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  1. 1. S. Mallikarjunaiah, S. Narayana Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 1, January -February 2013, pp.1207-1212 Design of PID Controller for Flexible Link Manipulator S. Mallikarjunaiah*, S. Narayana Reddy** *(Department of Electrical and Electronics Engineering, CR Engineering College, Tirupati, A.P., India) ** (Department of Electrical and Electronics Engineering, SV University, Tirupati, A.P., India)ABSTRACT This paper focuses on the end-point use of lightweight materials in large space structures.control of a single flexible link which rotates in The main difficulty associated with the flexible linkthe horizontal plane. The dynamic model is manipulator is the vibrations at the end effectors.derived using a Lagrangian assumed modes Various control strategies are available in themethod based on Euler–Bernoulli beam theory. literature to minimize the vibrations. A briefInitially the system is modeled as a voltage-input description of these methods is given below.model, and different controllers were applied to It is known that the flexible system is a highlycontrol the system. The position and trajectory nonlinear and heavily coupled system, ddifferentcontrol is performed by PID control methods for assumed modes method [1-3] is used for thethis system. The purpose of this study is to keep dynamic modeling and implemented via athe rotate angle of the link at desired position and commercially available symbolic manipulationto eliminate the oscillation angle of end effectors. program. Systematic method is developed toThe results were produced for mode 1 and mode symbolically derive the full nonlinear dynamic2 operation. The control blocks required for this equations.system are performed on MATLAB – High performance manipulators withSIMULINK. The simulated results of the system dynamic behavior in which the flexibility is anbased on PID controller are quite satisfactory. essential aspect are addressed [4]. The mathematical representations commonly used in modeling flexibleKeywords - Flexible Link Manipulator, PID arms and arms with flexible drives are examined.controller, High Frequency modes Dynamic deformations of the flexible arm were represented in a simple and compact form with useI. INTRODUCTION of the virtual coordinate systems in [5]. Using the An important advanced robotic system is assumed-modes approach it is possible to findthe flexible-link robot arm. The desire to improve the transfer function between the torque input androbot performance has led to the design of lighter the net tip deflection [6]. It is shown that when theflexible links. A light-flexible link robot arm has number of modes is increased for more accuratemany advantages compared to conventional rigid modeling, the relative degree of thelink robots such as lower power consumption, higher transfer function becomes ill-defined.payload-to-robot weight ratio, lower manufacturing In [8] the first method designs a stable pre-cost, and easier transportation to name a few. filter using the extended bandwidth zero phase errorBecause of the elasticity of the flexible robot arm, tracking control method. The second feed forwardthe controller algorithms are different from that of method adds delay to the inverse model and thenthe traditional rigid robot arm. uses common filter design techniques to A flexible link robot arm is a distributed approximate this delayed frequency response.parameter system of infinite order, Due to elastic PID type composite controller forproperties of flexible manipulators, the development controlling flexible arms modeled by the singularof a mathematical description and subsequent perturbation approach [9], and investigates a tuningmodel-based control of the system is a complicated method based on the proposed controller structure.task. This is made difficult by the presence of a large For the slow sub-controller, a PD plus disturbance(infinite) number of modes of vibration in the observer is used, which eventually takes on PIDsystem. The modes become significant in two ways: form, and for the fast sub-controller, modal feedbackfirstly, because the oscillations themselves prolong PID control is utilized. By using the Tchebyshevthe settling time and secondly, because attempts to representation of a discrete-time transfer functionactively control some modes result in instability of and some new results on root counting with respectthe other modes. This non-linear behavior of the to the unit circle[10], were shown that how thestructure at high speeds, firstly, degrades end-point digital PID stabilizing gains can be determined byaccuracy and secondly complicates controller solving sets of linear inequalities in two unknownsdevelopment. for a fixed value of the third parameter. The The flexible link systems play an important application of the H∞ and PID control synthesisrole in the industrial applications, mainly due to the method is used to develop the tip position control of 1207 | P a g e
  2. 2. S. Mallikarjunaiah, S. Narayana Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 1, January -February 2013, pp.1207-1212a flexible-link manipulator [11]. A modified PID payload at one end. The beam can bend freely in thecontrol (MPID) is proposed [12] which depend only horizontal plane but is considered stiff with respecton vibration feedback to improve the response of the to vertical bending and torsion. The model isflexible arm without the massive need for developed using the Lagrange formulation andmeasurements. model expansion method. The length of the The requirement of controllers with faster manipulator is assumed to be constant, andresponse and higher accuracy introduces a challenge deformation due to shear, rotary inertia and thethat the researchers have faced in different ways. effects of axial forces are neglected. The moment ofThe large mass and energy requirements of standard inertia about the hub O is denoted by and  is therigid link manipulators have led to a desire for linear mass density. The arm has length l , and theflexible link manipulators characterized by low-mass payload mass is given by . The control torque Tlinks and actuators with low power requirements.This is particularly desirable in certain applications, is applied at the hub of the manipulator by way ofsuch as space systems, where mass and energy the rotary actuator. The angular displacement of therequirements must be minimized for transport manipulator, moving in the xOy plane, is denoted bypurposes. Flexible link dynamics are also found in  . The width of the arm is assumed to be muchcertain mechanical pointing systems and in systems greater than it’s the thickness, thus allowing it towith links having high length-to-width ratios. These vibrate dominantly in the horizontal direction. Thedynamics make the system outputs such as tip shear deformation and rotary inertia effects areposition more difficult to control. Therefore, before ignored.flexible link manipulators can be surrealisticallyimplemented, it is necessary to study the nature offlexible link manipulators and determine effectivemethods for position control. This paper deals with the modeling and PIDcontrol of single flexible link manipulator. The PIDcontroller is a combination of PID controller andplant matrix controller that is adapted based on theoutput of PID controller. The paper aims at end-point control by using PID controllers, whichovercomes the disadvantage of the conventionalcontrollers such as more raise time and settling time. Fig. 1 Schematic representation of the flexible manipulator systemII. FLEXIBLE MANIPULATOR For an angular displacement  and an elastic The conventional approach to the design ofan automatic control system often involves the deflection y(x,t) the total displacement u(x,t) of aconstruction of a mathematical model which best point, measured at a distance x from the hub can bedescribes the dynamics behavior of the plant to be described as a function of the above, measured fromcontrolled, and the application of analytical the direction of Ox.techniques to this plant model to derive an (x, t) = u(x, t) +θ (t) x (1)appropriate control law. Usually, such a The kinetic energy of the system can be written asmathematical model consist of a set linear or non-linear differential equations, most of them arederived using some form of approximation and (2)simplification. The traditional model-based controltechniques break down, when a representative model In eqn. (2), the first term on the right handis difficult to obtain due to uncertainty or sheer side is due to the hub inertia, the second term is duecomplexity. It is known that robot system is highly to the rotation of the manipulator with respect to thenon-linear and heavily coupled system, and accurate origin, and the third term is due to the payload mass.mathematical model is difficult to obtain, thus it The potential energy is related to the bending ofmaking difficult to control using conventional manipulator. Since the width of the manipulatortechniques. This paper presents the mathematical under consideration is assumed to be significantlymodeling of a single link flexible manipulator. The larger than its thickness, the effects of the shearsystem is modeled by the Lagrange formulation and displacement can be neglected. In this way, themodel expansion method. potential energy of the manipulator can be written asIII. MATHEMATICAL MODELLING OF (3)FLEXIBLE MANIPULATOR where E is the modulus of the elasticity for the beam The manipulator is illustrated in fig. 1, and material, and I denote the second moment of area ofis modeled as a pinned-free flexible beam with the beam cross-section. 1208 | P a g e
  3. 3. S. Mallikarjunaiah, S. Narayana Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 1, January -February 2013, pp.1207-1212 The non-conservative work for the input torque T Table 1 Parametric values for the flexible can be written as manipulator Physical parameter Symbol Value W (4) Length L 0.61 m Section area A 310-5 m2 The Lagrangian for the system is formulated as Density Ρ 7.8103 kg/m3 Young modulus E 200109 N/m2 L=T-V (5) Second moment of area I 2.510-12 m4 To obtain the equation of the motion of the Payload M 31.710-3 kg manipulator, Hamilton extended principle is used as Moment of inertia of described in eqn. (6) hub J 4.310-3 kg-m2 (6) Substitution of eqn. (10) into eqn. (7) by apply boundary and initial conditions of eqn. (9), the where t1 and t2 are two arbitrary times and w following ordinary differential equations can be represents virtual work. Manipulating Eqn. (1) - (6) derived yields the equation of motion of the manipulator as (11) (7) (12) Where The dynamic equation of the manipulator is described as (8) Where The corresponding boundary and initial conditions denotes the frequency of vibration, which can are given by be determined from the above using the boundary conditions. The frequencies for the first and second modes of vibration and the corresponding terms are shown in table 2. Table 2 Mode dependent parameters Mode a bi i 2 -2 1 2.6178 1.003510 6.829210 4.496110-1 2 6.9626 5.0215103 9.653910-3 2.224810-1(9) Using the assumed mode method, the If X is assumed as state space variables, solution of the dynamic equation of motion of the manipulator can be obtained as linear combination and Y is assumed as the output, of the product of the admissible function i (x) and , eqn. (11) and eqn .(12) can the time-dependent generalized coordinates q i (t), as be written as follows, where (10) The admissible function  i (x) also called (13) the mode shape, is purely a function of the and displacement along the length of the manipulator and q i (t) is purely a function of the time and (14) includes an arbitrary, multiplicative constant. The parameter values for flexible manipulator are given thus a fourth order model is considered as a system. in Table 1. 1209 | P a g e
  4. 4. S. Mallikarjunaiah, S. Narayana Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 1, January -February 2013, pp.1207-1212IV. PROBLEM FORMULATION system from reaching its target value due to the Control strategies are designed for models control action.of a flexible link manipulator that are linearizedabout a particular operating point, that is, for a given 4.2 Turning of PID Controllerset of hub angle and elastic mode positions and If the PID controller parameters are chosenvelocities. The greater the variation of these in correctly, the controlled process input can bepositions and velocities from the operating point, the unstable, i.e. its output diverges, with or withoutgreater is the variation of the linearized model from oscillation and is limited only by saturation orthe actual system. This variation becomes more mechanical breakage. Tuning a control loop ispronounced for high-performance control systems adjustment of its control parameters to the optimumsince high performance implies rapid motion and values for the desired control response.therefore large departures of the hub angle and There are several methods for tuning a PIDelastic mode positions and velocities from their loop. The most effective methods are generallynominal values. System performance and stability involve the development of some form of processwill degrade unless the situation is addressed. The model, then choosing P, I and D based on theobjective is to control the angular displacement θ, dynamic model parameters. Manual tuning methodsand to minimize the vibrations at the final deflected can be relatively inefficient. The choice of theposition. method will depend largely on whether or not the loop can be taken offline for tuning, and the4.1 PID Controller response time of the system. If the system can be A proportional–integral–derivative taken offline, the best tuning method often involvecontroller (PID controller) is a generic loop feedback subjecting the system to a step change in input,(controller) widely used in industrial control measuring the output as a function of time, and usingsystems. A PID controller attempts to correct the this response to determine the control parameters.error between a measured process variable and adesired set point by calculating and then instigating a V. RESULTS AND ANALYSIScorrective action that can adjust the process By exerting this controller to the system,accordingly and rapidly, to keep the error minimal. the step response and the closed loop response in both modes will be as shown in Fig. 3 to 8. Fig. 2 Block diagram of PID Controller The PID controller calculation involvesthree separate parameters; the proportional, theintegral and derivative values as shown in fig. 2. The Fig. 3 Step response (alpha) of the system with PIDproportional value determines the reaction to the controller for mode 1current error, the integral value determines thereaction based on the sum of recent errors, and thederivative value determines the reaction based on therate at which the error has been changing. Theweighted sum of these three actions is used to adjustthe process via a control element such as the positionof a control valve or the power supply of a heatingelement.Some applications may require using only one ortwo modes to provide the appropriate systemcontrol. This is achieved by setting the gain ofundesired control outputs to zero. A PID controllerwill be called a PI, PD, P or I controller in the Fig. 4 Step response (theta) of the system with PIDabsence of the respective control actions. PI controller for mode 1controllers are particularly common, since derivativeaction is very sensitive to measurement noise, andthe absence of an integral value may prevent the 1210 | P a g e
  5. 5. S. Mallikarjunaiah, S. Narayana Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 1, January -February 2013, pp.1207-1212 Fig. 6 and 7 are shown the amplitude (alpha) and angular velocity (Theta) of the step response of the system PID controller without addition of disturbance for mode 2 operation where as fig. 8 shows the step response(alpha) of the PID controller with addition of disturbance under mode 2 operation. The operating modes are shown with different parameters in the table 3. Table 3 Results of different parameters of PID ControllerFig. 5 Step response (Alpha) of PID controller with Type of Settling time Peak over a disturbance for mode1 operation (Sec) Shoot (Amps) Mode -1 15.2 1.40 Fig. 3 and 4 are shown the amplitude Mode - 2 19.5 1.58(alpha) and angular velocity (Theta) of the stepresponse of the system PID controller without From the Fig. 3 to 8, it has observed thataddition of disturbance for mode 1 operation where the PID controller doesn’t create a suitable stepas fig. 5 shows the step response(alpha) of the PID response for the system. Intense the settling time iscontroller with addition of disturbance under mode 1 15.2 seconds for mode-1 operation and 19.5 secondsoperation. for mode -2 operation. It shows the Peak over shoot is 1.4 Amps for mode-1 operation and 1.58 Amps for mode-2 operation. VI. CONCLUSION PID Control of a Single Flexible Link Manipulator (SFLM) has been investigated in this paper. In this paper initially simulations have been carried out using PID controller, the step response and the closed-loop response in both modes are shown in fig 3 to 8. It is clear that the PID controllerFig. 6 Step response (theta) of PID controller with doesn’t create a suitable step response for the disturbance input system. Intense transient oscillation and high over shoot are the shortcoming of such controller. Moreover the parameters of this controller are constant, no adaption with system dynamical changes. Since the amplitude of oscillation persists and in order improves the transient response of the system, a PID controller has been designed and simulated along with the system.Fig. 7 Step response (Alpha) of the system with PID controller for mode 2Fig. 8 Step response (Theta) of the system with PID controller for mode 2 1211 | P a g e
  6. 6. S. Mallikarjunaiah, S. Narayana Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 3, Issue 1, January -February 2013, pp.1207-1212REFERENCES [11] Ming-Tzu Ho and Yi-Wei Tu, PID [1] S. Centikunt, B. Siciliano, and W. J. controller for flexible link manipulator, Book, Symbolic modeling and dynamic Proceedings of the 44th IEEE Conference analysis of flexible manipulators, on Decision and Control, and the Proceedings on IEEE International European Control Conference, 2005, Conference Systems, Man, and 6841-6846. Cybernetics, Oct. 1986, 798–803. [12] T. Mansour, A. Konno and M. Uchiyama, [2] G. G. Hastings and W. J. Book, A linear Modified PID Control of a Single- Link dynamic model for flexible robotic flexible Robot, Advanced Robotics, 22(4), manipulators, IEEE Control System 2008, 433-449. Magazine, 7, Apr. 1987, 61–64. [3] P. B. Usoro, R. Nadira, and S. S. Mahil, “A finite element/Lagrange approach to modeling lightweight flexible manipulators,” Journal of Dynamic Systems, Measurement, and Control, 108, 1986, 198–205. [4] B. Siciliano and W. J. Book, A singular perturbation approach to control of lightweight manipulators, International Journal on Robotic Research, 7(4), Aug. 1988, 79–90. [4] W. J. Book, Modeling, design, and control of flexible manipulator arms: A tutorial review, Proceedings on 29th IEEE Conference on Decision and Control, Dec. 1990, pp. 500–506. [5] H. Asada, Z.-D. Ma and H. Tokumaru, Inverse dynamics of flexible robot arms: modeling and computation for trajectory control, Journal of Dynamic Systems, Measurement, and Control, 112, June 1990, 177–185. [6] D. Wang and M. Vidyagasar, Transfer functions for a single flexible link, International Journal on Robotic Research, 10(5), Oct. 1991, 540–549. [7] M. W. Vandegrift, F. L. Lewis, and S. Zhu, Flexible-link robot arm control by a feedback linearization/singular perturbation approach, Journal on Robotic Systems, 11(7), 1994, 591–603. [8] D. E. Torfs, R.Vuerinckx, J. Sewevers and J. Schoukens, Comparison of two feed forward design methods aiming at accurate trajectory racking of the end point of a flexible robot arm, IEEE Transactions Control Systems Technology, 6(1), Jan. 1998, 2-14. [9] Joono Cheong, PID composite controller and its tuning for flexible link robots, IEEE/RSJ International Conference on Intelligent Robots and Systems, 3, 2002, 2122-2127. [10] L. H. Keel, J. I. Rego, and S. P. Bhattacharyya, A new approach to Digital PID Controller Design, IEEE Transactions on Automatic Control, 48(4), Apr. 2003, 687-692. 1212 | P a g e