Recent progress in exact geometric computation

AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters

Optimal Control of Fully Routed Air Traffic in the Presence of Uncertainty and Kinodynamic Constraints

A method is presented to extend current graph-based Air Traffic Management optimization frameworks. In general, Air Traffic Management is the process of guiding a finite set of aircraft, each along its pre-determined path within some local airspace, subject to various physical, policy, procedural and operational restrictions. This research addresses several limitations of current graph-based Air Traffic Management optimization methods by incorporating techniques to account for stochastic effects, physical inertia and variable arrival sequencing. In addition, this research provides insight into the performance of multiple methods for approximating non-differentiable air traffic constraints, and incorporates these methods into a generalized weighted-sum representation of the multi-objective Air Traffic Management optimization problem that minimizes the total time of flight, deviation from scheduled arrival time and fuel consumption of all aircraft. The methods developed and tested throughout this dissertation demonstrate the ability of graph-based optimization techniques to model realistic air traffic restrictions and generate viable control strategies

Pseudo approximation algorithms with applications to optimal motion planning

We introduce a technique for computing approximate solutions to optimization problems. If X is the set of feasible solutions, the standard goal of approximation algorithms is to compute x ∈ X that is an ε - approximate solution in the following sense: d(x) ≤ - (1+ε)d(x^*), where x^* ∈ X is an optimal solution, d : X → R ≥ 0 is the optimization function to be minimized, andε>0 is an input parameter. Our approach is first to devise algorithms that compute pseudo ε-approximate solutions satisfying the bound d(x) ≤ - d(x^*_R) +εR, where R>0 is a new input parameter. Here x^*_R denotes an optimal solution in the space X_R of R-constrained feasible solutions. The parameter R provides a stratification of X in the sense that (1) X_R ⊆ X_ for R < R’ and (2) X_R = X for R sufficiently large. We first describe a highly efficient scheme for converting a pseudo ε-approximation algorithm into a trueε-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than ε-approximation algorithms. Another benefit is that our algorithm is automatically precision-sensitive. We apply our technique to two problems in robotics: (A) Euclidean Shortest Path (3ESP), namely the shortest path for a point robot amidst polyhedral obstacles in 3D, and (B) d_1-optimal motion for a rod moving amidst planar obstacles (1ORM). Previously, no polynomial time ε-approximation algorithm for (B) was known. For (A), our new solution is simpler than previous solutions and has an exponentially smaller complexity in terms of the input precision