Available via license: CC BY

Content may be subject to copyright.

* Corresponding author: mlazarevic@mas.bg.ac.rs, mihailo.lazarevic@gmail.com

Open-closed Iterative Learning Control Algorithm for

Exoskeleton Rehabilitation Purposes

Mihailo Lazarević1*, and Nikola Živković1, D ar k o Radojević,Ph D 1

1Faculty of Mechanical Engineering, Kraljice Marije 16,11120 Belgrade,Serbia

Abstract. The p ap er d esigns an appropriate iterative learning control(ILC)

algorithm based on the trajectory characteristics of upper exoskeleton robotic

sy st em .The procedure of mathematical modelling of an exoskeleton system for rehabilitation is given and

synthesis of a control law with two loops. First (inner) loop represents exact linearization of a given system,

and the second (outer) loop is synthesis of a iterative learning control law which consists of two loops, open

and closed loop. In open loop ILC sgnPDD2 is applied, while in feedback classical PD control law is used.

Finally, a simulation example is presented to illustrate the feasibility and effectiveness of the proposed

advanced open-closed iterative learning control scheme.

1 Introduction

Stroke is second cause of mortality and disability in the

world, [1]. Patients who survive stroke are faced with

some degree of limb impairment, depending on the place

in brain structure and size of caused damage. It is widely

accepted that brain structure can be reorganized after

stroke and thus some functions can be fully or partially

recovered. Rehabilitative training plays crucial role in

recovery of lost functions, [2]. In order to enhance

therapy deliveredby therapists, use of robotics emerged

as aid in rehabilitation process,[3]. It is noted in [4] and

[5] that robot-aided sensorimotor training, especially in

upper limbs, shows that more activity leads to better

recovery and that recovered functions are sustained over

long period.

Rehabilitative robotics of upper limbs started with end-

effector robots research. End-effector robot supports

patients arm in one point of contact, usually patients

hand or forearm. End-effector robot joints movement

doesn’t coincide with movement of patients arm. These

drawbacks with end-effector robots influenced research

of exoskeleton rehabilitation robots. Exoskeletons

mitigate important flaws of end-effector robots

mentioned above [6]. Rehabilitation robots can be

developed to assist rehabilitation in individuals with

stroke.

Also, in the last three decades, iterative learning control

(ILC) has been extensively studied, achieves significant

progress in both theory and application, and becomes

one of the most active fields in intelligent control and

system control,[7-11].

Fig. 1. Schematics of end-effector robots (a) and exoskeleton

robots (b)

ILC is an intelligent control method for systems which

perform tasks repetitively over a finite time interval

where ILC approach is an imitation of a human learning

process. Intelligent beings tend to learn by performing a

trial (i.e. selecting a control input) and observing what

was the end result of this control input selection.

Emulating human learning, ILC uses knowledge

obtained from the previous trial to adjust the control

input for the current trial so that a better performance can

be achieved. The basic idea of ILC schemes is to refine

the control input to make better operation performance

of the system on the next trial by use of updated data of

the previous trial.

On the other side, rehabilitation training is a kind of

repetitive training. Body state of patients will improve

with an increase in the number of training while the

auxiliary level of robot and electrical stimulation will be

reduced.In ILC the control input is directly updated

between trials and it is this feature that makes it suitable

for exoskeleton robots (i.e robotic assisted stroke

rehabilitation),[12].

In this paper, a advanced robust open-closed iterative

learning control for exoskeleton rehabilitation robots is

,0 (2019)

MATEC Web of Conferences https://doi.org/10.1051/matecconf/2019

292

292010

10

CSCC 2019

10

108

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative

Commons Attribution License 4.0

(http://creativecommons.org/licenses/by/4.0/).

suggested and introduced. First, the procedure of

mathematical modeling of an exoskeleton robotic system

for rehabilitation is presented using the Rodrigues

approach, [13,14]. Then, we propose a joint space

trajectory tracking control system consisting of two

loops. First (inner) loop represents exact linearization of

a given system, and the second (outer) loop is synthesis

of a linear control law which consists of two branches,

feedforward and feedback branch of ILC control. In

feedforward ILC algorithm is applied, while in feedback

classical PD control law is applied. Finally, a simulation

example is presented to illustrate the feasibility and

effectiveness of the proposed advanced open-closed ILC

scheme.

2 Nonlinear mathematical model of

exoskeleton robot

Control object is modelled using Rodrigues approach.

This approach is more viable, than Denavit-Hartenberg

method, for setting up kinematics and dynamics of

biomechanical systems [7].Validation of control laws is

carried out using 3DOF biomechanical system, (Fig 2b).

The system consists of two bodies - links. Links are

simplified model of human arm attached to exoskeleton.

Links are modeled as two truncated cones (Fig.2a).

Fig. 2. a)Biomechanical system of upper limb as open chain of

the rigid bodies system b) Biomechanical system presented as

3DOFs system –Rodrigues approach

The mechanical structure of a proposed system consists

of a sequence of rigid bodies (or links) interconnected by

means of one-degree-of–freedom joints forming

kinematical pairs of the fifth class, [13]. The open chain

system of rigid bodies (V1), (V2),(V3) is shown in Fig.

2b.Two neighboring bodies (

1i

V

−

) and (

i

V

) are

connected with a joint

1, 2 , 3i=

, which allows rotation of

body (

i

V

) in respect to the body (

1i

V

−

). The values

, 1, 2 , 3

i

qi=

represent generalized coordinates and define

a configuration of the mechanical model, where

3n=

is

a number of bodies in the system. The reference frame

Oxyz is the inertial Cartesian frame, and the reference

frame

O

ξηζ

is a local body–frame which is associated

with the body (

i

V

). At an initial time, the corresponding

axes of reference frames were parallel, i.e. all the

variables

0, 1, 2, 3

i

qi= =

and the robotic system is in

reference position. Parameters

,,1 ,

ii i

ξξ ξ

= −

denote

parameters in general case for recognizing joints

between bodies (

1i

V

−

) and (

i

V

), (

1,

i

ξ

=

-prismatic,0-

cylindrical joint). The geometry of the system is defined

by the unit vectors i

e

and the position vectors

i

ρ

and ii

ρ

expressed in local coordinate systems

iii

i

C

ξηζ

are

connected to mass centers of bodies in a multibody

system [13,14]. Unit vectors

, 1, 2 , 3

i

ei=

are describing

the axis of rotation of the

1, 2 , 3i=

-th segment with

respect to the previous segment, and

ii

ρ

denotes a vector

between two neighboring joints in a multi body system,

while the position of the center of mass of i-th segment

is expressed by vectors

i

ρ

. For the entire determination

of this system, it is necessary to specify masses mi and

tensors of inertia

Ci

J

expressed in local coordinate

systems. Dynamic equations of motion for the robot

system can be obtained by applying Lagrange equations

of the second kind in the covariant form as follows:

(1)

( ) ( )

,

1 11

1,2,...

n nn

a qq qqq Q n

γα α αβ γ α β γ

α= α= β=

+ Γ = γ=

∑ ∑∑

where the coefficients

aa

γα αγ

=

are the covariant

coordinates of the basic metric tensor

[ ]

33

aR

×

γα

∈

and

,αβ γ

Γ

present Christoffel symbols of the first kind.

Coefficients of the metric tensor are defined as, [13]:

( )

( )

( )

{ }

( )

( )

[ ]

( )

{ }

1

,

n

ii i i

i Ci

i

a mT T J

αβ α β

αβ

=

= +Ω Ω

∑

(2)

where quasi-base vectors

( )

i

Tα

and

( )

iα

Ω

ar e

(3)

( )

( )

, ,

0, ,

i

k

k

kk i

k

ik

e eq e i

T

i

αα

αα

α=α

ξ × ρ +ξ +ρ +ξ ∀α≤

=

∀α >

∑

( )

, ,

0, ,

i

ei

i

α

α

α

ξ ∀α ≤

Ω=

∀α >

(4)

and Cristoffel symbols are (5)

,

1, , , 1, , .

2

aaa n

qqq

βγ γα αβ

αβ γ αβ γ

∂∂∂

Γ = + − αβ γ=

∂∂∂

The generalized forces

Q

γ

can be presented in the

following expression (6), wherein

, ,,, ,

ag vc f

QQ QQQ

γγ γγγ

denote the generalized control,

gravitational, viscous, spring and friction forces,

respectively.

, 1, 2,...,

a gv c f

Q QQ QQ n

γ γ γγ γ γ

γ

= + ++ + =

(6)

The robot arm dynamics can be written in compact

matrix form as (where in our case

, , 0,

vc f a

QQQ Q v= =

):

,0 (2019)

MATEC Web of Conferences https://doi.org/10.1051/matecconf/2019

292

292010

10

CSCC 2019

10

108

2

( )

( )

( ) ( )

(,) ,

g

A q q C qq Q A q q n qq v+ −= + =

(7)

where

( )

Aq

represents basic metric tensor (or inertia

matrix),

( )

,C qq

is a matrix that includes centrifugal and

Coriolis effects,, respectively, [15].

2.1. State-space representation

In our case, the state vector of the nonlinear robot arm

system is introduced as:

[ ]

( )

( )

23

1 2 1, 2, 3 1, 2 , 3

,, ,

T

Tx

x x x qq qq q qq q R= = = ∈

so

one can obtain (7) in state space form:

( ) ( ) ( ) ( )

xt Ax B xvt= +

(8a)

( ) ( ) ( )

( )

( )

( ) ( )

2

11

11

() 0

,

() ()

g

xt

Ax Bx

Ax t C xt Q Ax t

−−

= =

−−

(8b)

( )

[ ]

( )

( ( )) 1 0y t hxt x t= =

(8c)

3 Control Design

Given biomechanical system is nonlinear MIMO time

varying system, hence two levels of control laws are

applied. First level is Computed torque (Inverse

dynamics, Feedback linearization). Role of this control

law is to linearize given nonlinear system, so linear

control law can be applied afterwards.

3.1 Feedback Linearization (Computed Torque

control)

The idea of Computed torque control is to provide exact

linearization of all nonlinearities in biomechanical

system via closed loop. In other words direct linear

connection between input and output is achieved with

application of Computed torque, It is a special

application of feedback linearization technique used in

nonlinear control systems, [16]. Computed torque

controllers can be very effective, since they provide us

independent joint control, which can then be used

together with some classical and modern design

techniques,such as iterative learning control. The

nonlinear transformation (9) has converted a complicated

nonlinear controls design problem into a simple design

problem for a linear system consisting of n decoupled

subsystems. A nonlinear controller will be realized as:

( ) ( )

( ) ( )

1

1

... ,

rr

gf f

v L L h L h u Aq u n qq

−

−

= + = = ⋅+

(9)

where

,

gf

LL

denote corresponding Lie

derivatives,[17]

A schematic diagram of a feedback linearization

technique is illustrated in Fig. 3.

Fig. 3. Block diagram of exact feedback linearization

So, one can linearize the dynamics in ideal case, as

follows:

( ) ( )

qt ut=

(10)

where taking into account under the influence of model

uncertainties

( )

, 1, 2, 3

kk

ti

ηη

= =

we have:

( ) ( ) ( )

, 1, 2 , 3

k kk

q t u t ti

η

=+=

(11)

3.2 Advanced open-closed loop iterative learning

control

We investigate the problem where the exoskeleton robot

must repeatedly follow the desired trajectory

()

n

d

qt R∈

,

[ ]

, 0, ,t JJ T J R∈= ⊂

in the joint space under the

influence of model uncertainties,

( )

n

k

tR

η

∈

, where T is

the time duration, k denotes the iteration index. For the

linearized dynamics of the robot arm (11), the open-

closed ILC algorithm is suggested which comprises two

types of control laws: a feed-forward

2

sgn PDD

control

law and a PD feedback law, see Fig. 4.

Fig. 4. Block diagram of the open-cl o se d

2

sgn /PDD PD

type of ILC for a robotic system

( ) ( ) ( )

( ) ( ) ( )

1111

k ffk fbk

k kk kk kk

pk vk

ut u t u t

u sgn e M sgn e M sgn e M

K e Ke

−−−−

=+=

′ ′′

=+++

++

(12)

where

( )

() ()

k dk

e t yt yt= −

is the trajectory tracking

error in

k−

kth iteration,

( )

d

yt

denotes desired output

trajectory.

33

,

pv

KK R

×

∈

are closed-loop positive-

,0 (2019)

MATEC Web of Conferences https://doi.org/10.1051/matecconf/2019

292

292010

10

CSCC 2019

10

108

3

definite diagonal learning matrices and

,,,

kkk

MMM

′ ′′

are functions of

,,,

kkk

se se se

(Fig.5) respectively

where we define the error sums for the k-th iterative in

the form, [18]:

( )

0

T

kk

se e t=∑

,

( )

0

T

kk

se e t=∑

,

( )

0

T

kk

se e t=∑

(13)

Fig. 5. Scheme of function

( )

kk

M f se=

To reduce the computation and storage size for the

proposed ILC control functions are introduced as step

functions.

Also, the following assumptions on the system (11) are

imposed.

A1. The desired trajectories

( ) ( )

,

dd

ytxt

are

continuously differentiable on

[ ]

0,T

.

A2. The system (11) is causal. Specifically, for a given

desired output trajectory

()

d

yt

, there exists a unique

bounded control input

( )

d

ut

such that the system has a

unique bounded state

()

d

xt

and

()

d

yt

, i.e:

() (),

ddd

(t)= Ax t Bu t+

x

(14)

() (),

dd

tCt=yx

(15)

A3. The initial resetting conditions hold for all iterations,

i.e.

( ) ( )

0 0 , 1,2,3,......

kd

x xk= =

(16)

A4. The uncertainties

( )

,

3

k

tR

η

∈

are uniformly

bounded.

Convergence analysis of the proposed method is omitted

here, some more details, see [18].

4 Simulation results and discussion

Here, we used a main exoskeleton system due to

biomechanical system with revolute joints, with three

DOFs, Fig. 2, to solve the trajectory tracking problem in

joint space. For the simulation, we use the next model

parameters of robot arm

12 3

0 , 1.4 , 1.1m kg m kg m kg= = =

where first segment is

fictive due to decomposition [14]. Numerical simulations

were carried out to demonstrate the feasibility and

effectiveness of the proposed advanced ILC PDD2/PD

type. The desired trajectories are given as polynomial of

fifth order

( )

2345

012345

d

q t a at at at at at

=+++++

with constraints

123

(0) 0, 1, 2, 3,

( ) / 2, ( ) / 4, ( ) / 6,

(0) ( ) 0, 1,2, 3,

(0) ( ) 0, 1,2, 3,

dk

dd d

dk dk

dk dk

qk

qTqTqT

q qT k

q qT k

πππ

= =

= = =

= = =

= = =

(17)

Model of feedback linearized robotic system in state

space is given as:

(18)

( ) ( ) ( ) ( )

( )

[ ]

( )

01 0 0 , 1, 2, 3

00 1 1

10

k kkk

kk

xt xt ut tk

yt xt

η

= ++ =

=

where ( ) , 1, 2, 3

ktk

η

= are model uncertainties:

[ ]

12

3

3

(10 0.

5.

2)( ) 0.1 , ( ) ,

( ) 0 2 sin( t) 0,

t e) txp(1 - t t

tt

ηη

ηπ

−⋅=⋅=

=⋅ ⋅∈

(19)

For the elements of learning gain matrices,

,

pv

KK

the

following values are adopted:

{ } { }

10,25,2 , 4.6,15,6 ,

pv

K diag K diag= =

(20)

as well as: (21)

1,2 , 3

1 2 3 1,2 , 3

1,2 , 3

1,2 , 3

1 2 3 1,2 , 3

1,2 , 3

1,2 , 3

1 2 3 1, 2,3

0.09, 0.5

0.005, 0.5 0.1

0.001, 0.1

0.09, 0.5

0.05, 0.2 0.1

0.01, 0.1

0.01, 1

0.005, 1 0.5

kkk

kkk

kkk

se

M M M se

se

se

M M M se

se

se

M M M se

≥

=== ≥≥

≤

≥

′′′

=== ≥≥

≤

≥

′′ ′′′ ′′′

=== ≥≥

1,2 , 3

0.001, 0.5se

≤

Fig. 6

( )

,

kk

M se

in function of iteration number

,0 (2019)

MATEC Web of Conferences https://doi.org/10.1051/matecconf/2019

292

292010

10

CSCC 2019

10

108

4

Fi g u r e 6 sh o w s the maximum

,

k

M

k

se

from

iteration to

iteration.

Fig. 7 Maximum error bounds in each iteration

It is clear that the trajectory tracking error decreases

through the iterations. Also, we can find (see Fig. 7), that

proposed requirement of tracking performance is

achieved at th 50th iteration.

Fig. 8 shows that the ILC control law drives the

considered robotic system output on the interval

t∈[0,5] through the desired trajectory as closely as

possible after 100 iterations.Also on Fig. 9 presents

the tracking errors

( )

, 1, 2 , 3

i

eti=

of t h e sy st em

output after 100 iterations.

Fig.8 The tracking performance of t h e

system output (

( )

, 1, 2 , 3

di

q ti=

-solid line,

( )

,

, 1, 2 , 3

i

qti=

, (

.−

line))

Fig.9 The tracking errors

( )

, 1, 2 , 3

i

eti=

of t h e

sy st em

output

5 Conclusion

In this paper, we studied the tracking problem of

exoskeleton robotic system robot with revolute joints via

intelligent control which includes advanced ILC control.

First, a feedback linearization control technique is

applied on a given robotic system. Then, the proposed

intelligent control algorithm takes the advantages offered

by closed-loop control PD type and open-loop control

sgnPDD2 type of ILC. Suggested robust ILC algorithm

is applied to the linearized system to further enhance

tracking performance for repetitive tasks and deal with

the model uncertainties. Finally, a simulation example is

presented to illustrate the effectiveness of the proposed

robust ILC scheme for a robot arm.

Authors gratefully acknowledge the support of Ministry of

Education, Science and Technological Development of the

Republic of Serbia under the projects TR 35006,III41006.

References

1. Global Health Estimates. Geneva: World Health

Organization,2018,wttps://www.who.int/healthinfo/

global_burden_disease/en/

2. N.Lo sseff ,Ed., Neurological Rehabilitation of

Stroke, (Taylor&Francis,UK,2004,)

3. A.Rob er t , W.T easel l ,L.Kalra,AHA/ ASAJ.35

,2,3 (2004 )

4. B. T. Volpe, H. I. Krebs, N. Hogan,Adv. in Neuro.

92, (2003)

5. H.I. Krebs, B.T. Volpe, M.L. Aisen, N. Hogan, J. of

Rehab. Res. and Devel.37,6 (2000)

6. H.S.Lo, S. Q. Xie, Med. Eng.& Phys. 34, (2012)

7. S.Arimoto, Kawamura S., Miyazaki F., J.of Rob.

Sys., 2,1,(1984)

8. Ahn H.S., Moore K., Chen Y., Iterative learning

control robustness and monotonic convergence for

interval systems, Springer-Verlag London Limited,

London, (2007)

,0 (2019)

MATEC Web of Conferences https://doi.org/10.1051/matecconf/2019

292

292010

10

CSCC 2019

10

108

5

9. J.X Xu, S. K. Panda, T. H. Lee, Real-time Iterative

Learning Control, Design and Applications,

Springer-Verlag London, (2009)

10. M.Lazarević , Sci. Tech. Rev., 64,1, (2014)

11. M.Lazarević , Panagiotis T., J. of Vib. and Cont.,

22, 8, (2016)

12. T.C. Freeman, Rogers E., Burridge J. H., Hughes

Ann-Marie, Meadmore K.L., Iterative Learning

Control for Electrical Stimulation and Stroke

Rehabilitation, Springer Briefs in Control,

Automation and Robotics, (2015)

13. V.Čović, M. Lazarević, Robot Mechanics, Belgrade,

Fac. of Mech. Eng. (in Serbian), (2009)

14. M. Lazarević, FME Trans. 34, 2, (2006)

15. B.Siciliano, L. Sciavicco, Villani, L., Oriolo, G.:

Robotics, Springer-Verlag, London, (2009)

16. H. Khalil, Nonlinear Systems, Prentice Hall, Upper

Saddle River, (2002)

17. J.J.Slotine,Applied nonlinear control. Vol. 199:

Prentice-Hall Englewood Cliffs, NJ.,(1991)

18. Wang Y., Proc.of the 10th World Cong. on Int.

Cont.and Aut., (2012)

,0 (2019)

MATEC Web of Conferences https://doi.org/10.1051/matecconf/2019

292

292010

10

CSCC 2019

10

108

6