Related papers: Simplicial Complexity: piecewise linear motion pla…
We study an extensive class of movement minimization problems which arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents…
We present a centralized algorithmic framework for solving multi-robot path planning problems in general, two-dimensional, continuous environments while minimizing globally the task completion time. The framework obtains high levels of…
We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call…
We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data -- called a complex of groups -- amounts to a functor from (the…
Feedback motion planning over cell decompositions provides a robust method for generating collision-free robot motion with formal guarantees. However, existing algorithms often produce paths with unnecessary bending, leading to slower…
In this paper, we transfer the problem of measuring navigational complexity in topological spaces to the nearness theory. We investigate the most important component of this problem, the topological complexity number (denoted by TC), with…
The topological complexity ${\sf TC}(X)$ is a homotopy invariant of a topological space $X$, motivated by robotics, and providing a measure of the navigational complexity of $X$. The topological complexity of a connected sum of real…
In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system…
This is a continuation of our recent paper in which we developed the theory of sequential parametrized motion planning. A sequential parametrized motion planning algorithm produced a motion of the system which is required to visit a…
In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss…
Lattice-based planning techniques simplify the motion planning problem for autonomous vehicles by limiting available motions to a pre-computed set of primitives. These primitives are then combined online to generate more complex maneuvers.…
This paper displays complexity reduction techniques for calculations of path categories (or fundamental categories) P(K) for finite simplicial and cubical complexes K. The central technique involves identifying inclusions of complexes for…
We give simple upper bounds for rational sectional category and use them to compute invariants of the type of Farber's topological complexity of rational spaces. In particular we show that the sectional category of formal morphisms reaches…
We introduce higher simplicial complexity of a simplicial complex $K$ and higher combinatorial complexity of a finite space $P$ (i.e. $P$ is a finite poset). We relate higher simplicial complexity with higher topological complexity of $|K|$…
We compute the higher topological complexity of ordered configuration spaces of orientable surfaces, thus extending Cohen-Farber's description of the ordinary topological complexity of those spaces.
We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of…
We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably…
We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and…
Learning the topology of higher-order networks from data is a fundamental challenge in many signal processing and machine learning applications. Simplicial complexes provide a principled framework for modeling multi-way interactions, yet…
We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion.