stochastic dynamic modeling of human visuomotor tracking

Transactions of ISCIE, Vol. 31, No. 6, pp. 209–219, 2018 209

Paper

Stochastic Dynamic Modeling of HumanVisuomotor Tracking Task of an UnstableVirtual Object*

Shigeki Matsumoto†, Katsutoshi Yoshida‡ and Munehisa Sekikawa‡

We conducted an experiment of a visuomotor tracking task undertaken by human participants andcompared it with numerical simulations of the same task performed by a nonlinear stochastic modelcomprising additive and multiplicative white Gaussian noise, state feedback terms, and a deadbandfunction. We identified the model parameters using particle swarm optimization to minimize squaredresiduals between the probability density functions (PDFs) of the human and those of the model.All experimentally obtained PDFs were in close agreement with those simulated by the model. Wefinally propose a reduced model for system identification in order to decrease the number of modelparameters and demonstrate that it also reproduces accurate PDFs without prior knowledge of anexperimental system.

1. Introduction

Human balance motions arise during various hu-man tasks such as quiet standing [1–5], stick bal-ancing on the fingertip [6–9], and visuomotor track-ing [10, 11]. Although these balancing motions areproduced using different bodily functions such aslower leg control, upper limb control, and visuomotorcontrol, they commonly display random fluctuations.

In the previous studies [1–11], random fluctu-ations were considered as a means for developinghuman balancing control models. By way of ex-ample, it is confirmed that the probability densityfunction (PDF) of a change in the speed of the fin-gertip during stick balancing by skilled participantshad a broader tail compared with that by unskilledparticipants [8, 9]. The changes in the speed ofhand movements during stick balancing were best de-scribed by a truncated Levy distribution [8]. In otherstudies adopting PDFs, some researchers confirmedbimodal distributions in experiments involving quietstanding and replicated typical bimodal distributionsusing their control models [2–4]. In these studies,they reproduced PDFs primarily in a qualitativemanner. The generation of fluctuated motions playsan important role in reproducing natural human-likemotions [8, 12, 13]. According to standard stochastic

∗ Manuscript Received Date: March 6, 2017† Intelligent Vision & Image Systems Inc; Hongo OGI-

BLDG., Hongo 3-6-6, Bunkyo, Tokyo 113-0033, JAPAN‡ Department of Mechanical and Intelligent Engineering,

Utsunomiya University; Yoto 7-1-2, Utsunomiya city,

Tochigi 321-8585, JAPANKey Words: human dynamics, system identification,

visuomotor tracking, probability density function.

process theory, the properties of fluctuations aredescribed by PDFs, making their precise identifica-tion effective in reproducing human-like fluctuations.Reproduced PDFs can be utilized in many applica-tions such as generating humanoid motions, detectingunlikely postures, and automatically segmenting hu-man motions [14] as well as developing a human-likeartificial controller in cooperation with a humanoperator [15].

In this study, we construct a human controllermodel (HCM) that precisely reproduces the PDFsof human balance control. For this purpose, wefirst experimentally analyze a visuomotor trackingtask in which human participants manipulate anunstable virtual object in a numerical simulatorusing a pointing device. This visuomotor trackingtask reveals the underlying mechanisms supportingthe corrective adjustments when humans track amoving object based on visual information [10,11,16],such as stick balancing [6–9, 15] and car driving[17–19]. We then construct a four-dimensional linearmodel comprising additive and multiplicative whiteGaussian noise terms and state feedback terms.The parameters of the HCM are then identifiedusing particle swarm optimization (PSO) [20–22] tominimize the squared residuals between the PDFs ofthe human participants and those of the HCM. Theresulting linear HCM yields accurate PDFs for threeof the four variables, but produces inaccurate resultsfor the final variable, which exhibits intermittentbehavior; therefore, we modify the linear HCMinto a nonlinear model by introducing a deadbandfunction and demonstrate that the modified modelaccurately reproduces the intermittent behavior. Wefinally propose a reduced nonlinear HCM for system

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　Transactions of the Institute of Systems, Control and Information Engineers, Vol. 31, No. 6, pp. 209–219, 2018 209

210 Transactions of ISCIE, Vol. 31, No. 6 (2018)

Fig. 1 Experimental device and human operator

Fig. 2 Design of animation window. The thin and thicklines represent the cursor displacement x and thetracking target displacement x+y, respectively.

identification in order to decrease the number ofmodel parameters and demonstrate that it alsoreproduces accurate PDFs without prior knowledgeof an experimental system.

2. Experiment on Tracking Task

2.1 Experimental SetupFig. 1 shows a photograph of the experimental

device and a human operator. The experimental de-vice consists of three units: a numerical simulator, amonitor, and a pointing device. The pointing devicecomprises a Wii R⃝ remote controller and a wirelesssensor bar. Data transmission between the numericalsimulator and remote controller is obtained throughthe use of Bluetooth R⃝ communication. The humanoperator visually receives animated motions of vir-tual objects from a monitor displaying an animationwindow (Fig. 2) and operates the remote controller.Thin and thick lines, representing the cursor dis-placement x and tracking target displacement x+y,respectively, are displayed in the window; we referto the relative displacement y as the tracking error.The resolution and window refresh interval ∆t areset to 1200×600 pixels and 2×10−2 s, respectively.The horizontal range of pixels (1,1200) is mapped tothe range of displacement (−3,3) of the numericalsimulator.

The experimental setup described above can bemodeled as shown in Fig. 3; it comprises a numericalsimulator as a plant to be controlled and a humancontroller unit. In particular, the plant to be

Fig. 3 Model of the experimental setup. It comprises anumerical simulator as a plant to be controlledand a human controller unit.

controlled is provided as a numerical simulator of anunstable virtual object, given by the state vector

x=(x1,x2,x3,x4)T := (x,x,y,y)T , (1)

and the state equation

x=Ax+Bu (2)

: =

0 1 0 00 −γ −α 00 0 0 10 0 2α −γ

x+

010−1

u (3)

with the state feedback

u :=−K1{x−z(t)}−K2x, (4)

where α is a spring coefficient, γ is a viscous dampingcoefficient, u is a control input, and K1 and K2 arethe feedback gains of the cursor position and velocity,respectively. The state feedback term u allows thecursor position x to track the human output z(t).

Note that the cursor position x and the humanoutput z are distinct in our experimental setup,although x≈ z is assumed in our theoretical analysis(see Section 3). When x≈ z in this experimentalsetup, the state equations (2) and (3) have derivativesx and x, which may become jagged in the numericaldifferentiations.

Additionally, we consider a second order repre-sentation of the state equation (3):

x=−γx−αy+u, (5a)

y=−γy+2αy−u. (5b)

A mechanical interpretation of (5) is given in Fig. 4.Here x is the displacement of unit mass and force uacts on x; y is the relative displacement of anotherunit mass from x. These masses are connected by aspring of negative stiffness −α< 0 and subjected toviscous damping of γ.

As shown in Fig. 3, the numerical simulatorreceives the human output z(t) from the humancontroller unit and outputs the numerical solution ofx(t) in (3) at the sampling interval ∆t, the samevalue as the window refresh interval. The numerical

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Trans. of ISCIE, Vol. 31, No. 6 (2018)

Matsumoto, Yoshida and Sekikawa: Stochastic Dynamic Modeling of Human Visuomotor Tracking Task 211

Fig. 4 A mechanical interpretation of (5). x is displace-ment of unit mass and force u acts on x; y is therelative displacement of another unit mass fromx.

solution of x(t) is calculated by means of a fourth-order Runge–Kutta–Gill method with a time step of10−2 s, the half-length of the window refresh interval∆t. In the human controller unit, the human operatorvisually receives the state vector x(t) as an animatedmotion and outputs z(t) via a man–machine interfacecomprising the monitor and remote controller. Toenable feasible operation by human operators, weempirically set the system parameters of (3) and (4)at α=22, γ =6, K1 =5000, and K2 =200 and theinitial state vector x(0)= (0,0,0.1,0)

T.

2.2 Experimental ProcedureThe participants were healthy males in their early

twenties. They were first instructed on the operationof the experimental device, the number of trials,and the duration of each trial. The experimentwas performed according to the principles of theDeclaration of Helsinki and informed consent wasobtained.

In each trial, the participant watched the motionof the target on the monitor and manipulated theremote controller so that the cursor tracked thetarget, which was repelled from the cursor.

Several practice trials were performed prior tomeasurement. An audio signal began each trial, andthe participant attempted the visuomotor trackingtask during the interval 0≤ t≤ 340 s. The trial wasrepeated if the target or cursor exceeded the limitsof the window.

Therefore, the human participants were intendedto track the target with the cursor while maintainingboth virtual objects inside the window.

2.3 Experimental DataDuring the experiment, we measured the time

series of the state vector x(t) as

{x(s,n)H (t0),···,x(s,n)

H (ti),···,x(s,n)H (tI−1)

},

ti := i∆t, (1≤ s≤S,1≤n≤N), (6)

where ti is a discrete time with a sampling period ∆t(= 2×10−2 s), I is the length of the time series, sand S are an index and the number of participants,respectively, and n and N are an index and thenumber of trials, respectively.

For this study, we chose I=16384 so that the cor-responding physical data length (I−1)∆t=327.66 s

was covered by the experimental measurement timeof 340 s in Section 2.2. The number of participantswas S =5, and the number of trials undertaken byeach participant was N =20.

2.4 Construction of Human PDFsWe construct the PDF P

(s,n)H (xk) with respect

to{x(s,n)k (ti)

}I−1

i=0, the kth component of the time

series{x(s,n)H (ti)

}I−1

i=0in (6) for the nth trial of the

sth participant. In practice, we obtain P(s,n)H (xk)

by normalizing the histogram of the time series{x(s,n)k (ti)

}I−1

i=0with bin width ∆ϕk as

∆ϕk :=1

nϕ(ϕu

k−ϕlk), (7)

where nϕ is the number of the histogram bins andϕuk and ϕl

k are the upper and lower limits of xk,

respectively. Then, we take the average of P(s,n)H (xk)

(n=1,···,N) to obtain the sth participant’s PDF as

P(s)H (xk)=

1

N

N∑n=1

P(s,n)H (xk), (k=1,2,3,4). (8)

In this study, we set nϕ =400 and the respectivelimits to (ϕl

1,ϕu1 )= (−2,2), (ϕl

2,ϕu2 )= (−0.008,0.008),

(ϕl3,ϕ

u3 )= (−0.4,0.4), and (ϕl

4,ϕu4 )= (−1.5,1.5).

3. Linear Human Controller Model

Based on the PDF derived above, we develop anHCM that can replicate the experimental behaviorof a human controller that receives x(t) and outputsz(t).

3.1 State Feedback Model of z(t)We assume that z(t) is produced by the human

characteristics of state feedback. We model thischaracteristic as a linear feedback control model inthe following form:

u :=F1x+F2x+F3y+F4y, (9)

where x is the cursor position and y is the track-ing error (relative displacement from the cursor totarget).

We can also describe the human output z(t) asfollows. As it is possible to assume x(t)≈ z(t) forthe selected gains K1=5000 and K2=200 in (4), thecursor position x(t) in (5a) can be approximated byz(t) as

u= x+γx+αy≈ z+γz+αy. (10)

Equating (9) and (10) using u, we obtain a dynamicmodel of the human output z(t) in the followingform:

z=−γz+F1x+F2x+(F3−α)y+F4y. (11)

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Fig. 5 Block diagram of linear HCM for replicating the input x(t) and output z(t) of a human controller. The labelledblock “

∫” is an integrator and σ1ξ1(t) randomly perturbs the third component of the gain matrix.

3.2 Modeling FluctuationTo simulate fluctuations in human dynamics,

we modify the dynamic model of z(t) in (11) byintroducing both multiplicative and additive noise.The modified model is given by

z=−γz+F1x+F2x+µ{1+σ1ξ1(t)}y+F4y+σ2ξ2(t), (12)

where ξ1(t) and ξ2(t) are mutually independent whiteGaussian noises with zero mean and unit variance.Here, the tracking error coefficient (F3−α) in (11) isreplaced with µ{1+σ1ξ1(t)}, a white Gaussian noisewith mean µ and variance (µσ1)

2, similar to thecontrol models in [7, 23]. Furthermore, the additivenoise term σ2ξ2(t) is newly added in (12) as a whiteGaussian noise with zero mean and variance σ2

2 toavoid the problem discussed in [24], wherein themultiplicative fluctuations can vanish in the absenceof an additive noise term.

We refer to the differential equation (12) asa linear HCM whose block diagram is shown inFig. 5, where block “

∫” is an integrator and σ1ξ1(t)

randomly perturbs the third component of the gainmatrix.

In this manner, we derive the six-dimensionalunknown parameter vector of the linear HCM (12) as

p=plin := (F1,F2,µ,σ1,F4,σ2). (13)

3.3 Calculation of PDF in the HCMWe obtain N ′ samples of the simulated time series{

x(n′)A (ti;plin)

}I−1

i=0(n′=1,···,N ′) from the numerical

solution of the numerical simulator (3) and (4) withthe linear HCM (12) for p= plin. To generate theN ′ samples, N ′ different sequences of pseudo randomnumbers [25] are applied to the simulator.

From the kth component{x(n′)k (ti;plin)

}I−1

i=0of

the simulated time series{x(n′)A (ti;plin)

}I−1

i=0above,

we construct the PDF P(n′)A (xk;plin) with respect to

the n′th sample of the time series using the sameprocedure and conditions as applied when derivingthe human PDF in Section 2.4. Similar to the caseof the human PDF in (8), we take an average of

P(n′)A (xk;plin) (n

′ =1,···,N ′):

PA(xk;plin)=1

N ′

N ′∑n′=1

P(n′)A (xk;plin), (14)

where k=1,2,3,4. We regard the PA(xk;plin) to bethe PDF of the linear HCM, which is to be compared

with the human PDF P(s)H (xk).

4. Method of Parameter Identifica-tion

In this section, we formulate the problem ofidentifying an unknown parameter vector, p. Weassume the teacher data of this identification problem

to be P(s)H (xk) (k=1,2,3,4), as obtained in (8).

4.1 Parameter IdentificationLet PA(xk;p) be a PDF numerically simulated by

the HCM of interest using p. To estimate the p that

makes up PA(xk;p)≈P(s)H (xk), we solve the following

optimization problem:

Minimizep

E(p) :=

4∑k=1

akEk(p), (15)

Ek(p)=

∫ ϕuk

ϕlk

{PA(xk;p)−P(s)H (xk)}2dxk

∫ ϕuk

ϕlk

{P (s)H (xk)}2dxk

, (16)

where E(p) is a cost function and ak representsthe weight coefficient for the state variable xk.Here, Ek(p) is the integral of the squared residuals

between the numerical PA(xk;p) and human P(s)H (xk),

normalized by the integral of the squared values of

P(s)H (xk) (the teacher data).

4.2 Particle SwarmOptimization (PSO)We employ PSO to solve (15). PSO is a

population-based optimization tool used to solvefunction optimization problems or problems that canbe transformed into function optimization problems.It mimics the swarming behavior observed in flocksof birds, schools of fish, swarms of bees, and someaspects of human social behavior. We use a standardalgorithm [20–22] as follows.

Consider an optimization problem of the followingform:

Minimizep

E(p), (17)

where p := (p1,...,pj ,...,pM ) ∈D ⊂RM is an M -dimensional vector and the cost E(p) is assumed

– 4 –



Table 1 Identified vector components of plin of the linear HCM for all participants

Index of participant sComponents 1 2 3 4 5

F1 2.15 3.16 3.36 3.12 1.53F2 −9.88×10−1 3.50 6.02×10−1 −8.92×10−1 5.43×10−2

µ 3.75×10 2.50×10 3.40×10 3.55×10 3.15×10σ1 3.41 3.00 3.32 2.99 3.72F4 3.33 7.01 5.89 2.81 5.31σ2 6.06 5.71 5.17 7.49 6.68

to be a positive definite real-valued function. InPSO, a swarm is composed of Np candidate solutions{p1,...,pNp} called particles. The particles explorethe M -dimensional domain D in search of the globalsolution p given by

p=arg minp

E(p). (18)

The positions of the particles are recursively updatedby

pi(l+1)=pi(l)+vi(l), (19a)

vi(l+1)= ρ0(l)vi(l)+ρ1(l){P i(l)−pi(l)}+ρ2(l){G(l)−pi(l)}, (19b)

where pi(l) is the position of the ith particle atiteration l, vi(l) is the corresponding velocity, andρ0(l), ρ1(l), and ρ2(l) are random numbers. P i(l)is called the personal best; the ith particle positiontaking the lowest cost among pi(0),...,pi(l). G(l) iscalled the global best, which is the particle positionthat has the lowest cost among all the particles forall iterations. Therefore, the optimization solution pin (15) is approximated by

p≈G(l). (20)

In this study, ρ0(l), ρ1(l), and ρ2(l) are taken asuniform random numbers over the intervals [0.6,1.2],[0,0.12], and [0,0.06], respectively.

The initial search domain D0 ⊂D is taken as anM -dimensional hyper-rectangle:

D0 := [b1,c1]× ...× [bM ,cM ], (21)

where bj and cj are the minimum and maximumpoints of the initial parameters, respectively.

The initial particle positions pi(0) are taken asNp = nM

p uniform grid points on D0. Thus, Np

represents the number of particles. All the initialparticle velocities are set to zero.

5. Parameter Identification Resultsfor Linear HCM

5.1 Identification ConditionWe use a PSO of dimension M = 6 from (13)

and empirically select np =2; hence, the number ofparticles becomes Np =26 =64. The initial searchdomain in (21) is taken as

D0 = [0,3]× [−8,8]× [25,50]

×[0,6]× [−6,6]× [0,6]. (22)

The number of iterations l is 400. We set the weightcoefficients to a1 = a2 = a3 = a4 =1 in (15) and thenumber of samples in (14) to N ′ =20.

5.2 Identification ResultsTable 1 shows the identified vector components

of plin of the linear HCM (12) for all participantss=1,···,5. In what follows, we denote the identified

PDF as P(s)A (xk;plin), which corresponds to the sth

participant’s teacher data P(s)H (xk) obtained from the

human experiment in Section 2.Fig. 6 shows a comparison between the first par-

ticipant’s P(1)H (xk) and the P

(1)A (xk;plin) identified

from the linear HCM, denoted as a gray solid lineand a black broken line, respectively. Fig. 6 (a)–(d)show the results for the cursor position x1, the cursorvelocity x2, the tracking error x3, and the timederivative x4 of x3, respectively.

To evaluate the identification accuracy, we intro-duce a fitness measure in the following form:

Q(s)k (p) :=

(1−E

(s)k (p)

)×100%, (23)

where E(s)k (p) is the cost function for sth participant

in (16). Hereafter, we call Q(s)k (p) the model quality.

It is clear from Fig. 6 that the identified

P(1)A (xk;plin) is in close agreement with the first

participant’s P(1)H (xk) for k= 1,3, and 4, yielding

the model qualities Q(1)1 (plin) = 95.32%, Q

(1)3 (plin) =

98.82%, and Q(1)4 (plin) = 99.43%, respectively. How-

ever, as shown in Fig. 6 (b), P(1)A (x2;plin) of the

cursor velocity x2 results in the low model quality

Q(1)2 (plin)= 12.26%.Table 2 shows the model qualities for all partic-

ipants. All these results exhibit a similar tendency;

i.e., Q(s)k (plin) (k= 1,3,4) yield the high qualities,

while Q(s)2 (plin) yields low quality.

This low model quality can be explained by adiscrepancy between the human and identified cursorvelocities x2(t), typical examples of which for thefirst participant are shown in Figs. 7 and 8, respec-tively. Fig. 7 shows a sample of the first participant’s

x(1,18)2 (t) during the 18th trial; it intermittently sticks

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0

0.5

1

1.5

2

-2 -1 0 1 2

(a)

Q1 = 95.32%

P(x

1)

x1

0

20

40

60

80

-0.008 0 0.008

(b)

Q2 = 12.26%

P(x

2)

x2

0

3

6

9

-0.4 0 0.4

(c)

Q3 = 98.82%

P(x

3)

x3

0

1

2

3

-1.5 0 1.5

(d)

Q4 = 99.43%

P(x

4)

x4

Fig. 6 Comparison between the first participant’s P(1)H (xk) and the P

(1)A (xk;plin) identified from the linear HCM,

denoted as a gray solid line and a black broken line, respectively. (a)–(d) show the results for the cursor positionx1, the cursor velocity x2, the tracking error x3, and the time derivative x4 of x3, respectively.

Table 2 Model qualities Q(s)k (plin) (k=1,2,3,4) of the linear HCM for all participants

Index of participant sModel qualities 1 2 3 4 5

Q1 95.32% 99.05% 99.49% 99.13% 98.31%Q2 12.26% 8.74% 10.27% 16.99% 14.66%Q3 98.82% 98.99% 98.98% 98.44% 98.83%Q4 99.43% 99.37% 99.13% 99.31% 98.94%

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20

x2

t [s]

Fig. 7 Sample of the cursor velocity x(1,18)2

of the first participant

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20

x2

t [s]

Fig. 8 Sample of the cursor velocity x2 ofthe identified linear HCM

to x2 =0, implying the presence of certain deadbandcharacteristics. In contrast, a sample of the identifiedx2(t) in Fig. 8 does not stick to x2 = 0, owing tothe absence of deadband characteristics in the linearHCM in (12).

6. Nonlinear Modification of HCM

To reduce the discrepancy between the humanand identified cursor velocities, we modify the linearHCM by adding deadband characteristics to it.

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Fig. 9 Block diagram of the nonlinear HCM, with block “H(z′)” added to the linear HCM

Table 3 Identified vector components of pnon of the nonlinear HCM for all participants


F1 2.84 3.17 4.83 1.29 4.11F2 −7.72 −5.91 −7.36 −2.35 −8.09µ 4.42×10 4.97×10 4.86×10 4.02×10 4.94×10σ1 1.75 7.69×10−1 1.79 1.48 2.28F4 6.76×10 −1 −5.91 −2.27 −4.59 1.22σ2 5.96 5.44 4.93 5.23 5.59β1 1.62×10−1 1.55×10−1 1.46×10−1 1.28×10−1 1.63×10−1

β2 8.01 2.65 5.52 1.43 3.59

6.1 Introducing Deadband FunctionThe linear HCM (12) can be transformed into the

following first-order differential equations:

z= z′, (24a)

z′ =−γz′+F1x+F2x+µ{1+σ1ξ1(t)}y+F4y+σ2ξ2(t). (24b)

We replace z′ in (24a) with a piecewise-linear dead-band function H(z′):

z=H(z′), (25a)

z′ =−γz′+F1x+F2x+µ{1+σ1ξ1(t)}y+F4y+σ2ξ2(t), (25b)

where

H(z′)=

β2(z′+β1) if z ′ ≤−β1 ,

β2(z′−β1) if z ′ ≥β1 ,

0 otherwise,(26)

where β1 and β2 are the half deadband width andthe slope of the linear part of H(z′), respectively.

The parameter β1 is the threshold at which thehuman visually recognizes the speed of a virtualobject that starts to move. The parameter β2

represents the gain of the generated cursor velocityz based on the internal state variable z′. Notethat as all calculated β1 are positive in this study(see Section 6.2), the H(z′) properly represents theaforementioned deadband characteristics.

We refer to this nonlinear model in (25) and(26) as the nonlinear HCM. A block diagram of thismodel is shown in Fig. 9, wherein the block “H(z′)” isadded to the linear HCM. In this case, the unknownparameters of the nonlinear HCM in (25) and (26)are given by the following eight-dimensional vector:

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20

x2

t [s]

Fig. 10 Sample of cursor velocity x2 of the identifiednonlinear HCM for the first participant

p=pnon := (F1,F2,µ,σ1,F4,σ2,β1,β2). (27)

We numerically construct the PDF of the nonlin-ear HCM as PA(xk;pnon) using the same procedureas described in Section 3.3.

6.2 Parameter Identification Resultsfor Nonlinear HCM

We use a PSO of dimension M = 8 from (27)and empirically select np =2; hence, the number ofparticles becomes Np =28 =256. The initial searchdomain in (21) is taken as

D0 = [0,3]× [−8,8]× [25,50]× [0,6]

×[−6,6]× [0,6]× [0.1,0.7]× [1,11]. (28)

The other conditions are the same as in Section 5.1.Table 3 shows the identified vector components

of pnon of the nonlinear HCM (25) and (26) for allparticipants. In this case, a sample of the cursorvelocity x2(t) simulated by the nonlinear HCM withpnon for the first participant (s=1) is obtained as

– 7 –


0

0.5

1

1.5

2

-2 -1 0 1 2

(a)

Q1 = 95.26%

P(x

1)

x1

0

20

40

60

80

-0.008 0 0.008

(b)

Q2 = 97.12%

P(x

2)

x2

0

3

6

9

-0.4 0 0.4

(c)

Q3 = 99.85%

P(x

3)

x3

0

1

2

3

-1.5 0 1.5

(d)

Q4 = 99.76%

P(x

4)

x4

Fig. 11 Comparison between the first participant’s P(1)H (xk) and the P

(1)A (xk;pnon) identified from the nonlinear HCM,

denoted as a gray solid line and a black broken line, respectively. (a)–(d) show the results for the cursorposition x1, the cursor velocity x2, the tracking error x3, and the time derivative x4 of x3, respectively.

Table 4 Model qualities Q(s)k (pnon) (k=1,2,3,4) of the nonlinear HCM for all participants


Q1 95.26% 98.87% 98.72% 99.19% 98.58%Q2 97.12% 96.45% 96.87% 95.44% 95.30%Q3 99.85% 99.76% 99.86% 99.79% 99.76%Q4 99.76% 99.77% 99.62% 99.38% 99.67%

shown in Fig. 10. It is seen that x2(t) appropriatelyreproduces the sticking behavior found in the humancursor velocity in Fig. 7, in contrast to the resultfrom the linear HCM in Fig. 8.

Fig. 11 shows a comparison between the first par-

ticipant’s P(1)H (xk) and the identified P

(1)A (xk;pnon)

from the nonlinear HCM, denoted as a gray solidline and a black broken line, respectively. Fig. 11(a)–(d) show the results for the cursor position x1,the cursor velocity x2, the tracking error x3, and thetime derivative x4 of x3, respectively.

As evident in Fig. 11(b), the model quality of the

cursor velocity Q(1)2 (pnon) was highly improved, from

12.26% in the linear HCM to Q(1)2 (pnon) = 97.12%

in the nonlinear HCM. It is also clear from Fig. 11

that the other identified results of P(1)A (xk;pnon) for

k=1,3, and 4 maintain close agreement with that

of the first participant’s P(1)H (xk), yielding the model

qualities Q(1)1 (pnon) = 95.26%, Q

(1)3 (pnon) = 99.85%,

and Q(1)4 (pnon)=99.76%, respectively. Table 4 shows

the model qualities for all participants. It is clearly

seen from Table 4 that all the Q(s)k (pnon) (for all

s=1,···,5 and k=1,···,4) denote high model qualitiesof more than 95%.

The above results quantitatively suggest that theproposed nonlinear HCM can accurately parameterizethe human PDFs obtained from distinct participants.

6.3 Summary and DiscussionThe results obtained in this section are summa-

rized as follows:• The nonlinear modification of the linear HCMresulting from addition of the deadband char-acteristics highly improves model qualities;

• The resulting nonlinear HCM can accurately

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Fig. 12 Block diagram of reduced nonlinear HCM, where the feedback terms z′(t) and y are omitted from the previousHCM in Fig. 9. The second and fourth components of the gain matrix become F ′

2 and zero, respectively.

parameterize the human PDFs obtained fromdistinct participants.

This implies the presence of deadband characteristicsin human dynamics during the visuomotor trackingtask. As there is no designed deadband element inthe man–machine interface of the human controllerunit, it is reasonable to suppose that the deadbandcharacteristics are produced by the other element ofthe human controller unit, namely, the behavior ofthe human operator during the visuomotor trackingtask.

However, it is also observed in Table 3 thatthe values of F4 take positive and negative valuesdepending upon the participants in spite of their highmodel qualities. This possibly implies dependencyamong the components of the parameter vector pnon

and suggests that the number of model parameterscan be reduced while maintaining model qualities.

7. ReducedNonlinearHCM inTermsof System Identification

In this section, we describe a simplified nonlinearHCM in terms of system identification. In otherwords, we regard the human controller unit shownin Fig. 3 as a black box whose input and outputare provided as the state vector x(t) and the z(t),respectively. It is demonstrated that the numberof model parameters is reduced while maintainingmodel qualities comparable to the previous results inTable 4.

7.1 Reduced Nonlinear HCMFirst, we ignore the derivative feedback of the

tracking error, similar to the control model in [7, 10,13,23], as we omit F4y from (25), and obtain

z=H(z′), (29a)

z′ =−γz′+F1x+F2x

+µ{1+σ1ξ1(t)}y+σ2ξ2(t). (29b)

This reduces one dimension of the parameter vectoras follows:

p=(F1,F2,µ,σ1,σ2,β1,β2). (30)

However, this model contains the plant parameter γthat is required to be identified if the plant is alsoregarded as a black box. To avoid such potentialproblems, we approximate z′ in the right side of(29b) using x to obtain

z=H(z′), (31a)

z′ =F1x+F ′2x+µ{1+σ1ξ1(t)}y+σ2ξ2(t), (31b)

where F ′2 = (F2−γ). In this case, the parameter

vector to be identified becomes

p=p′non := (F1,F

′2,µ,σ1,σ2,β1,β2). (32)

This makes it possible for the HCM to be identifiedwithout prior knowledge of plant structure, and weemploy the model in (31) with (32) as the reducednonlinear HCM.

A block diagram of model (31) is shown in Fig. 12.The feedback terms z′(t) and y are omitted from theprevious HCM in Fig. 9 as stated above. The secondand fourth components of the gain matrix become F ′

2

and zero, respectively.

7.2 Parameter Identification Resultsfor Reduced Nonlinear HCM

We use a PSO of dimension M =7 from (32) andthe number of particles Np = nM

p = 27 = 128. Theinitial search domain in (21) is taken as

D0 = [0,3]× [−8,8]× [25,50]× [0,6]

×[0,6]× [0.1,0.7]× [1,11]. (33)

The other conditions are the same as in Section 6.2.Tables 5 and 6 show the identified vector compo-

nents of p′non and the model qualities of the reduced

nonlinear HCM (31) for all participants, respectively.

It is clearly seen in Table 6 that all the Q(s)k (p′

non) (forall s=1,···,5 and k=1,···,4) represent high modelqualities of more than 95%, similar to the previousresults in Table 4.

These results indicate that we have produced areduced nonlinear HCM (31) that accurately param-eterizes human PDFs using seven model parameterswithout requiring prior knowledge of the experimen-tal system.

8. Conclusion

In this study, we constructed HCMs that preciselyreproduce the PDFs of a human visuomotor trackingtask, wherein human participants manipulate anunstable virtual object in a numerical simulatorusing a pointing device.

We conducted an experiment from which weconstructed human PDFs from measured time seriesof the state vector. To simulate fluctuations in the

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Table 5 Identified vector components of p′non of the reduced nonlinear HCM for all participants


F1 1.72 2.96 2.95 1.59 1.26F ′2 −7.90 −7.98 −7.81 −5.13 −7.17µ 4.56×10 5.17×10 4.86×10 3.23×10 4.26×10σ1 2.90 1.99 2.48 2.55 2.89σ2 6.00 6.22 4.90 5.24 5.88β1 2.41×10−1 3.04×10−1 2.27×10−1 1.98×10−1 2.17×10−1

β2 1.02 8.84×10−1 9.98×10−1 1.03 9.25×10−1

Table 6 Model qualities Q(s)k (p′

non) of the reduced nonlinear HCM for all participants


Q1 95.57% 99.10% 99.25% 99.26% 98.32%Q2 98.31% 97.37% 98.73% 96.35% 97.06%Q3 99.56% 99.34% 99.69% 99.76% 99.82%Q4 99.85% 99.89% 99.92% 99.68% 99.86%

human time series, we constructed a dynamic modelusing both multiplicative and additive noise. Weidentified the relevant model parameters based onthe measured human PDFs using PSO, and obtainedthe following results:

• The proposed linear HCM could not accu-rately identify human PDFs of cursor velocitiesdisplaying intermittent behavior for any partic-ipant;

• The identification accuracy was highly im-proved by adding a deadband element to theHCM. The resulting nonlinear HCM accuratelyreproduced the PDFs for all participants withrespect to all the state variables including thecursor velocities, producing model qualities ofgreater than 95%.

The above results indicate the presence of deadbandcharacteristics in human dynamics during the vi-suomotor tracking task. As there was no designeddeadband element in the man–machine interface, itis reasonable to assume that the deadband charac-teristics originated from the human operator.

Furthermore, we constructed another nonlinearHCM that employs reduced system identification andobtained the following results:

• The reduced nonlinear HCM can decrease thenumber of model parameters while maintain-ing model qualities comparable to the abovedescribed nonlinear HCM;

• In addition, the reduced nonlinear HCM accu-rately parameterizes human PDFs with sevenmodel parameters without prior knowledge ofthe experimental system.

Applying our proposed nonlinear HCM with dead-band elements and additive and multiplicative noises,we will analyze the involuntary movements of stickbalancing [6–9, 15] and car driving [17–19] in future

work.

Acknowledgements

We wish to express our gratitude to the membersof the System Dynamics Laboratory at UtsunomiyaUniversity for their participation and cooperation asparticipants in this study.

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Authors

Shigeki Matsumoto (Member)

Shigeki Matsumoto received the B.Eng.

and M.Eng. degrees from the Grad-

uate School of Engineering at the

Utsunomiya University, Utsunomiya,

Japan, in 2011 and 2013, respectively.

His research interests are in nonlinear

dyanamics, stochastic mechanics and system identifica-

tion. He is a member of ISCIE.

Katsutoshi Yoshida

Katsutoshi Yoshida received the Ph. D.

degree in Engineering from the Grad-

uate School of Engineering at the


Japan in 1996. Since 2012, he is

Professor at the Graduate School of

Engineering, Utsunomiya University. His research inter-

ests are in nonlinear dyanamics, stochastic mechanics,

complex systems and multi-human dynamics. He is a

member of JSME and SICE.

Munehisa Sekikawa

Munehisa Sekikawa received the B. E.

and M. E. degrees in information sci-

ence and the Ph. D. degree in pro-

duction and information science from


Japan, in 2000, 2002, and 2005, re-

spectively. In 2013, he joined the Department of

Mechanical and Intelligent Engineering at Utsunomiya

University, Utsunomiya, Japan, where he is currently an

Associate Professor. His research interests are in non-

linear dynamics, synchronization phenomena, and chaos.

He is a member of IEEE and IEICE.

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stochastic dynamic modeling of human visuomotor tracking

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