  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.
