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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.
