Published Online November 2013 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijmecs.2013.10.07
Design Sliding Mode Modified Fuzzy Linear Controller with Application to Flexible Robot
Manipulator
Mahdi Mirshekaran
Research and Development Unit, SanatkadeheSabze Pasargad Company, (S.S.P. Co), Shiraz, Iran
Email: mahdi_mirshekaran@yahoo.com
Farzin Piltan
Senior Researcher at Research and Development Unit, SanatkadeheSabze Pasargad Company, (S.S.P. Co), Shiraz, Iran
Email: Piltan_f@iranssp.com; WWW.IRANSSP.COM
Zahra Esmaeili
Research and Development Unit, SanatkadeheSabze Pasargad Company, (S.S.P. Co), Shiraz, Iran
Email: zahra_esmaeili@rocketmail.com
Tannaz Khajeaian
Research and Development Unit, SanatkadeheSabze Pasargad Company, (S.S.P. Co), Shiraz, Iran
Email: tannazi.kh@gmail.com
Meysam Kazeminasab
Research and Development Unit, Electrical and Electronic Researcher, SSP. Co, Shiraz/Iran
Email: kazemi.m1990@ymail.com
Abstract — This paper studies the use of Modified advantages of flexible manipulators include small mass, Proportional-Integral-Derivative Sliding Mode Controller fast motion, and large force to mass ratio, which are (MPIDSMC) control used to control a flexible reflected directly in the reduced energy consumption, manipulator. The control gain in the MPIDSMC increased productivity, and enhanced payload capacity. controller has been determined in an empirical way so far. Unlike the rigid manipulators, the difficulties facing the It is a considerable time-consuming process because the usage of flexible manipulators are numerous. The control performance depends not only on the control gain modeling of the flexibility of the manipulator is one of but also on the other parameters such as the payload, the challenges, the non-minimum phase problem, which references and PID joint servo gains. Hence, the control appears from the modeling of the flexible manipulators, is gain must be tuned considering the other parameters. In also another challenge. The precise and availability of the order to find the optimal control gain for the MPIDSMC measured variables used in the control is the third controller, a fuzzy logic approach is proposed in this challenge. The control of flexible manipulators has been paper. The proposed fuzzy logic scheme finds an studied with great interest by many researchers over the optimum control gain that minimizes the tip vibration for past years due to its pronounced benefits. To find a the end effector of the flexible manipulator. Tuned gain controller that can achieve the end effector position of the response results are compared to results for other types of flexible manipulator in a short time in addition to a gains. The effectiveness of using the fuzzy logic appears suppression of its vibration to be able to achieve the tasks in the reduction of the computational time and the ability is the main goal of the control of flexible manipulator in to tune the gain with different loading condition and input the free space. Although significant progresses have been parameters. made in many aspects over the last two decades, many issues are not yet resolved yet, and simple, effective, and Index Terms — Modified PID Controller, Flexible reliable controls of flexible manipulators remain open manipulator, sliding mode controller, vibration of end requests [1-15].
There are several methods for controlling a flexible effector, fuzzy logic theory
robot manipulator, which all of them follow important
main goals, namely, acceptable performance [16-21]. However, the mechanical design of flexible robot I. INTRODUCTION
manipulator is very important to select the best controller
The flexible manipulator started to play an important to have a reduce vibration but in general two types part in many engineering applications nowadays. Major
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
54
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
schemes can be presented, namely, a joint space control schemes and an operation space control schemes[13-23]. One of the simplest ways to analysis control of multiple Degrees Of Freedom (DOF) flexible robot manipulator is analyzed each joint separately such as SISO systems and design an independent joint controller for each joint. In this methodology, the coupling effects between the joints are modeled as disturbance inputs. To make this controller, the inputs are modeled as: total velocity/displacement and disturbance. Design a controller with the same formulation and different coefficient, low cost hardware and simple structure controller are some of most important independent-joint space controller advantages. In the absence of robot knowledge, proportional-integral-derivative (PID) may be the best controller, because it is model-free, and its parameters can be adjusted easily and separately [24-36]. And it is the most used in flexible robot manipulators. In order to remove steady-state error caused by uncertainties and noise, the integrator gain has to be increased. This leads to worse transient performance, even destroys the stability. The integrator in a PID controller also reduces the bandwidth of the closed-loop system. PD control guarantees stability only when the PD gains tend to infinity, the tracking error does not tend to zero when friction and gravity forces are included in the flexible robot manipulator dynamics [38-69]. Model-based compensation for PD control is an alternative method to substitute PID control [10-28], such as adaptive gravity compensation [11-33], desired gravity compensation [16-49], and PD+ with position measurement [18-69]. They all needed structure information of the robot gravity. Some nonlinear PD controllers can also achieve asymptotic stability, for example PD control with time-varying gains [5], PD control with nonlinear gains [6], and PD control with feedback linearization compensation [8]. But these controllers are complex; many good properties of the linear PID control do not exist because these controllers do not have the same form as the industrial PID. Nonlinear controllers divided into six groups, namely, feedback linearization (computed-torque control), passivity-based control, sliding mode control (variable structure control), artificial intelligence control, Lyapunov-based control and adaptive control[10-69]. Sliding mode controller (SMC) is a powerful nonlinear controller which has been analyzed by many researchers especially in recent years. This theory was first proposed in the early 1950 by Emelyanov and several co-workers and has been extensively developed since then with the invention of high speed control devices [12]. The main reason to opt for this controller is its acceptable control performance in wide range and solves two most important challenging topics in control which names, stability and robustness [7, 17-48]. Sliding mode control theory for control of flexible robot manipulator was first proposed in 1978 by Young to solve the set point problem (𝒒𝒒𝒅𝒅=𝟎𝟎) by discontinuous method in the following form;
where 𝑆𝑆𝑖𝑖 is sliding surface (switching surface), 𝑖𝑖=1,2,……,𝑛𝑛 for n-DOF flexible robot manipulator, 𝜏𝜏𝑖𝑖(𝑞𝑞,𝑡𝑡) is the 𝑖𝑖𝑡𝑡ℎ torque of joint. In recent years, artificial intelligence theory has been used in sliding mode control systems. Neural network, fuzzy logic and neuro-fuzzy are synergically combined with nonlinear classical controller and used in nonlinear, time variant and uncertain plant (e.g., robot manipulator). Fuzzy logic controller (FLC) is one of the most important applications of fuzzy logic theory. This controller can be used to control nonlinear, uncertain, and noisy systems. This method is free of some model techniques as in model-based controllers. As mentioned that fuzzy logic application is not only limited to the modeling of nonlinear systems [31-45] but also this method can help engineers to design a model-free controller. Control robot arm manipulators using model-based controllers are based on manipulator dynamic model. These controllers often have many problems for modeling. Conventional controllers require accurate information of dynamic model of robot manipulator, but most of time these models are MIMO, nonlinear and partly uncertain therefore calculate accurate dynamic model is complicated [32]. The main reasons to use fuzzy logic methodology are able to give approximate recommended solution for uncertain and also certain complicated systems to easy understanding and flexible. Fuzzy logic provides a method to design a model-free controller for nonlinear plant with a set of IF-THEN rules [22-69]. Normal combinations of fuzzy logic methodology (FLM) and modified PID sliding mode controller (MPIDSMC) are to apply fuzzy parallel compensator at the same time [22-45], while FLM compensates the control error, MPIDSMC reduce the error of fuzzy inference system such that the final tracking error is asymptotically stable [12-38]. This paper is organized as follows; second part focuses on the modeling dynamic formulation based on Lagrange methodology, fuzzy logic methodology and computed torque controller to have a robust control. Third part is focused on the methodology which can be used to reduce the error, increase the performance quality and increase the robustness and stability. Simulation result and discussion is illustrated in forth part which based on trajectory following and disturbance rejection. The last part focuses on the conclusion and compare between this method and the other ones.
II．THEORY
Dynamic Formulation of Flexible Robot Manipulator: It can be evinced from the force expressions that the total input forces acting on each module can be resolved into an additive component along the direction of extension and a subtractive component that results in a torque. For the first module, there is an additional torque produced by forces in the third module.
𝝉𝝉(𝒒𝒒,𝒕𝒕)
(𝒒𝒒,𝒕𝒕) 𝒊𝒊𝒊𝒊 𝑺𝑺𝒊𝒊>0𝝉𝝉+
=�𝒊𝒊 −
𝝉𝝉𝒊𝒊(𝒒𝒒,𝒕𝒕) 𝒊𝒊𝒊𝒊 𝑺𝑺𝒊𝒊<0
(1)
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
55
The model resulting from the application of Lagrange’s equations of motion obtained for this system can be represented in the form
𝑮𝑮�𝒒𝒒�=
(6)
−𝑚𝑚1𝑔𝑔−𝑚𝑚2𝑔𝑔+𝑘𝑘11(𝑐𝑐1+(1⁄2)𝜃𝜃1−𝑐𝑐01)+𝑘𝑘21(𝑐𝑐1−(1⁄2)𝜃𝜃1−𝑐𝑐01)−𝑚𝑚3𝑔𝑔⎡⎤⎢⎥⎢−𝑚𝑚2𝑔𝑔𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1)+𝑘𝑘12(𝑐𝑐2+(1⁄2)𝜃𝜃2−𝑐𝑐02)+𝑘𝑘22(𝑐𝑐2−(1⁄2)𝜃𝜃2−𝑐𝑐02)−𝑚𝑚3𝑔𝑔𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1)⎥⎢⎥
−𝑚𝑚3𝑔𝑔𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1+𝜃𝜃2)+𝑘𝑘13(𝑐𝑐3+(1⁄2)𝜃𝜃3−𝑐𝑐03)+𝑘𝑘23(𝑐𝑐3−(1⁄2)𝜃𝜃3−𝑐𝑐03)⎢⎥
⎢⎥⎢𝑚𝑚2𝑐𝑐2𝑔𝑔𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1)+𝑚𝑚3𝑐𝑐3𝑔𝑔𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1+𝜃𝜃2)+𝑚𝑚3𝑐𝑐2𝑔𝑔𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1)+𝑘𝑘11(𝑐𝑐1+(1⁄2)𝜃𝜃1−𝑐𝑐01)(1⁄2)⎥⎢⎥+𝑘𝑘21(𝑐𝑐1−(1⁄2)𝜃𝜃1−𝑐𝑐01)(−1⁄2)⎢⎥⎢⎥
⁄⁄⁄⁄((1(1)()(1()())))2𝜃𝜃2−𝑐𝑐022𝜃𝜃2−𝑐𝑐02−12⎥2+𝑘𝑘22𝑐𝑐2−⎢𝑚𝑚3𝑐𝑐3𝑔𝑔𝑐𝑐𝑖𝑖𝑛𝑛𝜃𝜃1+𝜃𝜃2+𝑘𝑘12𝑐𝑐2+
⎢⎥⎣⎦𝑘𝑘13(𝑐𝑐3+(1⁄2)𝜃𝜃3−𝑐𝑐03)(1⁄2)+𝑘𝑘23(𝑐𝑐3−(1⁄2)𝜃𝜃3−𝑐𝑐03)(−1⁄2)
𝑭𝑭𝒄𝒄𝒄𝒄𝒄𝒄𝒊𝒊𝒊𝒊 𝝉𝝉=𝑫𝑫�𝒒𝒒�𝒒𝒒+𝑪𝑪�𝒒𝒒�𝒒𝒒+𝑮𝑮�𝒒𝒒�
(2)
where 𝜏𝜏 is a vector of input forces and q is a vector of
generalized co-ordinates. The force coefficient matrix 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 transforms the input forces to the generalized forces and torques in the system. The inertia matrix, 𝐷𝐷 is composed of four block matrices. The block matrices that correspond to pure linear accelerations and pure angular accelerations in the system (on the top left and on the bottom right) are symmetric. The matrix 𝐶𝐶 contains coefficients of the first order derivatives of the generalized co-ordinates. Since the system is nonlinear, many elements of 𝐶𝐶 contain first order derivatives of the generalized co-ordinates. The remaining terms in the dynamic equations resulting from gravitational potential energies and spring energies are collected in the matrix 𝐺𝐺. The coefficient matrices of the dynamic equations are given below [46-55],
(3) 𝑭𝑭𝒄𝒄𝒄𝒄𝒄𝒄𝒊𝒊𝒊𝒊 =
Design PID Controller: Design of a linear
methodology to control of flexible robot manipulator was
very straight forward. Since there was an output from the torque model, this means that there would be two inputs into the PID controller. Similarly, the outputs of the controller result from the two control inputs of the torque signal. In a typical PID method, the controller corrects the error between the desired input value and the measured value. Since the actual position is the measured signal. Figure 1 is shown linear PID methodology, applied to flexible robot manipulator [56-69].
(7) 𝒄𝒄(𝒕𝒕)=𝜽𝜽𝒂𝒂(𝒕𝒕)−𝜽𝜽𝒅𝒅(𝒕𝒕)
(8) 𝑼𝑼=𝑲𝑲𝒄𝒄+𝑲𝑲𝒄𝒄+𝑲𝑲�𝒄𝒄
𝑷𝑷𝑰𝑰𝑫𝑫
𝒑𝒑𝒂𝒂
𝑽𝑽𝒂𝒂
𝑰𝑰
11𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1)⎡001⎢000⎢
1⁄2⎢1⁄2−1⁄2
01⁄2⎢0
⎣000𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1)
10−1⁄2−1⁄20𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1+𝜃𝜃2)𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1+𝜃𝜃2)
⎤
𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)
⎥
11⎥
1⁄2+𝑐𝑐2𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)−1⁄2+𝑐𝑐2𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)⎥
1⁄2−1⁄2⎥
⎦1⁄2−1⁄2
𝒎𝒎𝟏𝟏+𝒎𝒎𝟐𝟐⎡
⎢+𝒎𝒎𝟑𝟑⎢⎢
𝒎𝒎𝟐𝟐𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟏𝟏)⎢⎢+𝒎𝒎𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟏𝟏)⎢⎢
𝒎𝒎𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟏𝟏+𝜽𝜽𝟐𝟐)⎢⎢
⎢−𝒎𝒎𝟐𝟐𝒄𝒄𝟐𝟐𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏)⎢−𝒎𝒎𝟑𝟑𝒄𝒄𝟐𝟐𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏)⎢−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏+𝜽𝜽𝟐𝟐)⎢⎢
⎢−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏+𝜽𝜽𝟐𝟐)⎢⎢⎣𝟎𝟎
𝑫𝑫�𝒒𝒒�=
(4)
𝒎𝒎𝟐𝟐𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟏𝟏)
+𝒎𝒎𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟏𝟏)𝒎𝒎𝟐𝟐+𝒎𝒎𝟑𝟑
𝒎𝒎𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟏𝟏+𝜽𝜽𝟐𝟐)𝒎𝒎𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟐𝟐)
𝒎𝒎𝟑𝟑
−𝒎𝒎𝟐𝟐𝒄𝒄𝟐𝟐𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏)−𝒎𝒎𝟑𝟑𝒄𝒄𝟐𝟐𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏)−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏+𝜽𝜽𝟐𝟐)−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟐𝟐)𝒎𝒎𝟐𝟐𝒄𝒄𝟐𝟐𝟐𝟐
𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟐𝟐)
−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟏𝟏+𝜽𝜽𝟐𝟐)−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟐𝟐)
𝟎𝟎
⎥
⎥⎥𝟎𝟎⎥⎥⎥𝟎𝟎⎥⎥⎥𝑰𝑰𝟑𝟑⎥⎥⎥⎥𝑰𝑰𝟑𝟑⎥⎥⎥𝑰𝑰𝟑𝟑⎦⎤𝟎𝟎⎥
𝒎𝒎𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟐𝟐)−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟐𝟐)−𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟐𝟐)
𝟎𝟎
𝒎𝒎𝟑𝟑𝒄𝒄𝟐𝟐𝒄𝒄𝒊𝒊𝒔𝒔(𝜽𝜽𝟐𝟐)
𝟎𝟎
+𝑰𝑰𝟏𝟏+𝑰𝑰𝟐𝟐
𝟐𝟐
+𝑰𝑰𝟑𝟑+𝒎𝒎𝟑𝟑𝒄𝒄𝟐𝟐𝟐𝟐+𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑+𝟐𝟐𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟐𝟐)𝒄𝒄𝟐𝟐
𝟐𝟐
+𝑰𝑰𝟑𝟑𝑰𝑰𝟐𝟐+𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑
+𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟐𝟐)𝒄𝒄𝟐𝟐𝑰𝑰
Fig 1: Block diagram of linear PID method
𝑰𝑰𝟐𝟐+𝒎𝒎𝟑𝟑𝒄𝒄𝟐𝟐𝟑𝟑+𝑰𝑰𝟑𝟑+𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽𝟐𝟐)𝒄𝒄𝟐𝟐
𝟐𝟐
𝑰𝑰𝟐𝟐+𝒎𝒎𝟑𝟑𝒄𝒄𝟑𝟑+𝑰𝑰𝟑𝟑
𝟎𝟎𝑰𝑰𝟑𝟑𝑰𝑰𝟑𝟑
𝑪𝑪�𝒒𝒒�=
⎡
⎢⎢
⎢𝑐𝑐11+𝑐𝑐21⎢⎢⎢⎢⎢⎢⎢⎢
0⎢⎢⎢⎢⎢⎢⎢0⎢⎢⎢⎢
⎢(1⁄2)⎢(𝑐𝑐11+𝑐𝑐21)⎢⎢⎢⎢0⎢⎢⎢
0⎣
−2𝑚𝑚2𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1)𝜃𝜃1−2𝑚𝑚3𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1)𝜃𝜃1
−2𝑚𝑚3𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1+𝜃𝜃2)
+𝜃𝜃2��𝜃𝜃1
(5)
−𝑚𝑚2𝑐𝑐2
�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1)�𝜃𝜃1
+(1⁄2)(𝑐𝑐11+𝑐𝑐21)
−𝑚𝑚3𝑐𝑐2
�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1)�𝜃𝜃1
−𝑚𝑚3𝑐𝑐3
�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃1+𝜃𝜃2)�𝜃𝜃1
�−𝑚𝑚3𝑐𝑐3�𝜃𝜃1
+(1⁄2)(𝑐𝑐12+𝑐𝑐22)
�−𝑚𝑚3𝑐𝑐2�𝜃𝜃1
−𝑚𝑚3𝑐𝑐3
�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)�𝜃𝜃1−𝑚𝑚3𝑐𝑐3𝑐𝑐2�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)�𝜃𝜃1�−𝑚𝑚3𝑐𝑐3�𝜃𝜃12𝑚𝑚3𝑐𝑐3𝑐𝑐2�𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)�𝜃𝜃2
+(12⁄4)(𝑐𝑐11+𝑐𝑐21)𝑚𝑚3𝑐𝑐3𝑐𝑐2
�𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)�𝜃𝜃1
0
⎤
⎥⎥0⎥
⎥⎥⎥⎥⎥⎥⎥⎥0⎥
⎥⎥⎥⎥⎥
(1⁄2)
⎥
(𝑐𝑐13+𝑐𝑐23)⎥
⎥⎥⎥⎥0
⎥⎥⎥⎥⎥0
⎥⎥
(12⁄4)⎥(𝑐𝑐13+𝑐𝑐23)⎦
−𝑚𝑚3𝑐𝑐3𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃1+𝜃𝜃2)
𝑐𝑐12+𝑐𝑐22
−2𝑚𝑚3𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)
+𝜃𝜃2��𝜃𝜃1
−2𝑚𝑚3𝑐𝑐3
�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)�𝜃𝜃1
−𝑚𝑚3𝑐𝑐3
�𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)�𝜃𝜃2�−2𝑚𝑚3𝑐𝑐3�𝜃𝜃1�−𝑚𝑚3𝑐𝑐3�𝜃𝜃2𝑚𝑚3𝑐𝑐3𝑐𝑐2�𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)�𝜃𝜃2(12⁄4)
(𝑐𝑐12+𝑐𝑐22)
0
�2𝑚𝑚3𝑐𝑐𝑖𝑖𝑛𝑛(𝜃𝜃2)�𝜃𝜃1�2𝑚𝑚3𝑐𝑐3𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)�𝜃𝜃1�−2𝑚𝑚3𝑐𝑐2�𝜃𝜃1(1⁄2)(𝑐𝑐12+𝑐𝑐22)+�2𝑚𝑚3𝑐𝑐3𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)�𝜃𝜃1
0�+2𝑚𝑚2𝑐𝑐2�𝜃𝜃1
𝑐𝑐13+𝑐𝑐23+𝜃𝜃2�2𝑚𝑚3𝑐𝑐3�𝜃𝜃1
−2𝑚𝑚3𝑐𝑐2𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2)
+𝜃𝜃2��𝜃𝜃1
2𝑚𝑚3𝑐𝑐3+𝜃𝜃2��𝜃𝜃1
(1⁄2)(𝑐𝑐13−𝑐𝑐23)
The model-free control strategy is based on the assumption that the joints of the manipulators are all independent and the system can be decoupled into a group of single-axis control systems [18-23]. Therefore, the kinematic control method always results in a group of individual controllers, each for an active joint of the manipulator. With the independent joint assumption, no a priori knowledge of robot manipulator dynamics is needed in the kinematic controller design, so the complex computation of its dynamics can be avoided and the controller design can be greatly simplified. This is suitable for real-time control applications when powerful processors, which can execute complex algorithms rapidly, are not accessible. However, since joints coupling is neglected, control performance degrades as operating speed increases and a manipulator controlled in this way is only appropriate for relatively slow motion [44, 46]. The fast motion requirement results in even higher dynamic coupling between the various robot joints,
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
56
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
which cannot be compensated for by a standard robot controller such as PID [50], and hence model-based control becomes the alternative.
Sliding mode controller: Consider a nonlinear single input dynamic system is defined by [6]:
(9) �⃗)+𝒃𝒃(𝒙𝒙�⃗)𝒖𝒖 𝒙𝒙(𝒔𝒔)=𝒊𝒊(𝒙𝒙
Where u is the vector of control input, 𝒙𝒙(𝒔𝒔) is the 𝒔𝒔𝒕𝒕𝒕𝒕 derivation of 𝒙𝒙, 𝒙𝒙=[𝒙𝒙,𝒙𝒙,𝒙𝒙,…,𝒙𝒙(𝒔𝒔−𝟏𝟏)]𝑻𝑻 is the state vector, 𝒊𝒊(𝒙𝒙) is unknown or uncertainty, and 𝒃𝒃(𝒙𝒙) is of known sign function. The main goal to design this controller is train to the desired state; 𝒙𝒙𝒅𝒅=[𝒙𝒙𝒅𝒅,𝒙𝒙𝒅𝒅,𝒙𝒙𝒅𝒅,…,𝒙𝒙𝒅𝒅(𝒔𝒔−𝟏𝟏)]𝑻𝑻, and trucking error vector is defined by [6]:
(𝒔𝒔−𝟏𝟏)𝑻𝑻(10) �=𝒙𝒙−𝒙𝒙𝒅𝒅=[𝒙𝒙�,…,𝒙𝒙�𝒙𝒙]
A time-varying sliding surface 𝒄𝒄(𝒙𝒙,𝒕𝒕) in the state space 𝒔𝒔
𝑹𝑹 is given by [6]:
𝒅𝒅(11)
�=𝟎𝟎 𝒄𝒄(𝒙𝒙,𝒕𝒕)=(+𝝀𝝀)𝒔𝒔−𝟏𝟏 𝒙𝒙
𝒅𝒅𝒕𝒕
where λ is the positive constant. To further penalize tracking error, integral part can be used in sliding surface part as follows [6]:
𝒕𝒕
𝑑𝑑(12) 𝒔𝒔−𝟏𝟏
�𝒅𝒅𝒕𝒕�=𝟎𝟎 𝒄𝒄(𝒙𝒙,𝒕𝒕)=(+𝝀𝝀) ��𝒙𝒙
𝑑𝑑𝑡𝑡𝟎𝟎The main target in this methodology is kept the sliding
surface slope 𝒄𝒄(𝒙𝒙,𝒕𝒕) near to the zero. Therefore, one of the common strategies is to find input 𝑼𝑼 outside of 𝒄𝒄(𝒙𝒙,𝒕𝒕) [6].
𝟏𝟏𝒅𝒅𝟐𝟐(13)
𝒄𝒄(𝒙𝒙,𝒕𝒕)≤−𝜻𝜻|𝒄𝒄(𝒙𝒙,𝒕𝒕)| 𝟐𝟐𝒅𝒅𝒕𝒕
where ζ is positive constant.
𝐝𝐝(14) If S(0)>0→𝐒𝐒(𝐝𝐝)≤−𝛇𝛇
𝐝𝐝𝐝𝐝
Equation (17) guarantees time to reach the sliding surface is smaller than outside of 𝑆𝑆(𝑡𝑡).
(18) 𝒊𝒊𝒊𝒊 𝑺𝑺𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕=𝑺𝑺(𝟎𝟎)→𝒄𝒄𝒓𝒓𝒓𝒓𝒄𝒄𝒓𝒓(𝒙𝒙−𝒙𝒙𝒅𝒅)=𝟎𝟎
suppose S is defined as
𝒅𝒅(19) �=(𝐱𝐱−𝐱𝐱𝐝𝐝)+𝒄𝒄(𝒙𝒙,𝒕𝒕)=(+𝝀𝝀) 𝒙𝒙𝒅𝒅𝒕𝒕
𝛌𝛌(𝐱𝐱−𝐱𝐱𝐝𝐝)
The derivation of S, namely, 𝑆𝑆 can be calculated as the following;
(20) 𝑺𝑺=(𝐱𝐱−𝐱𝐱𝐝𝐝)+𝛌𝛌(𝐱𝐱−𝐱𝐱𝐝𝐝)
suppose the second order system is defined as;
(21) 𝒙𝒙=𝒊𝒊+𝒖𝒖 →𝑺𝑺=𝒊𝒊+𝑼𝑼−𝒙𝒙𝒅𝒅
+𝛌𝛌(𝐱𝐱−𝐱𝐱𝐝𝐝)
Where 𝒊𝒊 is the dynamic uncertain, and also since
� is 𝑆𝑆=0 𝑟𝑟𝑛𝑛𝑑𝑑 𝑆𝑆=0, to have the best approximation ,𝑼𝑼
defined as
�+𝒙𝒙𝒅𝒅−𝝀𝝀(𝐱𝐱−𝐱𝐱𝐝𝐝) �=−𝒊𝒊(22) 𝑼𝑼
A simple solution to get the sliding condition when the dynamic parameters have uncertainty is the switching control law [52-53]:
�−𝑲𝑲(𝒙𝒙(23) �⃗,𝒕𝒕)∙𝐬𝐬𝐬𝐬𝐬𝐬(𝒄𝒄) 𝑼𝑼𝒅𝒅𝒊𝒊𝒄𝒄=𝑼𝑼
where the switching function 𝐬𝐬𝐬𝐬𝐬𝐬(𝐒𝐒) is defined as [1, 6]
𝟏𝟏 𝒄𝒄>0(24)
𝒄𝒄𝒔𝒔𝒔𝒔(𝒄𝒄)=�−𝟏𝟏 𝒄𝒄<0
𝟎𝟎 𝒄𝒄=𝟎𝟎
�⃗,𝒕𝒕) is the positive constant. Suppose by (25) and the 𝑲𝑲(𝒙𝒙
the following equation can be written as,
𝟏𝟏𝒅𝒅𝟐𝟐(25) �−𝑲𝑲𝐬𝐬𝐬𝐬𝐬𝐬(𝒄𝒄)�∙𝒄𝒄(𝒙𝒙,𝒕𝒕)=𝐒𝐒∙𝐒𝐒=�𝒊𝒊−𝒊𝒊𝟐𝟐𝒅𝒅𝒕𝒕
��∙𝑺𝑺−𝑲𝑲|𝑺𝑺| 𝑺𝑺=�𝒊𝒊−𝒊𝒊
and if the equation (17) instead of (16) the sliding surface can be calculated as
𝒕𝒕𝒅𝒅(26) �𝒅𝒅𝒕𝒕�=(𝐱𝐱−𝐱𝐱𝐝𝐝)+𝒄𝒄(𝒙𝒙,𝒕𝒕)=(+𝝀𝝀)𝟐𝟐 �∫𝒙𝒙𝟎𝟎
in this method the approximation of 𝑼𝑼 is computed as [6]
�+𝒙𝒙𝒅𝒅−𝟐𝟐𝝀𝝀(𝐱𝐱−𝐱𝐱𝐝𝐝)+𝛌𝛌𝟐𝟐(𝐱𝐱−𝐱𝐱𝐝𝐝) �=−𝒊𝒊(27) 𝑼𝑼
𝟐𝟐𝝀𝝀(𝐱𝐱−𝐱𝐱𝐝𝐝)−𝛌𝛌𝟐𝟐(𝐱𝐱−𝐱𝐱𝐝𝐝)
𝒅𝒅𝒕𝒕
|𝑺𝑺(𝟎𝟎)|𝜻𝜻
since the trajectories are
To eliminate the derivative term, it is used an integral term from t=0 to t=𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕
𝒕𝒕=𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕𝒅𝒅𝒕𝒕=𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕
𝑺𝑺(𝒕𝒕)≤−∫𝜼𝜼→∫𝒕𝒕=𝟎𝟎𝒕𝒕=𝟎𝟎𝒅𝒅𝒕𝒕
(15)
𝑺𝑺(𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕)−𝑺𝑺(𝟎𝟎)≤−𝜻𝜻(𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕−𝟎𝟎)
Where 𝑡𝑡𝑟𝑟𝑐𝑐𝑟𝑟𝑐𝑐ℎ is the time that trajectories reach to the sliding surface so, suppose S(𝑡𝑡𝑟𝑟𝑐𝑐𝑟𝑟𝑐𝑐ℎ=0) defined as;
𝑺𝑺(𝟎𝟎)(16)
𝟎𝟎−𝑺𝑺(𝟎𝟎)≤−𝜼𝜼(𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕)→𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕≤
𝜻𝜻And
𝒊𝒊𝒊𝒊 𝑺𝑺(𝟎𝟎)<0→0−𝑆𝑆(𝟎𝟎)≤−𝜼𝜼(𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕)→
|𝑺𝑺(𝟎𝟎)|𝑺𝑺(𝟎𝟎)≤−𝜻𝜻(𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕)→𝒕𝒕𝒓𝒓𝒄𝒄𝒂𝒂𝒄𝒄𝒕𝒕≤
𝜼𝜼
(17)
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
57
Based on above discussion, the sliding mode control law for multi degrees of freedom robot manipulator is written as [1, 6]:
(28) 𝝉𝝉=𝝉𝝉𝒄𝒄𝒒𝒒+𝝉𝝉𝒅𝒅𝒊𝒊𝒄𝒄
Where, the model-based component 𝝉𝝉𝒄𝒄𝒒𝒒 is the nominal dynamics of systems calculated as follows [1]:
(29) 𝝉𝝉𝒄𝒄𝒒𝒒=�𝑫𝑫−𝟏𝟏(𝒊𝒊+𝑪𝑪+𝑮𝑮)+𝑺𝑺�𝑫𝑫
and 𝝉𝝉𝒅𝒅𝒊𝒊𝒄𝒄 is computed as [1];
(30) 𝝉𝝉𝒅𝒅𝒊𝒊𝒄𝒄=𝑲𝑲∙𝐬𝐬𝐬𝐬𝐬𝐬(𝑺𝑺)
By (30) and (29) the sliding mode control of robot manipulator is calculated as;
(31) 𝝉𝝉=�𝑫𝑫−𝟏𝟏(𝒊𝒊+𝑪𝑪+𝑮𝑮)+𝑺𝑺�𝑫𝑫+𝑲𝑲∙𝐬𝐬𝐬𝐬𝐬𝐬(𝑺𝑺)
Proof of Stability: The lyapunov formulation can be written as follows,
𝟏𝟏(32) 𝑽𝑽=𝑺𝑺𝑻𝑻.𝑫𝑫.𝑺𝑺
𝟐𝟐
the derivation of 𝑉𝑉 can be determined as,
𝟏𝟏(33) .𝑺𝑺+𝑺𝑺𝑻𝑻 𝑫𝑫𝑺𝑺 𝑽𝑽= 𝑺𝑺𝑻𝑻.𝑫𝑫𝟐𝟐
the dynamic equation of robot manipulator can be written based on the sliding surface as
(34) 𝑴𝑴𝑺𝑺=−𝑽𝑽𝑺𝑺+𝑫𝑫𝑺𝑺+𝒊𝒊+𝑪𝑪+𝑮𝑮
it is assumed that
−𝟐𝟐𝒊𝒊+𝑪𝑪+𝑮𝑮�𝑺𝑺=𝟎𝟎 (35) 𝑺𝑺𝑻𝑻�𝑫𝑫
by substituting (34) in (35)
𝟏𝟏(36) 𝑺𝑺−𝑺𝑺𝑻𝑻𝒊𝒊+𝑪𝑪𝑺𝑺+𝑺𝑺𝑻𝑻�𝑫𝑫𝑺𝑺+𝒊𝒊+𝑽𝑽=𝑺𝑺𝑻𝑻𝑫𝑫𝟐𝟐
𝑪𝑪𝑺𝑺+𝑮𝑮)=𝑺𝑺𝑻𝑻�𝑫𝑫𝑺𝑺+𝒊𝒊+𝑪𝑪𝑺𝑺+𝑮𝑮�
suppose the control input is written as follows
−𝟏𝟏��=𝑼𝑼𝑵𝑵𝒄𝒄𝒔𝒔𝑵𝑵��(37) 𝑼𝑼𝒊𝒊𝒔𝒔𝒄𝒄𝒂𝒂𝒓𝒓+𝑼𝑼𝒅𝒅𝒊𝒊𝒄𝒄=�𝑫𝑫(𝒊𝒊+𝑪𝑪+
�+𝑲𝑲.𝒄𝒄𝒔𝒔𝒔𝒔(𝑺𝑺)+𝑫𝑫+𝑪𝑪𝑺𝑺+𝑮𝑮 𝑮𝑮)+𝑺𝑺�𝑫𝑫
by replacing the equation (40) in (32)
��𝑺𝑺−𝒊𝒊(38) 𝑽𝑽=𝑺𝑺𝑻𝑻(𝑫𝑫𝑺𝑺+𝒊𝒊+𝑪𝑪+𝑮𝑮−𝑫𝑫+𝑪𝑪𝑺𝑺+��𝑺𝑺+𝒊𝒊𝑮𝑮−𝑲𝑲𝒄𝒄𝒔𝒔𝒔𝒔(𝑺𝑺)=𝑺𝑺𝑻𝑻�𝑫𝑫+𝑪𝑪𝑺𝑺+𝑮𝑮−
and
𝑲𝑲𝒄𝒄𝒔𝒔𝒔𝒔(𝑺𝑺)�
the Lemma equation in robot arm system can be written as follows
�𝑺𝑺�+|𝒊𝒊+𝑪𝑪𝑺𝑺+𝑮𝑮|+𝜼𝜼� ,𝒊𝒊=(40) 𝑲𝑲𝒖𝒖=��𝑫𝑫𝒊𝒊
𝟏𝟏,𝟐𝟐,𝟑𝟑,𝟒𝟒,…
and finally;
𝒔𝒔
(41)
𝑽𝑽≤−�𝜼𝜼𝒊𝒊|𝑺𝑺𝒊𝒊|
𝒊𝒊=𝟏𝟏
���𝑺𝑺+𝒊𝒊�𝑺𝑺�+�𝒊𝒊�𝑫𝑫+𝑪𝑪𝑺𝑺+𝑮𝑮�≤�𝑫𝑫+𝑪𝑪𝑺𝑺+𝑮𝑮�
(39)
Figure 2 show the pure PID sliding mode controller applied to flexible robot.
Fig 2: Sliding Mode Controller
Fuzzy Logic Technique: Based on foundation of fuzzy logic methodology; fuzzy logic management has played important rule to design nonlinear management for nonlinear and uncertain systems [16-36]. However the application area for fuzzy control is really wide, the basic form for all command types of controllers consists of;Input fuzzification (binary-to-fuzzy [B/F] conversion), Fuzzy rule base (knowledge base), Inference engine and Output defuzzification (fuzzy-to-binary [F/B] conversion). Figure 3 shows the fuzzy controller part. The fuzzy inference engine offers a mechanism for transferring the rule base in fuzzy set which it is divided into two most important methods, namely, Mamdani method and Sugeno method. Mamdani method is one of the common fuzzy inference systems and he designed one of the first fuzzy managements to control of system engine. Mamdani’s fuzzy inference system is divided into four major steps: fuzzification, rule evaluation, aggregation of the rule outputs and defuzzification. Michio Sugeno uses a singleton as a membership function of the rule consequent part. The following definition shows the Mamdani and Sugeno fuzzy rule base [22-33]
𝒊𝒊𝒊𝒊 𝒙𝒙 𝒊𝒊𝒄𝒄 𝑨𝑨 𝒂𝒂𝒔𝒔𝒅𝒅 𝒚𝒚 𝒊𝒊𝒄𝒄 𝑩𝑩 𝒕𝒕𝒕𝒕𝒄𝒄𝒔𝒔 𝒛𝒛 𝒊𝒊𝒄𝒄 𝑪𝑪 ′𝒎𝒎𝒂𝒂𝒎𝒎𝒅𝒅𝒂𝒂𝒔𝒔𝒊𝒊′ 𝒊𝒊𝒊𝒊 𝒙𝒙 𝒊𝒊𝒄𝒄 𝑨𝑨 𝒂𝒂𝒔𝒔𝒅𝒅 𝒚𝒚 𝒊𝒊𝒄𝒄 𝑩𝑩 𝒕𝒕𝒕𝒕𝒄𝒄𝒔𝒔 𝒛𝒛 𝒊𝒊𝒄𝒄 𝒊𝒊(𝒙𝒙,𝒚𝒚)′𝒄𝒄𝒖𝒖𝒔𝒔𝒄𝒄𝒔𝒔𝒄𝒄′
When 𝑥𝑥 and 𝑦𝑦 have crisp values fuzzification calculates the membership degrees for antecedent part. Rule
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
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Design Sliding Mode Modified Fuzzy Linear Controller with Application
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evaluation focuses on fuzzy operation (𝐴𝐴𝐴𝐴𝐷𝐷/𝑂𝑂𝑂𝑂 ) in the antecedent of the fuzzy rules. The aggregation is used to calculate the output fuzzy set and several methodologies can be used in fuzzy logic controller aggregation, namely, Max-Min aggregation, Sum-Min aggregation, Max-bounded product, Max-drastic product, Max-bounded sum, Max-algebraic sum and Min-max. Defuzzification is the last step in the fuzzy inference system which it is used to transform fuzzy set to crisp set. Consequently defuzzification’s input is the aggregate output and the defuzzification’s output is a crisp number. Centre of gravity method (𝐶𝐶𝑂𝑂𝐺𝐺) and Centre of area method (𝐶𝐶𝑂𝑂𝐴𝐴) are two most common defuzzification methods [34-69].
Where, the model-based component 𝝉𝝉𝒄𝒄𝒒𝒒 is the nominal dynamics of systems and to position control of continuum robot manipulator 𝝉𝝉𝒄𝒄𝒒𝒒 can be calculate as follows: and 𝝉𝝉𝒄𝒄𝒂𝒂𝒕𝒕 is computed as; 𝝉𝝉𝒄𝒄𝒒𝒒=�𝑫𝑫
−𝟏𝟏(
𝒊𝒊+𝑪𝑪+𝑮𝑮)+𝑺𝑺�𝑫𝑫
by replace the formulation (47) in (45) the control output can be written as;
𝝉𝝉=𝝉𝝉𝒄𝒄𝒒𝒒+𝑲𝑲.𝐬𝐬𝐒𝐒𝐝𝐝�𝑺𝑺�∅�
𝝉𝝉𝒄𝒄𝒒𝒒+𝑲𝑲.𝐬𝐬𝐬𝐬𝐬𝐬(𝑺𝑺) ,|𝑺𝑺|≥∅
=�𝑺𝑺𝝉𝝉𝒄𝒄𝒒𝒒+𝑲𝑲.�∅ ,|𝑺𝑺|<∅
𝝉𝝉𝒄𝒄𝒂𝒂𝒕𝒕=𝑲𝑲∙𝐬𝐬𝐒𝐒𝐝𝐝�𝑺𝑺�∅�
Fig 3: Fuzzy Controller Part
III.
METHODOLOGY
Chattering phenomenon is one of the most important
challenges in pure sliding mode controller with regards to the certain and partly uncertain systems. To reduce or eliminate the chattering in sliding mode controller most of researcher are focused on boundary layer method. In boundary layer method the basic idea is replaced the discontinuous method by saturation (linear) method with small neighborhood of the switching surface.
(42) 𝑩𝑩(𝒕𝒕)={𝒙𝒙,|𝑺𝑺(𝒕𝒕)|≤∅};∅>0 where ∅ is the boundary layer thickness. Therefore the
saturation function 𝐒𝐒𝐒𝐒𝐝𝐝(𝐒𝐒�∅) is added to the control law as
(43) �⃗,𝒕𝒕)∙𝐒𝐒𝐒𝐒𝐝𝐝�𝑺𝑺�∅�𝑼𝑼=𝑲𝑲(𝒙𝒙
where 𝐒𝐒𝐒𝐒𝐝𝐝�𝐒𝐒�∅� can be defined as
(44) 𝐬𝐬𝐒𝐒𝐝𝐝�𝑺𝑺�∅�
𝒄𝒄
⎧𝟏𝟏 (�∅>1)⎪
=−𝟏𝟏 �𝒄𝒄�∅<−1� ⎨
⎪𝒄𝒄� (−𝟏𝟏<𝒄𝒄�<1)
∅⎩∅
Based on above discussion, the control law for multi degrees of freedom robot manipulator is written as:
(45) 𝝉𝝉=𝝉𝝉𝒄𝒄𝒒𝒒+𝝉𝝉𝒄𝒄𝒂𝒂𝒕𝒕
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
By (48) and (46) the sliding mode control of
continuum robot manipulator is calculated as;
(49) 𝝉𝝉=�𝑫𝑫−𝟏𝟏(𝒊𝒊+𝑪𝑪+𝑮𝑮)+𝑺𝑺�𝑫𝑫+𝑲𝑲∙
𝐬𝐬𝐒𝐒𝐝𝐝�𝑺𝑺�∅�
To solve the challenge of pure sliding mode controller based on nonlinear dynamic formulation and chattering phenomenon this research is focused on applied modified PID fuzzy methodology to estimate the nonlinear equivalent formulation and reduce the chattering. In this method; modified PID sliding surface slope calculate and applied to fuzzy logic system to modified nonlinear sliding surface slope. Dynamic nonlinear equivalent part is estimate by performance/error-based fuzzy logic controller which applied to previous sliding surface slope. In fuzzy error-based sliding mode controller; error based Mamdani’s fuzzy inference system has considered with one input, one output and totally 7 rules to design nonlinear PID modified sliding surface slope. Figure 4 shows the modified PID fuzzy applied to sliding mode controller. (50) 𝑺𝑺𝒎𝒎𝒄𝒄𝒅𝒅𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒅𝒅−𝒊𝒊𝒖𝒖𝒛𝒛𝒛𝒛𝒚𝒚=
Based on fuzzy logic methodology [𝜶𝜶𝒄𝒄+𝒄𝒄+()𝟐𝟐∑𝒄𝒄 ]∑𝑴𝑴𝑵𝑵=𝟏𝟏𝜽𝜽𝜻𝜻(𝒙𝒙)
𝟐𝟐𝜶𝜶𝑻𝑻
(46) (47)
(48)
where 𝜽𝜽𝑻𝑻 is adjustable parameter (gain updating factor) and 𝜻𝜻(𝒙𝒙) is defined by;
∑𝒊𝒊𝝁𝝁(𝒙𝒙𝒊𝒊)𝒙𝒙𝒊𝒊(52) 𝜻𝜻(𝒙𝒙)=∑
𝒊𝒊𝝁𝝁(𝒙𝒙𝒊𝒊)
𝑻𝑻
𝒊𝒊(𝒙𝒙)=𝑼𝑼𝒊𝒊𝒖𝒖𝒛𝒛𝒛𝒛𝒚𝒚=∑𝑴𝑴𝑵𝑵=𝟏𝟏𝜽𝜽𝜻𝜻(𝒙𝒙)
(51)
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
59
Fig 4: Block Diagram of Fuzzy PID Modified New Sliding Mode
Control
IV.
RESULTS AND DISCUSSION
In this section, we use a benchmark model, robot manipulator, to evaluate our control algorithms. We
compare the following methods: PD sliding mode Fig 6: Pure SMC, Fuzzy SMC and Proposed method trajectory
controller, conventional fuzzy sliding mode controller following in presence of disturbance
and fuzzy PID modified new sliding mode controller
Based on Figure 6; by comparison between SMC, which is proposed method in this paper. The simulation
fuzzy SMC and proposed method, proposed methodology was implemented based on MATLAB/SIMULINK.
Close loop response of robot manipulator trajectory is more stable and robust than pure nonlinear controller. planning: Figure 5 illustrates the tracking performance in these methodologies. V. CONCLUSION
The important contributions of this paper are; design
modified PID controller to pre adjust and tune the error, design fuzzy logic controller by one input and output to compensate nonlinear equivalent part in sliding mode controller and reduce the chattering phenomenon, partly linear term which is used to bypass the inertial term in nonlinear system and sliding mode controller which is a main nonlinear and robust controller. The structure of sliding mode controller with modified PID serial fuzzy inference compensator is new. We also propose parallel partly linear term structure compensator to reduce the error and increase the stability.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for their careful reading of this paper and for their helpful comments. This work was supported by the SSP Research and Development Corporation Program of Iran under grant no. 2013-Persian Gulf-1A.
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Based on Figure 5; pure SMC controller has a slight overshoot in all links, because this flexible robot is a highly nonlinear system and control of this system by pure nonlinear method is very difficult.
Close loop response of trajectory planning in presence of disturbance: Figure 6 demonstrates the power disturbance elimination in these controllers in presence of disturbance for flexible robot manipulator. The disturbance rejection is used to test the robustness in this nonlinear system.
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
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Design Sliding Mode Modified Fuzzy Linear Controller with Application
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[15] Farzin Piltan, Mohammad Mansoorzadeh, Saeed
Zare, Fatemeh Shahriarzadeh, Mehdi Akbari, “Artificial tune of fuel ratio: Design a novel siso fuzzy backstepping adaptive variable structure control”, International Journal of Electrical and Computer Engineering (IJECE), 3 (2): 183-204, 2013.
[16] Farzin Piltan, M. Bazregar, M. Kamgari, M. Akbari,
M. Piran, “Adjust the fuel ratio by high impact chattering free sliding methodology with application to automotive engine”, International Journal of Hybrid Information Technology (IJHIT), 6 (1): 13-24, 2013.
[17] Shahnaz Tayebi Haghighi, S. Soltani, Farzin Piltan,
M. Kamgari, S. Zare, “Evaluation Performance of IC Engine: linear tunable gain computed torque controller Vs. Sliding mode controller”, I. J. Intelligent system and application, 6 (6): 78-88, 2013.
[18] Farzin Piltan, N. Sulaiman, Payman Ferdosali & Iraj
Assadi Talooki, “Design Model Free Fuzzy Sliding Mode Control: Applied to Internal Combustion Engine”, International Journal of Engineering, 5 (4):302-312, 2011.
[19] Farzin Piltan, N. Sulaiman, A. Jalali & F. Danesh
Narouei, “Design of Model Free Adaptive Fuzzy Computed Torque Controller: Applied to Nonlinear Second Order System”, International Journal of Robotics and Automation, 2 (4):245-257, 2011
[20] A. Jalali, Farzin Piltan, M. Keshtgar, M. Jalali,
“Colonial Competitive Optimization Sliding Mode Controller with Application to Robot Manipulator”, International Journal of Intelligent Systems and Applications, 5(7), 2013.
[21] Farzin Piltan, Amin Jalali, N. Sulaiman, Atefeh
Gavahian & Sobhan Siamak, “Novel Artificial Control of Nonlinear Uncertain System: Design a Novel Modified PSO SISO Lyapunov Based Fuzzy Sliding Mode Algorithm”, International Journal of Robotics and Automation, 2 (5): 298-316, 2011. [22] Farzin Piltan, N. Sulaiman, Iraj Asadi Talooki &
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[23] Farzin Piltan, N. Sulaiman, S.Soltani, M. H.
Marhaban & R. Ramli, “An Adaptive Sliding Surface Slope Adjustment in PD Sliding Mode Fuzzy Control For Robot Manipulator”, International Journal of Control and Automation, 4 (3): 65-76, 2011.
[24] Farzin Piltan, N. Sulaiman, Mehdi Rashidi, Zahra
Tajpaikar & Payman Ferdosali, “Design and Implementation of Sliding Mode Algorithm: Applied to Robot Manipulator-A Review”, International Journal of Robotics and Automation, 2 (5):265-282, 2011.
[25] Farzin Piltan, N. Sulaiman , Arash Zargari,
Mohammad Keshavarz & Ali Badri, “Design PID-
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
61
Like Fuzzy Controller with Minimum Rule Base and Mathematical Proposed On-line Tunable Gain: Applied to Robot Manipulator”, International Journal of Artificial Intelligence and Expert System, 2 (4):184-195, 2011.
[26] Farzin Piltan, SH. Tayebi HAGHIGHI, N. Sulaiman,
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[29] Farzin Piltan, Nasri B Sulaiman, Iraj Asadi Talooki
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[30] Farzin Piltan, M.J. Rafaati, F. Khazaeni, A.
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[36] Farzin Piltan, M.A. Bairami, F. Aghayari, M.R.
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[37] Farzin Piltan, N. Sulaiman, Samaneh Roosta, Atefeh
Gavahian & Samira Soltani, “Evolutionary Design of Backstepping Artificial Sliding Mode Based Position Algorithm: Applied to Robot Manipulator”, International Journal of Engineering, 5 (5):419-434, 2011.
[38] Farzin Piltan, N. Sulaiman, Amin Jalali, Sobhan
Siamak & Iman Nazari, “Control of Robot Manipulator: Design a Novel Tuning MIMO Fuzzy Backstepping Adaptive Based Fuzzy Estimator Variable Structure Control”, International Journal of Control and Automation, 4 (4):91-110, 2011.
[39] Farzin Piltan, N. Sulaiman, Atefeh Gavahian,
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[40] Farzin Piltan, N. Sulaiman, Samira Soltani, Samaneh
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[41] Farzin Piltan, F. ShahryarZadeh ,M. Mansoorzadeh ,
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[42] Farzin Piltan, Sadeq Allahdadi, Mohammad
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Design Sliding Mode Modified Fuzzy Linear Controller with Application
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Dynamic Fuzzy Controller”, International Journal of Methods Using MATLAB/SIMULINK and Their Robotics and Automation, (1):13-26, 2012. Integration into Graduate Nonlinear Control and [46] Farzin Piltan, Sadeq Allahdadi, Mohammad MATLAB Courses”, International Journal of
A.Bairami & Hajar Nasiri, “Design Auto Adjust Robotics and Automation, 3(3): 167-191, 2012. Sliding Surface Slope: Applied to Robot [57] Farzin Piltan, Hossein Rezaie, Bamdad Boroomand, Manipulator”, International Journal of Robotics and Arman Jahed. “Design Robust Backstepping on-line Automation, 3 (1):27-44, 2012. Tuning Feedback Linearization Control Applied to [47] Farzin Piltan, F. Aghayari, M. Rashidian & M. IC Engine”, International Journal of Advance
Shamsodini, “A New Estimate Sliding Mode Fuzzy Science and Technology, 11:40-22, 2012.
Controller for RoboticManipulator”, International [58] Farzin Piltan, S. Siamak, M.A. Bairami and I. Journal of Robotics and Automation, 3 (1):45-60, Nazari. ” Gradient Descent Optimal Chattering Free 2012 Sliding Mode Fuzzy Control Design: Lyapunov [48] Farzin Piltan, Iman Nazari, Sobhan Siamak, Approach”, International Journal of Advanced
Payman Ferdosali, “Methodology of FPGA-Based Science and Technology, 43: 73-90, 2012.
Mathematical error-Based Tuning Sliding Mode [59] Farzin Piltan, M.R. Rashidian, M. Shamsodini and Controller”, International Journal of Control and S. Allahdadi. ” Effect of Rule Base on the Fuzzy-Automation, 5(1), 89-118, 2012. Based Tuning Fuzzy Sliding Mode Controller: [49] Farzin Piltan, Bamdad Boroomand, Arman Jahed & Applied to 2nd Order Nonlinear System”,
International Journal of Advanced Science and Hossein Rezaie, “Methodology of Mathematical
Technology, 46:39-70, 2012. Error-Based Tuning Sliding Mode Controller”,
International Journal of Engineering, 6 (2):96-117, [60] Farzin Piltan, A. Jahed, H. Rezaie and B.
Boroomand. ” Methodology of Robust Linear On-2012.
line High Speed Tuning for Stable Sliding Mode [50] Farzin Piltan, S. Emamzadeh, Z. Hivand, F.
Controller: Applied to Nonlinear System”, Shahriyari & Mina Mirazaei. ” PUMA-560 Robot
International Journal of Control and Automation, Manipulator Position Sliding Mode Control
5(3): 217-236, 2012. Methods Using MATLAB/SIMULINK and Their
Integration into Graduate/Undergraduate Nonlinear [61] Farzin Piltan, R. Bayat, S. Mehara and J.
Meigolinedjad. ”GDO Artificial Intelligence-Based Control, Robotics and MATLAB Courses”,
Switching PID Baseline Feedback Linearization International Journal of Robotics and Automation,
Method: Controlled PUMA Workspace”, 3(3):106-150, 2012.
International Journal of Information Engineering [51] Farzin Piltan, A. Hosainpour, E. Mazlomian,
and Electronic Business, 5: 17-26, 2012. M.Shamsodini, M.H Yarmahmoudi. ”Online
Tuning Chattering Free Sliding Mode Fuzzy [62] Farzin Piltan, B. Boroomand, A. Jahed and H.
Rezaie. ”Performance-Based Adaptive Gradient Control Design: Lyapunov Approach”,
Descent Optimal Coefficient Fuzzy Sliding Mode International Journal of Robotics and Automation,
Methodology”, International Journal of Intelligent 3(3):77-105, 2012.
Systems and Applications, 11: 40-52 2012. [52] Farzin Piltan, R. Bayat, F. Aghayari, B.
[63] Farzin Piltan, S. Mehrara, R. Bayat and S. Rahmdel. ” Boroomand. “Design Error-Based Linear Model-Design New Control Methodology of Industrial Free Evaluation Performance Computed Torque
Robot Manipulator: Sliding Mode Baseline Controller”, International Journal of Robotics and
Methodology”, International Journal of Hybrid Automation, 3(3):151-166, 2012.
Information Technology, 5(4):41-54, 2012. [53] Farzin Piltan, J. Meigolinedjad, S. Mehrara, S.
[64] AH Aryanfar, MR Khammar, Farzin Piltan, Rahmdel. ”Evaluation Performance of 2nd Order
“Design a robust self-tuning fuzzy sliding mode Nonlinear System: Baseline Control Tunable Gain
control for second order systems”, International Sliding Mode Methodology”, International Journal
Journal of Engineering Science REsearch, 3(4): of Robotics and Automation, 3(3): 192-211, 2012.
711-717, 2012. [54] Farzin Piltan, Mina Mirzaei, Forouzan Shahriari,
[65] Farzin Piltan, Shahnaz Tayebi Haghighi, “Design Iman Nazari, Sara Emamzadeh, “Design Baseline
Gradient Descent Optimal Sliding Mode Control of Computed Torque Controller”, International
Continuum Robots”, International Journal of Journal of Engineering, 6(3): 129-141, 2012.
Robotics and Automation, 1(4): 175-189, 2012. [55] Farzin Piltan, Sajad Rahmdel, Saleh Mehrara, Reza
[66] Farzin Piltan, A. Nabaee, M.M. Ebrahimi, M. Bayat , “Sliding Mode Methodology Vs. Computed
Bazregar, “Design Robust Fuzzy Sliding Mode Torque Methodology Using
Control Technique for Robot Manipulator Systems MATLAB/SIMULINK and Their Integration into
with Modeling Uncertainties”, International Journal Graduate Nonlinear Control Courses” ,
of Information Technology and Computer Science, International Journal of Engineering, 6(3): 142-177,
5(8), 2013. 2012.
[67] Farzin Piltan, M. Akbari, M. Piran , M. [56] Farzin Piltan , M.H. Yarmahmoudi, M. Shamsodini,
Bazregar. ”Design Model Free Switching Gain E.Mazlomian, A.Hosainpour. ”PUMA-560 Robot
Scheduling Baseline Controller with Application to Manipulator Position Computed Torque Control Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
Design Sliding Mode Modified Fuzzy Linear Controller with Application
to Flexible Robot Manipulator
63
Meysam Kazeminasab is Mechanical- Engineering from Islamic Azad University. He is currently working as a researcher in Dept. of Research &
Mahdi Mirshekaran is Electrical- Development at the Iranian Research Engineering from Islamic Azad and Development Company SSP.Co, University. He has received her BE Shiraz, Iran. His current research (Bachelor of Engineering) degrees in interests are in the area of Artificial Electrical Engineering. He is currently Intelligence, Robotics, Nonlinear control and Fuzzy logic working as a researcher in Dept. of theory and application. Research & Development at the Iranian Research and Development Company
SSP.Co, Shiraz, Iran. His current research interests are in the area of Artificial Intelligence, Robotics, Nonlinear control and Fuzzy logic theory and application.
Farzin Piltan was born on 1975, Shiraz, Iran. In 2004 he is jointed the research and development company, SSP Co, Shiraz, Iran. In addition to 7 textbooks, Farzin Piltan is the main author of more than 80 scientific papers in refereed journals. He is editorial board of international journal
of control and automation (IJCA), editorial board of International Journal of Intelligent System and Applications (IJISA), editorial board of IAES international journal of robotics and automation, editorial board of International Journal of Reconfigurable and Embedded Systems and reviewer of (CSC) international journal of robotics and automation. His main areas of research interests are nonlinear control, artificial control system and applied to FPGA, robotics and artificial nonlinear control and IC engine modeling and control.
Automotive Engine”, International Journal of Information Technology and Computer Science, 01:65-73, 2013.
[68] Farzin Piltan, M. Piran , M. Bazregar, M. Akbari,
“Design High Impact Fuzzy Baseline Variable Structure Methodology to Artificial Adjust Fuel Ratio”, International Journal of Intelligent Systems and Applications, 02: 59-70, 2013.
[69] Farzin Piltan, M. Mansoorzadeh, M. Akbari, S. Zare,
F. ShahryarZadeh “Management of Environmental Pollution by Intelligent Control of Fuel in an Internal Combustion Engine“ Global Journal of Biodiversity Science And Management, 3(1), 2013.
Tannaz Khajeaian is a Master in computer engineering from Islamic Azad University. She has received her BE (Bachelor of Engineering) degrees in Computer Engineering. She is currently working as a researcher in Dept. of Research & Development at
the Iranian Research and Development Company SSP.Co, Shiraz, Iran. Her current research interests are in the area of Artificial Intelligence, Robotics, Nonlinear control and Fuzzy logic theory and application.
Zahra Esmaeili is Electrical-Electronic Engineering from Islamic Azad University. She has received her BE (Bachelor of Engineering) degrees in Electronic Engineering. She is currently working as a researcher in Dept. of Research & Development at the Iranian
Research and Development Company SSP.Co, Shiraz, Iran. Her current research interests are in the area of Artificial Intelligence, Robotics, Nonlinear control and Fuzzy logic theory and application.
Copyright © 2013 MECS I.J. Modern Education and Computer Science, 2013, 10, 53-63
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