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ComparisonofPolePlacementandLQRAppliedtoSingleLinkFlexibleManipulatorCONFERENCEPAPERJANUARY2012DOI:10.1109/CSNT.2012.183

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Comparison of Pole Placement and LQR Applied to Single Link Flexible Manipulator

Subhash Chandra Saini Yagvalkya Sharma Electronics Deptt, UCE, Electronics Deptt, UCE, RTU, KOTA RTU, KOTA saini87totoday@gmail.com yagvalkya@gmail.com Manisha Bhandari Udit Satija

ASSOCIATE PROFESSOR Electronics Deptt. LNMIIT, JAIPUR Electronics Deptt, UCE, uditsatija007@gmail.com RTU, KOTA mb_eck@yahoo.com

Abstract this work presents a comparative study of two different control strategies for a flexible single-link manipulator. The dynamic model of the flexible manipulator involves modeling the rotational base and the flexible link as rigid bodies using the Euler Lagrange's method. The resulting system has one Degree-Of- Freedom (one DOF) and it provide freedom to increase the degree as well. Two types of regulators are studied, the State-Regulator using Pole Placement, and the Linear-Quadratic regulator (LQR). The LQR is obtained by resolving the Ricatti equation, in this work, we apply and compare two strategies to control the tip of the flexible link: state-feedback and linear quadratic regulator. These regulators are designed to reduce tip vibrations and increase system stability due to the flexibility of the arm . Keywords Euler Lagrange's method ,SLFM-Single Link Flexible Manipulator, Pole Placement, LQR- Linear Quadratic Regulator ,Ricatti equation.

I. INTRODUCTION Modelling and control of flexible manipulators have received a thorough attention during the past few decades. Because of high performance requirements such as lightweight robots, high speed operation and more efficient , consideration of structural flexibility in robots arms is a real challenge. Unfortunately, taking into account the flexibility of the arm leads to the appearance of oscillations at the tip of the link during vibration. These oscillations make the control task of such systems more challenging [1]. A manipulator is composed of a series of links connected to each other via joints. Each joint usually has an actuator (a motor for eg.) connected to it. These actuators are used to cause relative vibration between successive links. One end of the manipulator is usually connected to a stable base and the other end is used to deploy a tool.[1] In this work, we apply and compare two strategies to control the tip of the flexible link: state-feedback and linear quadratic regulator. These regulators are designed to reduce tip vibrations due to the flexibility of the arm. The first step consists of modelling the flexible manipulator consisting of a thin flexible arm and the servomotor representing the

rotational base. In this purpose, we considered modelling both parts as rigid bodies [10].

II. MODELLING THE SYSTEM This system is similar in nature to the control problems encountered in large geared robot joints where flexibility is exhibited in the gearbox. A rigid beam is mounted on a flexible joint that rotates via a DC motor. The joint deflection is measured using a sensor. Variable springs and mounting points are also provided to change system parameters. Rotating the base of the beam causes the entire beam to oscillate due to the joint flexibility introduced by the springs. Different springs can be used, to represent different flexibility effects. Also, a weight can be added to the beam. An optical encoder attached to the shaft of the DC motor is used to measure the angular position of the shaft . The joint deflection (t) is measured by an encoder located at the motor end of the beam. [1].

Fig.1 Rotary flexible link module[10] The objective is to design a feedback controller such that the tip of the beam tracks a desired command while minimizing link deflection and resonance in the system. The system is supplied with a state feedback controller and linear quadratic regulator.

A. Modelling the Servomotor The servomotor model can be developed with the following block diagram as shown in figure:[2,10]

2012 International Conference on Communication Systems and Network Technologies

978-0-7695-4692-6/12 $26.00 2012 IEEEDOI 10.1109/CSNT.2012.183

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2012 International Conference on Communication Systems and Network Technologies

978-0-7695-4692-6/12 $26.00 2012 IEEEDOI 10.1109/CSNT.2012.183

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2012 International Conference on Communication Systems and Network Technologies

978-0-7695-4692-6/12 $26.00 2012 IEEEDOI 10.1109/CSNT.2012.183

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Fig.2 block diagram of servomotor[1]

Neglecting the armature inductance Lm relative to the armature resistance, the transfer function of the servomotor is obtained from the block diagram, and has the form of the following expression:[1]

(1)

Where is the angle generated by the servomotor, Vm is the input voltage, Am is the motor gain given by :-[1].

(2)

and is the time constant defined by:[1].

= (3)

Where is the motor efficiency, is the gearbox efficiency, is the back-emf constant, is the high-gear total gearbox ratio, Rm is the motor armature resistance, Beq and Jeq are respectively the high-gear viscous damping coefficient and the equivalent high-gear moment of inertia without external load. The output torque can be expressed as follows:[1]

(4)

Where is the angular velocity generated by the servomotor at the load.[1].

B Modelling the Flexible Link Module: A schematic of the flexible link is shown in figure (3), where the base angle (t) and the tip angular deflection (t) relative to the undeformed link are the generalized variables.[2,10]

Fig.3 Schematic of the flexible link[10].

Let:

y(t) = and (t)= (5) the output angle and the velocity of the DC servomotor, respectively. In order to determine the Lagrangian of the system, we need to calculate first the total potential and kinetic energies. Neglecting the gravitational potential energy, the total potential energy of the system is only due to the elasticity of the flexible arm:[9]. V = (6) where Ks is the torsion spring stiffness. The kinetic energy is due to the rotation of the base and the link. It is expressed as follows[10].

T= (7)

where is the flexible link moment of inertia. The non-conservative forces corresponding to the generalized coordinates are and the viscous damping forces[1].

(8) (9)

where is the viscous damping force of the link and is neglected, i.e. = 0, and is expressed in (4). The Euler-Lagrange's equations can be written as:[11]

(10)

i= 1,2 where L = T V ,

= ] are the generalized coordinates, and Qi are the non-conservative forces. Substituting (6), (7), (8), and (9) in the Euler-Lagrange's equations and resolving (10), we obtain the following state-space system [7].

= Ax + Bu (11) Where

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Table (1) shows the numerical values of the different parameters of the system. by putting the above parameter in matrix A,B and C we get the following system matrix as: A=[0 0 1 0;0 0 0 1;0 673.07 -35.1667 0;0 -1023.07 35.1667 0] B=[0 0 61.7325 -61.7325]' C=[1 1 0 0] D=0 The state-space model is illustrated in figure (4) as a combination of two subsystems, one for the servomotor and one for the flexible link:

Fig.4 State space model [1]

III. STATE FEEDBACK CONTROLLER State-Feedback is the most important aspect of modern control system. Using an appropriate state-feedback, unstable systems can be stabilized or damping oscillatory can be improved. The approach studied in this paper is known as pole-placement design. Pole-placement design allows the displacement of the poles to specified locations, provided the system is controllable. The control is achieved by feeding back the state variables through a regulator with a constant gain vector K. The controllability of the system is conserved after integrating this regulator. In fact, the pole-placement approach consists on replacing the input u(t) of the initial system by a new entry[4]:

r(t)= u(t) Kx(t) (12) Since the considered model is controllable and observable, we can proceed to design the state feedback regulator. First, a state estimator is designed. Based on input and output signals, the estimator estimates the different states of the

model. The design of the state estimator consists on calculating the gain L that multiplies the difference between the real and the estimated outputs in order to converge to zero. The dynamic equation representing the state estimator is expressed by [2]

= (A LC) +Ly + Bu (13)

The values that we obtained for the gain L are the following:

L=1.0e+003 *[ 0.0467 -0.0279 0.3606 -1.3415]

The chosen eigenvalues of the estimators dynamic are the following:

{-40,-10,-2+10i,-2-10i}

The state-feedback regulator is then applied to the estimated states. The new state matrix takes the following form:[1].

(14)

In order to place the poles of the system, we need to calculate the difference between the desired eigenvalues and those of the state matrix A. The eigenvalues of the original system are the following:

{0,-17.2072,-8.9629+25.1850i, -8.9629-25.1850i }

Concerning the desired closed-loop eigenvalues, two cases were studied:

the desired closed-loop eigenvalues are chosen as:

{-40,-10,-3+10i,-3-10i}

The calculated constant gain K has the following values:

K1= 2.0179 5.4861 -0.2069 -0.5438

Above gain matrix are obtained using MATLAB.

Figure (5) illustrates a schematic block presentation of the controlled system.

Fig.5 Controlled system via feedback regulator [11]

IV. LINEAR QUADRATIC REGULATOR

This regulator is also known by Kalman Gain, the object in this case is to determine the optimal controller u(t) = - Kx(t) such that a given performance index is minimized:

J= dt (15)

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The performance index is selected to give the best performance. The choice of the elements of Q and R allows the relative weighting of individual state variables and individual control inputs. The feedback gain K is defined in the closed loop by the following expression: (16) where P is the positive defined symmetric matrix called The Riccati Matrix. It is determined by resolving the Riccati equation which general form is the following: (17) Being observable and controllable, the Riccati equation in steady state can be written as: (18) Matrices Q and R defining the performance index to minimize are assumed as shown below: Q=diag(15,15,0,0) and R=1. The resulting controller gain is: K= [3.8730 -0.2857 0.2656 0.1576] The closed-loop eigenvalues are then: {-10.1170 +25.0077i,-10.1170 -25.0077i, -12.0887,-9.5120} Above are the result of gain matrix and eigenvalue and obtained using MATLAB.

V. MATLAB SIMULATION AND RESULT Figure (6) shows the implementation of the corresponding model using Matlab/Simulink. The input reference is a step input, and the output is to the total angular displacement of the tip of the beam, that is y (t)=

CASE 2

output angle Y (t) case 2

input

estimator

y

uOut1

angle theta casStep

State -Space

x' = Ax+Bu y = Cx+Du

Gain 1

K*uve

Gain

1.4

Angle Conversion 2

rad deg

Angle Conversion 1

rad deg

Angle Conversion

deg rad

Fig.6 Matlab simulink of single link flexible manipulator.

Fig.7 Motor's angle state feedback Figure (7) and eigenvalue of the original system shows that single link flexible manipulator system is initially unstable. A State-Feedback controller and a Linear Quadratic Regulator using the tip deflection feedback measured by a strain gauge, is proposed to minimize vibrations due to the flexibility of the link. The proposed controllers are simulated and implemented, and experimental results showed that based on the tip deflection feedback, dynamic vibration of the flexible link is minimized. The closed loop Eigenvalue of system using Pole placement and LQR techniques show that the resultant system is stable.

Fig.8 output angle y(t) state feedback (Pole placement)

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Fig.9 output angle y(t) state feedback (LQR) The approach makes it possible in practice to conserve material and energy resources for many robotic applications, particularly for large installations demanding high operating speeds. Figure (8) and (9) shows the output position y(t) for State feedback regulator and LQR regulator respectively. from figure (8) and (9) we can say that system can be make stable at desired value of K that is obtained using MATLAB above.

VI. CONCLUSION This paper contains a study of two different control strategies for a system with a single flexible link. It includes modelling and controlling a rotating Quanser flexible beam. Using Lagrange dynamical equations, we obtained a linear dynamic model expressed by ordinary differential equations. A State-Feedback controller and a Linear Quadratic Regulator using the tip deflection feedback measured by a strain gauge, is proposed to minimize vibrations due to the flexibility of the link. The proposed controllers are simulated and implemented, and experimental results showed that based on the tip deflection feedback, dynamic vibration of the flexible link is significantly minimized. The approach makes it possible in practice to conserve material and energy resources for many robotic applications, particularly for large installations demanding high operating speeds. We obtained a linear dynamic model expressed by ordinary differential equations.

VII. FUTURE SCOPE:

In this paper we are presenting two control strategies for the vibration control of tip of single link flexible manipulator and the work suppose to be extends for multi-link flexible manipulator and enhance the phenomena of controllability and obseverability.

REFERENCES

[1] S. K. Tso, T. W. Yang, W. L. Xu, Z. Q. Sun, Vibration Control For a Flexible-Link Robot Arm with Deflection Feedback, International Journal of Non-Linear Mechanics 38, 2003, pp. 5162.

VIII. REFERENCES [1] J.-C. Piedboeuf, D. Dochain, R. Hurteau, K. Benameur, Optimal Control of the Tip of a Flexible Arm, Canadian Conference on Electrical and Computer Engineering, 1991, pp. 73.2.173.2.4. [2] Y. Aoustin, C. Chevallerau, A. Gulmineau, C. H. Moog, Experimental Results for the End-Effect or Control of a Single Flexible Robot Arm, IEEE: Transactions on Control Systems Technology, vol. 2. No.4. 1994, pp. 371381. [3]. M.A. Ahmad, Member, IEEE, M.S. Ramli, R.M.T. Raja Ismail, N. Hambali, and M.A. Zawawi The investigations of input shaping with optimal state feedback for vibration control of a flexible joint manipulator 2009 Conference on Innovative Technologies in Intelligent Systems and Industrial Applications (CITISIA 2009) [4] K. Cho, N. Hori, J. Angeles, On the Controllability and Observability of Flexible Beams under Rigid BodyBMotion, IEEE IECON '9, pp. 455460. [5] D. Popescu, D. Sendrescu, E. Bobasu, Modelling and Robust Control of a Flexible Beam Quanser Experiment, Acta Montanistica Slovaca, 2008, pp.127135. [6].M Baroudi1, M Saa, W Ghie 1 A. Kaddouri2 and H Ziade3Vibration Controllability and Observability of a Single-Link Flexible Manipulator 2010 7th International Multi-Conference on Systems, Signals and Devices [7] Yim, W., 2001. Adaptive Control of a Flexible Joint Manipulator. Proc. 2001 IEEE, International Robotics & Automation, Seoul, Korea, pp. 34413446 [8] Mohamed, Z., Chee, A.K., Mohd Hashim, A. W. I., Tokhi, M. O., Amin, S. H. M. and Mamat, R., 2006. Techniques for Vibration Control of a Flexible Manipulator. Robotica 24, pp. 499-511. [9] Quanser Student Handout, Rotary Flexible Joint Module [10] D. M. Rovner and R. H. Cannon, Experiments toward on-line identification and control of a very flexible one-link manipulator, Int. J. Robot. Res., vol. 6, pp. 319, 1987.

TABLE I NUMERICAL VALUES OF SYSTEM[1]

Symbol Description value

High gear Viscous damping coefficient

4.00E-03 N.m1(rad/s)

Equivalent high gear moment of inertia without external load

2.0SE-03 kg.m2

Motor efficiency 0.69

Gearbox efficiency 0.90

Back-emf constant 7.6SE-03V /(rad/s)

High-gear total gearbox ration

70

Motor armature resistance

2.6

Flexible link moment of inertia 0.004 kg.

Stiffness constant

1.4

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