Proceeings of the 5 th International Conference of Control, Dynamic Systems, an Robotics (CDSR'18) Niagara Falls, Canaa June 7 9, 218 Paer No. 139 DOI: 1.11159/csr18.139 Robust Control of Robot Maniulators Using Difference Equations as Universal Aroximator Morteza Rezazaeh Mehrjou Deartment of Aza University of Damghan Semnan Province, Damghan, Iran mortezamehrjo56@gmail.com Abstract - This aer resents a simle an robust non inversion-base erfect tracking control strategy for robot maniulators. The roose aroach is caable to eliminate the environmental roblems arising from classic feeforwar control esign an so guarantees an aroriate level of robustness of control system to uncertainties incluing external isturbances, un- moele ynamics, an arametric uncertainty. Extensive simulation results erforme using a two egree-of-freeom actuate elbow robot rove the effectiveness of the roose aroach. Using free moel of system in control law esign is a consierable oint in the fiel of robot maniulator control. Keywors: Moel Free, Robust Control, Robot Maniulator. 1. Introuction Conventional moel-base fee forwar control (FFC) fails to rouce goo trajectory tracking erformance, in resence of uncertainties such as system arameter variations, external isturbance, friction force an unmoele ynamics. Some of inherent weaknesses of this aroach have been mentione in reference [1]. More ever, alication of this control methoology as a stanar aroach for linear systems has mae it unsuitable for nonlinear systems. The main reason for this sensitivity refers to necessity for solving artial ifferential equations for obtaining the fee forwar ath signal [2]. On the other han, using of this aroach for igital control systems encounters with a few ifficulties. Because iscretization rocess by zero-orer hols usually leas to a iscrete-time system with at least one notorious unstable zero in out of unit circle [3], thus the fee forwar branch becomes unstable an consequently, its realization will become imossible. Existence of this zero, also leas to many other lateral roblems. As a samle we can mention significant hase errors over a broa range of frequencies, which cause roblems in aative controller esign [4]. Therefore, many attemts have been mae by researches to overcome on the mentione roblems over the last ecae [3-15]. However, all of these stuies are base-moel an nee solving the comlicate equations. This work attemts to aress a unifie igital fee forwar control scheme for a 2 egree of freeom robotic maniulator using linear state feeback an without neeing any aitional control metho. An analytical consieration for tracking roblem is resente incluing free moel of lant for controller esign. The aer is organize as follows: The motion equations of the system an moel-free igital control esign are constructe in section 2. Stability analysis an simulation results are resente in section 3 an 4 resectively an conclusions are rawn in section 5. 2. Moel-Free Digital Control Consier motion equations of an integrate actuator-robot system escribe in the joint sace as below [16] (1) where, q is the n 1 vector of generalize joint coorinates, Dq ( ) is the inertia matrix, is the vector of centrietal an Coriolios terms, Gq ( ) is the vector of gravitational torques, is a constant matrix an u is the control inut vector. In aition, Dq ( ) an are nonsingular matrices. It can easily be shown that, by introucing aroriate 139-1
state variables an simle maniulations Eq. 1, a non-linear, time-variant, continuous-time control system, can be escribe by the following moel where (2) y Cx (3) (4) where x is a 2n 1 state vector, y is the n 1 outut vector, an an I are the n n zero an ientity matrices, resectively. Also, A x,, A x,1 an B x, are efine as follows: (5) Easily can be shown that, Eq. 2 can be rewritten as where is the vector of uncertainties incluing external isturbance, an unmoele ynamics, an A, B an C matrices are given by (6) I A, B I (7) Discritization rocess by zero orer hol leas to a iscrete-time form of Eq. 3 an Eq. 6 as below x( k 1) Gx( k) Hu( k) ( k) (8) y( k) Cx( k) (9) Now, we esign a linear control law of the form u( k) kx( k) k r( k) (1) where k an k are constant. It must be note that rk ( ) is the robustifying control inut, such that it leas to minimization of the tracking error. By substituting Eq. 1 into Eq. 8 we will have: x( k 1) G Hk x( k) Hk r( k) ( k) (11) Now we evelo an algorithm to ajust rk ( ). Towar this en, suose that the esire close loo state equations are given by: x ( k 1) G Hk x ( k) + Hk r ( k) (12) 139-2
y ( k) Cx ( k) (13) where r ( k ) an y ( ) k are the esire trajectory an esire outut in joint sace, resectively. It must be note that, the coefficient vector k is esigne so that, y ( ) k agreeably follows r ( k ). For continuation of this subject, let us introuce the following transformation: ( k) @ y( k) y ( k) (14) By these assumtions, Eq. 11, 12 in new coorinates becomes e( k) @ x( k) x ( k) (15) v( k) @ r( k) r ( k) (16) e( k 1) G Hk e( k) + Hk v( k) ( k) (17) ( k) C e( k) (18) Now, by consiering linear system roerties, we arrange a ifference equation as follows ( k 1) G Hk ( k ) Hk ( k ) ( k ) bj( k j) j1 (19) Where ( k) e( k) b e( k j) j1 ( k) = v( k) b v( k j) j1 j j (2) Here, if we assume, ( k) can be moele by a -orer ifference equation as below, where orer reflects the ynamic structure of ( k), we will have ( k ) bj( k j) (21) j1 The continuous-time form of this assumtion has been aresse in [17]. In the next ste, we efine a igital control law as follows Substituting the last equation in Eq. 19 we obtain ( ) - j ( ) ( ) (22) k k j k j=1 139-3
( ) G Hk Hk ( ) k 1 k HkC je ( k j) j=1 (23) whereas () k, (k) are functions of tracking error e (k), therefore by roer selection of close-loo system oles we can guarantee that e (k) converges to zero asymtotically. Finally, we ajust r(k) in Eq. 11, from Eq. 16. Therefore, we use a 2-stage aroach for igital fee forwar control esign. First, we esigne an inner state feeback control for tracking of reference inut r (k) by outut y (k) (base on esire state esign). Then we utilize an outer state feeback control to suress effects of uncertainties. It is useful to note that, goo tracking accuracy can be achieve with orer =1 or 2 [19]. The block iagram of the roose scheme is eicte in Fig. 1. It is its turn now that, we show the roose aroach above, is cancelling the isturbances in a fee forwar combination an oes not nee any lateral control scheme. In other wor, by comleting the roose aroach instea of classic feeforwar form, we will have more tranquillity. Towar this en, the z transform of equation (11) is obtaine as follows 1 Hk x X ( z) ZI G () H k r( z) ( z) (24) where X( z ) is z transform of xk ( ). Furthermore the z transform of equation (12) is efine as -Z H k r ( z) ZI G Hk X ( z) x () (25) With multilication extremes of equation (16) in H k we will have: Fig. 1: Moel free igital fee forwar control scheme. H k r( z) H k v( z) + H k r ( z) (26) Also, z transform of equation (22) an (2) uner initial conition e()= is given by j j z bj z v( z) -Cjz e( z) j1 j=1 j z bj z e( z) v( z), e( z) j1 (27) where ez () an vz () are z transform of ek ( ) an vk ( ), resectively. Hence 139-4
j j - C z z bj z j j 1 v( z) j=1 e( z) j z bj z j1 v( k), e( k) j z bj z j1 (28) where ez () is efine as: In aition, equation (28) can be rewritten in the form e( z) X( z) X ( z) (29) where v( z) - ( z) e( z) ( k, z) (3) C z z b z j ( z) j=1 j z bj z j1 v( k), e( k) k, z j z bj z j1 j j j j1 (31) Multilication extremes of equation (3) in HK yiels H k v( z) -H k ( z) e( z) H k ( k, z) (32) Finally, using Eq. 25, Eq. 26 an Eq. 32, the fee forwar scheme becomes comlete. In roose aroach, the fee forwar branch is inversion of controlle rocess by state feeback theory, (ZI-G+HK). The equivalent block iagram of the roose scheme is shown in Fig 2. 3. Stability Analysis Here we will show the roose aroach above leas to a stable scheme in resence of isturbances. Towar this en, Substituting Eq. 25, Eq. 26, Eq. 32 into Eq. 24 leas to: 1 e( z) ( ZI G H k H k ( z)) ( ( z) H k ( k, z)) (33) By roer selection of eigenvalues for outer control loo, H k( k, z) term for rejecting of uncertainties, final theorem, an also notification of this oint of view that, goo tracking accuracy can be achieve with low uncertainty moel error (=1 or 2), thus the roose aroach is stable an tracking error tens to zero asymtotically[18]. 139-5
z1 1 e ( t ) lim(1 Z ). e( z) (34) ss 4. Simulation Results In orer to emonstrate usefulness of the roose controller, we use a 2-link elbow robot maniulator for igital simulation, uner 1 ms samling simulation time. The major stes of the roose algorithm can be summarize as bellow: - The esire trajectory is secifie as follows a cos( t) a, t (35) T where we set a.5ra, an T 2sec - Calculating the state feeback vector k as Table-1 - Moeling of uncertainty by a th-orer ifference equation, set the uncertainty equation to zero an finally obtain bj. In this ste, if we choose =1 for the uncertainty, we will have ( k 1) b( k) k (36) 1 Fig. 2: Equivalent form of Moel free igital fee forwar control scheme. Table 1: Gains of the Controllers. Joint k 1, 2 [221 189.5] By these assumtions, b 1 is set to zero an consequently, ( k) is obtaine as an arbitrary constant at the time zero. - Calculation of state feeback vector μ as Table-2 In the simulation, we set the masses an lengths of link 1, 2 as m 1 = 17.4kg, m 2 = 4.8kg, l 1 =.4318m, l 2 =.4318m, resectively. Also, the true actuator ynamic coefficients are efine as: R=1.86Ω, L=.1216H, k m =.189, k b1 =.189, b m =.2, j m =.5 an r=.2. Base on aforementione exressions, Fig. 3 eicts tracking error of all joints for assume moel-free system with Eq. 12. In this manner the motors voltage obtaine as Fig. 4. To show the ability of this aroach in resence of external isturbances (external loa torques on the motors shaft) an moel uncertainties, we obtaine the technical limits such as, torque limit, tracking error, voltage limit an control signal as Fig. 5 to Fig. 8, resectively. As can be seen, tracking error, shown by Fig. 6, is boune an so the roose aroach leas to asymtotic stability. Simulation results show that the robot can be effectively controlle an robustifie subject to uncertainties base on using a free moel of lant as seen in Fig.9. Table 2: Gains of the Controllers. Joint μ 1,2 [.65.148.3] 139-6
Torque(N.m) Voltage(volt) Tracking error(ra) 8 x 1-3 Desire tracking error 7 6 5 4 3 2 1-1.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Fig. 3: Tracking error. 3 25 2 15 1 5-5 -1-15 -2.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Fig. 4: Voltages of motors. 14 12 Loa torque Joint 1 Joint 2 1 8 6 4 2-2.2.4.6.8 1 1.2 1.4 1.6 1.8 2 139-7
Control signal(ra) Voltag(volt) Tracking error(ra) Fig. 5: loa torques. 8 x 1-3 7 Joint 1 Joint 2 6 5 4 3 2 1-1.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Fig. 6: Tracking error subject to isturbances. 6 5 Voltages of motors subject to isturbances Joint 1 Joint 2 4 3 2 1-1 -2.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Fig. 7: Voltages of motor subject to isturbances 1 x 1-4 8 control signal(v) Joint 1 Joint 2 6 4 2-2 -4.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Fig. 8: Control signal 139-8
Position(ra) System resonse using roose aroach in aearance of external isturbance 1.9.8.7.6.5.4.3.2.1.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Time(Sec) Fig. 9: System resonse 5. Conclusions A moel-free igital control scheme roose for motion tracking control of robotic maniulator with unstructure uncertainty. This controller esign is extene form of our revious wok in continuous-time systems. The main avantages of the roose aroach are simlicity, racticably, an low comutation buren of this metho to control robotic maniulator systems. References [1] E. J. Aam, J. L. Marchetti, Designing an tuning robust fee forwar controllers, ELSEVIER, Comuters an Chemical Engineering, vol. 28,. 1899-1911, 24. [2] A. Isiori, an C. I. Byrnes, Outut regulation of nonlinear systems, IEEE Transactions on Automatic Control, vol. 35,. 131-14, 199. [3] K. J. Åström, P. Hanganer, an J. Sternby, Zeros of samle system, Automatica, vol. 2, no. 1,. 31-38, 1984. [4] M. Tomizuka, Aative Zero hase error tracking algorithm for igital control, Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 19,. 65-68, 1987. [5] S. Devasia, D. Chen an B. Paen, Nonlinear inversion-base outut tracking, IEEE Transactions on Automatic Control, vol. 41,. 93-942, 1996. [6] A. Piazzi an A. Visioli, Otimal inversion-base control for the set-oint regulation of non minimum-hase uncertain scalar systems, IEEE Transactions on Automatic Control, vol. 46,. 1654-1659, 21. [7] C. G. L. Bianco an A. Piazzi, A servo control system esign using ynamic inversion, Control Engineering Practice, vol. 1,. 847-855, 22. [8] A. Izabakhsh, M. Masoumi, FAT-base Robust Aative Control of Flexible-Joint Robots: Singular Perturbation Aroach, Inustrial Technology (ICIT) IEEE International conference, 217. [9] M. Benosman an G. Le Vey, Stable inversion of SISO non minimum hase linear systems through outut lanning: an exerimental alication to the one link flexible maniulator, IEEE Transactions on Control Systems Technology, vol. 11,. 588-597, 23. [1] E. Gross, M. Tomizuka, an W. Messner, Cancellation of iscrete time unstable zeros by feeforwar control, Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 116,. 33-38, 1994. [11] V. Feliu, E. Pereira, I. M. Dýaz, P. Roncero, Feeforwar control of multimoe single-link flexible maniulators base on an otimal mechanical esign, Robotics an Autonomous Systems, vol. 54,. 651-666, 26. [12] C. S. Chiu, Mixe Feeforwar/Feeback Base Aative Fuzzy Control for a Class of MIMO Nonlinear Systems, IEEE Transactions On Fuzzy Systems, vol. 14, no. 6,. 716-727, 26. [13] H. Fujimoto, Y. Hori, an A. Kawamura, Perfect Tracking Control Base on Multirate Fee forwar Control with Generalize Samling Perios, IEEE Transactions on Inustrial Electronics, vol. 48, no. 3,. 633-644, 21. 139-9
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