MPC-PID comparison for controlling therapeutic upper limb rehabilitation robot under perturbed conditions

Full text

MPC-PID Compari
son for Controlling Therapeutic
Upper Limb Rehabilitation Robot Under Perturbed
Conditions
Athar Ali, Syed Faiz Ahmed, M. Kamran Joyo, Kushsairy.K
University Kuala Lumpur, British Malaysian Institute
Kuala Lumpur, Malaysia
athar.ali@s.unikl.edu.my
, syedfaiz@unikl.edu.my
Abstract— Increase in the number of stroke patients upsurges
the need of rehabilitation robots. It’s the ability of human
muscles to recover from stroke if it performs certain movements
repetitively and robots are the best way to perform repetitive
tasks. In this study, a 3 degree of freedom (3DOF) upper limb
rehabilitation robot has been developed to recover the patient
who have impaired limb, physical trauma or hit by a stroke. In
therapeutic exercise robots, the position accuracy and stability
are two major concerns, how effectively exercise is being
performed and how much stable robot is from external
disturbances. To ensure that, an analysis has been performed on
a comparison of PID (Proportional integral and derivative) and
MPC (Model Predictive Control) control algorithms to find out
which control algorithm is most suitable for upper limb
rehabilitation robots.
Keywords—Dynamic Modeling; Model Predictive Control
(MPC); Rehabilitation Robotics; Upper Limb.
I. INTRODUCTION
Neuromuscular deterioration can be a serious barrier for
someone to perform activities of daily life. Main reasons for
impairment deterioration are the repercussion of stroke, an
accident or a progressive neuromuscular disorder. Brain
regions motor control functions can be damaged during a
stroke. However, the brain has the ability to adapt for this so
that these skills are not lost forever but can be relearned in time
[1
-
3].
The Rehabilitation process is bit intensive, with the subject
repeats the same movement again and again. The idea of
rehabilitation was to free the therapists from repetitive tasks
and intensive training. Robots are ideal for carrying out the
repetitive. Over aging of the population (in 2030 around 25%
of people will be over 65; in 2050 this number will have more
than doubled), the rising cost of health care and the increasing
burden impairments have on daily life. Therefore, modern
technologies are becoming more and more applied in
rehabilitation. Rehabilitation robotics gains significant
attention in recent years and become the area of interest for the
researchers. [4]
Rehabilitation arm support (RAS) systems are intended to
assess the human arm impairments and to regain the arm
functionality by training. RAS systems should be able to assist
or even correct the user when he follows predefined
trajectories. Usually, Rehabilitation arm support (RAS)
systems are used in rehabilitation centers and are designed to
be stationary; hence, they are not intended to be applied in a
home situation. However, to provide rehabilitation possibilities
at home, devices are being developed [5-7]. These home RAS
systems should be lightweight, easy to carry, occupy a small
volume, and have a low power consumption, whereas most the
RAS systems used in rehabilitation centers are designed to
provide a large ROM and many different training possibilities.
The upper limb is essential for our daily functioning. It
enables us to grip, write, lift and throw among many other
movements. The upper limb has been shaped by evolution, into
a highly mobile part of the human body. This contrasts with the
lower limb, which has developed for stability.[8,9] In this
article, we will discuss the regions of the upper limb, as well as
the individual comp
onents and essential functions. This will be
joint by joint, with other structures mentioned throughout.
The upper limb device requires an appropriate control
algorithm to stabilize the movement and following the precise
trajectory. In this paper, a comparison is addressed for the
upper limb device using PID and MPC control algorithms. The
two control algorithms have different implications on the
system. The response of the system while PID is used as a
control is faster and stabilizes the system rapidly wh
ile the
response of MPC is slower. PID is better under perturbed
conditions and MPC is better when the system is affected by
noise [10,11].
II. SYSTEM DESIGN
The robot manipulator is portable and easy to setup in any
small hospital, where the upper limb rehabilitation robot is
designed in such a way that it can be easily assembled and
dissembled. The System architecture of robot manipulator
utilizes actuator concept. It has three phalanges position, the
position starts from MP Phalanges for bicep, PIP Phalanges for
elbow and DIP Phalanges for the wrist.

The manipulator’s mobility or degree of freedom can be
counted by the number of links and joints consisting the
mechanism. TABLE I provides the angular range of motion of
the three upper arm joints. This study focuses on elbow joint
with the range of motion 00 -1600 Flexion-Extension.
TABLE I: RANGE OF MOTION OF UPPER LIMB
Limb
Therapeutic Exercise
ROM of limb
Flexion-extension
-60/180
Shoulder
External-internal rotation
-90/50
Abduction-adduction
-45/90
Elbow
Flexion-extension
0o/160o
Radial-Ulnar deviation
-35/25
Wrist
Flexion-extension
-80/70
TABLE II: MOTOR PARAMETERS
A. System Modeling
The System model is composed of two parts: kinematic
model and motor model. The Kinematic model describes the
relationship between upper limb robot manipulator (position,
velocity, and acceleration) and joint torque; while motor model
aims to present the relationship of energy conversion from
electrical to mechanical i.e. voltages to torque.
Dynamic modeling of the multi-rigid body can be done
through several methods, such as Newton Euler equation,
Lagrange equation, Kane Equation, Hamilton theory and
Newton motional theory. Among all these methods, the
methods more widely used for controller design are Lagrange
equation and
Newton Euler equation. Langrage equation is long drawn
out from Newton Euler equation but centers on the energy
analysis of
the
whole system while Newton Euler equation
focuses on force analysis of the component. In this study,
Newton Euler equation method is used to establish the
mathematical model of the system.
B. Motor Modeling
DC motor control voltages obeys the following equation
Per Kirchhoff law.
where
ea(t) is motor input voltages, ia(t) is armature current,
La is motor inductance, Ra is motor resistance and eb(t) is the
back EMF of the motor.
Since DC motor output torque is proportional to the
armature current ia(t), motor torque can be obtained by
following equation:
Where KT is motor torque constant, motor current ia(s) is
given by equation.
The relationship between torque and input voltages of
motor is given by the following equation.
Motor parameters are shown in TABLE II. Where
K
t is motor
torque constant La is motor inductance, Ra is armature
resistance, Kb is motor backlash constant, n is gear ratio, Jm is
motor inertia and bm is motor friction.
C. System Transfer function:
The goal of this research is to develop a robot which can
perform therapeutic exercise of one joint at a time. Three
transfer functions have been developed for each joint of upper
limb robotic arm.
Shoulder Joint transfer function in s-domain is derived using
by designing the 3-D model upper limb rehabilitation robot in
SolidWorks and then imported the .xml file of that model in to
MATLAB SimScape where motor model is added to the
system
to derive the transfer function of the complete system.
Given below is the transfer function of the elbow joint, taking
voltage as an INPUT and position as an OUTPUT to the
system.
The transfer function is in continues domain, which cannot be
practically used as hardware
is in
discrete domain. Therefore,
transfer function of each joint is converted in to digital domain
using zero order hold method with sampling time Ts=0.1sec.
Discrete domain transfer function for elbow joint is also a
third order transfer function derived in the similar manner as
shoulder joint transfer function.
D. System Stability:
The stability system is one of the major concern when
developing a therapeutic exercise robot. Instability in the
Parameters
Shoulder
Elbow
Wrist
Kt (N-m/A)
.023
.023
.023
La (Henry)
0.23
0.23
0.23
Ra (Ohm)
1
1
1
Kb (V-s/rad)
0.023
0.023
0.023
n
1
1
1
Jm (kg·m²)
0.02
0.02
0.02
bm(N.sec/m)
0.03
0.03
0.03

system may cause in breaking the arm of patient. Therefore,
it’s important to do stability analysis of the system.
Pole zero graphs of each joint shows that system is marginally
stable. To stabilized the system a controller is needed which
can stabilize the system under uncertainties and external
disturbances.
III. CONTROLLER
Various controllers are being used to deal with system
uncertainties, external disturbances and sensory noises. In this
study, Model Predictive Control (MPC) and Proportional
Integral Derivative (PID) Control are being studied under
perturbed conditions.
A. Model Predictive Control:
The motive of using MPC is to compute a future control
sequence in a defined horizon in such a way that the prediction
of the plant output is closer to the reference. MPC is selected
for better controlling of the system under disturbance
conditions, it predicts control signal in such a way that it
minimizes a defined cost function which is error signal
between the output and desired value over specified prediction
horizon. Kalman filter estimates the disturbances, allowing
MPC to reject the effect of disturbance. MPC starts predicting
the future control action by choosing the suitable values of
control horizon M, prediction horizon P and control
-
weighting
factor R. Once MPC completes its prediction process, it
implements best control action and then at the next sampling
interval, the control estimation is repeated with the new
available information. Thus, the performance is increased.
The system dynamics in state space representation can be
expressed as:
The system states for elbow joint are
The prediction horizon can be defined as
The estimation error is uniformly fed back at every instant.
The state prediction is determined by:
The optimization cost function is defined by:
Where ‘Q’ is associated wi
th
the weight coefficient. The
optimal solution to the optimization plant is given by
Where the estimated state vector is ‘ ’ and optimal ‘ ’ is
shown while ‘ ’ and ‘ ’ are the control gains.
Fig.1 shows the complete model of MPC implementation in
MATLAB on upper limb elbow joint under perturbed
conditions.
Fig. 1. System model using MPC under noise and external disturbance.
Firstly, MPC behavior was checked under noisy condition.
White noise with three different power range (i.e.
0.1,0.01,0.001) was given to tuned MPC and results showed
that MPC behaves perfectly under noisy conditions the control
parameters for MPC are given in TABLE III.
TABLE III: MPC CONTROL PARAMETERS.
Noise power
Sample time
Prediction
horizon
Control
horizon
0.1
0.01
8
2
0.01
0.01
10
2
0.001
0.01
40
3
Fig.2. shows the system response under noisy conditions. The
graph, shows in fig.2. is position versus time. Red line shows
the system respo
nse to the white noise of power 0.1, blue is
system response to the white noise of power 0.01 and black is
the system response to the white noise of power 0.001. Fig.3.
shows the white noise of various power that was applied on
the system.
Fig.2. MPC Controller response under noisy condition

Fig.3.Sensor Noises with different powers (0.1,0.01,0.001) respectively.
Model predictive controller behaved fine under noisy
conditions. That wasn’t the same when external disturbances
were applied to the system. MPC could not handle external
disturbance very well. Fig.4. shows MPC response under
external disturbance of 10% of the input(Red) 50% of the
input(black) and 100% of input (blue). The graph shown in
fig.4. is between position versus time.
Fig.4. MPC Controller Response under Disturbance.
B. Proportional Integral Derivative Control
To deal with issue of disturbance, PID Controller is used
with elbow joint of the rehabilitation arm. The PID controller
is tuned firstly by auto-tuning method then by
fine
tuned and
finally by Nicolas Ziegler. The block diagram of the system
with PID control is shown in fig.5.
Fig.5. Block diagram of the system with PID
Firstly, disturbances of various amplitude on the system with
PID controller to handle the disturbances.Fig.6. shows the
response of the system under different disturbance. TABLE IV
shows th
e parameter and system response of the system under
study.
TABLE IV. PID PARAMETERS AND RESPONSE
Fig.6. PID response using auto tuned(red), fined tuned(blue) and Nicolas
Ziegler method(black).
PID handles external disturbance effectively. In this
study disturbances from 10% to 200% of the input signal were
applied to the system with PID and system handles the
disturbance very well. TABLE V shows the overshoot and
settling time of the system under disturbances. System has no
effect on stability, overshoot and settling time under
disturbance equals to 10% of input signal. While at external
disturbance of 200% of the input signal system has 1%of
overshoot and takes 1.5 sec to stabilized the system.
TABLE V. OVERSHOOT AND SETTLING TIME OF THE SYSTEM
UNDER DISTURBANCES.
Controller Parameters
Auto Tuned
Fine Tuned
Nicolas Ziegler
P
71020.1844
319.8875
150
I
142040368.73
597.8409
180
D
0
42.6716
20
N
100
1528.13
100
b
1
0.983
1
c
1
0.383
1
Performanc e and Robustness
Rise time
0.195 sec
0.416 sec
1.52 sec
Settling Time
0.75 sec
1.02 sec
1.15 sec
Overshoot
0.001%
0.04%
0.07%
Peak
0.999
1.05
1.02
Gain Margin
51.9db
@440rad/s
-
-
Phase Margin
87deg
@5 rad/s
64.5deg
@ 13.4rad/s
-
Close-loop
Stability
Stable
Stable
Stable
Elbow
Disturbance
Overshoot
Settling time
200%
1%
1.5 sec
100%
0.5%
1.27 sec
50%
0.2%
1.21 sec
10%
No effect
-

PID controller fails under noisy conditions. Actually
PID control system is trying to reduce error signal which is the
difference between reference signal and feedback signal, when
noise (which is an always random and uneven signal) is added
in PID based F.B loop control system, then the error signal
will also become uneven (alter in every instant of time) so fix
PID gains would not be able to settle it to minimize it.
While external disturbance
normally acts as some sort of
extended impulse and there is no unevenness in it, so PID can
easily overcome it and makes the system stable.
The White noise of power 0.1 (response is shown in
red), 0.01(response is shown
in blue) and 0.001 (response
is
shown in black.) was applied to the system feedback. Fig.7
shows PID response under the noisy conditions. The graph
shown in fig.7 is position versus time.
Fig. 7. System response with under noisy condition.
IV. CONCLUSION
Output response based on two control strategies, the control
effects of both controllers (MPC and PI
D) were similar in
terms of settling time and overshoot. However, under
disturbance PID control method shows better results as
compared to MPC. This was because the original nonlinear
model used by PID strategy included a derivative part that
eliminated the overshoot for the control response while the
process model used by MPC strategy still had some mismatch
compared to the original model. PID controller cannot handle
noise, while in MPC, noise can be modeled and can be
removed as MPC has a Kalman filter to remove output noise.
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