# Linear motion planning with controlled collisions and pure planar braids

## TL;DR

This paper calculates topological invariants of a configuration space related to pure planar braids, providing insights into motion planning with controlled collisions in robotics.

## Contribution

It determines the LS-category and higher topological complexity of the no-k-equal configuration space and applies these results to pure planar braids, offering new tools for motion planning.

## Key findings

- Computed LS-category and topological complexity of no-k-equal configuration space.
- Derived invariants for pure planar braid groups as an analogue of classical braid groups.
- Provided methods for optimal motion planners for small numbers of strands.

## Abstract

We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($TC_s$, $s\geq2$) of the "no-$k$-equal" configuration space Conf$_k(\mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity of Khovanov's group PP$_n$ of pure planar braids on $n$ strands, which is an $\mathbb{R}$-analogue of Artin's classical pure braid group on $n$ strands. Our methods can be used to describe optimal motion planners for PP$_n$ provided $n$ is small.