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  • 1.
    Krasnov, Yakov
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Tkachev, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Idempotent Geometry in Generic Algebras2018In: Advances in Applied Clifford Algebras, ISSN 0188-7009, E-ISSN 1661-4909, Vol. 28, no 5, article id UNSP 84Article in journal (Refereed)
    Abstract [en]

    Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

  • 2.
    Lian, Binbin
    KTH, School of Industrial Engineering and Management (ITM), Machine Design (Dept.). Tianjin University, China.
    Geometric Error Modeling of Parallel Manipulators Based on Conformal Geometric Algebra2018In: Advances in Applied Clifford Algebras, ISSN 0188-7009, E-ISSN 1661-4909, Vol. 28, no 1, article id 30Article in journal (Refereed)
    Abstract [en]

    An approach for geometric error modeling of parallel manipulators (PMs) based on the visual representation and direct calculation of conformal geometric algebra is introduced in this paper. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then the error sources are separated in terms of joint perturbations and geometric errors. Next, motions and constraints of PMs are analyzed visually by their reciprocal property. Finally geometric error model of PMs are formulated considering the actuations and constraints. The merits of this new approach are twofold: (1) complete and continuous geometric error modeling can be achieved since finite motions are considered, (2) visual and analytical computation of motions and constraints are applied for transferring geometric errors from the open-loop chain to the PM. A 2-DoF rotational PM is applied to demonstrate the geometric error modeling process. Comparisons between simulation and analytical models show that this approach is highly effective.

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