Sensor Network-Based Rigid Body Localization via Semi-Definite Relaxation Using Arrival Time and Doppler Measurements

This paper addresses the rigid body localization problem using a convex optimization approach. We propose a semi-definite relaxation (SDR) method for locating a stationary rigid body using arrival time measurements, and extend it for moving rigid body using both arrival time and Doppler measurements. Localization of a stationary (moving) rigid body involves not only the position (and velocity) but also the rotation angles (and angular velocity), making it a challenging optimization problem with nonlinear constraints. We approximate the maximum likelihood problem with a constrained weighted least-squares (CWLS) minimization and apply SDR to obtain a coarse estimate. The orthogonalization and refinement procedures are followed next to recover the performance loss caused by relaxation and approximation. It is shown analytically that the CWLS solution can achieve the Cramér–Rao lower bound accuracy for Gaussian noise when the noise level is not significant. The simulations show that the proposed method achieves better accuracy than the previously developed methods.

I. Introduction

Localization traditionally considers the object of interest is a point source whose location is defined by a unique position in a space. Typically, localization is achieved using a wireless sensor network (WSN), in which the sensors collect the signals from the source and transmit them to the fusion center for further processing. Point source localization has played an extremely important role in wireless communications, target tracking, surveillance, and others [1]–[22]. The literatures [23]–[25] give more details on the fundamentals and recent advances of point localization. Point location model is appropriate in many cases, nevertheless, some situations require additional location information of the object when it is viewed as a rigid body. Localization of a rigid body consists of not only the position of its center, but also the orientation as well. If the body is moving, additional parameters of translation and angular velocities are also desired. Rigid body localization is thus much more challenging than point localization, and has important applications in robotics, spacecraft, underwater vehicles, and many others [26]–[29].