Stochastic Observability and Uncertainty Characterization in Simultaneous Receiver and Transmitter Localization

The stochastic observability of simultaneous receiver and transmitter localization is studied. A mobile vehicle-mounted receiver is assumed to draw pseudorange measurements from multiple unknown radio frequency (RF) transmitters and to fuse these measurements through an extended Kalman filter (EKF) to simultaneously localize the receiver and transmitters together with estimating the receiver’s and transmitters’ clock errors. The receiver is assumed to have perfect a priori knowledge of its initial states, while the transmitters’ states are unknown. It is shown that the receiver’s and transmitters’ clock biases are stochastically unobservable and that their estimation error variances will diverge. A lower bound on the divergence rate of the estimation error variances of the receiver’s and transmitters’ clock biases is derived and demonstrated numerically. Simulation and experimental results are presented for an unmanned aerial vehicle (UAV) navigating without GPS signals, using pseudoranges made on unknown terrestrial transmitters. It is demonstrated that despite the receiver’s and transmitters’ clock biases being stochastically unobservable, the EKF produces bounded localization errors.