Semidefinite programming‐based localisation and tracking algorithm using Gaussian mixture modelling

In this study, the authors propose a semidefinite programming (SDP)-based localisation and tracking algorithm, which mitigates the non-line-of-sight (NLOS) error of range measurement and calibrates the accumulative error within the inertial sensing data. Both the range measurement in a mixed line-of-sight/NLOS environment and the step length estimated from inertial sensing information are approximated parametrically using Gaussian mixture modelling, and a maximum-likelihood estimator (MLE) is formulated to obtain the optimal position estimation. Since the Gaussian mixture models are non-linear functions of positions, the MLE is a non-convex problem, which global optimum is difficult to attain. Then, the non-convex MLE is transformed into an SDP-based localisation and tracking problem, relying on Jensen's inequality and semidefinite relaxation. Thus, a sub-optimal solution to the original MLE can be achieved. Moreover, the Cramer-Rao lower bound is also derived to serve as a performance indicator for localisation errors. The simulation and experimental results demonstrate the performance of the proposed algorithm. Compared with the existing algorithms, the proposed algorithm owns the best localisation accuracy, and can achieve a sub-metre level accuracy to a root mean square error of 0.46 m in the real deployments.