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Scientific Technical Review, 2020,Vol.70,No.3,pp.29-34 29

UDK: 025.4.03/06:007.52:519.87

COSATI: 12-01 doi: 10.5937/str2003029L

The advanced iterative learning control algorithm for

rehabilitation exoskeletons

Mihailo Lazarević 1)

Nikola Živković 2)

In this paper an advanced iterative learning control algorithm for rehabilitation exoskeletons is proposed. A simplified

biomechanical model is used as the control object to verify contro l al gor ith m fe asi bil ity . Th e control design is proposed as two

level controller consisting of inner and outer loop. In the inner loop the feedback linearization is applied to cancel out the

model nonlinearities. In the outer loop the advanced iterative learning control algorithm of sgnPDD2 type is a ppl ied as a fee d-

forward controller and classical PD controller as a feedback controller. Uncertainties are added in order to examine the

controller design robustness. Numerical simulation is carried out.

Key words: biomechanical model, iterative learning control, feedback linearization, uncertainty.

1) Military Technical Institute (VTI), Ratka Resanovića 1, 11132 Belgrade, SERBIA

2) Lola institute, Kneza Viseslava 70a,11030 Belgrade, SERBIA

Correspondence to: Milailo Lazarević; e-mail: mihailo.lazarevic@gmail.com

Introduction

XOSKELETON as a real-time interaction with the

wearer’s intelligent robot, in recent years, becomes a hot

topic mouth class research in the field of robotics. The term

‘exoskeleton’ was used in biology referring to the chitinous or

calcified external skeleton used by numerous animal taxa for

structural support and defense against predators [1, 2].

Wearable exoskeleton outside the body, combined with the

organic body, plays a role in the protection and support. By

wearing an exoskeleton robot, it is possible to expand the

wearer’s athletic ability, increase muscle endurance, and

enable the wearer to complete tasks that one cannot perform

under natural conditions as well as help for recovering of lost

human motor functions. Based on the above advantages, the

exoskeleton robot in military medical care and rehabilitation

has broad application prospects [2].

In the military field, the exoskeleton is very attractive

because it can effectively improve the individual combat

capability. For example, HULC is an exoskeleton robot that

can greatly increase the soldier’s ability to carry weight,

making it easy for soldiers to carry heavy loads of 90 kg.

Now, the exoskeletons are generally regarded as a

technology that extends, complements, substitutes or

enhances human function and capability or empowers the

human limb where it is worn [3]. Unlike other robots, the

operator of exoskeletons is a human who needs to make

decisions [4] and perform tasks with exoskeletons.

Stroke survivors are faced with some degree of limb

impairment, depending on the place in brain, structure and

size of caused damage. An increasing number of evidence

suggests that brain of a patient who survived stroke has

increased capacity for plastic change and thus some motor

functions can be fully or partially recovered. Rehabilitative

training plays an important role in recovery of lost motor

functions [5]. In order to enhance a therapy delivered by

therapists, the use of robotics emerged as an aid in the

rehabilitation process [6]. In [7] and [8] it is shown that the

robot-aided training, especially in the upper limbs, has

positive effects in patient stroke rehabilitation. Also, it has

been observed that robot-assisted rehabilitation has potential

benefits to patients even several years after the stroke.

Rehabilitative robotics of the upper limbs can be

differentiated into two types – rehabilitative robots with one

point of contact (end-effector robots) and rehabilitative robots

with multiple points of contact (exoskeletons). End-effector

robot supports patients arm in one point of contact, usually

patients hand or forearm. End-effector robot joints movement

does not coincide with the movement of the patients arm.

These drawbacks with the end-effector robots influenced the

research of exoskeleton rehabilitation robots. Exoskeletons

mitigate important flaws of the end-effector robots mentioned

above [9]. Rehabilitation robots can be developed to assist

rehabilitation in individuals with stroke.

Considering the repetitive nature of the patient`s

rehabilitation process, Iterative Learning Control (ILC)

algorithms emerged as suitable for this application. In the last

three decades, (ILC) has been extensively studied. ILC has

become one of the most active research and study topic in the

field of control theory and its applications [10], [11], [12],

[13], [14].

ILC is an intelligent control method for systems that

perform tasks repetitively over a finite time interval. This

method is similar to how humans learn – performing a certain

task and observing the outcome. The next performance of the

same task will be adjusted accordingly to the previously

observed outcome. ILC algorithm constructs current system

input by applying predetermined mathematical law on a

previous system output which was stored in the memory [11].

Loosely said ILC is using past experience to improve current

system behaviour. Basic ILC principle is illustrated in Fig.1.

E

30 LAZAREVIĆ,M., etc.: THE ADVANCED ITERATIVE LEARNING CONTROL ALGORITHM FOR REHABILITATION EXOSKELETONS

Figure 1. Basic ILC scheme

As it has been already said, rehabilitation training is a kind

of repetitive training. Patient’s motor function 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) [15].

In this paper, an advanced robust open-closed iterative

learning control for exoskeleton rehabilitation robots is

presented.

Control object model

In order to validate the proposed control algorithm

feasibility, the control object is mathematically modelled

using the Rodriguez approach. Human upper limb and

supporting exoskeleton are adopted as the control object in

this paper. The existing complex system is simplified to a

biomechanical model consisting of 3DOF robot arm with

truncated cone shaped links (Fig.2).

Structure of this biomechanical model is a sequence of two

rigid bodies – links, interconnected with joints where the first

link has two degrees of freedom and the second link has one

degree of freedom. In order to apply the Rodriguez approach

the first joint is decomposed from 2DOF to two links

interconnected with 1DOF joints. The first link is set to be

fictive, i.e. mass and length of this link is zero. Now, open

chain system of rigid bodies consists of three bodies with

joints between them starting from inertial reference frame

as shown in Figure . The values ,1,2,3

i

qi are

generalized coordinates that define a configuration of the

mechanical model. Each link is associated with local

reference frame C

.

Figure 2. a)-Simplified model of the exoskeleton-human arm, b) kinematic

scheme after joint decomposition

At initial time reference configuration is set by setting

corresponding axes of reference frames parallel to the

Cartesian inertial reference frame Oxyz . This convention

enables using the Rodriguez approach for obtaining dynamic

equations of motion.

Further, parameters ,1

ii i

are defined, which

denote whether a joint is prismatic or cylindrical. The

geometry of the model is defined by the unit vectors i

e

and

the position vectors i

ρ

and ii

ρ

expressed in local coordinate

frames iii i

C

. Origins of these local coordinate frames are

attached to the centre of mass of each link i [16], [17].

Dynamic equations of motion for the robot system can be

obtained by applying the Lagrange equations of the second

kind in the covariant form as follows:

,

111

nnn

aqq qq Q

(1)

where the coefficients

are the covariant

coordinates of the basic metric tensor and present the

Christoffel symbols of the first kind. Coefficients of the

metric tensor are defined as,

1

n

iCi

i

mJ

αi

β

iαi

β

i

TT Ω Ω

(2)

where quasi-base vectors

αi

T

and

αi

Ω

are:

,

0,

i

i

αα

αi

αi

eR e

T

(3)

,

0,

i

i

α

αi e

Ω

(4)

where

ik

k

kq

kk k i

αi

Rρeρ

. The Christoffel

symbols are:

,1

Γ, ,,1,,

2

aaa n

qqq

(5)

The generalized forces can be presented as:

a

g

vcf

γ γγγγγ

QQQQQQ (6)

Wherein ,,,

a

g

vc

γγγγ

QQQQ and f

γ

Q denote the generalized

control, gravitational, viscous, spring and friction forces,

respectively.

Biomechanical model dynamics can be written in compact

matrix form:

¨¨

,,aK a

g

u

q q qq Q q q c qq Q

(7)

where

aq is the inertia matrix,

,

K

qq

is matrix that

includes centrifugal and the Coriolis effects,

g

Q is vector of

gravitational forces and u

Q is vector of generalized control

forces [18]. It is assumed that viscous, spring and friction

forces are equal to zero in our case.

LAZAREVIĆ,M., etc.: THE ADVANCED ITERATIVE LEARNING CONTROL ALGORITHM FOR REHABILITATION EXOSKELETONS 31

Control design

The obtained mathematical model is multi-input multi-

output, nonlinear, time-varying system. Control law design in

this case consists of inner and outer control loop. Inner loop is

Feedback linearizing control law and outer loop is Iterative

Learning Control algorithm with classical PD feedback.

Feedback Linearization

Feedback linearization or Computed torque control

algorithm is introduced in order to linearize nonlinear

biomechanical model so the linear control laws can be applied

in the outer loop. This linearization is exact and it differs from

the Jacobian linearization which is linear approximation of

dynamics of given system. Feedback linearization can be

applied in general form [19], but here the concept of

Computed torque is applied out of the convenience. Given the

equations of motion (7),

¨,a

u

qq cqq Q

(8)

where input vector u

Q can be chosen as:

,a

u

Qqucqq

(9)

Where u is new input vector. A new linear decoupled

system is obtained in form of double integrator:

qu

(10)

Now u can be chosen conveniently as the linear control

law which is going to close the outer control loop.

Open-closed loop iterative learning control

After closing the inner loop control algorithm in the outer

loop an open-closed loop ILC algorithm is applied which

consists of feed-forward sgnPDD2 type control law and

feedback PD type control law as represented in the block

diagram (Fig.3).

Figure 3. Block diagram of an ILC sgnPDD2/PD type control algorithm

Control law can be written as:

'

11 1

''

1PD

sgn sgn

sgn K K

kk

k

ttt

M

M

M

kkffkfb

kk k

kkk

e

uu u

ue e

ee

(11)

where

kdk

et yt yt is trajectory tracking error and

d

yt

is desired output trajectory. P

K

and D

K

are closed-

loop positive definite diagonal learning matrices. Function

sgn(.) is sign function defined as:

1, 0

sgn 0, 0

1, 0

x

xx

x

(12)

and value k in index denotes the iteration number.

Advantages of the proposed approach in calculating open-

loop ILC law are: the controller needs only addition and

subtraction as well as the controller needs only to store the

values of corresponding variables, thus dramatically reduces

the demanding for storage size.

'

,

kk

M

M and ''

k

M

are the functions of ,,

kkk

s

esese

,

respectively [20]. These sums are defined as:

000

,,

TTT

kkk

se se se

kkk

et et et

(13)

Also, to reduce the computation and storage size for the

proposed method, parameters, '

,

kk

M

M and ''

k

M

can be

adapted as a simple step functions of previously defined error

sums (13). This is shown in Fig.4.

Figure 4. Step function Mk. Threshold values m1, m2, m3, s1, s2, s3 initially are

chosen arbitrarily and then tuned.

For example if sum k

s

e has value between 1

S and 2

S,

parameter k

M

is designated with value 2

m. Parameters k

M

and ''

k

M

are chosen analogously. Step function threshold

values are chosen purely by trial and error and currently no

specific recommendation can be given for the initial selection.

Simulation results and discussion

In this section the numerical simulation in MATLAB is

carried out in order to verify control law error convergence. A

method used to solve ordinary differential equations is the

Runge-Kutta with fixed step. Simplified exoskeleton/ human

arm model with 3DOF is used as a simulation object with

following parameters 10 kg, m

23

1.4 kg, 1.1 kg, mm

10 m, l

20.369 m, l

30.36 ml wh ere the firs t se gment

is fictive due to 2DOF joint decomposition [16]. Exoskeleton

system is tasked with desired trajectory tracking,

n

d

qt .

System response is observed over the time interval

0,tT

where 5sT

. Desired trajectories are given as the fifth order

32 LAZAREVIĆ,M., etc.: THE ADVANCED ITERATIVE LEARNING CONTROL ALGORITHM FOR REHABILITATION EXOSKELETONS

polynomials

2345

01 2 3 4 5d

qtaatatatatat with

constraints:

123

00, 1,2,3;

, , ;

246

00, 1,2,3;

00, 1,2,3;

dk

dd d

dk dk

dk dk

qk

qT q T qT

qqTk

qTkq

(14)

Let the error bounds be

2

max 0.5 10 rade

. Exact

feedback linearization of the given system left us with a linear

system. In order to verify the control law robustness

additional uncertainties are introduced in the system via η.

Uncertainty of the given biomechanical model can be

represented as a change of mass of the object. With that in

mind inertia matrix, centrifugal matrix and gravity vector can

be written as a sum of nominal and uncertain part [21].

Equations of motion (7) can be rewritten as:

Δ,Δ

NN

aaK K

ggu

N

qqqQΔQQq

(15)

Applying computed torque we can choose:

,

NN

aK

g

uN

QquqqQ

(16)

Equations now become:

1,

N

aa

qt q qut ηqq

(17)

where u is a new control signal and η is the model

uncertainty:

1

,,aq

g

ηqq ΔKqq ΔQ

(18)

A new control input u is selected as sgnPDD2/PD type

controller designed in the previous section. Equations of

motion can be represented in state-space form as:

,,

.

a

AB D

C

xt xt ut ηqq

yt xt

(19)

Where

1

0

00

, , ,

00

0.

nn

nn nn nn

a

nn nn nn

N

nn nn

I

ABD

I

aa

CI

(20)

Before applying the proposed ILC algorithm as input u, the

following assumptions will be made:

(i) Desired trajectories

d

qt are continuously

differentiable on

0,T,

(ii) Initial conditions for all iterations are

00, 1,2,,k

kd

xx (21)

(iii) Influence of changing the masses of links are negligible

on matrix

aq, so it follows that 1

N

aa I

and

0nn

a

nn

BBI

,

(iv) System (18) is causal [17].

Gain matrices for the feedback loop are chosen as

diag 161 01 0

P

K and

diag 8 5 7

D

K. Step values k

M

,

k

M

and k

M

and their bounds are chosen in the process of

trial and error:

1

2

3

0.9, 20

0.03,1 .5 20

0.001, 1.5

0.5, 15

0.05, 2 15

0.001, 2

0.1, 15

0.05,1 15

0.001, 1

k

kk

k

k

kk

k

k

kk

k

se

Mse

se

se

Mse

se

se

Mse

se

(22)

'1

'2

'3

0.01, 0.2

0.005, 0.1 0.2

0.001, 0.1

0.01, 0.5

0.005, 0.1 0.5

0.001, 0.1

0.009, 0.5

0.005, 0.25 0.5

0.001, 0.25

k

kk

k

k

kk

k

k

kk

k

se

Mse

se

se

Mse

se

se

Mse

se

(23)

''1

'' 2

'' 3

0.01, 0.5

0.005, 0.1 0.5

0.001, 0.1

0.09, 0.5

0.005, 0.1 0.5

0.001, 0.1

0.008, 0.5

0.005, 0.25 0.5

0.001, 0.25

k

kk

k

k

kk

k

k

kk

k

se

Mse

se

se

Mse

se

se

Mse

se

(24)

It can be seen from figures (Figures 1, 2 and 3) that the

system output reaches desired trajectory. Increasing mass

uncertainties results in slower system output error

convergence.

The first link (fictive) reaches proposed error bound in 35

iterations for 5% mass uncertainty, in 96 iterations for 10%

mass uncertainty and in 101 iterations for 15% mass

uncertainty. Two other links reach desired trajectory much

faster, within 20 iterations.

Also, it can be concluded that, although the system

response in some cases requires relatively large number of

iterations to reach the desired error bound, the initial

trajectory does not deviate too much from the desired

trajectory. In other words, maximum errors above the

proposed error bound are not drastically greater than

me ntio ned e rro r bou nd f or r eal wor ld a ppl ica tions, in t his case

- patient rehabilitation.

LAZAREVIĆ,M., etc.: THE ADVANCED ITERATIVE LEARNING CONTROL ALGORITHM FOR REHABILITATION EXOSKELETONS 33

Figure 5. Maximum error of the first link per iteration for different

percentages of link mass uncertainties

Figure 6. Maximum error of the second link per iteration for different

percentages of link mass uncertainties

Figure 7. Maximum error of third link per iteration for different percentages

of link mass uncertainties

Conclusion

In this paper sgnPDD2 type of feed-forward iterative

learning controller is investigated. In addition to feed-forward

controller, regular PD feedback controller is present. Control

law behaviour is tested, through simulation, on a linear system

with added bounded uncertainties. A simulation results show

convergence of the systems response and controller ability to

track the reference trajectory. System output in some cases

needs a relatively large number of iterations to reach the

desired trajectory.

Parameters k

M

, k

M

, k

M

and threshold values in

sgnPDD2 type of ILC algorithm are chosen arbitrarily and

tuned by trial and error, which can be time consuming.

Acknowledgment

The presented research was supported by the Ministry of

Education, Science and Technological Development of the

Republic of Serbia by contract no. 451-03-9/2021-14/200105

from 05.02.2021 and contract no. 451-03-9/2021-14/200066.

Literature

[1] SASHI S KOMMU Ed. Rehabilitation Robotics, I-Tech Education and

Publishing, Vienna Austria,2007.

[2] GUNASEKARA,J.M.P., GOPURA,R.A.R.C., JAYAWARDANE,T.S.S,.

LALITHARATHNE,S.T.: Control Methodologies for Upper Limb

Exoskeleton Robots, 2012 IEEE/SICE International Symposium on System

Integration (SII) Kyushu University, Fukuoka, Japan December 16-18, 19-

24 (2012).

[3] YANG,C.J., ZHANG,J.F., CHEN,Y., DONG,Y.M., ZHANG,Y.: A

Review of exoskeleton-type systems and their key technologies,

Proceedings of The Institution of Mechanical Engineers Part C-journal

of Mechanical Engineering Science 222, 1599-1612 (2008).

[4] ZHENG,Y., SONG,Q., LIU,J., SONG,Q., YUE,Q.: Research on

motion pattern recognition of exoskeleton robot based on multimodal

machine learning model,Neural Computing and Applications (2020)

32:1869–1877.

[5] LOSSEFF,N., THOMPSON,A.J., Eds.: Neurological Rehabilitation of

Stroke. Taylor & Francis, UK, 2004.

[6] TEASELL,R.W., LALIT,K.: What’s New in Stroke Rehabilitation,

AHAJ, no.35, pp.383–385, 2004, doi:

10.1161/01.STR.0000115937.94104.76.

[7] VOLPE,B., KREBS,H., HOGAN,N.: Robot-aided sensorimotor

training in stroke rehabilitation, Adv. Neurol., vol. 92, pp. 429–33,

Feb. 2003.

[8] KREBS,H., VOLPE,B., HOGAN,N., AISEN,M.: Increasing

productivity and quality of care : Robot-aided neuro-rehabilitation,

Journal of Rehabilitation Research and Development, vol. 37, no. 6, pp.

639–652, Dec. 2000.

[9] LO,H.S, XIE,S.Q.: Exoskeleton robots for upper-limb rehabilitation:

State of the art and future prospects, Med. Eng. Phys., vol. 34, no. 3,

pp. 261–268, Apr. 2012,doi: 10.1016 /j.medengphy. 2011.10.004.

[10] ARIMOTO,S.,KAWAMURA,S., MIYAZAKI,F.: Bettering operation

of Robots by learning, J. Robot. Syst., vol. 1, no. 2, pp. 123–140, 1984,

doi: https://doi.org/10.1002/rob.4620010203.

[11] AHN,H.S, MOORE,K.L, CHEN,Y.: Iterative Learning Control:

Robustness and Monotonic Convergence for Interval Systems. London:

Springer-Verlag, 2007.

[12] XU,J.X.S., PANDA,K., LEE,T.H.: Real-time Iterative Learning

Control: Design and Applications. London: Springer-Verlag, 2009.

[13] LAZAREVIĆ,M.: Iterative learning control of integer and noninteger

order: An overview, Sci. Tech. Rev., vol. 64, no. 1, pp. 35–47, 2014.

[14] LAZAREVIĆ,P.M., TZEKIS,P.: Robust second-order PDα type

iterative learning control for a class of uncertain fractional order

singular systems, J. Vib. Control, vol. 22, no. 8, pp. 2004–2018, May

2016, doi: 10.1177/1077546314562241.

[15] FREEMAN,C., ROGERS,E., BURRIDGE,J.H., HUGHES,A.M.,

MEADMORE,K.L.: Iterative Learning Control for Electrical

Stimulation and Stroke Rehabilitation. London: Springer-Verlag, 2015.

[16] ČOVIĆ,V.,LAZAREVIĆ,M.: Robot mechanics. Faculty of Mechanical

Engineering, Belgrade, 2007.

[17] LAZAREVIĆ,M.P.: Mechanics of Human Locomotor System, FME

Transactions, vol. 34, no. 2, pp. 105–114, 2006.

[18] SICILIANO,B., SCIAVICCO,L., VILLANI,L., ORIOLO,G.: Robotics:

Modelling, Planning and Control. London: Springer-Verlag, 2009

[19] SLOTINE,J.: Applied nonlinear control. Englewood Cliffs N.J.:

Prentice Hall, 1991.

[20] WANG,Y.: Iterative Learning Control Algorithm with Self-adaptive

Steps, Proceedings of the 10th World Congress on Intelligent Control

and Automation, Jul. 2012.

[21] LAZAREVIĆ,P.M, MANDIĆ,P., OSTOJIĆ,S.: Further results on

advanced robust iterative learning control and modeling of robotic

systems, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., p.

0954406220965996, Nov. 2020, doi: 10.1177/0954406220965996.

Received: 16.02.2021.

Accepted: 19.03.2021.

34 LAZAREVIĆ,M., etc.: THE ADVANCED ITERATIVE LEARNING CONTROL ALGORITHM FOR REHABILITATION EXOSKELETONS

Napredni iterativni algoritam upravljanja učenjem za

rehabilitacione egzoskelete

U ovom radu predložen je napredni iterativni algoritam kontrole učenja za rehabilitacione egzoskelete. Pojednostavljeni

biomehanički model koristi se kao objekt upravljanja kako bi se proverila izvodljivost algoritma upravljanja. Projektovano

upravljanje je predloženo kao dvostepeni regulator koji se sastoji od unutrašnje i spoljašnje petlje. U unutrašnjoj petlji

primjenjuje se linearizacija kako bi se eliminisala nelinearnost modela. U spoljašnjoj petlji napredni iterativni algoritam

upravljanja učenja tipa sgnPDD2 primjenjuje se kao kontroler u direktnoj grani, a klasični PD kontroler kao kontroler u

povratnoj grani. Dodaju se nesigurnosti kako bi se ispitala robusnost projektovanog kontrolera. Sprovedena je numerička

simulacija.

Ključne reči: biomehanički model, iterativna kontrola učenja, feedback linearizacija, neizvesnost.