RBF Neural Network based Trajectory Control and Impedance Control

of a Upper Limb Tele-rehabilitation Process

Ting Wang

1 a

and Yanfeng Pu

2 b

1

School of Instrument Science and Engineering, Southeast University, 2, Sipailou, Xuanwu District, Southeast University,

210096, Nanjing, China

2

Nanjing Customs District P. R. China, 360, Longpanzhong Road, Qinhuai District, Nanjing, China

Keywords:

Impedance Control, Upper Limb, Tele-rehabilitation.

Abstract:

In the passive tele-rehabilitation process, the safety is the most important thing for patients avoiding the sec-

ondary damage of the impaired upper limb. Aiming at adjusting the appropriated contact force in time during

the training exercises, an adaptive impedance control is proposed for the slave side. At the same time, the

trajectory control based on the Hamilton-Jacobi-Inequality theory and the RBF Neural network is performed

for the master manipulator operated by therapists. The stability is analyzed and numerical simulations show

the efﬁciencies and high performances of the proposed method.

1 INTRODUCTION

Recently, researchers publish their study in the New

England Journal of Medicine (NEJM) based on the

GBD data in 2016 (GBDStrokeCollaborators, 2018),

which calculates the lifetime stroke risk in various

countries and regions from 1990 to 2016. Results

demonstrate that the risk of the lifetime stroke in

adults aged 25 increases by 8.9% to 24.9% in the

past 26 years. What is more, the overall risk of

the lifetime stroke in China and Chinese men both

reach 40%. Scientists continue to point out that the

number of stroke in China accounts for the ﬁrst rank

in the world (GBDStrokeCollaborators, 2019)(Gore-

lick, 2019). After the participation of the Chinese re-

searchers, the latest research evidence and the speciﬁc

information in the ﬁeld of stroke prevention and treat-

ment in China are published in(S Wu, 2019). It re-

veals that the incidence of stroke in China is promoted

in the past 20 years. The prevention and treatment of

stroke in China is not balanced among regions. The

risk factors of stroke are higher in the countryside

than in the city (Brainin, 2019). With the continu-

ous improvement of the stroke treatment in China in

recent years, the mortality of stroke patients are not

enhanced signiﬁcantly in the past 20 years. However,

the incidence of stroke is still rising. That is, the bur-

a

https://orcid.org/0000-0001-7414-5390

b

https://orcid.org/0000-0003-0857-9048

den of the stroke is still heavy. In recent 10 years,

the prevalence of stroke among urban and rural resi-

dents has a stable trend, while the prevalence in rural

areas has increased signiﬁcantly (Z Li, 2019). Since

the China has the vast territory, many rural stroke pa-

tients live far away from the city, so that it is more

difﬁcult for them to go to hospital for rehabilitation

treatment than urban residents. Therefore, the tele-

rehabilitation training is a good way to solve the prob-

lem. In the meantime, it may decrease the cost of ex-

pensive therapies, and it also raise the efﬁciency of

the medical therapists.

The upper limb tele-rehabilitation refers that a

therapist operates a rehabilitative manipulator in the

master side, while another rehabilitation manipulator

simultaneously assists patients in the slave side so as

to achieve training exercises in the remote place. Via

the internet vision and the communication on both

sides, the therapist may connivently guide and adjust

the patient’s training exercises. Since it is an inter-

esting and a new active issue, many researchers de-

vote various study for the purpose of spreading the

tele-rehabilitation mode rehabilitation to remote pas-

sive therapists. On the basis of the wave variable the-

orem, Mendoza and his colleagues present a novel

bilateral tele-rehabilitation method in (M Mendoza,

2016). In the tele-rehabilitation system, the motion-

based adaptive impedance control is exerted on both

the master and slave robot manipulators for robot as-

sisted passive rehabilitation. Thinking of the time de-

Wang, T. and Pu, Y.

RBF Neural Network based Trajectory Control and Impedance Control of a Upper Limb Tele-rehabilitation Process.

DOI: 10.5220/0009143401090116

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 109-116

ISBN: 978-989-758-442-8

Copyright

c

2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

109

lay, the stability of the tele-rehabilitation system is

analyzed. Numerical simulations are performed and

results demonstrate that the proposed method may

ensure the stable humanrobot interaction as well as

compensate the position drift. The paper developed a

novel control method based on the estimation of the

forces on both the master and slave robots so as to re-

place the expensive force sensor (F Azimifar, 2017).

The stability is analyzed with the consideration of the

time delay. The proposed method is veriﬁed by posi-

tion tracking experiments. In order to enhance the pa-

tienttherapist interaction, Mojtaba and his colleagues

introduce a adaptive bilateral impedance controller

for the upper limb tele-rehabilitation training process

(M Shariﬁ, 2017). The proposed method is veriﬁed

by experiments on a nonlinear multi-DOF manipula-

tors. Thus, they continue to study the impedance con-

trol of robot assisted tele-operation process (M Shar-

iﬁ, 2018). They tune an adaptive law with bilat-

eral impedance control for the purpose of changing

impedance model parameters during the tele-surgery

of a beating heart. The advantage of their proposed

method is that it avoids affording expensive force sen-

sor’s device as well estimating the heart’s motion. Af-

ter the discuss of the stability, experimental results

demonstrate that the proposed bilateral impedance

control may increase the safety of patients and com-

pensate the motion of the beating heart.

In this paper, we focus on the bilateral impedance

control for the upper limb tele-rehabilitation training

process so as to achieve the protection of the safety of

impaired patients. Simultaneously, a trajectory con-

troller is proposed also on the basis of the Hamilton-

Jacobi-Inequality (HIJ) theory and the RBF Neural

network to ensure the normal training exercises. The

therapist may change the motion through regulating

parameters of the impedance model from the contact

force in the slave side. Rest of the paper is organized

as follows. The dynamic model and the impedance

model is introduced in section 2. Section 3 explains

the trajectory control and the impedance control in the

master side and the adaptive neural fuzzy impedance

control in the slave side. Numerical simulations are

demonstrated in section 4. Some conclusions are

given in the conclusion part.

2 THE DYNAMIC MODEL AND

THE IMPEDANCE MODEL OF

THE TELE-REHABILITATION

SYSTEM

The upper limb tele-rehabilitation process may be de-

scribed in the following picture as shown in Figure.1.

Figure 1: The upper limb tele-rehabilitation process.

The integral system may be regarded as a bilateral

tele-operation system involving (1)the therapist and

the master manipulator in the master side, (2) the pa-

tient and the slave manipulator in the slave side and

(3) a VR screen which exhibits the training process

by the internet. At the beginning of our study, the

time delay is simpliﬁed to zero in this paper although

it cannot be ignored in the reality. Therefore, the up-

per limb tele-rehabilitation system can be separated

into two subsystems as shown in Figure. 2. The ﬁrst

subsystem is the master trajectory control loop. As

the therapist designs the desired training exercises tra-

jectory, the master manipulator tracks it by the mas-

ter trajectory control. In the initial training exercises,

the slave manipulator follows the master manipulator

to push the upper limb of the patient. Since patients

have different degree of disability, they can bear dif-

ferent contact forces. The second subsystem is the

slave impedance control. With the changes of the con-

tact forces, the desired trajectory is adjusted through

the slave impedance control. In fact, the role of the

slave impedance control is a forward feedback using

to modify the training exercises trajectory depending

on the affordable conditions of patients.

Master

Manipulator

Therapist

Master

Trajectory

control

Path

planning

Slave

Manipulator

Environment

Slave

Trajectory

control

Slave

Impedance

Control

Figure 2: The composition of the tele-rehabilitation system.

Assuming that both the master and the slave take

the n DOF manipulator system, the dynamic model is

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

110

expressed as follows.

M

i

(q

i

(t)) ¨q

i

(t)+C

i

(q

m

(t), ˙q

i

(t)) ˙q

i

(t)+G

i

(q

i

)+∆(q

i

, ˙q

i

)+d

i

(t) = T

i

(t)

(1)

where M

i

∈ R

m×n

,C

i

∈ R

m×n

, G

i

∈ R

m×n

, i = m, s

stand for the inertial items, the Coriolis and centrifu-

gal effects items and the gravitational items. The

subscripts m and s represent respectively the master

and the slave. State variables in the joint space of

the master and the slave manipulators are described

as q

i

, ˙q

i

, ¨q

i

∈ R

n×n

. T

i

indicates the control inputs.

∆(q

i

, ˙q

i

) and d

i

(t) represent the model uncertainties

and the external disturbances.

The impedance model of the slave system is writ-

ten as follow.

H

s

¨x

s

(t) + B

s

˙x

s

(t) + K

s

x

s

= f

e

, (2)

where H

i

, B

i

and K

i

are respectively inertial,

damping and stiffness matrices. x

s

, ˙x

s

, ¨x

s

∈ R

n×n

are

state variables in the work space. The state variables

in the work space may transferred to the joint space

via following nonsingular Jacobian matrices.

˙x

i

= J( ˙q

i

)

¨x

i

=

˙

J(q

i

) +

˙

J ¨q

i

(3)

where

J =

−l

1

sin(q

i1

) − l

2

sin(q

i1

+ q

i2

) −l

2

sin(q

i1

+ q

i2

)

l

1

cos(q

i1

) + l

2

cos(q

i1

+ q

i2

) l

2

cos(q

i1

+ q

i2

)

The tele-rehabilitation system has following prop-

erties (J Zhang, 2018).

• The inertia matrices M

i

are symmetric positive

deﬁnite matrices, M

i

= M

T

i

, and there exist the

upper and lower boundedness, M

down

I ≤ |M

i

| ≤

M

up

I (M

down

and M

up

are two positive constants)

•

˙

M

i

(q

i

)−2C

i

(q

i

, ˙q

i

) are skew-symmetric matrices.

• For all q

i

(t), ˙q

i

(t) ∈ R

n×1

, there exists a positive

scalar c

i

may render C

i

(q

i

(t), ˙q

i

(t)) ≤ c

3

| ˙q

i

|(1 +

|q

i

|) ≤ c

3

| ˙q

i

|, in which c

1

, c

2

, c

3

> 0 and | · | rep-

resents the Euclidean matrix norm.

• The linear parameterizable dynamic model of the

tele-rehabilitation system (in Eq.(1)) may be ex-

pressed as M

i

(q

i

(t)) ¨q

i

(t) + C

i

(q

i

(t), ˙q

i

(t)) ˙q

i

(t) +

G

i

(q

i

(t)) = Y

i

(q

i

(t), ˙q

i

(t), ¨q

i

(t))θ

i

, where

Y

i

(q

i

(t), ˙q

i

(t), ¨q

i

(t)) ∈ R

n×p

are a certain function

and θ

i

∈ R

p

are physical parameter vectors of the

master and the slave manipulators.

3 THE TRAJECTORY CONTROL

BASED ON THE HJI THEORY

AND RBF NEURAL NETWORK

3.1 The HJI Theory

The bilateral thele-rehabilitation system (in Eq.1)

may be rewritten as the following form.

˙x

i

(t) = f (x

i

) + g(x)d(t)

z(t) = h (x

i

)

(4)

where x(t), u(t), z(t) are respectively the state vari-

able, the input and the system evaluation index. d is

the external disturbances causing by communication

time delay, noises and so on.

Deﬁnition For the disturbance signal, its norm is de-

ﬁned as kd(t)k

2

= {

R

∞

0

d

T

(t)d(t)dt}

1

2

, which may

measure the magnitude of the d(t) energy. In order

to evaluate the disturbance suppression ability of the

system, the performance index is deﬁned as follow

J = sup

kd(t)k6=0

=

k z k

2

kdk

2

, (5)

where J is the L

2

gain of the system, indicating the ro-

bust performance. The smaller J results better robust

performance of the system.

According to (Schaft, 1992), the Hamilton-Jacobian-

Inequality theorem can be described as follow. For

a positive number γ, if there exists a positive deﬁnite

and differentiable function L(x) ≥ 0, and

˙

L ≤

1

2

{γ

2

kdk

2

− kzk

2

}, (∀d), (6)

then J ≤ γ.

3.2 The Trajectory Control and Its

Analysis

Assuming there is no time delay, the master is the

same as the slave manipulator, and both of manipu-

lators have the same initial state variables. Due to the

limitation of the paper, we merely introduce the mas-

ter trajectory control, and the slave adopts the same

trajectory control method. The desired trajectory is

noted by q

d

, and the trajectory tracking error is de-

ﬁned as e = q − q

d

. The forwards feedback control

law is designed as

T = u + M ¨q

d

+C ˙q

d

+ G, (7)

where u is the feedback control law.

Substituting the Eq.(7) into the Eq. (1), the close

RBF Neural Network based Trajectory Control and Impedance Control of a Upper Limb Tele-rehabilitation Process

111

tele-rehabilitation system may change to the follow-

ing form,

M ¨e +C ˙e + ∆(q, ˙q) + d = u. (8)

Ordering ∆ f (q, ˙q)+d, we may get M ¨e+C ˙e+∆ f = u.

Taking the RBF neural network (RBFNN) to approx-

imate the ∆ f , the ∆ f is expressed as

∆ f = W

∗

f

σ

f

+ ε

f

, (9)

where ε

f

stands for the approximation error. σ

f

is the

RBF Gaussian function and W

∗

f

is the ideal weights

of the RBFNN. Combing Eq.(8) and Eq.(9), we may

acquire M ¨e +C ˙e +W

∗

f

σ

f

+ ε

f

= u.

Deﬁne

x

1

= e

x

2

= ˙e + αe, α > 0

(10)

then,

x

1

= x

2

− αx

1

M ˙x

2

= −Cx

2

+ ω −W

∗

f

− ε

f

+ u

ω = Mα ˙e +Cαe

(11)

Using the HJI inequality, rewritten the Eq. (11) to the

state space form as follow,

˙x = f (x) + g(x)d

z = h(x)

(12)

where f (x) =

x

2

− αx

1

1

M

(−Cx

2

+ ω −W

∗

f

σ

f

+ u)

,

g(x) =

0

1

M

, d = ε

f

. The evaluation index is

deﬁned as follow. Due to d = ε, the approximation

error may be regarded as the external disturbance.

Therefore, the evaluation index is denoted by

z = x

2

= ˙e + αe, and its L

2

gain calculated by

J = sup

kε

f

k6=0

=

kzk

2

kε

f

k

2

.

The adaptive law of the tele-rehabilitation system is

designed as

˙

ˆ

W

f

= −ηx

2

σ

T

f

. (13)

Thus, the feedback control is expressed as

u

i

= −ω −

1

2γ

2

x

2

+

ˆ

W

f

σ

f

−

1

2

x

2

, (14)

where

ˆ

W

f

and σ

f

respectively note the weight of the

RBFNN and the output of the Gaussian function. That

is, the tele-rehabilitation system satisﬁes J ≤ γ. Ac-

cording to (Y wang, 2009), the stability is analyzed

as follow. The Lyapunov function is selected as

L =

1

2

x

T

2

Mx

2

+

1

2η

tr(

˜

W

T

f

˜

W

f

),

where

˜

W

f

=

ˆ

W

f

− W

∗

f

. Due to the properties of the

tele-rehabilitation system and the proposed feedback

control law, we may get following relations.

˙

L = x

T

2

M ˙x

2

+

1

2

x

T

2

˙

Mx

2

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

)

= x

T

2

(−Cx

2

+ ω −W

∗

f

σ

f

− ε

f

+ u)

+

1

2

x

T

2

˙

Mx

2

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

)

= x

T

2

(−Cx

2

−W

∗

f

σ

f

− ε

f

−

1

2γ

2

x

2

+

ˆ

W

f

σ

f

−

1

2

x

2

)

+

1

2

x

T

2

˙

Mx

2

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

)

= x

T

2

(−ε

f

−

1

2γ

2

x

2

+

˜

W

f

σ

f

−

1

2

x

2

)

+

1

2

x

T

2

(

˙

M − 2C)x

2

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

)

= −x

T

2

ε

f

−

1

2γ

2

x

T

2

x

2

+ x

T

2

˜

W

f

σ

f

−

1

2

x

T

2

x

2

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

)

Order

H =

˙

L −

1

2

γ

2

kε

f

k

2

+

1

2

kzk

2

, (15)

then,

H = −x

T

2

ε

f

−

1

2γ

2

x

T

2

x

2

+ x

T

2

˜

W

f

σ

f

−

1

2

x

T

2

x

2

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

)

−

1

2

γ

2

kε

f

k

2

+

1

2

kzk

2

Considering the following conditions,

• −x

T

2

ε

f

−

1

2γ

2

x

T

2

x

2

−

1

2

γ

2

kε

f

k

2

= −

1

2

k

1

γ

x

2

+ γε

f

k

2

≤ 0,

• x

T

2

˜

W

f

σ

f

+

1

η

tr(

˙

˜

W

T

f

˜

W

f

) = 0,

• −

1

2

x

T

2

x

2

+

1

2

kzk

2

,

we may easily get H ≤ 0. Depending on the deﬁnition

of H, we have

˙

L ≤

1

2

γ

2

kε

f

k

2

−

1

2

kzk

2

. Due to Eq.(6)

of the HJI theorem,we may deduce J ≤ γ, so that kzk

satisﬁes the performance index. That is, the trajectory

tracking error e

i

and ˙e conform to the convergency

requirement.

3.3 Adaptive Slave Impedance Control

The impedance model and the contact force between

the patient and the slave manipulator can be written

as follow.

H( ¨x − ¨x

m

) + B( ˙x − x

m

) + K(x − x

m

) = f

e

, (16)

where f

e

is the desired contact force between the im-

paired upper limb and the slave manipulator.. Since

we assume that there is not the time delay, the an-

gle and the angular velocity of the master manipu-

lator q, ˙q, ¨q may lossless send to the slave manipula-

tor. Via the transfer of the Jacobian matrix, we may

acquire the modiﬁed angular accelerator of the slave

manipulator. H, B and K are the inertia, the stiffness

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

112

and damping matrices parameters of the impedance

model. x, ˙x, ¨x are the position under the appropriate

contact force of patients.

Assuming that the tracking error is deﬁned as ˜x =

x − x

m

, the reference model is expressed as follow,

¨x

m

+ λ

1

˙x

m

+ λ

2

x

m

= λ

2

f

e

, (17)

where λ

1

and λ

2

are positive numbers. For an uncer-

tain H, the adaptive law is set as follow,

U =

ˆ

H( ¨x

m

− 2λ

˙

˜x − λ

2

˜x), (18)

where λ is strictly positive and

ˆ

H is the estimation of

the inertial matrix.

Deﬁning v = ¨x

m

− 2λ

˙

˜x − λ

2

˜x, and substituting the

Eq.(18) into Eq.(16), we may get the following re-

lation,

H ¨x =

ˆ

H( ¨x

m

− 2λ

˙

˜x − λ

2

˜x) =

ˆ

Hv. (19)

Setting

˜

H =

ˆ

H − H, the Eq.(19) may be transferred to

H( ¨x − v) =

˜

Hv. (20)

Deﬁne the tracking error function s as

s =

˙

˜x + λ˜x. (21)

From Eq. (19), it can be seen that the convergence of

s implies the the convergence of the position tracking

error ˜x and the velocity tracking error

˙

˜x.

Due to

¨x − v = ¨x − ¨x

m

+ 2λ

˙

˜x + λ

2

˜x

=

¨

˜x + λ

˙

˜x + λ(

˙

˜x + λ ˜x)

= ˙s + λs

, the Eq.(20) changes into

H( ˙s + λs) =

˜

Hv. (22)

That is, Hs ˙s = −λHs

2

+

˜

Hvs. Deﬁning the Lyapunov

function as

V =

1

2

(Hs +

1

γ

˜m

2

), γ > 0. (23)

The derivation of V is as follows,

˙

V = Hs ˙s +

1

γ

˜

H

˙

˜

H

= −λHs

2

+

˜

Hvs +

1

γ

˜

H

˙

ˆ

H

= −λHs

2

+

˜

H(vs +

1

γ

˙

ˆ

H)

The adaptive law of the

ˆ

H is deﬁned as

˙

ˆ

H = −γvs.

Thus, it is obvious that

˙

V = −λms

2

≤ 0. Due to

V ≥ 0,

˙

V ≤ 0, S and

˜

H are bounded depending on

(J LaSalle, 1961)(Hassan, 2002). As

˙

V ≡ 0, s = 0.

According to LaSalle invariance principle, the close

system are asymptotically stable. As t → 0, s → 0,

˜x → 0,

˙

˜x → 0.

4 NUMERICAL SIMULATION

In the simulation, the dynamic model set as follow,

M

11

(q

i

) = (m

1

+ m

2

)r

2

1

+ m

2

r

2

2

+ 2m

2

r

1

r

2

cos(q

i2

),

M

22

(q

i

) = m

2

r

2

2

M

12

(q

i

) = M

21

(q

i

) = m

2

r

2

2

+ m

2

r

1

r

2

cos(q

i2

),

C

11

(q

i

, ˙q

i

) = −m

2

r

1

sin(q

i2

) ˙q

2

,

C

22

(q

i

, ˙q

i

) = 0

C

12

(q

i

, ˙q

i

) = −m

2

r

1

r

2

sin(q

i2

)( ˙q

1

+ ˙q

2

)

C

21

(q

i

, ˙q

i

) = m

2

r

1

r

2

sin(q

i2

) ˙q

2

g

1

(q

i1

) = (m

1

+ m

2

)r

1

cos(q

i2

) + m

2

r

2

cos(q

1

+ q

2

),

g

2

(q

i2

) = m

2

r

2

cos(q

1

+ q

2

),

where m

1

and m

2

are respectively masses of the

upper limb and the forearm. r

1

and r

2

are the lengths

of the two links (the upper limb and the forearm). The

model uncertainties ∆(q

i

, ˙q

i

) is the uncertainty part of

the tele-rehabilitation dynamic model while d

i

repre-

sents the external disturbances. Set D = ∆(q

i

, ˙q

i

)+d

i

,

where

d

i

=

30sgnq

1

30sgnq

2

.

Real values in the numerical simulations

are used as follows:m

1

= m

2

= 1.5kg, r

1

= 1m,

r

2

= 0.8m. Ideal tracking signals are selected as

q

1d

= sint, q

2d

= sint, γ = 0.05. The with and

the center of the RBF Gaussian function choose

c

i

= [−1.5, −1.0, −0.5, 0, 0.5, 1.0, 1.5] and b

i

= 10.

Other parameters list as follows:λ

1

= 10, λ

2

= 25,

λ = 6. Initial conditions are set as ˙x

i

(0) = 0, x

i

= 0.5.

Applying the proposed trajectory tracking control

and impedance control to the master and the slave ma-

nipulators of the tele-rehabilitation system, results are

illustrated in the following Figures. from 3 to 12. In

the Figure. 3, the position signal and the speed track-

ing result are displayed. The ideal position signal and

the ideal speed signal mark with red solid lines while

the position tracking results and the speed tracking re-

sult are used by black dotted lines.

The details of the position tracking show in the

Figure.4. The ideal position of the upper limb and the

forearm represent by red solid lines while tracking re-

sults are marked with black dotted lines. The speed

tracking results are displayed in Figure. 5. The ideal

speed of the upper limb and the forearm represent by

red solid lines while tracking results are marked with

black dotted lines. The position tracking error and the

speed tracking error are showed in Figure. 6. The

control inputs of the upper limb and the forearm are

RBF Neural Network based Trajectory Control and Impedance Control of a Upper Limb Tele-rehabilitation Process

113

0 1 2 3 4 5 6 7 8 9 10

time(s)

-1

-0.5

0

0.5

1

position signal

ideal position signal

position tracking

0 1 2 3 4 5 6 7 8 9 10

time(s)

-4

-2

0

2

4

Speed tracking

ideal speed signal

speed tracking

Figure 3: The upper limb tele-rehabilitation process.

0 5 10 15 20 25 30 35 40

time(s)

-0.2

-0.1

0

0.1

0.2

Position tracking for upper limb

ideal position

position tracking

0 5 10 15 20 25 30 35 40

time(s)

-0.1

-0.05

0

0.05

0.1

Position tracking for forearm

ideal position

position tracking

Figure 4: The position tracking of the tele-rehabilitation

process.

0 5 10 15 20 25 30 35 40

time(s)

-1

-0.5

0

0.5

1

Speed tracking for upper limb

ideal speed for upper limb

speed tracking for upper limb

0 5 10 15 20 25 30 35 40

time(s)

-0.2

0

0.2

0.4

Speed tracking for forearm

ideal speed for forearm

speed tracking for forearm

Figure 5: The speed tracking of the tele-rehabilitation pro-

cess.

presented in Figure. 7. The ﬁgure 8 shows the es-

timation of

ˆ

H by the adaptive impedance control of

0 1 2 3 4 5 6 7 8 9 10

time(s)

-0.1

-0.05

0

0.05

position tracking error(cm)

position tracking error

0 1 2 3 4 5 6 7 8 9 10

time(s)

-2

0

2

4

6

Speed tracking error(m/s)

speed tracking error

Figure 6: The tracking error of the tele-rehabilitation pro-

cess.

0 5 10 15 20 25 30 35 40

time(s)

0

20

40

60

control input of upper limb (N.m)

0 5 10 15 20 25 30 35 40

time(s)

0

5

10

15

20

control input of forearm

Figure 7: The control input of the tele-rehabilitation pro-

cess.

0 5 10 15 20 25 30 35 40

time(s)

0

10

20

30

40

50

60

70

Ideal and estimation of H

Figure 8: The estimation of the impedance parameter of the

tele-rehabilitation process.

the slave manipulator. The contact force between the

patient and the slave manipulator is displayed in ﬁg-

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114

0 1 2 3 4 5 6 7 8 9 10

time(s)

-100

-50

0

50

Contact force of the slave side(N)

Practice Force

Estimated Force

Figure 9: The contact force of between the slave manipula-

tor and the patient.

0 2 4 6 8 10 12 14 16 18 20

time(s)

0

2

4

6

Uncertanties of system

Practical uncertainties of upper limb

Estimation uncertainties of upper limb

0 2 4 6 8 10 12 14 16 18 20

time(s)

-4

-2

0

2

4

Uncertanties of system

Practical uncertainties of forearm

Estimation uncertainties of forearm

Figure 10: The model uncertainties of the tele-rehabilitation

process.

ure 9. Model uncertainties of the upper limb and the

forearm are explained in Figure. 10. Figure. 11 intro-

duces the estimation of the external disturbances.

From results of numerical simulations, it may be

concluded that the patient may follow the training ex-

ercises trajectory according to the therapist through

the master trajectory tracking control. In the mean-

time, the therapist may sense the contact force of the

patient through the slave adaptive impedance control,

so that the therapist can momentarily adjust the train-

ing trajectory according to the severity of the patient.

5 CONCLUSIONS

The tele-rehabilitation process is studied in this pa-

per. In order to guide the training exercises of remote

patients, the contact force is an important issue to let

the therapist know how to strengthen or attenuate the

training exercises. At the same time, the trajectory

0 5 10 15 20 25 30 35 40 45 50

time(s)

-40

-20

0

20

40

60

80

100

120

140

160

Disturbances and its estimation

Disturbances

Estimation of disturbances

Figure 11: The external disturbances of the tele-

rehabilitation process.

tracking is also important so that different strengths

must be realized by different kinds of training exer-

cises. Aiming to solve the problems, we propose a

trajectory control based on the HJI theorem and the

RBF neural network for the master manipulator. Si-

multaneously, an adaptive impedance control is de-

signed so as to get the appropriate contact force of

the patient, for the purpose of protecting the patient

not to be impaired twice. Both of the trajectory con-

trol and the adaptive impedance control are analyzed

by the Lyapunov terrorem. Although the impedance

control is simple, it is easily to achieve in the prac-

tice. The proposed control methods are implemented

by numerical simulations. Results show the efﬁciency

and high performances.

ACKNOWLEDGEMENTS

This work was supported by the National Nat-

ural Science Foundation of China [grant num-

bers No.61906086, No.61802428], National Natu-

ral Science Foundation of Jiangsu province [BK.

20171019].

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