Intermittent Control Explains Human Motor Remnant Without Additive

Noise.

Adamantia Mamma, Henrik Gollee, Peter J. Gawthrop and Ian D. Loram

Abstract— Early work on modelling the human motion con-trol system showed that only a part of the corresponding motionsignals could be described in terms of a deterministic linearcontinuous-time model of the human control system. It wassuggested that the unexplained part, called the remnant, couldbe modelled by adding a noise signal with a carefully chosenfrequency spectrum.

Intermittent control provides an alternative description of thehuman controller which includes a sampling mechanism. Thispaper suggests that the remnant can be explained by assumingthat this sampling mechanism is not uniform; the addition ofa noise signal is not required using this assumption.

The approach of this paper is to compare the remnantfrequency spectrum derived from experimental data with thatfrom equivalent simulated data using each of the two modelsof the human controller in turn. It is found that both of thesimulated models give similar remnant spectra to that of theexperimental data.

Further work is required to show which of the two modelsprovides the best physiological explanation of remnant.

Keywords: Intermittent control, human motor control, bio-logically inspired control systems, predictive control.

I. INTRODUCTION

When the human operator is asked to control a system,

which is excited over a range of discrete frequencies, the

output response contains information both at input frequen-

cies and their harmonics, the non-excited frequencies. The

control signal at non-input frequencies is called remnant. So

far, physiological systems are modelled as linear systems

which treat the remnant signal as a random noise signal

that is injected to the linear controller. This paper attempts

to answer, on a computational level, whether or not the

non-linear intermittent control model can be an alternative

approach to describe remnant in human manual control tasks.

A very early investigation of remnant was implemented

by Levison et al. [1] where a theoretical model of remnant

was developed and compared against manual control data.

The study was based on the fact that the remnant is the

controller’s random response shown at non-input frequencies

and it is not related to the linear function. Based on studies

that have supported that the power-spectra density of the

controller’s remnant is smooth in the frequency domain,

This work was supported by the UK Engineering and Physical Sci-ences Research Council, grant “Intermittent control of man and machine”(EP/F068514/1, EP/F069022/1 and EP/F06974X/1)

A. Mamma, H. Gollee and P. J. Gawthrop are with theSchool of Engineering, University of Glasgow, Scotland, U.K,a.mamma.1@research.gla.ac.uk, {henrik.gollee,peter.gawthrop}@glasgow.ac.uk

Ian D. Loram is with the Institute for Biomedical into Human Movementand Health, Manchester Metropolitan University, John Dalton Building,Oxford Road Manchester, U.K I.Loram@mmu.ac.uk

Levison et al. [1] described remnant as a white random

process, whose components are linearly independent from

each other, and are not related linearly to the system input

by the transfer function. This remnant model was used by

Kleinman et al. [2] and Baron et al. [3]. They have demon-

strated that the human operator could be described as a linear

predictive controller (PC), using a continuous-time Observer-

Predictor-Feedback (OPF) algorithm [4] with added random

white noise. The model was found to be consistent with

experimental data taken from simple compensatory tracking

tasks. The above model is prominent and considered as a

paradigm for modelling human motor control.

The predictive control approach used for systems with

pure time delays and based on prediction has been shown to

describe physiological systems, [5], [6]. On the other hand,

the approach that the remnant signal during manual control

tasks has white characteristics does not have a “proven”

physiological basis. There is a debate on the characteristics

of the inherent noise source that is revealed during manual

control tasks. There are some studies which support that

noise measured at the motor output tends to have structure

consistent with low pass filtering of white noise, [7], [8],

however others ( [9], chapter 1) reported that there are

studies showing that the remnant signal does not reveal a

white noise characteristic, and that it is based on structure

rather than on randomness. A different approach to the linear

time-invariant predictive control model was presented by

Metz, [10], in which he suggested that the variability in

human control tasks is due to the time-varying behaviour

of the human controller. Although the model presented an

alternative approach to describe the remnant signal, the

simulations results using that model were not consistent with

that derived from compensatory tracking experiments.

As discussed by Gawthrop et al. [11] an alternative

explanation of the human motor control mechanism is the

non-linear intermittent control approach. Neilson et al. [12]

were among the first to propose a theoretical intermittent

non-linear model of the human operator. Their controller

model is based on three processing stages: Sensory Analysis

(SA), Response Planning (RP), Response Execution (RE),

in which the first and the third stages operate in continuous-

time and the second intermittently as it needs an interval

time to preplan a movement before passing the information

to the next stage.

A similar theoretical intermittent model has been devel-

oped by [13] and continued by Gawthrop in a series of stud-

ies, [11], [14], [15], including frequency domain analysis,

[16]. The controller model is a feedback control design and

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a combination of continuous and discrete-time based on the

OPF model of [4]. In particular, the continuous-time state

observer is intermittently sampled over an interval of time

in which the control is open-loop and results in a non-linear

open-loop trajectory of the control signal. A generalised

hold element is used to reconstruct the sampled signal.

According to [11], the intermittent interval in which the

control is open-loop can either be periodic or non-periodic.

Loram et al. [17] showed experimentally that the subjects

during manual compensatory tracking in which they had been

tapping a joystick to control an unstable load, revealed a

non-uniform distribution of tapping time intervals. Because

of that, in this study, an intermittent controller (IC) with

non-uniform intermittent sampling is considered. The study

shows that the IC with non-uniform intermittent sampling

provides an alternative explanation of describing remnant

in manual control tasks, without the need to add random

noise to the controller. The study is based on modelling

and simulations in which theoretical results are compared

with experimental data taken from compensatory tracking

task experiments [15], where subjects are asked to control a

non-linear unstable pendulum with the dynamics of a human

during quiet standing.

II. METHODS

A. Methodology

Modelling and simulations were used to study the remnant

in the physiological control mechanism of human postural

balance. The mechanism can be described as a single input

single output time-delay system, with an unpredictable peri-

odic multisine signal, d(t) to disturb the controlled system,

which has the dynamics of a human in quiet standing, Fig. 1.

The control signal, u(t) was generated and analysed using

three different controller models to describe the “Human

operator”.

The following approaches of the “Human controller” were

investigated:

1) Continuous-time PC. The human controller is de-

scribed as a pure continuous-time linear predictive

model.

2) PC with added noise. The human controller is de-

scribed as a continuous-time linear predictive model

with an added random noise signal, v(t). Following

Levison et al. [1] the noise signal, v(t), is modelled

as random white noise.

3) IC with non-uniform open loop intermittent inter-

val. The human controller is described as a non-linear

intermittent controller which is non-uniformly sampled

to give an intermittent open-loop trajectory.

B. Human quiet standing modelling

For our analysis, the system to be controlled was a virtual

model of a human during quiet standing. According to

[18] and [19], the human body can be described as an

inverted pendulum that sways around the ankle joint via

active muscles, the calf muscles, which are responsible for

preventing forward toppling, through the Achilles tendon.

In our model we assumed that the input to the system,

θin(t) is the angular position which represents the stretch of

the calf muscles, together with the disturbance signal. The

output, θ(t), is the resulting angular position of the centre of

mass (CoM). The linearised second order equation of motion

in state space form is:

[

θ(t)

θ(t)

]

=

[

−cI

mgl−KI

1 0

] [

θ(t)θ(t)

]

+

[

KI0

]

θin(t) (1)

where K is the stiffness of the Achilles tendon, c is the

damping coefficient of the calf muscles, m represents the

mass of the pendulum, l is the length of the pendulum from

the joint to the CoM, mgl is the gravitational torque from

the CoM, and I is the moment of inertia of the pendulum

about the ankle joint. θ(t) is the output of the system and

corresponds to the new angular position of the pendulum,

and θin(t) = u0(t − ∆) + d(t) is the input to the system,

with u0(t−∆) being the scaled neuromuscular control signal

which leads to a change in angle with time delay, ∆. Other

muscle dynamics are neglected. The rate of stability of the

pendulum depends on the ratio, a = Kmgl . For a = 100% the

system is stable, for a < 100% the system is unstable and

requires stabilisation.

Following the equation of motion (1), the corresponding

state-space representation of the time-delay system is of the

form:

x(t) = Ax(t) +B[u(t) + d(t)] (2)

u(t) = u0(t−∆) (3)

y(t) = Cx(t) (4)

x(0) = x0 initial conditions (5)

with a state vector,

x(t) =

[

θ(t)θ(t)

]

(6)

where ∆ is the time-delay, d(t) the external disturbance

signal, A a 2× 2 state matrix, B a 2× 1 column vector, C

a row vector, u(t) the time delayed control signal and y(t)is the corresponding angular position.

C. Controller modeling

1) Predictive controller: The model of the “Human op-

erator” was based on the linear controller structure of [18],

following Kleinman et al. [2], Fig.1. The model consists of

a continuous-time observer, a predictor, a delay, and state

feedback.

More particularly, a state observer is used to generate an

approximation, x(t) of the states, x(t). The observer system

is given by :

˙x(t) = A0x(t) +Bu(t) + Ly(t) (7)

where A0 = A−LC, is the observer state matrix with the

observer gain, L, and the state estimate vector, x(t).

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In addition to the observer, a predictor is used to eliminate

the delay in the feedback. The optimum predictor is obtained

by following [4]. The state space system (2) is integrated

from time t to time t+∆ using the observed state, x(t) as

the initial state. Therefore,

xp(t) = eA∆x(t) +

∫ ∆

0

eAτBu0(t− τ)dτ (8)

where xp(t) = x(t + ∆) is the predicted state at time

t + ∆, based on data given at time t. The state feedback

control signal, u0(t), is calculated as:

u0(t) = −Kxp(t) (9)

Both the feedback gain, K, and observer gain, L,

were generated with the Linear Quadratic Regulator (LQR)

method [20]. It involves minimisation of the cost function

J(x(t0), t0) =

∫ t1

0

(xT (t)Qcx(t) + u(t)RuT (t))dt

t1 → ∞

(10)

with a 2× 2 diagonal weighting matrix, Qc and R = 1.

Due to the duality relationship between the estimation

and control problem, the observer gain, L, is defined by the

weighting matrix, Qo.

2) Intermittent controller: The intermittent control is

based on the sequence of open loop trajectories of the control

signal indexed by an integer i. Each control signal is based

on a new measurement of the state with an open loop interval

lasting for ∆ol sec, [14]. The IC model, Fig. 2 used in this

study is based on Gawthrop et al. [11], [14].

The OPF model of section II-C.1 remained the same.

However, the IC differs from the PC model in the following

ways:

Delay System Observer

PredictorState FB

)(td

)(tu )(ty

+

)(tx�

)( '�

�

tx

PC Controller

)(0 tu

)(tv

+

Fig. 1. Predictive controller model

Delay System Observer

PredictorState FB

)(td

)(0 tu )(ty )(tx�

Hold

�

)( itx�

)( '�

�

ip tx)(^

tx h

IC Controller

it

)(tu

Fig. 2. Intermittent controller model

1) The continuous time state observer vector, x(t) is sam-

pled by a sampling element which converts it into a

sample signal, x(ti) at the discrete instants, ti.

2) The predictor operates on the sampled signal giving the

sampled vector, xp(ti +∆).3) The sampled signal, xp(ti + ∆), is converted into a

continuous time vector signal, xh(t), using a generalised

hold element vector to reconstruct the sampled state

[14].

4) The state feedback is driven by the open loop state

estimate, xh(t), and not from the closed loop state

estimate, x(t).

The IC model has three different time frames:

Continuous-time, in which the controlled system (2)–(6)

is described.

Discrete-time at which feedback occurs. The state estimates

are given in this time frame indexed by i. Also the inter-

mittent interval, ∆ol, where the control signal is generated

from the open loop trajectories, (12), is operating between

discrete-times. The ith intermittent interval, ∆iol is defined

as:

∆iol = ti+1 − ti (11)

Intermittent-time is a continuous-time denoted by τ and

restarts at each intermittent interval. The open loop inter-

mittent control signal is equal to:

u(t) = u(ti + τ)

ti ≤ t ≤ ti+1

0 ≤ τ ≤ ti+1 − ti

(12)

a) Intermittent controller hold: A System Matched

Hold (SMH) model was used in this study. Following [16]

the state space form of the SMH model is given by:

dxh(τ)

dτ= AT

h xh(τ) (13)

u(ti + τ) = xTh (0)xh(τ) (14)

u(ti + τ) = ui (15)

xh(0) = xp(ti) (16)

where xh(τ) is the n × 1 state vector, Ah is the n × n

state matrix of the hold which was chosen to have the same

eigenvalues as the closed loop system matrix, Ac = A−BK

and u(ti + τ), the intermittent control signal computed at

each intermittent interval with xh(0) = K where K is the

feedback gain of the closed loop system.

b) Intermittent Predictor model: The predictor model

used is based on the continuous-time predictor, (8). It is

generated by differentiating (8) denoting ∆ = τ . The state

space form is then given by:

dxp(τ)

dτ= Axp(τ) +Bu(τ) (17)

xp(0) = x(ti) (18)

Combining equations (13)–(18) results in:

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dX(τ)

dτ=

[

A −BK

0n×n Ah

] [

xp(τ)xh(τ)

]

(19)

X(0) = Xi (20)

For this study the predictor was designed by integrating

(19) where τ = ∆, giving:

X(∆) = eAph∆Xi (21)

Aph =

[

A −BK

0n×n Ah

]

(22)

The prediction xp(ti) is extracted from (21) and 22,

giving:

xp(ti) = Eppx(ti) + Ephxh(ti) (23)

with the n × n matrices, Epp and Eph, being part of the

2n× 2n matrix E:

E =

[

Epp Eph

Eph Ehh

]

(24)

D. Simulation analysis

1) System and Controller properties: For our analysis,

we have assumed that the system is unstable with 85%stability; this means that the ratio of the stiffness of the

Achilles tendon over the gravitational torque per unit angle is

a = Kmgl = 0.85, cf. equation (1). From [15] and considering

85% stability, the equation of motion of the pendulum in

canonical form is given:

[

θ(t)

θ(t)

]

=

[

−0.0372 1.2311 0

] [

θ(t)θ(t)

]

+

[

6.9770

]

θin(t)

y(t) =[

0 1]

[

θ(t)θ(t)

]

(25)

The design properties used for the PC and IC controllers

were

∆ = 0.15 sec, Qc =

[

1 00 1

]

, Qo =

[

0 00 100

]

Qc, is the weighting matrix for the control design and Qo,

the weighting matrix for the observer design.

E. Experimental method

The data generated from simulations were compared with

experimental results obtained from manual compensatory

tracking task. The experimental procedures and methods that

were used are described in detail in Loram et al. [15]. One

subject was asked, by holding a contactless joystick, to keep

a virtual pendulum as close as possible to the centre of an

oscilloscope throughout the trial. The dynamic characteristics

of the virtual pendulum were identical to those used for the

simulations.

F. Data processing

The analysis of remnant was implemented in the frequency

domain, based on the approach described by Pintelon and

Schoukens [21]. The system was excited by a continuous-

time periodic multisine signal, d(t), for t = 200 sec which

contained the frequencies 0.1, 0.2, . . . , 10Hz

d(t) =

Nf∑

k=1

A(ωk) cos(ωkt+ φk) (26)

The signal, d(t) was sampled with a sample period

Ts = 0.01sec resulting in:

d(tn) =

Nf∑

k=1

A(ωk) cos(ωktn + φk) (27)

where ωk, is the kth excited discrete frequency with ωk =kω0, and fundamental frequency, ω0 = 2πf0 and Nf = 100.

The amplitude of the signal, A(ωk), was unity for all ωk, to

ensure that the frequencies were evenly excited. The phase,

φk was taken from a random distribution in the open interval

(0, 2π). The sampled signal, d(tn), perturbed the system and

the control signal, u(tn), was calculated for the simulations

whereas in the manual tracking task the experimental data,

u(tn), was measured. For the remnant analysis we were

interested in the spectra of the control signal, not only at

excited frequencies, ωk, but also in their harmonics, the non-

excited frequencies. For this reason, the period of the control

signal, u(tn), was increased by a factor n1 = 4, giving a

new period T1 = n1T0 = 40sec with fundamental frequency

ω1 = 2πf1, and f1 = 0.025 Hz. The first period (40sec)

of the analysis was discarded to remove transient behavior.

The discrete fourier transformation (DFT) of the extended

sampled signal, d(tn1), is given by

uDFT (jωk1) =

N1−1∑

n1=0

u(tn1)e−j2πn1k1/N1

k1 = 0, 1, 2....400

(28)

where ωk1= k1ω1 is the discrete frequencies of the

extended signal and N1 = T1

Ts= 4000 denotes the number of

sample points within one signal period. For a linear controller

the control signal, uDFT (jωk1), contains information only at

excited frequencies, ωk, whereas for a non-linear controller

the control signal, uDFT (jωk1), contains information both

at excited frequencies and non-excited frequencies.

III. RESULTS

All simulations and analysis algorithms were implemented

in Matlab (R2010a, The Mathworks, MA, USA). Simulations

were defined by the type of the controller, PC or IC, the noise

signal, v(t), and, for the IC, the intermittent interval, ∆ol,

and were compared with the experimental data described

in section II-E. Following Pintelon and Schoukens [21] the

control signal, uDFT , was averaged over the four periods.

The spectrum of the signal against the excited and non-

excited frequencies was generated.

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Figure 3, shows experimental results with one represen-

tative subject. It shows, that the remnant signal (dashed

line) was an important factor of the output response of the

subject during the experiment. The amplitude of the control

signal at non-excited frequencies is relatively high and above

1 Hz similar in amplitude to the control signal at excited

frequencies.

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

101

fk1

(Hz)

log |u

DF

Tm

|

Excited freqs

Non−excited freqs

Fig. 3. Experimental data: Control signal at excited and non-excitedfrequencies

As expected, in simulations using a pure linear PC the

spectrum of the control signal showed no remnant. Figure

4 shows simulation results obtained using the PC where a

random white noise with a distribution of mean µ = 0 and

standard deviation, σ = 5, is added.

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

101

fk1

(Hz)

log |u

DF

Tm

|

Excited freqs

Non−excited freqs

Fig. 4. PC with added noise: Control signal at excited and non-excitedfrequencies

Simulation results using the non-linear IC with a non-

uniform open-loop intermittent interval, ∆ol, that follows a

normal distribution with µ = 0.24 and standard deviation

σ = 0.5 are shown in Fig. 5 and Fig. 6. The minimum

intermittent interval had to be ∆ol ≤ ∆ = 0.15sec and the

maximum intermittent interval was chosen to be 2.25 sec,

corresponding to the maximum time interval the “Human

controller” needs to respond to a stimulus. Figure 6 shows

the distribution of the non-uniform intermittent intervals used

to generate the results of Fig. 5.

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

101

fk1

(Hz)

log |u

DF

Tm

|

Excited freqs

Non−excited freqs

Fig. 5. IC with non-uniform ∆ol: Control signal at excited and non-excitedfrequencies

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

35

40

Intermittent intervals ∆ol

(sec)

Num

ber

of

ele

ments

that

fall

in t

he g

roup

Fig. 6. Histogram of intermittent intervals

IV. DISCUSSION

Although the continuous-time linear PC model of Klein-

man [4] is used as a paradigm to explain the internal models

of human control, it is not adequate to describe the variability

in human manual tasks as there is no remnant in the control

signal when the human operator is described as a pure

linear PC. This could be interpreted in such a way that

the human operator preprograms his responses so well that

he behaves as a pure linear servomechanism. On the other

hand, the model of the PC with added random noise, with

white characteristics, is a better approach on modelling the

human controller. Levison et al. [1] have shown that the

model fitted experimental data quite well in simple tracking

tasks with linear controlled systems. In addition, in our

study remnant was shown using the above model, Fig. (4).

However, the qualitative construction of the random noise

which is added to the continuous-time PC depends strongly

on the properties of the disturbed signal, which according

to ( [9], chapter 1) are unknown in biological systems.

Also, there is no “proven” physiological evidence for an

added motor noise which has white characteristics. The IC

algorithm using non-uniform open loop intermittent intervals

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has shown that it is an appropriate model to describe remnant

in human motor tasks, without the need to add any external

signals to the controller, as in the case of the PC. The

open loop intermittent interval, ∆ol, defines only the periods

at which the control signal is generated and it does not

depend on properties such as disturbance signal or order

and stability of the controlled system. Moreover, comparing

the spectrum of the control signal, uDFTm, derived from

the subject, Fig. 3, with the corresponding spectra from the

simulations for both controllers, it is shown that the remnant

signal generated from the IC with the non-uniform ∆ol is

similar to the remnant signal derived from the subject. So

far, in engineering, the remnant which is depicted in the

control signal spectrum has been described as a random

white noise, which is injected to the linear human controller.

Although this approach seems to work and fit experimental

data quite well, it faces limitations such as an explanation of

the exact source of remnant in tracking tasks ( [9], chapter

1). However, our study has shown that the approach of

modelling the human operator as a non-linear IC with non-

uniform open loop intermittent intervals, is an alternative

model to describe the remnant signal in manual motor tasks

which may have a physiological basis. Future work is to

analyse experimental data from more subjects and derive

the system identification of the above models considering

the remnant during manual control tasks. Moreover, different

types of excitation signals to more complicated tracking tasks

such as double stimuli or by changing the time delays in the

controller structure could be a good approach on comparing

the two controllers.

V. CONCLUSION

In this study we have shown that the intermittent control

provides an alternative description of the human controller

which includes a sampling mechanism. It is suggested that

the remnant can be explained by assuming that this sampling

mechanism is not uniform; the addition of a noise signal is

not required using this assumption.

Moreover, the intermittent control model seems to be

consistent with studies, [22], [23], which support that the

human operator responds intermittently to a continuous series

of stimuli or to a continuous changing stimulus, [11]. Also,

this study may be an alternative description than [10]; it is

consistent with the experimental data and also more realistic

as no assumptions are made.

Although the continuous-time predictive control engineer-

ing model is used as a paradigm to explain physiological

systems, it is designed and considered to be used for high

bandwidth systems, whereas intermittent control is designed

for limited bandwidth applications which is more applicable

and natural to human control, as it is low bandwidth with

noisy sensors and actuators, [17].

Further work is required to show which of the two models

provides the best physiological explanation of remnant. The

non-uniform sampling in this paper uses a random model. In

fact, as discussed by [11], the sampling may in fact be event-

driven. Further work will compare and contrast random and

event-driven sampling in this context.

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