A Method for Determining the Number of Dependent Constraints Using Transformation Parameters of Coordinate Systems Korganbay Sholanov1, a*

Transcription

1 Iteratoal Coferece o Mechacs, Materals ad Structural Egeerg (ICMMSE 206) A Method for Determg the Number of Depedet Costrats Usg Trasformato Parameters of Coordate Systems Korgabay Sholaov, a* Karagada State Techcal Uversty, Mra blvd, 56, Karagada cty, 00027, Republc of Kazahsta a sholor@jadex.z Keywords: Trasformato parameters, Aalyss, Degrees of freedom, Depedet costrats. Abstract. Ths paper proposes a ew method whch the umber of depedet costrats s determed based o the trasformato parameters of the coordate systems assocated wth the ls of a mechasm; the umber of homoymous parameters that are equal to zero s subsequetly used to determe the overall moblty (umber of DOF) of the mechasm. Ths method ca be used to solve problems volvg the aalyss ad sythess of the topologcal structure of ay closed mechasm or parallel mapulator. Itroducto The umber of degrees of freedom (DOF) of a mechasm s determed by aalysg ts topologcal structure. Therefore, for stace, whe solvg a ematcs problem, the umber of DOF determes the umber of depedet parameters the posto fucto. I the dyamcs ad cotrol of mechasms ad mapulators, the umber of depedet dyamc equatos ad the umber of actuators s equal to the umber of DOF. Ths s why t s mportat to be able to accurately determe the umber of DOF for sgle-loop ad mult-loop mechasms, cludg parallel mapulator mechasms. However, the aalyss performed by Grgore Gogu [] of studes publshed the past 50 years dcates that to date, o theory or method for determg the umber of DOF that s applcable to ay arbtrary mechasm has bee developed. Therefore, the purpose of ths paper s to obta a uversal ad coveet method for determg the umber of depedet costrats for subsequet DOF calculatos cocerg closed mechasms ad parallel mapulators. The objectve s to use the coordate trasformato obtaed from the mechasm s ematcs ad ts dyamcal equato Choosg the Coordate Systems for the Ls Accordg to ematcs [2], the posto of a mechasm ad the relatve postos of ts ls ca homogeeous trasformato matrces. The elemets of the matrces be determed usg are the trasformato parameters or trgoometrc fuctos derved from them. The trasformato parameters are the values of the lear or agular dsplacemets that are requred to merge the coordate systems coected by jot-formg ls. Fg. shows the trasformato parameters, δl, θl, αl, α, b, ad β, that are ecessary to superpose the O-X-Y-Z- ad OXYZ coordate systems. Here АВ=a commo perpedcular to the arbtrarly oreted axs O-Z- ad O Z. I ths paper, the trasformato parameters are used to defe the postos of ls relatve to each other. It should be oted that ths paper ether specfcally defes the trasformato parameters of the homogeeous trasformato matrces or solves matrx equatos followg the tradto of ematcs. Istead, the purpose of ths paper s to demostrate that the umber of depedet costrats ca be determed drectly by aalysg the trasformato parameters. The soluto to the problem begs wth the recogto of the rules goverg the O-X-Y-Z- ad OXYZ 206. The authors - Publshed by Atlats Press 286

2 coordate systems assocated wth each par of ls, ad -, betwee whch (--) jots are formed (=,,, where s the total umber of moble ls). A fxed l (frame) s deoted by the umeral zero (0). The ma codto for selectg coordate systems s that all of the ematc varables descrbg the postos of the ls ad costat geometrc quattes (the sze of the ls, the twst agles, etc.) must be cluded the umber of trasformato parameters. B b b X O Z - a a q X - d A O - Z - Fg. The trasformato parameters. Whe choosg a coordate system, the followg are recommeded: The Z - axes should be oreted alog the axes of cyldrcal, screw, prsmatc, or revolute (-)- pars or perpedcular to faces wth slots for the fgers of two moble sphercal (-)- pars. The org of the O - X - Y - Z - coordate systems should be set at the cetre of a sphercal (-)- par. The drecto of the X 0 axs, whch s assocated wth the support frame, must be chose arbtrarly. The drecto of the X axs should be selected so that t s perpedcular to the plae that cotas the Z - ad Z axes f these axes tersect; f these axes do ot tersect, t should be perpedcular to the Z - ad Z axes. I a sphercal ematc par wth the fger, the X axs should be drected alog the axs of the fger. Thus, the mutual locatos of the ls formg the (-)- jot s uquely determed by sx trasformato parameters: three lear dsplacemets, d, a, ad b, alog the geerally o-coplaar Z -, AВ, ad Z axes ad three rotato agles θ, α, ad β aroud the same axes. The bass vectors of the Z -, AB, ad Z axes are the ut vectors r, e r, ad r, respectvely. The vector s used to provde vector desgatos for the trasformato parameters, where the dex r=,, 6 deotes the umber of the trasformato parameter ad the dex =,, deotes the umber of ls ad ematcal pars of the mechasm, where s the total umber of moble ls. Usg ths desgato, the trasformato parameter vectors ca be wrtte as follows: () It should be oted, that the cases where t s feasble, the Deavt Harteberg method for selectg ad trasformg a coordate system ca be used. As t s ow, ths case the four trasformato parameters are used. A aalyss of the Trasformato Parameters The trasformato parameters, d, θ ι, a, α, b, ad β, may be varable, costat, or equal to zero. Partcular ote should be tae of the parameters that are equal to zero because they characterze peculartes of the relatve locatos of the O - X - Y - Z - ad O X Y Z coordate systems. Therefore, the zero-valued parameters characterze peculartes of the cofgurato of jots. Table descrbes 287

3 these peculartes of the locatos of the coordate system whch the trasformato parameters equal to zero. Table. The parameters are zero ad the correspodg of the peculartes of the relatve postos of the coordate systems. Trasformato parameter equal to zero d θ a α b β Peculartes of the relatve postos of the assocated coordate systems O - X - Y - Z - ad O X Y Z. The org of the O - X - Y - Z - coordate system s located at pot A, amely, the tersecto of the O - Z - axs wth the mutual perpedcular AB The O - X - axs s oreted parallel to le AB or perpedcular to the Z axs The O - Z - ad O Z axes tersect The O - Z - ad O Z axes are parallel The org of the O X Y Z coordate system s located at pot B, amely, the tersecto of the O Z axs wth the mutual perpedcular AB The O X axs s parallel to le AB A Determato of the Number of Depedet Costrats ad DOF Let us cosder a mechacal system wth holoomc statoary costrats that s composed of members (=,, ). All of the forces actg o each l, cludg erta, are represeted by force compoets ad pars of forces wth momets (l=,2,) drected alog the vectors, e r, ad r (). The, all of the forces ad momets ca be represeted usg geeralzed force vectors. v r v r v v v r v r 2 v 2 v Q = F ; Q 2 = M ; Q = F e ; Q 4 = M e ; Q 5 = F ; Q 6 = M. (2) The geeral dyamcal equato for the gve system at a vrtual dsplacemet s 6 δ v p rq r = 0 (=,,), () r = where r r r r r r = Q Q Q Q Q Q ) s a colum vector whose elemets are the geeralzed force Qr r r r r r r vectors Q (2) ad r δ p ( ) T r = δp δp2 δp δp4 δp5 δp6 s a row vector whose elemets are the possble dsplacemet vectors of the trasformato parameters gve (). The matrx equato gve () s equvalet to a lear system of equatos of the form 2 2 δ d F + δθ M + δa F + δα M + δb F + δβ M = 0, (=,..,). (4) I ths equato, t s uderstood that the forces ad momets are varables ad that the possble chages the trasformato parameters ( δd δθ δa δα δb δβ ) are the coeffcets of a system of Eq. (4) that descrbes a lear space of o more tha 6 dmesos because the colum ra of the R matrx s lmted to 6 - R 6. If all of the ls are rgdly coected by meas 6 costrats mposed o each jot, the all of the equaltes Eq. (4) are satsfed, ad the equato system has a ra of 6. From ths, t follows that the system of equatos gve Eq. (4) govers a costrat space ad that the ra of the R matrx, whch defes the dmesoalty of ths space, correspods to the umber of depedet costrats. Chagg the dmeso of the space,.e., the umber of depedet costrats, s possble whe the trasformato parameters are depedet or zero. Cases whch the trasformato parameters are equal to zero due to the mutual oretato of the coordate axes are dscussed before. The possble depedecy of the trasformato parameters s dscussed here. Suppose the O - Z - ad O Z axes are parallel, the, the trasformato parameters d, b ad θ ι, β ι characterze the lear ad 288

4 agular dsplacemet the same drecto. Therefore, the expressos Eq. (4) are smlar ad as a result, the umber of smlar terms s reduced to four. I ths case, the duplcate terms should be set to zero (b =β =0), whch s equvalet to reducg the umber of summads Eq. (4). Coverso s also possble whe there are depedet parameters. For example, trasformg some pars of parameters s equvalet to movg the system alog a axs twce. The, Eq. () s reduced by oe summad because the expressos for the geeralzed forces of moto alog the same axs are smlar. I ths case, the depedet parameters are both repeated, ad oe of them s zero. I ths case, the system s coeffcets are equal to zero whe the correspodg trasformato parameters are equal to zero. I some cases, for example, the overcostraed lages there are repettve ematc chas, whch allow to derve addtoal equatos for further processg. These equatos reduce the umber of depedet Eq. (4), that s, every equato whch s ew to the system of four (4) equatos, reduces the sze of the costrat space. I order to defe the codtos for determg the umber of depedet costrats a closed ematc cha, we wll summarze the four (4) equatos the basc effects of smlar forces ad reacto force pars (a vertcal summato wll be performed). Thus, the followg equato s derved: = F δd M δθ F δa M δα F δb = = = = = To fd out whch of the summads a equato becomes zero, let s revew ay gve stadaloe summad, for example as follows: = F δd + F2 δd F δd + F δd = F δd = I ths example, the reacto forces caot equal zero based o costrats of the gve lage. Trasformato parameters each lage are depedet values. Hece the followg cocluso; ths equato ca oly be executed o codto that all homoymous trasformato parameters (d ths case) equal zero. Thus, f a umber of -S<6 zero value homoymous trasformato parameters s preset all of the four (4) system equatos, the the ra of the coeffcet matrx of the four (4) equatos, ad cosequetally the dmesos of space reduce by the value of S. I ths case, the maxmum dmesos of the space descrbed by the system (4) wll equal R = 6- S. Coeffcets that are equal to zero correspod to parameters that are equal to zero, ad t follows that the quatty S s the smallest umber of zero homoymous parameters preset each of the equatos the system. For example, f for some mechasm, the system of sx equatos of the form gve Eq. (4), of the parameters the st equato, 4 the 2 d, 4 the rd, the 4 th, the 5 th, ad 2 the 6 th are zero, the S=2 because o fewer tha 2 parameters are equal to zero ay of the system s equatos. I ths stace, the dmesoalty of the space s R=6-S=4. Therefore, the umber of depedet parameters defg costrats s 4, ad S=2 s the umber of depedet costrats. It follows that the umber of depedet costrats s equal to the smallest umber of zero homoymous parameters preset ay of the equatos of system (4). A aalyss of the system s coeffcet matrx (4) allows the umber of depedet costrats to be determed. From a practcal pot of vew, determg S for each closed loop mechasm or mapulator s suffcet to create a table (matrx) of the trasformato parameters. The, usg the table, the smallest umber of zero homoymous parameters ay row s determed. Ths determes the umber of depedet costrats, S. For a oe-loop mechasm, the umber of DOF s W = M 6 + S. (5) The umber of DOF of a mult-loop mechasm ca be calculated usg the followg relato: W = M 6 + S, (6) = 0 M δβ = 0 289

5 where s the umber of depedet loops ad S s the umber of depedet costrats o the -th loop. A depedecy smlar to the oe gve (5, 6) has bee reported referece []. The Steps Aalysg the Structure of a Mechasm Usg the Zero Trasformato Parameters The sequece of steps the aalyss of a mechasm s as follows:. Assg umbers to the moble ls. The frame should be labelled Defe the umber of depedet loops.. Choose a coordate system coected wth the ls accordace wth the recommedatos gve Part. 4. Determe the trasformato parameters ad mae a table of the trasformato parameters each loop by mergg the coordate systems sequece (Part 2). Exclude the depedet parameters. The, determe whch row has the fewest parameters equal to zero. Ths mmum umber of zero parameters s equal to the umber of depedet costrats o the loop uder aalyss. 5. Use formula (5) or (6) to determe the umber of DOF of the mechasm. A aalyss of the structures of mechasms for whch there s a dvergece betwee the actual ad calculated (usg tradtoal methods) umbers of DOF s preseted below. Samples Aalyses of Mechasms Structures Let us cosder a four-bar mechasm (Fg. 2) that s called the sphercal mechasm, whch the axes of revolute pars tersect at a pot O. Fg. 2 The sphercal mechasm. I Fg. 2, the ls are deoted 0- ad the Z axes assocated wth ls that are oreted alog the axes of revolute ematc pars are selected. The drecto of the X 0 axs s chose perpedcular to the plae Z 0 ОZ. The other X I axes are perpedcular to the plae cotag the Z - ad Z axes. To combe the coordate systems O 0 X 0 Z 0 ad O X Z, the followg steps are tae. The system O 0 X 0 Z 0 s shfted alog the O 0 Z 0 axs by a dstace d =O 0 O. The OX 0 axs s rotated through a agle θ to alg t wth the O X axs. The O X axs s ot shfted; therefore, a =0. From ths posto, the system s rotated through a agle α = O 0OO. The system s shfted a dstace b =OO alog the OZ. There s o eed to rotate the O Z axs; therefore, β =0. Smlar actos are performed to detfy the trasformato parameters of the other ematc pars. As a result of ths trasformato, 290

6 the depedet parameters, amely b =d 2, b 2 =d, ad b =d 4, b 4 =d are foud. I accordace wth the coclusos obtaed Part 4, we set b =b 2 =b =b 4 =0. The trasformato parameters for the ematc pars obtaed after elmatg the depedet parameters are summarzed Table 2. Table 2. The trasformato parameters for the sphercal mechasm. Jot formed by the Trasformato parameters specfed ls d θ a α b β 0- d = O 0 O θ 0 α d 2 = OO θ 2 0 α d = OO 2 θ 0 α d 4 = OO θ 4 0 α From a aalyss of Table 2, t s foud that M=4,.e., θ =Var, ad S=; therefore, accordg to formula (5), the umber of DOF of ths mechasm s W=. Coclusos Ths paper proposes a method of ematcal aalyss based o the use of the zero trasformato parameters of coordate systems to determe the umber of depedet costrats, gve the topologcal structure of a closed-loop mechasm or mapulator. It s prove that the umber of homoymous trasformato parameters that are zero depeds o the ra of the coeffcet matrx of the system of lear equatos derved from the geeral dyamcal equatos. It s also foud that the umber of depedet costrats s equal to the smallest umber of zero homoymous parameters (coeffcets equal to zero) ay equato of the system. Oce all of the depedet costrats have bee foud, the umber of DOF of a sgle-loop or mult-loop closed mechasm ca be determed based o the ow depedeces. Refereces [] G. Grgore, Moblty of mechasms: a crtcal revew. Mech. Mach. Theory, 40(9) (2005) [2] P. N. Sheth, J.J. Jr. Ucer, A geeralzed symbolc otato for mechasms, J. Eg. Id. Tras. ASME, Seres B, 9() (97) [] R. Voea ad M. Ataasu, Cotrbuto to the study of the structure of ematc chaes. Bull. Yst. Polteh. Bucurest, t. XXII, Fasc.,