On the complexity of parametrized motion planning algorithms

TL;DR

This paper introduces a probabilistic variant of the parametrized topological complexity to evaluate the difficulty of designing motion planning algorithms, revealing diverse behaviors across different fiber bundle structures.

Contribution

It defines a new invariant for parametrized motion planning, analyzes its properties using cohomology and equivariant homotopy theory, and compares it with existing invariants.

Findings

Invariant behaves like classical on certain fibrations

Invariant differs significantly on bundles with real projective fibers

Links with other topological robotics invariants

Abstract

We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.