Robot Motion Planning Under Topological Constraints

My thesis addresses the the problem of manipulation using multiple robots with cables. I study how robots with cables can tow objects in the plane, on the ground and on water, and how they can carry suspended payloads in the air. Specifically, I focus on planning optimal trajectories for robots. Path planning or trajectory generation for robotic systems is an active area of research in robotics. Many algorithms have been developed to generate path or trajectory for different robotic systems. One can classify planning algorithms into two broad categories. The first one is graph-search based motion planning over discretized configuration spaces. These algorithms are complete and quite efficient for finding optimal paths in cluttered 2-D and 3-D environments and are widely used [48]. The other class of algorithms are optimal control based methods. In most cases, the optimal control problem to generate optimal trajectories can be framed as a nonlinear and non convex optimization problem which is hard to solve. Recent work has attempted to overcome these shortcomings [68]. Advances in computational power and more sophisticated optimization algorithms have allowed us to solve more complex problems faster. However, our main interest is incorporating topological constraints. Topological constraints naturally arise when cables are used to wrap around objects. They are also important when robots have to move one way around the obstacles rather than the other way around. Thus I consider the optimal trajectory generation problem under topological constraints, and pursue problems that can be solved in finite-time, guaranteeing global optimal solutions. In my thesis, I first consider the problem of planning optimal trajectories around obstacles using optimal control methodologies. I then present the mathematical framework and algorithms for multi-robot topological exploration of unknown environments in which the main goal is to identify the different topological classes of paths. Finally, I address the manipulation and transportation of multiple objects with cables. Here I consider teams of two or three ground robots towing objects on the ground, two or three aerial robots carrying a suspended payload, and two boats towing a boom with applications to oil skimming and clean up. In all these problems, it is important to consider the topological constraints on the cable configurations as well as those on the paths of robot. I present solutions to the trajectory generation problem for all of these problems

A Topological Approach to Workspace and Motion Planning for a Cable-controlled Robot in Cluttered Environments

There is a rising demand for multiple-cable controlled robots in stadiums or warehouses due to its low cost, longer operation time, and higher safety standards. In a cluttered environment the cables can wrap around obstacles. Careful choice needs to be made for the initial cable congurations to ensure that the workspace of the robot is optimized. The presence of cables makes it imperative to consider the homotopy classes of the cables both in the design and motion planning problems. In this thesis we study the problem of workspace planning for multiple-cable controlled robots in an environment with polygonal obstacles. This goal of this thesis is to establish a relationship between the workspace\u27s boundary and cable congurations of such robots, and solve related optimization and motion planning problems. We rst analyze the conditions under which a conguration of a cable-controlled robot can be considered valid, then discuss the relationship between cable conguration, the robot\u27s workspace and its motion state, and finally use graph search based motion planning in h-augmented graph to perform workspace optimization and to compute optimal paths for the robot. We demonstrated corresponding algorithms in simulations

Optimal Path Planning in Distinct Topo-Geometric Classes using Neighborhood-augmented Graph and its Application to Path Planning for a Tethered Robot in 3D

Many robotics applications benefit from being able to compute multiple locally optimal paths in a given configuration space. Examples include path planning for of tethered robots with cable-length constraints, systems involving cables, multi-robot topological exploration & coverage, and, congestion reduction for mobile robots navigation without inter-robot coordination. Existing paradigm is to use topological path planning methods that can provide optimal paths from distinct topological classes available in the underlying configuration space. However, these methods usually require non-trivial and non-universal geometrical constructions, which are prohibitively complex or expensive in 3 or higher dimensional configuration spaces with complex topology. Furthermore, topological methods are unable to distinguish between locally optimal paths that belong to the same topological class but are distinct because of genus-zero obstacles in 3D or due to high-cost or high-curvature regions. In this paper we propose an universal and generalized approach to multi-class path planning using the concept of a novel neighborhood-augmented graph, search-based planning in which can compute paths in distinct topo-geometric classes. This approach can find desired number of locally optimal paths in a wider variety of configuration spaces without requiring any complex pre-processing or geometric constructions. Unlike the existing topological methods, resulting optimal paths are not restricted to distinct topological classes, thus making the algorithm applicable to many other problems where locally optimal and geometrically distinct paths are of interest. For the demonstration of an application of the proposed approach, we implement our algorithm to planning for shortest traversible paths for a tethered robot with cable-length constraint navigating in 3D and validate it in simulations & experiments.Comment: 18 pages, 17 figure

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field