An adaptive PID like controller using mix locally recurrent neural network for robotic manipulator with variable payload

Research Article
An adaptive PID like controller using mix locally recurrent neural
network for robotic manipulator with variable payload
Richa Sharma a
, Vikas Kumar a,n
, Prerna Gaur b
, A.P. Mittal b
a
Department of Electrical and Instrumentation Engineering, Thapar University Patiala, 147004, India
b
Instrumentation and Control Engineering Division, Netaji Subhas Institute of Technology, Dwarka, New Delhi 110078, India
a r t i c l e i n f o
Article history:
Received 27 November 2015
Received in revised form
13 January 2016
Accepted 25 January 2016
Available online 23 February 2016
Keywords:
Robotic manipulator
Cuckoo search algorithm
Artificial neural networks
Recurrent neural networks
On-line learning
a b s t r a c t
Being complex, non-linear and coupled system, the robotic manipulator cannot be effectively controlled
using classical proportional-integral-derivative (PID) controller. To enhance the effectiveness of the
conventional PID controller for the nonlinear and uncertain systems, gains of the PID controller should be
conservatively tuned and should adapt to the process parameter variations. In this work, a mix locally
recurrent neural network (MLRNN) architecture is investigated to mimic a conventional PID controller
which consists of at most three hidden nodes which act as proportional, integral and derivative node. The
gains of the mix locally recurrent neural network based PID (MLRNNPID) controller scheme are initi-
alized with a newly developed cuckoo search algorithm (CSA) based optimization method rather than
assuming randomly. A sequential learning based least square algorithm is then investigated for the on-
line adaptation of the gains of MLRNNPID controller. The performance of the proposed controller scheme
is tested against the plant parameters uncertainties and external disturbances for both links of the two
link robotic manipulator with variable payload (TL-RMWVP). The stability of the proposed controller is
analyzed using Lyapunov stability criteria. A performance comparison is carried out among MLRNNPID
controller, CSA optimized NNPID (OPTNNPID) controller and CSA optimized conventional PID (OPTPID)
controller in order to establish the effectiveness of the MLRNNPID controller.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In the past few years, the process industry and medical field are
significantly captured by the robotic manipulators as these sys-
tems have the ability of quick and accurate positioning. The precise
control of end-effector is the prime requirement for the use of
these systems. Over the time, several advanced control techniques
have been developed to control these systems. The conventional
PID controller is the widely deployed controller in the industry,
due to its simple and easy design, almost generalized training
rules and cost-effectiveness. The performance of conventional PID
controller falls short in applications which are non linear and
uncertain [1] while almost all industrial processes are nonlinear
and uncertain. Such applications demand real time tuning/adap-
tation of controller gains in an on-line manner and hence tuning of
PID controller in such situations is a challenging task. The con-
ventional tuning methods such as Zeigler–Nicholas and Coon–
Cohen [2] cannot be applied to a highly non-linear, coupled and
uncertain robotic manipulator. Moreover a 2-degree of freedom
(2-DOF) robotic manipulator is a coupled system and requires
tuning of two PID controllers simultaneously. With the advance-
ments in the computation capabilities of the microprocessors and
consequently computational techniques several authors have used
nature inspired and evolutionary optimization methods such as
Genetic Algorithm (GA) [3], Multi-Objective GA [4], Tabu Search
[5], Particle Swarm Optimization (PSO) [6] and CSA [7] etc. for
tuning of the conventional PID controller. These nature inspired
and evolutionary optimization methods provide fixed optimal
gains and using these fixed gains for uncertain and nonlinear
processes it is very difficult to obtain the optimum performance.
Therefore, an efficient and effective on-line tuning mechanism is
required for the precise position control of an end effector.
The introduction of fuzzy controllers has been significant in the
field of control engineering due to their capability of nonlinear
mapping; flexible and model-free design etc. These controllers are
also able to cope up with uncertainties and nonlinearities [8–9].
The performance of classical PID controller can be enhanced to
some extent due to its collaboration with fuzzy logic controllers
(FLC) the conventional PID controller along with FLC can
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.01.016
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail addresses: richasharma_7@yahoo.co.in (R. Sharma),
sainivika@gmail.com (V. Kumar), prernagaur@yahoo.com (P. Gaur),
mittalap@gmail.com (A.P. Mittal).
ISA Transactions 62 (2016) 258–267

effectively deal with linear, nonlinear, higher order and uncertain
systems [10–11].
In order to enhance the performance of conventional PID
controller several authors have used fuzzy logic method to obtain
the optimal gains of different structures of PID controller in an on-
line manner for different plants such as induction machine, ship
plant [12–14], AVR system [15], hybrid electric vehicle system [16]
and microgrid [17]. Various authors investigated the use of fuzzy
logic for tuning of different structures of PID controllers for robotic
manipulators [18-22].
From the literature, it is clear that the performance of con-
ventional PID or intelligent control techniques can be improved
signiﬁcantly by optimizing the critical parameters of these control
methods using evolutionary optimization methods. GA and PSO
are the mostly employed evolutionary optimization methods for
the optimization of various intelligent controller parameters.
Among the nature inspired optimization algorithms integrated
with intelligent techniques, recently developed CSA based opti-
mization method developed by Yang and Deb [23] has been
established as a better optimization technique in literature in
terms of convergence rate and optimization performance for the
robotic manipulators [24–26]. CSA requires lesser number of
parameters for the initial settings and the performance of this
algorithm is also independent of parameters chosen [25–26].
Application of ANN in the areas of dynamical control systems
has recently witnessed tremendous growth due to outstanding
learning, adaptation and generalization capabilities. However
implementation of fuzzy logic or ANN requires large number of
parameters to be determined off line prior to the implementation
[27]. Due to the unavailability of faster training algorithms for
ANN, and convergence to local minima, the application of ANN in
an on-line tuning of the controller remains less explored [28].
Cong and Liang proposed a mix locally recurrent neural network
based PID like controller for motion control systems [29] using the
conventional back propagation algorithm. The training time for
the conventional gradient/back propagation based method was
large and no systematic procedure for gain initialization was
proposed. Moreover, the conventional gradient based tuning
method suffers from certain limitations like convergence to a local
minima, very large training time, and stability. The training time of
the training algorithm depends on the learning rate as too small
value of learning rate results in large training time and too large
value of learning rate leads to stability problems. Based upon the
faster training algorithm proposed by Huang et. al. [30], a mix
locally recurrent neural network based PID like controller is
implemented by Kumar et. al. for the precise position control of
PMSM servo drives [31]. In their work the initial gains were not
randomly assumed but were calculated using Moore–Penrose
generalized inverse method which provides minimum norm least
square (MNLS) solution. The training time for the proposed
method is at least 100–1000 times faster than the conventional
gradient based method. MNLS solution suffers from overtraining
problems, as it tries to minimize the error for entire data set [30].
For an over trained network an on-line learning scheme cannot
ensure the best tracking error. Also, the stability analysis of the
proposed controller was not presented in both [30] and [31].
In order to overcome the limitations mentioned in [29] and [31],
in the presented work all parameters of the MLRNNPID controller are
optimized using CSA. Once the optimized parameters of the
MLRNNPID controller are obtained, the weights of the hidden layer
are re-adjusted using Moore Penrose generalized inverse. An on-line
sequential learning algorithm based on the recursive least square
solution is then derived for an on-line training of the MLRNNPID
controller. The on-line sequential learning algorithm is quite fast and
can never converge to a local minima as it is based on Moore–Pen-
rose generalized inverse.
The motivation of this work is to preserve the favorable char-
acteristics of the conventional PID controller and integrate them with
the learning and powerful approximation capabilities of the
MLRNNPID controller for trajectory tracking control of a highly
complex, non-linear and coupled 2- DOF robotic manipulator. Mix
locally recurrent NN provide a partial memory to the NNPID con-
troller, and hence the approximation and on-line learning capabilities
of the NNPID controller will increase manifolds. The initial gains i.e.
the weights of the hidden layer as well as output layer are calculated
using CSA based optimization method which is established as a
better optimization method as compared to GA and PSO [25]. An on-
line sequential learning algorithm is then derived to tune the output
weights of the NNPID controller in an on-line manner. The stability
analysis of the presented control scheme is investigated out using
Lyapunov's approach. Robustness testing of the proposed controller
is presented under parametric uncertainty and external disturbances
and ﬁnally a performance comparison is carried out among
MLRNNNNPID controller, CSA tuned NNPID (OPTNNPID) controller
and CSA tuned conventional PID (OPT PID) controller.
2. Mathematical model of TL-RMWVP
The dynamic model of SCARA type TL-RMWVP, expressed in Eq.
(1), has been given by Lin [32]. Fig. 1 represents the robotic
manipulator with two rigid links and a payload at the tip. Also,
Table 1 lists the parameters of TL-RMWVP plant for the simulation.
Q11 Q12
Q21 Q22
" #
€θ1
€θ2
" #
þ
P11
P21
" #
þ
f 1
f 2
" #
þ
g1q
g2q
" #
¼
τf ln1
τfln2
" #
ð1Þ
where
Q11 ¼ I1 þI2 þm1l
2
c1 þm2l
2
1 þm2l
2
c2 þ2m2l1lc2 cos θ2 þm33l
2
1 þm33l
2
2
þ2m33l1l2 cos θ2
Q12 ¼ I2 þm2l
2
c2 þm2l1lc2 cos θ2
À Á
þm33l
2
2 þm33l1l2 cos θ2
À Á
Q21 ¼ Q12
Q22 ¼ I2 þm2l
2
c2 þm33l
2
2
P11 ¼ Àm2l1lc2 2 _θ1 þ _θ2

_θ2 sin θ2 Àm33l1l2 2 _θ1 þ _θ2

_θ2 sin θ2
P21 ¼ m2l1
_θ
2
1lc2 sin θ2 þm33l1
_θ
2
1lc2 sin θ2
f 1 ¼ b1q
_θ1
f 2 ¼ b2q
_θ2
g1q ¼ m1lc1gcos θ1
À Á
þm2g lc2 cos θ1 þθ2
À Á
þl1 cos θ1
À ÁÀ Á
Fig. 1. Two-link planar robotic manipulator with payload attached.
R. Sharma et al. / ISA Transactions 62 (2016) 258–267 259

þm33g l2 cos θ1 þθ2
À Á
þl1 cos θ1
À ÁÀ Á
g2q ¼ m2lc2gcos θ1 þθ2
À Á
þm33lc2gcos θ1 þθ2
À Á
where θ1 and θ2 are the positions of links; τf ln1
and τfln2
represent
the control outputs; m1 and m2 are masses; l1 and l2 are the
lengths; lc1 and lc2 presents the distances from the joints to their
center of gravity; I1 and I2 represent lengthwise centroid
inertia;b1q and b2q are the coefficients of friction at joints; f1 and f2
represent the coefficients of dynamic friction of Link1 and Link2
respectively. Also, m33 represents the mass of variable payload and
its value varies between 0.5699 kg to 0.14172 kg as shown in Fig. 2.
3. MLRNN based PID controller
For the past few years, the dynamic neural networks have been
increasingly used in the area of robotic manipulator for designing
of adaptive controllers because of their enhanced prediction and
adaptation capabilities. Dynamic neural networks are classified as:
Elman NN, Feed-forward networks with filters, globally recurrent
NN, and locally recurrent NN. The MLRNN possess lesser number
of parameters as compared to other dynamic NN. Hence MLRNN
can be trained easily and training time is also small [33]. Apart
from the lesser number of tuning parameters, the local feedback in
mix locally recurrent NN provides partial memory to the network
which further increases the approximation and learning cap-
abilities of the network. Modeling of MLRNN-PID like controller
created using mix locally recurrent NN is presented in the fol-
lowing section.
3.1. MLRNN-PID controllers
The structure of MLRNN based PID like controller is shown in
Fig. 3. The structure consists of three nodes which act like pro-
portional, integral and derivative nodes.
The output of hidden layer neurons H1; H2 and H3 can be
expressed in terms of elink1 and elink2 as follows:
H1 kð Þ ¼ ϕ h11elink1 kð Þþh21elink2 kð Þð Þ
H2 kð Þ ¼ ϕ h12elink1 kð Þþh22elink2 kð Þð ÞþH2 kÀ1ð Þ
H3 kð Þ ¼ ϕ h13elink1ðkÞþh23elink2ðkÞþh13elink1ðkÀ1Þþh23elink2ðkÀ1Þð Þ
9
=
;
ð2Þ
where h ¼
h11h12h13
h21h22h23
#
; is the hidden layer weight matrix and
Q ¼
Q11Q12Q13
Q21Q22Q23
#
represents the output layer weights.
If zÀ1
is a unit delay operator, the output of the node 2 as well
as 3 can be represented using (2) as follows:
H2 kð Þ ¼
ϕ h12elink1ðkÞÞþh22elink2ðkÞð Þ
1ÀzÀ 1
ð3Þ
H3 kð Þ ¼ ϕ h13elink1ðkÞþh23elink2ðkÞð Þð1ÀzÀ1
Þ ð4Þ
Considering the single input and single output system, the
hidden layer output matrix can be expressed as:
H ¼ H1 H2 H3
Â Ã
The output weight matrix can be expressed as
Q ¼ Q1 Q2 Q3
Â Ã
The final output of the MLRNN-PID controller is
τ ¼ Q1H1 kð ÞþQ2H2 kð ÞþQ3H3 kð Þ
In matrix form, the above equation can be expressed in matrix
form as
τ ¼ HQ ð5Þ
3.2. On-line training algorithm
The training algorithm for locally recurrent NN is developed in
this section:
E ¼ RÀHQ; E 2
¼ ðRÀHQÞT
ðRÀHQÞ

E 2
¼ RT
RÀ2HRT
Q þQ2
HT
H

The training of MLRNN-PID controller is required to adjust Q
and H so that the error E 2

is minimized.
∂‖E‖2
∂Q
¼ À2RT
Hþ2QHT
H ð6Þ
∂‖E‖2
∂H
¼ À2Q
∂HT
∂w
RT
þ2Q2∂QT
∂x
Q
Table 1
Parameters for the SCARA type TL-RMWVP plant.
Parameters Link1 Link2
Mass 0.392924 kg 0.094403 kg
Acceleration due to gravity
(g)
9.81 m/s2
9.81 m/s2
Length 0.2032 m 0.1524 m
Distance from the joint to
its center of gravity
0.104648 m 0.081788 m
Lengthwise centroid inertia
of link
0.0011411 kg m2
0.0020247 kg m2
Friction at joints 0.141231 N-m/radian/s 0.3530776 N-m/radian/s
Fig. 2. Payload variations at tip.
Fig. 3. MLRNN PID controller.
R. Sharma et al. / ISA Transactions 62 (2016) 258–267260

The optimum value of H and Q is obtained when ∂‖E‖2
∂h
and ∂‖E‖2
∂Q ¼
0. From (6), the output weight matrix is calculated as
Q ¼
RT
H
HT
H
or Q ¼ Hf
T ð7Þ
where Hf
is the Moore–Penrose generalized inverse of matrix H,
which gives minimum norm least square solution. If the weights of
hidden layer are known; the weights of the output can be calcu-
lated analytically in a single step using (7). In a traditional gradient
based method the error has to back propagate recursively so the
training time for conventional gradient based method is quite large.
As the output weights are calculated in a single step the training
speed can be several 100–1000 times greater than the conventional
gradient based methods. In [30], and [31] the input weights are
assumed randomly, in this work initial weights used are calculated
using CSA. The CSA optimized output weights are used in
MLRNNPID controller as output weights and the hidden layer
weights are readjusted to obtain the minimum norm least square
solution. The optimized weights of the output layer as calculated
using CSA are used to reinitialize the hidden layer matrix using
H0 ¼ Qf
T ð8Þ
The hidden layer weights hij are re-adjusted using (2),
Once the hidden layer and output layer weights are obtained,
for each incoming data chunk or single data a sequential learning
algorithm which is based on the recursive least square solution is
employed to update the weights of the MLRNNPID controller. The
training algorithm is summarized in Fig. 4.
Q0 ¼ ðHT
0H0ÞÀ 1
HT
T ð9Þ
For the next chunk of data, the following function is to be
minimized
H0
H1
#
Q À
T0
T1
#

ð10Þ
From (9), taking both data samples into account, the output
weight matrix is represented as:
Q1 ¼ K À1
1
H0
H1
#T
T0
T1
#
ð11Þ
where
K1 ¼
H0
H1
#T
H0
H1
#
ð12Þ
For sequential learning we have to express Q1 as a function of
Q0. Using (11) K1 can be expressed as
K1 ¼ HT
0HT
1
h i H0
H1
#
ð13Þ
K1 ¼ K0 þHT
1H1 ð14Þ
H0
H1
#T
T0
T1
#
¼ HT
0T0 þHT
1T1
¼ K0Kð À 1Þ
0 HT
0T0 þHT
1T1
¼ K0Q0 þHT
1T1
¼ ðK1 ÀQT
1Q1ÞG0 þQT
1Y1
¼ K1Q0 ÀHT
1H1Q0 þHT
1T1 ð15Þ
Q1 can be written using (15) and (11)
Q1 ¼ K À1
1 K1Q0 ÀHT
1H1Q0 þHT
1T1

¼ Q0 þK À 1
1 HT
1 T1 ÀH1Q0ð Þ ð16Þ
From (14), the value of K for (kþ1)th data sample is given as
Kk þ 1 ¼ Kk þHT
k þ 1Hk þ 1 ð17Þ
K À 1
k þ 1 ¼ ðKk þHT
k þ1Hk þ 1ÞÀ 1
ð18Þ
Each incoming data samples the Kk þ1 can be updated using
(19) with the help of Woodbury formula [34]:
K À 1
k þ 1 ¼ K À 1
k ÀK À1
k HT
k þ 1ðIþHk þ1K À1
k HT
k þ 1ÞHk þ1K À 1
k ð19Þ
Using Pk þ 1 ¼ K À 1
k þ 1, (15) and (12) can be written in a simpler
form as follows:
Pkþ 1 ¼ Pk ÀPkHT
k þ1ðIþHk þ 1PkHT
k þ 1ÞÀ 1
Hk þ1Pk ð20Þ
Qk þ 1 ¼ Qk þPk þ 1HT
k þ 1ðTk þ1 ÀHkþ 1QkÞ ð21Þ
3.3. Optimization of MLRNNPID and conventional PID controller
using Cuckoo search algorithm
As robotic manipulator is a multi-input, multi-output plant,
therefore the conventional PID controllers require six parameters
Fig. 4. Flowchart for the implementation.

which are to be tuned before the implementation while the
implementation of MLRNNPID controllers require twelve para-
meters. Cuckoo search, a recently developed optimization method
which gives faster convergence performance is used to tune the
parameters of the MLRNNPID controller.
Yang and Deb proposed an excellent technique namely CSA for
optimization purpose and it is formed on the unique parasitic
breeding behavior of cuckoos [23]. It is designed on the cuckoo's
cunning planning of finding the other species nest. The cuckoos
are in search of a nest in which other host bird has just laid its eggs
[23,35]. Despite its nascent stage, CSA has emerged out as an
efficient optimization technique because of its invincible features.
The parameters used for the initialization of this algorithm are
lesser as compared to famous GA and PSO techniques. The con-
vergence rate is independent of all of its parameters [35–38]. The
large steps taken during the searching procedure can make it more
effective and powerful than other competitive techniques. It also
possesses the well-renowned and significant elitism feature.
The cuckoo birds have some unbelievable abilities such as
copying the call of host bird; copying the color and pattern of eggs
of hosts etc. [34–36]. CSA has been implemented to find different
cutting parameters for the milling operation and it has been found
effective than many other optimization methods such as hybrid
PSO, hybrid immune algorithm, feasible direction method, hand-
book recommendations, GA and ACO [37]. The algorithm is based
on the Lévy flight for searching the space [23]. The detailed
implementation of CSA can be obtained from [23,38].
The selection of appropriate objective function is necessary to
use any optimization technique. For the present work, the objec-
tive functions chosen are integral of absolute error (IAE) of both
Link1 and Link2 for minimization and are expressed by Eqs. (22)
and (23) respectively. The aggregate fitness function AOF is
designed as the weighted sum of IAE of both the links. The reason
behind the selection of these objective functions is to reduce the
error between actual and desired trajectories.
Of 1 ¼
Z
j e1ðtÞjdt ð22Þ
Of 2 ¼
Z
je2ðtÞjdt ð23Þ
AOF ¼ w1Of 1 þw2Of 2 ð24Þ
w1 and w2 are the weights assigned to objective functions Of1 and
Of2 respectively.
3.4. Stability analysis of the proposed controller
The stability analysis of the proposed controller is presented in
this section.
Assume the non linear dynamical system is given by the fol-
lowing equation
yk þ1 ¼ f xkð Þþg xkð Þuk ð25Þ
Assumption 1. : For the non-linear dynamical system described
by (25), ( a constant Q such that if uk ¼ HT
k Qk;[39] then
Tkþ 1 ÀXk þ1 ¼ Δk, where Tk þ 1 is the desired trajectory, Xkþ 1 is the
system output and Δk rεj

is a small positive constant.
The tracking position error can be expressed as:
ek þ1 ¼ Xk þ 1 ÀTkþ 1
or ek þ1 ¼ HT
Q þΔk ð26Þ
Referring to the recursive least square algorithm as described
above by (20)–(21)
Pk þ 1 ¼ Pk À
PkHT
k þ 1Hkþ 1Pk
IþHk þ 1PkHT
k þ 1
ð27Þ
Qk þ1 ¼ Qk þPkHT
k ek þ 1 ð28Þ
where P0 ¼ HT
0H0
T
and Q0 ¼ HT
0H0
À 1
HT
0T
If Δk oεj

, then the RLS algorithm as derived in the earlier
section can be modified using dead zone. We can rewrite (27) and
(28) as follows [40]:
Pk þ 1 ¼ Pk À
γkβkPkÀ 1HT
k HkPk
IþHkPkHT
k
ð29Þ
Qk þ1 ¼ Qk ÀγkβkPkþ 1HT
kþ 1ekþ 1 ð30Þ
where
βk ¼
1
1þγkHT
k PkHk
ð31Þ
γk ¼
1 ‖ek þ 1‖2
1þγkHkPk À 1HT
kð Þ
2 oϵ2
0 otherwise
8

:
ð32Þ
Desired
Trajectory
for link1
PID1/
NNPID1/
OPTNNPID1
Desired
Trajectory
for link2
2-DOFRobotic
Manipulator
(SCARA)
Torque1
Torque2
Theta 1
Theta 2
+ -
+
-
d1
d2
++
++
PID2/
NNPID2/
OPTNNPID2
Fig. 5. Block diagram of the proposed strategy.
Table 2
IAE and optimized parameters for implemented control schemes.
Parameters OPT PID (Kp, Ki, Kd) OPT PID MLRNNPID
Hidden layer Output layer Hidden layer Output layer
Link 1 400, 67.6405, 15.2305 0.984, 0.994,
1.0
400, 136.3997, 39.5656 1.469, 0.258, 2.31 395.99, 134.8173, 39.1254
Link 2 45.791, 20.28, 4.26 1.0, 0.82, 0.053 60.9486,12.3286, 4.1424 0.987,0.1037, 4.21 60.5018, 12.6301, 4.5960
IAE Link1 0.0097 0.005731 0.003049
Link2 0.01853 0.01467 0.006418

3.4.1. Stability analysis by Lyapunov method
According to Lyapunov stability criterion for non-linear and
uncertain systems, if Lyapunov functionΔVk o0, the closed loop
system is stable.
Theorem 1. : Under the assumption 1, the system described by (25)
is stable and the weights in the controller are updated as per (20)–
(21). Then a stable closed loop system will satisfy the following:
(i) The position tracking error between the reference input and
the system output converges to the small neighborhood
of zero.
(ii) Qk is bounded and Qk þ 1 ÀQkk

converges to zero.
(iii) All the weight matrices in the RLS algorithm remains bounded.
From (26) we can obtain
HT
k þ 1Qkþ 1 þΔk ¼ βk HT
k þ1Qk þ1 þΔk

¼ βkek þ1 ð33Þ
Using (33), (30) can be re-written as
Qk þ 1 ¼ Qk ÀγkPkHkðHT
k Qk þ1 þΔkÞ ð34Þ
Using following matrix property
P þMNð ÞÀ 1
¼ P À1
ÀPÀ 1
MðIþNPÀ 1
MÀ 1
ÞNPÀ 1
PÀ 1
k þ 1 can be calculated as
PÀ 1
kþ 1 ¼ P À1
k þγkHkHT
k ð35Þ
Considering the following Lyapunov function as inspired by [40]
Vk þ1 ¼ QT
k þ1PÀ 1
kþ 1Qk þ1 ð36Þ
Substituting the value of PÀ 1
k þ1 from (35) in (36), the Lyapunov
function can be written as:
¼ QT
k þ1 P À1
k þγkHkHT
k

Qk þ 1
¼ QT
k þ 1PÀ 1
k Qk þ1 þγk HT
k Qk þ1
2
¼ Qk À γkβkPkHT
k ek þ 1
T
P À1
k Qk À γkβkPkHT
k ek þ 1

þγk HT
k Qk þ 1
2
¼ Vk þγk HT
k Qk þ1
2
À2 γkHT
k Qk HT
k Qk þΔk

þγ2
k HkPk À1HT
k
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
5
10
Time (s)
Position(radian)
MLRNNPID Link 1
Ref. Link1
MLRNNPID Link2
Ref. Link2
Opt NNPID link1
Opt. NNPID Link2
Opt PID Link1
Opt. PID link2
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
-0.01
0
0.01
0.02
Time (s)
ErrorLink1(radian)
MLRNNPID
OPTNNPID
OPTPID
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
-0.01
0
0.01
0.02
Time(s)
ErrorLink2(radian)
MLRNNPID
OPTNNPID
PID Error
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
0
2
4
6
Time (s)
TorqueLink1(Nm)
NNPID
OPT NNPID
OPT PID
0 0.5 1,0 1.5 2.0 2.5 3.0 3.5 4.0
-0.5
0
0.5
1
1.5
Time (s)
TorqueLink2(Nm)
MLRNNPID
OPTNNPID
OPT PID
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
50
100
Time(s)
MLRNNPIDController
GainsVariationsLink2
Proportional Gain
Derivative Gain
Integral Gain
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
200
400
600
Time(s)
MLRNNPIDController
GainVariation(link1)
Proportional Gain
Derivative Gain
Integral Gain
Fig. 6. Performance comparison of implemented control schemes under nominal parameters and variable payload (a) trajectory tracking, (b) tracking error for link 1,
(c) tracking error for link 2, (d) controller output for link1, (e) controller output for link2, (f) variation in gains of MLRNNPID controller for link 2 and (g) variations in gains of
MLRNNPID controller for link 1.

Â HT
k Qk þ1 þΔk
2
rVk Àγk HT
k Qk þ 1
2
À2 γkHT
k Qk þ 1Δk
¼ Vk þγkΔ2
k Àγk HT
k Qk þ1
2
Àγk HT
k Qk þ 1 þΔk
2
¼ Vk þγkΔ2
k Àγk β2
k e2
k þ 1

¼ Vk Àγk β2
k e2
k þ1 ÀΔ2
k

Vk þ1 rVk
Àγk β2
k e2
kþ 1 ÀΔ2
k

ð37Þ
Using (33), it can be concluded that Vk þ1 rVk
Now from (32) and (36), we can obtain
1À
1
ε

γ2
k β2
k e2
k þ 1 rVkþ 1 ÀVk
Àγk
1
ε
β2
k e2
k þ 1 ÀΔk
2

rVk þ1 ÀVk
Summing up both sides from 0 to 1
1À
1
ε
X1
k ¼ 0
γkβ2
k e2
k þ1 rV0 ¼ QT
0P À1
0 Q0 o1 ð38Þ
From (38) it can be calculated
γkβ2
k e2
k þ 1→0; As k→∞ ð39Þ
From (32) and (39), γk ¼ 0; βk ¼ 18k4k0
From (38) we obtain
ek þ1 o
ffiffiffi
ε
p
Δ0 8k4k0

ð40Þ
(40) Implies that error is bounded.
Since Vk rV0 it follows that
γÀ1
Qk
2
¼ QT
k PÀ 1
0 Qk

rQT
k P À1
k Qk
rQT
0P À1
0 Q0 ¼ γÀ 1
Q0
2

ð41Þ
From (41), we can conclude that
Qk r Q0 ;k

Hence Qkk

is bounded.
4. Simulation results
The developed algorithm is implemented for two link rigid
SACARA robot. The control scheme is shown in Fig. 5. The per-
formance of the proposed controller scheme is compared with CSA
optimized NNPID controller which is optimized in an off line mode
and CSA optimized conventional PID controller. All simulations are
carried out using Matlab software 2009b version and fixed step
ODE 4 solver is used with a fixed time of 0.01 s. Payload of the
manipulator is changed as shown in Fig. 2. The optimized con-
troller parameters are summarized in Table 2.
For the presented work, the trajectory used is a cubic poly-
nomial type trajectory and is expressed as follows [21]:
θRef flci
tzð Þ ¼ zflc0
þzflc1
tzð Þþzflc2
tzð Þ2
þzflc3
tzð Þ3
ð42Þ
Table 3
IAE values for link 1 and link 2 for 5% decrease in parameters.
Parameter variation ( 5%) Decrease
MLRNNPID OPTNNPID OPT PID
Link 1 Link 2 Link 1 Link 2 Link 1 Link 2
Parameter 1: m1 0.002999 0.007621 0.005637 0.01853 0.008566 0.02472
Parameter 2: m2 0.003031 0.007624 0.005694 0.01853 0.008566 0.02465
Parameter 3:m1; m2 0.002981 0.007624 0.005601 0.01853 0.008513 0.02465
Parameter 4: b1q 0.003047 0.007621 0.005725 0.01853 0.008580 0.02474
Parameter 5: b2q 0.003047 0.007624 0.00573 0.01928 0.008598 0.02403
Parameter 6: b1q ; b2q 0.003045 0.007625 0.005725 0.01930 0.008579 0.02403
Parameter
7:b1q ; b2q ; m1; m2
0.002979 0.007625 0.005595 0.01932 0.008493 0.02395
Table 4
IAE values for link 1 and link 2 for 5% increase in parameters.
Parameter variation (5%) Increase
Parameter 1: m1 0.003098 0.007621 0.005825 0.01853 0.008643 0.02472
Parameter 2: m2 0.003066 0.007619 0.005767 0.01853 0.008643 0.02472
Parameter 3:m1; m2 0.003115 0.007619 0.005861 0.01853 0.008686 0.02479
Parameter 4: b1q 0.003050 0.007621 0.005736 0.01853 0.008618 0.02472
Parameter 5:b2q 0.003049 0.007623 0.005731 0.01928 0.008600 0.02542
Parameter 6: b1q; b2q 0.002979 0.007626 0.005736 0.01928 0.008619 0.02542
Parameter
7:b1q ; b2q ; m1; m2
0.002979 0.007626 0.005866 0.01928 0.008706 0.02549
Fig. 7. Disturbance applied at link 1 and link 2.

The constraints are:
_θRef flci
tzð Þ ¼ zflc1
þ2zflc2
tzð Þþ3zflc3
tzð Þ2
ð43Þ
€θRef flci
tzð Þ ¼ 2zflc2
þ6zflc3
tzð Þ ð44Þ
where θRef flci
represent the reference positions; i¼1, 2 represent
Link1 and Link2 respectively; θRefflci
¼ 1radian and θRefflc2
¼ 2radian
for tz ¼ 2s; θRef flc1
¼ 0:5radian and θRefflc2
¼ 4radian for tz ¼ 4s; _θRflci
¼ 0radian=s for both tz ¼ 2s and tz ¼ 4s.
4.1. Performance evaluation under nominal parameters and variable
payload
The comparison of the simulation results for reference trajec-
tory tracking, position error and controller output among the
implemented controllers for both links are as shown in Fig. 6. A
comparison of the IAE for all the three control schemes is sum-
marized in Table 2. The IAE for the proposed MLRNNPID for link1
and link2 are 0.003049 rad and 0.006418 rad respectively, IAE for
OPTNNPID controller is 0.005731 rad and 0.01467 rad respectively
and for OPT PID controller IAE values for link 1 and link 2 are
0.0097 and 0.01853 rad respectively. Clearly IAE values for the
proposed MLRNNPID controller are least and for OPT PID con-
troller are highest among the three implemented control schemes.
The proposed MLRNNPID control scheme has outperformed the
OPTNNPID and OPTPID controller in terms of tracking error and
IAE. The variations in the output weights of the MLRNNPID
controller for link1 and link 2 are shown in Fig. 6(f) and
(g) respectively. Some variations in the controller gains are seen
during initial portions of the reference trajectory; the gains are
maintained in the closed vicinity of the optimized gains as
obtained for OPTPID and OPTNNPID controller.
4.2. Robustness testing
In this section, the effect of various parameters variation on the
trajectory tracking performance has been observed by evaluating
the change in the IAE. Following variations in the parameter of the
robotic manipulator are considered in both links: (i) 5% decrease in
mass: link 1, link 2 and for both link 1 and link 2 simultaneously,
coefﬁcient of friction: link 1, link 2 and for both link 1 and link
2 simultaneously. (ii) 5% increase in the mass: link 1, link 2 and for
both link 1 and link 2 simultaneously. Same variations have been
introduced in coefﬁcient of friction: for link 1, link 2 and also for
both link 1 and link 2 simultaneously. The IAE values for both the
cases are summarized in Tables 3 and 4 respectively. Clearly the
change in IAE values is least for the proposed MLRNNPID con-
troller for all the cases considered. MLRNNPID controller being
adaptive in nature; is able to reject the effect of parameter varia-
tions in an effective manner as there are small variations in the IAE
values for the NNPID controller. The IAE values for OPT PID con-
troller are the most affected due to the change in the parameters
as considered above.
Fig. 8. Performance evaluation of implemented control schemes under the effect of disturbances (a) trajectory tracking, (b) IAE for for link 1, (c) IAE for link 2, (d) controller
output for link 1 and (e) controller output for link 2.

4.3. Disturbance rejection
The effect of external disturbances on trajectory tracking per-
formance of the robotic manipulator has been observed for both
links by adding a sinusoidal signal 1.0sin25tNm to the controller
output for both links. The disturbance signal applied to both the
links is shown in Fig. 7. The effect of external disturbance on the
trajectory tracking, IAE for both links and controller output and for
all implemented control schemes shown in Fig.8. The variations in
the output layer weights of the MLRNNPID controller for link 1 and
link 2 is shown in Fig. 9(a) and (b) respectively.
The IAE values under the effect of disturbances are summarized in
Table 5. For MLRNNPID controller the IAE values for disturbances in
both links are 0.003817 and 0.01412 for link 1 and link 2 respectively.
Clearly the MLRNNPID controller has outperformed conventional
OPTPID controller and OPTNNPID controller in terms of trajectory
tracking under nominal conditions as the IAE values of the
MLRNNPID controller is smallest among all controllers. Moreover the
MLRNNPID controller is more robust in comparison to OPTNNPID
and OPTPID controller, also the performance of the MLRNNPID con-
troller is least affected in the presence of disturbances.
5. Discussions
In the presented work, a mix locally recurrent neural network's
powerful approximation and learning capabilities are integrated
with simplicity and effectiveness of the conventional PID con-
troller to from an adaptive and robust MLRNNPID controller. A
systematic approach is proposed for an on-line learning of the
gains of the MLRNNPID controller. The initial gains are calculated
using CSA based optimization scheme. The optimized initial
weights are readjusted using Moore Penrose generalized inverse
as given in (8). The optimization and re-initialization ensured that
the mix locally recurrent neural network is free from convergence
to a local minima and over training problems. The weights of the
MLRNNPID controller are adjusted in an on-line manner using
sequential learning methods which is based on the least square
solution. The performance of the proposed controller is tested
against parametric uncertainties in mass and coefficient of friction
for both links. Moreover the external disturbances in sinusoidal
form are added to both links and the performance of implemented
controllers is compared taking IAE as the performance criteria. It is
verified through simulations outcomes that MLRNNPID controller
being adaptive in nature tends to minimize the effects of external
disturbances and parametric uncertainties in an effective manner
as compared to the conventional PID and optimized NNPID con-
troller. With the proposed tuning procedure, the MLRNNPID con-
troller is easy to implement and tune.
6. Conclusion
In this work, a mix locally recurrent neural network based PID like
controller scheme is implemented for the trajectory tracking control of
a robotic manipulator. The tuning of NN based PID controller does not
require the exact mathematical model of the process. The main
advantage of the MLRNNPID controller is its simple structure, tre-
mendous learning capabilities and more flexibility as it contains
twelve tunable parameters as compared to the six parameters for
conventional PID controller. The robotic manipulator is a nonlinear,
time-varying and uncertain system. Also, the payload attached at the
end is varying in nature. The on-line update of the parameters of
MLRNNPID controller is necessary to deal with these complexities. The
initial gains of MLRNNPID controller are calculated using cuckoo
search algorithm, a sequential learning algorithm is then derived to
tune weights of output layer in an on-line manner to deal with the
highly complex, time varying and nonlinear dynamics of the robotic
manipulator. It is verified through the results obtained that the per-
formance of the proposed MLRNNPID controller is least affected due
to parametric uncertainties and external disturbances.
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0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
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