For a path-connected topological space \(X\), the topological complexity \(TC(X)\) is the minimal number \(k\) such that there are \(k+1\) different “motion planning rules,” each defined on an open subset of \(X\times X\), so that each rule is continuous in the source and target configurations. Topological complexity, introduced by Michael Farber in 2003 in the context of motion planning in robotics, is a homotopy invariant.
In the parametrized setting, motion planning algorithms for collision-free control involve the motion of multiple objects (robots) in the presence of moving obstacles (external conditions). We discuss the topological complexity of algorithms solving this problem. The parametrized topological complexity can be significantly higher than the topological complexity (non-parametrized). Specifically, we analyze the parametrized topological complexity of obstacle-avoiding, collision-free motion of objects (robots) in three-dimensional space.
