arXiv subjects

Georg Nawratil

Publications and source records attributed to Georg Nawratil.

35 records · Page 2Linked to original sources

arXiv2021

On the snappability and singularity-distance of frameworks with bars and triangular plates

In a recent article the author presented a method to measure the snapping capability -- shortly called snappability -- of bar-joint frameworks based on the total elastic strain energy by computing the deformation of all bars using Hooke's law and the definition of Cauchy/Engineering strain. Within the paper at hand, we extend this approach to isostatic frameworks composed of bars and triangular plates by using the physical concept of Green-Lagrange strain. An intrinsic pseudometric based on the resulting total elastic strain energy density cannot only be used for evaluating the snappability but also for measuring the distance to the closest singular configuration. The presented methods are demonstrated on the basis of the 3-legged planar parallel manipulator.

cs.CG↗

arXiv2020

On homotopy continuation based singularity distance computations for 3-RPR manipulators

It is known that parallel manipulators suffer from singular configurations. Evaluating the distance between a given configuration to the closest singular one is of interest for industrial applications (e.g.\ singularity-free path planning). For parallel manipulators of Stewart-Gough type, geometric meaningful distance measures are known, which are used for the computation of the singularity distance as the global minimizer of an optimization problem. In the case of hexapods and linear pentapods the critical points of the corresponding polynomial Lagrange function cannot be found by the Grobner basis method due to the degree and number of unknowns. But this polynomial system of equations can be solved by software tools of numerical algebraic geometry relying on homotopy continuation. To gain experiences for the treatment of the mentioned spatial manipulators, this paper attempts to find minimal multi-homogeneous Bezout numbers for the homotopy continuation based singularity distance computation with respect to various algebraic motion representations of planar Euclidean/equiform kinematics. The homogenous and non-homogenous representations under study are compared and discussed based on the 3-RPR manipulator.

cs.CG↗

arXiv2020

Evaluating the snappability of bar-joint frameworks

It is well-known that there exist bar-joint frameworks (without continuous flexions) whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these snapping capability -- shortly called snappability -- based on the total elastic strain energy of the framework by computing the deformation of all bars using Hook's law. The presented theoretical results give further connections between shakiness and snapping beside the well-known technique of averaging and deaveraging.

cs.CG↗

arXiv2019

Variational Path Optimization of Linear Pentapods with a Simple Singularity Variety

The class of linear pentapods with a simple singularity variety is obtained by imposing architectural restrictions on the design in such a way that the manipulators singularity variety is linear in orientation position variables. It turns out that such simplification leads to crucial computational advantages while maintaining the machines applications in some fundamental industrial tasks such as five axis milling and laser cutting. We assume that a path between a given start and end pose of the end effector is known which is singularity free and within the manipulators workspace. An optimization process of the initial path is proposed in such a way that the parallel robot increases its distance to the singularity loci while the motion is being smoothed. In our case the computation time of the optimization is improved as we are dealing with pentapods having simple singularity varieties allowing a closed form solution for the local exterma of the singularity distance function. Formally this process is called variational path optimization which is the systematic optimization of a path by manipulating its variations of energy and distance to the obstacle which in this case is the singularity variety. In this process some physical limits of the mechanical joints are also taken into account.

math.OC↗

arXiv2018

Alternative interpretation of the Plücker quadric's ambient space and its application

It is well-known that there exists a bijection between the set of lines of the projective 3-dimensional space $P^3$ and all real points of the so-called Plücker quadric $Ψ$. Moreover one can identify each point of the Plücker quadric's ambient space with a linear complex of lines in $P^3$. Within this paper we give an alternative interpretation for the points of $P^5$ as lines of an Euclidean 4-space $E^4$, which are orthogonal to a fixed direction. By using the quaternionic notation for lines, we study straight lines in $P^5$ which correspond in the general case to cubic 2-surfaces in $E^4$. We show that these surfaces are geometrically connected with circular Darboux 2-motions in $E^4$, as they are basic surfaces of the underlying line-symmetric motions. Moreover we extend the obtained results to line-elements of the Euclidean 3-space $E^3$, which can be represented as points of a cone over $Ψ$ sliced along the 2-dimensional generator space of ideal lines. We also study straight lines of its ambient space $P^6$ and show that they correspond to ruled surface strips composed of the mentioned 2-surfaces with circles on it. Finally we present an application of this interpretation in the context of interactive design of ruled surfaces and ruled surface strips/patches based on the algorithm of De Casteljau.

cs.CG↗

arXiv2017

Linear Pentapods with a Simple Singularity Variety

There exists a bijection between the configuration space of a linear pentapod and all points $(u,v,w,p_x,p_y,p_z)\in\mathbb{R}^{6}$ located on the singular quadric $Γ: u^2+v^2+w^2=1$, where $(u,v,w)$ determines the orientation of the linear platform and $(p_x,p_y,p_z)$ its position. Then the set of all singular robot configurations is obtained by intersecting $Γ$ with a cubic hypersurface $Σ$ in $\mathbb{R}^{6}$, which is only quadratic in the orientation variables and position variables, respectively. This article investigates the restrictions to be imposed on the design of this mechanism in order to obtain a reduction in degree. In detail we study the cases where $Σ$ is (1) linear in position variables, (2) linear in orientation variables and (3) quadratic in total. The resulting designs of linear pentapods have the advantage of considerably simplified computation of singularity-free spheres in the configuration space. Finally we propose three kinematically redundant designs of linear pentapods with a simple singularity surface.

cs.RO↗

arXiv2017

Rational Parametrization of Linear Pentapod's Singularity Variety and the Distance to it

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via extendible legs to the base. This manipulator has five controllable degrees-of-freedom and the remaining one is a free rotation around the motion platform axis (which in fact is an axial spindle). In this paper we present a rational parametrization of the singularity variety of the linear pentapod. Moreover we compute the shortest distance to this rational variety with respect to a suitable metric. Kinematically this distance can be interpreted as the radius of the maximal singularity free-sphere.

cs.RO↗

arXiv2017

Kinematically Redundant Octahedral Motion Platform for Virtual Reality Simulations

We propose a novel design of a parallel manipulator of Stewart Gough type for virtual reality application of single individuals; i.e. an omni-directional treadmill is mounted on the motion platform in order to improve VR immersion by giving feedback to the human body. For this purpose we modify the well-known octahedral manipulator in a way that it has one degree of kinematical redundancy; namely an equiform reconfigurability of the base. The instantaneous kinematics and singularities of this mechanism are studied, where especially "unavoidable singularities" are characterized. These are poses of the motion platform, which can only be realized by singular configurations of the mechanism despite its kinematic redundancy.

cs.RO↗

arXiv2017

Kinematic interpretation of the Study quadric's ambient space

It is well known that real points of the Study quadric (sliced along a 3-dimensional generator space) correspond to displacements of the Euclidean 3-space. But we still lack of a kinematic meaning for the points of the ambient 7-dimensional projective space $P^7$. This paper gives one possible interpretation in terms of displacements of the Euclidean 4-space. From this point of view we also discuss the extended inverse kinematic map, motions corresponding to straight lines in $P^7$ and linear complexes of SE(3)-displacements. Moreover we present an application of this interpretation in the context of interactive motion design.

cs.CG↗

arXiv2016

Mobile Icosapods

Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected by a number of other rigid bodies, called legs, that are anchored via spherical joints. It is possible to prove that the maximal number of legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique to construct examples of such 20-pods, but could not constrain the legs to have base and platform points with real coordinates. We show that Borel's construction yields all mobile 20-pods, and that it is possible to construct examples with all real coordinates.

cs.RO↗

arXiv2016

Addendum to Pentapods with Mobility 2

In a foregoing publication the authors studied pentapods with mobility 2, where neither all platform anchor points nor all base anchor points are located on a line. It turned out that the given classification is incomplete. This addendum is devoted to the discussion of the missing cases resulting in additional solutions already known to Duporcq.

cs.RO↗

arXiv2015

On the line-symmetry of self-motions of linear pentapods

We show that all self-motions of pentapods with linear platform of Type 1 and Type 2 can be generated by line-symmetric motions. Thus this paper closes a gap between the more than 100 year old works of Duporcq and Borel and the extensive study of line-symmetric motions done by Krames in the 1930's. As a consequence we also get a new solution set for the Borel Bricard problem. Moreover we discuss the reality of self-motions and give a sufficient condition for the design of linear pentapods of Type 1 and Type 2, which have a self-motion free workspace.

cs.RO↗

arXiv2015

Self-motions of pentapods with linear platform

We give a full classification of all pentapods with linear platform possessing a self-motion beside the trivial rotation about the platform. Recent research necessitates a contemporary and accurate re-examination of old results on this topic given by Darboux, Mannheim, Duporcq and Bricard, which also takes the coincidence of platform anchor points into account. For our study we use bond theory with respect to a novel kinematic mapping for pentapods with linear platform, beside the method of singular-invariant leg-rearrangements. Based on our results we design pentapods with linear platform, which have a simplified direct kinematics concerning their number of (real) solutions.

cs.RO↗

arXiv2015

Liaison Linkages

The complete classification of hexapods - also known as Stewart Gough platforms - of mobility one is still open. To tackle this problem, we can associate to each hexapod of mobility one an algebraic curve, called the configuration curve. In this paper we establish an upper bound for the degree of this curve, assuming the hexapod is general enough. Moreover, we provide a construction of hexapods with curves of maximal degree, which is based on liaison, a technique used in the theory of algebraic curves.

cs.RO↗

arXiv2015

Möbius Photogrammetry

Motivated by results on the mobility of mechanical devices called pentapods, this paper deals with a mathematically freestanding problem, which we call Möbius Photogrammetry. Unlike traditional photogrammetry, which tries to recover a set of points in three-dimensional space from a finite set of central projection, we consider the problem of reconstructing a vector of points in $\mathbb{R}^3$ starting from its orthogonal parallel projections. Moreover, we assume that we have partial information about these projections, namely that we know them only up to Möbius transformations. The goal in this case is to understand to what extent we can reconstruct the starting set of points, and to prove that the result can be achieved if we allow some uncertainties in the answer. Eventually, the techniques developed in the paper allow us to show that for a pentapod with mobility at least two either some anchor points are collinear, or platform and base are similar, or they are planar and affine equivalent.

math.AG↗

arXiv2014

Pentapods with Mobility 2

In this paper we give a full classification of all pentapods with mobility 2, where neither all platform anchor points nor all base anchor points are located on a line. Therefore this paper solves the famous Borel-Bricard problem for 2-dimensional motions beside the excluded case of five collinear points with spherical trajectories. But even for this special case we present three new types as a side-result. Based on our study of pentapods, we also give a complete list of all non-architecturally singular hexapods with 2-dimensional self-motions.

cs.RO↗

arXiv2014

Bond theory for pentapods and hexapods

This paper deals with the old and classical problem of determining necessary conditions for the overconstrained mobility of some mechanical device. In particular, we show that the mobility of pentapods/hexapods implies either a collinearity condition on the anchor points, or a geometric condition on the normal projections of base and platform points. The method is based on a specific compactification of the group of direct isometries of $\mathbb{R}^3$.

cs.RO↗