Evolving an interval type-2 fuzzy PID controller for the redundant robotic manipulator

Highlights

• Implementation of IT2FPD+I controllers for redundant robot manipulator.

• Control of redundant robot and comparison with T1FPID and conventional PID.

• Computation of upper and lower search rang for optimization technique is presented.

• Robustness testing is also investigated for proposed controllers.

• The IT2FPID controllers are found to be superior for trajectory tracking problem.

Abstract

Robotic manipulators are a multi-input multi-output, dynamically coupled, highly time-varying, complex and highly nonlinear systems wherein the external disturbances, parameter variations, and random noise adversely affects the performance of the robotic system. Therefore, in order to deal with such complexities, however, an intriguing task for control researchers, these systems require an efficient and robust controller. In this paper, a novel application of genetic algorithms (GA) optimization approach to optimize the scaling factors of interval type-2 fuzzy proportional derivative plus integral (IT2FPD+I) controllers is proposed for 5-DOF redundant robot manipulator for trajectory tracking task. All five controllers' parameters are optimized simultaneously. Further, a procedure for selecting appropriate initial search space is also demonstrated. In order to make a fair comparison between different controllers, the tuning of each of the controllers' parameters is done with GA. This optimization technique uses the time domain optimal tuning while minimizing the fitness function as the sum of integral of multiplication of time with square error (ITSE) for each joint. To ascertain the effectiveness of IT2FPID controller, it is compared against type-1 fuzzy PID (T1FPID) and conventional PID controllers. Furthermore, robustness testing of developed IT2FPID controller for external disturbances, parameter variations, and random noise rejection is also investigated. Finally, the experimental study leads us to claim that our proposed controller can not only assure best trajectory tracking in joint and Cartesian space, but also improves the robustness of the systems for external disturbances, parameter variations, and random noise.

Introduction

Robots are mechanized devices that have a certain degree to duplicate the working of humans, whenever they need continuous operations. Further, they provide more accuracy, high strength and reduce the danger in case of medical fields, industrial operations and nuclear plants (Spong & Vidyasagar, 2008). Controlling the motion of a manipulator robot's joints is a running issue, whose study is of great importance in order to make its terminal effectors follow a predefined trajectory with the minimum deviation (Spong & Fujita, 2011). Robotic manipulator systems contain unavoidable uncertainties such as structured and unstructured uncertainties that degraded performance of manipulators. Correct dynamical model with parameter variations, sizes and mass distributions of payloads, difference in links properties are defined as structured uncertainties present in robotic manipulators. Unstructured uncertainties are defined as unmodelled dynamics, which contain the external disturbances, nonlinear frictions and random noise (Song, Yi, Zhao, & Li, 2005). To combat with such challenges, it is essential to develop a robust controller for these systems.

Controlling of robot manipulator requires the complex dynamic equation (mathematical modeling) and trajectory planning, which is computationally intensive. The conventional control theories such as PID have been successfully applied to areas where systems are well defined. However, when the system is complex, nonlinear, ill-structured process and with excessive parameter variations, the potency of conventional controller becomes poor. With the advancement of technology, there are imperative need to design controllers that are able to manage structured and unstructured uncertainties (Laxmidhar & Kar, 2010). The fuzzy set technique given by Zadeh has become a powerful modeling tool for solving severe real world problems with uncertain and unpredictable environment. The fuzzy logic controllers (FLCs) come in the category of intelligent control and as an intelligent controller, FLCs (type-1 fuzzy logic controllers, i.e., T1FLCs) parameter is easily tuned by non-expertise person. The operation of T1FLCs is based on human expertise, and the knowledge acquisition techniques to convert human expertise to appropriate if-then rules as well as a proper fuzzy membership functions (MFs) for each fuzzy variable. The T1FLC has some important advantages such as, (1) it provides a higher level of automation by incorporating expert knowledge, (2) it do not require exact knowledge of the dynamics model of the controlled system, (3) it plays a major role while controlling complex non-linear systems, (4) it reduces development and maintenance time.

There are some drawbacks of T1FLCs, i.e., it can't fully deal with or handle the linguistic and numerical uncertainties connected with dynamic structured or unstructured environments. Further, its performance is not satisfactory, because the ordinary T1FLCs have limited capabilities to directly handle data uncertainties (Mendel, 2007). There are generally five sources of uncertainties observed in T1FLC (Hagras, 2004, Hagras, 2007, Mendel and John, 2002), which are listed here (1) Uncertainties in the inputs to the T1FLC are produced by noise and change in environmental conditions of sensors. These uncertainties are translated into uncertainties in the antecedents MFs. In addition, the input sensors can be affected by the conditions of observation, i.e., their characteristics can be changed by the environmental conditions such as wind, sunshine, humidity, rain, etc. (2) The output of T1FLC is applied to actuators to control the plant. Change in actuators characteristics is generally due to internal and external changes such as wear, tear, and environmental changes, entails uncertainties in control output of T1FLC. Further, these uncertainties in control outputs of T1FLC are translated into uncertainties in the consequents MFs of the T1FLC. Once the type-1 MFs have been chosen, all the uncertainties could not be considered because type-1 MFs are totally precise. For example, a “fast speed” on a sunny day with the dry ground may be different from “fast speed” on a rainy day with the muddy ground as wheels may slip. (3) Linguistic uncertainties can be produced in linguistic labels used to define MFs. As the meaning of words defining the antecedents and consequents linguistic labels can be uncertain as words mean different things to different people. Therefore, experts do not always agree and often provide different consequents for the same antecedents. A survey of experts will usually lead to a histogram of possibilities for the consequent of a rule; this histogram represents the uncertainty about the consequent of a rule (4) As discussed above, uncertainties associated with the change in the operating conditions of the controller can also be translated into uncertainties in the antecedents and/or consequents MFs. For these uncertainties the chosen T1FLC, with precise MFs, might not be appropriate anymore and can become sub-optimal. (5) Uncertainties associated with the use of noisy training data that could be used to learn, tune or optimize the T1FLC. All of these mentioned uncertainties are not able to directly deal with T1FLC because their MFs are totally crisp (Mendel & John, 2002). Further, the concept of type-2 fuzzy sets (T2FSs), originally introduced by Zadeh (1975), is able to model such uncertainties because their MFs are themselves fuzzy. The concept of interval type-2 fuzzy sets (IT2FSs) is prolongation of the concept of type-1 fuzzy sets (T1FSs), essentially “fuzzy fuzzy” sets where the fuzzy degree of membership is T1FSs. Unlike T1FSs, where membership grade is a crisp number in [0, 1], IT2FSs are characterized by fuzzy MFs. Each element of IT2FSs is fuzzy sets in [0, 1]. The MFs of T1FSs are characterized by two-dimensional MFs, whereas IT2FSs are characterized by three-dimensional MFs. Lately, it is very useful in the situations where MFs of fuzzy sets are difficult to determine (Hagras, 2007, Maity and Sil, 2009, Mendel et al., 2006, Wagner and Hagras, 2007).

From the last two decades, a number of applications related to interval type-2 fuzzy logic controllers (IT2FLCs) have been increased in control theory due to its extra flexibility. Several authors have demonstrated IT2FLCs for various plants and various applications. A novel application on optimizing antecedent MFs through Big Bang–Big Crunch (BB–BC) optimization technique is proposed for IT2FPID controller in cascade control architecture. Several simulations and experimental studies are performed to evaluate the path tracking problem of mobile robot. The simulation results show that the optimized IT2FPID controller performance is enhanced over optimized PID and T1FPID controller even in the presence of uncertainties caused by the internal dynamics and disturbances (Kumbasar & Hagras, 2014). El-Nagar and El-Bardini proposed an analytical structure of the IT2FPID controller, which is a parallel combination of IT2FPD and IT2FPI controllers, implemented with new simplified method for type-reduction. Finally, in order to test the efficacy of the IT2FPID controller, it is applied to a coupled-tank system, and resulting controller performance is better than other type reduction methods (El-Nagar & El-Bardini, 2014). Several papers have investigated the implementation of IT2FLC for different applications such as thyristor control series compensator (TCSC) (Panda, Pillai, & Kumar, 2013), inverted pendulum system (El-Bardini & El-Nagar, 2014), magnetic levitation applications (Kumar & Kumar, 2015) and 3-DOF Helicopter (Mehndiratta, Kayacan, & Kumbasar, 2016) etc. Finally, this comprehensive survey of the recent literature evinced the variety of applications that have been achieved by IT2FLCs and it is also clear that IT2FLCs surpassed its type-1 counterparts in many applications.

The significance of this paper is to develop the IT2FPD+I controller from T1FPD+I controller and to implement IT2FPD+I controller for a redundant robot manipulator for trajectory tracking problem with robustness analysis. The proposed controller applied with IT2FSs, which are useful to represent the uncertainty about the membership grades and defined by the region between upper and lower MFs. Here, the MFs are three dimensional and include footprint of uncertainty, these provide additional degrees of freedom that make it possible to directly model and handle uncertainties. It is noted that the operation of robotic manipulator is adversely affected by various parameter uncertainties, external disturbances, and noise rejection. This is a staple motivation factor for designing a robust, flexible and effective controller called IT2FPID, which can suppress above complexities that deteriorate the performance of the robotic manipulator.

Major contributions of the presented study are as follow: (1) The successful development of T1FPD+I controller to IT2FPD+I controller. (2) The proposed controller is an exploration of IT2FSs for designing the IT2FLC, which introduces some extra degree of freedom for better tuning and robustness of the proposed controller. (3) The problem under consideration has five-link manipulator; each link is assigned a controller and each controller contains four control variable which results in a total of 20 controller parameters to be optimized simultaneously. It is a difficult task to tune such a large number of parameters simultaneously, therefore, an evolutionary algorithm called GA is used to achieve the optimal parameters of proposed controllers. (4) For controller optimization, the initial search range is a crucial factor since if the search space is not properly bounded, then there is very less chance to get a singular solution. Thus, in order to circumvent this issue, a procedure for selecting appropriate initial search space is adopted. The major advantage of this is that, it reduces a large number of experiments for achieving an optimal solution. (5) To show the efficacy of the IT2FPID controller, it is compared with tuned T1FPID and conventional PID controllers for trajectory tracking. (6) The robustness of a system tells about the resilience of that system towards changes. So, it is important to test the robustness of a system. In this regard, our proposed system is examined for external disturbance rejection, changing parameters of the manipulator, and random noise rejection.

In this paper a GA based optimally tuned IT2FPID controller which uses IT2FSs is designed for the SCARA type redundant robot. This paper is prepared as follows. A review of interval type-2 fuzzy logic system (IT2FLS) is presented in Section 2. In Section 3, modeling of the redundant manipulator is presented. Design and implementation of IT2FPID and the selection of upper bound and lower bound parameters for GA optimization technique are presented in Section 4. Simulation results are presented for tracking of the robotic system in joint and Cartesian space and robustness testing is also discussed in Section 5. In the last conclusion is discussed in Section 6.