Relative kinematics of an anchorless network

The estimation of the coordinates of nodes their proximity (or distance) measurements, is a principal challenge in numerous fields. Conventionally, when localizing a static network of immobile nodes, non-linear dimensionality reduction techniques are applied on the measured distances to obtain the relative coordinates up to a rotation and translation. In this article, we consider an anchorless network of mobile nodes, where the distance measurements between the mobile nodes are time-varying. In such an anchorless framework, where the absolute knowledge of any node position, motion or reference frame is absent, we aim to estimate the relative positions using the measured time-varying distances. To this end, we derive a data model which relates the time-varying distances to the time-varying relative positions of an anchorless network. Given this data model, we estimate the relative (position, 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 under certain immobility constraints, we propose closed-form solutions to recursively estimate the relative kinematics. For the sake of completeness, we also estimate the absolute node kinematics, given reference anchors. Theoretical bounds are derived, and simulations are conducted to analyse the performance of proposed solutions.