A fractional order fuzzy PD+I controller for three-link electrically driven rigid robotic manipulator system

Abstract

In this paper, a robust fractional order fuzzy proportional derivative plus fractional order integrator (FOFPD+FOI) control structure is proposed to effectively control a nonlinear, coupled, multi-input multi-output, electrically driven three-link rigid robotic manipulator (EDRRM) system. The FOFPD+FOI controller is realized by using non-integer order differentiator and integrating operators in the integer order fuzzy proportional derivative plus integer order integrator (IOFPD+IOI) controller. A comparative study is carried out to assessed the performance of FOFPD+FOI controller with IOFPD+IOI controller, fractional order proportional, integral and derivative (FOPID) controller and integer order PID controller for reference trajectory tracking, noise suppression, disturbance rejection and model uncertainty. The gains of the controllers were tuned using a meta-heuristic optimization technique cuckoo search algorithm for objective function which is defined as the weighted sum of integral of absolute error and integral of absolute change in controller output. The simulation studies reveal that proposed FOFPD+FOI controller offers much superior performance over PID, FOPID and IOFPD+IOI controllers.

Keywords

PID controller FOPID controller IOFPD+IOI controller FOFPD+FOI controller cuckoo search algorithm model uncertainty noise suppression

1 Introduction

Since last seven decades, research for controlling nonlinear, uncertain, multi-input multi-output (MIMO), coupled systems ascended to a great extent. As most of the practical systems applied in industrial environment are nonlinear and complex in nature, their effective control is a challenge for researchers and engineers. Earlier, linear proportional, integral and derivative (PID) controllers were used for controlling these types of complex systems but it has been reported in the literature that it fails to give satisfactory results. In view of this, different adaptive controllers such as gain scheduling, model reference adaptive control (MRAC), self-tuning regulator, sliding mode controller (SMC) etc. were developed to cope with these kinds of complex control problems [1]. These nonlinear controllers have certain drawbacks which further compel the researchers and scientists to explore more proper control solutions for such kind of systems. A complete design of MRAC, self-tuning and SMC required the exact mathematical model of the system. Design of gain scheduling becomes complex when the operating points of plant becomes very large [2, 3]. Therefore, to control a nonlinear, complex and poorly modelled system, these methods are not able to give satisfactory control solution. Some new methods are always desirable to investigate the new control techniques which can ease the plant operation and increase the profitability of industries. In this context, the evolution of artificial intelligence techniques and its combination with classical control methods have revolutionised a new trend to the researchers in the field of process control and automation [4].

The production in automobile industries, textile industries, chemical industries, food industries etc. is benefitted by the inclusion of artificial intelligent with classical control techniques. Nature of these industrial processes is highly nonlinear, uncertain, time-varying, coupled, MIMO and complex where an intelligent method is required for efficient operation. Out of these systems, robotic manipulators are extensively used nowadays in industries at inaccessible places where repetitive tasks are carried out in accurate time [5].

Robotic manipulators are nonlinear and MIMO systems which are mechanically coupled and normally containing arms. It consists of segment joints which grasp and move the material on a fixed path. Pick and place of material is one of the major applications of industrial robotic manipulators. Other applications of industrial manipulators are like welding, assembling, manufacturing, painting etc. in the field of automobile industries whereas these are also used in robotically assisted surgery, handling of radioactive and bio-hazardous materials etc. As these are complex and nonlinear systems, control and efficient operation of these systems are the challenging task for researchers and engineers [6]. Also, if the links of manipulator system will increase its control complexity will also increase. In the present work, an electrically driven three-link rigid robotic manipulator (EDRRM) system is considered for study [6]. For controlling EDRRM, conventional PID controllers fail to provide acceptable control solutions. After the development of intelligent techniques like fuzzy Logic by Prof. L. A. Zadeh, it was extensively used to design intelligent controllers for such a complex system [7]. It was based on the concept of linguistic variables which have given a powerful tool to design intelligent controller which can take actions on the basis of logics investigated by expert knowledge. The first successful application of fuzzy logic controller (FLC) of a laboratory scale process was reported by Mamdani et al. [8]. He suggested advances in linguistic synthesis of fuzzy controller. To design the FLC, it does not require exact mathematical model of the system which is the main reason of success of FLC in technical community. In the next section, a literature survey of FLC and its variants to control a nonlinear and complex system is presented.

Fuzzy logic scheme was added by researchers to enhance the robustness of controller due to itsability to convert the expert’s control action into rule base. A Fuzzy PID (FPID) controller and their stability analysis was described by Mishir et al. [9] which was applied to control a two-link flexible joint robot arm. A realistic sufficient condition for stability by graphical approach was designed to give the safe region for feedback system on which it gives the guaranteed stability. A fuzzy PD controller in which their gains and rule base are optimized by genetic algorithm (GA) was proposed by Alam and Tokhi [10] to reduce the end-point vibration of a single link flexible manipulator without sacrificing the speed of response. Simulation results showed the reduction in vibration at end point with satisfactory level of overshoot, rise time, settling time and steady-state error. A stable hybrid fuzzy adaptive robust controller was developed by Ho et al. [11] to control a two-link robotic manipulator with parameter uncertainties and external disturbances. A fuzzy adaptive algorithm was used for parameter identification of robotic system and then an integral SMC was added to remove the effect of external disturbances and uncertainties. The simulated results have been shown the validation of results. Mudi and Paul [12 –14] designed a robust self-tuning FPD and FPI controllers. In these works, the output gains of controller were adjusted online by another fuzzy logic scheme according to the current value of error and rate of change of error. This controller was applied on different linear and nonlinear second order processes. Performances are compared with conventional FLC and it was reported that self-tuning fuzzy logic controller remarkably improved the results. Further, a self-tuning robust fuzzy PD+I controller configuration was proposed by Malki et al. [15] to control a flexible joint robot arm with uncertainties from time-varying loads. In this case, gains of fuzzy controller vary at run-time. It was reported that due to these variable gains, robustness of fuzzy PD+I controller increased with faster response time and less overshoot than its conventional counterpart.

To improve the positional accuracy, usefulness and reachability, three-links robotic manipulators show better performance on two-link manipulators. In this context, a new output feedback tracking control approach is developed by Kim using adaptive fuzzy logic technique for three-link robotic manipulator system with model uncertainty [16]. The adaptive fuzzy allow to approximate the three-link robot model while another fuzzy system is used to implement the observer-controller structure of the output feedback robot system. Simulation results revealed the usefulness in case of trajectory tracking, disturbance rejection and payload uncertainty. Xiong et al. is presented a position control strategy based on energy attenuation for a planar three-link under-actuated manipulator in horizontal plane where simulation results show the validity of research [17]. Further, Hoffmann et al. proposed a nonlinear control of an industrial three degrees of freedom (DOF) robotic manipulator as a benchmark problem for controller synthesis method where guaranteed stability and performances are shown [18]. A three DOF parallel Maryland manipulator is controlled by fractional order PID (FOPID) controller and obtained performances were compared with classical PID controller. Simulated results are shown that FOPID controller outperforms on classical PID controller. An under-actuated spring-coupled three-link horizontal manipulator is controlled by Zhang et al. where a new control method was proposed that asymptotically stabilizes the system at origin. In this research work it is claimed that this control method reduces the cost of overall control system as well as it avoids the influence of velocity noise [19].

History of the fractional calculus, invented by Leibniz and Newton, is as old as ordinary differential calculus. During last three centuries it was a topic of mathematics but since last few years it was extensively used in the applied fields of science, engineering and economics. The concept of FOPID controller was proposed by Podlubny [20] to control a dynamic system with fractional order. A novel fractional order FPID (FOFPID) controller and their hybrid combinations have been proposed by Das et al. [21, 22]. The proposed control schemes have been tested on nonlinear time-delayed process, fractional order processes and fractional order time delayed processes. Gains of the controller was tuned by GA. Simulation results have been shown the usefulness and effectiveness of FOFPID controller. A FOFPID controller for a two-link planar rigid robotic manipulator for trajectory tracking problem was investigated by Sharma et al. in [23]. Robustness testing was performed by model uncertainties, disturbance rejection and noise suppression. The performance of FOFPID controller was compared by PID, FPID and FOPID controllers. Tuning of controller gains were performed by Cuckoo Search Algorithm (CSA). On the basis of numerical simulation results, it was clearly indicated the superiority of FOFPID controllers among all four investigated controllers.

To effectively handle the steady state error present in the system output, a fractional order fuzzy PD with integer order integral (FOFPD+IOI) control structure is proposed by Jesus Barbosa [24 –26]. This structure is considered from [27]. This controller is successfully applied to control three different systems, a fourth order system, a non-minimum phase system and a time delayed second order system. The result obtained by FOFPD+IOI controller is compared with integer order counter-part and it is claimed that FOFPD+IOI controller outperforms its integer order counter-part. In the present work, to increase the flexibility and to handle the closed-loop system more sensibly, a fractional order fuzzy proportional derivative plus fractional order integrator (FOFPD+FOI) type of control structure is proposed and successfully tested on an EDRRM system for trajectory tracking, noise suppression, disturbance rejection and uncertainty analysis. To investigate the efficacy of proposed controller, the obtained performances are compared with PID, FOPID and integer order fuzzy proportional derivative with integer order integrator (IOFPD+IOI) controllers. The gains of the controller are tuned offline by a meta-heuristic bio-inspired optimization algorithm, CSA which is given by Yang and Deb [28, 29]. The main contributions of the present work can be summarized as follows:

An intelligent fractional order fuzzy controller as FOFPD+FOI is proposed to control a nonlinear and complex EDRRM system. It shows the robust behaviour at runtime which makes it more effective for such kind of systems.

The robustness of proposed FOFPD+FOI controller is increased by incorporating the Grünwald Letnikov (GL) based fractional order approximation method which can easily implement in hardware [35].

The robustness of FOFPD+FOI controller is established by comparing its performance with PID, FOPID and IOFPD+IOI controllers for trajectory tracking, noise suppression, disturbance rejection, and model uncertainty.

The main advantage of the proposed controller is that it can be applied to a poorly known system.

The rest of the paper is organized as follows: after an introduction and literature survey in Section1, system description and mathematical model is elaborated in Section 2 where complexity of the plant is described and parameters of plant are also tabulated. Fractional order implementation is shown in Section 3 while detailed description of controllers is discussed in Section 4. Gains of the all the considered controllers are tuned by CSA and corresponding trajectory tracking performance are presented in Section 5. Detailed performance comparisons of all the controllers for noise suppression, disturbance rejection and model uncertainty studies are presented in Section 6. Finally, conclusions of the proposed work are shown in Section 7.

2 Dynamic model of EDRRM system

A three-links EDRRM system with three DOF is described in this section and shown in Fig. 1. The first link is mounted on a solid base and the second link is attached with a frictionless ball bearing to the end of first link. Input to the EDRRM is considered as the applied voltages V with range –12 V to 12 V. The dynamic equations of considered EDRRM system in the present work are taken from [6]. $\begin{matrix} [\begin{matrix} D_{11} & D_{12} & D_{13} \\ D_{21} & D_{22} & D_{23} \\ D_{31} & D_{32} & D_{33} \end{matrix}] [\begin{matrix} {\overset{˙˙}{θ}}_{1} \\ {\overset{˙˙}{θ}}_{2} \\ {\overset{˙˙}{θ}}_{3} \end{matrix}] + [\begin{matrix} P_{1} \\ P_{2} \\ P_{3} \end{matrix}] \\ + [\begin{matrix} R_{1} \\ R_{2} \\ R_{3} \end{matrix}] + [\begin{matrix} g_{1} \\ g_{2} \\ g_{3} \end{matrix}] = [\begin{matrix} τ_{1} \\ τ_{2} \\ τ_{3} \end{matrix}] \end{matrix}$ (1) $L \dot{I} + RI + K_{b} \dot{θ} = V$ (2) $\begin{matrix} [\begin{matrix} L_{1} & 0 & 0 \\ 0 & L_{2} & 0 \\ 0 & 0 & L_{3} \end{matrix}] [\begin{matrix} {\dot{I}}_{1} \\ {\dot{I}}_{2} \\ {\dot{I}}_{3} \end{matrix}] + [\begin{matrix} R_{1} & 0 & 0 \\ 0 & R_{2} & 0 \\ 0 & 0 & R_{3} \end{matrix}] [\begin{matrix} I_{1} \\ I_{2} \\ I_{3} \end{matrix}] \\ + [\begin{matrix} K_{b_{1}} & 0 & 0 \\ 0 & K_{b_{2}} & 0 \\ 0 & 0 & K_{b_{3}} \end{matrix}] [\begin{matrix} {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \\ {\dot{θ}}_{3} \end{matrix}] = [\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix}] \end{matrix}$ (3)

Fig.1

Three-link robotic manipulator system.

Equation (1) contains four terms. The first term comprises the second order derivative of generalised coordinates, i.e. angular position of three-links ${\overset{˙˙}{θ}}_{i} (i = 1, 2, 3)$ . The second term known as Centrifugal consist of a product of ${\overset{˙˙}{θ}}_{i}^{2} (i = 1, 2, 3)$ . The third types of terms involving a product of ${\dot{θ}}_{i} {\dot{θ}}_{j} (where i \neq j)$ are called Coriolis terms. The last term contains only θ_i but not its derivatives. It is derived by differentiating the potential energy stored in links of manipulator. The terms used in (1) is defined as: $\begin{matrix} D_{11} & = & (m_{1} + m_{2} + m_{3}) l_{1}^{2} + (m_{2} + m_{3}) l_{2}^{2} \\ + m_{3} l_{3}^{2} + 2 m_{3} l_{1} l_{3} cos (θ_{2} + θ_{3}) \\ + 2 (m_{2} + m_{3}) l_{1} l_{2} cos (θ_{2}) \\ + 2 m_{3} l_{2} l_{3} cos (θ_{3}) \end{matrix}$ (4) $\begin{matrix} D_{12} & = & (m_{2} + m_{3}) l_{2}^{2} + m_{3} l_{3}^{2} + m_{3} l_{1} l_{3} cos (θ_{2} + θ_{3}) \\ + (m_{2} + m_{3}) l_{1} l_{2} cos (θ_{2}) + 2 m_{3} l_{2} l_{3} cos (θ_{3}) \end{matrix}$ (5) $\begin{matrix} D_{13} & = & m_{3} l_{3}^{2} + m_{3} l_{1} l_{3} cos (θ_{2} + θ_{3}) \\ + m_{3} l_{2} l_{3} cos (θ_{3}) \end{matrix}$ (6) $\begin{matrix} D_{21} & = & (m_{2} l_{2}^{2} + m_{3} l_{2}^{2} + m_{3} l_{3}^{2} + m_{3} l_{1} l_{3} cos (θ_{2} + θ_{3}) \\ + m_{2} l_{1} l_{2} cos (θ_{2}) + m_{3} l_{1} l_{2} cos (θ_{2}) \\ + 2 m_{3} l_{2} l_{3} cos (θ_{3})) \end{matrix}$ (7) $D_{22} = (m_{2} l_{2}^{2} + m_{3} l_{2}^{2} + m_{3} l_{3}^{2} + 2 m_{3} l_{2} l_{3} cos (θ_{3}))$ (8) $D_{23} = (m_{3} l_{3}^{2} + m_{3} l_{2} l_{3} cos (θ_{3}))$ (9) $\begin{matrix} D_{31} & = & (m_{3} l_{3}^{2} + m_{3} l_{1} l_{3} cos (θ_{2} + θ_{3}) \\ + m_{3} l_{2} l_{3} cos (θ_{3})) \end{matrix}$ (10) $D_{32} = (m_{3} l_{3}^{2} + m_{3} l_{2} l_{3} cos (θ_{3}))$ (11) $D_{33} = (m_{3} l_{3}^{2})$ (12)

Centrifugal terms are defined as: $\begin{matrix} P_{1} & = & - l_{1} (m_{3} l_{3} sin (θ_{2} + θ_{3}) + m_{2} l_{2} sin (θ_{2}) \\ + m_{3} l_{2} sin (θ_{2})) {\dot{θ}}_{2}^{2} - m_{3} l_{3} (l_{1} sin (θ_{2} + θ_{3}) \\ + l_{2} sin (θ_{3})) {\dot{θ}}_{3}^{2} \end{matrix}$ (13) $\begin{matrix} P_{2} & = & l_{1} (m_{3} l_{3} sin (θ_{2} + θ_{3}) + m_{2} l_{2} sin (θ_{2}) \\ + m_{3} l_{2} sin (θ_{2})) {\dot{θ}}_{1}^{2} - m_{3} l_{2} l_{3} sin (θ_{3}) {\dot{θ}}_{3}^{2} \end{matrix}$ (14) $\begin{matrix} P_{3} & = & m_{3} l_{3} (l_{1} sin (θ_{2} + θ_{3}) + l_{2} sin (θ_{3})) {\dot{θ}}_{1}^{2} \\ + m_{3} l_{2} l_{3} sin (θ_{3}) {\dot{θ}}_{2}^{2} \end{matrix}$ (15)

Coriolis terms are defined as: $\begin{matrix} R_{1} & = & - 2 l_{1} (m_{3} l_{3} sin (θ_{2} + θ_{3}) \\ + (m_{2} + m_{3}) l_{2} sin (θ_{2})) {\dot{θ}}_{1} {\dot{θ}}_{2} \\ - 2 m_{3} l_{3} (l_{1} sin (θ_{2} + θ_{3}) + l_{2} sin (θ_{3})) \\ {\dot{θ}}_{2} {\dot{θ}}_{3} - 2 m_{3} l_{3} (l_{1} sin (θ_{2} + θ_{3}) \\ + l_{2} sin (θ_{3})) {\dot{θ}}_{1} {\dot{θ}}_{3} \end{matrix}$ (16) $\begin{matrix} R_{2} & = & - 2 m_{3} l_{2} l_{3} sin (θ_{3}) {\dot{θ}}_{1} {\dot{θ}}_{3} \\ - 2 m_{3} l_{2} l_{3} sin (θ_{3}) {\dot{θ}}_{3} {\dot{θ}}_{2} \end{matrix}$ (17) $R_{3} = 2 m_{3} l_{2} l_{3} sin (θ_{3}) {\dot{θ}}_{1} {\dot{θ}}_{2}$ (18)

Potential energy terms are defined as: $\begin{matrix} g_{1} & = & (m_{1} + m_{2} + m_{3}) {gl}_{1} cos (θ_{1}) \\ + (m_{2} + m_{3}) {gl}_{2} cos (θ_{1} + θ_{2}) \\ + m_{3} {gl}_{3} (θ_{1} + θ_{2} + θ_{3}) \end{matrix}$ (19) $\begin{matrix} g_{2} & = & (m_{3} + m_{2}) {gl}_{2} cos (θ_{1} + θ_{2}) \\ + m_{3} {gl}_{3} cos (θ_{1} + θ_{2} + θ_{3}) \end{matrix}$ (20) $g_{3} = m_{3} {gl}_{3} cos (θ_{1} + θ_{2} + θ_{3})$ (21)

In this simulation work the following system parameters have been taken: m₁ = 0.1 kg, mass of link-1; m₂ = 0.1 kg, mass of link-2; m₃ = 0.1 kg, mass of link-3; l₁ = 0.8 m, length of link one; l₂ = 0.4 m, length of link two; l₃ = 0.2 m, length of link three; g = 9.8 m/sec². In the present work, the motor parameters H₁ = 100N - m/A, H₂ = 100N - m/A, H₃ = 100N - m/A, L₁ = 0.025H, L₂ = 0.025H, L₃ = 0.025H, R₁ = 10Ω, R₂ = 10Ω, R₃ = 10Ω K_b1 = 1vol/rad/sec, K_b2 = 1vol/rad/sec and K_b3 = 1vol/rad/sec are considered to define the system suitably.

3 Implementation of fractional order

Fractional order mathematical operators and its combination with FLC, increase the DOF and as a result accurate solution can be obtained which have been used in the field of control in various applications [30 –34]. A fractional order differentiator and integrator of a function g (t) is represented as, D^αg (t) and D^-βg (t) respectively where 0 < α < 1 and 0 < β < 1 [35]. In the present work, GL method (22) has been used to implement the fractional order operators with a memory size of 100.

$\begin{matrix} {}_{a}D_{t}^{γ} g (t) \\ = lim_{h \to 0} \frac{1}{h^{γ}} \sum_{i = 0}^{[(t - a) / h]} {(- 1)}^{i} (\begin{matrix} γ \\ i \end{matrix}) g (t - ih) \end{matrix}$ (22)where, D signifies the derivative/integrator operator and γ (α, β) is the order of fractional operator. In the present work, α is chosen as the order of differentiator whereas β is the order of integrator, whose value is obtained by CSA.

4 Controllers design

A basic structure of three-input three-output EDRRM system is shown in Fig. 2 where a feedback loop is depicted in link-1. A FOFPD+FOI controller is used to perfectly control the EDRRM system. A saturator with limit –12 V to 12 V is incorporated after the controller output which mimics the behaviour of final control element. A disturbance is injected at the controller output to test the robust behaviour of the controllers. Random noise analysis is also performed to test the robustness of the controller. Various concepts about the design consideration of PID, FOPID, IOFPD+IOI and FOFPD+FOI controllers are described in the following subsections.

Fig.2

Feedback controlled diagram of three-link EDRRM system.

4.1 PID controller design

In this subsection, the basic design of PID controller is presented. In time domain, the structure of simple PID controller [23] can be stated as; $U_{PID} (t) = [k_{p} e (t) + k_{i} \int e (t) dt + k_{d} \frac{de (t)}{dt}]$ (23) where, the parameters k_p, k_i and k_d are the gains of proportional, integral and derivative controllers. Ins-domain, the output of PID controller can be expressed as [23]; $U_{PID} (s) = [k_{p} + k_{i} * \frac{1}{s} + k_{d} * s] E (s)$ (24)

4.2 FOPID controller design

In this subsection, the design of FOPID controller is presented. To control a nonlinear, coupled and uncertain system, a robust controller is needed. It has been mentioned in [23] that after incorporating the fractional order calculus in PID controller, the robustness of controller is increased. The simple structure of FOPID controller is stated in (25). For realization of fractional order differentiator and integrator, GL method is used which is elaborated in Section 3. $U_{PID} (t) = [k_{p} + k_{i} \frac{d^{- β}}{{dt}^{- β}} + k_{d} \frac{d^{α}}{{dt}^{α}}] e (t)$ (25)

4.3 IOFPD+IOI controller design

The structure of IOFPD+IOI controller is presented in this section and shown in Fig. 3 after considering α= 1 and β= 1 [24]. In this structure, addition of two different control methods, one is integer order fuzzy proportional and derivative (IOFPD) controller while other is integer order integrator (IOI) is implemented. The output u (t) of IOFPD+IOI controller can be expressed as; $u (t) = k_{u} (u_{1} (t) + u_{2} (t))$ (26)where, k_u is the output gain of IOFPD+IOI controller, u₁ is the output of IOFPD controller block which is explained in the later subsection and u₂ is the output of IOI controller block. u₂ can be expressed as; $u_{2} = k_{i} \int e (t) dt$ (27)

Fig.3

FOFPD+FOI controller block diagram.

In the present work, the gains of the IOFPD+IOI controller are tuned by CSA and for rest of the study these gains remain unchanged.

4.4 FOFPD+FOI controller design

An intelligent FOFPD+FOI controller is discussed in this section where a complete block diagram of same is shown in Fig. 3 [24]. Fractional order technique is considered in the present work because it gives the higher DOF and extra flexibility to design the controllers. In the proposed control structure, addition of two different control technique, one is fractional order fuzzy proportional and derivative (FOFPD) controller while other is fractional order integrator (FOI) is implemented. FOFPD controller block gives the control action according to the current position of error and rate of change of error while main task of fractional order integrator is to reduce the steady state error effectively. It is well known that the gains of the controller greatly affect performances of the controller therefore gains of the controller are tuned by a meta-heuristic algorithm namely, CSA and for rest of the analysis these gains are remain unaltered. There are four gains considered inside the FOFPD+FOI controller out of which k_p and k_d are input gains and k_u is the output gain of FOFPD controller whereas k_i is considered as the input gain of FOI controller block. The output u (t) of FOFPD+FOI controller can be expressed as; $u^{'} (t) = k_{u}^{'} (u_{1}^{'} (t) + u_{2}^{'} (t))$ (28) where, $k_{u}^{'}$ is the output gain of FOFPD+FOI controller, $u_{1}^{'}$ is the output of FOFPD controller block and $u_{2}^{'}$ is the output of FOI controller block. $u_{2}^{'}$ can be expressed as; $u_{2}^{'} (t) = k_{i} \frac{d^{- β} e (t)}{{dt}^{- β}}$ (29)

4.5 Details of FLC implementation

The structure of FLC scheme is shown in Fig. 4 which comprises fuzzification, fuzzy inference system, knowledge base and defuzzification blocks [36]. The FLC considered in the present work is constructed on two dimensional rule bases where triangular membership functions (MFs) are used. Other types of MFs like, Gaussian, trapezoidal, sigmoidal etc. can also be used to design an FLC but as triangular MF is easy to implement, it is considered here for simulation study. For

Fig.4

FLC block diagram.

input such as error and rate of change of error and for controller output, similar types of MFs are used as shown in Fig. 5 having the range between [–1, 1]. The acronyms of inputs and outputs MFs for FOFPD controllers are as, VN stands for ‘Very Negative’ NM stands for ‘Negative Medium, NS stands for ‘Negative Small’ ZE stands for ‘Zero’ PS stands for ‘Positive Small’ PM stands for ‘Positive Medium’ VP stands for ‘Very Positive’. A Mamdani type of inference mechanism and centre of gravity method for defuzzification is used here whereas for implication, minimum type of operator and for aggregation, maximum type of operator is used. Knowledge base which is the principal part of FLC design contains a set of rules for FOFPD controller which are tabulated in Table 1. This set of rules shows a three-dimensional surface plot for FOFPD controller which is shown in Fig. 6.

Fig.5

MFs for error, rate of change of error and output for FOFPD controller.

Fig.6

Surface plot for rule base of FPID controller.

Table 1

Input-output rule base of FOFPD controller

de\e	VN	NM	NS	ZE	PS	PM	VP
VN	VN	VN	VN	NM	NS	NS	ZE
NM	VN	NM	NM	NM	NS	ZE	PS
NS	VN	NM	NS	NS	ZE	PS	PM
ZE	VN	NM	NS	ZE	PS	PM	VP
PS	NM	NS	ZE	PS	PS	PM	VP
PM	NS	ZE	PS	PM	PM	PM	VP
VP	ZE	PS	PS	PM	VP	VP	VP

5 Optimization of controllers gains by CSA

To achieve the desired performance indices, the gains of the controllers must be tuned. CSA optimization algorithm is used in the present work to tuned gains of PID, FOPID, IOFPD+IOI and FOFPD+FOI controllers. The basic description of CSA is presented in the following subsection.

5.1 CSA description

CSA is a meta-heuristic optimization algorithm which is proposed by Yang and Deb [28, 29]. It is a bio-inspired algorithm based on bird Cuckoo and having the strong ability to find the solutions of multi-dimensional problems. It has efficient searching ability comparable to another optimization algorithms like, genetic algorithm, particle swarm optimization technique etc. The bird Cuckoo is a cleaver bird and it has pleasing sound and magnificent reproduction strategies. The development of CSA is based on the reproduction behaviour of bird Cuckoo. It laid her eggs in the nest of other bird’s nest which is random. The best solution (eggs) in the given search space is obtained by CSA from use of Lévy flight with proper step size. One of the best features of CSA is that the number of parameters and its convergence rate does not depend on its parameters and it is also better than other optimization algorithms for tuning multi-dimensional objective function. Parameter settings for CSA in the current work are shown in Table 2. The best solution is always kept in the search space provided by user by using elitism property. Basic fundamental rules proposed by Yang and Deb for the best optimizations by CSA are as follows:

Each Cuckoo lays one egg i.e. one solution at a time and dumps it in a random nest.

The best nest i.e. the nest which contains the highest quality eggs goes to next generation and other nests are discarded.

The available host nests are fixed and host can find the alien with a probability p_a ∈ [0, 1]. If the host can search and find out alien egg then it can either destroy or discard the nest so as to make an entirely new nest in a new place.

Table 2
Used parameters setting for CSA

Design Parameter Value

Number of Nests 25

Tolerance 1.0E-9

Step size 1.5

Probability of alien eggs 0.25

Maximum number of iteration 100

Design Parameter	Value
Number of Nests	25
Tolerance	1.0E-9
Step size	1.5
Probability of alien eggs	0.25
Maximum number of iteration	100

The most important attribute of CSA is the concept of Lévy flight, which is a complex random walk strategy that makes it prominent with the other optimization algorithms. The different parameters of CSA which are used to optimize gains of controllers are as follows:

5.2 Tuning of controller for trajectory tracking

All the existing simulations are carried out in MATLAB^®/SIMULINK^® (R2012a) environment on a personal computer having Intel core^TM i5 processor working at 3.33 GHz, 4 GB RAM with a 32-bit operating system. For ODE solver, fourth order Runge-Kutta method is used, working at 1 ms sampling time. It may be noted that throughout analysis, optimized gains stayed unchanged. The inputvoltage limitations for all three-links have been taken as [–12V, 12V]. The desired trajectories (θ_{r
₁}, θ_{r
₂} and θ_{r
₃}) for link-1, link-2 and link-3 have been given in (30), (31) and (32), respectively as follows, $θ_{r_{1}} = sin (0.4 π t)$ (30) $θ_{r_{2}} = sin (0.4 π t + \frac{π}{4})$ (31) $θ_{r_{3}} = sin (0.4 π t + \frac{π}{2})$ (32)

The objective function (OBF) has been taken as the weighted sum of integral of absolute error (IAE) and integral of absolute change in controller output (IACCO). $IACCO = \int | (u (t) - u (t - 1)) | dt$ (33) $OBF = (w_{1} * IAE) + (w_{2} * IACCO)$ (34)

The aggregate IAE is defined as the sum of the IAE values of all the three-links whereas the aggregate value of IACCO is defined as the sum of IACCO values of all three-links. The weightage parameters w₁ and w₂ are chosen as 0.999 and 0.001. Tuned gains are listed in Table 3 whereas the objective function vs. iteration curves are depicted in Fig. 7 for PID, FOPID, IOFPD+IOI and FOFPD+FOI controllers. In this figure, it can be clearly observed that FOFPD+FOI controller shows best convergence rate among four explored controllers. The objective functions are obtained as; 0.0426, 0.0244, 0.0230, and 0.0181 for PID, FOPID, IOFPD+IOI and FOFPD+FOI controllers, respectively. Table 4 replicates the IAE values of all three-links and objective function values obtained for all four controllers for trajectory tracking study. Figures 8 and 9 shows the performance for all four controllers in terms of trajectory tracking curves and controller outputs, respectively. Supremacy of FOFPD+FOI controller can be observed in all the investigations among four controllers where it shows 57.51% improvement to PID, 25.82% improvement to FOPID, 21.30% improvement to IOFPD+IOI controllers in objective function values for trajectory tracking analysis.

Fig.7

Objective function vs. iteration curve for PID, FOPID and FOFPD+FOI controller.

Fig.8

Trajectory tracking curves (a) Link-1 (b) Link-2 (c) Link-3.

Fig.9

Controller outputs for trajectory tracking (a) Link-1 (b) Link-2 (c) Link-3.

Table 3

Tuned gain parameters of PID, FOPID and FOFPD+FOI controllers

	PID			FOPID			IOFPD+IOI			FOFPD+FOI
	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3
k _p	167.95	366.95	578.23	998.25	38.16	708.9	15.91	40.99	2.22	0.854	1413.39	8.103
k _i	789.89	586.58	534.22	876.434	998.58	997.89	198.57	132.56	474.79	1672.47	0.703	1151.01
k _d	0.1	0.099	0.098	119.1	0.10	1.58	0.001	0.011	0.001	0.775	1991.51	1995.88
α	1	1	1	0.182	0.638	0.302	1	1	1	0.010	0.70	0.637
β	1	1	1	0.285	0.624	0.936	1	1	1	0.5	0.48	0.511
k _u	–	–	–	–	–	–	36.57	8.55	391.01	1355.26	0.10	0.307

Table 4

Obtained IAE values and OBFs for trajectory tracking study

Controllers	IAE Values			Objective Function
	Link-1	Link-2	Link-3
PID	0.0176	0.0138	0.0104	0.0426
FOPID	0.0043	0.0129	0.0071	0.0244
IOFPD+IOI	0.0116	0.0042	0.0068	0.0230
FOFPD+FOI	0.0031	0.0129	0.0014	0.0181

Table 5

Obtained IAE and objective function values for noise suppression study

Controllers	IAE Values			Objective Function
	Link-1	Link-2	Link-3
PID	0.0274	0.0391	0.0649	0.2350
FOPID	0.0192	0.0365	0.0412	0.1838
IOFPD+IOI	0.0290	0.0373	0.0640	0.2506
FOFPD+FOI	0.0232	0.0531	0.0251	0.1461

Table 6

IAE values for variation (increasing) in amplitude of sinusoidal disturbance signal

Amplitude of sinusoidal disturbance signal	IAE Values
	PID			FOPID			IOFPD+IOI			FOFPD+FOI
	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3
0.2	0.0181	0.0154	0.0119	0.0044	0.0144	0.0082	0.0119	0.0043	0.006	0.0032	0.0142	0.0016
0.4	0.0189	0.0171	0.0135	0.0046	0.0159	0.0093	0.0131	0.0045	0.0071	0.0033	0.0156	0.0018
0.6	0.0201	0.0189	0.0151	0.0049	0.0176	0.0105	0.0150	0.0048	0.0084	0.0035	0.0171	0.0019
0.8	0.0216	0.0208	0.0167	0.0053	0.0193	0.0116	0.0176	0.0055	0.0102	0.0040	0.0187	0.0022
1.0	0.0233	0.0227	0.0183	0.0057	0.0210	0.0127	0.0203	0.0062	0.0121	0.0044	0.0203	0.0024

Table 7

Objective function values for variation (increasing) in amplitude of sinusoidal disturbance signal

Objective function values
Amplitude of sinusoidal disturbance signal	PID	FOPID	IOFPD+IOI	FOFPD+FOI
0.2	0.0463	0.0271	0.0233	0.0197
0.4	0.0504	0.0300	0.0250	0.0214
0.6	0.0550	0.0330	0.0285	0.0233
0.8	0.0599	0.0362	0.0335	0.0257
1.0	0.0652	0.0395	0.0388	0.0280

Table 8

IAE values for variation (increasing) in amplitude of step disturbance signal

Amplitude of sinusoidal disturbance signal	IAE Values
	PID			FOPID			IOFPD+IOI			FOFPD+FOI
	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3	Link-1	Link-2	Link-3
0.2	0.0174	0.0147	0.0145	0.0043	0.0127	0.0070	0.0133	0.0049	0.0077	0.0031	0.0127	0.0014
0.4	0.0174	0.0146	0.0152	0.0042	0.0128	0.0069	0.0154	0.0056	0.0089	0.0030	0.0128	0.0014
0.6	0.0175	0.0146	0.0156	0.0042	0.0130	0.0070	0.0174	0.0063	0.0103	0.0031	0.0131	0.0014
0.8	0.0179	0.0148	0.0161	0.0043	0.0133	0.0071	0.0193	0.0070	0.0117	0.0031	0.0137	0.0015
1.0	0.0185	0.0152	0.0165	0.0044	0.0139	0.0074	0.0212	0.0077	0.0130	0.0033	0.0146	0.0015

Table 9

Objective function values for variation (increasing) in amplitude of step disturbance signal

Objective function values
Magnitude of step disturbance signal	PID	FOPID	IOFPD+IOI	FOFPD+FOI
0.2	0.0546	0.0241	0.0263	0.0179
0.4	0.0563	0.0240	0.0303	0.0180
0.6	0.0577	0.0243	0.0344	0.0184
0.8	0.0592	0.0249	0.0384	0.0192
1.0	0.0610	0.0259	0.0424	0.0203

6 Performance assessment

An intelligent controller must have the capability to completely eliminate the effect of sensor noise, measured or unmeasured disturbances and model uncertainties from the controlled variable. These phenomena can occur individually or in the combined form at the same time in the control loop and degrade the overall performance of closed loop controlsystem. Therefore, to further access the performance of considered controllers, following investigations such as noise suppression, disturbance rejection and parameter uncertainties have been carried out and presented in the following subsection.

6.1 Noise suppression

It is very difficult to avoid the noise in any closed loop control system. General, it is disruptive in nature which is non-distinguishable random fluctuations. Actually, it curb the real signal and try to make a feedback control system unstable. The noise can be appear in closed loop control system in feedforward path and feedback path as controller and sensor noise, respectively. Therefore, a controller must have sufficent robust to suppress the effect of noise in feedback loop. So, in order to investigate the robustness feature of all the considered controllers, a sensor noise was incorporated in the simulation study in feedback path as shown in Fig. 2 and mathematically described as follows: $θ_{s} (t) = (θ_{i} (t) + θ_{r_{i}} (t) * per * rand), i = 1, 2, 3;$ (35)

where, θ_i and θ_{r
_i} are the current and reference angular positions of link-1, link-2 and link-3, respectively; per represents the percentage of instant reference trajectory while rand generates the random number within the range [–1, 1].

For a typical case of 0.5% noise, attained values of IAE and objective functions for all three-links are listed in Table 5. Trajectory tracking and required voltage of all three-links for this particular case has been shown in Figs. 10 and 11. On the basis of obtained simulated results, it can be inferred that FOFPD+FOI controller performs best among the four controllers in the case of noise suppression. The improvements in objective function values by FOFPD+FOI controller with respect to PID, FOPID and IOFPD+IOI are estimated to be 37.82%, 20.51% and 41.69%, respectively in this case study.

Fig.10

Trajectory tracking curves for noise suppression study: (a) link-1 and (b) link-2 (c) link-3.

Fig.11

Controller outputs for noise suppression study: (a) link-1 (b) link-2 (c) link-3.

6.2 Disturbance rejection study

To track the required trajectory smoothly, the prime objective of a controller is to discard the effect of undesired disturbances inserted inside the feedback control loop. Under this context, two different types of disturbance signals are considered at controller output in the feedback control loop. One is sinusoidal while another is step disturbances. The mathematical expressions for sinusoidal and step disturbances are presented in (36) and (37), respectively. For sinusoidal disturbance signal ω₁ is considered as 0.4π rad/s. $d_{1} (t) = A_{1} sin (ω_{1} t)$ (36) $d_{2} (t) = A_{2} [u (t - 2) - u (t - 6)]$ (37)

For comparative study of robustness of all the controllers, total five cases of amplitude variations in sinusoidal as well as step disturbances are considered and the obtained results are depicted separately. Amplitude variations from 0.2 to 1.0 with a step of 0.2 in sinusoidal disturbance signal are performed and inserted at controller output separately and the corresponding IAE and objective function values are listed in Tables 6 and 7, respectively. Trajectory tracking and controller output curves for a case of sinusoidal disturbance of amplitude 1.0 are shown in Figs. 12 and 13, respectively. Similarly, magnitude variations from 0.2 to 1.0 with an increment of 0.2 in step disturbance signal is executed and injected at controller output independently and the obtained IAE and objective function values are listed in Tables 8 and 9, respectively. Trajectory tracking and controller output curves for a case of step disturbance of magnitude 1.0 are shown in Figs. 14 and 15, respectively. After careful observation of the obtained IAE and objective functions values for disturbance rejection cases, it can be easily noted that FOFPD+FOI controller outperforms among all four investigated controllers.

Fig.12

Trajectory tracking curves for sinusoidal disturbance rejection study at input of EDRRM (a) link-1 and (b) link-2 (c) link-3.

Fig.13

Controller outputs for sinusoidal disturbance rejection study at input of EDRRM (a) link-1 and (b) link-2 (c) link-3.

Fig.14

Trajectory tracking curves for step disturbance rejection study at input of EDRRM (a) link-1 and (b) link-2 (c) link-3.

Fig.15

Controller outputs for step disturbance rejection study at input of EDRRM (a) link-1 and (b) link-2 (c) link-3.

6.3 Robustness testing: Uncertainty in the mass of link

In industry, main task of the manipulator is to pick and place the object of different masses by its end-effector. After changing the mass of all three-links, a new system seems by the controller at run-time. An effective controller must nullify the effect of mass variation in all three-links. For robustness testing of controllers, mass of each link is increased as 10%. Obtained IAE and objective function values are listed in Table 10. After careful observation of results shown in Tables 10, it can be stated that FOFPD+FOI controller outperforms among four control techniques for the testing of uncertainty in mass of each link and it offered 90.57% improvements on PID controller, 84.83% improvements on FOPID controller and 58.97% improvements on IOFPD+IOI controller in objective functions values.

Table 10
Obtained IAE and objective function values for mass uncertainty study

Controllers IAE Values Objective Function

Link-1 Link-2 Link-3

PID 0.0182 0.0355 0.0826 0.2471

FOPID 0.0057 0.0258 0.0525 0.1536

IOFPD+IOI 0.0118 0.0078 0.0183 0.0568

FOFPD+FOI 0.0036 0.0169 0.0016 0.0233

Controllers	IAE Values	Objective Function
PID	0.0182	0.0355	0.0826	0.2471
FOPID	0.0057	0.0258	0.0525	0.1536
IOFPD+IOI	0.0118	0.0078	0.0183	0.0568
FOFPD+FOI	0.0036	0.0169	0.0016	0.0233

6.4 Robustness testing: Uncertainty in length of link

In this subsection, further for robustness testing of controllers, length of each link is increased as 1 cm. Acquired IAE and objective function values are shown in Table 11. After comparing the results obtained in Table 11, it can be observed that FOFPD+FOI controller shows much better performance in IAE values as well as objective function values among four control techniques. After investigating uncertainty in length of each link, it revealed 89.39% enhancements on PID controller, 74.81% enhancements on FOPID controller while 89.87% enhancements on IOFPD+IOI controller in objective functions values.

Table 11
Obtained IAE and objective function values for length uncertainty study

Controllers IAE Values Objective Function

Link-1 Link-2 Link-3

PID 0.0179 0.0331 0.0730 0.2234

FOPID 0.0050 0.0201 0.0312 0.0941

IOFPD+IOI 0.0117 0.0042 0.0068 0.2340

FOFPD+FOI 0.0043 0.0155 0.0021 0.0237

Controllers	IAE Values	Objective Function
PID	0.0179	0.0331	0.0730	0.2234
FOPID	0.0050	0.0201	0.0312	0.0941
IOFPD+IOI	0.0117	0.0042	0.0068	0.2340
FOFPD+FOI	0.0043	0.0155	0.0021	0.0237

7 Conclusion

Control of nonlinear, coupled, multi-input multi-output system like electrically driven rigid robotic manipulator (EDRRM) is always a challenging task for researchers and engineers. A robust intelligent control technique, fractional order fuzzy proportional and derivative with fractional order integrator (FOFPD+FOI) is proposed to control an EDRRM system. The performance of controller is tested for trajectory tracking, noise suppression, disturbance rejection and model uncertainties. To show the superiority of the proposed controller, its performance is compared with proportional, integral and derivative (PID) controller, fractional order PID (FOPID) controller as well as integer order fuzzy proportional and derivative with integer order integrator (IOFPD+IOI) controller. For trajectory tracking, 57.51%, 25.82% and 21.30% improvements have been observed in objective function values by FOFPD+FOI on PID, FOPID and IOFPD+IOI controllers, respectively. Further, for noise suppression investigation, 0.5% random noise signal of current trajectory is inserted to the sensor and 37.82%, 20.51% and 41.69% improvements have been observed in objective function values by FOFPD+FOI on PID, FOPID and IOFPD+IOI controllers, respectively. To examine the robust behaviour of the proposed controller, two types of disturbance signals, sinusoidal and step, are injected at the controller output inside the feedback control loop. Also, for model uncertainties, mass of each links are increased as 10% while length of each links are increased as 1.0 cm. After investigating the obtained IAE values for disturbance rejection and model uncertainties it has been clearly observed that FOFPD+FOI controller demonstrated much superior performances among the all considered controllers.

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de\e	VN	NM	NS	ZE	PS	PM	VP
VN	VN	VN	VN	NM	NS	NS	ZE
NM	VN	NM	NM	NM	NS	ZE	PS
NS	VN	NM	NS	NS	ZE	PS	PM
ZE	VN	NM	NS	ZE	PS	PM	VP
PS	NM	NS	ZE	PS	PS	PM	VP
PM	NS	ZE	PS	PM	PM	PM	VP
VP	ZE	PS	PS	PM	VP	VP	VP

de\e	VN	NM	NS	ZE	PS	PM	VP
VN	VN	VN	VN	NM	NS	NS	ZE
NM	VN	NM	NM	NM	NS	ZE	PS
NS	VN	NM	NS	NS	ZE	PS	PM
ZE	VN	NM	NS	ZE	PS	PM	VP
PS	NM	NS	ZE	PS	PS	PM	VP
PM	NS	ZE	PS	PM	PM	PM	VP
VP	ZE	PS	PS	PM	VP	VP	VP

A fractional order fuzzy PD+I controller for three-link electrically driven rigid robotic manipulator system

Abstract

Keywords

1 Introduction

2 Dynamic model of EDRRM system

5.1 CSA description

Table 2 Used parameters setting for CSA Design Parameter Value Number of Nests 25 Tolerance 1.0E-9 Step size 1.5 Probability of alien eggs 0.25 Maximum number of iteration 100

6.1 Noise suppression

Table 10 Obtained IAE and objective function values for mass uncertainty study Controllers IAE Values Objective Function Link-1 Link-2 Link-3 PID 0.0182 0.0355 0.0826 0.2471 FOPID 0.0057 0.0258 0.0525 0.1536 IOFPD+IOI 0.0118 0.0078 0.0183 0.0568 FOFPD+FOI 0.0036 0.0169 0.0016 0.0233

Table 11 Obtained IAE and objective function values for length uncertainty study Controllers IAE Values Objective Function Link-1 Link-2 Link-3 PID 0.0179 0.0331 0.0730 0.2234 FOPID 0.0050 0.0201 0.0312 0.0941 IOFPD+IOI 0.0117 0.0042 0.0068 0.2340 FOFPD+FOI 0.0043 0.0155 0.0021 0.0237

References

Table 2
Used parameters setting for CSA

Design Parameter Value

Number of Nests 25

Tolerance 1.0E-9

Step size 1.5

Probability of alien eggs 0.25

Maximum number of iteration 100

Table 10
Obtained IAE and objective function values for mass uncertainty study

Controllers IAE Values Objective Function

Link-1 Link-2 Link-3

PID 0.0182 0.0355 0.0826 0.2471

FOPID 0.0057 0.0258 0.0525 0.1536

IOFPD+IOI 0.0118 0.0078 0.0183 0.0568

FOFPD+FOI 0.0036 0.0169 0.0016 0.0233

Table 11
Obtained IAE and objective function values for length uncertainty study

Controllers IAE Values Objective Function

Link-1 Link-2 Link-3

PID 0.0179 0.0331 0.0730 0.2234

FOPID 0.0050 0.0201 0.0312 0.0941

IOFPD+IOI 0.0117 0.0042 0.0068 0.2340

FOFPD+FOI 0.0043 0.0155 0.0021 0.0237

de\e	VN	NM	NS	ZE	PS	PM	VP
VN	VN	VN	VN	NM	NS	NS	ZE
NM	VN	NM	NM	NM	NS	ZE	PS
NS	VN	NM	NS	NS	ZE	PS	PM
ZE	VN	NM	NS	ZE	PS	PM	VP
PS	NM	NS	ZE	PS	PS	PM	VP
PM	NS	ZE	PS	PM	PM	PM	VP
VP	ZE	PS	PS	PM	VP	VP	VP