Closure Polynomials for Strips of Tetrahedra

Transcription

1 Closure Polnomials for Strips of Tetrahedra Federico Thomas and Josep M. Porta Abstract A tetrahedral strip is a tetrahedron-tetrahedron truss where an tetrahedron has two neighbors ecept those in the etremes which have onl one. Unless an of the tetrahedra degenerate, such a truss is rigid. In this case, if the distance between the strip endpoints is imposed, an rod length in the truss is constrained b all the others to attain discrete values. In this paper, it is shown how to characterie these values as the roots of a closure polnomial whose derivation requires surprisingl no other tools than elementar algebraic manipulations. As an application of this result, the forward kinematics of two parallel platforms with closure polnomials of degree 1 and 1 is straightforwardl solved. Ke words: Position analsis, closed-form solutions, Distance Geometr, spatial linkages. 1 Introduction Let us consider the strip of tetrahedra in Fig. 1. An such strip has two endpoints. In this case, P a and P b. If the distance between these two points is imposed, the length of an rod cannot be freel chosen. This paper is essentiall devoted to obtain a closed-form solution for the length of an rod in a strip of tetrahedra, once the distance between its endpoints and the lengths of all other rods are known. Although closure polnomials have been tpicall obtained on a case-b-case analsis, a common pattern can be identified for most cases. First, a set of loop equations involving both translation and orientation variables is derived. Then, translation variables are eliminated resulting in a sstem of trigonometric equations that is algebraied using the tangent half-angle substitution. Finall, elimination theor is Federico Thomas Josep M. Porta Institut de Robòtica i Informàtica Industrial, CSIC-UPC, Barcelona, Spain {fthomas,porta}@iri.upc.edu 1

2 F. Thomas and J. M. Porta P a P b Fig. 1 A strip of eight tetrahedra whose endpoints are P a and P b. Observe that no triangular face is shared b more than two tetrahedra. used to obtain a univariate closure polnomial. Here we solve this problem departing from this standard approach. The proposed method can be summaried as follows. The distance between the strip endpoints is first derived b iterating a basic operation involving onl two neighboring tetrahedra over the whole strip. This leads to a scalar equation containing radical terms. We will see how clearing these radicals is a trivial task, and how the resulting polnomial contains, in general, factor terms that correspond to singularities of the formulation that depend on the chosen variable length. Since these terms can be easil spotted beforehand, their elimination is just a matter of iterative polnomial division until a no null remainder is obtained. The result is the sought-after univariate closure polnomial obtained without variable eliminations or trigonometric substitutions. Net, we detail this procedure and then we appl it to derive the minimal degree closure polnomial for two widel studied parallel platforms: the decoupled parallel platform, and a - platform with planar base and moving platform. Obtaining the Closure Polnomials Given a set of points, the valid distances between them can be characteried using the theor of Cale-Menger determinants [1,, ]. The Cale-Menger bideterminant of the two sets of points P i1,...,p in and P j1,...,p jn is defined as D(i 1,...,i n ; j 1,..., j n )= 1 n 1 s i1, j 1... s i1, j n s in, j 1... s in, j n, (1) where s i, j stands for the squared distance between P i and P j.

3 Closure Polnomials for Strips of Tetrahedra 3 If the two sets of points are the same, then D(i 1,...,i n )=D(i 1,...,i n ;i 1,...,i n ) is called the Cale-Menger determinant of the involved set of points. The Cale- Menger determinant D(i 1,...,i n ) is proportional to the squared volume of the simple spanned b P i1,...,p in in R n 1. P m P m P i P k P j P k P j ψ l,i, j,k,m P l P l Fig. Substitution rule. Now, let us suppose the two neighboring tetrahedra in Fig. -left belong to a strip of tetrahedra in R 3. The squared distance between P l and P m can be epressed as (see [7] for details): s l,m = D(i, j,k) D(i, j,k,l;i, j,k,m) ± p D(i, j,k,l) D(i, j,k,m) s l,m =!. () where the ± sign accounts for the two possible solutions depending on the relative orientation between the two tetrahedra. To lighten the notation, () will be simpl written as s l,m = Y l,i, j,k,m. If some of the distances involved in Y l,i, j,k,m are taken as variables, the will be made eplicit in parenthesis. For eample, if s i, j and s i,k are variables, we will write s l,m = Y l,i, j,k,m (s i, j,s i,k ). If one of the points in the set {P i,p j,p k } does not belong to an other tetrahedron in the strip, it can be removed from the strip provided that a rod connecting P l and P m is introduced with the double-valued length given b () [Fig. -right]. This reduces the number of tetrahedra in the strip b one. Then, b repeating this operation until the strip contains onl two tetrahedra, the distance between the tetrahedral strip endpoints is finall obtained as a n valued function, where n is the number of tetrahedra in the strip. To obtain the closure condition as a polnomial in terms of a given rod length, the first step consists in taking the numerator of the rational form of the obtained function and then clearing radicals. As radicals will appear nested, the are cleared using an iterative process starting from the outer one. At each step of this process, the epressions involving a radical will have the general form a + a 1 p r + a ( p r) + a 3 ( p r) 3 + =, (3)

4 F. Thomas and J. M. Porta which can be rewritten as (a + a r + a r +...)+ p r(a 1 + a 3 r + a r +...)=. () This equation can be unfolded into two equations, one for each sign of p r. Since we are interested in the roots of both equations, we obtain their product, which can be written as (a + a r + a r +...) r(a 1 + a 3 r + a r +...) =. () While clearing radicals as eplained above introduces no etraneous roots, one cannot epect for the obtained polnomial to be of minimal degree. This is due to the presence of singularities of the formulation. Indeed, let us suppose that the closure polnomial is epressed in terms of the squared rod length s i, j. If a rod with variable length belongs to a shared face, this face degenerates for some values of s i, j. When this happens, the three points defining the face get aligned and the tetrahedral strip can be decomposed into two parts so that one can freel rotate with respect the other about the ais defined b these three aligned points. As we will see, terms corresponding to these degenerate configurations will appear in the closure polnomial. The can be easil removed b iterativel dividing the closure polnomial b them until the remainder is not null. 3 Eamples Net, we appl the technique eplained above to solve the foward kinematics of a decoupled platform and a - platform with planar base and platform (see Fig. 3 and Fig., respectivel). The decoupled platform owes its name to the fact that three legs permit the rotation of the platform about a point whose location is controlled b 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 closing polnomial of degree on a squared variable was first derived in []. In [7], this derivation is simplified b using the closure polnomial of the so-called double banana. Despite the simpler derivation, variable eliminations were still necessar. For the chosen - platform with planar base and platform, a minimal 1th-degree closure polnomial was first derived in [3]. The derivation was far from trivial and applicable onl to this particular platform. To properl compare our results with those reported in [7] and [3], we use the same numerical eamples. First, let us consider the decoupled platform defined b the squared distance matri S appearing in Fig. 3, where s i, j = S(i, j). It can be topologicall described as the strip of tetrahedra shown in Fig. -left. Appling the substitution presented in the previous section three times (see Fig. ), we have

5 Closure Polnomials for Strips of Tetrahedra P P 7 P P 1 3 9?? ?? 9 1? 1? S = B? 1? A 1?? P Fig. 3 Decoupled parallel manipulator, with non-planar moving platform, used as eample. Ψ 3,,,,7 (s 3, ) Ψ 3,,,,7(s 3,) P P 7 P P 7 P 7 P P P P P P P Ψ,3,,,7 (s 3,7,s 3, ) Fig. The decoupled parallel platform in Fig. 3 can be topologicall described as the strip of four tetrahedra in which the distance between and P is variable and the distance between its endpoints, and P 7, is known. The application of the substitution rule presented in Section to this strip (left) permits to sequentiall eliminate P (center) and P (right). s 3,7 = Y 3,,,,7 (s 3, ), () s,7 = Y,3,,,7 (s 3,7,s 3, ), (7) s 1,7 = Y 1,,3,,7 (s,7,s 3,7 ). () The numerator of the rational form resulting from substituting () in (7), and the result in (), can be written as: where R 1 13.R + 799R s 3, + 1R 3 s 3, 33733s 3, + 71s 3, 3 R 3 s 3, + 77 =, q R 1 = 771R + 97R R 3 s 3, 7917R R 3 s 3, +..., q R = 1R 3 s 3, 19771R 3 s 3, R 3..., q R 3 = 31s 3, + 971s 3, 71. The full epressions for R 1 and R are not included here due to space limitations.

6 F. Thomas and J. M. Porta Now, clearing the radicals as described in Section, we obtain a polnomial of th-degree. It is not of minimal degree because the rod connecting and P belongs to the shared face defined b, P, and P which is singular when D(3,,) =, that is, when s 3, 1s 3, + 91 =. B iterativel dividing the obtained polnomial b this singular factor until the remainder is not null, we get s 1 3, 1. 1 s 1 3, s 1 3,.9 1 s 13 3, s 1 3, s 11 3, s 1 3, s 9 3, s 3, s 7 3, s 3,.1 1 s 3, s 3,. 1 s 3 3, s 3, s 3, , which coincides with the closure polnomial reported in [7], but obtained in a much simpler wa. P P P 1.19 P 7 P ??? s, 1.917??? ?? 1 s, 1.? ?? ? s,7.71 B?? C s,7.71 A.17??? Fig. - parallel manipulator used as eample. Since the base and the moving platform are conve planar quadrilaterals, s, and s,7 are unambiguousl determined b the other known distances. As a second eample, let us consider the - parallel platform appearing in Fig.. Its forward kinematics is known to have solutions [3]. However, the can be split in two sets that are smmetric with respect to the base. Since the distance-based formulation is invariant to this smmetr, we will get a 1th-degree closure polnomial. The two sets of configurations are obtained in the coordinatiation process using trilateration [,, ]. Appling the substitution presented in the previous section four times (see Fig. ), we have that s, = Y,,,7, (s, ), s 3, = Y 3,,,, (s,,s, ), s, = Y,3,,, (s 3,,s, ), s 1, = Y 1,,3,, (s,,s 3,,s, ). After a proper sequence of forward substitutions in the above four equations, s 1, can be epressed onl in terms of s,. Since this parallel platform has planar base and platform, Y,,,7, and Y 1,,3,, are single-valued functions. Onl Y 3,,,, and Y,3,,, contribute with square roots to the obtained closure condition. Eliminating them as eplained leads to a nd-degree polnomial in s,. In this case, the rod connecting P and P belongs to two shared faces (the ones defined b P P P and P P ), whose associated singular terms are s, 7.11s, , and

7 Closure Polnomials for Strips of Tetrahedra 7 P P 7 P P P P P P P P P Ψ,,,7, (s, ) Ψ 3,,,, (s,,s, ) Ψ 3,,,, (s,,s, ) P P P P P P P Ψ,,,7, (s, ) Ψ,,,7, (s, ) Ψ,3,,, (s 3,,s, ) Fig. The - parallel manipulator in Fig. can be topologicall described as the strip of five tetrahedra in which the distance between P and P is variable and the distance between its endpoints, and P, is known. The application of the substitution rule to this strip (top left) permits to sequentiall eliminate P 7 (top right), P (bottom left), and P (bottom right) s, After iterativel dividing the obtained polnomial b these two factors until the remainder is not null, the following 1th-degree polnomial is obtained s, s 1, s 11, s 1, s 9, s, s 7, s, s, s, s 3, s,.3 1 S This polnomial has si roots that lead to real configurations of the moving platform obtained b coordinatiation via trilaterations [,]. These roots and the corresponding configurations appear in Fig. 7. The coincide with the solutions reported in [3] obtained using an ad hoc intricate method. Conclusions It has been eplained how to obtain closure polnomials for tetrahedral strips in terms of the involved rod lengths and the distance between the strip endpoints, and how this technique can be applied to solve some position analsis problems. However, this technique cannot incorporate orientation constraints between tetrahedra at different parts of the strip. As a consequence, if applied to a case in which such

8 F. Thomas and J. M. Porta P P P P 1 1 P s, =. s, = 3.13 P 1 1 P P 1 1 P P 1 P s, = s, = 171. P 1 1 P 1 1 P P 1 P P s, = 13. s, = 13.3 P 1 1 Fig. 7 Forward kinematics solutions of the - manipulator used as eample. The mirror configurations with respect to the base are also solutions, but the are not represented. constraints are necessar, the obtained closure polnomial would not be of minimaldegree as some of its roots would violate these constraints. Despite this important limitation, it supersedes the method presented in [7] in scope and simplicit, thus providing a better starting point for a complete generaliation to three dimensions of the techniques developed for the position analsis of planar linkages using Distance Geometr.

9 Closure Polnomials for Strips of Tetrahedra 9 References 1. Havel, T.: Some eamples of the use of distances as coordinates for Euclidean geometr. Journal of Smbolic Computation 11(-), (1991). Innocenti, C., Parenti-Castelli, V.: Echelon form solution of direct kinematics for the general full-parallel spherical wrist. Mechanism and Machine Theor, 3 1 (1993) 3. Lin, W., Griffis, M., Duff, J.: Forward displacement analses of the - Stewart platforms. ASME Journal of Mechanical Design 11(3), (199). Porta, J.M., Ros, L., Thomas, F.: Inverse kinematics b distance matri completion. In: International Workshop on Computational Kinematics (). Porta, J.M., Ros, L., Thomas, F.: On the trilaterable si-degree-of-freedom parallel and serial manipulators. In: IEEE International Conference on Robotics and Automation, pp (). Porta, J.M., Ros, L., Thomas, F., Torras, C.: A branch-and-prune algorithm for solving sstems of distance constraints. In: IEEE International Conference on Robotics and Automation, pp. 3 3 (3) 7. Rojas, N., Thomas, F.: The closure condition of the double banana and its application to robot position analsis. In: International Conference on Robotics and Automation, pp. 1 (13). Thomas, F., Ros, L.: Revisiting trilateration for robot localiation. IEEE Transactions on Robotics 1(1), ()