Given a network of N static nodes in D-dimensional space and the pairwise distances between them, the challenge of estimating the coordinates of the nodes is a well-studied problem. However, for numerous application domains, the nodes are mobile and the estimation of relative kinematics (e.g., position, velocity and acceleration) is a challenge, which has received limited attention in literature. In this paper, we introduce a time-varying Grammian-based data model for estimating the relative kinematics of mobile nodes with polynomial trajectories, given the time-varying pairwise distance measurements between the nodes. Furthermore, we consider a scenario where the nodes have on-board accelerometers, and extend the proposed data model to include these accelerometer measurements. We propose closed-form solutions to estimate the relative kinematics, based on the proposed data models. We conduct simulations to showcase the performance of the proposed estimators, which show improvement against state-of-the-art methods.

For a network of mobile nodes, the problem of estimation of relative kinematics, given pairwise distances between the nodes, has received limited attention in literature. In this context, relative kinematics includes relative position, relative velocity and other higher order kinematic parameters defined with respect to a common frame of reference within the network. For numerous application domains in engineering, the nodes are highly dynamic, making the estimation task much harder. To solve the estimation problem uniquely, conventional methods either require the positions of some nodes of the mobile network to be known [2] or impose rigid body constraints on the mobile network [3]. These conditions limit the scope of proposed methods. Given a network of mobile nodes and time-varying pairwise distance measurements, we introduce a time-varying Grammian-based data model under the assumption that the mobile nodes have polynomial trajectories. Using the results in [4] and [5], estimators are proposed to estimate the relative kinematic parameters. Furthermore, we consider a scenario where the nodes have on-board accelerometers and the mobile nodes are holonomic. Under such assumtpions, the proposed data model is extended to include these accelerometer measurements, leading to improvements in relative kinematics estimation. We conduct simulations to showcase the performance of the proposed estimators, which show improvement against state-of-the-art methods.