Design of two-layered fractional order fuzzy logic controllers applied to robotic manipulator with variable payload

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Design of two-layered fractional order fuzzy logic controllers applied to robotic manipulator with variable payload

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2

3 4 5

Q1

Richa Sharma a,∗ , Prerna Gaur b , A.P. Mittal b a b

Department of Electrical & Instrumentation Engineering, Thapar University, Patiala 147004, India Instrumentation and Control Engineering Division, Netaji Subhas Institute of Technology, Dwarka, New Delhi 110078, India

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7 23

a r t i c l e

i n f o

a b s t r a c t

8 9 10 11 12 13

Article history: Received 14 November 2015 Received in revised form 24 April 2016 Accepted 30 May 2016 Available online xxx

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Keywords: Two-link rigid robotic manipulator Fractional order controller Two-layered fuzzy logic controller Cuckoo search algorithm PID controller Trajectory tracking Robustness testing

24

1. Introduction

15 16 17 18 19 20 21

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

The robotic manipulators are highly coupled and nonlinear systems wherein the time-varying parameters and uncertainties adversely affect the characteristics and response of these systems. Hence, these systems require an effective and robust controller to handle such complexities which is a difficult challenge for control engineers. This paper presents two-layered fractional order fuzzy logic controller (TL-FOFLC) scheme for a two-link planar rigid robotic manipulator with payload for trajectory tracking task. For the optimal design, the controller parameters of the proposed scheme are obtained with potential meta-heuristic technique named as cuckoo search algorithm (CSA). In order to ensure effectiveness, the performance of proposed TL-FOFLC is compared with that of its integer order design approach, i.e., twolayered FLC (TL-FLC), single-layered FLC (SL-FLC), and the conventional proportional-integral-derivative (PID) controllers. Further, the robustness testing is carried out for parameter variations and external disturbance rejection. © 2016 Elsevier B.V. All rights reserved.

Recently, the robotic manipulators have widely been used in the industrial, medical and space applications, especially, for the purpose of accurate positioning and path following. For the precise execution of the motion, it is essential to provide an effective control to the end-effector of a robotic manipulator. These systems are highly nonlinear, uncertain and interacting in nature. The operation of robotic manipulators is also affected by various parametric uncertainties and external disturbances. To deal with such uncertainties and complexities, there is a need of development of an effective and robust control scheme for these systems. The introduction of fuzzy logic, in the world of control theory, has remarkably enhanced the applicability to controllers to control the complex and nonlinear plants. The FLC offers new horizons in the field of control engineering due to its several advantages over classical approaches such as involvement of human expertise, model-free and flexible approach etc. Unlike the conventional PID controller, it can effectively deal with system uncertainties and nonlinearities [1–3]. Song et al. presented a hybrid scheme with

∗ Corresponding author. E-mail addresses: richasharma [email protected] (R. Sharma), [email protected] (P. Gaur), [email protected] (A.P. Mittal).

combination of classical computed torque control technique and fuzzy logic for the two-link planar robotic manipulator wherein the fuzzy logic was used as the compensator and worked effectively for uncertainties [4]. Sooraksa and Chen investigated fuzzy logic with PID controller for trajectory tracking as well as vibration suppression for a flexible-link robotic system [5]. Meza et al. presented real-time investigation of fuzzy self-tuning PID controller for a direct drive vertical robotic manipulator. The semiglobal asymptotic stability of the proposed scheme was obtained using Lyapunov theory [6]. Sanchez et al. presented a generalized Type-2 fuzzy control scheme for a mobile robot. It was found that the proposed control scheme outperforms Type-1 and interval Type-2 FLC in the presence of external noises [7]. Q2 In the recent times, several control engineers have been working towards different design structures of FLCs, to develop more efficient control techniques. Kim et al. proposed TL-FLC scheme for the systems having deadzone nonlinearities. The two-layered control scheme, consists of a fuzzy pre-compensator and a traditional FLC, has superior performance as compared to the traditional fuzzy Proportional-Derivative (PD) controller in terms of transient and steady-state response [8]. Pratumsuwan and Thongchai experimentally investigated the effectiveness of TL-FLC scheme for a proportional hydraulic system having the deadzone nonlinearities [9]. From the literature, it seems that the TL-FLC scheme may also

http://dx.doi.org/10.1016/j.asoc.2016.05.043 1568-4946/© 2016 Elsevier B.V. All rights reserved.

Please cite this article in press as: R. Sharma, et al., Design of two-layered fractional order fuzzy logic controllers applied to robotic manipulator with variable payload, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.05.043

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be applied to robotic manipulator systems having inherent nonlinearities and uncertainties. For the last few years, there have been overwhelming participation of fractional calculus in the field of control theory applications, and are applied to different plants such as aerofin control system [10], automatic voltage regulator [11], power system [12] and some fractional order plant [13] etc., in the form of fractional order PID (FOPID) controllers. The significant advantage of fractional order design is that the orders of both the conventional integrator and differentiator terms are indicated by non-integer values rather than integer ones. Efe presented the use of fractional calculus in the design of various control schemes such as FOPID, sliding mode control, backstepping control and adaptive control etc. [14]. Sharma et al. presented a two-degree of freedom FOPID controller for the robotic manipulator applications [15]. Recently, various authors have investigated fuzzy logic with fractional order mathematics for the design of controllers. Das et al. presented fractional order fuzzy controller for the delayed nonlinear systems and the open-loop unstable systems with time delay, and found that this controller is better than the other three controllers namely fuzzy PID, FOPID and traditional PID controllers [16]. A comparative study for the different design structures of the fractional order fuzzy PID (FOFPID) controller for the oscillatory fractional order systems with dead ti""me. The performance of different controllers were obtained for the set-point tracking, control effort and disturbance rejection was also investigated [17]. Sharma et al. presented the FOFPID controller for the two-link planar robotic manipulator for the trajectory tracking task, and found it to be superior to other conventional controllers for the trajectory tracking, disturbance rejection, parameter variations and noise suppression [18]. In [16–18], it was concluded that the performance of FOFPID scheme is superior to conventional FLC approaches. From the literature, it seems that the TL-FOFLC scheme have not been developed for control applications yet. Therefore, this scheme may be explored for the robotic manipulator applications for enhancing the performance and robustness of the controller. For providing an effective control, there is a need of optimal design parameters for the designated controller for any plant. The development of heuristic and nature-inspired optimization methods is a paradigm in the field of artificial intelligence. Several heuristic and evolutionary optimization techniques namely genetic algorithm (GA) [19], particle swam optimization (PSO) [20], differential evolution [21], simulated annealing [22] and bat algorithm [23] etc. have extensively been used in the literature for obtaining the parameters for different plants. Various authors have investigated the optimization of FLC schemes with different optimization techniques. Precup et al. presented a fairly good survey on the optimization of the different parameters of the FLC schemes [24]. Caraveo et al. presented a bee colony optimization method for obtaining the optimal parameters of FLC [25]. In [26], Bingul and Karahan proposed trajectory control of robotic manipulator using FLC and the tuning of FLC was done with PSO. In [27], Mahalakshmi and Sumathi presented fuzzy differential evolution algorithm for the 7-DOF serial link robotic system for trajectory optimization task. Gaxiola et al. presented the optimization of Type-2 fuzzy inference system using GA and PSO. The optimized Type-2 fuzzy weights of the back propagation neural networks [28]. Guerrero et al. presented new CSA algorithm wherein the fuzzy system is used to adapt its parameters. The proposed optimization method was found to be more effective than conventional CSA for a set of mathematical functions [29]. Sanchez et al. presented the information granules formation using the concept of uncertainty based information with the interval Type-2 fuzzy sets. The Takagi-Sugeno-Kang consequents are optimized with CSA [30]. Despite its nascent stage, CSA has been investigated for different applications. Civicioglu and Besdek presented a detailed compara-

tive study among various optimization techniques namely Cuckoo search, PSO, Differential evolution and artificial bee colony, for 50 different benchmark functions [31]. Manikandan et al. presented CSA for the clustering of data [32]. In [33], Patwardhan investigated a CSA based infinite impulse systems identification scheme. Yildiz presented the application of CSA for obtaining the cutting parameters for milling operation and the results obtained were claimed to be better than many optimization techniques such as GA, hybrid PSO, hybrid immune algorithm, feasible direction method, handbook recommendations and ant colony optimization [34]. The CSA can be investigated to mechanical problems such as speed reducer design, spring design, three-bar truss design and welded beam design etc. [35,36]. Bulatovic et al. proposed a novel CSA for finding the optimal parameters of a six-bar double dwell linkage [37]. In the short lifespan, the CSA has become popular among its competitive optimization methods. Now, as CSA is an efficient optimization tool and therefore, may be investigated for the robotic manipulator applications. The main contribution of this paper is to develop a TL-FOFLC approach for a two-link planar rigid robotic manipulator with variable payload for trajectory tracking task. The proposed scheme consists of two layers of FOFLCs: 1. Pre-compensator FOFLC layer 2. A traditional FOFLC layer. The first layer consists of a precompensator FOFLC and it has the ability to counteract the changes in the output response due to unknown parametric variations and disturbances. The second layer of proposed scheme consists of conventional FOFLC and it improves the performance of the controller further. As the operation of robotic manipulator is adversely affected by various parameter uncertainties and external disturbances, therefore the strong motivation behind this study is to design a robust and effective control approach namely TL-FOFLC which can counteract above such complexities that degrade the performance of the robotic manipulator. Other significances of the presented work are as follows: (1) It explores the advantages of the fractional order mathematics in combination with FLC which introduces some extra design parameters. (2) This design approach provides more choice to the control engineers for the selection of controller parameters. There are overall twenty parameters in the proposed control scheme that can be tuned by the user. (3) It is a difficult task to tune such large number of parameters, therefore, CSA is explored as an effective tuning method for obtaining the optimal parameters of proposed TL-FOFLC controller approach. (4) To witness the effectiveness, the performance of the proposed TL-FOFLC is compared with its integer order design TL-FLC scheme, SL-FLC scheme and the conventional PID controllers. (5) The robustness testing is carried out for parameter uncertainties and external disturbances to explore the effectiveness of the proposed TL-FOFLC approach. The paper is organized as follows: Following a detailed literature survey in the first Section, the mathematical model of a two-link planar robotic manipulator with payload is described in Section 2. In Section 3, the design and implementation of TL-FOFLC approach is presented. In Section 4, the implementation of CSA is presented in brief. In Section 5, the simulation results for the trajectory tracking performance, parameter uncertainties and disturbance rejection are presented in detail. Finally, the conclusion of the proposed work is drawn in Section 6.

2. Dynamic model of robotic manipulator The mathematical model of SCARA type two-link planar rigid robotic manipulator with payload, given by Eq. (1), has been described by Lin [38]. The Fig. 1 shows the robotic manipulator having two rigid links and a payload at the tip of the second link.

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Table 1

Q7 Parameters for a two-link planar rigid robotic manipulator. Parameters

Link1

Link2

Mass Acceleration due to gravity (g) Length Distance from the joint of link to its center of gravity Lengthwise centroid inertia of link Friction at joints

0.392924 kg 9.81 m/s2 0.2032 m 0.104648 m 0.0011411 kg m2 0.141231 N-m/radian/s

0.094403 kg 9.81 m/s2 0.1524 m 0.081788 m 0.0020247 kg m2 0.3530776 N-m/radian/s

Fig. 2. Payload variations at the tip. 212







gf2p = m2 lc2 gcos 1 + 2 + mpld lc2 gcos 1 + 2

Fig. 1. Two-link planar rigid robotic manipulator with payload at tip.

Also, Table 1 lists the relevant parameters of robotic manipulator Q3 used for simulation. 196 195



R11

R12

R21

R22





197

198

199

200

¨ 1 ¨ 2



+

V11

 

V21

+

vf 1 vf

  +

2

gf



1p

gf 1p



=

f 1



f 1

 213

where 1 and 2 are the positions; f1 and f2 are the controller outputs or torques; m1 and m2 represent masses; l1 and l2 express the lengths; I1 and I2 are lengthwise centroid inertia; lc1 and lc2 are distances from the joint of Link1 and Link2 to their center of gravity; b11 and b21 represent coefficients of friction at joints; vf1 and vf2 represent the coefficients of dynamic friction of Link1 and Link2 respectively. Also, mpld represents the mass of a payload at the end of the link and its value is varied from 0.56699 kg to 0.14174 kg in the entire 4 s as shown in Fig. 2.

214 215 216 217 218 219 220 221 222

(1) 3. Implementation of two-layered fractional order FLC scheme

223 224

where 2 2 R11 = I1 + I2 + m1 lc1 + m2 l12 + m2 lc2 + 2m2 l1 lc2 cos2 + mpld l12

+mpld l22 + 2mpld l1 l2 cos2

The design and implementation of TL-FOFLC scheme is presented in this Section. The technique used for fractional order design is also described in this Section.

225 226 227

201

202

203

204

R12 =

2 I2 + m2 lc2

 

+ m2 l1 lc2 cos 2

R22 =

2 I2 + m2 lc2





207

vf1 = b11 ˙ 1 vf1 = b21 ˙ 2

209

 







 

gf1p = m1 lc1 gcos 1 + m2 g lc2 cos 1 + 2 + l1 cos 1



211



V11 = −m2 l1 lc2 2˙ 1 + ˙ 2 ˙ 2 sin2 − mpld l1 l2 2˙ 1 + ˙ 2 ˙ 2 sin2 V21 = m2 l1 ˙ 12 lc2 sin2 + mpld l1 ˙ 12 lc2 sin2

210

+ mpld l1 l2 cos 2

+ mpld l22

206

208

 

R21 = R12



205

+ mpld l22





 

+mpld g l2 cos 1 + 2 + l1 cos 1

3.1. Design and implementation of TL-FOFLC approach

228

The design of the presented TL-FOFLC approach consists of two layers: the first layer is composed of the pre-compensator FOFLC, and the second layer contains a traditional FOFLC. The pre-compensator FOFLC is introduced to modify the controller output for compensating the changes in the output response due to unknown parametric uncertainties and external disturbances. The traditional FOFLC is used to improve the performance of the controller. The basic schematic diagram of the TL-FOFLC approach is shown in Fig. 3 in which there are two layers of FOFLC: first layer is named as pre-compensator FOFLC and the second layer is of traditional FOFLC. The gain KIi (i = 1, 2 for Link1 and Link2) is called as feed-forward gain and is applied here to eliminate the steady-state error [8]. The dynamics of the proposed TL-FOFLC scheme can be explained as follows: ei (t) = Ri (t) − f (t)

(2)

where ei (t) is the error between the desired trajectory Ri (t) (i = 1,2 for Link1 and Link2) and the actual trajectory f (t), (f = 1,2 for Link1 and Link2).

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Fig. 3. Basic design scheme of TL-FOFLC approach.

248

Now, the output of pre-compensator FOFLC is given as

 249

Ui (t) = FLCi

ei (t) ,

dıi ei (t) dt ıi

dıi ei (t)

where kou is gain, wfzu represents zeros and wfpu represents poles of the filter and these are obtained as follows [26,27]:

 , Ui (t − 1)

(3)

250

where the term

251

The termUi (t)is the output of the pre-compensator FLCi which has

252

the three inputs, i.e., ei (t) ,

253 254

255

256 257

258

dt

ıi

is the fractional order change in the error. dıi ei (t) dt ıi

and Ui (t − 1).

The output of the first layer (precompensator layer) Prei (t) is called the compensating term and is given below: Prei (t) = Ui (t) + Ri (t)

(4)

Now, the new error ej (t), the difference between Prei (t) and f (t) ej (t) = Prei (t) − f (t)

(5)

260

The final controller signal or torque f (t), which is the output of second layer, is given as follows

261

f (t) = FLCj

259



262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279

280

ej (t) ,

dıj ej (t) dt

ıj

 + KIi Prei (t)

(6)

The basic block diagram of implementation of TL-FOFLC approach for a two-link rigid robotic manipulator is shown in Fig. 4 wherein there are four FLCi (i = 1,2,3,4) are used, two for each link. For the first layer, the constants KP1 , KD1 , KU1 and ˛1 are the scaling gains for FLC1 and the constants KP2 , KD2 and KU2 are the scaling gains for FLC2. Similarly for the second layer, the constants KP3 , KD3 , KU3 and ˛2 are the scaling gains for FLC3 and the constants KP4 , KD4 and KU4 are the scaling gains for FLC4. There are overall twenty parameters in the proposed control scheme that can be tuned for generating effective control action. The fractional order mathematics is incorporated in the present scheme to enhance the design parameters for selecting the more appropriate controller parameters. In the present work, the fractional order differentiator is implemented with the Oustaloup’s approximation. It is based on the recursive distribution of zeroes and poles [39]. The approximating transfer function obtained with this method is equivalent to fractional operator sı where ı is the fraction power of s.

sı = kou

Nu  s + wfzu kou =−Nu

s + wfpu

(7)

wfpu = wbu (

whu ) wb u

 wfzu = wbu

whu wbu

kou +Nu + 1 + ı 2 2 2Nu +1



u

(8)

283

(9)

284

(10)

285

2Nu + 1 represents Thus, ı is the order of fractional differentiator;

the order of approximation; whu , wbu represents the frequency range [40]. The Oustaloup’s approximation is based on the value of Nu as there is always a conflict between the value of Nu and its performance [40,41]. The present method is preferred over others due to the possibilities of implementing it in real time in the form of higher-order infinite impulse response type digital or analog filters [40]. 3.2. Fuzzy logic controller In the TL-FOFLC approach, two layers of fuzzy inference mechanism is used. The FLCs of both layers consist of the four basic blocks namely fuzzification, rule base, inference engine and defuzzification [18]. For the implementation of any FLC, the pre-processing on the input and output variables is essential which means the inputs are multiplied by normalization factor to get it into normalized range, say [−1,1] and the output is multiplied with denormalization factor to get back it into actual range. The basic design of a general FLC is shown in Fig. 5 and is explained as follows: 1. Fuzzification: The conversion of crisp values into the linguistic or fuzzy values is called as fuzzification. In the present work, two FLCs are used for each layer for each link of the robotic manipulator. For the pre-compensator FLCs, there are inputs variables namely ei (t) ,

dt ıi

282

kou +Nu + 1 − ı 2 2 2Nu +1

kou = whı

dıi ei (t)

281

and Ui (t − 1) and a single output variable namely Ui (t).

The input variables are characterized by three membership functions (MFs) namely Negative (N), Zero (Z) and Positive (P) whereas the output variable is characterized by seven MFs namely Negative Large (NL), Negative Medium (NM), Negative Small (NS), Zero (ZO), Positive Small (PS), Positive Medium (PM) and Positive Large (PL). The 50% overlapping was used between all the input and output MFs. In the present work, Gaussian MFs are preferred for all inputs and output variables due to their significant advantages such as non-zero at all points, smooth functions and gives the actual information at every point [42,43]. The range of MFs for both inputs and output are [−1,1] as shown in Fig. 6(a).

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Fig. 4. Block diagram of implementation of TL-FOFLC approach for a two-link robotic manipulator.

Input Variable

Membership Functions Normalization

Scaling Factors

Fuzzification Fuzzifier

Membership Functions

Inference Method Fuzzy Inferencing Rule Base

Output Variable

Defuzzification

Denormalization

Defuzzifier

Scaling Factors

Fig. 5. Basic block diagram of a general FLC.

(a)

(b)

Fig. 6. Membership functions for (a) input variables for FLC of First Layer (b) Input/output variables for FLC of second layer or output variables of first layer.

320 321 322 323 324 325 326 327 328 329 330

For the second layer with traditional FOFLC, there are two input variables namely ej (t) and

ı d j ej (t) ı dt j

Uj (t) whereas there is single out-

put variable The input and output variables as shown in Fig. 6 are characterized by seven MFs namely Negative Large (NL), Negative Medium (NM), Negative Small (NS), Zero (ZO), Positive Small (PS), Positive Medium (PM) and Positive Large (PL). 2. Rule Base: The rule base is the core part for the design of a FLC and it is designed on the basis of expert’s knowledge and the plant dynamics. The rule base for the FLCs of the pre-compensator or first layer is presented in Table 2 wherein there are three input variables

and a single output variable having a total number of rules 3 × 3 × 3 or 27 [8]. The rules can be formulated with IF-THEN statements and a typical example is given below [8]: IF the ei (t) is N and

dıi ei (t) dt ıi

is N and Ui (t − 1) is N THEN the Ui (t)

is NS. For the second layer of FOFLC, the rule bases are formulated between two input variables ej (t) and

ı d j ej (t) ı dt j

with seven MFs and

the numbers of rules obtained are 49 as listed in Table 3. 3. Fuzzy inference and defuzzification: The fuzzy inference system provides the suitable control action on the basis of contribution of each rule. In this work, Mamdani Min-Max operators are used for

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6 Table 2 Rule base for the pre-compensating FLC of first layer.

4.2. Basis of the algorithm

IF

THEN

Error ei (t)

Fractional order change in error

N

N

ıi

d ei (t) ı dt i

Z

P

Z

N

Z

P

P

N

Z

P

Ui (t − 1)

Ui (t)

N Z P N Z P N Z P

NS ZE ZE PS ZE NS PM PS ZE

N Z P N Z P N Z P

ZO NS NS ZO ZO ZO PS PS ZO

N Z P N Z P N Z P

PM PS ZO PM PS ZO PL PS ZO

Table 3 Rule base for second layer. Error

Fractional order change of error

342 343 344 345

346

347

348 349 350 351 352 353 354 355 356 357 358

NL NM NS ZR PS PM PL

NL NL NL NL NL NM NS ZR

NM NL NL NL NM NS ZR PS

NS NL NL NM NS ZR PS PM

ZR NL NM NS ZR PS PM PL

PS NM NS ZR PS PM PL PL

PM NS ZR PS PM PL PL PL

PL ZR PS PM PL PL PL PL

inference engine. The output of FLC is converted into a single crisp value and this conversion from fuzzy to a crisp value is known as defuzzification. In the present work, the centre of gravity method is utilized to defuzzify the fuzzy data.

4. Cuckoo search algorithm 4.1. Properties of CSA CSA is a novel metaheuristic optimization technique based on the parasitic breeding behavior of cuckoos birds It was developed by Xin-She Yang and Suash Deb in 2009 [44]. CSA has some unique features which makes it a potential optimization technique among its competitive technique. The parameters used for initialization of CSA are lesser as compared to PSO and GA. It uses the large steps that make it more efficient than the other potential methods. The convergence rate of CSA is independent of the initialized parameters [44,35]. Tan et al. proposed that CSA is superior to both trademark techniques GA and PSO in finding the optimum solutions for the size and location of the distributed generation [45].

359

It is based on the cuckoo’s strategy of finding the other bird’s nest. The Cuckoos search for an optimal nest in which host bird has just laid its eggs [37]. The cuckoo birds have some unusual abilities such as mimic the call of host bird’s chick; imitate the pattern and color of eggs of the other host birds; cuckoo’s chicks throw the eggs of the host out from nest etc. Some host birds have the ability to find the invading eggs and throw them out or they vacate their own nest [36,46]. This algorithm is based on the searching for the best nest having optimal solutions. There are significant three rules of CSA which are the building blocks of this algorithm and are explained below [36,44]: 1. Each cuckoo bird lays only single egg and keeps it in an arbitrary nest 2. The best nests keep the optimal solution and forward it to the next step or count 3. The host nests present are limited and some host have ability to find invading eggs with a probability ‘Pcs’ between [0,1] range. For searching the new nest, the Lévy flight law is used and is given as follows [44]: xcs (tcs + 1) = xcs (tcs ) + cs ⊕ Lévy(cs )

(11)

wherecs (cs > 0) is the step size of Lévy. The Lévy flight is based on the random walk having Lévy distribution with an infinite mean and variance [44]. Eq. (11) presents a random walk which is based on the Markov chain with the next step depends on the current position and the probability of the transition. The significant steps used in designing the CSA are given in the flowchart as shown in Fig. 7 [36,44]. The constraints used for CSA for optimization are number of nests, abandon probability and iterations and their values are 25, 0.25 and 100 respectively. 4.3. Stepwise implementation for finding optimal parameters

f1 =

|ei (t)|dt

361 362 363 364 365 366 367 368 369 370

371 372 373 374 375 376

377 378

379

380 381 382 383 384 385 386 387 388

389

For obtaining the effective response, it is essential to find the optimal values of the controller parameters. There are overall twenty tunable parameters present in the proposed scheme which are unmanageable for such complex and coupled system. Therefore CSA is an effective alternate for finding the optimal values of these parameters. The objective functions chosen, for the CSA are Integral of Absolute Error (IAE) and Integral of Absolute Change in Control Output (IACCO) of both the links and are represented by (12) and (13) respectively. The combined objective function OF is the weighted algebraic sum of IAE and IACCO of both the links of the robotic manipulator. These objective functions are chosen to minimize the error between desired and actual positions and also, to provide the effective controller output.



360

390 391 392 393 394 395 396 397 398 399 400 401 402 403

(12)

404

(13)

405

(14)

406

i=1,2

f2 =



|f (t)|dt

f =1,2

OF = w1 f1 + w2 f2

where w1 and w2 are the weights assigned to fitness functions f1 and f2 respectively. The parameters KP1 , KD1 , KU1 , ˛1 , KP2 , KD2 , KU2 , KI1 , ı1 and ı2 for Link1 of TL-FOFLC scheme are required to be tuned. Similarly, the parameters KP3 , KD3 , KU3 , ˛2 , KP4 , KD4 , KU4 , KI2 , ı3 and ı4 for Link2 of

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Table 4 IAE for TL-FOFLC and TL-FLC for various controller parameters. Parameters

KP1 KD1 KU1 ˛1 KP2 KD2 KU2 KI1 ı1 ı2 IAE

Link1

Parameters

TL-FOFLC

TL-FLC

300.0000 0.0010 162.0096 0.0055 119.2088 0.3319 163.4644 0.0083 0.1434 0.0077 0.00006989

550.0090 0.0656 4.5823 0.0029 174.8816 0.0039 145.3750 0.0087 – – 0.0005568

KP3 KD3 KU3 ˛2 KP4 KD4 KU4 KI2 ı3 ı4 IAE

Link2 TL-FOFLC

TL-FLC

500.0010 0.2580 175.4845 0.0048 0.0784 0.8319 442.7679 0.0052 0.1493 0.1031 0.0006394

34.2583 0.0010 0.0100 3.7728 273.5833 0.3318 1.9100 0.0092 – – 0.004511

Step 4: Compare the obtained fitness values fa and fb , and if fa > fb , then the host bird nest u is replaced by the new found nest s. Step 5: A fraction ‘Psa’ of the invaders are gave up and new host nests zo are generated at new locations with the Lévy flight. Step 6: Obtain the fitness of all the new formed host nests. Step 7: Update the best nest for the on-going generation. Step 8: The best nest zffb found in the present generation is replaced by the best nest of the generation zff q , if the fitness value fzff of the best nest zffb is more than the calculated fitness value fzffq

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b

of the best nest zff q at that time. Step 9: Repeat steps 2–8 till the stopping criteria is met. The best nest found at the final count gives the optimal solution for this problem. 5. Simulation results

2

Ri (k) (tst ) = d0 + d1 (tst ) + d2 (tst ) + d3 (tst )

3

(15)

the constraints are

413 414 415 416 417 418 419 420 421 422 423

TL-FOFLC scheme need to be tuned. To get an idea of implementation of the proposed algorithm, the following steps are explained and are also presented in Fig. 7. Step1: Set an objective function as per Eq. (14). Arbitrarily initialize a population of A host nests zff i (wherei = 1, 2, . . .. . ..., 25). Set the stopping criteria as maximum number of generations ut = 100 and also fix Pcs = 0.25. Step 2: Randomly obtain a cuckoo s using the Lévy flight as per Eq. (11), and find its fitness fa in concern of minimization of the objective function chosen. fb Step 3: Arbitrarily find a nest u from the entire population A and obtain its fitness as

435 436

438 439 440 441 442 443 444 445 446 447 448

˙ Ri (k) (tst ) = d1 + 2d2 (tst ) + 3d3 (tst )2

(16)

449

¨ Ri (k) (tst ) = 2d2 + 6d3 (tst )

(17)

450

where Ri (k) is the desired position; i = 1, 2 for Link1 and Link2 respectively; R1 = 2radian and R2 = 3radian for tst = 2s; R1 = 0.5radian and R = 5radian for tst = 4s; ˙ R = 0radian/s for both 2

412

434

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In this Section, the results for trajectory tracking, parameter uncertainties and disturbance rejection for TL-FOFLC, TL-FLC, SLFLC and traditional PID controller schemes are presented. The simulations are obtained with Simulink/MATLAB version R2009b platform and has the sampling time 1 ms. The Oustaloup’s approxis used with Nou = 3 and range of frequency was wh =

imation 10−3 , 100 rad/s for the fractional order operator design. In this work, the desired trajectory chosen is a cubic polynomial and is given as follows [47]:

Fig. 7. Flowchart for the CSA implementation.

433

i (k)

tst = 2s and tst = 4s. The TL-FOFLC approach for a two-link planar rigid robotic manipulator is implemented as shown in Fig. 4. The TL-FLC approach is the integer order design of the proposed scheme and the fractional order operator ıi in the differentiator term for both the layers, is kept unity. The SL-FLC approach is implemented with the conventional FLCs in the proposed design. For demonstrating the effectiveness of proposed schemes, the performances of TL-FOFLC and TL-FLC schemes are also compared with SL-FLC scheme and the conventional PID controller. The IAE values for PID controllers for Link1 and Link2 are 0.009616 and 0.02769 respectively. The IAE values for SL-FLC approach are 0.0008512 and 0.005921. The controller parameters and IAE values for the TL-FOFLC and TL-FLC approaches are listed in Table 4. The IAE values for Link1 and Link2 are 0.00006989 and

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(b)

(a)

(d)

(c)

Fig. 8. Fitness value versus iterations graphs for (a) TL-FOFLC (b) TL-FLC (c) SL-FLC (d) PID controller schemes.

(a)

(b)

(d)

(c)

(e) Fig. 9. (a) Trajectory tracking (b) Controller outputs (c) Path tracked by end-effector (d) Position errors for SL-FLC, TL-FOFLC and TL-FLC approach for Link1 and Link2 (e) Positions errors for PID controlle.

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0.0006394 for TL-FOFLC approach for trajectory tracking respectively whereas the IAE values for TL-FLC for Link1 and Link2 are 0.0005568 and 0.004511 respectively. The fitness value versus iteration graphs for TL-FOFLC, TL-FLC, SL-FLC and conventional PID controller approaches are shown in Fig. 8. The graphs for trajectory tracking, controller output, path tracked by end-effector and posi-

tion errors for TL-FOFLC, TL-FLC, SL-FLC and PID controller schemes are presented in Fig. 9. For analysis of the performance of the proposed approach, the comparisons of the IAE values of Link1 and Link2 for four potential control approaches namely TL-FOFLC, TL-FLC, SL-FLC and PID controllers are shown in Fig. 10. From the graphs in Fig. 10, it

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Fig. 10. Comparison of IAE values for SL-FLC, TL-FOFLC and TL-FLC approach for Link1 and Link2.

Table 5 IAE values for Link1 and Link2 for ± 5% change in parameter values. Parameter variation (5%)

Decrease TL-FOFLC

TL-FLC

SL-FLC

PID

Link1

Link2

Link1

Link2

Link1

Link2

Link1

Link2

Parameter 1: m1 Parameter 2: m2 Parameter 3:m1 , m2 Parameter 4: b11 Parameter 5: b21 Parameter 6:b11 , b21

0.00006935 0.00006941 0.00006893 0.00006998 0.00006989 0.00006998

0.0006397 0.0006382 0.0006383 0.0006394 0.0006125 0.0006125

0.0005502 0.0005529 0.0005463 0.0005575 0.0005568 0.0005575

0.004511 0.004502 0.004502 0.004511 0.004315 0.004315

0.0008412 0.0008452 0.0008352 0.0008518 0.0008512 0.0008518

0.005921 0.005908 0.005908 0.005921 0.005663 0.005663

0.009509 0.009554 0.009446 0.009610 0.009615 0.009608

0.02769 0.02765 0.02765 0.02769 0.02649 0.02649

Parameter variation (5%)

Increase TL-FOFLC Link1

Link2

TL-FLC Link1

Link2

SL-FLC Link1

Link2

PID Link1

Link2

0.00007072 0.00007038 0.00007121 0.00006981 0.00006989 0.00006981

0.0006394 0.0006406 0.0006406 0.0006394 0.0006663 0.0006663

0.0005634 0.0005607 0.0005673 0.0005561 0.0005568 0.0005562

0.004512 0.004521 0.004521 0.004511 0.004508 0.004508

0.0008612 0.0008572 0.0008672 0.0008506 0.0008512 0.0008507

0.005921 0.005933 0.005933 0.005921 0.006179 0.006179

0.009724 0.009679 0.009787 0.009624 0.009618 0.009625

0.02769 0.02774 0.02774 0.02769 0.02889 0.02889

Parameter 1: m1 Parameter 2: m2 Parameter 3:m1 , m2 Parameter 4: b11 Parameter 5: b21 Parameter 6: b11 , b21

Fig. 11. Variation in IAE for 5% decrease in parameters for (a) Link1 (b) Link2.

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can be clearly visible that the TL-FOFLC approach has smallest IAE values for Link1 and Link2. Therefore, it can be concluded that the performance of proposed TL-FOFLC approach outperforms the other three controllers namely TL-FLC, SL-FLC and PID controllers.

5.1. Robustness testing: parameter uncertainties In this Section, the effectiveness of proposed controllers is investigated under parameter uncertainties incorporated to the robotic manipulator. The parameter variations include ± 5% change

Please cite this article in press as: R. Sharma, et al., Design of two-layered fractional order fuzzy logic controllers applied to robotic manipulator with variable payload, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.05.043

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Fig. 12. Variation in IAE for 5% increase in parameters for (a) Link1 (b) Link2.

(b)

(a)

(c)

(d)

(e) Fig. 13. (a) Trajectory tracking performance (b) Control output (c) Path tracked by end-effector (d) Position errors for TL-FOFLC, TL-FLC, SL-FLC and PID controller schemes(e) Position errors for PID controller, for adding disturbance 1.0 sin25 t N-m in both links.

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in parameters namely mass and coefficient of friction for Link1 as well as Link2, from their nominal values. The IAE values for TLFOFLC, TL-FLC, SL-FLC and PID controller schemes for 5% increase as well as decrease in parameters are listed in Table 5. The IAE variation for all the four controller schemes namely TL-FOFLC, TL-FLC, SL-FLC and PID controllers are shown in Figs. 11 and 12 for both the links under 5% decrease and increase in the parameter values respectively. From Figs. 11 and 12, it is inferred that IAE values of Link1 and Link2 for TL-FOFLC approach remain lesser as compared to TLFLC scheme, SL-FLC approach and traditional PID controllers for the model uncertainties. Therefore, the TL-FOFLC approach outperforms its integer order design, i.e., TL-FLC scheme, SL-FLC approach,

and traditional PID controllers in presence of parameter uncertainties. 5.2. Robustness testing: disturbance rejection This Section is presented to test the effectiveness of the proposed scheme for incorporation of the disturbances to the controller output. The sinusoidal disturbances 1.0 sin25 t N-m are added to the controller output for the entire time period in each link of the robotic manipulator. For the brevity, the graphs for trajectory tracking performance, controller output, path tracked by the endeffector and position errors for TL-FOFLC, TL-FLC, SL-FLC and PID controller approaches for addition of 1.0 sin25 t N-m disturbance

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Table 6 IAE variation of TL-FOFLC, TL-FLC, SL-FLC and PID controllers for disturbances. Disturbances (1.0sin25 t N-m)

Link1 Link2 Both Links

TL-FOFLC

TL-FLC

SL-FLC

PID

Link1

Link2

Link1

Link2

Link1

Link2

Link1

Link2

0.00009404 0.00006989 0.00009406

0.0006394 0.0009007 0.0009007

0.0007482 0.0005569 0.0007502

0.004511 0.008114 0.008119

0.0010760 0.0008513 0.0010780

0.005921 0.008704 0.008710

0.009804 0.009621 0.009822

0.02770 0.04930 0.04933

Fig. 14. Variation in IAE for (a) Link1 (b) Link2 for adding disturbance 1.0 sin25 t N-m in Link1, Link2 and both links.

530

to both the links are shown in Fig. 13. The IAE values for adding the disturbances to controller outputs for TL-FOFLC, TL-FLC, SL-FLC and PID controller schemes are presented in Table 6. The IAE variations for adding disturbances to all the four controllers namely TL-FOFLC, TL-FLC, SL-FLC and PID controllers are presented in Fig. 14. From Fig. 14, it is inferred that the IAE values remain smaller for TL-FOFLC approach as compared to the TL-FLC, SL-FLC and PID controller schemes for Link1 and Link2 for all cases of adding disturbances. Thus, the performance of proposed TL-FOFLC scheme is superior to other three mentioned controllers namely TL-FLC, SL-FLC and conventional PID controller schemes in presence of external disturbances. From the overall results above, it can be clearly investigated that TL-FOFLC approach is more effective and robust than its integer order design TL-FLC scheme, SL-FLC approach and conventional PID controller for trajectory tracking, parameter uncertainties and external disturbance rejection.

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6. Conclusions

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532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552

In this work, the TL-FOFLC approach is implemented for a twolink rigid planar robotic manipulator having payload for trajectory tracking problem. The design consists of a pre-compensator FOFLC followed by a traditional FOFLC. The pre-compensator FLC layer is introduced to compensate the changes in the controller signal due to changes in the model parameters and non-linearities involved during the operation of robotic manipulator. The incorporation of fractional order operators has enhanced the flexibility in choosing the controller parameters to the control engineers. A detailed comparative study of the TL-FOFLC scheme with its integer order design TL-FLC, SL-FLC approach and traditional PID controller for the trajectory tracking, parameter uncertainties and disturbance rejection has been investigated to witness the effectiveness and robustness of the proposed schemes. From the results, it can be clearly indicated that the TL-FOFLC scheme is more robust and effective than its integer order design TL-FLC scheme, SL-FLC approach and conventional PID controller for the trajectory tracking. The beauty of the proposed controller scheme is to effectively counteract the effects of model uncertainties and external disturbances. Therefore, the proposed controller scheme may also be used in process industries for the task such as welding and paint-

ing which require precise positioning of the end-effector of robotic manipulators and the environment where large parameter variations occur due to external disturbances such as nuclear plants. This work also proves the applicability of CSA to tune the controller parameters of highly nonlinear and coupled plants. The future scope of the proposed controller includes the hardware implementation of the proposed scheme to explore its practical applicability. The proposed scheme may also be beneficial for the flexible manipulator systems as these systems are more sensitive towards uncertainties and disturbances.

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