Compensation of backlash effects in an Electrical Actuator

Transcription

1 1 Compenation of backlah effect in an Electrical Actuator R. Merzouki, J. C. Caiou an N. M Siri LaboratoireeRobotiqueeVeraille 10-12, avenue e l Europe Vélizy merzouki@robot.uvq.fr Abtract We evelop for ytem with backlah a non linear oberver, bae on the etimation of the iturbe torque tranmitte ue the ea zone. Then we eign an aaptive controller uing thi non linear oberver in orer to compenate the iturbe torque on line. Simulation an experimantal reult applie on an elecrical actuator are given to upport the theoretical emontration. eywor: Backlah effect, Tranmitte torque, electrical actuator, ea zone, aaptive control. I. Introuction The preence of the ea zone in a mechanical ytem introuce an hyterei phenomenon between the input an the output poition. It ecribe the backlah phenomenon, which caue a non table behavior for the controlle ytem. Backlah i inherent in mechanical ytem, pecially when tarting the motion. But if it increae ue to the wear, it will iturb the performance of the ytem. In thi cae, we try to compenate it effect by uing a mechanic or an automatic metho. For a long time, the mechanical olution exite to eliminate thee iturber effect by changing all imperfect part on the ytem. Alo, ome control olution have been propoe, like in: Branenbourg & Schaffer[1] where they have tuie the influence an the partial compenation of imultaneouly acting backlah an coulomb friction in a pee an poition controller elatic two-ma ytem. Recker & al [4] an Tao & okotovic [2] have worke on the aaptive control of ytem with backlah. On thi ubject, ifferent mathematical moel are propoe like in: Tao & okotovic [2] have meloizetheinverebacklahmoelbaeonahyterei cycle. Caiou & M Siri [5] have evelope a ifferentiable moel bae on the ea zone characteritic. In thi paper, we inpire from the lat meullization to eign the backlah compenation an control. In mot application, the backlah non linearity coul not be accurately known an only an etimation of it effect coul be poible. Firt, we are going to eign a non linear oberver to etimate the iturber torque ecribing the backlah effect by uing a mathematical moel repreenting an invere igmoi. An then, we contruct an aaptive control uing a PD controller aociate to the lat torque oberver to compenate the ea zone effect. II. Definition & Backlah Meullization.1 Decription of the bench tet the experimental bench tet of Fig. 1 i an electrical actuator, ivie on two part: The firt one correpon to the motor part, riving by a DC motor. the econ repreent the reucer part which regroup three important mechanical imperfection. The firt imperfection i the tatic an the vicou friction, where there coefficient coul be change for ifferent application by uing many brake part like aluminium, metal,..etc., an ifferent lubriciou liqui. Thee coefficient coul be alo ientifie approximately by uing the claic ientification algorithm. The econ imperfection i the backlah an i repreente by a variable ea zone from 0 eg to 24 eg. Finally the tranmitte motion to the output axi i via a couple of tring with a variable tiffne. On thi bench tet, we can meaure the input an the output poition of the reucer axi by uing two incremental coer a it hown in Fig. 1. Fig. 2. Fig. 1. Bench Tet Bloc cheme of the bench tet.2 Decription of the backlah mechanim We can reume the backlah mechanim by ecribing the Fig. 3. The boy1 try to tranmit the motion to boy2

2 2 via a ea zone of magnitue 2.j 0. The tranmiion will be correct when the two boie are in contact, in thi cae there poition are ientical. Out of the contact, the tranmiion will be elaye by the preence of the ea zone where the relation between the boie poition will ecribe an hyterei cycle behavior. w Cr Co C Boy1 velocity Boy 2 j poition Fig. 4. The ea zone an the backlah torque Fig. 3. 2j 0 The backlah mecanim Now, referring to Fig. 4, we can notice that the tranmitte torque inie a ea zone coul have the repreentation of the torque Cr. Thi imple meullization of the tranmitte torque via a ea zone of magnitue 2.j 0 i contant an equal to zero inie thi zone but it i linear outie. Let put θ, the ifference between the input poition θ e an the output poition N 0.θ of the reucer part, with N 0 the reuction contant. Then we can formulate mathematically thi tranmitte torque a follow: if : j 0 < θ <j 0 then: Cr =0 (1) ele : Cr =.( θ j 0.ign( θ)) Thi moel ecribe the expreion of the tranmitte torque inie an outie a ea zone with a flexible part where correpon to the rigiity contant an give the linear repreentation to the lat torque Fig. 4. Thepreenceoftheignum function make the lat formulation of the tranmitte torque not erivable. For that, we have choen to write the characteritic of the tranmitte torque inie an outie the ea zone in function of a igmoi function, a follow: C =. µ θ 4.j 0. 1 e γ. θ (2) 1+e γ. θ where C i the approximative tranmitte torque repreente in Fig. 4. We can ecompoe the lat expreion of the tranmitte torque C to two part a follow: C = C 0 + w (3) with C 0 the linear tranmitte torque, given by: C 0 =. θ (4) an w i the iturber an nonlinear tranmitte torque, given by: w = 4..j 0. 1 e γ. θ 1+e γ. θ (5) We alo notice that the repreentation of w epen on the value of the contant γ, which it epen on the magnitue j 0. The value given in the lat repreentation, correpon to the bet approximation of the torque to be equal to zero inie [ j 0,j 0 ] an varying exponentially outie Fig. 4. Accoring to the imulation an experimental tet, we obtain a bet ecreaing of w inie [ j 0,j 0 ] after putting: γ = 1 (6) 2.j 0 For the next ection, the parameter γ i calculate on or line by giving an approximative an initial value to j 0. Then we etimate the torque w after oberving the evolution of the magnitue j 0 on line. III. Control of the bench tet incluing backlah We uppoe that the tatic friction in neglecte, then the mechanical moel of the bench tet incluing the backlah i ecribing by the equation ytem a follow: ( J. θ.. +f. θ. = C J m. θ.. e +f m. θ. (7) e +C = U J, f, θ.., θ. are ucceively: the inertia of the reucer part, the vicou output friction which i uppoe known, the output reucer acceleration an velocity. J m, f m, θ.. e, θ. e are ucceively: the inertia of the motor part, the vicou input friction which i uppoe known, the input reucer acceleration an velocity. U i the control torque, θ e an θ are the input an output meaure poition of the reucer part. C repreent the tranmitte torque to the loa via a flexible axi an a ea zone, it expree a follow: C =. θ + w (8) i the tiffne of the flexible part an w i the iturber torque ecribe before. We take: θ = θ e N 0.θ (9) which repreent the ifference between the input an output poition of the reucer part. N 0 i the reaction contant.

3 3 Then, the ytem (7) will be expree by the ytem (10): ( θ +f.. θ =.θ e.n 0.θ + w J m... θ e +f m.. θ e = U.θ e +.N 0.θ w We put: ε = θ θ ε e = θ e θ e z =.θ e.n 0.θ ε z = z z w = w ŵ (10) (11) θ i the eire output, θ e i the eire input, z i the ifference between the input an output poition of the reucer part an z it eire value. ŵ i the etimate iturber torque an w correpon to the etimate error of the iturber torque. Now, the ytem (10) coul be expree a follow: ( J. θ.. +f. θ. =.θ e.n 0.θ + w J m. θ.. e +f m. θ. e +J. θ.. +f. θ. (12) = U The control cheme to the bench tet i given by Fig. (5). θ - Fig. 5. ŵ θ & && θ H, 2 H 1 θ& &&, θ ε z, z & z, & z z θ C e + C A ctuator B acklah T orque O b e r v e r Control cheme of the bench tet incluing backlah In orer to linearize the firt equation of ytem (12), we replace the z an w value by there etimate value. So, we will obtain the following equation: θ +f.. θ = ε z + z + w +ŵ (13) For that, we chooe z equal to: z = θ +f.. θ ŵ (14) If we replace expreion (14) in (13), we will obtain: ε +f.. ε = ε z + w (15) which i the equation euce after linearization of the reucer part moel. Now, to linearize the econ equation of the ytem (12), we o thee variable affectation: θ e θ θ e = z + N 0.θ. θ e = ż + N 0. θ. (16).. θ e =.. z + N 0. θ.. So, the econ equation of the ytem (12) will become: J m. z.. + f m. ż +(J m.n 0 + J ). θ.. +(f m.n 0 + f ). θ. = U (17) Thu, we efine a control law U for the global ytem expree a follow: U = J µ m. z.. fm + + D1. ż + P 1.z (18) +(J m.n 0 + J ). θ.. +(f m.n 0 + f + D1.N 0 ). θ. + P 1.N 0.θ k 1.ε z D1. ε. e P 1.ε e (19) P 1 an D1 are the PD contant of the controller C 1 of Fig. 5. AccoringtothecontrolchemeofFig.5,wehave: θ e = D2. ε. + P 2.ε (20) with P 2 an D2 repreent the controller C 2 contant of Fig. 5. We replace expreion of U in equation (17), we obtain: + D1 ³ J m... ε z + fm +(J m.n 0 + J D1. D2 ).... ε. z + k 1 + P 1.εz +(f m.n 0 + f P 1. D2 D1. P 2 + D1.N 0 ).. ε +( P 1.N 0 P 1. P 2 ).ε =0 (21) Thi lat equation ecribe the linearization of the motor part of the bench tet. the torque etimator ha the ame formulation a it moel an given by the following expreion: ŵ = 4..ˆ 0. 1 e γ. θ 1+e γ. θ (22) with γ uppoe known. Let take the cae where θ > 0, then w will take the following expreion: w = 4.. j 0 (23) Now we chooe a backlah magnitue moel given by: j 0 t = 0 (24) with: j 0 the magnitue contant. Thu, we efine an oberver for thi magnitue epening on the output poition an velocity error, a follow: ˆ 0 t = k 2.. ε k 3.ε (25) k 2 an k 3 are a contant. ε

4 4 The etimation error of the backlah magnitue i given a follow: j 0 t = j 0 t ˆ 0 t = k 2. ε. +k 3.ε (26) So, the global ytem will be the combination of all thee equation: ε +f. ε. ³ = ε z + w J m... ε z + fm + D1. ε. z + k 1 + P 1.εz +(J m.n 0 + J D1. D2 )... ε +(f m.n 0 + f P 1. D2 D1. P 2 + D1.N 0 ). ε. +( P 1.N 0 P 1. P 2 ).ε =0 j 0 t = k 2. ε. +k 3.ε w = 4.. j 0 (27) Then, the characteritic equation of the global ytem (27), i efine by the following equation: 4..J.a 3.p (a 3.f + J.a 2 ).p 5 +(4..f.a 2 + k 2.a J.a b 2 ).p 4 +(a 3.k 3 + a 2.k f.a b 1 ).p 3 +(a 1.k 2 + a 2.k 3 +4.k.b 0 ).p 2 + a 1.k 3.p =0 (28) with b 2 = J m.n 0 + J D1. D2, b 1 = f m.n 0 + f P 1. D2 D1. P 2 + D1.N 0, b 0 = P 1.N 0 P 1. P 2, a 3 = J m, a 2 = f m + D1, a 1 = k 1 + P 1. Now, we etablih the impoe conition on the controller contant an on k 1, k 2 an k 3,fortheglobalytem to converge to the equilibrium tate (ε 0, ε z 0, w 0), after uing the Routh criterium. So, we obtain: ecribe the tracking of the output ignal before the compenation of the iturber torque. We notice that the real poition ignal i le eforme at the peak area ue to thepreenceoftheeazoneanflexibility effect. Thi eformation i compenate after introucing of the torque oberver a it i hown in Fig. 6(low), where the output real poition ignal i approache the eire one with a magnitue ifference ue the flexibility effect. Fig. 7 ecribe the hyterei cycle between the input an output reucer poition. After compenating the ea zone effect, the with of the cycle i reuce. knowing that the flexibility effect i preent in the meullization, o the relation between the input an output poition i not exactly linear after compenation. Fig. 8 ecribe the tracking of the input ignal before (top) an after (low) the iturber torque compenation. The tracking i perfect in the two cae but without the perturbation an more clear in the cae after compenation. Fig. 9 how the evolution of the error correponing to the output an input poition tracking. After aing the torque oberver, the output tracking error i reuce an uniforme for each perio. But a tatic error i till preent, ue to the preence of the flexibility effect. For the input tracking error, the convergence to zero i verifie inthetwocae. Thepreenceofthecon- troller C1 anc2 in the ytem of Fig. 5 are ufficient to make an error convergence in a finite time, but in the cae after compenation, the error ignal i le perturbe. Fig. 10 repreent the control ignal before an after compenation. The ignal before compenation i le clean than the after compenation one. Thi i ue to the backlah effect, efine by the iturber torque of Fig. 11, where the ea zone i preente on the tranmitte torque curve. After the compenation, we obtain a linear repreentation of the torque, tranmitte via a flexible axi. ½ k1 > 0 k 2 > a 3.C 6.k 3 +C 3.C 6 C 5.C 4 C 5.C 4 (29) C 5.a 3 a 2.C 6 with C 6 =4..J.a 3, C 5 =4.. (a 3.f + J.a 2 ) C 4 =4..f.a 2, C 4 =4..J.a b 2, C 3 =4..f.a b 1, C 2 =4.k.b 0, C 1 = a 1.k 3, A. Simulation reult The imulation tet are one on a mechanical moel repreentative of the bench tet of Fig. 1. The parameter of the moel for the imulation tet are: P 1 = 15, D1 = 0.5, P 2 = 15, D2 = 0.3, =1N.m/ra, J m =0, N.m 2, f m =0, N.m./ra, J =7, 5 N.m 2, f =16N.m./ra, j 0 =0.1 ra, k 1 =1,k 2 =0, 01, k 3 =1anN 0 = 59. For a eire output ignal θ (t) =0, 5. in(0, 2.π.t) applie on ytem Fig. 5, we obtain figure Fig. 6(top) which Fig. 6. Deire an Reel output ignal before an after compenation IV. Experimental reult The experimental tet have been applie on the bench tet of Fig. 1, with the following control parameter:

5 5 Fig. 7. Hyterii backlah behaviour before an after compenation Fig. 11. Compenation of the ea zone after compenation of the iturbe torque Fig. 8. Deire an Reel input ignal before an after compenation P 1 =1, D1 =0.01, P 2 = 10, D2 = 10, =1 N.m/ra, j 0 =0.16 ra, k 1 =1,k 2 =0, 01, k 3 =1. In thee tet, the motor-reucer wa require to move from the initial tatic output poition θ (0) = π/2 ra, an output velocity θ. (0) = 0 ra/ ec to the origin θ (0) = 0 ra, θ. (0) = 0 ra/ ec Fig.12 repreent the tracking output poition before (i.e. the regulation i mae by only a PD controller) an after applying the aaptive compenation. The tatic poition error i about 0, 32 ra.an it compenate after aing the etimator of the uneire ea zone torque. Fig.13 how the output velocity ignal, before an after the aaptive compenation. So, before the compenation cae, an uneire ocillation aroun θ. (0) = 0 ra/ ec are preent an repreent the non linearitie effect. Thee imperfection effect are compenate after applying the etimate iturber torque. The control ignal before an after compenating the backlah effect are hown in Fig.14. In the cae after compenation, the control ignal i cleaner than it equivalent before the compenation ue to the aaptive compenation of the iturbance effect an backlah. Finally, the uppoe iturber torque which introuce the ea zone i repreente in Fig. 15. Fig. 9. Poition output an input error ignal before an after compenation Fig. 10. Control Torque before an after compenation Fig. 12. Output poition before an after aaptive compenation

6 6 Fig. 13. Output velocity before an after aaptive compenation V. Concluion The preence of backlah in mechanical ytem make ifficult it control with high accuracy. So, we coul reuce the effect of thi lat imperfection by etimating the neceary iturber torque inie the ea zone, then aing it in the control law in orer to compenate the effect of the real torque. For that, a nonlinear an erivable mathematical moel for the iturber torque ha been choen an the ea zone magnitue ha been uppoe contant value. The etimation of the magnitue variation i oberve in function of the output poition an velocity error. A goo choice of the control ytem parameter make the convergence of the global ytem to the origin tate, a it ha been hown in the imulation an experimental reult. Reference [1] G. Branenburg, U. Schafer, Influence an partial compenation of imultaneouly acting backlah an coulomb friction in a pee an poition controlle elatic two-ma ytem, ICED 88, Proceeing of International Conference on electrical rive, Romania, pp. 1-12, [2] G.Tao, P.V. okotovic, Aaptive control of ytem with backlah, Automatica, Vol. 29, No. 2, pp , [3] G.Tao, P.V. okotovic, Aaptive control of plant with unknown hyterei, IEEE Tranaction on Automatic Control, 40(2), , [4] D.A. Recker, P.V. okotovic, D. Rhoe, J. Winkelman, Aaptive nonlinear control of ytem containing a ea zone, Proceeing of the 30 th conf. on eciion an control, Brighton, Englan, Vol. 3, pp , December [5] J.C.Caiou, an N..M Siri, Moelization an Analyi of a Sytem with Torque Tranmitte through a Backlah, 9 th worl congre on the theory of machine an mechanim, IFT.MM, Vol. 2, pp , Milan, Italy, Augut-September [6] R. Merzouki, J.C. Caiou, N.. M Siri, S. Femmam, A nonlinear Oberver an Aaptive Compenation Control of an Electrical Actuator with Backlah an friction IEEE, 16th IMACS Computat. Eng. in Sy. Appl., IMACS Multiconf. Lauane, Switzerlan, CDROM CPcoe 616-5, Augut 21-25, Fig. 14. Control ignal before an after aaptive compenation Fig. 15. The etimtion of the iturbe tranmitte torque