Lusternik-Schnirelmann category and based topological complexities of motion planning

Farber and Rudyak introduced topological complexity $\mathbf{TC}(X)$ of motion planning and its higher analogs $\mathbf{TC}_n(X)$ to measure the complexity of assigning paths to point tuples. Motivated by motion planning where a robotic system starts at the home configuration and possibly comes back after passing through a list of locations, we define three other classes of topological complexities $\mathbf{LTC}_n(X)$, $\mathbf{ltc}_n(X)$ and $\mathbf{tc}_n(X)$. We will compare these notions and compute the latter for some familiar classes of spaces.