Geometry-aware Manipulability Learning, Tracking and Transfer

Body posture influences human and robots performance in manipulation tasks, as appropriate poses facilitate motion or force exertion along different axes. In robotics, manipulability ellipsoids arise as a powerful descriptor to analyze, control and design the robot dexterity as a function of the articulatory joint configuration. This descriptor can be designed according to different task requirements, such as tracking a desired position or apply a specific force. In this context, this paper presents a novel \emph{manipulability transfer} framework, a method that allows robots to learn and reproduce manipulability ellipsoids from expert demonstrations. The proposed learning scheme is built on a tensor-based formulation of a Gaussian mixture model that takes into account that manipulability ellipsoids lie on the manifold of symmetric positive definite matrices. Learning is coupled with a geometry-aware tracking controller allowing robots to follow a desired profile of manipulability ellipsoids. Extensive evaluations in simulation with redundant manipulators, a robotic hand and humanoids agents, as well as an experiment with two real dual-arm systems validate the feasibility of the approach.Comment: Accepted for publication in the Intl. Journal of Robotics Research (IJRR). Website: https://sites.google.com/view/manipulability. Code: https://github.com/NoemieJaquier/Manipulability. 24 pages, 20 figures, 3 tables, 4 appendice

Gauge invariant finite size spectrum of the giant magnon

It is shown that the finite size corrections to the spectrum of the giant magnon solution of classical string theory, computed using the uniform light-cone gauge, are gauge invariant and have physical meaning. This is seen in two ways: from a general argument where the single magnon is made gauge invariant by putting it on an orbifold as a wrapped state obeying the level matching condition as well as all other constraints, and by an explicit calculation where it is shown that physical quantum numbers do not depend on the uniform light-cone gauge parameter. The resulting finite size effects are exponentially small in the $R$-charge and the exponent (but not the prefactor) agrees with gauge theory computations using the integrable Hubbard model.Comment: 12 pages, some clarifications, references adde

BCJ duality and double copy in the closed string sector

This paper is focused on the loop-level understanding of the Bern-Carrasco-Johansson double copy procedure that relates the integrands of gauge theory and gravity scattering amplitudes. At four points, the first non-trivial example of that construction is one-loop amplitudes in N=2 super-Yang-Mills theory and the symmetric realization of N=4 matter-coupled supergravity. Our approach is to use both field and string theory in parallel to analyze these amplitudes. The closed string provides a natural framework to analyze the BCJ construction, in which the left- and right-moving sectors separately create the color and kinematics at the integrand level. At tree level, in a five-point example, we show that the Mafra-Schlotterer-Stieberger procedure gives a new direct proof of the color-kinematics double copy. We outline the extension of that argument to n points. At loop level, the field-theoretic BCJ construction of N=2 SYM amplitudes introduces new terms, unexpected from the string theory perspective. We discuss to what extent we can relate them to the terms coming from the interactions between left- and right-movers in the string-theoretic gravity construction.Comment: 46 pages, 8 figures, 2 tables; v3 significantly revised published versio

Second Quantization of the Wilson Loop

Treating the QCD Wilson loop as amplitude for the propagation of the first quantized particle we develop the second quantization of the same propagation. The operator of the particle position $\hat{\cal X}_{\mu}$ (the endpoint of the "open string") is introduced as a limit of the large $N$ Hermitean matrix. We then derive the set of equations for the expectation values of the vertex operators \VEV{ V(k_1)\dots V(k_n)} . The remarkable property of these equations is that they can be expanded at small momenta (less than the QCD mass scale), and solved for expansion coefficients. This provides the relations for multiple commutators of position operator, which can be used to construct this operator. We employ the noncommutative probability theory and find the expansion of the operator $\hat{\cal X}_\mu$ in terms of products of creation operators $a_\mu^{\dagger}$. In general, there are some free parameters left in this expansion. In two dimensions we fix parameters uniquely from the symplectic invariance. The Fock space of our theory is much smaller than that of perturbative QCD, where the creation and annihilation operators were labelled by continuous momenta. In our case this is a space generated by $d = 4$ creation operators. The corresponding states are given by all sentences made of the four letter words. We discuss the implication of this construction for the mass spectra of mesons and glueballs.Comment: 41 pages, latex, 3 figures and 3 Mathematica files uuencode