Formal Foundations of 3D Geometry to Model Robot Manipulators

International audienceWe are interested in the formal specification of safety properties of robot manipulators down to the mathematical physics. To this end, we have been developing a formalization of the mathematics of rigid body transformations in the COQ proof-assistant. It can be used to address the forward kinematics problem, i.e., the computation of the position and orientation of the end-effector of a robot manipulator in terms of the link and joint parameters. Our formalization starts by extending the Mathematical Components library with a new theory for angles and by developing three-dimensional geometry. We use these theories to formalize the foundations of robotics. First, we formalize a comprehensive theory of three-dimensional rotations, including exponentials of skew-symmetric matrices and quaternions. Then, we provide a formalization of the various representations of rigid body transformations: isometries, homogeneous representation, the Denavit-Hartenberg convention, and screw motions. These ingredients make it possible to formalize robot manipulators: we illustrate this aspect by an application to the SCARA robot manipulator

NF-κB: A lesson in family values

A set of mobile robots (represented as points) is distributed in the Cartesian plane. The collection contains an unknown subset of byzantine robots which are indistinguishable from the reliable ones. The reliable robots need to gather, i.e., arrive to a configuration in which at the same time, all of them occupy the same point on the plane. The robots are equipped with GPS devices and at the beginning of the gathering process they communicate the Cartesian coordinates of their respective positions to the central authority. On the basis of this information, without the knowledge of which robots are faulty, the central authority designs a trajectory for every robot. The central authority aims to provide the trajectories which result in the shortest possible gathering time of the healthy robots. The efficiency of a gathering strategy is measured by its competitive ratio, i.e., the maximal ratio between the time required for gathering achieved by the given trajectories and the optimal time required for gathering in the offline case, i.e., when the faulty robots are known to the central authority in advance. The role of the byzantine robots, controlled by the adversary, is to act so that the gathering is delayed and the resulting competitive ratio is maximized. The objective of our paper is to propose efficient algorithms when the central authority is aware of an upper bound on the number of byzantine robots. We give optimal algorithms for collections of robots known to contain at most one faulty robot. When the proportion of byzantine robots is known to be less than one half or one third, we provide algorithms with small constant competitive ratios. We also propose algorithms with bounded competitive ratio in the case where the proportion of faulty robots is arbitrary

Extreme enantiomeric discrimination of fluoroalkanes using deuterium NMR in chiral liquid crystalline media

The enantiomeric assay of fluoroalkanes using H-2-NMR in a chiral liquid crystalline medium is demonstrated, and at its limit the enantiomers of [5-H-2]-5-fluorodecane were successfully resolved.</p

Enantiomeric and enantiotopic analysis of cone-shaped compounds with C(3) and C(3)(v) symmetry using NMR spectroscopy in chiral anisotropic solvents

We describe the enantiomeric and enantiotopic analysis of the NMR spectra of compounds derived from the functionalized cone-shaped core, cyclotriveratrylenes (CTV), dissolved in weakly oriented lyotropic chiral liquid crystals (CLCs) based on organic solutions of poly-γ-benzyl-l-glutamate. The CTV core lacks prostereogenic as well as stereogenic tetrahedral centers. However, depending on the pattern of substitution, chiral and achiral compounds with different symmetries can be obtained. Thus, symmetrically nonasubstituted CTVs (C3 symmetry) are optically active and exhibit enantiomeric isomers, while symmetrically hexasubstituted (C3v symmetry) derivatives are prochiral and possess enantiotopic elements. In the first part we use 2H and 13C NMR to study two nonasubstituted (−OH or −OCH3) CTVs, where the ring methylenes are fully deuterated, and show for the first time that the observation of enantiomeric discrimination of chiral molecules with a 3-fold symmetry axis is possible in a CLC. It is argued that this discrimination reflects different orientational ordering of the M and P isomers, rather than specific chiral short-range solvent−solute interactions that may affect differently the magnetic parameters of the enantiomers or even their geometry. In the second part we present similar measurements on hexasubstituted CTV with flexible side groups (−OC(O)CH3 and the, partially deuterated bidentate, −OCH2CH2O−), having on the average C3v symmetry. No spectral discrimination of enantiotopic sites was detected for the −OC(O)CH3 derivative. This is consistent with a recent theoretical work (J. Chem. Phys. 1999, 111, 6890) that indicates that in C3v molecules no chiral discrimination between enantiotopic elements, based on ordering, is possible. In contrast, a clear splitting was observed in the 2H spectra of the enantiotopic deuterons of the side groups in the tri(dioxyethylene)−CTV. It is argued that this discrimination reflects different ordering characteristics of the various, rapidly (on the NMR time scale) interconverting conformers of this compound. Assuming two twisted structures for each of the dioxyethylene side groups, four different conformers are expected, comprising two sets of enantiomeric pairs with, respectively, C3 and C1 symmetries. Differential ordering and/or fractional population imbalance of these enantiomeric pairs leads to the observed spectral discrimination of sites in the side chains that on average form enantiotopic pairs