Maria Alberich-Carramiñana

Flagged Parallel Manipulators

The conditions for a parallel manipulator to be flagged can be simply expressed in terms of linear dependencies between the coordinates of its leg attachments, both on the base and on the platform. These dependencies permit to describe the manipulator singularities in terms of incidences between two flags (hence, the name ldquoflaggedrdquo). Although these linear dependencies might look, at first glance, too restrictive, in this paper, the family of flagged manipulators is shown to contain large subfamilies of six-legged and three-legged manipulators. The main interest of flagged parallel manipulators is that their singularity loci admit a well-behaved decomposition with a unique topology irrespective of the metrics of each particular design. In this paper, this topology is formally derived and all the cells, in the configuration space of the platform, of dimension 6 (nonsingular) and dimension 5 (singular), together with their adjacencies, are worked out in detail.

Partially Flagged Parallel Manipulators: Singularity Charting and Avoidance

There are only three 6-SPS parallel manipulators with triangular base and platform, i.e., the octahedral, the flagged, and the partially flagged, which are studied in this paper. The forward kinematics of the octahedral manipulator is algebraically intricate, while those of the other two can be solved by three trilaterations. As an additional nice feature, the flagged manipulator is the only parallel platform for which a cell decomposition of its singularity locus has been derived. Here, we prove that the partially flagged manipulator also admits a well-behaved decomposition, technically called a stratification, some of whose strata are not topological cells, however. Remarkably, the adjacency diagram of the 5-D and 6-D strata (which shows what 5-D strata are contained in the closure of a 6-D one) is the same as for the flagged manipulator. The availability of such a decomposition permits devising a redundant 7-SPS manipulator, combining two partially flagged ones, which admits a control strategy that completely avoids singularities. Simulation results support these claims.

Stratifying the singularity loci of a class of parallel manipulators

Some in-parallel robots, such as the 3-2-1 and the 3/2 manipulators, have attracted attention because their forward kinematics can be solved by three consecutive trilaterations. In this paper, we identify a class of these robots, which we call flagged manipulators, whose singularity loci admit a well-behaved decomposition, i.e., a stratification, derived from that of the flag manifold. Two remarkable properties must be highlighted. First, the decomposition has the same topology for all members in the class, irrespective of the metric details of each particular robot instance. Thus, we provide explicitly all the singular strata and their connectivity, which apply to all flagged manipulators without any tailoring. Second, the strata can be easily characterized geometrically, because it is possible to assign local coordinates to each stratum (in the configuration space of the manipulator) that correspond to uncoupled rotations and/or translations in the workspace.

Multiplier ideals in two-dimensional local rings with rational singularities

The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is studied extensively.

The ultrametric space of plane branches

We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] ∩ ℚ is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.

An Algorithm for Computing the Singularity of the Generic Germ of a Pencil of Plane Curves

Abstract We give an algorithm that describes the singularity of all but finitely-many germs in a pencil generated by two arbitrary germs of plane curve.

Recovering epipolar direction from two affine views of a planar object

The mainstream approach to estimate epipolar geometry from two views requires matching the projections of at least four non-coplanar points in the scene, assuming a full projective camera model. Our work deviates from this in three respects: affine camera, planar scene and active contour tracking. A B-spline is fitted to a planar contour, which is tracked using a Kalman filter. The corresponding control points are used to compute the affine transformation between images. We prove that the affine epipolar direction can be computed as one of the eigenvectors of this affine transformation, provided camera motion is free of cyclorotation. A Staubli robot is used to obtain calibrated image streams, which are used as ground truth to evaluate the performance of the method, and to test its limiting conditions in practice. The fact that our method and the gold standard algorithm produce comparable results shows the potential of our proposal.

Affine epipolar direction from two views of a planar contour

Most approaches to camera motion estimation from image sequences require matching the projections of at least 4 non-coplanar points in the scene. The case of points lying on a plane has only recently been addressed, using mainly projective cameras. We here study what can be recovered from two uncalibrated views of a planar contour under affine viewing conditions. We prove that the affine epipolar direction can be recovered provided camera motion is free of cyclorotation. The proposed method consists of two steps: 1) computing the affinity between two views by tracking a planar contour, and 2) recovering the epipolar direction by solving a second-order equation on the affinity parameters. Two sets of experiments were performed to evaluate the accuracy of the method. First, synthetic image streams were used to assess the sensitivity of the method to controlled changes in viewing conditions and to image noise. Then, the method was tested under more realistic conditions by using a robot arm to obtain calibrated image streams, which permit comparing our results to ground truth.

On redundant flagged manipulators

Flagged in-parallel manipulators are attractive because their singularity loci admit a well-behaved decomposition, with a unique topology irrespective of the metrics of each particular design. In this paper, this topology is formally derived and all the cells, in the configuration space of the platform, of dimension 6 (non-singular) and dimension 5 (singular), together with their adjacencies, are worked out in detail. This characterization of the singularity loci is useful to come up with designs which admit control strategies free of singularities. In particular, it is shown that by adding an extra leg to any flagged manipulator, the resulting 7-leg structure admits a control strategy (by appropriately choosing which leg remains passive) that completely avoids singularities

Effective computation of base points of ideals in two-dimensional local rings

In this work we describe a minimal log-resolution of an ideal in a smooth complex surface from the minimal log-resolution of its generators.