Motion Planning for Unlabeled Discs with Optimality Guarantees

We study the problem of path planning for unlabeled (indistinguishable) unit-disc robots in a planar environment cluttered with polygonal obstacles. We introduce an algorithm which minimizes the total path length, i.e., the sum of lengths of the individual paths. Our algorithm is guaranteed to find a solution if one exists, or report that none exists otherwise. It runs in time $\tilde{O}(m^4+m^2n^2)$, where $m$ is the number of robots and $n$ is the total complexity of the workspace. Moreover, the total length of the returned solution is at most $\text{OPT}+4m$, where OPT is the optimal solution cost. To the best of our knowledge this is the first algorithm for the problem that has such guarantees. The algorithm has been implemented in an exact manner and we present experimental results that attest to its efficiency

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Algorithmic Motion Planning and Related Geometric Problems on Parallel Machines (Dissertation Proposal)

The problem of algorithmic motion planning is one that has received considerable attention in recent years. The automatic planning of motion for a mobile object moving amongst obstacles is a fundamentally important problem with numerous applications in computer graphics and robotics. Numerous approximate techniques (AI-based, heuristics-based, potential field methods, for example) for motion planning have long been in existence, and have resulted in the design of experimental systems that work reasonably well under various special conditions [7, 29, 30]. Our interest in this problem, however, is in the use of algorithmic techniques for motion planning, with provable worst case performance guarantees. The study of algorithmic motion planning has been spurred by recent research that has established the mathematical depth of motion planning. Classical geometry, algebra, algebraic geometry and combinatorics are some of the fields of mathematics that have been used to prove various results that have provided better insight into the issues involved in motion planning [49]. In particular, the design and analysis of geometric algorithms has proved to be very useful for numerous important special cases. In the remainder of this proposal we will substitute the more precise term of algorithmic motion planning by just motion planning

Unifying (Machine) Vision via Counterfactual World Modeling

Leading approaches in machine vision employ different architectures for different tasks, trained on costly task-specific labeled datasets. This complexity has held back progress in areas, such as robotics, where robust task-general perception remains a bottleneck. In contrast, "foundation models" of natural language have shown how large pre-trained neural networks can provide zero-shot solutions to a broad spectrum of apparently distinct tasks. Here we introduce Counterfactual World Modeling (CWM), a framework for constructing a visual foundation model: a unified, unsupervised network that can be prompted to perform a wide variety of visual computations. CWM has two key components, which resolve the core issues that have hindered application of the foundation model concept to vision. The first is structured masking, a generalization of masked prediction methods that encourages a prediction model to capture the low-dimensional structure in visual data. The model thereby factors the key physical components of a scene and exposes an interface to them via small sets of visual tokens. This in turn enables CWM's second main idea -- counterfactual prompting -- the observation that many apparently distinct visual representations can be computed, in a zero-shot manner, by comparing the prediction model's output on real inputs versus slightly modified ("counterfactual") inputs. We show that CWM generates high-quality readouts on real-world images and videos for a diversity of tasks, including estimation of keypoints, optical flow, occlusions, object segments, and relative depth. Taken together, our results show that CWM is a promising path to unifying the manifold strands of machine vision in a conceptually simple foundation

View-Invariant Regions and Mobile Robot Self-Localization

This paper addresses the problem of mobile robot self-localization given a polygonal map and a set of observed edge segments. The standard approach to this problem uses interpretation tree search with pruning heuristics to match observed edges to map edges. Our approach introduces a preprocessing step in which the map is decomposed into 'view-invariant regions' (VIRs). The VIR decomposition captures information about map edge visibility, and can be used for a variety of robot navigation tasks. Basing self-localization search on VIRs greatly reduces the branching factor of the search tree and thereby simplifies the search task. In this paper we define the VIR decomposition and give algorithms for its computation and for self-localization search. We present results of simulations comparing standard and VIR-based search, and discuss the application of the VIR decomposition to other problems in robot navigation

Model-Based Environmental Visual Perception for Humanoid Robots

The visual perception of a robot should answer two fundamental questions: What? and Where? In order to properly and efficiently reply to these questions, it is essential to establish a bidirectional coupling between the external stimuli and the internal representations. This coupling links the physical world with the inner abstraction models by sensor transformation, recognition, matching and optimization algorithms. The objective of this PhD is to establish this sensor-model coupling