Design of De oupled Sliding Mode Control for the PUMA 560 Robot Manipulator J. Mozaryn, _ J. E. Kurek Warsaw University of Te hnology Institute of Automati Control and Roboti s Warszawa, ul. Chodkiewi za 8, 02-525 J.Mozarynm htr.pw.edu.pl, jkurekm htr.pw.edu.pl Abstra t There is presented a design of a de upled sliding mode ontrol algorithm for the PUMA 560 robot arm position ontrol. The Lagrange-Euler model of the robot is used for al ulation of ontrol law. Computer sim- ulations of the robot with the de oupled sliding mode ontrol and exa t and non-exa t model are des ribed. The ontrol algorithm is simple and has a good robust- ness with respe t to the robot model un ertainty. 1 Introdu tion One of the fundamental problems for the robot manipulator is positioning of the robot arm endpoint. The aim of this paper is to present the design of sliding mode ontrol algorithm for the ontrol of robot arm. The prin iples of sliding mode ontrol an be found in many publi ations e.g.[2, 3, 4, 6℄. There are also presented di erent approa hes to problem of sliding mode ontrol of robot arm e.g.[3, 4, 6℄. The advantage of the sliding ontrol system is its robustness with respe t to model un ertainties and disturban es. Organization of the paper is as follows: model of PUMA 560 robot using Lagrange-Euler equations is presented in se tion 2, in se tion 3 the prin iples of the sliding mode ontrol and de oupled sliding mode ontrol algorithm of robot are des ribed and in se tion 4 the simulations of the proposed ontrol algorithm are presented. Finally, in se tion 5, on luding remarks are given. lows [5℄  = M () + V (; _) + G(); where  2 Rn is a ve tor of ontrol signals,  2 Rn is a ve tor of generalised joint oordinates, M () 2 Rnn is a robot inertia matrix, V (; _) 2 Rn is a ve tor of Coriolis and entrifugal e e ts, G() 2 Rn is a ve tor of gravity loading. The oordinate frames onne ted to joints were set a ording to Denavitt-Hartenberg notation [1℄. The physi al parameters of PUMA 560 used in omputer simulations base on the literature sour es [5, 9, 13, 12℄, see Tables 1 and 2. 3 De oupled sliding mode ontrol of PUMA 560 robot In order to minimize nonlinear e e ts we have deoupled the robot joints and then used sliding mode ontrol for robot arm position. This approa h signi antly simpli es the ontrol algorithm. 3.1 De oupling of PUMA 560 robot joints The de oupling te hnique for the n degree of freedom robot gives n independent equations of the se ond order subsystems instead of matrix equations (1). A ording to [5, 11℄ assume for a model of robot (1) the ve tor of ontrol signals as follows  2 Mathemati al model of PUMA 560 robot Model of robot with n degrees of freedom an be presented in the form of Lagrange{Euler equations as fol- (1) = P ()^ + R(; _); (2) where ^ is a ve tor of ontrol signals in de oupled system. Assuming P ( ) = M () and R(; _) = V (; _) + G() (3) one obtains from (1) and (2) M () + V (; _) + G() = P ()^  + R(; _): (4) If the model and robot parameters are equal, and matrix M () is nonsingular, we have n se ond order subsystems instead of (1) i = ^i ; i = 1; 2; : : : ; n: (5) The above model (5) an be presented in the state spa e form as follows x_i = Axi + B ^i ; i = 1; 2; : : : ; n; (6) yi = Cxi       where: xi = _i ; A = 00 10 ; B = 01 ; i   C= 1 0 3.2 Design of the sliding surfa e Let us denote 2 ~ =4 xi i ri _ _ i ri 3 2 5 =4 ei ei _ 3 5;i = 1; 2; : : : ; n; (7) where ri is the referen e oordinate of joint i . Then, based on equation (6) of i {th subsystem one an write the following model equation x ~_i = Ax~i + B ^i : (8) yi = C x ~i In sliding mode ontrol algorithm one has to design the sliding surfa e in the state spa e for every joint. The state of ea h subsystem shall rea h sliding surfa es and slide along them. Moreover, the system in sliding mode shall be stable. The sliding surfa e i for i {th subsystem onne ted with i {th joint an be des ribed in the following way xi 2 R2 : f (~ xi ) = 0g: (9) i = f~ For ea h subsystem, the swit hing fun tion is proposed as follows [13℄ f (~ xi ) = Li x ~i ; (10)   where Li = li 1 , and li is a tuning parameter for sliding mode ontrol for i{th subsystem. In so{ alled sliding mode we have f (xi ) = 0 ) f_(~ xi ) = Li x ~_ i = 0: (11) Hen e, using (8) and (11) the so{ alled equivalent ontrol eqi an be derived from the following equation Li x ~_ i = Li (Ax~i + Beqi ) = 0: (12) Thus, the equivalent ontrol an be al ulated a ording to the formula eqi = (Li B ) 1 Li Ax ~i = li e_ : (13) The equation of i{th subsystem (8) with equivalent ontrol (13) is x ~_ i = [A B (Li B ) 1 Li A℄~xi = [A BLi A℄~xi : (14) yi = C x ~i Cal ulating the hara teristi equation of i{th subsystem one obtains det[Is (A BLi A)℄ = s(s + li ): (15) Hen eforth, the eigenvalues of hara teristi equation are: si1 = 0; si2 = li . Clearly, if li > 0 the subsystem (14) is stable. 3.3 Sliding mode ontrol algorithm of robot The rea hing ondition for sliding surfa e i (9) an be des ribed as follows [13℄ f (~ xi )f_(~ xi ) < 0: (16) From (10), (7) and (5), we have f_(~ xi ) = li ei ri : (17) _ + ^i _ = li ei _ + e_ i The stru ture of ontrol signal ^i is given by ^i = sldi = eqi + swi ; (18) where swi is swit hing ontrol al ulated a ording to the formula swi = Ki sat(f (~ xi ) (19) and 8 fi > Æi < +1 if f (20) if f sat(fi ) = i j Æi j ; : Æ 1 if fi < Æi where Æi is a swit hing boundary value in joint i. This form of swit hing part should eliminate the high frequen y os illations alled hattering [6℄ in the sliding mode. Value of Ki should be adjusted in su h way, that the ondition (16) is satis ed. Then, one an note, that i i system (6) with ontrol (18) will rea h sliding surfa es and slide along them. Assuming that parameters of model and system are equal, the equivalent ontrol eqi is des ribed by (13). Condition (16) an be rewritten in the form  f (~ xi ) li e_ li e_ Ki sign[f (~ xi )℄  ri < 0: (21) Clearly, if Ki > 0, ondition (21) is always satis ed. 4 Simulations of PUMA 560 robot with de oupled sliding mode ontrol The introdu ed de oupled sliding mode ontrol algorithm was used in omputer simulations of the real robot manipulator PUMA 560. The robot was simulated using the following formulas for al ulation of the rst and se ond derivatives of the oordinates, Tp is sampling time _ (t)  (t) (t Tp Tp ) ; (t) _  (t + TTp ) p _(t) : (22) In omputer simulations the aim of ontrol system was to rea h the given referen e position. The referen e traje tory in every joint was the step fun tion with di erent values (Fig. 1, 2), thus _r (t) = 0; r (t) = 0. The start point angles and the referen e point angles for ea h joint are given in Table 3. In simulations the robustness of the system with respe t to model un ertainties was he ked. Robot model, used in al ulation of ontrol signal, was exa tly the same as simulated robot, overestimated (+5% in values of links masses and +10% in values of links inertias) and underestimated (-5% in values of links masses and -10% in values of links inertias). This approa h an tell how robust is the sliding mode ontrol method. The ontrol parameters were the same for all of the joints and were equal to: 1 Ki = 100[N m℄; Æi = 1[ rad s ℄; li = 10[ s ℄. The sampling time in all simulations was Tp = 0:01[s℄. The position error was al ulated as follows ei =j 1i j ri ; (23) where 1i is position of i{th joint in steady state. The positions of robot joints in transition pro ess for all the examples are shown on Fig. 1, 2. Moreover, there was also he ked how the hoi e of Tp in uen e the a ura y of positioning. In ase when Tp = 0:001[s℄ the a ura y of positioning in ea h joint (Table.4.) in reased twi e to ei = 0:0014[Æ℄; i = 1; 2; : : : ; n. 5 Con luding remarks The design of ontrol system for robot manipulator using the de oupled sliding mode ontrol algorithm. De upling enables simple design of the sliding mode ontrol for nonlinear system like robot. A ording to the simulation results of PUMA 560 robot manipulator one an see that the system with proposed algorithm works properly. The system is robust with respe t to the parameter ina ura ies in the model of robot, what was shown in simulations with model parameters hange of 5% and 10%. The errors in steady state shown in Table 4 have suÆ iently small values for some real appli ations. Referen es [1℄ J. Denavit, R. S. Hartenberg, \A kinemati notation for lower pair me hanisms based on matri es," Journal of Applied Me hani s , vol.77, pp.215-221, 1955. [2℄ V. I. Utkin, \Variable stru ture systems with sliding modes," IEEE Transa tions on Automati Control , vol.22, pp.212-222, 1977. [3℄ K. K. D. Young, \Controller design for a manipulator using theory of variable stru ture systems," IEEE Trans. Sys. Man. And Cyb. , vol.SMC-8, pp.101-109, 1978. [4℄ J. J. Slotine, S. S. Sastry, \Tra king ontrol of nonlinear systems using sliding surfa es with apli ation to robot manipulators," International Journal of Control , vol.38, no.2, pp.465-492, 1983. [5℄ K. S. Fu, R. C. Gonzalez, C. S. G. Lee, Roboti s: ontrol, sensing, vision, and inteligen e , M GrawHill Book Company, 1987. [6℄ W. Gao, J. C. Hung, \Variable Stru ture Control of Nonlinear Systems:A New Approa h," IEEE Transa tions on Industrial Ele troni s , vol.40, pp.45-56, 1993. [7℄ J. Y. Hung, W. Gao, J. C. Hung, \Variable Stru ture Control: A Survey" IEEE Transa tions on Industrial Ele troni s , vol.40, pp.2-22, 1993. [8℄ A. More ki, J. Knap zyk, Podstawy Robotyki, Teoria i elementy manipulator ow i robotow , WNT Warszawa, 1993. [9℄ P. I. Corke, B. Armstrong{Helouvry, \A sear h for onsensus among model pa-rameters reported for the PUMA 560 robot," Pro . IEEE Int. Conf. Roboti s and Automation , vol.1, pp.1608-1613, 1994. [10℄ P. I. Corke, The Unimation Puma servo system , CSIRO Australia, 1994. [11℄ J. J. Craig, Introdu tion to Roboti s , WNT Warszawa, 1995. [12℄ P. I. Corke, Symboli Algebra for Manipulator Dynami s , CSIRO Australia, 1996. [13℄ C. Edwards,S. K. Spurgeon, Sliding Mode Control: Theory and Appli ations , Taylor & Fran is, 1998. [14℄ P. I. Corke, Roboti s Toolbox (release 5) , CSIRO Australia, 1999. Table 1. Parameters of PUMA 560 robot (Denavit-Hartenberg notation). Link i i[ Æ℄ ai [ m ℄ i [ Æ ℄ di [m℄ Movement Range[Æ ℄ 1 90 0 0 0 -160 to 160 2 0 0.4318 0 0 -225 to 45 3 -90 0.0203 0 0.15005 -45 to 225 4 90 0 0 0.4318 -110 to 170 5 -90 0 0 0 -100 to 100 6 0 0 0 0 -266 to 266 Table 2. Inertia parameters of the PUMA 560 robot. Link i M[kg℄ rx [m℄ ry [m℄ r z [m℄ Ixx [kg
m2 ℄ Iyy [kg
m2 ℄ Izz [kg
m2 ℄ Ixy ; Iyz ; Ixz [kg
m2 ℄ 1 0 0 0 0 0 0.35 0 0 2 17.4 -0.3638 0.006 0.2275 0.13 0.524 0.539 0 3 4.8 -0.0203 0.0141 0.070 0.066 0.086 0.0125 0 4 0.82 0 0.019 0 0.0018 0.013 0.0018 0 5 0.34 0 0 0 0.0003 0.0004 0.0003 0 6 0.09 0 0 0.032 0.00015 0.00015 0.00004 0 Table 3. Start point angles and referen e point angles in ea h joint. Joint number Start point angle [Æ ℄ Referen e point angle [Æ ℄ 1 2 3 4 5 6 7 -30 30 -30 30 -30 30 -30 30 -30 30 -30 30 -30 30 Table 4. Position error in every joint in steady state . Error Exa t parameters Tp = 0:01[s℄ Exa t parameters Tp = 0:001[s℄ Overestimated parameters Underestimated parameters e1[Æ ℄ e2 [ Æ ℄ e3 [ Æ ℄ e4 [ Æ ℄ e5 [ Æ ℄ e6 [ Æ ℄ 0.0024 0.0024 0.0024 0.0024 0.0024 0.0024 0.0014 0.0014 0.0014 0.0014 0.0014 0.0014 0.052 0.006 0.029 0.032 0.178 0.0008 0.095 0.008 0.045 0.067 0.224 0.015 Figure 1: Simulation of PUMA 560 Robot manipulator with de oupled sliding mode ontrol and exa t model parameters. Figure 2: Simulation of PUMA 560 Robot manipulator with a.) de oupled sliding mode ontrol and overestimated model parameters, b.) de oupled sliding mode ontrol and underestimated model parameters.