Discrete Planning

This chapter provides introductory concepts that serve as an entry point into other parts of the book. The planning problems considered here are the simplest to describe because the state space will be finite in most cases. When it is not finite, it will at least be countably infinite (i.e., a unique integer may be assigned to every state). Therefore, no geometric models or differential equations will be needed to characterize the discrete planning problems. Furthermore, no forms of uncertainty will be considered, which avoids complications such as probability theory. All models are completely known and predictable. There are three main parts to this chapter. Sections 2.1 and 2.2 define and present search methods for feasible planning, in which the only concern is to reach a goal state. The search methods will be used throughout the book in numerous other contexts, including motion planning in continuous state spaces. Following feasible planning, Section 2.3 addresses the problem of optimal planning. The principle of optimality, or the dynamic programming principle, [1] provides a key insight that greatly reduces the computation effort in many planning algorithms

Bang-Bang Boosting of RRTs

This paper explores the use of time-optimal controls to improve the performance of sampling-based kinodynamic planners. A computationally efficient steering method is introduced that produces time-optimal trajectories between any states for a vector of double integrators. This method is applied in three ways: 1) to generate RRT edges that quickly solve the two-point boundary-value problems, 2) to produce an RRT (quasi)metric for more accurate Voronoi bias, and 3) to time-optimize a given collision-free trajectory. Experiments are performed for state spaces with up to 2000 dimensions, resulting in improved computed trajectories and orders of magnitude computation time improvements over using ordinary metrics and constant controls

Multi-agent Path Planning and Network Flow

This paper connects multi-agent path planning on graphs (roadmaps) to network flow problems, showing that the former can be reduced to the latter, therefore enabling the application of combinatorial network flow algorithms, as well as general linear program techniques, to multi-agent path planning problems on graphs. Exploiting this connection, we show that when the goals are permutation invariant, the problem always has a feasible solution path set with a longest finish time of no more than $n + V - 1$ steps, in which $n$ is the number of agents and $V$ is the number of vertices of the underlying graph. We then give a complete algorithm that finds such a solution in $O(nVE)$ time, with $E$ being the number of edges of the graph. Taking a further step, we study time and distance optimality of the feasible solutions, show that they have a pairwise Pareto optimal structure, and again provide efficient algorithms for optimizing two of these practical objectives.Comment: Corrected an inaccuracy on time optimal solution for average arrival tim

Equivalent Environments and Covering Spaces for Robots

This paper formally defines a robot system, including its sensing and actuation components, as a general, topological dynamical system. The focus is on determining general conditions under which various environments in which the robot can be placed are indistinguishable. A key result is that, under very general conditions, covering maps witness such indistinguishability. This formalizes the intuition behind the well studied loop closure problem in robotics. An important special case is where the sensor mapping reports an invariant of the local topological (metric) structure of an environment because such structure is preserved by (metric) covering maps. Whereas coverings provide a sufficient condition for the equivalence of environments, we also give a necessary condition using bisimulation. The overall framework is applied to unify previously identified phenomena in robotics and related fields, in which moving agents with sensors must make inferences about their environments based on limited data. Many open problems are identified.Comment: 34 pages, 8 figure

Extracting Visibility Information by Following Walls

This paper presents an analysis of a simple robot model, called Bitbot. The Bitbot has limited capabilities; it can reliably follow walls and sense a contact with a wall. Although the Bitbot does not have a range sensor or a camera, it is able to acquire visibility information from the environment, which is then used to solve a pursuit-evasion task. Our developments are centered on the characterization of the information the Bitbot acquires. At any given moment, due to the sensing uncertainty, the robot does not know the current state. In general, uncertainty in the state is one of the central issues in robotics; the Bitbot model serves as an example of how the notion of information space naturally handles uncertainty. We show that state estimation with the Bitbot is a challenging problem, related to the well-known open problem of characterizing visibility graphs in computational geometry. However, state estimation becomes unnecessary to the achievement of the Bitbot\u27s visibility tasks. We show how pursuit-evasion strategy is derived from a careful manipulation with histories of observations, and present analysis of the algorithm and experimental results

A Mathematical Characterization of Minimally Sufficient Robot Brains

This paper addresses the lower limits of encoding and processing the information acquired through interactions between an internal system (robot algorithms or software) and an external system (robot body and its environment) in terms of action and observation histories. Both are modeled as transition systems. We want to know the weakest internal system that is sufficient for achieving passive (filtering) and active (planning) tasks. We introduce the notion of an information transition system for the internal system which is a transition system over a space of information states that reflect a robot's or other observer's perspective based on limited sensing, memory, computation, and actuation. An information transition system is viewed as a filter and a policy or plan is viewed as a function that labels the states of this information transition system. Regardless of whether internal systems are obtained by learning algorithms, planning algorithms, or human insight, we want to know the limits of feasibility for given robot hardware and tasks. We establish, in a general setting, that minimal information transition systems exist up to reasonable equivalence assumptions, and are unique under some general conditions. We then apply the theory to generate new insights into several problems, including optimal sensor fusion/filtering, solving basic planning tasks, and finding minimal representations for modeling a system given input-output relations.Comment: arXiv admin note: text overlap with arXiv:2212.0052