Design of a Stewart Platform for General Machining Using Magnetic Bearings

Transcription

1 eign of a Stewart Platform for eneral Machining Uing Magnetic earing Jeff Pieper epartment of Mechanical and Manufacturing Engineering Univerity of algary algary lberta anada N N4 pieper@ucalgary.ca Preented are the detail of the deign of a Stewart Platform or hexapod to be ued for generalized machining. ecaue of the nature of the platform ix degree of freedom machining i poible with a fairly large range of motion and thu i applicable to a relatively large uite of manufacturing tak including drilling grinding milling welding and other. urther the platform i inherently well uited for preciion poitioning due to it tiffne. hi tiffne can be ignificantly enhanced by the ue of magnetic bearing. ogether thee feature make the deign of thi platform attractive a a precie poitioning tool for machining purpoe. Stewart Platform he Stewart Platform i ueful to tudy becaue it i a widely accepted deign for a motion control device largely due to it wide range of motion and accurate poitioning capability. t provide a large amount of tiffne enabling the ytem to provide a ignificant ource of poitional certainty. Kinematic of the Stewart platform are well known and provide a framework for thi advanced deign. he Stewart platform involve ix actuator to provide ix degree of freedom motion of an end effector while providing clear workpace to the tool. he Stewart Platform wa originally deigned a a flight imulator but it i alo commonly ued for automotive tranportation and machine tool technology. he Stewart Platform deign i alo ued for poitioning of atellite communication dihe and telecope and in application uch a hipbuilding and bridge contruction. he kinematic and the kinetic of the platform indicate the ue of magnetic bearing a an actuation technique for fine and precie poitioning while gro platform poiting i accomplihed via hydraulic ytem. y allowing general machining tak via the Stewart platform reduced number of part re-poitioning or cell change need be accomplihed. hi ave not only manufacturing time but alo uperviion time for the tak. More accurate machining and reduced vibration will greatly reduce the number of tool change required due to lower level of tool wear and alo tool breakage. lexibility in the ytem by allowing more than one machining tak enhance the value of the ytem. he deign of the platform capture thee key advantage in an indutrially relevant device for general machining in a manufacturing environment. t i etimated that the cot of the platform i comparable to that of current more tatic manufacturing ytem. Preentation of the deign i intended a a proof of concept tage for prototype development and eventual machining device.

2 raditionally a common method for deigning the controller for the Stewart Platform required manipulating complicated equation that modeled the phyical component ued to olve the mechanical equation. hen the engineer had to olve thee equation uing complex numerical integration technique. Magnetic earing Magnetic bearing a actuator can allow extremely fat and reponive control of vibration and diplacement in mechanical ytem. heir drawback i their inherent intability and related need for active feedback control. n addition in order to achieve high performance a fully multivariable and well-deigned control need to be in place. Offetting thi are reduced need for lubrication and ignificant vibration control. etail of the control deign are conidered with repect to robutne and nominal performance achievement. ontrol i conidered uing variou technique including modern -infinity method and uing tructured ingular value to account for unmodeled vibration mode. ontrol Sytem eign One way to decribe control ytem performance i in term of the ize of certain ignal of interet. or example the performance of a tracking ytem could be meaured by the ize of the error ignal. he and performance meaure are preented a ytem norm whoe uefulne for evaluating ytem performance i demontrated given their ignal interpretation. ontroller to achieve optimality with repect to thee norm will be developed. he Riccati operator i introduced a algebraic Riccati equation which are ued to compute tate pace realization for both and optimal control law. Optimization i introduced with the general problem tatement and review of optimization theory in optimal control of linear time invariant ytem with a quadratic performance criterion. Review of the general -optimal ynthei theory i alo treated together with the mall gain theorem. ctual engineering control problem invariably include a et of conflicting requirement on time repone frequency repone actuator activity diturbance rejection and robutne to changing condition and plant uncertainty. he key to ucceful application of optimal control i to contruct formal cot function that tend to caue thee genuine ytem requirement to be atified. lthough the -norm may be a meaningful performance meaure and although Q theory can give efficient deign compromie under certain diturbance and plant aumption the major limitation of the theory i the lack of formal treatment of uncertainty in the plant itelf. he reult i a controller adept at attenuating impule diturbance but with potentially weak robutne characteritic. Much of modern control theory relie upon a mathematical model of the plant to be controlled. n the real world there i alway uncertainty in any uch model; the actual repone of the plant may be quite different from the aumed model. urthermore the plant may change a it age or a operating condition vary. optimization focue on the minimization of ytem output energy to unknown but bounded energy input. control can be formulated to give robut tability to model

3 uncertainty. he -optimal controller reult in a highly robut ytem; however one which can be notably deficient in providing optimal nominal performance. he mixed / control deign methodology reviewed in thi paper allow and performance objective to be addreed imultaneouly. t etablihe a link between and -optimization allowing a trade-off between their objective. n thi manner it i deired to yntheize an admiible robut tate feedback controller that minimize an performance objective ubject to a contraint on the -norm of a different tranfer function matrix. here are many way in which plant uncertainty decription and feedback deign problem can be cat a optimization problem. t i very ueful therefore to have a general framework into which any particular problem can be manipulated. ll ytem conidered in thi paper are modelled a finite dimenional linear time-invariant operating in continuou time with a tate pace and tranfer function repreentation in the aplacian given by where in R n n in R n m in R l n and in R l m where R i j repreent a real matrix of dimenion i j. he ytem i aid to be table if the tate pace matrix ha no eigenvalue in the cloed right half-plane. he general framework to be ued here will be referred to a the eneral ontrol onfiguration. he objective i to deign a controller K for the plant uch that the input/output tranfer characteritic from the external input vector w to the external output vector z are deirable according to ome engineering pecification. he exogenou input w typically conit of command ignal diturbance and enor noie; u i the control ignal; the output to be controlled z which cutomarily conit of the o-called error ignal which are to be minimized in ome ene to meet control objective; and y the meaured output. he ytem i decribed by z w w y u u u K y with a tate pace realization of the generalized plant given by where. ij i j ij oth and K are real rational and proper. K i contrained to provide internal tability with tate model of and K defined. hen internal tability will mean that the tate of and K go to zero from all initial value when w. Since controller are retricted to being proper and real rational which are alo tabilizable and detectable thee propertie will be aumed throughout. hu the term controller will be taken to mean a controller that atifie thee propertie. ontroller that have the additional property of being

4 internally tabilizable will aid to be admiible. he realization of the tranfer matrix i taken to be of the form. Notice the pecial off-diagonal tructure of i aumed to be zero o that i trictly proper; alo i aumed to be zero in order to guarantee that the problem i properly poed. he following additional aumption are made for the output feedback problem i i tabilizable and i detectable; ii i tabilizable and i detectable; iii [ ] [ ]. iv. he firt aumption i made for a technical reaon together with ii it guarantee that the two amiltonian matrice aociated with the problem belong to domric. umption ii i neceary and ufficient for to be internally tabilizable. umption iii implie that x and u are orthogonal o that the penalty on z x u include a non-ingular normalized component on the control u. umption iv i dual to iii and concern how the exogenou ignal w enter. he problem tatement i then ummarized a Problem he control problem i to find a proper real rational controller K which tabilize internally and minimize the norm of the tranfer matrix K in w K z. y emma the amiltonian matrice J belong to domric and moreover Ric and Y RicJ are poitive emidefinite. efine Y and f c.

5 he unique optimal controller i K opt n practice it i uually not neceary to obtain an optimal controller for the problem and it i often computationally and theoretically impler to deign a uboptimal one. et min be the minimal value of K over all tabilizing controller K. he problem conidered in thi ubection i the uboptimal control problem Suboptimal Problem iven a > min find all admiible K uch that < K. or the with aumption i-iv the olution involve two amiltonian matrice J. here exit a tabilizing controller K uch that < K if and only if the following three condition hold i domric and Ric. ii J domric and J Ric Y. iii ρ < Y where ρm i the pectral radiu or maximum abolute valued eigenvalue of M. Moreover when thee condition hold one uch controller i Z K ub where. Y Z Y Z f full tate information i available the deign i implified. onider an open loop ytem tranfer matrix / S with the ame aumption inherited from the output feedback problem. here exit a tabilizing controller K uch that < K P if and only if domric and Ric. he reulting uboptimal controller in thi cae i then K ub.

6 Summary he problem of deigning a general machining ytem with a fairly large operating pace i addreed in a conceptual tage manner in thi work. Shown i the viability of a Stewart platform for thi tak and the aociated iue of uing uch a device. hee iue include kinematic and kinetic of the device and ubequent control. t i noted that the forward kinematic problem i relatively difficult while the invere kinematic ha a unique cloed form olution for mot operating condition. hi implie that the olution for a control i reaonable. evelopment of an optimal and robut control i conidered in theory by preenting and -optimal control theory. Progre toward manufacture of the device i ongoing. Reference i. 997 Modeling Simulation and ontrol of a Stewart Platform M.Sc. thei Univ. of ritih olumbia. iu K. itzgerald M awon. and ewi. 99 Modeling and ontrol of a Stewart Platform Manipulator SME yn. Sy. and ontrol iv. Proceeding page Nanua P. and Waldron K. 989 irect Kinematic Solution of a Stewart Platform EEE ranaction on Robotic and utomation page Rathbun. 985 Stewart Platform Six-xi Milling Machine evelopment ME hei Univerity of anterbury. Spong M. and M. Vidyaagar 989 Robot ynamic and ontrol Wiley. Stewart. 965 Platform with Six egree of reedom Proc. MechE page