DEVELOPMENT AND COMPARISON OF POLE PLACEMENT AND LINEAR QUADRATIC REGULATOR (LQR) FOR ROBOT ARM

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1 DEVELOPMENT AND COMPARISON OF POLE PLACEMENT AND LINEAR QUADRATIC REGULATOR (LQR) FOR ROBOT ARM Anis Pauzi Polytechnic of Sultan Salahuddin Abdul Aziz Shah Zakiah Md Saad Polytechnic of Sultan Salahuddin Abdul Aziz Shah Rokiah Hassan Polytechnic of Sultan Salahuddin Abdul Aziz Shah ABSTRACT Robotic arm system has many applications in many aspects not only in the industrial field but it also extends to surgery and aerospace. The problem with robot arm is that the dynamic response leads to uncertainties, nonlinearity and unstable output result. The researchers in this project have attempted to minimize this problem by designing various types of modern controllers for robotic arm. The modern controllers design in this research project is based on pole placement method and LQR. The design of digital controllers is simulation-based using MATLAB Simulink software. Comparisons between all the designs are made to find the best design among these digital controllers. The analyses of the results, shows that the digital controllers designed have given promising result whereby transient responses of the system are improved. The best design found was that of state feedback and feed forward controller. KEYWORDS: Robot Arm, Digital Controller, Pole Placement, Linear Quadratic Regulator 1. INTRODUCTION Robot arm has a wide application in manufacturing industry, particularly in electronics and automotive. It replaces human in accomplishing tasks such as packaging, installation, painting and more. Its application also has been extended to space exploration and surgery. A latest technology development in the robotics research has led to increasing demand. In order to implement task, the robot arm must be strong enough to generate the lateral forces required. Thus the performance of robot arm required fast transient response in a system. The control methods such as Pole Placement and Linear Quadratic Regulator (LQR) are proposed in this research project. This methodology solved the nonlinearities and the unstable problem. The proposed controller ensures fast and precise dynamic response with excellent steady state performance (Tiwari, 2011). Pole Placement and Linear Quadratic Regulator (LQR) is a type of digital controller. While, Frank L.Lewis et al., (2004) found that most robot controllers are designed in continuous time, they are implemented on actual robots digitally. This is what applied in the design of digital controller in this study. Another technique in controlling a robotic arm is to make a hybrid of fuzzy-logic and conventional control method to design 1

2 fuzzy logic controllers which pave to appropriate solution for controlling the robot manipulators (Nigam, 2008). 2. LITERATURE REVIEW 2.1 Robot Arm Robot arm is probably the most mathematically complex robot. The Degree of Freedom (DOF) is a very important term to understand. DOF is the joint of the arms, namely a place where it can bend, rotate and translate. DOF can be identified by the number of degrees of freedom by the number of actuators on the robot arm. The robot arm used in this research is illustrate in Figure 1 and it only have 2 link (2DOF). Arm of robot manipulator Embedded DC servo motor Figure 1: Robot arm The Denavit-Hartenberg (DH) Convention is the accepted technique of drawing robot arms in Free Body Diagram (FBD). There are only two motions a joint could make, there is atranslate and rotate. There are only three axes (x, y, and z). 2.2 Control approached for Robot Arm Robot arm can be controlled by various types of control, either in analog or digital. Several types of controllers designed by previous researchers discussed in this paper Digital Controller Although most robot controllers are designed in continuous time, they are employed on actual robots digitally. That is, the control signals are only efficient at discrete instants of time using a microprocessor. It is highly necessary to simulate it in its digitized or discretized form prior to actual implementation, to verify that a controller will operate as expected. Figure 2: Digital controller (Frank L.Lewis, Darren M.Dawson, Chaouki T.Abdallah, 2004) A digital control scheme is shown in Figure 2. The plant or system to be controlled is a continuous-time system, and K(z) is the dynamic digital controller, where z is the Z-transform variable. The digital controller K(z) is executed using software code in a digital signal processor (DSP). The reference input r(t) is the preferred trajectory that y(t) should follow, and ek is the (discrete) tracking error (Frank L.Lewis, Darren M.Dawson, Chaouki T.Abdallah, 2004) PID Controller 2

3 Proportional Integral - Derivative (PID) controllers have been widely used for speed and position control of robot arm. The block diagram of the robot arm control system is shown in Figure 3. Figure 3: Robot joint control system (Majed D. Youns, 2013) Ziegler-Nichols Rule is used for tuning PID Controller. Controller tuning involves the selection of the values of Kp, Ki and Kd. The value of Kp, Ki and Kd is shown in Table 1. Table 1: PID Controller Gain Values Gain Coefficient Kp Ki Kd Values Source: Majed D. Youns, 2013 From the above algorithm the step response of the system with conventionally PID controller is shown in Figure 4. Figure 4: Response of the System with Conventionally Tuned PID Controller (Majed D. Youns, 2013) From the above step response, the rise time is 0.2s. The maximum over shoot of the system is approximately 23.9 % and settling time is about 2s. The analysis show the system has not been tuned to its optimum. Thus, in order to achieve the system requirement, Genetic algorithm approach is proposed Neuro-Fuzzy Controller Adaptive network based fuzzy logic controller is applied for position control of robot arm. Established adaptive network based fuzzy inference system (ANFIS) uses a hybrid learning algorithm to identify parameters of Sugeno (type fuzzy inference system). It applies a combination of the least squares method and the back propagation gradient descent method 3

4 for training fuzzy inference system (FIS) membership function parameters to emulate a given training data set (Jafar Tavoosi, Majid Alaei, Behrouz Jahani, 2011). Figure 5: Simulink model of the neuro-fuzzy controller and robot (Jafar Tavoosi, Majid Alaei, Behrouz Jahani, 2011) The block diagram of robot arm and neuro-fuzzy controller is shown in Figure 5. In this block, two trained fuzzy neural networks are used that one of them is utilized for control of θ 1 and another is utilized for control of θ 2. The initial value of θ 1 = 30 and θ 2 = 40 and the final value of θ 1 = 30 and θ 2 = 40 (Jafar Tavoosi, Majid Alaei, Behrouz Jahani, 2011). Jafar Tavoosi et al., (2011) compared a performance between PID and Neuro-Fuzzy controller. The result give that Neuro Fuzzy controllers has provided best results for control of robotic manipulators as compared to the conventional control strategies. Simulation results show that the Neuro Fuzzy controller can achieve better accuracy and has less or no deviation from the trajectory compared to the PID controller. 3. METHODOLOGY 3.1 Modeling of Robot Arm The arm of a practical robot manipulator is affected by the gravitational field due to the mass of each arm. This lead mechanical rotational system be nonlinear and difficult analysis. However negative feedback works even for nonlinear systems and the particular arm is assumed to be placed horizontally without being affected by the gravitational field. In order to perform mathematic modeling for the system and the controller, specifications parameter for servo motor which embedded at the joint of the arm of robot need to be identified first. Thus, the specifications of the servo motor (embedded direct drive DC motor) are assumed as follows: Table 2: Servo Motor (embedded direct drive DC motor) Parameters Parameter Value General High initial torque, linear torque DC motor. Input Voltage ±28V maximum Output Torque Not specified Angular Velocity 3600rpm at 24V Output Power 1/20HP at 24V Torque Linearity Less than 1% within ±28V Input Current 4A at 24V Maximum Current 5A Voltage Gain Unity Input Impedance 100KΩ Source: K. Takaya,

5 The power supply used is a voltage source for which the output voltage is controlled by the input voltage. The mechanical specifications are known only in terms of the time constant associated with each arm. The particular arm of interest is quoted as 2 seconds. The basic equation describing the dynamics of the arm is a second order differential equation, Eq. 2: Jθ + fθ = τ Where, J is inertia of the arm, f is friction coefficient, θ is angle of rotation and τ is applied torque. From this differential equation, the transfer function can be derived as Eq. 3: G 0 (s) = θ(s) = 1 = 1/f T(s) s(js+f) s( J f s+1) Since J f = T, mechanical time constant, and the torque is a linear function of the input voltage (Ea), the transfer function from the input voltage (Ea) to the angle (θ) now becomes, Eq. 4:G(s) = θ(s) E a (s) = A f s( J f s+1) = K s(ts+1) The transfer function from the input voltage Ea to the angular velocity of the arm (ω) is also given by Eq. 5: G ω (s) = Ω(s) K = E a (s) Ts+1 If the arm is vertically placed instead of horizontally, the arm is affected by the gravitational field (K. Takaya, 2003). Thus, in this research, the system s equation needs to be modified by taking an additional torque into account. The distance from the pivot to the centre of the gravity is assumed equal to l, the differential equation is Eq. 6: Jθ + fθ = τ + Mgl cos(θ) This system is non-linear since the equation involves θ, θ and cos(θ) in a differential equation. A position sensor is used to sense the angular position of the robot arm. The simplest position detector is a potentiometer. A ±12V is supplied between two end terminals and the voltage of the brush relative to the ground is measured. As one revolution gives a voltage change of 24V, the sensor gain is 24V/revolution. The transfer function is Eq. 7:E p (s) = 24θ(s) +12V +12V + θ s(2s + 1) 12V 12V simplified θ r s(2s + 1) θ Figure 6: Block diagram of closed loop system (K. Takaya, 2003) 5

6 Incorporating two potentiometers, one is set as a reference angle (θ r ) and the other to measure the output angular position (θ). According to Fig. 6, the open loop transfer function is Eq. 8: θ θ r = 0.5 s s Figure 7: The output performance of open loop uncompensated system The output performance of Robot Arm in Figure 7 shows that Robot Arm produces an output which is 8 times greater than values that would occur because the final value of input signal is set to 12. This mean the output performance of Robot Arm is very unstable and it is clearly possess a quite large steady state error. Thus, modern controller is proposed to get a better performance. 3.2 Modern Control Approach Pole Placement Method Pole placement method is one of the classical control theories and has the advantage in the control system to obtain desired performance. Ideally, pole placement is to set the desired pole location and to move the pole location of the system to that desired pole location to get the desired system response. Mathematically, if the system s transfer function is defined, the desired transfer function must also define, so each coefficient in the same order in polynomial is compared to be the same. This pole placement control method results the desired system response and is easy to find the gain mathematically but the accuracy of system transfer function is considerably important and is costly to implement in the high order system. a. Full State Feedback Controller The purpose of the state feedback is to control the response characteristic of the system. For this research project, pole placement method is to place the poles at desired location, where the location of the poles is corresponds directly to eigenvalues of the system, which control the response characteristics of the system. Thus, by using pole placement method, feedback matrix gain K can be determine by the condition Eq. 9: λi A + BK = 0 For Robot Arm, the dominant pairs of closed loop poles that generated by MATLAB software are j and j as shown in Figure 8. 6

7 Figure 8: Closed Loop Poles The desired poles shall arbitrarily far into left hand s-plane of this dominant poles but not too far to avoid amplification of noise. Hence, for simplification purpose, only the real value of the left of s-plane is selected as an eigenvalues. By substituting all the matrix value of A and B into Eq. 9 gives Eq. 10: λi A + BK = [ λ K 1 K 2 1 λ ] From the above equation, the feedback gain matrix K is calculated by MATLAB software and gets the following value: Eq. 11: K = [ ] b. State Feedback and Feed forward Controller The state feedback and feed forward controller is an improvisation of state feedback controller. By combining the state feedback controller and feed forward controller, the performance of state feedback controller can be improved significantly. It can improve the control performance by adding a feed forward term to the feedback output so that the controller can respond more quickly to the command signal. The first step to implement this feed forward controller is find the feedback gain K, and then used the value of gain K to find feed forward gain N. By using MATLAB, feedback gain K is obtained as in Eq. 11. The feed forward gain N can be obtained by substituting the matrix A, B, C and D of the Robot Arm into Eq. 12 and substitute the value of feedback gain K into Eq. 10. Eq. 12: [ N x ] = [ A B N u C D ] 1 [ 0 I ] Eq. 13: N = N u + KN x Eq. 14 N = [80] c. Integral Controller The Robot Arm will be more robust if the tracking of Robot Arm is improved by using integral action. The integral action reduces the previous finite error to zero and give zero steady state error for a step input. The feedback error, e, was formed by adding a feedback path that is fed forward via an integrator to the controlled plant. Eq. 15 and Eq. 16 is used to calculate the value of matrix K and N that used in the integral control design. Eq. 15: [ x ] = [ A 0 v C 0 ] [x v ] + [ B D ] u + [0 I ] r Eq. 16: u = [K N] [ x v ] 7

8 By using the characteristic equation as in a pole placement technique which is λi B + AK, the result get is as follows: Eq. 17: K = [ ] Eq. 18: N = [800] Linear Quadratic Regulator (LQR) The optimal controller is inherently robust with respect to process uncertainty. Optimal control is a set of differential equations describing the control variable paths that minimize the cost function. The calculation to find the state feedback gain for a system is based on optimizing the performance index. The general performance index is as follows: Eq. 24: J = (x T Q x + U T R U )dt 0 In order to find the performance index, the value of R is selected to be 1 and the feedback gain of optimal control is calculated by: Eq. 25: K = R 1 B d T P Eq. 26: K = [ ] 4. RESULTS AND ANALYSIS All designed controllers are compared to determine the best controller for Robot Arm. Figure 10 shows the combination of all output controllers using the same scope. Therefore, the best performance output can be clearly seen and the block diagram is as follows: Figure 9: Combination of all designed controllers. 8

9 Figure 10: Combination of all designed controller output. From Figure 10, among all designed controller, state feedback with feed forward controller has the best rise time (Tr) and settling time (Ts). It then followed by integral controller. It can be seen clearly that state feedback with feed forward and integral has a better performance compare to the others, where these two types of controllers are faster in settling time and there are no overshoot (%OS) and steady state error (e ss )and very low steady state error for LQR. The results of all designed controllers are summarized in the Table 2. Table 3: Summarization of output controller performance Method Rise Time, Tr Settling Time, Ts Overshoot (%OS) Steady State Error, e ss State feedback controller No 0.99 (High) State feedback and feed forward No No controller Integral controller No No LQR controller % (Low) 0.01 (Low) From Table 3, the best controller for Robot Arm is state feedback with feed forward controller due to its faster settling time, no overshoot and steady state error. It is then followed by integral controller with the same characteristic of output performance. The state feedback and LQR controller is not suitable for Robot Arm due to its steady state error and overshoot. 5. CONCLUSION All the modern controller for Robot Arm has been developed by using MATLAB Simulink software. Simulink is capable of showing real time results by reduced simulation time. The analyses of the result are focused on the performance of the system, settling time, over shoot and steady state error. 9

10 A comparative study has been made of all the designed modern controller and observer. The study proves that the state feedback with feed forward controller is the best modern controller due to its faster settling time, no overshoot and steady state error. The findings of this research have proven that most of modern controller method can eliminate the steady state error and overshoot. REFERENCES Journal Ahmed M. Kassem, A. A. (2012). Performace Improvements of a Permanent Magnet Synchronous Machine via Functional Model Predictive Control. Journal of Control Science and Engineering, 1-8. Alavandar. S and Nigam. M. J. (2008). Inverse Kinematics Solution of 3DOF Planar Robot using ANFIS. International Journal of Computers, Communication & Control, Frank L.Lewis, Darren M.Dawson, Chaouki T.Abdallah. (2004). Robot Manipulator Control: Theory and Practice. New York: Marcel Decker. Gurpreet S. Sandhu and Kuldip S. Rattan. (1997). Design of a Neuro Fuzzy Controller: Systems, Man, and Cybernetics IEEE International Conference. Jafar Tavoosi, Majid Alaei, Behrouz Jahani. (2011). Neuro-Fuzzy Controller for Position Control of Robot Arm. 5th Symposium on Advance in Science & Technology. Mashhad, Iran. Li, W. (1998). Design of a Hybrid Fuzzy Logic Proportional Plus. IEEE trans. on Fuzzy Systems, Nigam, S. A. (2008). Adaptive Neuro-Fuzzy Inference System based control of six DOF robot manipulator. Journal of Engineering Science and Technology, Petar K., Yiannis D., Darwin G. C. (2016). Kinematic-free Position Control of a 2-DOF Planar Robot Arm. Proc. IEEE/RSJ Intelligent Robots and Systems (IROS 2016),. W.Chen, H. Li, X. Z. Zhang. (2015). Modeling and Analysis of Planar Robotic Arm Dynamics Based on an Improved Transfer Matrix Method for Multi-body Systems. The 14th IFToMM World Congress. Taipei. Book Nise, N. S. (2004). Control Systems Engineering. Danvers: John Wiley & Sons, Inc. Ogata, K. (2010). Modern Control Engineering. New Jersey: Pearson Education. Frank L. L., Darren M. D., Chaouki T. A. (2004). Robot Manipulator Control: Theory and Practice. New York: Marcel Decker. Electronic Source Benjamin M. O., Xinggang Y. (30 Disember, 2016). Retrieved from University of Kent: K. Takaya. (January, 2003). University of Saskatchewan. Retrieved from Review of Analog Controller Design. Majed D. Youns, S. M. (2013). Position Control of Robot Arm using Genetic Algorithm based PID Controller. (pp ). University of Mosul. Report Syaban, S. (2014). Modeling and Control of 6-DOF of Industrial Robot by using Neuro-Fuzzy Controller. Johor, Malaysia: Universiti Tun Hussein Onn Malaysia. 10

11 AUTHORS BIBLIOGRAPHY Anis Pauzi was born in Kuala Terengganu, Terengganu. She received the Bachelor s degrees from the Department of Electrical System, Universiti Malaysia Perlis (UniMaP) and Master s degrees from Faculty of Electrical Engineering, University of Malaya (UM). Since 2009, she has been as a lecturer at Electrical Engineering Department at Politeknik Sultan Azlan Shah, Behrang and since 2015, she transferred to Politeknik Sultan Salahuddin Abdul Aziz Shah, Shah Alam. She teaches courses in Power Electronics, Embedded Robotics and Circuit Analysis. Her areas of research interest are in control system, robotics, and autonomous. Zakiah Md Saad was born in Permatang Pauh, Pulau Pinang. She received the Bachelor s degrees from the University of Technology Malaysia (UTM), Skudai and Master s degrees from Universiti Teknologi MARA (UiTM), Shah Alam. She joined the Electrical Engineering Department at the Politeknik Sultan Salahuddin Abdul Aziz Shah, Shah Alam in Her areas of research interest are in control system, power electrics and communications. Rokiah Hassan was born in Sungai Petani, Kedah. She received the Bachelor s degrees from the University of Technology Malaysia (UTM), Skudai. She joined the Electrical Engineering Department at the Politeknik Sultan Salahuddin Abdul Aziz Shah, Shah Alam in Her areas of research interest are in control system and power electrics. 11