Performance analysis of two-degree of freedom fractional order PID controllers for robotic manipulator with payload.

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Performance analysis of two-degree of freedom fractional order PID controllers for robotic manipulator with payload Richa Sharma n, Prerna Gaur, A.P. Mittal Instrumentation and Control Engineering Division, Netaji Subhas Institute of Technology, Dwarka, New Delhi 110078, India

art ic l e i nf o

a b s t r a c t

Article history: Received 9 September 2014 Received in revised form 9 March 2015 Accepted 30 March 2015 This paper was recommended for publication by A.B. Rad.

The robotic manipulators are multi-input multi-output (MIMO), coupled and highly nonlinear systems. The presence of external disturbances and time-varying parameters adversely affects the performance of these systems. Therefore, the controller designed for these systems should effectively deal with such complexities, and it is an intriguing task for control engineers. This paper presents two-degree of freedom fractional order proportional-integral-derivative (2-DOF FOPID) controller scheme for a twolink planar rigid robotic manipulator with payload for trajectory tracking task. The tuning of all controller parameters is done using cuckoo search algorithm (CSA). The performance of proposed 2-DOF FOPID controllers is compared with those of their integer order designs, i.e., 2-DOF PID controllers, and with the traditional PID controllers. In order to show effectiveness of proposed scheme, the robustness testing is carried out for model uncertainties, payload variations with time, external disturbance and random noise. Numerical simulation results indicate that the 2-DOF FOPID controllers are superior to their integer order counterparts and the traditional PID controllers. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Two-link rigid robotic manipulator Fractional order PID controller Two-degree of freedom controller Cuckoo search algorithm Trajectory tracking Payload variation with time Robustness testing

1. Introduction For the past two decades, the robotic manipulators have become an integral part of various areas such as process industries, nuclear plants, medical fields, space applications etc. The accurate positioning is the significant advantage of these systems. The robotic manipulators are complex nonlinear and coupled systems. Therefore the control of end-effectors of robotic manipulator to execute the precise positioning is a challenging task. However, there have been substantial progressions in the field of control for these systems. The traditional PID controller is still the prior choice of control engineers due to its invincible features such as less cost, simple design, easy implementation etc. The advancement in the field of fractional calculus is a paradigm in the control theory applications. In fractional order controller design, the orders of traditional integrator and differentiator terms are indicated by non-integer values rather than integers. The FOPID controllers are the introductory controllers designed using incorporation of fractional mathematics with traditional PID controllers. Various authors presented FOPID

n

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

controllers for the robotic manipulator applications [1,2]. Silva et al. presented the FOPD controller applied to hexapod robot having joints at the legs which includes flexibility and viscous friction. The experimental investigations claimed that FOPD controller is better than conventional PD controller [1]. Bingul and Karahan presented FOPID controllers for the robotic manipulator using particle swarm optimization (PSO) and genetic algorithm (GA). The robustness testing was done for mass change, noise rejection and for different trajectories. The PSO-tuned FOPID controllers were superior to GA-tuned FOPID controllers for trajectory tracking as well as robustness testing [2]. Several other authors investigated the implementation of FOPID controller for various plants such as automatic voltage regulator [3,4], power system [5,6], active magnetic bearing system [7], aerofin control system [8], fractional order plant [9] etc. The FOPID controller was claimed to be more effective than the conventional PID controller for all these plants. From the literature above, it is clear that FOPID controllers provide superior responses to the conventional PID controllers for different plants. Moreover, the FOPID controllers have some extra flexibility for choosing the controller parameters. Both the conventional PID as well as FOPID controllers are single-DOF controllers which means that these controllers have only one closed-loop and cannot control both trajectory tracking and disturbance rejection simultaneously. For the past two decades, the control engineers have been motivated towards the

http://dx.doi.org/10.1016/j.isatra.2015.03.013 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Sharma R, et al. Performance analysis of two-degree of freedom fractional order PID controllers for robotic manipulator with payload. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.013i

R. Sharma et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

such as speed reducer design, spring design, gear train design, welded beam design, a three-bar truss design etc. [32,35]. Bulatovic et al. proposed a new CSA for finding the parameters of a six-bar double dwell linkage [36]. From the literature above, it is evident that CSA is a fairly effective optimization technique and can be investigated for the robotic manipulator applications. In this paper, the 2-DOF FOPID controller scheme is implemented for a two-link planar rigid robotic manipulator with payload for trajectory tracking task. The 2-DOF controller design provides effective control to both the trajectory tracking and external disturbances simultaneously. The motivation behind this study is to explore the advantages of fractional order mathematics in combination with 2-DOF PID controller which introduces some extra design parameters to the control engineers for enhancing the performance of the system. Therefore, the significant contribution of this work is to explore the application of 2-DOF FOPID controllers for two-link rigid robotic manipulator for trajectory tracking control. The increased flexibility in the design of controller parameters leads to difficulty in tuning these parameters. The tuning of parameters of the proposed controller becomes difficult as the numbers of parameters are enhanced. The problem under consideration is a two-link planar robotic manipulator which requires one controller for each link. Therefore, the number of variables that need to be tuned are 16, i.e. eight variables for each link. The other contribution of this work is to effectively use the CSA for tuning of all the 16 parameters of the proposed controllers. The performance of proposed 2-DOF FOPID controllers is compared with those of their integer order designs as well as traditional PID controllers. The robustness testing is carried out for model uncertainties, payload variations, external disturbances and random noise to demonstrate the effectiveness of the proposed 2-DOF FOPID controller scheme.

2. Mathematical model of robotic manipulator with payload The mathematical model of two-link planar rigid robotic manipulator with payload as shown in Fig. 1 has been described by Lee and Lee [37]. Eq. (1) shows the relationship between torque

Y

l2

mp

2

m2, I2

l1

θ2

lc

use of 2-DOF controller design. The controller with 2-DOF has advantages over single-DOF controller in terms of trajectory tracking as well as regulation in the presence of external disturbances [10]. Various authors have investigated 2-DOF PID controllers for different plants and applications. Sahu et al. presented 2-DOF PID controller for load frequency control of interconnected system with governor dead-band nonlinearity using a differential evolution (DE) technique. The simulation results were presented for two-area thermal system with nonlinearity. A new performance index chosen was the weighted combination of error, damping ratio of dominant eigen values, peak overshoot and settling times of frequency. The proposed controller was claimed to be superior to craziness based PSO tuned 2-DOF PID controller. The proposed controller was also claimed to be robust to parameter variations and varying operating loads [11]. Araki and Taguchi presented a detailed survey about the 2-DOF controller structure, its relation to PID and I-PD controllers and also investigated a tuning method for these controllers [12]. Ghosh et al. presented 2-DOF PID controller for magnetic levitation systems and the experimental results for the proposed controller were claimed to be robust for selected feed-forward gains [13]. Alfaro and Vilanova investigated a model reference optimization based 2DOF PI controller for the plants with first and second order plus dead time [14]. Vilanova et al. also proposed simple tuning rules for the 2-DOF PI controller including the robustness considerations [15]. The use of fractional order mathematics with 2-DOF PID controller enhances the flexibility in choice of controller parameters. Feng and Xiao-Ping presented the 2-DOF FOPID controller for pitch control of unmanned air vehicle. The controller was designed with a given value filter and FOPID controller. The Nelder-Mead's simplex method and PSO were used for the design of FOPID and given value filter respectively [16]. Debbarma et al. investigated the 2-DOF FOPID controller for automatic generation control of power systems. The simulation analysis was done for three unequal area thermal systems with reheat turbines and appropriate generation rate constraints. The controller parameters and speed regulation parameters of governor were optimized with firefly algorithm. The 2-DOF FOPID controller was claimed to be superior to the other controllers namely Integral controller, PI and PID controllers. The proposed 2-DOF FOPID controller was also robust to varying loading conditions and variations in coefficient of inertia [10]. From the literature survey conducted, it seems that the 2-DOF FOPID controller can be explored for robotic manipulator applications. The optimal parameters are the prime and foremost requirements for designing an effective control approach. With the advancement in the computational algorithms, the tuning of such parameters has become possible with ease. Several heuristic and evolutionary optimization techniques namely GA [17,18], simulated annealing [19], PSO [20,21], DE [22], tabu search [23], ant colony optimization (ACO) [24,25], artificial bee colony [26], bat algorithm [27], gravitational search algorithm [28], fruit fly optimization [29,30] etc. have been extensively used in the literature. In very short duration, CSA has emerged as a potential optimization technique due to its unique features. The parameters used in CSA are lesser as compared to PSO and GA and its convergence rate does not depend upon these parameters [31,32]. The large steps can make it more efficient than other techniques. It also has the well-renowned elitism property. Tan et al. proposed that CSA outperforms both GA and PSO for finding the optimal solutions for the location as well as size of the distributed generation [33]. Yildiz investigated the use of CSA for finding the cutting parameters for milling operation and it was better than many present optimization techniques such as hybrid immune algorithm, feasible direction method, hybrid PSO, GA, handbook recommendations and ACO [34]. The CSA can be applied to mechanical problems

m1, I1

g

l c1 θ1

τ2

2

τ1

X

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

Table 1 Parameters for a two-link planar rigid robotic manipulator. Parameters

Link1

Link2

Mass (kg) Acceleration due to gravity (g) (m/s2) Length (m) Distance from the joint of link to its center of gravity (m) Lengthwise centroid inertia of link (kg m2) Coefficient of viscous friction Coefficient of dynamic friction

1 9.81 1 0.5 0.2 0.1 0.1

1 9.81 1 0.5 0.2 0.1 0.1



3

Fig. 2. Basic control scheme of 2-DOF FOPID controller for a two-link robotic manipulator.

Fig. 3. Block diagram of 2-DOF FOPID controller for a two-link robotic manipulator.

ðτi Þ and link position ðθi Þ for both the links. Table 1 lists the parameters of two-link planar robotic manipulator used for simulation #" # " " #" # " # g 1p A11 A12 θ_ 1 bθ_ 2 bθ_ 1 bθ_ 2 θ€ 1 þ þ g 2p A21 A22 bθ_ 1 0 θ_ 2 θ€ 2 # " # " # " p1 sgn θ_ 1 τ1 v1 θ_ 1 ¼ þ þ ð1Þ _ τ2 p2 sgn θ_ 2 v2 θ2 where

2 2 2 A11 ¼ I 1 þ I 2 þm1 lc1 þ m2 l1 þ lc2 þ 2l1 lc2 cos ðθ2 Þ 2 2 þ mp l1 þl2 þ 2l1 l2 cos ðθ2 Þ 2 2 A12 ¼ I 2 þ m2 lc2 þ l1 lc2 cos ðθ2 Þ þ mp l2 þ l1 l2 cos ðθ2 Þ A21 ¼ A12 2 2 A22 ¼ I 2 þ m2 lc2 þ mp l2

b ¼ m2 l1 lc2 sin ðθ2 Þ

g 1p ¼ m1 lc1 g cos ðθ1 Þ þ m2 g ðlc2 cos ðθ1 þ θ2 Þ þ l1 cos ðθ1 ÞÞ g 2p ¼ m2 lc2 g cos ðθ1 þ θ2 Þ where θ1 and θ2 are the positions of Link1 and Link2 respectively; τ1 and τ2 are the control outputs or torques for Link1 and Link2; m1 and m2 represent masses of Link1 and Link2 respectively; l1 and l2 express the lengths of Link1 and Link2 respectively; I 1 and I 2 are lengthwise centroid inertia of Link1 and Link2; lc1 and lc2 are distances from the joint of Link1 and Link2 to their center of gravity; v1 and v2 represent coefficients of viscous friction of Link1 and Link2; p1 and p2 represent the coefficients of dynamic friction of Link1 and Link2 respectively. Also, mp represents the mass of a payload at the end of the link and its value is chosen as 0.5 kg.

3. Design structure of two-degree of freedom FOPID controller In this section, the basic design of 2-DOF FOPID controller is presented. The 2-DOF controller consists of two-closed loop



4

In s-domain, the single-DOF-FOPID controller is expressed as 1 U FOPIDi ðsÞ ¼ K P i þ K Ii γ þK Di sδi Ei ðsÞ ð3Þ si The PID controller, in general, provides faster response and improved stability as compared to PI controller. At one end, the derivative term improves the stability of the system. At another end, it makes the system more prone to disturbance and noise. The performance of the system can be maintained in the presence of the noise or disturbance by employing the derivative filter term. The basic control scheme of 2-DOF-FOPID design is shown in Fig. 2 wherein the C 2 DOF FOPIDi ðsÞ is the main closed loop transfer function and F 2 DOF FOPIDi ðsÞ is the filter transfer function and is expressed as [10]: C 2 DOF FOPIDi ðsÞ ¼ K Pi þ

K Ii K Di sδi þ γ s i 1 þ ðK Di sδi =K Pi Ni Þ

F 2 DOF FOPIDi ðsÞ ¼ K P i αi þ

K Ii K Di sδi β þ γ s i 1 þ ðK Di sδi =K P i N i Þ i

ð4Þ

ð5Þ

where N i is the filter coefficient; αi and βi are the weighting factors for the controller gains. Therefore, the overall control output can be expressed as follows [10]: KI K D sδi βi θRi ðsÞ θi ðsÞ U 2 DOF FOPID ðsÞ ¼ K P i αi θRi ðsÞ θi ðsÞ þ γ i ðei ðsÞÞ þ i si 1 þ ðK Di sδi =K P i N i Þ

ð6Þ

where ei ðsÞ ¼ θRi ðsÞ θi ðsÞ; θRi ðsÞ is the desired trajectory and θi ðsÞ is the actual trajectory. The basic block diagram of 2-DOF FOPID controller applied to a two-link planar rigid robotic manipulator for trajectory tracking task is shown in Fig. 3.

4. Implementation of fractional order operators In this section, the method used for implementing the fractional order integrators and differentiators is presented. For the implementation of fractional order operators, various authors have presented distinguished definitions such as Grunwald–Letnikov (G–L), Riemann and Liouville (R–L), Oustaloup's approximation, Caputo definition, Mittag–Leffler (M–L) etc. The basic fractional order differentiator and integrator can be expressed as follows [38]: 8 λ0 d > Rðλ0 Þ 4 0 > < dt λ0 λ0 1 Rðλ ð7Þ 0Þ ¼ 0 h0 D t ¼ R > > : t ðdτÞ λ0 Rðλ0 Þ o 0 h0

Fig. 4. Flowchart for the CSA implementation.

Table 2 Constraints used for CSA for simulation. Parameters

Value

Number of nests (Population) Abandon probability (Pcsa) Generations

25 0.25 100

transfer functions and provides effective control to both trajectory tracking and disturbance rejection simultaneously. The 2-DOF controller provides the control actions based on the weighted difference between the desired signal and actual output according to the chosen gain parameters [10]. In time domain the single-DOF-FOPID controller can be expressed as follows: U FOPIDi ðt Þ ¼ K P i ei ðtÞ þ K Ii

d

γi

ei ðtÞ

dt γ i

δ

þ K Di

d i ei ðtÞ dt δi

ð2Þ

where i¼1, 2 represent Link1 and Link2; K Pi is proportional gain; K Ii is integral gain; K Di is derivative gain; δi is the fractional derivative value; γ i is the fractional integral value; U FOPIDi ðt Þ is the output of FOPID controller; ei ðtÞ is the tracking error between desired trajectory and actual output.

where R stands for Real, h0 is concerned to initial conditions and λ0 is the desired fractional order operator that can be a complex number. In the present work, the fractional order is implemented with the Oustaloup's approximation. It is based on the recursive dispersion of zeroes and poles [39]. The approximating transfer function available is equivalent to fractional operator sγ where γ is the fractional power of s sγ ¼ kou

Nou

∏

s þ wf zo

kou ¼ Nou s þwf po

ð8Þ

where kou is gain, wf zo represents zeros and wf po represents poles of the filter and can be calculated as follows [40,41]: wf po ¼ wbou

wf zo ¼ wbou

þ ð1=2Þ þ ðγ=2Þ kou þ Nou 2Nou þ 1 whou wbou þ ð1=2Þ ðγ=2Þ kou þ Nou 2Nou þ 1 whou wbou

ð9Þ

ð10Þ



5

Table 3 IAE for 2-DOF FOPID and 2-DOF PID controllers for various controller parameters. Parameters

Link1

K P1 K I1 K D1 α1 β1 δ1 γ1 N1 IAE

Parameters

2-DOF FOPID

2-DOF PID

3999.5678 3523.8990 0.5575 0.9975 0.5577 0.9899 0.9335 458.9870 0.008787

3590.4879 2367.8956 0.5975 0.9969 0.2576 – – 350.0598 0.01807

K P2 K I2 K D2 α2 β2 δ2 γ2 N2 IAE

0.06 0.05

Fitness value

Fitness value

0.07

1

0.5

2-DOF PID

2251.4188 524.0400 0.9975 0.9999 0.7558 0.9999 0.9421 324.2426 0.01496

1501.4188 519.1522 3.5751 0.9990 0.8598 – – 432.2426 0.04695

0.1 0.095 0.09 0.085

0.04 0.03

2-DOF FOPID

0.105

1.5

0.08

Fitness value

Link2

0

20

40

60

80

0 0

100

20

40

60

80

0.08

100

0

20

40

Iterations

Iterations

60

80

100

Iterations

4

3.098 3.096 3.094 2.266 2.268 1.995 1.99 1.985

2 0

2.085

0

0.5

1

1.5

2 2.5 Time (s)

3

Y-Coordinate (m)

3 0.55

2.09

3.5

-0.77

-0.765

-0.76

1 0 -1

-0.5

0

0.5

1

1.5

2-DOF FOPID Link1 2-DOF FOPID Link2 2-DOF PID Link1 2-DOF PID Link2 PID Link1 PID Link2

5 0 -5 -10

4

Reference path 2-DOF FOPID 2-DOF PID PID

0.56

2

Control output (N-m)

10

Reference trajectory Link1 Reference trajectory Link2 2-DOF FOPID Link1 2-DOF FOPID Link2 2-DOF PID Link1 2-DOF PID Link2 PID Link1 PID Link2

6

X and Y-Coordinates (m)

Position (radian)

Fig. 5. Fitness value versus iterations (generations) obtained with CSA for (a) 2-DOF FOPID, (b) 2-DOF PID and (c) PID controllers.

0

6

1

1.5

2.5

0.84 1.19 1.2 1.21

-0.72

2

3

3.5

4

Reference X-Coordinate Reference Y-Coordinate 2-DOF-FOPID X-Coordinate 2-DOF-FOPID Y-Coordinate 2-DOF-PID X-Coordinate 2-DOF-PID Y-Coordinate PID X-Coordinate PID Y-Coordinate

0.86

4

-0.73 1.38 1.39 1.4

0 0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

X-Coordinate (m) 0.1

Error (radian)

2 Time (s)

0.88

0

2

0.5


0.05 0 -0.05

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s) Fig. 6. (a) Trajectory tracking, (b) control outputs, (c) path traced by end-effector, (d) X and Y versus time variations and (e) position errors for 2-DOF FOPID, 2-DOF PID and PID controllers.



6

Table 4 IAE values for Link1 and Link2 for 5% (a) decrease and (b) increase in parameter values. Parameter variation (5%)

2-DOF FOPID

2-DOF PID

PID

Link1

Link2

Link1

Link2

Link1

Link2

(a) Decrease Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter

1: m1 2: m2 3: m1 ; m2 4: l1 5: l2 6: l1 ; l2 7: lc1 8: lc2 9: lc1 ; lc2 10: I 1 11: I 2 12: I1 ; I 2 13: p1 14: p2 15: p1 ; p2

0.008764 0.008593 0.008572 0.008800 0.008790 0.008812 0.008777 0.008541 0.008536 0.008800 0.008794 0.008806 0.008793 0.008787 0.008793

0.01496 0.01501 0.01502 0.01505 0.01489 0.01498 0.01496 0.01499 0.01499 0.01496 0.01494 0.01494 0.01496 0.01497 0.01497

0.01808 0.01789 0.01790 0.01800 0.01808 0.01802 0.01809 0.01785 0.01787 0.01808 0.01808 0.01810 0.01808 0.01807 0.01808

0.04695 0.04728 0.04728 0.04718 0.04698 0.04720 0.04695 0.04726 0.04726 0.04695 0.04694 0.04694 0.04695 0.04698 0.04698

0.04354 0.04185 0.04152 0.04354 0.04377 0.04413 0.04358 0.04156 0.04129 0.04389 0.04378 0.04381 0.04392 0.04385 0.04392

0.06420 0.06176 0.06175 0.06420 0.06304 0.06301 0.06420 0.06151 0.06150 0.06420 0.06400 0.06399 0.06421 0.06417 0.06417

(b) Increase Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter

1: m1 2: m2 3: m1 ; m2 4: l1 5: l2 6: l1 ; l2 7: lc1 8: lc2 9: lc1 ; lc2 10: I 1 11: I 2 12: I1 ; I 2 13: p1 14: p2 15: p1 ; p2

0.008818 0.008984 0.009016 0.008777 0.008787 0.008784 0.008800 0.009038 0.009055 0.008775 0.008781 0.008769 0.008782 0.008788 0.008782

0.01496 0.01492 0.01492 0.01488 0.01505 0.01497 0.01496 0.01496 0.01495 0.01496 0.01498 0.01498 0.01496 0.01495 0.01495

0.01807 0.01825 0.01825 0.01813 0.01806 0.01813 0.01805 0.01829 0.01827 0.01806 0.01806 0.01805 0.01807 0.01807 0.01807

0.04696 0.04665 0.04665 0.04673 0.04695 0.04671 0.04696 0.04667 0.04667 0.04695 0.04697 0.04697 0.04695 0.04693 0.04693

0.04419 0.04587 0.04620 0.04355 0.04398 0.04366 0.04414 0.04616 0.04644 0.04381 0.04393 0.04389 0.04378 0.04385 0.04378

0.06421 0.06668 0.06669 0.06426 0.06547 0.06553 0.06422 0.06695 0.06696 0.06421 0.06442 0.06442 0.06421 0.06425 0.06425

kou ¼ wγhou

ð11Þ

Thus, γ is the order of fractional differentiator or integrator; 2Nou þ1 is the order of approximation; wbou ; whou is the frequency range [40]. This approximation is chosen over other methods due to its possibilities of implementing it in real hardware using higher-order infinite impulse response type digital or analog filters for the non-integer order differential-integrator [40].

1. Every cuckoo lays one egg and drops it in a randomly chosen nest. 2. The best nests have the optimal solution and carry forward it to the next step. 3. The host nests present are restricted in numbers. 4. A host bird have ability to find foreigner eggs with a probability ‘Pcsa’ between [0, 1] range. 5. For obtaining the new nest, the Lévy flight law is utilized and is presented as follows [31]: xcsa ðt 0 þ 1Þ ¼ xcsa ðt 0 Þ þ βcsa Le' vyðλcsa Þ

5. Cuckoo search algorithm In 2009, Xin-She Yang and Suash Deb proposed a new metaheuristic optimization technique namely CSA which is based on the parasitic breeding behavior of cuckoos species [31]. It imitates the cuckoo's strategy of finding the other species nest. The cuckoos hunt for a nest in which other bird has just laid its eggs [36]. The cuckoo birds have some unique abilities such as ability to mimic the call of host's chick; imitate the color and pattern of eggs of other birds; cuckoo's chicks throw the eggs of the host bird out from nest, craftiness to frequent calling for enhancing their share in food etc. Some host birds detect the foreigner eggs and throw these out or they vacate their nest and develop a new nest [31,36,42]. This algorithm is based on hunting for the best nest with optimal solutions. The significant rules, as explained below, are the cornerstones of CSA algorithm [35]:

ð12Þ

where βcsa (βcsa 4 0) is step size of Lévy. The Lévy flight is based on random walk with Lévy distribution with an infinite variance and means [35]. Eq. (12) expresses a random walk based on Markov chain wherein the next step depends on the present location and the transition probability. The general flowchart for implementation of CSA is shown in Fig. 4 [31,35]. For the presented work, the constraints used for CSA for optimization are presented in Table 2. The backbone of any optimization technique is the selection of appropriate objective function that need to be minimized or maximized. In the present work, the objective functions chosen for minimization are integral of absolute error (IAE) and integral of absolute change in control output (IACCO) of both the links and are represented by (13) and (14) respectively. The aggregate objective function O-F is the weighted sum of IAE and IACCO of both the links. The purpose for selecting these objective functions



are, to minimize the error between actual and reference trajectories and also, to suppress the change in the controller output Z Z j e1 ðtÞj dt þ j e2 ðtÞj dt ð13Þ Obj_f 1 ¼ Z Obj_f 2 ¼

Z j Δτ1 ðtÞj dt þ

j Δτ2 ðtÞj dt

ð14Þ

O F ¼ w1 Obj_f 1 þ w2 Obj_f 2

7

Step 7: Update the best nest of the present generation. Step 8: The best nest zb obtained in the present generation is replaced by the best nest of the generation zq , if the fitness f zb of the best nest zb is greater than the fitness f zq of the best nest zq of the generation. Step 9: Repeat steps 2–8 till the stopping criteria reached. The best nest obtained at the final generation gives the optimal solution for the problem under consideration.

ð15Þ

where e1(t) and e2(t) are the tracking errors for Link1 and Link2 respectively; Δτ1(t) and Δτ2(t) are changes in the control outputs for Link1 and Link2 respectively; w1 and w2 are the weights assigned to Obj_f1 and Obj_f2 respectively. From Eq. (6), it is clearly visible that there are overall eight flexible parameters in the proposed control scheme that can be tuned to make an effective controller. The problem under consideration is a two-link manipulator which means overall two controllers are required, i.e., single controller for each link. The operators K P 1 ; α1 ; K I1 ; γ 1 ; δ1 ; β1 ; K D1 and N 1 are the parameters for Link1 that need to be tuned, and the operators K P2 ; α2 ; K I2 ; γ 2 ; δ2 ; β2 ; K D2 and N 2 are the parameters for Link2 that need to be tuned. The stepwise implementation of CSA for obtaining the optimal solution is presented below: Step1: Set the objective function as expressed in Eq. (15). Initialize a population of d host nests zi ði ¼ 1; 2; …; 25Þ. Set the maximum number of generations¼100 as the stopping criteria. Step 2: Generate a cuckoo named as csa arbitrarily using the Lévy flight as presented in Eq. (12), and compute its fitness f csa according to the fitness objective chosen for the problem. Step 3: Choose a nest cu among the randomly generated population d and calculate its fitness as f cu . Step 4: Compare the fitness f csa and f cu , and if f csa 4 f cu , then the host nests cu is replaced by the new nests csa obtained with Lévy flight. Step 5: A fraction ‘Pcsa’ of the worst nests is abandoned and new nests zn are generated at new locations with the Lévy flight. Step 6: Calculate the fitness of all new generated nests.

6. Simulation results In this section, the results obtained for trajectory tracking, model uncertainties, payload variations, external disturbance rejection and random noise suppression for 2-DOF FOPID, 2-DOF PID and PID controllers are presented. The numerical simulations were obtained with MATLAB version R2009b. The algebraic solver used for simulink model developed by Eq. (1) was Runge–Kutta 4 method. The sampling time was kept as 1 ms and the torque constraints were limited to [ 20, 20] N m for the simulation. The fifth order Oustaloup's approximation was utilized using Nou ¼2 h i and range of frequency was wh ¼ 10 5 ; 100 rad/s for the fractional order design. In this work, the trajectory chosen was of cubic polynomial nature [43] and is given as follows with different desired points: θRðkÞ ðt c Þ ¼ c0 þ c1 ðt c Þ þ c2 ðt c Þ2 þ c3 ðt c Þ3

ð16Þ

the constraints are θ_ RðkÞ ðt c Þ ¼ c1 þ 2c2 ðt c Þ þ 3c3 ðt c Þ2

ð17Þ

θ€ RðkÞ ðt c Þ ¼ 2c2 þ 6c3 ðt c Þ

ð18Þ

where θRðkÞ is the reference position; k ¼1, 2 for Link1 and Link2 respectively; θRð1Þ ¼ 2 rad and θRð2Þ ¼ 3 rad for t c ¼ 2 s; θRð1Þ ¼ 0:5 rad and θRð2Þ ¼ 5 rad for t c ¼ 4 s; θ_ RðkÞ ¼ 0 rad=s for both t c ¼ 2 s and t c ¼ 4 s. For ensuring the effectiveness of the proposed scheme, its performance is compared with its integer order design as well as

Table 5 IAE values for change in coefficient of viscous friction for Link1 and Link2. Parameter variation

Case 1: v1 ¼ 0:5sign θ_ 1 ; v2 ¼ 0:1 Case 2: v1 ¼ 0:1; v2 ¼ 0:5sign θ_ 2 Case 3: v1 ¼ 0:5sign θ_ 1 ; v2 ¼ 0:5sign θ_ 2

2-DOF FOPID

2-DOF PID Link1

PID

Link1

Link2

Link2

Link1

Link2

0.008775

0.01496

0.01796

0.04695

0.04484

0.06425

0.008788

0.01418

0.01807

0.04595

0.04386

0.06522

0.008775

0.01418

0.01796

0.04569

0.04483

0.06526

Fig. 7. IAE variations for (a) Link1 and (b) Link2 for 2-DOF FOPID, 2-DOF PID and PID controllers for change in coefficient of viscous friction for case 1, case 2 and case 3.



8

5

0

10


0

0.5

1

4.966 4.964 4.962

2.002 2 1.998 1.996 1.994 1.995

1.5

Control Output (N-m)

Position (radian)

10

3.838

3.84

2

2

2.5

3

3.5

0

-20

4


-10

0

0.5

1

1.5

2

X and Y- Coordinate (m)

Y-Coordinate (m)

3

Reference Path 2-DOF FOPID 2-DOF PID PID

2

0.1

1

0.095 0.51 0.515 0.52

0 -1

-0.5

0

0.5

1

1.5

8

0.2 0.19 0.18

4 2

0

Payload variation (kg)

Error (radain)

0

1

1.5

2

4

1.08 1.06 0.89 0.895 0.9 1.04

1.141.151.16

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

0.05

0.5

3.5

0

2

2-DOF FOPID Link1 2-DOF FOPID Link2 2-DOF PID Link1 2-DOF PID Link2 PID Link1 PID LInk2

0

3

Reference X-Coordinate Reference Y-Coordinate 2-DOF FOPID X-Coordinate 2-DOF FOPID Y-Coordinate 2-DOF PID X-Coordinate 2-DOF PID Y-Coordinate PID Link1 PID Link2

6

X-Coordinate (m)

-0.05

2.5

Time (s)

Time (s)

2.5

3

3.5

4

1

Case 2 Case 1

0.5

0

0

1

Time (s)

2

3

4

Payload variations (kg)

Time (s)

3 2 1 0 -1

0

1

2

3

4

Time (s) Fig. 8. (a) Trajectory tracking, (b) control outputs, (c) path traced by end-effector, (d) X and Y versus time variations, (e) position errors for 2-DOF FOPID, 2-DOF PID and PID controllers for payload variation from nominal to null, (f) variation of payload mass for case 1 and case 2 and (g) variations of payload mass for case 3.

Table 6 IAE values for 2-DOF FOPID, 2-DOF PID and PID controllers for payload variations. Payload variations (mp)

Case 1 Case 2 Case 3

2-DOF FOPID

2-DOF PID

PID

Link1

Link1

Link1

Link2

Link2

Link2

0.008793 0.01524 0.01809 0.04674 0.04433 0.07303 0.009061 0.01482 0.01825 0.04731 0.04733 0.05639 0.009278 0.01632 0.01857 0.04781 0.05099 0.09058

with conventional PID controller. The proportional, integral and derivative gains for Link1 and Link2 of PID controllers are 250.59, 89.38, 50.0981, 170.067, 51.24 and 20. The controller parameters and IAE values for the 2-DOF FOPID and 2-DOF PID controllers are listed in Table 3. The fitness value versus iteration graphs for 2DOF FOPID, 2-DOF PID and PID controllers are shown in Fig. 5 and their values are 0.0390, 0.0695 and 0.086 respectively. The IAE values for PID controllers for Link1 and Link2 are 0.04385 and 0.06421 respectively. The IAE values for Link1 and Link2 are 0.008787 and 0.01496 for 2-DOF FOPID controller for

trajectory tracking respectively whereas the IAE values for 2-DOF PID controller for Link1 and Link2 are 0.01807 and 0.04695 respectively. Since the IAE values for 2-DOF FOPID controllers are smaller as compared to 2-DOF PID and conventional PID controllers for both Link1 and Link2, it is clear that 2-DOF FOPID controllers perform better than 2-DOF PID and conventional PID controllers for trajectory tracking. The graphs for trajectory tracking, controller output, path traced by end-effector, X and YCoordinate versus time variations and position errors for 2-DOF FOPID, 2-DOF PID and PID controller schemes are presented in Fig. 6. In the present work, the robotic manipulator used does not include the actuator dynamics. As there is a linear relation between torque and current of the actuator/motor used and it can be expressed as τ ¼ ki, where τ is the torque and i is the current; k is the transmission factor/matrix that relates the current to control torque. For real time applications point of view, this torque can be converted into proportional current. As the relation between torque and current is linear so the control signal is taken as torque in the presented work in accordance with the dynamics of robotic manipulator. However, if the actuator dynamics is



9

Fig. 9. IAE variations for (a) Link1 and (b) Link2 for 2-DOF FOPID, 2-DOF PID and PID controllers for variation in payload mass (in kg) for case 1, case 2 and case 3.

Table 7 IAE variation of 2-DOF FOPID, 2-DOF PID and PID controllers for disturbances. Disturbances (N m)

Link1

Link2

Both links

2-DOF FOPID

0.1sin 25t 0.2sin 25t 0.3sin 25t 0.4sin 25t 0.5sin 25t 0.6sin 25t 0.7sin 25t 0.8sin 25t 0.9sin 25t 1.0sin 25t 0.1sin 25t 0.2sin 25t 0.3sin 25t 0.4sin 25t 0.5sin 25t 0.6sin 25t 0.7sin 25t 0.8sin 25t 0.9sin 25t 1.0sin 25t 0.1sin 25t 0.2sin 25t 0.3sin 25t 0.4sin 25t 0.5sin 25t 0.6sin 25t 0.7sin 25t 0.8sin 25t 0.9sin 25t 1.0sin 25t

2-DOF PID

PID

Link1

Link2

Link1

Link2

Link1

Link2

0.008784 0.008783 0.008783 0.008787 0.008797 0.008810 0.008825 0.008842 0.008860 0.008890 0.008791 0.008796 0.008800 0.008804 0.008808 0.008813 0.008817 0.008821 0.008826 0.008830 0.008788 0.008790 0.008793 0.008797 0.008802 0.008809 0.008815 0.008823 0.008832 0.008843

0.01496 0.01496 0.01496 0.01496 0.01496 0.01496 0.01496 0.01496 0.01496 0.01496 0.01495 0.01494 0.01493 0.01492 0.01492 0.01491 0.01491 0.01490 0.01490 0.01490 0.01495 0.01494 0.01493 0.01492 0.01492 0.01491 0.01491 0.01490 0.01490 0.01490

0.01807 0.01807 0.01808 0.01808 0.01809 0.01809 0.01810 0.01811 0.01812 0.01813 0.01807 0.01807 0.01808 0.01808 0.01808 0.01808 0.01808 0.01808 0.01809 0.01809 0.01807 0.01808 0.01808 0.01809 0.01810 0.01810 0.01811 0.01812 0.01813 0.01814

0.04695 0.04695 0.04695 0.04695 0.04695 0.04695 0.04695 0.04695 0.04695 0.04695 0.04696 0.04696 0.04696 0.04696 0.04696 0.04697 0.04697 0.04697 0.04698 0.04698 0.04696 0.04696 0.04696 0.04696 0.04696 0.04696 0.04697 0.04697 0.04697 0.04697

0.04386 0.04386 0.04387 0.04388 0.04388 0.04389 0.04389 0.04390 0.04391 0.04392 0.04385 0.04385 0.04385 0.04385 0.04385 0.04385 0.04385 0.04385 0.04385 0.04385 0.04386 0.04386 0.04387 0.04388 0.04388 0.04389 0.04390 0.04391 0.04392 0.04393

0.06420 0.06420 0.06420 0.06420 0.06419 0.06419 0.06419 0.06419 0.06419 0.06419 0.06423 0.06425 0.06428 0.06431 0.06434 0.06437 0.06441 0.06444 0.06448 0.06451 0.06423 0.06425 0.06427 0.06430 0.06433 0.06436 0.06439 0.06443 0.06446 0.06450

included then the order of system is enhanced and thereby, increases the complexity of the controller. The complexity of the controller becomes more challenging in the presence of uncertainties and external disturbances. Therefore, the consideration of the proper actuator dynamics is crucial for better performance of the system. Also, the motor current and control effort should be used in acceptable limits during the uncertain and noisy conditions. The robustness investigations for the proposed controller approach are presented in the following sections.

6.1. Robustness testing: model uncertainties In this section, the effectiveness of proposed controllers is investigated under model uncertainties incorporated to the robotic manipulator. The model uncertainties include 75% change in

parameters namely mass, distance from link joint to its center of gravity, length, lengthwise centroid inertia, coefficient of dynamic friction for Link1 as well as Link2, from their nominal values. The IAE for 2-DOF FOPID, 2-DOF PID and PID controllers for 5% increase as well as decrease in parameters are listed in Table 4. Moreover, the coefficient of viscous friction was chosen as the function of velocity, i.e., θ_ ðt Þ and varies for links as case 1, case 2 and case 3. The IAE variations are listed in Table 5 for change in viscous friction. From Table 4, it is inferred that the variations in IAE values of Link1 and Link2 for 2-DOF FOPID controllers remain smaller as compared to 2-DOF PID as well as PID controllers for all cases of parameter variations. The variations in IAE for viscous friction coefficient for case 1, case 2 and case 3 for 2-DOF FOPID, 2DOF PID and PID controllers for Link1 and Link2 are also shown in Fig. 7. The comparison between IAE values in Fig. 7 clearly shows that all IAE values for 2-DOF FOPID controllers for Link1 and Link2


Position (radian)

6

Reference trajectory Link1 Reference trajectory Link2 2-DOF FOPID Link1 2-DOF FOPID Link2 2-DOF PID Link1 2-DOF PID Link2

4

4.3 4.295 4.29 4.285 3.195 3.2 3.205 1.351 1.35 1.349

2

0

0

0.5

1

C o n tr o l O u tp u t ( N - m )


10

1.5

2

2.5

3

2.911

2.912

3.5

5 0 -5 -10

4

2-DOF-FOPID Link1 2-DOF FOPID Link2 2-DOF PID Link1 2-DOF PID Link2 PID Link1 PID Link2

10

0

0.5

1

1.5

2

X and Y-Coordinate (m)

Y-Coordinate (m)

3

Reference Path 2-DOF FOPID 2-DOF PID PID

1

-0.135 -0.14 -0.145 1.47

1.475

1.48

0 -1

-0.5

0

0.5

1

1.5

2

6 4

0.91 0.9 0.89 0.72

0.73

0.6 0.58 0.56 1.29 1.3 1.31

2

2.5

3

3.5

4

3.5

4

Reference X-Coordinate Reference Y-Coordinate 2-DOF FOPID X-Coordinate 2-DOF FOPID Y-Coordinate 2-DOF PID X-Coordinate 2-DOF PID Y-Coordinate PID X-Coordinate PID Y-Coordinate

0 0

0.5

1

1.5

X-Coordinate (m)

Error (radian)

2

Time (s)

Time (s)

2

2.5

3

Time (s)


0.05

0

-0.05

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s) Fig. 10. (a) Trajectory tracking performance, (b) control output, (c) path traced by end-effector, (d) X and Y versus time variations and (e) position errors for 2-DOF FOPID, 2-DOF PID and PID controllers for disturbance 0.5sin 25t N m in both links.

Fig. 11. Variation in IAE for (a) Link1 with disturbances in Link1, (b) Link2 with disturbances in Link1, (c) Link1 with disturbances in Link2, (d) Link2 with disturbances in Link2, (e) Link1 with disturbances in both links and (f) Link2 with disturbances in both links for 2-DOF FOPID, 2-DOF PID and PID controllers.

are lesser as compared to those of 2-DOF PID and conventional PID controller schemes. Therefore, the 2-DOF FOPID controller performs better than both 2-DOF PID controller and conventional PID controller in the presence of model uncertainties.

6.2. Robustness testing: payload variations In this section, the robustness of the proposed controllers for the payload variations with time is investigated. The payload variations


Position (radian)

15


10 5 0

0

0.5

0.5 0.495 0.49 3.996

3.19 3.18 3.17 2.365

1

1.5

Control output (N-m)


3.998

2.37

2

2.5

3

3.5

11

80

2-DOF FOPID Link1 2-DOF FOPID Link2 2-DOF PID Link1 2-DOF PID Link2 PID LInk1 PID Link2

60 40 20 0 -20

4

0

0.5

1

1.5

2

Time (s)

2

1.548 1.55 1.552

1 0 -1 -1

-0.5

0

0.5 1 X-Coordinate (m)

1.5

0.2 Error (radian)

X and Y-Coordinate (m)

Reference path 2-DOF FOPID 2-DOF PID PID

-0.19 -0.195


0.02 0.01 0 -0.01

0.1

1

1.2

1.4

0

-0.1

0

0.5

1

1.5

2

2.5

3

3.5

8

3

3.5

4

4

Reference X-Coordinate Reference Y-Coordinate 2-DOF FOPID X-Coordinate 2-DOF FOPID Y-Coordinate 2-DOF PID X-Coordinate 2-DOF PID Y-Coordinate PID X-Coordinate PID Y-Coordinate

6 1.5 1.45 1.4 0.55

4

0.9 0.6

0.85

2

1.2 1.21 1.22

0

2

0

0.5

0

0.5

1

1.5

2 2.5 Time (s)

3

3.5

4

0.01 Amplitude (radian)

Y-Coordinate (m)

4 3

2.5

Time (s)

0.005 0 -0.005 -0.01

Time (s)

1

1.5

2

2.5

3

3.5

4

Time (s)

Fig. 12. (a) Trajectory tracking performance, (b) control output, (c) path traced by end-effector, (d) X and Y versus time variations, (e) position errors and (f) added noise profile for 2-DOF FOPID, 2-DOF PID and PID controllers for addition of noise in both links.

Table 8 IAE values for 2-DOF FOPID, 2-DOF PID and PID controllers for addition of noise. Random noise (radian)

Link1 Link2 Both links

2-DOF FOPID

2-DOF PID

PID

Link1

Link2

Link1

Link2

Link1

Link2

0.021390 0.008787 0.021390

0.01496 0.02413 0.02413

0.02672 0.01807 0.02672

0.04695 0.05027 0.05027

0.05398 0.04356 0.05363

0.06412 0.07784 0.07774

include the changes in mp with time from nominal value of 0.5 kg. For case 1, the mp was increased linearly at a rate of 0.125 kg/s from its nominal value to 1.0 kg in 4 s. For case 2, the payload mass was decreased linearly at a rate of 0.125 kg/s to null in 4 s. For case 3, a customized trajectory was formulated, as shown in Fig. 8(g). The IAE values for all three cases are listed in Table 6. The trajectory tracking, control output, path traced by the end-effector, X and YCoordinate versus time variations and position error variations for 2-DOF FOPID, 2-DOF PID and PID controllers for payload variation from nominal 0.5 kg to null are shown in Fig. 8. For payload variations as listed in Table 6, the IAE values for 2-DOF FOPID controllers for Link1 and Link2 are lesser as compared to those of 2DOF PID and conventional PID controllers for all three cases. To show the comparison, the IAE variations for all the three cases for 2DOF FOPID, 2-DOF PID and conventional PID controllers for Link1 and Link2 are also shown in Fig. 9 and it is inferred that the variations in IAE for Link1 and Link2 are smaller for 2-DOF FOPID controller as compared to 2-DOF PID and conventional PID controllers for all three cases of payload variations. Thus, the 2-DOF FOPID controllers are more robust than 2-DOF PID and traditional PID controllers for payload variations.

6.3. Robustness testing: external disturbance rejection This section presents the robustness testing for incorporating the disturbances to the control output to witness the performance of the proposed controllers. The disturbances mentioned in Table 7 were given to the control output for the entire time period in each link of the robotic manipulator. As a typical case, the trajectory tracking performance, control output, path traced by the endeffector, X and Y versus time variations and position errors for 2DOF FOPID, 2-DOF PID and conventional PID controllers for addition of 0.5sin 25t N m disturbance to both the links are shown in Fig. 10. The variations of IAE for the disturbances for 2-DOF FOPID, 2-DOF PID and PID controllers are also summarized in Table 7. The comparative analysis of IAE variations for adding disturbances to the all three controllers is also presented in Fig. 11 and it is indicated that the variations in IAE values remain smaller for Link1 and Link2 for 2-DOF FOPID controller as compared to 2DOF PID and traditional PID controllers for all cases of disturbances. Thus, the performance of proposed 2-DOF FOPID controllers is better than the other two controllers namely 2-DOF PID and PID controllers in the presence of external disturbances.



12

Fig. 13. IAE variations for (a) Link1 and (b) Link2 for 2-DOF FOPID, 2-DOF PID and PID controllers for adding noise to Link1, Link2 and both links.

6.4. Robustness testing: random noise suppression In this section, the effects of adding random noise are presented in order to demonstrate the effectiveness of proposed controller scheme. A random noise of maximum amplitude 70.01, as shown in Fig. 12(f), is added in the feedback path of the proposed controller schemes to Link1, Link2 and then both links simultaneously. The trajectory tracking performance, control output, path traced by the end-effector, X and Y versus time variations and position errors for 2-DOF FOPID, 2-DOF PID and conventional PID controllers for addition of random noise to both the links are shown in Fig. 12. The variations of IAE for addition of random noise for 2-DOF FOPID, 2-DOF PID and PID controllers are presented in Table 8. The comparative analysis of IAE variations for adding noise to the all three mentioned controllers is presented in Fig. 13 and it is indicated that the variations in IAE values are smaller for Link1 and Link2 for 2-DOF FOPID controller as compared to those of 2-DOF PID and traditional PID controllers for all cases. Thus, the performance of proposed 2-DOF FOPID controller scheme is better than the other two controllers namely 2-DOF PID and conventional PID controllers in the presence of random noise. From the overall results presented above, it can be clearly investigated that 2-DOF FOPID controller is more effective and robust than its integer order design as well as traditional PID controllers for trajectory tracking control.

7. Conclusions In this work, the 2-DOF FOPID controllers have been implemented for a two-link planar rigid robotic manipulator with payload for trajectory tracking problem. The incorporation of 2DOF design has enhanced the robustness of the controller towards both the external disturbances and trajectory tracking where as the incorporation of fractional order operators has increased the flexibility in selection of controller parameters to the control engineers. However, tuning of the large number of controller parameters has become the challenging task to obtain the effective performance. The CSA has been explored and effectively used for tuning of large number of controller parameters. An exhaustive comparative study of the 2-DOF FOPID controllers with 2-DOF PID controllers and conventional PID controllers to demonstrate the robustness for the model uncertainties, payload variations, external disturbance rejection and random noise suppression has been carried out. From the results obtained, it can be concluded that 2DOF FOPID controllers are more robust and effective than their integer order designs as well as traditional PID controllers. The proposed controller scheme can be used in precise positioning applications of robotic manipulators such as nuclear plants and process industries where large parameter variations occur due to external disturbances. This work also proves the applicability of

CSA to tune the controller parameters of highly uncertain and nonlinear plants. Furthermore, the proposed 2-DOF FOPID controller needs to be explored for real time implementation for robotic manipulator. The suitability of proposed method for the flexible-link and joints robotic manipulators can also be investigated as these systems are more prone to model uncertainties and disturbances due to the vibrations in end-effector.

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