Simple Wriggling is Hard unless You Are a Fat Hippo

We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length/width ratio is small, the shortest path can be computed in polynomial time.Comment: A shorter version is to be presented at FUN 201

Near-Optimal Min-Sum Motion Planning for Two Square Robots in a Polygonal Environment

Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment (i.e., a polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can translate inside $\mathcal{W}$. Given source and target placements $s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to compute a \emph{collision-free motion plan} $\mathbf{\pi}^*$, i.e., a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots, $\left|\mathbf{\pi}^*\right|$. Given $\mathcal{W}, s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an $n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm for this problem. We are not aware of any polynomial time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion planning problem involving two robots moving in a polygonal environment.Comment: The conference version of the paper is accepted to SODA 202

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Path planning for robotic truss assembly

A new Potential Fields approach to the robotic path planning problem is proposed and implemented. Our approach, which is based on one originally proposed by Munger, computes an incremental joint vector based upon attraction to a goal and repulsion from obstacles. By repetitively adding and computing these 'steps', it is hoped (but not guaranteed) that the robot will reach its goal. An attractive force exerted by the goal is found by solving for the the minimum norm solution to the linear Jacobian equation. A repulsive force between obstacles and the robot's links is used to avoid collisions. Its magnitude is inversely proportional to the distance. Together, these forces make the goal the global minimum potential point, but local minima can stop the robot from ever reaching that point. Our approach improves on a basic, potential field paradigm developed by Munger by using an active, adaptive field - what we will call a 'flexible' potential field. Active fields are stronger when objects move towards one another and weaker when they move apart. An adaptive field's strength is individually tailored to be just strong enough to avoid any collision. In addition to the local planner, a global planning algorithm helps the planner to avoid local field minima by providing subgoals. These subgoals are based on the obstacles which caused the local planner to fail. A best-first search algorithm A* is used for graph search

Placement and motion planning algorithms for robotic sensing systems

University of Minnesota Ph.D. dissertation. October 2014. Major: Computer Science. Advisor: Prof. Ibrahim Volkan Isler. I computer file (PDF); xxiii, 226 pages.Recent technological advances are making it possible to build teams of sensors and robots that can sense data from hard-to-reach places at unprecedented spatio-temporal scales. Robotic sensing systems hold the potential to revolutionize a diverse collection of applications such as agriculture, environmental monitoring, climate studies, security and surveillance in the near future. In order to make full use of this technology, it is crucial to complement it with efficient algorithms that plan for the sensing in these systems. In this dissertation, we develop new sensor planning algorithms and present prototype robotic sensing systems.In the first part of this dissertation, we study two problems on placing stationary sensors to cover an environment. Our objective is to place the fewest number of sensors required to ensure that every point in the environment is covered. In the first problem, we say a point is covered if it is seen by sensors from all orientations. The environment is represented as a polygon and the sensors are modeled as omnidirectional cameras. Our formulation, which builds on the well-known art gallery problem, is motivated by practical applications such as visual inspection and video-conferencing where seeing objects from all sides is crucial. In the second problem, we study how to deploy bearing sensors in order to localize a target in the environment. The sensors measure noisy bearings towards the target which can be combined to localize the target. The uncertainty in localization is a function of the placement of the sensors relative to the target. For both problems we present (i) lower bounds on the number of sensors required for an optimal algorithm, and (ii) algorithms to place at most a constant times the optimal number of sensors. In the second part of this dissertation, we study motion planning problems for mobile sensors. We start by investigating how to plan the motion of a team of aerial robots tasked with tracking targets that are moving on the ground. We then study various coverage problems that arise in two environmental monitoring applications: using robotic boats to monitor radio-tagged invasive fish in lakes, and using ground and aerial robots for data collection in precision agriculture. We formulate the coverage problems based on constraints observed in practice. We also present the design of prototype robotic systems for these applications. In the final problem, we investigate how to optimize the low-level motion of the robots to minimize their energy consumption and extend the system lifetime.This dissertation makes progress towards building robotic sensing systems along two directions. We present algorithms with strong theoretical performance guarantees, often by proving that our algorithms are optimal or that their costs are at most a constant factor away from the optimal values. We also demonstrate the feasibility and applicability of our results through system implementation and with results from simulations and extensive field experiments

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum