Controlling the Motion of a Planar 3-DOF Manipulator Using PID Controllers

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1 Controlling the Motion of a Planar -DOF Manipulator Using PID Controllers Thien Van NGUYEN*,1, Dan N. DUMITRIU 1,, Ion STROE 1 *Corresponding author *,1 POLITEHNICA University of Bucharest, Department of Mechanics, 1 Splaiul Independentei, Bucharest 06004, Romania, bangden468@gmail.com*, ion.stroe@gmail.com Institute of Solid Mechanics of the Romanian Academy, 15 Constantin Mille, Bucharest , Romania, dumitriu@imsar.bu.edu.ro DOI: / Received: 4 April 017/ Accepted: 17 October 017/ Published: December 017 Copyright 017. Published by INCAS. This is an open access article under the CC BY-NC-ND license ( Abstract: This paper mainly presents a method to model the -DOF manipulator with PID controller in Matlab-SimMechanics and the way to find out optimal parameters in the PID controller including: proportional gain, integral gain and derivative gain using the genetic algorithm (GA). For that aim, the -DOF manipulator will be simulated in order to perform a given task. The results achieved from the model show that difficulties in tuning parameters of PID controller can be solved using GA, while SimMechanics is considered as a good virtual experiment tool for simulating an arbitrary mechanism. Key Words: Manipulator, GA, SimMechanics, parameters, PID controller 1. INTRODUCTION Nowadays, the role of robots is more and more important thanks to their application in replacing people in hard and hazardous works. Therefore the study of manipulators modeling and control is very important. In recent years, the control of manipulators has attracted increased attention. For instance, Hossein Sadegh Lafmejani and Hassan Zarabadipour used a PID controller [4], which was linearized by feedback linearization to control the position of a -DOF articulated manipulator. Meanwhile, based on the approach proposed by Sciavicco and Siciliano [], Le Tien Dung, Hee-Jun Kang, Young-Shick Ro ([]) used a PD controller with online Gravity compensation to control the position of a -DOF manipulator. The control methods can be classified into three main categories: Traditional feedback controls (PD and PID controls) [6,7]; Adaptive control [8,9]; Iterative learning control (ILC) [15,16]. Besides, there are some other control methods: robust control, inverse dynamics control and switching control. Among those, the PID controllers are widely used thanks to their simplicity and transparence. However, one of the most common difficulties when using the PID controllers is to determine the optimal parameters, i.e: the proportional gain K, the integral gain K i, and the derivative gain K d because of the constantly changing of the system parameters in almost all processes. Thus, GA is one of the well-known approaches p INCAS BULLETIN, Volume 9, Issue 4/ 017, pp (P) ISSN , (E) ISSN

2 Thien Van NGUYEN, Dan N. DUMITRIU, Ion STROE 9 for solving that difficult problem. Many authors have used GA to adjust the parameters of PID controller in their own purposes. For instance: Wang and Kwok presented a ph neutralization process regulated by a PID controller with its parameters optimized using a GA [11]; The study given in [1] covers the optimization of two decoupled PID controllers for the propulsion and navigation of an oil platform supplying ship using a GA; In [1], a methodology for the optimal PID controller design using a GA is proposed to improve the transient stability of AC-DC transmission systems after faults; T. K. Teng, J. S. Shieh, C. S. Chen apply a simple GA method in a real-time experiment on a liquid level control system for online auto-tuning PID parameters [14]. Based on the case study done by Dan N. Dumitriu and Cornel Secara [1], where the - DOF manipulator is placed in the horizontal plane, i.e the components of gravity are not considered, this paper presents the case of a manipulator placed in vertical plane. In this case, it is absolutely necessary to consider the components of gravity in the equations of motion. Therefore, the control of manipulator when performing a given task is more complicated. For solving the problem, PID controllers are used to control input torques corresponding to the feedback signal. GA is also applied for determining the optimal parameters of the PID controllers in this specified case.. EQUATIONS OF MOTION Let us consider the -DOF articulated manipulator as shown in Fig.1. The first link (1) characterized by length l 1 and mass m 1 is subjected to the external torque 1, which is perpendicular to (Oxy) plane, at the point O. Analogously, the links () and () characterized by lengths l, l, and masses m, m, are actuated by the external torques, at the points A, B, respectively. C F (0; 0.5 m) C B C C1 A O S (0.5 m; 0) INCAS BULLETIN, Volume 9, Issue 4/ 017 Fig. 1 - The -DOF manipulator

3 9 Controlling the Motion of a Planar -DOF Manipulator by Using PID Controllers This mechanism has three degrees of freedom; thus let us choose,, as the 1 generalized coordinates representing the manipulator links positions with respect to time. Then the equations of the manipulator motion are written easily using Lagrange equation [5]: d E E U Qi, ( i1,, ), dt i i i where E is the total kinetic energy of mechanism, U is the force function (which has the same value with the potential energy but in opposite sign), * * Q i is the external generalized force acting on the link i, in this case Qi i, ( i 1,,). From geometric and kinematic relationship between the links of the manipulator, after computing all necessary terms then replacing them into equation (1), the equations of motion are obtained in matrix form as following: T (1) AθBτ, () where 1,, is the vector of second time derivative of the joint coordinates, T 1,, is the external torque vector acting on the links at the points O, A, and B, respectively, a11 a1 a1 A a1 a a, is the inertia matrix, and its components have specific values as a 1 a a m1 m m a11 m m 1 1 cos l m l m l l l ml l1 cos l cos, m m l l a a m l m l l m l l cos cos( ) cos, In equation (), b, b, b l1 l l a1 a1 ml cos cos, m l a m l m l l cos, l l a a m l cos, T 1 ml a. B stands for vector of Coriolis/ Centripetal and Gravity components, and is written in more detail as: () INCAS BULLETIN, Volume 9, Issue 4/ 017

4 Thien Van NGUYEN, Dan N. DUMITRIU, Ion STROE 94 m b1 m l1l 1 sin ml l1 1 sin l 1 sin m1 m m l g m m l1g cos1 m lg cos1 cos 1, m ml b m l1l 1 sin l1 1 sin l 1 sin m m l g m l g cos1 cos 1, m l g ml b l1 1 sin l 1 sin cos 1, (4). MODELLING THE -DOF MANIPULATOR IN SIMMECHANICS SimMechanics is based on Simulink environment in Matlab software and can be used to model and analyze an arbitrary mechanical system. Besides, the animation of the operation mechanism is displayed on the screen during the motion analysis. Thanks to that characteristic, the researchers can both analyze and adjust more easily the mechanism operation corresponding to their own aims. In addition, this property is also useful when researchers want to verify the fulfillment of a given task of the mechanism. Moreover, by removing and adding blocks in the SimMechanics diagram, the mechanical system structure can be easily changed, then it is very convenient for the robotics design field. Fig. - Program scheme of -DOF manipulator in SimMechanics INCAS BULLETIN, Volume 9, Issue 4/ 017

5 95 Controlling the Motion of a Planar -DOF Manipulator by Using PID Controllers A significant advantage of using SimMechanics in modeling and simulating a mechanical system is that it presents a simple way to obtain the dynamic model of the system regardless driving differential equations of the considered mechanism. Also, the results obtained using SimMechanics can be further used in forward kinematics, etc. The Toolbox in SimMechanics provides a large number of blocks corresponding to real systems components: bodies, joints, constraints, actuators, sensors. So that almost all mechanisms can be modeled and analyzed in that environment. For our case, the -DOF manipulator is created easily in SimMechanics by taking blocks in Toolbox and connecting them in correct rules, i.e consistent with geometric and kinematic relationship between interconnected links. Fig. shows the SimMechanics scheme for the considered manipulator from Fig DETERMINING THE OPTIMAL PARAMETERS OF PID CONTROLLERS USING GA To demonstrate our objective, the tip C of end-effector of the -DOF manipulator is required to perform a given task. More precisely, it is necessary to follow the straight line from point S(0.5m; 0) to point F(0; 0.5m) in the plane Oxy. This straight line is defined by: t t o t xc ( t) ,( m), Tf t o 0 t to yc ( t) t,( m), T t f o (5) where to 0, Tf 0( s) are the initial and final times of the analysis process. An additional requirement is that tip C should be kept perpendicular to the given straight line SF, that means in our case 1( t) ( t) ( t), (6) 4 For the simulation purpose, let s consider that the -DOF manipulator has the following inertial and geometric data: m1 m m 0.1( kg); l1 l 0.1( m); l 0.04 ( m) and that the initial state of the manipulator considered in the case study is ( t ) ( rad), ( t ) ( rad / s), 1 o 1 ( t ) ( rad), ( t ) ( rad / s), o ( t ) ( rad), ( t ) ( rad / s), o o o o Based on the approaches proposed by Sciavicco and Siciliano [], the online gravity compensation will be taken into account in the PID control law; so the input torque applied in the manipulator is written as following: dei () t i ( t) KPi. ei ( t) KIi. ei ( t) KDi Gi ( ), ( i 1,, ), (7) dt INCAS BULLETIN, Volume 9, Issue 4/ 017

6 Thien Van NGUYEN, Dan N. DUMITRIU, Ion STROE 96 where ei () t represents the tracking error, i.e the difference between the desired input value rt () and the actual output qt (), K Di are the PID controller proportional gain, integral gain and derivative gain, respectively, corresponding to the link i. Gi ( ) is online gravity compensation component corresponding to the link i, i () t is the input torque to control the link i. For finding out the appropriate torques to control the manipulator performing the given task, we have to determine the optimal values of K Di. That is a real hard work especially in complex mechanisms because of variable system parameters. Also with the random search property, GA is well-known as a good tool for PID parameter tuning. The block diagram of our control system using GA for determining the optimal PID parameters is shown in Fig. genet ic al gor it hm G gr avit y compensat ion d cont r ol l er s -dof manipul at or Fig. - PID tuning diagram with GA One of the most important steps in applying GA is to choose the objective functions that are used to evaluate the fitness of chromosomes. The sum of squared angle errors ( e1 e e ) is chosen as objective function, in this case. As we need to minimize the angle errors to obtain the desired trajectory, the fitness function is taken as inverse of the sum of squared angle errors. The fitness function will be used to evaluate the fitness of each individual in population. In this paper, K Di (i=1,, ) are considered as chromosomes and are encoded in each of 0 individuals of the initial population. Then they are randomly assigned a value in a specified variable range. Based on criteria of possible stability and position error of mechanism corresponding to the desirable trajectory, the chromosomes in initial population are evaluated and then all the best chromosomes are kept. After that, these chromosomes are reproduced and combine/combined with current population to create a new population. By applying crossover and mutation to chromosomes in the GA, after some generations, the best solutions satisfying the mentioned criteria will be determined, i.e. the values of K Di will be obtained. INCAS BULLETIN, Volume 9, Issue 4/ 017

7 97 Controlling the Motion of a Planar -DOF Manipulator by Using PID Controllers However, one should notice that the quality of solutions depends on the maximum number of generation in the GA program as well as on each time the GA program is carried out due to its stochastic property. The GA is implemented according to the following steps: - Step 1: Define the population size, the maximum number of generations for the GA program, the probabilities of crossover and mutation (noticing that their sum is equal to 1, and the probability of mutation is smaller than that of crossover to avoid destroying the fit properties of individuals); also randomly assign the value of K Di (i=1,, ) in a specified variable range, etc. - Step : Randomly generate the initial population. - Step : Evaluate the fitness of each individual in the population and save the individuals having the best fitness. - Step 4: Check if the pre-defined criterion (possible stability of manipulator) is satisfied or not. - Step 5: Reproduce the new population based on the previous best individuals. - Step 6: Implement crossover and mutation operations to improve the qualities in the next generation. - Step 7: Repeat step until the pre-defined convergence criterion is satisfied. Fig. 4 - Completed scheme of -DOF manipulator with PID controllers After using GA for tuning PID controllers in our system, the values of K Di (i=1,, ) are determined as: K ; K 1.450; K ; P1 I1 D1 K ; K ; K ; P I D K ; K ; K P I D The results obtained from SimMechanics model corresponding to the above value of PID parameters are shown in Fig.5, Fig.6 and Fig.7. INCAS BULLETIN, Volume 9, Issue 4/ 017

8 Thien Van NGUYEN, Dan N. DUMITRIU, Ion STROE 98,, Fig. 5 - Evolution of the joint coordinates 1 versus real time Fig. 6 - Evolution of the external torque 1,, versus real time With the aim of illustrating how well the manipulator performed the given task with computed value of PID parameters, the position error showing the difference between real position of the tip C and its desirable position is considered and is defined by: real des real des ( t) x ( t) x ( t) y ( t) y ( t), (8) C C C C C Fig. 7 - Position of tip C versus real time Fig. 8 - Position error of tip C versus real time As shown in Fig.8, the position error of tip C remains in the range of [0; 0. mm]. In practice, this error is due to accumulation errors and is acceptable, taking into account the length of trajectory of the -DOF serial manipulator, which is about 0,55 m. This error can be reduced by improving the GA program to give the better solutions of the PID parameters. 5. CONCLUSIONS This paper presents the process of designing the PID controllers with online gravity compensation components for a -DOF articulated manipulator. For this purpose, GA is used as a good method for determining these PID parameters. The paper also describes the way to model the dynamics of a -DOF manipulator in SimMechanics, which is considered as a virtual experiment tool used in robotic systems INCAS BULLETIN, Volume 9, Issue 4/ 017

9 99 Controlling the Motion of a Planar -DOF Manipulator by Using PID Controllers dynamic modeling and design. More precisely, SimMechanics is used to validate the results obtained by the integration of the motion equations presented in. REFERENCES [1] D. N. Dumitriu, C. Secara, Dynamics of a Planar -DOF Manipulator, SISOM 010 and Session of the Commission of Acoustics, Bucharest 7-8 May, 010, ISSN , pp [] D. Le Tien, K. Hee-Jun, R. Young-Shick, Robot manipulator modeling in Matlab-SimMechanics with PD control and online Gravity compensation, Proceeding of Strategic Technology (IFOST, pp ), 010. [] L. Sciavicco, B. Siciliano, Modeling and control of Robot manipulators, nd edition, London: Springer. [4] H. S. Lafmejani, H. Zarabadipour, Modeling, Simulation and Position Control of DOF Articulated Manipulator, Indonesian Journal of Electrical Engineering and Informatics (IJEEI), Vol., No., ISSN: 089-7, pp , September 014. [5] I. Stroe, S. Staicu, Calculus of joint forces using Lagrange equations and principle of virtual work, Proceedings of the Romanian Academy, Series A, Volume 11, pp. 5-60, 010. [6] J. J. Craig, Introduction to robotics: Mechanics and Control, rd edition, Addison-Wesley, [7] Q. Chen, H. Chen, Y. J. Wang, P. Y. Woo, Global stability analysis for some trajectory tracking control schemes of robotic manipulators, Journal of Robotic Systems, pp.69-75, 001. [8] J. Y. Choi, J. S. Lee, Adaptive iterative learning control of uncertain robotic systems, IEE Proceeding Control Theory Appl., pp.17-, 000. [9] J. J. Slotine, W. Li, On the adaptive control of robot manipulators, The International of Robotics Reasearch, pp.49-59, [10] A. Bagis, Determination of the PID controller Parameters by Modified Genetic Algorithm for Improved Performance, Journal of Information Science and Engineering, pp , 007. [11] P. Wang, D. P. Kwok, Optimal design of PID process controllers based on genetic algorithms, Control Engineering Practice, Vol., pp , [1] E. ALfaro-Cid, E. W. McGookin, D. J. Murray-Smith, GA-optimized PID and pole placement real and simulated performance when controlling the dynamics of a supply ship, IEE Proceeding on Control Theory Applications, Vol. 15, pp. 8-6, 006. [1] Y. P. Wang, N. R. Waston, H. H. Chong, Modified genetic algorithm approach to design of an optimal PID controller for AC-DC transmission systems, Electrical Power and Energy Systems, Vol. 4, 00, pp [14] T. K. Teng, J. S. Shieh, C. S. Chen, Genetic algorithm applied in online auto-tuning PID parameters of a liquid control system, Transactions of Institute of Measurement and Control, Vol. 5, 00, pp [15] A. Tayebi, Adaptive iterative learning control for robot manipulators, Proceeding of American control conference, Denver, CO, USA, June 00, 5: [16] T. Y. Kuc, K. Nam, J. S. Lee, An iterative learning control of robot manipulators, IEEE Trans Robot Automation, 7(6): 85-84, [17] D. T. Pham, D. Karaboga, Intelligent Optimization Techniques: Genetic Algorithms, Tabu search, Simulated Annealing and Neural Networks, Springer-Verlag, 000. INCAS BULLETIN, Volume 9, Issue 4/ 017