Trajectory tracking with high accuracy is a very challenging topic in direct drive robot control. This is due to the nonlinearities and input couplings present in the dynamics of the arm. This paper deals with the tracking control of a class of direct-drive robot manipulators. A robust Proportional-Integral (PI) sliding mode control law is derived so that the robot trajectory tracks a desired trajectory as closely as possible despite the highly non-linear and coupled dynamics. The controller is designed using the decentralized approaches. Application to a two degree of freedom direct drive robot arm is considered

Ahmad, Mohd. Noh

Bakar, Saleha

Kader Ibrahim, Babul Salam

Osman, Johari H. S.

English

Universiti Teknologi Malaysia Institutional Repository

A DECENTRALIZED PROPORTIONAL-INTEGRAL SLIDING MODE  TRACKING CONTROLLER  FOR A 2 D.O.F ROBOT ARM   BABUL SALAM KSM KADER IBRAHIM1, SALEHA BAKAR1, MOHAMAD NOH AHMAD2 and JOHARI.H.S. OSMAN2 1Dept. of Mechatronics and Robotics, Faculty of Electrical & Electronics Engineering,    Kolej Universiti Teknologi Tun Hussein Onn, 86400 Parit Raja, Batu Pahat.  Tel: +607-4536467, Fax: +607-4536467, Email: babul@kuittho.edu.my 2Dept. of Robotics and Mechatronics, Faculty of Electrical Engineering,  Universiti Teknologi Malaysia, 81310 UTM Skudai  Abstract  Trajectory tracking with high accuracy is a very challenging topic in direct drive robot control. This is due to the nonlinearities and input couplings present in the dynamics of the arm. This paper deals with the tracking control of a class of direct-drive robot manipulators.  A robust Proportional-Integral (PI) sliding mode control law is derived so that the robot trajectory tracks a desired trajectory as closely as possible despite the highly non-linear and coupled dynamics. The controller is designed using the decentralized approaches. Application to a two degree of freedom direct drive robot arm is considered.  1 Introduction  Variable Structure Control (VSC) with Sliding Mode Control (SMC) has been widely applied to system with uncertainties and/or input couplings [1]. The idea of the SMC is simple; first the desired system dynamics is defined on sliding mode surface. Then, controller is designed to drive the closed loop system to reach the sliding mode surface. In other words, the desired dynamics of the closed loop system is defined first and the state trajectory of the system is then forced to slide on this surface. This can be done through an appropriate switching of the control structures such that the system state will be attracted and stay there afterwards.   When a system is in the sliding mode, its dynamics is strictly determined by the dynamics of the sliding surfaces and hence insensitive to parameter variations and system disturbances.  Nevertheless, the system posses no such insensitivity properties during the reaching phase.  Therefore insensitivity cannot be ensured throughout the entire response and the robustness during the reaching phase is normally improved by designing the system in such a way that the reaching phase is as short as possible [1].  A variety of the SMC known as Integral Sliding Mode Control (ISMC) has also been reported in the literature [2].  Different from the conventional SMC design approaches, the order of the motion equation in ISMC is equal to the order of the original system, rather than reduced by the number of dimension of the control input.  Moreover, by using this approach, the robustness of the system can be guaranteed throughout the entire response of the system starting from the initial time instance. In this paper, the problem of robust tracking for robot manipulator is considered.  On the basis of sliding mode control theory, a class of VSC controllers for robust tracking of robot manipulators is proposed under decentralized approaches.  It is shown theoretically that for system with matched uncertainties, the tracking error is guaranteed to decrease asymptotically to zero and the system dynamics during the sliding phase can easily be shaped up using any conventional pole placement method.  2 Problem Formulation  Consider the dynamics of the robot as an uncertain composite system S defined by an N interconnected sub-systems NiiS ,...,2,1, = is  with each sub-system described by )()]([)()]([)()]([)()]([)(:,1,1tutBBtxtAAtutBBtxtAAtxSjijijNijjNijjjijijiiiiiiii∆++∆++∆++∆+=≠=≠=•     (1) where imRtiuinRtix ∈∈ )(,)(  represent the state and input of sub-system Si, respectively. Ai, Bi, Aij and Bij are constant nominal matrices.  ∆Ai, ∆Aij, ∆Bi and ∆Bij representing uncertainties present in the system, interconnection, input and coupling matrices, respectively. The following assumptions are introduced: (1) Every state vector xi(t) can be locally observed; (2) There exist continuous functions Hi(t), Hij(t), Ei(t) and Eij(t) such that for all NRX ∈  and all t:  ijijijiijiiiiiiijijijiijiiiiiitEtEBtBtEtEBtBtHtHBtAtHtHBtAββαα≤=∆≤=∆≤=∆≤=∆)(;)()()(;)()()(;)()()(;)()(            (2) (3) There exist a Lebesgue function Rti ∈Ω )( : )()()( tBtxAtx iidiidi Ω+=•             (3) where Ai and Bi are the i-th subsystem nominal system and input matrices, respectively; (4) The pair (Ai, Bi) is controllable.   The state vector of the composite system S is defined as [ ] iniTTNTT RtxtxtxtxtX ∈= )(;)(),...,(),()( 21         (4) Let Nnd iRtX ∈)(  be the desired state trajectory: [ ] indiTTdnTdTdd RtxtxtxtxtX ∈= )(;)(),...,(),()( 21             (5) Define the tracking error, zi(t) as )()()( txtxtz diii −=               (6) In view of equations (2), (3) and (6), equation (1) can be written as )()]([)()()()()]([)(tutEBBtBtdixtiBHtiztiHBiAtziiiiiiii++Ω−++=• ≠=++NijjtjxtijHBijA i,1)()]([≠=++NijjtjutijEBijB i,1)()]([                (7) Define the local PI sliding surface for Si as  +−=tiiiiiiiii dzKBCACtzCt0)(][)()( ττσ          (8) where ii nmi RC ×∈  and iinmi RK×∈  are constant matrices.  The matrix Ki satisfies  0)(max <+ iii KBAλ               (9) and Ci is chosen such that CiBi is nonsingular.  For this class of system, the sliding manifold can be described as TTNTTt ],...,,[)( 21 σσσσ =             (10) The control problem is to design a decentralized controller for each sub-system using the PI sliding mode (17) such that the system state trajectory Xi(t) tracks the desired state trajectory Xdi(t) as closely as possible for all t in spite of the uncertainties and non-linearities present in the system.    3 System Dynamics During Sliding Mode  Differentiating equation (8) and substitute equation (7) into it, and equating the resulting equation to zero gives the equivalent control, ueqi(t): )()()()())({()]([)( 1txtHttzKtHtEItudiiiiiiineqi i+Ω−−+−= −})()]([)()()]([)(,11,11≠=−≠=−++++NijjjijiijiiiNijjjijiijiiitutEBBCBCtxtHBACBC              (11) The system dynamics during sliding mode can be found by substituting the equivalent control (11) into the system error dynamics (7): )12(})()]([)()]([]{)([)(][)(,1,11≠=≠=−•+++−++=NijjjijiijNijjjijiijiiiiniiiiiitutEBBtxtHBACBCBItzKBAtz   Define   ])([ 1 iiiins CBCBIP ii −−∆                           (13) whereisP  is a projection operator and satisfies the following two equations [2]: 0and0 == issi BPPC ii              (14) In view of assumption (2), then it follows that by the projection property, equation (14) can be reduced as )(][)( tzKBAtz iiiii +=•                          (15) Hence if the matching condition is satisfied, the system error dynamics during sliding mode are independent of the interconnection between the subsystems and couplings between the inputs, and, insensitive to the parameter variations.  Equation (15) shows that the error dynamics during sliding mode can be specified by the designer through appropriate choice of the matrix Ki.   4 Sliding Mode Tracking Controller Design  The composite manifold (10) is asymptotically stable in the large, if the following hitting condition is held [3]: 0)())(/)((1<•= ttt iNiiTi σσσ              (16) As a proof, let the positive definite Lyapunov function be ==Nii ttV1)()( σ               (17) Then  )())(/)(()(1ttttV iNiiTi•=•= σσσ             (18) Following the Lyapunov stability theory, if equation (16) holds, then the sliding manifold )(tσ  is asymptotically stable in the large.    Theorem 4.2:  The global hitting condition (16) of the composite manifold (10) is satisfied if every local control ui(t) of the error system (7) is given by : )())((])()()()([)()(43211ttSGNttxtxtzBCtuiiiidiiiiiiiiiΩ+Ω+++−= −σγγγγ        (19) where ≠=−++++> NijjiijjjijijiiiiiiiiiiiiBCBCBCBCKBCBC,111)]}([)1{( ββαγ   (20) ≠=−≠=++++> NijjiijjjijijiiiiNijjjjjijijiBCBCBCBCBCAC,11,12)]}([)1{(][ββαγ   (21) ≠=−+++> NijjiijjjijijiiiiiiiiiBCBCBCBCBC,113)]}([)1{( ββαγ    (22) ≠=−≠=+++++> NijjiijjjijijiiiiNijjjjjijijiiiiiBCBCBCBCBCBCBC,11,14)]}([)1{(][ββββγ     (23) Proof:  See [4].    It is shown in [4] that the system (1) is stable in the sense of Lyapunov if the system is control by the input (19).  The structure of the Decentralized Integral Sliding Mode Controller is shown in Figure 1.  5 Simulation Example Consider a two-link manipulator with rigid links of nominally equal length l and mass m shown in Figure 2.  The dynamics of the manipulator is [5]:  [ ] [ ][ ]++ ++= 212 22 102.0)cos(084.0102.0 )cos(084.0102.0)cos(168.0351.221 qqq qq ττ ++−++−2122221222111122)sgn()sin(084.0)sin(084.0)sgn()sin(168.0qqqqfbqqqqqqfbqqcc [ ]11211211)sin(8247.1)sin(8247.1)sin(465.38qxqqqqqq++++       (24) Define [ ] [ ]322111654321)( qqqqqqxxxxxxtX T =∆         (25) [ ]TuutU 21)( ∆                 (26)  Then the plant can be represented in the form of  )()()()()( tUxBtXxAtX +=               (27)  where, =6665636261363533323101000000100000000100000010aaaaaaaaaaAand=6261323100000000bbbbB(28) the nonzero elements of matrices A and B are shown in the Appendix.  Suppose that the bounds of the )(tiθ  and )(ti•θ  are: 121211114380,1850,1500,550−•−−•−≤≤≤≤≤≤≤≤ssss             (29) It is assumed that each sub-system is required to track a pre-specified cycloidal function of the form: 		≤≤≤−∆+=tttttiiidiττθττpiτpipiθθ),(0)],2sin(2[2)0()(    (30) where  .2,1),0()( =−=∆ iiii θτθ   In this example, the input trajectory data used are as follows: Start time, t(0) = 0.0 s Final time, τ = 10.0 s Start positions, θ1(0) = 10 deg ;  θ2(0) = 15 deg Final positions, θ1(τ) = 50 deg ;  θ2(τ) = 60 deg  6      Results and Discussion  With the given bounds, the plant can be represented in the form of equation (1).  Each joint of the robot is treated as a sub-system with the nominal value of Ai, Aij, Bi and Bij is calculated as: ;55.466.146.101000101−−−=A−−=135.0081.001000102A  ;1383.00034.0000000012−=A=2913.00996.0974.000000021A ;55.0001=B ;934.0002=B  ;0179.00012−=B−=0365.00021B  Using equation (2), the bounds of Hi(t) and Ei(t) can be computed: 811.0)(8101.0)(0021.0)(0302.0)(2895.103)(4715.9)(;0225.0)(,838.9)(212112122211212112122211=≤=≤=≤=≤=≤=≤=≤=≤tEtEtEtEtHtHtHtH In this study, the gains are chosen as follows: ]7282.13527.138423.10[1 −=K    so   that }3,5.1,1{)( 111 −−−=+ KBAλ ; ]7435.55482.98175.4[2 =K       so   that }3,5.1,1{)( 222 −−−=+ KBAλ ; and  ]11525[1 =C  and  ]123[2 =C  Therefore, from equations (20)-(23): 3105.0;0140.0;1006.0;5837.7;3276.0;1853.2;1263.4;0253.62423222114131211>>>>>>>>γγγγγγγγFor simulation purposes, two sets of controller parameters are chosen: Set1:2.0;01.0;1.0;4;04.0;1.0;5.0;5.02423222114131211========γγγγγγγγ Set2:      9383.0;02.0;2.0;10;45.0;3;6;82423222114131211========γγγγγγγγ Set 1 contains the controller parameter selected to study the performance of the system if equations (20)-(23) are not met; while Set 2 contains the parameters satisfying the condition imposed on the controller are met.  It can be seen that the tracking performance for both subsystems when Set 1 parameters were used are unsatisfactory (Figures 3a and 3b).  The simulation was run again but this time with the decentralized controller parameter was supplied from Set 2 (Figures 3c and 3d).  As predicted theoretically, the tracking performance is good for both subsystems.    6 Conclusions  Precise trajectory tracking is important in the robotic control.  In this project, a Decentralized Integral Sliding Mode controller is designed and used to track the desired trajectory of direct drive robot arm.  It is shown mathematically that the error dynamics during sliding mode is stable and can easily be shaped-up using the conventional pole-placement technique.  Besides, the system stability is also guaranteed during the reaching phase.  Application to a two degree of freedom direct drive robot arm shows that this controller is a reliable solution to a robust tracking problem of uncertain dynamical systems.  References  [1]. Utkin, V., and Shi, J., Integral Sliding Mode in Systems Operating under Uncertainty Conditions, Proc. IEEE Conf. Decision and Control, Kobe, Japan, 1996, pp. 4591-4596. [2]. Edwards, C., and Spurgeon, S. K., Sliding Mode Control: Theory and Applications, Taylor & Francis, 1998. [3]. Yan, J. -J, Tsai, J. S-H, Kung, F.-C., Robust Decentralized Stabilization of Large-Scale Delay Systems Via Sliding Mode Control, Trans. ASME, Vol. 119, 1997, pp. 307-312.  [4].  Mohamad Noh Ahmad, “Modelling And          Control Of Direct Drive Robot Manipulators”,          PhD Thesis, 2003,  Universiti Teknologi Malaysia. [5].  Fernando Reyes and Rafeal Kelly, ”Experimental              Evaluation of Model-based Controller on a Direct-             drive Robot Arm.” Mechatronics, Vol 11, 2001,             pp. 267-282  )(1 tΩ)(2 tΩ)(1 tXd)(2 tXd)(1 tX)(2 tX)(1 tX)(2 tX)(1 tU)(2 tU Figure 1:  Block Diagram of Decentralized Integral                      Sliding Mode Controller       Figure 2:  A configuration of 2 DOF Direct Drive                         Robot Arm   (a) State x1(t) response for Set 1  (b) State x4(t) response for Set 1   (c) State x1(t) response for Set 2  (d) State x3(t) response for Set 2  Figure 3:  Simulation Results for Decentralized PI Sliding Mode Control.  Appendix  Elements of the matrices A and Bhgh a31 = 1.07/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+      0.06cos(x4))(-0.07(38.47sin(x1)+1.82sin(x1+x4))/x1-0.35 (38.47cos(x1)-3.65sin(x1)sin(x4)+3.65cos(x1) cos(x4))/x1)+         (-0.04-0.03cos(x4))/((-0.03-0.02cos(x4)) (0.1+0.08cos(x4)) +1.96+0.06cos(x4))(-0.28sin(x1+x4) /x1-0.71(-3.65sin(x1) sin(x4)+3.65cos(x1)cos(x4))/x1) a32 = 1.07/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+ 0.06cos(x4)) (-0.16+0.01sin(x4)x5-0.49/x2+0.06cos(x4) x5)+(-0.04-  0.03 cos(x4))/((-0.03-0.02cos(x4))(0.1+0.08 cos(x4)) +1.96+ 0.06cos(x4))(-.007sin(x4)x2-0.12sin(x4)) a33 = 1.07/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+ 0.06cos(x4))(-8.39-0.01cos(x4)+0.06sin(x4)+0.06sin(x4) x5)+    (-0.04-0.03cos(x4))/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+ 1.96+ 0.06cos(x4))(-0.02-0.01cos(x4)+ 0.06sin(x4)x2) a35 = 1.07/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+0.06cos(x4)) (0.01sin(x4)x5+0.03cos(x4)x5)+(-0.04-0.03cos(x4))/((-0.03-0.02 cos(x4))(0.1+0.08cos(x4))+1.96+0.06cos(x4))(-0.03-0.27/x5) a36 = 1.07/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+ 0.06cos(x4))(-0.01-0.01cos(x4)+0.03sin(x4)+0.06 sin(x4)x2)-7.99(-0.04-0.03cos(x4))/((-0.03-0.02cos(x4)) (0.1+0.08cos(x4)) +1.96+0.06cos(x4)) b31 = 1.07/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96               +0.06cos(x4)) b32 = (-0.04-0.03cos(x4))/((-0.03-0.02cos(x4))(0.1+             0.08cos(x4)) +1.96+0.06cos(x4)) a61 =  (-0.07-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+0.08 cos(x4))+1.96+0.06cos(x4))(-0.07(38.47sin(x1)+1.82 sin(x1+x4))/x1-0.35(38.47cos(x1)-3.65sin(x1)sin(x4)+ 3.65cos(x1)cos(x4))/x1)-(-1.82-0.06cos(x4))/((-0.03-2/95 cos(x4))(0.1+0.08cos(x4))+1.96+0.06cos(x4))(-0.2sin (x1 +x4)/x1-0.71(3.65sin(x1)sin(x4) +3.65cos(x1)cos(x4))/x1)                                                                                                                                                               a62 = (-0.07-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+ 0.08cos(x4))+1.96+0.06cos(x4))(-0.16+0.01sin(x4)x5-0.49/x2+0.06cos(x4)x5)-(-1.82-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+0.06cos(x4))      (-0.07sin(x4)x2-0.12sin(x4))    a63 =(-0.07-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+ 0.08cos(x4))+1.96+0.06cos(x4))(-8.39-0.01cos(x4) +0.06sin(x4) +0.06sin(x4)x5)-(-1.82-28/475cos(x4))/((-0.03-0.02cos(x4)) (0.1+0.08cos(x4))+1.96+ 0.06cos(x4) ) (-0.02-0.01cos(x4) +0.06sin(x4)x2) a65 = (-0.07-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+0.08 cos(x4))+1.96+0.06cos(x4))(1701/296875sin(x4)x5+14/475 cos(x4)x5)-(-1.82-28/475cos(x4))/((-0.03-0.02cos(x4)) (0.1+0.08cos(x4))+1.96+0.06cos(x4))(-0.03-0.27/x5) a66 = (-0.07-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+0.08 cos(x4))+1.96+0.06cos(x4))(-0.01-0.01cos(x4)+ 0.03 sin(x4)+0.06sin(x4)x2)+0.14(-1.82-28/475cos(x4))/ ((-0.03-0.02cos(x4))(0.1+0.08cos(x4))+1.96+0.06cos(x4)) b61=(-0.07-0.06cos(x4))/((-0.03-0.02cos(x4))(0.1+ 0.08cos(x4))+1.96+0.06cos(x4)) b62 =(1.82+0.06cos(x4))/((-0.03-0.02cos(x4)) (0.1+0.08cos(x4))   +1.96+0.06cos(x4)) 

A decentralized proportional-integral sliding mode tracking controller for a 2 D.O.F robot arm

http://eprints.utm.my/1844/1/babulsalam05_decentralized_propotional_integral.pdf

A decentralized proportional-integral sliding mode tracking controller for a 2 D.O.F robot arm

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