Comparison Between Pid Controllers for Gryphon Robot Optimized With Neuro-Fuzzy System and Three Intelligent Optimizationalgorithms

International Journal of Control Theory and Computer Modeling (IJCTCM) Vol.3, No.6, November 2013
DOI : 10.5121/ijctcm.2013.3604 39
COMPARISON BETWEEN PID CONTROLLERS FOR
GRYPHON ROBOT OPTIMIZED WITH NEURO-
FUZZY SYSTEM AND THREE INTELLIGENT
OPTIMIZATION ALGORITHMS
Somayyeh Nalan Ahmadabad1
and Maryam Kouzehgar2
and Fatemeh Masoudnia3
1,3
Faculty of Sofian Islamic Azad University Tabriz, Iran
2Control Engineering Department Faculty of Electrical & Computer Engineering Tabriz,
Iran
ABSTRACT
In this paper three intelligent evolutionary optimization approaches to design PID controller for a
Gryphon Robot are presented and compared to the results of a neuro-fuzzy system applied. The three
applied approaches are artificial bee colony, shuffled frog leaping algorithm and particle swarm
optimization. The design goal is to minimize the integral absolute error and reduce transient response by
minimizing overshoot, settling time and rise time of step response. An Objective function of these indexes is
defined and minimized applying the four optimization methods mentioned above. After optimization of the
objective function, the optimal parameters for the PID controller are adjusted. Simulation results show that
FNN has a remarkable effect on decreasing the amount of settling time and rise-time and eliminating of
steady-state error while the SFL algorithm performs better on steady-state error and the ABC algorithm is
better on decreasing of overshoot. On the other hand PSO sounds to perform well on steady-state error
only. In steady state manner all of the methods react robustly to the disturbance, but FNN shows more
stability in transient response.
KEYWORDS
Robotics, PID controller, Gryphon, dynamic equations, shuffled frog leaping, artificial bee colony, particle
swarm optimization, neuro-fuzzy system
1. INTRODUCTION
PID controller (proportional-integral-derivative controller) benefiting from major advantages
such as simple structure and wide-domain performance, is one of the most frequently used
controllers. There are three parameters, on the basis of which, the PID controller is defined, i.e.
proportional gain , integral gain , and derivative gain . The traditional procedure to assign
these parameters is a sophisticated and time-consuming try and error process. Thus several new
methods are recommended to reduce the complexity of tuning PIDs. These methods, having the
biological or social inspiration as a common theme, are accomplished within evolutionary

40
algorithms such as binary and continuous genetic algorithms [1], ant colony optimization [2],
continuous and discrete particle swarm optimization [3], [4] and different types of honey bee
colony algorithms [5],[6].
In this approach, working on Gryphon robot which includes 5 joints, we must design five PID
controllers; each for one joint. Thus we must first verify the dynamic equations for Gryphon
robot. Then we will formulate the PID controller designation problem as an optimization problem
with an objective function which is supposed to be minimized by adjusting the four performance
indexes, i.e. the maximum overshoot, the rise time, the settling time and the integral absolute
error of step response. Finally we will minimize the defined objective function for each joint by
applying shuffled frog leaping (SFL) algorithm, artificial bee colony (ABC) algorithm, particle
swarm optimization (PSO) and neuro-fuzzy system (FNN).
The paper is organized in 6 sections to illustrate the case. Section 2 includes a brief description of
the system under-investigate and presents the verification of the dynamic equations of the robot.
Section 3 reviews the utilized intelligent algorithms and FNN and in section 4 the problem
formulation is explained. Section 5 presents the simulation results and analyses them.
Consequently section 6 concludes the paper.
2. A BRIEF DESCRIPTION OF THE SYSTEM
As shown in Figure 1, Gryphon is a robot with 5 revolute joints. Thus this robot benefits from 5
degrees of freedom. The first three joints which are called shoulder elbow and wrist respectively,
are supposed to determine the position of the end-effecter while the last two joints are responsible
for the orientation of the end-effecter. Fast and smooth movement while maintaining the high
precision is one of the major characteristics of this robot. This robot is controlled by four micro-
processors, one of which is to determine the position of the axis, and two are motor controllers
and the last one is supposed to support the others and communicate with the host computer. Each
joint is moved by a stepper motor while there is feedback on the corresponding encoder to realize
the closed-loop control. In this robot the utilized gear ratio is high enough to assume all joints
independent from one another [7].
As with [8], we obtained the dynamic equations of all five joints. Table 1 shows transfer
functions of all five joints that are obtained from the so-called dynamic equations, adding the
process of obtaining which sounds to need a vast space for complicated equations and that is why
we have avoided them here. In the transfer functions in Table 1, α stands for the amount of
disturbance exerted on the system. For an undisturbed system α is set to zero.
Figure 1: Joint model for Gryphon

41
3. THE UTILIZED INTELLIGENT ALGOTITHMS
3.1 The Shuffled Frog Leaping Algorithm (SFL)
In the SFL, the population consists of a set of frogs (solutions) that is partitioned into subsets
referred to as memeplexes. The different memeplexes are considered as different cultures of
frogs, each performing a local search. Within each memeplex, the individual frogs hold ideas, that
can be influenced by the ideas of other frogs, and evolve through a process of memetic evolution.
After a defined number of memetic evolution steps, ideas are passed among memeplexes in a
shuffling process [9]. The local search and the shuffling processes continue until defined
convergence criteria are satisfied [10].
Tble 1. Transfer Functions of Gryphon Joints
Joint Number Transfer Function
Joint 1
Joint 2
Joint 3
Joint 4
Joint 5
An initial population of P frogs is created randomly. For S-dimensional problems (S variables), a
frog i is represented as . Afterwards, the frogs are sorted in a scending order
according to their fitness. Then, the entire population is divided into m memeplexes, each
containing n frogs (i.e. ). In this process, the first frog goes to the first memeplex, the
second frog goes to the second memeplex, frog (m) goes to the mthmemeplex, and frog (m+1)
goes back to the first memeplex, etc.
Within each memeplex, the frogs with the best and the worst fitnesses are identified as and ,
respectively. Also, the frog with the global best fitness is identified as . Then, a process is
applied to improve only the frog with the worst fitness (not all frogs) in each cycle. Accordingly,
the position of the frog with the worst fitness is adjusted as follows:
(2)
Where rand is a random number between 0 and 1; and is the maximum allowed change in a
frog’s position.
If this process produces a better solution, it replaces the worst frog. Otherwise, the calculations in
Eqs (1) and (2) are repeated but with respect to the global best frog (i.e. replaces ). If no

42
improvement becomes possible in this case, then a new solution is randomly generated to replace
that frog. The calculations then continue for a specific number of iterations [8]. Accordingly, the
main parameters of SFL are: number of frogs P; number of memeplexes; number of generation
for each memeplex before shuffling; number of shuffling iterations; and maximum step size.
3.2 The Artificial Bee Colony Algorithm (ABC)
Karaboga analyzes the foraging behavior of honey bee swarm and proposes a new algorithm
simulating this behavior for solving multi-dimensional and multi-modal optimization problems,
called Artificial Bee Colony (ABC) [5]. The main steps of the algorithm are:
1) send the employed bees onto the food sources and determine their nectar amounts.
2) calculate the probability value of the sources with which they are preferred by the
onlooker bees.
3) stop the exploitation process of the sources abandoned by the bees.
4) send the scouts into the search area for discovering new food sources, randomly.
5) memorize the best food source found so far.
In the algorithm, an artificial bee colony consists of three groups of bees: employed bees,
onlookers and scouts. Employed bees are associated with a particular food source which they are
currently exploiting. They carry the information about this particular source and share this
information with a certain probability by waggle dance. Unemployed bees seek a food source to
exploit. There are two types of unemployed bees: scouts and onlookers. Scouts search the
environment for new food sources without any guidance. Occasionally, the scouts can
accidentally discover rich, entirely unknown food sources. On the other hand onlookers observe
the waggle dance and so are placed on the food sources by using a probability based selection
process. As the nectar amount of a food source increases, the probability value with which the
food source is preferred by onlookers increases, too. In the ABC algorithm the first half of the
colony consists of the employed bees and the second half includes the onlookers. For every food
source, there is only one employed bee. Another issue that is considered in the algorithm is that
the employed bee whose food source has been exhausted by the bees becomes a scout. In other
words, if a solution representing a food source is not improved by a predetermined number of
trials, then the food source is abandoned by its employed bee and the employed bee is converted
to a scout.
3.3 Particle Swarm Optimization
PSO is an evolutionary computation technique developed by Kennedy and Eberhart in 1995[11].
In this algorithm each solution is regarded as a particle which is defined by its position and the
fitness calculated based on the position. Also
There is a speed vector which specifies the direction in which the particle is moving. Other
parameters which are determined during the run, are as follows:
- -the personal best: Each particle remembers the best position that it has visited so far. This
best position is known as .

43
- - the global best: The best of all positions explored by all particles.
- -the neighborhood best: For each particle is the best position of the particles in the
neighborhood of particle.
To apply the algorithm, first the particles are distributed randomly in the search space. Then the
cost function is evaluated for each particle, afterwards are updated. At the end
by applying (3) the positions and speeds are updated. Eventually the algorithm checks the
stopping criteria and loops until they are satisfied.
Where is the velocity of particle, shows the position of the particle, w is the inertia
weight, utilized to avoid premature convergence and usually is set to 0.5. Separate random
numbers are generated to accelerate through and . and are acceleration constants
both equal to 2; these parameters change the amount of tension in the system, i.e. weighting the
stochastic acceleration terms that pull the particle towards or . In some iterations
may be substituted by .
Particle velocities are clamped to a maximum value of , thus serve a constraint on the global
exploration ability. is routinely adjusted at about 10-20% of the dynamic range of the
variable on each dimension [3].
3.4 The Nero-Fuzzy System (FNN)
Neural Networks, using a system based on human brain structure, have the ability to learn how to
face new challenges. These types of systems are able to adapt themselves to learn how to become
compatible with new conditions which have not formerly experienced. Fuzzy logic, based on
rules that includes a knowledgebase, is formed byfuzzy rules of "if - then". These rules are made
of simple and understandable words. The experts who know the system behavior define these
rules by means of natural language. Through combination of Neural Networks and Fuzzy logic
Neuro-Fuzzy systems are accessible. Any Neuro-Fuzzy system is a Neural Network which learns
how to classify data using Fuzzy rules and Fuzzy sets.
A Neuro-Fuzzy network is a network with feedback comprising five layers. These layers are:
1) Input layer.
2) Input fuzzymembership functions.
3) Fuzzy rules.
4) Output fuzzymembership functions.
5) Output layer.
Figure 2 shows a general view of this network. In this figure the network has two deterministic
inputs, T and H and it produces a deterministic output, Q. In a system, number of inputs and
outputs are changeable according to our requirement. In this paper there are four inputs(e,ė) and
three outputs( )

44
Figure 2. A general view of a nero-fuzzy network with five layers.
First layer is only supposed totransmitthe deterministic inputs to the second layer of network. In
the second layer, deterministic inputs turn tofuzzy values (in this paper, we used triangular
membership functions for fuzzification). Third layer includes aset of fuzzy rules. In this layer,
each neuron is a fuzzy rule. System can be set with some initial fuzzy rules and the network
adjusts the weights in a way which gives the best response. In some cases it is possible to start the
system without initial rules. In such cases system learns the rules by itself and adjusts the weights
based on them [12].
Forth layer in the network includes the neurons presenting the membership functions for different
and possible outputs of fuzzy rules. Fifth and the last layer combines and defuzzyfiesdifferent
tuned outputs.
FNN method hereissimulated by ANFIS, thetrain data for ANFIS was obtained from several
hand-tunings of PID parameters[13]. Train data loaded to ANFIS and was use for FNN learning.
After learning, by loading ANFIS into simulation environment, the PID parameters will be set.
4. PROBLEM FORMULATION TO DESIGN PID CONTROLLERS
The transfer function of the PID controller is considered as follows.
(4)
Where , and are proportional, integral and derivative gains respectively. In the time
domain the characteristics of step response such as rise time , overshoot , settling time and
steady state error are considered as a performance criterion for PID controller. We use a
function of these indexes as the objective function which is illustrated by Equation (5)[14].
(5)
Where K is a vector whose elements are the PID controller gains. With which the step response
of the system is simulated and the values of Mp, Ess, ts and tr are obtained to evaluate f(k).

45
(6)
And β is a weighting factor whose value effects the four indexes. This value, if set to be smaller
than 0.7, results in reduction of and . Otherwise, if set to values larger than 0.7, end in a
decrease in and while setting β=0.7 will cause the four parameters to effect similarly on
objective function.
On both of the compared algorithms, the vector K is defined as a member of population. The
algorithms start from a random population. During each iteration of the algorithm, the elements of
each member of population are substituted in the controller transfer function and the step
response is calculated for each member of the population, on the basis of which, , , and
are calculated and substituted in the defined objective function. The goal of all the algorithms is
to minimize the objective function through the strategies illustrated above.
5. SIMULATION RESUTLS
5.1. Parameter Evaluation on the Algorithms
On both of the algorithms, all of the controller parameters are restricted to the interval [0, 30].
For both of the algorithms, 30 runs , with 5000 function evaluations included in each run, have
been carried out to have the average answer which is more reliable. To determine the robustness
of the system, all simulations have been run under the effect of a disturbance factor. The
parameter evaluation for each utilized algorithm is as follows.
5.1.1. SFL Algorithm
Population size=50, The number of memeplexes=10, Memetic evolution steps before
shuffling=10, maximum number of function evaluation=5000.
5.1.2 ABC Algorithm
Population size=20, The number of employee bees=0.3 population size, The number of on-looker
bees=0.3 population size, The number of scout bees=0.4 population size, maximum number of
function evaluation=5000.
5.1.2 PSO Algorithm
Population size=30, neighbourhood size=6, w=0.8, c1=c2=2, Maximum number of function
evaluation=5000.
5.2. Parameter Evaluation on FNN

46
FNN was generated by grid partition with 3 membership functions and was trained by hybrid
method [13] .
Number of MFs=3, Input MF type=gbellmf, Output MF type=constant
5.3. The results
Tables 2, 3 and 4 respectively present the simulation results of SFL, ABC and PSO algorithm for
all of the five joints with β=0.7. Table 5 indicates the same values for FNN. The step responses
for each of the five motors are shown on figures 3 to 22. In these pictures, the graph shown with
direct line indicates the response of the undisturbed system while the other two show the
disturbed systems with different disturbances (α=0.1 and α= -0.1).
Applying the kp, ki, kd values obtained from the intelligent algorithms, simulation results show
that FNN has a remarkable effect on decreasing the amount of settling time and rise-time
and eliminating of steady-state error because considering the average value of the 5 joints
according to Tables 2 to 5, it is obviously seen in Table 6 that the average settling-time, the
average rise-time and the average steady-state error is far less in FNN in comparison to ABC and
SFL while the SFL algorithm compared to ABC performs slightly better on steady-state error (
0.0006 vs. 0.0009) and the ABC algorithm is better on decreasing of overshoot . On the other
hand PSO to some extent performs well only on steady-state error. In steady state manner all of
the methods react robustly to the disturbance, but FNN shows more stability in transient response.
Table 2. Simulation Results applying SFL algorithm
Joints PID Parameters Step Response Parameters
Joint 1 9.8832 23.4645 6.0537 .0417 .0000 1.3371 6.7864
Joint 2 2.0628 26.3914 14.1972 .0861 .0000 .5354 4.0012
Joint 3 12.2736 24.8578 3.3813 .0253 .0001 1.7084 7.8000
Joint 4 2.2865 22.9050 14.4540 .1038 .0000 .5998 3.8516
Joint 5 1.8861 21.5737 .6692 .0130 .0029 1.1362 5.3366
Table 3. Simulation Results applying ABC algorithm
Joint 1 3.1838 22.8034 1.2721 .0172 .0012 1.1903 5.2351
Joint 2 5.5132 29.9019 3.0487 .0196 .0001 1.0494 5.0728
Joint 3 .8966 20.3226 1.1469 .0201 .0013 .9203 7.1673
Joint 4 1.4114 27.2506 1.0234 .0118 .0019 .7417 2.7938
Joint 5 11.7983 24.4511 3.9375 .0287 .0000 1.6203 7.7243
Table 4. Simulation Results applying PSO algorithm
Joint 1 7.4025 23.8861 4.4924 .0351 .0000 1.2592 7.2114
Joint 2 3.9689 26.8548 12.0161 .0744 .0000 0.7927 4.9169
Joint 3 10.6848 23.9077 2.9857 .0194 .0001 1.5611 5.8308
Joint 4 1.8521 23.5006 10.1128 .1426 .0000 1.2944 6.8835
Joint 5 1.7355 22.7348 1.9102 .0393 .0008 2.5943 16.9618

47
Table 5. Simulation Results applying FNN algorithm
Joint 1 2.1666 52.5094 8.2394 .0421 .0001 .2822 1.7952
Joint 2 2.1665 52.5157 8.2425 .0422 .0001 .2822 1.7954
Joint 3 2.5001 52.5080 12.7716 .0421 .0000 .2876 2.8411
Joint 4 1.9811 52.5017 15.1483 .0416 .0000 .2665 2.9366
Joint 5 1.3549 38.9996 19.6747 .0839 .0000 .3273 3.3340
Table 6. The Average Values For Comparison
Average settling-time Average rise-time Average steady-state error Average overshoot
FNN 2.54046 sec 0.28916 sec 0.00004 0.05038
ABC 4.59866 sec 1.1044 sec 0.00090 0.01948
SFL 5.55516 sec 1.06328 sec 0.00060 0.05398
PSO 8.36088 sec 1.50034 sec 0.00018 0.06216
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
Fig 3. Step responses with SFL applied on motor 1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1

48
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
timeStepRespance
alfa=0
alfa=0.1
alfa=-0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
Fig 8. Step responses with ABS applied on motor 1

49
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1

50
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1
Fig 13. Step responses with PSO applied on motor 1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa = 0
alfa = 0.1
alfa = - 0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa = 0
alfa = 0.1
alfa = - 0.1
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa = 0
alfa = 0.1
alfa = - 0.1

51
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
time
StepRespance alfa = 0
alfa = 0.1
alfa = - 0.1
0 5 10 15 20 25
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1
Fig 18. Step responses with FNN applied on motor 1
0 5 10 15 20 25
0.6
0.7
0.8
0.9
1
1.1
time
StepRespance
alfa=0
alfa=0.1
alfa=-0.1
0 5 10 15 20 25
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1

52
0 5 10 15 20 25
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1
0 5 10 15 20 25
0.4
0.6
0.8
1
1.2
time
StepRespance
alfa=0
alfa=0.1
alfa= -0.1
5. CONCLUSION, DISCUSSION AND FUTURE WORKS
This research begins by introducing the Grypon robot as case-study. Then four methods were
applied to design an intelligent PID controller (SFL, ABC, PSO and FNN)which, unfortunately,
are not fast enough to be utilized in on-line applications. The results of applied simulations are
reported in the included tables and figures.
Comparing the contents of tables and figures leads to introducing the FNN strong enough in
elimination of the steady state error and reduction of the settling time and rise time. On the other
hand the ABC algorithm acts better on decreasing the overshoot. Also the act of SFL and
PSOalgorithm is good in elimination of the steady state error.
From the robustness point of view, while applying the disturbance factor, the SFL algorithm and
FNN perform stronger in the transient part, however all of the methods detect the main graph in
the steady-state manner.
Designing PID controllers with other intelligent algorithms and changing the present controller to
an on-line one may be our future field ofstudy.

53
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