Robot control using a sliding mode

p zyxwvutsrqponmlkjihgfed Robot Control using a sliding mode M. BELHOCINE, M. HAMERLAIN and K. BOUYOUCEF Laboratoire de Robotique et d’hrtelligence Artificielle, CDTA. 128, Chemin Mohamed Gacem, El-Madania, BP. No 245.16075 ALGER. ALGERIE Tel : 213 2 67 73 25, Fax :213 2 66 26 89, e-mail : [email protected] Abstract : The dynamical model of manipulator robot is represented by equations system which are nonlinear and strongly coupled. Furthermore, the inertial parameters of the manipulator depend on the payload which is often unknown and variable. So, to avoid these problems we studied variable structure system which is well suited for robotics arms. To this end, an application of the sliding mode control based on variable structure system for a four degrees of freedom robot is described in this paper. This technique suppresses the uncertainties due to parametric variations, external disturbances and variable payloads. To prove these advantages, this technique is applied to the regulation (point to point) control of the SCARA robot. So the aim of this work is to show the practical realization and to demonstrate the robustness and the validity of this control law on the robot manipulator via experimental results obtained and discussed in the end. Keywords : Variable structure systems, sliding mode, robot control, robust control, non linear system. 1. Introduction linear and free system [lo]. The sliding mode control technique has already been used for robot control [4], either without decoupling of the robot equations or combined with a model of the robot used to estimate the torques necessary at the joints. The dynamic of the robots is described by coupled second nonlinear differential equations and the inertial parameters depends on the payload which is often unknown and changes during the task. Usually, in a classical control we must have an accurate model, so to avoid this constraint we decide to implement a robust controller based on variable structure systems (VSS). The proposed sliding mode controller is realized by linearizing the equations system by the MATLAB software; and the objective of the controller is to avoid using velocity signal. So, only measured position signal of the joints is used to control the robot arm. Like this, shaft encoder is used to sense the output position a 12 bit A/D converter provides the required signal. To illustrate the application of the sliding mode on robotics manipulator, we implement the controller on a SCARA robot which has four degrees of freedom. But, this work is concerned with the three degrees of freedom because the fourth degree is the translation in only two positions of the end effector. So, in this paper, the practical realization of a sliding mode controller is described. After introducing the VSS theory, we show the experimental robot with its model identification and the linear system obtained; so after we develop the calculation of the control method. In the last, we discuss the experimental results obtained and confirm the validity of the approach. The theory of VSS has been developed firstly in Soviet Union by Emelyanov [2], introduced after by Utkin [9] and more recently studied by several authors [5], [8], [4]. In the last years, many applications of VSS were and proposed : in motion control, DC-servo-motor robotics manipulator [6], [7], [12]. From these applications, the conclusion is that the robust nature of VSS is proved by the sliding mode. When the sliding mode occurs, the system will be forced to slide along or near the vicinity of the switching surface. The system became then robust and insensitive to the interactions, disturbances and variations. In addition, this does not require an accurate model of the robot (plant) : it is only necessary to know the boundaries of the parameter variations and load disturbances. The dynamics of the system is submerged in the dynamics of the reduced 1 2. Control methodology 1 u eq = g-Yx,o Variable structure systems are referred to systems which their structure changes. This kind of systems has an attractive feature for control applications, which consists in a sliding mode [ 111: This mode occurs on switching surface, and the system remains insensitive to parameter variations and disturbance. So, this mode allows also elimination of interactions among the joints of the manipulator. The aim of this paper is the implementation of, the sliding mode control on the linkage manipulator. A general type of the motion equation is represented in the space state by : x = f-(x, Where functions t) + g(x, f(X, t) So, the discontinuous written : ui = t) are nonlinear and not in (2) is uieq + Au; si < 0 (4) control (low frequency) and term (high frequency). U* eq = u,, + Aueq (5) As we explain the formula of discontinuous control input in (4), the term of high frequency can be expressed in different manners [l] [4]; and for our experiments we choose the equation proposed by Harashima [4] : The control input is : zq(x,t) if Si(X,f)>O 1 uz:(x, t) if si (x, t) < 0 given As we are in the practical case, the equivalent control is known by estimated value due to error modelisation and variation of the parameters. So, this yields to : zyxwvutsrqponm known exactly. ui (x, t) = input u,, + Au+ si > 0 Aui the discontinuous (1) x is the output, and the and g(x, control Where U, is the equivalent t). u u is the control input, (3) 4u - f(w) [ zyxwvutsrqponmlkjihgfedcbaZYXW (2) Where ui is the zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ith component of u Si (x, t) = 0 is the zth component hypersurfaces S(X, t) = of the m switching 3. Design procedure 0 S E R” . As in our experiments, we use the manipulator model issued horn testing the system to identify its parameters, we present in this section the identification of the manipulator model and the different steps of the controller calculation. This system with discontinuous control is called variable structure system, since the control structure switches alternatively according to the state of the system. The sliding mode occurs on a switching surface S(X) = 0, which forces the original system to behave as linear time 3.1 Manipulator model invariant system, which can be considered to be stable. In our study the surfaces are taken to be linear and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Usually the manipulator dynamics are obtained from the n Lagrangian equation, they have the following form : zyxwvutsrqponm Si(x)=xn +~li.xi zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA written as : i=l w4+ The condition for the sliding mode to exist on the ti surface is given by the equation : Limii s,+o+ Which is equivalent of to I is an inertial matrix, C represents coriolis and centrifugal g is the torque due to the gravity, Si Si < 0 in the neighborhood Si (x) = 0, when all the trajectories = I- (6) Where : and /iy~SsO , ~0 C(a d+ m move towards forces, I? is the torque input vector, the switching surface. hr the ideal sliding control is the mode equivalent on Si , the corresponding control issued Tom (q,q, 4) represent and acceleration the equation (1) and given by the equation for s = 0 : 2 the generalized vector. position, velocity The (*) is related to the estimated term because of the modelisation error and variation of the parameters. We must control the system by holding it in the sliding surface. The surface has the equation : These equations are coupled and nonlinear; since the manipulator is controlled via a computer as PC, so we can approximate the dynamic model by an identified model which is linear. This identification is done using test input signals which excite the system and the least square method is used with the measured input and output signals to estimate system parameters. The equation (6) can be written in the following form : Si = A,.e, where : R, is a positive ei , ii q+ Z-‘(q).H(q,i)i+ I-‘(qhG(q1.q Where H(q,q).q = C(q,q) and = I-‘(q1.r G(q1.q and (7) + ii and ii are the position, acceleration Si = 0 and Si = 0 , Corn where we extract the equivalent control input equation This work proposes an experimental study of the three =~[~id+ai,.qi+ai2.qi ‘ieq degrees of freedom of the SCARA robot and its I identification is achieved using the MATLAB software. The estimated equivalent control is : By the end, we obtain a linear approximated and decoupled model dynamics which is represented by the u:e, =ijt;[4,d+U~l.qi+U~2.qi following system : zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 4+ A,.q+ A,.q = B.U Where The calculaiion (8) A, = I-‘(q).H(q,q) -lj.ei] (10) - &- ii] (11) : zyxwvuts yields to : Meq = di,. ei+ diz. ei + di3.qid+ di4.qid+ dis.qid = Diag[ail] A, = I-‘(q).G(q) velocity errors The sliding mode is for = g(q) parameter = Diag[a,;!] with: di, = di3 The control - /Z,.d,, B = Diag[b,] gains are obtained from the sliding q=[q,,q2,q31Ttheangularrotationvector, condition The results identified obtained for the parameters : ai, , ai, , bi ; in the prediction from ARX method on the MATLAB in the table 1. ail 201.0 560.0 413.5 i 1 2 3 ai -2.4 -5.4 -117.5 to be Si S i < 0. So, after calculation, Si Si = bi(SiAUieq error method we obtain : + S~AU~) Which can be written : software are shown Si Si = bi(SiA~ieq bi 0.64 0.50 20.00 + (ailgil II II + pi pi + yi)Sgn(Si)Si) zyxwvutsrqpo Si Si = bi(SiAUieq + (CtileiI + pi ei + ~i)l~iI) Then : Si Si I IbiIISil(l Au,ql+ (aileil+ Pi ei + Yi) Table 1 : Identified parameters zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I I And we substitute A u e 4 by its expression, we obtain : 3.2 Control calculation As showed in the previous section, the equation identified model yields to : ~i+ai,4i+ai2qi For each joint i, the input control ui = uliq + Aui ; where and Aui 11i13 =biui =[ai/ei/ +piiiil u,:, + of the I~i~14id+ldi41~id+ldis14id +Yi> (9) is noted To satisfy the sliding condition to take : as : = uieq + Auieq SS < 0 , it is sticient Id,,l~i~+ Id,4IGid+ ldislqtd + Yi <O ai + Idi < 0 yi].sgn(Si) Pi +Idill<O 3 From where we deduce : zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA generalized variable structure system to avoid chattering, which appears in the sliding mode, by the switching on the highest derivative of the input [3]. “i=-SuP(ldizI); Pi=-SuP(ldill); I j’i = -SUP( zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 6. References dij.qid + did*iid + dis*qid > zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA [l] : H.Asada et J.J.E Slotine : Robot Analysis and Control , A Wiley-Interscience Publication John Wiley and Sons, 1986. [2] : T. Emelyanov : (( Sur une classe de systemes de 4. Experimental results regulation automatique a structure variable )), Journal de l’academie des sciences d’URSS, Energetique et This section describes experimental results of automatique, N”3, 1962. applying sliding mode on the robot in point to point [3] : M. Hamerlain : (( The new robust control using the motion (regulation). The three degrees of freedom move theory of generalized variable structure )), IEEE respectively from the initial position International Symposium on Industrial Electronics, loqi&2.78,1.05,0.51)rd to the desired final position 14 July 1995, Athens, Greece. q&l .05,2.78,5.23)rd with the following parameters of [4] : F. Harashima, H. Hashimoto, K. Maruyama : the surfaces 70,140 and 110 for each joint. “Practical robust control of robot arm using variable structure system”, Proc. of IEEE, Int. Conf. On Robotics In the figures 1, 2 and 3 the controller is tested without and Automation, San Francisco, 532-538, 1986. payload in the figures 4, 5 and 6 show the case where [5] : H. Hashimoto : “A variable structure system with the robot is disturbed which consist to fight the arm of an invariant trajectory”, Power electronics, Tokyo, the robot with a force and to see if it doesn’t make the Vo1.2, 1983. robot disable to go the desired position. [6] : S. Nouri, M. Hamerlain, C. Mira, P. Lopez : “Variable structure model reference adaptive control The first figures (1, 2 and 3) give the error position for using only input and output measurements for two oneeach joint, the velocity and the corresponding control link manipulators” IEEE-SMC, Le Touquet 1993. input evolution. These results show the good behavior of [7] : H. Sira Ramirez, S. Ahmad, M. Zribi : “Dynamical the control algorithm. The steady states are reached after feedback control of robotics manipulators with joint 2.5 seconds for the first joint, 2 seconds for the second flexibility”, TR.EE 90-70, December 1990, School of and 1 second for the third joint. The position errors are Electrical Engineering, Purdue University, West around 0.02 rd. Lafayette, Indiana 47907. In the figures 4, 5 and 6, we see when the robot is [8] : J.J.E Slotine & J.A. Coetsee : “Adaptive sliding disturbed by an external force, that it will come back to controller synthesis for non linear systems”, Int. Jou. the desired position with an error equal to 0.02 rd. This Control, Vo143, N”6, 1986. demonstrate the robustness of sliding mode controller [9] : V.1 Utkin : (( Sliding mode and their application in against the disturbance. The steady states are reached variable structure systems )), Moscow, 1978. after 6 seconds for the first joint, 6 seconds for the [lo] : V.I. utkin : “Sliding modes in control and second and 4 seconds for the third joint. optimization”, Edition Springer Verlag, 1992. [l l] : K.S. Yeung & Y.P. Chen : G Sliding mode controller design of a single-link flexible manipulator 5. Conclusion under gravity )), International Journal on Control, Vo1.52, N”1, pp.lOl-117, 1990. A practical realization of robust controller using [ 121 : K.K. Young : “Variable structure control for sliding mode is proposed in this paper. It was applied to robotics and aerospace applications”, Elsevier, London, the three degrees of freedom of the SCARA robot which New York, Tokyo, 1993. has three degrees of freedom, and shows that nonlinear dynamic interactions of the manipulator joints are suppressed and the system is insensitive to the parameters variations. The experimental results shows also its performances against the payloads variations. In our future study, it is interested to test our algorithm in the case of trajectory tracking. Further, it will be more interested to develop an algorithm based on 4 Ax is 1 Ax is 1 zyxwvutsrqponmlkjihgfedcbaZ gy:$-jfF~ i-1:: -2 ’ 0 I 10 5 T im e -1 .5 (se c .) Ax is ’ 0 I 10 5 T im e 1 (se c .> -3 0 0 0 0 1 2 3 4 T im e 5 <se c .) 6 7 8 9 Figure 1: VSS without disturbance Ax is 0 2 Ax is 5 T im e 4000 IO 0 [se c .) I Ax is I I I I ----T ----<-----:m -- ;= .E5 5 T im e 2 I 2 IO c se c .) I --& ---- I ’ ____- ->----- - -- ----t ----i-----i-----i-----i-----t -----4 0 0 0 0 1 2 3 4 T im e 5 [se c .) 6 Figure 2 : VSS Without disturbance Ax is 7 8 , 3 Ax is d; 9 3 sgq=j 0 5 T im e 10 0 (se c .) Ax is 5 T im e 3 10 [se c .) 3000 -1 0 0 0 0 1 2 3 4 T im e 5 [se c .) Figure 3 : VSS without disturbance 5 6 7 8 9 Ax is 1 Ax is 1 1 8 1 //l-e -+; ____ 0 -1 . 5 T im e 0 Q.----J @;;;i. IO 0 [se c .) 1 Ax is 2 3 4 T im e 5 T im e 1 5 [se c .] 6 IO [se c .) 7 8 9 Figure 1 : VSS with disturbance Ax is 2 Ax is 2 -5 5 T im e IO 0 [se c .> I I 1 2 Ax is 1 5 T im e 2 I IO (se c .] I I I I 5 fse c .l 6 7 8 -4 0 0 0 0 3 4 T im e 9 Figure 2 : VSS with disturbance Ax is -1000 ' 0 1 2 3 4 T im e zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG 5 (se c .> Figure 3 : VSS with disturbance 6 6 7 8 I 9