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Ramamurthy V. Dwivedula Principal Ideal Institute of Technology, Kakinada 533 003, India e-mail: ram_ Prabhakar R. Pagilla1 Professor School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078-016 e-mail: Effect of Compliance and Backlash on the Output Speed of a Mechanical Transmission System A dynamic model to describe the effect of compliance in a transmission system is pre- sented. Analysis of this model shows that it is desirable to use feedback from driver-side of the transmission system. This model is extended to include the effects of both compli- ance and backlash in a mechanical transmission system. The proposed model considers compliance (which may be either due to the elasticity of the shafts or belt in a belt-pulley transmission system) and backlash appearing in series in a drive system. In contrast to the classical backlash model which considers both input and output to the backlash as displacements, the proposed model considers (torque) force as input to the backlash and (angular velocity) velocity of the driven member as the output of the backlash. Thus, the proposed model does not assume that the load is stationary when contact is lost due to backlash width, i.e., momentum of the load is taken into account. Using the proposed model, a bound on the speed error due to the presence of backlash is derived. Experi- ments were conducted on a rectilinear mass-spring system platform, which has a provi- sion to change the backlash width by a known value. Experiments were conducted with different backlash widths and a velocity error bound was computed. The error bound obtained from the experimental results agrees with the theoretically computed bound. DOI: 10.1115/1.4005493 1Introduction Backlash is one of the most commonly encountered nonlinearities in drive systems employing gears or ball-screws and indicates the play between adjacent moveable parts. Since the action of two mat- ing gears can be represented by the action of one pair of teeth, back- lash is commonly represented by the schematic shown in Fig. 1. When used in the context of mechanical engineering, backlash denotes two salient features as shown in Fig. 2: (i) a mechanical hysteresis due to the presence of a clearance (D), and (ii) impact phenomena between the surfaces of the masses (Mmand ML). In Fig. 1, Mmand MLare the masses (inertias) of the driving and driven members, xmand xLare the linear (angular) displacements of the driving and driven members, respectively, from a fi xed refer- ence position, and Fmand FLare the driving and load forces (tor- ques). It is a common practice to lump all the mass (inertia) on the driving side into one quantity, Mm, and refer to it as the “motor” and lump all the mass (inertia) on the driven side, and refer to it as the “load.” The classical backlash model considers the schematic shown in Fig. 1 with input to the backlash as the displacement xmand the output of the backlash as the displacement xL. The inputoutput characteristics of the backlash are represented by Fig. 2. The slopes of lines GBC and FED are equal to the speed ra- tio of the gearing in the case of rotary systems. The closed curve BCDEFGB in Fig. 2 represents mechanical hysteresis due to the presence of clearance D. At points B, D, and G in Fig. 2, the two masses impact and near these points, the input output plot may not be straight but may “oscillate” with a small amplitude. However, impact may be considered to be suffi ciently plastic so that points on these lines lie along a curve bounded by the dotted circles shown, before they resume to lie on the straight lines. The classical backlash model resorts to this simplifi cation mainly because in large industrial machines, which operate at steady state and do not reverse direction, impact does not arise exceptduringstarting/stoppingconditions.Also,insmaller machines, the gear and impact energy are very small. Thus, a plas- tic impact is considered to be a reasonable assumption. Since large industrial machines do not reverse direction many times during their operation, the lines CDE and FGB in Fig. 2 are ignored. And this prompted many researchers to erroneously consider the input output graph of backlash to be represented by the curve FEABC, which is the inputoutput graph for dead-zone nonlinearity. Also, it may be observed that the backlash characteristics shown in Fig. 2 consider the input to the backlash to be the displacement of the Fig. 1Schematic of backlash 1Corresponding author. Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OFDYNAMICSYSTEMS, MEASUREMENT,ANDCONTROL. Manuscript received January 11, 2008; fi nal manuscript received October 14, 2011; published online April 3, 2012. Assoc. Editor: YangQuan Chen. Journal of Dynamic Systems, Measurement, and ControlMAY 2012, Vol. 134 / 031010-1 CopyrightV C2012 by ASME Downloaded From: / on 02/23/2014 Terms of Use: /terms motor (xmin Fig. 1) and output of the backlash to be the displace- ment of the load (xLin Fig. 1). However, in actual practice, the input is a force (torque) on the motor and the output is either the displacement or the velocity of the load. In such situations, the plot between xmand xLis different from that shown in Fig. 2 and depends on the nature of the applied force. For example, if a force as shown in Fig. 3 is applied on the motor (Mm5 kg, ML10 kg, bmbL0.5 N-s/m) in Fig. 1, the plot between xmand xLmay be obtained as shown in Fig. 4. Thus, the describing function approaches based on the inputoutput plot as shown in Fig. 2 are not applicable for this case. Research on modeling backlash and its effects date back to the 1960s. Much of this research focused on the method of describ- ing functions to investigate limit cycles and deriving stability cri- teria for systems containing backlash 14. A rectilinear model called “impact pair” was presented in Refs. 5, 6 based on which the dynamic behavior of meshing gears was studied in Ref. 7. As a further development, a rotary model for spur gears was developed in Ref. 8 and the contact-spring rate and a time de- pendent damping for a pair of standard spur gears (pressure angle15 deg or 20 deg) was computed. Several nonlinear feed- back models of hysteresis were discussed in Ref. 9. Using the models developed, a number of researchers investi- gated control strategies to compensate for the effects of backlash 1019. These control strategies may be grouped into two main categories: (i) strategies for controlling the displacement of the driven member, and (ii) strategies for controlling the velocity of the driven member. A comprehensive survey of various such strat- egies is reported in Ref. 10. A delayed output feedback controller is proposed for a backlash-free plant in Ref. 11 to compensate for the effects of backlash with displacement as its input and output. However, it was not clear as to how the delayed feedback controller stabilizes the system. In Ref. 12, an “adaptive right-backlash-inverse” was proposed, and it was shown that all closed-loop signals are bounded. Similar work on dynamic inversion using neural net- works was reported in Refs. 13,14. Though stability of the sys- tem using these inversion schemes is shown through simulations, it is reported in Ref. 10 that “the adaptive control seems to yield bad transients during adaptation, while after adaptation, the gain, and hence the bandwidth of the adaptive control system is lower than the gain of the robust linear system.” Also, a study to experi- mentally evaluate the dynamic inversion schemes is presented in Ref. 15, wherein the “backlash inverters” were found to actually degrade performance in the experiments. Besides, these inversion schemes pertain to position controlled drive systems and are not directly applicable to speed controlled systems. In Ref. 17, a nonlinear controller with “soft switching” is proposed. A gear tor- que compensation scheme using a proportional-integral-derivative (PID) speed controller is proposed in Ref. 16. A backlash com- pensation scheme using an open-loop modifi cation of the input trajectory was proposed in Ref. 19. The proposed velocity com- pensation method is most effi cient only for low operating speeds and large mounting allowance between gears. Quantitative design of a class of nonlinear systems with param- eter uncertainty was considered in Ref. 20. The nonlinearities yN(x) considered are such that they can be expressed as yKxg(x) where jg(x)j : (11) With the kinetic energy and potential energy defi ned in Eq. (11), the EulerLagrange dynamics for the system shown in Fig. 9(b), ignoring the inertias of the spring and the shaft, may be written as Mm xm bm_ xm wxm;xL Fm ML xL bL_ xL? wxm;xL FL (12) Fig. 9Linear analog: (a) without backlash and (b) with backlash Fig. 8Schematic of a gear drive 031010-4 / Vol. 134, MAY 2012Transactions of the ASME Downloaded From: / on 02/23/2014 Terms of Use: /terms where wxm;xL kxm? xL DifEq: (8) holds kxm? xL? DifEq: (9) holds 0ifEq: (10) holds 8 : (23) Notice that Eqs. (21) and (22) are identical except for the extra term, /(xm, xL), present in Eq. (22). And this extra term, because of the condition in Eq. (10), is bounded by j/(xm, xL)j ? kD for all xm, xL 2 R. Defi ning the state-variables zm1 xm, zm2 vm _ xm, zL1 xL, zL2 vL _ xL, and z z m1 ;z m2 ;z L1 ;z L2 ?, a state space representation of the system shown in Fig. 9(b) is obtained asFig. 10Schematic of a belt-pulley transmission system Journal of Dynamic Systems, Measurement, and ControlMAY 2012, Vol. 134 / 031010-5 Downloaded From: / on 02/23/2014 Terms of Use: /terms _ z Apz BpFm CpFL bDpxm;xL y Lpz (24) where Ap 0100 ? k Mm ? bm Mm k Mm 0 0001 k ML 0? k ML ? bL ML 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ;Bp 0 1 Mm 0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ; Cp 0 0 0 1 ML 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ;Dpxm;xL 0 /xm;xL Mm 0 ? /xm;xL ML 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ; Lp 0100 0001 “# D Lp1 Lp2 “# (25) and b is zero if the backlash gap is zero and unity otherwise. Thus, with b0, Eq. (24) is a state space representation of the system as shown in Fig. 9(a) and with b1, Eq. (24) is a state space representation of the system shown in Fig. 9(b). Equation (24) represents a system with two inputs (the actuating force, Fm and the load force, FL) and two outputs (the motor speed, zm2and the load speed, zL2) as shown in Fig. 11. It may be noted, here, that, though the actuating force, Fmis known, the load force (or the “disturbance force”), FL, is seldom known/measured. The idea behind the analysis presented below is to capture the effect of compliance/backlash in a given situation, that is, under the infl u- ence of a given disturbance force, FL. In the control scheme shown in Fig. 11, the controller, G, uses feedback from the motor-side (zm2). As already explained in Sec. 2, it is not desirable to use load-side speed as feedback when a compliant member is present in the transmission system. Another intuitive interpretation is that, during the no-contact period, the motor has no control over the speed of the load and hence using load-speed during this period is not desirable. This might have been the reason for specifi c lack of interest in using the load side feedback (as noted in Ref 10, Sec. 3). Suppose the controller, G, has a state-space representation _ xc Acxc Bce Fm Ccxc Dce (26) It may be noted that for the case of a PI controller, Ac0, Bc1, CcKi, and DcKp. Then, the state-space representation for the closed-loop system as shown in Fig. 11 may be obtained as _ zcl Aclzcl CclFL Wvr Dclxm;xL zb L2 Lczcl (27) where zcl zx c ?, W D cB p B c hi , Lc0 0 0 1 0, and Acl Ap? BpDcLp1BpCc ?BcLp1Ac “# ;Ccl Cp 0 “# ; Dclxm;xL bDpxm;xL 0 “# (28) The superscript in zb L2indicates the output in the presence of back- lash. If the backlash was to be absent, b0 and the state-space representation of the closed-loop system may be written as _ vcl Aclvcl CclFL Wvr z0 L2 Lcvcl (29) where the matrices Acl, Ccl, W are given in Eq. (28) and z0 L2is the output in the absence of backlash. Equations (27) and (29) are similar except for the last term in the state equation in Eq. (27), and the deviation zb L2? z 0 L2j Lc zcl? vcljj ? ? represents the effect of backlash. For a given reference velocity vr, and the disturbance force FL, solution of the state equation in Eq. (29) is obtained as vclt eAcltv0 cl t 0 eAclt?sFLs Wvrs?ds D /vr;FL;t (30) where vcl0 v0 cl is the initial condition. Then, taking the initial condition to be zcl0 z0 cl v 0 cl, the solution of the state equation in Eq. (27) may be written as zclt /vr;FL;t t 0 eAclt?sDclxms;xLsds(31) Thus, the deviation in state variable due to the effect of backlash may be written as kzcl? vclkt t 0 eAclt?sDclxms;xLsds ? ? ? ? ? ? ? ? ? kD t 0 eAclt?sD1ds ? ? ? ? ? ? ? ? D db (32) where D1 01=Mm0 ? 1=ML0?. 4.2BoundonErrorduetoBacklashandBelt Compliance. This section considers the schematic of the trans- mission system as shown in Fig. 10 and presents a bound on the error due to the presence of backlash. Defi ning the state- variables zm1 hm, zm2 xm _ hm, zL1 hL, zL2 xL _ hL, and z z m1 ;z m2 ;z L1 ;z L2 ?, and using Eqs. (14), (15), a state- space representation of the system shown in Fig. 10 is obtained as _ z Apz Bpsm CpsL bDphm;hL y Lpz (33) Fig. 11Block diagram of a controller for system with backlash 031010-6 / Vol. 134, MAY 2012Transactions of the ASME Downloaded From: / on 02/23/2014 Terms of Use: /terms where Ap 0100 ? KbR2 1 Jm ? bm Jm KbR1a1 Jm 0 0001 Kba1R1 JL 0? Kba2 1 JL ? bL JL 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ;Bp 0 1 Jm 0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ; Cp 0 0 0 1 JL 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ;Dpxm;xL 0 - R1/hm;hL Jm 0 a1/hL;hL JL 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ; Lp 0100 0001 “# D Lp1 Lp2 “# (34) and b is zero if the backlash gap is zero and unity otherwise and /(hm, hL ) is defi ned as /hm;hL Kb ?a2DifEq: (16a) holds a2DifEq: (16b) holds a1hL? R1hmifEq: (16c) holds 8 and a2R2/Rg1. If a2is small, the bound dbis also small and so it is advantageous to have R2=Rg1? 1. 5Experiments This section presents the experiments conducted to verify the error bound due to backlash, given in Eq. (32) and also to compare the measured velocities and simulated velocities. The experimen- tal setup, shown in Fig. 12, consists of two masses mounted on carriages which are free to slide. A spring is used to represent the compliance k shown in Fig. 9. That is the system shown in Fig. 9 is realized as masses 1 and 2 connected by a spring so that MmM1and MLM2. Fig. 12Rectilinear system Fig. 13Backlash gap in experiments Fig. 14Closed-loop experiment with backlash of 1.55 mm Fig. 15Closed-loop experiment with backlash of 1.55 mm Journal of Dynamic Systems, Measurement, and ControlMAY 2012, Vol. 134 / 031010-7 Downloaded From: / on 02/23/2014 Terms of Use: /terms The position of each of the masses is measured by a high reso- lution encoder. Nominal values of the masses are M12.28 kg and M22.55 kg, respectively. Nominal stiffness of the spring is k200 N/m. The damping present at masses, as estimated by a preliminary identifi cation procedure, are bmbL0.05 N-s/m. Backlash is introduced into the system shown in Fig. 12 between the spring and mass 1. Figure 13 shows a close-up view of the sys- tem showing the backlash gap. Through a simple screw-and-lock- ing-nut arrangement, the length of the backlash gap can be adjusted. A PI controller was implemented to impart a prescribed veloc- ity to mass 1. Since our objective was to capture the effect of backlash for a given set of gains, no attempt was made to “optimize” the gains in any way. The reference velocity for the load mass was specifi ed as a sinusoid of 0.04 m/s amplitude and 3 cycles/s frequency. There was no disturbance force on mass M2so that FL0. Positions and velocities of the masses 1 and 2 were acquired with and without backlash present in the system. From each set of experiments, the difference between the load velocity (velocity of mass 2) with and without backlash was computed using the experimental data. Figure 14 shows the experimental load velocity for a backlash gap of 1.55 mm. In Fig. 14, two plots are shown: (i) the solid line shows the load velocity with a backlash of 1.55 mm (2D =1.55 mm) present, and (ii) the dotted line shows the load velocity when there is no backlash. The difference between these two plots is computed and is plotted in Fig. 15. In Fig. 15, the continuous red line shows the difference between the two variables as shown in Fig. 14 and the dashed blue line shows the bound computed using Eq. (32). It can be seen that the experimental deviation lies below the bound obtained using Eq. (32). As a further check, the system shown in Fig. 12 is simulated with a backlash gap of 1.55 mm and with zero backlash. Figure 16 shows the simulated values of load speed. Table 1 summarizes the results of experiments conducted with two other backlash gap values, 3.56 mm and 5.38 mm. It can be observed that the experimental error amplitudes are less than the theoretical error amplitudes computed using Eq. (32). 6Summary and Future Work Dynamic models for describing the behavior of a mechanical transmission system that contains backlash and/or compliance are developed. Analysis of the model for compliance reveals that it is a good idea to use feedback from the driver side of the transmission sy