  Modelling the trajectorial motion of humans along the ground is a
foundational task in the quantitative analysis of sports like association
football. Most existing models of football player motion have not been
validated yet with respect to actual data. One of the reasons for this lack is
that performing such a validation is not straightforward, because models of
player motion are usually phrased in a way that emphasises possibly reachable
positions rather than expected positions. Since positional data of football
players typically contains outliers, this data may misrepresent the range of
actually reachable positions.
  This paper proposes a validation routine for trajectorial motion models that
measures and optimises the ability of a motion model to accurately predict all
possibly reachable positions by favoring the smallest predicted area of
reachable positions that encompasses all observed reached positions up to a
manually defined threshold. We demonstrate validation and optimisation on four
different motion models, assuming (a) motion with constant speed, (b) motion
with constant acceleration, (c) motion with constant acceleration with a speed
limit, and (d) motion along two segments with constant speed. Our results show
that assuming motion with constant speed or constant acceleration without a
limit on the achievable speed is particularly inappropriate for an accurate
distinction between reachable and unreachable locations. Motion along two
segments of constant speed provides by far the highest accuracy among the
tested models and serves as an efficient and accurate approximation of
real-world player motion.
