Estimation of Relative Kinematic Parameters of an Anchorless Network

Estimation of the relative positions of $N$ static nodes in $D$-dimensional space given the pairwise distances between them is a well-studied problem in literature. However, for a network of mobile nodes, the existing solutions proposed in literature rely either on the knowledge of absolute positions of some nodes or enforce constraints on the motion of individual nodes to achieve a unique solution. In this work, we consider an anchorless environment and propose a time-varying Grammian-based data model which relates the relative positions of the mobile nodes to the pairwise distances between them. Given the data model, we propose algorithms to estimate the relative positions, velocity and other higher order derivatives, referred to as relative kinematics, associated with the network of mobile nodes. We further consider a scenario where accelerometers are on-board on all the mobile nodes, and investigate the inclusion the accelerometer measurements in the proposed model. The Cramér-Rao lower bound for the proposed data models are derived and compared with the performance of the estimators using Monte-Carlo simulations. We further compare and analyze the performance of the proposed estimators against the state-of-the-art methods, and present research directions for future work to further improve the proposed approach.