Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

Weighted skeletons and fixed-share decomposition

AbstractWe introduce the concept of weighted skeleton of a polygon and present various decomposition and optimality results for this skeletal structure when the underlying polygon is convex

A Novel Approach for Extraction of Polygon Regions

This paper presents a new algorithm to find out whether a polygon exists around a reference point given within the graphical domain. The algorithm is based on creating discrete line segments and then searching them using the orientations formed at segments intersections. The computational complexity of the searching algorithm has been determined as O( n2

2D numerical simulation of vessels advancing in level ice and ice management operations

This thesis provides a hybrid, numerical and semi-empirical model that predicts ship-ice interaction in level ice. In addition, the model is integrated into an existing software solution, allowing for the simulation of ice management operations and vessel control in sea ice environments. The software implementation numerically solves for contact surfaces between the ship and ice in the time domain and utilizes semi-empirical formulas to estimate the ice forces generated in the interaction. A new method that does not remove rubble ice from the simulation is proposed. The addition of broken ice floes and increased ice shape complexity is facilitated by applying polygon partitioning and simplification algorithms to allow for a an adjustable trade-off between accuracy and performance. The software solution is validated using the data from model-scale tests and-full scale trials. Finally, recommendations for future work to improve upon the proposed model are provided

High-Speed Obstacle Avoidance at the Dynamic Limits for Autonomous Ground Vehicles

Enabling autonomy of passenger-size and larger vehicles is becoming increasingly important in both military and commercial applications. For large autonomous ground vehicles (AGVs), the vehicle dynamics are critical to consider to ensure vehicle safety during obstacle avoidance maneuvers especially at high speeds. This research is concerned with large-size high-speed AGVs with high center of gravity that operate in unstructured environments. The term `unstructured' in this context denotes that there are no lanes or traffic rules to follow. No map of the environment is available a priori. The environment is perceived through a planar light detection and ranging sensor. The mission of the AGV is to move from its initial position to a given target position safely and as fast as possible. In this dissertation, a model predictive control (MPC)-based obstacle avoidance algorithm is developed to achieve the objectives through an iterative simultaneous optimization of the path and the corresponding control commands. MPC is chosen because it offers a rigorous and systematic approach for taking vehicle dynamics and safety constraints into account. Firstly, this thesis investigates the level of model fidelity needed for an MPC-based obstacle avoidance algorithm to be able to safely and quickly avoid obstacles even when the vehicle is close to its dynamic limits. Five different representations of vehicle dynamics models are considered. It is concluded that the two Degrees-of-Freedom (DoF) representation that accounts for tire nonlinearities and longitudinal load transfer is necessary for the MPC-based obstacle avoidance algorithm to operate the vehicle at its limits within an environment that includes large obstacles. Secondly, existing MPC formulations for passenger vehicles in structured environments do not readily apply to this context. Thus, a novel nonlinear MPC formulation is developed. First, a new cost function formulation is used that aims to find the shortest path to the target position. Second, a region partitioning approach is used in conjunction with a multi-phase optimal control formulation to accommodate the complicated forms of obstacle-free regions from an unstructured environment. Third, the no-wheel-lift-off condition is established offline using a fourteen DoF vehicle dynamics model and is included in the MPC formulation. The formulation can simultaneous optimize both steering angle and reference longitudinal speed commands. Simulation results show that the proposed algorithm is capable of safely exploiting the dynamic limits of the vehicle while navigating the vehicle through sensed obstacles of different size and number. Thirdly, in the algorithm, a model of the vehicle is used explicitly to predict and optimize future actions, but in practice, the model parameter values are not exactly known. It is demonstrated that using nominal parameter values in the algorithm leads to safety issues in about one fourth of the evaluated scenarios with the considered parametric uncertainty distributions. To improve the robustness of the algorithm, a novel double-worst-case formulation is developed. Results from simulations with stratified random scenarios and worst-case scenarios show that the double-worst-case formulation considering both the most likely and less likely worst-case scenarios renders the algorithm robust to all uncertainty realizations tested. The trade-off between the robustness and the task completion performance of the algorithm is also quantified. Finally, in addition to simulation-based validation, preliminary experimental validation is also performed. These results demonstrate that the developed algorithm is promising in terms of its capability of avoiding obstacles. Limitations and potential improvements of the algorithm are discussed.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135770/1/ljch_1.pd

Reactive Planning With Legged Robots In Unknown Environments

Unlike the problem of safe task and motion planning in a completely known environment, the setting where the obstacles in a robot\u27s workspace are not initially known and are incrementally revealed online has so far received little theoretical interest, with existing algorithms usually demanding constant deliberative replanning in the presence of unanticipated conditions. Moreover, even though recent advances show that legged platforms are becoming better at traversing rough terrains and environments, legged robots are still mostly used as locomotion research platforms, with applications restricted to domains where interaction with the environment is usually not needed and actively avoided. In order to accomplish challenging tasks with such highly dynamic robots in unexplored environments, this research suggests with formal arguments and empirical demonstration the effectiveness of a hierarchical control structure, that we believe is the first provably correct deliberative/reactive planner to engage an unmodified general purpose mobile manipulator in physical rearrangements of its environment. To this end, we develop the mobile manipulation maneuvers to accomplish each task at hand, successfully anchor the useful kinematic unicycle template to control our legged platforms, and integrate perceptual feedback with low-level control to coordinate each robot\u27s movement. At the same time, this research builds toward a useful abstraction for task planning in unknown environments, and provides an avenue for incorporating partial prior knowledge within a deterministic framework well suited to existing vector field planning methods, by exploiting recent developments in semantic SLAM and object pose and triangular mesh extraction using convolutional neural net architectures. Under specific sufficient conditions, formal results guarantee collision avoidance and convergence to designated (fixed or slowly moving) targets, for both a single robot and a robot gripping and manipulating objects, in previously unexplored workspaces cluttered with non-convex obstacles. We encourage the application of our methods by providing accompanying software with open-source implementations of our algorithms