Control and Intelligent Systems, Vol. 38, No. 1, 2010

CONTROL OF A SCARA ROBOT:

PSO-PID APPROACH

M.T. Das∗ and L.C. Dulger∗∗

Abstract

This paper presents an approach where the Particle swarm optimiza-

tion (PSO) is employed to get optimum PID controller parameters

to control of a SCARA robot. PSO is an evolutionary technique

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

and optimum PID parameters. Different trajectories are studied.

Performances of the traditional controllers; PD, PID and the ap-

plication of PSO-PID controller are then compared. Simulation

results are presented. The numerical results definitely enforced the

potential use of PSO-PID controller herein.

Key Words

Particle swarm optimization (PSO), PID (proportional derivative

integral control, SCARA robot, optimum control, PSO-PID con-

troller

Notation

e(t) the error signal

Ia1, Ia2 the armature current for motor 1 and

2 (A)

J1, J2 inertias of the main and the fore arm

(kgm2)

Jm, Jm1, Jm2 the motor and equivalent inertias (kgm2)

Jg1, Jg2 inertias of the gearbox 1 and 2 (kgm2)

La1, La2 the armature inductances for motor 1 and

2 (H)

Ke1,Ke2 the back emf constants for motor 1 and

2 (V/rad/s)

Kt1,Kt2 the torque constants for motor 1 and

2 (Nm/A)

m1,m2 masses of the main(shoulder) and the

fore arm (elbow)(kg)

N1,N2 the gearbox ratios for motor 1 and 2

∗ Roketsan Company, Elmadag-Ankara, Turkey; e-mail:tdas@roketsan.com.tr

∗∗ Gaziantep University, Faculty of Engineering, Department ofMechanical Engineering, Gaziantep, Turkey; e-mail: dulger@gantep.edu.tr

Recommended by Prof. K.K. Tan(paper no. 201-2151)

r1,r2 lengths of the main arm and the fore arm

(m)

Ra1,Ra2 the armature resistances for motor 1 and

2 (Ω)

τ1, τ2 the motor torques (Nm)

xB , yB the coordinates of point B (kg)

θ1,.

θ1,..

θ1 the angular disp., vel. and acceleration of

the main arm (rad, rad/s, rad/s2)

θ2,.

θ2,..

θ2 the angular disp., vel. and acceleration of

the fore arm (rad, rad/s, rad/s2)

θ1ref , θ2ref the reference positions for the main and

fore arm

1. Introduction

SCARA (selective compliant articulated robot arm) con-sists of three planar revolute joints as shoulder, elbow andwrist and a prismatic joint which operates in vertical planefor proper positioning of the work piece. It offers espe-cially a good choice for assembly tasks. Many differentstudies are seen on modelling and control of SCARA typerobot. Some of them are referred in the study and a briefoverview is given. Das [1] has studied on motion control ofa SCARA robot with a PLC unit, traditional controllersare included. Ali et al. [2] have studied on the Hinfinitycontrol of robotic system (SCARA) with linear parametervarying (LPV) representation. Belhocine et al. [3] haveapplied sliding mode in robot control. They have usedthis technique to control of a SCARA robot and its iden-tification is done by using MATLAB. Lin et al. [4] havepresented a new optimal control approach to robust controlof robot manipulators. This approach has been explainedusing a SCARA robot.

In recent studies on mechanical design problems andon control issues, many different evolutionary optimiza-tion algorithms are used. Among evolutionary algorithms(EAs), PSO is quite commonly revealed for solution of dif-ferent engineering optimization problems. Some of themare noted here to show its application in solution of engi-neering problems. The algorithm can be used with neuralnetworks (NN) as PSO-NN. It can also be used with backpropagation algorithm (BP) to train a NN. Prempain et al.[5] have presented an improved particle swarm optimizer(PSO) for solving mechanical design problems; spring de-sign, vessel design for example. Gaing [6] has then deter-

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mined the optimal PID parameters of an automatic volt-age regulator (AVR) system using PSO algorithm. Thismethod 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 forPID Controller. Wang et al. [8] have presented an activequeue management (AQM) model with the PSO-PID con-troller. Simulation results of PSO-PID controller for AQMrouters are given. Kao et al. [9] have achieved a studyon the self-tuning PID control in a slider-crank mecha-nism system by using PSO algorithm. Nasri et al. [10]have presented a study on PSO-based optimum design ofPID controller for a linear brushless dc motor (BLDC). Acomparative study is given GA and Linear Quadratic Reg-ulator (LQR) method with PSO method. Das and Dulger[11] have then performed a study on control of a four-barmechanism using the proposed PSO-PD controller.

The contribution of this paper is that a control systembased on PSO-PID is designed and implemented on aSCARA robot (Serpent 1) which has taken 2 degreesof freedom (dof) with RR configuration. This paper isprepared in five sections. A mathematical backgroundon PSO algorithm and the proposed controller details ispresented in Section 2. SCARA robot in laboratory isdescribed together with its mechanical and electrical datain Section 3. The mathematical structure of robot withits actuator dynamics is included in Section 4. Finally,numerical examples are given to show performance of PSO-PID controller for position control of a SCARA robot inSection 4. Conclusions are then presented in Section 5.

2. Background

2.1 Particle SwarmOptimization Algorithm (PSO)

PSO is an evolutionary algorithm based on population.PSO algorithm first introduced by Kennedy and Eberhartis based on simulation of social system behaviour in 1995[12]. Since then a lot of research study has been carried onthe algorithm. In a recent study, Eberhart and Shi havepresented a bibliography on the algorithm [13]. They havepresented the original algorithm with developments andvarious applications. Objectives, priorities and resourcesare changed for a real world. Potential applications are seenin system design, pattern recognition, and identification,robotic applications, image segmentation, time frequencyanalysis, robot path planning, signal processing and controlpurposes. Eberhart and Shi [14] have studied particleswarms with a review on tracking and optimizing dynamicsystems. Three kinds of dynamic systems are defined andstudied giving many answers for the algorithm in futureinvestigations.

In this algorithm, the “individual” is used to replacethe “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 mindividuals in a population resulting in m× 3 population.Increasing the number of swarms increase in the numberof error function evaluations. Each particle updates itsposition and velocity by following the two “best” values.

The first one is the best solution pbest,i that a particle hasachieved so far. The other one is the global best value gbestthat is obtained by any particle in the population. Afterfinding the two best values pbest,i and gbest, each particleupdates its velocity and position according to the following(1) and (2) below.

vk+1i = wvki + c1r1(pbest,i − xk

i ) + c2r2(gbest − xki ) (1)

xk+1i = xk

i + vk+1i (2)

where,

vki the velocity of the particle

xki the current position of the particle (solution)

w the inertia weight factor

pbest 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 byusing (3) where iter is the iteration number in a particularinstance and itermax is the maximum number of iterations.

w = wmax − (wmax − wmin)

itermax· iter where 0.4 < w < 0.9

(3)Equations (1) to (3) given in [6, 9, 11] are used while

performing optimization of PID controller parameters.

2.2 PSO-PID Controller Implementation

An overview on PID control is performed by Cuminosand Munro [15] where different tuning methods are pre-sented with their offered advantages and disadvantages.Mann et al. [16] have presented a time response-baseddesign methodology for PID controllers. Many approacheshave been applied previously to determine PID parame-ters. Ziegler–Nichols (ZN) method, Kapa-Tau tuning, poleplacement, use of gain and phase margins and partitioning,OLDP method and GA can be given as some of the meth-ods. The ZN method is applied to set bounds of the PIDcontroller parameters. Ang et al. [17] have presented anoverview on tuning methods, software packages and com-mercial hardware modules. Tuning methods for PID con-trollers are summarized as analytical, heuristic, frequencyresponse and optimization methods. Then Li et al. [18]have studied on PID design objectives and methods. Ac-cording to design methods, here evolutionary computationis used with optimization tools, and PSO algorithm is ap-plied. The PSO-PID controller is then developed to searchoptimal PID parameters. In time domain, the output ofPID is given by:

u(t) = Kpnen(t) +Kin

t∫0

en(t)dt+Kdnden(t)

dt(4)

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Figure 1. Structure of PSO-PID controller.

Figure 2. Serpent 1(top view).

where en(t) represents the positional error for each axis,for n=1, 2. Kpn is the proportional gain, Kin is theintegral gain and Kdn is the derivative gain. The errorsignal can be defined for each axis as; e1(t)= θ1ref (t)− θ1and e2(t)= θ2ref (t)− θ2.

Performance of the controller can be evaluated eitherin frequency domain or time domain. In this study, perfor-mance of PID controller is analyzed in time domain withthe overshoot (Mp), rise time (tr), settling time (ts) andsteady-state error (ess). The performance criterion W (K)is defined as:

W (K) = (1− e−β)(MP + ess) + e−β(ts − tr) (5)

where the parameter K includes the controller parametersKp, Kd and Ki. In this study, β is chosen 0.7 representingthe weighting factor. The fitness function f of the PSOalgorithm is defined as the reciprocal of W (K). Thesmaller W (K) the value of individual K, the higher its

evaluation value [6, 9, 11].

f =1

W (K)(6)

Equations (5) and (6) are applied together with theassumptions in time domain; no overshoot, considerablysmall settling time and introducing minimum steady-stateerror. Details of convergence analysis and parameter se-lection of PSO are included in Section 5. The proposedPSO-PID structure can be given in Fig. 1.

3. SCARA Robot

The SCARA robot (Serpent 1) is available in DynamicSystems Laboratory, Gaziantep University, andMechanicalEngineering Department. The photograph is shown inFig. 2 for the given configuration. Technical specificationsof the robot are included in Table 1 [20]. Mechanical andelectrical data for Serpent 1 are also presented in Table 2

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Table 1Specifications of Serpent 1

Main arm (shoulder) length (r1) 250mm

Fore arm length (elbow) (r2) 150mm

Shoulder movement (θ1 axis) 200◦

Elbow movement (θ2 axis) 250◦

Wrist rotation (roll axis) 450◦

Up and down (z axis) 75mm

Maximum tip velocity 550mm/s

Capacity 2.0 kg

Table 2Mechanical and Electrical Data for Serpent 1

J1 =0.0980 kgm2, J2 =0.0115 kgm2

m1 =1.90 kg, m2 =0.93 kg

Jm=3.3× 10−6 kgm2

Ke1 =Ke2 =0.047 V/rad/s

Kt1 =Kt2 =0.047Nm/A

Ra1 =Ra2 =3.5Ω

La1 =La2 =1.3mH

N1 =90, N2 =220

Jg1 =0.0002 kgm2, Jg2 =0.0005 kgm2

[20]. A real trajectory is chosen by looking at PNP (Pickand Place) job specified.

3.1 Working Envelope and Trajectory

Working envelope of Serpent 1 and dimensional limitationsare given in Fig. 3(a). Different sizes of work pieces canbe assembled. In system, SCARA has been applied toplace 3 different sizes of work pieces (shown by differentcolours as red-small, blue-medium, green-big). Here, thetrajectory is taken while performing picking operation for58 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 pickand place operation as X–Y locus which is performed in3 s. Trajectory data points are taken during an operationof the robot in Dynamic System Laboratory, MechanicalEngineering Department, Gaziantep University. Theseexperimental trajectory points are then characterized aspolynomials using curve fitting for use as the model inputdata during simulation.

θ1 = −0.1798t2 − 0.1787t+ 3.0227

θ2 = −0.2458t2 + 0.1303t+ 1.6902(7)

Figure 3. (a) Working envelope of Serpent 1 and (b) Atrajectory example for PNP job.

4. Manipulator and Actuator Model

SCARA (Serpent 1) is an industrial robot with 4 dof,where three axes are rotational (electrical drives) and one istranslational (pneumatic). The simulation example of theSCARA robot is considered here with two axes, so 2 dof.The robot dynamics (referred to Fig. 2) are obtained fromLagrange’s equations. Das and Dulger have derived themathematical model in a previous study with a traditionalPD controller [19]. So the nonlinear system equations aredirectly taken with inclusion of actuator dynamics andnecessary gearing. Two second order nonlinear coupledequations are resulted in. The general arm equations havethe following form;

M(q)q + C(q, q)q +G(q) = Q (8)

where q includes the joint variables; positions, velocitiesand accelerations. Q is the input generalized forces ortorques, M(q) is the inertia matrix and C(q, q) is thecoriolis/centripetal forces and G(q) is the gravity force ortorque. The configuration of SCARA and its parametersare shown with two joint variables as the generalized coor-

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Figure 4. Convergence rates for both axes.

dinates, q1 = θ1 and q2 = θ2. So the system model can bewritten as in matrix form:

⎡⎣M11

M21

M12

M22

⎤⎦⎡⎣ θ1

θ2

⎤⎦+

⎡⎣C11

C21

C12

C22

⎤⎦⎡⎣ θ1

θ2

⎤⎦ =

⎡⎣ τ1

τ2

⎤⎦ (9)

where;

M11 =

[Jm1 +

(J1 + J2)

N21

+(m1r

21 +m2r

22 + 4m2r

21)

4N21

]

+

[m2r1r2N2

1

]

](cos(θ1 + θ2) cos θ1 + sin(θ1 + θ2) sin θ1)

]

(9a)

M12 =

[J2

N1N2+

m2r22

4N1N2+

m2r1r22N1N2

(cos(θ1 + θ2)

× cos θ1 + sin(θ1 + θ2) sin θ1)

](9b)

M21 =

[J2

N1N2+

m2r22

N1N2+

m2r1r22N1N2

(cos(θ1 + θ2)

× cos θ1 + sin(θ1 + θ2) sin θ1)

](9c)

M22 =

[Jm2 +

J2N2

2

+m2r

22

4N1N2

](9d)

C11 =m2r1r2N1

θ2[cos(θ1 + θ2) sin θ1 − sin(θ1 + θ2) cos θ1]

(9e)

C12 =m2r1r22N1

θ2[cos(θ1 + θ2) sin θ1 − sin(θ1 + θ2) cos θ1]

(9f)

C21 =m2r1r22N2

θ1[sin(θ1 + θ2) cos θ1 − cos(θ1 + θ2) sin θ1]

(9g)

C22 = 0 (9h)

τ1 and τ2 are the applied actuator torques and θ1 =θ1mN1

and θ2 =θ2mN2

respectively. The equations representing the

motors electrical structure can be expressed in a matrixform as given below.

⎡⎣La1

0

0

La2

⎤⎦⎡⎣ I1

I2

⎤⎦+

⎡⎣Ra1

0

0

Ra2

⎤⎦⎡⎣ I1

I2

⎤⎦

+

⎡⎣Ke1

0

0

Ke2

⎤⎦⎡⎣ θ1m

θ2m

⎤⎦ =

⎡⎣ V1

V2

⎤⎦ (10)

Equation (10) represents electrical structure where V1

and V2 are the applied armature voltages. They are calcu-lated by using PID controller parameters found applyingPSO algorithm. Other motor data are taken from themanufacturer’s data sheet [20].

5. Numerical Simulation

In this section, the performance of PSO-PID controlleris examined. Different examples are studied. Initially alinearized model is derived representing the motor, gearboxand a constant load for each axis. A stability analysisis performed on this linearized model. Systems transferfunction is obtained with inclusion of PID controller. PSOprograms are implemented in MATLAB using the system’stransfer function. The mathematical model of SCARAwith actuators is simulated in Pascal. The nonlinear robot-actuator model is then used in simulation. Having giventhe system measures in time domain, the optimizationprocedure is completed; PID parameters found in PSOapplication are implemented in the model referred. Timedomain measures are then found in terms of the system’sparameters. They are used for calculations of (5) and (6)in Matlab environment. The performance of the algorithmis then tested by using the systems transfer function. The

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Table 3Performance 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 thevalues of Kp, Kv and Ki. Then these parameters are tunedby using PSO algorithm with the required time domainperformance. A set of good parameters for PID controllerobviously is resulted in a good step response also satisfyingtime domain requirements.

5.1 Parameter Selection and Convergence Analysis

In this approach, PSO-PID, some analysis can also be per-formed on convergence analysis and parameter selection.Some suggestions are taken from previous studies on pa-rameter selection [12–14, 21]. From the early experimentson PSO, the acceleration constants, population sizes and

Figure 5. Numerical results for SCARA robot. (a) Response of the main arm (shoulder), (b) Positional error for the mainarm, (c) Response of the fore arm (elbow), and (d) Positional error for the fore arm.

number of iterations are studied in detail. For example,Eberhart et al. [13] have specified c1 and c2 as 2. Popula-tion sizes are generally taken between 20 and 50 and the in-ertia weight is taken as 0.4<w< 0.9 during a run. Tuningrange for 6 parameters for both controllers are taken be-tween 0 and 200. The random numbers are usually chosenas [0, 1]. Previous studies on SCARA have given an insightfor values range in PSO [19]. So the random numbers aremultiplied by 200 to get PID values here. Trelea [21] haspresented a detailed study on some important issues forPSO parameter tuning. The population sizes are generallyreported as 15, 30 and 60. Iterations to convergence rateare important to note during tests referring to “slow” and“quick” convergence. It has been specified that “quickconvergence” can be obtained nearly with 900 iterations.So 1,000 iterations are allowed here. Different trials havebeen performed on the algorithm. Convergence rate forboth axes are given in Fig. 4(a) and (b). Stable conver-gence characteristics are seen for both axes in PSO methodhere.

Consequently, the number of particles in the swarminfluences the convergence of the algorithm. During theoptimization, the number of particles in the swarm, the

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objective function and the parameters c1 and c2 are im-portant. The following parameters in PSO are taken aspopulation 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 thanPID algorithm alone. Better dynamic response is seencompared to traditional PD and PID controller. Table 3shows performance of conventional algorithms in terms ofmaximum positional errors obtained in the trajectory withrespect to PSO-PID algorithm given in Fig. 3(a) and (b)while picking and placing the work piece specified.

5.2 Example Trajectory

The inputs required for the simulation study are the initialvalues of the system; initial current, angular displacementand velocity, the parameters of SCARA and the motorsdata (Tables 1 and 2), the step length for simulationand the total response time. In addition to above, theoptimum PID parameters are required as the controllersettings. Three different trajectories are designed forSerpent 1. Here one of them is only presented. Simulationsand optimization studies are also performed for the otherpicking and placing jobs. Simulation results for both axesof SCARA are given in Fig. 5. Figure 5(a) and (b)shows numerical responses for axis 1 representing the mainarm (shoulder) and available positional error in radians.Figure 5(c) and (d) represents numerical responses for axis2 for the main arm (shoulder) and the positional errorin radians. Using PSO-PID controller, the parametersare obtained as Kp1 =198, Kd1 =102 and Ki1 =51.28 forthe 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 foundare then set into the mathematical model.

6. Conclusions

PID controller parameters are determined with the ap-plication of PSO method and a practical application ofPSO-PID controller is presented in this study. The algo-rithm was applied to 2 dof point-to-point control case in aSCARA robot, Serpent1. There are nonlinear dynamic in-teractions on the manipulator joints. It is obvious from thesimulations that PSO-PID improved positional responsesfor both the main and the fore arm. Simulations on bothaxes have shown less error and better convergence charac-teristics. PSO algorithm includes many tuning parametersthat have a great effect on its performance. Some time isdevoted for tuning PSO parameters. In this paper, refer-ence studies are taken into consideration while adjustingPSO algorithm for SCARA system. The algorithm candefinitely be applied to any system of interest because ofits simple integration to the systems. Acceptable computa-tional results and convergence efficiency are certainly seenin the numerical examples.

References

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Biographies

M.T.Das received hisB.S. degreein Mechanical Engineering fromGaziantep University, Turkey in2000, his M.S. degree in 2003 andPh.D. degree in 2008. He was withGaziantep University’s Mechani-cal Engineering Department from2000–2008 as a research assistant.Since December 2008, he has beenwith the Roketsan Company El-madag, Ankara.

L.C. Dulger received her B.S.degree in Mechanical Engineer-ing from Middle East TechnicalUniversity (METU) in 1986, herM.S. degree in METU 1988 andPh.D. in Mechanical Engineeringfrom Liverpool Polytechnic Eng-land 1992, respectively. She iscurrently professor of MechanicalEngineering at Gaziantep Univer-sity, Mechanical Engineering De-partment.

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