Open-closed Iterative Learning Control Algorithm for Exoskeleton Rehabilitation Purposes

Abstract

The paper designs an appropriate iterative learning control (ILC) algorithm based on the trajectory characteristics of upper exosk el eton robotic system. 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.

Full text

* 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
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© 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= =
):
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( )
( )
( ) ( )
(,) ,
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-
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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
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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.
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