  We study computationally-hard fundamental motion planning problems where the
goal is to translate $k$ axis-aligned rectangular robots from their initial
positions to their final positions without collision, and with the minimum
number of translation moves. Our aim is to understand the interplay between the
number of robots and the geometric complexity of the input instance measured by
the input size, which is the number of bits needed to encode the coordinates of
the rectangles' vertices. We focus on axis-aligned translations, and more
generally, translations restricted to a given set of directions, and we study
the two settings where the robots move in the free plane, and where they are
confined to a bounding box. We obtain fixed-parameter tractable (FPT)
algorithms parameterized by $k$ for all the settings under consideration. In
the case where the robots move serially (i.e., one in each time step) and
axis-aligned, we prove a structural result stating that every problem instance
admits an optimal solution in which the moves are along a grid, whose size is a
function of $k$, that can be defined based on the input instance. This
structural result implies that the problem is fixed-parameter tractable
parameterized by $k$. We also consider the case in which the robots move in
parallel (i.e., multiple robots can move during the same time step), and which
falls under the category of Coordinated Motion Planning problems. Finally, we
show that, when the robots move in the free plane, the FPT results for the
serial motion case carry over to the case where the translations are restricted
to any given set of directions.
