  Estimating the location of N coordinates in a P dimensional Euclidean space
from pairwise distances (or proximity measurements), is a principal challenge
in a wide variety of fields. Conventionally, when localizing a static network
of immobile nodes, non-linear dimensional reduction techniques are applied on
the measured Euclidean distance matrix (EDM) to obtain the relative coordinates
upto a rotation and translation. In this article, we focus on an anchorless
network of mobile nodes, where the distance measurements between the mobile
nodes are time-varying in nature. Furthermore, in an anchorless network the
absolute knowledge of any node positions, motion or reference frame is absent.
We derive a novel data model which relates the time-varying EDMs to the
time-varying relative positions of an anchorless network. Using this data
model, we estimate the relative position, relative velocity and higher order
derivatives, which are collectively termed as the relative kinematics of the
anchorless network. The derived data model is inherently ill-posed, however can
be solved using certain relative immobility constraints. We propose elegant
closed form solutions to recursively estimate the relative kinematics of the
network. For the sake of completeness, estimators are also proposed to find the
absolute kinematics of the nodes, given known reference anchors. Cramer-Rao
bounds are derived for the new data model and simulations are performed to
analyze the performance of the proposed solutions.
