Myard’s first five-bar linkage as a degeneracy of a plane-symmetric six-bar loop.doc

available online at sciencedirectmechanism and machine theory 43 (2008) 391-399mechanismandmachine t/locate/mechmtmyards first five-bar linkage as a degeneracy of a plane-symmetric six-bar loopj. eddie bakerschool of information technologies, the university of sydney, nsw 2006, australiareceived 18 september 2006; received in revised form 15 november 2007; accepted 19 november 2007available online 6 february 2008abstractamong the best known of overconstrained kinematic loops, goldbergs generalised five-bar is also recognised as a parent” linkage to myards first five-bar of 10 years earlier in appearance. somewhat oddly, despite both having been synthes-ised from bennett chains, the latter is plane-symmetric and the former essentially asymmetric in character. explored in this paper is the notion that myards linkage might be alternatively sought as a derivative of a plane-symmetric six-bar. whilst there is high expectation of success in this instance, the approach may offer broader application. as well, the analysis highlights technical features of significance to workers in the field of linkage kinematics. 2007 elsevier ltd. all rights reserved.keywords: linkage kinematics; overconstrained linkages; linkage mobility; displacement-closure equations; screw-vector algebra; spatial linkages1. introductionnotwithstanding the many recorded solutions 13 of the six-revolute kinematic chain, for very few are there available minimal sets of displacementclosure relationships. this is so because, first, most cases with mobility one were revealed through intuition or synthesis rather than systematic means and, second, the general forms of governing equations are exceedingly difficult to manipulate into a definitive set of five independent ones. the situation is effectively the same for the governing relationships of five-revolute chains. if myard 4 and goldberg 5 had not synthesised their offerings from bennett loops, there would possibly be no 5r linkages to discuss.the myard five- and six-bar loops are related analytically to goldbergs solutions in ref. 6, and goldbergs five- and six-bar chains are accorded detailed attention in refs. 79. goldbergs (generalised) five-bar has remained the only known all-inclusive five-revolute linkage family. in searching for other cases, lee 10 obtains a set of necessary algebraic conditions which are to be satisfied, but their complicated character makes analytical solutions very unlikely. karger 11 builds upon ref. 10 with the assistance ofe-mail address: .au0094-114x/s - see front matter 2007 elsevier ltd. all rights reserved. doi:10.1016/j.mechmachtheory.2007.11.003392j.e. baker/mechanism and machine theory 43 (2008) 391399computer-software routines, distinguishing between the generic grouping with zero joint offsets, namely, goldbergs basic five-bar (but incorrectly regarded there as the totality of goldbergs five-bar synthesis), and all others. he claims to have found exhaustive solution sets, but in such a convoluted fashion that their natures are hidden. the only example of a particular solution provided in ref. 11 can be shown, using dietmaiers 9 formulas, to be a very special instance of goldbergs generalised five-bar in which the kink-angle” is 83.62. the diligent reader is alerted to an apparent typographical error there; the second d5 should be given instead as d2. ref. 9 includes a description of a numerical search procedure for five-revolute solutions outside goldbergs family, dietmaier concluding that any are highly improbable.it is not uncommon to find published accounts, in various formats, of sets of conditions to be satisfied by the link parameters of potential solutions to a family of overconstrained linkages. in spite of the relatively well developed requirements presented in refs. 10,11, it must be conceded that such criteria are of little assistance in disclosing the categories of solutions which must be recognised for an appreciation of their overall motion characteristics and our level of knowledge concerning the possibility of cases remaining to be identified. now, goldbergs five-bar family is essentially asymmetric. myard observes this fact in synthesising his second five-bar from two identical bennett loops, realising thereby a special case of goldbergs basic linkage. his first five-bar, a particular case of goldbergs generalised loop, imposes plane symmetry upon the latter in the only viable way, by virtue of the contributing bennett chains being rectangular” and the adjacent links of the composition having complementary skew angles. hence, there arises the question of a provenance for myards plane-symmetric solution other than the device of synthesising it from a pair of four-bars.given the abovementioned difficulty of a systematic search, one is led to the conjecture that myards first five-bar is derivable from a mobile six-bar with one joint locked. testing this proposition calls for employment of the latters displacement-closure equations which, as already stated, are usually unknown. one available avenue is offered by waldrons 12 plane-symmetric r-2h-p-2h-loop proved mobile via screw-system algebra. on the basis of its established motion capability it is feasible to determine a minimal set of governing equations for its r-derivative. our objective, then, is to find under what conditions this latter linkage, whilst retaining its plane symmetry, will remain mobile when the prismatic joint is fixed.fig. 1. typical variables and parameters of a spatial linkage.our notation for linkage elements is as exemplified in fig. 1, and use is made of the abbreviations c for cosine, s for sine, and t for tangent. the character x stands for 1. for the readers benefit, three of a generalj.e. baker/mechanism and machine theory 43 (2008) 391399393set of 12 six-bar displacement-closure relationships is reproduced in the appendix to this paper. although it is more usual in an investigation in this area to choose appropriate equations from a standard grouping of nine equations of orientation and three of position, there is sometimes the valid alternative of employing fewer relationships from the set through the device of reversing the sense of traverse of a kinematic loop or that of using a different starting-point. we have the freedom to do so here, whereby eqs. (a1)-(a3) are sufficient for our analysis.it is also convenient to invoke dual screw-vector terminology for (relative) motion about and along instantaneous screw axis (isa) $i, for which there must be specified an orientation i and a location ri in some frame of reference, together with screw pitch hi. the corresponding unit dual vector is then defined byst = tti + sm*, wheres2 = 0 andm* = hiiii + ri x iti = hiiii + mi.conversely, once provided with the unit vector, the pitch and orthogonal position of the isa are determinable from the equationsht = iti m*, pi = iti x m*.as well, the unit vector along the common normal directed from joint axisj to joint axis i is denoted by ni/j. finally, we need to refer to the rate of rotation about joint axis i, indicating it by(i = 9i.2. waldrons loopthe six-bar linkage in its most general form, as illustrated schematically in fig. 2, requires that the symmetrical pairs of helical articulations have pitches of equal magnitude and opposite sign. as will become evident, we can restrict consideration to the special instance of zero-valued pitches. then the consequent 5r-p-loop is subject to the dimensional conditions34 + 23 = p! 45 + 12 = 2p = 061 + 056,t3 = 0; t 4 = t2; t5 = t1 96 = p. two closure equations arising directly from the plane symmetry areo4 + 62 = 2p = 95 + 91.(1,2)it is stressed that we are not engaged in a systematic reduction of the full complement of closure equations to a minimal set of five. because the linkages mobility is otherwise known 12, we simply select relationships appropriate to our purpose.under the foregoing constraints and the accompanying simplifications of eqs. (1,2), equation of orientation (a2) with indices advanced by 5 is reduced tos01s02so61so23 c01c02so61co12so23 c91s 061so12c023 c02co61 so12s023 + c061co12c023 = 0.(3)similarly, equation of position (a3) with indices advanced by 2 becomesfl23(s01c02so61 +c01s02so61co12 + sco61so12) +fl12s01so61 + 71c061 + (co61co12 c01so61so12) +t6/2 = 0.(4)394j.e. baker/mechanism and machine theory 43 (2008) 391399h-76rs6fig. 2. the plane-symmetric r-2h-p-2h-loop (after ref. 12).the latter relationship is seen to represent a resolution of linear displacements parallel to the prismatic joint. one more independent equation is needed to complete a set for the linkage. a suitable candidate is yielded from eq. (a2) when indices are advanced by 4, givingsf?2sf?3sa12sa23 cf?2cf?3sa12sa23ca23 + cf?2sa12sa23ca23 cf?3ca12s2a23 ca12c2a23= 2cf?1sa61ca61sa12 s2a61ca12 + c2a61ca12.(5)3. the 5r degeneracywe seek a five-revolute linkage of the apposite form by fixing t6. the variable 92 can then be eliminated between eqs. (3) and (4) to yield a relationship in 91 alone, namely,fl23s2a23s2f?1s2a61 + c2f?1s2a61c2a12 + c2a61s2a12 + 2cf?1sa61ca61sa12ca12= a23c01sa61sa12 co61ca12 c cx23 + s a232,2(c01sa61sa12 co61ca12) a12sf?1sa61 t1co61 t6/2 .now, we know from ref. 1 that there can be no proper solution for which sa23 = 0, and refs. 13,14 preclude any for which a23 = 0. this putative identity in 91 can be simplified toa23s2a23 c2a61c2a12 + 2cf?1sa61ca61sa12ca12 c2f?1s2a61s2a12= s2a23j,2(c01sa61sa12 - ca61ca12)2 + a22s291s2a61 + t1ca61 + t6/2)2 - 2t2a12s91sa61(c91sa61sa12- ca61ca12) - 2t2(t1ca61 + t6/2)(c91sa61sa12 - ca61ca12) + 2a12s01sa61(7,1ca61 + t6/2).(6)in examining the coefficient of s01c01, we may, by the findings of refs. 1,13,14, disallow the possibilities of sa12 and a12 being zero. if sa61 were zero, revolutes 5 and 1 would be coaxial, resulting in part-chain mobility. consequently, we infer thatt2 = 0.j.e. baker/mechanism and machine theory 43 (2008) 391399395from the coefficient of s#1 we conclude thatt1ca61 + t6/2 = 0.from the remainder of eq. (6) we deduce the following requirements:c2$1: a223s2a12 = a22s2a23,c91: ca61 =0 or ca12 = 0,c 91: a23(s a23 c a61c a12) = a12s a61s2 0(23.if ca61 were zero, t6 would also vanish; the five-bar would degenerate further to a special (equilateral) bennett linkage with an additional, coaxial revolute. hence, we are led to the further constraints for our solution given byp a12 = 7; 23 = fl12s0(23; scx23 = s61 2as we are free to assume that 0 a61, (x12, a23 p. closure eq. (3) is simplified tos01s02s23 c$1ca23 tc02co123 = 0(7)and eq. (4), equivalently, tos01c02s23 + s01 + ts02co(23 = 0.(8)eq. (5) is reduced tos02s03 c02c03ca23 + c02co123 = 2xc01c23.(9)we have hereby isolated the only proper degeneracy of the six-bar for which the slider is locked. it is clear that the screw joints of the most general form of the original loop cannot be accommodated, because offsets t1 and t2 have been found to be strictly constant in value. it remains to delineate the five-bar fully by re-defining the joint angles and offsets on $5 and $1; the link length between them, of course, isfl51 = 0.we are free to allow1/5 = fl6/5(= 1*1/6).then 95 may be retained for the five-bar. consequently, for the derived loop, the angular displacement on $1 is given by9 = 01+p.(10)in order to investigate geometrical properties of the derived loop, we mount reference axes as depicted in fig. 3. then, after some manipulation,fig. 3. the cartesian frame of reference defined.396j.e. baker/mechanism and machine theory 43 (2008) 3913993 = k; r?, = 0 = a/3,0 sa23; r2 ca23 s0200 i02a/2 = fl23 ca23c$2 / 0a/ = fll2(l + c02s23) i sa23ca23sa23;c02co(23-c02sa23s03sa23c03sa23 -ca231r = ri an s02ca23s02sa23c03/ s03ca234;/?4 = fl23 s03 ; a/4 = fl23 c03ca230 / sa23s02c03 + c02s03ca23 / c$2c$3 s02s3ca235 = s02s$3 c02c03ca23 ; r5 = r4 + 12 c$2s$3 + s02c03c23c02sa23/-s02sa23(s03sa23 c03sa23 -ca23(c$i c$2 c$is02ca23 + s0isa23 c0is02sa23 + s0ica23now, under the constraints established above, displacement-closure eq. (al) with indices advanced by 4 is reduced to(11)s202c$3 + 2s02c02s$3ca23 c202c03c223 c2$2s223 = s223 + c223-hence,5 i = c(2a23).as well,(-s02c#2sa23s03 - c202sa23ca23l - c93 s02c02so(231 + cot, + c202sa23ca23s$3 (c202c2a23 s202)s03 + 2s92c92ca23c93advancing indices by 4 in displacement-closure eq. (a3) simplifies it to(12)c02s03 + s02c$3ca23 s02co123 + s03s23 = 0,whences03s02co(23 1 c$3 c$2 + s0c23but, from eqs. (5) and (6), it is seen thats02co123to i = z.co 2 + scx23consequently,(13)03 = p 2q.j.e. baker/mechanism and machine theory 43 (2008) 391399397with the assistance of eqs. (7) and (8) we can re-express(c92(c92 + sa23) -s92c92ca23 s02(1 +c02sa23)application of eqs. (7)-(9), (12) and (13) and considerable manipulation allow us to re-write also/ -c92(c92 + sa23) it5 x it1 =s92c92ca23= xs(2a23)1/5.1 + c92sa23 ii-s02(1 +c02sa23)/deployment of eq. (12) permits us to easily establishr5 = r1, so that the relevant offsets on the derived five-bar are given by t 5 = 0 = t 1. differentiation of relationships (1), (2), (7), (8), (13) leads to the results4 + m2 = 0 = a5 + w1, to)1s92 - o)2s91 = 0, 3 + 2a1 = 0.velocity closure around the loop is thereby confirmed through the dual equation5 2_) mini + mj) = 0. i=1more significant here is a consideration of groupings within the equation. with the aid of eqs. (7), (8), (12), (13) we find that the projections on the plane of symmetry of $4, $2 and $5, $1 intersect $3 at the point12 c23t01 /that is, the simplified screw system” 12 of the five-bar, regardless of the value of x, consists of a planar pencil of zero-pitch screws.4. the solutionsthe case x = 1for a reason soon to be clear, let us reverse the sense of $5, indicating the consequently modified quantities by the symbol . gathering the dimensional constraints for the five-bar, we havefl34 = fl23 = fl12s0(23; 45 = cl12 51 = 0,a34 + a23 = p; 0t45 = = &12, t 1 t = 2 .2 2 23 p1398j.e. baker / mechanism and machine theory 43 (2008) 391-399a comparison with the relevant results stated in ref. 6 establishes an exact match, and so the present case is confirmed as myards first five-bar. thus, the approach introduced here is found to be effective, and perhaps applicable elsewhere as an alternative to a direct search for a mobile linkage of a certain form.the foregoing analysis shows, however, the necessity for great care in determining relationships among link parameters and joint variables in the conversion from six links to five. the same finding is yielded, but at greater length, when the initial loop is bricards plane-symmetric six-bar; for that chain a minimal set of closure equations has been derived in ref. 15. perhaps obvious, but significant nonetheless, is the observation that the six-bar which possesses or can manifest the special dimensional conditions of the myard loop could have, depending upon its particular circumstances, two degrees of gross mobility (one of them passive) or configurations of supramobility” 16, positions of possible bifurcation.the case x = 1for the sake of consistency we again reverse the sense of $5. the link parameters are subject to the conditionsfl34 = fl23 = fl12s23; 45 = 12! 51 = 0,34 + 23 = p; %5 = = &12 x51 = p 2a23, 2t2 = t3 = t4 = 0 = t5 = tand an apposite set of closure equations is6*5 = 6 + p, 64 + 62 = 2p,63 + 201 = p,62 sp |23 6 t =p2 t 1.2 s p/2 2superficially, these relationships appear to define a solution distinct from the myard chain. because $5 and $1 are concurrent, however, the sense of their notional common normal may be reversed; doing so changes a*51, 6*5, 6 in such a way that the two cases become identical.5. closing remarksit is not unusual to encounter and subsequently discard degeneracies when seeking mobile solutions to a linkage of a certain type. indeed, such instances are noted above. the present study is novel in demonstrating that one class of overconstrained linkage is derivable from another appropriate one of greater connectivity sum. the analysis draws attention to the technical subtleties required of such a conversion and the resolution of apparent perplexities by the device of reversing the sense of a joint axis or notional link vector. misunderstanding of these latter issues has led, in some publications, to the announcement of spurious solutions to classes of overconstrained loops.appendixs04c05s06sa61 + c04s$5s06ca45sa61 + s6 4s6 5c6 6ca56sa61 c64c65c66ca45ca56sci61 + c04c06sa45sa56sa61+ s$4s05sa56ca61 c&4c05ca45sa56ca61 c$4sa45ca56ca61= s02s03sa12ca34 + c02c03sa12ca23ca34 + c03ca12sa23ca34 c02sa12s23sa34 + ca12ca23sa34,(a1)s05s06sa45sa61 c05c06sa45ca56sa61 c06ca45sa56sa61 c05sa45sa56ca61 + ca45ca56ca61= s02s03sa12sa34 c02c03sa12ca23sa34 c03ca12sa23sa34 csa12so23co34 + ca12ca23ca34,(a2)j.e. baker/mechanism and machine th