P. Dietmaier. The Stewart-Gough platform of general geometry can have 40 real postures. In ARK, pages 716, Strobl, 1998.  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 ....

P. Dietmaier, The Stewart-Gough platform of general geometry can have 40 real postures, in Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, 1998, pp. 1-10.

....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 distances l 1 ; l 2 ; l 6 with all 40 positions real. Theorem 3. 5 (Dietmaier [14]) All 40 positions can be real. Dietmaier s method will nd future applications to other problems of this kind. He began with a formulation of the problem as a system of equations depending upon the distances l 1 ; l 2 ; l 6 . An initial solution for a given instance of the distances ....

P. Dietmaier, The Stewart-Gough platform of general geometry can have 40 real postures, in Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, 1998, pp. 1-10.

....of determining all possible endplate locations given the leg lengths has a long history. For generic choices of the mechanical parameters, the problem has forty isolated solutions, a fact rst established by continuation [18] and later proven analytically [24, 13] One of the more recent results [7] involved the demonstration (obtained by methods of numerical homotopy continuation) that platforms exist that have the forty real solutions. For special choices of the parameters, a Stewart Gough platform may have solution curves or other higher dimensional solution components, instead of only ....

P. Dietmaier. The Stewart-Gough platform of general geometry can have 40 real postures. In J. Lenarcic and M.L. Husty, editors, Advances in Robot Kinematics: Analysis and Control, pages 1-10. Kluwer Academic Publishers, Dordrecht, 1998.

....is that there will always be a robot geometry and a set of articular coordinates such that the number of real solutions of the direct kinematics will reach the maximum. 1 This has been shown for particular architectures [14, 15] and recently for the most general case of general Gough platform [5]. Still for many architectures the conjecture has still to be verified [6] Problem 5 is quite important in practice: for real applications solving the direct kinematics means to determine the current, real pose of the end effector of the robot. Consequently computing all the solutions is only ....

Dietmaier P. The Stewart-Gough platform of general geometry can have 40 real postures. In ARK, pages 7--16, Strobl, June 29- July 4, 1998. 2

....[12] and later with Pragacz reiterated this question in the context of ag manifolds [14] Interestingly, in every known case all solutions may be real. These include the classical problem of 3264 plane conics tangent to 5 plane conics [26] the 40 positions of the Stewart platform of robotics [5], the 12 lines mutually tangent to 4 spheres [36] the 12 rational plane cubics meeting 8 points in the plane [19] all problems of enumerating linear subspaces of a vector space satisfying special Schubert conditions [31] and certain problems of enumerating rational curves in Grassmannians [32] ....

P. Dietmaier, The Stewart-Gough platform of general geometry can have 40 real postures, in Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, 1998, pp. 1-10.

.... having specified position with respect to some general fixed figures [5] For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real [10] Recently, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real [2]. Similarly, given any problem of enumerating lines in projective space incident on some general fixed linear subspaces, there are real fixed subspaces such that each of the (finitely many) incident lines are real [13] Other examples are shown in [12, 14] and the case of 462 4 planes meeting 12 ....

P. Dietmaier, The Stewart-Gough platform of general geometry can have 40 real postures, in Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, 1998, pp. 1--10. CMP 99:01

....We refer to [2] 38] 64] and [60, 61] for a discussion of numerical techniques to deal with quadratic turning points. A remarkable application of real homotopies in the real world consists in the finding of the relevant parameters of a polynomial system to maximize the number of real roots, see [18] for the 40 real solutions for the Stewart Gough platform in mechanics. In [93] the use of homotopy continuation to deal with overdetermined and components of solutions is discussed. Geometrically one slices the components of solutions with as many random hyperplanes as the dimension of the ....

.... SAGBI homotopies for verifying a conjecture in real algebraic geometry [98] With the input planes chosen to osculate a rational normal curve, an instance with all 1,430 solutions real and isolated was solved in [114] The problem is relevant to control theory [90] stewgou40: mechanism design [18]) Whether the Stewart Gough parallel platform in robotics could have all its 40 solutions real was a notorious open problem, until recently, as it was solved by numerical continuation methods [18] The problem formulation in [18] is highly deficient: the mixed volume equals 1,536 whereas only 40 ....

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P. Dietmaier. The Stewart-Gough platform of general geometry can have 40 real postures. In J. Lenarcic and M.L. Husty, editors, Advances in Robot Kinematics: Analysis and Control, pages 1--10. Kluwer Academic Publishers, Dordrecht, 1998.

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P. Dietmaier. The Stewart-Gough platform of general geometry can have 40 real postures. In ARK, pages 716, Strobl, 1998.

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Peter Dietmaier. The stewart-gough platform of general geometry can have 40 real postures. In Advances in Robot Kinematics: Analysis and Control, pages 1-- 10, 1998.

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Dietmaier, P. 1998. The Stewart--Gough platform of general geometry can have 40 real postures. Proceedings of ARK, Strobl, Austria, June 29--July 4, pp. 7--16.

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P. Dietmaier. The stewart-gough platform of general geometry can have 40 real postures. In Advances in Robot Kinematics: Analysis and Control, pages 1--10, 1998.

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Peter Dietmaier. The stewart-gough platform of general geometry can have 40 real postures. In Advances in Robot Kinematics: Analysis and Control, pages 110, 1998.

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Peter Dietmaier. The stewart-gough platform of general geometry can have 40 real postures. In Advances in Robot Kinematics: Analysis and Control, pages 110, 1998.

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P. Dietmaier. The Stewart-Gough platform of general geometry can have 40 real postures. In ARK, pages 716, Strobl, 1998.

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