Advances in Robot Kinematics: Analysis and Control, 1998
We study architecturally shaky Stewart-Gough platforms (SGP) and characterize the geometry of its... more We study architecturally shaky Stewart-Gough platforms (SGP) and characterize the geometry of its anchor points. The main tool is the connection of geometric objects with mappings which preserve the structure of the problem. Here, the geometric way is the use of linear manifolds of correlations and quadratic transformations. By these methods we show that the anchor points have to be conjugate points with respect to 3-dimensional linear manifolds of correlations. This result is used to give all possible configurations of anchor points of architecturally shaky SGP.
Advances in Robot Kinematics: Analysis and Control, 1998
A Stewart-Gough platform (SGP) is architecturally shaky iff it remains shaky while subject to arb... more A Stewart-Gough platform (SGP) is architecturally shaky iff it remains shaky while subject to arbitrary displacements of the planes e and φ, wherein the anchor points of the platform are situated. We show that a smaller subset of displacements of e and φ exists which characterises the architectural shakiness of the platform. This subset consists of those displacements which transform e and φ into two fixed orthogonal planes.
Abstract We interpolate the helical motion σ by rational axial motions of degree three and four ζ... more Abstract We interpolate the helical motion σ by rational axial motions of degree three and four ζ. This will be done by interpolating one point path σ( p 0) by ζ( p 0). Then all point paths of σ are interpolated in the same way if we use the method given here.
Fibonacci numbers and the Golden Mean are numbers and thus 0-dimensional objects. Usually, they a... more Fibonacci numbers and the Golden Mean are numbers and thus 0-dimensional objects. Usually, they are visualized in the Euclidean plane using squares and rectangles in a spiral arrangement. The Golden Mean, as a ratio, is an affine geometric concept and therefore Euclidean visualizations are not mandatory. There are attempts to visualize the Fibonacci number sequence and Golden Spirals in higher dimensions [11], in Minkowski planes [12], [4] and in hyperbolic planes (again [4]). The latter has to replace the not existing squares by sequences of touching circles. This article aims at visualizations in all Cayley-Klein planes and makes use of three different visualization ideas: nested sets of squares, sets of touching circles and sets of triangles that are related to Euclidean right angled triangles.
There are many interesting publications with fascinating figures dealing with such ornaments (see... more There are many interesting publications with fascinating figures dealing with such ornaments (see [1]-[9]). We will present an approach to teach regular polyhedra and their ornaments even for undergraduates. It is an interesting topic to visualize the action of G with CAD-packages (see [10], [11]). We will work with an hierarchic block structure as implementation of the corresponding group of symmetry G. Additionally, the design of the motives trains geometric modeling and needs familiarity with spatial congruence transformations. All considerations can be performed directly in the 3-dimensional space. Former troubles to draw these ornaments by hand are replaced by the use of a CAD-package.1 In this paper we restrict our examples to the group G = O: This group O is the set of direct automorphic displacements of a regular octahedron (or equivalently a cube).
The paper is devoted to the generation of frameworks of polyhedra as ornaments of the octahedral ... more The paper is devoted to the generation of frameworks of polyhedra as ornaments of the octahedral group O .A n hierarchical block structure is used to implement the action of O in a CAD-package. The framework is generated by a starting (prismatic) rod as the motif. We will provide a range of examples and discuss the symmetry of the corresponding ornaments.
Advances in Robot Kinematics: Analysis and Control, 1998
We study architecturally shaky Stewart-Gough platforms (SGP) and characterize the geometry of its... more We study architecturally shaky Stewart-Gough platforms (SGP) and characterize the geometry of its anchor points. The main tool is the connection of geometric objects with mappings which preserve the structure of the problem. Here, the geometric way is the use of linear manifolds of correlations and quadratic transformations. By these methods we show that the anchor points have to be conjugate points with respect to 3-dimensional linear manifolds of correlations. This result is used to give all possible configurations of anchor points of architecturally shaky SGP.
Advances in Robot Kinematics: Analysis and Control, 1998
A Stewart-Gough platform (SGP) is architecturally shaky iff it remains shaky while subject to arb... more A Stewart-Gough platform (SGP) is architecturally shaky iff it remains shaky while subject to arbitrary displacements of the planes e and φ, wherein the anchor points of the platform are situated. We show that a smaller subset of displacements of e and φ exists which characterises the architectural shakiness of the platform. This subset consists of those displacements which transform e and φ into two fixed orthogonal planes.
Abstract We interpolate the helical motion σ by rational axial motions of degree three and four ζ... more Abstract We interpolate the helical motion σ by rational axial motions of degree three and four ζ. This will be done by interpolating one point path σ( p 0) by ζ( p 0). Then all point paths of σ are interpolated in the same way if we use the method given here.
Fibonacci numbers and the Golden Mean are numbers and thus 0-dimensional objects. Usually, they a... more Fibonacci numbers and the Golden Mean are numbers and thus 0-dimensional objects. Usually, they are visualized in the Euclidean plane using squares and rectangles in a spiral arrangement. The Golden Mean, as a ratio, is an affine geometric concept and therefore Euclidean visualizations are not mandatory. There are attempts to visualize the Fibonacci number sequence and Golden Spirals in higher dimensions [11], in Minkowski planes [12], [4] and in hyperbolic planes (again [4]). The latter has to replace the not existing squares by sequences of touching circles. This article aims at visualizations in all Cayley-Klein planes and makes use of three different visualization ideas: nested sets of squares, sets of touching circles and sets of triangles that are related to Euclidean right angled triangles.
There are many interesting publications with fascinating figures dealing with such ornaments (see... more There are many interesting publications with fascinating figures dealing with such ornaments (see [1]-[9]). We will present an approach to teach regular polyhedra and their ornaments even for undergraduates. It is an interesting topic to visualize the action of G with CAD-packages (see [10], [11]). We will work with an hierarchic block structure as implementation of the corresponding group of symmetry G. Additionally, the design of the motives trains geometric modeling and needs familiarity with spatial congruence transformations. All considerations can be performed directly in the 3-dimensional space. Former troubles to draw these ornaments by hand are replaced by the use of a CAD-package.1 In this paper we restrict our examples to the group G = O: This group O is the set of direct automorphic displacements of a regular octahedron (or equivalently a cube).
The paper is devoted to the generation of frameworks of polyhedra as ornaments of the octahedral ... more The paper is devoted to the generation of frameworks of polyhedra as ornaments of the octahedral group O .A n hierarchical block structure is used to implement the action of O in a CAD-package. The framework is generated by a starting (prismatic) rod as the motif. We will provide a range of examples and discuss the symmetry of the corresponding ornaments.
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Papers by Sybille Mick