Morse-Bott functions on orthogonal groups

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B$. We give a simplified treatment of T. Frankel's analysis of the linear trace function on $SO(n)$, as well as a combinatorial explanation of the relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysis. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod $2$ Betti numbers of $SO(n)$.Comment: 28 page

Assembly as a noncooperative game of its pieces: analysis of 1D sphere assemblies

We propose an event-driven algorithm for the control of simple robot assembly problems based on noncooperative game theory. We examine rigorously the simplest setting — three bodies with one degree of freedom and offer extensive simulations for the 2 DOF extension. The initial analysis and the accompanying simulations suggest that this approach may indeed, offer an attractive means of building robust event driven assembly systems

Assembly as a noncooperative game of its pieces: the case of endogeneous disk assemblies

We propose an event-driven approach to planning and control of robot assembly problems using ideas from non-cooperative game theory. We report on the results of an extensive simulation study for a very simple two degree of freedom case - the arrangement of disks on a plane by a disk shaped robot

Event Driven Parts Moving in 2D Endogenuous Environments

This paper is concerned with the parts’ moving problem based on an event-driven planning and control. We are interested in developing feedback based approaches to the automatic generation of actuator commands that cause the robot to move a set of parts from an arbitrary initial disassembled configuration to a specif ed final configuration. In the Phase I of this project, a composite algorithm that reactively switches between different feedback controllers has been shown to induce a noncooperative game being played among the parts being manipulated. This paper describes experimental results with EDAR - Event-Driven Assembler Robot - developed for moving parts based on feedback techniques. For more information: Kod*La

Feedback-Based Event-Driven Parts Moving

A collection of unactuated disk-shaped parts must be brought by an actuated manipulator robot into a specified configuration from arbitrary initial conditions. The task is cast as a noncooperative game played among the parts—which in turn yields a feedback-based event-driven approach to plan generation and execution. The correctness of this approach, an open question, has been demonstrated in simpler settings and is further suggested by the extensive experiments reported here using an actual working implementation with EDAR—a mobile robot operating in a purely feedback-based event-driven manner. These results verify the reliability of this approach against uncertainties in sensory information and unanticipated changes in workspace configuration

On the Coordinated Navigation of Multiple Independent Disk-Shaped Robots

This paper addresses the coordinated navigation of multiple independently actuated disk-shaped robots - all placed within the same disk-shaped workspace. Assuming perfect sensing, shared centralized communications and computation, as well as perfect actuation, we encode complete information about the goal, obstacles and workspace boundary using an artificial potential function over the cross product space of the robots’ simultaneous configurations. The closed-loop dynamics governing the motion of each robot take the form of the appropriate projection of the gradient of this function. We show, with some reasonable restrictions on the allowable goal positions, that this function is an essential navigation function - a special type of artificial potential function that is ensured of connecting the kinematic planning with the dynamic execution in a manner that guarantees collision-free navigation of each robot to its destination from almost all initial free placements. We summarize the results of an extensive simulation study investigating such practical issues as average resulting trajectory length and robustness against simulated sensor noise

EDAR - mobile robot for parts moving based on a game-theoretic approach

EDAR (event-driven assembler robot) — a mobile robot capable of moving a collection of disk-shaped parts located on a two-dimensional workspace from an arbitrary initial configuration to a desired configuration while avoiding collisions in a purely reactive manner, is presented. Since EDAR uses a higher-level scheduler to switch among the subtasks of moving individual parts, it is viewed as mediating a noncooperative game played among the parts

Coordinated Navigation of Multiple Independent Disk-Shaped Robots

This paper addresses the coordinated navigation of multiple independently actuated disk-shaped robots-all placed within the same disk-shaped workspace. Assuming perfect sensing, shared-centralized communications and computation, as well as perfect actuation, we encode complete information about the goal, obstacles, and workspace boundary using an artificial potential function over the configuration space of the robots’ simultaneous nonoverlapping positions. The closed-loop dynamics governing the motion of each (velocity-controlled) robot take the form of the appropriate projection of the gradient of this function. We impose (conservative) restrictions on the allowable goal positions that yield sufficient conditions for convergence: We prove that this construction is an essential navigation function that guarantees collision-free motion of each robot to its destination from almost all initial free placements. The results of an extensive simulation study investigate practical issues such as average resulting trajectory length and robustness against simulated sensor noise. For more information: Kod*La

IEEE ACCESS SPECIAL SECTION EDITORIAL: REAL-TIME MACHINE LEARNING APPLICATIONS IN MOBILE ROBOTICS

In the last ten years, advances in machine learning methods have brought tremendous developments to the field of robotics. The performance in many robotic applications such as robotics grasping, locomotion, human–robot interaction, perception and control of robotic systems, navigation, planning, mapping, and localization has increased since the appearance of recent machine learning methods. In particular, deep learning methods have brought significant improvements in a broad range of robot applications including drones, mobile robots, robotics manipulators, bipedal robots, and self-driving cars. The availability of big data and more powerful computational resources, such as graphics processing units (GPUs), has made numerous robotic applications feasible which were not possible previously