PDS Based Bilateral Control System of Flexi- ble Master

Transactions of ISCIE, Vol. 30, No. 1, pp. 1–9, 2017 1

Paper

PDS Based Bilateral Control System of Flexi-ble Master-Slave Arms with Random DelayUsing Kalman Filter*

Masaharu Yagi†, Kengo Kimura‡ and Yuichi Sawada‡

In this paper, a PDS (Proportional-Derivative-Strain) based bilateral control system for flexiblemaster–slave arms with random delay is discussed. The proposed PDS based bilateral control systemconsists of a rigid master and flexible slave arms. The flexible slave arm is actuated by a high-gearedservomotor. Furthermore, the communication network which causes random delay connects botharms. The random delay is defined as the sum of the average time delay and a white Gaussian noise.A Kalman filter is designed to estimate the signals affected by the random delay. The controllersfor the rigid master arm and the flexible slave arm are designed as the PD- and PDS-controllers,respectively. Numerical simulations are accomplished to confirm the performance of the proposedKalman filter for the proposed PDS based bilateral control system.

1. Introduction

In recent years, teleoperation technologies drawmuch attention as technologies for robotic systems toprovide human skills for remote locations. The aimof teleoperation technologies is useful in general envi-ronments such as offices and houses without specialequipments. Therefore, these technologies should berealized by using existing communication networks,such as local area networks (LAN), wide area net-works (WAN), and wireless LAN. In this research,a PDS based control system of flexible master-slavearms is discussed as one type of teleoperation tech-nology. This system is constructed by a rigid mas-ter and a flexible slave arms, and a communicationnetwork. The flexible slave arm tracks the rigid mas-ter arm and the communication network causes ran-dom delay. State signals and observation signals ofthis system are interacted through the communica-tion network. Thus, the PDS based control systemwith the communication network can be regarded asa type of feedback system with a communication net-work. Furthermore, since the communication network

∗ Manuscript Received Date: June 10, 2016The material of this paper was partially presented at

the 47th ISCIE International Symposium on Stochastic

Systems Theory and Its Applications (SSS’15) which

was held in December, 2015.† Prefectural Government Buildings Division, Kyoto Pre-

fecture; Shinmachi, Kamigyo, Kyoto 602-8570, JAPAN‡ Division of Mechanical and System Engineering, Ky-

oto Institute of Technology; Matsugasaki, Sakyo, Kyoto

606-8585, JAPANKey Words: PDS based control system, flexible arm,

random time delay, Kalman filter.

causes random delay, the signals transmitted throughthis network become noisy. Then, the random delaymight cause instability of this system.

A lot of researchers have studied bilateral con-trol system with time delay and flexible manipula-tors. Hoshino et al. and Mori et al. studied a sym-metric bilateral system of flexible master-slave manip-ulators and proved the passivity of this system[1],[2].Namerikawa researched a control scheme for teleop-eration systems with time-varying delay and a simplePD-type control method[3],[4]. Matsuno et al. pro-posed a PDS feedback controller to control the two-link flexible beams and discussed the stability of theclosed-loop system[5]. The flexible beams was mod-eled as the two interconnected flexible beams whichwas regarded as an element of the truss structure inthe study [5]. However, these researches did not con-sider random delay.

Furthermore, state estimation problems for a net-worked system with random delay have been discussedby many researchers. Guo et al. researches the ex-ponential stability for a type of discrete-time systemwith random delay[6], and Wu et al. proved the mean-square exponential stability for a networked system[7].Liu et al. discussed the model predictive control prob-lem for a closed-loop networked control system whichcauses random delay defined as a Markov chain[8].Schenato studied the design of optimal estimatorswith random delay and packet loss[9],[10]. However,itwas considered that these studies of random delay didnot well model actual random delay because the ran-dom delay in these researches was not defined stochas-tically (e.g., as Gaussian noise). To reduce the effectof the time delay which is occurred by the commu-nication network, the time delay should be modeled

– 1 –

Transactions of ISCIE, Vol. 30, No. 1, pp. 1–9, 2017 1

Paper

PDS Based Bilateral Control System of Flexi-ble Master-Slave Arms with Random DelayUsing Kalman Filter*

Masaharu Yagi†, Kengo Kimura‡ and Yuichi Sawada‡

In this paper, a PDS (Proportional-Derivative-Strain) based bilateral control system for flexiblemaster–slave arms with random delay is discussed. The proposed PDS based bilateral control systemconsists of a rigid master and flexible slave arms. The flexible slave arm is actuated by a high-gearedservomotor. Furthermore, the communication network which causes random delay connects botharms. The random delay is defined as the sum of the average time delay and a white Gaussian noise.A Kalman filter is designed to estimate the signals affected by the random delay. The controllersfor the rigid master arm and the flexible slave arm are designed as the PD- and PDS-controllers,respectively. Numerical simulations are accomplished to confirm the performance of the proposedKalman filter for the proposed PDS based bilateral control system.

1. Introduction

In recent years, teleoperation technologies drawmuch attention as technologies for robotic systems toprovide human skills for remote locations. The aimof teleoperation technologies is useful in general envi-ronments such as offices and houses without specialequipments. Therefore, these technologies should berealized by using existing communication networks,such as local area networks (LAN), wide area net-works (WAN), and wireless LAN. In this research,a PDS based control system of flexible master-slavearms is discussed as one type of teleoperation tech-nology. This system is constructed by a rigid mas-ter and a flexible slave arms, and a communicationnetwork. The flexible slave arm tracks the rigid mas-ter arm and the communication network causes ran-dom delay. State signals and observation signals ofthis system are interacted through the communica-tion network. Thus, the PDS based control systemwith the communication network can be regarded asa type of feedback system with a communication net-work. Furthermore, since the communication network

∗ Manuscript Received Date: June 10, 2016The material of this paper was partially presented at

the 47th ISCIE International Symposium on Stochastic

Systems Theory and Its Applications (SSS’15) which

was held in December, 2015.† Prefectural Government Buildings Division, Kyoto Pre-

fecture; Shinmachi, Kamigyo, Kyoto 602-8570, JAPAN‡ Division of Mechanical and System Engineering, Ky-

oto Institute of Technology; Matsugasaki, Sakyo, Kyoto

606-8585, JAPANKey Words: PDS based control system, flexible arm,

random time delay, Kalman filter.

causes random delay, the signals transmitted throughthis network become noisy. Then, the random delaymight cause instability of this system.

A lot of researchers have studied bilateral con-trol system with time delay and flexible manipula-tors. Hoshino et al. and Mori et al. studied a sym-metric bilateral system of flexible master-slave manip-ulators and proved the passivity of this system[1],[2].Namerikawa researched a control scheme for teleop-eration systems with time-varying delay and a simplePD-type control method[3],[4]. Matsuno et al. pro-posed a PDS feedback controller to control the two-link flexible beams and discussed the stability of theclosed-loop system[5]. The flexible beams was mod-eled as the two interconnected flexible beams whichwas regarded as an element of the truss structure inthe study [5]. However, these researches did not con-sider random delay.

Furthermore, state estimation problems for a net-worked system with random delay have been discussedby many researchers. Guo et al. researches the ex-ponential stability for a type of discrete-time systemwith random delay[6], and Wu et al. proved the mean-square exponential stability for a networked system[7].Liu et al. discussed the model predictive control prob-lem for a closed-loop networked control system whichcauses random delay defined as a Markov chain[8].Schenato studied the design of optimal estimatorswith random delay and packet loss[9],[10]. However,itwas considered that these studies of random delay didnot well model actual random delay because the ran-dom delay in these researches was not defined stochas-tically (e.g., as Gaussian noise). To reduce the effectof the time delay which is occurred by the commu-nication network, the time delay should be modeled

– 1 –

Transactions of the Institute of Systems, Control and Information Engineers

2 Transactions of ISCIE, Vol. 30, No. 1 (2017)

and explicitly treated in the observation system.In our previous researches, a bilateral control sys-

tem with a communication network which causes time-varying delay was discussed. The system consisted ofa rigid master arm, a flexible slave arm and a commu-nication network. The internal stability and the pas-sivity were proved by using the Lyapunov theorem,and the performance of the proposed bilateral con-trol system was evaluated through numerical simula-tions[11]. However, the time delay was modeled as thesinusoidal wave in the study[11]. Furthermore, stateestimation problems for a networked system with ran-dom delay were investigated. A Kalman filter wasdesigned as a state estimator for a time-invariant lin-ear system and the observation system that was af-fected by the random delay. The effect of the pro-posed Kalman filter was confirmed through numericalsimulations[12]. However, the system was defined asthe general time-invariant system in the study[12].

In this paper, a PDS based bilateral control sys-tem for flexible master–slave arms with random delayis investigated. By using Hamilton’s principle, themathematical models of the master arm and the slavearm are derived. The random delay is defined as thesum of the average time delay and a Gaussian noise.A novel Kalman filter is designed to estimate the statesignal and the observation signal that are affected bythe random delay. The PD and PDS controllers aredesigned for generating the reaction torque for therigid master arm and the reference signal for the flexi-ble slave arm, respectively. The state estimates whichare estimated by the novel Kalman filter are used forthese controllers. Numerical simulations are demon-strated to confirm the effectiveness of the proposedKalman filter for the proposed system. The proposedsystem is formulated as the suitable stochastic sys-tem since the random delay is defined as the stochas-tic process. This study is an unique research whichmeans that previous researches are not done such thisresearch.

2. Mathematical Models

The considered PDS based bilateral control sys-tem consists of a rigid master arm and a flexible slavearm, which are connected through a communicationnetwork (shown in Fig. 1). The flexible slave arm isactuated by a high-geared servomotor, and the com-munication network causes random delay.

In this section, mathematical models of the pro-posed control system are derived to demonstrate nu-merical simulations[13].

2.1 Rigid Master ArmThe rigid master arm is constructed by an uni-

form rigid rod and a hub that is rotated by a DCmotor (shown in Fig. 2). OmXmYm denotes the iner-tial Cartesian coordinate system. mm and ℓm repre-sent the mass and the length of the rigid master arm,respectively. The operating force, Fh(t) acts at the

Network

PDScontroller

Flexible slave arm

Rigid master arm

Stateestimator

Stateestimator

PDcontroller

Human operator

High-gearedservomotor

+-

+-

Fig. 1 Block diagram of considered flexible master-slavearms

Fig. 2 Schematic drawing of the rigid master arm

end of this arm and generates an operational torque,τh(t), i.e., τh(t) :=Fh(t)ℓm. θm(t) and τm(t) representthe rotational angle and the reaction torque, respec-tively. The physical parameters of the rigid arm are

– 2 –

Trans. of ISCIE, Vol. 30, No. 1 (2017)

Yagi, Kimura and Sawada: PDS Based Bilateral Control System with Random Delay 3

PD

controllerDC motor

Reference input

High-geared servomotor

+-

OutputVoltage

Fig. 3 Block diagram of the high-geared servomotor

as follows: Jh is the moment of inertia of the motor’srotor and Jm is the moment of inertia of the rigidmaster arm. µm is the coefficient of friction of themotor shaft. The mathematical model of the rigidmaster arm is obtained as follows:

(Jh+Jm+

1

4mmℓ2m

)θm(t)

=−µmθm(t)+τh(t)+τm(t)+gmγm(t), (1)

where γm(t) is the system noise which defined as awhite Gaussian noise process. gm is the coefficientfor the system noise.

Eq. (1) should be rewritten as a state space modelto design a state estimator. xm(t) := [ θm(t) θm(t) ]T

represents a state vector. The state space model ofthe rigid master arm can be written as

xm(t)=Amxm(t)+Bmum(t)+Gmγm(t). (2)

2.2 High-geared ServomotorIn this research, the flexible slave arm is rotated by

a high-geared servomotor in the horizontal plane. Theblock diagram of the high-geared servomotor is shownin Fig. 3. As seen in this figure, Θcom(s) and Θ(s)are, respectively, the Laplace transforms of θcom(t)and θ(t). θcom(t) and θ(t) represent the reference an-gle and the output angle, respectively. The transferfunction from Θcom(s) to Θ(s) can be calculated [14].By applying the inverse Laplace transform, the con-trol torque, τ(t), is obtained as follows:

τ(t)=kτKP

Rθcom(t)−θ(t)

+kτKD

R

θcom(t)− θ(t)

− kekτ

Rθ(t), (3)

where kτ denotes the torque constant and ke denotesthe back electromotive force constant. R denotes theinternal resistance of the coil in the DC motor. KP

and KD are the proportional and derivative gains ofthe PD controller, respectively. This torque drivesthe flexible slave arm.

2.3 Flexible Slave ArmThe parallel-structured single-link flexible arm is

employed as the slave arm in this study (shown inFig. 4). This arm is the experimental device of ourgroup. In this research, the mathematical model ofthe flexible arm is derived by considering this arm.This arm is constructed by a pair of uniform Euler-Bernoulli beams. Both ends of the flexible beam clampan unit hub and a tip-mass. Since the mathematical

Fig. 4 Schematic drawing of the parallel-structured flex-ible slave arm

Fig. 5 Schematic drawing of the simplified model of theflexible slave arm

model of this arm is represented by using the highlycomplex nonlinear differential equations, this arm isapproximated as a single-link flexible arm in this re-search (shown in Fig. 5). This simplified flexible armhas the same boundary conditions as the original arm[15].

A simplified single-link flexible arm has a length, ℓ.OXY is defined as the inertial Cartesian coordinatesystem and Oxy is defined as the rotating coordinatesystem. u(t,x) denotes the transverse displacement ofthe flexible beam from the x−axis. θ(t) denotes therotational angle between the OX− and the Ox−axes.The physical parameters of the flexible arm are as fol-lows: ρ and S are the uniform mass density and thecross section, respectively; EI is the uniform flexiblerigidity (where E is Young’s modulus and I is thesecond moment of the cross-sectional area); cD is thecoefficient of Kelvin–Voigt-type damping and µ is thecoefficient of friction of the slave motor shaft; J0 de-notes the inertia moment of the motor shaft of theslave arm; and m denotes the mass of the tip-mass,assuming that the tip-mass is a point mass.

The total kinetic energy T (t) and the potentialenergy U(t) are expressed as follows:

T (t) = Th(t)+Tb(t)+Ttm(t) (4)

U(t) =

∫ ℓ

0

1

2EIu′′(t,x)2dx (5)

– 3 –


Th(t) =1

2J0θ

2(t) (6)

Tb(t) =

∫ ℓ

0

1

2ρS

xθ(t)+ u(t,x)

2

dx (7)

Ttm(t) =1

2mℓθ(t)+ ˙u(t)

2

, (8)

where u(t) is defined as u(t,ℓ). Th(t) denotes the rota-tional kinetic energy of the unit of the hub. Tb(t) andTtm(t) denote the translational kinetic energy of theflexible beam and the tip-mass, respectively. U(t) de-notes the bending strain energy of the flexible beam.The prime represents the derivative with respect tox, i.e., ·′ = ∂ ·/∂x. The virtual work δW (t) is cal-culated as

δW (t)=−∂F0(t)

∂θδθ−

∫ ℓ

0

∂F (t,x)

∂u′′ δu′′dx+τ(t)δθ,

(9)

where F0(t)(:=µθ2/2) is the dissipation function dueto the rotation of the slave motor, and F (t,x) (:=

cDIu′′(t,x)2/2) is the dissipation function due tothe internal damping of the slave arm.

By applying Hamilton’s principle, the mathemat-ical model of the flexible arm for the motor shaft andthe flexible beam are expressed as follows:

J0θ(t)+µθ(t)−cDIu′′(t,0)− u′′(t,ℓ)−τ(t)

−EI u′′(t,0)−u′′(t,ℓ)−∫ ℓ

0

gsxγs(t,x)dx=0

(10)

ρSu(t,x)+cDIu′′′′(t,x)+EIu′′′′(t,x)+ρSxθ(t)

+m ℓθ(t)+ ¨u(t)δ(x−ℓ)−gsγs(t,x)= 0, (11)

where γs(t,x) is the system noise, which is defined asa white Gaussian noise. gs is the coefficient of thesystem noise.

The boundary and initial conditions are given by

B.C. :u(t,0)=u′(t,0)=u′(t,ℓ)=u′′′(t,ℓ)= 0 (12)

I.C. :u(0,x)= u(0,x)= 0. (13)

Eq. (11) should be rewritten as a finite-dimensionallinear system in the modal representation to design astate estimator. xs(t) := [ θ(t) u(t) ··· uN (t) θ(t) u(t)··· uN (t) ]T is expressed as the state vector. The statespace model of the flexible arm can be rewritten as

xs(t)=Asxs(t)+Bsus(t)+Gsγs(t). (14)

3. Modeling of Random Delay

The communication network causes the randomdelay. It is known that the actual random delay haslog-normal distribution. However, the random delayis approximated by the sum of an average time de-lay and a random process in this research since afunction with log-normal distribution has not beenconstructed. The approximated random delay is rep-

resented as T +γo(t), where T and γo(t) denote theaverage time delay and the random process, respec-tively. The random delay is greater than zero and isdefined by

γo(t)=

γ(t) for γo(t)>−T

0 otherwise,(15)

where γ(t) is the white Gaussian noise. Its mean andcovariance are Eγ(t)=0 and E

γ2(t)

=:R, respec-

tively. It is assumed that the covariance of γ(t) holdsR≪T 2. Therefore, γo(t) can be approximated by thewhite Gaussian noise, γ(t), i.e., γ(t)∼= γo(t).

4. Synthesis of Kalman filter

A Kalman filter is designed as the state estimatorto reduce the effect of the random delay in this re-search. An ordinary time-invariant linear stochasticsystem with random delay is discussed for designingthe Kalman filter [13].

dx(t) =Ax(t)dt+Bu(t)dt+Gdws(t) (16)

dy(t) =Cx(t−(T +γ(t)))dt, (17)

where x(t)∈ℜm is the state vector and y(t)∈ℜn is theobservation vector; u(t)∈ℜ1 is a control input; anddws(t) is an increment of the Wiener process withzero-mean and covariance Qs(:= Edw2

s(t)). dws(t)is expressed as γs(t)dt, where γs(t) denotes the systemnoise, which is defined as a white Gaussian noise.

To process the random delay clearly, the Taylorseries expansion form is applied to the state vectorx(t−(T+γ(t))) around ts(:= t−T ) in the observationsystem. Thus, eq. (17) is rewritten as

dy(t)∼=Cx(ts)dt−CAx(ts)+Bu(ts)dwo(t)

−CG γs(ts)γ(t)dt. (18)

The high-order terms with respect to dt are neglectedand γs(ts)γ(t) can be ignored. Hence, the observationsystem can be rewritten as

dy(t)=Cx(ts)dt−CAx(ts)+Bu(ts)dwo(t), (19)

where dwo(t) is defined as γ(t)dt and denotes an in-crement of the Wiener process with zero-mean andcovariance R(:= Edw2

o(t)). In eq. (19), the secondterm of the right-hand side expresses the multiplica-tive noise that is expressed as a product of the state,x(ts), and the random variable, γ(t).

To reduce the effect of the multiplicative noise, theKalman filter should be designed for the stochasticsystem described by eqs. (16), (19). An innovationprocess, dν(t), is given as

dν(t) = dy(t)−Cx(ts|ts)dt=Ce(ts)dt−CAx(ts)+Bu(ts)dwo(t), (20)

where x(ts|ts) denotes the state estimate. e(ts) isrepresented as the error between x(ts) and x(ts|ts),i.e., e(ts) := x(ts)− x(ts|ts). The correlation function

– 4 –



of dν(t) is given as

Edν(t)dνT(t)

=C

[AΠ(ts)A

T+BQuBT]RCT,(21)

whereΠ(ts)(:=Ex(ts)xT(ts)) is the autocorrelationmatrix value of the state and Qu(:=Eu2(ts)) is theautocorrelation value of the input. Assuming that thestate, x(ts), and the input, u(ts), are uncorrelated,the correlation functions of these variables can be con-sidered zero, i.e., Ex(ts)uT(ts)= Eu(ts)xT(ts)=0. The state estimate, x(ts|ts), can be calculated byapplying the orthogonal projection[16]. The estima-tion error covariance, P (ts|ts), is calculated by theRiccati differential equation and the autocorrelationfunction of the state, Π(ts), is calculated by the Lya-punov differential equation. Eventually, the equationsof x(ts|ts), P (ts|ts), and Π(ts|ts) are given as follows:

dx(ts|ts)=Ax(ts|ts)dt+Bu(t−(T +γ(t)))dt

+P (ts|ts)CT[CAΠ(ts)A

T+BQuBTRCT

]−1

×dy(t)−Cx(ts|ts)dt (22)

P (ts|ts)=AP (ts|ts)+P (ts|ts)AT+GQsGT

−P (ts|ts)CT[CAΠ(ts)A

T+BQuBTRCT

]−1

×CP (ts|ts) (23)

Π(ts)=AΠ(ts)+Π(ts)AT+GQsG

T. (24)

The control input should be used as u(t−(T +γ(t)))because the control input, u(ts) cannot be obtained.

5. Synthesis of Controllers

The controllers to generate the reaction torque forthe rigid master arm and the reference angle for thehigh-geared servomotor are designed in reference toour previous studies in this research[11]. The PDcontroller is derived to provide the reaction torqueand the PDS controller is derived to calculate the ref-erence angle, respectively. These controllers are ex-pressed as follows:

τm(t)=KPm

θ(ts|ts)−θm(t)

+KDm

ˆθ(ts|ts)− θm(t)

(25)

f(t)=R

kτKD

[KPs

θm(ts|ts)−θ(t)

+KDs

ˆθm(ts|ts)− θ(t)

+KS

[cDIu′′(t,0)− u′′(t,ℓ)

+EIu′′(t,0)−u′′(t,ℓ)]

−kτKP

Rθcom(t)−θ(t)+ (ke+KD)kτ

Rθ(t)

], (26)

where KPi(i=m,s) is the proportional gain, KDi(i=m,s) is the derivative gain, and KS is the strain gain.The control input, f(t), is defined as θcom(t), i.e.,f(t) := θcom(t). Therefore, by integrating f(t), the

Table 1 Physical parameters of the rigid master arm

Symbol ValueJh 0.70 [kg ·m2]µm 3.03×10−2 [kg ·m2 ·s]mm 29.05×10−3 [kg]ℓm 0.30 [m]Jm 6.54×10−4 [kg ·m2]

Table 2 Physical parameters of the flexible slave arm

Symbol ValueJ0 0.70 [kg ·m2]µ 3.03×10−2 [kg ·m2 ·s]ℓ 0.30 [m]ρ 8.8×103 [kg/m3]S 4.0×10−5 [m2]E 1.1×1011 [Pa]I 8.33×10−13 [m4]cD 1.93×109 [N ·s/m2]m 0.245 [kg]

Table 3 Physical parameters of the high-geared servo-motor

Symbol ValueKP 32 [V/rad]KD 32 [V/(rad/s)]ke 2.6 [V/(rad/s)]kτ 2.6 [N/A]R 1.73 [Ω]

reference angle, θcom(t), can be calculated.

6. Numerical Simulations

The performance of the proposed system is demon-strated through numerical simulations in this research.It is confirm that the angle of the flexible slave armtracks that of the rigid master arm and the proposedKalman filter can reduce the effect of the random de-lay.

6.1 SetupIt is considered that the flexible slave arm is made

of phosphor bronze in the numerical simulations. Thelength, thickness, and width of the flexible slave armis ℓ=0.3[m], 1.0×10−3[m], and 4.0×10−2[m], respec-tively. Tables 1-3 list the physical parameters of therigid master arm, the flexible slave arm, and the high-geared servomotor.

The number of modes of the proposed system isset as ten. Matlab and Simulink are used to demon-strate the numerical simulations. The initial con-ditions of the proposed system are set as θm(0) =0[rad], θm(0)=0[rad/s], θ(0)=0[rad], θ(0)=0[rad/s],u(0,x)= 0[m], and u(0,x)= 0[m/s]. Furthermore, theaverage time delay of the random delay, T , is set asT =0.25,0.50[s] (shown in Fig. 6(a),(b)). Fig. 7 showsthe sample simulation results of the tip’s displacement

– 5 –


Tim

e d

elay[s

]

Time[s]0 5 10 15

0.1

0.2

0.3

0.4

(a) T =0.25[s]

Tim

e d

elay[s

]

Time[s]0 5 10 15

0.3

0.4

0.5

0.6

0.7

(b) T =0.50[s]

Fig. 6 Random time delay

Time[s]0 5 10 15

-10

-5

0

5

(a) T =0.25[s]

Time[s]0 5 10 15

-10

-5

0

5

(b) T =0.50[s]

Fig. 7 The tip’s displacement, u(t,ℓ) (Sample simulationresults without the system noise)

of the slave arm without the system noise. As seenin this figure, the tip’s displacement does not vibrate.Since states of the master and slave arms are only af-fected by the random delay, this figure indicates thevalidity of the proposed Kalman filter. The simu-lation results in next subsection are affected by therandom delay and the system noise. In this paper,the filter for reducing the effect of the system noisedoes not designed. However, because this filter is the

An

gle

[rad

]

Time[s]0 5 10 15

0

0.5

1

(a) Angles, θm(t), θm(ts|ts) and θ(t)

Time[s]0 5 10 15

-10

-5

0

5

(b) The tip’s displacement, u(t,ℓ)

Torque[Nm]

Time[s]0 5 10 15

-1.5

-1

-0.5

0

0.5

1

(c) Torque, τm(t) and τ(t)

Fig. 8 Numerical simulation results (T =0.25[s])

usual filter, it is considered that the effect of the sys-tem noise can be reduced.

6.2 ResultsFigs. 8-11 show the results of the numerical simu-

lations. Figs. 8, 9 show the results when the averagetime delay, T , is 0.25[s] and Figs. 10, 11 show the re-sults when the average time delay, T , is 0.50[s]. Asseen in each figures, the top figures depict the angle ofthe rigid master arm, θm(t), and the estimated angle

of the rigid master arm, θm(ts|ts), and the angle of theflexible slave arm, θ(t); the middle figures depict thetip displacement of the flexible slave arm, u(t,ℓ); andthe bottom figures depict the reaction torque, τm(t),and the control torque, τ(t).

In Fig. 8(a)-Fig. 11(a), the estimated angle of the

rigid master arm, θm(ts|ts), is well estimated by us-ing the proposed Kalman filter because the effect ofthe random delay is reduced. Furthermore, the angleof the flexible slave arm, θ(t), tracks the angle of therigid master arm well. In Fig. 8(b),Fig. 10(b), theovershoot appears to reach the desired state. How-ever, due to the PDS controller, it quickly converges

– 6 –



Angle[rad]

Time[s]0 5 10 15

-2

-1

0

1

2


Time[s]0 5 10 15

-2

-1

0

1

2


Torque[Nm]

Time[s]0 5 10 15

-4

-2

0

2

4



Table 4 Improvement rates of the rotational angle of theflexible slave arm [%]Input signal

Average delay0.25 0.50

Step function 98.9 99.3Sinusoidal wave 95.9 97.5

to zero. In Fig. 9(b),Fig. 11(b), the tip displacementis not diverge because of the effect of the PDS con-troller. Therefore, the control aim is accomplished.Furthermore, in Fig. 8(c)-Fig. 11(c), the reaction torquefor the rigid master arm, τm(t), and the control torquefor the flexible slave arm, τ(t) are generated by thedifferences of the angle and angular velocity and thenoise affects. Therefore, since the reaction torque be-comes noisy, the operator may feel uncomfortable.

Table 4 represents the improvement rates of therotational angle of the flexible slave arm when thestates are interacted through the communication net-work. These value are calculated by means of themean squared error. True values are defined as thestates which are affected by the average time delay.

Angle[rad]

Time[s]0 5 10 15

0

0.5

1


Time[s]0 5 10 15

-10

-5

0

5


Torque[Nm]

Time[s]0 5 10 15

-1.5

-1

-0.5

0

0.5

1



Measurement values are defined as the states whichare affected by the random delay. Moreover, thesevalues are measured by using the proposed Kalmanfilter or not. The mean squared errors are calculatedby using these values. Furthermore, the improvementrates are calculated as follow:

Improvement rate=MSE1−MSE2

MSE1×100[%]

(27)

where MSE1 is the mean squared error without theproposed Kalman filter andMSE2 is the mean squarederror with the proposed Kalman filter. The improve-ment rate indicates the reduced effect for the randomdelay. It is found that the proposed Kalman filter canreduce the effect of the random delay.

7. Conclusions

A PDS based bilateral control system for flexi-ble master-slave arms with random delay was inves-tigated in this paper. By using Hamilton’s princi-ple, the mathematical model of the rigid master andflexible slave arms were derived. Because the flexible

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Angle[rad]

Time[s]0 5 10 15-2

-1

0

1

2


Time[s]0 5 10 15

-2

-1

0

1

2


Torque[Nm]

Time[s]0 5 10 15

-4

-2

0

2

4



slave arm was actuated by the high-geared servomo-tor, the transfer function from the reference input tothe torque was calculated through the proper model.The random delay was defined as a sum of the averagetime delay and a white Gaussian noise. The averagetime delay was assumed to be known in this research.The state space models including the system noise ofthe rigid master arm and the flexible slave arm werederived to design the novel Kalman filter. The obser-vation system included the random delay. To treatthe random delay explicitly, the Taylor series expan-sion form was applied to the observation system. Bythese procedures, the approximated observation sys-tem showed multiplicative noise that consisted of theproduct of the states and the randomness of the ran-dom delay. The proposed Kalman filter had threeequations to estimate the state and to reduce the ef-fect of the random delay: the estimate state system,the Riccati equation for the estimation error covari-ance, and the Lyapunov differential equation for theautocorrelation of the states. The PD controller wasdesigned to generate the reaction torque for the rigidmaster arm and the PDS controller was designed to

generate the control input for the flexible slave arm.Numerical simulations were accomplished to confirmthe performance of the proposed control system andthe proposed Kalman filter. As seen in the numeri-cal simulation results, the proposed Kalman filter wasa well-designed filter since the effect of the randomdelay was reduced. Furthermore, the PD and PDScontrollers were also well-designed since the reactiontorque and control input were generated, and the an-gle of the flexible slave arm tracks the estimated angleof the rigid master arm well.

Acknowledgments

This work was supported by JSPS KAKENHI (C)Grant Number 25420184.

References

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[2] T. Mori, Y. Morita, et al.: A study on passivityof flexible master-slave manipulators; SICE AnnualConference 2004, Vol. 3, pp. 2407–2410 (2004)

[3] T. Namerikawa: Bilateral control with constant feed-back gains for teleoperation with time varying delay;Proc. of the 48th IEEE Conference on Decision andControl, 2009 held jointly with the 2009 28th ChineseControl Conference, pp. 7527–7532 (2009)

[4] H. Fujita and T. Namerikawa: Delay-independentstabilization for teleoperation with time varying de-lay; 2009 American Control Conference, pp. 5459–5464 (2009)

[5] F. Mastuno, T. Ohno, et al.: PDS (ProportionalDerivative and Strain) feedback control of a flexiblestructure with closed-loop mechanism; Proc. of the38th Conference on Decision and Control, pp. 4331–4335 (1999)

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[11] M. Yagi and Y. Sawada: Bilateral control of flexi-

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ble master-slave arms using high-geared servomotorwith time-varying delay; SICE Annual Conference,pp. 180–187 (2013)

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AuthorsMasaharu Yagi (Member)

He received his Ph.D. degree from the

Kyoto Institute of Technology, Japan,

in 2014. In 2015, he joined Kyoto Pre-

fecture. His research interests include

network control systems, modeling and

vibration control of flexible structures,

and optimal filters. He is a member of RSJ and SICE.

Kengo Kimura

He received his B.S. degree from the

university of Shiga Prefecture, Japan, in

2015. He is currently an M.S. student

at Kyoto Institute of Technology. His

research interests include network con-

trol systems. He is a student member of

JSME.

Yuichi Sawada (Member)

He received his Ph.D. degree from the

Kyoto Institute of Technology, Japan,

in 1994. In 1995, he joined the Kyoto

Institute of Technology as an Assistant

Professor in the Division of Mechanical

and System Engineering. In 2015, he

became a Professor in the Division of Mechanical and Sys-

tem Engineering. His research interests include modeling

and vibration control of flexible structures, and optimal

filters. He is a member of IEEE, IMechE, SICE, RSJ, and

JSME.

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Documents

PDS Based Bilateral Control System of Flexi- ble Master