The final publication is available at link.springer.comA tetrahedral strip is a tetrahedron-tetrahedron truss where any tetrahedron has two neighbors except those in the extremes which have only one. Unless any of the tetrahedra degenerate, such a truss is rigid. In this case, if the distance between the strip endpoints is imposed, any rod length in the truss is constrained by all the others to attain discrete values. In this paper, it is shown how to characterize these values as the roots of a closure polynomial whose derivation requires surprisingly no other tools than elementary algebraic manipulations. As an application of this result, the forward kinematics of two parallel platforms with closure polynomials of degree 16 and 12 is straightforwardly solved.Peer ReviewedPostprint (author's final draft

Porta Pleite, Josep Maria

Thomas, Federico

The final publication is available at link.springer.comA tetrahedral strip is a tetrahedron-tetrahedron truss where any tetrahedron has two neighbors except those in the extremes which have only one. Unless any of the tetrahedra degenerate, such a truss is rigid. In this case, if the distance between the strip endpoints is imposed, any rod length in the truss is constrained by all the others to attain discrete values. In this paper, it is shown how to characterize these values as the roots of a closure polynomial whose derivation requires surprisingly no other tools than elementary algebraic manipulations. As an application of this result, the forward kinematics of two parallel platforms with closure polynomials of degree 16 and 12 is straightforwardly solved.Peer Reviewe

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Closure polynomials for strips of tetrahedra

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Closure Polynomials for Strips of TetrahedraFederico Thomas and Josep M. PortaAbstractA tetrahedral strip is a tetrahedron-tetrahedron truss where any tetrahedronhas two neighbors except those in the extremes which have only one. Unless any ofthe tetrahedra degenerate, such a truss is rigid. In this case, if the distance betweenthe strip endpoints is imposed, any rod length in the truss is constrained by all theothers to attain discrete values. In this paper, it is shown how to characterize thesevalues as the roots of a closure polynomial whose derivation requires surprisinglyno other tools than elementary algebraic manipulations. As an application of thisresult, the forward kinematics of two parallel platforms with closure polynomials ofdegree 16 and 12 is straightforwardly solved.Key words: Position analysis, closed-form solutions, Distance Geometry, spatiallinkages.1 IntroductionLet us consider the strip of tetrahedra in Fig. 1. Any such strip has two endpoints. Inthis case, Pa and Pb. If the distance between these two points is imposed, the lengthof any rod cannot be freely chosen. This paper is essentially devoted to obtain aclosed-form solution for the length of any rod in a strip of tetrahedra, once thedistance between its endpoints and the lengths of all other rods are known.Although closure polynomials have been typically obtained on a case-by-caseanalysis, a common pattern can be identified for most cases. First, a set of loop equa-tions involving both translation and orientation variables is derived. Then, transla-tion variables are eliminated resulting in a system of trigonometric equations that isalgebraized using the tangent half-angle substitution. Finally, elimination theory isFederico Thomas · Josep M. PortaInstitut de Robo`tica i Informa`tica Industrial, CSIC-UPC, Barcelona, Spaine-mail: {fthomas,porta}@iri.upc.edu12 F. Thomas and J. M. PortaPaPbFig. 1 A strip of eight tetrahedra whose endpoints are Pa and Pb. Observe that no triangular face isshared by more than two tetrahedra.used to obtain a univariate closure polynomial. Here we solve this problem departingfrom this standard approach. The proposed method can be summarized as follows.The distance between the strip endpoints is first derived by iterating a basic opera-tion involving only two neighboring tetrahedra over the whole strip. This leads to ascalar equation containing radical terms. We will see how clearing these radicals is atrivial task, and how the resulting polynomial contains, in general, factor terms thatcorrespond to singularities of the formulation that depend on the chosen variablelength. Since these terms can be easily spotted beforehand, their elimination is justa matter of iterative polynomial division until a no null remainder is obtained. Theresult is the sought-after univariate closure polynomial obtained without variableeliminations or trigonometric substitutions.Next, we detail this procedure and then we apply it to derive the minimal degreeclosure polynomial for two widely studied parallel platforms: the decoupled parallelplatform, and a 4-4 platform with planar base and moving platform.2 Obtaining the Closure PolynomialsGiven a set of points, the valid distances between them can be characterized us-ing the theory of Cayley-Menger determinants [1, 6, 8]. The Cayley-Menger bi-determinant of the two sets of points Pi1 , . . . ,Pin and Pj1 , . . . ,Pjn is defined asD(i1, . . . , in; j1, . . . , jn) = 2(− 12)n∣∣∣∣∣∣∣∣∣0 1 . . . 11 si1, j1 . . . si1, jn1.... . ....1 sin, j1 . . . sin, jn∣∣∣∣∣∣∣∣∣ , (1)Closure Polynomials for Strips of Tetrahedra 3where si, j stands for the squared distance between Pi and Pj.If the two sets of points are the same, then D(i1, . . . , in) = D(i1, . . . , in; i1, . . . , in)is called the Cayley-Menger determinant of the involved set of points. The Cayley-Menger determinant D(i1, . . . , in) is proportional to the squared volume of the sim-plex spanned by Pi1 , . . . ,Pin in Rn−1.PiPjPjPkPkPlPlPmPmψl,i, j,k,mFig. 2 Substitution rule.Now, let us suppose the two neighboring tetrahedra in Fig. 2-left belong to a stripof tetrahedra in R3. The squared distance between Pl and Pm can be expressed as(see [7] for details):sl,m =2D(i, j,k)(D(i, j,k, l; i, j,k,m)∣∣∣∣sl,m=0±√D(i, j,k, l)D(i, j,k,m)). (2)where the ± sign accounts for the two possible solutions depending on the relativeorientation between the two tetrahedra. To lighten the notation, (2) will be simplywritten as sl,m =Ψl,i, j,k,m. If some of the distances involved inΨl,i, j,k,m are taken asvariables, they will be made explicit in parenthesis. For example, if si, j and si,k arevariables, we will write sl,m =Ψl,i, j,k,m(si, j,si,k).If one of the points in the set {Pi,Pj,Pk} does not belong to any other tetrahedronin the strip, it can be removed from the strip provided that a rod connecting Pl and Pmis introduced with the double-valued length given by (2) [Fig. 2-right]. This reducesthe number of tetrahedra in the strip by one. Then, by repeating this operation untilthe strip contains only two tetrahedra, the distance between the tetrahedral stripendpoints is finally obtained as a 2n−2−valued function, where n is the number oftetrahedra in the strip.To obtain the closure condition as a polynomial in terms of a given rod length,the first step consists in taking the numerator of the rational form of the obtainedfunction and then clearing radicals. As radicals will appear nested, they are clearedusing an iterative process starting from the outer one. At each step of this process,the expressions involving a radical will have the general form4 F. Thomas and J. M. PortaP1P2P3P4P5P6P7S=0 34 49 62 ? ? 10834 0 41 58 108 ? ?49 41 0 68 ? 126 ?62 58 68 0 38 91 34? 108 ? 38 0 85 74? ? 126 91 85 0 197108 ? ? 34 74 197 0Fig. 3 Decoupled parallel manipulator, with non-planar moving platform, used as example.α0 +α1√r+α2(√r)2 +α3(√r)3 + · · ·= 0, (3)which can be rewritten as(α0 +α2r+α4r2 + . . .)+√r(α1 +α3r+α5r2 + . . .) = 0. (4)This equation can be unfolded into two equations, one for each sign of√r. Sincewe are interested in the roots of both equations, we obtain their product, which canbe written as(α0 +α2r+α4r2 + . . .)2− r(α1 +α3r+α5r2 + . . .)2 = 0. (5)While clearing radicals as explained above introduces no extraneous roots, onecannot expect for the obtained polynomial to be of minimal degree. This is dueto the presence of singularities of the formulation. Indeed, let us suppose that theclosure polynomial is expressed in terms of the squared rod length si, j. If a rodwith variable length belongs to a shared face, this face degenerates for some valuesof si, j. When this happens, the three points defining the face get aligned and thetetrahedral strip can be decomposed into two parts so that one can freely rotatewith respect the other about the axis defined by these three aligned points. As wewill see, terms corresponding to these degenerate configurations will appear in theclosure polynomial. They can be easily removed by iteratively dividing the closurepolynomial by them until the remainder is not null.3 ExamplesNext, we apply the technique explained above to solve the foward kinematics ofa decoupled platform and a 4-4 platform with planar base and platform (see Fig. 3and Fig. 5, respectively). The decoupled platform owes its name to the fact that threelegs permit the rotation of the platform about a point whose location is controlledClosure Polynomials for Strips of Tetrahedra 5by the other three. Since the forward kinematics for the translational part is trivial,the interest of this linkage lies in the spherical part for which a minimal closingpolynomial of degree 8 on a squared variable was first derived in [2]. In [7], thisderivation is simplified by using the closure polynomial of the so-called doublebanana. Despite the simpler derivation, variable eliminations were still necessary.For the chosen 4-4 platform with planar base and platform, a minimal 12th-degreeclosure polynomial was first derived in [3]. The derivation was far from trivial andapplicable only to this particular platform. To properly compare our results withthose reported in [7] and [3], we use the same numerical examples.First, let us consider the decoupled platform defined by the squared distancematrix S appearing in Fig. 3, where si, j = S(i, j). It can be topologically describedas the strip of tetrahedra shown in Fig. 4-left. Applying the substitution presented inthe previous section three times (see Fig. 4), we haves3,7 =Ψ3,4,5,6,7(s3,5), (6)s2,7 =Ψ2,3,4,5,7(s3,7,s3,5), (7)s1,7 =Ψ1,2,3,4,7(s2,7,s3,7). (8)The numerator of the rational form resulting from substituting (6) in (7), and theresult in (8), can be written as:R1−1346.0R2 +7899650R3 +24942632734s3,5 +1402R3 s3,52−323070338s3,52 +741658s3,53−208500R3 s3,5 +528767086008 = 0,whereR1 =√−2027718R22 +4695768R2R3 s23,5−729124704R2R3 s3,5 + ... ,R2 =√100464R23 s23,5−19847712R23 s3,5 +115799664R23− ... ,R3 =√−3481450s23,5 +806976100s3,5−27440188650 .The full expressions for R1 and R2 are not included here due to space limitations.Now, clearing the radicals as described in Section 2, we obtain a polynomialof 24th-degree. It is not of minimal degree because the rod connecting P3 and P5belongs to the shared face defined by P3, P4, and P5 which is singular whenD(3,4,5) = 0, that is, when s23,5− 214s3,5 + 961 = 0. By iteratively dividing theobtained polynomial by this singular factor until the remainder is not null, we gets163,5−1.6652 ·104 s153,5 +1.2722 ·106 s143,5−5.8952 ·108 s133,5 +1.8487 ·1011 s123,5−4.1525 ·1013 s113,5 +6.9146 ·1015 s103,5−8.7384 ·1017 s93,5 +8.5338 ·1019 s83,5−6.5533 ·1021 s73,5 +4.0715 ·1023 s63,5−2.1848 ·1025 s53,5 +1.1165 ·1027 s43,5−5.4256 ·1028 s33,5 +2.0923 ·1030 s23,5−5.0066 ·1031 s3,5 +5.2479 ·1032,6 F. Thomas and J. M. PortaP1P1P1P2P2P2P3P3P3P4P4P4P5P5P6P7P7P7Ψ3,4,5,6,7(s3,5)Ψ3,4,5,6,7(s3,5)Ψ2,3,4,5,7(s3,7,s3,5)Fig. 4 The decoupled parallel platform in Fig. 3 can be topologically described as the strip offour tetrahedra in which the distance between P3 and P5 is variable and the distance between itsendpoints, P1 and P7, is known. The application of the substitution rule presented in Section 2 tothis strip (left) permits to sequentially eliminate P6 (center) and P5 (right).Closure Polynomials for Strips of Tetrahedra 7P1P2P3P4P5P6P7 P80 10.1982 19.7992 182 ? ? ? 15.1657210.1982 0 16.49242 s2,4 14.29172 ? ? ?19.7992 16.49242 0 14.56022 11.87702 10.85452 ? ?182 s2,4 14.56022 0 ? 15.17192 15.79072 ?? 14.29172 11.87702 ? 0 62 s5,7 4.47212? ? 10.85452 15.17192 62 0 4.47212 5.65692? ? ? 15.79072 s5,7 4.47212 0 2215.16572 ? ? ? 4.47212 5.65692 22 0Fig. 5 4-4 parallel manipulator used as example. Since the base and the moving platform are con-vex planar quadrilaterals, s2,4 and s5,7 are unambiguously determined by the other known distances.which coincides with the closure polynomial reported in [7], but obtained in a muchsimpler way.As a second example, let us consider the 4-4 parallel platform appearing in Fig. 5.Its forward kinematics is known to have 24 solutions [3]. However, they can be splitin two sets that are symmetric with respect to the base. Since the distance-basedformulation is invariant to this symmetry, we will get a 12th-degree closure poly-nomial. The two sets of configurations are obtained in the coordinatization processusing trilateration [4, 5, 8].Applying the substitution presented in the previous section four times (seeFig. 6), we have thats4,8 =Ψ4,5,6,7,8(s4,5),s3,8 =Ψ3,4,5,6,8(s4,8,s4,5),s2,8 =Ψ2,3,4,5,8(s3,8,s4,8),s1,8 =Ψ1,2,3,4,8(s2,8,s3,8,s4,8).After a proper sequence of forward substitutions in the above four equations, s1,8can be expressed only in terms of s4,5. Since this parallel platform has planar baseand platform,Ψ4,5,6,7,8 andΨ1,2,3,4,8 are single-valued functions. OnlyΨ3,4,5,6,8 andΨ2,3,4,5,8 contribute with square roots to the obtained closure condition. Eliminatingthem as explained leads to a 52nd-degree polynomial in s4,5. In this case, the rodconnecting P4 and P5 belongs to two shared faces (the ones defined by P4P5P6 andP3P4P5), whose associated singular terms are s24,5−706.1251s4,5 +5031.9580, ands4,52− 532.3731s4,5 + 37708.4160. After iteratively dividing the obtained polyno-8 F. Thomas and J. M. PortaP1P1P1P1P2P2P2P2P3P3P3P3P4P4P4P4P5P5P5P6 P6P7P8P8P8P8Ψ4,5,6,7,8(s4,5)Ψ4,5,6,7,8(s4,5)Ψ4,5,6,7,8(s4,5)Ψ3,4,5,6,8(s4,8,s4,5)Ψ3,4,5,6,8(s4,8,s4,5)Ψ2,3,4,5,8(s3,8,s4,8)Fig. 6 The 4-4 parallel manipulator in Fig. 5 can be topologically described as the strip of fivetetrahedra in which the distance between P4 and P5 is variable and the distance between its end-points, P1 and P8, is known. The application of the substitution rule to this strip (top left) permitsto sequentially eliminate P7 (top right), P6 (bottom left), and P5 (bottom right).mial by these two factors until the remainder is not null, the following 12th-degreepolynomial is obtaineds124,5−0.676 ·103 s114,5−3.873 ·106 s104,5 +5.400 ·109 s94,5−9.858 ·1012 s84,5−2.327 ·1015 s74,5 +1.967 ·1018 s64,5−7.316 ·1020 s54,5 +1.518 ·1023 s44,5−1.834 ·1025 s34,5 +1.257 ·1027 s24,5−4.432 ·1028 S45 +6.171 ·1029.This polynomial has six roots that lead to real configurations of the moving platformobtained by coordinatization via trilaterations [5,8]. These roots and the correspond-ing configurations appear in Fig. 7. They coincide with the solutions reported in [3]obtained using an ad hoc intricate method.4 ConclusionsIt has been explained how to obtain closure polynomials for tetrahedral strips interms of the involved rod lengths and the distance between the strip endpoints, andClosure Polynomials for Strips of Tetrahedra 9  P2  P5  P8  P1  P7  P6  P3  P4xyz0002455−5681010151520  P2  P5  P1  P8  P7  P3  P6  P4xyz0002455−5681010151520s4,5 = 420.405 s4,5 = 388.143  P2  P1  P5  P6  P8  P7  P3  P4xyz0002455−5681010151520  P2  P1  P8  P7  P3  P5  P6  P4xyz0002455−5681010151520s4,5 = 339.179 s4,5 = 171.428  P2  P1  P3  P6  P7  P8  P5  P4xyz0002455−5681010151520  P2  P1  P6  P7  P8  P3  P5  P4xyz0002455−5681010151520s4,5 = 136.826 s4,5 = 136.300Fig. 7 Forward kinematics solutions of the 4-4 manipulator used as example. The mirror configu-rations with respect to the base are also solutions, but they are not represented.how this technique can be applied to solve some position analysis problems. How-ever, this technique cannot incorporate orientation constraints between tetrahedraat different parts of the strip. As a consequence, if applied to a case in which suchconstraints are necessary, the obtained closure polynomial would not be of minimal-degree as some of its roots would violate these constraints. Despite this importantlimitation, it supersedes the method presented in [7] in scope and simplicity, thusproviding a better starting point for a complete generalization to three dimensions10 F. Thomas and J. M. Portaof the techniques developed for the position analysis of planar linkages using Dis-tance Geometry.Acknowledgements This work has been partially supported by by the Spanish Ministry of Econ-omy and Competitiveness under project DPI2014-57220-C2-2-P.References1. Havel, T.: Some examples of the use of distances as coordinates for Euclidean geometry. Journalof Symbolic Computation 11(5-6), 579–593 (1991)2. Innocenti, C., Parenti-Castelli, V.: Echelon form solution of direct kinematics for the generalfully-parallel spherical wrist. Mechanism and Machine Theory 28, 553–561 (1993)3. Lin, W., Griffis, M., Duffy, J.: Forward displacement analyses of the 4-4 Stewart platforms.ASME Journal of Mechanical Design 114(3), 444–450 (1992)4. Porta, J.M., Ros, L., Thomas, F.: Inverse kinematics by distance matrix completion. In: Inter-national Workshop on Computational Kinematics (2005)5. Porta, J.M., Ros, L., Thomas, F.: On the trilaterable six-degree-of-freedom parallel and serialmanipulators. In: IEEE International Conference on Robotics and Automation, pp. 960–967(2005)6. Porta, J.M., Ros, L., Thomas, F., Torras, C.: A branch-and-prune algorithm for solving systemsof distance constraints. In: IEEE International Conference on Robotics and Automation, pp.342–348 (2003)7. Rojas, N., Thomas, F.: The closure condition of the double banana and its application to robotposition analysis. In: International Conference on Robotics and Automation, pp. 4641–4646(2013)8. Thomas, F., Ros, L.: Revisiting trilateration for robot localization. IEEE Transactions onRobotics 21(1), 93–101 (2005)

Closure polynomials for strips of tetrahedra

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