Moving Rigid Body Localization Using TOA Only

Moving rigid body localization requires the estimation of not only the position vector and rotation angles but also its translational and angular velocities, which leads to a very challenging problem. This paper demonstrates the feasibility and develops a semidefinite relaxation (SDR) method for locating a moving rigid body with constant translational and angular velocities using time-of-arrival (TOA) measurements only collected at several successive instants over a time period. More specifically, we analyze the localizability and determine the minimum numbers of anchors, sensors, and observation samples required for this particular problem. Afterward, beginning with a proposed model by transforming the TOA relation with the unknowns, we construct a constrained weighted least squares (CWLS) problem by introducing auxiliary variables. The CWLS problem is nonlinear and complicated to tackle owing to the large number of variables and the special orthogonal group constraint on the rotation matrix. To solve this difficult problem, we use SDR and transform it into a convex semidefinite programming problem to obtain a preliminary estimate. Subsequently, a more accurate solution is achieved through orthogonalization and refinement to compensate for the performance loss caused by approximations and relaxation. Theoretical mean square error analysis shows that the proposed method is able to achieve the Cramér-Rao lower bound performance under small Gaussian noise. Simulation results also demonstrate the effectiveness and robustness of the method proposed.