Gaussian process state-space models are a widely used modeling paradigm for learning and estimation in dynamical systems. Reduced-rank Gaussian process state-space models combine spectral characterization of dynamical systems with Hilbert space methods to enable learning, which scale linearly with the length of the time series. However, the current state of the art algorithms struggle to deal efficiently with the dimensionality of the state-space itself. In this work, we propose a novel algorithm, referred to as Domain-Aware reduced-rank Gaussian Process State-Space Model (DA-GPSSM), which exploits the relationship between state dimensions to model only necessary dynamics resulting in reduced computational cost, by potentially orders of magnitude in comparison to the state-of-the-art. The proposed approach grants modeling flexibility while maintaining comparable performance and thus increasing the applicability of these models. We present implications of the proposed approach and discuss applications where DA-GPSSM can be beneficial. Finally, we conduct simulations to demonstrate the performance and reduced computational cost of our proposed method, compared to the state-of-the-art learning method, and propose future research directions.

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.

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.