1 100101

−270

−180

−90

0

90

180

270

360

450

ω, rad/s

6Hot, deg

(c)

10−1 100101

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

pursuit

preview

(d)

Fig. 8. Operator DFs for double integrator CE: Subject 3.

A. CE Output Response

For all evaluated CE dynamics, |Hox(jω)|is similar in the

pursuit and preview conditions. The slope is -1 in the mid-

frequency range for gain CE dynamics, Fig. 7b, which can be

modeled with a gain, Kx, and a lag term, Tl,x. For the single

and double integrator CE dynamics, Fig. 6b, d and 8b show

that |Hox(jω)|has a slope of 0 and +1, respectively. These

can be modeled by a gain Kxand a combination of a gain Kx

with a lead term TL,x. The HC’s adaption to the CE dynamics

is thus similar as in compensatory tracking [5], Section II-B.

Additionally, the decrease in |Hox(jω)|at the highest input

frequencies indicates the presence of NMS dynamics. This

is not clearly visible for gain CE dynamics though, possibly

because here the subjects adopted a NMS break frequency well

above the highest measured frequency.

For all conditions, 6Hox(jω)at high frequencies shows

the decreasing phase lag that characterizes a pure time delay;

the HC’s response time delay, τv. The mentioned phenomena

combined yield our model for Hox(jω), see Table III.

B. Target Response

The decrease of |Hot(jω)|at high frequencies indicates the

presence of the same NMS dynamics as in Hox(jω). Contrary

to Hox(jω), the shape of Hot(jω)is not similar in preview

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 7

TABLE III. OPERATOR DESCRIBING FUNCTION Hox(j ω).

CE dynamics Operator describing function

Kce KxHnms(j ω)e−τvjω /(1 + Tl,x jω)

Kce/j ω KxHnms(jω)e−τvjω

Kce/(j ω)2KxHnms(j ω)e−τvjω (1 + TL,x jω)

and pursuit tracking tasks; within preview tracking tasks even

variations exist between subjects, i.e., for the single integrator

CE dynamics, see Fig. 6a, c, e and g. Hot(jω)will therefore

be modeled step by step.

1) Pursuit Tracking Tasks: For all considered CE dynamics,

the shape of |Hot(jω)|is very similar to that observed for

|Hox(jω)|. They possibly differ in magnitude though, which

we model by the target weighting gain,Kf. The phase shift

at the highest input frequencies is approximately equal in

Hot(jω)and Hox(jω), see, for instance, Fig. 8c and d,

suggesting that τvis equally large in both responses. So, in

pursuit tracking tasks, Hot(jω)is modeled with:

HP S

ot(jω) = KfHox(jω),(16)

with Hox(jω)as given in Table III.

2) Preview Tracking Tasks: For Subject 2 and single in-

tegrator CE dynamics, Hot(jω)is, at low frequencies, ap-

proximately equal to the pursuit condition, see Fig. 6c, d,

g and h. Above 2 rad/s, |Hot(jω)|reduces strongly with a

break frequency that is too low to be caused by the NMS.

Possibly parts of the target signal’s high frequencies were

purposely ignored, facilitated by the available preview. We

model this with a low-pass ﬁlter with lag time constant Tl,f

in a ﬁltering function Hof(jω), which also incorporates the

previously deﬁned target weighting gain Kf:

Hof(jω) = Kf

1

1 + Tl,f jω .(17)

The distinctive increasing phase lead for higher frequencies,

Fig. 6g, further indicates the presence of a negative time delay,

suggesting that the subject is responding to the previewed tar-

get somewhere ahead. This far-viewpoint is located τfseconds

ahead; subscript fis added to all parameters associated with

it. As the HC’s response delay is now lumped with the negative

‘look-ahead’ time delay, it can no longer be uniquely deter-

mined from the DFs, which only capture the total input-output

delay. We therefore assume that τvis equal to that in Hox(jω)

as was also found in the pursuit conditions. Summarizing, for

Subject 2, Hot(jω)is modeled by:

HSI -P R,S2

ot(jω) = Hox(jω)Hof(jω)eτfjω .(18)

Remember that the NMS dynamics and τvare incorporated

in Hox(jω), Table III. Alternatively, (18) is referred to as the

far-viewpoint response,Hot,f (jω).

Equation (18) can only partially capture the DFs for Subject

1, see Fig. 6a and e. Instead of the magnitude drop and

increasing phase that were observed at high frequencies for

Subject 2, here, a magnitude peak appears and the response

phase ﬂattens. This reveals the presence of more complex

dynamics, i.e., a summation of (at least) two additive or parallel

responses, each with its own negative time delay. This subject

clearly uses multiple points of the previewed target for control.

The ﬁrst response is assumed to be equivalent to Subject 2’s

far-viewpoint response, see (18). The ﬂattening of 6Hot(jω)

at high frequencies, Fig. 6e, indicates that the look-ahead time

for the second response is lower than τf, hence it is named

the near-viewpoint response,Hot,n (jω). The point responded

to is located τnseconds ahead.

As opposed to the low-pass ﬁltered far-viewpoint response,

the Hot,n (jω)needs to be high-pass ﬁltered, so it has a

contribution only in the high-frequency region where the ﬁrst

response insufﬁciently describes the observed behavior. Again,

we ﬁrst deﬁne the near-viewpoint ﬁlter Hon(jω):

Hon(jω) = Kn

jω

1 + Tl,nj ω ,(19)

with gain Knand lag time constant Tln. The NMS dynamics

and visual response time-delay are again assumed to be com-

mon with the other responses, therefore, Hot,n (jω)becomes:

Hot,n (jω) = Hon(jω)e(τn−τv)jω Hnm(jω).(20)

Finally, the total target DF Hot(jω)for Subject 1 is a sum-

mation of the near- and far-viewpoint responses:

HSI -P R,S1

ot(jω) = Hot,n (jω) + Hot,f (jω).(21)

When Hot,n (jω) = 0, (21) is simply Hot,f (jω), given by

(18). Additionally, if Tl,f =τf= 0, (18) further reduces into

(16). Equation (21) thus simply extends the simpler models

previously found for Hot(jω).

For gain CE dynamics, no structural differences are observed

between subjects. At high frequencies a similar magnitude

peak and phase lead occurs as for Subject 1 for single

integrator CE dynamics, see Fig. 7a and c. The combination of

the near- and far-viewpoint responses in (21) is well capable

to capture both these phenomena. Note that Hox(jω)in the

far-viewpoint response accounts for the HC’s adaption to the

CE dynamics, Table III.

Similar considerations apply for double integrator CE dy-

namics. For Subject 3, presented in Fig. 8, the ever-increasing

phase at high frequencies suggests that only the far viewpoint

is utilized. For a few other subjects the phase seems to

ﬂatten somewhat at high frequencies, although not as clearly

as for conditions with single integrator CE dynamics. The

derived two-point model can deﬁnitely capture these variations

between subjects.

C. Model Restructuring

The two-channel model structure in Fig. 4, combined with

the modeled Hox(jω)and Hot(jω)in Table III and (16)

to (21), respectively, yields a generic HC model for pursuit

and preview tracking, for gain, single integrator and double

integrator CE dynamics. Fig. 9 (top) shows the complete model

with all dynamics substituted into the control diagram. HC lim-

itations, i.e., the NMS dynamics and time delay, are separated

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 8

e⋆(t)

ft(t+τn)

u(t)x(t)

n(t)

Operator

fd(t)

ft(t+τf)Hnme−τvjω

Hof

ft(t)u(t)x(t)

n(t)

Operator

Honeτnjω +Hce

fd(t)

Hofeτfjω Hox

HoxHnme−τvjω

Hnme−τvjω

Hox

=Hoe⋆

f⋆

t,f (t)

Hon

Hoe⋆

f⋆

t,f ft

e⋆(t)

++

−

+

+

+

+

+

+

+

+

+

+

−

Hce

m

Fig. 9. Control diagram of the derived HC model, restructured into a more intuitive form.

from Hox(jω)for clarity. Moving all common elements to the

right of the summation point and separating the summation in

Hot(jω)into two parallel blocks, the model structure at the

bottom of Fig. 9 is obtained. Its structure intuitively explains

how HCs perform pursuit and preview tracking tasks.

The bottom ﬁgure reveals that Hox(jω)is in fact a response

to the difference between the ﬁltered far-viewpoint f⋆

t,f (t)and

x(t), hence to an internally calculated error e⋆(t) = f⋆

t,f (t)−

x(t).Hox(jω)is therefore renamed to Hoe⋆(jω). The target to

steer to, f⋆

t,f (t)is obtained by low-pass ﬁltering and weighting

the far-viewpoint with Hof(jω), (17); a possible visualization

is illustrated in the display in Fig. 9.

D. Discussion

1) Preview Tracking: From the restructured model, the two

governing mechanisms underlying the HC’s control actions

appear to be feedback and pure feedforward, as commonly

suggested before [3], [7], [10], [13], [14], [16], [18], although

never with all the key terms introduced here. The two responses

do not only separate the HC behavior into two spatial regions,

but also into two frequency regions, i.e., low-pass ﬁltered

feedback control with respect to a far-viewpoint and high-pass

ﬁltered feedforward control with respect to a near-viewpoint.

The feedforward Hon(jω)is an open-loop control action,

at the high frequencies facilitated because substantial parts of

full periods of the target sine-components are instantly visible

on the display. The expected function of the inverse of the

CE dynamics, which would result in perfect target-tracking

performance [21]–[23], is not found here though, possibly

because the NMS dynamics also become effective at those

frequencies where the contribution of the feedforward is large,

hence interfering with it.

The preview visualizes a negligible portion of full periods of

the target sine-components at low frequencies, so anticipation

to these is more difﬁcult and feedback control is exerted. The

mechanism behind this control is somewhat like the crossover

model for compensatory tracking [5] with a similar adaptation

to the CE dynamics. The deﬁnition of the error the HC

responds to is rather different, however. The structure of the

feedback response is more similar to the HC model as proposed

by Ito and Ito [4], which is also an extension of the crossover

model.

Previous ﬁndings that a minimum of two points of the

target are needed as model inputs to adequately model the

measured HC behavior [2], [4], [8], [12], [15], [17], [19] are

only partially conﬁrmed. Depending on the subject and the

condition, either a response to a single point or to two points is

initiated. This substantial difference between subjects is easily

captured by the model though.

2) Pursuit Tracking: Recall that the pursuit tracking task is

a special case of preview tracking task with τp= 0 s. The fact

that the derived model for pursuit tracking is also a reduced

version of the model for preview tracking is therefore very

intuitive. Hon(jω) = 0 and Hof(jω) = Kfin pursuit con-

ditions. The model thus explains what already followed from

the identiﬁed DFs, namely that no pure open-loop response is

initiated and that the feedback response is now based on to

the current target, weighted by Kf. So, similar as in preview

tracking, the HC is responding to an internally calculated error,

e⋆(t).

3) Compensatory Tracking: The derived model ﬁts very well

in the crossover model framework for compensatory tracking,

derived by McRuer et al. [5]. In compensatory tracking, the

HC can respond to the true error only, so Kf= 1 by deﬁnition.

Substitution in Fig. 9 yields the exact same control model as

in Fig. 3.

VI. PARAMETER ESTIMATION AND MODEL VALIDATION

In this section the derived model will be validated, starting

with a justiﬁcation for the use of a quasi-linear HC model.

The ability of the model to describe the measured HC control

actions is quantiﬁed, and all model parameters are estimated

and analyzed for consistency.

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 9

A. Methods

1) Coherence: The coherence is a measure for the linearity

between two signals, it has a value between 0 (completely

nonlinear) and 1 (perfectly linear). A highly linear relation

between the input forcing functions and the HC control actions

can justify the use of a quasi-linear operator model. The

coherence between the target input forcing function and the

HC’s control actions is given by [30]:

Γ(˜ωft) = s|˜

Sftu(˜ωft)|2

˜

Sftft(˜ωft)˜

Suu( ˜ωft).(22)

The tilde indicates the average power-spectral density ( ˜

S) at

the average frequency (˜ω) between the neighboring frequencies

in a double band. The coherences at the disturbance input

signal frequencies are calculated similarly.

2) Model Fitting/Parameter Estimation: The parameters are

estimated by ﬁtting the model to the DFs in the frequency-

domain, using the two-channel ‘T X’ model structure. The

normalized error between the two, ǫ, is a measure for the

quality of the ﬁt. For each parameter set Θit is given by:

ǫ(Θ) =

2Nf

X

i=1

W|HDF

ot(jωi)−Hmod

ot(jωi|Θ)|2

|HDF

ot(jωi)|2

+

2Nf

X

i=1

W|HDF

ox(jωi)−Hmod

ox(jωi|Θ)|2

|HDF

ox(jωi)|2,(23)

with 2Nfthe total number of input frequencies and Wa

weighting vector [0 0 w3. . . wN−20 0]T. The two zeros

at the upper and lower end of Wensure that the unreliable,

extrapolated DF components at the lowest and highest input

frequencies cannot affect the results. All other weights, w3to

wN−2, were determined based on the estimation reliability of

the respective DF component. Weighting penalties were added

to (23) for any negative parameter estimates and values of the

NMS break frequency outside the measured frequency range.

The parameter set that describes the operator’s control behavior

best was then calculated by minimizing the cost function ǫ:

Θopt =arg min

Θ

ǫ(Θ).(24)

3) Variance Accounted For: The VAF is a measure for the

similarity between two signals, its highest value of 100%

indicates that the signal are exactly equal. Applied to compare

the measured control inputs u(k)and the modeled control

inputs ˆu(k), it serves as a measure for the ability of the model

to capture the HC’s behavior. It is calculated by:

VAF = 1−PN

k=1 |u(k)−ˆu(k)|2

PN

k=1 u2(k)!×100%,(25)

with Nthe number of samples in the time series. ˆu(k)is

obtained from time-domain simulation of the derived model

with the measured target and CE output signals as inputs. The

VAF is calculated per run, as time averaging is not possible

due to the ﬁve different target signals presented to the HC.

B. Results

1) Coherence: Fig. 10 shows the average coherence and the

standard deviations, calculated per run for all twelve subjects.

The coherence between ft(t)and u(t)is lower in preview

tracking than in pursuit tracking at high frequencies. This

corresponds to [1] and suggests that HCs exert less linear,

perhaps time-varying control when preview is available. Except

for the lowest input frequency, the mean coherence is well

above 0.75 in all evaluated conditions, justifying the use of a

quasi-linear model.

2) Model Fits: The ﬁtted models are drawn with solid lines

in the DF Bode plots in Fig. 11, 12 and 13, for the same CE

dynamics and subjects as in Section IV. The model clearly

captures the shape of the DFs for all CE dynamics.

For the preview condition, the HC’s response to the far-

viewpoint, (18), and near-viewpoint, (20), are also plotted

individually to clarify their contributions to the total target

response. Indeed, the magnitude plots show that the far-

viewpoint response (black dashed line) always dominates

Hot(jω)at low frequencies. Depending on the condition and

subject, the near-viewpoint response (gray dash-dotted line)

can become the dominant contributor at high frequencies.

At low frequencies, the DFs are not very well captured by

the (linear) model, especially for the double integrator CE.

This is consistent with the fact that the coherence is also

comparatively low at these frequencies, Fig. 10, indicating that

the contribution of the remnant is high. As such, the weights wi

in (23) were low, resulting in a less tight ﬁt to the DFs at these

frequencies. Even for the double integrator CE the differences

remain quite small, considering that they are visually magniﬁed

by the logarithmic scale of the Bode plots.

3) Parameter Estimates: The estimated operator model pa-

rameters are given in Table IV for the same combination of

subjects and conditions as the Bode plots in the previous

sections. Estimates of the HC parameters that also present

10−1 100101

0.5

0.75

1

ω, rad/s

Γft,u, -

(a)

10−1 100101

0.5

0.75

1

ω, rad/s

Γfd,u, -

(b)

10−1 100101

0.5

0.75

1

ω, rad/s

Γft,u, -

(c)

10−1 100101

0.5

0.75

1

ω, rad/s

Γfd,u, -

(d)

10−1 100101

0.5

0.75

1

ω, rad/s

Γft,u, -

(e)

10−1 100101

0.5

0.75

1

ω, rad/s

Γfd,u, -

pursuit

preview

(f)

Fig. 10. Coherence between uand the ft(left) and fd(right) inputs for

gain (a,b), single integrator (c,d) and double integrator CE dynamics (e,f). To

clarify, the pursuit data are shifted slightly to the left.

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 10

10−1 100101

10−1

100

101

ω, rad/s

|Hot|, -

(a)

10−1 100101

10−1

100

101

ω, rad/s

|Hox|, -

(b)

10−1 100101

−360

−270

−180

−90

0

90

180

270

360

450

ω, rad/s

6Hot, deg

pursuit DF

pursuit model

preview DF

preview model

Hot,n(jω )

Hot,f(j ω)

(e)

10−1 100101

−360

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

(f)

10−1 100101

10−1

100

101

ω, rad/s

|Hot|, -

(c)

10−1 100101

10−1

100

101

ω, rad/s

|Hox|, -

(d)

10−1 100101

−360

−270

−180

−90

0

90

180

270

360

450

ω, rad/s

6Hot, deg

pursuit DF

pursuit model

preview DF

preview model

Hot,n(jω )

Hot,f(j ω)

(g)

10−1 100101

−360

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

(h)

Fig. 11. Model ﬁts for single integrator CE dynamics; Subject 1 (a,b,e,f) and Subject 2 (c,d,g,h).

10−1 100101

10−1

100

101

ω, rad/s

|Hot|, -

pursuit DF

pursuit model

preview DF

preview model

Hot,n(jω )

Hot,f(j ω)

(a)

10−1 100101

10−1

100

101

ω, rad/s

|Hox|, -

(b)

10−1 100101

−270

−180

−90

0

90

ω, rad/s

6Hot, deg

(c)

10−1 100101

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

(d)

Fig. 12. Model ﬁts for gain CE dynamics; Subject 3.

10−1 100101

10−2

10−1

100

ω, rad/s

|Hot|, -

(a)

10−1 100101

10−2

10−1

100

ω, rad/s

|Hox|, -

(b)

10−1 100101

−270

−180

−90

0

90

180

270

360

450

ω, rad/s

6Hot, deg

pursuit DF

pursuit model

preview DF

preview model

Hot,n(jω )

Hot,f(j ω)

(c)

10−1 100101

−270

−180

−90

0

90

ω, rad/s

6Hox, deg

(d)

Fig. 13. Model ﬁts for double integrator CE dynamics; Subject 3.

in the model for compensatory tracking, namely Ke⋆,Tl,e⋆,

TL,e⋆,τv,ωnms and ζnm, see Table I, are consistent with

previous HC behavior studies, for example [6], [21], [22]. For

conditions with gain CE dynamics, ωnms is indeed well above

the highest input frequency. Therefore it can not be estimated

accurately and its estimate approaches our deﬁned limit of 18

rad/s.

For pursuit tracking tasks, the far-viewpoint low-pass ﬁlter

was kept in the model during the estimation procedure. Ta-

ble IV shows that Tl,f is estimated to be zero for all four

cases, which is strong evidence that the ﬁlter can indeed be

omitted and that Hof(jω) = Kf.

For single integrator CE dynamics, the average of the gain

Kfis close to 1 in the pursuit condition. As the derived

model closely resembles the crossover model for compensatory

tracking when Kf= 1, the present study thus conﬁrms

previous results that HC behavior is very similar in pursuit

and compensatory tracking tasks for single integrator CE

dynamics [23], [25]. For conditions with double integrator CE

dynamics, Kfis substantially lower than 1. Kfrepresents the

HC’s relative weighting of the target and the CE output in

the calculation of e⋆(t), so this suggests that HC’s prioritize

stabilizing the CE output over tracking of the target.

Without being restricted in the estimation procedure, the

look-ahead times τnand τfare both estimated to be lower

than 1 s, so within the visually presented preview. They both

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 11

0

20

40

60

80

100

replacements

Kce

Kce Kce/sKce/s Kce /s2

Kce/s2

pursuitpursuitpursuit previewpreviewpreview

VAF, %

Fig. 14. VAF per subject (gray bars), mean and standard deviation per condition (errorbars) and grand mean (dotted line).

TABLE IV. ESTIMATED OPERATOR MODEL PARAMETERS.1

Kce Kce/s Kce /s2

PS PR PS PR PS PR PS PR

Subject # 3 3 1 1 2 2 3 3

Ke⋆, - 3.85 6.62 1.43 1.11 0.85 1.07 0.14 0.14

Tl,e⋆, s 2.06 2.39 - - - - - -

TL,e⋆, s - - - - - - 2.54 2.22

τv, s 0.18 0.16 0.23 0.18 0.21 0.25 0.28 0.31

ωnms, rad/s 17.9 18.0 11.2 10.2 7.89 7.66 6.15 5.33

ζnm, - 0.18 0.37 0.30 0.26 0.55 0.58 0.67 0.50

Kn, - - 0.06 - 0.18 - 0.01 - 0.32

τn, s - 0.08 - 0.34 - 0.00 - 0.00

Tl,n, s - 0.06 - 0.04 - 0.94 - 5.89

Kf, - 1.21 1.11 0.95 1.12 1.01 0.93 0.54 0.63

τf, s - 0.55 - 0.70 - 0.97 - 0.99

Tl,f , s 0.00 0.26 0.00 0.38 0.00 0.93 0.00 0.59

VAF, % 82.0 77.4 67.7 67.6 73.6 70.6 70.1 70.3

1PS and PR are abbreviations for the pursuit and preview conditions, respectively.

increase when the CE dynamics become more difﬁcult to

control, supplying the HC with more phase lead to compensate

for the CE dynamics inherent phase lag.

Especially in the preview tracking tasks with single integra-

tor CE dynamics, the identiﬁed DFs were different between

subjects. For Subject 2, Knis indeed estimated to be ap-

proximately zero, conﬁrming that this subject only uses the

far-viewpoint for control. Subjects 1 clearly responds to an

additional near-viewpoint, indicated by a non-zero Kn.

4) Variance Accounted For: The VAFs are generally be-

tween 65% and 85% for all twelve subjects in all conditions,

see Fig. 14, indicating that the model output matches the mea-

sured control actions fairly well. Considered that the models

are estimated on averaged frequency-domain data, while the

related VAFs are calculated for each individual run in the

time-domain, before averaging, makes these values particularly

impressive. They roughly correspond to similar manual control

modeling studies where data were averaged ﬁrst to mitigate

remnant effects, yielding higher VAFs [21], [22].

VII. DISCUSSION

In this paper, a generic HC model for preview tracking

tasks was derived from measurement data. For the ﬁrst time,

rigorous system identiﬁcation was applied that allowed the

identiﬁcation of two independent operator responses. Based

on the DFs, these two responses were modeled, followed by

a restructuring of the HC model into a more intuitive form.

The model parameters were estimated, after which the VAF

and frequency-domain ﬁts served as measures for the model

validity. Also, the use of a quasi-linear operator model was

justiﬁed by calculation of the coherence.

The resulting model is well capable of capturing the shape of

the identiﬁed DFs for all subjects in all measured conditions.

It describes the HC adaption to both the CE dynamics, by

equalization in Hoe⋆(jω), and to the display type, by optional

feedforward control with a negative time delay. Additionally,

the model captures the considerable variability in control be-

havior between subjects, which we demonstrated in particular

for tasks with preview and single integrator CE dynamics.

The model helps to gain deeper insight in the underlying

control mechanisms of manual tracking. A completely new

view of HC behavior in pursuit tracking task emerges, which,

according to the model, is very similar to compensatory

tracking. Gain Kfby itself completely explains the difference,

relatively weighting the contribution of the target and the CE

output in HCs calculation of the internal error. Values of Kf

lower than 1, as found for conditions with double integrator

CE dynamics, suggest that HCs prioritize stabilizing the CE

over tracking of the target.

In conditions with preview HC behavior becomes much

more advanced. At low frequencies of the target, feedback

control is applied with respect to the far-viewpoint, while some

subjects apply additional feedforward control with respect to

the near-viewpoint at high frequencies of the target. At high

frequencies, entire periods of the target sinusoids are instantly

observable. Recognizing these oscillations as such enables the

HC to apply open-loop control with approximately the right

timing to track the target. At low frequencies the periods of

the target sinusoids are longer, so HCs can no longer recognize

these as the oscillations they are and they revert to a feedback

control strategy. By basing their feedback control on the target

ahead, HCs do utilize the displayed preview though, generating

extra phase lead to compensate for any lags in the loop.

Derived from measurements, our empirical model is not

based on any previously proposed models. In hindsight, some

remarkable similarities appear however. In Ito and Ito’s pre-

view model [4] for example, the previewed target is also

weighted to internally calculate a current error to compensate

for, but their model does not separate the low and high frequen-

cies and no feedforward control is incorporated. Car drivers are

commonly modeled using a combination of feedforward and

feedback [10], [14], e.g., in the two-point model [18], which

SUBMITTED TO IEEE TRANS. ON CYBERNETICS 12

explicitly incorporates a near- and far-viewpoint. These models

also lack the low and high frequency separation, as well as the

relative target/CE output weighting. The similarities with other

models that incorporate preview are generally much smaller.

The model derived here surpasses other preview models on

one other point: identiﬁability of the physically meaningful

parameters. For the ﬁrst time, it is possible to explicitly identify

how the HC utilizes preview information for control.

VIII. CONCLUSION

A human operator model for preview tracking tasks is

derived from measurement data by application of a black-

box, instrumental variable identiﬁcation method. The derived

model is an extension of the quasi-linear operator model for

compensatory tracking tasks, with two points of the previewed

target as inputs to the operator. The model is capable of

describing the measured control behavior in conditions with

both zero (pursuit) and one second of preview, and with

gain, single integrator and double integrator controlled element

dynamics. It also allows for considerable between-subject con-

trol behavior variations. The model provides a deeper insight

into how humans utilize information on the future target for

control. Two very distinct mechanisms split the response to

the target both spatially and in frequency regions. Feedforward

control is exerted with respect to a near-viewpoint at the higher

frequencies, while feedback control is exerted with respect to

a far-viewpoint at lower frequencies.

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SUBMITTED TO IEEE TRANS. ON CYBERNETICS 13

Kasper van der El received the M.Sc. degree

in aerospace engineering (cum laude) from Delft

University of Technology, Delft, The Netherlands,

in 2013, for his research on manual control behavior

in preview tracking tasks.

He is currently pursuing the Ph.D. degree on

manual control behavior in general control tasks that

involve preview, with the Control and Simulation

Division, Faculty of Aerospace Engineering, Delft.

His current research interests include cybernetics,

human motion perception, mathematical modeling,

and system identiﬁcation and parameter estimation.

Daan M. Pool received the M.Sc. and Ph.D. degrees

(cum laude) from Delft University of Technology,

Delft, The Netherlands, in 2007 and 2012, respec-

tively. His Ph.D. research focused on the devel-

opment of an objective method for optimization

of ﬂight simulator motion cueing ﬁdelity based on

measurements of pilot control behavior. He is cur-

rently an Assistant Professor with the Control and

Simulation Division, Faculty of Aerospace Engineer-

ing, Delft University of Technology. His research

interests include cybernetics, manual vehicle control,

ﬂight simulation, human motion perception, and mathematical modeling,

identiﬁcation, and optimization techniques.

Herman Damveld received the M.Sc. degree in

aeroservoelasticity and the Ph.D. degree in handling

qualities of aeroelastic aircraft from Delft University

of Technology, Delft, The Netherlands, in 2001 and

2009, respectively. In the past, he was a Develop-

ment Engineer with Lange Aviation, Zweibruecken,

Germany, and a Postdoctoral Research Fellow and

Researcher with Delft University of Technology. He

is currently an Assistant Professor with the Con-

trol and Simulation Division, Faculty of Aerospace

Engineering, Delft University of Technology. His

research interests include human factors, cybernetics, car and aircraft simula-

tion, simulator motion cueing, aeroservoelasticity, and avionics. Dr. Damveld

received the Best Paper Award from the American Institute of Aeronautics

and Astronautics Modeling and Simulation Technologies conference in 2009.

Marinus (Ren´

e) M. van Paassen (M’08, SM’15)

received the M.Sc. and Ph.D. degrees from Delft

University of Technology (TU Delft), Delft, The

Netherlands, in 1988 and 1994, respectively, for his

studies on the role of the neuromuscular system of

the pilot’s arm in manual control.

He is currently an Associate Professor with the

Faculty of Aerospace Engineering, TU Delft, work-

ing on human-machine interaction and aircraft sim-

ulation. His work on human-machine interaction

ranges from studies of perceptual processes and

human manual control to complex cognitive systems. In the latter ﬁeld,

he applies cognitive systems engineering analysis (abstraction hierarchy and

multilevel ﬂow modeling) and ecological interface design to the work domain

of vehicle control.

Dr. van Paassen is an Associate Editor of the IEEE TRANSACTIONS ON

HUMAN-MACHINE SYSTEMS.

Max Mulder (M’14) received the M.Sc. degree and

Ph.D. degree (cum laude) in aerospace engineering

from Delft University of Technology, Delft, The

Netherlands, in 1992 and 1999, respectively, for

his work on the cybernetics of tunnel-in-the-sky

displays.

He is currently Full Professor and Head of

the Control and Simulation Division, Faculty of

Aerospace Engineering, Delft University of Technol-

ogy. His research interests include cybernetics and its

use in modeling human perception and performance,

and cognitive systems engineering and its application in the design of

“ecological” human-machine interfaces.