D. Lazard. Generalized Stewart Platform: How to compute with rigid motions? In IMACS Symp. on Symbolic Computation, pages 8588, Lille, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from algebraic geometry. One strength of this book is that it will provide a coherent presentation of work in these areas that is presently dispersed throughout the literature. We also plan to cover other topics such as eigenvalue inequalities [35, 30] parallel manipulators (the Stewart platform) [36, 16], and some problems from computational vision that may be studied with symbolic computation. Our focus on applications will keep the presentation of algebraic geometry straightforward. We plan to rely on accessible treatments of algebraic geometry (such as [15] for proofs of some results and ....

D. Lazard, Generalized Stewart platform: how to compute with rigid motions, in IMACS-SC'93, 1993.

....Question 3.4. For a given Stewart platform, how many (complex) positions are there for a generic choice of the distances l 1 ; l 2 ; l 6 How many of these can be real In the early 1990 s, several approaches (numerical experimentation [59] intersection theory [62] Gr obner bases [48], resultants [53] and algebra [54] each showed that there 14 FRANK SOTTILE are 40 complex positions of a general Stewart platform. The obviously practical question of how many positions could be real remained open until 1998, when Dietmaier introduced a novel method to nd a value of the ....

D. Lazard, Generalized Stewart platform: how to compute with rigid motions?, in IMACS-SC'93, 1993.

....other problems. Indeed we cannot use directly the results stated in [Laz83] since they require a lot of monomials to appear in the system, and we cannot ensure that these monomials are actually present. 4. 2 Algebraic parameterization of the FKP We use a formulation of the FKP proposed by Lazard [Laz93], because it introduces a small number of monomials and some linear equations. Some joints on the platform are used to form a basis of this platform. A set B of 4 joints is needed in the case of a three dimensional platform (another formulation using only 3 points is also proposed in [Laz93] but ....

....Lazard [Laz93] because it introduces a small number of monomials and some linear equations. Some joints on the platform are used to form a basis of this platform. A set B of 4 joints is needed in the case of a three dimensional platform (another formulation using only 3 points is also proposed in [Laz93], but it introduces equations of larger degree) while a set B with only 3 joints is enough in the case of a planar platform. The unknowns will be the coordinates of the vectors A i B i for i 2 B: 8i 2 B A i B i = 0 x i y i z i 1 A For a planar platform B = f1; 2; 3g and there are 9 ....

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D. Lazard. Generalized Stewart Platform: How to compute with rigid motions? In IMACS, pages 8588, Lille, France, May 1993.

....has to be done only once, is less than 3 seconds. It thus appears that the camera motion reconstruction can be done on line in real time using the generalized Dixon resultants. 6. 4 Kinematics: Stewart s Platform The Stewart platform problem is a well known benchmark from robotics and kinematics [27, 43, 45, 21]. It is a parallel manipulator with six prismatic joints connecting two platforms, in which the base platform is fixed, while the top platform, or end effector, is moving in 3 dimensional space, controlled by the lengths of the joints. The quaternion formulation of the Stewart platform presented ....

.... operator from the above formulation in total computer time of less than 10 minutes (273:5 seconds to construct the Dixon matrix and 256:79 seconds to interpolate a projection operator from it) The projection operator is a polynomial of degree 40, consistent with similar results reported in [51, 43, 45]. The generalized Dixon resultant formulation is thus able to compute the exact resultant, without any extraneous factors. To our knowledge, our results are the first successful attempt to compute the resultant for the Stewart platform problem using multivariate elimination methods in which no a ....

Lazard D., Generalized Stewart platform: How to compute with rigid motions? Proc. IMACS-SC, Lille, June 1993, 85-88.

....of C [D] is at most 40. The proof we are giving here is based on simple but magic computations on monomials. It is yet another proof for this problem. The first one by F. Ronga and T. Vust (see [16] used Intersection Theory on vector bundles, Chern classes, etc. The second approach (see [14]) used Grobner bases to compute very quickly the polynomials we have shown in the previous section and to deduce the Hilbert function. The third one (see [15] uses an explicit computation of a resultant (by a Computer Algebra System) to conclude that the dimension is 40. The proof we are giving ....

D. Lazard. Generalized Stewart platform: How to compute with rigid motions? In IMACS -SC'93, 1993.

....in the past few years, which is exact because there exist particular instances with as many complex solutions. Ronga and Vust [RV92] offered the first proof of 40 being an upper bound by applying intersection theory on vector bundles and using Chern classes. Simpler proofs were proposed in [Mou93, Laz93] and a probabilistic one in [Rag93] A second proof by Mourrain [Mou94] views this problem as well as the computation of the camera motion from 5 point matches as special cases of a general problem on displacements. Wampler [Wam94] arrives at the upper bound by applying the multi homogeneous ....

D. Lazard. Generalized Stewart platform: How to compute with rigid motions? In Proc. IMACS-SC, 1993.

....polynomial systems for which the generalized Dixon formulation computes the exact resultant. In this direction, we identify families of systems, both unmixed and mixed, for which the projection operator contains no extraneous factors. As an example, we discuss the Stewart Platform problem [12]. We show how the generalized Dixon method can be used to compute the exact resultant of degree 40 of the Stewart Platform problem. This is accomplished by establishing the existence of a variable ordering under which the projection operator is of degree 40 and hence, has no extraneous factor [15, ....

....[12] We show how the generalized Dixon method can be used to compute the exact resultant of degree 40 of the Stewart Platform problem. This is accomplished by establishing the existence of a variable ordering under which the projection operator is of degree 40 and hence, has no extraneous factor [15, 12, 13], and is exactly its resultant. To the best of our knowledge, it is the first time that a general resultant method has been directly applied to this problem without producing any extraneous factor. The results reported in this paper extend our earlier results about the generalized Dixon ....

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Lazard D., Generalized Stewart Platform: How to compute with rigid motions?, Proc. IMACS-SC, 1993.

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D. Lazard. Generalized Stewart Platform: How to compute with rigid motions? In IMACS Symp. on Symbolic Computation, pages 8588, Lille, 1993.

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D. Lazard. Generalized Stewart Platform: How to compute with rigid motions? In IMACS Symp. on Symbolic Computation, pages 8588, Lille, 1993.

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