  The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion of an
extended test body in general relativity. This system of equations, though, is
underdetermined and has to be accompanied by constraining supplementary
conditions, even in its simplest version, which is the pole-dipole
approximation corresponding to a spinning test body. In particular, imposing a
spin supplementary condition (SSC) fixes the center of the mass of the spinning
body, i.e. the centroid of the body. In the present study, we examine whether
characteristic features of the centroid of a spinning test body, moving in a
circular equatorial orbit around a massive black hole, are preserved under the
transition to another centroid of the same physical body, governed by a
different SSC. For this purpose, we establish an analytical algorithm for
deriving the orbital frequency of a spinning body, moving in the background of
an arbitrary, stationary, axisymmetric spacetime with reflection symmetry, for
the Tulzcyjew-Dixon, the Mathisson-Pirani and the Ohashi-Kyrian-Semer\'{a}k
SSCs. Then, we focus on the Schwarzschild black hole background and a power
series expansion method is developed, in order to investigate the discrepancies
in the orbital frequencies expanded in power series of the spin among the
different SSCs. Lastly, by employing the fact that the position of the centroid
and the measure of the spin alters under the centroid's transition, we impose
proper corrections to the power expansion of the orbital frequencies, which
allows to improve the convergence between the SSCs. Our concluding argument is
that when we shift from one circular equatorial orbit to another in the
Schwarzschild background, under the change of a SSC, the convergence between
the SSCs holds only up to certain powers in the spin expansion, and it cannot
be achieved for the whole power series.
