control of a scara robot: pso-pid approach - Semantic Scholar
Control and Intelligent Systems, Vol. 38, No. 1, 2010
CONTROL OF A SCARA ROBOT: PSO-PID APPROACH M.T. Da¸s∗ and L.C. D¨ ulger∗∗
Abstract
r1 ,r2
This paper presents an approach where the Particle swarm optimiza-
Ra1 ,Ra2
tion (PSO) is employed to get optimum PID controller parameters to control of a SCARA robot. PSO is an evolutionary technique
τ1 , τ2 xB , yB . .. θ1 , θ 1 , θ 1
which is originated from swarm intelligence. A practical application of the PSO-PID controller is performed with the system dynamics. A simulation study is also included with the proposed controller by using the systems dynamic equations with the actuator dynamics
.
..
θ2 , θ 2 , θ 2
and optimum PID parameters. Different trajectories are studied. Performances of the traditional controllers; PD, PID and the application of PSO-PID controller are then compared.
Simulation
θ1ref , θ2ref
results are presented. The numerical results definitely enforced the
lengths of the main arm and the fore arm (m) the armature resistances for motor 1 and 2 (Ω) the motor torques (N m) the coordinates of point B (kg) the angular disp., vel. and acceleration of the main arm (rad, rad/s, rad/s2 ) the angular disp., vel. and acceleration of the fore arm (rad, rad/s, rad/s2 ) the reference positions for the main and fore arm
potential use of PSO-PID controller herein.
1. Introduction
Key Words
SCARA (selective compliant articulated robot arm) consists of three planar revolute joints as shoulder, elbow and wrist and a prismatic joint which operates in vertical plane for proper positioning of the work piece. It offers especially a good choice for assembly tasks. Many different studies are seen on modelling and control of SCARA type robot. Some of them are referred in the study and a brief overview is given. Da¸s [1] has studied on motion control of a SCARA robot with a PLC unit, traditional controllers are included. Ali et al. [2] have studied on the Hinfinity control of robotic system (SCARA) with linear parameter varying (LPV) representation. Belhocine et al. [3] have applied sliding mode in robot control. They have used this technique to control of a SCARA robot and its identification is done by using MATLAB. Lin et al. [4] have presented a new optimal control approach to robust control of robot manipulators. This approach has been explained using a SCARA robot. In recent studies on mechanical design problems and on control issues, many different evolutionary optimization algorithms are used. Among evolutionary algorithms (EAs), PSO is quite commonly revealed for solution of different engineering optimization problems. Some of them are noted here to show its application in solution of engineering problems. The algorithm can be used with neural networks (NN) as PSO-NN. It can also be used with back propagation algorithm (BP) to train a NN. Prempain et al. [5] have presented an improved particle swarm optimizer (PSO) for solving mechanical design problems; spring design, vessel design for example. Gaing [6] has then deter-
Particle swarm optimization (PSO), PID (proportional derivative integral control, SCARA robot, optimum control, PSO-PID controller
Notation e(t) Ia1 , Ia2 J1 , J2 Jm , Jm1 , Jm2 Jg1 , Jg2 La1 , La2 Ke1 , Ke2 Kt1 , Kt2 m1 ,m2 N1 ,N2
the error signal the armature current for motor 1 and 2 (A) inertias of the main and the fore arm (kg m2 ) the motor and equivalent inertias (kg m2 ) inertias of the gearbox 1 and 2 (kg m2 ) the armature inductances for motor 1 and 2 (H) the back emf constants for motor 1 and 2 (V/rad/s) the torque constants for motor 1 and 2 (N m/A) masses of the main(shoulder) and the fore arm (elbow)(kg) the gearbox ratios for motor 1 and 2
∗
Roketsan Company, Elmada˘ g-Ankara, Turkey; e-mail: [email protected] ∗∗ Gaziantep University, Faculty of Engineering, Department of Mechanical Engineering, Gaziantep, Turkey; e-mail: dulger@ gantep.edu.tr Recommended by Prof. K.K. Tan (paper no. 201-2151)
24
The first one is the best solution pbest,i that a particle has achieved so far. The other one is the global best value gbest that is obtained by any particle in the population. After finding the two best values pbest,i and gbest , each particle updates its velocity and position according to the following (1) and (2) below.
mined the optimal PID parameters of an automatic voltage regulator (AVR) system using PSO algorithm. This method has been compared with the genetic algorithm (GA) and the numerical results are presented. Liu et al. [7] have studied on an optimization design based on PSO for PID Controller. Wang et al. [8] have presented an active queue management (AQM) model with the PSO-PID controller. Simulation results of PSO-PID controller for AQM routers are given. Kao et al. [9] have achieved a study on the self-tuning PID control in a slider-crank mechanism system by using PSO algorithm. Nasri et al. [10] have presented a study on PSO-based optimum design of PID controller for a linear brushless dc motor (BLDC). A comparative study is given GA and Linear Quadratic Regulator (LQR) method with PSO method. Da¸s and D¨ ulger [11] have then performed a study on control of a four-bar mechanism using the proposed PSO-PD controller. The contribution of this paper is that a control system based on PSO-PID is designed and implemented on a SCARA robot (Serpent 1) which has taken 2 degrees of freedom (dof) with RR configuration. This paper is prepared in five sections. A mathematical background on PSO algorithm and the proposed controller details is presented in Section 2. SCARA robot in laboratory is described together with its mechanical and electrical data in Section 3. The mathematical structure of robot with its actuator dynamics is included in Section 4. Finally, numerical examples are given to show performance of PSOPID controller for position control of a SCARA robot in Section 4. Conclusions are then presented in Section 5.
vik+1 = wvik + c1 r1 (pbest,i − xki ) + c2 r2 (gbest − xki )
xk+1 = xki + vik+1 i
(1)
(2)
where, vik xki w pbest
the velocity of the particle the current position of the particle (solution) the inertia weight factor the best solution value among the particle found in a particular iteration gbest global best solution achieved so far r1 , r2 the random numbers between [0, 1] c1 , c2 acceleration constants where; c1 = c2 = 2 The inertia weight factor w in (1) can be calculated by using (3) where iter is the iteration number in a particular instance and iter max is the maximum number of iterations. w = wmax −
(wmax − wmin ) · iter itermax
where
0.4 < w < 0.9
2. Background
(3) Equations (1) to (3) given in [6, 9, 11] are used while performing optimization of PID controller parameters.
2.1 Particle Swarm Optimization Algorithm (PSO)
2.2 PSO-PID Controller Implementation
PSO is an evolutionary algorithm based on population. PSO algorithm first introduced by Kennedy and Eberhart is based on simulation of social system behaviour in 1995 [12]. Since then a lot of research study has been carried on the algorithm. In a recent study, Eberhart and Shi have presented a bibliography on the algorithm [13]. They have presented the original algorithm with developments and various applications. Objectives, priorities and resources are changed for a real world. Potential applications are seen in system design, pattern recognition, and identification, robotic applications, image segmentation, time frequency analysis, robot path planning, signal processing and control purposes. Eberhart and Shi [14] have studied particle swarms with a review on tracking and optimizing dynamic systems. Three kinds of dynamic systems are defined and studied giving many answers for the algorithm in future investigations. In this algorithm, the “individual” is used to replace the “particle” and the “population” is used to replace the “group” [13]. There are three parameters Kp , Ki and Kd with three members as K = [Kp , Ki ,Kd ]. So there are m individuals in a population resulting in m × 3 population. Increasing the number of swarms increase in the number of error function evaluations. Each particle updates its position and velocity by following the two “best” values.
An overview on PID control is performed by Cuminos and Munro [15] where different tuning methods are presented with their offered advantages and disadvantages. Mann et al. [16] have presented a time response-based design methodology for PID controllers. Many approaches have been applied previously to determine PID parameters. Ziegler–Nichols (ZN) method, Kapa-Tau tuning, pole placement, use of gain and phase margins and partitioning, OLDP method and GA can be given as some of the methods. The ZN method is applied to set bounds of the PID controller parameters. Ang et al. [17] have presented an overview on tuning methods, software packages and commercial hardware modules. Tuning methods for PID controllers are summarized as analytical, heuristic, frequency response and optimization methods. Then Li et al. [18] have studied on PID design objectives and methods. According to design methods, here evolutionary computation is used with optimization tools, and PSO algorithm is applied. The PSO-PID controller is then developed to search optimal PID parameters. In time domain, the output of PID is given by: t en (t)dt + Kdn
u(t) = Kpn en (t) + Kin 0
25
den (t) dt
(4)
Figure 1. Structure of PSO-PID controller.
Figure 2. Serpent 1(top view). evaluation value [6, 9, 11].
where en (t) represents the positional error for each axis, for n = 1, 2. Kpn is the proportional gain, Kin is the integral gain and Kdn is the derivative gain. The error signal can be defined for each axis as; e1 (t) = θ1ref (t) − θ1 and e2 (t) = θ2ref (t) − θ2 . Performance of the controller can be evaluated either in frequency domain or time domain. In this study, performance of PID controller is analyzed in time domain with the overshoot (Mp ), rise time (tr ), settling time (ts ) and steady-state error (ess ). The performance criterion W (K) is defined as: W (K) = (1 − e−β )(MP + ess ) + e−β (ts − tr )
f=
1 W (K)
(6)
Equations (5) and (6) are applied together with the assumptions in time domain; no overshoot, considerably small settling time and introducing minimum steady-state error. Details of convergence analysis and parameter selection of PSO are included in Section 5. The proposed PSO-PID structure can be given in Fig. 1. 3. SCARA Robot
(5)
The SCARA robot (Serpent 1) is available in Dynamic Systems Laboratory, Gaziantep University, and Mechanical Engineering Department. The photograph is shown in Fig. 2 for the given configuration. Technical specifications of the robot are included in Table 1 [20]. Mechanical and electrical data for Serpent 1 are also presented in Table 2
where the parameter K includes the controller parameters Kp , Kd and Ki . In this study, β is chosen 0.7 representing the weighting factor. The fitness function f of the PSO algorithm is defined as the reciprocal of W (K). The smaller W (K) the value of individual K, the higher its 26
Table 1 Specifications of Serpent 1 Main arm (shoulder) length (r1 ) 250 mm Fore arm length (elbow) (r2 )
150 mm
Shoulder movement (θ1 axis)
200◦
Elbow movement (θ2 axis)
250◦
Wrist rotation (roll axis)
450◦
Up and down (z axis)
75 mm
Maximum tip velocity
550 mm/s
Capacity
2.0 kg
Table 2 Mechanical and Electrical Data for Serpent 1 J1 = 0.0980 kg m2 , J2 = 0.0115 kg m2 m1 = 1.90 kg, m2 = 0.93 kg Jm = 3.3 × 10−6 kg m2 Ke1 = Ke2 = 0.047 V/rad/s Kt1 = Kt2 = 0.047 Nm/A Ra1 = Ra2 = 3.5 Ω La1 = La2 = 1.3 mH N1 = 90, N2 = 220 Jg1 = 0.0002 kg m2 , Jg2 = 0.0005 kg m2 Figure 3. (a) Working envelope of Serpent 1 and (b) A trajectory example for PNP job. [20]. A real trajectory is chosen by looking at PNP (Pick and Place) job specified.
4. Manipulator and Actuator Model
3.1 Working Envelope and Trajectory
SCARA (Serpent 1) is an industrial robot with 4 dof, where three axes are rotational (electrical drives) and one is translational (pneumatic). The simulation example of the SCARA robot is considered here with two axes, so 2 dof. The robot dynamics (referred to Fig. 2) are obtained from Lagrange’s equations. Da¸s and D¨ ulger have derived the mathematical model in a previous study with a traditional PD controller [19]. So the nonlinear system equations are directly taken with inclusion of actuator dynamics and necessary gearing. Two second order nonlinear coupled equations are resulted in. The general arm equations have the following form;
Working envelope of Serpent 1 and dimensional limitations are given in Fig. 3(a). Different sizes of work pieces can be assembled. In system, SCARA has been applied to place 3 different sizes of work pieces (shown by different colours as red-small, blue-medium, green-big). Here, the trajectory is taken while performing picking operation for 58 mm diameter and 50 mm length cylinder work piece (big one) between two stations (picking–placing point). Figure 3(b) shows its followed trajectory during a pick and place operation as X–Y locus which is performed in 3 s. Trajectory data points are taken during an operation of the robot in Dynamic System Laboratory, Mechanical Engineering Department, Gaziantep University. These experimental trajectory points are then characterized as polynomials using curve fitting for use as the model input data during simulation. θ1 = −0.1798t2 − 0.1787t + 3.0227 θ2 = −0.2458t2 + 0.1303t + 1.6902
M (q)¨ q + C(q, q) ˙ q˙ + G(q) = Q
(8)
where q includes the joint variables; positions, velocities and accelerations. Q is the input generalized forces or torques, M (q) is the inertia matrix and C(q, q) is the coriolis/centripetal forces and G(q) is the gravity force or torque. The configuration of SCARA and its parameters are shown with two joint variables as the generalized coor-
(7) 27
Figure 4. Convergence rates for both axes. dinates, q1 = θ1 and q2 = θ2 . So the system model can be written as in matrix form: ⎤⎡
⎡ ⎣
M11 M12 M21 M22
⎦⎣
θ¨1 θ¨2
⎤
⎤⎡
⎡
⎦+⎣
C11 C12 C21 C22
⎦⎣
θ˙1 θ˙2
⎤
⎦=⎣
τ1
⎦
(9)
τ2
θ1m N1
⎤⎡
⎡ ⎣
La1 0 0
La2 ⎡
+⎣
Ke1 0
M21 =
τ1 and τ2 are the applied actuator torques and θ1 =
θ2m respectively. The equations representing the N2 motors electrical structure can be expressed in a matrix form as given below.
(J1 + J2 ) (m1 r12 + m2 r22 + 4m2 r12 ) M11 = Jm1 + + N12 4N12 m2 r1 r2 ] (cos(θ + θ ) cos θ + sin(θ + θ ) sin θ ) + 1 2 1 1 2 1 N12 (9a)
m2 r22 m2 r1 r2 J2 + + (cos(θ1 + θ2 ) N1 N2 4N1 N2 2N1 N2 × cos θ1 + sin(θ1 + θ2 ) sin θ1 ) (9b)
(9h)
and θ2 =
where;
M12 =
C22 = 0
⎤
⎡
m2 r1 r2 ˙ θ2 [cos(θ1 + θ2 ) sin θ1 − sin(θ1 + θ2 ) cos θ1 ] N1 (9e)
C12 =
m2 r1 r2 ˙ θ2 [cos(θ1 + θ2 ) sin θ1 − sin(θ1 + θ2 ) cos θ1 ] 2N1 (9f)
C21 =
m2 r1 r2 ˙ θ1 [sin(θ1 + θ2 ) cos θ1 − cos(θ1 + θ2 ) sin θ1 ] 2N2 (9g)
⎤
⎤⎡
⎡ Ra1 0
I1
⎦⎣ ⎦ ⎦+⎣ I2 0 Ra2 I˙2 ⎤ ⎡ ⎤ ⎤⎡ V θ˙ 0 ⎦⎣ 1m ⎦ = ⎣ 1 ⎦ V2 θ˙2m Ke2
⎦⎣
(10)
5. Numerical Simulation In this section, the performance of PSO-PID controller is examined. Different examples are studied. Initially a linearized model is derived representing the motor, gearbox and a constant load for each axis. A stability analysis is performed on this linearized model. Systems transfer function is obtained with inclusion of PID controller. PSO programs are implemented in MATLAB using the system’s transfer function. The mathematical model of SCARA with actuators is simulated in Pascal. The nonlinear robotactuator model is then used in simulation. Having given the system measures in time domain, the optimization procedure is completed; PID parameters found in PSO application are implemented in the model referred. Time domain measures are then found in terms of the system’s parameters. They are used for calculations of (5) and (6) in Matlab environment. The performance of the algorithm is then tested by using the systems transfer function. The
(9d)
C11 =
⎤
Equation (10) represents electrical structure where V1 and V2 are the applied armature voltages. They are calculated by using PID controller parameters found applying PSO algorithm. Other motor data are taken from the manufacturer’s data sheet [20].
J2 m2 r22 m2 r1 r2 + + (cos(θ1 + θ2 ) N1 N2 N1 N2 2N1 N2 × cos θ1 + sin(θ1 + θ2 ) sin θ1 ) (9c) J2 m2 r22 M22 = Jm2 + 2 + N2 4N1 N2
I˙1
28
Table 3 Performance of the Algorithm Maximum Error (rad)
PD
PID
PSO-PID
Axis 1 (main arm)
−0.0775 −0.0728 −0.0297
Axis 2 (fore arm)
−0.0397 −0.0250 −0.0128
robot system is initially tuned ZN method realizing the values of Kp , Kv and Ki . Then these parameters are tuned by using PSO algorithm with the required time domain performance. A set of good parameters for PID controller obviously is resulted in a good step response also satisfying time domain requirements. 5.1 Parameter Selection and Convergence Analysis In this approach, PSO-PID, some analysis can also be performed on convergence analysis and parameter selection. Some suggestions are taken from previous studies on parameter selection [12–14, 21]. From the early experiments on PSO, the acceleration constants, population sizes and
number of iterations are studied in detail. For example, Eberhart et al. [13] have specified c1 and c2 as 2. Population sizes are generally taken between 20 and 50 and the inertia weight is taken as 0.4 < w < 0.9 during a run. Tuning range for 6 parameters for both controllers are taken between 0 and 200. The random numbers are usually chosen as [0, 1]. Previous studies on SCARA have given an insight for values range in PSO [19]. So the random numbers are multiplied by 200 to get PID values here. Trelea [21] has presented a detailed study on some important issues for PSO parameter tuning. The population sizes are generally reported as 15, 30 and 60. Iterations to convergence rate are important to note during tests referring to “slow ” and “quick ” convergence. It has been specified that “quick convergence ” can be obtained nearly with 900 iterations. So 1,000 iterations are allowed here. Different trials have been performed on the algorithm. Convergence rate for both axes are given in Fig. 4(a) and (b). Stable convergence characteristics are seen for both axes in PSO method here. Consequently, the number of particles in the swarm influences the convergence of the algorithm. During the optimization, the number of particles in the swarm, the
Figure 5. Numerical results for SCARA robot. (a) Response of the main arm (shoulder), (b) Positional error for the main arm, (c) Response of the fore arm (elbow), and (d) Positional error for the fore arm. 29
objective function and the parameters c1 and c2 are important. The following parameters in PSO are taken as population size is 45, m = 1,000 iterations, c1 = c2 = 2, wmax = 0.9 and wmin = 0.4 in this paper. As expected, PSO_PID algorithm has introduced less settling time than PID algorithm alone. Better dynamic response is seen compared to traditional PD and PID controller. Table 3 shows performance of conventional algorithms in terms of maximum positional errors obtained in the trajectory with respect to PSO-PID algorithm given in Fig. 3(a) and (b) while picking and placing the work piece specified.
Department, Gaziantep, 2003. [2] H.S. Ali, L.B. Badas, Y.B. Aubry, & M. Darouach, Hinfinity control of a Scara robot using polytopic LPV approach, IEEE Transactions on Industrial Electronics, 41(2), 1994, 173–181. [3] M. Belhocine, M. Hamerlain, & K. Bouyoucef, Robot control using a sliding mode, Proc. 12th IEEE International Symposium on 1ntelligent Control, Turkey, 1997. [4] F. Lin & R.D. Brandt, An optimum control approach to robust control of robot manipulators, IEEE Transactions on Robotics and Automation, 14(1), 1998, 69–77. [5] S. He, E. Prempain, & Q.H. Wu, An improved particle swarm optimizer for mechanical design optimization problems, Engineering Optimization, 36(5), 2004, 585–605.
5.2 Example Trajectory
[6] Z.L. Gaing, A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Transactions on Energy conversion, 19(2), 2004, 384–391.
The inputs required for the simulation study are the initial values of the system; initial current, angular displacement and velocity, the parameters of SCARA and the motors data (Tables 1 and 2), the step length for simulation and the total response time. In addition to above, the optimum PID parameters are required as the controller settings. Three different trajectories are designed for Serpent 1. Here one of them is only presented. Simulations and optimization studies are also performed for the other picking and placing jobs. Simulation results for both axes of SCARA are given in Fig. 5. Figure 5(a) and (b) shows numerical responses for axis 1 representing the main arm (shoulder) and available positional error in radians. Figure 5(c) and (d) represents numerical responses for axis 2 for the main arm (shoulder) and the positional error in radians. Using PSO-PID controller, the parameters are obtained as Kp1 = 198, Kd1 = 102 and Ki1 = 51.28 for the axis 1, the main arm (shoulder) and Kp2 = 126.62, Kd2 = 25.84 and Ki2 = 0.83 for the axis 2, the fore arm (elbow) respectively. These optimum parameters found are then set into the mathematical model.
[7] Y. Liu, J. Zhang, & S. Wang, Optimization design based on PSO algorithm for PID controller, Proc. 5th World Congress on Intelligent Control and Automation, China, June 15–19, 2004. [8] X. Wang, Y. Wang, H. Zhou, & X Huai, PSO-PID: A novel controller for AQM routers, Proc. IEEE/IFIP on Wireless and Optical Communication Networks, 2006. [9] C.C. Kao, C.W. Chuang, & R.F. Fung, The self tuning PID control in a slider-crank mechanism system by applying particle swarm optimization approach, Mechatronics, 16, 2006, 513–522. [10] M. Nasri, H. Nezamadi-Pour, & M. Maghfoori, A PSO-based optimum design of PID controller for a linear brushless dc motor, PWASET, 20, 2007, 211–215. [11] M.T. Da¸s & L.C. D¨ ulger, Particle swarm optimization (PSO) algorithm: Control of a four bar mechanism, Proc. EU/ME 2007, ‘Metaheuristics in Service Industry, Germany, 96–101. [12] J. Kennedy & R.Eberhart, Particle swarm optimization, Proc. IEEE Int. Conf. Neural Networks, 4, Australia, 1995, 1942– 1948. [13] R.C. Eberhart & Y. Shi, Particle swarm optimization: Developments, applications and resources, IEEE 2001, 81–86. [14] R.C. Eberhart & Y. Shi, Tracking and optimizing dynamic systems with particle swarms, IEEE -2001, 94–100.
6. Conclusions
[15] P. Cuminos & N. Munro, PID Controllers: Recent tuning methods and design to specification, IEE Proceedings Control, an Theory, and Application, 149(1), 2002, 46–56.
PID controller parameters are determined with the application of PSO method and a practical application of PSO-PID controller is presented in this study. The algorithm was applied to 2 dof point-to-point control case in a SCARA robot, Serpent1. There are nonlinear dynamic interactions on the manipulator joints. It is obvious from the simulations that PSO-PID improved positional responses for both the main and the fore arm. Simulations on both axes have shown less error and better convergence characteristics. PSO algorithm includes many tuning parameters that have a great effect on its performance. Some time is devoted for tuning PSO parameters. In this paper, reference studies are taken into consideration while adjusting PSO algorithm for SCARA system. The algorithm can definitely be applied to any system of interest because of its simple integration to the systems. Acceptable computational results and convergence efficiency are certainly seen in the numerical examples.
[16] G.K.I. Mann, B.G. Hu, & R.G. Gosine, ‘Time-domain based design and analysis of new PID tuning rules’, IEE Proceedings an Control, Theory and Application, 148(3), 2001, 251–261. [17] K.H. Ang, G.C.Y. Chong, & Y. Li, PID Control System Analysis, Design and Technology, IEEE Transactions on Control Systems Technology, 13(4), 2005, 559–576. [18] Y. Li, K.H. Ang, & G.C.Y. Chong, PID control system analysis and design, IEEE Control Systems Magazine, February 2006, 32–41. [19] M.T. Da¸s & L.C. D¨ ulger, Mathematical modeling, simulation and experimental verification of a SCARA robot, Simulation Modeling Practice and Theory, 13, 2005, 257–271. [20] Wall I Serpent Manual, 1993. [21] I.C. Trelea, The particle swarm optimization algorithm: Convergence analysis and parameter selection, Information Processing Letters, 85, 2003, 317–325. [22] S. Skoczowski, S. Domek, & K.Pietrusewicz, Robust PID model following control, Control and Intelligent Systems, 34(3), 2006, 1544–1550. [23] Z. Al Hamous, S.F. Faisal, & S. Al. Sharif, Application of particle swarm optimization algorithm for optimal reactive power planning, Control and Intelligent Systems, 35(1), 2007, 1642–1653.
References [1] M.T. Da¸s, Motion control of a SCARA robot with a PLC unit, M.Sc.Thesis, Gaziantep University, Mechanical Engineering
30
L.C. D¨ ulger received her B.S. degree in Mechanical Engineering from Middle East Technical University (METU) in 1986, her M.S. degree in METU 1988 and Ph.D. in Mechanical Engineering from Liverpool Polytechnic England 1992, respectively. She is currently professor of Mechanical Engineering at Gaziantep University, Mechanical Engineering Department.
Biographies M.T. Da¸s received his B.S. degree in Mechanical Engineering from Gaziantep University, Turkey in 2000, his M.S. degree in 2003 and Ph.D. degree in 2008. He was with Gaziantep University’s Mechanical Engineering Department from 2000–2008 as a research assistant. Since December 2008, he has been with the Roketsan Company Elmadaˇg, Ankara.
31
CONTROL OF A SCARA ROBOT: PSO-PID APPROACH M.T. Da¸s∗ and L.C. D¨ ulger∗∗
Abstract
r1 ,r2
This paper presents an approach where the Particle swarm optimiza-
Ra1 ,Ra2
tion (PSO) is employed to get optimum PID controller parameters to control of a SCARA robot. PSO is an evolutionary technique
τ1 , τ2 xB , yB . .. θ1 , θ 1 , θ 1
which is originated from swarm intelligence. A practical application of the PSO-PID controller is performed with the system dynamics. A simulation study is also included with the proposed controller by using the systems dynamic equations with the actuator dynamics
.
..
θ2 , θ 2 , θ 2
and optimum PID parameters. Different trajectories are studied. Performances of the traditional controllers; PD, PID and the application of PSO-PID controller are then compared.
Simulation
θ1ref , θ2ref
results are presented. The numerical results definitely enforced the
lengths of the main arm and the fore arm (m) the armature resistances for motor 1 and 2 (Ω) the motor torques (N m) the coordinates of point B (kg) the angular disp., vel. and acceleration of the main arm (rad, rad/s, rad/s2 ) the angular disp., vel. and acceleration of the fore arm (rad, rad/s, rad/s2 ) the reference positions for the main and fore arm
potential use of PSO-PID controller herein.
1. Introduction
Key Words
SCARA (selective compliant articulated robot arm) consists of three planar revolute joints as shoulder, elbow and wrist and a prismatic joint which operates in vertical plane for proper positioning of the work piece. It offers especially a good choice for assembly tasks. Many different studies are seen on modelling and control of SCARA type robot. Some of them are referred in the study and a brief overview is given. Da¸s [1] has studied on motion control of a SCARA robot with a PLC unit, traditional controllers are included. Ali et al. [2] have studied on the Hinfinity control of robotic system (SCARA) with linear parameter varying (LPV) representation. Belhocine et al. [3] have applied sliding mode in robot control. They have used this technique to control of a SCARA robot and its identification is done by using MATLAB. Lin et al. [4] have presented a new optimal control approach to robust control of robot manipulators. This approach has been explained using a SCARA robot. In recent studies on mechanical design problems and on control issues, many different evolutionary optimization algorithms are used. Among evolutionary algorithms (EAs), PSO is quite commonly revealed for solution of different engineering optimization problems. Some of them are noted here to show its application in solution of engineering problems. The algorithm can be used with neural networks (NN) as PSO-NN. It can also be used with back propagation algorithm (BP) to train a NN. Prempain et al. [5] have presented an improved particle swarm optimizer (PSO) for solving mechanical design problems; spring design, vessel design for example. Gaing [6] has then deter-
Particle swarm optimization (PSO), PID (proportional derivative integral control, SCARA robot, optimum control, PSO-PID controller
Notation e(t) Ia1 , Ia2 J1 , J2 Jm , Jm1 , Jm2 Jg1 , Jg2 La1 , La2 Ke1 , Ke2 Kt1 , Kt2 m1 ,m2 N1 ,N2
the error signal the armature current for motor 1 and 2 (A) inertias of the main and the fore arm (kg m2 ) the motor and equivalent inertias (kg m2 ) inertias of the gearbox 1 and 2 (kg m2 ) the armature inductances for motor 1 and 2 (H) the back emf constants for motor 1 and 2 (V/rad/s) the torque constants for motor 1 and 2 (N m/A) masses of the main(shoulder) and the fore arm (elbow)(kg) the gearbox ratios for motor 1 and 2
∗
Roketsan Company, Elmada˘ g-Ankara, Turkey; e-mail: [email protected] ∗∗ Gaziantep University, Faculty of Engineering, Department of Mechanical Engineering, Gaziantep, Turkey; e-mail: dulger@ gantep.edu.tr Recommended by Prof. K.K. Tan (paper no. 201-2151)
24
The first one is the best solution pbest,i that a particle has achieved so far. The other one is the global best value gbest that is obtained by any particle in the population. After finding the two best values pbest,i and gbest , each particle updates its velocity and position according to the following (1) and (2) below.
mined the optimal PID parameters of an automatic voltage regulator (AVR) system using PSO algorithm. This method has been compared with the genetic algorithm (GA) and the numerical results are presented. Liu et al. [7] have studied on an optimization design based on PSO for PID Controller. Wang et al. [8] have presented an active queue management (AQM) model with the PSO-PID controller. Simulation results of PSO-PID controller for AQM routers are given. Kao et al. [9] have achieved a study on the self-tuning PID control in a slider-crank mechanism system by using PSO algorithm. Nasri et al. [10] have presented a study on PSO-based optimum design of PID controller for a linear brushless dc motor (BLDC). A comparative study is given GA and Linear Quadratic Regulator (LQR) method with PSO method. Da¸s and D¨ ulger [11] have then performed a study on control of a four-bar mechanism using the proposed PSO-PD controller. The contribution of this paper is that a control system based on PSO-PID is designed and implemented on a SCARA robot (Serpent 1) which has taken 2 degrees of freedom (dof) with RR configuration. This paper is prepared in five sections. A mathematical background on PSO algorithm and the proposed controller details is presented in Section 2. SCARA robot in laboratory is described together with its mechanical and electrical data in Section 3. The mathematical structure of robot with its actuator dynamics is included in Section 4. Finally, numerical examples are given to show performance of PSOPID controller for position control of a SCARA robot in Section 4. Conclusions are then presented in Section 5.
vik+1 = wvik + c1 r1 (pbest,i − xki ) + c2 r2 (gbest − xki )
xk+1 = xki + vik+1 i
(1)
(2)
where, vik xki w pbest
the velocity of the particle the current position of the particle (solution) the inertia weight factor the best solution value among the particle found in a particular iteration gbest global best solution achieved so far r1 , r2 the random numbers between [0, 1] c1 , c2 acceleration constants where; c1 = c2 = 2 The inertia weight factor w in (1) can be calculated by using (3) where iter is the iteration number in a particular instance and iter max is the maximum number of iterations. w = wmax −
(wmax − wmin ) · iter itermax
where
0.4 < w < 0.9
2. Background
(3) Equations (1) to (3) given in [6, 9, 11] are used while performing optimization of PID controller parameters.
2.1 Particle Swarm Optimization Algorithm (PSO)
2.2 PSO-PID Controller Implementation
PSO is an evolutionary algorithm based on population. PSO algorithm first introduced by Kennedy and Eberhart is based on simulation of social system behaviour in 1995 [12]. Since then a lot of research study has been carried on the algorithm. In a recent study, Eberhart and Shi have presented a bibliography on the algorithm [13]. They have presented the original algorithm with developments and various applications. Objectives, priorities and resources are changed for a real world. Potential applications are seen in system design, pattern recognition, and identification, robotic applications, image segmentation, time frequency analysis, robot path planning, signal processing and control purposes. Eberhart and Shi [14] have studied particle swarms with a review on tracking and optimizing dynamic systems. Three kinds of dynamic systems are defined and studied giving many answers for the algorithm in future investigations. In this algorithm, the “individual” is used to replace the “particle” and the “population” is used to replace the “group” [13]. There are three parameters Kp , Ki and Kd with three members as K = [Kp , Ki ,Kd ]. So there are m individuals in a population resulting in m × 3 population. Increasing the number of swarms increase in the number of error function evaluations. Each particle updates its position and velocity by following the two “best” values.
An overview on PID control is performed by Cuminos and Munro [15] where different tuning methods are presented with their offered advantages and disadvantages. Mann et al. [16] have presented a time response-based design methodology for PID controllers. Many approaches have been applied previously to determine PID parameters. Ziegler–Nichols (ZN) method, Kapa-Tau tuning, pole placement, use of gain and phase margins and partitioning, OLDP method and GA can be given as some of the methods. The ZN method is applied to set bounds of the PID controller parameters. Ang et al. [17] have presented an overview on tuning methods, software packages and commercial hardware modules. Tuning methods for PID controllers are summarized as analytical, heuristic, frequency response and optimization methods. Then Li et al. [18] have studied on PID design objectives and methods. According to design methods, here evolutionary computation is used with optimization tools, and PSO algorithm is applied. The PSO-PID controller is then developed to search optimal PID parameters. In time domain, the output of PID is given by: t en (t)dt + Kdn
u(t) = Kpn en (t) + Kin 0
25
den (t) dt
(4)
Figure 1. Structure of PSO-PID controller.
Figure 2. Serpent 1(top view). evaluation value [6, 9, 11].
where en (t) represents the positional error for each axis, for n = 1, 2. Kpn is the proportional gain, Kin is the integral gain and Kdn is the derivative gain. The error signal can be defined for each axis as; e1 (t) = θ1ref (t) − θ1 and e2 (t) = θ2ref (t) − θ2 . Performance of the controller can be evaluated either in frequency domain or time domain. In this study, performance of PID controller is analyzed in time domain with the overshoot (Mp ), rise time (tr ), settling time (ts ) and steady-state error (ess ). The performance criterion W (K) is defined as: W (K) = (1 − e−β )(MP + ess ) + e−β (ts − tr )
f=
1 W (K)
(6)
Equations (5) and (6) are applied together with the assumptions in time domain; no overshoot, considerably small settling time and introducing minimum steady-state error. Details of convergence analysis and parameter selection of PSO are included in Section 5. The proposed PSO-PID structure can be given in Fig. 1. 3. SCARA Robot
(5)
The SCARA robot (Serpent 1) is available in Dynamic Systems Laboratory, Gaziantep University, and Mechanical Engineering Department. The photograph is shown in Fig. 2 for the given configuration. Technical specifications of the robot are included in Table 1 [20]. Mechanical and electrical data for Serpent 1 are also presented in Table 2
where the parameter K includes the controller parameters Kp , Kd and Ki . In this study, β is chosen 0.7 representing the weighting factor. The fitness function f of the PSO algorithm is defined as the reciprocal of W (K). The smaller W (K) the value of individual K, the higher its 26
Table 1 Specifications of Serpent 1 Main arm (shoulder) length (r1 ) 250 mm Fore arm length (elbow) (r2 )
150 mm
Shoulder movement (θ1 axis)
200◦
Elbow movement (θ2 axis)
250◦
Wrist rotation (roll axis)
450◦
Up and down (z axis)
75 mm
Maximum tip velocity
550 mm/s
Capacity
2.0 kg
Table 2 Mechanical and Electrical Data for Serpent 1 J1 = 0.0980 kg m2 , J2 = 0.0115 kg m2 m1 = 1.90 kg, m2 = 0.93 kg Jm = 3.3 × 10−6 kg m2 Ke1 = Ke2 = 0.047 V/rad/s Kt1 = Kt2 = 0.047 Nm/A Ra1 = Ra2 = 3.5 Ω La1 = La2 = 1.3 mH N1 = 90, N2 = 220 Jg1 = 0.0002 kg m2 , Jg2 = 0.0005 kg m2 Figure 3. (a) Working envelope of Serpent 1 and (b) A trajectory example for PNP job. [20]. A real trajectory is chosen by looking at PNP (Pick and Place) job specified.
4. Manipulator and Actuator Model
3.1 Working Envelope and Trajectory
SCARA (Serpent 1) is an industrial robot with 4 dof, where three axes are rotational (electrical drives) and one is translational (pneumatic). The simulation example of the SCARA robot is considered here with two axes, so 2 dof. The robot dynamics (referred to Fig. 2) are obtained from Lagrange’s equations. Da¸s and D¨ ulger have derived the mathematical model in a previous study with a traditional PD controller [19]. So the nonlinear system equations are directly taken with inclusion of actuator dynamics and necessary gearing. Two second order nonlinear coupled equations are resulted in. The general arm equations have the following form;
Working envelope of Serpent 1 and dimensional limitations are given in Fig. 3(a). Different sizes of work pieces can be assembled. In system, SCARA has been applied to place 3 different sizes of work pieces (shown by different colours as red-small, blue-medium, green-big). Here, the trajectory is taken while performing picking operation for 58 mm diameter and 50 mm length cylinder work piece (big one) between two stations (picking–placing point). Figure 3(b) shows its followed trajectory during a pick and place operation as X–Y locus which is performed in 3 s. Trajectory data points are taken during an operation of the robot in Dynamic System Laboratory, Mechanical Engineering Department, Gaziantep University. These experimental trajectory points are then characterized as polynomials using curve fitting for use as the model input data during simulation. θ1 = −0.1798t2 − 0.1787t + 3.0227 θ2 = −0.2458t2 + 0.1303t + 1.6902
M (q)¨ q + C(q, q) ˙ q˙ + G(q) = Q
(8)
where q includes the joint variables; positions, velocities and accelerations. Q is the input generalized forces or torques, M (q) is the inertia matrix and C(q, q) is the coriolis/centripetal forces and G(q) is the gravity force or torque. The configuration of SCARA and its parameters are shown with two joint variables as the generalized coor-
(7) 27
Figure 4. Convergence rates for both axes. dinates, q1 = θ1 and q2 = θ2 . So the system model can be written as in matrix form: ⎤⎡
⎡ ⎣
M11 M12 M21 M22
⎦⎣
θ¨1 θ¨2
⎤
⎤⎡
⎡
⎦+⎣
C11 C12 C21 C22
⎦⎣
θ˙1 θ˙2
⎤
⎦=⎣
τ1
⎦
(9)
τ2
θ1m N1
⎤⎡
⎡ ⎣
La1 0 0
La2 ⎡
+⎣
Ke1 0
M21 =
τ1 and τ2 are the applied actuator torques and θ1 =
θ2m respectively. The equations representing the N2 motors electrical structure can be expressed in a matrix form as given below.
(J1 + J2 ) (m1 r12 + m2 r22 + 4m2 r12 ) M11 = Jm1 + + N12 4N12 m2 r1 r2 ] (cos(θ + θ ) cos θ + sin(θ + θ ) sin θ ) + 1 2 1 1 2 1 N12 (9a)
m2 r22 m2 r1 r2 J2 + + (cos(θ1 + θ2 ) N1 N2 4N1 N2 2N1 N2 × cos θ1 + sin(θ1 + θ2 ) sin θ1 ) (9b)
(9h)
and θ2 =
where;
M12 =
C22 = 0
⎤
⎡
m2 r1 r2 ˙ θ2 [cos(θ1 + θ2 ) sin θ1 − sin(θ1 + θ2 ) cos θ1 ] N1 (9e)
C12 =
m2 r1 r2 ˙ θ2 [cos(θ1 + θ2 ) sin θ1 − sin(θ1 + θ2 ) cos θ1 ] 2N1 (9f)
C21 =
m2 r1 r2 ˙ θ1 [sin(θ1 + θ2 ) cos θ1 − cos(θ1 + θ2 ) sin θ1 ] 2N2 (9g)
⎤
⎤⎡
⎡ Ra1 0
I1
⎦⎣ ⎦ ⎦+⎣ I2 0 Ra2 I˙2 ⎤ ⎡ ⎤ ⎤⎡ V θ˙ 0 ⎦⎣ 1m ⎦ = ⎣ 1 ⎦ V2 θ˙2m Ke2
⎦⎣
(10)
5. Numerical Simulation In this section, the performance of PSO-PID controller is examined. Different examples are studied. Initially a linearized model is derived representing the motor, gearbox and a constant load for each axis. A stability analysis is performed on this linearized model. Systems transfer function is obtained with inclusion of PID controller. PSO programs are implemented in MATLAB using the system’s transfer function. The mathematical model of SCARA with actuators is simulated in Pascal. The nonlinear robotactuator model is then used in simulation. Having given the system measures in time domain, the optimization procedure is completed; PID parameters found in PSO application are implemented in the model referred. Time domain measures are then found in terms of the system’s parameters. They are used for calculations of (5) and (6) in Matlab environment. The performance of the algorithm is then tested by using the systems transfer function. The
(9d)
C11 =
⎤
Equation (10) represents electrical structure where V1 and V2 are the applied armature voltages. They are calculated by using PID controller parameters found applying PSO algorithm. Other motor data are taken from the manufacturer’s data sheet [20].
J2 m2 r22 m2 r1 r2 + + (cos(θ1 + θ2 ) N1 N2 N1 N2 2N1 N2 × cos θ1 + sin(θ1 + θ2 ) sin θ1 ) (9c) J2 m2 r22 M22 = Jm2 + 2 + N2 4N1 N2
I˙1
28
Table 3 Performance of the Algorithm Maximum Error (rad)
PD
PID
PSO-PID
Axis 1 (main arm)
−0.0775 −0.0728 −0.0297
Axis 2 (fore arm)
−0.0397 −0.0250 −0.0128
robot system is initially tuned ZN method realizing the values of Kp , Kv and Ki . Then these parameters are tuned by using PSO algorithm with the required time domain performance. A set of good parameters for PID controller obviously is resulted in a good step response also satisfying time domain requirements. 5.1 Parameter Selection and Convergence Analysis In this approach, PSO-PID, some analysis can also be performed on convergence analysis and parameter selection. Some suggestions are taken from previous studies on parameter selection [12–14, 21]. From the early experiments on PSO, the acceleration constants, population sizes and
number of iterations are studied in detail. For example, Eberhart et al. [13] have specified c1 and c2 as 2. Population sizes are generally taken between 20 and 50 and the inertia weight is taken as 0.4 < w < 0.9 during a run. Tuning range for 6 parameters for both controllers are taken between 0 and 200. The random numbers are usually chosen as [0, 1]. Previous studies on SCARA have given an insight for values range in PSO [19]. So the random numbers are multiplied by 200 to get PID values here. Trelea [21] has presented a detailed study on some important issues for PSO parameter tuning. The population sizes are generally reported as 15, 30 and 60. Iterations to convergence rate are important to note during tests referring to “slow ” and “quick ” convergence. It has been specified that “quick convergence ” can be obtained nearly with 900 iterations. So 1,000 iterations are allowed here. Different trials have been performed on the algorithm. Convergence rate for both axes are given in Fig. 4(a) and (b). Stable convergence characteristics are seen for both axes in PSO method here. Consequently, the number of particles in the swarm influences the convergence of the algorithm. During the optimization, the number of particles in the swarm, the
Figure 5. Numerical results for SCARA robot. (a) Response of the main arm (shoulder), (b) Positional error for the main arm, (c) Response of the fore arm (elbow), and (d) Positional error for the fore arm. 29
objective function and the parameters c1 and c2 are important. The following parameters in PSO are taken as population size is 45, m = 1,000 iterations, c1 = c2 = 2, wmax = 0.9 and wmin = 0.4 in this paper. As expected, PSO_PID algorithm has introduced less settling time than PID algorithm alone. Better dynamic response is seen compared to traditional PD and PID controller. Table 3 shows performance of conventional algorithms in terms of maximum positional errors obtained in the trajectory with respect to PSO-PID algorithm given in Fig. 3(a) and (b) while picking and placing the work piece specified.
Department, Gaziantep, 2003. [2] H.S. Ali, L.B. Badas, Y.B. Aubry, & M. Darouach, Hinfinity control of a Scara robot using polytopic LPV approach, IEEE Transactions on Industrial Electronics, 41(2), 1994, 173–181. [3] M. Belhocine, M. Hamerlain, & K. Bouyoucef, Robot control using a sliding mode, Proc. 12th IEEE International Symposium on 1ntelligent Control, Turkey, 1997. [4] F. Lin & R.D. Brandt, An optimum control approach to robust control of robot manipulators, IEEE Transactions on Robotics and Automation, 14(1), 1998, 69–77. [5] S. He, E. Prempain, & Q.H. Wu, An improved particle swarm optimizer for mechanical design optimization problems, Engineering Optimization, 36(5), 2004, 585–605.
5.2 Example Trajectory
[6] Z.L. Gaing, A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Transactions on Energy conversion, 19(2), 2004, 384–391.
The inputs required for the simulation study are the initial values of the system; initial current, angular displacement and velocity, the parameters of SCARA and the motors data (Tables 1 and 2), the step length for simulation and the total response time. In addition to above, the optimum PID parameters are required as the controller settings. Three different trajectories are designed for Serpent 1. Here one of them is only presented. Simulations and optimization studies are also performed for the other picking and placing jobs. Simulation results for both axes of SCARA are given in Fig. 5. Figure 5(a) and (b) shows numerical responses for axis 1 representing the main arm (shoulder) and available positional error in radians. Figure 5(c) and (d) represents numerical responses for axis 2 for the main arm (shoulder) and the positional error in radians. Using PSO-PID controller, the parameters are obtained as Kp1 = 198, Kd1 = 102 and Ki1 = 51.28 for the axis 1, the main arm (shoulder) and Kp2 = 126.62, Kd2 = 25.84 and Ki2 = 0.83 for the axis 2, the fore arm (elbow) respectively. These optimum parameters found are then set into the mathematical model.
[7] Y. Liu, J. Zhang, & S. Wang, Optimization design based on PSO algorithm for PID controller, Proc. 5th World Congress on Intelligent Control and Automation, China, June 15–19, 2004. [8] X. Wang, Y. Wang, H. Zhou, & X Huai, PSO-PID: A novel controller for AQM routers, Proc. IEEE/IFIP on Wireless and Optical Communication Networks, 2006. [9] C.C. Kao, C.W. Chuang, & R.F. Fung, The self tuning PID control in a slider-crank mechanism system by applying particle swarm optimization approach, Mechatronics, 16, 2006, 513–522. [10] M. Nasri, H. Nezamadi-Pour, & M. Maghfoori, A PSO-based optimum design of PID controller for a linear brushless dc motor, PWASET, 20, 2007, 211–215. [11] M.T. Da¸s & L.C. D¨ ulger, Particle swarm optimization (PSO) algorithm: Control of a four bar mechanism, Proc. EU/ME 2007, ‘Metaheuristics in Service Industry, Germany, 96–101. [12] J. Kennedy & R.Eberhart, Particle swarm optimization, Proc. IEEE Int. Conf. Neural Networks, 4, Australia, 1995, 1942– 1948. [13] R.C. Eberhart & Y. Shi, Particle swarm optimization: Developments, applications and resources, IEEE 2001, 81–86. [14] R.C. Eberhart & Y. Shi, Tracking and optimizing dynamic systems with particle swarms, IEEE -2001, 94–100.
6. Conclusions
[15] P. Cuminos & N. Munro, PID Controllers: Recent tuning methods and design to specification, IEE Proceedings Control, an Theory, and Application, 149(1), 2002, 46–56.
PID controller parameters are determined with the application of PSO method and a practical application of PSO-PID controller is presented in this study. The algorithm was applied to 2 dof point-to-point control case in a SCARA robot, Serpent1. There are nonlinear dynamic interactions on the manipulator joints. It is obvious from the simulations that PSO-PID improved positional responses for both the main and the fore arm. Simulations on both axes have shown less error and better convergence characteristics. PSO algorithm includes many tuning parameters that have a great effect on its performance. Some time is devoted for tuning PSO parameters. In this paper, reference studies are taken into consideration while adjusting PSO algorithm for SCARA system. The algorithm can definitely be applied to any system of interest because of its simple integration to the systems. Acceptable computational results and convergence efficiency are certainly seen in the numerical examples.
[16] G.K.I. Mann, B.G. Hu, & R.G. Gosine, ‘Time-domain based design and analysis of new PID tuning rules’, IEE Proceedings an Control, Theory and Application, 148(3), 2001, 251–261. [17] K.H. Ang, G.C.Y. Chong, & Y. Li, PID Control System Analysis, Design and Technology, IEEE Transactions on Control Systems Technology, 13(4), 2005, 559–576. [18] Y. Li, K.H. Ang, & G.C.Y. Chong, PID control system analysis and design, IEEE Control Systems Magazine, February 2006, 32–41. [19] M.T. Da¸s & L.C. D¨ ulger, Mathematical modeling, simulation and experimental verification of a SCARA robot, Simulation Modeling Practice and Theory, 13, 2005, 257–271. [20] Wall I Serpent Manual, 1993. [21] I.C. Trelea, The particle swarm optimization algorithm: Convergence analysis and parameter selection, Information Processing Letters, 85, 2003, 317–325. [22] S. Skoczowski, S. Domek, & K.Pietrusewicz, Robust PID model following control, Control and Intelligent Systems, 34(3), 2006, 1544–1550. [23] Z. Al Hamous, S.F. Faisal, & S. Al. Sharif, Application of particle swarm optimization algorithm for optimal reactive power planning, Control and Intelligent Systems, 35(1), 2007, 1642–1653.
References [1] M.T. Da¸s, Motion control of a SCARA robot with a PLC unit, M.Sc.Thesis, Gaziantep University, Mechanical Engineering
30
L.C. D¨ ulger received her B.S. degree in Mechanical Engineering from Middle East Technical University (METU) in 1986, her M.S. degree in METU 1988 and Ph.D. in Mechanical Engineering from Liverpool Polytechnic England 1992, respectively. She is currently professor of Mechanical Engineering at Gaziantep University, Mechanical Engineering Department.
Biographies M.T. Da¸s received his B.S. degree in Mechanical Engineering from Gaziantep University, Turkey in 2000, his M.S. degree in 2003 and Ph.D. degree in 2008. He was with Gaziantep University’s Mechanical Engineering Department from 2000–2008 as a research assistant. Since December 2008, he has been with the Roketsan Company Elmadaˇg, Ankara.
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