8 problems

Topological complexity conjecture. If $\Gamma$ has at least two essential vertices and $n\geq 2m(\Gamma)$, then

Let $C_n(\mathbf R^m)$ denote the configuration space of $n$ ordered distinct points in $\mathbf R^m$, and let ${\rm\bf {TC}}$ denote topological complexity. Conjecture. For even…

Let $r$ denote the parameter in the spherical Dubins vehicle problem, and let $C$ denote a circular path segment. For $r > \frac{1}{2}$, the candidate list of paths may contain mor…

Optimal-planner conjecture. These effectual motion planners are optimal, namely

Let a planning problem be given by $\{\boldsymbol{q}_{init},\mathcal{Q}_{goal}, \mathcal{X}_{obs}, \mathbf{F}, J\}$ together with a collision-checker. Here $\boldsymbol{q}_{init}$…

A rational motion is represented by a rational curve on the Study quadric, and exceptional degree reduction is characterized by the geometric criterion established in the paper. Th…

Let a sampling-based motion planner construct a roadmap, and let the corresponding obstacle-free roadmap model be an underlying random geometric graph (RGG). A property is said to…

Let $F(\mathbf{R}^m,n)$ be the configuration space of $n$ distinct points in Euclidean $m$-space, with $m,n\geq 2$, and let ${\sf {TC}}(X)$ denote the topological complexity of a s…