Tools for User Modification of Optimal Roadmaps

Robotic motion planning is a ubiquitous field of study, with innumerable applications in sci- ence, engineering, and beyond. At its core, however, motion planning is infeasible for many complex problems. Sampling-based algorithms address this issue by building an approximate model of the planning space, while optimal planners extend this to provide desirable guaran- tees on solution features (e.g., shortest paths). Unfortunately, these guarantees can require the creation of dense, cumbersome planning graphs. Automatic refinement algorithms can help to sparsify these dense graphs, though they may be costly themselves if they affect the quality of the original solution. In another direction, harnessing human intuition with user- guided planning strategies has also shown promise. In this research, we seek to combine the unique strengths of human and machine reasoning with a foundational, interactive toolset for graph modification and, thus, to overcome some weaknesses inherent in either alone. We provide a visual interface that allows the user to modify a pre-computed planning graph by adding, removing, and adjusting vertices and edges as desired, with reciprocal feedback from the planner on the feasibility of each operation. This provides a more adaptable way to improve graph quality–e.g., by sparsifying particular areas based on the unique dynamics of the environment, which are easily and naturally conceptualized by human instinct. In experiments, we found our tools to be quite helpful in improving some measures of graph quality, while their benefits on others dependended on the intuitiveness of the user interface

The Critical Radius in Sampling-based Motion Planning

We develop a new analysis of sampling-based motion planning in Euclidean space with uniform random sampling, which significantly improves upon the celebrated result of Karaman and Frazzoli (2011) and subsequent work. Particularly, we prove the existence of a critical connection radius proportional to ${\Theta(n^{-1/d})}$ for $n$ samples and ${d}$ dimensions: Below this value the planner is guaranteed to fail (similarly shown by the aforementioned work, ibid.). More importantly, for larger radius values the planner is asymptotically (near-)optimal. Furthermore, our analysis yields an explicit lower bound of ${1-O( n^{-1})}$ on the probability of success. A practical implication of our work is that asymptotic (near-)optimality is achieved when each sample is connected to only ${\Theta(1)}$ neighbors. This is in stark contrast to previous work which requires ${\Theta(\log n)}$ connections, that are induced by a radius of order ${\left(\frac{\log n}{n}\right)^{1/d}}$. Our analysis is not restricted to PRM and applies to a variety of PRM-based planners, including RRG, FMT* and BTT. Continuum percolation plays an important role in our proofs. Lastly, we develop similar theory for all the aforementioned planners when constructed with deterministic samples, which are then sparsified in a randomized fashion. We believe that this new model, and its analysis, is interesting in its own right