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Identiﬁcation of Nonlinear Motion Perception

Dynamics Using Time-Domain Pilot Modeling

D.M. Pool,

∗

A.R. Valente Pais,

†

A.M. de Vroome,

‡

M.M. van Paassen,

§

and M. Mulder

¶

Delft University of Technology, Delft, The Netherlands

This paper describes and tests a method for the estimation of nonlinear dynamics in

human manual control behavior, here applied to estimating the magnitude of vestibular

thresholds, from measurements of tracking behavior. The proposed method exploits the

marked effect of physical motion feedback that is observed for such control tasks and

the fact that human manual control behavior can be modeled successfully with multi-

modal pilot models. A three-step identiﬁcation procedure is proposed that uses time-

domain measurements of human manual control behavior to identify the parameters of

an augmented multimodal pilot model that includes a nonlinear model for an absolute

motion perception threshold. The proposed identiﬁcation procedure is evaluated using

both experimental data and pilot model simulations. To vary the fraction of the applied

simulator motion that is below threshold and hence possibly the effect of the vestibular

threshold on the adopted control strategy, for both data sets values of 0.25, 0.5, 0.75, and

1 for the motion cueing gain were evaluated. For the experimental as well as the simu-

lated data, the identiﬁcation procedure is found to yield the most consistent and accurate

results if the gain of the supplied motion cues is unity. This indicates that for retriev-

ing reliable estimates of human motion perception thresholds from active control task

measurements with the proposed method sufﬁcient excitation of the vestibular modality

is required, which is controlled by the magnitude of the applied forcing function signals

and the motion cueing gain.

∗

Ph.D. student, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058, 2600GB

Delft, The Netherlands; d.m.pool@tudelft.nl. Student member AIAA.

†

Ph.D. student, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058, 2600GB

Delft, The Netherlands; a.r.valentepais@tudelft.nl. Student member AIAA.

‡

M.Sc. student, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058, 2600GB

Delft, The Netherlands; adevroome@gmail.com.

§

Associate Professor,Control and Simulation Division, Faculty of Aerospace Engineering,P.O. Box 5058, 2600GB

Delft, The Netherlands; m.m.vanpaassen@tudelft.nl. Member AIAA.

¶

Professor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058, 2600GB Delft,

The Netherlands; m.mulder@tudelft.nl. Senior member AIAA.

1 of 36

Daan M. Pool, Ana Rita Valente Pais, Aniek M. de Vroome, Marinus M. van Paassen and

Max Mulder, Identification of Nonlinear Motion Perception Dynamics Using Time-Domain

Pilot Modeling (2012), in: Journal of Guidance, Control, and Dynamics, 35:3(749-763)

dx.doi.org/10.2514/1.56236

Nomenclature

A

d,t

Sinusoid amplitude, rad

D Lilliefors test statistic

e Tracking error signal, rad

F

n

Remnant power fraction

f Probability density function

f

d

Disturbance forcing function, rad

f

t

Target forcing function, rad

H

n

Remnant shaping ﬁlter

H

nm

Neuromuscular system dynamics

H

p

v

Pilot visual response

H

p

m

Pilot motion response

H

scc

Semi-circular canal dynamics

H

c

Controlled element dynamics

i Forcing function sinusoid index

i

∆

Threshold input signal, impulses

per unit of time (IPUT)

j Imaginary unit

K

c

Controlled element gain

K

m

Pilot motion gain, rad/IPUT

K

n

Remnant ﬁlter gain

K

scc

Semicircular canal dynamics

gain, IPUT s

2

/rad

K

v

Pilot visual gain

K

δ,u

Control input gain

K

θ

Pitch motion gain

L Likelihood function

N Number of data points

N

d,t

Number of sinusoids

N

50%

Number of ﬁts within 50%

of true value of ∆

abs

n Remnant signal, rad

n

d,t

Forcing function frequency

integer factor

o

∆

Threshold output signal, IPUT

T

0

Measurement duration, s

T

n

Remnant ﬁlter time constant, s

T

vl

Pilot lead time constant, s

T

scc

1,2,3

Semicircular canals dynamics

time constants, s

t Time, s

u Pilot control signal, rad

Symbols

ˆ Estimate

∆

abs

Absolute threshold, IPUT

δ Control input, rad

ǫ Output error

Θ Parameter vector

θ Pitch angle, rad

¨

θ Pitch acceleration, rad/s

2

¨

θ

m

Pitch motion acceleration, rad/s

2

µ Distribution mean

φ

d,t

Sinusoid phase shift, rad

σ Standard deviation

σ

2

Variance

τ

m

Pilot motion time delay, s

τ

v

Pilot visual time delay, s

ω Frequency, rad/s

ω

0

Measurement base frequency, rad/s

ω

d,t

Sinusoid frequency, rad/s

ω

nm

Neuromuscular frequency, rad/s

ζ

nm

Neuromuscular damping

2 of 36

I. Introduction

The modeling of human manual control behavior has signiﬁcantly increased our understanding

of human adaptation to differences in controlled element dynamics,

1–3

the effects of physical mo-

tion cues on manual control,

4–7

and how simulator motion washout algorithms affect motion cue

utilization.

4,5,8,9

Not only have these modeling efforts revealed much about the control-theoretic

characteristics of human controllers during manual control tasks, they have also led to increased

insight in the underlying physiological, multisensor integration, and neuromuscular activation pro-

cesses.

6,10,11

Modeling human manual control behavior has been the most successful for continuous and sta-

tionary control tasks (tracking). For such tasks, quasi-linear pilot models as proposed by McRuer

et al.

1

can accurately describe measured human control behavior. Such quasi-linear models typi-

cally consist of a set of linear (transfer function) models that describe pilots’ responses to perceived

variables, and a remnant signal that represents the cumulative sum of all nonlinear contributions to

the observed control behavior, that is, those that are not explained by the linear part of the model.

As for instance argued by McRuer and Jex,

2

the most important contributions to this remnant are

considered to be (1) pure noise injected by the human operator, (2) nonlinear operations by the

human operator such as perception thresholds or control rate saturation, and (3) nonsteady and

time-varying operator control behavior. For typical measurements of human tracking behavior us-

ing quasi-random forcing function signals, manual control behavior is often found to be sufﬁciently

linear to yield only a comparatively minor effect of remnant. For instance, Ref. 12 shows that for

tracking tasks with physical motion feedback around 80% of measured control behavior could be

explained with linear transfer function models, leaving only 20% to be accounted for by remnant.

For increased understanding of some of the nonlinear contributions to manual control behav-

ior, it may be worthwhile to extend current quasi-linear models of human manual control behavior

with terms that can capture part of the observed nonlinearities, and to ﬁt these models to measured

data. For instance, in human motion perception research there has always been much interest in

determining the characteristics of human motion perception thresholds,

13–16

which are an inher-

ently nonlinear aspect of the human sensory system. It is known that the magnitudes of human

vestibular motion perception thresholds are affected by workload

13,14

and the presence of other vi-

sual and physical motion cues.

17,18

As opposed to the pure sensory motion perception thresholds,

threshold values under such workload or multisensory cueing conditions are referred to as indif-

ference thresholds.

18–21

It is hypothesized, but not yet fully supported by experimental evidence,

that the magnitudes of motion perception thresholds are affected by performing active control.

This paper proposes and evaluates an approach to extending the linear responses of quasi-linear

pilot models with additional nonlinear terms that can capture and model part of the nonlinear con-

tributions to human manual control behavior. One of the key issues for such an approach is that

3 of 36

this introduction of nonlinear terms in the structural part of the model causes commonly applied

pilot model identiﬁcation techniques – such as frequency-domain describing function techniques

4

and linear time-domain parameter estimation methods

12

– to be no longer applicable. In this paper,

this approach is developed for the estimation of the magnitude of vestibular thresholds from mea-

surements of manual control behavior during tracking tasks with motion feedback. The proposed

method makes use of a multimodal pilot model similar to those used in many previous investi-

gations into human multimodal control behavior.

5–7,9,10

A time-invariant but nonlinear vestibular

threshold model is included in the vestibular channel of the otherwise linear pilot model, and an

extension to the time-domain multimodal pilot model identiﬁcation approach described in Ref. 12

is proposed for identiﬁcation of this now nonlinear model.

Due to the limited effect of nonlinear contributions on typical measurements of tracking behav-

ior, the estimation of vestibular thresholds with the proposed model-based approach is anticipated

to be difﬁcult. For any identiﬁcation approach, sufﬁcient information on the phenomenon that

needs to be modeled should be available in the measurements to yield reliable model identiﬁcation

results and parameter estimates. To assess the viability of the proposed approach, it is tested using

data from the experiment described by Valente Pais et al.

22

In this experiment eight subjects per-

formed pitch attitude tracking tasks with physical pitch motion feedback for varying pitch motion

cueing gains (ranging from 1-to-1 to 0.25), with the speciﬁc objective of estimating the magnitude

of motion perception thresholds during active manual control. The controlled element dynamics

in this experiment were a double integrator, for which many previous experiments have shown a

signiﬁcant inﬂuence of motion feedback on manual control behavior.

4,6,7,23,24

In addition, Valente

Pais et al. considered control tasks that involved both compensatory target following and distur-

bance rejection, because of the different use of motion feedback for these two types of control

task.

6,7,24,25

The present paper focuses on issues regarding the reliability of threshold estimates

that can be obtained with the proposed method by evaluating the vestibular threshold estimates

obtained for the different experimental settings tested in Ref. 22.

The structure of this paper is as follows. First, a detailed overview is given of the modeling

of multimodal control behavior using quasi-linear pilot models and the selected approach for in-

cluding the nonlinear motion perception threshold dynamics in such a model. In Section III, the

time-domain parameter estimation algorithm used for identifying all model parameters, including

the absolute motion perception threshold, is explained. In addition, a description of the two data

sets that used for evaluating the proposed method, the experiment data from Ref. 22 and a set of

corresponding pilot model simulation data, is provided. Section IV then presents the results of

the application of the proposed method to the experimental measurements from the experiment

of Ref. 22. Furthermore, the reliability of the obtained results is evaluated with the pilot model

simulation data. The paper ends with a discussion of all results and conclusions.

4 of 36

II. Pilot Modeling

II.A. Control Task

e

Figure 1. Compensatory display.

The work described in this paper builds on the considerable knowl-

edge of human manual control behavior in compensatory tracking

tasks.

1,2,4–7,9,26,27

Such tasks involve the continuous minimization

of a tracking error e from a visual display like the one depicted in

Fig. 1.

This paper focuses on compensatory attitude tracking tasks

where physical motion feedback on the controlled variable is avail-

able in addition to the tracking error that is presented on a visual display. Previous experiments

that investigated the effects of physical motion feedback during compensatory attitude tracking

using multimodal pilot modeling have indicated that pilot control behavior is signiﬁcantly affected

by both the presence and quality – which may, for instance, be inﬂuenced by the presence of time

delays and washout ﬁlters – of the supplied motion feedback.

4–7,9

A schematic representation of

such a compensatory tracking task where physical motion feedback is available is shown in Fig. 2.

f

t

+

e

¨

θ

−

¨

θ

m

K

δ,u

u

n

H

p

v

(jω)

H

p

m

(jω)

−

pilot

+

+

¨

θ

f

d

δ

+

+

K

θ

K

c

1

(jω)

2

θ

controlled element, H

c

(jω)

u

v

u

m

cueing

Figure 2. Schematic representation of a compensatory pitch attitude tracking task with double integrator

controlled element dynamics.

Fig. 2 shows a schematic representation of the pitch attitude tracking task described in more

detail in Ref. 22, which is used in this paper for evaluating the proposed method for identifying

nonlinear motion perception dynamics. Fig. 2 shows a combined target-following and disturbance-

rejection task with a double integrator controlled element. Due to the increased requirement for

pilot lead equalization that is typically required for controlled elements with decreasing inherent

stability,

23

highly signiﬁcant effects of vestibular motion feedback on pilot control have been re-

ported for control tasks with double integrator controlled elements.

4,6,7,24

For this reason, double

integrator controlled element dynamics were considered in Ref. 22.

Pilot control action is induced with quasi-random target and disturbance forcing function sig-

nals, indicated by f

t

and f

d

, respectively. The target forcing function deﬁnes a reference signal for

the system pitch attitude θ that needs to be followed. The disturbance forcing function acts on the

controlled element as an external disturbance that needs to be attenuated. Pilot control inputs (u in

5 of 36

Fig. 2) are scaled with a gain K

δ,u

and added to the disturbance forcing function signal before be-

ing fed to the controlled element dynamics. As will be explained in more detail in Section III.B.1,

to manipulate the fraction of the supplied pitch motion that was above and below threshold level,

Valente Pais et al. collected measurements of pilot tracking behavior under a variation in the pitch

motion cueing gain K

θ

.

Pilot control during compensatory tracking with physical motion feedback, as depicted in

Fig. 2, is generally modeled as the sum of parallel pilot visual and motion (typically vestibu-

lar) responses.

4–7,9,27

These responses to tracking errors e and physical pitch motion feedback

¨

θ

m

are indicated in Fig. 2 as H

p

v

(jω) and H

p

m

(jω), respectively. The remnant signal n is added to

the summed output of these responses to account for nonlinear and noise contributions to the pilot

control inputs u.

II.B. Nonlinear Motion Perception Dynamics

Analysis ofmeasured pilot behaviorin previous investigationsinto manual attitude tracking tasks

6,7

has suggested that the pilot motion response H

p

m

(jω) (see Fig. 2) is dominated by vestibular

motion perception with the semicircular canals. The semicircular canals are sensitive to angular

acceleration and their dynamics can be described by:

6,28

H

scc

(jω) = K

scc

1 + T

scc

1

jω

(1 + T

scc

2

jω)(1 + T

scc

3

jω)

≈ K

scc

1 + T

scc

1

jω

1 + T

scc

2

jω

(1)

The time constants of the semicircular canal dynamics model given by Eq. (1) are equal to

T

scc

1

= 0.11 s, T

scc

2

= 5.9 s, and T

scc

3

= 0.005 s, respectively. These values have been taken

from previous research.

6

It should be remarked that the high-frequency pole of the semicircular

canals transfer function (T

scc

3

= 0.005 s) is omitted here because of its limited effect on the canal

dynamics in the frequency range of interest to this study (0.1-20 rad/s). Similar to the approach

taken in Ref. 18, the semicircular canal dynamics used for modeling vestibular motion perception

in this paper have been normalized using a value for the gain K

scc

that ensures the absolute value of

H

scc

(jω) at 1 rad/s is unity. This choice of K

scc

gives a model of the semicircular canal dynamics

that relates a rotational acceleration input (in rad/s

2

) to an output proportional to, but not equal to,

the afferent neuron ﬁring rate (in impulses per second). For this reason, unit of the output of the

semicircular canal dynamics model given by Eq. (1) is deﬁned here as a number of impulses per

unit of time (IPUT). For the reduced semicircular canal dynamics model a value of K

scc

= 5.97

IPUT s

2

/rad ensures a unity gain of H

scc

(jω) at 1 rad/s.

A notable nonlinear characteristic of vestibular motion perception is the existence of percep-

tual thresholds, that is, vestibular motion stimulation below a certain threshold value will remain

undetected by the human vestibular sensors. Research into the dynamics of these motion percep-

tion thresholds, and measuring the actual values of these thresholds under varying conditions, has

6 of 36

received a signiﬁcant amount of attention.

13–15,17,18,20,21,29

Fig. 3 depicts a schematic representa-

tion of the angular pitch acceleration perception process in the semicircular canals. As proposed in

many publications on human motion perception and perception thresholds,

4,26,29,30

the perceptual

threshold associated with angular motion perception with the semicircular canals is assumed to

operate on the semicircular canal output afferent ﬁring rate i

∆

. In this paper, as also for instance

proposed in Refs. 26 and 29, the perception threshold is modeled as a nonlinear dead zone op-

eration on the time-domain output of the semicircular canals. Using the deﬁnition from Ref. 22,

the threshold magnitude – that is, the magnitude of i

∆

below which the output of the threshold

operation (o

∆

) is zero – is deﬁned by the absolute threshold parameter ∆

abs

. Note that both the

signals i

∆

and o

∆

, as well as the absolute threshold parameter ∆

abs

, have unit IPUT. As can be

veriﬁed from Fig. 3, the perceived pitch accelerations corresponding to the this nonlinear model

of rotational motion perception by the semicircular canals can be evaluated by passing o

∆

through

the inverse semicircular canal dynamics.

applied

¨

θ

m

H

scc

(jω)

i

∆

o

∆

∆

abs

perceived

¨

θ

m

H

−1

scc

(jω)

i

∆

o

∆

−∆

abs

Figure 3. The relation between simulator and perceived pitch accelerations with a dead zone threshold opera-

tion acting on the semicircular canal output.

II.C. Augmented Multimodal Pilot Model

Fig. 4 depicts the multimodal pilot model used in this paper. Much like the pilot models used in

previous investigations,

6,7,9,27

the model shown in Fig. 4 consists of parallel visual and vestibular

motion channels, as also depicted in Fig. 2. The inputs to these separate channels are the tracking

error e and the perceivable angular acceleration

¨

θ

m

, provided by the simulator motion system in

the experiment of Ref. 22, respectively. For both the visual and vestibular channels of the pilot

model, the contributions of sensory dynamics, pilot equalization, and inherent limitations such as

perceptual delays and neuromuscular actuation dynamics are modeled separately.

As proposed by McRuer et al.,

1

the visual (compensatory) contributionto pilot control behavior

for acceleration control can be modeled by using a gain-lead element K

v

(1 + jωT

vl

) and a time

delay (τ

v

). For control of a system with double integrator dynamics, pilots will need to generate

lead to achieve satisfactory characteristics around the pilot-vehicle system crossover frequency.

1

The selected visual equalization dynamics allow for modeling this lead generation. As can be

veriﬁed from Fig. 4, the pilot vestibular response is modeled as a gain equalization on the output

of the semicircular canals – which includes the effects of the absolute threshold operator ∆

abs

–

delayed by a pure time delay τ

m

, as proposed by Hosman

6

and Van der Vaart.

7

Note that as K

m

7 of 36

u

¨

θ

m

n

H

p

v

(jω)

e

sensory

dynamics

K

v

(1 + jωT

v l

)

H

scc

(jω)

K

m

e

−jωτ

v

e

−jωτ

m

H

nm

(jω)

H

p

m

(jω)

+

+

−

perception

delays

pilot

equalization

actuation

dynamics

i

∆

o

∆

∆

abs

u

v

u

m

Figure 4. Multimodal pilot model including a nonlinear absolute vestibular motion perception threshold oper-

ation.

is downstream of the threshold block in the model and the output of the motion response channel

(u

m

) is in rad, the pilot motion gain has the unit rad/IPUT.

Finally, the dynamics of the neuromuscular actuation (H

nm

(jω)) required for producing a con-

trol deﬂection u, which represent a lumped model of the interaction between the sidestick manip-

ulator and the human neuromuscular actuation dynamics, are modeled as a second-order mass-

spring-damper system:

H

nm

(jω) =

ω

2

nm

(jω)

2

+ 2ζ

nm

ω

nm

jω + ω

2

nm

(2)

The augmented multimodal pilot model shown in Fig. 4 has a total of eight model parameters,

of which one (∆

abs

) deﬁnes the magnitude of the threshold operation in the vestibular model chan-

nel. In addition, the equalization dynamics’ parameters (K

v

, T

vl

, and K

m

), the perceptual time

delays (τ

v

and τ

m

), and the neuromuscular system natural frequency ω

nm

and damping ratio ζ

nm

are also free parameters of the pilot model. This gives the following vector of free parameters for

the augmented pilot model shown in Fig. 4:

Θ =

h

K

v

T

vl

K

m

τ

v

τ

m

ω

nm

ζ

nm

∆

abs

i

T

(3)

III. Identiﬁcation Procedure

III.A. Parameter Estimation Algorithm

Estimating the parameter vector given by Eq. (3) from measurements of e,

¨

θ

m

, and u (see Fig. 4)

is an identiﬁcation problem that is similar to the one evaluated in Ref. 12. There it was shown that

estimating the parameters of a two-channel model as shown in Fig. 4 is a nonlinear identiﬁcation

problem, due to the overdetermined model structure resulting from the exchangeable lead contri-

8 of 36

butions of the visual and vestibular model channels. The identiﬁcation problem investigated here

is further complicated by the inclusion of a nonlinear dead zone threshold model in the vestibular

sensory dynamics portion of the model. In this paper, a time-domain maximum likelihood algo-

rithm based on the output-error identiﬁcation procedure of Ref. 12 is proposed for estimating the

parameters of the augmented pilot model of Fig. 4.

Maximum likelihood estimation involves the minimization of the negative logarithm of the

likelihood function L as a function of the estimated set of parameters,

ˆ

Θ:

−ln L(

ˆ

Θ) =

N

2

ln σ

2

ǫ

(

ˆ

Θ) +

1

2σ

2

ǫ

(

ˆ

Θ)

N

X

k=1

ǫ

2

(k|

ˆ

Θ) (4)

The output error ǫ in Eq. (4) is deﬁned as the difference between the measured and modeled

pilot model output, ǫ(

ˆ

Θ) = u − ˆu(

ˆ

Θ), where both u and ˆu consist of N data points and ˆu(

ˆ

Θ)

is obtained by simulating the model with the parameters from

ˆ

Θ and the measured model inputs

e and

¨

θ

m

. The symbol σ

2

ǫ

indicates the covariance of this output error. As can be veriﬁed from

Eq. (4), the negative log-likelihood function as deﬁned here consists of two separate terms: the ﬁrst

(N/2 ln σ

2

ǫ

) penalizes output error covariance, while the other ensures lower −ln L with reducing

output errors ǫ. For further details and the derivation of Eq. (4), the reader is referred to Ref. 12.

The task is now to ﬁnd the optimal set of multimodal pilot model parameters

ˆ

Θ according to

the cost function given by Eq. (4). The method proposed for this in Ref. 12 combines a genetic

algorithm (GA) and unconstrained Gauss-Newton (steepest-descent) optimization to yield param-

eter estimates that have a high probability of representing the global minimum of the multimodal

pilot model identiﬁcation problem, which is close to overdetermined because of the similarity of

the contributions of the visual lead term and the vestibular channel to the total model output.

12

The

parameter estimation procedure that is proposed here for estimating the parameters of the pilot

model including the nonlinear threshold model is depicted in Fig. 5.

Genetic Algorithm

optimization

(7+1 parameters)

Gauss-Newton

optimization

(7 parameters)

ﬁnal threshold

optimization

(1 parameter)

Step I

z

}| {

pilot model

parameters:

K

v

T

L

K

m

τ

v

τ

m

ω

nm

ζ

nm

∆

abs

measured

time traces:

e

¨

θ

m

u

Step II

z

}| {

Step III

z

}| {

nonlinear threshold operation parameter, ∆

abs

linear model parameters, K

v

, T

vl

, K

m

, τ

v

, τ

m

, ω

nm

, ζ

nm

Figure 5. Block diagram of the three-step estimation algorithm.

9 of 36

The ﬁrst two steps of the estimation algorithm as depicted in Fig. 5 are the same as those

described in Ref. 12. The steepest-descent optimization in Step II utilizes Jacobian matrices of the

pilot model with respect to the model parameters. As explained in Ref. 12, for all model parameters

except ∆

abs

the Jacobian matrices can be calculated analytically by transforming the pilot model to

state-space form. For ∆

abs

numerical approximation of the Jacobian matrices is found to be highly

sensitive to the selected size of the parameter perturbation, often leading to diverging estimates of

∆

abs

. Therefore, a strategy is chosen that relies on the genetic algorithm (Step I) for determining

an initial estimate of the threshold value. As indicated in Fig. 5, this threshold estimate is then kept

ﬁxed during the Gauss-Newton optimization of the remainder of the model parameters. Then, in a

ﬁnal step, it is veriﬁed if the estimated value of ∆

abs

can still be reﬁned more when the parameter

optimization of Step II is taken into account. As Step III represents a one-dimensional optimization

problem, for this step simply the model likelihood, as calculated from Eq. (4), was evaluated over

the full range of values considered for ∆

abs

(see Table 1), from which the value of

ˆ

∆

abs

that gave

the lowest value of −ln L was then selected.

The initial genetic algorithm population size was set to 160 (20 times the number of estimated

model parameters) and the algorithm was allowed to run for 100 iterations. The probabilities of

gene crossover and gene mutation were set to typical values of 0.7 and 0.01,

12

respectively. Table 1

lists the upper and lower bounds that were implemented in the genetic algorithm for estimation of

the pilot model parameters given by Eq. (3). These bounds are based on ﬁndings from previous

human-in-the-loop experiments

12

and most notably those of Ref. 22. Note that the model parame-

ters were only restricted to these upper and lower bounds during Steps I and III of the estimation

algorithm. No such constraints were present on the parameter values during the steepest-descent

optimization step (II, see Fig. 5). For performing gene crossover and gene mutation in the genetic

algorithm, model parameters were encoded to a 20 -bit binary representation, yielding a parameter

resolution of, for instance, 10

−5

for K

v

, T

vl

, and K

m

and 10

−7

for ∆

abs

.

Table 1. Genetic algorithm parameter upper and lower bounds.

K

v

T

vl

τ

v

K

m

τ

m

ω

nm

ζ

nm

∆

abs

− s s rad/IPUT s rad/s − IPUT

Estimation lower boundary 0.0 0.0 0.0 0.0 0.0 5.0 0.0 0.0

Estimation upper boundary 10.0 10.0 1.0 10.0 1.0 30.0 1.0 0.1

Due to the inherent randomness of genetic algorithms there is no guaranteed convergence to a

correct solution for a single evaluation of the algorithm.

12

Similar to the approach taken in Ref. 12,

20 repeated evaluations of the estimation procedure depicted in Fig. 5 were run on each evaluated

data set. For the estimation of ∆

abs

, the average of the ﬁve evaluations of the genetic algorithm

(Step I) that yielded the model ﬁts with the ﬁve best cost function values (lowest likelihoods) were

averaged to yield a single estimate of the threshold parameter for each data set. This estimated

threshold value was then utilized, and kept ﬁxed, in Step II.

10 of 36

III.B. Identiﬁcation Data Sets

III.B.1. Experimental Data

To verify the proposed method for retrieving threshold values from measurements of manual con-

trol behavior, two different data sets are considered. First, the experimental measurements from

the human-in-the-loop experiment described by Valente Pais et al.

22

are used for illustrating and

evaluating the performance of the proposed augmented pilot model identiﬁcation approach. These

experimental measurements will be used to illustrate the proposed model identiﬁcation procedure

and its typical results. In addition, pilot model simulations of the same control tasks as con-

sidered in Ref. 22 are used to verify the identiﬁability of the absolute threshold parameter (see

Section III.B.2).

In the experiment described in Ref. 22 the gain of the double integrator controlled element,

K

c

, was set to −4, yielding exactly the same double integrator dynamics as also used in previous

experiments.

6,7,24

The scaling gain between stick deﬂection u and control input δ (K

δ,u

) was

set to −0.69. Combined target-following and disturbance-rejection tasks as presented in Fig. 2

were performed. The presence of both forcing function signals was required to allow for reliable

identiﬁcation of both the pilot visual and motion responses H

p

v

(jω) and H

p

m

(jω) using spectral

4

or time-domain identiﬁcation techniques.

12

The disturbance and target forcing function signals

considered for the pitch attitude control tasks were both sums of 10 different sinusoids (N

d,t

= 10),

as given by:

f

d,t

(t) =

N

d,t

X

i=1

A

d,t

(i) sin [ω

0

n

d,t

(i)t + φ

d,t

(i)] (5)

The sinusoid frequencies were chosen as integer multiples of the experimental measurement

time base frequency ω

0

to allow for frequency domain describing function measurements.

4

Sinu-

soid frequencies are therefore directly related to the measurement duration T

0

through ω

0

= 2π/T

0

.

The measurement duration was chosen at 81.92 seconds, yielding a sinusoid base frequency ω

0

of

0.0767 rad/s. The integer factors n

d,t

were used to ensure that f

d

and f

t

had power at interleaving

frequencies over the frequency range of interest (0.1-20 rad/s).

Forcing function amplitude distributions A

d,t

had the low-passcharacteristic deﬁned in Ref. 27,

yielding reduced signal power at higher frequencies. Sinusoid phases φ

d,t

were selected to (1)

yield signals with an approximately Gaussian signal distribution and (2) avoid excessive peaking

or cresting in the time-domain realizations of f

d

and f

t

.

31

As the disturbance signal was inserted

before the controlled element dynamics (see Fig. 2), the disturbance signal amplitudes and phases

were preshaped with the inverse controlled element dynamics to yield the designed effect of the

disturbance signal on the pitch attitude θ. The numerical values of the disturbance and target

forcing function parameters are listed in Table 2.

11 of 36

Table 2. Forcing function properties.

disturbance, f

d

target, f

t

n

d

ω

d

A

d

φ

d

n

t

ω

t

A

t

φ

t

− rad/s rad rad − rad/s rad rad

5 0.3835 0.0031 -1.2726 6 0.4602 0.0876 6.0641

8 0.6136 0.0071 -3.0424 9 0.6903 0.0764 3.3242

11 0.8437 0.0115 0.1229 13 0.9971 0.0613 6.2580

17 1.3039 0.0193 1.5607 19 1.4573 0.0430 5.7018

28 2.1476 0.0284 -1.0470 29 2.2243 0.0250 1.6352

46 3.5282 0.0363 2.3832 47 3.6049 0.0120 4.7074

59 4.5252 0.0407 -0.4468 61 4.6786 0.0081 2.3684

82 6.2893 0.0488 1.0625 83 6.3660 0.0052 1.3033

106 8.1301 0.0590 -1.5272 107 8.2068 0.0038 0.7234

137 10.5078 0.0755 -1.0093 139 10.6612 0.0029 0.1830

178 13.6524 0.1033 -2.8089 179 13.7291 0.0024 0.5258

211 16.1835 0.1308 -2.3834 213 16.3369 0.0021 2.9077

Finally, as explained in detail in Ref. 25, the relative magnitude of the disturbance and target

forcing function signals can be manipulated to yield predominantly disturbance-rejection or target-

following tasks. Due to the different function of vestibular motion feedback in both these types

of tracking tasks,

6,7,13,24

different manual control behavior is adopted in the presence of physical

motion cues, which would in turn affect the measurable effect of motion perception thresholds

on control. Therefore, as detailed in Ref. 22, two settings were evaluated: (1) mainly disturbance-

rejection, where the target forcing function was scaled down with a factor 0.5 and (2) mainly target-

following, with a full-power target signal and a disturbance forcing function signal scaled down

with a factor 0.5. These two forcing function settings will be referred to as “disturbance-rejection”

and “target-following” or with the symbols “FD” and “FT”, respectively, in the remainder of this

paper.

In addition to this variation in target and disturbance forcing function setting, Valente Pais et

al. considered four different values of the pitch motion cueing gain K

θ

(see Fig. 2): 0.25, 0.5, 0.75

and 1. As will be illustrated in Section IV.B.1 using pilot model simulation data, lowering the value

of K

θ

reduces the magnitude of the supplied pitch motion cues (

¨

θ

m

, see Fig. 2), thereby bringing a

larger portion of the signal sensed by the semicircular canals below the threshold value. Together

with the variation in forcing function settings, these four different settings of K

θ

yielded eight

different experimental conditions that were evaluated in Ref. 22. Valente Pais et al. collected data

from eight subjects, with ﬁve repeated measurements for each experimental condition per subject,

yielding a total of 40 data sets for each combination of control task and K

θ

.

III.B.2. Pilot Model Simulation Data

The experimental data from Ref. 22 are used to illustrate the typical results of the identiﬁcation

procedure, while the simulation data, for which the true pilot model parameters are known, are

12 of 36

used to evaluate identiﬁcation reliability and accuracy. To generate the simulation data, simula-

tions of the closed-loop control task depicted in Fig. 2 with the experimental settings described in

Section III.B.1, including an implementation of the pilot model shown in Fig. 4, were performed.

Pilot model parameters were set at values identiﬁed from the experiments of Ref. 22 and both

the disturbance-rejection and target-following tasks were simulated, using different sets of pilot

model parameters for each task. The numerical values of all pilot model parameters used for the

simulations are listed in Table 3.

Table 3. Pilot model simulation parameters.

K

v

T

vl

τ

v

K

m

τ

m

ω

nm

ζ

nm

∆

abs

T

n

− s s rad/IPUT s rad/s − IPUT s

Disturbance-rejection task 1.057 0.409 0.279 0.532 0.203 12.00 0.220 0.0075 0.0578

Target-following task 0.875 0.630 0.290 0.425 0.186 11.14 0.259 0.0075 0.0645

To be able to assess model identiﬁcation bias and variance for the simulation data, pilot model

simulations were performed for 100 different realizations of the pilot remnant signal n. Remnant

noise was generated by ﬁltering white noise through the fourth-order low-pass ﬁlter given by:

H

n

(jω) =

K

n

(T

n

jω + 1)

4

(6)

The time constant of the remnant ﬁlter was determined from measurement data from Ref. 22

for both the disturbance-rejection and target-following tasks. The values for T

n

that were used in

the simulations are listed in the ﬁnal column of Table 3. The remnant ﬁlter gain K

n

was used to

set the fraction of the control signal variance that is caused by remnant, F

n

:

F

n

=

σ

2

n

σ

2

u

(7)

From previous experiments it is known that F

n

is around 20-25% for typical measurements of

pilot control behavior during tracking.

12

As identiﬁed by McRuer and Jex,

2

the remnant traveling

through the control loop is the result of a number of different nonlinear processes internal to the

human operator. The nonlinear threshold element as included in the pilot model in this study (see

Section II.C) also accounts for a portion of the total nonlinearity in the (simulated) measurements.

Therefore, it is likely that the presence of additional remnant noise, and its power relative to the

effect of the threshold operation, will affect the estimation of the threshold parameter ∆

abs

. To

investigate this, simulation data were generated with F

n

equal to 0 (no remnant), 0.1, 0.2, 0.25,

and 0.3. The values of K

n

used to achieve these remnant fractions for the different control tasks

and values of K

θ

are listed in Table 4.

13 of 36

Table 4. Pilot model simulation remnant gains K

n

for different settings of F

n

and K

θ

.

F

n

disturbance rejection target following

K

θ

= 0.25 0.5 0.75 1 0.25 0.5 0.75 1

0 0 0 0 0 0 0 0 0

0.1 0.080 0.060 0.058 0.063 0.073 0.062 0.058 0.057

0.2 0.148 0.101 0.096 0.104 0.129 0.106 0.098 0.096

0.25 0.209 0.128 0.119 0.129 0.172 0.135 0.122 0.119

0.3 0.370 0.165 0.149 0.162 0.248 0.177 0.156 0.151

IV. Results

IV.A. Identiﬁcation on Experiment Data

IV.A.1. Step I: Genetic Algorithm

As explained in Section III.A, the ﬁrst step in the identiﬁcation procedure proposed in this paper

(see Fig. 5) consists of using a genetic algorithm to ﬁnd initial estimates of all pilot model pa-

rameters, including the absolute threshold ∆

abs

. For the determination of threshold values from

measurement data, the success of this ﬁrst step is very important, as it is the main means of esti-

mating the value of ∆

abs

.

Fig. 6 depicts the ﬁnal estimates of ∆

abs

that result from Step I of the estimation algorithm for

both forcing function settings (FD and FT, rows in Fig. 6) and for all four settings of K

θ

(columns

in Fig. 6). Figures 6(a)-(d) show the identiﬁed threshold values for the disturbance-rejection task,

while Figures 6(e)-(h) present the corresponding target-following task data. The data presented in

Fig. 6, and also further data presented in this section, were collected for subject 1 of the experiment

described in Ref. 22. Equivalent results were obtained for the other participants, but these are not

presented here for the sake of brevity.

Each graph in Fig. 6 depicts the ﬁnal value of ∆

abs

(that is, after the 100 genetic algorithm

iterations) for 20 repetitions of running the genetic algorithm on the same data (variation along

the longitudinal axis). The estimated threshold parameters are presented in ascending order of the

corresponding parameter sets’ likelihood values, with the estimated values of ∆

abs

from the best

sets of estimated model parameters (lowest likelihoods) at left. Furthermore, each graph depicts

these threshold estimates for ﬁve sets of data, corresponding to the ﬁve measurements taken for

each condition during the experiment for Ref. 22. Finally, a horizontal gray line is used to indicate

the perceptual threshold value measured for angular pitch motion by Groen et al.

18

in a passive

perception experiment, ∆

abs

= 0.00734 IPUT, for reference.

Fig. 6 shows effects of both the variation in the motion gain K

θ

and the type of control task –

that is, disturbance-rejection or target-following, indicated with FD and FT in Fig. 6, respectively

– on the obtained estimates of ∆

abs

from measured data. First, for the lowest considered values

of K

θ

, the estimated absolute threshold parameters are very high, on average almost one order

14 of 36

ˆ

∆

abs

, IPUT

GA repetition

(a) subj. 1, FD, K

θ

= 0.25

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(b) subj. 1, FD, K

θ

= 0.5

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(c) subj. 1, FD, K

θ

= 0.75

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(d) subj. 1, FD, K

θ

= 1

ˆ

∆

abs

, Step I

∆

abs

, Ref. 18

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(e) subj. 1, FT, K

θ

= 0.25

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(f) subj. 1, FT, K

θ

= 0.5

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(g) subj. 1, FT, K

θ

= 0.75

0 10 20

0.00

0.05

0.10

ˆ

∆

abs

, IPUT

GA repetition

(h) subj. 1, FT, K

θ

= 1

0 10 20

0.00

0.05

0.10

Figure 6. Estimated absolute threshold parameters for subject 1 for all experimental conditions. Results for

ﬁve measurement runs are shown in each graph.

of magnitude higher than those found for K

θ

= 0.7 5 or 1. In addition, there is markedly less

consistency in the estimates of ∆

abs

over both the data from different runs and the repetitions of

the genetic algorithm for the lowest considered values of K

θ

.

Fig. 6 also shows a difference between the threshold parameters estimates for the disturbance-

rejection and target-following tasks. For the data from the target-following task for both K

θ

= 0.25

and 0.5, a small number of repetitions of the genetic algorithm are found to converge to a value of

∆

abs

that is close to the absolute threshold that are estimated consistently for the higher values of

K

θ

. Note that these solutions always represent the leftmost estimates depicted in Figures 6(e) and

(f), indicating that they correspond to the best estimates provided by the genetic algorithm, that

is, those with the lowest values of the likelihood function. Comparison with Figures 6(a) and (b)

shows that such a limited number of threshold estimates that are consistent with the results for the

higher values of K

θ

are not found for the disturbance-rejection task data.

To further explain the results shown in Fig. 6, Fig. 7 depicts the variation in the negative log-

arithm of the likelihood function −ln L, the pilot visual and motion gains (K

v

and K

m

, respec-

tively), the pilot motion delay τ

m

, and the absolute threshold parameter ∆

abs

as a function of the

100 iterations of the genetic algorithm. Each graph shows the changes in either of these parameters

for three repeated genetic algorithm evaluations (separate lines). Each column of graphs in Fig. 7

corresponds to the estimation results for one of the values of the simulator motion gain K

θ

. Note

that only data for the disturbance-rejection task (FD) are depicted in Fig. 7. Highly similar results

were obtained for the target-following task data. The vertical axes of the graphs that depict the

15 of 36

−ln L, -

GA iteration

(a) subj. 1, FD, K

θ

= 0.25

×10

4

0 50 100

0.6

0.8

1.0

1.2

1.4

−ln L, -

GA iteration

(b) subj. 1, FD, K

θ

= 0.5

×10

4

0 50 100

0.6

0.8

1.0

1.2

1.4

−ln L, -

GA iteration

(c) subj. 1, FD, K

θ

= 0.75

×10

4

0 50 100

0.6

0.8

1.0

1.2

1.4

−ln L, -

GA iteration

(d) subj. 1, FD, K

θ

= 1

rep. 1

rep. 2

rep. 3

×10

4

0 50 100

0.6

0.8

1.0

1.2

1.4

ˆ

K

v

, −

GA iteration

(e) subj. 1, FD, K

θ

= 0.25

0 50 100

0

2

4

6

8

10

ˆ

K

v

, −

GA iteration

(f) subj. 1, FD, K

θ

= 0.5

0 50 100

0

2

4

6

8

10

ˆ

K

v

, −

GA iteration

(g) subj. 1, FD, K

θ

= 0.75

0 50 100

0

2

4

6

8

10

ˆ

K

v

, −

GA iteration

(h) subj. 1, FD, K

θ

= 1

rep. 1

rep. 2

rep. 3

0 50 100

0

2

4

6

8

10

ˆ

K

m

, rad/IPUT

GA iteration

(i) subj. 1, FD, K

θ

= 0.25

0 50 100

0

2

4

6

8

10

ˆ

K

m

, rad/IPUT

GA iteration

(j) subj. 1, FD, K

θ

= 0.5

0 50 100

0

2

4

6

8

10

ˆ

K

m

, rad/IPUT

GA iteration

(k) subj. 1, FD, K

θ

= 0.75

0 50 100

0

2

4

6

8

10

ˆ

K

m

, rad/IPUT

GA iteration

(l) subj. 1, FD, K

θ

= 1

rep. 1

rep. 2

rep. 3

0 50 100

0

2

4

6

8

10

ˆτ

m

, s

GA iteration

(m) subj. 1, FD, K

θ

= 0.25

0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

ˆτ

m

, s

GA iteration

(n) subj. 1, FD, K

θ

= 0.5

0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

ˆτ

m

, s

GA iteration

(o) subj. 1, FD, K

θ

= 0.75

0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

ˆτ

m

, s

GA iteration

(p) subj. 1, FD, K

θ

= 1

rep. 1

rep. 2

rep. 3

0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

ˆ

∆

abs

, IPUT

GA iteration

(q) subj. 1, FD, K

θ

= 0.25

0 50 100

0.00

0.02

0.04

0.06

0.08

0.10

ˆ

∆

abs

, IPUT

GA iteration

(r) subj. 1, FD, K

θ

= 0.5

0 50 100

0.00

0.02

0.04

0.06

0.08

0.10

ˆ

∆

abs

, IPUT

GA iteration

(s) subj. 1, FD, K

θ

= 0.75

0 50 100

0.00

0.02

0.04

0.06

0.08

0.10

ˆ

∆

abs

, IPUT

GA iteration

(t) subj. 1, FD, K

θ

= 1

rep. 1

rep. 2

rep. 3

0 50 100

0.00

0.02

0.04

0.06

0.08

0.10

Figure 7. Likelihood and parameter value variation during the genetic optimization of Step I. Presented data

are from measurement run 1 of the disturbance-rejection tasks performed by subject 1. Separate lines indicate

different repetitions of the Step I optimization.

16 of 36

estimated values of pilot model parameters have been scaled to show the range between the lower

and upper bounds speciﬁed for each parameter in Table 1.

The ﬁrst rowof graphs in Fig. 7 shows that for all valuesof K

θ

, the genetic algorithm achieves a

continuously decreasing trend in likelihood over the 100 iterations, which stabilizes after about 50-

75 iterations. A similar convergence of the genetic algorithm estimate is observed for the estimates

of the visual gain K

v

for all values of K

θ

. Such convergence of the genetic algorithm is expected

for identiﬁcation of a model that is appropriate for a given set of identiﬁcation data.

On the other hand, Fig. 7 shows a marked difference in the variation of the parameters of the

pilot vestibular response (K

m

, τ

m

, and ∆

abs

) during the 100 iterations of the genetic algorithm for

the highest and lowest two settings of K

θ

. For K

θ

= 0.75 and 1, the three depicted repetitions

of the genetic algorithm estimation process stabilize around approximately equal values for these

three parameters after about 50 iterations.

As can be veriﬁed from Fig. 7, this convergence of the estimates of the parameters of the

pilot model vestibular channel for K

θ

= 0.75 and 1 is not visible for the two lower values of

the simulator motion gain. Especially for K

m

and τ

m

the estimated parameter values are seen to

vary between the speciﬁed upper and lower bounds for K

θ

= 0.25 and 0.5, without showing any

convergence. The values of ∆

abs

show the same erratic behavior over the GA iterations, but do

not cover the full range between the speciﬁed upper and lower bounds. A comparison with the

magnitude of the semicircular canal output i

∆

shows that the comparatively high values of ∆

abs

estimated for K

θ

= 0.25 and 0.5 will almost completely disable the vestibular channel of the

pilot model. On average, the standard deviation of the time-domain signal i

∆

was around 0.012

and 0.023 IPUT for K

θ

= 0.25 and 0.5, respectively, for both target-following and disturbance

rejection. The corresponding maximum peak values of i

∆

were around 0.04 and 0.07 IPUT for the

same values of K

θ

. Comparison with the identiﬁed threshold values shown in Figs. 6 and 7 shows

that for these conditions very little of the signal i

∆

will pass the threshold operation. A further

consequence of this is that this yields o

∆

≈ 0, which explains the large variation in observed in the

estimates of K

m

and τ

m

. When the threshold operation blocks nearly all of the signal entering the

pilot model vestibular channel, these parameters can take on any value without affecting the ﬁnal

model output.

In conclusion, the results of Step I of the estimation process indicate that determination of ∆

abs

seems only viable from measurements for which K

θ

is 0.7 5 or higher, for both the disturbance-

rejection and target-followingtasks. For lower valuesof the simulator motion gain, the large spread

in the ﬁnal absolute threshold estimates over different GA repetitions (Fig. 6) and the absence of

convergence over the GA iterations (Fig. 7) suggest too little effect of vestibular feedback on pilot

control to allow for successful identiﬁcation of the model of Fig. 4. This issue will be addressed in

more detail when analyzing the results of Steps II and III of the identiﬁcation procedure.

17 of 36

As illustrated by Figures 6 and 7, the stochastic nature of genetic algorithmscauses the obtained

parameter estimates to differ each time the algorithm is run.

12

As the genetic algorithm of Step I

is the main means of estimating the absolute threshold parameter ∆

abs

from measured data, it is

desirable to limit the effect of the algorithm’s inherent randomness on the ﬁnal results. For this

reason, the genetic algorithm was ran on all data sets a total of 20 times, yielding 20 repeated

estimates of ∆

abs

per data set. As an additional means of removing part of the variance in the

obtained estimates of ∆

abs

, the ﬁnal absolute threshold estimate of Step I was taken to be the

average over the best ﬁve estimates provided by the genetic algorithm, that is, the ﬁve left-most

values depicted in each graph in Fig. 6.

IV.A.2. Step II: Gauss-Newton Algorithm

As illustrated by Fig. 5, Step II of the proposed identiﬁcation procedure involves the further opti-

mization of all pilot model parameters except the absolute threshold ∆

abs

using an unconstrained

steepest-descent Gauss-Newton optimization algorithm as described in Ref. 12. Fig. 8 depicts the

ﬁnal model likelihoods for the 20 repeated estimates performed on the ﬁrst measurement run of

the disturbance-rejection task data from subject 1 for all four values of the simulator motion gain

K

θ

. Only data for one measurement run of the disturbance-rejection task measurements are de-

picted here for brevity, but for all other experimental data comparable results were obtained. The

likelihoods of the parameter sets obtained from Step I are depicted in gray, while those that are ob-

tained after further reﬁnement of all pilot model parameters except ∆

abs

during Step II are shown

in black. Furthermore, a white circular marker indicates the set of parameters that was taken as the

ﬁnal parameter estimate after Step II for the data from each condition.

Fig. 8 shows that there is considerable variability in the likelihoodsof the 20 different parameter

estimates obtained after the optimization with the genetic algorithm during Step I. The parameters

estimated using this genetic algorithm have a high probability of being close to the (global) op-

timum of the identiﬁcation problem, but require further reﬁnement to attain a true minimum of

the likelihood function.

12

The Gauss-Newton optimization performed for Step II has been shown

capable of providing this required reﬁnement of the parameter estimates obtained from the genetic

algorithm.

The data for K

θ

= 0.75 and 1 presented in Figures 8(c) and (d), respectively, showoptimization

results that are consistent with those described in Ref. 12. The large variation in the likelihoods

of the parameter estimates obtained from Step I is reduced to a single likelihood value for all 20

identiﬁcation repetitions for K

θ

= 0.75, indicating that all Step I parameter estimates converge

to the same minimum of the likelihood function. For K

θ

= 1, the different parameter estimates

are found to converge to two distinct likelihood minima, of which one corresponds to a clearly

lower, that is, more optimal value of −ln L. For these two conditions the ﬁnal Step II estimate can

therefore be selected straightforwardly from the 20 repeated estimates.

18 of 36

−ln L, -

GA repetition

(a) subj. 1, FD, K

θ

= 0.25

Step I

Step II

Step II, ﬁnal

×10

3

1 10 20

2.9

3.0

3.1

3.2

3.3

3.4

−ln L, -

GA repetition

(b) subj. 1, FD, K

θ

= 0.5

×10

3

1 10 20

1

2

3

4

5

−ln L, -

GA repetition

(c) subj. 1, FD, K

θ

= 0.75

×10

3

1 10 20

1.4

1.6

1.8

2.0

2.2

−ln L, -

GA repetition

(d) subj. 1, FD, K

θ

= 1

×10

3

1 10 20

0.2

0.4

0.6

0.8

1.0

1.2

Figure 8. Optimization of pilot model likelihood during Step II of the optimization process for subject 1 for the

disturbance-rejection task. Each graph depicts the likelihoods of the 20 repeated estimates for data from the

ﬁrst measurement run for one of the settings of the simulator motion gain K

θ

.

19 of 36

Fig. 8(a) shows that this convergence of the parameter estimates to a limited number of dif-

ferent solutions is not achieved for the data from the condition with the lowest simulator motion

gain, K

θ

= 0.25. For this condition, it is clear that the Gauss-Newton optimization hardly suc-

ceeds in further reﬁnement of the estimates from Step I and would typically terminate after only

a few iterations without converging to a proper minimum of the likelihood function. This is what

would be expected for an apparent mismatch between model and data as present here due to the

fact that the high estimated value of ∆

abs

for this condition (see Fig. 6) effectively disables the

vestibular part of the pilot model and hence causes the vestibular channel parameters K

m

and τ

m

to be unidentiﬁable.

As can be veriﬁed from Fig. 8(b), the success of Step II of the identiﬁcation procedure for the

presented data from K

θ

= 0.5 is in between the results shown for K

θ

= 0.25 and the two highest

settings of the simulator motion gain. The Step II results depicted in Fig. 8 therefore conﬁrm the

main results from Step I, that is, the fact that the proposed pilot model can only be accurately

estimated for data from conditions with K

θ

= 0.75 and higher.

IV.A.3. Step III: Final Threshold Optimization

The combination of Steps I and II of the identiﬁcation procedure depicted in Fig. 5 provides an

estimate of all pilot model parameters that has a high probability of representing a solution close

to the optimum of the likelihood function for the given model and data set. Fig. 9 depicts the

negative logarithm of the likelihood function as a function of the value of the absolute threshold

parameter ∆

abs

(all other parameters are ﬁxed to their value from Step II) for the four different

values of the simulator motion gain K

θ

. Fig. 9 shows these likelihood curves for the data from

the ﬁrst measurement run performed by subject 1. Note that the range of absolute threshold values

considered in Fig. 9 is consistent with the upper and lower bounds set for this parameter in Table 1.

Furthermore, it should be remarked that for this evaluation of the relation between ∆

abs

and −ln L

the other pilot model parameters were set to the reﬁned estimates from Step II (see Section IV.A.2).

Fig. 9 shows typical data from the disturbance-rejection task performed by subject 1 of the

experiment of Ref. 22. The solid curves in the graphs in Fig. 9 represent the variation in likelihood

with ∆

abs

, while the 20 estimates of ∆

abs

from Step I are depicted with crosses on this curve.

The single averaged estimate of ∆

abs

from Step II is depicted with a vertical solid black line and a

white circular marker on the likelihood curve. Thetrue minimum of −ln L for this one-dimensional

optimization problem is depicted with a second vertical solid black line and a black circular marker

on the likelihood curve.

Fig. 9 shows that further optimization of

ˆ

∆

abs

after Steps I and II is typically possible, as slight

reductions in −ln L can still be achieved compared to the solutions obtained after Step II (white

markers). The 20 estimates of ∆

abs

found from Step I (crosses in Fig. 9), however, are found to

be around the ﬁnal likelihood curve minimum, especially for the higher values of the simulator

20 of 36

∆

abs

, IPUT

−ln L, -

(a) subj. 1, FD, K

θ

= 0.25, run 1

×10

3

−ln L(∆

abs

)

ˆ

∆

abs

, Step I (20)

ˆ

∆

abs

, Step II (1)

ˆ

∆

abs

, Step III (1)

0 0.02 0.04 0.06 0.08 0.1

3.0

3.5

4.0

4.5

5.0

5.5

∆

abs

, IPUT

−ln L, -

(b) subj. 1, FD, K

θ

= 0.5, run 1

×10

4

0 0.02 0.04 0.06 0.08 0.1

0.0

0.5

1.0

1.5

∆

abs

, IPUT

−ln L, -

(c) subj. 1, FD, K

θ

= 0.75, run 1

×10

3

0 0.02 0.04 0.06 0.08 0.1

0

1

2

3

4

5

6

∆

abs

, IPUT

−ln L, -

(d) subj. 1, FD, K

θ

= 1, run 1

×10

3

0 0.02 0.04 0.06 0.08 0.1

0

1

2

3

4

5

6

Figure 9. Optimized estimates of the absolute threshold parameter ∆

abs

from Step III compared to estimates

from Steps I and II. Each graph shows the likelihood variation as a function of ∆

abs

for the ﬁrst measurement

run forone of the settings of the simulator motion gain taken fromthe disturbance-rejection task measurements

of subject 1.

21 of 36

motion gain. This indicates that the averaged threshold estimate calculated for evaluating Step

II seems to provide an acceptable estimate of ∆

abs

. Furthermore, note that the likelihood curves

depicted in Figures 9(c) and (d) are similar to those shown in Fig. 13 for pilot model simulation

data. These graphs show that increasing ∆

abs

to values above 0.02 IPUT yields an increase in

likelihood, indicating degraded model ﬁt.

For simulator motion gains of 0.25 and 0.5, Figures 9(a) and (b) show opposite trends in the

likelihood curves. For those lower values of K

θ

, reduction of ∆

abs

below a certain value yield

signiﬁcant degradation of the model ﬁt. This is consistent with the analysis of the Step I estimation

results in Section IV.A.1 and indicates no signiﬁcant contribution of the pilot model vestibular

channel for these conditions.

To amend the fact that not all model parameters can be optimized during Step II, identiﬁca-

tion Steps II and III can be performed recursively until no further changes in both ∆

abs

and the

remaining pilot model parameters are observed. Despite the apparent improvement in ﬁt obtained

from Step III (see Fig. 9), such recursive execution of Steps II and III was not found to provide

signiﬁcant further reduction in −ln L and pilot model parameter values for the experiment data

from Ref. 22.

IV.A.4. Threshold Parameter Estimation Results

Fig. 10 shows the ﬁnal estimates of the threshold parameter ∆

abs

obtained after the three steps

of the estimation procedure. Each graph shows a histogram of the estimated thresholds for one

evaluated setting of K

θ

for either the disturbance-rejection task or target-following task data. Note

that each graph shows a total of 40 identiﬁed values of ∆

abs

: one for each of the ﬁve data sets

collected for each of the eight subjects for all experimental conditions. The mean of the distribution

is depicted with a vertical solid black line and the numerical values of the mean µ and standard

deviation σ of the histogram are listed in each graph.

Fig. 10 shows that estimated values of ∆

abs

become increasingly more consistent with increas-

ing K

θ

, for both the disturbance-rejection and target-following task data. For K

θ

= 0.25 and 0.5

wide distributions of the estimated threshold parameter values are observed, and the individual

bins of the histograms (spaced 0.005 IPUT apart) for these values of K

θ

are found to never hold

more than 10 estimates. As also visible from the decreasing values of the standard deviations with

increasing pitch motion cueing gains listed in all graphs in Fig. 10, increasingly more consistent

estimated values of ∆

abs

are obtained for the data from K

θ

= 0.75 and 1. For K

θ

= 1, around

60% of the estimated threshold parameter values are within the two bins closest to the mean of

the distribution, while for the lower values of K

θ

this percentage can be seen to be lower from the

presented distributions.

In addition to the decreasing standard deviations observed in Fig. 10, also the means µ of the

presented histograms are seen to decrease with increasing K

θ

. Note that there is a factor 2-5 dif-

22 of 36

(a) FD, K

θ

= 0.25

f(

ˆ

∆

abs

), -

µ= 0.0326

σ= 0.0133

0 0.05 0.1

0

5

10

15

(b) FD, K

θ

= 0.5

µ= 0.0481

σ= 0.0230

0 0.05 0.1

0

5

10

15

(c) FD, K

θ

= 0.75

µ= 0.0125

σ= 0.0135

0 0.05 0.1

0

5

10

15

(d) FD, K

θ

= 1

Distribution

Mean, µ

µ= 0.0089

σ= 0.0080

0 0.05 0.1

0

5

10

15

replacements

(e) FT, K

θ

= 0.25

f(

ˆ

∆

abs

), -

ˆ

∆

abs

, IPUT

µ= 0.0215

σ= 0.0118

0 0.05 0.1

0

5

10

15

(f) FT, K

θ

= 0.5

ˆ

∆

abs

, IPUT

µ= 0.0201

σ= 0.0201

0 0.05 0.1

0

5

10

15

(g) FT, K

θ

= 0.75

ˆ

∆

abs

, IPUT

µ= 0.0112

σ= 0.0135

0 0.05 0.1

0

5

10

15

(h) FT, K

θ

= 1

ˆ

∆

abs

, IPUT

µ= 0.0077

σ= 0.0077

0 0.05 0.1

0

5

10

15

Figure 10. Estimated absolute threshold distributions for all experiment data sets. Histograms show compiled

data for eight subjects and 5 repeated measurements per subject (N = 40).

ference between the mean threshold estimates obtained for K

θ

= 1 and those for the lowest values

of K

θ

. As also shown for data from one subject in Fig. 6, the average estimated absolute thresh-

old parameters for the highest pitch motion gains are found to be close to the absolute thresholds

reported by Groen et al.

18

based on the experimental data of Heerspink et al.

16

(0.00734 IPUT).

Furthermore, also the comparatively high values of ∆

abs

shown for K

θ

= 0.25 and 0.5 in Fig. 6 are

seen to be conﬁrmed by the averages of the distributions in Fig. 10. Also given the large spread ob-

served in the threshold parameter estimates, especially those for K

θ

= 0.5, the average estimated

threshold results show that under these conditions reliable estimation ∆

abs

is not possible.

For veriﬁcation of the ﬁts of the augmented multimodal pilot model used for estimating the

vestibular threshold values presented in Fig. 10, the same pilot model with the threshold operation

removed was ﬁt to the same data. Fig. 11 shows the ﬁnal negative log-likelihoods – that is, the

values of the cost function that was minimized during the estimation procedure – for both the pilot

models with and without the threshold operation ﬁtted to the data for all four values of K

θ

and

the two different tracking tasks. Again, each graph shows the compiled results for all subjects and

measurement runs (N = 40). A 1-to-1 line is depicted in gray in each graph for reference, where

data points below this line represent data sets for which the model with the threshold operation

yielded a better ﬁt to the data (lower −ln L). Data points above the 1-to-1 line represent data sets

for which the identiﬁed model without the threshold was found to yield a better ﬁt. Finally, each

graph shows the average log-likelihood for both models over the 40 data sets, in addition to the

results of a t-test performed to compare the ﬁnal likelihoods of the ﬁts of both models.

Comparison of the mean (µ) likelihoods for the models with and without the threshold oper-

ation listed in each graph in Fig. 11 shows that on average lower values of −ln L for the model

23 of 36

(a) FD, K

θ

= 0.25

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −20032.5

µ(no ∆

abs

)= −20195.8

t(78) = −0.204, p = 0.84

Data, N = 40

1-to-1 line

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(b) FT, K

θ

= 0.25

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −21441.0

µ(no ∆

abs

)= −21714.3

t(78) = −0.353, p = 0.73

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(c) FD, K

θ

= 0.5

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −20594.4

µ(no ∆

abs

)= −20840.0

t(78) = −0.346, p = 0.73

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(d) FT, K

θ

= 0.5

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −22196.6

µ(no ∆

abs

)= −22323.7

t(78) = −0.188, p = 0.85

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(e) FD, K

θ

= 0.75

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −20704.0

µ(no ∆

abs

)= −20692.8

t(78) = 0.017, p = 0.99

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(f) FT, K

θ

= 0.75

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −22490.3

µ(no ∆

abs

)= −22489.6

t(78) = 0.001, p = 1.00

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(g) FD, K

θ

= 1

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −20660.1

µ(no ∆

abs

)= −20643.1

t(78) = 0.028, p = 0.98

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

(h) FT, K

θ

= 1

−ln L [no ∆

abs

], -

−ln L [with ∆

abs

], -

µ(∆

abs

) = −22332.8

µ(no ∆

abs

)= −22325.2

t(78) = 0.014, p = 0.99

×10

4

×10

4

-3.0 -2.5 -2.0 -1.5

-3.0

-2.5

-2.0

-1.5

Figure 11. Comparison of the ﬁnal likelihood of pilot model ﬁts obtained using multimodal pilot models with

and without vestibular threshold operation in the model.

24 of 36

including the threshold were only found for K

θ

= 0.75 and 1. For the two lowest settings of

K

θ

, for both the disturbance-rejection and target-following data the model without the threshold is

on average found to yield a better ﬁt, indicating that the extension of the model with a threshold

would actually not be appropriate for these data sets. As can be veriﬁed from the data presented

in Figs. 11(e) to (h), despite the fact that on average the better ﬁts of the model that included the

threshold were obtained for K

θ

= 0.75 and 1, this was not so for all 40 considered data sets. Fur-

thermore, as can be observed from the t-test results listed in all graphs in Fig. 11, the difference in

the likelihood of both models was not found to be signiﬁcant in any of the considered cases. This

indicates that for all considered data sets, the measurable effect of the threshold operation on the

control behavioral measurements considered in the selected approach, if even present, is conﬁrmed

to be comparatively minor, as anticipated.

From a modeling standpoint, the data shown in Fig. 11 therefore suggest that the addition of

a threshold operation to multimodal pilot models is perhaps not needed for accurately capturing,

explaining, and predicting manual control behavior. However, the comparatively small effect of

the addition of a threshold operation in a multimodal pilot model does not necessarily preclude the

fact that threshold values can still be estimated, with some accuracy, using the proposed modeling

approach. The data presented in Fig. 11 do suggest that if this is indeed possible, the results with

the highest accuracy can be expected for the data for the highest values of K

θ

. Furthermore, as

also concluded from the Step I results presented in Section IV.A.1, the results obtained for the

target-following task data appear to be slightly more consistent (lower σ). The magnitude of the

measurable effect of a vestibular threshold operation, which is of course important for the proposed

model identiﬁcation approach, and the reliability with which the magnitude of vestibularthresholds

can be estimated from control task measurements is further assessed in Section IV.B using pilot

model simulation data.

IV.B. Identiﬁcation on Simulation Data

IV.B.1. Quantiﬁcation of Threshold Effect for Varying Motion Gain

As already pointed out in Section II.A, for a given set of forcing function signals (f

d

and f

t

)

the magnitude of the effect of the motion perception threshold on pilot control behavior and the

measured control signal u will likely depend on the value of the simulator motion scaling gain K

θ

.

As an example, Fig. 12 depicts the in- and output signals of the absolute threshold block of the

pilot model, i

∆

and o

∆

, for two values of the simulator motion gain. For reference, the value of

the absolute threshold parameter used for the pilot model simulations – ∆

abs

= 0.0075 IPUT, see

Table 3 – is depicted in Fig. 12 using dashed lines. Note the different scaling of the vertical axis

for both graphs.

25 of 36

(a) K

θ

= 0.25

t, s

i

∆

, o

∆

, IPUT

+∆

abs

−∆

abs

threshold input, i

∆

threshold output, o

∆

0 2 4 6 8 10

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

(b) K

θ

= 1

t, s

i

∆

, o

∆

, IPUT

+∆

abs

−∆

abs

0 2 4 6 8 10

-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

Figure 12. Example time traces of i

∆

and o

∆

from pilot model simulations for different settings of pitch motion

gain K

θ

(disturbance-rejection, F

n

= 0.25).

As illustrated by Fig. 12, for lower values of K

θ

the magnitude of i

∆

will also be lower. Hence,

for a given value of ∆

abs

this means that the magnitude of the dead zone threshold model becomes

relatively larger with respect to the magnitude of i

∆

, yielding more effect of the threshold on o

∆

,

and hence also on u, than for K

θ

= 1. This further implies that the effect of the threshold will

diminish for high-amplitude physical motion, as then nearly all of i

∆

will be larger than ∆

abs

,

yielding o

∆

≈ i

∆

. Table 5 lists the probability of i

∆

being above the selected value of ∆

abs

for

the pilot model simulations (0.0075 IPUT, see Table 3) for the four settings of K

θ

and both types

of tracking task. The presented numbers represent the mean ± standard deviation over the 100

generated realizations of pilot model simulation data. As can be veriﬁed from Table 5, i

∆

is on

average above threshold around 85% of the time for both disturbance rejection and target following

for K

θ

= 1. This percentage is seen to drop to around 60-70% for K

θ

= 0.25.

Table 5. Probability of i

∆

being abovethreshold for pilot sim-

ulation data for different values of K

θ

, F

n

= 0.25.

K

θ

disturbance rejection target following

0.25 0.73 ± 0.0072 0.64 ± 0.0074

0.5 0.77 ± 0.0045 0.75 ± 0.0050

0.75 0.82 ± 0.0029 0.81 ± 0.0042

1 0.86 ± 0.0024 0.84 ± 0.0034

Intuitively, this change in the impact the threshold operation has on the simulated model output

for the different settings of K

θ

will affect the precision with which it can be identiﬁed with the

proposed model identiﬁcation procedure (see Section III.A). Furthermore, the lower the relative

contribution of the vestibular response to the observed control behavior, which is observed with

reducing K

θ

for the experiment data from Ref. 22, the lower the sensitivity of the optimization

26 of 36

cost function to changes in the vestibular channel parameters. These contradictory effects suggest

a trade-off between maximizing the effect of the threshold on measured behavior and retaining suf-

ﬁcient excitation of the vestibular response to allow for reliable parameter estimation. To illustrate

this effect on the likelihood function as given by Eq. (4), Fig. 13 depicts −ln L as a function of

∆

abs

for data from the pilot model simulations for two values of the simulator motion gain K

θ

and

three of the evaluated values of the remnant fraction F

n

: 0, 0.1 and, 0.25.

In each graph, ten different curves are depicted, representing different realizations of the rem-

nant signal n. Note that Fig. 13 depicts the variation in likelihood that occurs if only the value

of ∆

abs

is varied; the other pilot model parameters are ﬁxed to their true values as listed in Ta-

ble 3. Finally, note that the vertical dashed lines in Fig. 13 depict the value of ∆

abs

that was used

to generate the simulation data (0.0075 IPUT, see Table 3) and that circular markers indicate the

minimum of the likelihood function, and hence the optimal estimate of ∆

abs

, for each data set.

−ln L, -

∆

abs

, IPUT

(a) K

θ

= 0.25, F

n

= 0

×10

−1

0 0.02 0.04 0.06 0.08 0.1

-2.8

-2.6

-2.4

-2.2

−ln L, -

∆

abs

, IPUT

(b) K

θ

= 0.25, F

n

= 0.1

×10

−1

0 0.02 0.04 0.06 0.08 0.1

-2.8

-2.6

-2.4

-2.2

−ln L, -

∆

abs

, IPUT

(c) K

θ

= 0.25, F

n

= 0.25

×10

−1

0 0.02 0.04 0.06 0.08 0.1

-2.8

-2.6

-2.4

-2.2

−ln L, -

∆

abs

, IPUT

(d) K

θ

= 1, F

n

= 0

×10

−1

0 0.02 0.04 0.06 0.08 0.1

-2.8

-2.6

-2.4

-2.2

−ln L, -

∆

abs

, IPUT

(e) K

θ

= 1, F

n

= 0.1

×10

−1

0 0.02 0.04 0.06 0.08 0.1

-2.8

-2.6

-2.4

-2.2

−ln L, -

∆

abs

, IPUT

(f) K

θ

= 1, F

n

= 0.25

true ∆

abs

likelihood

min. likelihood

×10

−1

0 0.02 0.04 0.06 0.08 0.1

-2.8

-2.6

-2.4

-2.2

Figure 13. Likelihood variation as a function of absolute threshold parameter value for two settings of the

motion gain K

m

and three settings of the remnant fraction F

n

(disturbance-rejection task). Insets depict a

magniﬁed view of the likelihood curves in the gray-shaded areas.

When no additional remnant is present (F

n

= 0), the true value of ∆

abs

indeed represents a

distinct minimum of the likelihood function, as can be veriﬁed from Figs. 13(a) and (d). Both

increasing and decreasing the value of ∆

abs

yields inferior model ﬁts and increased values of

−ln L. Note from the insets in both ﬁgures, which depict a magniﬁed view of the likelihood

27 of 36

curves for ∆

abs

varying from 0 to 0.0225 IPUT, that the difference between setting ∆

abs

to zero

(no threshold) and its true value are more pronounced for the lowermotion gain setting, K

θ

= 0.25.

This conﬁrms the effect shown in Fig. 12, illustrating the larger impact of the threshold for lower

motion magnitudes.

Figs. 13(a) and (d) illustrate one further effect of the value of K

θ

on the relation between the

likelihood function and ∆

abs

. For K

θ

= 0.25, the likelihood curves level off at absolute threshold

values above 0.04 IPUT. This can be easily explained by referring to Fig. 12(a), which shows

that the maximum absolute value of i

∆

is around 0.04 for this setting of the motion gain. Hence,

for high values of ∆

abs

nearly all of the signal i

∆

is blocked by the threshold block, yielding no

contribution of the pilot vestibular response H

p

m

(jω) to the total model response. The likelihood

plateau in Fig. 13(a) therefore corresponds to pilot model ﬁts in which the vestibular channel has

been disabled and where only the visual part of the model (H

p

v

(jω)) is active.

Fig. 13(d) shows that this effect does not occur for K

θ

= 1 in the reasonable range of threshold

values considered here. In addition, note that the likelihood increases more compared to the min-

imum value for higher values of ∆

abs

than observed for K

θ

= 0.25. This indicates a more stable

minimum for higher motion gains, as for K

θ

= 0.25 the difference in likelihood between the true

minimum and the plateau that effectively yields H

p

m

(jω) ≈ 0 is comparatively small.

It can be seen from Fig. 13 that as remnant magnitude increases, so does the likelihood for all

values of ∆

abs

, for both values of K

θ

. This could be expected since for increasing F

n

the fraction

of u that is captured by the pilot model decreases, yielding increased −ln L. The presence of n is

further seen to affect the average magnitude of −ln L and thereby to induce vertical offsetsbetween

the likelihood curves that are obtained for the different realizations of n. Furthermore, as can be

seen from Fig. 13(b), the remnant noise yields less smooth likelihood curves, even introducing

additional local minima of the likelihood function for some remnant realizations. Finally, note

that the global minima of the likelihood curves, indicated with the circular markers, no longer

correspond to ∆

abs

= 0.0075, the true value of the threshold parameter used for the simulations.

As could be expected, these effects of n on −ln L become more pronounced for increased remnant

levels (compare the data presented for F

n

= 0.1 and F

n

= 0.25 in Fig. 13).

The increased stability of the minimum in the likelihood function for K

θ

= 1 compared to

K

θ

= 0.25 seen in Figs. 13(a) and (d) is found to persist when additional remnant is present. Es-

pecially for the higher remnant levels, this results in decreased spread of the likelihood minima

obtained for the different realizations of n when compared to K

θ

= 0.25 (compare Figs. 13(c)

and (f)). This suggests that despite the reduced effect of the absolute threshold parameter ∆

abs

as

illustrated by Fig. 12 for higher values of K

θ

, the spread in the estimates of ∆

abs

from measure-

ments with values of F

n

typical for human manual control also reduces. The reduced spread in the

minima of −ln L therefore suggests that a more reliable determination of ∆

abs

might be possible

for still higher values of K

θ

or forcing function signals of higher magnitude than those considered

28 of 36

here. Still, Fig. 13 illustrates that the effect of the threshold parameter on the model likelihood is,

at best, comparatively small and that estimation of ∆

abs

can only be performed with a certain accu-

racy, especially for remnant noise levels representative for experimental measurements of tracking

behavior.

IV.B.2. Threshold Estimation Accuracy Veriﬁcation

The experimental results indicate that the success of the approach to determining human motion

perception thresholds as proposed in this paper is affected by choices in the design of the con-

sidered control task. For instance, a larger effect of the perception threshold on manual control

behavior was expected for low-magnitude physical motion feedback (see Section IV.B.1). Exper-

imental measurements, however, suggested that for these low values of the simulator motion gain

K

θ

the measurable contribution of H

p

m

(jω) to the control signal u was very limited. This made

measurement of the pilot vestibular dynamics, and hence the estimated values of the absolute mo-

tion perception threshold, inaccurate. This degrading accuracy in the obtained multimodal pilot

model estimation results was found to be less pronounced for the data from the target-following

task as compared to the disturbance-rejection task data, indicating that the power and design of the

forcing function signals also affect the accuracy with which thresholds can be estimated with the

proposed method.

In a similar format as used in Fig. 10, Figures 14 and 15 show the histograms of the estimated

absolute threshold parameters for the simulated disturbance-rejection and target-following task

data, respectively. Note that only data for F

n

= 0, 0.2, and 0.3 are considered here for brevity.

In addition to the histogram, which shows the distribution of the estimated values of ∆

abs

for the

100 realizations of the remnant noise considered for all cases, each plot depicts the true absolute

threshold value (∆

abs

= 0.0075 IPUT, see Table 3) as a vertical black line. Furthermore, all

ﬁgures list the numerical values of the means µ and sample standard deviations σ of the presented

distributions. Finally, each ﬁgure also lists the number of realizations out of 100 for which the

estimate of the threshold parameter ∆

abs

is within ±50% of its true value, to provide an intuitive

measure of estimation accuracy. The margins of this 50% range are indicated with the dashed black

lines shown on both sides of the depicted true threshold value.

The absolute threshold parameters identiﬁed from the pilot model simulation data that are

depicted in Figures 14 and 15 show the same effect of the variation in the simulator motion gain

K

θ

as observed from the threshold estimates obtained from the experimental data, see Fig. 10. For

the lower values of K

θ

(graphs at left in Figures 14 and 15), the estimates of ∆

abs

are found to

be, on average, around one order of magnitude higher than the true value of the absolute threshold

parameter used for the simulations. Despite the fact that pilot adaptation to this variation in K

θ

is

not taken into account for the data shown in Figures 14 and 15 – as explained in Section III.B.2,

the parameters listed in Table 3 were used for the pilot model simulations for all variations in K

θ

29 of 36

(a) K

θ

= 0.25, F

n

= 0

f(

ˆ

∆

abs

), -

µ = 0.0671

σ = 0.0109

N

50%

= 0

0 0.05 0.1

0

10

20

(b) K

θ

= 0.5, F

n

= 0

µ = 0.0666

σ = 0.0117

N

50%

= 0

0 0.05 0.1

0

10

20

(c) K

θ

= 0.75, F

n

= 0

µ = 0.0527

σ = 0.0187

N

50%

= 2

0 0.05 0.1

0

10

20

(d) K

θ

= 1, F

n

= 0

Distribution

True value

50% margin

µ = 0.0056

σ = 0.0015

N

50%

= 93

0 0.05 0.1

0

10

20

30

40

50

60

(e) K

θ

= 0.25, F

n

= 0.2

f(

ˆ

∆

abs

), -

µ = 0.0680

σ = 0.0127

N

50%

= 0

0 0.05 0.1

0

10

20

(f) K

θ

= 0.5, F

n

= 0.2

µ = 0.0710

σ = 0.0141

N

50%

= 0

0 0.05 0.1

0

10

20

(g) K

θ

= 0.75, F

n

= 0.2

µ = 0.0287

σ = 0.0210

N

50%

= 24

0 0.05 0.1

0

10

20

(h) K

θ

= 1, F

n

= 0.2

µ = 0.0088

σ = 0.0049

N

50%

= 55

0 0.05 0.1

0

10

20

30

(i) K

θ

= 0.25, F

n

= 0.3

f(

ˆ

∆

abs

), -

ˆ

∆

abs

, IPUT

µ = 0.0626

σ = 0.0218

N

50%

= 1

0 0.05 0.1

0

10

20

(j) K

θ

= 0.5, F

n

= 0.3

ˆ

∆

abs

, IPUT

µ = 0.0644

σ = 0.0189

N

50%

= 1

0 0.05 0.1

0

10

20

(k) K

θ

= 0.75, F

n

= 0.3

ˆ

∆

abs

, IPUT

µ = 0.0186

σ = 0.0172

N

50%

= 27

0 0.05 0.1

0

10

20

(l) K

θ

= 1, F

n

= 0.3

ˆ

∆

abs

, IPUT

µ = 0.0112

σ = 0.0072

N

50%

= 42

0 0.05 0.1

0

10

20

Figure 14. Estimated absolute threshold distributions for simulated disturbance-rejection task data.

(a) K

θ

= 0.25, F

n

= 0

f(

ˆ

∆

abs

), -

µ = 0.0594

σ = 0.0124

N

50%

= 0

0 0.05 0.1

0

10

20

(b) K

θ

= 0.5, F

n

= 0

µ = 0.0589

σ = 0.0161

N

50%

= 0

0 0.05 0.1

0

10

20

(c) K

θ

= 0.75, F

n

= 0

µ = 0.0337

σ = 0.0200

N

50%

= 14

0 0.05 0.1

0

10

20

(d) K

θ

= 1, F

n

= 0

Distribution

True value

50% margin

µ = 0.0095

σ = 0.0031

N

50%

= 88

0 0.05 0.1

0

10

20

30

40

50

(e) K

θ

= 0.25, F

n

= 0.2

f(

ˆ

∆

abs

), -

µ = 0.0596

σ = 0.0144

N

50%

= 0

0 0.05 0.1

0

10

20

(f) K

θ

= 0.5, F

n

= 0.2

µ = 0.0548

σ = 0.0170

N

50%

= 0

0 0.05 0.1

0

10

20

(g) K

θ

= 0.75, F

n

= 0.2

µ = 0.0189

σ = 0.0151

N

50%

= 32

0 0.05 0.1

0

10

20

(h) K

θ

= 1, F

n

= 0.2

µ = 0.0090

σ = 0.0047

N

50%

= 65

0 0.05 0.1

0

10

20

30

(i) K

θ

= 0.25, F

n

= 0.3

f(

ˆ

∆

abs

), -

ˆ

∆

abs

, IPUT

µ = 0.0587

σ = 0.0136

N

50%

= 0

0 0.05 0.1

0

10

20

(j) K

θ

= 0.5, F

n

= 0.3

ˆ

∆

abs

, IPUT

µ = 0.0543

σ = 0.0216

N

50%

= 4

0 0.05 0.1

0

10

20

(k) K

θ

= 0.75, F

n

= 0.3

ˆ

∆

abs

, IPUT

µ = 0.0141

σ = 0.0129

N

50%

= 37

0 0.05 0.1

0

10

20

(l) K

θ

= 1, F

n

= 0.3

ˆ

∆

abs

, IPUT

µ = 0.0098

σ = 0.0070

N

50%

= 40

0 0.05 0.1

0

10

20

30

Figure 15. Estimated absolute threshold distributions for simulated target-following task data.

30 of 36

and F

n

– this shows that reducing K

θ

below 0.75 already yields such a limited the contribution of

the pilot model vestibular channel that the identiﬁcation of this part of the model, and hence ∆

abs

,

is affected.

For K

θ

= 1, both the disturbance-rejection and target-following task data show estimates of

∆

abs

that give an approximately Gaussian distribution around the true value of ∆

abs

= 0.0 075

IPUT, as would be expected for a parameter that can be estimated accurately from noisy measure-

ments. Furthermore, note that increasing the power of the remnant noise is found to affect the

spread in the estimated values of ∆

abs

for K

θ

, as the distributions shown in Figures 14 and 15

are found to become wider with increasing F

n

. This is also reﬂected in the number of estimated

thresholds that are found to be within 50% of the true threshold value: N

50%

is seen to decrease

from around 90 for the case where there is no remnant to around 40 for F

n

= 0.3. Given the mod-

est measurable effect of ∆

abs

on the model output and the likelihood function, this is an expected

result, as increasing remnant noise typically yields larger bias and spread in estimated pilot model

parameter results.

12

Still, note that even for a representative remnant noise level of F

n

= 0.2 ,

which is typically reported for tracking task measurements,

12

the pilot model simulation data show

that reliable identiﬁcation of ∆

abs

is possible for K

θ

= 1.

Both the presented histograms and the corresponding values of N

50%

for K

θ

= 0.75 show

threshold parameter estimates that become increasingly more consistent with increasing levels of

remnant noise. Note that this is opposite to what is observed for K

θ

= 1 and to what would be

expected for increasing noise on identiﬁcation data sets. This effect can be explained by consider-

ing that for pilot model simulation data, the remnant signal n provides excitation of the simulated

pilot vehicle system in a similar way to the target and disturbance forcing functions f

t

and f

d

. For

cases where the latter do not provide sufﬁcient excitation for identifying a portion of the model, the

addition of more high-powered remnant noise may provide part of the lacking model excitation.

As also observed for the threshold parameter estimation results for the experiment data, for the rep-

resentative remnant noise level condition F

n

= 0.2 the estimates of ∆

abs

are found to be slightly

more consistent for the simulated target-following task data than for the disturbance-rejection task

data with N

50%

= 65 and 55, respectively. Thereby, this analysis of the identiﬁcation of the abso-

lute threshold parameter from pilot model simulation data conﬁrms the observation made from the

experimental data, that is, that the most accurate estimates of the absolute threshold parameter can

be achieved for the excitation provided by the forcing functions during a target-following task, for

a motion cueing gain K

θ

equal to 1.

V. Discussion

This paper proposed an experimental method, and a tailored mathematical identiﬁcation pro-

cedure, for the estimation of nonlinear human motion perception thresholds from measurements

31 of 36

of pilot control behavior during active control tasks with visual and vestibular motion feedback.

Using data from a recent ﬂight simulator experiment,

22

in which eight participants performed a

pitch attitude tracking task in the presence of simulator pitch motion cues, it is found that depend-

ing on the magnitude of the supplied physical motion feedback, the magnitude of human motion

perception thresholds during active control can be estimated.

The experimental and off-line simulation results indicate that there is a strong inﬂuence of the

magnitude of the presented angular simulator motion on the success with which the perception

thresholds can be estimated. For a given controlled element, the magnitude of the perceivable

simulator motion is determined by the magnitude of the applied forcing function signals and the

gain with which simulator motion cues are scaled compared to the controlled element attitude.

Even though it would be expected that motion magnitudes should not become too large, as then

nearly all physical motion is above threshold, yielding no measurable effect of the threshold on

human control behavior, the data from the experiment of Valente Pais et al.

22

show that thresh-

olds can be measured with the highest accuracy for the experimental conditions in which motion

magnitudes were largest. The reason for this is thought to be that for the lowest considered motion

cueing levels, not enough use of motion feedback was made in the experiment to allow for reliable

identiﬁcation of the vestibular response of the pilot model, including the threshold parameter.

Experiments with higher magnitude forcing function signals than those considered in Ref. 22

should be performed to attempt to further increase the accuracy of the threshold measurements that

can be obtained with the proposed method. In addition, as there is a strong relation between the

characteristics of the applied forcing function signals and the accuracy with which pilot models

can be identiﬁed from experimental measurements,

25,31

further optimization of the characteristics

of the forcing function signals themselves – that is, the number of sinusoids and the sinusoid

frequency, magnitude, and phase distributions – should also be performed.

For any modeling effort, the importance of selecting an appropriate model for the phenomenon

under consideration is paramount. The selection of an invalid model structure can cause problems

in the ﬁtting of the model to experimental measurements and the interpretation of the modeling

results. Multimodal pilot models like the one used in this paper for modeling manual control be-

havior for the pitch attitude control tasks from the experiment of Ref. 22 have been shown to be

applicable to such multimodal control tasks in many previous investigations.

6,7,12,25,27

For the dead

zone absolute threshold model included in the pilot model, however, such extensive experimental

validation is yet to be performed. Rather than a crisp dead zone, human motion perception thresh-

old dynamics could be more complex. Some investigations have even suggested that a distinction

should be made between perceived