Mechanism and Machine Theory

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1 Mechanim and Machine Theory 46 (011) Content lit available at ScienceDirect Mechanim and Machine Theory journal homepage: Ditance-baed poition analyi of the three even-link Aur kinematic chain Nicolá Roja, Federico Thoma Intitut de Robòtica i Informàtica Indutrial (CSIC-UPC), Lloren Artigue 4-6, 0808 Barcelona, Spain article info abtract Article hitory: Received 11 May 010 Received in revied form 7 October 010 Accepted 8 October 010 Available online 4 November 010 Keyword: Aur kinematic chain Poition analyi Ditance-baed formulation Cayley Menger determinant Bilateration The poition analyi of planar linkage ha been dominated by reultant elimination and tangent-half-angle ubtitution technique applied to et of kinematic loop equation. Thi analyi i thu reduced to finding the root of a polynomial in one variable, the characteritic polynomial of the linkage. In thi paper, by uing a new ditance-baed technique, it i hown that thi tandard approach become unnecearily involved when applied to the poition analyi of the three even-link Aur kinematic chain. Indeed, it i hown that the characteritic polynomial of thee linkage can be traightforwardly derived without relying on variable elimination nor trigonometric ubtitution, and uing no other tool than elementary algebra. 010 Elevier Ltd. All right reerved. 1. Introduction A planar linkage i a et of rigid bodie, alo called link, pairwie articulated through revolute or lider joint, all lying in a plane. A linkage configuration i an aignment of poition and orientation to all link that repect the kinematic contraint impoed by all joint. The poition analyi of a linkage conit in obtaining a complete characterization of it valid configuration. At the beginning of the twentieth century, the Ruian mathematician L.W. Aur propoed a tructural claification of planar linkage baed on the mallet kinematic chain which, when added to, or ubtracted from a linkage, reult in a linkage that ha the ame mobility. Thereafter, thee elementary linkage have been called Aur group. The relevance of thee chain become evident when analyzing a complex planar linkage, becaue it i alway poible to decompoe it into Aur group which can be analyzed one-by-one. A linkage, with no mobility, from which an Aur kinematic group i obtained by removing any one of it link i defined a an Aur kinematic chain (AKC) or Baranov tru when no lider joint are conidered. Hence an AKC correpond to multiple Aur group. Conidering only revolute pair a primatic pair can be modeled a a limit cae of a revolute pair, the implet AKC i the well-known triad, a one-loop tructure with three link and two aembly mode. There i one AKC with two loop, the pentad, a five-link tructure whoe poition analyi lead to up to 6 aembly mode. E. Peyah i credited to be the firt reearcher in obtaining an analytic form olution for thi problem in 1985 [1], the ame reult wa obtained independently at leat by G. Pennock and D. Kaner [], K. Wohlhart [3], and C. Goelin et al. [4]. More recently, N. Roja and F. Thoma [5] howed that thi reult can be obtained, in a traightforward way, uing bilateration. Regarding three loop, or even link, there are three type of AKC (ee Fig. 1), namely, I) a linkage with three binary link and four ternary link with one ternary link connected to the other three, II) a linkage with three binary link and four erially-connected ternary link, and III) a linkage with four binary link, two ternary link, and one quaternary link. The poition analyi of thee linkage lead to up to 14, 16, and 18 aembly mode, repectively. C. Innocenti, in Ref. [6 8], obtained thee reult uing reultant elimination technique. Alternatively, for the type I even-link AKC, Correponding author. addree: nroja@iri.upc.edu (N. Roja), fthoma@iri.upc.edu (F. Thoma) X/$ ee front matter 010 Elevier Ltd. All right reerved. doi: /j.mechmachtheory

2 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Type I Type II Type III Fig. 1. The three even-link Aur kinematic chain. a olution baed on homotopy continuation wa preented by A. Liu and T. Yang in Ref. [9]. We how herein, by extending the idea preented in Ref. [5], that a formulation baed on bilateration lead to the ame reult preented by C. Innocenti in hi three article in a more traightforward way. The poition analyi of planar linkage ha been dominated by reultant elimination and tangent-half-angle ubtitution technique applied to et of kinematic loop equation. Thi analyi i thu reduced to finding the root of a polynomial in one variable, the characteritic polynomial of the linkage. A. Dhingra and colleague ued reduced Gröbner bae and Sylveter' elimination to obtain thee polynomial [10]. J. Nielen and B. Roth alo gave an elimination-baed method that ue Dixon' reultant to derive the lowet degree characteritic polynomial [11]. Thi technique wa later improved by C. Wampler [1], who ued a complex-plane formulation to reduce the ize of the final eigenvalue problem by half. The poition analyi of planar linkage ha alo been tackled uing general continuation-baed olver [13,14], that tart with a ytem whoe olution are known, and then tranform it gradually into the ytem whoe olution are ought, while tracking all olution path along the way [15]. Interval-baed method have alo been uccefully applied to olve the equation reulting from poition analyi problem [16,17]. Herein, we propoe an alternative approach baed on bilateration. The ue of bilateration reduce the number of equation to the point in which no variable elimination i required for the poition analyi of the three even-link Aur kinematic chain. Moreover, ince a bilateration operation i entirely poed and olved in term of ditance, no tangent-half-angle ubtitution are needed. The ret of the paper i organized a follow. A coordinate-free formula for bilateration expreed in term of Cayley Menger determinant i preented in Section. InSection 3, it i hown how the ditance ratio between any two couple of vertice in a tree of triangle can be obtained by a et of bilateration. Thi reult i ued in Section 4 6 to derive a ditance-baed characteritic polynomial for the even-link Aur kinematic chain of type I, II, and III, repectively. Finally, Section 7 ummarize the main point and give propect for further reearch.. Cayley Menger determinant and bilateration Let P i and p i denote a point and it poition vector in a given reference frame, repectively. Then, let u define Di ð 1 ; ; i n ; j 1 ; ; j n Þ = 1 n 1 i1 ; j 1 i1 ; j n 1 in ; j 1 in ; j n ð1þ with i;j = d i;j = p ij, where p ij = p j p i = P i P j. Thi determinant i known a the Cayley Menger bi-determinant of the point equence P i1,,p in, and P j1,,p jn and it geometric interpretation play a fundamental role in ditance geometry, the analytical tudy of Euclidean geometry in term of invariant. When the two point equence are the ame, it i convenient to abbreviate D(i 1,, i n ;i 1,,i n )byd(i 1,,i n ), which i imply called the Cayley Menger determinant of the involved point. The evaluation of D(i 1,,i n )giveðn 1Þ! time the quared hypervolume of the implex panned by the point P i1,,p in in R n 1. Therefore, the quared ditance between P i and P j can be expreed a D(i,j) and the igned area 1 of the triangle P i P j P k a 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di; ð j; kþ. It can alo be verified that D(i 1,i ;j 1,j ) i equivalent to the dot product between the vector p i p i1 and pj p j1. Then, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coðþ= θ Di; ð j; i; kþ= Di; ð jþdi; ð kþ, θ being P j P i P k. For a brief review of the propertie of Cayley Menger determinant, ee Ref. [18] and the reference therein. Many geometric problem can be elegantly formulated uing Cayley Menger determinant. The bilateration problem i one of them. It conit of finding the feaible location of a point, ay P k, given it ditance to two other point, ay P i and P j, whoe 1 For a triangle P i P j P k in the Euclidean plane with area A, the igned area i defined a +A (repectively, A) if the point P j i to the right (repectively to the left) of the line P i P k, when going from P i to P k.

3 114 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) P k p k θ h y p i Pi p p j P P j x Fig.. The bilateration problem in R. location are known. Then, according to Fig., the poition vector of the orthogonal projection of P k onto the line defined by P i P j can be expreed a ffiffiffiffiffiffiffiffiffiffiffiffiffi Di; ð kþ p = p i + Di; ð jþ coθ p j p i = p i + Di; ð j; i; k Þ p Di; ð jþ j p i : ðþ Moreover, the poition vector of P k can be expreed a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di; ð j; kþ p k = p S p Di; ð jþ j p i where S = 0 1 and the ± ign account for the two mirror ymmetric location of P 1 0 k with repect to the line upporting the egment defined by P i P j. Then, ubtituting Eq. () in Eq. (3) and expreing the reult in matrix form, we obtain ð3þ p ik = Z i;j;k p ij ð4þ where Z i;j;k = 1 p Di; ð j; i; kþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Di; ð j; kþ Di; ð jþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Di; ð j; kþ Di; ð j; i; kþ which will be called bilateration matrix. Now, it i important to oberve that thi kind of matrice contitute an Abelian group under product and addition and if v = Zw, where Z i a bilateration matrix, then v = detðzþ w. In what follow, in order to implify the notation, we will abbreviate the product Z i;j;k Z i;k;l by Ω i, j, k, l. 3. Ditance ratio in tree of triangle A tree of triangle i defined a a et of triangle that are connected by their edge uch that any two triangle are connected by a ingle trip of triangle, i.e. a erie of connected triangle that hare one edge with one neighbor and another with the next [Fig. 3 (left)]. Note that thi definition include cae in which edge are hared by more than two triangle. In a tree of triangle, it i traightforward to find the ratio between any two ditance involving any couple of vertice uing equence of bilateration. Thi i better explained through an example. Let u uppoe that we are intereted in finding 5, 7 / 1, in the tree of triangle in Fig. 3 (left). The correponding edge are connected by the trip of triangle {P 1 P P 3,P 1 P 3 P 4,P 1 P 4 P 5,P 5 P 4 P 6, P 5 P 6 P 7 }. Then, taking the egment P 1 P a reference, we can perform the following equence of bilateration: p 1;3 = Z 1;;3 p 1; p 1;4 = Z 1;3;4 p 1;3 = Z 1;3;4 Z 1;;3 p 1; p 1;5 = Z 1;4;5 p 1;4 = Z 1;4;5 Z 1;3;4 Z 1;;3 p 1; p 5;6 = Z 5;4;6 p 5;4 = Z 5;4;6 p 1;4 p 1;5 = Z 5;4;6 I Z 1;4;5 Z 1;3;4 Z 1;;3 p 1; ð5þ ð6þ ð7þ ð8þ ð9þ

4 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) P 3 P 1 P P 4 P 5 P 6 P 7 Fig. 3. Left: in a tree of triangle the ditance ratio between two couple of vertice can be obtained by a et of bilateration (ee text for detail). Right: the ame i poible for arbitrary couple of vertice belonging to two tree of triangle haring two vertice. p 5;7 = Z 5;6;7 p 5;6 = Z 5;6;7 Z 5;4;6 I Z 1;4;5 Z 1;3;4 Z 1;;3 p 1; : ð10þ A a conequence, 5;7 = det Z 5;6;7 Z 5;4;6 I Z 1;4;5 Z 1;3;4 Z 1;;3 : 1; Now, let u uppoe that we want to compute ;7. In thi cae P P 7 i not an edge of any triangle but clearly 1; p ;7 = p 1; + p 1;5 + p 5;7 : Then, the ubtitution of Eq. (7) and (10) in the above equation yield: p ;7 = I + Z 1;4;5 + Z 5;6;7 Z 5;4;6 I Z 1;4;5 Z 1;3;4 Z 1;;3 p 1; : Therefore, ;7 = det I + Z 1;4;5 + Z 5;6;7 Z 5;4;6 I Z 1;4;5 Z 1;3;4 Z 1;;3 : 1; By proceeding in a imilar way, it i poible to obtain the ditance ratio between any two couple of point. The poibility of computing ditance ratio that involve arbitrary couple of vertice, uing equence of bilateration, i not limited to tree of triangle. Oberve how thi can alo be applied to two triangular tree haring any two vertice [ee Fig. 3 (right)]. Thi i the cae of the even-link Aur kinematic chain of type I, II, and III, a hown in the next three ection. 4. Poition analyi of the type I even-link AKC Fig. 4 how the general even-link AKC of type I. If the central ternary link i aumed to be connected to the ground, the center of it revolute pair define the bae triangle P 3 P 5 P 4, the revolute pair center of the other three ternary link define the moving triangle P 5 P 8 P 9, P 4 P 7 P, and P 3 P 1 P 6. The poition analyi problem for thi linkage conit in, given the dimenion of every link and the poition of the center P 3, P 4, and P 5, calculating the Carteian poe of the moving ternary link. Next, a coordinate-free formula for the poition analyi of thi linkage, without uing trigonometrical function nor reultant method, i derived. To thi end, intead of directly computing the Carteian poe of the moving ternary link, we will compute the et of value of, 3 compatible with the binary and ternary link ide length. Thu, thi procedure i entirely poed in term of ditance. Actually, we will how how thi boil down to compute the ditance ratio 6;8 which can be obtained by conidering the two three ;3 of triangle defined by {P 1 P 3 P 6,P 1 P P 3,P P 4 P 3,P P 7 P 4,P 3 P 4 P 5 } and {P 5 P 7 P 9,P 5 P 9 P 8 }.

5 116 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) p 5,8 = Ω 5,7,9,8 p 5,7 P 8 P 9 p 5,7 = (Ω 3,,4,5 + Ω,3,4,7 I) p 3, p 3,5 = Ω 3,,4,5 p 3, P 5 P 7 P 6 P 3 Alpha P 4 p,7 = Ω,3,4,7 p 3, p 3,6 = Ω 3,,1,6 p 3, p 3, P P 1 Fig. 4. The general even-link AKC of type I. p 6, 8 can be expreed in function of p 3, by computing eight bilateration Scalar equation derivation According to Fig. 4, we have p ;7 = Z ;4;7 p ;4 = Z ;4;7 Z ;3;4 p 3; = Ω ;3;4;7 p 3; p 3;5 = Z 3;4;5 p 3;4 = Z 3;4;5 Z 3;;4 p 3; = Ω 3;;4;5 p 3; p 3;6 = Z 3;1;6 p 3;1 = Z 3;1;6 Z 3;;1 p 3; = Ω 3;;1;6 p 3; p 5;8 = Z 5;9;8 p 5;9 = Z 5;9;8 Z 5;7;9 p 5;7 = Ω 5;7;9;8 p 5;7 : ð11þ ð1þ ð13þ ð14þ Since p 6;8 = p 3;6 + p 3;5 + p 5;8 ð15þ and p 5;7 = p 3;5 + p 3; + p ;7 ; ð16þ then p 6;8 = Ω 3;;1;6 p 3; + Ω 3;;4;5 p 3; + Ω 5;7;9;8 p h 5;7 i = Ω 3;;1;6 + Ω 3;;4;5 Ω 5;7;9;8 Ω 3;;4;5 + Ω ;3;4;7 I p 3; : Therefore, det Ω 3;;1;6 + Ω 3;;4;5 Ω 5;7;9;8 Ω 3;;4;5 + Ω ;3;4;7 I = 6;8 ;3 : The left hand ide of the above equation i a function of the unknown quared ditance, 3 and 5, 7. Since, from Eq. (16), 5;7 = det Ω 3;;4;5 + Ω ;3;4;7 I ;3 ; ð17þ ð18þ then the ubtitution of Eq. (18) in Eq. (17) yield a calar equation in a ingle variable:, 3. The root of thi equation, in the range in which the igned area of the triangle P 1 P 3 P and P 3 P P 4 are real, that i, the range ; ; max d 1; d 1;3 d;4 d 3;4 ; min d 1; + d 1;3 d;4 + d 3;4 ;

6 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) determine the aembly mode of the even-link AKC of type I. Thee root can be readily obtained for the four poible combination of ign for the igned area of the triangle P 1 P 3 P and P 3 P P 4 uing, for example, an interval Newton method. For each of thee root, we can determine the Carteian poition of the ix revolute pair center of the moving ternary link uing Eq. (11) (14) and the equation p ;3 = Z 3;4; p 3;4. Thi proce lead up to eight combination of location for P 6 and P 8, and at leat one of them mut atify the ditance impoed by the binary link connecting them. If a polynomial repreentation i preferred, depite the previou derivation completely olve the poition analyi problem, we can proceed a decribed next. 4.. Polynomial derivation By expanding all the Cayley Menger determinant involved in Eq. (18), we get 5;7 = Γ 1 + Γ A 3;;4 ð19þ where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 3;;4 = 1 ;3 ;3 d ;4 d 3;4 d ;4 + d 3;4 ; and Γ 1,Γ are polynomial in, 3 whoe coefficient are algebraic function of the known quared ditance 3, 4, 3, 5, 4, 5, 4, 7,, 7, and, 4. On the other hand, by expanding Eq. (17), we obtain ϒ 1 + ϒ A 3;;1 + ϒ 3 A 3;;4 + ϒ 4 A 5;7;9 + ϒ 5 A 3;;1 A 3;;4 + ϒ 6 A 3;;1 A 5;7;9 + ϒ 7 A 3;;4 A 5;7;9 + ϒ 8 A 3;;1 A 3;;4 A 5;7;9 6;8 ;3 5;7 =0 ð0þ where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 3;;1 = 1 ;3 ;3 d 1; d 1;3 d 1; + d 1;3 ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 5;7;9 = 1 5;7 5;7 d 5;9 d 7;9 d 5;9 + d 7;9 and ϒ i, i=1,,8, are polynomial in, 3 and 5, 7 whoe coefficient are algebraic function of known ditance. Now, by properly quaring Eq. (0), we obtain a polynomial equation in 5, 7 whoe coefficient are radical expreion in, 3. Therefore, by replacing Eq. (19) into thi polynomial equation, we get Φ 1 + Φ A 3;;1 + Φ 3 A 3;;4 + Φ 4 A 3;;1 A 3;;4 =0 ð1þ where Φ 1, Φ, Φ 3 and Φ 4 are polynomial in, 3 of degree 6, 5, 5, and 4, repectively. Finally, the quare root in Eq. (1) can be eliminated by properly twice quaring it. Thi operation yield Φ 4 4A 4 3;;1A 4 3;;4 +Φ 4Φ A 4 3;;1A 3;;4 +Φ 4Φ 3A 3;;1A 4 3;;4 Φ 4 A 4 3;;1 + Φ Φ 3 8Φ Φ 3 Φ 4 Φ 1 +Φ 4 Φ 1 A 3;;1 A 3;;4 Φ4 3 A4 3;;4 +Φ 1Φ A 3;;1 +Φ 1Φ 3A 3;;4 Φ 4 1 =0 ðþ which, when fully expanded, lead to 4 ;3 5;7Δ I =0 ð3þ where Δ I i a 14th-degree polynomial in, 3. The extraneou root at, 3 =0 and 5, 7 =0 were introduced when clearing denominator to obtain Eq. (0), o they can be dropped. The degree of polynomial Δ I concur with the reult preented by C. Innocenti in Ref. [6].

7 118 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Example According to the notation ued in Fig. 4, let u uppoe that 1, =101, 1, 3 =17, 1, 6 =34,, 4 =5,, 7 =36, 3, 4 =37, 3, 5 =5, 3, 6 =17, 4, 5 =0, 4, 7 =13, 5, 8 =5, 5, 9 =0, 6, 8 =61, 7, 9 =45, and 8, 9 =5. Subtituting thee value in Eq. (1), we obtain Φ 1 + Φ A 3;;1 + Φ 3 A 3;;4 + Φ 4 A 3;;1 A 3;;4 =0 ð4þ where Φ 1 =3: ;3 8: ;3 6: ;3 +1: ;3 1: ;3 +: ;3 5: Φ = 9: ;3 +3: ;3 3: ;3 +1: ;3 1: ;3 +6: Φ 3 =9: ;3 4: ;3 +6: ;3 3: ;3 +6: ;3 +4: Φ 4 =1: ;3 3: ;3 +3: ;3 1: ;3 +: A 3;;1 = 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;3 00: :166 ;3 A 3;;4 = 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;3 1:876 1:174 ;3 Eq. (4) i a calar equation in, 3 which can be numerically olved for the four poible combination of ign of the two involved quared root. Alternatively, ubtituting the above value in Δ I, the following characteritic polynomial i obtained: 119: ;3 13: ;3 +67: ;3 0: ;3 +4: ;3 654: ;3 +71: ;3 5: ; : ;3 15: ;3 + 49: ;3 11: ; : ;3 1: ;3 +5: : The real root of thi polynomial are , , , , , , , and The correponding configuration for the cae in which p 3 = ð0; 0Þ T, p 4 = ð6; 1Þ T, and p 5 = ð4; 3Þ T appear in Fig Poition analyi of the type II even-link AKC Fig. 6 how the general even-link AKC of type II. If the central ternary link i aumed to be connected to the ground, the center of it revolute pair define the bae triangle P P 4 P 5, the revolute pair center of the other three ternary link define the moving triangle P 4 P 6 P 7, P 7 P 9 P 8, and P P 3 P 1. The poition analyi problem for thi linkage conit in, given the dimenion of every link and the poition of the center P, P 4, and P 5, calculating the Carteian poe of the moving ternary link. Next, following the ame trategy a the one ued in the previou ection, a polynomial in 4, 8 i derived Scalar equation derivation According to Fig. 6, we have p 4; = Z 4;5; p 4;5 = Z 4;5; Z 4;8;5 p 8;4 = Ω 4;8;5; p 8;4 p 4;6 = Z 4;7;6 p 4;7 = Z 4;7;6 Z 4;8;7 p 8;4 = Ω 4;8;7;6 p 8;4 p 8;9 = Z 8;7;9 p 8;7 = Z 8;7;9 Z 8;4;7 p 8;4 = Ω 8;4;7;9 p 8;4 p ;1 = Z ;3;1 p ;3 = Z ;3;1 Z ;9;3 p ;9 = Ω ;9;3;1 p ;9 : ð5þ ð6þ ð7þ ð8þ

8 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Fig. 5. The aembly mode of the analyzed type I even-link AKC.

9 10 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) p 4,6 = Ω 4,8,7,6 p 8,4 P 7 P 6 p 4, = Ω 4,8,5, p 8,4 P 4 p 8,4 P 8 P 9 p 8,9 = Ω 8,4,7,9 p 8,4 P 5 p,1 = Ω,9,3,1 p,9 P P 1 p,9 =(Ω 4,8,5, + Ω 8,4,7,9 I) p 8,4 P 3 Fig. 6. The general even-link AKC of type II. p 1, 6 can be expreed in function of p 8, 4 by computing eight bilateration. Since p 1;6 = p ;1 p 4; + p 4;6 ð9þ and p ;9 = p 4; p 8;4 + p 8;9 ð30þ then p 1;6 = Ω ;9;3;1 p ;9 + Ω 4;8;5; p 8;4 Ω 4;8;7;6 p h 8;4 i = Ω ;9;3;1 Ω 4;8;5; + Ω 8;4;7;9 I + Ω 4;8;5; Ω 4;8;7;6 p 8;4 : Therefore, det Ω ;9;3;1 Ω 4;8;5; + Ω 8;4;7;9 I + Ω 4;8;5; Ω 4;8;7;6 = 1;6 4;8 : ð31þ The left hand ide of the above equation i a function of the unknown quared ditance, 9 and 4, 8. Since, from Eq. (30), ;9 = det Ω 4;8;5; + Ω 8;4;7;9 I 4;8 ; ð3þ then the ubtitution of Eq. (3) in Eq. (31) yield a calar equation in 4, 8 whoe root in the range in which the igned area of the triangle P 4 P 8 P 5 and P 8 P 4 P 7 are real, that i, the range ; ; max d 5;8 d 4;5 d4;7 d 7;8 ; min d 5;8 + d 4;5 d4;7 + d 7;8 ; determine the aembly mode of the analyzed linkage. Thee root can be obtained, a in the previou ection, for the four poible combination of ign for the igned area of the triangle P 4 P 8 P 5 and P 8 P 4 P 7. For each real root, we can determine the Carteian poition of the ix revolute pair center of the moving ternary link uing Eq. (5) (8) and the equation p 4;8 = Z 4;5;8 p 4;5. Thi proce lead up to eight combination of location for P 1 and P 6, and at leat one of them mut atify the ditance impoed by the binary link connecting them. If a polynomial repreentation i till preferred, we can proceed a decribed next.

10 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Polynomial derivation By expanding all the Cayley Menger determinant involved in Eq. (3), we get ;9 = 1 Γ 1 + Γ A 4;8;5 + Γ 3 A 8;4;7 + Γ 4 A 4;8;5 A 8;4;7 4;8 ð33þ where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 4;8;5 = 1 4;8 4;8 d 5;8 d 4;5 d 5;8 + d 4;5 ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 8;4;7 = 1 4;8 4;8 d 4;7 d 7;8 d 4;7 + d 7;8 ; and Γ i, i=1,,4, are polynomial in 4, 8 whoe coefficient are algebraic function of the known quared ditance, 4,, 5, 4, 5, 4, 7, 5, 8, 7, 8, 7, 9, and 8, 9. On the other hand, by expanding Eq. (31), we obtain ϒ 1 + ϒ A ;9;3 + ϒ 3 A 4;8;5 + ϒ 4 A 8;4;7 + ϒ 5 A ;9;3 A 4;8;5 + ϒ 6 A ;9;3 A 8;4;7 + ϒ 7 A 4;8;5 A 8;4;7 + ϒ 8 A ;9;3 A 4;8;5 A 8;4;7 1;6 ;9 4;8 =0 ð34þ where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ;9;3 = 1 ;9 ;9 d 3;9 d ;3 d 3;9 + d ;3 and ϒ i, i=1,,4, are polynomial in the unknown ditance, 9 and 4, 8 whoe coefficient are algebraic function of known quared ditance. Now, by properly quaring Eq. (34), we obtain a polynomial equation in, 9 whoe coefficient are radical expreion in 4, 8. Therefore, by replacing Eq. (33) in thi polynomial equation, we get 1 4;8 Φ 1 + Φ A 4;8;5 + Φ 3 A 8;4;7 + Φ 4 A 4;8;5 A 8;4;7 =0; ð35þ where Φ 1, Φ, Φ 3, and Φ 4 are polynomial in 4, 8 of degree 8, 7, 7, and 6, repectively. Finally, to obtain a polynomial equation, the quare root in Eq. (35) can be eliminated by properly twice quaring it. When the reult i fully expanded, we obtain 4;8 ;9 Δ II =0; ð36þ where Δ II i a 16th-degree polynomial in 4, 8. The extraneou root at, 9 =0 and 4, 8 =0 were introduced when clearing denominator to obtain Eq. (34), o they can be dropped. The degree of polynomial Δ II concur with the reult preented by C. Innocenti in Ref. [7] Example According to the notation ued in Fig. 6, let u uppoe that 1, =5, 1, 3 =100, 1, 6 =97,, 3 =45,, 4 =13,, 5 =36, 3, 9 =97, 4, 5 =5, 4, 6 =13, 4, 7 =0, 5, 8 =16, 6, 7 =17, 7, 8 =13, 7, 9 =37, and 8, 9 =0. Subtituting thee value in Δ II, the following characteritic polynomial i obtained 18: ;8 5: ; : ;8 64: ;8 +3: ;8 108: ;8 +: ;8 41: ;8 + 45: ;8 3: ;8 +1: ;8 8: ;8 +38: ;8 36: ;8 +5: ;8 15: ;8 +5: : The real root of thi polynomial are , 1.00, , , , , 5.981, , and The correponding configuration for the cae in which p = ð0; 0Þ T, p 4 = ð; 3Þ T, and p 5 = ð6; 0Þ T appear in Fig. 7.

11 1 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Fig. 7. The aembly mode of the analyzed type II even-link AKC.

12 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) p 5,6 = (Ω 4,1,3,5 + Ω 1,4,,6 I) p 4,1 p 4,5 = Ω 4,1,3,5 p 4,1 P 5 P 3 p 5,9 = Ω 5,6,8,9 p 5,6 P 8 P 9 P 4 p 1,6 = Ω 1,4,,6 p 4,1 p 4,1 P P 6 P 1 p 1,7 = Ω 1,4,,7 p 4,1 P 7 Fig. 8. The even-link AKC of type III. p 7, 9 can be expreed in function of p 1, 4 by computing eight bilateration. 6. Poition analyi of the type III even-link AKC Fig. 8 how the general even-link AKC of type III. If the quaternary link i aumed to be connected to the ground, the center of it revolute pair define the bae quadrilateral P 1 P P 6 P 7, the revolute pair center of the other two ternary link define the moving triangle P 3 P 5 P 4, and P 8 P 5 P 9. The poition analyi problem for thi linkage conit in, given the dimenion of every link and the poition of the center P 1, P, P 6, and P 7, calculating the Carteian poe of the moving ternary link Scalar equation derivation According to Fig. 8, we have p 1;6 = Z 1;;6 p 1; = Z 1;;6 Z 1;4; p 4;1 = Ω 1;4;;6 p 4;1 ; p 1;7 = Z 1;;7 p 1; = Z 1;;7 Z 1;4; p 4;1 = Ω 1;4;;7 p 4;1 ; p 4;5 = Z 4;3;5 p 4;3 = Z 4;3;5 Z 4;1;3 p 4;1 = Ω 4;1;3;5 p 4;1 ; p 5;8 = Z 5;6;8 p 5;6 = Z 5;8;9 Z 5;6;8 p 5;6 = Ω 5;6;8;9 p 5;6 : ð37þ ð38þ ð39þ ð40þ Since p 7;9 = p 1;6 p 1;7 p 5;6 + p 5;9 ð41þ and p 5;6 = p 4;5 + p 4;1 + p 1;6 ; ð4þ then p 7;9 = Ω 1;4;;6 p 4;1 + Ω 1;4;;7 p 4;1 p 5;6 + Ω 5;6;8;9 p h 5;6 i = Ω 1;4;;6 + Ω 1;4;;7 Ω 5;6;8;9 I Ω 4;1;3;5 + Ω 1;4;;6 I p 4;1 : Therefore, det Ω 1;4;;6 + Ω 1;4;;7 Ω 5;6;8;9 I Ω 4;1;3;5 + Ω 1;4;;6 I = 7;9 1;4 : The left hand ide of the above equation i a function of the unknown quared ditance 1, 4 and 5, 6. Since, from Eq. (4), 5;6 = det Ω 4;1;3;5 + Ω 1;4;;6 I 1;4 ; ð43þ ð44þ

13 14 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) then the ubtitution of Eq. (44) in Eq. (43) yield a calar equation in 1, 4 whoe root in the range in which the igned area of the triangle P 1 P 4 P and P 4 P 1 P 3 are real, that i, the range ; ; max d ;4 d 1; d1;3 d 3;4 ; min d ;4 + d 1; d1;3 + d 3;4 ; determine the aembly mode of the analyzed linkage. A mentioned in previou ection, thee root can be obtained for the four poible combination of ign for the igned area of the triangle P 1 P 4 P and P 4 P 1 P 3 but, if a polynomial repreentation i preferred, we can proceed a decribed next. 6.. Polynomial derivation Following the procedure decribed in the previou ection for the polynomial derivation, from Eq. (43) and (44), we obtain 1 1;4 Φ 1 + Φ A 1;4; + Φ 3 A 4;1;3 + Φ 4 A 1;4; A 4;1;3 =0 ð45þ where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1;4; = 1 1;4 1;4 d ;4 d 1; d ;4 + d 1; ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 4;1;3 = 1 1;4 1;4 d 1;3 d 3;4 d 1;3 + d 3;4 ; and Φ 1, Φ, Φ 3, and Φ 4 are polynomial in 1, 4 of degree 8, 7, 7, and 6, repectively. Finally, by properly twice quaring the above equation, we get 1;4 5;6 Δ III =0 ð46þ where Δ III i an 18th-degree polynomial in 1, 4. The extraneou root at 1, 4 =0 and 5, 6 =0 were introduced when clearing denominator in thi polynomial derivation, o they can be dropped. The degree of polynomial Δ III concur with the reult preented by C. Innocenti in [8]. Each of real root of Δ III determine the Carteian poition of the five revolute pair center of the moving ternary link uing Eq. (37) (40) and the equation p 1;4 = Z 1;;4 p 1;. Thi proce lead to up eight combination of location for P 7 and P 9, and at leat one of them mut atify the ditance impoed by the binary link connecting them Example According to the notation ued in Fig. 8, let u uppoe that 1, =0, 1, 3 =40, 1, 6 =65, 1, 7 =144,, 4 =13,, 6 =17,, 7 =68, 3, 4 =17, 3, 5 =40, 4, 5 =13, 5, 8 =18, 5, 9 =9, 6, 7 =17, 6, 8 =5, 7, 9 =37, and 8, 9 =5. Subtituting thee value in Δ III, the following characteritic polynomial i obtained 70: ; : ;4 16: ;4 +1: ;4 : ;4 + 1: ;4 1: ; : ;4 19: ; : ;4 10: ; : ;4 : ;4 +: ;4 171: ; : ;4 3: ;4 +5: ;4 5: : The real root of thi polynomial are 5.357, 6.730, , , , , , and The correponding configuration for the cae in which p 1 = ð0; 0Þ T, p = ð4; Þ T, p 6 = ð8; 1Þ T,and p 7 = ð1; 0Þ T appear in Fig. 9.

14 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Fig. 9. The aembly mode of the analyzed type III even-link AKC.

15 16 N. Roja, F. Thoma / Mechanim and Machine Theory 46 (011) Concluion The three even-link Aur kinematic chain contain three independent kinematic loop. The tandard approach for the poition analyi of thee chain conit in deriving the cloure condition for thee three loop and obtaining an algebraic reultant. Neverthele, formulating the poition analyi in term of kinematic loop equation introduce a major diadvantage: the reulting equation involve tranlation and rotation imultaneouly. We have preented a different approach in which, intead of dealing with the Carteian poe of the involved link, the poition analyi problem i fully poed in term of ditance. Then, under thi approach, the cloure condition boil down to a ingle ditance ratio computable by bilateration. An important implification i thu obtained. How the propoed approach cale to more complex Aur kinematic chain i an open problem. It i well-known that there are 8 AKC with four loop or nine link [19]. The poition analyi of ome of thee AKC, baed on either reultant elimination [0 3] or homotopy continuation method [9,4], ha already been reported in the literature. It can be checked that the required ditance ratio for mot of thee chain can alo be obtained by conidering tree of triangle haring only two vertice. Thu, the application of the propoed technique to the poition analyi of the AKC with nine link eem advantageou but thi i certainly a point that deerve further attention. Acknowledgment We gratefully acknowledge the financial upport of the Autonomou Government of Catalonia through the VALTEC program, cofinanced with FEDER fund, and the Colombian Minitry of Communication and Colfuturo through the ICT National Plan of Colombia. Reference [1] E. Peiach, Determination of the poition of the member of three-joint and two-joint four member Aur group with rotational pair (in Ruian), Machinowedenie (5) (1985) [] G. Pennock, D. 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Innocenti, Analytical-form poition analyi of the 7-link Aur kinematic chain with four erially-connected ternary link, Journal of Mechanical Deign 116 () (1994) [8] C. Innocenti, Polynomial olution to the poition analyi of the 7-link Aur kinematic chain with one quaternary link, Mechanim and Machine Theory 30 (8) (1995) [9] A. Liu, T. Yang, Diplacement analyi of planar complex mechanim uing continuation method, Mechanical Science and Technology 13 () (1994) [10] A. Dhingra, A. Almadi, D. Kohli, Cloed-form diplacement and coupler curve analyi of planar multi-loop mechanim uing Gröbner bae, Mechanim and Machine Theory 36 () (001) [11] J. Nielen, B. Roth, Solving the input/output problem for planar mechanim, Journal of Mechanical Deign 11 () (1999) [1] C. Wampler, Solving the kinematic of planar mechanim by Dixon determinant and a complex-plane formulation, Journal of Mechanical Deign 13 (3) (001) [13] J. 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Yao, Topological characteritic and automatic generation of tructure analyi and ynthei of plane mechanim, part i theory, part ii application, Proceeding of the ASME Mechanim Conference, Vol.15. ASME (American Society Of Mechanical Engineer), Kiimmee, FL, 1988, pp [0] L. Han, Q. Liao, C. Liang, A kind of algebraic olution for the poition analyi of a planar baic kinematic chain, Journal of Machine Deign 16 (3) (1999) [1] L. Han, Q. Liao, C. Liang, Cloed-form diplacement analyi for a nine-link Barranov tru or an eight-link Aur group, Mechanim and Machine Theory 35 (3) (000) [] P. Wang, Q. Liao, Y. Zhuang, S. Wei, A method for poition analyi of a kind of nine-link Barranov tru, Mechanim and Machine Theory 4 (10) (007) [3] P. Wang, Q. Liao, Y. Zhuang, S. Wei, Reearch on poition analyi of a kind of nine-link Barranov tru, Journal of Mechanical Deign 130 (1) (008) [4] L. Hang, Q. Jin, T. 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