  We aim at performing a kinematic census of young stellar objects (YSOs) in
the Rho Ophiuchi F core and partially in the E core of the L1688 dark cloud. We
run a proper motion program at the ESO New Technology Telescope (NTT) with the
Son of ISAAC (SOFI) instrument over nine years in the near-infrared. We
complemented these observations with various public image databases to enlarge
the time base of observations and the field of investigation to 0.5 deg X 0.5
deg. We derived positions and proper motions for 2213 objects. From these, 607
proper motions were derived from SOFI observations with a ~1.8 mas/yr accuracy
while the remaining objects were measured only from auxiliary data with a mean
precision of about ~3 mas/yr. We performed a kinematic analysis of the most
accurate proper motions derived in this work, which allowed us to separate
cluster members from field stars and to derive the mean properties of the
cluster. From the kinematic analysis we derived a list of 68 members and 14
candidate members, comprising 26 new objects with a high membership
probability. These new members are generally fainter than the known ones. We
measured a mean proper motion of (mu_RA*, mu_DEC)=(-8.2, -24.3)+/-0.8 mas/yr
for the L1688 dark cloud. A supervised classification was applied to
photometric data of members to allocate a spectral energy distribution (SED)
classification to the unclassified members.} We kinematically confirmed that
the 56 members that were known from previous studies of the Rho Ophiuchi F
cluster and that were also part of our survey are members of the cluster, and
we added 26 new members. We defined the evolutionary status of the unclassified
members of the cluster. We showed that a large part (23) of these new members
are probably brown dwarfs, which multiplies the number of known substellar
objects in the cluster by a factor of 3.3.
