New Methods for Characterizing Phases of 2D Supersymmetric Gauge Theories

We study the physics of two-dimensional N=(2,2) gauged linear sigma models (GLSMs) via the two-sphere partition function. We show that the classical phase boundaries separating distinct GLSM phases, which are described by the secondary fan construction for abelian GLSMs, are completely encoded in the analytic structure of the partition function. The partition function of a non-abelian GLSM can be obtained as a limit from an abelian theory; we utilize this fact to show that the phases of non-abelian GLSMs can be obtained from the secondary fan of the associated abelian GLSM. We prove that the partition function of any abelian GLSM satisfies a set of linear differential equations; these reduce to the familiar A-hypergeometric system of Gel'fand, Kapranov, and Zelevinski for GLSMs describing complete intersections in toric varieties. We develop a set of conditions that are necessary for a GLSM phase to admit an interpretation as the low-energy limit of a non-linear sigma model with a Calabi-Yau threefold target space. Through the application of these criteria we discover a class of GLSMs with novel geometric phases corresponding to Calabi-Yau manifolds that are branched double-covers of Fano threefolds. These criteria provide a promising approach for constructing new Calabi-Yau geometries.Comment: 25 pages + references, appendices. v2: references added, typos corrected. v3: two small typos correcte

Abstraction and Control for Groups of Robots

This paper addresses the general problem of controlling a large number of robots required to move as a group. We propose an abstraction based on the definition of a map from the configuration space Q of the robots to a lower dimensional manifold A, whose dimension is independent of the number of robots. In this paper, we focus on planar fully actuated robots. We require that the manifold has a product structure A = G x S, where G is a Lie group, which captures the position and orientation of the ensemble in the chosen world coordinate frame, and S is a shape manifold, which is an intrinsic characterization of the team describing the “shape” as the area spanned by the robots. We design decoupled controllers for the group and shape variables. We derive controllers for individual robots that guarantee the desired behavior on A. These controllers can be realized by feedback that depends only on the current state of the robot and the state of the manifold A. This has the practical advantage of reducing the communication and sensing that is required and limiting the complexity of individual robot controllers, even for large numbers of robots

An SVD-Based Projection Method for Interpolation on \u3ci\u3eSE\u3c/i\u3e(3)

This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA+(n), a subgroup of the affine group in Rn and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA+(n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods

Object Closure and Manipulation by Multiple Cooperating Mobile Robots

We address the manipulation of planar objects by multiple cooperating mobile robots using the concept of Object Closure. In contrast to Form or Force Closure, Object Closure is a condition under which the object is trapped so that there is no feasible path for the object from the given position to any position that is beyond a specified threshold distance. Once Object Closure is achieved, the robots can cooperatively drag or flow the trapped object to the desired goal. In this paper, we define object closure and develop a set of decentralized algorithms that allow the robots to achieve and maintain object closure. We show how simple, first-order, potential field based controllers can be used to implement multirobot manipulation tasks

Workspace delineation of cable-actuated parallel manipulators

While there is extensive literature available on parallel manipulators in general, there has been much less attention given to cable-driven parallel manipulators. In this paper, we address the problem of analyzing the reachable workspace using the tools of semi-definite programming. We build on earlier work [1,2] done using similar techniques by deriving limiting conditions that allow us to compute analytic expressions for the boundary of the reachable workspace. We illustrate this computation for a planar parallel manipulator with four actuators

Motion generation for groups of robots: a centralized, geometric approach

We develop a method for generating smooth trajectories for a set of mobile robots. We show that, given two end configurations of the set of robots, by tuning one parameter, the user can choose an interpolating trajectory from a continuum of curves varying from that corresponding to maintaining a rigid formation to motion of the robots toward each other. The idea behind this method is to change the original constant kinetic energy metric in the configuration space and can be summarized into three steps. First, the energy of the motion as a rigid structure is decoupled from the energy of motion along directions that violate the rigid constraints. Second, the metric is shaped by assigning different weights to each term, and, third, geodesic flow is constructed for the modified metric. The optimal motions generated on the manifolds of rigid body displacements in 3-D space (SE(3)) or in plane (SE(2)) and the uniform rectilinear motion of each robot corresponding to a totally uncorrelated approach are particular cases of our general treatment

An Approach to Simultaneous Control of Trajectory and Interaction Forces in Dual-Arm Configurations

Multiple arm systems, multifingered grippers, and walking vehicles all have two common features. In each case, more than one actively coordinated articulation interacts with a passive object, thus forming one or more closed chains. For example, when two arms grasp an object simultaneously, the arms together with the object and the ground (base) form a closed chain. This induces kinematic and dynamic constraints and the resulting equations of motion are extremely nonlinear and coupled. Furthermore, the number of actuators exceeds the kinematic mobility of the chain in a typical case, which results in an underdetermined system of equations. An approach to control such constrained dynamic systems is described in this short paper. The basic philosophy is to utilize a minimal set of inputs to control the trajectory and the surplus inputs to control the constraint or interaction forces and moments in the closed chain. A dynamic control model is derived for the closed chain that is suitable for designing a controller, in which the trajectory as well as the interaction forces and moments are explicitly controlled. Nonlinear feedback techniques derived from differential geometry are then applied to linearize and decouple the nonlinear model. In this paper, these ideas are illustrated through a planar example in which two arms are used for cooperative manipulation. Results from a simulation are used to illustrate the efficacy of the method

Dynamics and Generation of Gaits for a Planar Rollerblader

We develop the dynamic model for a planar ROLLERBLADER. The robot consists of a rigid platform and two planar, two degree-of-freedom legs with in-line skates at the foot. The dynamic model consists of two unicycles coupled through the rigid body dynamics of the planar platform. We derive the Lagrangian reduction for the ROLLERBLADING robot. We show the generation of some simple gaits that allow the platform to move forward and rotate by using cyclic motions of the two legs