  We study the competition between $\sqrt{3} \times \sqrt{3}$ (RT3) and $q=0$
(Q0) magnetic orders in spin-one and spin-$3/2$ Kagome-lattice XXZ
antiferromagnets with varying XY anisotropy parameter $\Delta$, using series
expansion methods. The Hamiltonian is split into two parts: an $H_0$ which
favors the classical order in the desired pattern and an $H_1$, which is
treated in perturbation theory by a series expansion. We find that the ground
state energy series for the RT3 and Q0 phases are identical up to sixth order
in the expansion, but ultimately a selection occurs, which depends on spin and
the anisotropy $\Delta$. Results for ground state energy and the magnetization
are presented. These results are compared with recent spin-wave theory and
coupled-cluster calculations. The series results for the phase diagram are
close to the predictions of spin-wave theory. For the spin-one model at the
Heisenberg point ($\Delta=1$), our results are consistent with a vanishing
order parameter, that is an absence of a magnetically ordered phase. We also
develop series expansions for the ground state energy of the spin-one
Heisenberg model in the trimerized phase. We find that the ground state energy
in this phase is lower than those of magnetically ordered ones, supporting the
existence of a spontaneously trimerized phase in this model.
