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1、Dynamic feed-forward control of a parallel kinematic machineJinsong Wang, Jun Wu*, Liping Wang, Zheng YouDepartment of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, PR Chinaa r t i c l ei n f oArticle history:Received 29 November 2007Accepted 4 November 2008Keywords:Dyn
2、amic feed-forward controlParallel kinematic machineClosed-loop control subsystemDynamic control subsystema b s t r a c tThis paper deals with the dynamic feed-forward control of a 6-UPS parallel kinematic machine (PKM).The control system consists of the closed-loop control subsystem and the dynamic
3、control subsystem.The closed-loop control subsystem is constructed by using the PD control and low-pass filter. Moreover,zero phase error tracking control (ZPETC) is introduced to the closed-loop control subsystem to improveits response capability. Based on the rigid-body dynamic model of the PKM, t
4、he dynamic control subsys-tem is designed by using the computed torque control and feed-forward control. Since the phase lag ofthe closed-loop control subsystem is almost eliminated by ZPETC and PD control, the motion precisionof the closed-loop control subsystem is improved largely. The influence o
5、f the dynamic characteristicof the PKM on the motion precision is decreased due to the introduction of rigid-body dynamic modelinto the control system. As a result, the motion precision of the moving platform is greatly improved.? 2008 Elsevier Ltd. All rights reserved.1. IntroductionDue to their pr
6、ecision, stiffness and dynamics, parallel manipu-lators are becoming more and more interesting in the field of ma-chine tools and robots. There has been a great amount ofresearches on kinematics and dynamics of parallel manipulators13, but studies on control are relatively few. For the highly non-li
7、near and coupled dynamic behavior of the PKM, pure linear con-trollers do not provide satisfactory results. For example, Honeggeret al. 4 pointed out that for the parallel kinematic machine Hex-aglide simple linear joint controllers yield tracking errors rapidlyincreasing with speed, though they are
8、 sufficiently accurate withvery low velocities.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 t
9、hecontrol 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 approachis computationally intensive. But, it is su
10、pposed 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 fe
11、edback 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-el-based control provides a possibility for PKMs to obtain a bettermotion precision, there is still much work on the model-b
12、ased 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
13、 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 s
14、ubsystem. The closed-loop control subsystem controlsthe 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
15、-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-trol subsystem can be regarded as indepe
16、ndent 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 PK
17、M, 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 or0957-4158/$ - see front matter ? 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechat
18、ronics.2008.11.004* Corresponding author. Tel.: +86 10 62772633; fax: +86 10 62782351.E-mail address: wu- (J. Wu).Mechatronics 19 (2009) 313324Contents lists available at ScienceDirectMechatronicsjournal homepage: computed torque control with other control approaches1618. Computed torque control is
19、 a method that designs thecontroller strictly 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 modelof a PKM. In order to improve the accuracy of dynamic models,computed torque
20、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
21、 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 th
22、atthe 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 w
23、as 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 characteris
24、tic 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. Sincetheproport
25、ional-integral-differential(PID)controlhastheseadvantages, 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.Althou
26、gh 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 contr
27、ol 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 cont
28、rol 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 s
29、ubsystem 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
30、 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 dynam
31、ic model affects the control precision. How-ever, 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
32、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
33、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 accura
34、te 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, res
35、pectively. 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 platfor
36、m,respectively; A4Gu, A5FuI1, A5FuI2, A6MuI1and A6MuI2are 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, tang
37、ential 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 A1Gp A2FpI1 A2FpI2 A2FpI3 A3MpI1 A3MpI2 A4Gu A5FuI1
38、 A5FuI2 A6MuI1 A6MuI2 A7Gd A8FdI1 A8FdI2 A8FdI3 A8FdI4 A9MdI1 A9MdI21where A1, A2, A3are 6 ? 3 matrices, A4, A5,.,A9are 6 ? 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 classifie
39、d 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
40、 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 A1Gp A4Gu A7Gd2In middle velocity case, the dynamic model is simplified asF A1Gp A2FpI3 A3MpI2 A4Gu A6MuI1 A7Gd A8FdI23
41、orF A1Gp A2FpI3 A3MpI2 A4Gu A6MuI1 A7Gd A8FdI44In high velocity case, the simplified dynamic model can be ex-pressed asF A1Gp A2FpI3 A3MpI2 A4Gu A6MuI1 A7Gd A8FdI1 A8FdI2 A8FdI45When the moving platform moves with an acceleration or deceler-ation, the dynamic model for low acceleration or decelerati
42、on caseis simplified asF A1Gp A2FpI1 A2FpI2 A3MpI1 A4Gu A6MuI1 A7Fd6314J. Wang et al./Mechatronics 19 (2009) 313324In middle acceleration or deceleration case, the simplified dynamicmodel is given byF A1Gp A2FpI1 A2MpI2 A3GpI1 A3MpI2 A4Gu A6MuI1 A7Gd A8FdI2 A8FdI47orF A1Gp A2FpI1 A2MpI2 A2GpI3 A3MpI
43、1 A4Gu A6MuI1 A7Gd A8FdI2 A9FdI18In high acceleration or deceleration case, the dynamic model is sim-plified asF A1Gp A2FpI1 A2MpI2 A3GpI1 A3MpI2 A4Gu A6MuI1 A7Gd A8FdI1 A8FdI2 A8FdI4 A9MdI193. Control schemeThe original control system of XNZ63 PKM is designed based onthe kinematic model. Taking the
44、 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 contro
45、lling accurately the servo-system ofeach chain. Thus, the feed-forward control system of the singlechain servo-system should be designed for XNZ63 PKM. Each chainservo-system of XNZ63 PKM consists of servo-driver, motor andscrew. The servo-driver can provide position-loop controller,velocity-loop co
46、ntroller, 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
47、screware regarded as the control object. The scheme for the dynamicfeed-forward control is proposed in Fig. 1, where M-S denotes mo-tor 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 c
48、ontrol 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 de
49、signed 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
50、 in Section 2, the computed tor-que 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
51、 assumed that the velocity-loop and current-loop are unit feedback loops and the friction inthe screw is negligible. Thus, the model of the control object canbe expressed aswhere Cs3= JKipTvi+ KeKtTiiTvi+ KtKipKvpTiiTvi, and other parametersare given in Table 1.Since the theoretical model in Eq. (10
52、) 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 controlobje
53、ct is a sinusoidal signal, the steady-state output is also a sinu-soidal signal with the same frequency. Then, the Bode diagram canPosition output Closed-loop control subsystem Control object Dynamic control subsystem Inverse kinematics Computed torque controllerFeed-forward controller P-LC M-S ZPET
54、C + +V-LC C-LC + Fig. 1. Scheme for dynamic feed-forward control.Gs K1K2KtKipKvpTiiTvis2 Tii Tvis 1?JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tvis2 KtKipKvps10J. Wang et al./Mechatronics 19 (2009) 313324315be obtained by recording the amplitudes and phases of sinusoidalinput and output in differen
55、t 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 ca
56、n 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 objectis reasonable.Fig. 3b is the phase difference between the real phase obtainedby test and theoretical phase obtained by theoretical mo
57、del, andthe phase difference changes linearly. It can be concluded that atime delay exists in the control object, and the time delay can beexpressed asGds e?Tds11where 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
58、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 controller
59、4.1.1. Analysis of original control systemThe 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 sel
60、f-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 express
61、ed 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 capabilityof the closed-loop control subsystem is lower. Although the pro-portional coefficient of positi
62、on 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. Th
63、us, the position-loop controller should beredesigned 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
64、that aproportional 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 bigg
65、er 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 mechan
66、ical 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 asGpcs Kpp KpdsT2s 1 KcT1s 1T2s 114Kc
67、 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 freq
68、uency re-sponse method can be used to determine the parameters of the po-Table 1Parameters of the control object.ParameterDefinitionValueUnitK1Conversion coefficient from linear velocity to angular velocity50p/3rad/(V s)K2Conversion coefficient from angular position to linear position3/pmm/radKipPro
69、portional coefficient of current-loop30V/ATiiIntegral coefficient of current-loop0.002sKvpProportional coefficient of velocity-loop0.9A s/radTviIntegral coefficient of velocity-loop0.005sJinertia of motor rotor and screw0.001149Kg m2LWinding inductance0.023HRWinding resistance4.55XKtMagnetic torque
70、coefficient1.38N m/AKeBack electromotive force coefficient of motor1.198437V s/radGs GdsK1K2KtKipKvpTiiTvis2 Tii Tvis 1?JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tvis2 KtKipKvps12Ts KppGdsK1K2KtKipKvpTiiTvis2 Tii Tvis 1?JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tvis2 KtKipKvps13316J. Wang et al.
71、/Mechatronics 19 (2009) 313324sition-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-loop controller is 2 V/mm. Second, under the condition that thegain
72、of position-loop controller is 2 V/mm and the phase marginis 75?, the design results can be obtained by using the design pro-cedure of standard phase advance controller to design the position-loop controller.Kc Kpp 2V=mmKpd 0:019862V ? s=mmT1 0:009931sT2 0:006992s8:17According to Eqs. (12), (14), an
73、d (17), the open-loop transfer func-tion of the closed-loop control subsystem can be expressed asBased 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-t
74、em, 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 Kp
75、pchanges 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 ofT1increases. But the oscillation is enhanced and the adj
76、ustable 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.
77、 68, the coefficient for differential control of the po-sition loop controller is determined byT1 0:008 sKpd 0:016 V s=mm?194.1.3. Stability analysis of position-loop controllerInSection4.1.2,theclosed-loopcontrolsubsystemisredesigned.The stability of the new closed-loop control subsystem should bei
78、nvestigated. 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)
79、from the kine-matic 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 asFig. 3. AFC and phase difference.sinusoidal positionfeedback signal Control object Industrial computer PMAC c
80、ard Servo driver (velocity loop, current loop) screw motorsinusoidal velocity signal Fig. 2. Test principle for identifying the control object.Ts GpcsGdsK1K2KtKipKvpTiiTvis2 Tii Tvis 1?JLTiiTvis5 JTiiTviR Kips4 Cs3s3 KtKipKvpTii Tvis2 KtKipKvps18Fig. 4. Real PFC and modified PFC in theory.J. Wang et
81、 al./Mechatronics 19 (2009) 313324317where 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,G1s Gpcs KcT1s 1T2s 1;G2s K1;G3s Gds e?T
82、ds;G4s KvpTvis KvpTvis;G5s KipTiis KipTiis;G6s 1Ls R;G7s Kt;G8s 1Js;G9s 1s;G10s K2;Gts G1sG2sG3sG4sG5sG6sG7sG8sG9sG10s;G11s Ke: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) c
83、an be expressed ass1 ?1264:899166s2 ?671:811884s3 ?181:194983s4 ?70:424128s5 ?540:566023 1263:137993is6 ?540:566023 ? 1263:137993is7 ?187:866151 309:925275is8 ?187:866151 309:925275i8:23T1s YsI1sG1sG2sG3sG4sG5sG6sG7sG8sG9sG10s1 G5sG6s G6sG7sG8sG11s G4sG5sG6sG7sG8s Gts20T2s YsI2sG5sG6sG7sG8sG9sG10s1
84、G5sG6s G6sG7sG8sG11s G4sG5sG6sG7sG8s Gts21T3s Ys?TdsG8sG9sG10s1 G5sG6s G6sG7sG8sG11s G4sG5sG6sG7sG8s Gts22Fig. 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 the closed-loop control subsystem.
85、318J. Wang et al./Mechatronics 19 (2009) 313324Thus, 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
86、-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 responsecurve. It can be seen that the
87、 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 ofchain 1 is given for an example. It can
88、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 nece
89、ssary to improve the response capability of the closed-loopcontrol subsystem more for higher accuracy. Thus, the zero phaseerror tracking control is designed in Section 4.2.4.2. Zero phase error tracking control4.2.1. Design of controllerIn the control system, the closed-loop control subsystem is be
90、-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
91、 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 asGoz BozAoz24where Ao(z) and Bo(z) are the denominator polynomial and numer-ator pol
92、ynomial of the pulse transfer function, respectively, andFig. 10. AFC and step response with amplitude 0.5 mm.Fig. 8. Closed-loop Bode diagram of the closed-loop control subsystem.velocity command dynamic control command external disturbance kinematic control command position output P-LC Control obj
93、ect Fig. 9. Block diagram of closed-loop control subsystem.J. Wang et al./Mechatronics 19 (2009) 313324319Boz 0:002821z80:004077z70:035975z60:038158z5?0:058446z4?0:054939z30:034457z20:019032z?0:008479;Aoz z8?2:776104z73:315748z6?2:564700z51:666464z4?0:853194z30:249685z2?0:025361z0:000117The stable z
94、ero 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 byz4 ?0:999996z5 ?0:165696 3:725175iz6 ?0:165696 ? 3:725175iz7 ?1:000002 0:000003iz8 ?1:000002 ? 0:000003i8:26Based on Eq. (25)
95、 and (26), Bo(z) can be decomposed asBoz BaozBuoz27whereBaoz ?0:000610 0:003213z ? 0:005320z2 0:002821z3;Buoz 13:904384 42:044544z 43:707328z2 17:898560z3 3:331392z4 z5:Based on Eqs. (24) and (27), the zero phase error tracking controllercan be expressed asGcz BczAcz28whereBcz 0:000117z?5? 0:024971z
96、?4 0:167292z?3? 0:470207z?2 2:189600z?1? 2:435600 ? 2:546534z 2:801768z2? 2:794778z3 13:905127z4 0:311011z5? 26:909265z6 3:444528z7 13:904384z8;Acz ?9:060835 47:736163z ? 79:042108z2 41:909475z3:After the introduction of ZPETC, the closed-loop control subsystemis given in Fig. 12. According to Eqs.
97、(24) and (28), the pulse transferfunction of the closed-loop control system is given byTz GczGoz294.2.2. 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
98、 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 su
99、bsystem keep zero in therange of frequency, but also improves greatly the closed-loopcut-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-l
100、oop 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.Fig. 11. Trac
101、king error of chain 1.velocity command dynamic control command external disturbance kinematic control command position output P-LC Control object ZPETC Fig. 12. Closed-loop control subsystem with ZPETC.320J. Wang et al./Mechatronics 19 (2009) 3133245. Design of dynamic control subsystem5.1. Computed
102、 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.
103、 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 c
104、ontrol effect, but the controller based on the taskspace has a better real-time performance. Thus, in this section,the 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 gre
105、atly 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
106、torque controller based on the task workspace canbe expressed asM Fp2p30where M M1M2M3M4M5M6?Tis 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
107、 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 eachchain at the input point of load torque, the effect of
108、 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 obtainedGfs 1Gos31Fig. 13. Bode diagram of the closed-loop control subsystem.Fig. 14. Tracking error of the closed-loop control subsy
109、stem.Fig. 15. Principle of dual channel compensation.J. Wang et al./Mechatronics 19 (2009) 313324321where 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
110、 velocity of the current-loop is so fastthat the time delay in the current-loop can be neglected. Thus,Go(s) can be expressed asGos JKtKipTiis 1JLTiis2 JTiiR Kips JKip KeKtTii32Since there is no unstable zero point in Go(s), the dynamic feed-for-ward controller can be designed asGfs JLTiis2 JTiiR Ki
111、ps JKip KeKtTiiJKtKipTiis 1335.3. Performance analysis of dynamic control subsystemIn 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 platformmovesfromthepoint(?0.150,?0.
112、150,0.400 m,?27.056?,7.461?, 57.484?) to another point (0.150, 0.150, 0.520 m, 26.938?,?7.907?, 58.367?) 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 mode
113、l 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. I
114、t 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
115、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 t
116、orque 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
117、.By 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 track
118、ing 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
119、 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 me
120、thod 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 platform
121、should 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 bythe dynamic characteri
122、stic and response performance of controlFig. 16. Load torque of chain 1: (a) low velocity case; (b) high velocity case.Fig. 17. Tracking error caused by load torque in middle velocity case.322J. Wang et al./Mechatronics 19 (2009) 313324system, increases with the velocity and acceleration of the movi
123、ngplatform. 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 fromthepointwiththecoordinate(?0.150,?0.150,0.400 m,?27.056?, 7.461?, 57.484?) to another point with the coord
124、inate(0.150,0.150,0.520 m,26.938?,?7.907?,58.367?)isstillconsidered.In high speed, the original control system and the dynamicfeed-forward control system designed in this paper are simulated,respectively. The tracking error caused by the response perfor-mance of the control system and the tracking e
125、rror 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
126、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 controlsub
127、system 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 investigat
128、ed in this article. From this investiga-tion, the following conclusions can be drawn:(1) To improvethe responseperformanceofthe closed-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-l
129、oop control subsystemshow that the motion precision is improved largely.Fig. 18. Tracking error caused by load torque in middle velocity case.Fig. 19. Tracking error of moving platform caused by load torque and response.Fig. 20. Tracking error of the moving platform in X direction.J. Wang et al./Mec
130、hatronics 19 (2009) 313324323(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 con
131、trol 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)
132、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 (Gr
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