外文翻译--并联机床的动态前馈控制 英文版【优秀】

dynamicofoncompsubsystemprecisiona result,1. Introductiondynamics,interestingbeenoffew.of thekinematicyieldare sufficientlyis computationally intensive. But, it is supposed to provide goodtracking accuracy. By augmenting the nonlinear model-based con-troller with feedback laws that include models of the PKM flexibil-ities and disturbances, higher robustness against these undesirableeffects can be achieved. Model-based control can be used for thedesign of both feedback and feed-forward control laws. However,the research on the model-based control of PKMs is insufficient,and only a few researches can get ideal results. Although the mod-trol subsystem can be regarded as independent systems such thatthey can be designed independently. Therefore, dynamic feed-forward control is widely applied to the motion control of PKMs912.Many control methods have been used to design the dynamiccontrol subsystem. By computing the control torque from the ri-gid-body dynamic model of a PKM, the dynamic control subsystemimplements feed-forward compensation for the dynamic charac-teristic of the PKM. Some researchers 1315 designed thedynamic control subsystem by using computed torque control or* Corresponding author. Tel.: +86 10 62772633； fax: +86 10 62782351.Mechatronics 19 (2009) 313324Contents lists availableE-mail address: (J. Wu).In order to maximize the performance of PKMs in high-speedmotion, model-based control 5,6 is essential and effective. Thedynamic characteristics of PKMs have a great effect on the motionprecision, especially in high speed. It is well accepted 7,8 that thecontrol system based on the dynamic model is necessary fordecreasing the influence of the dynamic characteristics on the mo-tion precision. Model-based control is realized by integrating thenonlinear dynamics into the control design. This control approachthe motion of the chain servo-system. Based on the rigid-body dy-namic model of the PKM, the dynamic control subsystem compen-sates for the dynamic characteristics of the PKM. Although thedynamic feed-forward control increases the number of indepen-dent control systems, it has some advantages: (1) anti-disturbancecapability of the closed-loop control subsystem can be used to de-crease the influence of inaccurate dynamic model on the motionprecision； (2) the dynamic control subsystem and closed-loop con-Due to their precision, stiffness andlators are becoming more and morechine tools and robots. There hasresearches on kinematics and dynamics13, but studies on control are relativelylinear and coupled dynamic behaviortrollers do not provide satisfactory results.et al. 4 pointed out that for the parallelaglide simple linear joint controllersincreasing with speed, though theyvery low velocities.0957-4158/$ - see front matter C211 2008 Elsevier Ltd. Alldoi:10.1016/j.mechatronics.2008.11.004parallel manipu-in the field of ma-a great amount ofparallel manipulatorsFor the highly non-PKM, pure linear con-For example, Honeggermachine Hex-tracking errors rapidlyaccurate withel-based control provides a possibility for PKMs to obtain a bettermotion precision, there is still much work on the model-based con-trol of PKMs to be done to transfer the possibility to practicality.Model-based control is generally realized by using the dynamicfeed-forward control. In dynamic feed-forward control, each chainof a PKM is regarded as an independent control object, and the con-trol system for each chain is designed based on the dynamic modelof the chain. Moreover, the feed-forward compensator is added tothe control system of each chain to compensate for the load torqueof each chain. The dynamic feed-forward control system of eachchain consists of the closed-loop control subsystem and dynamiccontrol subsystem. The closed-loop control subsystem controlsDynamic feed-forward control of a parallelJinsong Wang, Jun Wu*, Liping Wang, Zheng YouDepartment of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084,article infoArticle history:Received 29 November 2007Accepted 4 November 2008Keywords:Dynamic feed-forward controlParallel kinematic machineClosed-loop control subsystemDynamic control subsystemabstractThis paper deals with theThe control system consistsThe closed-loop control subsystemzero phase error tracking controlits response capability. Basedtem is designed by using thethe closed-loop control subsystemof the closed-loop controlof the PKM on the motioninto the control system. AsMechatronjournal homepage: www.elsevirights reserved.kinematic machinePR Chinafeed-forward control of a 6-UPS parallel kinematic machine (PKM).the closed-loop control subsystem and the dynamic control subsystem.is constructed by using the PD control and low-pass filter. Moreover,(ZPETC) is introduced to the closed-loop control subsystem to improvethe rigid-body dynamic model of the PKM, the dynamic control subsys-uted torque control and feed-forward control. Since the phase lag ofis almost eliminated by ZPETC and PD control, the motion precisionis improved largely. The influence of the dynamic characteristicis decreased due to the introduction of rigid-body dynamic modelthe motion precision of the moving platform is greatly improved.C211 2008 Elsevier Ltd. All rights reserved.at ScienceD/locate/mechatronicsof a PKM. In order to improve the accuracy of dynamic models,computed torque controller can be combined with the learningalgorithm. Therefore, the dynamic control subsystem has thelearning capability to modify the dynamic model in real time. Gengand Haynes 16 introduced Cerebella model arithmetic computer(CMAC) to the dynamic control subsystem. But the learning algo-rithm is so complex that the real-time performance of the controlsystem is affected. In addition, some researchers combined theadaptive control approach with the computed torque control. Thedynamic model is modified on line by identifying the dynamicparameters. However, the dynamic parameters are so many thatthe real-time performance of control system is debased and thecontrol system becomes more complex. To overcome these prob-lems, Honegger et al. 17 studied the dynamic parameter identifi-cation of a PKM first. Based on the minimization of the trackingerror, a non-linear adaptive control algorithm was used to identifythe dynamic parameters. Then, they designed the dynamic controlsubsystem with the friction coefficient identified in on-line form.The closed-loop control subsystem controls the motion of thechain servo-system. At the same time, it suppresses the influenceof the dynamic characteristic that the dynamic control subsystemcannot compensate (can be regarded as the dynamic disturbanceon the closed-loop control subsystem). Therefore, the closed-loopcontrol subsystem should have not only steady, fast and preciseperformance, but also higher anti-disturbance capability. Sincethe proportional-integral-differential (PID) control has theseadvantages, it is usually employed to design the closed-loop con-trol subsystem. Furthermore, fuzzy control 19, sliding mode var-iable structure control and model reference adaptive control arealso used to design the closed-loop control subsystem.Although many different control approaches have been used,the motion precision in high-speed motion is not ideal. Only afew researches can get satisfactory results. The reason for thesesuccesses is that a more accurate dynamic model and linear motorwith high response capability are introduced to the control system.It means that an accurate dynamic model and the control systemwith high response capability are helpful for high-precision motionof a PKM.In this paper, based on the dynamic model of a 6-UPS Stewartplatform-based parallel kinematic machine (XNZ63 PKM) whichwas simplified for real-time control in literature 20, the dynamicfeed-forward control of XNZ63 PKM is investigated, the closed-loop control subsystem and dynamic control subsystem aredesigned, respectively. The closed-loop control subsystem is con-structed by using the ZPETC 21, PD control and low-pass filter.The dynamic control subsystem is designed by using the computedtorque control and feed-forward control. The phase lag of theclosed-loop control subsystem is almost eliminated by ZPETC,and the motion precision of XNZ63 PKM is improved largely. Theinfluence of the dynamic characteristic of XNZ63 PKM on themotion precision is decreased due to the introduction of moreaccurate rigid-body dynamic model into the control system.2. Simplified dynamic model for real-time controlIt is well accepted that the control system based on the dynamicmodel should be developed for PKMs in a high-speed environment.The accuracy of dynamic model affects the control precision. How-combining computed torque control with other control approaches1618. Computed torque control is a method that designs thecontrollerstrictly based on the mathematical model. Thus, an accu-rate dynamic model is indispensable for computed torque control.However, it is difficult to obtain the accurate mathematical model314 J. Wang et al./Mechatronicsever, it is very difficult to derive the accurate dynamic model of aPKM. The reason is (1) in modeling, some factors are neglected ordifficult to be considered such as the joint clearance； (2) it is diffi-cult to obtain the accurate dynamic parameter even if the technol-ogy of dynamic parameter identification is developed.In addition, the dynamics of a PKM is non-linear and coupled. Ittakes a long time to solve the inverse dynamics, and cannot meetthe real-time requirement of the control system. To apply the dy-namic model to the control of PKM, the computational efficiency ofinverse dynamics should be improved. Therefore, the dynamicmodel is simplified to meet the real-time application. From Refs.4,13,14, one may see that an ideal motion precision can be ob-tained even if a more simplified dynamic model is used. In this pa-per, the more accurate dynamic model is used in the controlsystem. It is expected to obtain a good motion precision.In literature 20, the simplified methods of the dynamic modelof XNZ63 PKM were proposed for real-time control in constantvelocity movement and accelerating/decelerating movement ofthe moving platform, respectively. In this paper, we cite directlythe simplified results. In the dynamic model, A1Gp, A2FpI1, A2FpI2,A2FpI3, A3MpI1and A3MpI2are the gravity, following inertia force,tangential inertia force, norm inertia force, tangential inertia mo-ment, and norm inertia moment terms for the moving platform,respectively；A4Gu,A5FuI1,A5FuI2,A6MuI1andA6MuI2are the gravity,the tangential inertia force, norm inertia force, tangential inertiamoment, and norm inertia moment term for the upper parts oflegs； A7Gd, A8FdI1, A8FdI2, A8FdI3, A8FdI4, A9MdI1and A9MdI2are thegravity, sliding inertia force, tangential inertia force, norm inertiaforce, Coriolis inertia force, tangential inertia moment and norminertia moment terms for the lower parts of legs.The dynamic model of XNZ63 PKM without external load andjoint friction was decomposed into 18 termsF A1GpA2FpI1A2FpI2A2FpI3A3MpI1A3MpI2A4GuA5FuI1A5FuI2A6MuI1A6MuI2A7GdA8FdI1A8FdI2A8FdI3A8FdI4A9MdI1A9MdI21where A1, A2, A3are 6 C2 3 matrices, A4, A5,.,A9are 6 C2 18 matri-ces, and A1, A2,.,A9are related to the position and orientation ofthe moving platform.The moving type of the moving platform with a constant veloc-ity can be classified into low velocity case, middle velocity case,and high velocity case. The movement of the moving platform withan acceleration or deceleration is divided into low acceleration ordeceleration case, middle acceleration or deceleration case, andhigh acceleration or deceleration case. The criterions for distin-guishing these moving cases are given in literature 20. Whenthe moving platform moves with a constant velocity, the dynamicmodel for low velocity case can be simplified asF A1GpA4GuA7Gd2In middle velocity case, the dynamic model is simplified asF A1GpA2FpI3A3MpI2A4GuA6MuI1A7GdA8FdI23orF A1GpA2FpI3A3MpI2A4GuA6MuI1A7GdA8FdI44In high velocity case, the simplified dynamic model can be ex-pressed asF A1GpA2FpI3A3MpI2A4GuA6MuI1A7GdA8FdI1A8FdI2A8FdI45When the moving platform moves with an acceleration or deceler-ation, the dynamic model for low acceleration or deceleration caseis simplified as19 (2009) 313324F A1GpA2FpI1A2FpI2A3MpI1A4GuA6MuI1A7Fd6In middle acceleration or deceleration case, the simplified dynamicmodel is given byF A1GpA2FpI1A2MpI2A3GpI1A3MpI2A4GuA6MuI1A7GdA8FdI2A8FdI47orF A1GpA2FpI1A2MpI2A2GpI3A3MpI1A4GuA6MuI1A7GdA8FdI2A9FdI18In high acceleration or deceleration case, the dynamic model is sim-plified asF A1GpA2FpI1A2MpI2A3GpI1A3MpI2A4GuA6MuI1A7GdA8FdI1A8FdI2A8FdI4A9MdI193. Control schemeeach chain. Thus, the feed-forward control system of the singlechain servo-system should be designed for XNZ63 PKM. Each chaintor and screw, P-LC, V-LC and C-LC denote position-loop controller,velocity-loop controller and current-loop controller, respectively.3.2. Procedure for designing the control systemBased on the original control system of XNZ63 PKM, the follow-ing procedure can be proposed for designing the dynamic feed-for-ward control system.(1) Based on the identification of control object and analysis ofthe original control system of XNZ63 PKM, the position-loopcontroller is designed to improve the response capability ofthe closed-loop control subsystem.(2) Based on the identified results of the closed-loop controlsubsystem, zero phase error tracking controller is designedto improve the response capability of the closed-loop controlsubsystem.(3) Based on the dynamic model in Section 2, the computed tor-the screw is negligible. Thus, the model of the control object canbe expressed asV-LC GsK1K2KtKipKvpTiiTvis2Tii Tvis 1C1385 4 3 2tKip10J. Wang et al./Mechatronics 19 (2009) 313324 315Closed-loop control subsystem Control object Dynamic control subsystem Inverse kinematics Computed torque controllerFeed-forward controller P-LC ZPETC + +servo-system of XNZ63 PKM consists of servo-driver, motor andscrew. The servo-driver can provide position-loop controller,velocity-loop controller, and current-loop controller. In order toenable the closed-loop control subsystem to have a great capabilityof anti-disturbance, the velocity-loop and current-loop keep un-changed in designing the dynamic feed-forward control system.In this paper, the velocity-loop, current-loop, motor and screware regarded as the control object. The scheme for the dynamicfeed-forward control is proposed in Fig. 1, where M-S denotes mo-JLTiiTvis JTiiTviR Kips Cs3s KtKipKvpTii Tvis KThe original control system of XNZ63 PKM is designed based onthe kinematic model. Taking the high-speed and high-precisionapplication of the PKM into account, a dynamic feed-forward con-trol system is designed for XNZ63 PKM in this paper.3.1. Determination of the scheme for dynamic feed-forward controlThe dynamic feed-forward control realizes the accurate motioncontrol of the PKM by controlling accurately the servo-system ofFig. 1. Scheme for dynamicwhere Cs3= JKipTvi+ KeKtTiiTvi+ KtKipKvpTiiTvi, and other parametersare given in Table 1.Since the theoretical model in Eq. (10) does not consider thetime delay and it is difficult to determine all parameters of the con-trol object, especially the parameters of the time delay, a frequencyresponse test is done to identify the model of the control object.The test principle is shown in Fig. 2. When the input of the controlobject is a sinusoidal signal, the steady-state output is also a sinu-soidal signal with the same frequency. Then, the Bode diagram canPosition output M-S +C-LC + Kvpsque controller is designed.(4) Selecting the current-loop as the control object, the feed-for-ward controller is designed.3.3. Identification of the model for control objectTo design a control system, the model of the control objectshould be determined first 22. It is assumed that the velocity-loop and current-loop are unit feedback loops and the friction infeed-forward control.is reasonable.Fig. 3b is the phase difference between the real phase obtainedby test and theoretical phase obtained by theoretical model, andthe phase difference changes linearly. It can be concluded that aredesigned to improve the response capability. Based on the PIDcontrol strategy, proportional controller, PI controller, or propor-tional-integral-differential (PID) controller can be designed forthe position-loop. In Section 4.1.1, it has been investigated that aTable 1Parameters of the control object.Parameter Definition Value UnitK1Conversion coefficient from linear velocity to angular velocity 50p/3 rad/(V s)K2Conversion coefficient from angular position to linear position 3/p mm/rads2time delay exists in the control object, and the time delay can beexpressed asGdseC0Tds11where Td= 0.003 s is the time coefficient of the time delay.Thus, the time delay should be added to the theoretical modelin Eq. (10). The modified model can be expressed asThe real phase frequency characteristic (PFC) obtained by test andthe theoretical PFC obtained by Eq. (12) are shown in Fig. 4. It canbe seen that PFC obtained by Eq. (12) is similar to real PFC.4. Design of closed-loop control subsystem4.1. Position-loop controller4.1.1. Analysis of original control systemGsGdsK1K2KtKipKvpTiiTvis2Tii Tvis 1C138JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tvibe obtained by recording the amplitudes and phases of sinusoidalinput and output in different frequencies.The magnitude-frequency characteristic, which is obtained bythe theoretical model, is compared with that obtained by the fre-quency response test. The magnitude-frequency characteristicsare shown in Fig. 3a. In Fig. 3a, AFC denotes magnitude-frequencycharacteristic. From Fig. 3a, it can be seen that the theoretical mag-nitude-frequency characteristic is similar to that obtained by thetest. It shows that the assumption for modeling the control objectKipProportional coefficient of current-loopTiiIntegral coefficient of current-loopKvpProportional coefficient of velocity-loopTviIntegral coefficient of velocity-loopJ inertia of motor rotor and screwL Winding inductanceR Winding resistanceKtMagnetic torque coefficientKeBack electromotive force coefficient of motor316 J. Wang et al./MechatronicsThe servo drivers of the original control system provide propor-tional control for position loop, proportional-integral (PI) controlfor velocity-loop, and PI control for current-loop. The original con-trol system of XNZ63 PKM was created by utilizing the self-regula-tion of the servo drivers. After self-regulation, the proportionalcoefficient Kppof position loop is 1.333 V/mm, and the values ofparameters in velocity-loop and current-loop are given in Table1. Based on Eq. (12), the open-loop transfer function of the originalcontrol system can be expressed asBased on Eq. (13), the closed-loop Bode diagram of the original sys-tem is obtained as shown in Fig. 5. It can be seen that the closed-loop cut-off frequency is about 15Hz, and the response capabilityTsKppGdsK1K2KtKipKvpTiiTvis2Tii Tvis 1C138JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tviproportional controller has some defects for the position loop.Comparing the PD control with PI control and PID control, the PDcontrol has the best response capability. Moreover, since the differ-ential control of PD controller improves the damp of the system,the control system can have a bigger open-loop gain to improvethe steady-state accuracy. Thus, in this section, PD control isadopted to design the position-loop controller. However, PD con-trol amplifies the high-frequency signal, which not only amplifiesthe high-frequency noise signal, but also can excite the vibrationof the mechanical system. Therefore, a low-pass filter in series withthe PD controller is employed to suppress the effect of PD control-ler on the high-frequency signal.Therefore, the position-loop controller, which consists of PDcontrol and low-pass filter, can be expressed asKpp Kpds T1s 1KtKipKvps12of the closed-loop control subsystem is lower. Although the pro-portional coefficient of position loop can be augmented to improvethe response capability, the stability of the control system will beaffected.4.1.2. Design of position-loop controllerIn order to realize high-speed and high-precision motion of thePKM, the closed-loop control subsystem should have a better re-sponse capability. Thus, the position-loop controller should be30 V/A0.002 s0.9 A s/rad0.005 s0.001149 Kg m20.023 H4.55 X1.38 N m/A1.198437 V s/rad19 (2009) 313324GpcsT2s 1 KcT2s 114Kc Kpp15T1KpdKpp16where Kppis the proportional coefficient, Kpdis the differential coef-ficient, T2is the time coefficient of the low-pass filter, and Kcis thegain of position loop.From Eq. (14), it can be seen that the position-loop controller issimilar to the phase advance controller. Thus, the frequency re-sponse method can be used to determine the parameters of the po-s2 KtKipKvps13sition-loop controller. First, the gain of position-loop controller isdetermined by the open-loop gain of the control object. Consider-ing that augment the open-loop gain one time, the gain of position-tion of the closed-loop control subsystem can be expressed asT1 0:008 sKpd 0:016 V s=mm19matic control command to position output, T2(s) from the dynamiccontrol command to position output, and T3(s) from the externaldisturbance to position output can be expressed assinusoidal positionfeedback signal Control object Industrial computer PMAC card Servo driver (velocity loop, current loop) screw motorsinusoidal velocity signal Fig. 2. Test principle for identifying the control object.TsGpcsGdsK1K2KtKipKvpTiiTvis2Tii Tvis 1C138JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tvis2 KtKipKvps18J. Wang et al./Mechatronics 19 (2009) 313324 317Based on Eq. (18), the unit step response of the closed-loop controlsubsystem is obtained as shown in Fig. 6. Although there is no over-shoot in the unit step response of the closed-loop control subsys-tem, the oscillation occurs due to the differential control. Due tothe oscillation, the control effect is affected. However, if the differ-ential control is adjusted, the stability, response capability and sta-ble accuracy are affected. From Eq. (16), it can be seen that T1onlyrelates with Kpdwhen Kppchanges in a range. Thus, the effect of dif-ferential control on the performance of closed-loop control subsys-tem can be discussed by changing the value of T1. From Fig. 6, onemay see that the response capability is improved when the value ofloop controller is 2 V/mm. Second, under the condition that thegain of position-loop controller is 2 V/mm and the phase marginis 75C176, the design results can be obtained by using the design pro-cedure of standard phase advancecontroller to design the position-loop controller.Kc Kpp 2V=mmKpd 0:019862 V C1 s=mmT1 0:009931 sT2 0:006992 s8:17According to Eqs. (12), (14), and (17), the open-loop transfer func-Fig. 3. AFC and phase4.1.3. Stability analysis of position-loop controllerInSection4.1.2,theclosed-loopcontrolsubsystemisredesigned.The stability of the new closed-loop control subsystem should beinvestigated. The block diagram of the closed-loop control subsys-tem is shown in Fig. 9. The system has three inputs: kinematic con-trolcommand,dynamiccontrolcommandandexternaldisturbance.The stability should be investigated based on the three inputs.According to Fig. 9, the transfer function T1(s) from the kine-T1increases. But the oscillation is enhanced and the adjustable timeis prolonged.Figs. 7 and 8 give the open-loop Bode diagram and closed-loopBode diagram of the closed-loop control subsystem, respectively. Itcan be seen that the stability, response capability and stable accu-racy is improved when the value of T1increases. Considering theresults in Figs. 68, the coefficient for differential control of the po-sition loop controller is determined byC26Fig. 4. Real PFC and modified PFC in theory.difference.where Y(s) is the output function of one chain position； I1(s) is theinput function for kinematic control command； I2(s) is the inputfunction for dynamic control command； Td(s) is the input functionfor external disturbance,T1sYsI1sG1sG2sG3sG4sG5sG6sG7sG8sG9sG10s1 G5sG6sG6sG7sG8sG11sG4sG5sG6sG7sG8sGts20T2sYsI2sG5sG6sG7sG8sG9sG10s1 G5sG6sG6sG7sG8sG11sG4sG5sG6sG7sG8sGts21T3sYsC0TdsG8sG9sG10s1 G5sG6sG6sG7sG8sG11sG4sG5sG6sG7sG8sGts22Fig. 6. Unit step response of the closed-loop control subsystem.Fig. 5. Magnitude-frequency characteristic of original system.Fig. 7. Open-loop Bode diagram of the318 J. Wang et al./Mechatronics 19 (2009) 313324G1sGpcsKcT1s 1T2s 1；G2sK1；G3sGdseC0Tds；G4sKvpTvis KvpTvis；G5sKipTiis KipTiis；G6s1Ls R；G7sKt；G8s1Js；G9s1s；G10sK2；GtsG1sG2sG3sG4sG5sG6sG7sG8sG9sG10s；G11sKe:One may see that T1(s), T2(s) and T3(s) have the same closed-loop poles. By taking two-order Poisson approximation of Gd(s),the closed-loop poles of T1(s), T2(s) and T3(s) can be expressed ass1C01264:899166s2C0671:811884s3C0181:194983s4C070:424128s5C0540:566023 1263:137993is6C0540:566023 C0 1263:137993is7C0187:866151 309:925275is8C0187:866151 309:925275i8:23closed-loop control subsystem.Thus, for the input of kinematic control command, dynamic controlcommand and external disturbance, the closed-loop control subsys-tem is stable.4.1.4. Response and tracking error of the closed-loop control subsystemThe main aim of redesigning the closed-loop control subsystemis to improve the response performance such that the tracking er-ror of the chain is reduced. In this section, the response perfor-mance and tracking error are inspected. Fig. 10 shows themagnitude-frequency characteristic and 0.5 mm step responseFig. 8. Closed-loop Bode diagram of theJ. Wang et al./Mechatronics 19 (2009) 313324 319curve. It can be seen that the closed-loop cut-off frequency is im-proved to 40 Hz from 15 Hz. Thus, the response capability of thecontrol system is improved greatly.Fig. 11 gives the tracking error of chain 1 with the velocity12 m/min. For convenience, in this paper, the tracking error ofvelocity command dynamic control command external disturbance kinematic control command position output P-LC Control object Fig. 10. AFC and step responseFig. 9. Block diagram of closed-loop control subsystem.chain 1 is given for an example. It can be seen that the tracking er-ror of chain 1 of the new closed-loop control subsystem is smallerthan that of the original system. The reason is that the responsecapability is improved by using PD control and low-pass filter.However, the tracking error is still a litter bigger in high speed. Itis necessary to improve the response capability of the closed-loopcontrol subsystem more for higher accuracy. Thus, the zero phaseerror tracking control is designed in Section . Zero phase error tracking control4.2.1. Design of controllerIn the control system, the closed-loop control subsystem is be-tween the kinematic command and position output of the chain.Due to the limitation of response performance and time delay inthe closed-loop control subsystem, the position output of the chainlags behind the input. In this paper, zero phase error tracking con-trol is introduced to compensate for the lag.Under the condition that the control period is 0.002 s, by takingtwo-order Poisson approximation and bilinear transform of Eq.(20), the pulse transfer function of the closed-loop control subsys-tem is obtained asGozBozAoz24closed-loop control subsystem.where Ao(z) and Bo(z) are the denominator polynomial and numer-ator polynomial of the pulse transfer function, respectively, andwith amplitude 0.5 mm.After the introduction of ZPETC, the closed-loop control subsystemerror320 J. Wang et al./Mechatronics 19 (2009) 313324Boz0:002821z80:004077z70:035975z60:038158z5C00:058446z4C00:054939z30:034457z20:019032zC00:008479；Aozz8C02:776104z73:315748z6C02:564700z51:666464z4C00:853194z30:249685z2C00:025361z0:000117The stable zero point of the pulse transfer function is given byz1 0:800687z2 0:699091z3 0:3862428:25The unstable zero point and the zero point that nears the unit circleare determined byz4C00:999996z5C00:165696 3:725175iz6C00:165696 C0 3:725175iz7C01:000002 0:000003iz8C01:000002 C0 0:000003i8:26Based on Eq. (25) and (26), Bo(z) can be decomposed asBozBaozBuoz27whereFig. 11. TrackingBaozC00:000610 0:003213z C0 0:005320z2 0:002821z3；Buoz13:904384 42:044544z 43:707328z2 17:898560z3 3:331392z4 z5:Based on Eqs. (24) and (27), the zero phase error tracking controllercan be expressed asGczBczAcz28kinematic control command P-LC ZPETC Fig. 12. Closed-loop controlis given in Fig. 12. According to Eqs. (24) and (28), the pulse transferfunction of the closed-loop control system is given byTzGczGoz294.2.2. Performance analysis of ZPETCThe zero phase error tracking control is introduced to reducethe phase lag of the closed-loop control subsystem. In this section,the response performance and tracking error of the closed-loopcontrol subsystem with ZPETC are inspected, respectively.Fig. 13 shows the Bode diagrams of the closed-loop controlsubsystem without ZPETC and the designed closed-loop controlsubsystem with ZPETC. It can be seen that ZPETC not only makethe phase of the closed-loop control subsystem keep zero in therange of frequency, but also improves greatly the closed-loopwhereBcz0:000117zC05C0 0:024971zC04 0:167292zC03C0 0:470207zC02 2:189600zC01C0 2:435600 C0 2:546534z 2:801768z2C0 2:794778z3 13:905127z4 0:311011z5C0 26:909265z6 3:444528z7 13:904384z8；AczC09:060835 47:736163z C0 79:042108z2 41:909475z3:of chain 1.cut-off frequency. When the screw moves linearly with thevelocity 12 m/min, the tracking performance of the closed-loopcontrol subsystem is simulated and tested, respectively. Boththe tracking errors of the closed-loop control subsystems withZPETC and without ZPETC by simulation are shown in Fig. 14.It can be seen that the tracking error of the chain is improveby three times due to the introduction of ZPETC. The reason isthat the response capability of the closed-loop control subsystemis improved.velocity command dynamic control command external disturbance position output Control object subsystem with ZPETC.closed-loopJ. Wang et al./Mechatronics 19 (2009) 313324 321Fig. 13. Bode diagram of the5. Design of dynamic control subsystem5.1. Computed torque controllerThe dynamic model of a PKM can be derived either by the jointspace in terms of the pose of the legs, or by the task space in termsof the pose of the moving platform. Accordingly, the computed tor-que controller can be designed based on the dynamic model injoint space or task space. Dasgupata and Mruthyunjaya 23, Kanget al. 24, Ting et al. 25 pointed out that the dynamic model injoint space is more complex and make the computation of the dy-namic model more expensive. Kang 23 compared the controllerdesigned in task space with that in joint space. Both controllershave similar control effect, but the controller based on the taskspace has a better real-time performance. Thus, in this section,Fig. 14. Tracking error of the closed-loop control subsystem.Fig. 15. Principle of dual channelthe computed torque controller is designed based on the dynamicmodel in task space.In order to design a computed torque controller, the dynamicmodel of the PKM is usually simplified greatly to satisfy the real-time demand of the control system. However, the control effectof the dynamic control subsystem is debased such that the motionprecision of the PKM is reduced. For this problem, the solution wasgiven in literature 20. Based on the simplified results in Section 2,the computed torque controller based on the task workspace canbe expressed asM Fp2p30where M M1M2M3M4M5M6C138Tis the output of torquecommand of the computed torque controller, and p is the pitch oflead screw.5.2. Dynamic feed-forward control subsystemThe output command of the computed torque controller can notinput directly to the current-loop. A feed-forward controller isneeded to transform the command. Based on the principle of dualchannel compensation, if the torque produced by the torque com-mand can compensate approximately for the load torque of eachcontrol subsystem.chain at the input point of load torque, the effect of the dynamiccharacteristic of PKM on the output of control object is reducedgreatly (see Fig. 15).In order to compensate the load torque, the following equationcan be obtainedGfs1Gos31compensation.where Gf(s) is the transfer function of the feed-forward controller,and Go(s) is the transfer function from the input of dynamic controlcommand to the output of control torque.In general, the response velocity of the current-loop is so fastthat the time delay in the current-loop can be neglected. Thus,Go(s) can be expressed asGosJKtKipTiis 1JLTiis2 JTiiR Kips JKip KeKtTii32Since there is no unstable zero point in Go(s), the dynamic feed-for-ward controller can be designed asGfsJLTiis2 JTiiR Kips JKip KeKtTiiJKtKipTiis 1335.3. Performance analysis of dynamic control subsystemthe dynamic characteristic and response performance of controlFig. 17. Tracking error caused by load torque in middle velocity case.322 J. Wang et al./Mechatronics 19 (2009) 313324In Sections 5.1 and 5.2, the dynamic control subsystem is de-signed. In this section, the control effect of the dynamic controlsubsystem is investigated. The motion that the moving platformmoves from the point (C00.150, C00.150, 0.400 m, C027.056C176,7.461C176, 57.484C176) to another point (0.150, 0.150, 0.520 m, 26.938C176,C07.907C176, 58.367C176) is still considered.In different moving types of the PKM, the load torque of onechain in motion is computed by the original model, simplifyingmodel and previous model, respectively. Original model denotesthe dynamic model given in Eq. (1), simplified model is the simpli-fied dynamic model based on the method presented in literature20, and previous model is the dynamic model with the gravita-tional forces and inertial forces of legs neglected. The load torqueof chain 1 is shown in Fig. 16. It can be seen that the load torquecomputed by the simplified dynamic model approximately equalsto that computed by the complete model. The load torque acts onchain 1 changes greatly, and the load torque increases with thevelocity and acceleration of moving platform.The tracking errors caused by the load torque can be obtained asshown in Fig. 17 by simulating the closed-loop control subsystemof chain 1 without the dynamic feed-forward compensation, andthe input is the load torque which is obtained by the complete dy-namic model of XNZ63 PKM. From Fig. 17, it can be seen that theeffect of load torque on the chain motion can be eliminated bythe velocity-loop when the load torque changes smoothly. How-ever, when the load torque changes abruptly, the PI control ofvelocity-loop cannot eliminate the effect of load torque. Thus, thefeed-forward control is needed to compensate for the load torque.Fig. 16. Load torque of chain 1: (a) lowBy using the dynamic control subsystem designed in Sections5.1 and 5.2 to compensate for the load torque that acts on chain1, the tracking error can be obtained as shown in Fig. 18a. By usingthe previous dynamic model without considering the mass of chainto compensate for the load torque, the tracking error is shown inFig. 18b. From Fig. 18a and b, one may see that the feed-forwardcontrol based on the simplified dynamic model proposed in litera-ture 20 almost eliminates the effect of the dynamic characteristicon the chain motion since the simplified dynamic model can de-scribe the load torque more accurately. Moreover, it can be con-cluded that the simplified strategy proposed in literature 20 iseffective for simplifying the dynamic model of XNZ63 PKM.6. Performance analysis of dynamic feed-forward controlsystemIn essence, the control system based on the dynamic feed-for-ward control method separates the control of XNZ63 PKM intoindependent control of each chain. A higher motion precision ofthe moving platform is expected by controlling each chain accu-rately. Therefore, not only is the motion precision of the chaininvestigated, but also the motion precision of the moving platformshould be investigated. The motion precision of the chain has beenverified above, in this section, the precision of the moving platformis investigated.According to the results in Sections 4.1 and 4.2, it can be con-cluded that the tracking error of the chain, which is caused byvelocity case； (b) high velocity case.J. Wang et al./Mechatronicssystem, increases with the velocity and acceleration of the movingplatform. Thus, the motion of the moving platform in high speed issimulated to investigate the tracking error of the moving platform.The motion that the moving platform of XNZ63 PKM moves fromthe point with the coordinate (C00.150, C00.150, 0.400 m,C027.056C176, 7.461C176, 57.484C176) to another point with the coordinate(0.150, 0.150, 0.520 m, 26.938C176, C07.907C176, 58.367C176) is stillconsidered.In high speed, the original control system and the dynamicfeed-forward control system designed in this paper are simulated,Fig. 18. Tracking error caused by loadFig. 19. Tracking error of moving platformFig. 20. Tracking error of the moving platform in X direction.torque in middle velocity case.19 (2009) 313324 323respectively. The tracking error caused by the response perfor-mance of the control system and the tracking error caused by thedynamic characteristic are shown in Figs. 19a and b, respectively.The total tracking error is shown in Fig. 20. Fig. 19a shows thatthe effect of the dynamic characteristic on the motion precisionof the platform movement is very small since the load torque hasa smaller influence on each chain motion for the introduction ofmore accurate dynamic model. From Fig. 19b, it can be seen thatthe tracking error caused by response performance of control sys-tem is reduced since the tracking error of each chain is reduced forthe enhanced response performance of the closed-loop controlsubsystem of each chain. From Fig. 20, it can be seen that the track-ing error of the moving platform is reduced about four times.Therefore, the dynamic feed-forward control is effective forXNZ63 PKM.7. ConclusionsThe dynamic feed-forward control of a 6-UPS parallel kinematicmachine has been investigated in this article. From this investiga-tion, the following conclusions can be drawn:(1) Toimprovetheresponseperformanceoftheclosed-loopcon-trol subsystem, the position controller is designed by usingPD control and low-pass filter. Both the simulation and testabout the performance of the closed-loop control subsystemshow that the motion precision is improved largely.caused by load torque and response.(2) The phase lag of the closed-loop control system is almosteliminated by the ZPETC, and the response performance isimproved. As a result, the motion precision of the closed-loop control subsystem is improved by three times.(3) The control performance of the dynamic control subsystemis simulated. The simulation shows that the dynamic controlsubsystem can compensate for the load torque, and theeffect of the dynamic characteristic of the PKM on themotion precision is almost eliminated due to the introduc-tion of more accurate dynamic model to the control system.(4) The tracking errors of the XNZ63 PKM are reduced signifi-cantly since the response performance of the closed-loopcontrol subsystem is improved and more accurate dynamicfeed-forward compensation is realized.AcknowledgementsThis work is supported by the National Nature Science Founda-tion of China (Grant No. 50775117), and the 973” key fundamen-tal programs of China (Grant No. 2006CB705400).9 Grotjahn M, Heimann B. Model-based feed-forward control in industrialrobotics. Int J Robotic Res 2002；21(1):99114