  This work investigates whether an extended test body obeying the
Mathisson-Papapetrou- Dixon equations under the Ohashi-Kyrian-Semerak spin
supplementary condition can follow geodesic trajectories in curved spacetimes.
In particular, we explore what are the requirements under which pole-dipole and
pole-dipole-quadrupole approximated bodies moving in the Schwarzschild or Kerr
spacetimes can follow equatorial geodesic trajectories. We do this exploration
thoroughly in the pole-dipole case, while we focus just on particular
trajectories in the pole-dipole-quadrupole case. Using the
Ohashi-Kyrian-Semerak spin supplementary condition to fix the center of the
mass of a pole-dipole body has the advantage that the hidden momentum is
eliminated. This allows the four-velocity to be parallel to the four-momentum,
which provides a convenient framework for our investigation. We discuss how
this feature can be recovered at a pole-dipole-quadrupole approximation and
what are the consequences.
